Subgroups — C₃₁×#3546 #16843479

Order 3968.

1150 conjugacy classes of subgroups.

The subgroup lattice has 482 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#560	C₂	1	4	yes	maximal, complemented
2	C₃₁×#556	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#553	C₂	1	2	yes	maximal, complemented
4	C₃₁×#550	C₂	1	2	yes	maximal, complemented
5	C₃₁×#550	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#547	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#546	C₂	1	2	yes	maximal, complemented
8	C₃₁×#546	C₂	1	2	yes	maximal, complemented
9	D₄²×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#542	C₂	1	2	yes	maximal, complemented
11	C₃₁×#538	C₂	1	4	yes	maximal, complemented
12	C₃₁×#537	C₂	1	2	yes	maximal, complemented
13	C₃₁×#535	C₂	1	2	yes	maximal, complemented
14	C₃₁×#535	C₂	1	2	yes	maximal, complemented
15	C₃₁×#532	C₂	1	1	yes	maximal, characteristic, complemented
16	C₃₁×#532	C₂	1	1	yes	maximal, characteristic, complemented
17	C₃₁×#529	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#41×C₃₁	C₂²	1	2	yes	complemented
19	C₂×#41×C₃₁	C₂²	1	2	yes	complemented
20	C₂×#41×C₃₁	C₂²	1	2	yes	complemented
21	C₂×#41×C₃₁	C₂²	1	2	yes	complemented
22	C₂²×D₄×C₃₁	C₂²	1	2	yes	—
23	C₂²×D₄×C₃₁	C₂²	1	2	yes	—
24	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
25	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
26	C₃₁×#127	C₂²	1	1	yes	characteristic
27	C₃₁×#127	C₂²	1	1	yes	characteristic
28	C₃₁×#127	C₂²	1	2	yes	complemented
29	C₃₁×#127	C₂²	1	4	yes	complemented
30	C₃₁×#126	C₂²	1	2	yes	complemented
31	C₃₁×#126	C₂²	1	4	yes	complemented
32	C₃₁×#126	C₂²	1	4	yes	complemented
33	C₃₁×#125	C₂²	1	1	yes	characteristic, complemented
34	C₃₁×#124	C₂²	1	1	yes	characteristic
35	C₃₁×#124	C₂²	1	2	yes	—
36	C₃₁×#124	C₂²	1	2	yes	complemented
37	C₃₁×#124	C₂²	1	2	yes	complemented
38	C₃₁×#124	C₂²	1	4	yes	complemented
39	C₃₁×#123	C₂²	1	2	yes	complemented
40	C₃₁×#123	C₂²	1	4	yes	complemented
41	C₃₁×#123	C₂²	1	4	yes	complemented
42	C₃₁×#123	C₂²	1	4	yes	complemented
43	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic, complemented
44	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	complemented
45	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	complemented
46	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	complemented
47	C₃₁×#121	C₂²	1	1	yes	characteristic
48	C₃₁×#121	C₂²	1	1	yes	characteristic
49	C₃₁×#121	C₂²	1	2	yes	—
50	C₃₁×#121	C₂²	1	2	yes	—
51	C₃₁×#121	C₂²	1	2	yes	—
52	C₃₁×#121	C₂²	1	2	yes	complemented
53	C₃₁×#121	C₂²	1	1	yes	characteristic, complemented
54	C₃₁×#121	C₂²	1	2	yes	complemented
55	C₃₁×#121	C₂²	1	2	yes	complemented
56	C₃₁×#121	C₂²	1	4	yes	complemented
57	C₃₁×#121	C₂²	1	2	yes	complemented
58	C₃₁×#121	C₂²	1	4	yes	complemented
59	C₃₁×#121	C₂²	1	4	yes	complemented
60	C₃₁×#121	C₂²	1	4	yes	complemented
61	C₃₁×#121	C₂²	1	4	yes	complemented
62	C₃₁×#121	C₂²	1	4	yes	complemented
63	C₃₁×#120	C₂²	1	4	yes	—
64	C₃₁×#120	C₂²	1	4	yes	complemented
65	C₃₁×#120	C₂²	1	4	yes	complemented
66	C₃₁×#120	C₂²	1	4	yes	complemented
67	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic, complemented
68	C₄×Q₈×C₃₁	C₂²	1	2	yes	complemented
69	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
70	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
71	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
72	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
73	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
