Subgroups — C₂×C₃₁×#1961 #21477957

Order 7936.

2286 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₃₁×#127	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₃₁×#102	C₂	1	4	yes	maximal, complemented
3	C₂²×C₃₁×#97	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₃₁×#382	C₂	1	1	yes	maximal, characteristic, complemented
5	C₃₁×#1961	C₂	1	8	yes	maximal, direct product factor
6	C₂³×D₄×C₃₁	—	2	2	no	—
7	C₂³×D₄×C₃₁	C₂²	1	2	yes	—
8	C₂³×C₈×C₃₁	C₂²	1	2	yes	—
9	C₂×C₃₁×#127	—	2	2	no	—
10	C₂×C₃₁×#127	—	2	4	no	—
11	C₂×C₃₁×#127	C₄	1	8	yes	complemented
12	C₂×C₃₁×#127	C₂²	1	4	yes	—
13	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
14	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
15	C₂×C₃₁×#102	—	2	16	no	—
16	C₂×C₃₁×#102	C₂²	1	16	yes	complemented
17	C₂×C₃₁×#97	—	2	4	no	—
18	C₂×C₃₁×#97	C₂²	1	4	yes	complemented
19	C₃₁×#382	—	2	2	no	—
20	C₃₁×#382	C₂²	1	4	yes	complemented
21	C₂×C₃₁×#95	—	2	2	no	—
22	C₂⁵×C₃₁	—	2	2	no	—
23	C₂²×D₄×C₃₁	—	2	16	no	—
24	C₂²×D₄×C₃₁	—	2	8	no	—
25	C₂²×D₄×C₃₁	—	2	2	no	—
26	C₂²×D₄×C₃₁	—	2	4	no	—
27	C₂²×D₄×C₃₁	—	2	8	no	—
28	C₂²×D₄×C₃₁	—	2	8	no	—
29	C₂²×D₄×C₃₁	—	4	16	no	—
30	C₂²×D₄×C₃₁	C₂×C₄	1	8	yes	—
31	C₂²×D₄×C₃₁	C₂³	1	4	yes	—
32	C₂³×C₄×C₃₁	—	2	2	no	—
33	C₂³×C₄×C₃₁	D₄	1	2	yes	—
34	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₃₁	—	2	4	no	—
36	C₂²×C₈×C₃₁	—	2	4	no	—
37	C₂²×C₈×C₃₁	D₄	1	8	yes	complemented
38	C₂²×C₈×C₃₁	C₂³	1	4	yes	—
39	C₃₁×#127	—	2	16	no	—
40	C₃₁×#127	—	2	8	no	—
41	C₃₁×#127	C₂×C₄	1	16	yes	complemented
42	C₂×C₄⋊C₄×C₃₁	—	2	8	no	—
43	C₂×C₄⋊C₄×C₃₁	—	2	2	no	—
44	C₂×C₄⋊C₄×C₃₁	C₂³	1	4	yes	—
45	C₂×C₄²×C₃₁	C₂×C₄	1	4	yes	—
46	C₂×C₄²×C₃₁	D₄	1	2	yes	—
47	C₂×C₄²×C₃₁	D₄	1	4	yes	—
48	C₂×C₄²×C₃₁	C₂³	1	2	yes	—
49	C₃₁×#102	—	2	32	no	—
50	C₃₁×#97	—	2	8	no	—
51	C₃₁×#95	—	2	8	no	—
52	C₂⁴×C₃₁	—	2	8	no	—
53	C₂⁴×C₃₁	—	2	4	no	—
54	C₂⁴×C₃₁	—	4	4	no	—
55	C₂⁴×C₃₁	—	4	4	no	—
56	C₂⁴×C₃₁	—	4	16	no	—
57	C₂⁴×C₃₁	C₂×D₄	1	1	yes	characteristic
58	C₂×D₄×C₃₁	—	2	8	no	—
59	C₂×D₄×C₃₁	—	2	32	no	—
60	C₂×D₄×C₃₁	—	2	32	no	—
61	C₂×D₄×C₃₁	—	2	8	no	—
62	C₂×D₄×C₃₁	—	2	4	no	—
63	C₂×D₄×C₃₁	—	2	8	no	—
