Group #27150112 — subgroups

Subgroups — #27150112

Order 12416.

504 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 181 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×#15690809	C₂	1	2	yes	maximal
2	#20260639	C₂	1	8	yes	maximal
3	#20260601	C₂	1	1	yes	maximal, characteristic
4	#20260600	C₂	1	2	yes	maximal
5	C₄²×Dic₉₇	C₂	1	1	yes	maximal, characteristic
6	#382×C₉₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₉₄	C₂²	1	4	yes	—
8	#15690809	—	2	8	no	—
9	C₂×C₄⋊Dic₉₇	C₂²	1	2	yes	—
10	C₂×#13095490	C₂²	1	4	yes	—
11	C₂×C₄×Dic₉₇	C₂²	1	4	yes	—
12	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
13	C₂×C₄×Dic₉₇	C₂²	1	1	yes	characteristic
14	#15690717	C₂²	1	8	yes	—
15	#15690717	C₂²	1	4	yes	—
16	C₂×C₄⋊C₄×C₉₇	C₂²	1	2	yes	—
17	C₂×C₄²×C₉₇	C₂²	1	1	yes	characteristic
18	#95×C₉₇	C₂²	1	4	yes	—
19	C₂×Dic₁₉₄	—	2	8	no	—
20	C₂×Dic₁₉₄	—	2	16	no	—
21	C₂²×Dic₉₇	C₂³	1	4	yes	—
22	C₂²×Dic₉₇	C₂³	1	2	yes	—
23	C₂²×Dic₉₇	C₂³	1	2	yes	—
24	C₄⋊Dic₉₇	—	2	4	no	—
25	#13095490	—	2	8	no	—
26	C₄×Dic₉₇	—	2	8	no	—
27	C₄×Dic₉₇	D₄	1	4	yes	—
28	C₄×Dic₉₇	Q₈	1	4	yes	complemented
29	C₂²×C₄×C₉₇	C₂³	1	4	yes	—
30	C₂²×C₄×C₉₇	C₂³	1	2	yes	—
31	C₂²×C₄×C₉₇	C₂³	1	1	yes	characteristic
32	C₄⋊C₄×C₉₇	—	2	4	no	—
33	C₄²×C₉₇	Q₈	1	4	yes	complemented
34	C₂³×C₉₇	C₂⁴	1	1	yes	characteristic
35	C₂×C₄×C₉₇	—	2	4	no	—
36	C₂×C₄×C₉₇	—	2	4	no	—
37	C₂×C₄×C₉₇	—	2	8	no	—
38	C₂×C₄×C₉₇	C₂×D₄	1	2	yes	—
39	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
40	C₂×C₄×C₉₇	C₂×Q₈	1	4	yes	—
41	C₂×C₄×C₉₇	C₂×Q₈	1	2	yes	—
42	C₂×Dic₉₇	—	2	4	no	—
43	C₂×Dic₉₇	—	2	4	no	—
44	C₂×Dic₉₇	—	2	8	no	—
45	C₂×Dic₉₇	C₂×D₄	1	4	yes	—
46	C₂×Dic₉₇	C₂×Q₈	1	4	yes	—
47	C₂×Dic₉₇	#41	1	8	yes	—
48	Dic₁₉₄	—	4	16	no	—
49	C₂²×C₉₇	C₂²×D₄	1	1	yes	characteristic
50	C₂²×C₉₇	C₂²×Q₈	1	1	yes	characteristic
51	C₂²×C₉₇	C₂²×Q₈	1	1	yes	characteristic
52	C₂²×C₉₇	C₂×#41	1	4	yes	—
53	C₄×C₉₇	—	2	4	no	—
54	C₄×C₉₇	—	4	4	no	—
55	C₄×C₉₇	C₄⋊Q₈	1	4	yes	—
56	Dic₉₇	—	2	4	no	—
