Subgroups — C₃₁×#19238 #21475997

Order 7936.

722 conjugacy classes of subgroups.

The subgroup lattice has 478 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3246	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3208	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3151	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2962	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1688	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1686	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1668	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1666	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1565	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1540	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#331	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
13	C₂×C₃₁×AGL(1,ℤ/8)	—	2	1	no	—
14	C₃₁×#533	C₂²	1	1	yes	characteristic
15	C₃₁×#532	C₂²	1	1	yes	characteristic
16	C₂×C₃₁×#127	C₂²	1	1	yes	characteristic
17	C₃₁×#496	—	2	2	no	—
18	C₃₁×#493	C₂²	1	2	yes	—
19	C₃₁×#492	—	2	1	no	—
20	C₃₁×#492	C₂²	1	2	yes	—
21	C₃₁×#486	—	2	1	no	—
22	C₃₁×#476	—	2	1	no	—
23	C₃₁×#475	C₂²	1	2	yes	complemented
24	C₃₁×#465	—	2	2	no	—
25	C₃₁×#461	—	2	1	no	—
26	C₃₁×#460	C₂²	1	2	yes	—
27	C₃₁×#460	C₂²	1	2	yes	—
28	C₃₁×#460	C₂²	1	2	yes	complemented
29	C₃₁×#450	—	2	2	no	—
30	C₃₁×#446	—	2	2	no	—
31	C₃₁×#446	C₂²	1	2	yes	complemented
32	C₈×Q₈×C₃₁	C₂²	1	2	yes	complemented
33	C₃₁×#431	C₂²	1	1	yes	characteristic
34	C₃₁×#424	—	2	1	no	—
35	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₃₁×#405	—	2	1	no	—
38	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
39	C₃₁×#335	C₂²	1	2	yes	complemented
40	C₃₁×#334	C₂²	1	2	yes	complemented
41	C₃₁×#332	C₂²	1	1	yes	characteristic, complemented
42	C₃₁×#332	C₂²	1	1	yes	characteristic, complemented
43	C₃₁×#331	C₂²	1	2	yes	—
44	C₃₁×#331	C₂²	1	2	yes	—
45	C₃₁×#331	C₂²	1	1	yes	characteristic, complemented
46	C₃₁×#331	C₂²	1	1	yes	characteristic, complemented
47	C₂×#41×C₃₁	—	2	1	no	—
48	C₂²×D₄×C₃₁	—	2	1	no	—
49	C₂²×D₄×C₃₁	—	2	1	no	—
50	C₂²×D₄×C₃₁	—	2	1	no	—
51	C₃₁×AGL(1,ℤ/8)	—	4	4	no	—
52	C₃₁×AGL(1,ℤ/8)	—	4	4	no	—
53	C₂×SD₁₆×C₃₁	—	2	2	no	—
54	C₂×SD₁₆×C₃₁	—	2	2	no	—
55	C₂×SD₁₆×C₃₁	—	2	2	no	—
56	C₂×SD₁₆×C₃₁	—	2	2	no	—
57	C₂×SD₁₆×C₃₁	—	2	2	no	—
58	C₂×D₈×C₃₁	—	2	2	no	—
59	C₂×D₈×C₃₁	—	2	2	no	—
60	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
61	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
62	C₂²×C₈×C₃₁	—	2	1	no	—
63	C₂²×C₈×C₃₁	—	2	1	no	—
64	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
65	C₃₁×C₄⋊Q₈	D₄	1	1	yes	characteristic
66	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
67	C₃₁×#127	—	2	1	no	—
68	C₃₁×#127	—	2	1	no	—
69	C₃₁×#127	—	2	1	no	—
70	C₃₁×#127	C₂³	1	2	yes	—
71	C₃₁×#127	C₂³	1	1	yes	characteristic
72	C₃₁×#125	—	2	1	no	—
