Subgroups — C₃₁×#3515 #16843448

Order 3968.

1014 conjugacy classes of subgroups.

The subgroup lattice has 434 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#557	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#554	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#554	C₂	1	1	yes	maximal, characteristic, complemented
4	C₃₁×#553	C₂	1	2	yes	maximal, complemented
5	C₃₁×#552	C₂	1	2	yes	maximal, complemented
6	C₃₁×#552	C₂	1	4	yes	maximal, complemented
7	C₃₁×#552	C₂	1	2	yes	maximal, complemented
8	C₃₁×#547	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#547	C₂	1	2	yes	maximal, complemented
10	C₃₁×#541	C₂	1	2	yes	maximal, complemented
11	C₃₁×#539	C₂	1	4	yes	maximal, complemented
12	C₃₁×#533	C₂	1	1	yes	maximal, characteristic, complemented
13	C₃₁×#529	C₂	1	2	yes	maximal, complemented
14	C₃₁×#529	C₂	1	1	yes	maximal, characteristic, complemented
15	C₂×C₃₁×#125	C₂	1	2	yes	maximal, complemented
16	C₃₁×#520	C₂	1	2	yes	maximal, complemented
17	C₄×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
18	C₂×#41×C₃₁	C₂²	1	1	yes	characteristic
19	C₂×#41×C₃₁	C₂²	1	2	yes	—
20	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
21	C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
22	C₃₁×C₄⋊Q₈	C₂²	1	2	yes	—
23	C₃₁×#126	C₂²	1	4	yes	—
24	C₃₁×#126	C₂²	1	2	yes	—
25	C₃₁×#126	C₂²	1	4	yes	—
26	C₃₁×#126	C₂²	1	4	yes	—
27	C₃₁×#125	C₂²	1	1	yes	characteristic
28	C₃₁×#125	C₂²	1	2	yes	—
29	C₃₁×#125	C₂²	1	4	yes	—
30	C₃₁×#125	C₂²	1	4	yes	—
31	C₃₁×#125	C₂²	1	2	yes	—
32	C₃₁×#125	C₂²	1	2	yes	—
33	C₃₁×#125	C₂²	1	4	yes	—
34	C₃₁×#125	C₂²	1	4	yes	—
35	C₃₁×#125	C₂²	1	2	yes	—
36	C₃₁×#124	C₂²	1	1	yes	characteristic
37	C₃₁×#123	C₂²	1	2	yes	—
38	C₃₁×#123	C₂²	1	4	yes	—
39	C₃₁×#123	C₂²	1	4	yes	—
40	C₃₁×#123	C₂²	1	4	yes	—
41	C₃₁×#123	C₂²	1	4	yes	—
42	C₃₁×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
43	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
44	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
45	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
46	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
47	C₃₁×C₂²⋊Q₈	C₂²	1	4	yes	—
48	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
49	C₃₁×C₂²⋊Q₈	C₂²	1	2	yes	—
50	C₃₁×#121	C₂²	1	2	yes	—
51	C₃₁×#121	C₂²	1	1	yes	characteristic
52	C₃₁×#121	C₂²	1	4	yes	—
53	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
54	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
55	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
56	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
57	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
58	C₄×Q₈×C₃₁	C₂²	1	1	yes	characteristic
59	C₄×Q₈×C₃₁	C₂²	1	2	yes	—
60	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
61	C₄×D₄×C₃₁	C₂²	1	2	yes	—
62	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
63	C₄×D₄×C₃₁	C₂²	1	2	yes	—
64	C₄×D₄×C₃₁	C₂²	1	4	yes	—
65	C₄×D₄×C₃₁	C₂²	1	2	yes	—
66	C₄×D₄×C₃₁	C₂²	1	4	yes	—
67	C₄×D₄×C₃₁	C₂²	1	2	yes	—
68	C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
69	C₄×D₄×C₃₁	C₂²	1	2	yes	—
70	C₃₁×#117	C₂²	1	1	yes	characteristic
71	C₃₁×#117	C₂²	1	2	yes	—
72	C₃₁×#117	C₂²	1	2	yes	—
73	C₃₁×#117	C₂²	1	4	yes	—
74	C₃₁×#117	C₂²	1	4	yes	—
75	C₃₁×#117	C₂²	1	2	yes	—
76	C₂×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
77	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
78	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
79	C₂×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
80	C₂×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
81	C₂×C₄²×C₃₁	C₂²	1	2	yes	—
82	C₂×C₄²×C₃₁	C₂²	1	1	yes	characteristic
83	#41×C₃₁	—	2	4	no	—
84	#41×C₃₁	—	2	8	no	—
85	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
86	C₂×Q₈×C₃₁	C₂³	1	1	yes	characteristic
87	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