74	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
75	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic, complemented
76	C₄×D₄×C₃₁	C₂²	1	4	yes	complemented
77	C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
78	C₃₁×#117	C₂²	1	2	yes	complemented
79	C₃₁×#117	C₂²	1	2	yes	complemented
80	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
81	C₂×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	complemented
82	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic, complemented
83	C₂⁴×C₃₁	C₂³	1	4	yes	—
84	#41×C₃₁	—	4	4	no	—
85	#41×C₃₁	—	4	4	no	—
86	#41×C₃₁	—	4	4	no	—
87	#41×C₃₁	—	4	4	no	—
88	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
89	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
90	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
91	C₂×D₄×C₃₁	—	4	4	no	—
92	C₂×D₄×C₃₁	—	4	4	no	—
93	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
94	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
95	C₂×D₄×C₃₁	C₂³	1	2	yes	—
96	C₂×D₄×C₃₁	C₂³	1	2	yes	—
97	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
98	C₂×D₄×C₃₁	C₂³	1	2	yes	—
99	C₂×D₄×C₃₁	C₂³	1	2	yes	—
100	C₂×D₄×C₃₁	C₂³	1	2	yes	—
101	C₂×D₄×C₃₁	C₂³	1	4	yes	—
102	C₂×D₄×C₃₁	C₂³	1	4	yes	—
103	C₂×D₄×C₃₁	C₂³	1	4	yes	—
104	C₂×D₄×C₃₁	C₂³	1	2	yes	—
105	C₂×D₄×C₃₁	C₂³	1	4	yes	—
106	C₂×D₄×C₃₁	C₂³	1	2	yes	—
107	C₂×D₄×C₃₁	C₂³	1	4	yes	—
108	C₂×D₄×C₃₁	C₂³	1	4	yes	—
109	C₂×D₄×C₃₁	C₂³	1	2	yes	—
110	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
111	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
112	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
113	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
114	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
115	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
116	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
117	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
118	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
119	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
120	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
121	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
122	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
123	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
124	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
125	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
126	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
127	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
128	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
129	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
130	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
131	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
132	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
133	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
134	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
135	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
136	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
137	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
138	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
139	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
140	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
141	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
142	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
143	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
