Group #27138285 — subgroups

Subgroups — C₉₇×#2327 #27138285

Order 12416.

516 conjugacy classes of subgroups; only the 296 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 296 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×C₉₇	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Q₈×C₉₇	C₂	1	2	yes	maximal
3	#391×C₉₇	C₂	1	2	yes	maximal
4	#387×C₉₇	C₂	1	8	yes	maximal
5	#382×C₉₇	C₂	1	1	yes	maximal, characteristic
6	#376×C₉₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Q₈×C₉₇	C₂²	1	2	yes	—
8	C₄⋊Q₈×C₉₇	C₄	1	8	yes	complemented
9	C₄×Q₈×C₉₇	—	2	8	no	—
10	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄×C₉₇	C₂²	1	4	yes	—
12	C₂×C₄⋊C₄×C₉₇	C₂²	1	4	yes	—
13	C₂×C₄²×C₉₇	C₂²	1	4	yes	—
14	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
15	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
16	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
17	#95×C₉₇	C₂²	1	4	yes	—
18	#95×C₉₇	C₂²	1	4	yes	—
19	#95×C₉₇	C₂²	1	8	yes	—
20	C₂×Q₈×C₉₇	—	2	4	no	—
21	C₂×Q₈×C₉₇	—	2	4	no	—
22	C₂×Q₈×C₉₇	C₂×C₄	1	8	yes	—
23	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
24	C₂²×C₄×C₉₇	C₂³	1	4	yes	—
25	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
26	C₂²×C₄×C₉₇	C₂³	1	4	yes	—
27	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
28	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₉₇	—	2	8	no	—
30	C₄⋊C₄×C₉₇	—	2	4	no	—
31	C₄⋊C₄×C₉₇	C₂×C₄	1	16	yes	—
32	C₄²×C₉₇	—	2	8	no	—
33	C₄²×C₉₇	—	2	2	no	—
34	C₄²×C₉₇	C₂×C₄	1	4	yes	—
35	C₄²×C₉₇	D₄	1	4	yes	—
36	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
37	Q₈×C₉₇	—	2	16	no	—
38	C₂×C₄×C₉₇	—	2	4	no	—
39	C₂×C₄×C₉₇	—	2	4	no	—
40	C₂×C₄×C₉₇	—	2	4	no	—
41	C₂×C₄×C₉₇	—	2	4	no	—
42	C₂×C₄×C₉₇	—	2	4	no	—
43	C₂×C₄×C₉₇	—	2	2	no	—
44	C₂×C₄×C₉₇	—	2	8	no	—
45	C₂×C₄×C₉₇	—	2	4	no	—
46	C₂×C₄×C₉₇	C₂²×C₄	1	4	yes	—
47	C₂×C₄×C₉₇	C₂²×C₄	1	8	yes	—
48	C₂×C₄×C₉₇	C₂²×C₄	1	2	yes	—
49	C₂×C₄×C₉₇	C₂×D₄	1	4	yes	—
50	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₉₇	#41	1	2	yes	—
52	C₂²×C₉₇	C₂³×C₄	1	1	yes	characteristic
53	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
54	C₂²×C₉₇	C₂×#41	1	1	yes	characteristic
55	C₂²×C₉₇	2₋¹⁺⁴	1	4	yes	—
56	C₄×C₉₇	—	2	8	no	—
57	C₄×C₉₇	—	2	4	no	—
