Subgroups — C₃₁×#60808 #21517567

Order 7936.

2162 conjugacy classes of subgroups.

The subgroup lattice has 460 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#540	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3449	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3221	C₂	1	8	yes	maximal, complemented
4	C₃₁×#3220	C₂	1	8	yes	maximal, complemented
5	C₃₁×#3151	C₂	1	2	yes	maximal, complemented
6	C₃₁×#3142	C₂	1	2	yes	maximal, complemented
7	C₃₁×#3036	C₂	1	4	yes	maximal, complemented
8	C₃₁×#3006	C₂	1	4	yes	maximal, complemented
9	C₃₁×#2921	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×#41×C₃₁	C₂²	1	2	yes	—
11	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	8	yes	complemented
12	C₃₁×#540	—	2	4	no	—
13	C₃₁×#540	C₂²	1	8	yes	complemented
14	C₃₁×#537	—	2	2	no	—
15	C₃₁×#536	C₂²	1	4	yes	complemented
16	C₃₁×#535	C₂²	1	4	yes	complemented
17	C₃₁×#534	—	2	2	no	—
18	C₃₁×#532	C₂²	1	2	yes	—
19	C₃₁×#532	C₂²	1	2	yes	—
20	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic, complemented
21	C₂×C₃₁×#125	C₂²	1	1	yes	characteristic, complemented
22	C₃₁×#525	—	2	4	no	—
23	C₂×C₃₁×#123	—	2	2	no	—
24	C₂×C₃₁×#121	—	2	2	no	—
25	C₂×C₃₁×#121	C₂²	1	2	yes	—
26	C₂×C₄×D₄×C₃₁	C₂²	1	2	yes	complemented
27	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
28	C₃₁×#496	C₂²	1	8	yes	complemented
29	C₃₁×#493	C₂²	1	4	yes	complemented
30	C₃₁×#492	C₂²	1	4	yes	complemented
31	C₃₁×#491	C₂²	1	4	yes	complemented
32	C₃₁×#490	C₂²	1	4	yes	complemented
33	C₃₁×#488	C₂²	1	8	yes	complemented
34	C₃₁×#463	C₂²	1	16	yes	complemented
35	C₃₁×#461	C₂²	1	8	yes	complemented
36	C₃₁×#459	C₂²	1	8	yes	complemented
37	C₃₁×#449	C₂²	1	16	yes	complemented
38	C₃₁×#448	C₂²	1	8	yes	complemented
39	C₃₁×#447	C₂²	1	8	yes	complemented
40	C₃₁×#433	C₂²	1	8	yes	complemented
41	C₃₁×#423	C₂²	1	2	yes	complemented
42	C₃₁×#417	C₂²	1	8	yes	complemented
43	C₃₁×#407	C₂²	1	2	yes	—
44	C₄×M₁₆×C₃₁	C₂²	1	2	yes	complemented
45	C₂×#41×C₃₁	—	2	2	no	—
46	C₂×#41×C₃₁	—	2	2	no	—
47	C₂×#41×C₃₁	—	4	8	no	—
48	C₂×#41×C₃₁	C₂³	1	4	yes	—
49	C₂×#41×C₃₁	C₂³	1	4	yes	—
50	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
51	C₂²×D₄×C₃₁	—	2	2	no	—
52	C₂²×D₄×C₃₁	—	2	2	no	—
53	C₂²×D₄×C₃₁	—	2	2	no	—
54	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
55	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
56	C₂³×C₄×C₃₁	—	2	2	no	—
57	C₂³×C₄×C₃₁	—	2	2	no	—
58	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
