Subgroups — C₂×C₃₁×#2401 #21491308

Order 7936.

2294 conjugacy classes of subgroups.

The subgroup lattice has 284 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₃₁×#2401	C₂	1	16	yes	maximal, direct product factor
2	C₂×C₃₁×#401	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₃₁×#400	C₂	1	3	yes	maximal, complemented
4	C₂×C₃₁×#397	C₂	1	3	yes	maximal
5	C₂×C₃₁×#396	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×C₄⋊C₄×C₃₁	C₂	1	3	yes	maximal, complemented
7	C₂×C₄×C₂²⋊C₄×C₃₁	C₂	1	3	yes	maximal
8	C₂×C₃₁×#376	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
10	C₂²×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
11	C₂²×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
12	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	3	yes	—
13	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	3	yes	—
14	C₂²×C₄²×C₃₁	C₂²	1	3	yes	—
15	C₂²×C₄²×C₃₁	C₂²	1	3	yes	—
16	C₃₁×#401	C₂²	1	8	yes	complemented
17	C₃₁×#400	C₂²	1	24	yes	complemented
18	C₃₁×#397	C₂²	1	24	yes	—
19	C₃₁×#396	C₂²	1	8	yes	—
20	C₄×C₄⋊C₄×C₃₁	C₂²	1	24	yes	complemented
21	C₄×C₂²⋊C₄×C₃₁	C₂²	1	24	yes	—
22	C₃₁×#376	C₂²	1	8	yes	complemented
23	C₂×C₃₁×#95	C₂²	1	1	yes	characteristic
24	C₂×C₃₁×#95	C₂²	1	3	yes	—
25	C₂×C₃₁×#95	C₂²	1	3	yes	—
26	C₂×C₃₁×#95	C₂²	1	3	yes	—
27	C₂×C₃₁×#95	C₂²	1	3	yes	—
28	C₂×C₃₁×#95	C₂²	1	3	yes	—
29	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
30	C₂³×C₄×C₃₁	—	2	12	no	—
31	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
32	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
33	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
34	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
35	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
36	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
37	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
38	C₂×C₄⋊C₄×C₃₁	—	2	12	no	—
39	C₂×C₄⋊C₄×C₃₁	—	2	12	no	—
40	C₂×C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
41	C₂×C₄⋊C₄×C₃₁	C₂³	1	12	yes	—
42	C₂×C₂²⋊C₄×C₃₁	—	2	12	no	—
43	C₂×C₂²⋊C₄×C₃₁	—	2	12	no	—
44	C₂×C₂²⋊C₄×C₃₁	C₂³	1	12	yes	—
45	C₂×C₂²⋊C₄×C₃₁	C₂³	1	12	yes	—
46	C₂×C₄²×C₃₁	—	2	12	no	—
47	C₂×C₄²×C₃₁	—	2	12	no	—
48	C₂×C₄²×C₃₁	C₂³	1	12	yes	—
49	C₂×C₄²×C₃₁	C₂³	1	12	yes	—
50	C₃₁×#95	C₂³	1	4	yes	—
51	C₃₁×#95	C₂³	1	12	yes	—
52	C₃₁×#95	C₂³	1	12	yes	—
53	C₃₁×#95	C₂³	1	12	yes	—
54	C₃₁×#95	C₂³	1	12	yes	—
55	C₃₁×#95	C₂³	1	12	yes	—
56	C₂⁴×C₃₁	—	2	6	no	—
57	C₂⁴×C₃₁	—	2	6	no	—
58	C₂⁴×C₃₁	#41	1	4	yes	—
59	C₂⁴×C₃₁	C₂⁴	1	2	yes	—
60	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
61	C₂²×C₄×C₃₁	—	2	6	no	—
62	C₂²×C₄×C₃₁	—	2	6	no	—
63	C₂²×C₄×C₃₁	—	2	6	no	—
64	C₂²×C₄×C₃₁	—	2	6	no	—
65	C₂²×C₄×C₃₁	—	2	6	no	—
66	C₂²×C₄×C₃₁	—	2	6	no	—
