Subgroups — C₂²×C₃₁×#541 #21519440

Order 7936.

7110 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₃₁×#541	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₃₁×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₃₁×#126	C₂	1	4	yes	maximal
4	C₂²×C₃₁×#125	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×C₃₁×#125	C₂	1	4	yes	maximal, complemented
6	C₂²×C₃₁×C₂²⋊Q₈	C₂	1	2	yes	maximal
7	C₂²×C₄×Q₈×C₃₁	C₂	1	2	yes	maximal, complemented
8	C₂²×C₃₁×#117	C₂	1	1	yes	maximal, characteristic
9	C₂³×Q₈×C₃₁	C₂²	1	2	yes	—
10	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
11	C₃₁×#541	C₂²	1	256	yes	direct product factor
12	C₂×C₃₁×C₄⋊Q₈	C₂²	1	24	yes	complemented
13	C₂×C₃₁×#126	C₂²	1	96	yes	—
14	C₂×C₃₁×#125	C₂²	1	24	yes	complemented
15	C₂×C₃₁×#125	C₂²	1	96	yes	complemented
16	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	48	yes	—
17	C₂×C₄×Q₈×C₃₁	C₂²	1	48	yes	complemented
18	C₂×C₃₁×#117	C₂²	1	24	yes	—
19	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
20	C₂²×C₄⋊C₄×C₃₁	C₂²	1	2	yes	—
21	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
22	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
23	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
24	C₂²×C₄⋊C₄×C₃₁	C₂²	1	4	yes	—
25	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	2	yes	—
26	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	4	yes	—
27	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
28	C₂²×C₄²×C₃₁	C₂²	1	1	yes	characteristic
29	C₂²×C₄²×C₃₁	C₂²	1	4	yes	—
30	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
31	C₂²×Q₈×C₃₁	—	2	16	no	—
32	C₂²×Q₈×C₃₁	C₂³	1	24	yes	—
33	C₂³×C₄×C₃₁	—	2	8	no	—
34	C₂³×C₄×C₃₁	C₂³	1	12	yes	—
35	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
36	C₂³×C₄×C₃₁	C₂³	1	2	yes	—
37	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
38	C₂³×C₄×C₃₁	C₂³	1	4	yes	—
39	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
40	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
41	C₃₁×C₄⋊Q₈	C₂³	1	64	yes	complemented
42	C₃₁×#126	C₂³	1	256	yes	—
43	C₃₁×#125	C₂³	1	64	yes	complemented
44	C₃₁×#125	C₂³	1	256	yes	complemented
45	C₃₁×C₂²⋊Q₈	C₂³	1	128	yes	—
46	C₄×Q₈×C₃₁	C₂³	1	128	yes	complemented
47	C₃₁×#117	C₂³	1	64	yes	—
48	C₂×C₄⋊C₄×C₃₁	C₂³	1	24	yes	—
49	C₂×C₄⋊C₄×C₃₁	C₂³	1	24	yes	—
50	C₂×C₄⋊C₄×C₃₁	C₂³	1	48	yes	—
51	C₂×C₄⋊C₄×C₃₁	C₂³	1	48	yes	—
52	C₂×C₄⋊C₄×C₃₁	C₂³	1	48	yes	—
53	C₂×C₄⋊C₄×C₃₁	C₂³	1	48	yes	—
54	C₂×C₂²⋊C₄×C₃₁	C₂³	1	24	yes	—
55	C₂×C₂²⋊C₄×C₃₁	C₂³	1	48	yes	—
56	C₂×C₄²×C₃₁	C₂³	1	12	yes	—
57	C₂×C₄²×C₃₁	C₂³	1	12	yes	—
58	C₂×C₄²×C₃₁	C₂³	1	48	yes	—
59	C₂⁴×C₃₁	—	2	4	no	—
60	C₂⁴×C₃₁	—	2	8	no	—
61	C₂⁴×C₃₁	C₂⁴	1	6	yes	—
62	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
63	C₂×Q₈×C₃₁	—	2	96	no	—
64	C₂×Q₈×C₃₁	C₂⁴	1	32	yes	—
65	C₂²×C₄×C₃₁	—	2	4	no	—
66	C₂²×C₄×C₃₁	—	2	16	no	—