144	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
145	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
146	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
147	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
148	C₄²×C₃₁	C₂³	1	1	yes	characteristic
149	C₄²×C₃₁	C₂³	1	1	yes	characteristic
150	C₄²×C₃₁	C₂³	1	1	yes	characteristic
151	C₄²×C₃₁	C₂³	1	1	yes	characteristic
152	C₄²×C₃₁	C₂³	1	2	yes	—
153	C₄²×C₃₁	C₂³	1	4	yes	—
154	C₂³×C₃₁	—	4	4	no	—
155	C₂³×C₃₁	—	4	4	no	—
156	C₂³×C₃₁	—	4	4	no	—
157	C₂³×C₃₁	C₂⁴	1	2	yes	—
158	C₂³×C₃₁	C₂⁴	1	2	yes	—
159	C₂³×C₃₁	C₂⁴	1	4	yes	—
160	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
161	C₂³×C₃₁	C₂⁴	1	2	yes	—
162	Q₈×C₃₁	—	2	2	no	—
163	Q₈×C₃₁	—	2	4	no	—
164	Q₈×C₃₁	—	4	2	no	—
165	D₄×C₃₁	—	2	2	no	—
166	D₄×C₃₁	—	2	2	no	—
167	D₄×C₃₁	—	2	4	no	—
168	D₄×C₃₁	—	2	4	no	—
169	D₄×C₃₁	—	2	2	no	—
170	D₄×C₃₁	—	4	4	no	—
171	D₄×C₃₁	—	4	4	no	—
172	D₄×C₃₁	—	4	2	no	—
173	D₄×C₃₁	—	4	2	no	—
174	D₄×C₃₁	—	4	2	no	—
175	D₄×C₃₁	—	4	4	no	—
176	D₄×C₃₁	—	4	2	no	—
177	D₄×C₃₁	—	4	2	no	—
178	D₄×C₃₁	—	4	2	no	—
179	D₄×C₃₁	—	4	4	no	—
180	D₄×C₃₁	—	4	4	no	—
181	D₄×C₃₁	—	4	4	no	—
182	C₂×C₄×C₃₁	—	2	4	no	—
183	C₂×C₄×C₃₁	—	2	4	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	2	4	no	—
186	C₂×C₄×C₃₁	—	2	2	no	—
187	C₂×C₄×C₃₁	—	4	1	no	—
188	C₂×C₄×C₃₁	—	4	4	no	—
189	C₂×C₄×C₃₁	—	4	4	no	—
190	C₂×C₄×C₃₁	—	4	4	no	—
191	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
192	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
193	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
194	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
195	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
196	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
197	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
198	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
199	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
200	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
201	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
202	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
203	#3546	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
204	C₂²×C₃₁	—	2	1	no	—
205	C₂²×C₃₁	—	2	1	no	—
206	C₂²×C₃₁	—	2	2	no	—
207	C₂²×C₃₁	—	2	2	no	—
208	C₂²×C₃₁	—	2	2	no	—
209	C₂²×C₃₁	—	2	2	no	—
210	C₂²×C₃₁	—	2	4	no	—
211	C₂²×C₃₁	—	2	4	no	—
212	C₂²×C₃₁	—	2	2	no	—
213	C₂²×C₃₁	—	2	2	no	—
214	C₂²×C₃₁	—	2	1	no	—
215	C₂²×C₃₁	—	2	2	no	—
216	C₂²×C₃₁	—	2	4	no	—
217	C₂²×C₃₁	—	2	2	no	—
218	C₂²×C₃₁	—	2	2	no	—
219	C₂²×C₃₁	—	8	8	no	—
220	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
221	C₄×C₃₁	—	2	1	no	—
222	C₄×C₃₁	—	2	2	no	—
223	C₄×C₃₁	—	2	2	no	—
224	C₄×C₃₁	—	2	2	no	—
225	C₄×C₃₁	—	2	1	no	—
226	C₄×C₃₁	—	2	4	no	—
227	C₄×C₃₁	—	2	2	no	—
228	C₄×C₃₁	—	2	1	no	—
229	C₄×C₃₁	—	2	1	no	—
230	C₄×C₃₁	—	2	2	no	—
231	C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
232	C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
233	#560	C₂×C₃₁	1	4	yes	complemented
234	#556	C₂×C₃₁	1	1	yes	characteristic, complemented
235	#553	C₂×C₃₁	1	2	yes	complemented
236	#550	C₂×C₃₁	1	2	yes	complemented
237	#550	C₂×C₃₁	1	1	yes	characteristic, complemented
238	#547	C₂×C₃₁	1	1	yes	characteristic, complemented
239	#546	C₂×C₃₁	1	2	yes	complemented
240	#546	C₂×C₃₁	1	2	yes	complemented
241	D₄²	C₂×C₃₁	1	1	yes	characteristic, complemented
242	#542	C₂×C₃₁	1	2	yes	complemented
243	#538	C₂×C₃₁	1	4	yes	complemented