64	C₂×D₄×C₃₁	—	4	32	no	—
65	C₂×D₄×C₃₁	—	4	64	no	—
66	C₂×D₄×C₃₁	—	4	16	no	—
67	C₂×D₄×C₃₁	—	4	4	no	—
68	C₂×D₄×C₃₁	—	4	4	no	—
69	C₂×D₄×C₃₁	C₂²×C₄	1	8	yes	—
70	C₂²×C₄×C₃₁	—	2	8	no	—
71	C₂²×C₄×C₃₁	—	2	4	no	—
72	C₂²×C₄×C₃₁	—	4	4	no	—
73	C₂²×C₄×C₃₁	—	4	8	no	—
74	C₂²×C₄×C₃₁	C₂²⋊C₄	1	8	yes	—
75	C₂²×C₄×C₃₁	C₂²×C₄	1	2	yes	—
76	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
77	C₂²×C₄×C₃₁	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₃₁	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₃₁	C₂×D₄	1	4	yes	—
80	C₂²×C₄×C₃₁	#41	1	4	yes	—
81	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
82	C₂×C₈×C₃₁	—	2	16	no	—
83	C₂×C₈×C₃₁	—	2	4	no	—
84	C₂×C₈×C₃₁	—	2	4	no	—
85	C₂×C₈×C₃₁	C₂×D₄	1	8	yes	complemented
86	C₄⋊C₄×C₃₁	—	2	8	no	—
87	C₄⋊C₄×C₃₁	—	4	4	no	—
88	C₄²×C₃₁	C₂²⋊C₄	1	8	yes	—
89	C₄²×C₃₁	C₂²×C₄	1	4	yes	—
90	C₄²×C₃₁	C₂×D₄	1	4	yes	—
91	C₂×#1961	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
92	C₂³×C₃₁	—	2	8	no	—
93	C₂³×C₃₁	—	4	4	no	—
94	C₂³×C₃₁	—	4	4	no	—
95	C₂³×C₃₁	—	4	16	no	—
96	C₂³×C₃₁	—	4	16	no	—
97	C₂³×C₃₁	—	4	32	no	—
98	C₂³×C₃₁	—	8	16	no	—
99	C₂³×C₃₁	—	8	16	no	—
100	C₂³×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
101	C₂³×C₃₁	#124	1	2	yes	—
102	C₂³×C₃₁	#127	1	2	yes	—
103	C₂³×C₃₁	AGL(1,ℤ/8)	1	8	yes	—
104	C₂³×C₃₁	C₂²×D₄	1	1	yes	characteristic
105	D₄×C₃₁	—	2	16	no	—
106	D₄×C₃₁	—	4	64	no	—
107	D₄×C₃₁	—	4	8	no	—
108	D₄×C₃₁	—	4	16	no	—
109	C₂×C₄×C₃₁	—	2	8	no	—
110	C₂×C₄×C₃₁	—	2	16	no	—
111	C₂×C₄×C₃₁	—	4	4	no	—
112	C₂×C₄×C₃₁	—	4	8	no	—
113	C₂×C₄×C₃₁	—	8	8	no	—
114	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	8	yes	—
115	C₂×C₄×C₃₁	C₂×C₂²⋊C₄	1	2	yes	—
116	C₂×C₄×C₃₁	C₄×D₄	1	8	yes	—
117	C₂×C₄×C₃₁	#121	1	8	yes	—
118	C₂×C₄×C₃₁	#124	1	2	yes	—
119	C₂×C₄×C₃₁	#127	1	2	yes	—
120	C₂×C₄×C₃₁	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₃₁	C₂²×D₄	1	2	yes	—
122	C₂×C₄×C₃₁	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₃₁	C₂×#41	1	2	yes	—
124	C₈×C₃₁	—	2	8	no	—
125	C₂²×#127	C₂×C₃₁	1	1	yes	characteristic
126	C₂²×#102	C₂×C₃₁	1	4	yes	complemented
127	C₂²×#97	C₂×C₃₁	1	1	yes	characteristic, complemented
