Subgroups — C₃₁×#19169 #21475928

Order 7936.

728 conjugacy classes of subgroups.

The subgroup lattice has 524 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3141	C₂	1	1	yes	maximal, characteristic, complemented
2	C₃₁×#3100	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#461	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄×M₁₆×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1701	C₂	1	2	yes	maximal, complemented
6	C₃₁×#1699	C₂	1	2	yes	maximal, complemented
7	C₃₁×#1681	C₂	1	2	yes	maximal, complemented
8	C₃₁×#1679	C₂	1	2	yes	maximal, complemented
9	C₃₁×#1565	C₂	1	1	yes	maximal, characteristic, complemented
10	C₃₁×#1549	C₂	1	1	yes	maximal, characteristic, complemented
11	C₃₁×#1547	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×SD₁₆×C₃₁	—	2	1	no	—
13	C₂²×M₁₆×C₃₁	—	2	1	no	—
14	C₃₁×#532	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₂×C₄×D₄×C₃₁	C₂²	1	1	yes	characteristic
17	C₃₁×#489	—	2	2	no	—
18	C₃₁×#487	C₂²	1	2	yes	complemented
19	C₃₁×#486	C₂²	1	2	yes	complemented
20	C₃₁×#480	—	2	1	no	—
21	C₃₁×#474	C₂²	1	1	yes	characteristic
22	C₃₁×#474	C₂²	1	1	yes	characteristic
23	C₃₁×#461	—	2	1	no	—
24	C₃₁×#461	—	2	1	no	—
25	C₃₁×#461	C₂²	1	2	yes	—
26	C₃₁×#461	C₂²	1	2	yes	—
27	C₃₁×#459	C₂²	1	1	yes	characteristic
28	C₃₁×#459	C₂²	1	1	yes	characteristic
29	C₃₁×#447	—	2	1	no	—
30	C₄×D₈×C₃₁	—	2	1	no	—
31	C₄×D₈×C₃₁	—	2	1	no	—
32	C₃₁×#436	C₂²	1	2	yes	—
33	C₃₁×#435	—	2	2	no	—
34	C₈×D₄×C₃₁	C₂²	1	2	yes	—
35	C₃₁×#431	C₂²	1	1	yes	characteristic
36	C₃₁×#423	C₂²	1	1	yes	characteristic
37	C₂×C₃₁×C₄⋊C₈	C₂²	1	1	yes	characteristic
38	C₃₁×#417	—	2	1	no	—
39	C₃₁×#417	—	2	1	no	—
40	C₂×C₃₁×#103	—	2	1	no	—
41	C₃₁×#407	—	2	1	no	—
42	C₄×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
43	C₃₁×#335	C₂²	1	2	yes	complemented
44	C₃₁×#334	C₂²	1	2	yes	complemented
45	C₃₁×#332	C₂²	1	2	yes	complemented
46	C₃₁×#332	C₂²	1	2	yes	complemented
47	C₃₁×#332	C₂²	1	2	yes	complemented
48	C₃₁×#331	C₂²	1	2	yes	complemented
49	C₂×#41×C₃₁	—	2	1	no	—
50	C₂²×Q₈×C₃₁	—	2	1	no	—
51	C₂²×D₄×C₃₁	D₄	1	1	yes	characteristic
52	C₂³×C₄×C₃₁	—	2	1	no	—
53	C₂×SD₁₆×C₃₁	—	2	2	no	—
54	C₂×SD₁₆×C₃₁	—	2	2	no	—
55	C₂×SD₁₆×C₃₁	—	4	4	no	—
56	C₂×D₈×C₃₁	—	2	1	no	—