244	#537	C₂×C₃₁	1	2	yes	complemented
245	#535	C₂×C₃₁	1	2	yes	complemented
246	#535	C₂×C₃₁	1	2	yes	complemented
247	#532	C₂×C₃₁	1	1	yes	characteristic, complemented
248	#532	C₂×C₃₁	1	1	yes	characteristic, complemented
249	#529	C₂×C₃₁	1	1	yes	characteristic, complemented
250	C₂×C₃₁	—	4	1	no	—
251	C₂×C₃₁	—	4	2	no	—
252	C₂×C₃₁	—	4	4	no	—
253	C₂×C₃₁	—	4	2	no	—
254	C₂×C₃₁	—	4	2	no	—
255	C₂×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
256	C₂×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
257	C₂×C₃₁	#585	1	1	yes	characteristic
258	C₂×#41	C₂²×C₃₁	1	2	yes	complemented
259	C₂×#41	C₂²×C₃₁	1	2	yes	complemented
260	C₂×#41	C₂²×C₃₁	1	2	yes	complemented
261	C₂×#41	C₂²×C₃₁	1	2	yes	complemented
262	C₂²×D₄	C₂²×C₃₁	1	2	yes	—
263	C₂²×D₄	C₂²×C₃₁	1	2	yes	—
264	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
265	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
266	#127	C₂²×C₃₁	1	1	yes	characteristic
267	#127	C₂²×C₃₁	1	1	yes	characteristic
268	#127	C₂²×C₃₁	1	2	yes	complemented
269	#127	C₂²×C₃₁	1	4	yes	complemented
270	#126	C₂²×C₃₁	1	2	yes	complemented
271	#126	C₂²×C₃₁	1	4	yes	complemented
272	#126	C₂²×C₃₁	1	4	yes	complemented
273	#125	C₂²×C₃₁	1	1	yes	characteristic, complemented
274	#124	C₂²×C₃₁	1	1	yes	characteristic
275	#124	C₂²×C₃₁	1	2	yes	—
276	#124	C₂²×C₃₁	1	2	yes	complemented
277	#124	C₂²×C₃₁	1	2	yes	complemented
278	#124	C₂²×C₃₁	1	4	yes	complemented
279	#123	C₂²×C₃₁	1	2	yes	complemented
280	#123	C₂²×C₃₁	1	4	yes	complemented
281	#123	C₂²×C₃₁	1	4	yes	complemented
282	#123	C₂²×C₃₁	1	4	yes	complemented
283	C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
284	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	complemented
285	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
286	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	complemented
287	#121	C₂²×C₃₁	1	1	yes	characteristic
288	#121	C₂²×C₃₁	1	1	yes	characteristic
289	#121	C₂²×C₃₁	1	2	yes	—
290	#121	C₂²×C₃₁	1	2	yes	—
291	#121	C₂²×C₃₁	1	2	yes	—
292	#121	C₂²×C₃₁	1	2	yes	complemented
293	#121	C₂²×C₃₁	1	1	yes	characteristic, complemented
294	#121	C₂²×C₃₁	1	2	yes	complemented
295	#121	C₂²×C₃₁	1	2	yes	complemented
296	#121	C₂²×C₃₁	1	4	yes	complemented
297	#121	C₂²×C₃₁	1	2	yes	complemented
298	#121	C₂²×C₃₁	1	4	yes	complemented
299	#121	C₂²×C₃₁	1	4	yes	complemented
300	#121	C₂²×C₃₁	1	4	yes	complemented
301	#121	C₂²×C₃₁	1	4	yes	complemented
302	#121	C₂²×C₃₁	1	4	yes	complemented
303	#120	C₂²×C₃₁	1	4	yes	—
304	#120	C₂²×C₃₁	1	4	yes	complemented
305	#120	C₂²×C₃₁	1	4	yes	complemented
306	#120	C₂²×C₃₁	1	4	yes	complemented
307	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic, complemented
308	C₄×Q₈	C₂²×C₃₁	1	2	yes	complemented
309	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
310	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
311	C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
312	C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
313	C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
314	C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
315	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic, complemented
316	C₄×D₄	C₂²×C₃₁	1	4	yes	complemented
317	C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
318	#117	C₂²×C₃₁	1	2	yes	complemented
319	#117	C₂²×C₃₁	1	2	yes	complemented
320	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
321	C₂×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	complemented
