Subgroups — C₂³×C₃₁×C₄⋊Q₈ #21519424

Order 7936.

12190 conjugacy classes of subgroups.

The subgroup lattice has 120 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
2	C₂²×C₃₁×C₄⋊Q₈	C₂	1	56	yes	maximal, direct product factor
3	C₂³×C₄⋊C₄×C₃₁	C₂	1	4	yes	maximal
4	C₂³×C₄²×C₃₁	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₃₁	—	2	32	no	—
6	C₂³×Q₈×C₃₁	C₂²	1	56	yes	—
7	C₂⁴×C₄×C₃₁	C₂²	1	4	yes	—
8	C₂⁴×C₄×C₃₁	C₂²	1	1	yes	characteristic
9	C₂⁴×C₄×C₃₁	C₂²	1	2	yes	—
10	C₂×C₃₁×C₄⋊Q₈	C₂²	1	448	yes	direct product factor
11	C₂²×C₄⋊C₄×C₃₁	C₂²	1	112	yes	—
12	C₂²×C₄²×C₃₁	C₂²	1	28	yes	—
13	C₂⁵×C₃₁	C₂³	1	1	yes	characteristic
14	C₂²×Q₈×C₃₁	—	2	448	no	—
15	C₂²×Q₈×C₃₁	C₂³	1	224	yes	—
16	C₂³×C₄×C₃₁	—	2	32	no	—
17	C₂³×C₄×C₃₁	D₄	1	16	yes	—
18	C₂³×C₄×C₃₁	Q₈	1	32	yes	complemented
19	C₂³×C₄×C₃₁	C₂³	1	56	yes	—
20	C₂³×C₄×C₃₁	C₂³	1	14	yes	—
21	C₂³×C₄×C₃₁	C₂³	1	28	yes	—
22	C₃₁×C₄⋊Q₈	C₂³	1	512	yes	direct product factor
23	C₂×C₄⋊C₄×C₃₁	C₂³	1	448	yes	—
24	C₂×C₄²×C₃₁	C₂³	1	112	yes	—
25	C₂⁴×C₃₁	C₂×D₄	1	8	yes	—
26	C₂⁴×C₃₁	C₂×Q₈	1	16	yes	—
27	C₂⁴×C₃₁	C₂⁴	1	7	yes	—
28	C₂×Q₈×C₃₁	—	2	896	no	—
29	C₂×Q₈×C₃₁	C₂⁴	1	128	yes	—
30	C₂²×C₄×C₃₁	—	2	224	no	—
31	C₂²×C₄×C₃₁	C₂×D₄	1	112	yes	—
32	C₂²×C₄×C₃₁	C₂×Q₈	1	224	yes	complemented
33	C₂²×C₄×C₃₁	C₂⁴	1	112	yes	—
34	C₂²×C₄×C₃₁	C₂⁴	1	28	yes	—
35	C₂²×C₄×C₃₁	C₂⁴	1	56	yes	—
36	C₄⋊C₄×C₃₁	C₂⁴	1	256	yes	—
37	C₄²×C₃₁	C₂⁴	1	64	yes	—
38	C₂³×C₄⋊Q₈	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
39	C₂³×C₃₁	C₄⋊Q₈	1	64	yes	direct product factor
40	C₂³×C₃₁	C₂²×D₄	1	28	yes	—
41	C₂³×C₃₁	C₂²×Q₈	1	56	yes	—
42	C₂³×C₃₁	C₂⁵	1	7	yes	—
43	Q₈×C₃₁	—	2	256	no	—
44	C₂×C₄×C₃₁	—	2	224	no	—
45	C₂×C₄×C₃₁	C₂²×D₄	1	112	yes	—
46	C₂×C₄×C₃₁	C₂²×Q₈	1	224	yes	complemented