73	C₃₁×#124	—	2	1	no	—
74	C₃₁×C₂²⋊Q₈	—	2	1	no	—
75	C₃₁×C₂²⋊Q₈	D₄	1	2	yes	—
76	C₃₁×#121	—	2	1	no	—
77	C₃₁×#121	—	2	1	no	—
78	C₄×Q₈×C₃₁	—	2	1	no	—
79	C₄×Q₈×C₃₁	C₂³	1	2	yes	—
80	C₄×D₄×C₃₁	—	2	1	no	—
81	C₄×D₄×C₃₁	—	2	1	no	—
82	C₃₁×#117	—	2	1	no	—
83	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
84	C₃₁×#106	—	2	2	no	—
85	C₃₁×C₄⋊C₈	—	2	2	no	—
86	C₃₁×C₄⋊C₈	—	2	2	no	—
87	C₃₁×C₄⋊C₈	—	2	1	no	—
88	C₃₁×C₄⋊C₈	—	2	1	no	—
89	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
90	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
91	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
92	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
93	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
94	C₃₁×#103	—	2	2	no	—
95	C₃₁×#102	—	2	2	no	—
96	C₃₁×#102	—	2	2	no	—
97	C₃₁×#102	—	2	2	no	—
98	C₃₁×C₂²⋊C₈	—	2	1	no	—
99	C₃₁×#97	—	2	2	no	—
100	C₃₁×#97	D₄	1	2	yes	—
101	C₄×C₈×C₃₁	—	2	2	no	—
102	C₄×C₈×C₃₁	—	2	1	no	—
103	C₄×C₈×C₃₁	C₂³	1	2	yes	—
104	C₂⁴×C₃₁	—	4	1	no	—
105	#41×C₃₁	—	4	2	no	—
106	#41×C₃₁	—	4	2	no	—
107	C₂×Q₈×C₃₁	—	2	1	no	—
108	C₂×Q₈×C₃₁	—	2	1	no	—
109	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
110	C₂×D₄×C₃₁	—	2	1	no	—
111	C₂×D₄×C₃₁	—	2	2	no	—
112	C₂×D₄×C₃₁	—	2	1	no	—
113	C₂×D₄×C₃₁	—	2	2	no	—
114	C₂×D₄×C₃₁	—	2	1	no	—
115	C₂×D₄×C₃₁	—	2	2	no	—
116	C₂×D₄×C₃₁	—	4	1	no	—
117	C₂×D₄×C₃₁	—	4	1	no	—
118	C₂×D₄×C₃₁	—	4	1	no	—
119	C₂×D₄×C₃₁	—	4	1	no	—
120	C₂×D₄×C₃₁	—	4	2	no	—
121	C₂×D₄×C₃₁	—	4	2	no	—
122	C₂×D₄×C₃₁	—	8	1	no	—
123	C₂×D₄×C₃₁	—	8	1	no	—
124	C₂×D₄×C₃₁	—	8	2	no	—
125	C₂²×C₄×C₃₁	—	4	1	no	—
126	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
129	SD₁₆×C₃₁	—	4	4	no	—
130	SD₁₆×C₃₁	—	4	4	no	—
131	SD₁₆×C₃₁	—	8	2	no	—
132	SD₁₆×C₃₁	—	8	2	no	—
133	SD₁₆×C₃₁	—	8	2	no	—
134	D₈×C₃₁	—	4	4	no	—
135	D₈×C₃₁	—	4	4	no	—
136	M₁₆×C₃₁	—	2	2	no	—
137	M₁₆×C₃₁	SD₁₆	1	4	yes	complemented
138	C₂×C₈×C₃₁	—	2	1	no	—
139	C₂×C₈×C₃₁	—	2	1	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	1	no	—
142	C₂×C₈×C₃₁	—	2	2	no	—
143	C₂×C₈×C₃₁	—	4	2	no	—
144	C₂×C₈×C₃₁	—	8	1	no	—
145	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
146	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
147	C₄⋊C₄×C₃₁	—	2	1	no	—
148	C₄⋊C₄×C₃₁	—	2	1	no	—
149	C₄⋊C₄×C₃₁	—	2	1	no	—
150	C₄⋊C₄×C₃₁	—	2	2	no	—
151	C₄⋊C₄×C₃₁	—	2	2	no	—
152	C₄⋊C₄×C₃₁	—	2	1	no	—
153	C₄⋊C₄×C₃₁	—	2	1	no	—
154	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
155	C₂²⋊C₄×C₃₁	—	2	2	no	—
156	C₂²⋊C₄×C₃₁	—	4	1	no	—
157	C₂²⋊C₄×C₃₁	—	4	1	no	—
158	C₄²×C₃₁	—	2	2	no	—
159	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
160	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
161	C₄²×C₃₁	#41	1	1	yes	characteristic
162	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
163	#19238	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
164	C₂³×C₃₁	—	4	1	no	—