88	C₂×Q₈×C₃₁	C₂³	1	2	yes	—
89	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
90	C₂×D₄×C₃₁	C₂³	1	2	yes	—
91	C₂×D₄×C₃₁	C₂³	1	4	yes	—
92	C₂×D₄×C₃₁	C₂³	1	2	yes	—
93	C₂×D₄×C₃₁	C₂³	1	1	yes	characteristic
94	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
95	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
96	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
99	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
100	C₂²×C₄×C₃₁	C₂³	1	2	yes	—
101	C₂²×C₄×C₃₁	C₂³	1	1	yes	characteristic
102	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
103	C₂²×C₄×C₃₁	C₂³	1	4	yes	—
104	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
106	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
107	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
108	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
109	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
110	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
111	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
112	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
113	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
114	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
115	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
116	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
117	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
118	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
119	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
120	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
121	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
122	C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
123	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
124	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
125	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
126	C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
127	C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
128	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
129	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
130	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
131	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
132	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
133	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
134	C₂²⋊C₄×C₃₁	C₂³	1	2	yes	—
135	C₂²⋊C₄×C₃₁	C₂³	1	4	yes	—
136	C₂²⋊C₄×C₃₁	C₂³	1	4	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	C₄²×C₃₁	C₂³	1	1	yes	characteristic
141	C₄²×C₃₁	C₂³	1	2	yes	—
142	C₄²×C₃₁	C₂³	1	1	yes	characteristic
143	C₄²×C₃₁	C₂³	1	1	yes	characteristic
144	C₄²×C₃₁	C₂³	1	2	yes	—
145	C₄²×C₃₁	C₂³	1	1	yes	characteristic
146	C₄²×C₃₁	C₂³	1	2	yes	—
147	C₄²×C₃₁	C₂³	1	4	yes	—
148	C₄²×C₃₁	C₂³	1	4	yes	—
149	C₄²×C₃₁	C₂³	1	2	yes	—
150	C₂³×C₃₁	C₂⁴	1	1	yes	characteristic
151	C₂³×C₃₁	C₂⁴	1	2	yes	—
152	C₂³×C₃₁	C₂⁴	1	2	yes	—
153	Q₈×C₃₁	—	2	2	no	—
154	Q₈×C₃₁	—	2	4	no	—
155	Q₈×C₃₁	—	2	2	no	—
156	Q₈×C₃₁	—	4	2	no	—
157	D₄×C₃₁	—	2	4	no	—
158	D₄×C₃₁	—	2	8	no	—
159	D₄×C₃₁	—	2	2	no	—
160	D₄×C₃₁	—	4	2	no	—
161	D₄×C₃₁	#41	1	4	yes	—
162	C₂×C₄×C₃₁	—	2	4	no	—
163	C₂×C₄×C₃₁	—	2	8	no	—
164	C₂×C₄×C₃₁	—	2	2	no	—
165	C₂×C₄×C₃₁	—	2	4	no	—
166	C₂×C₄×C₃₁	—	2	4	no	—
167	C₂×C₄×C₃₁	—	2	2	no	—
168	C₂×C₄×C₃₁	—	4	2	no	—
169	C₂×C₄×C₃₁	—	4	4	no	—
170	C₂×C₄×C₃₁	—	4	1	no	—
171	C₂×C₄×C₃₁	—	4	4	no	—
172	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
173	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
174	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
175	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
176	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
177	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
