Subgroups — C₃₁×#19266 #21476025

Order 7936.

786 conjugacy classes of subgroups.

The subgroup lattice has 448 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#492	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3079	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3038	C₂	1	1	yes	maximal, characteristic
4	C₈×#41×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1666	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1665	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1566	C₂	1	1	yes	maximal, characteristic, complemented
8	C₃₁×#1549	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1545	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
11	C₂²×SD₁₆×C₃₁	—	2	2	no	—
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
14	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
15	C₃₁×#492	—	2	1	no	—
16	C₃₁×#492	—	2	1	no	—
17	C₃₁×#492	C₂²	1	4	yes	—
18	C₃₁×#477	—	2	1	no	—
19	C₃₁×#476	—	2	1	no	—
20	C₃₁×#475	C₂²	1	2	yes	complemented
21	C₃₁×#474	C₂²	1	2	yes	complemented
22	C₃₁×#460	—	2	1	no	—
23	C₃₁×#460	C₂²	1	2	yes	—
24	C₃₁×#459	—	2	1	no	—
25	C₃₁×#459	C₂²	1	2	yes	—
26	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
27	C₈×D₄×C₃₁	—	2	2	no	—
28	C₈×D₄×C₃₁	C₂²	1	2	yes	complemented
29	C₃₁×#431	—	2	1	no	—
30	C₃₁×#431	C₂²	1	1	yes	characteristic
31	C₃₁×#428	—	2	1	no	—
32	C₃₁×#423	C₂²	1	1	yes	characteristic
33	C₃₁×#423	C₂²	1	1	yes	characteristic
34	C₃₁×#418	—	2	1	no	—
35	C₃₁×#417	—	2	1	no	—
36	C₂×C₄×C₈×C₃₁	—	2	1	no	—
37	C₂×C₄×C₈×C₃₁	C₂²	1	1	yes	characteristic
38	C₃₁×#334	C₂²	1	4	yes	complemented
39	C₃₁×#332	C₂²	1	4	yes	complemented
40	C₃₁×#331	C₂²	1	4	yes	—
41	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
42	C₂²×Q₈×C₃₁	—	2	1	no	—
43	C₂²×D₄×C₃₁	—	2	1	no	—
44	C₂²×D₄×C₃₁	—	2	1	no	—
45	C₂²×D₄×C₃₁	—	2	1	no	—
46	C₃₁×AGL(1,ℤ/8)	—	4	4	no	—
47	C₂×SD₁₆×C₃₁	—	2	2	no	—
48	C₂×SD₁₆×C₃₁	—	2	4	no	—
49	C₂×SD₁₆×C₃₁	—	2	4	no	—
50	C₂×SD₁₆×C₃₁	—	4	8	no	—
51	C₂×D₈×C₃₁	—	2	2	no	—
52	C₂×M₁₆×C₃₁	—	2	1	no	—
53	C₂×M₁₆×C₃₁	—	2	1	no	—
54	C₂²×C₈×C₃₁	—	2	2	no	—
55	C₂²×C₈×C₃₁	D₄	1	2	yes	—
56	C₃₁×C₄⋊Q₈	—	2	1	no	—
57	C₃₁×C₄⋊Q₈	—	2	1	no	—
58	C₃₁×C₄⋊Q₈	—	2	1	no	—
59	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
60	C₃₁×#127	—	2	1	no	—
61	C₃₁×#127	—	2	1	no	—
62	C₃₁×#127	—	2	1	no	—
63	C₃₁×#127	C₂³	1	2	yes	—
64	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
65	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
66	C₄×D₄×C₃₁	—	2	1	no	—
67	C₄×D₄×C₃₁	—	2	1	no	—
68	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
69	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
70	C₃₁×#117	—	2	1	no	—
71	C₃₁×#117	D₄	1	1	yes	characteristic
