Subgroups — C₃₁×#21012 #21477771

Order 7936.

1732 conjugacy classes of subgroups.

The subgroup lattice has 610 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#3449	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#3428	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#421	C₂	1	2	yes	maximal, complemented
4	C₂×C₃₁×#413	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1912	C₂	1	4	yes	maximal, complemented
6	C₃₁×#1911	C₂	1	4	yes	maximal, complemented
7	C₂×C₃₁×H(3,ℤ/4)	C₂	1	2	yes	maximal, complemented
8	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
9	C₂²×#41×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×M₁₆×C₃₁	C₂²	1	2	yes	—
11	C₃₁×#537	—	2	2	no	—
12	C₃₁×#536	C₄	1	2	yes	complemented
13	C₃₁×#536	C₂²	1	2	yes	—
14	C₃₁×#535	C₄	1	2	yes	complemented
15	C₃₁×#535	C₂²	1	2	yes	—
16	C₃₁×#534	—	2	2	no	—
17	C₃₁×#532	C₄	1	4	yes	complemented
18	C₃₁×#525	—	2	2	no	—
19	C₃₁×#525	—	2	2	no	—
20	C₃₁×#520	—	2	4	no	—
21	C₃₁×#519	C₂²	1	2	yes	complemented
22	C₃₁×#518	—	2	2	no	—
23	C₃₁×#518	C₂²	1	2	yes	complemented
24	C₄×#41×C₃₁	—	2	4	no	—
25	C₂×C₃₁×#117	—	2	2	no	—
26	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
27	C₂×C₃₁×#117	C₂²	1	2	yes	—
28	C₃₁×#421	—	2	4	no	—
29	C₃₁×#421	C₂²	1	8	yes	—
30	C₂×C₃₁×#104	—	2	4	no	—
31	C₃₁×#413	—	2	2	no	—
32	C₃₁×#413	—	2	2	no	—
33	C₂×C₃₁×#101	C₂²	1	2	yes	complemented
34	C₂×C₃₁×#100	C₂²	1	2	yes	complemented
35	C₃₁×H(3,ℤ/4)	C₂²	1	8	yes	complemented
36	C₂×#41×C₃₁	—	2	2	no	—
37	C₂×#41×C₃₁	—	2	2	no	—
38	C₂×#41×C₃₁	—	2	1	no	—
39	C₂×#41×C₃₁	—	2	1	no	—
40	C₂×#41×C₃₁	—	2	2	no	—
41	C₂×#41×C₃₁	—	2	2	no	—
42	C₂×#41×C₃₁	—	2	2	no	—
43	C₂×#41×C₃₁	—	2	2	no	—
44	C₂×#41×C₃₁	—	2	2	no	—
45	C₂×#41×C₃₁	—	2	2	no	—
46	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
47	C₂×#41×C₃₁	C₂×C₄	1	2	yes	—
48	C₂×#41×C₃₁	C₂×C₄	1	2	yes	complemented
49	C₂×#41×C₃₁	D₄	1	2	yes	—
50	C₂×#41×C₃₁	D₄	1	2	yes	—
51	C₂×#41×C₃₁	C₂³	1	2	yes	—
52	C₂²×Q₈×C₃₁	C₂×C₄	1	1	yes	characteristic, complemented
53	C₂²×Q₈×C₃₁	C₂³	1	1	yes	characteristic
54	C₂²×D₄×C₃₁	—	2	1	no	—
55	C₂²×D₄×C₃₁	C₂×C₄	1	1	yes	characteristic, complemented
56	C₂²×D₄×C₃₁	D₄	1	2	yes	—
57	C₂²×D₄×C₃₁	C₂³	1	1	yes	characteristic
58	C₂³×C₄×C₃₁	—	2	1	no	—
59	C₂³×C₄×C₃₁	D₄	1	2	yes	—
60	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
61	C₂×M₁₆×C₃₁	—	2	2	no	—
62	C₂×M₁₆×C₃₁	—	2	2	no	—
63	C₂×M₁₆×C₃₁	—	2	2	no	—
64	C₂×M₁₆×C₃₁	—	2	2	no	—
65	C₂×M₁₆×C₃₁	—	2	2	no	—
66	C₂×M₁₆×C₃₁	C₂³	1	4	yes	—
67	C₂²×C₈×C₃₁	—	2	2	no	—
68	C₃₁×C₄⋊Q₈	C₂×C₄	1	4	yes	complemented
69	C₃₁×#127	C₂×C₄	1	4	yes	complemented
70	C₃₁×#124	C₂×C₄	1	4	yes	complemented
71	C₃₁×#124	C₂×C₄	1	4	yes	complemented