59	C₃₁×AGL(1,ℤ/8)	—	4	16	no	—
60	C₂×SD₁₆×C₃₁	C₂³	1	16	yes	—
61	C₂×D₈×C₃₁	C₂³	1	16	yes	—
62	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
63	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
64	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
65	C₃₁×#127	—	2	2	no	—
66	C₃₁×#127	C₂³	1	2	yes	—
67	C₃₁×#127	C₂³	1	2	yes	—
68	C₃₁×#127	C₂³	1	4	yes	—
69	C₃₁×#125	—	2	2	no	—
70	C₃₁×#125	C₂³	1	4	yes	—
71	C₃₁×#124	C₂³	1	4	yes	—
72	C₃₁×#124	C₂³	1	4	yes	—
73	C₃₁×#123	—	2	16	no	—
74	C₃₁×#123	—	2	8	no	—
75	C₃₁×C₂²⋊Q₈	—	2	8	no	—
76	C₃₁×#121	—	2	4	no	—
77	C₃₁×#121	—	2	4	no	—
78	C₃₁×#121	—	2	4	no	—
79	C₃₁×#121	—	2	8	no	—
80	C₃₁×#121	—	2	16	no	—
81	C₃₁×#121	C₂³	1	8	yes	—
82	C₃₁×#120	—	2	8	no	—
83	C₄×D₄×C₃₁	—	2	4	no	—
84	C₄×D₄×C₃₁	—	2	4	no	—
85	C₄×D₄×C₃₁	—	2	4	no	—
86	C₄×D₄×C₃₁	C₂³	1	8	yes	—
87	C₃₁×#117	D₄	1	4	yes	—
88	C₃₁×#117	C₂³	1	4	yes	—
89	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
90	C₂×C₄⋊C₄×C₃₁	D₄	1	2	yes	—
91	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
92	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
93	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
94	C₂×C₂²⋊C₄×C₃₁	D₄	1	2	yes	—
95	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
96	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
97	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
98	C₃₁×#103	C₂³	1	16	yes	—
99	C₃₁×#102	C₂³	1	16	yes	—
100	C₃₁×C₂²⋊C₈	C₂³	1	8	yes	—
101	C₃₁×#97	C₂³	1	4	yes	—
102	C₄×C₈×C₃₁	C₂³	1	4	yes	—
103	C₂⁴×C₃₁	—	2	2	no	—
104	C₂⁴×C₃₁	—	2	2	no	—
105	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
106	#41×C₃₁	—	2	16	no	—
107	#41×C₃₁	—	4	8	no	—
108	#41×C₃₁	—	4	8	no	—
109	#41×C₃₁	—	4	8	no	—
110	C₂×Q₈×C₃₁	—	2	2	no	—
111	C₂×Q₈×C₃₁	—	4	4	no	—
112	C₂×Q₈×C₃₁	C₂⁴	1	4	yes	—
113	C₂×D₄×C₃₁	—	2	4	no	—
114	C₂×D₄×C₃₁	—	2	4	no	—
115	C₂×D₄×C₃₁	—	2	8	no	—
116	C₂×D₄×C₃₁	—	2	2	no	—
117	C₂×D₄×C₃₁	—	2	2	no	—
118	C₂×D₄×C₃₁	—	2	4	no	—
119	C₂×D₄×C₃₁	—	2	4	no	—
120	C₂×D₄×C₃₁	—	2	8	no	—
121	C₂×D₄×C₃₁	—	2	8	no	—
122	C₂×D₄×C₃₁	—	4	4	no	—
123	C₂×D₄×C₃₁	—	4	8	no	—
124	C₂×D₄×C₃₁	—	8	4	no	—
125	C₂×D₄×C₃₁	—	8	4	no	—
126	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
127	C₂×D₄×C₃₁	C₂⁴	1	4	yes	—
128	C₂²×C₄×C₃₁	—	2	4	no	—
129	C₂²×C₄×C₃₁	—	2	2	no	—
130	C₂²×C₄×C₃₁	—	2	2	no	—
131	C₂²×C₄×C₃₁	—	2	4	no	—
132	C₂²×C₄×C₃₁	—	2	4	no	—
133	C₂²×C₄×C₃₁	—	2	8	no	—
134	C₂²×C₄×C₃₁	—	4	8	no	—
135	C₂²×C₄×C₃₁	—	8	4	no	—
136	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
