Group #27137582 — subgroups

Subgroups — C₉₇×#1624 #27137582

Order 12416.

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

normal subgroups only

This group has 240 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	#555×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
2	#492×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
3	#459×C₉₇	C₂	1	2	yes	maximal, complemented
4	#333×C₉₇	C₂	1	1	yes	maximal, characteristic, complemented
5	D₄⋊C₈×C₉₇	C₂	1	2	yes	maximal, complemented
6	C₂×SD₁₆×C₉₇	—	2	2	no	—
7	C₂×D₈×C₉₇	—	2	2	no	—
8	C₄⋊Q₈×C₉₇	C₂²	1	1	yes	characteristic
9	#127×C₉₇	C₂²	1	1	yes	characteristic
10	#124×C₉₇	—	2	2	no	—
11	#121×C₉₇	—	2	2	no	—
12	C₄×D₄×C₉₇	C₂²	1	2	yes	—
13	#117×C₉₇	—	2	2	no	—
14	C₄⋊C₈×C₉₇	C₂²	1	2	yes	—
15	#102×C₉₇	—	2	2	no	—
16	C₄×C₈×C₉₇	C₂²	1	1	yes	characteristic
17	C₂×Q₈×C₉₇	D₄	1	2	yes	—
18	C₂×D₄×C₉₇	—	2	2	no	—
19	C₂×D₄×C₉₇	—	2	2	no	—
20	C₂×D₄×C₉₇	—	2	1	no	—
21	C₂×D₄×C₉₇	D₄	1	2	yes	—
22	C₂²×C₄×C₉₇	—	2	2	no	—
23	SD₁₆×C₉₇	—	4	4	no	—
24	D₈×C₉₇	—	4	4	no	—
25	C₂×C₈×C₉₇	—	2	1	no	—
26	C₂×C₈×C₉₇	—	2	2	no	—
27	C₄⋊C₄×C₉₇	—	2	2	no	—
28	C₄⋊C₄×C₉₇	D₄	1	2	yes	—
29	C₂²⋊C₄×C₉₇	—	2	4	no	—
30	C₂²⋊C₄×C₉₇	—	2	2	no	—
31	C₄²×C₉₇	—	2	2	no	—
32	C₄²×C₉₇	C₂³	1	1	yes	characteristic
33	C₂³×C₉₇	—	2	2	no	—
34	C₂³×C₉₇	—	4	1	no	—
35	Q₈×C₉₇	—	4	2	no	—
36	D₄×C₉₇	—	2	2	no	—
37	D₄×C₉₇	—	2	2	no	—
38	D₄×C₉₇	—	4	2	no	—
39	D₄×C₉₇	—	4	2	no	—
40	D₄×C₉₇	—	4	2	no	—
41	D₄×C₉₇	—	8	2	no	—
42	C₂×C₄×C₉₇	—	2	2	no	—
43	C₂×C₄×C₉₇	—	2	2	no	—
44	C₂×C₄×C₉₇	—	4	2	no	—
45	C₂×C₄×C₉₇	—	4	2	no	—
46	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
48	C₈×C₉₇	—	2	2	no	—
49	C₈×C₉₇	—	4	2	no	—
50	C₂²×C₉₇	—	4	2	no	—
51	C₂²×C₉₇	—	4	2	no	—
52	C₂²×C₉₇	—	4	2	no	—
53	C₂²×C₉₇	—	8	2	no	—
54	C₂²×C₉₇	—	8	1	no	—
55	C₂²×C₉₇	#120	1	1	yes	characteristic
56	C₄×C₉₇	—	2	1	no	—
57	C₄×C₉₇	—	4	2	no	—
58	C₄×C₉₇	—	4	2	no	—
59	C₄×C₉₇	#135	1	2	yes	—
60	C₄×C₉₇	AGL(1,ℤ/8)	1	2	yes	—
61	C₂×C₉₇	—	8	2	no	—
62	C₂×C₉₇	—	16	1	no	—
63	C₂×C₉₇	#449	1	2	yes	—
64	C₂×C₉₇	#453	1	1	yes	characteristic
65	#1624	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₉₇	#1624	1	1	yes	characteristic, direct product factor, Sylow
67	#555	C₂×C₉₇	1	1	yes	characteristic, complemented
68	#492	C₂×C₉₇	1	1	yes	characteristic, complemented
69	#459	C₂×C₉₇	1	2	yes	complemented
70	#333	C₂×C₉₇	1	1	yes	characteristic, complemented
71	D₄⋊C₈	C₂×C₉₇	1	2	yes	complemented
72	C₂×SD₁₆	—	2	2	no	—
73	C₂×D₈	—	2	2	no	—
74	C₄⋊Q₈	C₂²×C₉₇	1	1	yes	characteristic
75	#127	C₂²×C₉₇	1	1	yes	characteristic
76	#124	—	2	2	no	—
77	#121	—	2	2	no	—
78	C₄×D₄	C₂²×C₉₇	1	2	yes	—
79	#117	—	2	2	no	—
80	C₄⋊C₈	C₂²×C₉₇	1	2	yes	—
81	#102	—	2	2	no	—
82	C₄×C₈	C₂²×C₉₇	1	1	yes	characteristic
83	C₂×Q₈	D₄×C₉₇	1	2	yes	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	2	no	—
86	C₂×D₄	—	2	1	no	—
87	C₂×D₄	D₄×C₉₇	1	2	yes	—
88	C₂²×C₄	—	2	2	no	—
89	SD₁₆	—	4	4	no	—
90	D₈	—	4	4	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	2	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	D₄×C₉₇	1	2	yes	—
95	C₂²⋊C₄	—	2	4	no	—
96	C₂²⋊C₄	—	2	2	no	—
97	C₄²	—	2	2	no	—
98	C₄²	C₂³×C₉₇	1	1	yes	characteristic
99	C₂³	—	2	2	no	—
100	C₂³	—	4	1	no	—
101	Q₈	—	4	2	no	—
102	D₄	—	2	2	no	—
103	D₄	—	2	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	4	2	no	—
106	D₄	—	4	2	no	—
107	D₄	—	8	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	4	2	no	—
111	C₂×C₄	—	4	2	no	—
112	C₂×C₄	C₂×D₄×C₉₇	1	1	yes	characteristic
113	C₂×C₄	C₂×D₄×C₉₇	1	2	yes	—
114	C₈	—	2	2	no	—
115	C₈	—	4	2	no	—
116	C₂²	—	4	2	no	—
117	C₂²	—	4	2	no	—
118	C₂²	—	4	2	no	—
119	C₂²	—	8	2	no	—
120	C₂²	—	8	1	no	—
121	C₂²	#120×C₉₇	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	4	2	no	—
124	C₄	—	4	2	no	—
125	C₄	#135×C₉₇	1	2	yes	—
126	C₄	AGL(1,ℤ/8)×C₉₇	1	2	yes	—
127	C₂	—	8	2	no	—
128	C₂	—	16	1	no	—
129	C₂	#449×C₉₇	1	2	yes	—
130	C₂	#453×C₉₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#555×C₉₇

