Group #27142677 — subgroups

Subgroups — #27142677

Order 12416.

280 conjugacy classes of subgroups; only the 121 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 121 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⋊#11715895	C₂	1	1	yes	maximal, characteristic
2	C₂×#15690721	C₂	1	1	yes	maximal, characteristic
3	#20260490	C₂	1	4	yes	maximal
4	C₂×C₄×Q₈×C₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11715895	C₂²	1	2	yes	—
6	C₂²×#11715895	C₂²	1	2	yes	—
7	C₄⋊#11715895	—	2	2	no	—
8	#15690721	—	2	2	no	—
9	C₂²×Q₈×C₉₇	C₄	1	1	yes	characteristic
10	C₄×Q₈×C₉₇	—	2	2	no	—
11	C₄×Q₈×C₉₇	—	2	2	no	—
12	C₂×C₄⋊C₄×C₉₇	C₄	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₉₇	C₄	1	2	yes	—
14	C₂×C₄²×C₉₇	C₂²	1	2	yes	—
15	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
16	C₂×#11715895	—	2	2	no	—
17	C₂×#11715895	—	2	2	no	—
18	C₂×#11715895	—	2	2	no	—
19	C₂×#11715895	—	2	4	no	—
20	C₂×#11715895	Q₈	1	4	yes	complemented
21	C₂×Q₈×C₉₇	—	2	2	no	—
22	C₂×Q₈×C₉₇	—	2	4	no	—
23	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₉₇	C₂×C₄	1	1	yes	characteristic
26	C₂²×C₄×C₉₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₉₇	—	2	4	no	—
29	C₄⋊C₄×C₉₇	—	2	2	no	—
30	C₄²×C₉₇	—	2	2	no	—
31	C₄²×C₉₇	—	2	2	no	—
32	C₄²×C₉₇	D₄	1	2	yes	—
33	C₄²×C₉₇	D₄	1	2	yes	—
34	C₂³×C₉₇	C₂²×C₄	1	1	yes	characteristic
35	Q₈×C₉₇	—	2	8	no	—
36	C₂×C₄×C₉₇	—	2	2	no	—
37	C₂×C₄×C₉₇	—	2	2	no	—
38	C₂×C₄×C₉₇	—	2	2	no	—
39	C₂×C₄×C₉₇	—	2	2	no	—
40	C₂×C₄×C₉₇	—	2	2	no	—
41	C₂×C₄×C₉₇	—	2	4	no	—
42	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₉₇	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₉₇	M₁₆	1	4	yes	—
45	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₉₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
48	C₂×C₄×C₉₇	#41	1	2	yes	—
49	#11715895	—	2	4	no	—
50	#11715895	—	4	2	no	—
51	C₂²×C₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂²×C₉₇	C₄×Q₈	1	2	yes	—
53	C₂²×C₉₇	C₂×M₁₆	1	2	yes	—
54	C₂²×C₉₇	#131	1	2	yes	—
55	C₄×C₉₇	—	2	2	no	—
56	C₄×C₉₇	—	2	4	no	—
57	C₄×C₉₇	—	2	2	no	—
58	C₄×C₉₇	—	4	2	no	—
59	C₄×C₉₇	C₂²⋊Q₈	1	2	yes	—
60	C₄×C₉₇	#124	1	1	yes	characteristic
61	C₄×C₉₇	C₄⋊Q₈	1	1	yes	characteristic
