Subgroups — C₃₁×#19414 #21476173

Order 7936.

662 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3227	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3079	C₂	1	1	yes	maximal, characteristic, complemented
3	C₃₁×#3035	C₂	1	1	yes	maximal, characteristic
4	C₃₁×#2988	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1678	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1677	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1672	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1671	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1570	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1547	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₃₁×#333	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₃₁×#135	—	2	1	no	—
13	C₃₁×#532	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₄×#41×C₃₁	C₂²	1	1	yes	characteristic
16	C₃₁×#498	C₂²	1	2	yes	—
17	C₃₁×#486	C₂²	1	2	yes	—
18	C₃₁×#481	—	2	2	no	—
19	C₃₁×#477	C₂²	1	2	yes	—
20	C₃₁×#476	—	2	1	no	—
21	C₃₁×#476	—	2	1	no	—
22	C₃₁×#476	C₂²	1	2	yes	—
23	C₃₁×#475	—	2	1	no	—
24	C₃₁×#474	—	2	1	no	—
25	C₃₁×#464	—	2	1	no	—
26	C₃₁×#463	—	2	1	no	—
27	C₃₁×#462	C₂²	1	1	yes	characteristic
28	C₃₁×#461	C₂²	1	1	yes	characteristic
29	C₃₁×#460	C₂²	1	1	yes	characteristic
30	C₃₁×#459	C₂²	1	1	yes	characteristic
31	C₈×Q₈×C₃₁	C₂²	1	2	yes	—
32	C₈×D₄×C₃₁	—	2	2	no	—
33	C₈×D₄×C₃₁	C₂²	1	2	yes	—
34	C₃₁×#431	—	2	1	no	—
35	C₃₁×#431	C₂²	1	1	yes	characteristic
36	C₃₁×#428	—	2	1	no	—
37	C₃₁×#423	C₂²	1	1	yes	characteristic
38	C₂×C₃₁×C₄⋊C₈	—	2	1	no	—
39	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
40	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
41	C₃₁×#417	—	2	1	no	—
42	C₃₁×#416	—	2	1	no	—
43	C₃₁×#334	C₂²	1	2	yes	—
44	C₃₁×#334	C₂²	1	2	yes	—
45	C₃₁×#333	C₂²	1	2	yes	—
46	C₃₁×#333	C₂²	1	2	yes	—
47	C₃₁×#332	C₂²	1	2	yes	—
48	C₃₁×#331	C₂²	1	2	yes	—
49	C₂×#41×C₃₁	—	2	1	no	—
50	C₂×#41×C₃₁	D₄	1	1	yes	characteristic
51	C₂²×Q₈×C₃₁	—	2	1	no	—
52	C₃₁×#135	—	4	4	no	—
53	C₂×Q₁₆×C₃₁	—	2	1	no	—
54	C₂×SD₁₆×C₃₁	—	2	1	no	—
55	C₂×SD₁₆×C₃₁	—	2	1	no	—
56	C₂×D₈×C₃₁	—	2	1	no	—
57	C₂×M₁₆×C₃₁	—	2	1	no	—
58	C₂²×C₈×C₃₁	—	2	1	no	—
59	C₂²×C₈×C₃₁	—	2	1	no	—
60	C₂²×C₈×C₃₁	—	2	2	no	—
61	C₃₁×C₄⋊Q₈	—	2	1	no	—
62	C₃₁×C₄⋊Q₈	—	2	1	no	—
63	C₃₁×C₄⋊Q₈	—	2	1	no	—
64	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
65	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
66	C₃₁×#127	C₂³	1	1	yes	characteristic
67	C₃₁×#124	—	2	1	no	—
68	C₃₁×#121	—	2	1	no	—
69	C₃₁×#121	—	2	1	no	—
70	C₄×Q₈×C₃₁	D₄	1	1	yes	characteristic
71	C₄×Q₈×C₃₁	C₂³	1	1	yes	characteristic
72	C₄×D₄×C₃₁	—	2	1	no	—
73	C₄×D₄×C₃₁	—	2	1	no	—
74	C₄×D₄×C₃₁	—	2	1	no	—
75	C₄×D₄×C₃₁	—	2	1	no	—