57	C₂×D₈×C₃₁	—	2	1	no	—
58	C₂×M₁₆×C₃₁	—	2	1	no	—
59	C₂×M₁₆×C₃₁	—	2	2	no	—
60	C₂×M₁₆×C₃₁	—	4	4	no	—
61	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
62	C₂×M₁₆×C₃₁	D₄	1	1	yes	characteristic
63	C₂²×C₈×C₃₁	—	2	1	no	—
64	C₂²×C₈×C₃₁	—	2	2	no	—
65	C₃₁×C₄⋊Q₈	—	2	1	no	—
66	C₃₁×C₄⋊Q₈	—	2	1	no	—
67	C₃₁×C₄⋊Q₈	—	2	1	no	—
68	C₃₁×C₄⋊Q₈	C₂³	1	2	yes	—
69	C₃₁×C₄⋊Q₈	C₂³	1	1	yes	characteristic
70	C₃₁×#127	C₂³	1	1	yes	characteristic
71	C₃₁×#124	—	2	1	no	—
72	C₃₁×#121	—	2	1	no	—
73	C₃₁×#121	—	2	1	no	—
74	C₄×D₄×C₃₁	—	2	1	no	—
75	C₄×D₄×C₃₁	—	2	1	no	—
76	C₄×D₄×C₃₁	—	2	1	no	—
77	C₄×D₄×C₃₁	—	2	1	no	—
78	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
79	C₄×D₄×C₃₁	D₄	1	1	yes	characteristic
80	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
81	C₄×D₄×C₃₁	C₂³	1	1	yes	characteristic
82	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
83	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
84	C₂×C₄⋊C₄×C₃₁	D₄	1	1	yes	characteristic
85	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
86	C₂×C₄²×C₃₁	C₂³	1	1	yes	characteristic
87	C₃₁×#107	—	2	1	no	—
88	C₃₁×#107	—	2	1	no	—
89	C₃₁×C₄⋊C₈	—	2	1	no	—
90	C₃₁×C₄⋊C₈	—	2	1	no	—
91	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
92	C₃₁×C₄⋊C₈	C₂³	1	2	yes	—
93	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
94	C₃₁×C₄⋊C₈	C₂³	1	1	yes	characteristic
95	C₃₁×#103	—	2	2	no	—
96	C₃₁×#103	—	2	2	no	—
97	C₃₁×#103	—	2	2	no	—
98	C₃₁×#103	—	2	2	no	—
99	C₃₁×#102	—	2	1	no	—
100	C₃₁×#102	—	2	1	no	—
101	C₃₁×#102	—	2	2	no	—
102	C₃₁×#102	—	2	2	no	—
103	C₃₁×#102	—	2	1	no	—
104	C₃₁×#102	—	2	1	no	—
105	C₃₁×C₂²⋊C₈	—	2	1	no	—
106	C₃₁×C₂²⋊C₈	—	2	2	no	—
107	C₃₁×C₂²⋊C₈	—	2	2	no	—
108	C₃₁×#97	D₄	1	2	yes	—
109	C₄×C₈×C₃₁	—	2	1	no	—
110	C₄×C₈×C₃₁	C₂³	1	2	yes	—
111	C₂⁴×C₃₁	—	2	1	no	—
112	#41×C₃₁	—	4	2	no	—
113	#41×C₃₁	—	4	2	no	—
114	C₂×Q₈×C₃₁	—	2	1	no	—
115	C₂×Q₈×C₃₁	—	2	2	no	—
116	C₂×Q₈×C₃₁	—	4	2	no	—
117	C₂×Q₈×C₃₁	—	4	2	no	—
118	C₂×Q₈×C₃₁	—	4	1	no	—
119	C₂×D₄×C₃₁	—	2	2	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	—	2	1	no	—
124	C₂×D₄×C₃₁	—	4	1	no	—
125	C₂×D₄×C₃₁	—	4	1	no	—
126	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
127	C₂×D₄×C₃₁	C₂×D₄	1	1	yes	characteristic