72	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
73	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
74	C₂×C₄²×C₃₁	—	2	1	no	—
75	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
76	C₃₁×#107	—	2	2	no	—
77	C₃₁×#106	—	2	2	no	—
78	C₃₁×C₄⋊C₈	—	2	2	no	—
79	C₃₁×C₄⋊C₈	—	2	1	no	—
80	C₃₁×C₄⋊C₈	—	2	1	no	—
81	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
82	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
83	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
84	C₃₁×#103	—	2	2	no	—
85	C₃₁×#102	—	2	2	no	—
86	C₃₁×#102	—	2	2	no	—
87	C₃₁×#102	—	2	2	no	—
88	C₃₁×C₂²⋊C₈	—	2	1	no	—
89	C₃₁×C₂²⋊C₈	—	2	2	no	—
90	C₄×C₈×C₃₁	—	2	2	no	—
91	C₄×C₈×C₃₁	—	2	1	no	—
92	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
93	C₄×C₈×C₃₁	D₄	1	1	yes	characteristic
94	C₄×C₈×C₃₁	C₂³	1	2	yes	—
95	C₂⁴×C₃₁	—	4	1	no	—
96	#41×C₃₁	—	2	2	no	—
97	#41×C₃₁	SD₁₆	1	4	yes	complemented
98	C₂×Q₈×C₃₁	—	2	2	no	—
99	C₂×Q₈×C₃₁	—	4	4	no	—
100	C₂×Q₈×C₃₁	—	4	1	no	—
101	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
102	C₂×D₄×C₃₁	—	2	1	no	—
103	C₂×D₄×C₃₁	—	2	2	no	—
104	C₂×D₄×C₃₁	—	2	2	no	—
105	C₂×D₄×C₃₁	—	2	2	no	—
106	C₂×D₄×C₃₁	—	4	4	no	—
107	C₂×D₄×C₃₁	—	4	1	no	—
108	C₂×D₄×C₃₁	—	4	1	no	—
109	C₂×D₄×C₃₁	—	4	1	no	—
110	C₂×D₄×C₃₁	—	4	2	no	—
111	C₂×D₄×C₃₁	—	4	2	no	—
112	C₂×D₄×C₃₁	—	8	2	no	—
113	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
114	C₂²×C₄×C₃₁	—	2	1	no	—
115	C₂²×C₄×C₃₁	—	2	1	no	—
116	C₂²×C₄×C₃₁	—	2	1	no	—
117	C₂²×C₄×C₃₁	—	4	1	no	—
118	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
120	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
121	SD₁₆×C₃₁	—	4	8	no	—
122	SD₁₆×C₃₁	—	4	4	no	—
123	SD₁₆×C₃₁	—	4	8	no	—
124	D₈×C₃₁	—	4	4	no	—
125	M₁₆×C₃₁	—	4	2	no	—
126	M₁₆×C₃₁	—	4	2	no	—
127	C₂×C₈×C₃₁	—	2	4	no	—
128	C₂×C₈×C₃₁	—	2	2	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	—	2	2	no	—
131	C₂×C₈×C₃₁	—	4	4	no	—
132	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
133	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
134	C₄⋊C₄×C₃₁	—	2	1	no	—
135	C₄⋊C₄×C₃₁	—	2	1	no	—
136	C₄⋊C₄×C₃₁	—	2	2	no	—
137	C₄⋊C₄×C₃₁	—	2	2	no	—
138	C₄⋊C₄×C₃₁	—	4	1	no	—
139	C₄⋊C₄×C₃₁	—	4	1	no	—
140	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
141	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
142	C₂²⋊C₄×C₃₁	—	2	1	no	—
143	C₂²⋊C₄×C₃₁	—	2	1	no	—
144	C₂²⋊C₄×C₃₁	—	2	1	no	—
145	C₄²×C₃₁	—	2	1	no	—
146	C₄²×C₃₁	—	2	1	no	—
147	C₄²×C₃₁	—	2	1	no	—
148	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
149	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
150	C₄²×C₃₁	#41	1	1	yes	characteristic
151	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
152	#19266	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