62	C₂×C₉₇	#391	1	1	yes	characteristic
63	C₂×C₉₇	#407	1	1	yes	characteristic
64	C₂×C₉₇	#408	1	1	yes	characteristic
65	C₂×C₉₇	#446	1	4	yes	—
66	#1985	—	97	1	no	maximal, Sylow
67	C₉₇	#1985	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₄×Q₈	D₉₇	1	1	yes	characteristic
69	C₂×C₄⋊C₈	—	97	1	no	—
70	C₂×#97	—	97	1	no	—
71	#336	—	97	4	no	—
72	C₂²×Q₈	Dic₉₇	1	1	yes	characteristic
73	C₂²×C₈	—	97	2	no	—
74	C₂²×C₈	—	97	2	no	—
75	C₄×Q₈	—	2	2	no	—
76	C₄×Q₈	—	2	2	no	—
77	C₂×C₄⋊C₄	Dic₉₇	1	1	yes	characteristic
78	C₂×C₄⋊C₄	Dic₉₇	1	2	yes	—
79	C₂×C₄²	C₂×D₉₇	1	2	yes	—
80	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
81	C₄⋊C₈	—	194	2	no	—
82	#97	—	194	2	no	—
83	C₂×Q₈	—	2	2	no	—
84	C₂×Q₈	—	2	4	no	—
85	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
86	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
87	C₂²×C₄	C₂×Dic₉₇	1	1	yes	characteristic
88	C₂²×C₄	C₂×Dic₉₇	1	2	yes	—
89	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
90	C₂×C₈	—	97	4	no	—
91	C₂×C₈	—	194	2	no	—
92	C₂×C₈	—	194	4	no	—
93	C₂×C₈	—	194	2	no	—
94	C₂×C₈	—	194	2	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄²	—	2	2	no	—
98	C₄²	—	2	2	no	—
99	C₄²	C₂²⋊D₉₇	1	2	yes	—
100	C₄²	C₂²⋊D₉₇	1	2	yes	—
101	C₂³	C₂²×Dic₉₇	1	1	yes	characteristic
102	Q₈	—	2	8	no	—
103	C₂×C₄	—	2	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	C₂²⋊Dic₉₇	1	2	yes	—
110	C₂×C₄	C₂²⋊Dic₉₇	1	2	yes	—
111	C₂×C₄	#13095495	1	4	yes	—
112	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
113	C₂×C₄	C₂×C₂²⋊D₉₇	1	1	yes	characteristic
114	C₂×C₄	Q₈×D₉₇	1	2	yes	—
115	C₂×C₄	#13095512	1	2	yes	—
116	C₈	—	388	2	no	—
117	C₈	—	194	4	no	—
118	C₂²	C₂×C₂²⋊Dic₉₇	1	1	yes	characteristic
119	C₂²	Q₈×Dic₉₇	1	2	yes	—
120	C₂²	C₂×#13095495	1	2	yes	—
121	C₂²	#15690817	1	2	yes	—
122	C₄	—	2	2	no	—
123	C₄	—	2	4	no	—
124	C₄	—	2	2	no	—
125	C₄	—	4	2	no	—
126	C₄	#15690786	1	2	yes	—
127	C₄	#15690799	1	1	yes	characteristic
128	C₄	#15690808	1	1	yes	characteristic
129	C₂	#20260642	1	1	yes	characteristic
130	C₂	#20260699	1	1	yes	characteristic
131	C₂	#20260702	1	1	yes	characteristic
132	C₂	#20260866	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#11715895