72	C₃₁×#123	—	2	4	no	—
73	C₃₁×#123	—	2	4	no	—
74	C₃₁×C₂²⋊Q₈	—	2	4	no	—
75	C₃₁×C₂²⋊Q₈	—	2	4	no	—
76	C₃₁×#121	—	2	8	no	—
77	C₃₁×#121	—	2	4	no	—
78	C₃₁×#121	—	2	4	no	—
79	C₃₁×#120	—	2	4	no	—
80	C₃₁×#120	—	2	4	no	—
81	C₄×Q₈×C₃₁	—	2	8	no	—
82	C₄×D₄×C₃₁	—	2	16	no	—
83	C₄×D₄×C₃₁	—	2	8	no	—
84	C₄×D₄×C₃₁	—	2	8	no	—
85	C₃₁×#117	—	2	4	no	—
86	C₃₁×#117	—	2	4	no	—
87	C₃₁×#117	—	2	2	no	—
88	C₃₁×#117	—	2	4	no	—
89	C₃₁×#117	—	2	2	no	—
90	C₃₁×#117	—	2	1	no	—
91	C₃₁×#117	—	2	1	no	—
92	C₃₁×#117	C₂×C₄	1	2	yes	complemented
93	C₃₁×#117	D₄	1	4	yes	complemented
94	C₃₁×#117	C₂³	1	2	yes	—
95	C₃₁×#117	C₂³	1	4	yes	—
96	C₂×C₄⋊C₄×C₃₁	—	2	4	no	—
97	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
98	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
99	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
100	C₂×C₂²⋊C₄×C₃₁	—	2	2	no	—
101	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
102	C₂×C₄²×C₃₁	—	2	4	no	—
103	C₂×C₄²×C₃₁	C₂×C₄	1	2	yes	complemented
104	C₂×C₄²×C₃₁	D₄	1	4	yes	complemented
105	C₃₁×#104	—	2	16	no	—
106	C₃₁×#101	—	2	4	no	—
107	C₃₁×#100	—	2	4	no	—
108	C₂⁴×C₃₁	—	2	1	no	—
109	C₂⁴×C₃₁	C₂×D₄	1	2	yes	—
110	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
111	#41×C₃₁	—	2	4	no	—
112	#41×C₃₁	—	2	4	no	—
113	#41×C₃₁	—	2	4	no	—
114	#41×C₃₁	—	2	4	no	—
115	#41×C₃₁	—	2	4	no	—
116	#41×C₃₁	—	2	4	no	—
117	#41×C₃₁	—	2	4	no	—
118	#41×C₃₁	—	2	4	no	—
119	#41×C₃₁	—	2	4	no	—
120	#41×C₃₁	—	2	4	no	—
121	#41×C₃₁	—	2	4	no	—
122	#41×C₃₁	—	2	4	no	—
123	#41×C₃₁	—	4	1	no	—
124	#41×C₃₁	—	4	1	no	—
125	#41×C₃₁	—	4	2	no	—
126	#41×C₃₁	—	4	2	no	—
127	#41×C₃₁	—	4	1	no	—
128	#41×C₃₁	—	4	1	no	—
129	C₂×Q₈×C₃₁	—	2	1	no	—
130	C₂×Q₈×C₃₁	—	2	2	no	—
131	C₂×Q₈×C₃₁	—	2	2	no	—
132	C₂×Q₈×C₃₁	—	2	2	no	—
133	C₂×Q₈×C₃₁	—	2	2	no	—
134	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
135	C₂×Q₈×C₃₁	C₂²×C₄	1	2	yes	—
136	C₂×Q₈×C₃₁	C₂×D₄	1	2	yes	—
137	C₂×D₄×C₃₁	—	2	4	no	—
138	C₂×D₄×C₃₁	—	2	4	no	—
139	C₂×D₄×C₃₁	—	2	2	no	—
140	C₂×D₄×C₃₁	—	2	2	no	—
141	C₂×D₄×C₃₁	—	2	8	no	—
142	C₂×D₄×C₃₁	—	2	1	no	—
143	C₂×D₄×C₃₁	—	2	2	no	—
144	C₂×D₄×C₃₁	—	2	2	no	—
145	C₂×D₄×C₃₁	—	2	2	no	—
146	C₂×D₄×C₃₁	—	2	2	no	—
147	C₂×D₄×C₃₁	—	4	2	no	—
148	C₂×D₄×C₃₁	—	4	2	no	—
149	C₂×D₄×C₃₁	C₂²⋊C₄	1	4	yes	—
150	C₂×D₄×C₃₁	C₂²⋊C₄	1	4	yes	—
151	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
152	C₂×D₄×C₃₁	C₂²×C₄	1	2	yes	—
153	C₂×D₄×C₃₁	C₂×D₄	1	2	yes	—
154	C₂²×C₄×C₃₁	—	2	2	no	—
155	C₂²×C₄×C₃₁	—	2	2	no	—
156	C₂²×C₄×C₃₁	—	2	2	no	—
157	C₂²×C₄×C₃₁	—	2	2	no	—
158	C₂²×C₄×C₃₁	—	2	1	no	—
159	C₂²×C₄×C₃₁	—	2	8	no	—
160	C₂²×C₄×C₃₁	—	2	2	no	—
161	C₂²×C₄×C₃₁	—	2	4	no	—
162	C₂²×C₄×C₃₁	—	2	2	no	—
163	C₂²×C₄×C₃₁	—	2	2	no	—