47	C₂×C₄×C₃₁	C₂⁵	1	32	yes	—
48	C₂×C₄×C₃₁	C₂⁵	1	8	yes	—
49	C₂×C₄×C₃₁	C₂⁵	1	16	yes	—
50	C₂⁴×Q₈	C₂×C₃₁	1	2	yes	—
51	C₂²×C₄⋊Q₈	C₂×C₃₁	1	56	yes	direct product factor
52	C₂³×C₄⋊C₄	C₂×C₃₁	1	4	yes	—
53	C₂³×C₄²	C₂×C₃₁	1	1	yes	characteristic
54	C₂²×C₃₁	C₂×C₄⋊Q₈	1	112	yes	direct product factor
55	C₂²×C₃₁	C₂³×D₄	1	14	yes	—
56	C₂²×C₃₁	C₂³×Q₈	1	28	yes	—
57	C₂²×C₃₁	C₂⁶	1	1	yes	characteristic
58	C₄×C₃₁	—	2	32	no	—
59	C₄×C₃₁	C₂³×D₄	1	16	yes	—
60	C₄×C₃₁	C₂³×Q₈	1	32	yes	complemented
61	C₂³×Q₈	—	2	32	no	—
62	C₂³×Q₈	C₂²×C₃₁	1	56	yes	—
63	C₂⁴×C₄	C₂²×C₃₁	1	4	yes	—
64	C₂⁴×C₄	C₂²×C₃₁	1	1	yes	characteristic
65	C₂⁴×C₄	C₂²×C₃₁	1	2	yes	—
66	C₂×C₄⋊Q₈	C₂²×C₃₁	1	448	yes	direct product factor
67	C₂²×C₄⋊C₄	C₂²×C₃₁	1	112	yes	—
68	C₂²×C₄²	C₂²×C₃₁	1	28	yes	—
69	C₂×C₃₁	C₂²×C₄⋊Q₈	1	28	yes	direct product factor
70	C₂×C₃₁	C₂⁴×D₄	1	1	yes	characteristic
71	C₂×C₃₁	C₂⁴×Q₈	1	2	yes	—
72	C₂⁵	C₂³×C₃₁	1	1	yes	characteristic
73	C₂²×Q₈	—	2	448	no	—
74	C₂²×Q₈	C₂³×C₃₁	1	224	yes	—
75	C₂³×C₄	—	2	32	no	—
76	C₂³×C₄	D₄×C₃₁	1	16	yes	—
77	C₂³×C₄	Q₈×C₃₁	1	32	yes	complemented
78	C₂³×C₄	C₂³×C₃₁	1	56	yes	—
79	C₂³×C₄	C₂³×C₃₁	1	14	yes	—
80	C₂³×C₄	C₂³×C₃₁	1	28	yes	—
81	C₄⋊Q₈	C₂³×C₃₁	1	512	yes	direct product factor
82	C₂×C₄⋊C₄	C₂³×C₃₁	1	448	yes	—
83	C₂×C₄²	C₂³×C₃₁	1	112	yes	—
84	C₃₁	C₂³×C₄⋊Q₈	1	1	yes	characteristic, direct product factor, Sylow
85	C₂⁴	C₂×D₄×C₃₁	1	8	yes	—
86	C₂⁴	C₂×Q₈×C₃₁	1	16	yes	—
87	C₂⁴	C₂⁴×C₃₁	1	7	yes	—
88	C₂×Q₈	—	2	896	no	—
89	C₂×Q₈	C₂⁴×C₃₁	1	128	yes	—
90	C₂²×C₄	—	2	224	no	—
91	C₂²×C₄	C₂×D₄×C₃₁	1	112	yes	—
92	C₂²×C₄	C₂×Q₈×C₃₁	1	224	yes	complemented
93	C₂²×C₄	C₂⁴×C₃₁	1	112	yes	—
94	C₂²×C₄	C₂⁴×C₃₁	1	28	yes	—
95	C₂²×C₄	C₂⁴×C₃₁	1	56	yes	—
96	C₄⋊C₄	C₂⁴×C₃₁	1	256	yes	—
97	C₄²	C₂⁴×C₃₁	1	64	yes	—