76	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
77	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
78	C₃₁×#117	—	2	1	no	—
79	C₃₁×#117	D₄	1	1	yes	characteristic
80	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
81	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
82	C₂×C₄²×C₃₁	—	2	1	no	—
83	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
84	C₃₁×#107	—	2	2	no	—
85	C₃₁×#106	—	2	2	no	—
86	C₃₁×#106	—	2	2	no	—
87	C₃₁×#106	—	2	2	no	—
88	C₃₁×C₄⋊C₈	—	2	2	no	—
89	C₃₁×C₄⋊C₈	—	2	1	no	—
90	C₃₁×C₄⋊C₈	—	2	1	no	—
91	C₃₁×C₄⋊C₈	—	2	1	no	—
92	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
93	C₃₁×C₄⋊C₈	D₄	1	1	yes	characteristic
94	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
95	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
96	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
97	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
98	C₃₁×#103	—	2	2	no	—
99	C₃₁×#103	—	2	1	no	—
100	C₃₁×#103	—	2	1	no	—
101	C₃₁×#102	—	2	2	no	—
102	C₃₁×#102	—	2	2	no	—
103	C₃₁×#102	—	2	2	no	—
104	C₃₁×#102	—	2	1	no	—
105	C₃₁×#102	—	2	1	no	—
106	C₃₁×C₂²⋊C₈	—	2	2	no	—
107	C₃₁×C₂²⋊C₈	D₄	1	2	yes	—
108	C₄×C₈×C₃₁	—	2	2	no	—
109	C₄×C₈×C₃₁	C₂³	1	2	yes	—
110	#41×C₃₁	—	2	2	no	—
111	#41×C₃₁	—	4	2	no	—
112	#41×C₃₁	—	4	2	no	—
113	#41×C₃₁	SD₁₆	1	4	yes	—
114	C₂×Q₈×C₃₁	—	2	1	no	—
115	C₂×Q₈×C₃₁	—	2	2	no	—
116	C₂×Q₈×C₃₁	—	4	1	no	—
117	C₂×Q₈×C₃₁	—	8	2	no	—
118	C₂×Q₈×C₃₁	C₂×D₄	1	1	yes	characteristic
119	C₂×D₄×C₃₁	—	2	1	no	—
120	C₂×D₄×C₃₁	—	2	1	no	—
121	C₂×D₄×C₃₁	—	2	1	no	—
122	C₂×D₄×C₃₁	—	2	1	no	—
123	C₂×D₄×C₃₁	—	4	1	no	—
124	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
125	C₂²×C₄×C₃₁	—	2	1	no	—
126	C₂²×C₄×C₃₁	—	2	1	no	—
127	C₂²×C₄×C₃₁	—	2	1	no	—
128	C₂²×C₄×C₃₁	—	4	1	no	—
129	C₂²×C₄×C₃₁	—	4	1	no	—
130	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
131	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
132	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
133	Q₁₆×C₃₁	—	4	2	no	—
134	SD₁₆×C₃₁	—	4	2	no	—
135	SD₁₆×C₃₁	—	4	2	no	—
136	D₈×C₃₁	—	4	2	no	—
137	M₁₆×C₃₁	—	4	2	no	—
138	C₂×C₈×C₃₁	—	2	2	no	—
139	C₂×C₈×C₃₁	—	2	2	no	—
140	C₂×C₈×C₃₁	—	2	1	no	—
141	C₂×C₈×C₃₁	—	2	1	no	—
142	C₂×C₈×C₃₁	—	4	2	no	—
143	C₂×C₈×C₃₁	—	8	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	2	no	—
150	C₄⋊C₄×C₃₁	—	2	2	no	—
151	C₄⋊C₄×C₃₁	—	2	1	no	—
152	C₄⋊C₄×C₃₁	—	2	1	no	—
153	C₄⋊C₄×C₃₁	—	4	1	no	—
154	C₄⋊C₄×C₃₁	—	4	1	no	—
155	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
156	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
157	C₂²⋊C₄×C₃₁	—	2	1	no	—
158	C₂²⋊C₄×C₃₁	—	2	1	no	—
159	C₂²⋊C₄×C₃₁	—	2	1	no	—
160	C₂²⋊C₄×C₃₁	—	4	1	no	—
161	C₂²⋊C₄×C₃₁	—	4	1	no	—
162	C₄²×C₃₁	—	2	1	no	—
163	C₄²×C₃₁	—	2	1	no	—
164	C₄²×C₃₁	—	2	1	no	—