67	C₂²×C₄×C₃₁	—	2	8	no	—
68	C₂²×C₄×C₃₁	—	2	8	no	—
69	C₂²×C₄×C₃₁	—	2	16	no	—
70	C₂²×C₄×C₃₁	—	2	48	no	—
71	C₂²×C₄×C₃₁	#41	1	8	yes	—
72	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
73	C₂²×C₄×C₃₁	C₂⁴	1	12	yes	—
74	C₂²×C₄×C₃₁	C₂⁴	1	24	yes	—
75	C₂²×C₄×C₃₁	C₂⁴	1	24	yes	—
76	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
77	C₂²×C₄×C₃₁	C₂⁴	1	16	yes	—
78	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
79	C₄⋊C₄×C₃₁	C₂⁴	1	32	yes	—
80	C₄⋊C₄×C₃₁	C₂⁴	1	32	yes	—
81	C₄⋊C₄×C₃₁	C₂⁴	1	64	yes	—
82	C₄⋊C₄×C₃₁	C₂⁴	1	64	yes	—
83	C₄⋊C₄×C₃₁	C₂⁴	1	64	yes	—
84	C₄⋊C₄×C₃₁	C₂⁴	1	64	yes	—
85	C₂²⋊C₄×C₃₁	C₂⁴	1	32	yes	—
86	C₂²⋊C₄×C₃₁	C₂⁴	1	64	yes	—
87	C₄²×C₃₁	C₂⁴	1	16	yes	—
88	C₄²×C₃₁	C₂⁴	1	16	yes	—
89	C₄²×C₃₁	C₂⁴	1	64	yes	—
90	C₂²×#541	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
91	C₂³×C₃₁	—	2	12	no	—
92	C₂³×C₃₁	—	2	24	no	—
93	C₂³×C₃₁	—	4	16	no	—
94	C₂³×C₃₁	C₂×#41	1	4	yes	—
95	C₂³×C₃₁	2₋¹⁺⁴	1	8	yes	—
96	C₂³×C₃₁	C₂⁵	1	4	yes	—
97	C₂³×C₃₁	C₂⁵	1	3	yes	—
98	Q₈×C₃₁	—	2	64	no	—
99	C₂×C₄×C₃₁	—	2	12	no	—
100	C₂×C₄×C₃₁	—	2	48	no	—
101	C₂×C₄×C₃₁	—	2	24	no	—
102	C₂×C₄×C₃₁	—	2	24	no	—
103	C₂×C₄×C₃₁	—	2	48	no	—
104	C₂×C₄×C₃₁	—	2	32	no	—
105	C₂×C₄×C₃₁	C₂×#41	1	24	yes	—
106	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
107	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
108	C₂×C₄×C₃₁	C₂⁵	1	16	yes	—
109	C₂×C₄×C₃₁	C₂⁵	1	16	yes	—
110	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
111	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
112	C₂×#541	C₂×C₃₁	1	48	yes	direct product factor
113	C₂²×C₄⋊Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
114	C₂²×#126	C₂×C₃₁	1	4	yes	—
115	C₂²×#125	C₂×C₃₁	1	4	yes	complemented
116	C₂²×#125	C₂×C₃₁	1	1	yes	characteristic, complemented
117	C₂²×C₂²⋊Q₈	C₂×C₃₁	1	2	yes	—
118	C₂²×C₄×Q₈	C₂×C₃₁	1	2	yes	complemented
119	C₂²×#117	C₂×C₃₁	1	1	yes	characteristic
120	C₂²×C₃₁	—	2	4	no	—
121	C₂²×C₃₁	—	2	8	no	—
122	C₂²×C₃₁	—	4	24	no	—
123	C₂²×C₃₁	#541	1	16	yes	direct product factor
124	C₂²×C₃₁	C₂²×#41	1	6	yes	—
125	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	12	yes	—
126	C₂²×C₃₁	C₂⁶	1	1	yes	characteristic
127	C₄×C₃₁	—	2	4	no	—
128	C₄×C₃₁	—	2	16	no	—
129	C₄×C₃₁	—	2	8	no	—
130	C₄×C₃₁	—	2	8	no	—
131	C₄×C₃₁	—	2	16	no	—
132	C₄×C₃₁	C₂²×#41	1	8	yes	—
133	C₂³×Q₈	C₂²×C₃₁	1	2	yes	—
134	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
135	#541	C₂²×C₃₁	1	256	yes	direct product factor
136	C₂×C₄⋊Q₈	C₂²×C₃₁	1	24	yes	complemented
137	C₂×#126	C₂²×C₃₁	1	96	yes	—
138	C₂×#125	C₂²×C₃₁	1	24	yes	complemented
139	C₂×#125	C₂²×C₃₁	1	96	yes	complemented
140	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	48	yes	—
141	C₂×C₄×Q₈	C₂²×C₃₁	1	48	yes	complemented
142	C₂×#117	C₂²×C₃₁	1	24	yes	—
143	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
144	C₂²×C₄⋊C₄	C₂²×C₃₁	1	2	yes	—
