Subgroups — C₃₁×#61409 #21518168

Order 7936.

2078 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3508	C₂	1	2	yes	maximal, complemented
2	C₃₁×#3399	C₂	1	8	yes	maximal, complemented
3	C₃₁×#3396	C₂	1	4	yes	maximal, complemented
4	C₃₁×#3391	C₂	1	4	yes	maximal, complemented
5	C₃₁×#3155	C₂	1	2	yes	maximal, complemented
6	C₂×C₃₁×#493	C₂	1	1	yes	maximal, characteristic, complemented
7	C₃₁×#3041	C₂	1	4	yes	maximal, complemented
8	C₃₁×#3038	C₂	1	2	yes	maximal, complemented
9	C₃₁×#2960	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#2948	C₂	1	2	yes	maximal, complemented
11	C₄×C₃₁×#135	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×AGL(1,ℤ/8)	C₂²	1	4	yes	—
13	C₂×C₃₁×#135	C₂²	1	1	yes	characteristic
14	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
15	C₂²×D₈×C₃₁	C₂²	1	2	yes	—
16	C₃₁×#559	C₂²	1	8	yes	complemented
17	C₃₁×#550	C₂²	1	8	yes	complemented
18	C₃₁×#546	—	2	8	no	—
19	C₃₁×#540	—	2	4	no	—
20	C₃₁×#535	C₂²	1	4	yes	—
21	C₃₁×#533	C₂²	1	1	yes	characteristic
22	C₃₁×#532	—	2	4	no	—
23	C₂×C₃₁×#127	C₂²	1	2	yes	—
24	C₃₁×#518	C₂²	1	4	yes	—
25	C₄×#41×C₃₁	C₂²	1	2	yes	—
26	C₃₁×#496	C₂²	1	8	yes	complemented
27	C₃₁×#495	C₂²	1	4	yes	complemented
28	C₃₁×#493	C₂²	1	4	yes	—
29	C₃₁×#493	C₂²	1	4	yes	—
30	C₃₁×#493	C₂²	1	4	yes	complemented
31	C₃₁×#463	C₂²	1	16	yes	complemented
32	C₃₁×#460	C₂²	1	8	yes	complemented
33	C₃₁×#459	C₂²	1	8	yes	complemented
34	C₃₁×#459	C₂²	1	16	yes	complemented
35	C₃₁×#446	C₂²	1	4	yes	complemented
36	C₈×Q₈×C₃₁	C₂²	1	4	yes	complemented
37	C₃₁×#442	C₂²	1	8	yes	complemented
38	C₄×Q₁₆×C₃₁	C₂²	1	2	yes	complemented
39	C₄×SD₁₆×C₃₁	C₂²	1	4	yes	complemented
40	C₄×D₈×C₃₁	C₂²	1	2	yes	complemented
41	C₄×D₈×C₃₁	C₂²	1	8	yes	complemented
42	C₃₁×#427	C₂²	1	1	yes	characteristic
43	C₃₁×#424	C₂²	1	2	yes	—
44	C₃₁×#423	C₂²	1	1	yes	characteristic
45	C₃₁×#418	C₂²	1	4	yes	—
46	C₃₁×#416	C₂²	1	2	yes	—
47	C₃₁×#405	C₂²	1	2	yes	—
48	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
49	C₂×#41×C₃₁	—	2	8	no	—
50	C₂×#41×C₃₁	C₂³	1	2	yes	—
51	C₂²×D₄×C₃₁	—	2	4	no	—
52	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
53	C₃₁×AGL(1,ℤ/8)	—	4	8	no	—
54	C₃₁×#135	—	2	4	no	—
55	C₂×Q₁₆×C₃₁	C₂³	1	1	yes	characteristic
56	C₂×SD₁₆×C₃₁	C₂³	1	2	yes	—