165	C₂³×C₃₁	—	4	2	no	—
166	C₂³×C₃₁	—	8	2	no	—
167	C₂³×C₃₁	—	8	2	no	—
168	C₂³×C₃₁	—	8	2	no	—
169	C₂³×C₃₁	#121	1	1	yes	characteristic
170	Q₈×C₃₁	—	2	2	no	—
171	Q₈×C₃₁	—	2	2	no	—
172	Q₈×C₃₁	—	4	1	no	—
173	Q₈×C₃₁	—	4	1	no	—
174	Q₈×C₃₁	—	8	1	no	—
175	D₄×C₃₁	—	4	1	no	—
176	D₄×C₃₁	—	4	2	no	—
177	D₄×C₃₁	—	4	1	no	—
178	D₄×C₃₁	—	4	2	no	—
179	D₄×C₃₁	—	4	1	no	—
180	D₄×C₃₁	—	4	2	no	—
181	D₄×C₃₁	—	4	2	no	—
182	D₄×C₃₁	—	4	1	no	—
183	D₄×C₃₁	—	8	1	no	—
184	D₄×C₃₁	—	8	2	no	—
185	D₄×C₃₁	—	8	2	no	—
186	D₄×C₃₁	—	8	2	no	—
187	D₄×C₃₁	—	8	2	no	—
188	D₄×C₃₁	—	8	2	no	—
189	C₂×C₄×C₃₁	—	2	1	no	—
190	C₂×C₄×C₃₁	—	2	1	no	—
191	C₂×C₄×C₃₁	—	2	2	no	—
192	C₂×C₄×C₃₁	—	2	2	no	—
193	C₂×C₄×C₃₁	—	4	1	no	—
194	C₂×C₄×C₃₁	—	4	1	no	—
195	C₂×C₄×C₃₁	—	8	2	no	—
196	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
197	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
198	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
199	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
200	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
201	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
202	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
203	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
204	C₈×C₃₁	—	2	2	no	—
205	C₈×C₃₁	—	4	1	no	—
206	C₈×C₃₁	—	4	1	no	—
207	C₈×C₃₁	—	4	2	no	—
208	C₈×C₃₁	—	4	1	no	—
209	C₈×C₃₁	—	4	1	no	—
210	C₈×C₃₁	C₂×SD₁₆	1	4	yes	—
211	#3246	C₂×C₃₁	1	1	yes	characteristic, complemented
212	#3208	C₂×C₃₁	1	1	yes	characteristic, complemented
213	#3151	C₂×C₃₁	1	1	yes	characteristic
214	#2962	C₂×C₃₁	1	1	yes	characteristic, complemented
215	#1688	C₂×C₃₁	1	2	yes	complemented
216	#1686	C₂×C₃₁	1	2	yes	complemented
217	#1668	C₂×C₃₁	1	2	yes	complemented
218	#1666	C₂×C₃₁	1	2	yes	complemented
219	#1565	C₂×C₃₁	1	1	yes	characteristic, complemented
220	#1540	C₂×C₃₁	1	1	yes	characteristic, complemented
221	C₂×#331	C₂×C₃₁	1	1	yes	characteristic, complemented
222	C₂²×C₃₁	—	2	1	no	—
223	C₂²×C₃₁	—	2	1	no	—
224	C₂²×C₃₁	—	8	1	no	—
225	C₂²×C₃₁	—	8	2	no	—
226	C₂²×C₃₁	—	8	2	no	—
227	C₂²×C₃₁	—	8	1	no	—
228	C₂²×C₃₁	—	8	2	no	—
229	C₂²×C₃₁	—	8	1	no	—
230	C₂²×C₃₁	—	16	4	no	—
231	C₂²×C₃₁	#460	1	2	yes	—
232	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
233	C₄×C₃₁	—	2	1	no	—
234	C₄×C₃₁	—	2	1	no	—
235	C₄×C₃₁	—	2	1	no	—
236	C₄×C₃₁	—	4	2	no	—
237	C₄×C₃₁	—	4	2	no	—
238	C₄×C₃₁	—	8	1	no	—
239	C₄×C₃₁	#460	1	2	yes	—
240	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
241	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
242	C₄×C₃₁	#576	1	1	yes	characteristic
243	C₄×C₃₁	#577	1	1	yes	characteristic
244	C₂×AGL(1,ℤ/8)	—	2	1	no	—
245	C₂×AGL(1,ℤ/8)	—	2	1	no	—
246	#533	C₂²×C₃₁	1	1	yes	characteristic
247	#532	C₂²×C₃₁	1	1	yes	characteristic
248	C₂×#127	C₂²×C₃₁	1	1	yes	characteristic
249	#496	—	2	2	no	—
