Group #38755390 — subgroups

Subgroups — #38755390

Order 25216.

336 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 336 conjugacy classes of subgroups — too many to draw. Only its 141 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	#27161808	C₂	1	1	yes	maximal, characteristic
2	#27161431	C₂	1	3	yes	maximal
3	#27161371	C₂	1	3	yes	maximal
4	#27161289	C₂	1	3	yes	maximal
5	Q₈×#13096802	C₂	1	1	yes	maximal, characteristic
6	C₄×#15691348	C₂	1	3	yes	maximal
7	Q₈²×C₁₉₇	C₂	1	1	yes	maximal, characteristic
8	C₂×#15691348	C₂²	1	3	yes	—
9	#20263326	C₂²	1	3	yes	—
10	#20263317	—	2	3	no	—
11	Q₈×Dic₁₉₇	—	2	1	no	—
12	C₄×Dic₃₉₄	C₂²	1	3	yes	—
13	#20263259	C₂²	1	3	yes	—
14	C₄⋊#13096802	C₂²	1	3	yes	—
15	#20263250	C₂²	1	1	yes	characteristic
16	#20263249	C₂²	1	6	yes	—
17	#20263249	C₂²	1	3	yes	—
18	C₄×#13096802	C₂²	1	3	yes	—
19	C₄⋊Q₈×C₁₉₇	—	2	3	no	—
20	C₄⋊Q₈×C₁₉₇	C₂²	1	3	yes	—
21	C₄×Q₈×C₁₉₇	—	2	1	no	—
22	C₄×Q₈×C₁₉₇	C₂²	1	3	yes	—
23	C₄×Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
24	C₂×Dic₃₉₄	C₂³	1	3	yes	—
25	#15691348	—	2	6	no	—
26	C₂×#13096802	C₂³	1	1	yes	characteristic
27	C₂×#13096802	C₂³	1	3	yes	—
28	C₄⋊Dic₁₉₇	—	2	3	no	—
29	C₄⋊Dic₁₉₇	—	2	3	no	—
30	C₄⋊Dic₁₉₇	C₂³	1	1	yes	characteristic
31	#15691337	—	2	3	no	—
32	C₄×Dic₁₉₇	—	2	3	no	—
33	C₂×Q₈×C₁₉₇	—	2	3	no	—
34	C₂×Q₈×C₁₉₇	D₄	1	1	yes	characteristic
35	C₂×Q₈×C₁₉₇	C₂³	1	1	yes	characteristic
36	C₄⋊C₄×C₁₉₇	—	2	3	no	—
37	C₄⋊C₄×C₁₉₇	—	2	3	no	—
38	C₄⋊C₄×C₁₉₇	D₄	1	3	yes	—
39	C₄⋊C₄×C₁₉₇	C₂³	1	3	yes	—
40	C₄²×C₁₉₇	—	2	3	no	—
41	C₄²×C₁₉₇	C₂³	1	3	yes	—
42	Dic₃₉₄	—	2	3	no	—
43	Dic₃₉₄	—	2	3	no	—
44	C₂×Dic₁₉₇	—	2	1	no	—
45	C₂×Dic₁₉₇	—	2	3	no	—
46	#13096802	—	2	3	no	—
47	#13096802	—	2	1	no	—
48	Q₈×C₁₉₇	—	2	1	no	—
49	Q₈×C₁₉₇	—	8	3	no	—
50	Q₈×C₁₉₇	Q₁₆	1	4	yes	complemented
51	Q₈×C₁₉₇	#41	1	2	yes	—
52	C₂×C₄×C₁₉₇	—	2	1	no	—
53	C₂×C₄×C₁₉₇	—	2	3	no	—
54	C₂×C₄×C₁₉₇	C₂×D₄	1	3	yes	—
55	C₂×C₄×C₁₉₇	C₂×D₄	1	3	yes	—
56	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
57	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₁₉₇	—	2	2	no	—
59	C₄×C₁₉₇	—	2	3	no	—
60	C₄×C₁₉₇	—	4	3	no	—
61	C₄×C₁₉₇	C₂×Q₁₆	1	6	yes	—
62	C₄×C₁₉₇	C₂×#41	1	1	yes	characteristic
63	C₄×C₁₉₇	2₋¹⁺⁴	1	1	yes	characteristic