128	C₂×#382	C₂×C₃₁	1	1	yes	characteristic, complemented
129	#1961	C₂×C₃₁	1	8	yes	direct product factor
130	C₂²×C₃₁	—	4	16	no	—
131	C₂²×C₃₁	—	4	8	no	—
132	C₂²×C₃₁	—	8	8	no	—
133	C₂²×C₃₁	—	8	8	no	—
134	C₂²×C₃₁	—	8	32	no	—
135	C₂²×C₃₁	#390	1	4	yes	—
136	C₂²×C₃₁	#418	1	8	yes	—
137	C₂²×C₃₁	#490	1	8	yes	—
138	C₂²×C₃₁	#496	1	8	yes	—
139	C₂²×C₃₁	C₂²×C₂²⋊C₄	1	1	yes	characteristic
140	C₂²×C₃₁	C₂×#124	1	1	yes	characteristic
141	C₂²×C₃₁	C₂×#127	1	1	yes	characteristic
142	C₂²×C₃₁	C₂×AGL(1,ℤ/8)	1	4	yes	—
143	C₄×C₃₁	—	2	8	no	—
144	C₄×C₃₁	—	8	4	no	—
145	C₄×C₃₁	#390	1	4	yes	—
146	C₄×C₃₁	C₂×C₄×D₄	1	2	yes	—
147	C₄×C₃₁	C₂×#121	1	2	yes	—
148	C₂³×D₄	—	2	2	no	—
149	C₂³×D₄	C₂²×C₃₁	1	2	yes	—
150	C₂³×C₈	C₂²×C₃₁	1	2	yes	—
151	C₂×#127	—	2	2	no	—
152	C₂×#127	—	2	4	no	—
153	C₂×#127	C₄×C₃₁	1	8	yes	complemented
154	C₂×#127	C₂²×C₃₁	1	4	yes	—
155	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
156	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
157	C₂×#102	—	2	16	no	—
158	C₂×#102	C₂²×C₃₁	1	16	yes	complemented
159	C₂×#97	—	2	4	no	—
160	C₂×#97	C₂²×C₃₁	1	4	yes	complemented
161	#382	—	2	2	no	—
162	#382	C₂²×C₃₁	1	4	yes	complemented
163	C₂×#95	—	2	2	no	—
164	C₂×C₃₁	—	8	8	no	—
165	C₂×C₃₁	#1961	1	8	yes	direct product factor
166	C₂×C₃₁	C₂×#390	1	1	yes	characteristic
167	C₂×C₃₁	C₂×#418	1	2	yes	—
168	C₂×C₃₁	C₂×#490	1	2	yes	—
169	C₂×C₃₁	C₂×#496	1	2	yes	—
170	C₂⁵	—	2	2	no	—
171	C₂²×D₄	—	2	16	no	—
172	C₂²×D₄	—	2	8	no	—
173	C₂²×D₄	—	2	2	no	—
174	C₂²×D₄	—	2	4	no	—
175	C₂²×D₄	—	2	8	no	—
176	C₂²×D₄	—	2	8	no	—
177	C₂²×D₄	—	4	16	no	—
178	C₂²×D₄	C₂×C₄×C₃₁	1	8	yes	—
179	C₂²×D₄	C₂³×C₃₁	1	4	yes	—
180	C₂³×C₄	—	2	2	no	—
181	C₂³×C₄	D₄×C₃₁	1	2	yes	—
182	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
183	C₂²×C₈	—	2	4	no	—
184	C₂²×C₈	—	2	4	no	—
185	C₂²×C₈	D₄×C₃₁	1	8	yes	complemented
186	C₂²×C₈	C₂³×C₃₁	1	4	yes	—
187	#127	—	2	16	no	—
188	#127	—	2	8	no	—
189	#127	C₂×C₄×C₃₁	1	16	yes	complemented
190	C₂×C₄⋊C₄	—	2	8	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	C₂³×C₃₁	1	4	yes	—
193	C₂×C₄²	C₂×C₄×C₃₁	1	4	yes	—
194	C₂×C₄²	D₄×C₃₁	1	2	yes	—