137	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
138	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
139	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
140	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
141	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
142	SD₁₆×C₃₁	—	4	16	no	—
143	D₈×C₃₁	—	4	16	no	—
144	M₁₆×C₃₁	—	2	8	no	—
145	C₂×C₈×C₃₁	C₂⁴	1	8	yes	—
146	C₄⋊C₄×C₃₁	—	2	8	no	—
147	C₄⋊C₄×C₃₁	—	2	2	no	—
148	C₄⋊C₄×C₃₁	—	2	4	no	—
149	C₄⋊C₄×C₃₁	—	2	4	no	—
150	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
151	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
152	C₂²⋊C₄×C₃₁	—	2	8	no	—
153	C₂²⋊C₄×C₃₁	—	2	8	no	—
154	C₂²⋊C₄×C₃₁	—	2	8	no	—
155	C₂²⋊C₄×C₃₁	—	2	8	no	—
156	C₂²⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
157	C₄²×C₃₁	C₂×D₄	1	2	yes	—
158	C₄²×C₃₁	C₂×D₄	1	2	yes	—
159	C₄²×C₃₁	C₂⁴	1	2	yes	—
160	C₄²×C₃₁	C₂⁴	1	2	yes	—
161	#60808	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
162	C₂³×C₃₁	—	2	2	no	—
163	C₂³×C₃₁	—	2	4	no	—
164	C₂³×C₃₁	—	2	4	no	—
165	C₂³×C₃₁	—	4	2	no	—
166	C₂³×C₃₁	—	4	4	no	—
167	C₂³×C₃₁	—	4	4	no	—
168	C₂³×C₃₁	—	4	4	no	—
169	C₂³×C₃₁	—	4	4	no	—
170	C₂³×C₃₁	—	8	4	no	—
171	C₂³×C₃₁	C₂²×D₄	1	2	yes	—
172	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
173	Q₈×C₃₁	—	2	8	no	—
174	Q₈×C₃₁	—	4	4	no	—
175	D₄×C₃₁	—	2	8	no	—
176	D₄×C₃₁	—	2	8	no	—
177	D₄×C₃₁	—	4	4	no	—
178	D₄×C₃₁	—	4	8	no	—
179	D₄×C₃₁	—	4	8	no	—
180	D₄×C₃₁	—	4	4	no	—
181	D₄×C₃₁	—	8	4	no	—
182	D₄×C₃₁	—	8	4	no	—
183	D₄×C₃₁	—	8	8	no	—
184	D₄×C₃₁	—	8	8	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	2	4	no	—
187	C₂×C₄×C₃₁	—	2	4	no	—
188	C₂×C₄×C₃₁	—	2	4	no	—
189	C₂×C₄×C₃₁	—	2	4	no	—
190	C₂×C₄×C₃₁	—	4	2	no	—
191	C₂×C₄×C₃₁	—	4	2	no	—
192	C₂×C₄×C₃₁	—	4	8	no	—
193	C₂×C₄×C₃₁	—	4	4	no	—
194	C₂×C₄×C₃₁	—	4	4	no	—
195	C₂×C₄×C₃₁	—	4	8	no	—
196	C₂×C₄×C₃₁	—	8	8	no	—
197	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
198	C₂×C₄×C₃₁	C₂²×D₄	1	4	yes	—
199	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
200	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
201	C₂×C₄×C₃₁	H(5,𝔽₂)	1	4	yes	—
202	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
203	C₈×C₃₁	—	2	8	no	—
204	C₂×#540	C₂×C₃₁	1	1	yes	characteristic, complemented
205	#3449	C₂×C₃₁	1	1	yes	characteristic, complemented
206	#3221	C₂×C₃₁	1	8	yes	complemented
207	#3220	C₂×C₃₁	1	8	yes	complemented
208	#3151	C₂×C₃₁	1	2	yes	complemented
209	#3142	C₂×C₃₁	1	2	yes	complemented