322	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic, complemented
323	C₃₁	#3546	1	1	yes	characteristic, direct product factor, Sylow
324	C₂⁴	C₂³×C₃₁	1	4	yes	—
325	#41	—	4	4	no	—
326	#41	—	4	4	no	—
327	#41	—	4	4	no	—
328	#41	—	4	4	no	—
329	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
330	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
331	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
332	C₂×D₄	—	4	4	no	—
333	C₂×D₄	—	4	4	no	—
334	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
335	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
336	C₂×D₄	C₂³×C₃₁	1	2	yes	—
337	C₂×D₄	C₂³×C₃₁	1	2	yes	—
338	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
339	C₂×D₄	C₂³×C₃₁	1	2	yes	—
340	C₂×D₄	C₂³×C₃₁	1	2	yes	—
341	C₂×D₄	C₂³×C₃₁	1	2	yes	—
342	C₂×D₄	C₂³×C₃₁	1	4	yes	—
343	C₂×D₄	C₂³×C₃₁	1	4	yes	—
344	C₂×D₄	C₂³×C₃₁	1	4	yes	—
345	C₂×D₄	C₂³×C₃₁	1	2	yes	—
346	C₂×D₄	C₂³×C₃₁	1	4	yes	—
347	C₂×D₄	C₂³×C₃₁	1	2	yes	—
348	C₂×D₄	C₂³×C₃₁	1	4	yes	—
349	C₂×D₄	C₂³×C₃₁	1	4	yes	—
350	C₂×D₄	C₂³×C₃₁	1	2	yes	—
351	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
352	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
353	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
354	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
355	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
356	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
357	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
358	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
359	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
360	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
361	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
362	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
363	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
364	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
365	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
366	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
367	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
368	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
369	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
370	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
371	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
372	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
373	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
374	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
375	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
376	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
377	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
378	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
379	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
380	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
381	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
382	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
383	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
384	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
385	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
386	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
387	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
388	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
389	C₄²	C₂³×C₃₁	1	1	yes	characteristic
390	C₄²	C₂³×C₃₁	1	1	yes	characteristic
391	C₄²	C₂³×C₃₁	1	1	yes	characteristic
