Subgroups — C₃₁×#3565 #16843498

Order 3968.

930 conjugacy classes of subgroups.

The subgroup lattice has 616 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#563	C₂	1	2	yes	maximal, complemented
2	C₃₁×#562	C₂	1	2	yes	maximal, complemented
3	C₃₁×#557	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#556	C₂	1	2	yes	maximal, complemented
5	C₃₁×#556	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#555	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#554	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#553	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#553	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#552	C₂	1	2	yes	maximal, complemented
11	C₃₁×#552	C₂	1	1	yes	maximal, characteristic, complemented
12	C₃₁×#552	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#544	C₂	1	1	yes	maximal, characteristic, complemented
14	C₃₁×#544	C₂	1	1	yes	maximal, characteristic, complemented
15	C₃₁×#542	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#542	C₂	1	1	yes	maximal, characteristic, complemented
17	C₃₁×#542	C₂	1	2	yes	maximal, complemented
18	C₃₁×#542	C₂	1	2	yes	maximal, complemented
19	C₃₁×#541	C₂	1	2	yes	maximal, complemented
20	C₃₁×#539	C₂	1	2	yes	maximal, complemented
21	C₃₁×#533	C₂	1	1	yes	maximal, characteristic, complemented
22	C₃₁×#529	C₂	1	1	yes	maximal, characteristic, complemented
23	C₃₁×#529	C₂	1	1	yes	maximal, characteristic, complemented
24	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
25	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
26	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
27	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
28	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
29	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
30	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
31	C₃₁×#126	C₂²	1	2	yes	—
32	C₃₁×#126	C₂²	1	1	yes	characteristic
33	C₃₁×#126	C₂²	1	1	yes	characteristic
34	C₃₁×#126	C₂²	1	2	yes	—
35	C₃₁×#126	C₂²	1	2	yes	—
36	C₃₁×#126	C₂²	1	2	yes	—
37	C₃₁×#126	C₂²	1	2	yes	—
38	C₃₁×#126	C₂²	1	2	yes	—
39	C₃₁×#126	C₂²	1	2	yes	—
40	C₃₁×#126	C₂²	1	2	yes	—
41	C₃₁×#126	C₂²	1	2	yes	—
42	C₃₁×#125	C₂²	1	1	yes	characteristic
43	C₃₁×#125	C₂²	1	1	yes	characteristic
44	C₃₁×#125	C₂²	1	1	yes	characteristic
45	C₃₁×#125	C₂²	1	1	yes	characteristic
46	C₃₁×#125	C₂²	1	2	yes	—
47	C₃₁×#125	C₂²	1	2	yes	—
48	C₃₁×#125	C₂²	1	2	yes	—
49	C₃₁×#125	C₂²	1	2	yes	—
50	C₃₁×#125	C₂²	1	2	yes	—
51	C₃₁×#125	C₂²	1	1	yes	characteristic
52	C₃₁×#125	C₂²	1	2	yes	—
53	C₃₁×#125	C₂²	1	1	yes	characteristic
54	C₃₁×#124	C₂²	1	1	yes	characteristic
55	C₃₁×#124	C₂²	1	1	yes	characteristic
56	C₃₁×#124	C₂²	1	2	yes	—
57	C₃₁×#124	C₂²	1	2	yes	—
58	C₃₁×#124	C₂²	1	2	yes	—
59	C₃₁×#124	C₂²	1	2	yes	—
60	C₃₁×#124	C₂²	1	2	yes	—
61	C₃₁×#123	C₂²	1	1	yes	characteristic
62	C₃₁×#123	C₂²	1	1	yes	characteristic
63	C₃₁×#123	C₂²	1	2	yes	—
64	C₃₁×#123	C₂²	1	2	yes	—
65	C₃₁×#123	C₂²	1	2	yes	—
66	C₃₁×#123	C₂²	1	2	yes	—
67	C₃₁×#123	C₂²	1	2	yes	—
68	C₃₁×#123	C₂²	1	2	yes	—
69	C₃₁×#123	C₂²	1	2	yes	—
70	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
71	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
72	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
73	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
74	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
75	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
76	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
77	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
78	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
79	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
80	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
81	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
82	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
83	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
84	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
85	C₃₁×#121	C₂²	1	1	yes	characteristic
86	C₃₁×#121	C₂²	1	1	yes	characteristic
87	C₃₁×#121	C₂²	1	2	yes	—
88	C₃₁×#121	C₂²	1	1	yes	characteristic
89	C₃₁×#121	C₂²	1	1	yes	characteristic
90	C₃₁×#121	C₂²	1	2	yes	—
91	C₃₁×#121	C₂²	1	2	yes	—
92	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
93	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
94	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
95	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
96	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
97	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
98	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
99	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
100	C₄×D₄×C₃₁	C₂²	1	2	yes	—
101	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
102	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
103	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
104	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
105	C₄×D₄×C₃₁	C₂²	1	2	yes	—