153	C₂³×C₃₁	—	2	1	no	—
154	C₂³×C₃₁	—	4	2	no	—
155	C₂³×C₃₁	—	8	2	no	—
156	C₂³×C₃₁	—	8	2	no	—
157	C₂³×C₃₁	—	8	2	no	—
158	C₂³×C₃₁	#121	1	1	yes	characteristic
159	Q₈×C₃₁	—	2	1	no	—
160	Q₈×C₃₁	—	4	4	no	—
161	Q₈×C₃₁	—	8	2	no	—
162	Q₈×C₃₁	C₂×SD₁₆	1	2	yes	complemented
163	D₄×C₃₁	—	2	4	no	—
164	D₄×C₃₁	—	2	1	no	—
165	D₄×C₃₁	—	4	2	no	—
166	D₄×C₃₁	—	4	2	no	—
167	D₄×C₃₁	—	4	4	no	—
168	D₄×C₃₁	—	8	2	no	—
169	D₄×C₃₁	—	8	2	no	—
170	D₄×C₃₁	—	8	2	no	—
171	D₄×C₃₁	—	8	2	no	—
172	D₄×C₃₁	C₂×SD₁₆	1	2	yes	complemented
173	C₂×C₄×C₃₁	—	2	1	no	—
174	C₂×C₄×C₃₁	—	2	2	no	—
175	C₂×C₄×C₃₁	—	2	4	no	—
176	C₂×C₄×C₃₁	—	2	2	no	—
177	C₂×C₄×C₃₁	—	2	1	no	—
178	C₂×C₄×C₃₁	—	2	1	no	—
179	C₂×C₄×C₃₁	—	4	2	no	—
180	C₂×C₄×C₃₁	—	4	2	no	—
181	C₂×C₄×C₃₁	—	4	2	no	—
182	C₂×C₄×C₃₁	—	8	2	no	—
183	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
184	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
185	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
186	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
187	C₂×C₄×C₃₁	#135	1	2	yes	—
188	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
189	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
190	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
191	C₈×C₃₁	—	2	2	no	—
192	C₈×C₃₁	—	4	2	no	—
193	C₈×C₃₁	—	4	2	no	—
194	C₈×C₃₁	—	4	2	no	—
195	C₈×C₃₁	AGL(1,ℤ/8)	1	4	yes	complemented
196	C₂×#492	C₂×C₃₁	1	1	yes	characteristic, complemented
197	#3079	C₂×C₃₁	1	1	yes	characteristic, complemented
198	#3038	C₂×C₃₁	1	1	yes	characteristic
199	C₈×#41	C₂×C₃₁	1	1	yes	characteristic, complemented
200	#1666	C₂×C₃₁	1	4	yes	complemented
201	#1665	C₂×C₃₁	1	4	yes	complemented
202	#1566	C₂×C₃₁	1	1	yes	characteristic, complemented
203	#1549	C₂×C₃₁	1	1	yes	characteristic, complemented
204	#1545	C₂×C₃₁	1	1	yes	characteristic, complemented
205	C₂²×C₃₁	—	2	1	no	—
206	C₂²×C₃₁	—	2	1	no	—
207	C₂²×C₃₁	—	2	2	no	—
208	C₂²×C₃₁	—	4	1	no	—
209	C₂²×C₃₁	—	4	1	no	—
210	C₂²×C₃₁	—	8	2	no	—
211	C₂²×C₃₁	—	8	2	no	—
212	C₂²×C₃₁	—	8	2	no	—
213	C₂²×C₃₁	—	16	4	no	—
214	C₂²×C₃₁	#465	1	2	yes	—
215	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
216	C₄×C₃₁	—	2	1	no	—
217	C₄×C₃₁	—	2	2	no	—
218	C₄×C₃₁	—	2	1	no	—
219	C₄×C₃₁	—	4	1	no	—
220	C₄×C₃₁	—	4	1	no	—
221	C₄×C₃₁	—	8	2	no	—
222	C₄×C₃₁	#465	1	2	yes	—
223	C₄×C₃₁	#471	1	2	yes	—
224	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
225	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
226	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
227	C₂×AGL(1,ℤ/8)	—	2	1	no	—
228	C₂²×SD₁₆	—	2	2	no	—
229	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
230	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