195	C₂×C₄²	D₄×C₃₁	1	4	yes	—
196	C₂×C₄²	C₂³×C₃₁	1	2	yes	—
197	#102	—	2	32	no	—
198	#97	—	2	8	no	—
199	#95	—	2	8	no	—
200	C₃₁	C₂×#1961	1	1	yes	characteristic, direct product factor, Sylow
201	C₂⁴	—	2	8	no	—
202	C₂⁴	—	2	4	no	—
203	C₂⁴	—	4	4	no	—
204	C₂⁴	—	4	4	no	—
205	C₂⁴	—	4	16	no	—
206	C₂⁴	C₂×D₄×C₃₁	1	1	yes	characteristic
207	C₂×D₄	—	2	8	no	—
208	C₂×D₄	—	2	32	no	—
209	C₂×D₄	—	2	32	no	—
210	C₂×D₄	—	2	8	no	—
211	C₂×D₄	—	2	4	no	—
212	C₂×D₄	—	2	8	no	—
213	C₂×D₄	—	4	64	no	—
214	C₂×D₄	—	4	32	no	—
215	C₂×D₄	—	4	16	no	—
216	C₂×D₄	—	4	4	no	—
217	C₂×D₄	—	4	4	no	—
218	C₂×D₄	C₂²×C₄×C₃₁	1	8	yes	—
219	C₂²×C₄	—	2	8	no	—
220	C₂²×C₄	—	2	4	no	—
221	C₂²×C₄	—	4	4	no	—
222	C₂²×C₄	—	4	8	no	—
223	C₂²×C₄	C₂²⋊C₄×C₃₁	1	8	yes	—
224	C₂²×C₄	C₂²×C₄×C₃₁	1	2	yes	—
225	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
226	C₂²×C₄	C₂×D₄×C₃₁	1	1	yes	characteristic
227	C₂²×C₄	C₂×D₄×C₃₁	1	2	yes	—
228	C₂²×C₄	C₂×D₄×C₃₁	1	4	yes	—
229	C₂²×C₄	#41×C₃₁	1	4	yes	—
230	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
231	C₂×C₈	—	2	16	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	C₂×D₄×C₃₁	1	8	yes	complemented
235	C₄⋊C₄	—	2	8	no	—
236	C₄⋊C₄	—	4	4	no	—
237	C₄²	C₂²⋊C₄×C₃₁	1	8	yes	—
238	C₄²	C₂²×C₄×C₃₁	1	4	yes	—
239	C₄²	C₂×D₄×C₃₁	1	4	yes	—
240	C₂³	—	2	8	no	—
241	C₂³	—	4	32	no	—
242	C₂³	—	4	4	no	—
243	C₂³	—	4	4	no	—
244	C₂³	—	4	16	no	—
245	C₂³	—	4	16	no	—
246	C₂³	—	8	16	no	—
247	C₂³	—	8	16	no	—
248	C₂³	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
249	C₂³	C₃₁×#124	1	2	yes	—
250	C₂³	C₃₁×#127	1	2	yes	—
251	C₂³	C₃₁×AGL(1,ℤ/8)	1	8	yes	—
252	C₂³	C₂²×D₄×C₃₁	1	1	yes	characteristic
253	D₄	—	2	16	no	—
254	D₄	—	4	64	no	—
255	D₄	—	4	8	no	—
256	D₄	—	4	16	no	—
257	C₂×C₄	—	2	16	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	8	no	—
261	C₂×C₄	—	8	8	no	—
262	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	8	yes	—
263	C₂×C₄	C₂×C₂²⋊C₄×C₃₁	1	2	yes	—
264	C₂×C₄	C₄×D₄×C₃₁	1	8	yes	—
265	C₂×C₄	C₃₁×#121	1	8	yes	—
266	C₂×C₄	C₃₁×#124	1	2	yes	—
267	C₂×C₄	C₃₁×#127	1	2	yes	—
268	C₂×C₄	C₂³×C₄×C₃₁	1	1	yes	characteristic
269	C₂×C₄	C₂²×D₄×C₃₁	1	2	yes	—