106	C₄×D₄×C₃₁	C₂²	1	2	yes	—
107	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
108	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
109	C₃₁×#117	C₂²	1	1	yes	characteristic
110	C₃₁×#117	C₂²	1	1	yes	characteristic
111	C₃₁×#117	C₂²	1	2	yes	—
112	C₃₁×#117	C₂²	1	1	yes	characteristic
113	C₃₁×#117	C₂²	1	1	yes	characteristic
114	C₃₁×#117	C₂²	1	2	yes	—
115	C₃₁×#117	C₂²	1	2	yes	—
116	C₃₁×#117	C₂²	1	2	yes	—
117	C₃₁×#117	C₂²	1	1	yes	characteristic
118	C₃₁×#117	C₂²	1	1	yes	characteristic
119	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
120	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
121	C₂×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
122	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
123	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
124	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
125	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
126	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
127	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
128	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
129	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
130	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
131	C₂×D₄×C₃₁	C₂³	1	2	yes	—
132	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
133	C₂×D₄×C₃₁	C₂³	1	2	yes	—
134	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
135	C₂×D₄×C₃₁	C₂³	1	2	yes	—
136	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
137	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
138	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
139	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
140	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
141	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
142	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
143	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
144	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
145	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
146	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
147	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
148	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
149	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
150	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
151	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
152	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
153	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
154	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
155	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
156	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
157	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
158	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
159	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
160	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
161	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
162	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
163	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
164	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
165	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
166	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
167	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
168	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
169	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
170	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
171	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
172	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
173	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
174	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
175	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
176	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
177	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
178	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
179	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
180	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
181	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
182	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
183	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
184	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
185	C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
186	C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
187	C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
188	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
189	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
190	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
191	C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
192	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
193	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
194	C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
