Group #21519471 — subgroups

Subgroups — C₂²×Q₈²×C₃₁ #21519471

Order 7936.

7846 conjugacy classes of subgroups; only the 6958 normal subgroups are shown.

normal subgroups only

This group has 7846 conjugacy classes of subgroups — too many to draw. Only its 6958 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×Q₈²×C₃₁	C₂	1	48	yes	maximal, direct product factor
2	C₂²×C₃₁×C₄⋊Q₈	C₂	1	9	yes	maximal
3	C₂²×C₄×Q₈×C₃₁	C₂	1	6	yes	maximal
4	C₂³×Q₈×C₃₁	C₂²	1	6	yes	—
5	C₂³×Q₈×C₃₁	C₂²	1	2	yes	—
6	Q₈²×C₃₁	C₂²	1	256	yes	direct product factor
7	C₂×C₃₁×C₄⋊Q₈	C₂²	1	216	yes	—
8	C₂×C₄×Q₈×C₃₁	C₂²	1	144	yes	—
9	C₂²×C₄⋊C₄×C₃₁	C₂²	1	18	yes	—
10	C₂²×C₄²×C₃₁	C₂²	1	9	yes	—
11	C₂²×Q₈×C₃₁	—	4	24	no	—
12	C₂²×Q₈×C₃₁	Q₈	1	32	yes	direct product factor
13	C₂²×Q₈×C₃₁	C₂³	1	72	yes	—
14	C₂²×Q₈×C₃₁	C₂³	1	24	yes	—
15	C₂³×C₄×C₃₁	C₂³	1	9	yes	—
16	C₂³×C₄×C₃₁	C₂³	1	6	yes	—
17	C₃₁×C₄⋊Q₈	C₂³	1	576	yes	—
18	C₄×Q₈×C₃₁	C₂³	1	384	yes	—
19	C₂×C₄⋊C₄×C₃₁	C₂³	1	216	yes	—
20	C₂×C₄²×C₃₁	C₂³	1	108	yes	—
21	C₂⁴×C₃₁	C₂⁴	1	1	yes	characteristic
22	C₂×Q₈×C₃₁	—	4	144	no	—
23	C₂×Q₈×C₃₁	C₂×Q₈	1	192	yes	direct product factor
24	C₂×Q₈×C₃₁	C₂⁴	1	96	yes	—
25	C₂×Q₈×C₃₁	C₂⁴	1	32	yes	—
26	C₂²×C₄×C₃₁	—	2	36	no	—
27	C₂²×C₄×C₃₁	C₂×Q₈	1	48	yes	—
28	C₂²×C₄×C₃₁	C₂⁴	1	54	yes	—
29	C₂²×C₄×C₃₁	C₂⁴	1	36	yes	—
30	C₄⋊C₄×C₃₁	C₂⁴	1	288	yes	—
31	C₄²×C₃₁	C₂⁴	1	144	yes	—
32	C₂²×Q₈²	C₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
33	C₂³×C₃₁	C₂²×Q₈	1	8	yes	—
34	C₂³×C₃₁	H(5,𝔽₂)	1	4	yes	—
35	C₂³×C₃₁	C₂⁵	1	3	yes	—
36	Q₈×C₃₁	—	4	96	no	—
37	Q₈×C₃₁	C₂²×Q₈	1	128	yes	direct product factor
38	C₂×C₄×C₃₁	—	2	108	no	—
39	C₂×C₄×C₃₁	C₂²×Q₈	1	144	yes	—
40	C₂×C₄×C₃₁	C₂⁵	1	36	yes	—
41	C₂×C₄×C₃₁	C₂⁵	1	24	yes	—
42	C₂×Q₈²	C₂×C₃₁	1	48	yes	direct product factor
43	C₂²×C₄⋊Q₈	C₂×C₃₁	1	9	yes	—
44	C₂²×C₄×Q₈	C₂×C₃₁	1	6	yes	—
45	C₂²×C₃₁	Q₈²	1	16	yes	direct product factor
