Subgroups — C₃₁×#21060 #21477819

Order 7936.

1342 conjugacy classes of subgroups.

The subgroup lattice has 488 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#428	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₃₁×#418	C₂	1	1	yes	maximal, characteristic
3	C₂×C₃₁×#418	C₂	1	1	yes	maximal, characteristic
4	C₂×C₃₁×#390	C₂	1	1	yes	maximal, characteristic
5	C₄×C₃₁×#117	C₂	1	1	yes	maximal, characteristic, complemented
6	C₃₁×#1926	C₂	1	4	yes	maximal, complemented
7	C₃₁×#1925	C₂	1	4	yes	maximal, complemented
8	C₃₁×#1736	C₂	1	1	yes	maximal, characteristic, complemented
9	C₃₁×#1728	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
12	C₂²×M₁₆×C₃₁	C₂²	1	1	yes	characteristic
13	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
14	C₂×C₃₁×#117	C₂²	1	1	yes	characteristic
15	C₂²×C₄⋊C₄×C₃₁	C₂²	1	1	yes	characteristic
16	C₂²×C₂²⋊C₄×C₃₁	—	2	2	no	—
17	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
18	C₃₁×#428	—	2	4	no	—
19	C₂×C₃₁×#107	C₂²	1	2	yes	complemented
20	C₂×C₃₁×#106	C₂²	1	2	yes	complemented
21	C₃₁×#418	—	2	4	no	—
22	C₃₁×#418	C₄	1	8	yes	complemented
23	C₂×C₃₁×#102	C₂²	1	2	yes	—
24	C₂×C₃₁×#102	C₂²	1	2	yes	—
25	C₂×C₃₁×#102	C₂²	1	4	yes	—
26	C₃₁×#390	—	2	1	no	—
27	C₃₁×#390	—	2	1	no	—
28	C₃₁×#390	C₂²	1	4	yes	—
29	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
30	C₄×C₄⋊C₄×C₃₁	C₂²	1	2	yes	complemented
31	C₄×C₂²⋊C₄×C₃₁	—	2	2	no	—
32	C₃₁×#376	—	2	1	no	—
33	C₄³×C₃₁	—	2	1	no	—
34	C₃₁×#340	C₂²	1	4	yes	complemented
35	C₃₁×#336	C₂²	1	4	yes	complemented
36	C₂⁵×C₃₁	—	2	1	no	—
37	C₂²×D₄×C₃₁	—	2	2	no	—
38	C₂²×D₄×C₃₁	—	2	1	no	—
39	C₂²×D₄×C₃₁	—	2	2	no	—
40	C₂²×D₄×C₃₁	—	2	2	no	—
41	C₂²×D₄×C₃₁	—	2	2	no	—
42	C₂²×D₄×C₃₁	—	2	2	no	—
43	C₂²×D₄×C₃₁	C₂×C₄	1	4	yes	—
44	C₂²×D₄×C₃₁	C₂³	1	2	yes	—
45	C₂³×C₄×C₃₁	—	2	1	no	—
46	C₂³×C₄×C₃₁	D₄	1	2	yes	—
47	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
48	C₂×M₁₆×C₃₁	—	2	2	no	—
49	C₂×M₁₆×C₃₁	—	2	2	no	—
50	C₂×M₁₆×C₃₁	—	2	2	no	—
51	C₂×M₁₆×C₃₁	—	2	2	no	—
52	C₂×M₁₆×C₃₁	—	2	2	no	—
53	C₂×M₁₆×C₃₁	C₂×C₄	1	4	yes	—
54	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
55	C₂²×C₈×C₃₁	C₂³	1	2	yes	—
56	C₃₁×#117	—	2	2	no	—
57	C₃₁×#117	—	2	2	no	—
58	C₃₁×#117	C₂×C₄	1	4	yes	—
59	C₃₁×#117	D₄	1	4	yes	—
60	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
61	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
62	C₂×C₄⋊C₄×C₃₁	—	2	1	no	—