64	Dic₁₉₇	—	4	3	no	—
65	Dic₁₉₇	—	4	1	no	—
66	C₂×C₁₉₇	#548	1	1	yes	characteristic
67	C₂×C₁₉₇	C₂²×Q₁₆	1	1	yes	characteristic
68	C₂×C₁₉₇	#577	1	1	yes	characteristic
69	C₁₉₇	#3363	1	1	yes	characteristic, complemented, Sylow
70	#3363	—	197	1	no	maximal, Sylow
71	Q₈²	D₁₉₇	1	1	yes	characteristic
72	#557	—	197	1	no	—
73	#487	—	197	3	no	—
74	#477	—	197	3	no	—
75	#462	—	197	3	no	—
76	C₈×Q₈	—	197	1	no	—
77	C₄×Q₁₆	—	197	3	no	—
78	C₂×Q₁₆	—	197	3	no	—
79	C₄⋊Q₈	—	2	3	no	—
80	C₄⋊Q₈	—	197	3	no	—
81	C₄⋊Q₈	C₂×D₁₉₇	1	3	yes	—
82	#125	—	394	3	no	—
83	C₄×Q₈	—	2	1	no	—
84	C₄×Q₈	—	394	1	no	—
85	C₄×Q₈	—	197	3	no	—
86	C₄×Q₈	C₂×D₁₉₇	1	3	yes	—
87	C₄×Q₈	C₂×D₁₉₇	1	1	yes	characteristic
88	#107	—	197	3	no	—
89	C₄⋊C₈	—	197	3	no	—
90	#103	—	197	1	no	—
91	#103	—	197	3	no	—
92	#103	—	197	6	no	—
93	C₄×C₈	—	197	3	no	—
94	C₂×Q₈	—	2	3	no	—
95	C₂×Q₈	—	197	3	no	—
96	C₂×Q₈	C₂²⋊D₁₉₇	1	1	yes	characteristic
97	C₂×Q₈	C₂²×D₁₉₇	1	1	yes	characteristic
98	Q₁₆	—	394	6	no	—
99	C₂×C₈	—	197	1	no	—
100	C₂×C₈	—	197	3	no	—
101	C₄⋊C₄	—	2	3	no	—
102	C₄⋊C₄	—	2	3	no	—
103	C₄⋊C₄	—	394	3	no	—
104	C₄⋊C₄	—	394	3	no	—
105	C₄⋊C₄	—	394	3	no	—
106	C₄⋊C₄	—	197	1	no	—
107	C₄⋊C₄	C₂²⋊D₁₉₇	1	3	yes	—
108	C₄⋊C₄	C₂²×D₁₉₇	1	3	yes	—
109	C₄²	—	2	3	no	—
110	C₄²	—	394	3	no	—
111	C₄²	C₂²×D₁₉₇	1	3	yes	—
112	Q₈	—	2	1	no	—
113	Q₈	—	8	3	no	—
114	Q₈	—	394	3	no	—
115	Q₈	—	394	3	no	—
116	Q₈	#15691348	1	4	yes	complemented
117	Q₈	#15691359	1	2	yes	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	2	3	no	—
120	C₂×C₄	—	394	1	no	—
121	C₂×C₄	—	394	3	no	—
122	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	3	yes	—
123	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	3	yes	—
124	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
125	C₈	—	394	3	no	—
126	C₈	—	394	1	no	—
127	C₂²	C₂²×C₂²⋊D₁₉₇	1	1	yes	characteristic
128	C₄	—	2	3	no	—
129	C₄	—	2	2	no	—
130	C₄	—	4	3	no	—
131	C₄	—	788	1	no	—
132	C₄	—	788	3	no	—
133	C₄	C₂×#15691348	1	6	yes	—
134	C₄	C₂×#15691359	1	1	yes	characteristic
135	C₄	#20263374	1	1	yes	characteristic
136	C₂	#27161740	1	1	yes	characteristic
137	C₂	C₂²×#15691348	1	1	yes	characteristic
138	C₂	#27161890	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27161808