57	C₂×SD₁₆×C₃₁	C₂³	1	8	yes	—
58	C₂×D₈×C₃₁	—	2	8	no	—
59	C₂×D₈×C₃₁	—	4	4	no	—
60	C₂×D₈×C₃₁	C₂³	1	4	yes	—
61	C₂×D₈×C₃₁	C₂³	1	1	yes	characteristic
62	C₂×D₈×C₃₁	C₂³	1	4	yes	—
63	C₂×D₈×C₃₁	C₂³	1	8	yes	—
64	C₂×D₈×C₃₁	C₂³	1	8	yes	—
65	C₂×M₁₆×C₃₁	C₂³	1	2	yes	—
66	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
67	C₂²×C₈×C₃₁	C₂³	1	1	yes	characteristic
68	C₃₁×C₄⋊Q₈	D₄	1	2	yes	—
69	C₃₁×#127	—	2	2	no	—
70	C₃₁×#127	—	2	2	no	—
71	C₃₁×#127	C₂³	1	8	yes	—
72	C₃₁×#127	C₂³	1	8	yes	—
73	C₃₁×#125	D₄	1	2	yes	—
74	C₃₁×#124	—	2	8	no	—
75	C₃₁×#124	C₂³	1	8	yes	—
76	C₃₁×#123	—	2	16	no	—
77	C₃₁×C₂²⋊Q₈	D₄	1	4	yes	—
78	C₃₁×#121	—	2	8	no	—
79	C₃₁×#121	—	2	8	no	—
80	C₃₁×#121	—	2	16	no	—
81	C₃₁×#120	—	2	8	no	—
82	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
83	C₄×Q₈×C₃₁	C₂³	1	4	yes	—
84	C₄×D₄×C₃₁	—	2	4	no	—
85	C₄×D₄×C₃₁	—	2	8	no	—
86	C₄×D₄×C₃₁	—	2	8	no	—
87	C₄×D₄×C₃₁	C₂³	1	4	yes	—
88	C₄×D₄×C₃₁	C₂³	1	16	yes	—
89	C₃₁×#117	—	2	2	no	—
90	C₃₁×#117	C₂³	1	2	yes	—
91	C₃₁×#117	C₂³	1	2	yes	—
92	C₂×C₂²⋊C₄×C₃₁	—	2	8	no	—
93	C₂×C₄²×C₃₁	—	2	2	no	—
94	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
95	C₃₁×#107	C₂³	1	2	yes	—
96	C₃₁×#106	C₂³	1	2	yes	—
97	C₃₁×C₄⋊C₈	C₂³	1	4	yes	—
98	C₃₁×C₄⋊C₈	C₂³	1	8	yes	—
99	C₃₁×#103	C₂³	1	4	yes	—
100	C₃₁×#102	C₂³	1	4	yes	—
101	C₃₁×#102	C₂³	1	16	yes	—
102	C₃₁×#97	C₂³	1	4	yes	—
103	C₄×C₈×C₃₁	C₂³	1	2	yes	—
104	C₄×C₈×C₃₁	C₂³	1	2	yes	—
105	C₄×C₈×C₃₁	C₂³	1	4	yes	—
106	C₂⁴×C₃₁	—	2	4	no	—
107	#41×C₃₁	—	2	4	no	—
108	#41×C₃₁	—	2	4	no	—
109	#41×C₃₁	—	8	16	no	—
110	C₂×Q₈×C₃₁	C₂×D₄	1	4	yes	—
111	C₂×Q₈×C₃₁	C₂⁴	1	2	yes	—
112	C₂×D₄×C₃₁	—	2	2	no	—
113	C₂×D₄×C₃₁	—	2	4	no	—
114	C₂×D₄×C₃₁	—	2	8	no	—
115	C₂×D₄×C₃₁	—	2	16	no	—
116	C₂×D₄×C₃₁	—	2	8	no	—
117	C₂×D₄×C₃₁	—	2	8	no	—
118	C₂×D₄×C₃₁	—	2	8	no	—
119	C₂×D₄×C₃₁	—	8	8	no	—
120	C₂×D₄×C₃₁	C₂⁴	1	8	yes	—
121	C₂×D₄×C₃₁	C₂⁴	1	2	yes	—
122	C₂²×C₄×C₃₁	—	2	2	no	—
123	C₂²×C₄×C₃₁	—	2	4	no	—
124	C₂²×C₄×C₃₁	—	2	16	no	—
125	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
126	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
127	Q₁₆×C₃₁	—	2	2	no	—
128	SD₁₆×C₃₁	—	2	4	no	—
129	SD₁₆×C₃₁	—	4	8	no	—
130	D₈×C₃₁	—	2	8	no	—
131	D₈×C₃₁	—	2	8	no	—
132	D₈×C₃₁	—	2	2	no	—
133	D₈×C₃₁	—	4	8	no	—
134	D₈×C₃₁	—	4	8	no	—
135	M₁₆×C₃₁	—	2	4	no	—