178	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
179	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
180	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
181	C₂×C₄×C₃₁	C₂⁴	1	4	yes	—
182	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
183	C₂×C₄×C₃₁	C₂⁴	1	2	yes	—
184	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
186	#3515	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
187	C₂²×C₃₁	—	2	2	no	—
188	C₂²×C₃₁	—	2	2	no	—
189	C₂²×C₃₁	—	2	1	no	—
190	C₂²×C₃₁	—	2	2	no	—
191	C₂²×C₃₁	—	2	1	no	—
192	C₂²×C₃₁	—	2	2	no	—
193	C₂²×C₃₁	—	2	2	no	—
194	C₂²×C₃₁	—	2	1	no	—
195	C₂²×C₃₁	C₂×#41	1	4	yes	—
196	C₂²×C₃₁	C₂⁵	1	1	yes	characteristic
197	C₄×C₃₁	—	2	1	no	—
198	C₄×C₃₁	—	2	2	no	—
199	C₄×C₃₁	—	2	2	no	—
200	C₄×C₃₁	—	2	4	no	—
201	C₄×C₃₁	—	2	1	no	—
202	C₄×C₃₁	—	2	2	no	—
203	C₄×C₃₁	—	2	2	no	—
204	C₄×C₃₁	—	2	1	no	—
205	C₄×C₃₁	—	2	2	no	—
206	C₄×C₃₁	—	2	2	no	—
207	C₄×C₃₁	—	2	4	no	—
208	C₄×C₃₁	—	2	1	no	—
209	C₄×C₃₁	C₂×#41	1	2	yes	—
210	C₄×C₃₁	2₋¹⁺⁴	1	2	yes	—
211	#557	C₂×C₃₁	1	1	yes	characteristic, complemented
212	#554	C₂×C₃₁	1	1	yes	characteristic, complemented
213	#554	C₂×C₃₁	1	1	yes	characteristic, complemented
214	#553	C₂×C₃₁	1	2	yes	complemented
215	#552	C₂×C₃₁	1	2	yes	complemented
216	#552	C₂×C₃₁	1	4	yes	complemented
217	#552	C₂×C₃₁	1	2	yes	complemented
218	#547	C₂×C₃₁	1	1	yes	characteristic, complemented
219	#547	C₂×C₃₁	1	2	yes	complemented
220	#541	C₂×C₃₁	1	2	yes	complemented
221	#539	C₂×C₃₁	1	4	yes	complemented
222	#533	C₂×C₃₁	1	1	yes	characteristic, complemented
223	#529	C₂×C₃₁	1	2	yes	complemented
224	#529	C₂×C₃₁	1	1	yes	characteristic, complemented
225	C₂×#125	C₂×C₃₁	1	2	yes	complemented
226	#520	C₂×C₃₁	1	2	yes	complemented
227	C₄×#41	C₂×C₃₁	1	1	yes	characteristic, complemented
228	C₂×C₃₁	—	2	4	no	—
229	C₂×C₃₁	—	4	1	no	—
230	C₂×C₃₁	—	4	2	no	—
231	C₂×C₃₁	C₂²×#41	1	1	yes	characteristic
232	C₂×C₃₁	C₂×2₋¹⁺⁴	1	1	yes	characteristic
233	C₂×C₃₁	#585	1	1	yes	characteristic
234	C₂×#41	C₂²×C₃₁	1	1	yes	characteristic
235	C₂×#41	C₂²×C₃₁	1	2	yes	—
236	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
237	C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
238	C₄⋊Q₈	C₂²×C₃₁	1	2	yes	—
239	#126	C₂²×C₃₁	1	4	yes	—
240	#126	C₂²×C₃₁	1	2	yes	—
241	#126	C₂²×C₃₁	1	4	yes	—
242	#126	C₂²×C₃₁	1	4	yes	—
243	#125	C₂²×C₃₁	1	1	yes	characteristic
244	#125	C₂²×C₃₁	1	2	yes	—
245	#125	C₂²×C₃₁	1	4	yes	—
246	#125	C₂²×C₃₁	1	4	yes	—
247	#125	C₂²×C₃₁	1	2	yes	—
248	#125	C₂²×C₃₁	1	2	yes	—
249	#125	C₂²×C₃₁	1	4	yes	—
250	#125	C₂²×C₃₁	1	4	yes	—
251	#125	C₂²×C₃₁	1	2	yes	—
252	#124	C₂²×C₃₁	1	1	yes	characteristic
253	#123	C₂²×C₃₁	1	2	yes	—
254	#123	C₂²×C₃₁	1	4	yes	—
255	#123	C₂²×C₃₁	1	4	yes	—
256	#123	C₂²×C₃₁	1	4	yes	—
257	#123	C₂²×C₃₁	1	4	yes	—
258	C₂²⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
259	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
260	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
261	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
262	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
263	C₂²⋊Q₈	C₂²×C₃₁	1	4	yes	—
264	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
265	C₂²⋊Q₈	C₂²×C₃₁	1	2	yes	—
266	#121	C₂²×C₃₁	1	2	yes	—
267	#121	C₂²×C₃₁	1	1	yes	characteristic
268	#121	C₂²×C₃₁	1	4	yes	—
269	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
270	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
271	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
272	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
273	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
274	C₄×Q₈	C₂²×C₃₁	1	1	yes	characteristic
275	C₄×Q₈	C₂²×C₃₁	1	2	yes	—
276	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
277	C₄×D₄	C₂²×C₃₁	1	2	yes	—
278	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
