Subgroups — C₂²×D₄×Q₈×C₃₁ #21519455

Order 7936.

11798 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁴×Q₈×C₃₁	C₂	1	2	yes	maximal, complemented
2	C₂×D₄×Q₈×C₃₁	C₂	1	48	yes	maximal, direct product factor
3	C₂²×C₃₁×C₄⋊Q₈	C₂	1	3	yes	maximal, complemented
4	C₂²×C₃₁×C₂²⋊Q₈	C₂	1	6	yes	maximal, complemented
5	C₂²×C₄×Q₈×C₃₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×C₄×D₄×C₃₁	C₂	1	3	yes	maximal
7	C₂³×Q₈×C₃₁	—	2	32	no	—
8	C₂³×Q₈×C₃₁	C₂²	1	6	yes	—
9	C₂³×Q₈×C₃₁	C₂²	1	1	yes	characteristic
10	C₂³×Q₈×C₃₁	C₂²	1	48	yes	complemented
11	C₂³×D₄×C₃₁	C₂²	1	1	yes	characteristic
12	C₂⁴×C₄×C₃₁	C₂²	1	6	yes	—
13	D₄×Q₈×C₃₁	C₂²	1	256	yes	direct product factor
14	C₂×C₃₁×C₄⋊Q₈	C₂²	1	72	yes	complemented
15	C₂×C₃₁×C₂²⋊Q₈	C₂²	1	144	yes	complemented
16	C₂×C₄×Q₈×C₃₁	C₂²	1	24	yes	complemented
17	C₂×C₄×D₄×C₃₁	C₂²	1	72	yes	—
18	C₂²×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
19	C₂²×C₄⋊C₄×C₃₁	C₂²	1	3	yes	—
20	C₂²×C₄⋊C₄×C₃₁	C₂²	1	6	yes	—
21	C₂²×C₂²⋊C₄×C₃₁	C₂²	1	6	yes	—
22	C₂²×C₄²×C₃₁	C₂²	1	3	yes	—
23	C₂⁵×C₃₁	C₂³	1	2	yes	—
24	C₂²×Q₈×C₃₁	—	2	384	no	—
25	C₂²×Q₈×C₃₁	—	2	48	no	—
26	C₂²×Q₈×C₃₁	D₄	1	16	yes	direct product factor
27	C₂²×Q₈×C₃₁	C₂³	1	72	yes	—
28	C₂²×Q₈×C₃₁	C₂³	1	12	yes	—
29	C₂²×Q₈×C₃₁	C₂³	1	128	yes	complemented
30	C₂²×D₄×C₃₁	Q₈	1	16	yes	direct product factor
31	C₂²×D₄×C₃₁	C₂³	1	12	yes	—
32	C₂³×C₄×C₃₁	—	2	48	no	—
33	C₂³×C₄×C₃₁	C₂³	1	72	yes	—
34	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
35	C₂³×C₄×C₃₁	C₂³	1	6	yes	—
36	C₂³×C₄×C₃₁	C₂³	1	3	yes	—
37	C₂³×C₄×C₃₁	C₂³	1	1	yes	characteristic
38	C₃₁×C₄⋊Q₈	C₂³	1	192	yes	complemented
39	C₃₁×C₂²⋊Q₈	C₂³	1	384	yes	complemented
40	C₄×Q₈×C₃₁	C₂³	1	64	yes	complemented
41	C₄×D₄×C₃₁	C₂³	1	192	yes	—
42	C₂×C₄⋊C₄×C₃₁	C₂³	1	36	yes	—
43	C₂×C₄⋊C₄×C₃₁	C₂³	1	36	yes	—
44	C₂×C₄⋊C₄×C₃₁	C₂³	1	72	yes	—
45	C₂×C₂²⋊C₄×C₃₁	C₂³	1	72	yes	—
46	C₂×C₄²×C₃₁	C₂³	1	36	yes	—
47	C₂⁴×C₃₁	—	2	8	no	—
48	C₂⁴×C₃₁	—	2	8	no	—
49	C₂⁴×C₃₁	C₂×Q₈	1	16	yes	complemented
50	C₂⁴×C₃₁	C₂⁴	1	12	yes	—
51	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
52	C₂×Q₈×C₃₁	—	2	512	no	—