165	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
166	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
167	C₄²×C₃₁	#41	1	1	yes	characteristic
168	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
169	#19414	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
170	C₂³×C₃₁	—	2	1	no	—
171	C₂³×C₃₁	—	4	1	no	—
172	C₂³×C₃₁	#121	1	1	yes	characteristic
173	Q₈×C₃₁	—	2	1	no	—
174	Q₈×C₃₁	—	4	1	no	—
175	Q₈×C₃₁	—	4	1	no	—
176	Q₈×C₃₁	—	8	2	no	—
177	Q₈×C₃₁	—	8	2	no	—
178	Q₈×C₃₁	C₂×SD₁₆	1	2	yes	—
179	D₄×C₃₁	—	2	4	no	—
180	D₄×C₃₁	—	2	1	no	—
181	D₄×C₃₁	—	4	1	no	—
182	D₄×C₃₁	—	4	1	no	—
183	D₄×C₃₁	—	4	1	no	—
184	D₄×C₃₁	—	4	1	no	—
185	D₄×C₃₁	—	8	1	no	—
186	D₄×C₃₁	—	8	2	no	—
187	D₄×C₃₁	C₂×SD₁₆	1	2	yes	—
188	C₂×C₄×C₃₁	—	2	1	no	—
189	C₂×C₄×C₃₁	—	2	1	no	—
190	C₂×C₄×C₃₁	—	2	1	no	—
191	C₂×C₄×C₃₁	—	2	4	no	—
192	C₂×C₄×C₃₁	—	2	1	no	—
193	C₂×C₄×C₃₁	—	2	1	no	—
194	C₂×C₄×C₃₁	—	4	1	no	—
195	C₂×C₄×C₃₁	—	4	1	no	—
196	C₂×C₄×C₃₁	—	4	1	no	—
197	C₂×C₄×C₃₁	—	4	1	no	—
198	C₂×C₄×C₃₁	—	4	2	no	—
199	C₂×C₄×C₃₁	—	8	1	no	—
200	C₂×C₄×C₃₁	—	8	2	no	—
201	C₂×C₄×C₃₁	—	8	2	no	—
202	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
203	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
204	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
205	C₂×C₄×C₃₁	C₂×SD₁₆	1	2	yes	—
206	C₂×C₄×C₃₁	#135	1	2	yes	—
207	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
208	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
209	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
210	C₈×C₃₁	—	2	2	no	—
211	C₈×C₃₁	—	2	2	no	—
212	C₈×C₃₁	—	4	1	no	—
213	C₈×C₃₁	—	4	1	no	—
214	C₈×C₃₁	—	4	2	no	—
215	C₈×C₃₁	—	4	2	no	—
216	#3227	C₂×C₃₁	1	1	yes	characteristic, complemented
217	#3079	C₂×C₃₁	1	1	yes	characteristic, complemented
218	#3035	C₂×C₃₁	1	1	yes	characteristic
219	#2988	C₂×C₃₁	1	1	yes	characteristic, complemented
220	#1678	C₂×C₃₁	1	2	yes	complemented
221	#1677	C₂×C₃₁	1	2	yes	complemented
222	#1672	C₂×C₃₁	1	2	yes	complemented
223	#1671	C₂×C₃₁	1	2	yes	complemented
224	#1570	C₂×C₃₁	1	1	yes	characteristic, complemented
225	#1547	C₂×C₃₁	1	1	yes	characteristic, complemented
226	C₂×#333	C₂×C₃₁	1	1	yes	characteristic, complemented
227	C₂²×C₃₁	—	2	1	no	—
228	C₂²×C₃₁	—	2	1	no	—
229	C₂²×C₃₁	—	2	2	no	—
230	C₂²×C₃₁	—	4	1	no	—
231	C₂²×C₃₁	—	4	1	no	—
232	C₂²×C₃₁	—	8	1	no	—
233	C₂²×C₃₁	—	8	1	no	—
234	C₂²×C₃₁	—	8	1	no	—
235	C₂²×C₃₁	#465	1	2	yes	—
236	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
237	C₄×C₃₁	—	2	1	no	—
238	C₄×C₃₁	—	2	2	no	—
239	C₄×C₃₁	—	2	1	no	—
240	C₄×C₃₁	—	2	1	no	—
241	C₄×C₃₁	—	4	1	no	—
242	C₄×C₃₁	—	4	1	no	—
243	C₄×C₃₁	—	8	1	no	—
244	C₄×C₃₁	—	8	2	no	—
245	C₄×C₃₁	#465	1	2	yes	—
246	C₄×C₃₁	C₂²×SD₁₆	1	1	yes	characteristic