164	C₂²×C₄×C₃₁	—	4	2	no	—
165	C₂²×C₄×C₃₁	—	4	2	no	—
166	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
167	C₂²×C₄×C₃₁	C₂²⋊C₄	1	4	yes	—
168	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
169	C₂²×C₄×C₃₁	C₂²×C₄	1	1	yes	characteristic
170	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
171	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
172	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
173	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
174	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
175	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
176	C₂²×C₄×C₃₁	#41	1	2	yes	—
177	C₂²×C₄×C₃₁	#41	1	2	yes	—
178	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
179	M₁₆×C₃₁	—	2	4	no	—
180	M₁₆×C₃₁	—	2	4	no	—
181	M₁₆×C₃₁	—	4	2	no	—
182	M₁₆×C₃₁	—	4	2	no	—
183	C₂×C₈×C₃₁	—	2	4	no	—
184	C₂×C₈×C₃₁	—	4	2	no	—
185	C₂×C₈×C₃₁	—	4	2	no	—
186	C₄⋊C₄×C₃₁	—	2	8	no	—
187	C₄⋊C₄×C₃₁	—	2	2	no	—
188	C₄⋊C₄×C₃₁	—	2	8	no	—
189	C₄⋊C₄×C₃₁	—	2	4	no	—
190	C₄⋊C₄×C₃₁	—	2	4	no	—
191	C₄⋊C₄×C₃₁	—	2	2	no	—
192	C₂²⋊C₄×C₃₁	—	2	8	no	—
193	C₂²⋊C₄×C₃₁	—	2	2	no	—
194	C₂²⋊C₄×C₃₁	—	2	8	no	—
195	C₂²⋊C₄×C₃₁	—	2	4	no	—
196	C₂²⋊C₄×C₃₁	—	2	4	no	—
197	C₂²⋊C₄×C₃₁	—	2	2	no	—
198	C₂²⋊C₄×C₃₁	—	2	8	no	—
199	C₄²×C₃₁	—	2	8	no	—
200	C₄²×C₃₁	—	2	2	no	—
201	C₄²×C₃₁	—	2	4	no	—
202	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
203	C₄²×C₃₁	C₂×D₄	1	8	yes	—
204	#21012	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
205	C₂³×C₃₁	—	2	2	no	—
206	C₂³×C₃₁	—	2	2	no	—
207	C₂³×C₃₁	—	2	2	no	—
208	C₂³×C₃₁	—	4	2	no	—
209	C₂³×C₃₁	—	8	1	no	—
210	C₂³×C₃₁	—	8	2	no	—
211	C₂³×C₃₁	—	8	2	no	—
212	C₂³×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
213	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
214	C₂³×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
215	C₂³×C₃₁	#120	1	2	yes	—
216	C₂³×C₃₁	#123	1	2	yes	—
217	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
218	Q₈×C₃₁	—	2	4	no	—
219	Q₈×C₃₁	—	2	4	no	—
220	Q₈×C₃₁	—	2	4	no	—
221	Q₈×C₃₁	—	4	1	no	—
222	Q₈×C₃₁	—	4	1	no	—
223	D₄×C₃₁	—	2	8	no	—
224	D₄×C₃₁	—	2	8	no	—
225	D₄×C₃₁	—	2	4	no	—
226	D₄×C₃₁	—	2	4	no	—
227	D₄×C₃₁	—	2	4	no	—
228	D₄×C₃₁	—	4	2	no	—
229	D₄×C₃₁	—	4	2	no	—
230	D₄×C₃₁	—	4	4	no	—
231	D₄×C₃₁	—	4	1	no	—
232	D₄×C₃₁	—	4	1	no	—
233	D₄×C₃₁	—	8	8	no	—
234	C₂×C₄×C₃₁	—	2	2	no	—
235	C₂×C₄×C₃₁	—	2	2	no	—
236	C₂×C₄×C₃₁	—	2	4	no	—
237	C₂×C₄×C₃₁	—	2	4	no	—
238	C₂×C₄×C₃₁	—	2	4	no	—
239	C₂×C₄×C₃₁	—	2	4	no	—
240	C₂×C₄×C₃₁	—	2	4	no	—
241	C₂×C₄×C₃₁	—	2	2	no	—
242	C₂×C₄×C₃₁	—	2	8	no	—
243	C₂×C₄×C₃₁	—	4	2	no	—
244	C₂×C₄×C₃₁	—	4	4	no	—
245	C₂×C₄×C₃₁	—	4	2	no	—
246	C₂×C₄×C₃₁	—	4	2	no	—