67	C₂²×C₄×C₃₁	—	2	6	no	—
68	C₂²×C₄×C₃₁	—	2	6	no	—
69	C₂²×C₄×C₃₁	—	2	6	no	—
70	C₂²×C₄×C₃₁	—	2	6	no	—
71	C₂²×C₄×C₃₁	—	2	6	no	—
72	C₂²×C₄×C₃₁	—	2	6	no	—
73	C₂²×C₄×C₃₁	—	2	24	no	—
74	C₂²×C₄×C₃₁	—	4	16	no	—
75	C₂²×C₄×C₃₁	#41	1	12	yes	—
76	C₂²×C₄×C₃₁	#41	1	12	yes	—
77	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
78	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
79	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
80	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
81	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
82	C₂²×C₄×C₃₁	C₂⁴	1	2	yes	—
83	C₄⋊C₄×C₃₁	—	2	24	no	—
84	C₄⋊C₄×C₃₁	—	2	24	no	—
85	C₂²⋊C₄×C₃₁	—	2	24	no	—
86	C₂²⋊C₄×C₃₁	—	2	24	no	—
87	C₄²×C₃₁	—	2	24	no	—
88	C₄²×C₃₁	—	2	24	no	—
89	C₂×#2401	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
90	C₂³×C₃₁	—	2	24	no	—
91	C₂³×C₃₁	—	2	6	no	—
92	C₂³×C₃₁	—	2	6	no	—
93	C₂³×C₃₁	—	4	4	no	—
94	C₂³×C₃₁	—	4	12	no	—
95	C₂³×C₃₁	C₂×#41	1	4	yes	—
96	C₂³×C₃₁	C₂×#41	1	6	yes	—
97	C₂³×C₃₁	C₂×#41	1	2	yes	—
98	C₂³×C₃₁	C₂×#41	1	6	yes	—
99	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
100	C₂×C₄×C₃₁	—	2	24	no	—
101	C₂×C₄×C₃₁	—	2	6	no	—
102	C₂×C₄×C₃₁	—	2	6	no	—
103	C₂×C₄×C₃₁	—	2	6	no	—
104	C₂×C₄×C₃₁	—	2	6	no	—
105	C₂×C₄×C₃₁	—	2	6	no	—
106	C₂×C₄×C₃₁	—	2	6	no	—
107	C₂×C₄×C₃₁	—	2	6	no	—
108	C₂×C₄×C₃₁	—	2	6	no	—
109	C₂×C₄×C₃₁	—	2	6	no	—
110	C₂×C₄×C₃₁	—	2	6	no	—
111	C₂×C₄×C₃₁	—	2	6	no	—
112	C₂×C₄×C₃₁	—	2	6	no	—
113	C₂×C₄×C₃₁	—	4	4	no	—
114	C₂×C₄×C₃₁	—	4	12	no	—
115	C₂×C₄×C₃₁	—	4	12	no	—
116	C₂×C₄×C₃₁	—	4	12	no	—
117	C₂×C₄×C₃₁	—	4	16	no	—
118	C₂×C₄×C₃₁	#126	1	16	yes	—
119	C₂×C₄×C₃₁	C₂×#41	1	12	yes	—
120	C₂×C₄×C₃₁	C₂×#41	1	12	yes	—
121	#2401	C₂×C₃₁	1	16	yes	direct product factor
122	C₂×#401	C₂×C₃₁	1	1	yes	characteristic, complemented
123	C₂×#400	C₂×C₃₁	1	3	yes	complemented
124	C₂×#397	C₂×C₃₁	1	3	yes	—
125	C₂×#396	C₂×C₃₁	1	1	yes	characteristic
126	C₂×C₄×C₄⋊C₄	C₂×C₃₁	1	3	yes	complemented
127	C₂×C₄×C₂²⋊C₄	C₂×C₃₁	1	3	yes	—
128	C₂×#376	C₂×C₃₁	1	1	yes	characteristic, complemented
129	C₂²×C₃₁	—	2	12	no	—
130	C₂²×C₃₁	—	4	16	no	—
131	C₂²×C₃₁	—	4	2	no	—
132	C₂²×C₃₁	—	4	6	no	—
133	C₂²×C₃₁	C₂×#126	1	4	yes	—
134	C₂²×C₃₁	#529	1	12	yes	—
135	C₂²×C₃₁	#529	1	12	yes	—
136	C₂²×C₃₁	C₂²×#41	1	3	yes	—
137	C₂²×C₃₁	C₂²×#41	1	1	yes	characteristic
138	C₂²×C₃₁	C₂²×#41	1	3	yes	—
139	C₄×C₃₁	—	2	12	no	—
140	C₄×C₃₁	—	4	2	no	—
141	C₄×C₃₁	—	4	6	no	—
142	C₄×C₃₁	—	4	6	no	—
143	C₄×C₃₁	—	4	6	no	—
144	C₄×C₃₁	C₂×#126	1	8	yes	—
145	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
146	C₂²×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
147	C₂²×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
148	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	3	yes	—