279	C₄×D₄	C₂²×C₃₁	1	2	yes	—
280	C₄×D₄	C₂²×C₃₁	1	4	yes	—
281	C₄×D₄	C₂²×C₃₁	1	2	yes	—
282	C₄×D₄	C₂²×C₃₁	1	4	yes	—
283	C₄×D₄	C₂²×C₃₁	1	2	yes	—
284	C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
285	C₄×D₄	C₂²×C₃₁	1	2	yes	—
286	#117	C₂²×C₃₁	1	1	yes	characteristic
287	#117	C₂²×C₃₁	1	2	yes	—
288	#117	C₂²×C₃₁	1	2	yes	—
289	#117	C₂²×C₃₁	1	4	yes	—
290	#117	C₂²×C₃₁	1	4	yes	—
291	#117	C₂²×C₃₁	1	2	yes	—
292	C₂×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
293	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
294	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
295	C₂×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
296	C₂×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
297	C₂×C₄²	C₂²×C₃₁	1	2	yes	—
298	C₂×C₄²	C₂²×C₃₁	1	1	yes	characteristic
299	C₃₁	#3515	1	1	yes	characteristic, direct product factor, Sylow
300	#41	—	2	4	no	—
301	#41	—	2	8	no	—
302	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
303	C₂×Q₈	C₂³×C₃₁	1	1	yes	characteristic
304	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
305	C₂×Q₈	C₂³×C₃₁	1	2	yes	—
306	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
307	C₂×D₄	C₂³×C₃₁	1	2	yes	—
308	C₂×D₄	C₂³×C₃₁	1	4	yes	—
309	C₂×D₄	C₂³×C₃₁	1	2	yes	—
310	C₂×D₄	C₂³×C₃₁	1	1	yes	characteristic
311	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
312	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
313	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
314	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
315	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
316	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
317	C₂²×C₄	C₂³×C₃₁	1	2	yes	—
318	C₂²×C₄	C₂³×C₃₁	1	1	yes	characteristic
319	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
320	C₂²×C₄	C₂³×C₃₁	1	4	yes	—
321	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
322	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
323	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
324	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
325	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
326	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
327	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
328	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
329	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
330	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
331	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
332	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
333	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
334	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
335	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
336	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
337	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
338	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
339	C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
340	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
341	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
342	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
343	C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
344	C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
345	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
346	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
347	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
348	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
349	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
350	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
351	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
352	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
353	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