53	C₂×Q₈×C₃₁	—	2	288	no	—
54	C₂×Q₈×C₃₁	C₂×D₄	1	96	yes	direct product factor
55	C₂×Q₈×C₃₁	C₂⁴	1	96	yes	—
56	C₂×Q₈×C₃₁	C₂⁴	1	16	yes	—
57	C₂×D₄×C₃₁	C₂×Q₈	1	96	yes	direct product factor
58	C₂×D₄×C₃₁	C₂⁴	1	16	yes	—
59	C₂²×C₄×C₃₁	—	2	24	no	—
60	C₂²×C₄×C₃₁	—	2	12	no	—
61	C₂²×C₄×C₃₁	—	2	288	no	—
62	C₂²×C₄×C₃₁	C₂×D₄	1	24	yes	—
63	C₂²×C₄×C₃₁	C₂×Q₈	1	8	yes	complemented
64	C₂²×C₄×C₃₁	C₂⁴	1	18	yes	—
65	C₂²×C₄×C₃₁	C₂⁴	1	36	yes	—
66	C₂²×C₄×C₃₁	C₂⁴	1	96	yes	—
67	C₂²×C₄×C₃₁	C₂⁴	1	18	yes	—
68	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
69	C₄⋊C₄×C₃₁	C₂⁴	1	48	yes	—
70	C₄⋊C₄×C₃₁	C₂⁴	1	48	yes	—
71	C₄⋊C₄×C₃₁	C₂⁴	1	96	yes	—
72	C₂²⋊C₄×C₃₁	C₂⁴	1	96	yes	—
73	C₄²×C₃₁	C₂⁴	1	48	yes	—
74	C₂²×D₄×Q₈	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₃₁	—	2	64	no	—
76	C₂³×C₃₁	—	2	24	no	—
77	C₂³×C₃₁	—	2	24	no	—
78	C₂³×C₃₁	C₂²×D₄	1	4	yes	—
79	C₂³×C₃₁	C₂²×Q₈	1	4	yes	—
80	C₂³×C₃₁	C₂²×Q₈	1	48	yes	complemented
81	C₂³×C₃₁	2₋¹⁺⁴	1	4	yes	—
82	C₂³×C₃₁	C₂⁵	1	8	yes	—
83	C₂³×C₃₁	C₂⁵	1	3	yes	—
84	Q₈×C₃₁	—	2	192	no	—
85	Q₈×C₃₁	C₂²×D₄	1	64	yes	direct product factor
86	D₄×C₃₁	C₂²×Q₈	1	64	yes	direct product factor
87	C₂×C₄×C₃₁	—	2	72	no	—
88	C₂×C₄×C₃₁	—	2	36	no	—
89	C₂×C₄×C₃₁	—	2	192	no	—
90	C₂×C₄×C₃₁	C₂²×D₄	1	72	yes	—
91	C₂×C₄×C₃₁	C₂²×Q₈	1	24	yes	complemented
92	C₂×C₄×C₃₁	C₂⁵	1	12	yes	—
93	C₂×C₄×C₃₁	C₂⁵	1	24	yes	—
94	C₂×C₄×C₃₁	C₂⁵	1	12	yes	—
95	C₂×C₄×C₃₁	C₂⁵	1	4	yes	—
96	C₂⁴×Q₈	C₂×C₃₁	1	2	yes	complemented
97	C₂×D₄×Q₈	C₂×C₃₁	1	48	yes	direct product factor
98	C₂²×C₄⋊Q₈	C₂×C₃₁	1	3	yes	complemented
99	C₂²×C₂²⋊Q₈	C₂×C₃₁	1	6	yes	complemented
100	C₂²×C₄×Q₈	C₂×C₃₁	1	1	yes	characteristic, complemented
101	C₂²×C₄×D₄	C₂×C₃₁	1	3	yes	—
102	C₂²×C₃₁	—	2	8	no	—
103	C₂²×C₃₁	—	2	8	no	—
104	C₂²×C₃₁	—	2	96	no	—
105	C₂²×C₃₁	D₄×Q₈	1	16	yes	direct product factor
106	C₂²×C₃₁	C₂³×D₄	1	6	yes	—
107	C₂²×C₃₁	C₂³×Q₈	1	6	yes	—
108	C₂²×C₃₁	C₂³×Q₈	1	16	yes	complemented
109	C₂²×C₃₁	C₂×2₋¹⁺⁴	1	6	yes	—
110	C₂²×C₃₁	C₂⁶	1	1	yes	characteristic
111	C₄×C₃₁	—	2	24	no	—
112	C₄×C₃₁	—	2	12	no	—
113	C₄×C₃₁	C₂³×D₄	1	24	yes	—
114	C₄×C₃₁	C₂³×Q₈	1	8	yes	complemented