247	C₂×C₄×C₃₁	—	8	4	no	—
248	C₂×C₄×C₃₁	—	8	8	no	—
249	C₂×C₄×C₃₁	—	8	2	no	—
250	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	4	yes	—
251	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
252	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
253	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
254	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
255	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
256	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
257	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
258	C₂×C₄×C₃₁	#120	1	1	yes	characteristic
259	C₂×C₄×C₃₁	#121	1	2	yes	—
260	C₂×C₄×C₃₁	#121	1	2	yes	—
261	C₂×C₄×C₃₁	#121	1	2	yes	—
262	C₂×C₄×C₃₁	#121	1	2	yes	—
263	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
264	C₂×C₄×C₃₁	#123	1	1	yes	characteristic
265	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
266	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
267	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
268	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
269	C₈×C₃₁	—	4	4	no	—
270	#3449	C₂×C₃₁	1	1	yes	characteristic
271	#3428	C₂×C₃₁	1	1	yes	characteristic, complemented
272	C₂×#421	C₂×C₃₁	1	2	yes	complemented
273	C₂×#413	C₂×C₃₁	1	1	yes	characteristic, complemented
274	#1912	C₂×C₃₁	1	4	yes	complemented
275	#1911	C₂×C₃₁	1	4	yes	complemented
276	C₂×H(3,ℤ/4)	C₂×C₃₁	1	2	yes	complemented
277	C₂²×C₃₁	—	2	4	no	—
278	C₂²×C₃₁	—	4	1	no	—
279	C₂²×C₃₁	—	4	2	no	—
280	C₂²×C₃₁	—	4	1	no	—
281	C₂²×C₃₁	—	4	1	no	—
282	C₂²×C₃₁	—	4	2	no	—
283	C₂²×C₃₁	—	4	4	no	—
284	C₂²×C₃₁	—	4	4	no	—
285	C₂²×C₃₁	—	4	4	no	—
286	C₂²×C₃₁	—	4	2	no	—
287	C₂²×C₃₁	—	8	2	no	—
288	C₂²×C₃₁	—	8	4	no	—
289	C₂²×C₃₁	#386	1	2	yes	—
290	C₂²×C₃₁	#386	1	2	yes	—
291	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
292	C₂²×C₃₁	C₂×#120	1	1	yes	characteristic
293	C₂²×C₃₁	C₂×#123	1	1	yes	characteristic
294	C₄×C₃₁	—	2	4	no	—
295	C₄×C₃₁	—	4	8	no	—
296	C₄×C₃₁	—	4	4	no	—
297	C₄×C₃₁	—	4	2	no	—
298	C₄×C₃₁	#386	1	1	yes	characteristic
299	C₄×C₃₁	#386	1	1	yes	characteristic
300	C₄×C₃₁	#386	1	1	yes	characteristic
301	C₄×C₃₁	#386	1	1	yes	characteristic
302	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
303	C₄×C₃₁	C₂×#121	1	2	yes	—
304	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
305	C₂²×#41	C₂²×C₃₁	1	1	yes	characteristic
306	C₂²×M₁₆	C₂²×C₃₁	1	2	yes	—
307	#537	—	2	2	no	—
308	#536	C₄×C₃₁	1	2	yes	complemented
309	#536	C₂²×C₃₁	1	2	yes	—
310	#535	C₄×C₃₁	1	2	yes	complemented
311	#535	C₂²×C₃₁	1	2	yes	—
312	#534	—	2	2	no	—
313	#532	C₄×C₃₁	1	4	yes	complemented
314	#525	—	2	2	no	—
315	#525	—	2	2	no	—
316	#520	—	2	4	no	—