58	C₄×C₉₇	—	2	4	no	—
59	C₄×C₉₇	—	4	4	no	—
60	C₄×C₉₇	—	4	2	no	—
61	C₄×C₉₇	C₄×D₄	1	4	yes	—
62	C₂×C₉₇	C₂×C₄×D₄	1	1	yes	characteristic
63	C₂×C₉₇	#519	1	2	yes	—
64	C₂×C₉₇	#536	1	2	yes	—
65	C₂×C₉₇	#541	1	2	yes	—
66	#2327	C₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₉₇	#2327	1	1	yes	characteristic, direct product factor, Sylow
68	C₂×C₄⋊Q₈	C₂×C₉₇	1	1	yes	characteristic
69	C₂×C₄×Q₈	C₂×C₉₇	1	2	yes	—
70	#391	C₂×C₉₇	1	2	yes	—
71	#387	C₂×C₉₇	1	8	yes	—
72	#382	C₂×C₉₇	1	1	yes	characteristic
73	#376	C₂×C₉₇	1	1	yes	characteristic
74	C₂²×Q₈	C₂²×C₉₇	1	2	yes	—
75	C₄⋊Q₈	C₄×C₉₇	1	8	yes	complemented
76	C₄×Q₈	—	2	8	no	—
77	C₂×C₄⋊C₄	C₂²×C₉₇	1	2	yes	—
78	C₂×C₄⋊C₄	C₂²×C₉₇	1	4	yes	—
79	C₂×C₄⋊C₄	C₂²×C₉₇	1	4	yes	—
80	C₂×C₄²	C₂²×C₉₇	1	4	yes	—
81	C₂×C₄²	C₂²×C₉₇	1	1	yes	characteristic
82	C₂×C₄²	C₂²×C₉₇	1	1	yes	characteristic
83	C₂×C₄²	C₂²×C₉₇	1	1	yes	characteristic
84	#95	C₂²×C₉₇	1	4	yes	—
85	#95	C₂²×C₉₇	1	4	yes	—
86	#95	C₂²×C₉₇	1	8	yes	—
87	C₂×Q₈	—	2	4	no	—
88	C₂×Q₈	—	2	4	no	—
89	C₂×Q₈	C₂×C₄×C₉₇	1	8	yes	—
90	C₂²×C₄	C₂³×C₉₇	1	2	yes	—
91	C₂²×C₄	C₂³×C₉₇	1	4	yes	—
92	C₂²×C₄	C₂³×C₉₇	1	2	yes	—
93	C₂²×C₄	C₂³×C₉₇	1	4	yes	—
94	C₂²×C₄	C₂³×C₉₇	1	2	yes	—
95	C₂²×C₄	C₂³×C₉₇	1	1	yes	characteristic
96	C₄⋊C₄	—	2	8	no	—
97	C₄⋊C₄	—	2	4	no	—
98	C₄⋊C₄	C₂×C₄×C₉₇	1	16	yes	—
99	C₄²	—	2	8	no	—
100	C₄²	—	2	2	no	—
101	C₄²	C₂×C₄×C₉₇	1	4	yes	—
102	C₄²	D₄×C₉₇	1	4	yes	—
103	C₂³	C₂⁴×C₉₇	1	1	yes	characteristic
104	Q₈	—	2	16	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	4	no	—
110	C₂×C₄	—	2	2	no	—
111	C₂×C₄	—	2	8	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	C₂²×C₄×C₉₇	1	4	yes	—
114	C₂×C₄	C₂²×C₄×C₉₇	1	8	yes	—
115	C₂×C₄	C₂²×C₄×C₉₇	1	2	yes	—
116	C₂×C₄	C₂×D₄×C₉₇	1	4	yes	—
117	C₂×C₄	C₂×D₄×C₉₇	1	2	yes	—
118	C₂×C₄	#41×C₉₇	1	2	yes	—
119	C₂²	C₂³×C₄×C₉₇	1	1	yes	characteristic
120	C₂²	C₂²×D₄×C₉₇	1	1	yes	characteristic
121	C₂²	C₂×#41×C₉₇	1	1	yes	characteristic
122	C₂²	2₋¹⁺⁴×C₉₇	1	4	yes	—
123	C₄	—	2	8	no	—
124	C₄	—	2	4	no	—
125	C₄	—	2	4	no	—
126	C₄	—	4	4	no	—
127	C₄	—	4	2	no	—
128	C₄	C₄×D₄×C₉₇	1	4	yes	—
129	C₂	C₂×C₄×D₄×C₉₇	1	1	yes	characteristic
130	C₂	#519×C₉₇	1	2	yes	—
131	C₂	#536×C₉₇	1	2	yes	—
132	C₂	#541×C₉₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Q₈×C₉₇