250	#493	C₂²×C₃₁	1	2	yes	—
251	#492	—	2	1	no	—
252	#492	C₂²×C₃₁	1	2	yes	—
253	#486	—	2	1	no	—
254	#476	—	2	1	no	—
255	#475	C₂²×C₃₁	1	2	yes	complemented
256	#465	—	2	2	no	—
257	#461	—	2	1	no	—
258	#460	C₂²×C₃₁	1	2	yes	—
259	#460	C₂²×C₃₁	1	2	yes	—
260	#460	C₂²×C₃₁	1	2	yes	complemented
261	#450	—	2	2	no	—
262	#446	—	2	2	no	—
263	#446	C₂²×C₃₁	1	2	yes	complemented
264	C₈×Q₈	C₂²×C₃₁	1	2	yes	complemented
265	#431	C₂²×C₃₁	1	1	yes	characteristic
266	#424	—	2	1	no	—
267	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
268	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
269	#405	—	2	1	no	—
270	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
271	#335	C₂²×C₃₁	1	2	yes	complemented
272	#334	C₂²×C₃₁	1	2	yes	complemented
273	#332	C₂²×C₃₁	1	1	yes	characteristic, complemented
274	#332	C₂²×C₃₁	1	1	yes	characteristic, complemented
275	#331	C₂²×C₃₁	1	2	yes	—
276	#331	C₂²×C₃₁	1	2	yes	—
277	#331	C₂²×C₃₁	1	1	yes	characteristic, complemented
278	#331	C₂²×C₃₁	1	1	yes	characteristic, complemented
279	C₂×C₃₁	—	2	2	no	—
280	C₂×C₃₁	—	16	1	no	—
281	C₂×C₃₁	—	16	2	no	—
282	C₂×C₃₁	C₂×#460	1	1	yes	characteristic
283	C₂×C₃₁	#3062	1	1	yes	characteristic
284	C₂×C₃₁	#3068	1	1	yes	characteristic
285	C₂×#41	—	2	1	no	—
286	C₂²×D₄	—	2	1	no	—
287	C₂²×D₄	—	2	1	no	—
288	C₂²×D₄	—	2	1	no	—
289	AGL(1,ℤ/8)	—	4	4	no	—
290	AGL(1,ℤ/8)	—	4	4	no	—
291	C₂×SD₁₆	—	2	2	no	—
292	C₂×SD₁₆	—	2	2	no	—
293	C₂×SD₁₆	—	2	2	no	—
294	C₂×SD₁₆	—	2	2	no	—
295	C₂×SD₁₆	—	2	2	no	—
296	C₂×D₈	—	2	2	no	—
297	C₂×D₈	—	2	2	no	—
298	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
299	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
300	C₂²×C₈	—	2	1	no	—
301	C₂²×C₈	—	2	1	no	—
302	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
303	C₄⋊Q₈	D₄×C₃₁	1	1	yes	characteristic
304	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
305	#127	—	2	1	no	—
306	#127	—	2	1	no	—
307	#127	—	2	1	no	—
308	#127	C₂³×C₃₁	1	2	yes	—
309	#127	C₂³×C₃₁	1	1	yes	characteristic
310	#125	—	2	1	no	—
311	#124	—	2	1	no	—
312	C₂²⋊Q₈	—	2	1	no	—
313	C₂²⋊Q₈	D₄×C₃₁	1	2	yes	—
314	#121	—	2	1	no	—
315	#121	—	2	1	no	—
316	C₄×Q₈	—	2	1	no	—
317	C₄×Q₈	C₂³×C₃₁	1	2	yes	—
318	C₄×D₄	—	2	1	no	—
319	C₄×D₄	—	2	1	no	—
320	#117	—	2	1	no	—
321	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
322	#106	—	2	2	no	—
323	C₄⋊C₈	—	2	2	no	—
324	C₄⋊C₈	—	2	2	no	—
325	C₄⋊C₈	—	2	1	no	—
326	C₄⋊C₈	—	2	1	no	—
327	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
328	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
329	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
330	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
331	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
332	#103	—	2	2	no	—
333	#102	—	2	2	no	—
334	#102	—	2	2	no	—
335	#102	—	2	2	no	—
336	C₂²⋊C₈	—	2	1	no	—
337	#97	—	2	2	no	—