317	#519	C₂²×C₃₁	1	2	yes	complemented
318	#518	—	2	2	no	—
319	#518	C₂²×C₃₁	1	2	yes	complemented
320	C₄×#41	—	2	4	no	—
321	C₂×#117	—	2	2	no	—
322	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
323	C₂×#117	C₂²×C₃₁	1	2	yes	—
324	#421	—	2	4	no	—
325	#421	C₂²×C₃₁	1	8	yes	—
326	C₂×#104	—	2	4	no	—
327	#413	—	2	2	no	—
328	#413	—	2	2	no	—
329	C₂×#101	C₂²×C₃₁	1	2	yes	complemented
330	C₂×#100	C₂²×C₃₁	1	2	yes	complemented
331	H(3,ℤ/4)	C₂²×C₃₁	1	8	yes	complemented
332	C₂×C₃₁	—	2	1	no	—
333	C₂×C₃₁	—	2	1	no	—
334	C₂×C₃₁	—	2	1	no	—
335	C₂×C₃₁	—	2	2	no	—
336	C₂×C₃₁	—	2	2	no	—
337	C₂×C₃₁	—	4	4	no	—
338	C₂×C₃₁	—	8	2	no	—
339	C₂×C₃₁	C₂×#386	1	1	yes	characteristic
340	C₂×#41	—	2	2	no	—
341	C₂×#41	—	2	2	no	—
342	C₂×#41	—	2	1	no	—
343	C₂×#41	—	2	1	no	—
344	C₂×#41	—	2	2	no	—
345	C₂×#41	—	2	2	no	—
346	C₂×#41	—	2	2	no	—
347	C₂×#41	—	2	2	no	—
348	C₂×#41	—	2	2	no	—
349	C₂×#41	—	2	2	no	—
350	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
351	C₂×#41	C₂×C₄×C₃₁	1	2	yes	—
352	C₂×#41	C₂×C₄×C₃₁	1	2	yes	complemented
353	C₂×#41	D₄×C₃₁	1	2	yes	—
354	C₂×#41	D₄×C₃₁	1	2	yes	—
355	C₂×#41	C₂³×C₃₁	1	2	yes	—
356	C₂²×Q₈	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
357	C₂²×Q₈	C₂³×C₃₁	1	1	yes	characteristic
358	C₂²×D₄	—	2	1	no	—
359	C₂²×D₄	C₂×C₄×C₃₁	1	1	yes	characteristic, complemented
360	C₂²×D₄	D₄×C₃₁	1	2	yes	—
361	C₂²×D₄	C₂³×C₃₁	1	1	yes	characteristic
362	C₂³×C₄	—	2	1	no	—
363	C₂³×C₄	D₄×C₃₁	1	2	yes	—
364	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
365	C₂×M₁₆	—	2	2	no	—
366	C₂×M₁₆	—	2	2	no	—
367	C₂×M₁₆	—	2	2	no	—
368	C₂×M₁₆	—	2	2	no	—
369	C₂×M₁₆	—	2	2	no	—
370	C₂×M₁₆	C₂³×C₃₁	1	4	yes	—
371	C₂²×C₈	—	2	2	no	—
372	C₄⋊Q₈	C₂×C₄×C₃₁	1	4	yes	complemented
373	#127	C₂×C₄×C₃₁	1	4	yes	complemented
374	#124	C₂×C₄×C₃₁	1	4	yes	complemented
375	#124	C₂×C₄×C₃₁	1	4	yes	complemented
376	#123	—	2	4	no	—
377	#123	—	2	4	no	—
378	C₂²⋊Q₈	—	2	4	no	—
379	C₂²⋊Q₈	—	2	4	no	—
380	#121	—	2	8	no	—
381	#121	—	2	4	no	—
382	#121	—	2	4	no	—
383	#120	—	2	4	no	—
384	#120	—	2	4	no	—
385	C₄×Q₈	—	2	8	no	—
386	C₄×D₄	—	2	16	no	—
387	C₄×D₄	—	2	8	no	—
388	C₄×D₄	—	2	8	no	—
389	#117	—	2	4	no	—
390	#117	—	2	4	no	—
391	#117	—	2	2	no	—
392	#117	—	2	4	no	—
393	#117	—	2	2	no	—
394	#117	—	2	1	no	—
395	#117	—	2	1	no	—
396	#117	C₂×C₄×C₃₁	1	2	yes	complemented
397	#117	D₄×C₃₁	1	4	yes	complemented
398	#117	C₂³×C₃₁	1	2	yes	—
399	#117	C₂³×C₃₁	1	4	yes	—
400	C₂×C₄⋊C₄	—	2	4	no	—
401	C₂×C₄⋊C₄	—	2	2	no	—
402	C₂×C₄⋊C₄	—	2	1	no	—
403	C₂×C₂²⋊C₄	—	2	4	no	—
404	C₂×C₂²⋊C₄	—	2	2	no	—
405	C₂×C₂²⋊C₄	—	2	1	no	—
406	C₂×C₄²	—	2	4	no	—
407	C₂×C₄²	C₂×C₄×C₃₁	1	2	yes	complemented
408	C₂×C₄²	D₄×C₃₁	1	4	yes	complemented