128	C₂²×C₄×C₃₁	—	2	1	no	—
129	C₂²×C₄×C₃₁	—	2	1	no	—
130	C₂²×C₄×C₃₁	—	2	1	no	—
131	C₂²×C₄×C₃₁	—	2	1	no	—
132	C₂²×C₄×C₃₁	—	2	1	no	—
133	C₂²×C₄×C₃₁	—	4	2	no	—
134	C₂²×C₄×C₃₁	—	4	2	no	—
135	C₂²×C₄×C₃₁	—	4	1	no	—
136	C₂²×C₄×C₃₁	—	4	1	no	—
137	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
138	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
139	C₂²×C₄×C₃₁	#41	1	1	yes	characteristic
140	SD₁₆×C₃₁	—	4	4	no	—
141	SD₁₆×C₃₁	—	4	4	no	—
142	D₈×C₃₁	—	4	2	no	—
143	D₈×C₃₁	—	4	2	no	—
144	M₁₆×C₃₁	—	2	2	no	—
145	M₁₆×C₃₁	—	2	2	no	—
146	M₁₆×C₃₁	—	4	4	no	—
147	M₁₆×C₃₁	—	4	2	no	—
148	C₂×C₈×C₃₁	—	2	1	no	—
149	C₂×C₈×C₃₁	—	2	1	no	—
150	C₂×C₈×C₃₁	—	2	2	no	—
151	C₂×C₈×C₃₁	—	2	2	no	—
152	C₂×C₈×C₃₁	—	4	4	no	—
153	C₂×C₈×C₃₁	—	4	2	no	—
154	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
155	C₂×C₈×C₃₁	C₂×D₄	1	2	yes	—
156	C₄⋊C₄×C₃₁	—	2	1	no	—
157	C₄⋊C₄×C₃₁	—	2	2	no	—
158	C₄⋊C₄×C₃₁	—	2	1	no	—
159	C₄⋊C₄×C₃₁	—	2	2	no	—
160	C₄⋊C₄×C₃₁	—	2	1	no	—
161	C₄⋊C₄×C₃₁	—	4	1	no	—
162	C₄⋊C₄×C₃₁	—	4	1	no	—
163	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
164	C₄⋊C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
165	C₂²⋊C₄×C₃₁	—	2	1	no	—
166	C₂²⋊C₄×C₃₁	—	2	1	no	—
167	C₂²⋊C₄×C₃₁	—	2	1	no	—
168	C₂²⋊C₄×C₃₁	—	2	1	no	—
169	C₂²⋊C₄×C₃₁	—	4	1	no	—
170	C₂²⋊C₄×C₃₁	—	4	1	no	—
171	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
172	C₄²×C₃₁	C₂×D₄	1	1	yes	characteristic
173	C₄²×C₃₁	#41	1	1	yes	characteristic
174	C₄²×C₃₁	C₂⁴	1	1	yes	characteristic
175	#19169	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
176	C₂³×C₃₁	—	2	1	no	—
177	C₂³×C₃₁	—	2	1	no	—
178	C₂³×C₃₁	—	4	2	no	—
179	C₂³×C₃₁	—	4	2	no	—
180	C₂³×C₃₁	—	4	1	no	—
181	C₂³×C₃₁	—	8	1	no	—
182	C₂³×C₃₁	#121	1	1	yes	characteristic
183	Q₈×C₃₁	—	4	1	no	—
184	Q₈×C₃₁	—	4	2	no	—
185	Q₈×C₃₁	—	4	1	no	—
186	Q₈×C₃₁	—	4	2	no	—
187	Q₈×C₃₁	—	8	2	no	—
188	D₄×C₃₁	—	2	1	no	—
189	D₄×C₃₁	—	2	1	no	—
190	D₄×C₃₁	—	2	1	no	—
191	D₄×C₃₁	—	4	2	no	—
192	D₄×C₃₁	—	4	1	no	—
193	D₄×C₃₁	—	4	1	no	—
194	D₄×C₃₁	—	4	1	no	—
195	D₄×C₃₁	—	4	1	no	—
196	D₄×C₃₁	—	8	1	no	—
197	D₄×C₃₁	—	8	2	no	—
198	D₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
199	C₂×C₄×C₃₁	—	2	1	no	—
200	C₂×C₄×C₃₁	—	2	1	no	—
201	C₂×C₄×C₃₁	—	2	1	no	—