C₂

yes

maximal, characteristic, complemented

#492×C₉₇

C₂

yes

maximal, characteristic, complemented

#459×C₉₇

C₂

yes

maximal, complemented

#333×C₉₇

C₂

yes

maximal, characteristic, complemented

D₄⋊C₈×C₉₇

C₂

yes

maximal, complemented

C₂×SD₁₆×C₉₇

—

C₂×D₈×C₉₇

—

C₄⋊Q₈×C₉₇

C₂²

yes

characteristic

#127×C₉₇

C₂²

yes

characteristic

#124×C₉₇

—

#121×C₉₇

—

C₄×D₄×C₉₇

C₂²

yes

—

#117×C₉₇

—

C₄⋊C₈×C₉₇

C₂²

yes

—

#102×C₉₇

—

C₄×C₈×C₉₇

C₂²

yes

characteristic

C₂×Q₈×C₉₇

D₄

yes

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

—

C₂×D₄×C₉₇

D₄

yes

—

C₂²×C₄×C₉₇

—

SD₁₆×C₉₇

—

D₈×C₉₇

—

C₂×C₈×C₉₇

—

C₂×C₈×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

D₄

yes

—

C₂²⋊C₄×C₉₇

—

C₂²⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₄²×C₉₇

C₂³

yes

characteristic

C₂³×C₉₇

—

C₂³×C₉₇

—

Q₈×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

D₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₈×C₉₇

—

C₈×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

—

C₂²×C₉₇

#120

yes

characteristic

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

#135

yes

—

C₄×C₉₇

AGL(1,ℤ/8)