136	C₂×C₈×C₃₁	—	2	2	no	—
137	C₂×C₈×C₃₁	#41	1	4	yes	complemented
138	C₂×C₈×C₃₁	C₂⁴	1	2	yes	—
139	C₂×C₈×C₃₁	C₂⁴	1	4	yes	—
140	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
141	C₂×C₈×C₃₁	C₂⁴	1	1	yes	characteristic
142	C₄⋊C₄×C₃₁	—	2	4	no	—
143	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
144	C₄⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
145	C₄⋊C₄×C₃₁	C₂×D₄	1	8	yes	—
146	C₄⋊C₄×C₃₁	C₂⁴	1	4	yes	—
147	C₂²⋊C₄×C₃₁	—	2	2	no	—
148	C₂²⋊C₄×C₃₁	—	2	16	no	—
149	C₂²⋊C₄×C₃₁	—	2	4	no	—
150	C₂²⋊C₄×C₃₁	—	2	16	no	—
151	C₂²⋊C₄×C₃₁	C₂×D₄	1	4	yes	—
152	C₄²×C₃₁	—	2	2	no	—
153	C₄²×C₃₁	—	2	4	no	—
154	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
155	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
156	C₄²×C₃₁	C₂⁴	1	2	yes	—
157	C₄²×C₃₁	C₂⁴	1	4	yes	—
158	#61409	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
159	C₂³×C₃₁	—	2	8	no	—
160	C₂³×C₃₁	—	2	2	no	—
161	C₂³×C₃₁	—	4	8	no	—
162	C₂³×C₃₁	—	8	8	no	—
163	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
164	Q₈×C₃₁	—	2	8	no	—
165	Q₈×C₃₁	—	2	2	no	—
166	Q₈×C₃₁	—	2	2	no	—
167	D₄×C₃₁	—	2	8	no	—
168	D₄×C₃₁	—	2	2	no	—
169	D₄×C₃₁	—	2	2	no	—
170	D₄×C₃₁	—	2	8	no	—
171	D₄×C₃₁	—	4	4	no	—
172	D₄×C₃₁	—	4	8	no	—
173	D₄×C₃₁	—	4	16	no	—
174	D₄×C₃₁	—	8	8	no	—
175	D₄×C₃₁	—	8	16	no	—
176	C₂×C₄×C₃₁	—	2	4	no	—
177	C₂×C₄×C₃₁	—	2	2	no	—
178	C₂×C₄×C₃₁	—	2	4	no	—
179	C₂×C₄×C₃₁	—	4	4	no	—
180	C₂×C₄×C₃₁	—	4	8	no	—
181	C₂×C₄×C₃₁	—	8	16	no	—
182	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
183	C₂×C₄×C₃₁	C₂²×D₄	1	8	yes	—
184	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
185	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
186	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
187	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
188	C₂×C₄×C₃₁	C₂⁵	1	1	yes	characteristic
189	C₈×C₃₁	—	2	2	no	—
190	C₈×C₃₁	—	2	4	no	—
191	C₈×C₃₁	C₂×#41	1	2	yes	—
192	C₈×C₃₁	C₂×#41	1	2	yes	—
193	#3508	C₂×C₃₁	1	2	yes	complemented
194	#3399	C₂×C₃₁	1	8	yes	complemented
195	#3396	C₂×C₃₁	1	4	yes	complemented
196	#3391	C₂×C₃₁	1	4	yes	complemented
197	#3155	C₂×C₃₁	1	2	yes	complemented
198	C₂×#493	C₂×C₃₁	1	1	yes	characteristic, complemented
199	#3041	C₂×C₃₁	1	4	yes	complemented
200	#3038	C₂×C₃₁	1	2	yes	complemented