149	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	3	yes	—
150	C₂²×C₄²	C₂²×C₃₁	1	3	yes	—
151	C₂²×C₄²	C₂²×C₃₁	1	3	yes	—
152	#401	C₂²×C₃₁	1	8	yes	complemented
153	#400	C₂²×C₃₁	1	24	yes	complemented
154	#397	C₂²×C₃₁	1	24	yes	—
155	#396	C₂²×C₃₁	1	8	yes	—
156	C₄×C₄⋊C₄	C₂²×C₃₁	1	24	yes	complemented
157	C₄×C₂²⋊C₄	C₂²×C₃₁	1	24	yes	—
158	#376	C₂²×C₃₁	1	8	yes	complemented
159	C₂×#95	C₂²×C₃₁	1	1	yes	characteristic
160	C₂×#95	C₂²×C₃₁	1	3	yes	—
161	C₂×#95	C₂²×C₃₁	1	3	yes	—
162	C₂×#95	C₂²×C₃₁	1	3	yes	—
163	C₂×#95	C₂²×C₃₁	1	3	yes	—
164	C₂×#95	C₂²×C₃₁	1	3	yes	—
165	C₂×C₃₁	—	4	4	no	—
166	C₂×C₃₁	#2401	1	8	yes	direct product factor
167	C₂×C₃₁	C₂²×#126	1	1	yes	characteristic
168	C₂×C₃₁	C₂×#529	1	3	yes	—
169	C₂×C₃₁	C₂×#529	1	3	yes	—
170	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
171	C₂³×C₄	—	2	12	no	—
172	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
173	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
174	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
175	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
176	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
177	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
178	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
179	C₂×C₄⋊C₄	—	2	12	no	—
180	C₂×C₄⋊C₄	—	2	12	no	—
181	C₂×C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
182	C₂×C₄⋊C₄	C₂³×C₃₁	1	12	yes	—
183	C₂×C₂²⋊C₄	—	2	12	no	—
184	C₂×C₂²⋊C₄	—	2	12	no	—
185	C₂×C₂²⋊C₄	C₂³×C₃₁	1	12	yes	—
186	C₂×C₂²⋊C₄	C₂³×C₃₁	1	12	yes	—
187	C₂×C₄²	—	2	12	no	—
188	C₂×C₄²	—	2	12	no	—
189	C₂×C₄²	C₂³×C₃₁	1	12	yes	—
190	C₂×C₄²	C₂³×C₃₁	1	12	yes	—
191	#95	C₂³×C₃₁	1	4	yes	—
192	#95	C₂³×C₃₁	1	12	yes	—
193	#95	C₂³×C₃₁	1	12	yes	—
194	#95	C₂³×C₃₁	1	12	yes	—
195	#95	C₂³×C₃₁	1	12	yes	—
196	#95	C₂³×C₃₁	1	12	yes	—
197	C₃₁	C₂×#2401	1	1	yes	characteristic, direct product factor, Sylow
198	C₂⁴	—	2	6	no	—
199	C₂⁴	—	2	6	no	—
200	C₂⁴	#41×C₃₁	1	4	yes	—
201	C₂⁴	C₂⁴×C₃₁	1	2	yes	—
202	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
203	C₂²×C₄	—	2	6	no	—
204	C₂²×C₄	—	2	6	no	—
205	C₂²×C₄	—	2	6	no	—
206	C₂²×C₄	—	2	6	no	—
207	C₂²×C₄	—	2	6	no	—
208	C₂²×C₄	—	2	6	no	—
209	C₂²×C₄	—	2	6	no	—
210	C₂²×C₄	—	2	6	no	—
211	C₂²×C₄	—	2	6	no	—
212	C₂²×C₄	—	2	6	no	—
213	C₂²×C₄	—	2	6	no	—
214	C₂²×C₄	—	2	6	no	—
215	C₂²×C₄	—	2	24	no	—
216	C₂²×C₄	—	4	16	no	—
217	C₂²×C₄	#41×C₃₁	1	12	yes	—
218	C₂²×C₄	#41×C₃₁	1	12	yes	—
219	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
220	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
221	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
222	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
223	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
