Group #38742564 — subgroups

Subgroups — #38742564

Order 25216.

292 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 137 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×#20263255	C₂	1	2	yes	maximal
2	C₂×#20263236	C₂	1	2	yes	maximal
3	C₄²×Dic₁₉₇	C₂	1	1	yes	maximal, characteristic
4	#27160834	C₂	1	1	yes	maximal, characteristic
5	#336×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×#13096802	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₉₇	C₄	1	2	yes	—
8	C₂×C₄×Dic₁₉₇	C₄	1	2	yes	—
9	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
11	#20263255	C₄	1	8	yes	complemented
12	#20263236	—	2	4	no	—
13	C₂²×C₈×C₁₉₇	C₂²	1	2	yes	—
14	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
15	C₂²×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₁₉₇	C₂×C₄	1	2	yes	—
17	C₂²×Dic₁₉₇	C₂×C₄	1	1	yes	characteristic
18	C₂×#13096802	—	2	2	no	—
19	C₂×#13096802	—	2	2	no	—
20	C₂×#13096802	C₂×C₄	1	4	yes	—
21	C₄×Dic₁₉₇	—	2	4	no	—
22	C₄×Dic₁₉₇	—	2	4	no	—
23	C₄×Dic₁₉₇	C₂×C₄	1	4	yes	—
24	C₄×Dic₁₉₇	D₄	1	2	yes	—
25	C₄×Dic₁₉₇	Q₈	1	2	yes	—
26	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
27	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₁₉₇	—	2	2	no	—
29	C₂×C₈×C₁₉₇	—	2	2	no	—
30	C₂×C₈×C₁₉₇	C₂×C₄	1	4	yes	—
31	C₄²×C₁₉₇	—	2	2	no	—
32	C₂×Dic₁₉₇	—	2	2	no	—
33	C₂×Dic₁₉₇	—	2	2	no	—
34	C₂×Dic₁₉₇	—	2	2	no	—
35	C₂×Dic₁₉₇	C₄²	1	2	yes	—
36	C₂×Dic₁₉₇	C₄²	1	2	yes	—
37	C₂×Dic₁₉₇	C₄⋊C₄	1	4	yes	—
38	C₂×Dic₁₉₇	M₁₆	1	4	yes	—
39	#13096802	—	2	4	no	—
40	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₉₇	—	2	2	no	—
42	C₂×C₄×C₁₉₇	—	2	2	no	—
43	C₂×C₄×C₁₉₇	M₁₆	1	4	yes	—
44	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
45	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
47	C₂×C₄×C₁₉₇	#41	1	2	yes	—
48	C₈×C₁₉₇	—	2	4	no	—
49	C₂²×C₁₉₇	C₂×C₄²	1	1	yes	characteristic
50	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₉₇	#117	1	1	yes	characteristic
52	C₂²×C₁₉₇	C₂×M₁₆	1	2	yes	—
53	C₂²×C₁₉₇	C₂×M₁₆	1	2	yes	—
54	C₄×C₁₉₇	—	2	4	no	—
55	C₄×C₁₉₇	C₄×D₄	1	2	yes	—
56	C₄×C₁₉₇	C₄×Q₈	1	2	yes	—
57	Dic₁₉₇	—	2	2	no	—
58	Dic₁₉₇	—	2	4	no	—
59	Dic₁₉₇	#97	1	4	yes	complemented