201	#2960	C₂×C₃₁	1	1	yes	characteristic, complemented
202	#2948	C₂×C₃₁	1	2	yes	complemented
203	C₄×#135	C₂×C₃₁	1	1	yes	characteristic, complemented
204	C₂²×C₃₁	—	2	2	no	—
205	C₂²×C₃₁	—	4	2	no	—
206	C₂²×C₃₁	—	4	8	no	—
207	C₂²×C₃₁	—	4	4	no	—
208	C₂²×C₃₁	—	4	16	no	—
209	C₂²×C₃₁	—	16	8	no	—
210	C₂²×C₃₁	#550	1	2	yes	—
211	C₂²×C₃₁	C₂³×D₄	1	1	yes	characteristic
212	C₄×C₃₁	—	2	2	no	—
213	C₄×C₃₁	—	2	8	no	—
214	C₄×C₃₁	—	4	2	no	—
215	C₄×C₃₁	—	4	4	no	—
216	C₄×C₃₁	#550	1	2	yes	—
217	C₄×C₃₁	#576	1	4	yes	—
218	C₄×C₃₁	C₂²×#41	1	1	yes	characteristic
219	C₄×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
220	C₂×AGL(1,ℤ/8)	C₂²×C₃₁	1	4	yes	—
221	C₂×#135	C₂²×C₃₁	1	1	yes	characteristic
222	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
223	C₂²×D₈	C₂²×C₃₁	1	2	yes	—
224	#559	C₂²×C₃₁	1	8	yes	complemented
225	#550	C₂²×C₃₁	1	8	yes	complemented
226	#546	—	2	8	no	—
227	#540	—	2	4	no	—
228	#535	C₂²×C₃₁	1	4	yes	—
229	#533	C₂²×C₃₁	1	1	yes	characteristic
230	#532	—	2	4	no	—
231	C₂×#127	C₂²×C₃₁	1	2	yes	—
232	#518	C₂²×C₃₁	1	4	yes	—
233	C₄×#41	C₂²×C₃₁	1	2	yes	—
234	#496	C₂²×C₃₁	1	8	yes	complemented
235	#495	C₂²×C₃₁	1	4	yes	complemented
236	#493	C₂²×C₃₁	1	4	yes	—
237	#493	C₂²×C₃₁	1	4	yes	—
238	#493	C₂²×C₃₁	1	4	yes	complemented
239	#463	C₂²×C₃₁	1	16	yes	complemented
240	#460	C₂²×C₃₁	1	8	yes	complemented
241	#459	C₂²×C₃₁	1	8	yes	complemented
242	#459	C₂²×C₃₁	1	16	yes	complemented
243	#446	C₂²×C₃₁	1	4	yes	complemented
244	C₈×Q₈	C₂²×C₃₁	1	4	yes	complemented
245	#442	C₂²×C₃₁	1	8	yes	complemented
246	C₄×Q₁₆	C₂²×C₃₁	1	2	yes	complemented
247	C₄×SD₁₆	C₂²×C₃₁	1	4	yes	complemented
248	C₄×D₈	C₂²×C₃₁	1	2	yes	complemented
249	C₄×D₈	C₂²×C₃₁	1	8	yes	complemented
250	#427	C₂²×C₃₁	1	1	yes	characteristic
251	#424	C₂²×C₃₁	1	2	yes	—
252	#423	C₂²×C₃₁	1	1	yes	characteristic
253	#418	C₂²×C₃₁	1	4	yes	—
254	#416	C₂²×C₃₁	1	2	yes	—
255	#405	C₂²×C₃₁	1	2	yes	—
256	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
257	C₂×C₃₁	—	2	2	no	—
258	C₂×C₃₁	—	8	2	no	—
259	C₂×C₃₁	—	8	8	no	—
260	C₂×C₃₁	C₂×#550	1	1	yes	characteristic
261	C₂×C₃₁	C₂×#576	1	2	yes	—
262	C₂×#41	—	2	8	no	—
263	C₂×#41	C₂³×C₃₁	1	2	yes	—
264	C₂²×D₄	—	2	4	no	—
265	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