409	#104	—	2	16	no	—
410	#101	—	2	4	no	—
411	#100	—	2	4	no	—
412	C₃₁	#21012	1	1	yes	characteristic, direct product factor, Sylow
413	C₂⁴	—	2	1	no	—
414	C₂⁴	C₂×D₄×C₃₁	1	2	yes	—
415	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
416	#41	—	2	4	no	—
417	#41	—	2	4	no	—
418	#41	—	2	4	no	—
419	#41	—	2	4	no	—
420	#41	—	2	4	no	—
421	#41	—	2	4	no	—
422	#41	—	2	4	no	—
423	#41	—	2	4	no	—
424	#41	—	2	4	no	—
425	#41	—	2	4	no	—
426	#41	—	2	4	no	—
427	#41	—	2	4	no	—
428	#41	—	4	1	no	—
429	#41	—	4	1	no	—
430	#41	—	4	2	no	—
431	#41	—	4	2	no	—
432	#41	—	4	1	no	—
433	#41	—	4	1	no	—
434	C₂×Q₈	—	2	1	no	—
435	C₂×Q₈	—	2	2	no	—
436	C₂×Q₈	—	2	2	no	—
437	C₂×Q₈	—	2	2	no	—
438	C₂×Q₈	—	2	2	no	—
439	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
440	C₂×Q₈	C₂²×C₄×C₃₁	1	2	yes	—
441	C₂×Q₈	C₂×D₄×C₃₁	1	2	yes	—
442	C₂×D₄	—	2	4	no	—
443	C₂×D₄	—	2	4	no	—
444	C₂×D₄	—	2	2	no	—
445	C₂×D₄	—	2	2	no	—
446	C₂×D₄	—	2	8	no	—
447	C₂×D₄	—	2	1	no	—
448	C₂×D₄	—	2	2	no	—
449	C₂×D₄	—	2	2	no	—
450	C₂×D₄	—	2	2	no	—
451	C₂×D₄	—	2	2	no	—
452	C₂×D₄	—	4	2	no	—
453	C₂×D₄	—	4	2	no	—
454	C₂×D₄	C₂²⋊C₄×C₃₁	1	4	yes	—
455	C₂×D₄	C₂²⋊C₄×C₃₁	1	4	yes	—
456	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
457	C₂×D₄	C₂²×C₄×C₃₁	1	2	yes	—
458	C₂×D₄	C₂×D₄×C₃₁	1	2	yes	—
459	C₂²×C₄	—	2	2	no	—
460	C₂²×C₄	—	2	2	no	—
461	C₂²×C₄	—	2	2	no	—
462	C₂²×C₄	—	2	2	no	—
463	C₂²×C₄	—	2	1	no	—
464	C₂²×C₄	—	2	8	no	—
465	C₂²×C₄	—	2	2	no	—
466	C₂²×C₄	—	2	4	no	—
467	C₂²×C₄	—	2	2	no	—
468	C₂²×C₄	—	2	2	no	—
469	C₂²×C₄	—	4	2	no	—
470	C₂²×C₄	—	4	2	no	—
471	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
472	C₂²×C₄	C₂²⋊C₄×C₃₁	1	4	yes	—
473	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
474	C₂²×C₄	C₂²×C₄×C₃₁	1	1	yes	characteristic
475	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
476	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
477	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
478	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
479	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
480	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
481	C₂²×C₄	#41×C₃₁	1	2	yes	—
482	C₂²×C₄	#41×C₃₁	1	2	yes	—
483	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
484	M₁₆	—	2	4	no	—
485	M₁₆	—	2	4	no	—
486	M₁₆	—	4	2	no	—
487	M₁₆	—	4	2	no	—
488	C₂×C₈	—	2	4	no	—
489	C₂×C₈	—	4	2	no	—
490	C₂×C₈	—	4	2	no	—
491	C₄⋊C₄	—	2	8	no	—
492	C₄⋊C₄	—	2	2	no	—
493	C₄⋊C₄	—	2	8	no	—
494	C₄⋊C₄	—	2	4	no	—
495	C₄⋊C₄	—	2	4	no	—
496	C₄⋊C₄	—	2	2	no	—
497	C₂²⋊C₄	—	2	8	no	—
498	C₂²⋊C₄	—	2	2	no	—
499	C₂²⋊C₄	—	2	8	no	—
500	C₂²⋊C₄	—	2	4	no	—
501	C₂²⋊C₄	—	2	4	no	—
502	C₂²⋊C₄	—	2	2	no	—
503	C₂²⋊C₄	—	2	8	no	—