115	C₂³×Q₈	—	2	32	no	—
116	C₂³×Q₈	C₂²×C₃₁	1	6	yes	—
117	C₂³×Q₈	C₂²×C₃₁	1	1	yes	characteristic
118	C₂³×Q₈	C₂²×C₃₁	1	48	yes	complemented
119	C₂³×D₄	C₂²×C₃₁	1	1	yes	characteristic
120	C₂⁴×C₄	C₂²×C₃₁	1	6	yes	—
121	D₄×Q₈	C₂²×C₃₁	1	256	yes	direct product factor
122	C₂×C₄⋊Q₈	C₂²×C₃₁	1	72	yes	complemented
123	C₂×C₂²⋊Q₈	C₂²×C₃₁	1	144	yes	complemented
124	C₂×C₄×Q₈	C₂²×C₃₁	1	24	yes	complemented
125	C₂×C₄×D₄	C₂²×C₃₁	1	72	yes	—
126	C₂²×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
127	C₂²×C₄⋊C₄	C₂²×C₃₁	1	3	yes	—
128	C₂²×C₄⋊C₄	C₂²×C₃₁	1	6	yes	—
129	C₂²×C₂²⋊C₄	C₂²×C₃₁	1	6	yes	—
130	C₂²×C₄²	C₂²×C₃₁	1	3	yes	—
131	C₂×C₃₁	—	2	16	no	—
132	C₂×C₃₁	C₂×D₄×Q₈	1	12	yes	direct product factor
133	C₂×C₃₁	C₂⁴×D₄	1	1	yes	characteristic
134	C₂×C₃₁	C₂⁴×Q₈	1	1	yes	characteristic
135	C₂×C₃₁	C₂²×2₋¹⁺⁴	1	1	yes	characteristic
136	C₂⁵	C₂³×C₃₁	1	2	yes	—
137	C₂²×Q₈	—	2	384	no	—
138	C₂²×Q₈	—	2	48	no	—
139	C₂²×Q₈	D₄×C₃₁	1	16	yes	direct product factor
140	C₂²×Q₈	C₂³×C₃₁	1	72	yes	—
141	C₂²×Q₈	C₂³×C₃₁	1	12	yes	—
142	C₂²×Q₈	C₂³×C₃₁	1	128	yes	complemented
143	C₂²×D₄	Q₈×C₃₁	1	16	yes	direct product factor
144	C₂²×D₄	C₂³×C₃₁	1	12	yes	—
145	C₂³×C₄	—	2	48	no	—
146	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
147	C₂³×C₄	C₂³×C₃₁	1	6	yes	—
148	C₂³×C₄	C₂³×C₃₁	1	72	yes	—
149	C₂³×C₄	C₂³×C₃₁	1	3	yes	—
150	C₂³×C₄	C₂³×C₃₁	1	1	yes	characteristic
151	C₄⋊Q₈	C₂³×C₃₁	1	192	yes	complemented
152	C₂²⋊Q₈	C₂³×C₃₁	1	384	yes	complemented
153	C₄×Q₈	C₂³×C₃₁	1	64	yes	complemented
154	C₄×D₄	C₂³×C₃₁	1	192	yes	—
155	C₂×C₄⋊C₄	C₂³×C₃₁	1	36	yes	—
156	C₂×C₄⋊C₄	C₂³×C₃₁	1	36	yes	—
157	C₂×C₄⋊C₄	C₂³×C₃₁	1	72	yes	—
158	C₂×C₂²⋊C₄	C₂³×C₃₁	1	72	yes	—
159	C₂×C₄²	C₂³×C₃₁	1	36	yes	—
160	C₃₁	C₂²×D₄×Q₈	1	1	yes	characteristic, direct product factor, Sylow
161	C₂⁴	—	2	8	no	—
162	C₂⁴	—	2	8	no	—
163	C₂⁴	C₂×Q₈×C₃₁	1	16	yes	complemented
164	C₂⁴	C₂⁴×C₃₁	1	12	yes	—
165	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
166	C₂×Q₈	—	2	512	no	—
167	C₂×Q₈	—	2	288	no	—
168	C₂×Q₈	C₂×D₄×C₃₁	1	96	yes	direct product factor
169	C₂×Q₈	C₂⁴×C₃₁	1	96	yes	—
170	C₂×Q₈	C₂⁴×C₃₁	1	16	yes	—