392	C₄²	C₂³×C₃₁	1	1	yes	characteristic
393	C₄²	C₂³×C₃₁	1	2	yes	—
394	C₄²	C₂³×C₃₁	1	4	yes	—
395	C₂³	—	4	4	no	—
396	C₂³	—	4	4	no	—
397	C₂³	—	4	4	no	—
398	C₂³	C₂⁴×C₃₁	1	2	yes	—
399	C₂³	C₂⁴×C₃₁	1	2	yes	—
400	C₂³	C₂⁴×C₃₁	1	4	yes	—
401	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
402	C₂³	C₂⁴×C₃₁	1	2	yes	—
403	Q₈	—	2	2	no	—
404	Q₈	—	2	4	no	—
405	Q₈	—	4	2	no	—
406	D₄	—	2	2	no	—
407	D₄	—	2	2	no	—
408	D₄	—	2	4	no	—
409	D₄	—	2	4	no	—
410	D₄	—	2	2	no	—
411	D₄	—	4	4	no	—
412	D₄	—	4	4	no	—
413	D₄	—	4	2	no	—
414	D₄	—	4	2	no	—
415	D₄	—	4	2	no	—
416	D₄	—	4	4	no	—
417	D₄	—	4	2	no	—
418	D₄	—	4	2	no	—
419	D₄	—	4	2	no	—
420	D₄	—	4	4	no	—
421	D₄	—	4	4	no	—
422	D₄	—	4	4	no	—
423	C₂×C₄	—	2	4	no	—
424	C₂×C₄	—	2	4	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	4	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	4	1	no	—
429	C₂×C₄	—	4	4	no	—
430	C₂×C₄	—	4	4	no	—
431	C₂×C₄	—	4	4	no	—
432	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
433	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
434	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
435	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
436	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
437	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
438	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
439	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
440	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
441	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
442	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
443	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
444	C₂²	—	2	1	no	—
445	C₂²	—	2	1	no	—
446	C₂²	—	2	2	no	—
447	C₂²	—	2	2	no	—
448	C₂²	—	2	2	no	—
449	C₂²	—	2	2	no	—
450	C₂²	—	2	4	no	—
451	C₂²	—	2	4	no	—
452	C₂²	—	2	2	no	—
453	C₂²	—	2	2	no	—
454	C₂²	—	2	1	no	—
455	C₂²	—	2	2	no	—
456	C₂²	—	2	4	no	—
457	C₂²	—	2	2	no	—
458	C₂²	—	2	2	no	—
459	C₂²	—	8	8	no	—
460	C₂²	C₂⁵×C₃₁	1	1	yes	characteristic
461	C₄	—	2	1	no	—
462	C₄	—	2	2	no	—
463	C₄	—	2	2	no	—
464	C₄	—	2	2	no	—
465	C₄	—	2	1	no	—
466	C₄	—	2	4	no	—
467	C₄	—	2	2	no	—
468	C₄	—	2	1	no	—
469	C₄	—	2	1	no	—
470	C₄	—	2	2	no	—
471	C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
472	C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
473	C₂	—	4	2	no	—
474	C₂	—	4	2	no	—
475	C₂	—	4	4	no	—
476	C₂	—	4	1	no	—
477	C₂	—	4	2	no	—
478	C₂	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
479	C₂	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
480	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×#560	C₂	semidirect
2. C₃₁×#556	C₂	semidirect
3. C₃₁×#553	C₂	semidirect
4. C₃₁×#550	C₂	semidirect
5. C₃₁×#550	C₂	semidirect
6. C₃₁×#547	C₂	semidirect
7. C₃₁×#546	C₂	semidirect
8. C₃₁×#546	C₂	semidirect
9. D₄²×C₃₁	C₂	semidirect
10. C₃₁×#542	C₂	semidirect
11. C₃₁×#538	C₂	semidirect
12. C₃₁×#537	C₂	semidirect
13. C₃₁×#535	C₂	semidirect
14. C₃₁×#535	C₂	semidirect
15. C₃₁×#532	C₂	semidirect
16. C₃₁×#532	C₂	semidirect
17. C₃₁×#529	C₂	semidirect
18. C₂×#41×C₃₁	C₂²	semidirect
19. C₂×#41×C₃₁	C₂²	semidirect
20. C₂×#41×C₃₁	C₂²	semidirect
21. C₂×#41×C₃₁	C₂²	semidirect
24. C₃₁×C₄⋊Q₈	C₂²	semidirect
25. C₃₁×C₄⋊Q₈	C₂²	semidirect
28. C₃₁×#127	C₂²	semidirect
29. C₃₁×#127	C₂²	semidirect
30. C₃₁×#126	C₂²	semidirect
31. C₃₁×#126	C₂²	semidirect
32. C₃₁×#126	C₂²	semidirect
33. C₃₁×#125	C₂²	semidirect
36. C₃₁×#124	C₂²	semidirect
37. C₃₁×#124	C₂²	semidirect
38. C₃₁×#124	C₂²	semidirect