210	#3036	C₂×C₃₁	1	4	yes	complemented
211	#3006	C₂×C₃₁	1	4	yes	complemented
212	#2921	C₂×C₃₁	1	1	yes	characteristic, complemented
213	C₂²×C₃₁	—	2	2	no	—
214	C₂²×C₃₁	—	2	4	no	—
215	C₂²×C₃₁	—	2	2	no	—
216	C₂²×C₃₁	—	4	4	no	—
217	C₂²×C₃₁	—	4	4	no	—
218	C₂²×C₃₁	—	4	8	no	—
219	C₂²×C₃₁	—	4	4	no	—
220	C₂²×C₃₁	—	4	4	no	—
221	C₂²×C₃₁	—	8	2	no	—
222	C₂²×C₃₁	—	8	8	no	—
223	C₂²×C₃₁	—	8	8	no	—
224	C₂²×C₃₁	#535	1	2	yes	—
225	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
226	C₄×C₃₁	—	2	4	no	—
227	C₄×C₃₁	—	2	4	no	—
228	C₄×C₃₁	—	2	2	no	—
229	C₄×C₃₁	—	4	4	no	—
230	C₄×C₃₁	—	4	4	no	—
231	C₄×C₃₁	#535	1	2	yes	—
232	C₄×C₃₁	C₂×H(5,𝔽₂)	1	2	yes	—
233	C₂²×#41	C₂²×C₃₁	1	2	yes	—
234	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	8	yes	complemented
235	#540	—	2	4	no	—
236	#540	C₂²×C₃₁	1	8	yes	complemented
237	#537	—	2	2	no	—
238	#536	C₂²×C₃₁	1	4	yes	complemented
239	#535	C₂²×C₃₁	1	4	yes	complemented
240	#534	—	2	2	no	—
241	#532	C₂²×C₃₁	1	2	yes	—
242	#532	C₂²×C₃₁	1	2	yes	—
243	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic, complemented
244	C₂×#125	C₂²×C₃₁	1	1	yes	characteristic, complemented
245	#525	—	2	4	no	—
246	C₂×#123	—	2	2	no	—
247	C₂×#121	—	2	2	no	—
248	C₂×#121	C₂²×C₃₁	1	2	yes	—
249	C₂×C₄×D₄	C₂²×C₃₁	1	2	yes	complemented
250	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
251	#496	C₂²×C₃₁	1	8	yes	complemented
252	#493	C₂²×C₃₁	1	4	yes	complemented
253	#492	C₂²×C₃₁	1	4	yes	complemented
254	#491	C₂²×C₃₁	1	4	yes	complemented
255	#490	C₂²×C₃₁	1	4	yes	complemented
256	#488	C₂²×C₃₁	1	8	yes	complemented
257	#463	C₂²×C₃₁	1	16	yes	complemented
258	#461	C₂²×C₃₁	1	8	yes	complemented
259	#459	C₂²×C₃₁	1	8	yes	complemented
260	#449	C₂²×C₃₁	1	16	yes	complemented
261	#448	C₂²×C₃₁	1	8	yes	complemented
262	#447	C₂²×C₃₁	1	8	yes	complemented
263	#433	C₂²×C₃₁	1	8	yes	complemented
264	#423	C₂²×C₃₁	1	2	yes	complemented
265	#417	C₂²×C₃₁	1	8	yes	complemented
266	#407	C₂²×C₃₁	1	2	yes	—
267	C₄×M₁₆	C₂²×C₃₁	1	2	yes	complemented
268	C₂×C₃₁	—	2	2	no	—
269	C₂×C₃₁	—	4	2	no	—
270	C₂×C₃₁	—	8	4	no	—
271	C₂×C₃₁	—	8	4	no	—
272	C₂×C₃₁	C₂×#535	1	1	yes	characteristic
273	C₂×C₃₁	#3585	1	2	yes	—
274	C₂×#41	—	2	2	no	—
275	C₂×#41	—	2	2	no	—
276	C₂×#41	—	4	8	no	—
277	C₂×#41	C₂³×C₃₁	1	4	yes	—
278	C₂×#41	C₂³×C₃₁	1	4	yes	—
279	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
280	C₂²×D₄	—	2	2	no	—
281	C₂²×D₄	—	2	2	no	—
282	C₂²×D₄	—	2	2	no	—
283	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