yes

—

C₂×C₉₇

—

C₂×C₉₇

—

C₂×C₉₇

#449

yes

—

C₂×C₉₇

#453

yes

characteristic

#1624

C₉₇

yes

maximal, characteristic, direct product factor, Sylow

C₉₇

#1624

yes

characteristic, direct product factor, Sylow

#555

C₂×C₉₇

yes

characteristic, complemented

#492

C₂×C₉₇

yes

characteristic, complemented

#459

C₂×C₉₇

yes

complemented

#333

C₂×C₉₇

yes

characteristic, complemented

D₄⋊C₈

C₂×C₉₇

yes

complemented

C₂×SD₁₆

—

C₂×D₈

—

C₄⋊Q₈

C₂²×C₉₇

yes

characteristic

#127

C₂²×C₉₇

yes

characteristic

#124

—

#121

—

C₄×D₄

C₂²×C₉₇

yes

—

#117

—

C₄⋊C₈

C₂²×C₉₇

yes

—

#102

—

C₄×C₈

C₂²×C₉₇

yes

characteristic

C₂×Q₈

D₄×C₉₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

D₄×C₉₇

yes

—

C₂²×C₄

—

SD₁₆

—

D₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

D₄×C₉₇

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂³×C₉₇

yes

characteristic

C₂³

—

100

C₂³

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

106

D₄

—

107

D₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

C₂×D₄×C₉₇

yes

characteristic

113

C₂×C₄

C₂×D₄×C₉₇

yes

—

114

C₈

—

115

C₈

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

119

C₂²

—

120

C₂²

—

121

C₂²

#120×C₉₇

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#135×C₉₇

yes

—

126

C₄

AGL(1,ℤ/8)×C₉₇

yes

—

127

C₂

—

128

C₂

—

129

C₂

#449×C₉₇

yes

—

130

C₂

#453×C₉₇

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₉₇	#120
commutator, Frattini, cosocle	C₄²	C₂³×C₉₇
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₉₇ ◁ C₄²×C₉₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₉₇

#120

commutator, Frattini, cosocle

C₄²

C₂³×C₉₇

LCS

G ▷ C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₉₇ ◁ C₄²×C₉₇ ◁ G

N	H ≅ G/N	type
1. #555×C₉₇	C₂	semidirect
2. #492×C₉₇	C₂	semidirect
3. #459×C₉₇	C₂	semidirect
4. #333×C₉₇	C₂	semidirect
5. D₄⋊C₈×C₉₇	C₂	semidirect
65. #1624	C₉₇	direct (N × H)
66. C₉₇	#1624	direct (N × H)
67. #555	C₂×C₉₇	semidirect
68. #492	C₂×C₉₇	semidirect
69. #459	C₂×C₉₇	semidirect
70. #333	C₂×C₉₇	semidirect
71. D₄⋊C₈	C₂×C₉₇	semidirect

H ≅ G/N

type

1. #555×C₉₇

C₂

semidirect

2. #492×C₉₇

C₂

semidirect

3. #459×C₉₇

C₂

semidirect

4. #333×C₉₇

C₂

semidirect

5. D₄⋊C₈×C₉₇

C₂

semidirect

65. #1624

C₉₇

direct (N × H)

66. C₉₇

#1624

direct (N × H)

67. #555

C₂×C₉₇

semidirect

68. #492

C₂×C₉₇

semidirect

69. #459

C₂×C₉₇

semidirect

70. #333

C₂×C₉₇

semidirect

71. D₄⋊C₈

C₂×C₉₇

semidirect

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

subgroup	index
#555×C₉₇	2
#492×C₉₇	2
#459×C₉₇	2
#333×C₉₇	2
D₄⋊C₈×C₉₇	2
#1624	97

subgroup

index

#555×C₉₇

#492×C₉₇

#459×C₉₇

#333×C₉₇

D₄⋊C₈×C₉₇

#1624

Subgroups — C₉₇×#1624 #27137582

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C97×#1624 #27137582

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#1624 #27137582