354	C₂²⋊C₄	C₂³×C₃₁	1	4	yes	—
355	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
356	C₂²⋊C₄	C₂³×C₃₁	1	2	yes	—
357	C₄²	C₂³×C₃₁	1	1	yes	characteristic
358	C₄²	C₂³×C₃₁	1	2	yes	—
359	C₄²	C₂³×C₃₁	1	1	yes	characteristic
360	C₄²	C₂³×C₃₁	1	1	yes	characteristic
361	C₄²	C₂³×C₃₁	1	2	yes	—
362	C₄²	C₂³×C₃₁	1	1	yes	characteristic
363	C₄²	C₂³×C₃₁	1	2	yes	—
364	C₄²	C₂³×C₃₁	1	4	yes	—
365	C₄²	C₂³×C₃₁	1	4	yes	—
366	C₄²	C₂³×C₃₁	1	2	yes	—
367	C₂³	C₂⁴×C₃₁	1	1	yes	characteristic
368	C₂³	C₂⁴×C₃₁	1	2	yes	—
369	C₂³	C₂⁴×C₃₁	1	2	yes	—
370	Q₈	—	2	2	no	—
371	Q₈	—	2	4	no	—
372	Q₈	—	2	2	no	—
373	Q₈	—	4	2	no	—
374	D₄	—	2	4	no	—
375	D₄	—	2	8	no	—
376	D₄	—	2	2	no	—
377	D₄	—	4	2	no	—
378	D₄	#41×C₃₁	1	4	yes	—
379	C₂×C₄	—	2	4	no	—
380	C₂×C₄	—	2	8	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	2	4	no	—
383	C₂×C₄	—	2	4	no	—
384	C₂×C₄	—	2	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	4	4	no	—
387	C₂×C₄	—	4	1	no	—
388	C₂×C₄	—	4	4	no	—
389	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
390	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
391	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
392	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
393	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
394	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
395	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
396	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
397	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
398	C₂×C₄	C₂⁴×C₃₁	1	4	yes	—
399	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
400	C₂×C₄	C₂⁴×C₃₁	1	2	yes	—
401	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
402	C₂×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
403	C₂²	—	2	2	no	—
404	C₂²	—	2	2	no	—
405	C₂²	—	2	1	no	—
406	C₂²	—	2	2	no	—
407	C₂²	—	2	1	no	—
408	C₂²	—	2	2	no	—
409	C₂²	—	2	2	no	—
410	C₂²	—	2	1	no	—
411	C₂²	C₂×#41×C₃₁	1	4	yes	—
412	C₂²	C₂⁵×C₃₁	1	1	yes	characteristic
413	C₄	—	2	1	no	—
414	C₄	—	2	2	no	—
415	C₄	—	2	2	no	—
416	C₄	—	2	4	no	—
417	C₄	—	2	1	no	—
418	C₄	—	2	2	no	—
419	C₄	—	2	2	no	—
420	C₄	—	2	1	no	—
421	C₄	—	2	2	no	—
422	C₄	—	2	2	no	—
423	C₄	—	2	4	no	—
424	C₄	—	2	1	no	—
425	C₄	C₂×#41×C₃₁	1	2	yes	—
426	C₄	C₃₁×2₋¹⁺⁴	1	2	yes	—
427	C₂	—	2	4	no	—
428	C₂	—	4	1	no	—
429	C₂	—	4	2	no	—
430	C₂	C₂²×#41×C₃₁	1	1	yes	characteristic
431	C₂	C₂×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic
432	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×#557	C₂	semidirect
2. C₃₁×#554	C₂	semidirect
3. C₃₁×#554	C₂	semidirect
4. C₃₁×#553	C₂	semidirect
5. C₃₁×#552	C₂	semidirect
6. C₃₁×#552	C₂	semidirect
7. C₃₁×#552	C₂	semidirect
8. C₃₁×#547	C₂	semidirect
9. C₃₁×#547	C₂	semidirect
10. C₃₁×#541	C₂	semidirect
11. C₃₁×#539	C₂	semidirect
12. C₃₁×#533	C₂	semidirect
13. C₃₁×#529	C₂	semidirect
14. C₃₁×#529	C₂	semidirect
15. C₂×C₃₁×#125	C₂	semidirect
16. C₃₁×#520	C₂	semidirect
17. C₄×#41×C₃₁	C₂	semidirect
186. #3515	C₃₁	direct (N × H)
211. #557	C₂×C₃₁	semidirect
212. #554	C₂×C₃₁	semidirect
213. #554	C₂×C₃₁	semidirect
214. #553	C₂×C₃₁	semidirect
215. #552	C₂×C₃₁	semidirect
216. #552	C₂×C₃₁	semidirect
217. #552	C₂×C₃₁	semidirect
218. #547	C₂×C₃₁	semidirect
219. #547	C₂×C₃₁	semidirect
220. #541	C₂×C₃₁	semidirect
221. #539	C₂×C₃₁	semidirect
222. #533	C₂×C₃₁	semidirect
223. #529	C₂×C₃₁	semidirect
224. #529	C₂×C₃₁	semidirect
225. C₂×#125	C₂×C₃₁	semidirect
226. #520	C₂×C₃₁	semidirect
227. C₄×#41	C₂×C₃₁	semidirect
299. C₃₁	#3515	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×#557	2
C₃₁×#554	2
C₃₁×#553	2
C₃₁×#552	2
C₃₁×#547	2
C₃₁×#541	2
C₃₁×#539	2
C₃₁×#533	2
C₃₁×#529	2
C₂×C₃₁×#125	2
C₃₁×#520	2
C₄×#41×C₃₁	2
#3515	31

Subgroups — C31×#3515 #16843448

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#3515 #16843448