284	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
285	C₂³×C₄	—	2	2	no	—
286	C₂³×C₄	—	2	2	no	—
287	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
288	AGL(1,ℤ/8)	—	4	16	no	—
289	C₂×SD₁₆	C₂³×C₃₁	1	16	yes	—
290	C₂×D₈	C₂³×C₃₁	1	16	yes	—
291	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
292	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
293	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
294	#127	—	2	2	no	—
295	#127	C₂³×C₃₁	1	4	yes	—
296	#127	C₂³×C₃₁	1	2	yes	—
297	#127	C₂³×C₃₁	1	2	yes	—
298	#125	—	2	2	no	—
299	#125	C₂³×C₃₁	1	4	yes	—
300	#124	C₂³×C₃₁	1	4	yes	—
301	#124	C₂³×C₃₁	1	4	yes	—
302	#123	—	2	16	no	—
303	#123	—	2	8	no	—
304	C₂²⋊Q₈	—	2	8	no	—
305	#121	—	2	4	no	—
306	#121	—	2	4	no	—
307	#121	—	2	4	no	—
308	#121	—	2	8	no	—
309	#121	—	2	16	no	—
310	#121	C₂³×C₃₁	1	8	yes	—
311	#120	—	2	8	no	—
312	C₄×D₄	—	2	4	no	—
313	C₄×D₄	—	2	4	no	—
314	C₄×D₄	—	2	4	no	—
315	C₄×D₄	C₂³×C₃₁	1	8	yes	—
316	#117	D₄×C₃₁	1	4	yes	—
317	#117	C₂³×C₃₁	1	4	yes	—
318	C₂×C₄⋊C₄	—	2	2	no	—
319	C₂×C₄⋊C₄	D₄×C₃₁	1	2	yes	—
320	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
321	C₂×C₂²⋊C₄	—	2	2	no	—
322	C₂×C₂²⋊C₄	—	2	2	no	—
323	C₂×C₂²⋊C₄	D₄×C₃₁	1	2	yes	—
324	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
325	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
326	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
327	#103	C₂³×C₃₁	1	16	yes	—
328	#102	C₂³×C₃₁	1	16	yes	—
329	C₂²⋊C₈	C₂³×C₃₁	1	8	yes	—
330	#97	C₂³×C₃₁	1	4	yes	—
331	C₄×C₈	C₂³×C₃₁	1	4	yes	—
332	C₃₁	#60808	1	1	yes	characteristic, direct product factor, Sylow
333	C₂⁴	—	2	2	no	—
334	C₂⁴	—	2	2	no	—
335	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
336	#41	—	2	16	no	—
337	#41	—	4	8	no	—
338	#41	—	4	8	no	—
339	#41	—	4	8	no	—
340	C₂×Q₈	—	2	2	no	—
341	C₂×Q₈	—	4	4	no	—
342	C₂×Q₈	C₂⁴×C₃₁	1	4	yes	—
343	C₂×D₄	—	2	4	no	—
344	C₂×D₄	—	2	4	no	—
345	C₂×D₄	—	2	8	no	—
346	C₂×D₄	—	2	2	no	—
347	C₂×D₄	—	2	2	no	—
348	C₂×D₄	—	2	4	no	—
349	C₂×D₄	—	2	4	no	—
350	C₂×D₄	—	2	8	no	—
351	C₂×D₄	—	2	8	no	—
352	C₂×D₄	—	4	4	no	—
353	C₂×D₄	—	4	8	no	—
354	C₂×D₄	—	8	4	no	—
355	C₂×D₄	—	8	4	no	—
356	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
357	C₂×D₄	C₂⁴×C₃₁	1	4	yes	—
358	C₂²×C₄	—	2	4	no	—
359	C₂²×C₄	—	2	2	no	—
360	C₂²×C₄	—	2	2	no	—
361	C₂²×C₄	—	2	4	no	—
362	C₂²×C₄	—	2	4	no	—
363	C₂²×C₄	—	2	8	no	—
364	C₂²×C₄	—	4	8	no	—
365	C₂²×C₄	—	8	4	no	—
366	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