39. C₃₁×#123	C₂²	semidirect
40. C₃₁×#123	C₂²	semidirect
41. C₃₁×#123	C₂²	semidirect
42. C₃₁×#123	C₂²	semidirect
43. C₃₁×C₂²⋊Q₈	C₂²	semidirect
44. C₃₁×C₂²⋊Q₈	C₂²	semidirect
45. C₃₁×C₂²⋊Q₈	C₂²	semidirect
46. C₃₁×C₂²⋊Q₈	C₂²	semidirect
52. C₃₁×#121	C₂²	semidirect
53. C₃₁×#121	C₂²	semidirect
54. C₃₁×#121	C₂²	semidirect
55. C₃₁×#121	C₂²	semidirect
56. C₃₁×#121	C₂²	semidirect
57. C₃₁×#121	C₂²	semidirect
58. C₃₁×#121	C₂²	semidirect
59. C₃₁×#121	C₂²	semidirect
60. C₃₁×#121	C₂²	semidirect
61. C₃₁×#121	C₂²	semidirect
62. C₃₁×#121	C₂²	semidirect
64. C₃₁×#120	C₂²	semidirect
65. C₃₁×#120	C₂²	semidirect
66. C₃₁×#120	C₂²	semidirect
67. C₄×Q₈×C₃₁	C₂²	semidirect
68. C₄×Q₈×C₃₁	C₂²	semidirect
71. C₄×D₄×C₃₁	C₂²	semidirect
72. C₄×D₄×C₃₁	C₂²	semidirect
73. C₄×D₄×C₃₁	C₂²	semidirect
74. C₄×D₄×C₃₁	C₂²	semidirect
75. C₄×D₄×C₃₁	C₂²	semidirect
76. C₄×D₄×C₃₁	C₂²	semidirect
77. C₄×D₄×C₃₁	C₂²	semidirect
78. C₃₁×#117	C₂²	semidirect
79. C₃₁×#117	C₂²	semidirect
80. C₂×C₄⋊C₄×C₃₁	C₂²	semidirect
81. C₂×C₂²⋊C₄×C₃₁	C₂²	semidirect
82. C₂×C₄²×C₃₁	C₂²	semidirect
203. #3546	C₃₁	direct (N × H)
233. #560	C₂×C₃₁	semidirect
234. #556	C₂×C₃₁	semidirect
235. #553	C₂×C₃₁	semidirect
236. #550	C₂×C₃₁	semidirect
237. #550	C₂×C₃₁	semidirect
238. #547	C₂×C₃₁	semidirect
239. #546	C₂×C₃₁	semidirect
240. #546	C₂×C₃₁	semidirect
241. D₄²	C₂×C₃₁	semidirect
242. #542	C₂×C₃₁	semidirect
243. #538	C₂×C₃₁	semidirect
244. #537	C₂×C₃₁	semidirect
245. #535	C₂×C₃₁	semidirect
246. #535	C₂×C₃₁	semidirect
247. #532	C₂×C₃₁	semidirect
248. #532	C₂×C₃₁	semidirect
249. #529	C₂×C₃₁	semidirect
258. C₂×#41	C₂²×C₃₁	semidirect
259. C₂×#41	C₂²×C₃₁	semidirect
260. C₂×#41	C₂²×C₃₁	semidirect
261. C₂×#41	C₂²×C₃₁	semidirect
264. C₄⋊Q₈	C₂²×C₃₁	semidirect
265. C₄⋊Q₈	C₂²×C₃₁	semidirect
268. #127	C₂²×C₃₁	semidirect
269. #127	C₂²×C₃₁	semidirect
270. #126	C₂²×C₃₁	semidirect
271. #126	C₂²×C₃₁	semidirect
272. #126	C₂²×C₃₁	semidirect
273. #125	C₂²×C₃₁	semidirect
276. #124	C₂²×C₃₁	semidirect
277. #124	C₂²×C₃₁	semidirect
278. #124	C₂²×C₃₁	semidirect
279. #123	C₂²×C₃₁	semidirect
280. #123	C₂²×C₃₁	semidirect
281. #123	C₂²×C₃₁	semidirect
282. #123	C₂²×C₃₁	semidirect
283. C₂²⋊Q₈	C₂²×C₃₁	semidirect
284. C₂²⋊Q₈	C₂²×C₃₁	semidirect
285. C₂²⋊Q₈	C₂²×C₃₁	semidirect
286. C₂²⋊Q₈	C₂²×C₃₁	semidirect
292. #121	C₂²×C₃₁	semidirect
293. #121	C₂²×C₃₁	semidirect
294. #121	C₂²×C₃₁	semidirect
295. #121	C₂²×C₃₁	semidirect
296. #121	C₂²×C₃₁	semidirect
297. #121	C₂²×C₃₁	semidirect
298. #121	C₂²×C₃₁	semidirect
299. #121	C₂²×C₃₁	semidirect
300. #121	C₂²×C₃₁	semidirect
301. #121	C₂²×C₃₁	semidirect
302. #121	C₂²×C₃₁	semidirect
304. #120	C₂²×C₃₁	semidirect
305. #120	C₂²×C₃₁	semidirect
306. #120	C₂²×C₃₁	semidirect
307. C₄×Q₈	C₂²×C₃₁	semidirect
308. C₄×Q₈	C₂²×C₃₁	semidirect
311. C₄×D₄	C₂²×C₃₁	semidirect
312. C₄×D₄	C₂²×C₃₁	semidirect
313. C₄×D₄	C₂²×C₃₁	semidirect
314. C₄×D₄	C₂²×C₃₁	semidirect
315. C₄×D₄	C₂²×C₃₁	semidirect
316. C₄×D₄	C₂²×C₃₁	semidirect
317. C₄×D₄	C₂²×C₃₁	semidirect
318. #117	C₂²×C₃₁	semidirect
319. #117	C₂²×C₃₁	semidirect
320. C₂×C₄⋊C₄	C₂²×C₃₁	semidirect
321. C₂×C₂²⋊C₄	C₂²×C₃₁	semidirect
322. C₂×C₄²	C₂²×C₃₁	semidirect
323. C₃₁	#3546	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×#560	2
C₃₁×#556	2
C₃₁×#553	2
C₃₁×#550	2
C₃₁×#547	2
C₃₁×#546	2
D₄²×C₃₁	2
C₃₁×#542	2
C₃₁×#538	2
C₃₁×#537	2
C₃₁×#535	2
C₃₁×#532	2
C₃₁×#529	2
#3546	31

Subgroups — C31×#3546 #16843479

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3546 #16843479