Group #27137646 — subgroups

Subgroups — C₉₇×#1688 #27137646

Order 12416.

184 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 68 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	#493×C₉₇	C₂	1	1	yes	maximal, characteristic
2	#460×C₉₇	C₂	1	2	yes	maximal, complemented
3	#446×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
4	#334×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	#331×C₉₇	C₂	1	2	yes	maximal, complemented
6	C₂×SD₁₆×C₉₇	—	2	2	no	—
7	C₂×D₈×C₉₇	—	2	1	no	—
8	C₂×D₈×C₉₇	—	2	1	no	—
9	#127×C₉₇	C₂²	1	2	yes	—
10	C₄×Q₈×C₉₇	C₂²	1	1	yes	characteristic
11	C₄⋊C₈×C₉₇	—	2	1	no	—
12	C₄⋊C₈×C₉₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₉₇	C₂²	1	2	yes	—
14	#102×C₉₇	—	2	2	no	—
15	#97×C₉₇	—	2	1	no	—
16	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₉₇	D₄	1	1	yes	characteristic
18	C₂×D₄×C₉₇	—	2	2	no	—
19	C₂×D₄×C₉₇	—	2	2	no	—
20	C₂×D₄×C₉₇	—	2	2	no	—
21	SD₁₆×C₉₇	—	4	4	no	—
22	D₈×C₉₇	—	4	2	no	—
23	D₈×C₉₇	—	4	2	no	—
24	C₂×C₈×C₉₇	—	2	1	no	—
25	C₂×C₈×C₉₇	—	2	2	no	—
26	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
27	C₂×C₈×C₉₇	D₄	1	1	yes	characteristic
28	C₄⋊C₄×C₉₇	—	2	1	no	—
29	C₄⋊C₄×C₉₇	D₄	1	1	yes	characteristic
30	C₄²×C₉₇	—	2	1	no	—
31	C₄²×C₉₇	C₂³	1	1	yes	characteristic
32	C₂³×C₉₇	—	4	2	no	—
33	Q₈×C₉₇	—	2	2	no	—
34	D₄×C₉₇	—	4	2	no	—
35	D₄×C₉₇	—	4	2	no	—
36	D₄×C₉₇	—	4	2	no	—
37	D₄×C₉₇	—	4	2	no	—
38	D₄×C₉₇	—	8	2	no	—
39	C₂×C₄×C₉₇	—	2	1	no	—
40	C₂×C₄×C₉₇	—	2	1	no	—
41	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₉₇	#41	1	1	yes	characteristic
44	C₈×C₉₇	—	2	1	no	—
45	C₈×C₉₇	—	4	1	no	—
46	C₈×C₉₇	—	4	2	no	—
47	C₈×C₉₇	SD₁₆	1	2	yes	complemented
48	C₂²×C₉₇	—	8	2	no	—
49	C₂²×C₉₇	—	8	2	no	—
50	C₂²×C₉₇	—	8	2	no	—
51	C₂²×C₉₇	#121	1	1	yes	characteristic
52	C₄×C₉₇	—	2	1	no	—
53	C₄×C₉₇	—	4	1	no	—
54	C₄×C₉₇	—	4	1	no	—
55	C₄×C₉₇	C₂×SD₁₆	1	1	yes	characteristic
56	C₄×C₉₇	AGL(1,ℤ/8)	1	1	yes	characteristic
57	C₄×C₉₇	AGL(1,ℤ/8)	1	2	yes	—
58	C₂×C₉₇	—	16	2	no	—