57	Dic₉₇	—	4	4	no	—
58	Dic₉₇	#124	1	8	yes	—
59	C₂×C₉₇	C₂×#124	1	2	yes	—
60	C₂×C₉₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
61	C₂×C₉₇	#533	1	2	yes	—
62	C₂×C₉₇	#533	1	2	yes	—
63	#2862	—	97	1	no	maximal, Sylow
64	C₉₇	#2862	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₄⋊Q₈	—	97	2	no	—
66	#391	—	97	8	no	—
67	#382	—	97	2	no	—
68	#382	—	97	1	no	—
69	#382	D₉₇	1	1	yes	characteristic
70	C₄³	—	97	1	no	—
71	C₂²×Q₈	—	97	4	no	—
72	C₄⋊Q₈	—	194	8	no	—
73	C₂×C₄⋊C₄	—	97	4	no	—
74	C₂×C₄⋊C₄	—	97	2	no	—
75	C₂×C₄⋊C₄	C₂×D₉₇	1	2	yes	—
76	C₂×C₄²	—	97	1	no	—
77	C₂×C₄²	—	97	1	no	—
78	C₂×C₄²	—	97	4	no	—
79	C₂×C₄²	C₂×D₉₇	1	1	yes	characteristic
80	#95	—	97	8	no	—
81	#95	—	97	4	no	—
82	#95	C₂×D₉₇	1	4	yes	—
83	C₂×Q₈	—	194	8	no	—
84	C₂×Q₈	—	194	16	no	—
85	C₂²×C₄	—	97	2	no	—
86	C₂²×C₄	—	97	2	no	—
87	C₂²×C₄	—	97	4	no	—
88	C₂²×C₄	C₂²×D₉₇	1	4	yes	—
89	C₂²×C₄	C₂²×D₉₇	1	2	yes	—
90	C₂²×C₄	C₂²×D₉₇	1	1	yes	characteristic
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	194	8	no	—
93	C₄⋊C₄	—	194	4	no	—
94	C₄²	—	97	4	no	—
95	C₄²	—	97	4	no	—
96	C₄²	—	194	8	no	—
97	C₄²	Dic₁₉₄	1	4	yes	complemented
98	C₂³	C₂³×D₉₇	1	1	yes	characteristic
99	Q₈	—	388	16	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	8	no	—
103	C₂×C₄	—	97	4	no	—
104	C₂×C₄	—	97	8	no	—
105	C₂×C₄	—	97	4	no	—
106	C₂×C₄	—	194	4	no	—
107	C₂×C₄	—	194	8	no	—
108	C₂×C₄	—	194	4	no	—
109	C₂×C₄	D₄×D₉₇	1	2	yes	—
110	C₂×C₄	Q₈×D₉₇	1	2	yes	—
111	C₂×C₄	C₂×Dic₁₉₄	1	4	yes	—
112	C₂×C₄	C₂×Dic₁₉₄	1	2	yes	—
113	C₂²	C₂×D₄×D₉₇	1	1	yes	characteristic
114	C₂²	C₂×Q₈×D₉₇	1	1	yes	characteristic
115	C₂²	C₂²×Dic₁₉₄	1	1	yes	characteristic
116	C₂²	#41×D₉₇	1	4	yes	—
117	C₄	—	2	4	no	—
118	C₄	—	4	4	no	—
119	C₄	—	97	8	no	—
120	C₄	—	388	4	no	—
121	C₄	—	194	4	no	—
122	C₄	#15690809	1	4	yes	—
123	C₂	#124×D₉₇	1	2	yes	—
124	C₂	C₂×#15690809	1	1	yes	characteristic
125	C₂	#20261288	1	2	yes	—
126	C₂	#20261291	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#15690809