247	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
248	C₄×C₃₁	#577	1	1	yes	characteristic
249	C₄×C₃₁	#578	1	1	yes	characteristic
250	C₂×#135	—	2	1	no	—
251	#532	C₂²×C₃₁	1	1	yes	characteristic
252	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
253	C₄×#41	C₂²×C₃₁	1	1	yes	characteristic
254	#498	C₂²×C₃₁	1	2	yes	—
255	#486	C₂²×C₃₁	1	2	yes	—
256	#481	—	2	2	no	—
257	#477	C₂²×C₃₁	1	2	yes	—
258	#476	—	2	1	no	—
259	#476	—	2	1	no	—
260	#476	C₂²×C₃₁	1	2	yes	—
261	#475	—	2	1	no	—
262	#474	—	2	1	no	—
263	#464	—	2	1	no	—
264	#463	—	2	1	no	—
265	#462	C₂²×C₃₁	1	1	yes	characteristic
266	#461	C₂²×C₃₁	1	1	yes	characteristic
267	#460	C₂²×C₃₁	1	1	yes	characteristic
268	#459	C₂²×C₃₁	1	1	yes	characteristic
269	C₈×Q₈	C₂²×C₃₁	1	2	yes	—
270	C₈×D₄	—	2	2	no	—
271	C₈×D₄	C₂²×C₃₁	1	2	yes	—
272	#431	—	2	1	no	—
273	#431	C₂²×C₃₁	1	1	yes	characteristic
274	#428	—	2	1	no	—
275	#423	C₂²×C₃₁	1	1	yes	characteristic
276	C₂×C₄⋊C₈	—	2	1	no	—
277	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
278	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
279	#417	—	2	1	no	—
280	#416	—	2	1	no	—
281	#334	C₂²×C₃₁	1	2	yes	—
282	#334	C₂²×C₃₁	1	2	yes	—
283	#333	C₂²×C₃₁	1	2	yes	—
284	#333	C₂²×C₃₁	1	2	yes	—
285	#332	C₂²×C₃₁	1	2	yes	—
286	#331	C₂²×C₃₁	1	2	yes	—
287	C₂×C₃₁	—	2	2	no	—
288	C₂×C₃₁	—	4	2	no	—
289	C₂×C₃₁	—	16	1	no	—
290	C₂×C₃₁	#3046	1	1	yes	characteristic
291	C₂×C₃₁	C₂×#465	1	1	yes	characteristic
292	C₂×C₃₁	#3068	1	1	yes	characteristic
293	C₂×#41	—	2	1	no	—
294	C₂×#41	D₄×C₃₁	1	1	yes	characteristic
295	C₂²×Q₈	—	2	1	no	—
296	#135	—	4	4	no	—
297	C₂×Q₁₆	—	2	1	no	—
298	C₂×SD₁₆	—	2	1	no	—
299	C₂×SD₁₆	—	2	1	no	—
300	C₂×D₈	—	2	1	no	—
301	C₂×M₁₆	—	2	1	no	—
302	C₂²×C₈	—	2	1	no	—
303	C₂²×C₈	—	2	1	no	—
304	C₂²×C₈	—	2	2	no	—
305	C₄⋊Q₈	—	2	1	no	—
306	C₄⋊Q₈	—	2	1	no	—
307	C₄⋊Q₈	—	2	1	no	—
308	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
309	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
310	#127	C₂³×C₃₁	1	1	yes	characteristic
311	#124	—	2	1	no	—
312	#121	—	2	1	no	—
313	#121	—	2	1	no	—
314	C₄×Q₈	D₄×C₃₁	1	1	yes	characteristic
315	C₄×Q₈	C₂³×C₃₁	1	1	yes	characteristic
316	C₄×D₄	—	2	1	no	—
317	C₄×D₄	—	2	1	no	—
318	C₄×D₄	—	2	1	no	—
319	C₄×D₄	—	2	1	no	—
320	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
321	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
322	#117	—	2	1	no	—
323	#117	D₄×C₃₁	1	1	yes	characteristic
324	C₂×C₄⋊C₄	—	2	1	no	—
325	C₂×C₄⋊C₄	—	2	1	no	—
326	C₂×C₄²	—	2	1	no	—
327	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
328	#107	—	2	2	no	—
329	#106	—	2	2	no	—
330	#106	—	2	2	no	—
331	#106	—	2	2	no	—
332	C₄⋊C₈	—	2	2	no	—
333	C₄⋊C₈	—	2	1	no	—
334	C₄⋊C₈	—	2	1	no	—
335	C₄⋊C₈	—	2	1	no	—