145	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
146	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
147	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
148	C₂²×C₄⋊C₄	C₂²×C₃₁	1	4	yes	—
149	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	2	yes	—
150	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	4	yes	—
151	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
152	C₂²×C₄²	C₂²×C₃₁	1	1	yes	characteristic
153	C₂²×C₄²	C₂²×C₃₁	1	4	yes	—
154	C₂×C₃₁	—	4	4	no	—
155	C₂×C₃₁	C₂×#541	1	12	yes	direct product factor
156	C₂×C₃₁	C₂³×#41	1	1	yes	characteristic
157	C₂×C₃₁	C₂²×2₋¹⁺⁴	1	2	yes	—
158	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
159	C₂²×Q₈	—	2	16	no	—
160	C₂²×Q₈	C₂³×C₃₁	1	24	yes	—
161	C₂³×C₄	—	2	8	no	—
162	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
163	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
164	C₂³×C₄	C₂³×C₃₁	1	4	yes	—
165	C₂³×C₄	C₂³×C₃₁	1	2	yes	—
166	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
167	C₂³×C₄	C₂³×C₃₁	1	12	yes	—
168	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
169	C₄⋊Q₈	C₂³×C₃₁	1	64	yes	complemented
170	#126	C₂³×C₃₁	1	256	yes	—
171	#125	C₂³×C₃₁	1	64	yes	complemented
172	#125	C₂³×C₃₁	1	256	yes	complemented
173	C₂²⋊Q₈	C₂³×C₃₁	1	128	yes	—
174	C₄×Q₈	C₂³×C₃₁	1	128	yes	complemented
175	#117	C₂³×C₃₁	1	64	yes	—
176	C₂×C₄⋊C₄	C₂³×C₃₁	1	24	yes	—
177	C₂×C₄⋊C₄	C₂³×C₃₁	1	24	yes	—
178	C₂×C₄⋊C₄	C₂³×C₃₁	1	48	yes	—
179	C₂×C₄⋊C₄	C₂³×C₃₁	1	48	yes	—
180	C₂×C₄⋊C₄	C₂³×C₃₁	1	48	yes	—
181	C₂×C₄⋊C₄	C₂³×C₃₁	1	48	yes	—
182	C₂×C₂²⋊C₄	C₂³×C₃₁	1	24	yes	—
183	C₂×C₂²⋊C₄	C₂³×C₃₁	1	48	yes	—
184	C₂×C₄²	C₂³×C₃₁	1	12	yes	—
185	C₂×C₄²	C₂³×C₃₁	1	12	yes	—
186	C₂×C₄²	C₂³×C₃₁	1	48	yes	—
187	C₃₁	C₂²×#541	1	1	yes	characteristic, direct product factor, Sylow
188	C₂⁴	—	2	4	no	—
189	C₂⁴	—	2	8	no	—
190	C₂⁴	C₂⁴×C₃₁	1	6	yes	—
191	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
192	C₂×Q₈	—	2	96	no	—
193	C₂×Q₈	C₂⁴×C₃₁	1	32	yes	—
194	C₂²×C₄	—	2	4	no	—
195	C₂²×C₄	—	2	16	no	—
196	C₂²×C₄	—	2	8	no	—
197	C₂²×C₄	—	2	16	no	—
198	C₂²×C₄	—	2	8	no	—
199	C₂²×C₄	—	2	48	no	—
200	C₂²×C₄	#41×C₃₁	1	8	yes	—
201	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
202	C₂²×C₄	C₂⁴×C₃₁	1	12	yes	—
203	C₂²×C₄	C₂⁴×C₃₁	1	24	yes	—
204	C₂²×C₄	C₂⁴×C₃₁	1	24	yes	—
205	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
206	C₂²×C₄	C₂⁴×C₃₁	1	16	yes	—
207	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
208	C₄⋊C₄	C₂⁴×C₃₁	1	32	yes	—
209	C₄⋊C₄	C₂⁴×C₃₁	1	32	yes	—
210	C₄⋊C₄	C₂⁴×C₃₁	1	64	yes	—
211	C₄⋊C₄	C₂⁴×C₃₁	1	64	yes	—
212	C₄⋊C₄	C₂⁴×C₃₁	1	64	yes	—
213	C₄⋊C₄	C₂⁴×C₃₁	1	64	yes	—
214	C₂²⋊C₄	C₂⁴×C₃₁	1	32	yes	—
215	C₂²⋊C₄	C₂⁴×C₃₁	1	64	yes	—
216	C₄²	C₂⁴×C₃₁	1	16	yes	—
217	C₄²	C₂⁴×C₃₁	1	16	yes	—
218	C₄²	C₂⁴×C₃₁	1	64	yes	—
219	C₂³	—	2	12	no	—
220	C₂³	—	2	24	no	—