171	C₂×D₄	C₂×Q₈×C₃₁	1	96	yes	direct product factor
172	C₂×D₄	C₂⁴×C₃₁	1	16	yes	—
173	C₂²×C₄	—	2	24	no	—
174	C₂²×C₄	—	2	12	no	—
175	C₂²×C₄	—	2	288	no	—
176	C₂²×C₄	C₂×D₄×C₃₁	1	24	yes	—
177	C₂²×C₄	C₂×Q₈×C₃₁	1	8	yes	complemented
178	C₂²×C₄	C₂⁴×C₃₁	1	18	yes	—
179	C₂²×C₄	C₂⁴×C₃₁	1	36	yes	—
180	C₂²×C₄	C₂⁴×C₃₁	1	96	yes	—
181	C₂²×C₄	C₂⁴×C₃₁	1	18	yes	—
182	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
183	C₄⋊C₄	C₂⁴×C₃₁	1	48	yes	—
184	C₄⋊C₄	C₂⁴×C₃₁	1	48	yes	—
185	C₄⋊C₄	C₂⁴×C₃₁	1	96	yes	—
186	C₂²⋊C₄	C₂⁴×C₃₁	1	96	yes	—
187	C₄²	C₂⁴×C₃₁	1	48	yes	—
188	C₂³	—	2	64	no	—
189	C₂³	—	2	24	no	—
190	C₂³	—	2	24	no	—
191	C₂³	C₂²×D₄×C₃₁	1	4	yes	—
192	C₂³	C₂²×Q₈×C₃₁	1	4	yes	—
193	C₂³	C₂²×Q₈×C₃₁	1	48	yes	complemented
194	C₂³	C₃₁×2₋¹⁺⁴	1	4	yes	—
195	C₂³	C₂⁵×C₃₁	1	8	yes	—
196	C₂³	C₂⁵×C₃₁	1	3	yes	—
197	Q₈	—	2	192	no	—
198	Q₈	C₂²×D₄×C₃₁	1	64	yes	direct product factor
199	D₄	C₂²×Q₈×C₃₁	1	64	yes	direct product factor
200	C₂×C₄	—	2	72	no	—
201	C₂×C₄	—	2	36	no	—
202	C₂×C₄	—	2	192	no	—
203	C₂×C₄	C₂²×D₄×C₃₁	1	72	yes	—
204	C₂×C₄	C₂²×Q₈×C₃₁	1	24	yes	complemented
205	C₂×C₄	C₂⁵×C₃₁	1	12	yes	—
206	C₂×C₄	C₂⁵×C₃₁	1	24	yes	—
207	C₂×C₄	C₂⁵×C₃₁	1	12	yes	—
208	C₂×C₄	C₂⁵×C₃₁	1	4	yes	—
209	C₂²	—	2	8	no	—
210	C₂²	—	2	8	no	—
211	C₂²	—	2	96	no	—
212	C₂²	D₄×Q₈×C₃₁	1	16	yes	direct product factor
213	C₂²	C₂³×D₄×C₃₁	1	6	yes	—
214	C₂²	C₂³×Q₈×C₃₁	1	6	yes	—
215	C₂²	C₂³×Q₈×C₃₁	1	16	yes	complemented
216	C₂²	C₂×C₃₁×2₋¹⁺⁴	1	6	yes	—
217	C₂²	C₂⁶×C₃₁	1	1	yes	characteristic
218	C₄	—	2	24	no	—
219	C₄	—	2	12	no	—
220	C₄	C₂³×D₄×C₃₁	1	24	yes	—
221	C₄	C₂³×Q₈×C₃₁	1	8	yes	complemented
222	C₂	—	2	16	no	—
223	C₂	C₂×D₄×Q₈×C₃₁	1	12	yes	direct product factor
224	C₂	C₂⁴×D₄×C₃₁	1	1	yes	characteristic
225	C₂	C₂⁴×Q₈×C₃₁	1	1	yes	characteristic
226	C₂	C₂²×C₃₁×2₋¹⁺⁴	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₃₁	C₂⁴
commutator, Frattini, cosocle	C₂²	C₂⁶×C₃₁
prime radical	C₂³×D₄×C₃₁	C₂²
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. C₂⁴×Q₈×C₃₁	C₂	semidirect
2. C₂×D₄×Q₈×C₃₁	C₂	direct (N × H)
3. C₂²×C₃₁×C₄⋊Q₈	C₂	semidirect
4. C₂²×C₃₁×C₂²⋊Q₈	C₂	semidirect