336	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
337	C₄⋊C₈	D₄×C₃₁	1	1	yes	characteristic
338	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
339	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
340	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
341	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
342	#103	—	2	2	no	—
343	#103	—	2	1	no	—
344	#103	—	2	1	no	—
345	#102	—	2	2	no	—
346	#102	—	2	2	no	—
347	#102	—	2	2	no	—
348	#102	—	2	1	no	—
349	#102	—	2	1	no	—
350	C₂²⋊C₈	—	2	2	no	—
351	C₂²⋊C₈	D₄×C₃₁	1	2	yes	—
352	C₄×C₈	—	2	2	no	—
353	C₄×C₈	C₂³×C₃₁	1	2	yes	—
354	C₃₁	#19414	1	1	yes	characteristic, direct product factor, Sylow
355	#41	—	2	2	no	—
356	#41	—	4	2	no	—
357	#41	—	4	2	no	—
358	#41	SD₁₆×C₃₁	1	4	yes	—
359	C₂×Q₈	—	2	1	no	—
360	C₂×Q₈	—	2	2	no	—
361	C₂×Q₈	—	4	1	no	—
362	C₂×Q₈	—	8	2	no	—
363	C₂×Q₈	C₂×D₄×C₃₁	1	1	yes	characteristic
364	C₂×D₄	—	2	1	no	—
365	C₂×D₄	—	2	1	no	—
366	C₂×D₄	—	2	1	no	—
367	C₂×D₄	—	2	1	no	—
368	C₂×D₄	—	4	1	no	—
369	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
370	C₂²×C₄	—	2	1	no	—
371	C₂²×C₄	—	2	1	no	—
372	C₂²×C₄	—	2	1	no	—
373	C₂²×C₄	—	4	1	no	—
374	C₂²×C₄	—	4	1	no	—
375	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
376	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
377	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
378	Q₁₆	—	4	2	no	—
379	SD₁₆	—	4	2	no	—
380	SD₁₆	—	4	2	no	—
381	D₈	—	4	2	no	—
382	M₁₆	—	4	2	no	—
383	C₂×C₈	—	2	2	no	—
384	C₂×C₈	—	2	2	no	—
385	C₂×C₈	—	2	1	no	—
386	C₂×C₈	—	2	1	no	—
387	C₂×C₈	—	4	2	no	—
388	C₂×C₈	—	8	2	no	—
389	C₂×C₈	—	8	1	no	—
390	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
391	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
392	C₄⋊C₄	—	2	1	no	—
393	C₄⋊C₄	—	2	1	no	—
394	C₄⋊C₄	—	2	2	no	—
395	C₄⋊C₄	—	2	2	no	—
396	C₄⋊C₄	—	2	1	no	—
397	C₄⋊C₄	—	2	1	no	—
398	C₄⋊C₄	—	4	1	no	—
399	C₄⋊C₄	—	4	1	no	—
400	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
401	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
402	C₂²⋊C₄	—	2	1	no	—
403	C₂²⋊C₄	—	2	1	no	—
404	C₂²⋊C₄	—	2	1	no	—
405	C₂²⋊C₄	—	4	1	no	—
406	C₂²⋊C₄	—	4	1	no	—
407	C₄²	—	2	1	no	—
408	C₄²	—	2	1	no	—
409	C₄²	—	2	1	no	—
410	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
411	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
412	C₄²	#41×C₃₁	1	1	yes	characteristic
413	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
414	C₂³	—	2	1	no	—
415	C₂³	—	4	1	no	—
416	C₂³	C₃₁×#121	1	1	yes	characteristic
417	Q₈	—	2	1	no	—
418	Q₈	—	4	1	no	—
419	Q₈	—	4	1	no	—
420	Q₈	—	8	2	no	—
421	Q₈	—	8	2	no	—
422	Q₈	C₂×SD₁₆×C₃₁	1	2	yes	—
423	D₄	—	2	4	no	—
424	D₄	—	2	1	no	—
425	D₄	—	4	1	no	—
426	D₄	—	4	1	no	—
427	D₄	—	4	1	no	—
