Group #30663317 — subgroups

Subgroups — C₂×#491×C₁₃₁ #30663317

Order 16768.

384 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 200 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×C₄⋊Q₈×C₁₃₁	C₂	1	1	yes	maximal, characteristic
2	C₂×#125×C₁₃₁	C₂	1	1	yes	maximal, characteristic
3	#491×C₁₃₁	C₂	1	8	yes	maximal, direct product factor
4	C₂×#103×C₁₃₁	C₂	1	4	yes	maximal
5	C₂×#97×C₁₃₁	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₈×C₁₃₁	C₂²	1	2	yes	—
7	C₂²×C₈×C₁₃₁	C₂²	1	2	yes	—
8	C₄⋊Q₈×C₁₃₁	—	2	2	no	—
9	C₄⋊Q₈×C₁₃₁	C₂²	1	4	yes	—
10	#125×C₁₃₁	—	2	2	no	—
11	#125×C₁₃₁	C₂²	1	4	yes	—
12	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
13	C₂×C₄⋊C₄×C₁₃₁	—	2	2	no	—
14	C₂×C₄⋊C₄×C₁₃₁	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₃₁	C₂²	1	1	yes	characteristic
16	#103×C₁₃₁	C₂²	1	16	yes	—
17	#97×C₁₃₁	C₂²	1	4	yes	—
18	C₂×Q₈×C₁₃₁	—	2	2	no	—
19	C₂×Q₈×C₁₃₁	—	2	8	no	—
20	C₂×Q₈×C₁₃₁	C₂³	1	4	yes	—
21	C₂²×C₄×C₁₃₁	—	2	2	no	—
22	C₂²×C₄×C₁₃₁	—	2	2	no	—
23	C₂²×C₄×C₁₃₁	D₄	1	2	yes	—
24	C₂²×C₄×C₁₃₁	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₃₁	—	2	4	no	—
26	C₂×C₈×C₁₃₁	C₂³	1	4	yes	—
27	C₄⋊C₄×C₁₃₁	—	2	8	no	—
28	C₄⋊C₄×C₁₃₁	—	2	8	no	—
29	C₄⋊C₄×C₁₃₁	—	2	2	no	—
30	C₄⋊C₄×C₁₃₁	C₂³	1	4	yes	—
31	C₄²×C₁₃₁	D₄	1	2	yes	—
32	C₄²×C₁₃₁	C₂³	1	2	yes	—
33	C₂³×C₁₃₁	C₂×D₄	1	1	yes	characteristic
34	Q₈×C₁₃₁	—	2	8	no	—
35	Q₈×C₁₃₁	—	4	4	no	—
36	C₂×C₄×C₁₃₁	—	2	4	no	—
37	C₂×C₄×C₁₃₁	—	2	4	no	—
38	C₂×C₄×C₁₃₁	—	2	4	no	—
39	C₂×C₄×C₁₃₁	—	4	4	no	—
40	C₂×C₄×C₁₃₁	—	4	4	no	—
41	C₂×C₄×C₁₃₁	C₂×D₄	1	4	yes	—
42	C₂×C₄×C₁₃₁	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₃₁	#41	1	4	yes	—
44	C₂×C₄×C₁₃₁	C₂⁴	1	1	yes	characteristic
45	C₈×C₁₃₁	—	2	4	no	—
46	C₂²×C₁₃₁	#124	1	2	yes	—
47	C₂²×C₁₃₁	#137	1	4	yes	—
48	C₂²×C₁₃₁	C₂²×D₄	1	1	yes	characteristic
49	C₄×C₁₃₁	—	2	4	no	—
50	C₄×C₁₃₁	—	4	4	no	—
51	C₄×C₁₃₁	—	4	4	no	—
52	C₄×C₁₃₁	#124	1	2	yes	—
53	C₄×C₁₃₁	C₂×#41	1	2	yes	—
54	C₂×C₁₃₁	#491	1	4	yes	direct product factor