266	AGL(1,ℤ/8)	—	4	8	no	—
267	#135	—	2	4	no	—
268	C₂×Q₁₆	C₂³×C₃₁	1	1	yes	characteristic
269	C₂×SD₁₆	C₂³×C₃₁	1	2	yes	—
270	C₂×SD₁₆	C₂³×C₃₁	1	8	yes	—
271	C₂×D₈	—	2	8	no	—
272	C₂×D₈	—	4	4	no	—
273	C₂×D₈	C₂³×C₃₁	1	4	yes	—
274	C₂×D₈	C₂³×C₃₁	1	4	yes	—
275	C₂×D₈	C₂³×C₃₁	1	1	yes	characteristic
276	C₂×D₈	C₂³×C₃₁	1	8	yes	—
277	C₂×D₈	C₂³×C₃₁	1	8	yes	—
278	C₂×M₁₆	C₂³×C₃₁	1	2	yes	—
279	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
280	C₂²×C₈	C₂³×C₃₁	1	1	yes	characteristic
281	C₄⋊Q₈	D₄×C₃₁	1	2	yes	—
282	#127	—	2	2	no	—
283	#127	—	2	2	no	—
284	#127	C₂³×C₃₁	1	8	yes	—
285	#127	C₂³×C₃₁	1	8	yes	—
286	#125	D₄×C₃₁	1	2	yes	—
287	#124	—	2	8	no	—
288	#124	C₂³×C₃₁	1	8	yes	—
289	#123	—	2	16	no	—
290	C₂²⋊Q₈	D₄×C₃₁	1	4	yes	—
291	#121	—	2	8	no	—
292	#121	—	2	8	no	—
293	#121	—	2	16	no	—
294	#120	—	2	8	no	—
295	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
296	C₄×Q₈	C₂³×C₃₁	1	4	yes	—
297	C₄×D₄	—	2	4	no	—
298	C₄×D₄	—	2	8	no	—
299	C₄×D₄	—	2	8	no	—
300	C₄×D₄	C₂³×C₃₁	1	4	yes	—
301	C₄×D₄	C₂³×C₃₁	1	16	yes	—
302	#117	—	2	2	no	—
303	#117	C₂³×C₃₁	1	2	yes	—
304	#117	C₂³×C₃₁	1	2	yes	—
305	C₂×C₂²⋊C₄	—	2	8	no	—
306	C₂×C₄²	—	2	2	no	—
307	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
308	#107	C₂³×C₃₁	1	2	yes	—
309	#106	C₂³×C₃₁	1	2	yes	—
310	C₄⋊C₈	C₂³×C₃₁	1	4	yes	—
311	C₄⋊C₈	C₂³×C₃₁	1	8	yes	—
312	#103	C₂³×C₃₁	1	4	yes	—
313	#102	C₂³×C₃₁	1	4	yes	—
314	#102	C₂³×C₃₁	1	16	yes	—
315	#97	C₂³×C₃₁	1	4	yes	—
316	C₄×C₈	C₂³×C₃₁	1	2	yes	—
317	C₄×C₈	C₂³×C₃₁	1	2	yes	—
318	C₄×C₈	C₂³×C₃₁	1	4	yes	—
319	C₃₁	#61409	1	1	yes	characteristic, direct product factor, Sylow
320	C₂⁴	—	2	4	no	—
321	#41	—	2	4	no	—
322	#41	—	2	4	no	—
323	#41	—	8	16	no	—
324	C₂×Q₈	C₂×D₄×C₃₁	1	4	yes	—
325	C₂×Q₈	C₂⁴×C₃₁	1	2	yes	—
326	C₂×D₄	—	2	2	no	—
327	C₂×D₄	—	2	4	no	—
328	C₂×D₄	—	2	8	no	—
329	C₂×D₄	—	2	16	no	—
330	C₂×D₄	—	2	8	no	—
331	C₂×D₄	—	2	8	no	—
332	C₂×D₄	—	2	8	no	—
333	C₂×D₄	—	8	8	no	—
334	C₂×D₄	C₂⁴×C₃₁	1	8	yes	—
335	C₂×D₄	C₂⁴×C₃₁	1	2	yes	—
336	C₂²×C₄	—	2	2	no	—
337	C₂²×C₄	—	2	4	no	—
338	C₂²×C₄	—	2	16	no	—
339	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
340	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
341	Q₁₆	—	2	2	no	—
342	SD₁₆	—	2	4	no	—
343	SD₁₆	—	4	8	no	—
344	D₈	—	2	8	no	—