221	C₂³	—	4	16	no	—
222	C₂³	C₂×#41×C₃₁	1	4	yes	—
223	C₂³	C₃₁×2₋¹⁺⁴	1	8	yes	—
224	C₂³	C₂⁵×C₃₁	1	4	yes	—
225	C₂³	C₂⁵×C₃₁	1	3	yes	—
226	Q₈	—	2	64	no	—
227	C₂×C₄	—	2	12	no	—
228	C₂×C₄	—	2	48	no	—
229	C₂×C₄	—	2	24	no	—
230	C₂×C₄	—	2	24	no	—
231	C₂×C₄	—	2	48	no	—
232	C₂×C₄	—	2	32	no	—
233	C₂×C₄	C₂×#41×C₃₁	1	24	yes	—
234	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
235	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
236	C₂×C₄	C₂⁵×C₃₁	1	16	yes	—
237	C₂×C₄	C₂⁵×C₃₁	1	16	yes	—
238	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
239	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
240	C₂²	—	2	4	no	—
241	C₂²	—	2	8	no	—
242	C₂²	—	4	24	no	—
243	C₂²	C₃₁×#541	1	16	yes	direct product factor
244	C₂²	C₂²×#41×C₃₁	1	6	yes	—
245	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	12	yes	—
246	C₂²	C₂⁶×C₃₁	1	1	yes	characteristic
247	C₄	—	2	4	no	—
248	C₄	—	2	16	no	—
249	C₄	—	2	8	no	—
250	C₄	—	2	8	no	—
251	C₄	—	2	16	no	—
252	C₄	C₂²×#41×C₃₁	1	8	yes	—
253	C₂	—	4	4	no	—
254	C₂	C₂×C₃₁×#541	1	12	yes	direct product factor
255	C₂	C₂³×#41×C₃₁	1	1	yes	characteristic
256	C₂	C₂²×C₃₁×2₋¹⁺⁴	1	2	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₃₁
prime radical	C₂⁵×C₃₁	C₂³
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂⁴×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₃₁×#541	C₂	direct (N × H)
2. C₂²×C₃₁×C₄⋊Q₈	C₂	semidirect
4. C₂²×C₃₁×#125	C₂	semidirect
5. C₂²×C₃₁×#125	C₂	semidirect
7. C₂²×C₄×Q₈×C₃₁	C₂	semidirect
11. C₃₁×#541	C₂²	direct (N × H)
12. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
14. C₂×C₃₁×#125	C₂²	semidirect
15. C₂×C₃₁×#125	C₂²	semidirect
17. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
41. C₃₁×C₄⋊Q₈	C₂³	semidirect
43. C₃₁×#125	C₂³	semidirect
44. C₃₁×#125	C₂³	semidirect
46. C₄×Q₈×C₃₁	C₂³	semidirect
90. C₂²×#541	C₃₁	direct (N × H)
112. C₂×#541	C₂×C₃₁	direct (N × H)
113. C₂²×C₄⋊Q₈	C₂×C₃₁	semidirect
115. C₂²×#125	C₂×C₃₁	semidirect
116. C₂²×#125	C₂×C₃₁	semidirect
118. C₂²×C₄×Q₈	C₂×C₃₁	semidirect
123. C₂²×C₃₁	#541	direct (N × H)
135. #541	C₂²×C₃₁	direct (N × H)
136. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
138. C₂×#125	C₂²×C₃₁	semidirect
139. C₂×#125	C₂²×C₃₁	semidirect
141. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
155. C₂×C₃₁	C₂×#541	direct (N × H)
169. C₄⋊Q₈	C₂³×C₃₁	semidirect
171. #125	C₂³×C₃₁	semidirect
172. #125	C₂³×C₃₁	semidirect
174. C₄×Q₈	C₂³×C₃₁	semidirect
187. C₃₁	C₂²×#541	direct (N × H)
243. C₂²	C₃₁×#541	direct (N × H)
254. C₂	C₂×C₃₁×#541	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×#541	2
C₂²×C₃₁×C₄⋊Q₈	2
C₂²×C₃₁×#126	2
C₂²×C₃₁×#125	2
C₂²×C₃₁×C₂²⋊Q₈	2
C₂²×C₄×Q₈×C₃₁	2
C₂²×C₃₁×#117	2
C₂²×#541	31

Subgroups — C22×C31×#541 #21519440

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×C₃₁×#541 #21519440