55	C₂×C₁₃₁	C₂×#124	1	1	yes	characteristic
56	C₂×C₁₃₁	C₂×#137	1	2	yes	—
57	C₁₃₁	C₂×#491	1	1	yes	characteristic, direct product factor, Sylow
58	C₂×#491	C₁₃₁	1	1	yes	maximal, characteristic, direct product factor, Sylow
59	C₂×C₄⋊Q₈	C₂×C₁₃₁	1	1	yes	characteristic
60	C₂×#125	C₂×C₁₃₁	1	1	yes	characteristic
61	#491	C₂×C₁₃₁	1	8	yes	direct product factor
62	C₂×#103	C₂×C₁₃₁	1	4	yes	—
63	C₂×#97	C₂×C₁₃₁	1	1	yes	characteristic
64	C₂²×Q₈	C₂²×C₁₃₁	1	2	yes	—
65	C₂²×C₈	C₂²×C₁₃₁	1	2	yes	—
66	C₄⋊Q₈	—	2	2	no	—
67	C₄⋊Q₈	C₂²×C₁₃₁	1	4	yes	—
68	#125	—	2	2	no	—
69	#125	C₂²×C₁₃₁	1	4	yes	—
70	C₂×C₄⋊C₄	—	2	2	no	—
71	C₂×C₄⋊C₄	—	2	2	no	—
72	C₂×C₄⋊C₄	C₂²×C₁₃₁	1	2	yes	—
73	C₂×C₄²	C₂²×C₁₃₁	1	1	yes	characteristic
74	#103	C₂²×C₁₃₁	1	16	yes	—
75	#97	C₂²×C₁₃₁	1	4	yes	—
76	C₂×Q₈	—	2	2	no	—
77	C₂×Q₈	—	2	8	no	—
78	C₂×Q₈	C₂³×C₁₃₁	1	4	yes	—
79	C₂²×C₄	—	2	2	no	—
80	C₂²×C₄	—	2	2	no	—
81	C₂²×C₄	D₄×C₁₃₁	1	2	yes	—
82	C₂²×C₄	C₂³×C₁₃₁	1	1	yes	characteristic
83	C₂×C₈	—	2	4	no	—
84	C₂×C₈	C₂³×C₁₃₁	1	4	yes	—
85	C₄⋊C₄	—	2	8	no	—
86	C₄⋊C₄	—	2	8	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	C₂³×C₁₃₁	1	4	yes	—
89	C₄²	D₄×C₁₃₁	1	2	yes	—
90	C₄²	C₂³×C₁₃₁	1	2	yes	—
91	C₂³	C₂×D₄×C₁₃₁	1	1	yes	characteristic
92	Q₈	—	2	8	no	—
93	Q₈	—	4	4	no	—
94	C₂×C₄	—	2	4	no	—
95	C₂×C₄	—	2	4	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	4	4	no	—
98	C₂×C₄	—	4	4	no	—
99	C₂×C₄	C₂×D₄×C₁₃₁	1	4	yes	—
100	C₂×C₄	C₂×D₄×C₁₃₁	1	1	yes	characteristic
101	C₂×C₄	#41×C₁₃₁	1	4	yes	—
102	C₂×C₄	C₂⁴×C₁₃₁	1	1	yes	characteristic
103	C₈	—	2	4	no	—
104	C₂²	#124×C₁₃₁	1	2	yes	—
105	C₂²	#137×C₁₃₁	1	4	yes	—
106	C₂²	C₂²×D₄×C₁₃₁	1	1	yes	characteristic
107	C₄	—	2	4	no	—
108	C₄	—	4	4	no	—
109	C₄	—	4	4	no	—
110	C₄	#124×C₁₃₁	1	2	yes	—
111	C₄	C₂×#41×C₁₃₁	1	2	yes	—
112	C₂	#491×C₁₃₁	1	4	yes	direct product factor
113	C₂	C₂×#124×C₁₃₁	1	1	yes	characteristic
114	C₂	C₂×#137×C₁₃₁	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Q₈×C₁₃₁