C₂

yes

maximal, characteristic

C₂×C₄×Q₈×C₉₇

C₂

yes

maximal

#391×C₉₇

C₂

yes

maximal

#387×C₉₇

C₂

yes

maximal

#382×C₉₇

C₂

yes

maximal, characteristic

#376×C₉₇

C₂

yes

maximal, characteristic

C₂²×Q₈×C₉₇

C₂²

yes

—

C₄⋊Q₈×C₉₇

C₄

yes

complemented

C₄×Q₈×C₉₇

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂×C₄²×C₉₇

C₂²

yes

characteristic

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#95×C₉₇

C₂²

yes

—

#95×C₉₇

C₂²

yes

—

#95×C₉₇

C₂²

yes

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

—

C₄²×C₉₇

—

C₄²×C₉₇

C₂×C₄

yes

—

C₄²×C₉₇

D₄

yes

—

C₂³×C₉₇

C₂⁴

yes

characteristic

Q₈×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂²×C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

—

C₂×C₄×C₉₇

C₂²×C₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

#41

yes

—

C₂²×C₉₇

C₂³×C₄

yes

characteristic

C₂²×C₉₇

C₂²×D₄

yes

characteristic

C₂²×C₉₇

C₂×#41

yes

characteristic

C₂²×C₉₇

2₋¹⁺⁴

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₄×D₄

yes

—

C₂×C₉₇

C₂×C₄×D₄

yes

characteristic

C₂×C₉₇

#519

yes

—

C₂×C₉₇

#536

yes

—

C₂×C₉₇

#541

yes

—

#2327

C₉₇

yes

maximal, characteristic, direct product factor, Sylow

C₉₇

#2327

yes

characteristic, direct product factor, Sylow

C₂×C₄⋊Q₈

C₂×C₉₇

yes

characteristic

C₂×C₄×Q₈

C₂×C₉₇

yes

—

#391

C₂×C₉₇

yes

—

#387

C₂×C₉₇

yes

—

#382

C₂×C₉₇

yes

characteristic

#376

C₂×C₉₇

yes

characteristic

C₂²×Q₈

C₂²×C₉₇

yes

—

C₄⋊Q₈

C₄×C₉₇

yes

complemented

C₄×Q₈

—

C₂×C₄⋊C₄

C₂²×C₉₇

yes

—

C₂×C₄⋊C₄

C₂²×C₉₇

yes

—

C₂×C₄⋊C₄

C₂²×C₉₇

yes

—

C₂×C₄²

C₂²×C₉₇

yes

—

C₂×C₄²

C₂²×C₉₇

yes

characteristic

C₂×C₄²

C₂²×C₉₇

yes

characteristic

C₂×C₄²

C₂²×C₉₇

yes

characteristic

#95

C₂²×C₉₇

yes

—

#95

C₂²×C₉₇

yes

—

#95

C₂²×C₉₇

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×C₄×C₉₇

yes

—

C₂²×C₄

C₂³×C₉₇

yes

—

C₂²×C₄

C₂³×C₉₇

yes

—

C₂²×C₄

C₂³×C₉₇

yes

—

C₂²×C₄

C₂³×C₉₇

yes

—

C₂²×C₄

C₂³×C₉₇

yes

—

C₂²×C₄

C₂³×C₉₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×C₄×C₉₇

yes

—

C₄²

—

100

C₄²

—

101

C₄²

C₂×C₄×C₉₇

yes

—

102

C₄²

D₄×C₉₇

yes

—

103

C₂³

C₂⁴×C₉₇

yes

characteristic

104

Q₈

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂²×C₄×C₉₇

yes

—

114

C₂×C₄

C₂²×C₄×C₉₇

yes

—

115

C₂×C₄

C₂²×C₄×C₉₇

yes

—

116

C₂×C₄

C₂×D₄×C₉₇

yes

—

117

C₂×C₄

C₂×D₄×C₉₇

yes

—

118

C₂×C₄

#41×C₉₇

yes

—

119

C₂²

C₂³×C₄×C₉₇

yes

characteristic

120

C₂²

C₂²×D₄×C₉₇

yes

characteristic

121

C₂²

C₂×#41×C₉₇

yes

characteristic

122

C₂²

2₋¹⁺⁴×C₉₇

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

C₄×D₄×C₉₇

yes

—

129

C₂

C₂×C₄×D₄×C₉₇

yes

characteristic

130

C₂

#519×C₉₇

yes

—

131

C₂

#536×C₉₇

yes

—

132

C₂

#541×C₉₇

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₉₇	C₂⁴
commutator	C₂²	C₂³×C₄×C₉₇
Frattini, cosocle	C₂³	C₂⁴×C₉₇
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₉₇ ◁ G

subgroup

G/H

center, socle, prime radical

C₂³×C₉₇

C₂⁴

commutator

C₂²

C₂³×C₄×C₉₇

Frattini, cosocle

C₂³

C₂⁴×C₉₇

LCS

G ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₉₇ ◁ G

N	H ≅ G/N	type
8. C₄⋊Q₈×C₉₇	C₄	semidirect
66. #2327	C₉₇	direct (N × H)
67. C₉₇	#2327	direct (N × H)
75. C₄⋊Q₈	C₄×C₉₇	semidirect

H ≅ G/N

type

8. C₄⋊Q₈×C₉₇

C₄

semidirect

66. #2327

C₉₇

direct (N × H)

67. C₉₇

#2327

direct (N × H)

75. C₄⋊Q₈

C₄×C₉₇

semidirect

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

subgroup	index
C₂×C₄⋊Q₈×C₉₇	2
C₂×C₄×Q₈×C₉₇	2
#391×C₉₇	2
#387×C₉₇	2
#382×C₉₇	2
#376×C₉₇	2
#2327	97

subgroup

index

C₂×C₄⋊Q₈×C₉₇

C₂×C₄×Q₈×C₉₇

#391×C₉₇

#387×C₉₇

#382×C₉₇

#376×C₉₇

#2327

Subgroups — C₉₇×#2327 #27138285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C97×#2327 #27138285

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₉₇×#2327 #27138285