195	C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
196	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
197	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
198	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
199	C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
200	C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
201	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
202	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
203	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
204	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
205	C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
206	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
207	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
208	C₂²⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
209	C₄²×C₃₁	C₂³	1	1	yes	characteristic
210	C₄²×C₃₁	C₂³	1	1	yes	characteristic
211	C₄²×C₃₁	C₂³	1	1	yes	characteristic
212	C₄²×C₃₁	C₂³	1	1	yes	characteristic
213	C₄²×C₃₁	C₂³	1	2	yes	—
214	C₄²×C₃₁	C₂³	1	2	yes	—
215	C₄²×C₃₁	C₂³	1	2	yes	—
216	C₄²×C₃₁	C₂³	1	2	yes	—
217	C₄²×C₃₁	C₂³	1	1	yes	characteristic
218	C₄²×C₃₁	C₂³	1	1	yes	characteristic
219	C₄²×C₃₁	C₂³	1	2	yes	—
220	C₄²×C₃₁	C₂³	1	2	yes	—
221	C₄²×C₃₁	C₂³	1	2	yes	—
222	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
223	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
224	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
225	C₂³×C₃₁	C₂⁴	1	2	yes	—
226	Q₈×C₃₁	—	2	2	no	—
227	Q₈×C₃₁	—	4	1	no	—
228	Q₈×C₃₁	—	4	2	no	—
229	Q₈×C₃₁	—	4	2	no	—
230	Q₈×C₃₁	—	4	2	no	—
231	Q₈×C₃₁	—	4	2	no	—
232	D₄×C₃₁	—	2	2	no	—
233	D₄×C₃₁	—	4	1	no	—
234	D₄×C₃₁	—	4	1	no	—
235	D₄×C₃₁	—	4	2	no	—
236	D₄×C₃₁	—	4	2	no	—
237	D₄×C₃₁	—	4	1	no	—
238	D₄×C₃₁	—	4	2	no	—
239	C₂×C₄×C₃₁	—	2	2	no	—
240	C₂×C₄×C₃₁	—	2	2	no	—
241	C₂×C₄×C₃₁	—	2	2	no	—
242	C₂×C₄×C₃₁	—	4	2	no	—
243	C₂×C₄×C₃₁	—	4	1	no	—
244	C₂×C₄×C₃₁	—	4	2	no	—
245	C₂×C₄×C₃₁	—	4	2	no	—
246	C₂×C₄×C₃₁	—	4	2	no	—
247	C₂×C₄×C₃₁	—	4	2	no	—
248	C₂×C₄×C₃₁	—	4	1	no	—
249	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
250	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
251	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
252	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
253	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
254	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
255	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
256	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
257	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
258	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
259	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
260	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
261	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
262	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
263	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
264	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
265	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
266	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
267	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
268	#3565	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
269	C₂²×C₃₁	—	2	1	no	—
270	C₂²×C₃₁	—	2	1	no	—
271	C₂²×C₃₁	—	2	1	no	—
272	C₂²×C₃₁	—	2	2	no	—
273	C₂²×C₃₁	—	2	1	no	—
274	C₂²×C₃₁	—	2	1	no	—
275	C₂²×C₃₁	—	2	2	no	—
276	C₂²×C₃₁	—	2	1	no	—
277	C₂²×C₃₁	—	2	1	no	—
278	C₂²×C₃₁	—	2	2	no	—
279	C₂²×C₃₁	—	2	1	no	—
280	C₂²×C₃₁	—	2	1	no	—
281	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
282	C₄×C₃₁	—	2	1	no	—
283	C₄×C₃₁	—	2	1	no	—
284	C₄×C₃₁	—	2	1	no	—
285	C₄×C₃₁	—	2	2	no	—
286	C₄×C₃₁	—	2	1	no	—
287	C₄×C₃₁	—	2	1	no	—
288	C₄×C₃₁	—	2	2	no	—
289	C₄×C₃₁	—	2	1	no	—
290	C₄×C₃₁	—	2	2	no	—
291	C₄×C₃₁	—	2	2	no	—
292	C₄×C₃₁	—	2	2	no	—
293	C₄×C₃₁	—	2	1	no	—
294	C₄×C₃₁	—	2	1	no	—
295	C₄×C₃₁	—	2	2	no	—
296	C₄×C₃₁	—	2	1	no	—
297	C₄×C₃₁	—	2	1	no	—
298	C₄×C₃₁	—	2	2	no	—
299	C₄×C₃₁	—	2	1	no	—
300	C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
301	#563	C₂×C₃₁	1	2	yes	complemented
302	#562	C₂×C₃₁	1	2	yes	complemented
303	#557	C₂×C₃₁	1	1	yes	characteristic, complemented
304	#556	C₂×C₃₁	1	2	yes	complemented
305	#556	C₂×C₃₁	1	1	yes	characteristic, complemented
306	#555	C₂×C₃₁	1	1	yes	characteristic, complemented
307	#554	C₂×C₃₁	1	1	yes	characteristic, complemented
308	#553	C₂×C₃₁	1	1	yes	characteristic, complemented
309	#553	C₂×C₃₁	1	1	yes	characteristic, complemented
310	#552	C₂×C₃₁	1	2	yes	complemented
311	#552	C₂×C₃₁	1	1	yes	characteristic, complemented
312	#552	C₂×C₃₁	1	1	yes	characteristic, complemented
313	#544	C₂×C₃₁	1	1	yes	characteristic, complemented
314	#544	C₂×C₃₁	1	1	yes	characteristic, complemented
315	#542	C₂×C₃₁	1	1	yes	characteristic, complemented
316	#542	C₂×C₃₁	1	1	yes	characteristic, complemented
317	#542	C₂×C₃₁	1	2	yes	complemented
318	#542	C₂×C₃₁	1	2	yes	complemented