345	D₈	—	2	8	no	—
346	D₈	—	2	2	no	—
347	D₈	—	4	8	no	—
348	D₈	—	4	8	no	—
349	M₁₆	—	2	4	no	—
350	C₂×C₈	—	2	2	no	—
351	C₂×C₈	#41×C₃₁	1	4	yes	complemented
352	C₂×C₈	C₂⁴×C₃₁	1	2	yes	—
353	C₂×C₈	C₂⁴×C₃₁	1	4	yes	—
354	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
355	C₂×C₈	C₂⁴×C₃₁	1	1	yes	characteristic
356	C₄⋊C₄	—	2	4	no	—
357	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
358	C₄⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
359	C₄⋊C₄	C₂×D₄×C₃₁	1	8	yes	—
360	C₄⋊C₄	C₂⁴×C₃₁	1	4	yes	—
361	C₂²⋊C₄	—	2	2	no	—
362	C₂²⋊C₄	—	2	16	no	—
363	C₂²⋊C₄	—	2	4	no	—
364	C₂²⋊C₄	—	2	16	no	—
365	C₂²⋊C₄	C₂×D₄×C₃₁	1	4	yes	—
366	C₄²	—	2	2	no	—
367	C₄²	—	2	4	no	—
368	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
369	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
370	C₄²	C₂⁴×C₃₁	1	2	yes	—
371	C₄²	C₂⁴×C₃₁	1	4	yes	—
372	C₂³	—	2	8	no	—
373	C₂³	—	2	2	no	—
374	C₂³	—	4	8	no	—
375	C₂³	—	8	8	no	—
376	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
377	Q₈	—	2	8	no	—
378	Q₈	—	2	2	no	—
379	Q₈	—	2	2	no	—
380	D₄	—	2	8	no	—
381	D₄	—	2	2	no	—
382	D₄	—	2	2	no	—
383	D₄	—	2	8	no	—
384	D₄	—	4	4	no	—
385	D₄	—	4	8	no	—
386	D₄	—	4	16	no	—
387	D₄	—	8	8	no	—
388	D₄	—	8	16	no	—
389	C₂×C₄	—	2	4	no	—
390	C₂×C₄	—	2	2	no	—
391	C₂×C₄	—	2	4	no	—
392	C₂×C₄	—	4	4	no	—
393	C₂×C₄	—	4	8	no	—
394	C₂×C₄	—	8	16	no	—
395	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
396	C₂×C₄	C₂²×D₄×C₃₁	1	8	yes	—
397	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
398	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
399	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
400	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
401	C₂×C₄	C₂⁵×C₃₁	1	1	yes	characteristic
402	C₈	—	2	2	no	—
403	C₈	—	2	4	no	—
404	C₈	C₂×#41×C₃₁	1	2	yes	—
405	C₈	C₂×#41×C₃₁	1	2	yes	—
406	C₂²	—	2	2	no	—
407	C₂²	—	4	2	no	—
408	C₂²	—	4	8	no	—
409	C₂²	—	4	4	no	—
410	C₂²	—	4	16	no	—
411	C₂²	—	16	8	no	—
412	C₂²	C₃₁×#550	1	2	yes	—
413	C₂²	C₂³×D₄×C₃₁	1	1	yes	characteristic
414	C₄	—	2	2	no	—
415	C₄	—	2	8	no	—
416	C₄	—	4	2	no	—
417	C₄	—	4	4	no	—
418	C₄	C₃₁×#550	1	2	yes	—
419	C₄	C₃₁×#576	1	4	yes	—
420	C₄	C₂²×#41×C₃₁	1	1	yes	characteristic
421	C₄	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