C₂

yes

maximal, characteristic

#27161431

C₂

yes

maximal

#27161371

C₂

yes

maximal

#27161289

C₂

yes

maximal

Q₈×#13096802

C₂

yes

maximal, characteristic

C₄×#15691348

C₂

yes

maximal

Q₈²×C₁₉₇

C₂

yes

maximal, characteristic

C₂×#15691348

C₂²

yes

—

#20263326

C₂²

yes

—

#20263317

—

Q₈×Dic₁₉₇

—

C₄×Dic₃₉₄

C₂²

yes

—

#20263259

C₂²

yes

—

C₄⋊#13096802

C₂²

yes

—

#20263250

C₂²

yes

characteristic

#20263249

C₂²

yes

—

#20263249

C₂²

yes

—

C₄×#13096802

C₂²

yes

—

C₄⋊Q₈×C₁₉₇

—

C₄⋊Q₈×C₁₉₇

C₂²

yes

—

C₄×Q₈×C₁₉₇

—

C₄×Q₈×C₁₉₇

C₂²

yes

—

C₄×Q₈×C₁₉₇

C₂²

yes

characteristic

C₂×Dic₃₉₄

C₂³

yes

—

#15691348

—

C₂×#13096802

C₂³

yes

characteristic

C₂×#13096802

C₂³

yes

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

C₂³

yes

characteristic

#15691337

—

C₄×Dic₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

D₄

yes

characteristic

C₂×Q₈×C₁₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

D₄

yes

—

C₄⋊C₄×C₁₉₇

C₂³

yes

—

C₄²×C₁₉₇

—

C₄²×C₁₉₇

C₂³

yes

—

Dic₃₉₄

—

Dic₃₉₄

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

#13096802

—

#13096802

—

Q₈×C₁₉₇

—

Q₈×C₁₉₇

—

Q₈×C₁₉₇

Q₁₆

yes

complemented

Q₈×C₁₉₇

#41

yes

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂⁴

yes

characteristic

C₂²×C₁₉₇

C₂²×D₄

yes

characteristic

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

C₂×Q₁₆

yes

—

C₄×C₁₉₇

C₂×#41

yes

characteristic

C₄×C₁₉₇

2₋¹⁺⁴

yes

characteristic

Dic₁₉₇

—

Dic₁₉₇

—

C₂×C₁₉₇

#548

yes

characteristic

C₂×C₁₉₇

C₂²×Q₁₆

yes

C₂×C₁₉₇

yes

C₁₉₇

yes

characteristic, complemented, Sylow

#3363

—

197

maximal, Sylow

Q₈²

D₁₉₇

yes

characteristic

#557

—

197

—

#487

—

197

—

#477

—

197

—

#462

—

197

—

C₈×Q₈

—

197

—

C₄×Q₁₆

—

197

—

C₂×Q₁₆

—

197

—

C₄⋊Q₈

—

C₄⋊Q₈

—

197

—

C₄⋊Q₈

C₂×D₁₉₇

yes

—

#125

—

394

—

C₄×Q₈

—

C₄×Q₈

—

394

—

C₄×Q₈

—

197

—

C₄×Q₈

C₂×D₁₉₇

yes

—

C₄×Q₈

C₂×D₁₉₇

yes

characteristic

#107

—

197

—

C₄⋊C₈

—

197

—

#103

—

197

—

#103

—

197

—

#103

—

197

—

C₄×C₈

—

197

—

C₂×Q₈

—

C₂×Q₈

—

197

—

C₂×Q₈

C₂²⋊D₁₉₇

yes

characteristic

C₂×Q₈

C₂²×D₁₉₇

yes

characteristic

Q₁₆

—

394

—

C₂×C₈

—

197

—

100

C₂×C₈

—

197

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

394

—

104

C₄⋊C₄

—

394

—

105

C₄⋊C₄

—

394

—

106

C₄⋊C₄

—

197

—

107

C₄⋊C₄

C₂²⋊D₁₉₇

yes

—

108

C₄⋊C₄

C₂²×D₁₉₇

yes

—

109

C₄²

—

110

C₄²

—

394

—

111

C₄²

C₂²×D₁₉₇

yes

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

394

—

115

Q₈

—

394

—

116

Q₈

#15691348

yes

complemented

117

Q₈

#15691359

yes

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

394

—

121

C₂×C₄

—

394

—

122

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

—

123

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

—

124

C₂×C₄

C₂³×D₁₉₇

yes

characteristic

125

C₈

—

394

—

126

C₈

—

394

—

127

C₂²

C₂²×C₂²⋊D₁₉₇

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

—

788

—

132

C₄

—

788

—

133

C₄

C₂×#15691348

yes

—

134

C₄

C₂×#15691359

yes

characteristic

135

C₄

#20263374

yes

characteristic

136

C₂

#27161740

yes

characteristic

137

C₂

C₂²×#15691348

yes

characteristic

138

C₂

#27161890

yes

characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₉₇
commutator, cosocle	C₂×C₄×C₁₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting	Q₈²×C₁₉₇	C₂
socle, prime radical	C₂²×C₁₉₇	C₂²×D₄
hypercenter	Q₈²	D₁₉₇
hypocenter	C₁₉₇	#3363
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

subgroup

G/H

center

C₂²

C₂²×C₂²⋊D₁₉₇

commutator, cosocle

C₂×C₄×C₁₉₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉₇

Fitting

Q₈²×C₁₉₇

C₂

socle, prime radical

C₂²×C₁₉₇

C₂²×D₄

hypercenter

Q₈²

D₁₉₇

hypocenter

C₁₉₇

#3363

LCS

G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂² ◁ C₄×Q₈ ◁ Q₈²

N	H ≅ G/N	type
50. Q₈×C₁₉₇	Q₁₆	semidirect
69. C₁₉₇	#3363	semidirect
116. Q₈	#15691348	semidirect

H ≅ G/N

type

50. Q₈×C₁₉₇

Q₁₆

semidirect

69. C₁₉₇

#3363

semidirect

116. Q₈

#15691348

semidirect

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

subgroup	index
#27161808	2
#27161431	2
#27161371	2
#27161289	2
Q₈×#13096802	2
C₄×#15691348	2
Q₈²×C₁₉₇	2
#3363	197

subgroup

index

Q₈²×C₁₉₇

197

Subgroups — #38755390

Conjugacy classes of subgroups

Complements

Maximal subgroups