338	#97	D₄×C₃₁	1	2	yes	—
339	C₄×C₈	—	2	2	no	—
340	C₄×C₈	—	2	1	no	—
341	C₄×C₈	C₂³×C₃₁	1	2	yes	—
342	C₃₁	#19238	1	1	yes	characteristic, direct product factor, Sylow
343	C₂⁴	—	4	1	no	—
344	#41	—	4	2	no	—
345	#41	—	4	2	no	—
346	C₂×Q₈	—	2	1	no	—
347	C₂×Q₈	—	2	1	no	—
348	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
349	C₂×D₄	—	2	1	no	—
350	C₂×D₄	—	2	2	no	—
351	C₂×D₄	—	2	1	no	—
352	C₂×D₄	—	2	2	no	—
353	C₂×D₄	—	2	1	no	—
354	C₂×D₄	—	2	2	no	—
355	C₂×D₄	—	4	1	no	—
356	C₂×D₄	—	4	1	no	—
357	C₂×D₄	—	4	1	no	—
358	C₂×D₄	—	4	1	no	—
359	C₂×D₄	—	4	2	no	—
360	C₂×D₄	—	4	2	no	—
361	C₂×D₄	—	8	1	no	—
362	C₂×D₄	—	8	1	no	—
363	C₂×D₄	—	8	2	no	—
364	C₂²×C₄	—	4	1	no	—
365	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
366	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
367	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
368	SD₁₆	—	4	4	no	—
369	SD₁₆	—	4	4	no	—
370	SD₁₆	—	8	2	no	—
371	SD₁₆	—	8	2	no	—
372	SD₁₆	—	8	2	no	—
373	D₈	—	4	4	no	—
374	D₈	—	4	4	no	—
375	M₁₆	—	2	2	no	—
376	M₁₆	SD₁₆×C₃₁	1	4	yes	complemented
377	C₂×C₈	—	2	1	no	—
378	C₂×C₈	—	2	1	no	—
379	C₂×C₈	—	2	1	no	—
380	C₂×C₈	—	2	1	no	—
381	C₂×C₈	—	2	2	no	—
382	C₂×C₈	—	4	2	no	—
383	C₂×C₈	—	8	1	no	—
384	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
385	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
386	C₄⋊C₄	—	2	1	no	—
387	C₄⋊C₄	—	2	1	no	—
388	C₄⋊C₄	—	2	1	no	—
389	C₄⋊C₄	—	2	2	no	—
390	C₄⋊C₄	—	2	2	no	—
391	C₄⋊C₄	—	2	1	no	—
392	C₄⋊C₄	—	2	1	no	—
393	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
394	C₂²⋊C₄	—	2	2	no	—
395	C₂²⋊C₄	—	4	1	no	—
396	C₂²⋊C₄	—	4	1	no	—
397	C₄²	—	2	2	no	—
398	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
399	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
400	C₄²	#41×C₃₁	1	1	yes	characteristic
401	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
402	C₂³	—	4	1	no	—
403	C₂³	—	4	2	no	—
404	C₂³	—	8	2	no	—
405	C₂³	—	8	2	no	—
406	C₂³	—	8	2	no	—
407	C₂³	C₃₁×#121	1	1	yes	characteristic
408	Q₈	—	2	2	no	—
409	Q₈	—	2	2	no	—
410	Q₈	—	4	1	no	—
411	Q₈	—	4	1	no	—
412	Q₈	—	8	1	no	—
413	D₄	—	4	1	no	—
414	D₄	—	4	2	no	—
415	D₄	—	4	1	no	—
416	D₄	—	4	2	no	—
417	D₄	—	4	1	no	—
418	D₄	—	4	2	no	—
419	D₄	—	4	2	no	—
420	D₄	—	4	1	no	—
421	D₄	—	8	1	no	—
422	D₄	—	8	2	no	—
423	D₄	—	8	2	no	—
424	D₄	—	8	2	no	—
425	D₄	—	8	2	no	—
426	D₄	—	8	2	no	—
427	C₂×C₄	—	2	1	no	—
428	C₂×C₄	—	2	1	no	—
429	C₂×C₄	—	2	2	no	—
430	C₂×C₄	—	2	2	no	—
431	C₂×C₄	—	4	1	no	—
432	C₂×C₄	—	4	1	no	—
433	C₂×C₄	—	8	2	no	—
434	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
435	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
436	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
437	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
438	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