202	C₂×C₄×C₃₁	—	2	1	no	—
203	C₂×C₄×C₃₁	—	4	1	no	—
204	C₂×C₄×C₃₁	—	4	2	no	—
205	C₂×C₄×C₃₁	—	4	1	no	—
206	C₂×C₄×C₃₁	—	4	1	no	—
207	C₂×C₄×C₃₁	—	4	1	no	—
208	C₂×C₄×C₃₁	—	4	1	no	—
209	C₂×C₄×C₃₁	—	4	2	no	—
210	C₂×C₄×C₃₁	—	4	2	no	—
211	C₂×C₄×C₃₁	—	4	1	no	—
212	C₂×C₄×C₃₁	—	4	2	no	—
213	C₂×C₄×C₃₁	—	8	2	no	—
214	C₂×C₄×C₃₁	—	8	2	no	—
215	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
216	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
217	C₂×C₄×C₃₁	#121	1	1	yes	characteristic
218	C₂×C₄×C₃₁	#135	1	2	yes	—
219	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	2	yes	—
220	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
221	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
222	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
223	C₈×C₃₁	—	2	2	no	—
224	C₈×C₃₁	—	2	2	no	—
225	C₈×C₃₁	—	4	2	no	—
226	C₈×C₃₁	—	4	2	no	—
227	C₈×C₃₁	—	4	1	no	—
228	C₈×C₃₁	—	4	1	no	—
229	#3141	C₂×C₃₁	1	1	yes	characteristic, complemented
230	#3100	C₂×C₃₁	1	1	yes	characteristic
231	C₂×#461	C₂×C₃₁	1	1	yes	characteristic, complemented
232	D₄×M₁₆	C₂×C₃₁	1	1	yes	characteristic, complemented
233	#1701	C₂×C₃₁	1	2	yes	complemented
234	#1699	C₂×C₃₁	1	2	yes	complemented
235	#1681	C₂×C₃₁	1	2	yes	complemented
236	#1679	C₂×C₃₁	1	2	yes	complemented
237	#1565	C₂×C₃₁	1	1	yes	characteristic, complemented
238	#1549	C₂×C₃₁	1	1	yes	characteristic, complemented
239	#1547	C₂×C₃₁	1	1	yes	characteristic, complemented
240	C₂²×C₃₁	—	2	1	no	—
241	C₂²×C₃₁	—	2	1	no	—
242	C₂²×C₃₁	—	2	2	no	—
243	C₂²×C₃₁	—	4	1	no	—
244	C₂²×C₃₁	—	4	1	no	—
245	C₂²×C₃₁	—	4	1	no	—
246	C₂²×C₃₁	—	4	1	no	—
247	C₂²×C₃₁	—	4	1	no	—
248	C₂²×C₃₁	—	8	4	no	—
249	C₂²×C₃₁	—	8	1	no	—
250	C₂²×C₃₁	—	8	1	no	—
251	C₂²×C₃₁	—	8	1	no	—
252	C₂²×C₃₁	#463	1	2	yes	—
253	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
254	C₄×C₃₁	—	2	1	no	—
255	C₄×C₃₁	—	2	1	no	—
256	C₄×C₃₁	—	2	1	no	—
257	C₄×C₃₁	—	4	1	no	—
258	C₄×C₃₁	—	4	1	no	—
259	C₄×C₃₁	—	8	1	no	—
260	C₄×C₃₁	—	8	2	no	—
261	C₄×C₃₁	#463	1	2	yes	—
262	C₄×C₃₁	C₂×#135	1	1	yes	characteristic
263	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
264	C₄×C₃₁	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
265	C₄×C₃₁	#575	1	1	yes	characteristic