367	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
368	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
369	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
370	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
371	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
372	SD₁₆	—	4	16	no	—
373	D₈	—	4	16	no	—
374	M₁₆	—	2	8	no	—
375	C₂×C₈	C₂⁴×C₃₁	1	8	yes	—
376	C₄⋊C₄	—	2	8	no	—
377	C₄⋊C₄	—	2	2	no	—
378	C₄⋊C₄	—	2	4	no	—
379	C₄⋊C₄	—	2	4	no	—
380	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
381	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
382	C₂²⋊C₄	—	2	8	no	—
383	C₂²⋊C₄	—	2	8	no	—
384	C₂²⋊C₄	—	2	8	no	—
385	C₂²⋊C₄	—	2	8	no	—
386	C₂²⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
387	C₄²	C₂×D₄×C₃₁	1	2	yes	—
388	C₄²	C₂×D₄×C₃₁	1	2	yes	—
389	C₄²	C₂⁴×C₃₁	1	2	yes	—
390	C₄²	C₂⁴×C₃₁	1	2	yes	—
391	C₂³	—	2	2	no	—
392	C₂³	—	2	4	no	—
393	C₂³	—	2	4	no	—
394	C₂³	—	4	2	no	—
395	C₂³	—	4	4	no	—
396	C₂³	—	4	4	no	—
397	C₂³	—	4	4	no	—
398	C₂³	—	4	4	no	—
399	C₂³	—	8	4	no	—
400	C₂³	C₂²×D₄×C₃₁	1	2	yes	—
401	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
402	Q₈	—	2	8	no	—
403	Q₈	—	4	4	no	—
404	D₄	—	2	8	no	—
405	D₄	—	2	8	no	—
406	D₄	—	4	4	no	—
407	D₄	—	4	8	no	—
408	D₄	—	4	8	no	—
409	D₄	—	4	4	no	—
410	D₄	—	8	4	no	—
411	D₄	—	8	4	no	—
412	D₄	—	8	8	no	—
413	D₄	—	8	8	no	—
414	C₂×C₄	—	2	2	no	—
415	C₂×C₄	—	2	4	no	—
416	C₂×C₄	—	2	4	no	—
417	C₂×C₄	—	2	4	no	—
418	C₂×C₄	—	2	4	no	—
419	C₂×C₄	—	4	2	no	—
420	C₂×C₄	—	4	2	no	—
421	C₂×C₄	—	4	8	no	—
422	C₂×C₄	—	4	4	no	—
423	C₂×C₄	—	4	4	no	—
424	C₂×C₄	—	4	8	no	—
425	C₂×C₄	—	8	8	no	—
426	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
427	C₂×C₄	C₂²×D₄×C₃₁	1	4	yes	—
428	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
429	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
430	C₂×C₄	C₃₁×H(5,𝔽₂)	1	4	yes	—
431	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
432	C₈	—	2	8	no	—
433	C₂²	—	2	2	no	—
434	C₂²	—	2	2	no	—
435	C₂²	—	2	4	no	—
436	C₂²	—	4	4	no	—
437	C₂²	—	4	4	no	—
438	C₂²	—	4	4	no	—
439	C₂²	—	4	4	no	—
440	C₂²	—	4	8	no	—
441	C₂²	—	8	2	no	—
442	C₂²	—	8	8	no	—
443	C₂²	—	8	8	no	—
444	C₂²	C₃₁×#535	1	2	yes	—
445	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
446	C₄	—	2	2	no	—
447	C₄	—	2	4	no	—
448	C₄	—	2	4	no	—
449	C₄	—	4	4	no	—
450	C₄	—	4	4	no	—
451	C₄	C₃₁×#535	1	2	yes	—
452	C₄	C₂×C₃₁×H(5,𝔽₂)	1	2	yes	—
453	C₂	—	2	2	no	—
454	C₂	—	4	2	no	—
455	C₂	—	8	4	no	—
456	C₂	—	8	4	no	—
