Subgroups — C₄×Q₈²×C₃₁ #21490929

Order 7936.

2324 conjugacy classes of subgroups.

The subgroup lattice has 176 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈²×C₃₁	C₂	1	1	yes	maximal, characteristic
2	C₃₁×#2369	C₂	1	9	yes	maximal
3	Q₈×C₄⋊C₄×C₃₁	C₂	1	6	yes	maximal
4	C₄×C₃₁×C₄⋊Q₈	C₂	1	9	yes	maximal
5	C₄²×Q₈×C₃₁	C₂	1	6	yes	maximal
6	Q₈²×C₃₁	C₄	1	16	yes	direct product factor
7	C₂×C₃₁×C₄⋊Q₈	C₂²	1	9	yes	—
8	C₂×C₄×Q₈×C₃₁	C₂²	1	6	yes	—
9	C₂×C₄×Q₈×C₃₁	C₂²	1	6	yes	—
10	C₂×C₄×Q₈×C₃₁	C₂²	1	6	yes	—
11	C₂×C₄×Q₈×C₃₁	C₂²	1	2	yes	—
12	C₃₁×#391	C₂²	1	18	yes	—
13	C₃₁×#389	C₂²	1	36	yes	—
14	C₃₁×#384	C₂²	1	9	yes	—
15	C₄×C₄⋊C₄×C₃₁	C₂²	1	18	yes	—
16	C₄×C₄⋊C₄×C₃₁	C₂²	1	18	yes	—
17	C₄×C₄⋊C₄×C₃₁	C₂²	1	18	yes	—
18	C₄³×C₃₁	C₂²	1	9	yes	—
19	C₂²×Q₈×C₃₁	C₂³	1	6	yes	—
20	C₂²×Q₈×C₃₁	C₂³	1	2	yes	—
21	C₃₁×C₄⋊Q₈	C₂×C₄	1	72	yes	—
22	C₄×Q₈×C₃₁	—	2	24	no	—
23	C₄×Q₈×C₃₁	—	4	12	no	—
24	C₄×Q₈×C₃₁	C₂×C₄	1	48	yes	—
25	C₄×Q₈×C₃₁	Q₈	1	16	yes	direct product factor
26	C₂×C₄⋊C₄×C₃₁	C₂³	1	18	yes	—
27	C₂×C₄⋊C₄×C₃₁	C₂³	1	18	yes	—
28	C₂×C₄⋊C₄×C₃₁	C₂³	1	18	yes	—
29	C₂×C₄⋊C₄×C₃₁	C₂³	1	6	yes	—
30	C₂×C₄²×C₃₁	C₂³	1	18	yes	—
31	C₂×C₄²×C₃₁	C₂³	1	9	yes	—
32	C₂×C₄²×C₃₁	C₂³	1	9	yes	—
33	C₂×C₄²×C₃₁	C₂³	1	9	yes	—
34	C₂×C₄²×C₃₁	C₂³	1	6	yes	—
35	C₃₁×#95	C₂³	1	36	yes	—
36	C₂×Q₈×C₃₁	—	4	6	no	—
37	C₂×Q₈×C₃₁	—	4	6	no	—
38	C₂×Q₈×C₃₁	C₂²×C₄	1	24	yes	—
39	C₂×Q₈×C₃₁	C₂²×C₄	1	8	yes	—
40	C₂×Q₈×C₃₁	C₂×Q₈	1	8	yes	—
41	C₂×Q₈×C₃₁	#41	1	8	yes	—
42	C₂²×C₄×C₃₁	C₂⁴	1	9	yes	—
43	C₂²×C₄×C₃₁	C₂⁴	1	9	yes	—
44	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
45	C₂²×C₄×C₃₁	C₂⁴	1	6	yes	—
46	C₂²×C₄×C₃₁	C₂⁴	1	1	yes	characteristic
47	C₄⋊C₄×C₃₁	—	2	36	no	—
48	C₄⋊C₄×C₃₁	—	4	18	no	—
49	C₄⋊C₄×C₃₁	C₂²×C₄	1	72	yes	—
50	C₄⋊C₄×C₃₁	C₂×Q₈	1	24	yes	—
51	C₄²×C₃₁	—	2	36	no	—
52	C₄²×C₃₁	—	2	18	no	—
53	C₄²×C₃₁	—	4	9	no	—
54	C₄²×C₃₁	C₂²×C₄	1	36	yes	—
55	C₄²×C₃₁	C₂×Q₈	1	24	yes	—
56	C₄×Q₈²	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂³×C₃₁	C₂⁵	1	1	yes	characteristic
58	Q₈×C₃₁	—	4	24	no	—
59	Q₈×C₃₁	C₄×Q₈	1	32	yes	direct product factor
60	C₂×C₄×C₃₁	—	2	6	no	—
61	C₂×C₄×C₃₁	—	2	9	no	—
62	C₂×C₄×C₃₁	—	2	6	no	—
63	C₂×C₄×C₃₁	—	2	9	no	—
64	C₂×C₄×C₃₁	—	2	18	no	—
65	C₂×C₄×C₃₁	—	2	9	no	—