266	C₂²×SD₁₆	—	2	1	no	—
267	C₂²×M₁₆	—	2	1	no	—
268	#532	C₂²×C₃₁	1	1	yes	characteristic
269	C₂×C₄⋊Q₈	C₂²×C₃₁	1	1	yes	characteristic
270	C₂×C₄×D₄	C₂²×C₃₁	1	1	yes	characteristic
271	#489	—	2	2	no	—
272	#487	C₂²×C₃₁	1	2	yes	complemented
273	#486	C₂²×C₃₁	1	2	yes	complemented
274	#480	—	2	1	no	—
275	#474	C₂²×C₃₁	1	1	yes	characteristic
276	#474	C₂²×C₃₁	1	1	yes	characteristic
277	#461	—	2	1	no	—
278	#461	—	2	1	no	—
279	#461	C₂²×C₃₁	1	2	yes	—
280	#461	C₂²×C₃₁	1	2	yes	—
281	#459	C₂²×C₃₁	1	1	yes	characteristic
282	#459	C₂²×C₃₁	1	1	yes	characteristic
283	#447	—	2	1	no	—
284	C₄×D₈	—	2	1	no	—
285	C₄×D₈	—	2	1	no	—
286	#436	C₂²×C₃₁	1	2	yes	—
287	#435	—	2	2	no	—
288	C₈×D₄	C₂²×C₃₁	1	2	yes	—
289	#431	C₂²×C₃₁	1	1	yes	characteristic
290	#423	C₂²×C₃₁	1	1	yes	characteristic
291	C₂×C₄⋊C₈	C₂²×C₃₁	1	1	yes	characteristic
292	#417	—	2	1	no	—
293	#417	—	2	1	no	—
294	C₂×#103	—	2	1	no	—
295	#407	—	2	1	no	—
296	C₄×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
297	#335	C₂²×C₃₁	1	2	yes	complemented
298	#334	C₂²×C₃₁	1	2	yes	complemented
299	#332	C₂²×C₃₁	1	2	yes	complemented
300	#332	C₂²×C₃₁	1	2	yes	complemented
301	#332	C₂²×C₃₁	1	2	yes	complemented
302	#331	C₂²×C₃₁	1	2	yes	complemented
303	C₂×C₃₁	—	2	2	no	—
304	C₂×C₃₁	—	4	2	no	—
305	C₂×C₃₁	—	8	1	no	—
306	C₂×C₃₁	—	16	1	no	—
307	C₂×C₃₁	C₂×#463	1	1	yes	characteristic
308	C₂×C₃₁	#3055	1	1	yes	characteristic
309	C₂×C₃₁	#3069	1	1	yes	characteristic
310	C₂×#41	—	2	1	no	—
311	C₂²×Q₈	—	2	1	no	—
312	C₂²×D₄	D₄×C₃₁	1	1	yes	characteristic
313	C₂³×C₄	—	2	1	no	—
314	C₂×SD₁₆	—	2	2	no	—
315	C₂×SD₁₆	—	2	2	no	—
316	C₂×SD₁₆	—	4	4	no	—
317	C₂×D₈	—	2	1	no	—
318	C₂×D₈	—	2	1	no	—
319	C₂×M₁₆	—	2	1	no	—
320	C₂×M₁₆	—	2	2	no	—
321	C₂×M₁₆	—	4	4	no	—
322	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
323	C₂×M₁₆	D₄×C₃₁	1	1	yes	characteristic
324	C₂²×C₈	—	2	1	no	—
325	C₂²×C₈	—	2	2	no	—
326	C₄⋊Q₈	—	2	1	no	—
327	C₄⋊Q₈	—	2	1	no	—
328	C₄⋊Q₈	—	2	1	no	—
329	C₄⋊Q₈	C₂³×C₃₁	1	2	yes	—
330	C₄⋊Q₈	C₂³×C₃₁	1	1	yes	characteristic
331	#127	C₂³×C₃₁	1	1	yes	characteristic
332	#124	—	2	1	no	—
333	#121	—	2	1	no	—
334	#121	—	2	1	no	—
335	C₄×D₄	—	2	1	no	—
336	C₄×D₄	—	2	1	no	—
337	C₄×D₄	—	2	1	no	—
338	C₄×D₄	—	2	1	no	—