46	C₂²×C₃₁	C₂³×Q₈	1	12	yes	—
47	C₂²×C₃₁	C₂×H(5,𝔽₂)	1	6	yes	—
48	C₂²×C₃₁	C₂⁶	1	1	yes	characteristic
49	C₄×C₃₁	—	2	36	no	—
50	C₄×C₃₁	C₂³×Q₈	1	48	yes	—
51	C₂³×Q₈	C₂²×C₃₁	1	6	yes	—
52	C₂³×Q₈	C₂²×C₃₁	1	2	yes	—
53	Q₈²	C₂²×C₃₁	1	256	yes	direct product factor
54	C₂×C₄⋊Q₈	C₂²×C₃₁	1	216	yes	—
55	C₂×C₄×Q₈	C₂²×C₃₁	1	144	yes	—
56	C₂²×C₄⋊C₄	C₂²×C₃₁	1	18	yes	—
57	C₂²×C₄²	C₂²×C₃₁	1	9	yes	—
58	C₂×C₃₁	C₂×Q₈²	1	12	yes	direct product factor
59	C₂×C₃₁	C₂⁴×Q₈	1	2	yes	—
60	C₂×C₃₁	C₂²×H(5,𝔽₂)	1	1	yes	characteristic
61	C₂²×Q₈	—	4	24	no	—
62	C₂²×Q₈	Q₈×C₃₁	1	32	yes	direct product factor
63	C₂²×Q₈	C₂³×C₃₁	1	72	yes	—
64	C₂²×Q₈	C₂³×C₃₁	1	24	yes	—
65	C₂³×C₄	C₂³×C₃₁	1	9	yes	—
66	C₂³×C₄	C₂³×C₃₁	1	6	yes	—
67	C₄⋊Q₈	C₂³×C₃₁	1	576	yes	—
68	C₄×Q₈	C₂³×C₃₁	1	384	yes	—
69	C₂×C₄⋊C₄	C₂³×C₃₁	1	216	yes	—
70	C₂×C₄²	C₂³×C₃₁	1	108	yes	—
71	C₃₁	C₂²×Q₈²	1	1	yes	characteristic, direct product factor, Sylow
72	C₂⁴	C₂⁴×C₃₁	1	1	yes	characteristic
73	C₂×Q₈	—	4	144	no	—
74	C₂×Q₈	C₂×Q₈×C₃₁	1	192	yes	direct product factor
75	C₂×Q₈	C₂⁴×C₃₁	1	96	yes	—
76	C₂×Q₈	C₂⁴×C₃₁	1	32	yes	—
77	C₂²×C₄	—	2	36	no	—
78	C₂²×C₄	C₂×Q₈×C₃₁	1	48	yes	—
79	C₂²×C₄	C₂⁴×C₃₁	1	54	yes	—
80	C₂²×C₄	C₂⁴×C₃₁	1	36	yes	—
81	C₄⋊C₄	C₂⁴×C₃₁	1	288	yes	—
82	C₄²	C₂⁴×C₃₁	1	144	yes	—
83	C₂³	C₂²×Q₈×C₃₁	1	8	yes	—
84	C₂³	C₃₁×H(5,𝔽₂)	1	4	yes	—
85	C₂³	C₂⁵×C₃₁	1	3	yes	—
86	Q₈	—	4	96	no	—
87	Q₈	C₂²×Q₈×C₃₁	1	128	yes	direct product factor
88	C₂×C₄	—	2	108	no	—
89	C₂×C₄	C₂²×Q₈×C₃₁	1	144	yes	—
90	C₂×C₄	C₂⁵×C₃₁	1	36	yes	—
91	C₂×C₄	C₂⁵×C₃₁	1	24	yes	—
92	C₂²	Q₈²×C₃₁	1	16	yes	direct product factor
93	C₂²	C₂³×Q₈×C₃₁	1	12	yes	—
94	C₂²	C₂×C₃₁×H(5,𝔽₂)	1	6	yes	—
95	C₂²	C₂⁶×C₃₁	1	1	yes	characteristic
96	C₄	—	2	36	no	—
97	C₄	C₂³×Q₈×C₃₁	1	48	yes	—
98	C₂	C₂×Q₈²×C₃₁	1	12	yes	direct product factor
99	C₂	C₂⁴×Q₈×C₃₁	1	2	yes	—
100	C₂	C₂²×C₃₁×H(5,𝔽₂)	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈²×C₃₁