231	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
232	#492	—	2	1	no	—
233	#492	—	2	1	no	—
234	#492	C₂²×C₃₁	1	4	yes	—
235	#477	—	2	1	no	—
236	#476	—	2	1	no	—
237	#475	C₂²×C₃₁	1	2	yes	complemented
238	#474	C₂²×C₃₁	1	2	yes	complemented
239	#460	—	2	1	no	—
240	#460	C₂²×C₃₁	1	2	yes	—
241	#459	—	2	1	no	—
242	#459	C₂²×C₃₁	1	2	yes	—
243	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
244	C₈×D₄	—	2	2	no	—
245	C₈×D₄	C₂²×C₃₁	1	2	yes	complemented
246	#431	—	2	1	no	—
247	#431	C₂²×C₃₁	1	1	yes	characteristic
248	#428	—	2	1	no	—
249	#423	C₂²×C₃₁	1	1	yes	characteristic
250	#423	C₂²×C₃₁	1	1	yes	characteristic
251	#418	—	2	1	no	—
252	#417	—	2	1	no	—
253	C₂×C₄×C₈	—	2	1	no	—
254	C₂×C₄×C₈	C₂²×C₃₁	1	1	yes	characteristic
255	#334	C₂²×C₃₁	1	4	yes	complemented
256	#332	C₂²×C₃₁	1	4	yes	complemented
257	#331	C₂²×C₃₁	1	4	yes	—
258	C₂×C₃₁	—	2	2	no	—
259	C₂×C₃₁	—	4	2	no	—
260	C₂×C₃₁	—	16	2	no	—
261	C₂×C₃₁	#3038	1	1	yes	characteristic
262	C₂×C₃₁	C₂×#465	1	1	yes	characteristic
263	C₂×C₃₁	C₂×#471	1	1	yes	characteristic
264	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
265	C₂²×Q₈	—	2	1	no	—
266	C₂²×D₄	—	2	1	no	—
267	C₂²×D₄	—	2	1	no	—
268	C₂²×D₄	—	2	1	no	—
269	AGL(1,ℤ/8)	—	4	4	no	—
270	C₂×SD₁₆	—	2	2	no	—
271	C₂×SD₁₆	—	2	4	no	—
272	C₂×SD₁₆	—	2	4	no	—
273	C₂×SD₁₆	—	4	8	no	—
274	C₂×D₈	—	2	2	no	—
275	C₂×M₁₆	—	2	1	no	—
276	C₂×M₁₆	—	2	1	no	—
277	C₂²×C₈	—	2	2	no	—
278	C₂²×C₈	D₄×C₃₁	1	2	yes	—
279	C₄⋊Q₈	—	2	1	no	—
280	C₄⋊Q₈	—	2	1	no	—
281	C₄⋊Q₈	—	2	1	no	—
282	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
283	#127	—	2	1	no	—
284	#127	—	2	1	no	—
285	#127	—	2	1	no	—
286	#127	C₂³×C₃₁	1	2	yes	—
287	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
288	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
289	C₄×D₄	—	2	1	no	—
290	C₄×D₄	—	2	1	no	—
291	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
292	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
293	#117	—	2	1	no	—
294	#117	D₄×C₃₁	1	1	yes	characteristic
295	C₂×C₄⋊C₄	—	2	1	no	—
296	C₂×C₄⋊C₄	—	2	1	no	—
297	C₂×C₄²	—	2	1	no	—
298	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
299	#107	—	2	2	no	—
300	#106	—	2	2	no	—
301	C₄⋊C₈	—	2	2	no	—
302	C₄⋊C₈	—	2	1	no	—
303	C₄⋊C₈	—	2	1	no	—
304	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
305	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
306	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
307	#103	—	2	2	no	—
308	#102	—	2	2	no	—
309	#102	—	2	2	no	—
310	#102	—	2	2	no	—
311	C₂²⋊C₈	—	2	1	no	—
312	C₂²⋊C₈	—	2	2	no	—
313	C₄×C₈	—	2	2	no	—
314	C₄×C₈	—	2	1	no	—
315	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
316	C₄×C₈	D₄×C₃₁	1	1	yes	characteristic
317	C₄×C₈	C₂³×C₃₁	1	2	yes	—