63	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
64	C₂×C₄⋊C₄×C₃₁	C₂³	1	2	yes	—
65	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
66	C₂×C₄⋊C₄×C₃₁	C₂³	1	1	yes	characteristic
67	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
68	C₂×C₂²⋊C₄×C₃₁	—	2	1	no	—
69	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
70	C₂×C₂²⋊C₄×C₃₁	—	2	4	no	—
71	C₂×C₂²⋊C₄×C₃₁	—	4	8	no	—
72	C₂×C₄²×C₃₁	—	2	1	no	—
73	C₂×C₄²×C₃₁	—	2	1	no	—
74	C₂×C₄²×C₃₁	—	2	2	no	—
75	C₂×C₄²×C₃₁	—	2	2	no	—
76	C₂×C₄²×C₃₁	—	2	2	no	—
77	C₂×C₄²×C₃₁	—	2	2	no	—
78	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
79	C₂×C₄²×C₃₁	D₄	1	1	yes	characteristic
80	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
81	C₃₁×#107	—	2	4	no	—
82	C₃₁×#106	—	2	4	no	—
83	C₃₁×#102	—	2	4	no	—
84	C₃₁×#102	—	2	4	no	—
85	C₃₁×#102	C₂×C₄	1	16	yes	complemented
86	C₃₁×#95	—	2	2	no	—
87	C₃₁×#95	—	2	2	no	—
88	C₂⁴×C₃₁	—	2	4	no	—
89	C₂⁴×C₃₁	—	2	2	no	—
90	C₂⁴×C₃₁	—	4	4	no	—
91	C₂⁴×C₃₁	—	4	4	no	—
92	C₂⁴×C₃₁	—	4	2	no	—
93	C₂⁴×C₃₁	—	4	2	no	—
94	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
95	C₂×D₄×C₃₁	—	2	8	no	—
96	C₂×D₄×C₃₁	—	2	8	no	—
97	C₂×D₄×C₃₁	—	2	4	no	—
98	C₂×D₄×C₃₁	—	2	2	no	—
99	C₂×D₄×C₃₁	—	2	2	no	—
100	C₂×D₄×C₃₁	—	2	2	no	—
101	C₂×D₄×C₃₁	—	2	2	no	—
102	C₂×D₄×C₃₁	—	2	2	no	—
103	C₂×D₄×C₃₁	—	4	2	no	—
104	C₂×D₄×C₃₁	—	4	2	no	—
105	C₂×D₄×C₃₁	—	4	4	no	—
106	C₂×D₄×C₃₁	—	4	4	no	—
107	C₂×D₄×C₃₁	C₂²×C₄	1	4	yes	—
108	C₂²×C₄×C₃₁	—	2	4	no	—
109	C₂²×C₄×C₃₁	—	2	2	no	—
110	C₂²×C₄×C₃₁	—	2	1	no	—
111	C₂²×C₄×C₃₁	—	2	1	no	—
112	C₂²×C₄×C₃₁	—	2	4	no	—
113	C₂²×C₄×C₃₁	—	2	2	no	—
114	C₂²×C₄×C₃₁	—	4	2	no	—
115	C₂²×C₄×C₃₁	—	4	4	no	—
116	C₂²×C₄×C₃₁	—	4	2	no	—
117	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
118	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
119	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
120	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
121	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
122	C₂²×C₄×C₃₁	#41	1	2	yes	—
123	C₂²×C₄×C₃₁	#41	1	2	yes	—
124	C₂²×C₄×C₃₁	#41	1	2	yes	—
125	C₂²×C₄×C₃₁	#41	1	2	yes	—
126	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
127	M₁₆×C₃₁	—	2	8	no	—
128	M₁₆×C₃₁	—	4	4	no	—
129	C₂×C₈×C₃₁	—	2	2	no	—
130	C₂×C₈×C₃₁	—	2	2	no	—
131	C₂×C₈×C₃₁	—	2	2	no	—
132	C₂×C₈×C₃₁	—	2	2	no	—
133	C₂×C₈×C₃₁	—	2	2	no	—
134	C₂×C₈×C₃₁	C₂²×C₄	1	4	yes	—
135	C₄⋊C₄×C₃₁	—	2	2	no	—
136	C₄⋊C₄×C₃₁	—	2	1	no	—
137	C₄⋊C₄×C₃₁	—	2	1	no	—