C₂

yes

maximal, direct product factor

C₂²×C₃₁×C₄⋊Q₈

C₂

yes

maximal

C₂²×C₄×Q₈×C₃₁

C₂

yes

maximal

C₂³×Q₈×C₃₁

C₂²

yes

—

C₂³×Q₈×C₃₁

C₂²

yes

—

Q₈²×C₃₁

C₂²

256

yes

direct product factor

C₂×C₃₁×C₄⋊Q₈

C₂²

216

yes

—

C₂×C₄×Q₈×C₃₁

C₂²

144

yes

—

C₂²×C₄⋊C₄×C₃₁

C₂²

yes

—

C₂²×C₄²×C₃₁

C₂²

yes

—

C₂²×Q₈×C₃₁

—

C₂²×Q₈×C₃₁

Q₈

yes

direct product factor

C₂²×Q₈×C₃₁

C₂³

yes

—

C₂²×Q₈×C₃₁

C₂³

yes

—

C₂³×C₄×C₃₁

C₂³

yes

—

C₂³×C₄×C₃₁

C₂³

yes

—

C₃₁×C₄⋊Q₈

C₂³

576

yes

—

C₄×Q₈×C₃₁

C₂³

384

yes

—

C₂×C₄⋊C₄×C₃₁

C₂³

216

yes

—

C₂×C₄²×C₃₁

C₂³

108

yes

—

C₂⁴×C₃₁

C₂⁴

yes

characteristic

C₂×Q₈×C₃₁

—

144

—

C₂×Q₈×C₃₁

C₂×Q₈

192

yes

direct product factor

C₂×Q₈×C₃₁

C₂⁴

yes

—

C₂×Q₈×C₃₁

C₂⁴

yes

—

C₂²×C₄×C₃₁

—

C₂²×C₄×C₃₁

C₂×Q₈

yes

—

C₂²×C₄×C₃₁

C₂⁴

yes

—

C₂²×C₄×C₃₁

C₂⁴

yes

—

C₄⋊C₄×C₃₁

C₂⁴

288

yes

—

C₄²×C₃₁

C₂⁴

144

yes

—

C₂²×Q₈²

C₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂³×C₃₁

C₂²×Q₈

yes

—

C₂³×C₃₁

H(5,𝔽₂)

yes

—

C₂³×C₃₁

C₂⁵

yes

—

Q₈×C₃₁

—

Q₈×C₃₁

C₂²×Q₈

128

yes

direct product factor

C₂×C₄×C₃₁

—

108

—

C₂×C₄×C₃₁

C₂²×Q₈

144

yes

—

C₂×C₄×C₃₁

C₂⁵

yes

—

C₂×C₄×C₃₁

C₂⁵

yes

—

C₂×Q₈²

C₂×C₃₁

yes

direct product factor

C₂²×C₄⋊Q₈

C₂×C₃₁

yes

—

C₂²×C₄×Q₈

C₂×C₃₁

yes

—

C₂²×C₃₁

Q₈²

yes

direct product factor

C₂²×C₃₁

C₂³×Q₈

yes

—

C₂²×C₃₁

C₂×H(5,𝔽₂)

yes

—

C₂²×C₃₁

C₂⁶

yes

characteristic

C₄×C₃₁

—

C₄×C₃₁

C₂³×Q₈

yes

—

C₂³×Q₈

C₂²×C₃₁

yes

—

C₂³×Q₈

C₂²×C₃₁

yes

—

Q₈²

C₂²×C₃₁

256

yes

direct product factor

C₂×C₄⋊Q₈

C₂²×C₃₁

216

yes

—

C₂×C₄×Q₈

C₂²×C₃₁

144

yes

—

C₂²×C₄⋊C₄

C₂²×C₃₁

yes

—

C₂²×C₄²

C₂²×C₃₁

yes

—

C₂×C₃₁

C₂×Q₈²

yes

direct product factor

C₂×C₃₁

C₂⁴×Q₈

yes

—

C₂×C₃₁

C₂²×H(5,𝔽₂)

yes

characteristic

C₂²×Q₈

—

C₂²×Q₈

Q₈×C₃₁

yes

direct product factor

C₂²×Q₈

C₂³×C₃₁

yes

—

C₂²×Q₈

C₂³×C₃₁

yes

—

C₂³×C₄

C₂³×C₃₁

yes

—

C₂³×C₄

C₂³×C₃₁

yes

—

C₄⋊Q₈

C₂³×C₃₁

576

yes

—

C₄×Q₈

C₂³×C₃₁

384

yes

—

C₂×C₄⋊C₄

C₂³×C₃₁

216

yes

—

C₂×C₄²

C₂³×C₃₁

108

yes

—

C₃₁

C₂²×Q₈²

yes

characteristic, direct product factor, Sylow

C₂⁴

C₂⁴×C₃₁

yes

characteristic

C₂×Q₈

—

144

—

C₂×Q₈

C₂×Q₈×C₃₁

192

yes

direct product factor

C₂×Q₈

C₂⁴×C₃₁

yes

—

C₂×Q₈

C₂⁴×C₃₁

yes

—

C₂²×C₄

—

C₂²×C₄

C₂×Q₈×C₃₁

yes

—

C₂²×C₄

C₂⁴×C₃₁

yes

—

C₂²×C₄

C₂⁴×C₃₁

yes

—

C₄⋊C₄

C₂⁴×C₃₁

288

yes

—

C₄²

C₂⁴×C₃₁

144

yes

—

C₂³

C₂²×Q₈×C₃₁

yes

—

C₂³

C₃₁×H(5,𝔽₂)