428	D₄	—	4	1	no	—
429	D₄	—	8	1	no	—
430	D₄	—	8	2	no	—
431	D₄	C₂×SD₁₆×C₃₁	1	2	yes	—
432	C₂×C₄	—	2	1	no	—
433	C₂×C₄	—	2	4	no	—
434	C₂×C₄	—	2	1	no	—
435	C₂×C₄	—	2	1	no	—
436	C₂×C₄	—	2	1	no	—
437	C₂×C₄	—	2	1	no	—
438	C₂×C₄	—	4	1	no	—
439	C₂×C₄	—	4	1	no	—
440	C₂×C₄	—	4	1	no	—
441	C₂×C₄	—	4	1	no	—
442	C₂×C₄	—	4	2	no	—
443	C₂×C₄	—	8	1	no	—
444	C₂×C₄	—	8	2	no	—
445	C₂×C₄	—	8	2	no	—
446	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
447	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
448	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
449	C₂×C₄	C₂×SD₁₆×C₃₁	1	2	yes	—
450	C₂×C₄	C₃₁×#135	1	2	yes	—
451	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
452	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
453	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
454	C₈	—	2	2	no	—
455	C₈	—	2	2	no	—
456	C₈	—	4	1	no	—
457	C₈	—	4	1	no	—
458	C₈	—	4	2	no	—
459	C₈	—	4	2	no	—
460	C₂²	—	2	1	no	—
461	C₂²	—	2	1	no	—
462	C₂²	—	2	2	no	—
463	C₂²	—	4	1	no	—
464	C₂²	—	4	1	no	—
465	C₂²	—	8	1	no	—
466	C₂²	—	8	1	no	—
467	C₂²	—	8	1	no	—
468	C₂²	C₃₁×#465	1	2	yes	—
469	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
470	C₄	—	2	1	no	—
471	C₄	—	2	2	no	—
472	C₄	—	2	1	no	—
473	C₄	—	2	1	no	—
474	C₄	—	4	1	no	—
475	C₄	—	4	1	no	—
476	C₄	—	8	1	no	—
477	C₄	—	8	2	no	—
478	C₄	C₃₁×#465	1	2	yes	—
479	C₄	C₂²×SD₁₆×C₃₁	1	1	yes	characteristic
480	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
481	C₄	C₃₁×#577	1	1	yes	characteristic
482	C₄	C₃₁×#578	1	1	yes	characteristic
483	C₂	—	2	2	no	—
484	C₂	—	4	2	no	—
485	C₂	—	16	1	no	—
486	C₂	C₃₁×#3046	1	1	yes	characteristic
487	C₂	C₂×C₃₁×#465	1	1	yes	characteristic
488	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×#3035	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×#3227	C₂	semidirect
2. C₃₁×#3079	C₂	semidirect
4. C₃₁×#2988	C₂	semidirect
5. C₃₁×#1678	C₂	semidirect
6. C₃₁×#1677	C₂	semidirect
7. C₃₁×#1672	C₂	semidirect
8. C₃₁×#1671	C₂	semidirect
9. C₃₁×#1570	C₂	semidirect
10. C₃₁×#1547	C₂	semidirect
11. C₂×C₃₁×#333	C₂	semidirect
169. #19414	C₃₁	direct (N × H)
216. #3227	C₂×C₃₁	semidirect
217. #3079	C₂×C₃₁	semidirect
219. #2988	C₂×C₃₁	semidirect
220. #1678	C₂×C₃₁	semidirect
221. #1677	C₂×C₃₁	semidirect
222. #1672	C₂×C₃₁	semidirect
223. #1671	C₂×C₃₁	semidirect
224. #1570	C₂×C₃₁	semidirect
225. #1547	C₂×C₃₁	semidirect
226. C₂×#333	C₂×C₃₁	semidirect
354. C₃₁	#19414	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×#3227	2
C₃₁×#3079	2
C₃₁×#3035	2
C₃₁×#2988	2
C₃₁×#1678	2
C₃₁×#1677	2
C₃₁×#1672	2
C₃₁×#1671	2
C₃₁×#1570	2
C₃₁×#1547	2
C₂×C₃₁×#333	2
#19414	31

Subgroups — C31×#19414 #21476173

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19414 #21476173