439	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
440	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
441	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
442	C₈	—	2	2	no	—
443	C₈	—	4	1	no	—
444	C₈	—	4	1	no	—
445	C₈	—	4	2	no	—
446	C₈	—	4	1	no	—
447	C₈	—	4	1	no	—
448	C₈	C₂×SD₁₆×C₃₁	1	4	yes	—
449	C₂²	—	2	1	no	—
450	C₂²	—	2	1	no	—
451	C₂²	—	8	1	no	—
452	C₂²	—	8	2	no	—
453	C₂²	—	8	2	no	—
454	C₂²	—	8	1	no	—
455	C₂²	—	8	2	no	—
456	C₂²	—	8	1	no	—
457	C₂²	—	16	4	no	—
458	C₂²	C₃₁×#460	1	2	yes	—
459	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
460	C₄	—	2	1	no	—
461	C₄	—	2	1	no	—
462	C₄	—	2	1	no	—
463	C₄	—	4	2	no	—
464	C₄	—	4	2	no	—
465	C₄	—	8	1	no	—
466	C₄	C₃₁×#460	1	2	yes	—
467	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
468	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
469	C₄	C₃₁×#576	1	1	yes	characteristic
470	C₄	C₃₁×#577	1	1	yes	characteristic
471	C₂	—	2	2	no	—
472	C₂	—	16	2	no	—
473	C₂	—	16	1	no	—
474	C₂	C₂×C₃₁×#460	1	1	yes	characteristic
475	C₂	C₃₁×#3062	1	1	yes	characteristic
476	C₂	C₃₁×#3068	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3151	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. C₃₁×#3246	C₂	semidirect
2. C₃₁×#3208	C₂	semidirect
4. C₃₁×#2962	C₂	semidirect
5. C₃₁×#1688	C₂	semidirect
6. C₃₁×#1686	C₂	semidirect
7. C₃₁×#1668	C₂	semidirect
8. C₃₁×#1666	C₂	semidirect
9. C₃₁×#1565	C₂	semidirect
10. C₃₁×#1540	C₂	semidirect
11. C₂×C₃₁×#331	C₂	semidirect
23. C₃₁×#475	C₂²	semidirect
28. C₃₁×#460	C₂²	semidirect
31. C₃₁×#446	C₂²	semidirect
32. C₈×Q₈×C₃₁	C₂²	semidirect
39. C₃₁×#335	C₂²	semidirect
40. C₃₁×#334	C₂²	semidirect
41. C₃₁×#332	C₂²	semidirect
42. C₃₁×#332	C₂²	semidirect
45. C₃₁×#331	C₂²	semidirect
46. C₃₁×#331	C₂²	semidirect
137. M₁₆×C₃₁	SD₁₆	semidirect
163. #19238	C₃₁	direct (N × H)
211. #3246	C₂×C₃₁	semidirect
212. #3208	C₂×C₃₁	semidirect
214. #2962	C₂×C₃₁	semidirect
215. #1688	C₂×C₃₁	semidirect
216. #1686	C₂×C₃₁	semidirect
217. #1668	C₂×C₃₁	semidirect
218. #1666	C₂×C₃₁	semidirect
219. #1565	C₂×C₃₁	semidirect
220. #1540	C₂×C₃₁	semidirect
221. C₂×#331	C₂×C₃₁	semidirect
255. #475	C₂²×C₃₁	semidirect
260. #460	C₂²×C₃₁	semidirect
263. #446	C₂²×C₃₁	semidirect
264. C₈×Q₈	C₂²×C₃₁	semidirect
271. #335	C₂²×C₃₁	semidirect
272. #334	C₂²×C₃₁	semidirect
273. #332	C₂²×C₃₁	semidirect
274. #332	C₂²×C₃₁	semidirect
277. #331	C₂²×C₃₁	semidirect
278. #331	C₂²×C₃₁	semidirect
342. C₃₁	#19238	direct (N × H)
376. M₁₆	SD₁₆×C₃₁	semidirect

Maximal subgroups

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

subgroup	index
C₃₁×#3246	2
C₃₁×#3208	2
C₃₁×#3151	2
C₃₁×#2962	2
C₃₁×#1688	2
C₃₁×#1686	2
C₃₁×#1668	2
C₃₁×#1666	2
C₃₁×#1565	2
C₃₁×#1540	2
C₂×C₃₁×#331	2
#19238	31

Subgroups — C31×#19238 #21475997

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19238 #21475997