C₂

yes

maximal, characteristic

C₂×#15690721

C₂

yes

maximal, characteristic

#20260490

C₂

yes

maximal

C₂×C₄×Q₈×C₉₇

C₂

yes

maximal, characteristic

C₂²×#11715895

C₂²

yes

—

C₂²×#11715895

C₂²

yes

—

C₄⋊#11715895

—

#15690721

—

C₂²×Q₈×C₉₇

C₄

yes

characteristic

C₄×Q₈×C₉₇

—

C₄×Q₈×C₉₇

—

C₂×C₄⋊C₄×C₉₇

C₄

yes

characteristic

C₂×C₄⋊C₄×C₉₇

C₄

yes

—

C₂×C₄²×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

—

—

—

—

Q₈

yes

complemented

C₂×Q₈×C₉₇

—

C₂×Q₈×C₉₇

—

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₉₇

C₂×C₄

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

—

C₄²×C₉₇

—

C₄²×C₉₇

D₄

yes

—

C₄²×C₉₇

D₄

yes

—

C₂³×C₉₇

C₂²×C₄

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

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₉₇

M₁₆

yes

—

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×C₄×C₉₇

#41

yes

—

#11715895

—

#11715895

—

C₂²×C₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₉₇

C₄×Q₈

yes

—

C₂²×C₉₇

C₂×M₁₆

yes

—

C₂²×C₉₇

#131

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₂²⋊Q₈

yes

—

C₄×C₉₇

#124

yes

characteristic

C₄×C₉₇

C₄⋊Q₈

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

C₂×C₉₇

yes

—

#1985

—

maximal, Sylow

C₉₇

#1985

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

D₉₇

yes

characteristic

C₂×C₄⋊C₈

—

C₂×#97

—

#336

—

C₂²×Q₈

Dic₉₇

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

Dic₉₇

yes

characteristic

C₂×C₄⋊C₄

Dic₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

—

C₂×C₄²

C₂×D₉₇

yes

characteristic

C₄⋊C₈

—

194

—

#97

—

194

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂×Dic₉₇

yes

characteristic

C₂²×C₄

C₂×Dic₉₇

yes

characteristic

C₂²×C₄

C₂×Dic₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₂×C₈

—

194

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

C₄²

—

C₄²

C₂²⋊D₉₇

yes

—

100

C₄²

C₂²⋊D₉₇

yes

—

101

C₂³

C₂²×Dic₉₇

yes

characteristic

102

Q₈

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂²⋊Dic₉₇

yes

—

110

C₂×C₄

C₂²⋊Dic₉₇

yes

—

111

C₂×C₄

#13095495

yes

—

112

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

113

C₂×C₄

C₂×C₂²⋊D₉₇

yes

characteristic

114

C₂×C₄

Q₈×D₉₇

yes

—

115

C₂×C₄

#13095512

yes

—

116

C₈

—

388

—

117

C₈

—

194

—

118

C₂²

C₂×C₂²⋊Dic₉₇

yes

characteristic

119

C₂²

Q₈×Dic₉₇

yes

—

120

C₂²

C₂×#13095495

yes

—

121

C₂²

#15690817

yes

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#15690786

yes

—

127

C₄

#15690799

yes

characteristic

128

C₄

#15690808

yes

characteristic

129

C₂

#20260642

yes

characteristic

130

C₂

#20260699

yes

characteristic

131

C₂

#20260702

yes

characteristic

132

C₂

#20260866

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₉₇
commutator, socle, prime radical	C₂³×C₉₇	C₂²×C₄
Fitting	C₂×C₄×Q₈×C₉₇	C₂
cosocle	C₂²×C₄×C₉₇	C₂³
hypercenter	C₂×C₄×Q₈	D₉₇
hypocenter	C₉₇	#1985
LCS	G ▷ C₂³×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₉₇

commutator, socle, prime radical

C₂³×C₉₇

C₂²×C₄

Fitting

C₂×C₄×Q₈×C₉₇

C₂

cosocle

C₂²×C₄×C₉₇

C₂³

hypercenter

C₂×C₄×Q₈

D₉₇

C₉₇

G ▷ C₂³×C₉₇ ▷ C₉₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄×Q₈

N	H ≅ G/N	type
20. C₂×#11715895	Q₈	semidirect
67. C₉₇	#1985	semidirect

H ≅ G/N

type

20. C₂×#11715895

Q₈

semidirect

67. C₉₇

#1985

semidirect

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

subgroup	index
C₂×C₄⋊#11715895	2
C₂×#15690721	2
#20260490	2
C₂×C₄×Q₈×C₉₇	2
#1985	97

subgroup

index

C₂×C₄×Q₈×C₉₇

Subgroups — #27142677

Conjugacy classes of subgroups

Complements

Maximal subgroups