98	C₂³	C₃₁×C₄⋊Q₈	1	64	yes	direct product factor
99	C₂³	C₂²×D₄×C₃₁	1	28	yes	—
100	C₂³	C₂²×Q₈×C₃₁	1	56	yes	—
101	C₂³	C₂⁵×C₃₁	1	7	yes	—
102	Q₈	—	2	256	no	—
103	C₂×C₄	—	2	224	no	—
104	C₂×C₄	C₂²×D₄×C₃₁	1	112	yes	—
105	C₂×C₄	C₂²×Q₈×C₃₁	1	224	yes	complemented
106	C₂×C₄	C₂⁵×C₃₁	1	32	yes	—
107	C₂×C₄	C₂⁵×C₃₁	1	8	yes	—
108	C₂×C₄	C₂⁵×C₃₁	1	16	yes	—
109	C₂²	C₂×C₃₁×C₄⋊Q₈	1	112	yes	direct product factor
110	C₂²	C₂³×D₄×C₃₁	1	14	yes	—
111	C₂²	C₂³×Q₈×C₃₁	1	28	yes	—
112	C₂²	C₂⁶×C₃₁	1	1	yes	characteristic
113	C₄	—	2	32	no	—
114	C₄	C₂³×D₄×C₃₁	1	16	yes	—
115	C₄	C₂³×Q₈×C₃₁	1	32	yes	complemented
116	C₂	C₂²×C₃₁×C₄⋊Q₈	1	28	yes	direct product factor
117	C₂	C₂⁴×D₄×C₃₁	1	1	yes	characteristic
118	C₂	C₂⁴×Q₈×C₃₁	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂²×C₃₁×C₄⋊Q₈	C₂	direct (N × H)
10. C₂×C₃₁×C₄⋊Q₈	C₂²	direct (N × H)
18. C₂³×C₄×C₃₁	Q₈	semidirect
22. C₃₁×C₄⋊Q₈	C₂³	direct (N × H)
32. C₂²×C₄×C₃₁	C₂×Q₈	semidirect
38. C₂³×C₄⋊Q₈	C₃₁	direct (N × H)
39. C₂³×C₃₁	C₄⋊Q₈	direct (N × H)
46. C₂×C₄×C₃₁	C₂²×Q₈	semidirect
51. C₂²×C₄⋊Q₈	C₂×C₃₁	direct (N × H)
54. C₂²×C₃₁	C₂×C₄⋊Q₈	direct (N × H)
60. C₄×C₃₁	C₂³×Q₈	semidirect
66. C₂×C₄⋊Q₈	C₂²×C₃₁	direct (N × H)
69. C₂×C₃₁	C₂²×C₄⋊Q₈	direct (N × H)
77. C₂³×C₄	Q₈×C₃₁	semidirect
81. C₄⋊Q₈	C₂³×C₃₁	direct (N × H)
84. C₃₁	C₂³×C₄⋊Q₈	direct (N × H)
92. C₂²×C₄	C₂×Q₈×C₃₁	semidirect
98. C₂³	C₃₁×C₄⋊Q₈	direct (N × H)
105. C₂×C₄	C₂²×Q₈×C₃₁	semidirect
109. C₂²	C₂×C₃₁×C₄⋊Q₈	direct (N × H)
115. C₄	C₂³×Q₈×C₃₁	semidirect
116. C₂	C₂²×C₃₁×C₄⋊Q₈	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
C₂²×C₃₁×C₄⋊Q₈	2
C₂³×C₄⋊C₄×C₃₁	2
C₂³×C₄²×C₃₁	2
C₂³×C₄⋊Q₈	31

Subgroups — C23×C31×C4⋊Q8 #21519424

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂³×C₃₁×C₄⋊Q₈ #21519424