339	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
340	C₄×D₄	D₄×C₃₁	1	1	yes	characteristic
341	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
342	C₄×D₄	C₂³×C₃₁	1	1	yes	characteristic
343	C₂×C₄⋊C₄	—	2	1	no	—
344	C₂×C₄⋊C₄	—	2	1	no	—
345	C₂×C₄⋊C₄	D₄×C₃₁	1	1	yes	characteristic
346	C₂×C₂²⋊C₄	—	2	1	no	—
347	C₂×C₄²	C₂³×C₃₁	1	1	yes	characteristic
348	#107	—	2	1	no	—
349	#107	—	2	1	no	—
350	C₄⋊C₈	—	2	1	no	—
351	C₄⋊C₈	—	2	1	no	—
352	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
353	C₄⋊C₈	C₂³×C₃₁	1	2	yes	—
354	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
355	C₄⋊C₈	C₂³×C₃₁	1	1	yes	characteristic
356	#103	—	2	2	no	—
357	#103	—	2	2	no	—
358	#103	—	2	2	no	—
359	#103	—	2	2	no	—
360	#102	—	2	1	no	—
361	#102	—	2	1	no	—
362	#102	—	2	2	no	—
363	#102	—	2	2	no	—
364	#102	—	2	1	no	—
365	#102	—	2	1	no	—
366	C₂²⋊C₈	—	2	1	no	—
367	C₂²⋊C₈	—	2	2	no	—
368	C₂²⋊C₈	—	2	2	no	—
369	#97	D₄×C₃₁	1	2	yes	—
370	C₄×C₈	—	2	1	no	—
371	C₄×C₈	C₂³×C₃₁	1	2	yes	—
372	C₃₁	#19169	1	1	yes	characteristic, direct product factor, Sylow
373	C₂⁴	—	2	1	no	—
374	#41	—	4	2	no	—
375	#41	—	4	2	no	—
376	C₂×Q₈	—	2	1	no	—
377	C₂×Q₈	—	2	2	no	—
378	C₂×Q₈	—	4	2	no	—
379	C₂×Q₈	—	4	2	no	—
380	C₂×Q₈	—	4	1	no	—
381	C₂×D₄	—	2	2	no	—
382	C₂×D₄	—	2	1	no	—
383	C₂×D₄	—	2	1	no	—
384	C₂×D₄	—	2	1	no	—
385	C₂×D₄	—	2	1	no	—
386	C₂×D₄	—	4	1	no	—
387	C₂×D₄	—	4	1	no	—
388	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
389	C₂×D₄	C₂×D₄×C₃₁	1	1	yes	characteristic
390	C₂²×C₄	—	2	1	no	—
391	C₂²×C₄	—	2	1	no	—
392	C₂²×C₄	—	2	1	no	—
393	C₂²×C₄	—	2	1	no	—
394	C₂²×C₄	—	2	1	no	—
395	C₂²×C₄	—	4	2	no	—
396	C₂²×C₄	—	4	2	no	—
397	C₂²×C₄	—	4	1	no	—
398	C₂²×C₄	—	4	1	no	—
399	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
400	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
401	C₂²×C₄	#41×C₃₁	1	1	yes	characteristic
402	SD₁₆	—	4	4	no	—
403	SD₁₆	—	4	4	no	—
404	D₈	—	4	2	no	—
405	D₈	—	4	2	no	—
406	M₁₆	—	2	2	no	—
407	M₁₆	—	2	2	no	—
408	M₁₆	—	4	4	no	—
409	M₁₆	—	4	2	no	—
410	C₂×C₈	—	2	1	no	—
411	C₂×C₈	—	2	1	no	—
412	C₂×C₈	—	2	2	no	—
413	C₂×C₈	—	2	2	no	—
414	C₂×C₈	—	4	4	no	—
415	C₂×C₈	—	4	2	no	—
416	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
417	C₂×C₈	C₂×D₄×C₃₁	1	2	yes	—