66	C₂×C₄×C₃₁	C₂³×C₄	1	18	yes	—
67	C₂×C₄×C₃₁	C₂³×C₄	1	12	yes	—
68	C₂×C₄×C₃₁	C₂²×Q₈	1	12	yes	—
69	C₂×C₄×C₃₁	C₂²×Q₈	1	12	yes	—
70	C₂×C₄×C₃₁	C₂²×Q₈	1	4	yes	—
71	C₂×C₄×C₃₁	C₂×#41	1	12	yes	—
72	C₂×C₄×C₃₁	H(5,𝔽₂)	1	2	yes	—
73	C₂×Q₈²	C₂×C₃₁	1	1	yes	characteristic
74	#2369	C₂×C₃₁	1	9	yes	—
75	Q₈×C₄⋊C₄	C₂×C₃₁	1	6	yes	—
76	C₄×C₄⋊Q₈	C₂×C₃₁	1	9	yes	—
77	C₄²×Q₈	C₂×C₃₁	1	6	yes	—
78	C₂²×C₃₁	C₂⁴×C₄	1	1	yes	characteristic
79	C₂²×C₃₁	C₂³×Q₈	1	2	yes	—
80	C₂²×C₃₁	C₂²×#41	1	2	yes	—
81	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	1	yes	characteristic
82	C₂²×C₃₁	#585	1	1	yes	characteristic
83	C₄×C₃₁	—	2	18	no	—
84	C₄×C₃₁	—	2	12	no	—
85	C₄×C₃₁	—	4	9	no	—
86	C₄×C₃₁	C₂×C₄×Q₈	1	24	yes	—
87	C₄×C₃₁	Q₈²	1	4	yes	direct product factor
88	Q₈²	C₄×C₃₁	1	16	yes	direct product factor
89	C₂×C₄⋊Q₈	C₂²×C₃₁	1	9	yes	—
90	C₂×C₄×Q₈	C₂²×C₃₁	1	6	yes	—
91	C₂×C₄×Q₈	C₂²×C₃₁	1	6	yes	—
92	C₂×C₄×Q₈	C₂²×C₃₁	1	6	yes	—
93	C₂×C₄×Q₈	C₂²×C₃₁	1	2	yes	—
94	#391	C₂²×C₃₁	1	18	yes	—
95	#389	C₂²×C₃₁	1	36	yes	—
96	#384	C₂²×C₃₁	1	9	yes	—
97	C₄×C₄⋊C₄	C₂²×C₃₁	1	18	yes	—
98	C₄×C₄⋊C₄	C₂²×C₃₁	1	18	yes	—
99	C₄×C₄⋊C₄	C₂²×C₃₁	1	18	yes	—
100	C₄³	C₂²×C₃₁	1	9	yes	—
101	C₂×C₃₁	C₂²×C₄×Q₈	1	2	yes	—
102	C₂×C₃₁	C₄×H(5,𝔽₂)	1	1	yes	characteristic
103	C₂×C₃₁	C₂×Q₈²	1	1	yes	characteristic
104	C₂×C₃₁	Q₈×#41	1	2	yes	—
105	C₂×C₃₁	#3483	1	1	yes	characteristic
106	C₂²×Q₈	C₂³×C₃₁	1	6	yes	—
107	C₂²×Q₈	C₂³×C₃₁	1	2	yes	—
108	C₄⋊Q₈	C₂×C₄×C₃₁	1	72	yes	—
109	C₄×Q₈	—	2	24	no	—
110	C₄×Q₈	—	4	12	no	—
111	C₄×Q₈	C₂×C₄×C₃₁	1	48	yes	—
112	C₄×Q₈	Q₈×C₃₁	1	16	yes	direct product factor
113	C₂×C₄⋊C₄	C₂³×C₃₁	1	18	yes	—
114	C₂×C₄⋊C₄	C₂³×C₃₁	1	18	yes	—
115	C₂×C₄⋊C₄	C₂³×C₃₁	1	18	yes	—
116	C₂×C₄⋊C₄	C₂³×C₃₁	1	6	yes	—
117	C₂×C₄²	C₂³×C₃₁	1	9	yes	—
118	C₂×C₄²	C₂³×C₃₁	1	18	yes	—
119	C₂×C₄²	C₂³×C₃₁	1	9	yes	—
120	C₂×C₄²	C₂³×C₃₁	1	9	yes	—
121	C₂×C₄²	C₂³×C₃₁	1	6	yes	—
122	#95	C₂³×C₃₁	1	36	yes	—
123	C₃₁	C₄×Q₈²	1	1	yes	characteristic, direct product factor, Sylow
124	C₂×Q₈	—	4	6	no	—
125	C₂×Q₈	—	4	6	no	—
126	C₂×Q₈	C₂²×C₄×C₃₁	1	24	yes	—
127	C₂×Q₈	C₂²×C₄×C₃₁	1	8	yes	—
128	C₂×Q₈	C₂×Q₈×C₃₁	1	8	yes	—
129	C₂×Q₈	#41×C₃₁	1	8	yes	—
130	C₂²×C₄	C₂⁴×C₃₁	1	9	yes	—
131	C₂²×C₄	C₂⁴×C₃₁	1	9	yes	—