318	C₃₁	#19266	1	1	yes	characteristic, direct product factor, Sylow
319	C₂⁴	—	4	1	no	—
320	#41	—	2	2	no	—
321	#41	SD₁₆×C₃₁	1	4	yes	complemented
322	C₂×Q₈	—	2	2	no	—
323	C₂×Q₈	—	4	4	no	—
324	C₂×Q₈	—	4	1	no	—
325	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
326	C₂×D₄	—	2	1	no	—
327	C₂×D₄	—	2	2	no	—
328	C₂×D₄	—	2	2	no	—
329	C₂×D₄	—	2	2	no	—
330	C₂×D₄	—	4	4	no	—
331	C₂×D₄	—	4	1	no	—
332	C₂×D₄	—	4	1	no	—
333	C₂×D₄	—	4	1	no	—
334	C₂×D₄	—	4	2	no	—
335	C₂×D₄	—	4	2	no	—
336	C₂×D₄	—	8	2	no	—
337	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
338	C₂²×C₄	—	2	1	no	—
339	C₂²×C₄	—	2	1	no	—
340	C₂²×C₄	—	2	1	no	—
341	C₂²×C₄	—	4	1	no	—
342	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
343	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
344	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
345	SD₁₆	—	4	8	no	—
346	SD₁₆	—	4	4	no	—
347	SD₁₆	—	4	8	no	—
348	D₈	—	4	4	no	—
349	M₁₆	—	4	2	no	—
350	M₁₆	—	4	2	no	—
351	C₂×C₈	—	2	4	no	—
352	C₂×C₈	—	2	2	no	—
353	C₂×C₈	—	2	2	no	—
354	C₂×C₈	—	2	2	no	—
355	C₂×C₈	—	4	4	no	—
356	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
357	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
358	C₄⋊C₄	—	2	1	no	—
359	C₄⋊C₄	—	2	1	no	—
360	C₄⋊C₄	—	2	2	no	—
361	C₄⋊C₄	—	2	2	no	—
362	C₄⋊C₄	—	4	1	no	—
363	C₄⋊C₄	—	4	1	no	—
364	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
365	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
366	C₂²⋊C₄	—	2	1	no	—
367	C₂²⋊C₄	—	2	1	no	—
368	C₂²⋊C₄	—	2	1	no	—
369	C₄²	—	2	1	no	—
370	C₄²	—	2	1	no	—
371	C₄²	—	2	1	no	—
372	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
373	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
374	C₄²	#41×C₃₁	1	1	yes	characteristic
375	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
376	C₂³	—	2	1	no	—
377	C₂³	—	4	2	no	—
378	C₂³	—	8	2	no	—
379	C₂³	—	8	2	no	—
380	C₂³	—	8	2	no	—
381	C₂³	C₃₁×#121	1	1	yes	characteristic
382	Q₈	—	2	1	no	—
383	Q₈	—	4	4	no	—
384	Q₈	—	8	2	no	—
385	Q₈	C₂×SD₁₆×C₃₁	1	2	yes	complemented
386	D₄	—	2	4	no	—
387	D₄	—	2	1	no	—
388	D₄	—	4	2	no	—
389	D₄	—	4	2	no	—
390	D₄	—	4	4	no	—
391	D₄	—	8	2	no	—
392	D₄	—	8	2	no	—
393	D₄	—	8	2	no	—
394	D₄	—	8	2	no	—
395	D₄	C₂×SD₁₆×C₃₁	1	2	yes	complemented
396	C₂×C₄	—	2	1	no	—
397	C₂×C₄	—	2	4	no	—
398	C₂×C₄	—	2	2	no	—
399	C₂×C₄	—	2	2	no	—
400	C₂×C₄	—	2	1	no	—
401	C₂×C₄	—	2	1	no	—
402	C₂×C₄	—	4	2	no	—
403	C₂×C₄	—	4	2	no	—
404	C₂×C₄	—	4	2	no	—
405	C₂×C₄	—	8	2	no	—
406	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
407	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
408	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
409	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