5. C₂²×C₄×Q₈×C₃₁	C₂	semidirect
10. C₂³×Q₈×C₃₁	C₂²	semidirect
13. D₄×Q₈×C₃₁	C₂²	direct (N × H)
14. C₂×C₃₁×C₄⋊Q₈	C₂²	semidirect
15. C₂×C₃₁×C₂²⋊Q₈	C₂²	semidirect
16. C₂×C₄×Q₈×C₃₁	C₂²	semidirect
26. C₂²×Q₈×C₃₁	D₄	direct (N × H)
29. C₂²×Q₈×C₃₁	C₂³	semidirect
30. C₂²×D₄×C₃₁	Q₈	direct (N × H)
38. C₃₁×C₄⋊Q₈	C₂³	semidirect
39. C₃₁×C₂²⋊Q₈	C₂³	semidirect
40. C₄×Q₈×C₃₁	C₂³	semidirect
49. C₂⁴×C₃₁	C₂×Q₈	semidirect
54. C₂×Q₈×C₃₁	C₂×D₄	direct (N × H)
57. C₂×D₄×C₃₁	C₂×Q₈	direct (N × H)
63. C₂²×C₄×C₃₁	C₂×Q₈	semidirect
74. C₂²×D₄×Q₈	C₃₁	direct (N × H)
80. C₂³×C₃₁	C₂²×Q₈	semidirect
85. Q₈×C₃₁	C₂²×D₄	direct (N × H)
86. D₄×C₃₁	C₂²×Q₈	direct (N × H)
91. C₂×C₄×C₃₁	C₂²×Q₈	semidirect
96. C₂⁴×Q₈	C₂×C₃₁	semidirect
97. C₂×D₄×Q₈	C₂×C₃₁	direct (N × H)
98. C₂²×C₄⋊Q₈	C₂×C₃₁	semidirect
99. C₂²×C₂²⋊Q₈	C₂×C₃₁	semidirect
100. C₂²×C₄×Q₈	C₂×C₃₁	semidirect
105. C₂²×C₃₁	D₄×Q₈	direct (N × H)
108. C₂²×C₃₁	C₂³×Q₈	semidirect
114. C₄×C₃₁	C₂³×Q₈	semidirect
118. C₂³×Q₈	C₂²×C₃₁	semidirect
121. D₄×Q₈	C₂²×C₃₁	direct (N × H)
122. C₂×C₄⋊Q₈	C₂²×C₃₁	semidirect
123. C₂×C₂²⋊Q₈	C₂²×C₃₁	semidirect
124. C₂×C₄×Q₈	C₂²×C₃₁	semidirect
132. C₂×C₃₁	C₂×D₄×Q₈	direct (N × H)
139. C₂²×Q₈	D₄×C₃₁	direct (N × H)
142. C₂²×Q₈	C₂³×C₃₁	semidirect
143. C₂²×D₄	Q₈×C₃₁	direct (N × H)
151. C₄⋊Q₈	C₂³×C₃₁	semidirect
152. C₂²⋊Q₈	C₂³×C₃₁	semidirect
153. C₄×Q₈	C₂³×C₃₁	semidirect
160. C₃₁	C₂²×D₄×Q₈	direct (N × H)
163. C₂⁴	C₂×Q₈×C₃₁	semidirect
168. C₂×Q₈	C₂×D₄×C₃₁	direct (N × H)
171. C₂×D₄	C₂×Q₈×C₃₁	direct (N × H)
177. C₂²×C₄	C₂×Q₈×C₃₁	semidirect
193. C₂³	C₂²×Q₈×C₃₁	semidirect
198. Q₈	C₂²×D₄×C₃₁	direct (N × H)
199. D₄	C₂²×Q₈×C₃₁	direct (N × H)
204. C₂×C₄	C₂²×Q₈×C₃₁	semidirect
212. C₂²	D₄×Q₈×C₃₁	direct (N × H)
215. C₂²	C₂³×Q₈×C₃₁	semidirect
221. C₄	C₂³×Q₈×C₃₁	semidirect
223. C₂	C₂×D₄×Q₈×C₃₁	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂⁴×Q₈×C₃₁	2
C₂×D₄×Q₈×C₃₁	2
C₂²×C₃₁×C₄⋊Q₈	2
C₂²×C₃₁×C₂²⋊Q₈	2
C₂²×C₄×Q₈×C₃₁	2
C₂²×C₄×D₄×C₃₁	2
C₂²×D₄×Q₈	31

Subgroups — C22×D4×Q8×C31 #21519455

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×D₄×Q₈×C₃₁ #21519455