138	C₄⋊C₄×C₃₁	—	2	2	no	—
139	C₄⋊C₄×C₃₁	—	2	2	no	—
140	C₄⋊C₄×C₃₁	—	4	2	no	—
141	C₄⋊C₄×C₃₁	C₂²×C₄	1	4	yes	—
142	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
143	C₄⋊C₄×C₃₁	C₂×D₄	1	2	yes	—
144	C₂²⋊C₄×C₃₁	—	2	4	no	—
145	C₂²⋊C₄×C₃₁	—	2	4	no	—
146	C₂²⋊C₄×C₃₁	—	4	8	no	—
147	C₂²⋊C₄×C₃₁	—	4	8	no	—
148	C₄²×C₃₁	—	2	4	no	—
149	C₄²×C₃₁	—	2	4	no	—
150	C₄²×C₃₁	—	2	2	no	—
151	C₄²×C₃₁	—	2	2	no	—
152	C₄²×C₃₁	—	2	2	no	—
153	C₄²×C₃₁	—	2	2	no	—
154	C₄²×C₃₁	—	4	4	no	—
155	C₄²×C₃₁	—	4	4	no	—
156	#21060	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
157	C₂³×C₃₁	—	2	2	no	—
158	C₂³×C₃₁	—	2	2	no	—
159	C₂³×C₃₁	—	2	4	no	—
160	C₂³×C₃₁	—	4	16	no	—
161	C₂³×C₃₁	—	4	2	no	—
162	C₂³×C₃₁	—	4	2	no	—
163	C₂³×C₃₁	—	4	4	no	—
164	C₂³×C₃₁	—	4	4	no	—
165	C₂³×C₃₁	—	4	8	no	—
166	C₂³×C₃₁	—	8	8	no	—
167	C₂³×C₃₁	—	8	4	no	—
168	C₂³×C₃₁	—	8	4	no	—
169	C₂³×C₃₁	C₄×D₄	1	2	yes	—
170	C₂³×C₃₁	#121	1	2	yes	—
171	C₂³×C₃₁	#123	1	2	yes	—
172	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
173	D₄×C₃₁	—	2	4	no	—
174	D₄×C₃₁	—	2	4	no	—
175	D₄×C₃₁	—	4	2	no	—
176	D₄×C₃₁	—	4	2	no	—
177	D₄×C₃₁	—	4	8	no	—
178	C₂×C₄×C₃₁	—	2	2	no	—
179	C₂×C₄×C₃₁	—	2	2	no	—
180	C₂×C₄×C₃₁	—	2	4	no	—
181	C₂×C₄×C₃₁	—	2	2	no	—
182	C₂×C₄×C₃₁	—	2	2	no	—
183	C₂×C₄×C₃₁	—	2	2	no	—
184	C₂×C₄×C₃₁	—	2	2	no	—
185	C₂×C₄×C₃₁	—	2	2	no	—
186	C₂×C₄×C₃₁	—	4	8	no	—
187	C₂×C₄×C₃₁	—	4	2	no	—
188	C₂×C₄×C₃₁	—	4	2	no	—
189	C₂×C₄×C₃₁	—	4	2	no	—
190	C₂×C₄×C₃₁	—	4	2	no	—
191	C₂×C₄×C₃₁	—	4	2	no	—
192	C₂×C₄×C₃₁	—	4	4	no	—
193	C₂×C₄×C₃₁	—	8	4	no	—
194	C₂×C₄×C₃₁	#117	1	4	yes	—
195	C₂×C₄×C₃₁	C₄×D₄	1	4	yes	—
196	C₂×C₄×C₃₁	C₄×D₄	1	2	yes	—
197	C₂×C₄×C₃₁	#121	1	2	yes	—
198	C₂×C₄×C₃₁	#123	1	2	yes	—
199	C₂×C₄×C₃₁	#124	1	4	yes	—
200	C₂×C₄×C₃₁	#126	1	4	yes	—
201	C₂×C₄×C₃₁	AGL(1,ℤ/8)	1	4	yes	—
202	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
203	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
204	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
205	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
206	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
207	C₂×C₄×C₃₁	C₂×#41	1	1	yes	characteristic
208	C₈×C₃₁	—	2	4	no	—
209	C₈×C₃₁	—	4	2	no	—
210	C₂×#428	C₂×C₃₁	1	1	yes	characteristic, complemented
211	C₂×#418	C₂×C₃₁	1	1	yes	characteristic
212	C₂×#418	C₂×C₃₁	1	1	yes	characteristic
213	C₂×#390	C₂×C₃₁	1	1	yes	characteristic
214	C₄×#117	C₂×C₃₁	1	1	yes	characteristic, complemented