60	C₂×C₁₉₇	C₄×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₁₉₇	C₂×#97	1	1	yes	characteristic
62	C₂×C₁₉₇	C₄×M₁₆	1	1	yes	characteristic
63	C₂×C₁₉₇	#423	1	2	yes	—
64	C₂×C₁₉₇	#432	1	2	yes	—
65	C₁₉₇	#1767	1	1	yes	characteristic, complemented, Sylow
66	#1767	—	197	1	no	maximal, Sylow
67	C₂×C₄⋊C₈	—	197	2	no	—
68	C₂×#97	—	197	2	no	—
69	C₄³	—	197	1	no	—
70	#336	—	197	1	no	—
71	#336	D₁₉₇	1	1	yes	characteristic
72	C₂²×C₈	—	197	2	no	—
73	C₂²×C₈	C₂×D₁₉₇	1	2	yes	—
74	C₂×C₄²	—	197	2	no	—
75	C₂×C₄²	—	197	2	no	—
76	C₂×C₄²	—	197	1	no	—
77	C₂×C₄²	—	197	1	no	—
78	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
79	C₄⋊C₈	—	197	8	no	—
80	#97	—	394	4	no	—
81	C₂²×C₄	—	197	2	no	—
82	C₂²×C₄	—	197	1	no	—
83	C₂²×C₄	—	197	1	no	—
84	C₂²×C₄	C₄×D₁₉₇	1	2	yes	—
85	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
86	C₂×C₈	—	2	2	no	—
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	394	2	no	—
89	C₂×C₈	—	394	2	no	—
90	C₂×C₈	—	197	4	no	—
91	C₂×C₈	C₄×D₁₉₇	1	4	yes	—
92	C₄²	—	2	2	no	—
93	C₄²	—	394	4	no	—
94	C₄²	—	394	4	no	—
95	C₄²	—	197	4	no	—
96	C₄²	—	197	2	no	—
97	C₄²	—	197	2	no	—
98	C₂³	C₂×C₄×D₁₉₇	1	1	yes	characteristic
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	394	2	no	—
102	C₂×C₄	—	394	2	no	—
103	C₂×C₄	—	394	2	no	—
104	C₂×C₄	—	197	2	no	—
105	C₂×C₄	—	197	4	no	—
106	C₂×C₄	—	197	2	no	—
107	C₂×C₄	—	197	4	no	—
108	C₂×C₄	#15691341	1	4	yes	—
109	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
110	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
111	C₂×C₄	Q₈×D₁₉₇	1	1	yes	characteristic
112	C₂×C₄	#15691357	1	2	yes	—
113	C₈	—	2	4	no	—
114	C₈	—	394	4	no	—
115	C₂²	C₄²×D₁₉₇	1	1	yes	characteristic
116	C₂²	C₄⋊C₄×D₁₉₇	1	1	yes	characteristic
117	C₂²	#20263282	1	1	yes	characteristic
118	C₂²	M₁₆×D₁₉₇	1	2	yes	—
119	C₂²	C₂×#15691341	1	2	yes	—
120	C₄	—	2	4	no	—
121	C₄	—	394	4	no	—
122	C₄	—	394	2	no	—
123	C₄	—	197	4	no	—
124	C₄	#20263286	1	2	yes	—
125	C₄	#20263291	1	2	yes	—
126	C₂	#27160930	1	1	yes	characteristic
127	C₂	#97×D₁₉₇	1	1	yes	characteristic
128	C₂	C₄×#15691341	1	1	yes	characteristic
129	C₂	#27161101	1	2	yes	—
130	C₂	#27161133	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263255