504	C₄²	—	2	8	no	—
505	C₄²	—	2	2	no	—
506	C₄²	—	2	4	no	—
507	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
508	C₄²	C₂×D₄×C₃₁	1	8	yes	—
509	C₂³	—	2	2	no	—
510	C₂³	—	2	2	no	—
511	C₂³	—	2	2	no	—
512	C₂³	—	4	2	no	—
513	C₂³	—	8	1	no	—
514	C₂³	—	8	2	no	—
515	C₂³	—	8	2	no	—
516	C₂³	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
517	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
518	C₂³	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
519	C₂³	C₃₁×#120	1	2	yes	—
520	C₂³	C₃₁×#123	1	2	yes	—
521	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
522	Q₈	—	2	4	no	—
523	Q₈	—	2	4	no	—
524	Q₈	—	2	4	no	—
525	Q₈	—	4	1	no	—
526	Q₈	—	4	1	no	—
527	D₄	—	2	8	no	—
528	D₄	—	2	8	no	—
529	D₄	—	2	4	no	—
530	D₄	—	2	4	no	—
531	D₄	—	2	4	no	—
532	D₄	—	4	2	no	—
533	D₄	—	4	2	no	—
534	D₄	—	4	4	no	—
535	D₄	—	4	1	no	—
536	D₄	—	4	1	no	—
537	D₄	—	8	8	no	—
538	C₂×C₄	—	2	2	no	—
539	C₂×C₄	—	2	2	no	—
540	C₂×C₄	—	2	4	no	—
541	C₂×C₄	—	2	4	no	—
542	C₂×C₄	—	2	4	no	—
543	C₂×C₄	—	2	4	no	—
544	C₂×C₄	—	2	4	no	—
545	C₂×C₄	—	2	2	no	—
546	C₂×C₄	—	2	8	no	—
547	C₂×C₄	—	4	2	no	—
548	C₂×C₄	—	4	4	no	—
549	C₂×C₄	—	4	2	no	—
550	C₂×C₄	—	4	2	no	—
551	C₂×C₄	—	8	4	no	—
552	C₂×C₄	—	8	8	no	—
553	C₂×C₄	—	8	2	no	—
554	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	4	yes	—
555	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
556	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
557	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
558	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
559	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
560	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
561	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
562	C₂×C₄	C₃₁×#120	1	1	yes	characteristic
563	C₂×C₄	C₃₁×#121	1	2	yes	—
564	C₂×C₄	C₃₁×#121	1	2	yes	—
565	C₂×C₄	C₃₁×#121	1	2	yes	—
566	C₂×C₄	C₃₁×#121	1	2	yes	—
567	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
568	C₂×C₄	C₃₁×#123	1	1	yes	characteristic
569	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
570	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
571	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
572	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
573	C₈	—	4	4	no	—
574	C₂²	—	2	4	no	—
575	C₂²	—	4	1	no	—
576	C₂²	—	4	2	no	—
577	C₂²	—	4	1	no	—
578	C₂²	—	4	1	no	—
579	C₂²	—	4	2	no	—
580	C₂²	—	4	4	no	—
581	C₂²	—	4	4	no	—
582	C₂²	—	4	4	no	—
583	C₂²	—	4	2	no	—
584	C₂²	—	8	2	no	—
585	C₂²	—	8	4	no	—
586	C₂²	C₃₁×#386	1	2	yes	—
587	C₂²	C₃₁×#386	1	2	yes	—
588	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
589	C₂²	C₂×C₃₁×#120	1	1	yes	characteristic