215	#1926	C₂×C₃₁	1	4	yes	complemented
216	#1925	C₂×C₃₁	1	4	yes	complemented
217	#1736	C₂×C₃₁	1	1	yes	characteristic, complemented
218	#1728	C₂×C₃₁	1	1	yes	characteristic, complemented
219	C₂²×C₃₁	—	2	4	no	—
220	C₂²×C₃₁	—	2	4	no	—
221	C₂²×C₃₁	—	2	2	no	—
222	C₂²×C₃₁	—	2	2	no	—
223	C₂²×C₃₁	—	4	8	no	—
224	C₂²×C₃₁	—	4	4	no	—
225	C₂²×C₃₁	—	8	16	no	—
226	C₂²×C₃₁	—	8	4	no	—
227	C₂²×C₃₁	—	8	2	no	—
228	C₂²×C₃₁	—	8	2	no	—
229	C₂²×C₃₁	#388	1	4	yes	—
230	C₂²×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
231	C₂²×C₃₁	C₂×#121	1	1	yes	characteristic
232	C₂²×C₃₁	C₂×#123	1	1	yes	characteristic
233	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	2	yes	—
234	C₂²×C₃₁	#575	1	2	yes	—
235	C₄×C₃₁	—	2	4	no	—
236	C₄×C₃₁	—	4	2	no	—
237	C₄×C₃₁	—	4	4	no	—
238	C₄×C₃₁	—	4	2	no	—
239	C₄×C₃₁	—	4	2	no	—
240	C₄×C₃₁	—	8	2	no	—
241	C₄×C₃₁	#388	1	4	yes	—
242	C₄×C₃₁	C₂×#117	1	1	yes	characteristic
243	C₄×C₃₁	C₂×C₄×D₄	1	1	yes	characteristic
244	C₄×C₃₁	C₂×#124	1	1	yes	characteristic
245	C₄×C₃₁	C₂×#126	1	1	yes	characteristic
246	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
247	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
248	C₂²×M₁₆	C₂²×C₃₁	1	1	yes	characteristic
249	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
250	C₂×#117	C₂²×C₃₁	1	1	yes	characteristic
251	C₂²×C₄⋊C₄	C₂²×C₃₁	1	1	yes	characteristic
252	C₂²×C₂²⋊C₄	—	2	2	no	—
253	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
254	#428	—	2	4	no	—
255	C₂×#107	C₂²×C₃₁	1	2	yes	complemented
256	C₂×#106	C₂²×C₃₁	1	2	yes	complemented
257	#418	—	2	4	no	—
258	#418	C₄×C₃₁	1	8	yes	complemented
259	C₂×#102	C₂²×C₃₁	1	2	yes	—
260	C₂×#102	C₂²×C₃₁	1	2	yes	—
261	C₂×#102	C₂²×C₃₁	1	4	yes	—
262	#390	—	2	1	no	—
263	#390	—	2	1	no	—
264	#390	C₂²×C₃₁	1	4	yes	—
265	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
266	C₄×C₄⋊C₄	C₂²×C₃₁	1	2	yes	complemented
267	C₄×C₂²⋊C₄	—	2	2	no	—
268	#376	—	2	1	no	—
269	C₄³	—	2	1	no	—
270	#340	C₂²×C₃₁	1	4	yes	complemented
271	#336	C₂²×C₃₁	1	4	yes	complemented
272	C₂×C₃₁	—	2	4	no	—
273	C₂×C₃₁	—	8	4	no	—
274	C₂×C₃₁	C₂×#388	1	1	yes	characteristic
275	C₂×C₃₁	C₄×AGL(1,ℤ/8)	1	2	yes	—
276	C₂×C₃₁	#3038	1	1	yes	characteristic
277	C₂×C₃₁	#3040	1	1	yes	characteristic
278	C₂×C₃₁	#3094	1	2	yes	—
279	C₂⁵	—	2	1	no	—
280	C₂²×D₄	—	2	2	no	—
281	C₂²×D₄	—	2	1	no	—
282	C₂²×D₄	—	2	2	no	—
283	C₂²×D₄	—	2	2	no	—
284	C₂²×D₄	—	2	2	no	—