59	C₂×C₉₇	#460	1	1	yes	characteristic
60	C₂×C₉₇	#469	1	1	yes	characteristic
61	C₂×C₉₇	#472	1	1	yes	characteristic
62	#1688	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₉₇	#1688	1	1	yes	characteristic, direct product factor, Sylow
64	#493	C₂×C₉₇	1	1	yes	characteristic
65	#460	C₂×C₉₇	1	2	yes	complemented
66	#446	C₂×C₉₇	1	1	yes	characteristic, complemented
67	#334	C₂×C₉₇	1	1	yes	characteristic, complemented
68	#331	C₂×C₉₇	1	2	yes	complemented
69	C₂×SD₁₆	—	2	2	no	—
70	C₂×D₈	—	2	1	no	—
71	C₂×D₈	—	2	1	no	—
72	#127	C₂²×C₉₇	1	2	yes	—
73	C₄×Q₈	C₂²×C₉₇	1	1	yes	characteristic
74	C₄⋊C₈	—	2	1	no	—
75	C₄⋊C₈	C₂²×C₉₇	1	1	yes	characteristic
76	C₄⋊C₈	C₂²×C₉₇	1	2	yes	—
77	#102	—	2	2	no	—
78	#97	—	2	1	no	—
79	C₄×C₈	C₂²×C₉₇	1	1	yes	characteristic
80	C₂×Q₈	D₄×C₉₇	1	1	yes	characteristic
81	C₂×D₄	—	2	2	no	—
82	C₂×D₄	—	2	2	no	—
83	C₂×D₄	—	2	2	no	—
84	SD₁₆	—	4	4	no	—
85	D₈	—	4	2	no	—
86	D₈	—	4	2	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	D₄×C₉₇	1	1	yes	characteristic
90	C₂×C₈	D₄×C₉₇	1	1	yes	characteristic
91	C₄⋊C₄	—	2	1	no	—
92	C₄⋊C₄	D₄×C₉₇	1	1	yes	characteristic
93	C₄²	—	2	1	no	—
94	C₄²	C₂³×C₉₇	1	1	yes	characteristic
95	C₂³	—	4	2	no	—
96	Q₈	—	2	2	no	—
97	D₄	—	4	2	no	—
98	D₄	—	4	2	no	—
99	D₄	—	4	2	no	—
100	D₄	—	4	2	no	—
101	D₄	—	8	2	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
105	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
106	C₂×C₄	#41×C₉₇	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	4	2	no	—
109	C₈	—	4	1	no	—
110	C₈	SD₁₆×C₉₇	1	2	yes	complemented
111	C₂²	—	8	2	no	—
112	C₂²	—	8	2	no	—
113	C₂²	—	8	2	no	—
114	C₂²	#121×C₉₇	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	4	1	no	—
117	C₄	—	4	1	no	—
118	C₄	C₂×SD₁₆×C₉₇	1	1	yes	characteristic
119	C₄	AGL(1,ℤ/8)×C₉₇	1	1	yes	characteristic
120	C₄	AGL(1,ℤ/8)×C₉₇	1	2	yes	—
121	C₂	—	16	2	no	—
122	C₂	#460×C₉₇	1	1	yes	characteristic
123	C₂	#469×C₉₇	1	1	yes	characteristic
124	C₂	#472×C₉₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#493×C₉₇