457	C₂	C₂×C₃₁×#535	1	1	yes	characteristic
458	C₂	C₃₁×#3585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂³×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₃₁×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#540	C₂	semidirect
2. C₃₁×#3449	C₂	semidirect
3. C₃₁×#3221	C₂	semidirect
4. C₃₁×#3220	C₂	semidirect
5. C₃₁×#3151	C₂	semidirect
6. C₃₁×#3142	C₂	semidirect
7. C₃₁×#3036	C₂	semidirect
8. C₃₁×#3006	C₂	semidirect
9. C₃₁×#2921	C₂	semidirect
11. C₂×C₃₁×AGL(1,ℤ/8)	C₂²	semidirect
13. C₃₁×#540	C₂²	semidirect
15. C₃₁×#536	C₂²	semidirect
16. C₃₁×#535	C₂²	semidirect
20. C₂×C₃₁×#127	C₂²	semidirect
21. C₂×C₃₁×#125	C₂²	semidirect
26. C₂×C₄×D₄×C₃₁	C₂²	semidirect
28. C₃₁×#496	C₂²	semidirect
29. C₃₁×#493	C₂²	semidirect
30. C₃₁×#492	C₂²	semidirect
31. C₃₁×#491	C₂²	semidirect
32. C₃₁×#490	C₂²	semidirect
33. C₃₁×#488	C₂²	semidirect
34. C₃₁×#463	C₂²	semidirect
35. C₃₁×#461	C₂²	semidirect
36. C₃₁×#459	C₂²	semidirect
37. C₃₁×#449	C₂²	semidirect
38. C₃₁×#448	C₂²	semidirect
39. C₃₁×#447	C₂²	semidirect
40. C₃₁×#433	C₂²	semidirect
41. C₃₁×#423	C₂²	semidirect
42. C₃₁×#417	C₂²	semidirect
44. C₄×M₁₆×C₃₁	C₂²	semidirect
161. #60808	C₃₁	direct (N × H)
204. C₂×#540	C₂×C₃₁	semidirect
205. #3449	C₂×C₃₁	semidirect
206. #3221	C₂×C₃₁	semidirect
207. #3220	C₂×C₃₁	semidirect
208. #3151	C₂×C₃₁	semidirect
209. #3142	C₂×C₃₁	semidirect
210. #3036	C₂×C₃₁	semidirect
211. #3006	C₂×C₃₁	semidirect
212. #2921	C₂×C₃₁	semidirect
234. C₂×AGL(1,ℤ/8)	C₂²×C₃₁	semidirect
236. #540	C₂²×C₃₁	semidirect
238. #536	C₂²×C₃₁	semidirect
239. #535	C₂²×C₃₁	semidirect
243. C₂×#127	C₂²×C₃₁	semidirect
244. C₂×#125	C₂²×C₃₁	semidirect
249. C₂×C₄×D₄	C₂²×C₃₁	semidirect
251. #496	C₂²×C₃₁	semidirect
252. #493	C₂²×C₃₁	semidirect
253. #492	C₂²×C₃₁	semidirect
254. #491	C₂²×C₃₁	semidirect
255. #490	C₂²×C₃₁	semidirect
256. #488	C₂²×C₃₁	semidirect
257. #463	C₂²×C₃₁	semidirect
258. #461	C₂²×C₃₁	semidirect
259. #459	C₂²×C₃₁	semidirect
260. #449	C₂²×C₃₁	semidirect
261. #448	C₂²×C₃₁	semidirect
262. #447	C₂²×C₃₁	semidirect
263. #433	C₂²×C₃₁	semidirect
264. #423	C₂²×C₃₁	semidirect
265. #417	C₂²×C₃₁	semidirect
267. C₄×M₁₆	C₂²×C₃₁	semidirect
332. C₃₁	#60808	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#540	2
C₃₁×#3449	2
C₃₁×#3221	2
C₃₁×#3220	2
C₃₁×#3151	2
C₃₁×#3142	2
C₃₁×#3036	2
C₃₁×#3006	2
C₃₁×#2921	2
#60808	31

Subgroups — C31×#60808 #21517567

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#60808 #21517567