285	C₂²×D₄	—	2	2	no	—
286	C₂²×D₄	C₂×C₄×C₃₁	1	4	yes	—
287	C₂²×D₄	C₂³×C₃₁	1	2	yes	—
288	C₂³×C₄	—	2	1	no	—
289	C₂³×C₄	D₄×C₃₁	1	2	yes	—
290	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
291	C₂×M₁₆	—	2	2	no	—
292	C₂×M₁₆	—	2	2	no	—
293	C₂×M₁₆	—	2	2	no	—
294	C₂×M₁₆	—	2	2	no	—
295	C₂×M₁₆	—	2	2	no	—
296	C₂×M₁₆	C₂×C₄×C₃₁	1	4	yes	—
297	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
298	C₂²×C₈	C₂³×C₃₁	1	2	yes	—
299	#117	—	2	2	no	—
300	#117	—	2	2	no	—
301	#117	C₂×C₄×C₃₁	1	4	yes	—
302	#117	D₄×C₃₁	1	4	yes	—
303	C₂×C₄⋊C₄	—	2	2	no	—
304	C₂×C₄⋊C₄	—	2	2	no	—
305	C₂×C₄⋊C₄	—	2	1	no	—
306	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
307	C₂×C₄⋊C₄	C₂³×C₃₁	1	2	yes	—
308	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
309	C₂×C₄⋊C₄	C₂³×C₃₁	1	1	yes	characteristic
310	C₂×C₂²⋊C₄	—	2	1	no	—
311	C₂×C₂²⋊C₄	—	2	1	no	—
312	C₂×C₂²⋊C₄	—	2	4	no	—
313	C₂×C₂²⋊C₄	—	2	4	no	—
314	C₂×C₂²⋊C₄	—	4	8	no	—
315	C₂×C₄²	—	2	1	no	—
316	C₂×C₄²	—	2	1	no	—
317	C₂×C₄²	—	2	2	no	—
318	C₂×C₄²	—	2	2	no	—
319	C₂×C₄²	—	2	2	no	—
320	C₂×C₄²	—	2	2	no	—
321	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
322	C₂×C₄²	D₄×C₃₁	1	1	yes	characteristic
323	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
324	#107	—	2	4	no	—
325	#106	—	2	4	no	—
326	#102	—	2	4	no	—
327	#102	—	2	4	no	—
328	#102	C₂×C₄×C₃₁	1	16	yes	complemented
329	#95	—	2	2	no	—
330	#95	—	2	2	no	—
331	C₃₁	#21060	1	1	yes	characteristic, direct product factor, Sylow
332	C₂⁴	—	2	4	no	—
333	C₂⁴	—	2	2	no	—
334	C₂⁴	—	4	4	no	—
335	C₂⁴	—	4	4	no	—
336	C₂⁴	—	4	2	no	—
337	C₂⁴	—	4	2	no	—
338	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
339	C₂×D₄	—	2	8	no	—
340	C₂×D₄	—	2	8	no	—
341	C₂×D₄	—	2	4	no	—
342	C₂×D₄	—	2	2	no	—
343	C₂×D₄	—	2	2	no	—
344	C₂×D₄	—	2	2	no	—
345	C₂×D₄	—	2	2	no	—
346	C₂×D₄	—	2	2	no	—
347	C₂×D₄	—	4	2	no	—
348	C₂×D₄	—	4	2	no	—
349	C₂×D₄	—	4	4	no	—
350	C₂×D₄	—	4	4	no	—
351	C₂×D₄	C₂²×C₄×C₃₁	1	4	yes	—
352	C₂²×C₄	—	2	4	no	—
353	C₂²×C₄	—	2	2	no	—
354	C₂²×C₄	—	2	1	no	—
355	C₂²×C₄	—	2	1	no	—
356	C₂²×C₄	—	2	4	no	—
357	C₂²×C₄	—	2	2	no	—
358	C₂²×C₄	—	4	2	no	—
359	C₂²×C₄	—	4	4	no	—
360	C₂²×C₄	—	4	2	no	—
361	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
362	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
363	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
364	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