319	#541	C₂×C₃₁	1	2	yes	complemented
320	#539	C₂×C₃₁	1	2	yes	complemented
321	#533	C₂×C₃₁	1	1	yes	characteristic, complemented
322	#529	C₂×C₃₁	1	1	yes	characteristic, complemented
323	#529	C₂×C₃₁	1	1	yes	characteristic, complemented
324	C₂×C₃₁	—	4	1	no	—
325	C₂×C₃₁	—	4	1	no	—
326	C₂×C₃₁	—	4	1	no	—
327	C₂×C₃₁	—	4	2	no	—
328	C₂×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
329	C₂×C₃₁	#585	1	1	yes	characteristic
330	C₂×C₃₁	#585	1	1	yes	characteristic
331	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
332	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
333	C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
334	C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
335	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
336	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
337	C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
338	#126	C₂²×C₃₁	1	2	yes	—
339	#126	C₂²×C₃₁	1	1	yes	characteristic
340	#126	C₂²×C₃₁	1	1	yes	characteristic
341	#126	C₂²×C₃₁	1	2	yes	—
342	#126	C₂²×C₃₁	1	2	yes	—
343	#126	C₂²×C₃₁	1	2	yes	—
344	#126	C₂²×C₃₁	1	2	yes	—
345	#126	C₂²×C₃₁	1	2	yes	—
346	#126	C₂²×C₃₁	1	2	yes	—
347	#126	C₂²×C₃₁	1	2	yes	—
348	#126	C₂²×C₃₁	1	2	yes	—
349	#125	C₂²×C₃₁	1	1	yes	characteristic
350	#125	C₂²×C₃₁	1	1	yes	characteristic
351	#125	C₂²×C₃₁	1	1	yes	characteristic
352	#125	C₂²×C₃₁	1	1	yes	characteristic
353	#125	C₂²×C₃₁	1	2	yes	—
354	#125	C₂²×C₃₁	1	2	yes	—
355	#125	C₂²×C₃₁	1	2	yes	—
356	#125	C₂²×C₃₁	1	2	yes	—
357	#125	C₂²×C₃₁	1	2	yes	—
358	#125	C₂²×C₃₁	1	1	yes	characteristic
359	#125	C₂²×C₃₁	1	2	yes	—
360	#125	C₂²×C₃₁	1	1	yes	characteristic
361	#124	C₂²×C₃₁	1	1	yes	characteristic
362	#124	C₂²×C₃₁	1	1	yes	characteristic
363	#124	C₂²×C₃₁	1	2	yes	—
364	#124	C₂²×C₃₁	1	2	yes	—
365	#124	C₂²×C₃₁	1	2	yes	—
366	#124	C₂²×C₃₁	1	2	yes	—
367	#124	C₂²×C₃₁	1	2	yes	—
368	#123	C₂²×C₃₁	1	1	yes	characteristic
369	#123	C₂²×C₃₁	1	1	yes	characteristic
370	#123	C₂²×C₃₁	1	2	yes	—
371	#123	C₂²×C₃₁	1	2	yes	—
372	#123	C₂²×C₃₁	1	2	yes	—
373	#123	C₂²×C₃₁	1	2	yes	—
374	#123	C₂²×C₃₁	1	2	yes	—
375	#123	C₂²×C₃₁	1	2	yes	—
376	#123	C₂²×C₃₁	1	2	yes	—
377	C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
378	C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
379	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
380	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
381	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
382	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
383	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
384	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
385	C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
386	C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
387	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
388	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
389	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
390	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
391	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
392	#121	C₂²×C₃₁	1	1	yes	characteristic
393	#121	C₂²×C₃₁	1	1	yes	characteristic
394	#121	C₂²×C₃₁	1	2	yes	—
395	#121	C₂²×C₃₁	1	1	yes	characteristic
396	#121	C₂²×C₃₁	1	1	yes	characteristic
397	#121	C₂²×C₃₁	1	2	yes	—
398	#121	C₂²×C₃₁	1	2	yes	—
399	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
400	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
401	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
402	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
403	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
404	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
405	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
406	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
407	C₄×D₄	C₂²×C₃₁	1	2	yes	—
408	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
409	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
410	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
411	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
412	C₄×D₄	C₂²×C₃₁	1	2	yes	—
413	C₄×D₄	C₂²×C₃₁	1	2	yes	—
414	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
415	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
416	#117	C₂²×C₃₁	1	1	yes	characteristic
417	#117	C₂²×C₃₁	1	1	yes	characteristic
418	#117	C₂²×C₃₁	1	2	yes	—
419	#117	C₂²×C₃₁	1	1	yes	characteristic
420	#117	C₂²×C₃₁	1	1	yes	characteristic
421	#117	C₂²×C₃₁	1	2	yes	—
422	#117	C₂²×C₃₁	1	2	yes	—
423	#117	C₂²×C₃₁	1	2	yes	—
424	#117	C₂²×C₃₁	1	1	yes	characteristic
425	#117	C₂²×C₃₁	1	1	yes	characteristic
426	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
427	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
428	C₂×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
429	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
430	C₃₁	#3565	1	1	yes	characteristic, direct product factor, Sylow
431	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