C₂

yes

maximal, characteristic

#460×C₉₇

C₂

yes

maximal, complemented

#446×C₉₇

C₂

yes

maximal, characteristic, complemented

#334×C₉₇

C₂

yes

maximal, characteristic, complemented

#331×C₉₇

C₂

yes

maximal, complemented

C₂×SD₁₆×C₉₇

—

C₂×D₈×C₉₇

—

C₂×D₈×C₉₇

—

#127×C₉₇

C₂²

yes

—

C₄×Q₈×C₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₉₇

—

C₄⋊C₈×C₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₉₇

C₂²

yes

—

#102×C₉₇

—

#97×C₉₇

—

C₄×C₈×C₉₇

C₂²

yes

characteristic

C₂×Q₈×C₉₇

D₄

yes

characteristic

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

SD₁₆×C₉₇

—

D₈×C₉₇

—

D₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

D₄

yes

characteristic

C₂×C₈×C₉₇

D₄

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

D₄

yes

characteristic

C₄²×C₉₇

—

C₄²×C₉₇

C₂³

yes

characteristic

C₂³×C₉₇

—

Q₈×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

#41

yes

characteristic

C₈×C₉₇

—

C₈×C₉₇

—

C₈×C₉₇

—

C₈×C₉₇

SD₁₆

yes

complemented

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

#121

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂×SD₁₆

yes

C₄×C₉₇

yes

C₄×C₉₇

yes

—

C₂×C₉₇

—

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₉₇

yes

maximal, characteristic, direct product factor, Sylow

C₉₇

#1688

yes

characteristic, direct product factor, Sylow

#493

C₂×C₉₇

yes

characteristic

#460

C₂×C₉₇

yes

complemented

#446

C₂×C₉₇

yes

characteristic, complemented

#334

C₂×C₉₇

yes

characteristic, complemented

#331

C₂×C₉₇

yes

complemented

C₂×SD₁₆

—

C₂×D₈

—

C₂×D₈

—

#127

C₂²×C₉₇

yes

—

C₄×Q₈

C₂²×C₉₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₉₇

yes

characteristic

C₄⋊C₈

C₂²×C₉₇

yes

—

#102

—

#97

—

C₄×C₈

C₂²×C₉₇

yes

characteristic

C₂×Q₈

D₄×C₉₇

yes

characteristic

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

SD₁₆

—

D₈

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×C₉₇

yes

characteristic

C₂×C₈

D₄×C₉₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

D₄×C₉₇

yes

characteristic

C₄²

—

C₄²

C₂³×C₉₇

yes

characteristic

C₂³

—

Q₈

—

D₄

—

D₄

—

D₄

—

100

D₄

—

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×D₄×C₉₇

yes

characteristic

105

C₂×C₄

C₂×D₄×C₉₇

yes

characteristic

106

C₂×C₄

#41×C₉₇

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

110

C₈

SD₁₆×C₉₇

yes

complemented

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

#121×C₉₇

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

C₂×SD₁₆×C₉₇

yes

characteristic

119

C₄

AGL(1,ℤ/8)×C₉₇

yes

characteristic

120

C₄

AGL(1,ℤ/8)×C₉₇

yes

—

121

C₂

—

122

C₂

#460×C₉₇

yes

characteristic

123

C₂

#469×C₉₇

yes

characteristic

124

C₂

#472×C₉₇

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₉₇	#121
commutator, Frattini, cosocle	C₄²	C₂³×C₉₇
prime radical	#493×C₉₇	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₉₇ ◁ C₄²×C₉₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₉₇

#121

commutator, Frattini, cosocle

C₄²

C₂³×C₉₇

prime radical

#493×C₉₇

C₂

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₉₇ ◁ C₄²×C₉₇ ◁ G

N	H ≅ G/N	type
2. #460×C₉₇	C₂	semidirect
3. #446×C₉₇	C₂	semidirect
4. #334×C₉₇	C₂	semidirect
5. #331×C₉₇	C₂	semidirect
47. C₈×C₉₇	SD₁₆	semidirect
62. #1688	C₉₇	direct (N × H)
63. C₉₇	#1688	direct (N × H)
65. #460	C₂×C₉₇	semidirect
66. #446	C₂×C₉₇	semidirect
67. #334	C₂×C₉₇	semidirect
68. #331	C₂×C₉₇	semidirect
110. C₈	SD₁₆×C₉₇	semidirect

H ≅ G/N

type

2. #460×C₉₇

C₂

semidirect

3. #446×C₉₇

C₂

semidirect

4. #334×C₉₇

C₂

semidirect

5. #331×C₉₇

C₂

semidirect

47. C₈×C₉₇

SD₁₆

semidirect

62. #1688

C₉₇

direct (N × H)

63. C₉₇

#1688

direct (N × H)

65. #460

C₂×C₉₇

semidirect

66. #446

C₂×C₉₇

semidirect

67. #334

C₂×C₉₇

semidirect

68. #331

C₂×C₉₇

semidirect

110. C₈

SD₁₆×C₉₇

semidirect

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

subgroup	index
#493×C₉₇	2
#460×C₉₇	2
#446×C₉₇	2
#334×C₉₇	2
#331×C₉₇	2
#1688	97

subgroup

index

#493×C₉₇

#460×C₉₇

#446×C₉₇

#334×C₉₇

#331×C₉₇

#1688

Subgroups — C₉₇×#1688 #27137646

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C97×#1688 #27137646

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#1688 #27137646