422	C₂	—	2	2	no	—
423	C₂	—	8	8	no	—
424	C₂	—	8	2	no	—
425	C₂	C₂×C₃₁×#550	1	1	yes	characteristic
426	C₂	C₂×C₃₁×#576	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 ◁ C₃₁×#533 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#3508	C₂	semidirect
2. C₃₁×#3399	C₂	semidirect
3. C₃₁×#3396	C₂	semidirect
4. C₃₁×#3391	C₂	semidirect
5. C₃₁×#3155	C₂	semidirect
6. C₂×C₃₁×#493	C₂	semidirect
7. C₃₁×#3041	C₂	semidirect
8. C₃₁×#3038	C₂	semidirect
9. C₃₁×#2960	C₂	semidirect
10. C₃₁×#2948	C₂	semidirect
11. C₄×C₃₁×#135	C₂	semidirect
16. C₃₁×#559	C₂²	semidirect
17. C₃₁×#550	C₂²	semidirect
26. C₃₁×#496	C₂²	semidirect
27. C₃₁×#495	C₂²	semidirect
30. C₃₁×#493	C₂²	semidirect
31. C₃₁×#463	C₂²	semidirect
32. C₃₁×#460	C₂²	semidirect
33. C₃₁×#459	C₂²	semidirect
34. C₃₁×#459	C₂²	semidirect
35. C₃₁×#446	C₂²	semidirect
36. C₈×Q₈×C₃₁	C₂²	semidirect
37. C₃₁×#442	C₂²	semidirect
38. C₄×Q₁₆×C₃₁	C₂²	semidirect
39. C₄×SD₁₆×C₃₁	C₂²	semidirect
40. C₄×D₈×C₃₁	C₂²	semidirect
41. C₄×D₈×C₃₁	C₂²	semidirect
137. C₂×C₈×C₃₁	#41	semidirect
158. #61409	C₃₁	direct (N × H)
193. #3508	C₂×C₃₁	semidirect
194. #3399	C₂×C₃₁	semidirect
195. #3396	C₂×C₃₁	semidirect
196. #3391	C₂×C₃₁	semidirect
197. #3155	C₂×C₃₁	semidirect
198. C₂×#493	C₂×C₃₁	semidirect
199. #3041	C₂×C₃₁	semidirect
200. #3038	C₂×C₃₁	semidirect
201. #2960	C₂×C₃₁	semidirect
202. #2948	C₂×C₃₁	semidirect
203. C₄×#135	C₂×C₃₁	semidirect
224. #559	C₂²×C₃₁	semidirect
225. #550	C₂²×C₃₁	semidirect
234. #496	C₂²×C₃₁	semidirect
235. #495	C₂²×C₃₁	semidirect
238. #493	C₂²×C₃₁	semidirect
239. #463	C₂²×C₃₁	semidirect
240. #460	C₂²×C₃₁	semidirect
241. #459	C₂²×C₃₁	semidirect
242. #459	C₂²×C₃₁	semidirect
243. #446	C₂²×C₃₁	semidirect
244. C₈×Q₈	C₂²×C₃₁	semidirect
245. #442	C₂²×C₃₁	semidirect
246. C₄×Q₁₆	C₂²×C₃₁	semidirect
247. C₄×SD₁₆	C₂²×C₃₁	semidirect
248. C₄×D₈	C₂²×C₃₁	semidirect
249. C₄×D₈	C₂²×C₃₁	semidirect
319. C₃₁	#61409	direct (N × H)
351. C₂×C₈	#41×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3508	2
C₃₁×#3399	2
C₃₁×#3396	2
C₃₁×#3391	2
C₃₁×#3155	2
C₂×C₃₁×#493	2
C₃₁×#3041	2
C₃₁×#3038	2
C₃₁×#2960	2
C₃₁×#2948	2
C₄×C₃₁×#135	2
#61409	31

Subgroups — C31×#61409 #21518168

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#61409 #21518168