365	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
366	C₂²×C₄	#41×C₃₁	1	2	yes	—
367	C₂²×C₄	#41×C₃₁	1	2	yes	—
368	C₂²×C₄	#41×C₃₁	1	2	yes	—
369	C₂²×C₄	#41×C₃₁	1	2	yes	—
370	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
371	M₁₆	—	2	8	no	—
372	M₁₆	—	4	4	no	—
373	C₂×C₈	—	2	2	no	—
374	C₂×C₈	—	2	2	no	—
375	C₂×C₈	—	2	2	no	—
376	C₂×C₈	—	2	2	no	—
377	C₂×C₈	—	2	2	no	—
378	C₂×C₈	C₂²×C₄×C₃₁	1	4	yes	—
379	C₄⋊C₄	—	2	2	no	—
380	C₄⋊C₄	—	2	1	no	—
381	C₄⋊C₄	—	2	1	no	—
382	C₄⋊C₄	—	2	2	no	—
383	C₄⋊C₄	—	2	2	no	—
384	C₄⋊C₄	—	4	2	no	—
385	C₄⋊C₄	C₂²×C₄×C₃₁	1	4	yes	—
386	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
387	C₄⋊C₄	C₂×D₄×C₃₁	1	2	yes	—
388	C₂²⋊C₄	—	2	4	no	—
389	C₂²⋊C₄	—	2	4	no	—
390	C₂²⋊C₄	—	4	8	no	—
391	C₂²⋊C₄	—	4	8	no	—
392	C₄²	—	2	4	no	—
393	C₄²	—	2	4	no	—
394	C₄²	—	2	2	no	—
395	C₄²	—	2	2	no	—
396	C₄²	—	2	2	no	—
397	C₄²	—	2	2	no	—
398	C₄²	—	4	4	no	—
399	C₄²	—	4	4	no	—
400	C₂³	—	2	2	no	—
401	C₂³	—	2	2	no	—
402	C₂³	—	2	4	no	—
403	C₂³	—	4	16	no	—
404	C₂³	—	4	2	no	—
405	C₂³	—	4	2	no	—
406	C₂³	—	4	4	no	—
407	C₂³	—	4	4	no	—
408	C₂³	—	4	8	no	—
409	C₂³	—	8	8	no	—
410	C₂³	—	8	4	no	—
411	C₂³	—	8	4	no	—
412	C₂³	C₄×D₄×C₃₁	1	2	yes	—
413	C₂³	C₃₁×#121	1	2	yes	—
414	C₂³	C₃₁×#123	1	2	yes	—
415	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
416	D₄	—	2	4	no	—
417	D₄	—	2	4	no	—
418	D₄	—	4	2	no	—
419	D₄	—	4	2	no	—
420	D₄	—	4	8	no	—
421	C₂×C₄	—	2	2	no	—
422	C₂×C₄	—	2	2	no	—
423	C₂×C₄	—	2	4	no	—
424	C₂×C₄	—	2	2	no	—
425	C₂×C₄	—	2	2	no	—
426	C₂×C₄	—	2	2	no	—
427	C₂×C₄	—	2	2	no	—
428	C₂×C₄	—	2	2	no	—
429	C₂×C₄	—	4	8	no	—
430	C₂×C₄	—	4	2	no	—
431	C₂×C₄	—	4	2	no	—
432	C₂×C₄	—	4	2	no	—
433	C₂×C₄	—	4	2	no	—
434	C₂×C₄	—	4	2	no	—
435	C₂×C₄	—	4	4	no	—
436	C₂×C₄	—	8	4	no	—
437	C₂×C₄	C₃₁×#117	1	4	yes	—
438	C₂×C₄	C₄×D₄×C₃₁	1	4	yes	—
439	C₂×C₄	C₄×D₄×C₃₁	1	2	yes	—
440	C₂×C₄	C₃₁×#121	1	2	yes	—
441	C₂×C₄	C₃₁×#123	1	2	yes	—
442	C₂×C₄	C₃₁×#124	1	4	yes	—
443	C₂×C₄	C₃₁×#126	1	4	yes	—
444	C₂×C₄	C₃₁×AGL(1,ℤ/8)	1	4	yes	—
445	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
446	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
447	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
448	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
449	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
450	C₂×C₄	C₂×#41×C₃₁	1	1	yes	characteristic
451	C₈	—	2	4	no	—
452	C₈	—	4	2	no	—
453	C₂²	—	2	4	no	—
454	C₂²	—	2	4	no	—
455	C₂²	—	2	2	no	—