224	C₂²×C₄	C₂⁴×C₃₁	1	2	yes	—
225	C₄⋊C₄	—	2	24	no	—
226	C₄⋊C₄	—	2	24	no	—
227	C₂²⋊C₄	—	2	24	no	—
228	C₂²⋊C₄	—	2	24	no	—
229	C₄²	—	2	24	no	—
230	C₄²	—	2	24	no	—
231	C₂³	—	2	24	no	—
232	C₂³	—	2	6	no	—
233	C₂³	—	2	6	no	—
234	C₂³	—	4	4	no	—
235	C₂³	—	4	12	no	—
236	C₂³	C₂×#41×C₃₁	1	4	yes	—
237	C₂³	C₂×#41×C₃₁	1	6	yes	—
238	C₂³	C₂×#41×C₃₁	1	2	yes	—
239	C₂³	C₂×#41×C₃₁	1	6	yes	—
240	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
241	C₂×C₄	—	2	24	no	—
242	C₂×C₄	—	2	6	no	—
243	C₂×C₄	—	2	6	no	—
244	C₂×C₄	—	2	6	no	—
245	C₂×C₄	—	2	6	no	—
246	C₂×C₄	—	2	6	no	—
247	C₂×C₄	—	2	6	no	—
248	C₂×C₄	—	2	6	no	—
249	C₂×C₄	—	2	6	no	—
250	C₂×C₄	—	2	6	no	—
251	C₂×C₄	—	2	6	no	—
252	C₂×C₄	—	2	6	no	—
253	C₂×C₄	—	2	6	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	12	no	—
256	C₂×C₄	—	4	12	no	—
257	C₂×C₄	—	4	12	no	—
258	C₂×C₄	—	4	16	no	—
259	C₂×C₄	C₃₁×#126	1	16	yes	—
260	C₂×C₄	C₂×#41×C₃₁	1	12	yes	—
261	C₂×C₄	C₂×#41×C₃₁	1	12	yes	—
262	C₂²	—	2	12	no	—
263	C₂²	—	4	16	no	—
264	C₂²	—	4	2	no	—
265	C₂²	—	4	6	no	—
266	C₂²	C₂×C₃₁×#126	1	4	yes	—
267	C₂²	C₃₁×#529	1	12	yes	—
268	C₂²	C₃₁×#529	1	12	yes	—
269	C₂²	C₂²×#41×C₃₁	1	3	yes	—
270	C₂²	C₂²×#41×C₃₁	1	1	yes	characteristic
271	C₂²	C₂²×#41×C₃₁	1	3	yes	—
272	C₄	—	2	12	no	—
273	C₄	—	4	2	no	—
274	C₄	—	4	6	no	—
275	C₄	—	4	6	no	—
276	C₄	—	4	6	no	—
277	C₄	C₂×C₃₁×#126	1	8	yes	—
278	C₂	—	4	4	no	—
279	C₂	C₃₁×#2401	1	8	yes	direct product factor
280	C₂	C₂²×C₃₁×#126	1	1	yes	characteristic
281	C₂	C₂×C₃₁×#529	1	3	yes	—
282	C₂	C₂×C₃₁×#529	1	3	yes	—

	subgroup	G/H
center	C₂³×C₄×C₃₁	C₂³
commutator, Frattini, cosocle	C₂³	C₂⁵×C₃₁
socle	C₂⁴×C₃₁	C₂⁴
prime radical	C₂⁵×C₃₁	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₃₁×#2401	C₂	direct (N × H)
2. C₂×C₃₁×#401	C₂	semidirect
3. C₂×C₃₁×#400	C₂	semidirect
6. C₂×C₄×C₄⋊C₄×C₃₁	C₂	semidirect
8. C₂×C₃₁×#376	C₂	semidirect
16. C₃₁×#401	C₂²	semidirect
17. C₃₁×#400	C₂²	semidirect
20. C₄×C₄⋊C₄×C₃₁	C₂²	semidirect
22. C₃₁×#376	C₂²	semidirect
89. C₂×#2401	C₃₁	direct (N × H)
121. #2401	C₂×C₃₁	direct (N × H)
122. C₂×#401	C₂×C₃₁	semidirect
123. C₂×#400	C₂×C₃₁	semidirect
126. C₂×C₄×C₄⋊C₄	C₂×C₃₁	semidirect
128. C₂×#376	C₂×C₃₁	semidirect
152. #401	C₂²×C₃₁	semidirect
153. #400	C₂²×C₃₁	semidirect
156. C₄×C₄⋊C₄	C₂²×C₃₁	semidirect
158. #376	C₂²×C₃₁	semidirect
166. C₂×C₃₁	#2401	direct (N × H)
197. C₃₁	C₂×#2401	direct (N × H)
279. C₂	C₃₁×#2401	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×#2401	2
C₂×C₃₁×#401	2
C₂×C₃₁×#400	2
C₂×C₃₁×#397	2
C₂×C₃₁×#396	2
C₂×C₄×C₄⋊C₄×C₃₁	2
C₂×C₄×C₂²⋊C₄×C₃₁	2
C₂×C₃₁×#376	2
C₂×#2401	31

Subgroups — C2×C31×#2401 #21491308

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₃₁×#2401 #21491308