C₂

yes

maximal, characteristic

C₂×#125×C₁₃₁

C₂

yes

maximal, characteristic

#491×C₁₃₁

C₂

yes

maximal, direct product factor

C₂×#103×C₁₃₁

C₂

yes

maximal

C₂×#97×C₁₃₁

C₂

yes

maximal, characteristic

C₂²×Q₈×C₁₃₁

C₂²

yes

—

C₂²×C₈×C₁₃₁

C₂²

yes

—

C₄⋊Q₈×C₁₃₁

—

C₄⋊Q₈×C₁₃₁

C₂²

yes

—

#125×C₁₃₁

—

#125×C₁₃₁

C₂²

yes

—

C₂×C₄⋊C₄×C₁₃₁

—

C₂×C₄⋊C₄×C₁₃₁

—

C₂×C₄⋊C₄×C₁₃₁

C₂²

yes

—

C₂×C₄²×C₁₃₁

C₂²

yes

characteristic

#103×C₁₃₁

C₂²

yes

—

#97×C₁₃₁

C₂²

yes

—

C₂×Q₈×C₁₃₁

—

C₂×Q₈×C₁₃₁

—

C₂×Q₈×C₁₃₁

C₂³

yes

—

C₂²×C₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

—

C₂²×C₄×C₁₃₁

D₄

yes

—

C₂²×C₄×C₁₃₁

C₂³

yes

characteristic

C₂×C₈×C₁₃₁

—

C₂×C₈×C₁₃₁

C₂³

yes

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

—

C₄⋊C₄×C₁₃₁

C₂³

yes

—

C₄²×C₁₃₁

D₄

yes

—

C₄²×C₁₃₁

C₂³

yes

—

C₂³×C₁₃₁

C₂×D₄

yes

characteristic

Q₈×C₁₃₁

—

Q₈×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

—

C₂×C₄×C₁₃₁

C₂×D₄

yes

characteristic

C₂×C₄×C₁₃₁

#41

yes

—

C₂×C₄×C₁₃₁

C₂⁴

yes

characteristic

C₈×C₁₃₁

—

C₂²×C₁₃₁

#124

yes

—

C₂²×C₁₃₁

#137

yes

—

C₂²×C₁₃₁

C₂²×D₄

yes

characteristic

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

—

C₄×C₁₃₁

#124

yes

—

C₄×C₁₃₁

C₂×#41

yes

—

C₂×C₁₃₁

#491

yes

direct product factor

C₂×C₁₃₁

C₂×#124

yes

characteristic

C₂×C₁₃₁

C₂×#137

yes

—

C₁₃₁

C₂×#491

yes

characteristic, direct product factor, Sylow

C₂×#491

C₁₃₁

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄⋊Q₈

C₂×C₁₃₁

yes

characteristic

C₂×#125

C₂×C₁₃₁

yes

characteristic

#491

C₂×C₁₃₁

yes

direct product factor

C₂×#103

C₂×C₁₃₁

yes

—

C₂×#97

C₂×C₁₃₁

yes

characteristic

C₂²×Q₈

C₂²×C₁₃₁

yes

—

C₂²×C₈

C₂²×C₁₃₁

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂²×C₁₃₁

yes

—

#125

—

#125

C₂²×C₁₃₁

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²×C₁₃₁

yes

—

C₂×C₄²

C₂²×C₁₃₁

yes

characteristic

#103

C₂²×C₁₃₁

yes

—

#97

C₂²×C₁₃₁

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂³×C₁₃₁

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

D₄×C₁₃₁

yes

—

C₂²×C₄

C₂³×C₁₃₁

yes

characteristic

C₂×C₈

—

C₂×C₈

C₂³×C₁₃₁

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂³×C₁₃₁

yes

—

C₄²

D₄×C₁₃₁

yes

—

C₄²

C₂³×C₁₃₁

yes

—

C₂³

C₂×D₄×C₁₃₁

yes

characteristic

Q₈

—

Q₈

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂×D₄×C₁₃₁

yes

—

100

C₂×C₄

C₂×D₄×C₁₃₁

yes

characteristic

101

C₂×C₄

#41×C₁₃₁

yes

—

102

C₂×C₄

C₂⁴×C₁₃₁

yes

characteristic

103

C₈

—

104

C₂²

#124×C₁₃₁

yes

—

105

C₂²

#137×C₁₃₁

yes

—

106

C₂²

C₂²×D₄×C₁₃₁

yes

characteristic

107

C₄

—

108

C₄

—

109

C₄

—

110

C₄

#124×C₁₃₁

yes

—

111

C₄

C₂×#41×C₁₃₁

yes

—

112

C₂

#491×C₁₃₁

yes

direct product factor

113

C₂

C₂×#124×C₁₃₁

yes

characteristic

114

C₂

C₂×#137×C₁₃₁

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃₁	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃₁
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₁₃₁

C₂×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₁₃₁

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂³×C₁₃₁ ◁ C₂×C₄²×C₁₃₁ ◁ G

N	H ≅ G/N	type
3. #491×C₁₃₁	C₂	direct (N × H)
54. C₂×C₁₃₁	#491	direct (N × H)
57. C₁₃₁	C₂×#491	direct (N × H)
58. C₂×#491	C₁₃₁	direct (N × H)
61. #491	C₂×C₁₃₁	direct (N × H)
112. C₂	#491×C₁₃₁	direct (N × H)

H ≅ G/N

type

3. #491×C₁₃₁

C₂

direct (N × H)

54. C₂×C₁₃₁

#491

direct (N × H)

57. C₁₃₁

C₂×#491

direct (N × H)

58. C₂×#491

C₁₃₁

direct (N × H)

61. #491

C₂×C₁₃₁

direct (N × H)

112. C₂

#491×C₁₃₁

direct (N × H)

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

subgroup	index
C₂×C₄⋊Q₈×C₁₃₁	2
C₂×#125×C₁₃₁	2
#491×C₁₃₁	2
C₂×#103×C₁₃₁	2
C₂×#97×C₁₃₁	2
C₂×#491	131

subgroup

index

C₂×C₄⋊Q₈×C₁₃₁

C₂×#125×C₁₃₁

#491×C₁₃₁

C₂×#103×C₁₃₁

C₂×#97×C₁₃₁

C₂×#491

131

Subgroups — C₂×#491×C₁₃₁ #30663317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#491×C131 #30663317

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#491×C₁₃₁ #30663317