418	C₄⋊C₄	—	2	1	no	—
419	C₄⋊C₄	—	2	2	no	—
420	C₄⋊C₄	—	2	1	no	—
421	C₄⋊C₄	—	2	2	no	—
422	C₄⋊C₄	—	2	1	no	—
423	C₄⋊C₄	—	4	1	no	—
424	C₄⋊C₄	—	4	1	no	—
425	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
426	C₄⋊C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
427	C₂²⋊C₄	—	2	1	no	—
428	C₂²⋊C₄	—	2	1	no	—
429	C₂²⋊C₄	—	2	1	no	—
430	C₂²⋊C₄	—	2	1	no	—
431	C₂²⋊C₄	—	4	1	no	—
432	C₂²⋊C₄	—	4	1	no	—
433	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
434	C₄²	C₂×D₄×C₃₁	1	1	yes	characteristic
435	C₄²	#41×C₃₁	1	1	yes	characteristic
436	C₄²	C₂⁴×C₃₁	1	1	yes	characteristic
437	C₂³	—	2	1	no	—
438	C₂³	—	2	1	no	—
439	C₂³	—	4	2	no	—
440	C₂³	—	4	2	no	—
441	C₂³	—	4	1	no	—
442	C₂³	—	8	1	no	—
443	C₂³	C₃₁×#121	1	1	yes	characteristic
444	Q₈	—	4	1	no	—
445	Q₈	—	4	2	no	—
446	Q₈	—	4	1	no	—
447	Q₈	—	4	2	no	—
448	Q₈	—	8	2	no	—
449	D₄	—	2	1	no	—
450	D₄	—	2	1	no	—
451	D₄	—	2	1	no	—
452	D₄	—	4	2	no	—
453	D₄	—	4	1	no	—
454	D₄	—	4	1	no	—
455	D₄	—	4	1	no	—
456	D₄	—	4	1	no	—
457	D₄	—	8	1	no	—
458	D₄	—	8	2	no	—
459	D₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
460	C₂×C₄	—	2	1	no	—
461	C₂×C₄	—	2	1	no	—
462	C₂×C₄	—	2	1	no	—
463	C₂×C₄	—	2	1	no	—
464	C₂×C₄	—	4	1	no	—
465	C₂×C₄	—	4	2	no	—
466	C₂×C₄	—	4	2	no	—
467	C₂×C₄	—	4	2	no	—
468	C₂×C₄	—	4	1	no	—
469	C₂×C₄	—	4	1	no	—
470	C₂×C₄	—	4	1	no	—
471	C₂×C₄	—	4	1	no	—
472	C₂×C₄	—	4	1	no	—
473	C₂×C₄	—	4	2	no	—
474	C₂×C₄	—	8	2	no	—
475	C₂×C₄	—	8	2	no	—
476	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
477	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
478	C₂×C₄	C₃₁×#121	1	1	yes	characteristic
479	C₂×C₄	C₃₁×#135	1	2	yes	—
480	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	2	yes	—
481	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
482	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
483	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
484	C₈	—	2	2	no	—
485	C₈	—	2	2	no	—
486	C₈	—	4	2	no	—
487	C₈	—	4	2	no	—
488	C₈	—	4	1	no	—
489	C₈	—	4	1	no	—
490	C₂²	—	2	1	no	—
491	C₂²	—	2	1	no	—
492	C₂²	—	2	2	no	—
493	C₂²	—	4	1	no	—
494	C₂²	—	4	1	no	—
495	C₂²	—	4	1	no	—
496	C₂²	—	4	1	no	—
497	C₂²	—	4	1	no	—
498	C₂²	—	8	4	no	—
499	C₂²	—	8	1	no	—
500	C₂²	—	8	1	no	—
501	C₂²	—	8	1	no	—