410	C₂×C₄	C₃₁×#135	1	2	yes	—
411	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
412	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
413	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
414	C₈	—	2	2	no	—
415	C₈	—	4	2	no	—
416	C₈	—	4	2	no	—
417	C₈	—	4	2	no	—
418	C₈	C₃₁×AGL(1,ℤ/8)	1	4	yes	complemented
419	C₂²	—	2	1	no	—
420	C₂²	—	2	1	no	—
421	C₂²	—	2	2	no	—
422	C₂²	—	4	1	no	—
423	C₂²	—	4	1	no	—
424	C₂²	—	8	2	no	—
425	C₂²	—	8	2	no	—
426	C₂²	—	8	2	no	—
427	C₂²	—	16	4	no	—
428	C₂²	C₃₁×#465	1	2	yes	—
429	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
430	C₄	—	2	1	no	—
431	C₄	—	2	2	no	—
432	C₄	—	2	1	no	—
433	C₄	—	4	1	no	—
434	C₄	—	4	1	no	—
435	C₄	—	8	2	no	—
436	C₄	C₃₁×#465	1	2	yes	—
437	C₄	C₃₁×#471	1	2	yes	—
438	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
439	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
440	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
441	C₂	—	2	2	no	—
442	C₂	—	4	2	no	—
443	C₂	—	16	2	no	—
444	C₂	C₃₁×#3038	1	1	yes	characteristic
445	C₂	C₂×C₃₁×#465	1	1	yes	characteristic
446	C₂	C₂×C₃₁×#471	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3038	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₃₁ ◁ C₂×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#492	C₂	semidirect
2. C₃₁×#3079	C₂	semidirect
4. C₈×#41×C₃₁	C₂	semidirect
5. C₃₁×#1666	C₂	semidirect
6. C₃₁×#1665	C₂	semidirect
7. C₃₁×#1566	C₂	semidirect
8. C₃₁×#1549	C₂	semidirect
9. C₃₁×#1545	C₂	semidirect
20. C₃₁×#475	C₂²	semidirect
21. C₃₁×#474	C₂²	semidirect
26. C₈×Q₈×C₃₁	C₂²	semidirect
28. C₈×D₄×C₃₁	C₂²	semidirect
38. C₃₁×#334	C₂²	semidirect
39. C₃₁×#332	C₂²	semidirect
97. #41×C₃₁	SD₁₆	semidirect
152. #19266	C₃₁	direct (N × H)
162. Q₈×C₃₁	C₂×SD₁₆	semidirect
172. D₄×C₃₁	C₂×SD₁₆	semidirect
195. C₈×C₃₁	AGL(1,ℤ/8)	semidirect
196. C₂×#492	C₂×C₃₁	semidirect
197. #3079	C₂×C₃₁	semidirect
199. C₈×#41	C₂×C₃₁	semidirect
200. #1666	C₂×C₃₁	semidirect
201. #1665	C₂×C₃₁	semidirect
202. #1566	C₂×C₃₁	semidirect
203. #1549	C₂×C₃₁	semidirect
204. #1545	C₂×C₃₁	semidirect
237. #475	C₂²×C₃₁	semidirect
238. #474	C₂²×C₃₁	semidirect
243. C₈×Q₈	C₂²×C₃₁	semidirect
245. C₈×D₄	C₂²×C₃₁	semidirect
255. #334	C₂²×C₃₁	semidirect
256. #332	C₂²×C₃₁	semidirect
318. C₃₁	#19266	direct (N × H)
321. #41	SD₁₆×C₃₁	semidirect
385. Q₈	C₂×SD₁₆×C₃₁	semidirect
395. D₄	C₂×SD₁₆×C₃₁	semidirect
418. C₈	C₃₁×AGL(1,ℤ/8)	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#492	2
C₃₁×#3079	2
C₃₁×#3038	2
C₈×#41×C₃₁	2
C₃₁×#1666	2
C₃₁×#1665	2
C₃₁×#1566	2
C₃₁×#1549	2
C₃₁×#1545	2
#19266	31

Subgroups — C31×#19266 #21476025

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19266 #21476025