590	C₂²	C₂×C₃₁×#123	1	1	yes	characteristic
591	C₄	—	2	4	no	—
592	C₄	—	4	8	no	—
593	C₄	—	4	4	no	—
594	C₄	—	4	2	no	—
595	C₄	C₃₁×#386	1	1	yes	characteristic
596	C₄	C₃₁×#386	1	1	yes	characteristic
597	C₄	C₃₁×#386	1	1	yes	characteristic
598	C₄	C₃₁×#386	1	1	yes	characteristic
599	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
600	C₄	C₂×C₃₁×#121	1	2	yes	—
601	C₂	—	2	1	no	—
602	C₂	—	2	1	no	—
603	C₂	—	2	1	no	—
604	C₂	—	2	2	no	—
605	C₂	—	2	2	no	—
606	C₂	—	4	4	no	—
607	C₂	—	8	2	no	—
608	C₂	C₂×C₃₁×#386	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₃₁	#386
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
socle	C₂×C₃₁	C₂×#386
prime radical	C₃₁×#3449	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₃₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₃₁×#3428	C₂	semidirect
3. C₂×C₃₁×#421	C₂	semidirect
4. C₂×C₃₁×#413	C₂	semidirect
5. C₃₁×#1912	C₂	semidirect
6. C₃₁×#1911	C₂	semidirect
7. C₂×C₃₁×H(3,ℤ/4)	C₂	semidirect
12. C₃₁×#536	C₄	semidirect
14. C₃₁×#535	C₄	semidirect
17. C₃₁×#532	C₄	semidirect
21. C₃₁×#519	C₂²	semidirect
23. C₃₁×#518	C₂²	semidirect
33. C₂×C₃₁×#101	C₂²	semidirect
34. C₂×C₃₁×#100	C₂²	semidirect
35. C₃₁×H(3,ℤ/4)	C₂²	semidirect
48. C₂×#41×C₃₁	C₂×C₄	semidirect
52. C₂²×Q₈×C₃₁	C₂×C₄	semidirect
55. C₂²×D₄×C₃₁	C₂×C₄	semidirect
68. C₃₁×C₄⋊Q₈	C₂×C₄	semidirect
69. C₃₁×#127	C₂×C₄	semidirect
70. C₃₁×#124	C₂×C₄	semidirect
71. C₃₁×#124	C₂×C₄	semidirect
92. C₃₁×#117	C₂×C₄	semidirect
93. C₃₁×#117	D₄	semidirect
103. C₂×C₄²×C₃₁	C₂×C₄	semidirect
104. C₂×C₄²×C₃₁	D₄	semidirect
204. #21012	C₃₁	direct (N × H)
271. #3428	C₂×C₃₁	semidirect
272. C₂×#421	C₂×C₃₁	semidirect
273. C₂×#413	C₂×C₃₁	semidirect
274. #1912	C₂×C₃₁	semidirect
275. #1911	C₂×C₃₁	semidirect
276. C₂×H(3,ℤ/4)	C₂×C₃₁	semidirect
308. #536	C₄×C₃₁	semidirect
310. #535	C₄×C₃₁	semidirect
313. #532	C₄×C₃₁	semidirect
317. #519	C₂²×C₃₁	semidirect
319. #518	C₂²×C₃₁	semidirect
329. C₂×#101	C₂²×C₃₁	semidirect
330. C₂×#100	C₂²×C₃₁	semidirect
331. H(3,ℤ/4)	C₂²×C₃₁	semidirect
352. C₂×#41	C₂×C₄×C₃₁	semidirect
356. C₂²×Q₈	C₂×C₄×C₃₁	semidirect
359. C₂²×D₄	C₂×C₄×C₃₁	semidirect
372. C₄⋊Q₈	C₂×C₄×C₃₁	semidirect
373. #127	C₂×C₄×C₃₁	semidirect
374. #124	C₂×C₄×C₃₁	semidirect
375. #124	C₂×C₄×C₃₁	semidirect
396. #117	C₂×C₄×C₃₁	semidirect
397. #117	D₄×C₃₁	semidirect
407. C₂×C₄²	C₂×C₄×C₃₁	semidirect
408. C₂×C₄²	D₄×C₃₁	semidirect
412. C₃₁	#21012	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×#3449	2
C₃₁×#3428	2
C₂×C₃₁×#421	2
C₂×C₃₁×#413	2
C₃₁×#1912	2
C₃₁×#1911	2
C₂×C₃₁×H(3,ℤ/4)	2
#21012	31

Subgroups — C31×#21012 #21477771

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21012 #21477771