yes

—

C₂³

C₂⁵×C₃₁

yes

—

Q₈

—

Q₈

C₂²×Q₈×C₃₁

128

yes

direct product factor

C₂×C₄

—

108

—

C₂×C₄

C₂²×Q₈×C₃₁

144

yes

—

C₂×C₄

C₂⁵×C₃₁

yes

—

C₂×C₄

C₂⁵×C₃₁

yes

—

C₂²

Q₈²×C₃₁

yes

direct product factor

C₂²

C₂³×Q₈×C₃₁

yes

—

C₂²

C₂×C₃₁×H(5,𝔽₂)

yes

—

C₂²

C₂⁶×C₃₁

yes

characteristic

C₄

—

C₄

C₂³×Q₈×C₃₁

yes

—

C₂

C₂×Q₈²×C₃₁

yes

direct product factor

C₂

C₂⁴×Q₈×C₃₁

yes

—

100

C₂

C₂²×C₃₁×H(5,𝔽₂)

yes

characteristic

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

subgroup

G/H

center, socle, prime radical

C₂⁴×C₃₁

C₂⁴

commutator, Frattini, cosocle

C₂²

C₂⁶×C₃₁

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂⁴×C₃₁ ◁ G

N	H ≅ G/N	type
1. C₂×Q₈²×C₃₁	C₂	direct (N × H)
6. Q₈²×C₃₁	C₂²	direct (N × H)
12. C₂²×Q₈×C₃₁	Q₈	direct (N × H)
23. C₂×Q₈×C₃₁	C₂×Q₈	direct (N × H)
32. C₂²×Q₈²	C₃₁	direct (N × H)
37. Q₈×C₃₁	C₂²×Q₈	direct (N × H)
42. C₂×Q₈²	C₂×C₃₁	direct (N × H)
45. C₂²×C₃₁	Q₈²	direct (N × H)
53. Q₈²	C₂²×C₃₁	direct (N × H)
58. C₂×C₃₁	C₂×Q₈²	direct (N × H)
62. C₂²×Q₈	Q₈×C₃₁	direct (N × H)
71. C₃₁	C₂²×Q₈²	direct (N × H)
74. C₂×Q₈	C₂×Q₈×C₃₁	direct (N × H)
87. Q₈	C₂²×Q₈×C₃₁	direct (N × H)
92. C₂²	Q₈²×C₃₁	direct (N × H)
98. C₂	C₂×Q₈²×C₃₁	direct (N × H)

H ≅ G/N

type

1. C₂×Q₈²×C₃₁

C₂

direct (N × H)

6. Q₈²×C₃₁

C₂²

direct (N × H)

12. C₂²×Q₈×C₃₁

Q₈

direct (N × H)

23. C₂×Q₈×C₃₁

C₂×Q₈

direct (N × H)

32. C₂²×Q₈²

C₃₁

direct (N × H)

37. Q₈×C₃₁

C₂²×Q₈

direct (N × H)

42. C₂×Q₈²

C₂×C₃₁

direct (N × H)

45. C₂²×C₃₁

Q₈²

direct (N × H)

53. Q₈²

C₂²×C₃₁

direct (N × H)

58. C₂×C₃₁

C₂×Q₈²

direct (N × H)

62. C₂²×Q₈

Q₈×C₃₁

direct (N × H)

71. C₃₁

C₂²×Q₈²

direct (N × H)

74. C₂×Q₈

C₂×Q₈×C₃₁

direct (N × H)

87. Q₈

C₂²×Q₈×C₃₁

direct (N × H)

92. C₂²

Q₈²×C₃₁

direct (N × H)

98. C₂

C₂×Q₈²×C₃₁

direct (N × H)

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

subgroup	index
C₂×Q₈²×C₃₁	2
C₂²×C₃₁×C₄⋊Q₈	2
C₂²×C₄×Q₈×C₃₁	2
C₂²×Q₈²	31

subgroup

index

C₂×Q₈²×C₃₁

C₂²×C₃₁×C₄⋊Q₈

C₂²×C₄×Q₈×C₃₁

C₂²×Q₈²

Subgroups — C₂²×Q₈²×C₃₁ #21519471

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C22×Q82×C31 #21519471

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂²×Q₈²×C₃₁ #21519471