132	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
133	C₂²×C₄	C₂⁴×C₃₁	1	6	yes	—
134	C₂²×C₄	C₂⁴×C₃₁	1	1	yes	characteristic
135	C₄⋊C₄	—	2	36	no	—
136	C₄⋊C₄	—	4	18	no	—
137	C₄⋊C₄	C₂²×C₄×C₃₁	1	72	yes	—
138	C₄⋊C₄	C₂×Q₈×C₃₁	1	24	yes	—
139	C₄²	—	2	36	no	—
140	C₄²	—	2	18	no	—
141	C₄²	—	4	9	no	—
142	C₄²	C₂²×C₄×C₃₁	1	36	yes	—
143	C₄²	C₂×Q₈×C₃₁	1	24	yes	—
144	C₂³	C₂⁵×C₃₁	1	1	yes	characteristic
145	Q₈	—	4	24	no	—
146	Q₈	C₄×Q₈×C₃₁	1	32	yes	direct product factor
147	C₂×C₄	—	2	6	no	—
148	C₂×C₄	—	2	9	no	—
149	C₂×C₄	—	2	6	no	—
150	C₂×C₄	—	2	9	no	—
151	C₂×C₄	—	2	18	no	—
152	C₂×C₄	—	2	9	no	—
153	C₂×C₄	C₂³×C₄×C₃₁	1	18	yes	—
154	C₂×C₄	C₂³×C₄×C₃₁	1	12	yes	—
155	C₂×C₄	C₂²×Q₈×C₃₁	1	12	yes	—
156	C₂×C₄	C₂²×Q₈×C₃₁	1	12	yes	—
157	C₂×C₄	C₂²×Q₈×C₃₁	1	4	yes	—
158	C₂×C₄	C₂×#41×C₃₁	1	12	yes	—
159	C₂×C₄	C₃₁×H(5,𝔽₂)	1	2	yes	—
160	C₂²	C₂⁴×C₄×C₃₁	1	1	yes	characteristic
161	C₂²	C₂³×Q₈×C₃₁	1	2	yes	—
162	C₂²	C₂²×#41×C₃₁	1	2	yes	—
163	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
164	C₂²	C₃₁×#585	1	1	yes	characteristic
165	C₄	—	2	18	no	—
166	C₄	—	2	12	no	—
167	C₄	—	4	9	no	—
168	C₄	C₂×C₄×Q₈×C₃₁	1	24	yes	—
169	C₄	Q₈²×C₃₁	1	4	yes	direct product factor
170	C₂	C₂²×C₄×Q₈×C₃₁	1	2	yes	—
171	C₂	C₄×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic
172	C₂	C₂×Q₈²×C₃₁	1	1	yes	characteristic
173	C₂	Q₈×#41×C₃₁	1	2	yes	—
174	C₂	C₃₁×#3483	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄×C₃₁	C₂⁴
commutator	C₂²	C₂⁴×C₄×C₃₁
Frattini, cosocle	C₂³	C₂⁵×C₃₁
socle, prime radical	C₂³×C₃₁	C₂⁵
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₃₁ ◁ G

Complements

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

N	H ≅ G/N	type
6. Q₈²×C₃₁	C₄	direct (N × H)
25. C₄×Q₈×C₃₁	Q₈	direct (N × H)
56. C₄×Q₈²	C₃₁	direct (N × H)
59. Q₈×C₃₁	C₄×Q₈	direct (N × H)
87. C₄×C₃₁	Q₈²	direct (N × H)
88. Q₈²	C₄×C₃₁	direct (N × H)
112. C₄×Q₈	Q₈×C₃₁	direct (N × H)
123. C₃₁	C₄×Q₈²	direct (N × H)
146. Q₈	C₄×Q₈×C₃₁	direct (N × H)
169. C₄	Q₈²×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
C₂×Q₈²×C₃₁	2
C₃₁×#2369	2
Q₈×C₄⋊C₄×C₃₁	2
C₄×C₃₁×C₄⋊Q₈	2
C₄²×Q₈×C₃₁	2
C₄×Q₈²	31

Subgroups — C4×Q82×C31 #21490929

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×Q₈²×C₃₁ #21490929