C₂

yes

maximal

C₂×#20263236

C₂

yes

maximal

C₄²×Dic₁₉₇

C₂

yes

maximal, characteristic

#27160834

C₂

yes

maximal, characteristic

#336×C₁₉₇

C₂

yes

maximal, characteristic

C₂²×#13096802

C₂²

yes

—

C₂×C₄×Dic₁₉₇

C₄

yes

—

C₂×C₄×Dic₁₉₇

C₄

yes

—

C₂×C₄×Dic₁₉₇

C₂²

yes

characteristic

C₂×C₄×Dic₁₉₇

C₂²

yes

characteristic

#20263255

C₄

yes

complemented

#20263236

—

C₂²×C₈×C₁₉₇

C₂²

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₂²×Dic₁₉₇

C₂×C₄

yes

characteristic

C₂²×Dic₁₉₇

C₂×C₄

yes

—

C₂²×Dic₁₉₇

C₂×C₄

yes

—

—

C₂×C₄

yes

—

C₄×Dic₁₉₇

—

C₄×Dic₁₉₇

—

C₄×Dic₁₉₇

C₂×C₄

yes

—

C₄×Dic₁₉₇

D₄

yes

—

C₄×Dic₁₉₇

Q₈

yes

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

C₂×C₄

yes

—

C₄²×C₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

C₄²

yes

—

C₂×Dic₁₉₇

C₄²

yes

—

C₂×Dic₁₉₇

C₄⋊C₄

yes

—

C₂×Dic₁₉₇

M₁₆

yes

—

#13096802

—

C₂³×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

M₁₆

yes

—

C₂×C₄×C₁₉₇

C₂²×C₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₈×C₁₉₇

—

C₂²×C₁₉₇

C₂×C₄²

yes

characteristic

C₂²×C₁₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₉₇

#117

yes

characteristic

C₂²×C₁₉₇

C₂×M₁₆

yes

—

C₂²×C₁₉₇

C₂×M₁₆

yes

—

C₄×C₁₉₇

—

C₄×C₁₉₇

C₄×D₄

yes

—

C₄×C₁₉₇

C₄×Q₈

yes

—

Dic₁₉₇

—

Dic₁₉₇

—

Dic₁₉₇

#97

yes

complemented

C₂×C₁₉₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₉₇

C₂×#97

yes

characteristic

C₂×C₁₉₇

C₄×M₁₆

yes

characteristic

C₂×C₁₉₇

#423

yes

—

C₂×C₁₉₇

#432

yes

—

C₁₉₇

#1767

yes

characteristic, complemented, Sylow

#1767

—

197

maximal, Sylow

C₂×C₄⋊C₈

—

197

—

C₂×#97

—

197

—

C₄³

—

197

—

#336

—

197

—

#336

D₁₉₇

yes

characteristic

C₂²×C₈

—

197

—

C₂²×C₈

C₂×D₁₉₇

yes

—

C₂×C₄²

—

197

—

C₂×C₄²

—

197

—

C₂×C₄²

—

197

—

C₂×C₄²

—

197

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

C₄⋊C₈

—

197

—

#97

—

394

—

C₂²×C₄

—

197

—

C₂²×C₄

—

197

—

C₂²×C₄

—

197

—

C₂²×C₄

C₄×D₁₉₇

yes

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₂×C₈

—

197

—

C₂×C₈

C₄×D₁₉₇

yes

—

C₄²

—

C₄²

—

394

—

C₄²

—

394

—

C₄²

—

197

—

C₄²

—

197

—

C₄²

—

197

—

C₂³

C₂×C₄×D₁₉₇

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

394

—

102

C₂×C₄

—

394

—

103

C₂×C₄

—

394

—

104

C₂×C₄

—

197

—

105

C₂×C₄

—

197

—

106

C₂×C₄

—

197

—

107

C₂×C₄

—

197

—

108

C₂×C₄

#15691341

yes

—

109

C₂×C₄

C₂×C₄×D₁₉₇

yes

—

110

C₂×C₄

D₄×D₁₉₇

yes

characteristic

111

C₂×C₄

Q₈×D₁₉₇

yes

characteristic

112

C₂×C₄

#15691357

yes

—

113

C₈

—

114

C₈

—

394

—

115

C₂²

C₄²×D₁₉₇

yes

characteristic

116

C₂²

C₄⋊C₄×D₁₉₇

yes

characteristic

117

C₂²

#20263282

yes

characteristic

118

C₂²

M₁₆×D₁₉₇

yes

—

119

C₂²

C₂×#15691341

yes

—

120

C₄

—

121

C₄

—

394

—

122

C₄

—

394

—

123

C₄

—

197

—

124

C₄

#20263286

yes

—

125

C₄

#20263291

yes

—

126

C₂

#27160930

yes

characteristic

127

C₂

#97×D₁₉₇

yes

characteristic

128

C₂

C₄×#15691341

yes

characteristic

129

C₂

#27161101

yes

—

130

C₂

#27161133

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₉₇
commutator	C₂²×C₁₉₇	C₂×C₄²
Fitting	#336×C₁₉₇	C₂
socle, prime radical	C₂³×C₁₉₇	C₂²×C₄
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	#336	D₁₉₇
hypocenter	C₁₉₇	#1767
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₉₇

commutator

C₂²×C₁₉₇

C₂×C₄²

Fitting

#336×C₁₉₇

C₂

socle, prime radical

C₂³×C₁₉₇

C₂²×C₄

cosocle

C₂²×C₄×C₁₉₇

C₂³

D₁₉₇

C₁₉₇

G ▷ C₂²×C₁₉₇ ▷ C₁₉₇

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
11. #20263255	C₄	semidirect
59. Dic₁₉₇	#97	semidirect
65. C₁₉₇	#1767	semidirect

H ≅ G/N

type

11. #20263255

C₄

semidirect

59. Dic₁₉₇

#97

semidirect

65. C₁₉₇

#1767

semidirect

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

subgroup	index
C₂×#20263255	2
C₂×#20263236	2
C₄²×Dic₁₉₇	2
#27160834	2
#336×C₁₉₇	2
#1767	197

subgroup

index

C₄²×Dic₁₉₇

#336×C₁₉₇

197

Subgroups — #38742564

Conjugacy classes of subgroups

Complements

Maximal subgroups