432	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
433	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
434	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
435	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
436	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
437	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
438	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
439	C₂×D₄	C₂³×C₃₁	1	2	yes	—
440	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
441	C₂×D₄	C₂³×C₃₁	1	2	yes	—
442	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
443	C₂×D₄	C₂³×C₃₁	1	2	yes	—
444	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
445	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
446	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
447	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
448	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
449	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
450	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
451	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
452	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
453	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
454	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
455	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
456	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
457	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
458	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
459	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
460	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
461	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
462	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
463	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
464	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
465	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
466	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
467	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
468	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
469	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
470	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
471	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
472	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
473	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
474	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
475	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
476	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
477	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
478	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
479	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
480	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
481	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
482	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
483	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
484	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
485	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
486	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
487	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
488	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
489	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
490	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
491	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
492	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
493	C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
494	C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
495	C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
496	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
497	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
498	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
499	C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
500	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
501	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
502	C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
503	C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
504	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
505	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
506	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
507	C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
508	C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
509	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
510	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
511	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
512	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
513	C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
514	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
515	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
516	C₂²⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
517	C₄²	C₂³×C₃₁	1	1	yes	characteristic
518	C₄²	C₂³×C₃₁	1	1	yes	characteristic
519	C₄²	C₂³×C₃₁	1	1	yes	characteristic
520	C₄²	C₂³×C₃₁	1	1	yes	characteristic
521	C₄²	C₂³×C₃₁	1	2	yes	—
522	C₄²	C₂³×C₃₁	1	2	yes	—
523	C₄²	C₂³×C₃₁	1	2	yes	—
524	C₄²	C₂³×C₃₁	1	2	yes	—
525	C₄²	C₂³×C₃₁	1	1	yes	characteristic
526	C₄²	C₂³×C₃₁	1	1	yes	characteristic
527	C₄²	C₂³×C₃₁	1	2	yes	—
528	C₄²	C₂³×C₃₁	1	2	yes	—
529	C₄²	C₂³×C₃₁	1	2	yes	—
530	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
531	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
532	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
533	C₂³	C₂⁴×C₃₁	1	2	yes	—
534	Q₈	—	2	2	no	—
535	Q₈	—	4	1	no	—
536	Q₈	—	4	2	no	—
537	Q₈	—	4	2	no	—
538	Q₈	—	4	2	no	—
539	Q₈	—	4	2	no	—
540	D₄	—	2	2	no	—
541	D₄	—	4	1	no	—
542	D₄	—	4	1	no	—
543	D₄	—	4	2	no	—
544	D₄	—	4	2	no	—
545	D₄	—	4	1	no	—
546	D₄	—	4	2	no	—
547	C₂×C₄	—	2	2	no	—
548	C₂×C₄	—	2	2	no	—
549	C₂×C₄	—	2	2	no	—
550	C₂×C₄	—	4	2	no	—
551	C₂×C₄	—	4	1	no	—
552	C₂×C₄	—	4	2	no	—
553	C₂×C₄	—	4	2	no	—
554	C₂×C₄	—	4	2	no	—
555	C₂×C₄	—	4	2	no	—
556	C₂×C₄	—	4	1	no	—
557	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
558	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
559	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
560	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
561	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
562	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
563	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
564	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
565	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
566	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
567	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
568	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
569	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
570	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
571	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
572	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
573	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
574	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
575	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
576	C₂²	—	2	1	no	—
577	C₂²	—	2	1	no	—
578	C₂²	—	2	1	no	—
579	C₂²	—	2	2	no	—
580	C₂²	—	2	1	no	—
581	C₂²	—	2	1	no	—
582	C₂²	—	2	2	no	—
583	C₂²	—	2	1	no	—
584	C₂²	—	2	1	no	—
585	C₂²	—	2	2	no	—
586	C₂²	—	2	1	no	—
587	C₂²	—	2	1	no	—
588	C₂²	C₂⁵×C₃₁	1	1	yes	characteristic
589	C₄	—	2	1	no	—
590	C₄	—	2	1	no	—
591	C₄	—	2	1	no	—
592	C₄	—	2	2	no	—
593	C₄	—	2	1	no	—
594	C₄	—	2	1	no	—
595	C₄	—	2	2	no	—
596	C₄	—	2	1	no	—
597	C₄	—	2	2	no	—
598	C₄	—	2	2	no	—
599	C₄	—	2	2	no	—
600	C₄	—	2	1	no	—
601	C₄	—	2	1	no	—
602	C₄	—	2	2	no	—
603	C₄	—	2	1	no	—
604	C₄	—	2	1	no	—
605	C₄	—	2	2	no	—
606	C₄	—	2	1	no	—
607	C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
608	C₂	—	4	1	no	—
609	C₂	—	4	1	no	—
610	C₂	—	4	1	no	—
611	C₂	—	4	2	no	—
612	C₂	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
613	C₂	C₃₁×#585	1	1	yes	characteristic
614	C₂	C₃₁×#585	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₃₁
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ G

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
1. C₃₁×#563	C₂	semidirect
2. C₃₁×#562	C₂	semidirect
3. C₃₁×#557	C₂	semidirect
4. C₃₁×#556	C₂	semidirect
5. C₃₁×#556	C₂	semidirect
6. C₃₁×#555	C₂	semidirect
7. C₃₁×#554	C₂	semidirect
8. C₃₁×#553	C₂	semidirect
9. C₃₁×#553	C₂	semidirect
10. C₃₁×#552	C₂	semidirect
11. C₃₁×#552	C₂	semidirect
12. C₃₁×#552	C₂	semidirect
13. C₃₁×#544	C₂	semidirect
14. C₃₁×#544	C₂	semidirect
15. C₃₁×#542	C₂	semidirect
16. C₃₁×#542	C₂	semidirect
17. C₃₁×#542	C₂	semidirect
18. C₃₁×#542	C₂	semidirect
19. C₃₁×#541	C₂	semidirect
20. C₃₁×#539	C₂	semidirect
21. C₃₁×#533	C₂	semidirect
22. C₃₁×#529	C₂	semidirect
23. C₃₁×#529	C₂	semidirect
268. #3565	C₃₁	direct (N × H)
301. #563	C₂×C₃₁	semidirect
302. #562	C₂×C₃₁	semidirect
303. #557	C₂×C₃₁	semidirect
304. #556	C₂×C₃₁	semidirect
305. #556	C₂×C₃₁	semidirect
306. #555	C₂×C₃₁	semidirect
307. #554	C₂×C₃₁	semidirect
308. #553	C₂×C₃₁	semidirect
309. #553	C₂×C₃₁	semidirect
310. #552	C₂×C₃₁	semidirect
311. #552	C₂×C₃₁	semidirect
312. #552	C₂×C₃₁	semidirect
313. #544	C₂×C₃₁	semidirect
314. #544	C₂×C₃₁	semidirect
315. #542	C₂×C₃₁	semidirect
316. #542	C₂×C₃₁	semidirect
317. #542	C₂×C₃₁	semidirect
318. #542	C₂×C₃₁	semidirect
319. #541	C₂×C₃₁	semidirect
320. #539	C₂×C₃₁	semidirect
321. #533	C₂×C₃₁	semidirect
322. #529	C₂×C₃₁	semidirect
323. #529	C₂×C₃₁	semidirect
430. C₃₁	#3565	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₃₁×#563	2
C₃₁×#562	2
C₃₁×#557	2
C₃₁×#556	2
C₃₁×#555	2
C₃₁×#554	2
C₃₁×#553	2
C₃₁×#552	2
C₃₁×#544	2
C₃₁×#542	2
C₃₁×#541	2
C₃₁×#539	2
C₃₁×#533	2
C₃₁×#529	2
#3565	31

Subgroups — C31×#3565 #16843498

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3565 #16843498