270	C₂×C₄	C₂²×D₄×C₃₁	1	1	yes	characteristic
271	C₂×C₄	C₂×#41×C₃₁	1	2	yes	—
272	C₈	—	2	8	no	—
273	C₂²	—	4	16	no	—
274	C₂²	—	4	8	no	—
275	C₂²	—	8	8	no	—
276	C₂²	—	8	8	no	—
277	C₂²	—	8	32	no	—
278	C₂²	C₃₁×#390	1	4	yes	—
279	C₂²	C₃₁×#418	1	8	yes	—
280	C₂²	C₃₁×#490	1	8	yes	—
281	C₂²	C₃₁×#496	1	8	yes	—
282	C₂²	C₂²×C₂²⋊C₄×C₃₁	1	1	yes	characteristic
283	C₂²	C₂×C₃₁×#124	1	1	yes	characteristic
284	C₂²	C₂×C₃₁×#127	1	1	yes	characteristic
285	C₂²	C₂×C₃₁×AGL(1,ℤ/8)	1	4	yes	—
286	C₄	—	2	8	no	—
287	C₄	—	8	4	no	—
288	C₄	C₃₁×#390	1	4	yes	—
289	C₄	C₂×C₄×D₄×C₃₁	1	2	yes	—
290	C₄	C₂×C₃₁×#121	1	2	yes	—
291	C₂	—	8	8	no	—
292	C₂	C₃₁×#1961	1	8	yes	direct product factor
293	C₂	C₂×C₃₁×#390	1	1	yes	characteristic
294	C₂	C₂×C₃₁×#418	1	2	yes	—
295	C₂	C₂×C₃₁×#490	1	2	yes	—
296	C₂	C₂×C₃₁×#496	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂×D₄
commutator	C₂×C₄	C₂³×C₄×C₃₁
Frattini, cosocle	C₂²×C₄	C₂⁴×C₃₁
prime radical	C₂²×C₃₁×#127	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ C₂²×C₄²×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃₁×#102	C₂	semidirect
3. C₂²×C₃₁×#97	C₂	semidirect
4. C₂×C₃₁×#382	C₂	semidirect
5. C₃₁×#1961	C₂	direct (N × H)
11. C₂×C₃₁×#127	C₄	semidirect
16. C₂×C₃₁×#102	C₂²	semidirect
18. C₂×C₃₁×#97	C₂²	semidirect
20. C₃₁×#382	C₂²	semidirect
37. C₂²×C₈×C₃₁	D₄	semidirect
41. C₃₁×#127	C₂×C₄	semidirect
85. C₂×C₈×C₃₁	C₂×D₄	semidirect
91. C₂×#1961	C₃₁	direct (N × H)
126. C₂²×#102	C₂×C₃₁	semidirect
127. C₂²×#97	C₂×C₃₁	semidirect
128. C₂×#382	C₂×C₃₁	semidirect
129. #1961	C₂×C₃₁	direct (N × H)
153. C₂×#127	C₄×C₃₁	semidirect
158. C₂×#102	C₂²×C₃₁	semidirect
160. C₂×#97	C₂²×C₃₁	semidirect
162. #382	C₂²×C₃₁	semidirect
165. C₂×C₃₁	#1961	direct (N × H)
185. C₂²×C₈	D₄×C₃₁	semidirect
189. #127	C₂×C₄×C₃₁	semidirect
200. C₃₁	C₂×#1961	direct (N × H)
234. C₂×C₈	C₂×D₄×C₃₁	semidirect
292. C₂	C₃₁×#1961	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₃₁×#127	2
C₂²×C₃₁×#102	2
C₂²×C₃₁×#97	2
C₂×C₃₁×#382	2
C₃₁×#1961	2
C₂×#1961	31

Subgroups — C2×C31×#1961 #21477957

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#1961 #21477957