502	C₂²	C₃₁×#463	1	2	yes	—
503	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
504	C₄	—	2	1	no	—
505	C₄	—	2	1	no	—
506	C₄	—	2	1	no	—
507	C₄	—	4	1	no	—
508	C₄	—	4	1	no	—
509	C₄	—	8	1	no	—
510	C₄	—	8	2	no	—
511	C₄	C₃₁×#463	1	2	yes	—
512	C₄	C₂×C₃₁×#135	1	1	yes	characteristic
513	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
514	C₄	C₂×C₃₁×AGL(1,ℤ/8)	1	1	yes	characteristic
515	C₄	C₃₁×#575	1	1	yes	characteristic
516	C₂	—	2	2	no	—
517	C₂	—	4	2	no	—
518	C₂	—	8	1	no	—
519	C₂	—	16	1	no	—
520	C₂	C₂×C₃₁×#463	1	1	yes	characteristic
521	C₂	C₃₁×#3055	1	1	yes	characteristic
522	C₂	C₃₁×#3069	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₃₁	C₂×#121
commutator, Frattini, cosocle	C₄²	C₂⁴×C₃₁
prime radical	C₃₁×#3100	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×#3141	C₂	semidirect
3. C₂×C₃₁×#461	C₂	semidirect
4. D₄×M₁₆×C₃₁	C₂	semidirect
5. C₃₁×#1701	C₂	semidirect
6. C₃₁×#1699	C₂	semidirect
7. C₃₁×#1681	C₂	semidirect
8. C₃₁×#1679	C₂	semidirect
9. C₃₁×#1565	C₂	semidirect
10. C₃₁×#1549	C₂	semidirect
11. C₃₁×#1547	C₂	semidirect
18. C₃₁×#487	C₂²	semidirect
19. C₃₁×#486	C₂²	semidirect
43. C₃₁×#335	C₂²	semidirect
44. C₃₁×#334	C₂²	semidirect
45. C₃₁×#332	C₂²	semidirect
46. C₃₁×#332	C₂²	semidirect
47. C₃₁×#332	C₂²	semidirect
48. C₃₁×#331	C₂²	semidirect
175. #19169	C₃₁	direct (N × H)
229. #3141	C₂×C₃₁	semidirect
231. C₂×#461	C₂×C₃₁	semidirect
232. D₄×M₁₆	C₂×C₃₁	semidirect
233. #1701	C₂×C₃₁	semidirect
234. #1699	C₂×C₃₁	semidirect
235. #1681	C₂×C₃₁	semidirect
236. #1679	C₂×C₃₁	semidirect
237. #1565	C₂×C₃₁	semidirect
238. #1549	C₂×C₃₁	semidirect
239. #1547	C₂×C₃₁	semidirect
272. #487	C₂²×C₃₁	semidirect
273. #486	C₂²×C₃₁	semidirect
297. #335	C₂²×C₃₁	semidirect
298. #334	C₂²×C₃₁	semidirect
299. #332	C₂²×C₃₁	semidirect
300. #332	C₂²×C₃₁	semidirect
301. #332	C₂²×C₃₁	semidirect
302. #331	C₂²×C₃₁	semidirect
372. C₃₁	#19169	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×#3141	2
C₃₁×#3100	2
C₂×C₃₁×#461	2
D₄×M₁₆×C₃₁	2
C₃₁×#1701	2
C₃₁×#1699	2
C₃₁×#1681	2
C₃₁×#1679	2
C₃₁×#1565	2
C₃₁×#1549	2
C₃₁×#1547	2
#19169	31

Subgroups — C31×#19169 #21475928

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#19169 #21475928