456	C₂²	—	2	2	no	—
457	C₂²	—	4	8	no	—
458	C₂²	—	4	4	no	—
459	C₂²	—	8	16	no	—
460	C₂²	—	8	4	no	—
461	C₂²	—	8	2	no	—
462	C₂²	—	8	2	no	—
463	C₂²	C₃₁×#388	1	4	yes	—
464	C₂²	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
465	C₂²	C₂×C₃₁×#121	1	1	yes	characteristic
466	C₂²	C₂×C₃₁×#123	1	1	yes	characteristic
467	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
468	C₂²	C₃₁×#575	1	2	yes	—
469	C₄	—	2	4	no	—
470	C₄	—	4	2	no	—
471	C₄	—	4	4	no	—
472	C₄	—	4	2	no	—
473	C₄	—	4	2	no	—
474	C₄	—	8	2	no	—
475	C₄	C₃₁×#388	1	4	yes	—
476	C₄	C₂×C₃₁×#117	1	1	yes	characteristic
477	C₄	C₂×C₄×D₄×C₃₁	1	1	yes	characteristic
478	C₄	C₂×C₃₁×#124	1	1	yes	characteristic
479	C₄	C₂×C₃₁×#126	1	1	yes	characteristic
480	C₂	—	2	4	no	—
481	C₂	—	8	4	no	—
482	C₂	C₂×C₃₁×#388	1	1	yes	characteristic
483	C₂	C₄×C₃₁×AGL(1,ℤ/8)	1	2	yes	—
484	C₂	C₃₁×#3038	1	1	yes	characteristic
485	C₂	C₃₁×#3040	1	1	yes	characteristic
486	C₂	C₃₁×#3094	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₃₁	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂³×D₄×C₃₁	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#428	C₂	semidirect
5. C₄×C₃₁×#117	C₂	semidirect
6. C₃₁×#1926	C₂	semidirect
7. C₃₁×#1925	C₂	semidirect
8. C₃₁×#1736	C₂	semidirect
9. C₃₁×#1728	C₂	semidirect
19. C₂×C₃₁×#107	C₂²	semidirect
20. C₂×C₃₁×#106	C₂²	semidirect
22. C₃₁×#418	C₄	semidirect
29. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
30. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
34. C₃₁×#340	C₂²	semidirect
35. C₃₁×#336	C₂²	semidirect
85. C₃₁×#102	C₂×C₄	semidirect
156. #21060	C₃₁	direct (N × H)
210. C₂×#428	C₂×C₃₁	semidirect
214. C₄×#117	C₂×C₃₁	semidirect
215. #1926	C₂×C₃₁	semidirect
216. #1925	C₂×C₃₁	semidirect
217. #1736	C₂×C₃₁	semidirect
218. #1728	C₂×C₃₁	semidirect
255. C₂×#107	C₂²×C₃₁	semidirect
256. C₂×#106	C₂²×C₃₁	semidirect
258. #418	C₄×C₃₁	semidirect
265. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
266. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
270. #340	C₂²×C₃₁	semidirect
271. #336	C₂²×C₃₁	semidirect
328. #102	C₂×C₄×C₃₁	semidirect
331. C₃₁	#21060	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₃₁×#428	2
C₂×C₃₁×#418	2
C₂×C₃₁×#390	2
C₄×C₃₁×#117	2
C₃₁×#1926	2
C₃₁×#1925	2
C₃₁×#1736	2
C₃₁×#1728	2
#21060	31

Subgroups — C31×#21060 #21477819

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₃₁×#21060 #21477819