C₂

yes

maximal

#20260639

C₂

yes

maximal

#20260601

C₂

yes

maximal, characteristic

#20260600

C₂

yes

maximal

C₄²×Dic₉₇

C₂

yes

maximal, characteristic

#382×C₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₉₄

C₂²

yes

—

#15690809

—

C₂×C₄⋊Dic₉₇

C₂²

yes

—

C₂×#13095490

C₂²

yes

—

C₂×C₄×Dic₉₇

C₂²

yes

—

C₂×C₄×Dic₉₇

C₂²

yes

characteristic

C₂×C₄×Dic₉₇

C₂²

yes

characteristic

#15690717

C₂²

yes

—

#15690717

C₂²

yes

—

C₂×C₄⋊C₄×C₉₇

C₂²

yes

—

C₂×C₄²×C₉₇

C₂²

yes

characteristic

#95×C₉₇

C₂²

yes

—

C₂×Dic₁₉₄

—

C₂×Dic₁₉₄

—

C₂²×Dic₉₇

C₂³

yes

—

C₂²×Dic₉₇

C₂³

yes

—

C₂²×Dic₉₇

C₂³

yes

—

C₄⋊Dic₉₇

—

#13095490

—

C₄×Dic₉₇

—

C₄×Dic₉₇

D₄

yes

—

C₄×Dic₉₇

Q₈

yes

complemented

C₂²×C₄×C₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

C₂³

yes

—

C₂²×C₄×C₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₉₇

—

C₄²×C₉₇

Q₈

yes

complemented

C₂³×C₉₇

C₂⁴

yes

characteristic

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

—

C₂×C₄×C₉₇

C₂×D₄

yes

—

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×C₄×C₉₇

C₂×Q₈

yes

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

—

C₂×Dic₉₇

C₂×D₄

yes

—

C₂×Dic₉₇

C₂×Q₈

yes

—

C₂×Dic₉₇

#41

yes

—

Dic₁₉₄

—

C₂²×C₉₇

C₂²×D₄

yes

characteristic

C₂²×C₉₇

C₂²×Q₈

yes

characteristic

C₂²×C₉₇

C₂²×Q₈

yes

characteristic

C₂²×C₉₇

C₂×#41

yes

—

C₄×C₉₇

—

C₄×C₉₇

—

C₄×C₉₇

C₄⋊Q₈

yes

—

Dic₉₇

—

Dic₉₇

—

Dic₉₇

#124

yes

—

C₂×C₉₇

C₂×#124

yes

—

C₂×C₉₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₉₇

#533

yes

—

C₂×C₉₇

#533

yes

—

#2862

—

maximal, Sylow

C₉₇

#2862

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

—

—

—

D₉₇

yes

characteristic

C₄³

—

C₂²×Q₈

—

C₄⋊Q₈

—

194

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₉₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₉₇

yes

—

—

C₂×D₉₇

yes

—

C₂×Q₈

—

194

—

C₂×Q₈

—

194

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

—

C₂²×C₄

C₂²×D₉₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

194

—

C₄⋊C₄

—

194

—

C₄²

—

C₄²

—

C₄²

—

194

—

C₄²

Dic₁₉₄

yes

complemented

C₂³

C₂³×D₉₇

yes

characteristic

Q₈

—

388

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

194

—

107

C₂×C₄

—

194

—

108

C₂×C₄

—

194

—

109

C₂×C₄

D₄×D₉₇

yes

—

110

C₂×C₄

Q₈×D₉₇

yes

—

111

C₂×C₄

C₂×Dic₁₉₄

yes

—

112

C₂×C₄

C₂×Dic₁₉₄

yes

—

113

C₂²

C₂×D₄×D₉₇

yes

characteristic

114

C₂²

C₂×Q₈×D₉₇

yes

characteristic

115

C₂²

C₂²×Dic₁₉₄

yes

characteristic

116

C₂²

#41×D₉₇

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

388

—

121

C₄

—

194

—

122

C₄

#15690809

yes

—

123

C₂

#124×D₉₇

yes

—

124

C₂

C₂×#15690809

yes

characteristic

125

C₂

#20261288

yes

—

126

C₂

#20261291

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₉₇
commutator, socle, prime radical, cosocle	C₂³×C₉₇	C₂⁴
Fitting	#382×C₉₇	C₂
hypercenter	#382	D₉₇
hypocenter	C₉₇	#2862
LCS	G ▷ C₂³×C₉₇ ▷ C₉₇
UCS	C₁ ◁ C₂³ ◁ #382

subgroup

G/H

center, Frattini

C₂³

C₂³×D₉₇

commutator, socle, prime radical, cosocle

C₂³×C₉₇

C₂⁴

Fitting

#382×C₉₇

C₂

D₉₇

C₉₇

G ▷ C₂³×C₉₇ ▷ C₉₇

C₁ ◁ C₂³ ◁ #382

N	H ≅ G/N	type
28. C₄×Dic₉₇	Q₈	semidirect
33. C₄²×C₉₇	Q₈	semidirect
64. C₉₇	#2862	semidirect
97. C₄²	Dic₁₉₄	semidirect

H ≅ G/N

type

28. C₄×Dic₉₇

Q₈

semidirect

33. C₄²×C₉₇

Q₈

semidirect

64. C₉₇

#2862

semidirect

97. C₄²

Dic₁₉₄

semidirect

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

subgroup	index
C₂×#15690809	2
#20260639	2
#20260601	2
#20260600	2
C₄²×Dic₉₇	2
#382×C₉₇	2
#2862	97

subgroup

index

C₄²×Dic₉₇

#382×C₉₇

#2862

Subgroups — #27150112

Conjugacy classes of subgroups

Complements

Maximal subgroups