Group #38738480 — subgroups

Subgroups — #1546×C₁₉₇ #38738480

Order 25216.

240 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 112 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×C₁₉₇	C₂	1	1	yes	maximal, characteristic
2	#431×C₁₉₇	C₂	1	1	yes	maximal, characteristic
3	#423×C₁₉₇	C₂	1	1	yes	maximal, characteristic
4	#332×C₁₉₇	C₂	1	4	yes	maximal, complemented
5	C₂²×Q₈×C₁₉₇	—	2	1	no	—
6	C₂×M₁₆×C₁₉₇	—	2	1	no	—
7	C₄⋊Q₈×C₁₉₇	—	2	1	no	—
8	C₄⋊Q₈×C₁₉₇	—	2	1	no	—
9	C₄⋊Q₈×C₁₉₇	C₄	1	2	yes	—
10	C₄⋊Q₈×C₁₉₇	C₂²	1	2	yes	—
11	C₂×C₄⋊C₄×C₁₉₇	—	2	1	no	—
12	C₂×C₄⋊C₄×C₁₉₇	C₄	1	2	yes	—
13	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₁₉₇	—	2	1	no	—
15	C₄⋊C₈×C₁₉₇	C₂²	1	2	yes	—
16	C₄⋊C₈×C₁₉₇	C₂²	1	2	yes	—
17	C₂²⋊C₈×C₁₉₇	—	2	1	no	—
18	C₄×C₈×C₁₉₇	—	2	1	no	—
19	C₂×Q₈×C₁₉₇	—	2	2	no	—
20	C₂×Q₈×C₁₉₇	—	2	2	no	—
21	C₂×Q₈×C₁₉₇	—	4	2	no	—
22	C₂×Q₈×C₁₉₇	—	4	2	no	—
23	C₂²×C₄×C₁₉₇	—	2	2	no	—
24	C₂²×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
27	M₁₆×C₁₉₇	—	4	2	no	—
28	C₂×C₈×C₁₉₇	—	2	2	no	—
29	C₂×C₈×C₁₉₇	—	2	2	no	—
30	C₄⋊C₄×C₁₉₇	—	2	2	no	—
31	C₄⋊C₄×C₁₉₇	—	2	2	no	—
32	C₄⋊C₄×C₁₉₇	—	2	2	no	—
33	C₄⋊C₄×C₁₉₇	C₂×C₄	1	4	yes	—
34	C₄²×C₁₉₇	C₂×C₄	1	1	yes	characteristic
35	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
36	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
37	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
38	C₂³×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
39	Q₈×C₁₉₇	—	4	4	no	—
40	Q₈×C₁₉₇	—	4	2	no	—
41	Q₈×C₁₉₇	—	4	2	no	—
42	C₂×C₄×C₁₉₇	—	2	2	no	—
43	C₂×C₄×C₁₉₇	—	2	2	no	—
44	C₂×C₄×C₁₉₇	—	2	4	no	—
45	C₂×C₄×C₁₉₇	—	4	4	no	—
46	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₁₉₇	D₈	1	2	yes	—
50	C₂×C₄×C₁₉₇	SD₁₆	1	2	yes	—
51	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
53	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
54	C₈×C₁₉₇	—	4	2	no	—
55	C₈×C₁₉₇	—	4	2	no	—
56	C₂²×C₁₉₇	—	2	1	no	—
57	C₂²×C₁₉₇	—	2	1	no	—
58	C₂²×C₁₉₇	#102	1	2	yes	—
59	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
60	C₄×C₁₉₇	—	2	1	no	—
61	C₄×C₁₉₇	—	2	1	no	—
62	C₄×C₁₉₇	—	4	4	no	—
63	C₄×C₁₉₇	#101	1	2	yes	—
64	C₄×C₁₉₇	#102	1	2	yes	—
65	C₄×C₁₉₇	C₂×D₈	1	1	yes	characteristic
66	C₄×C₁₉₇	C₂×SD₁₆	1	1	yes	characteristic
67	C₄×C₁₉₇	#137	1	2	yes	—
68	C₂×C₁₉₇	—	2	2	no	—
69	C₂×C₁₉₇	C₂×#101	1	1	yes	characteristic
70	C₂×C₁₉₇	C₂×#102	1	1	yes	characteristic
71	C₂×C₁₉₇	#419	1	1	yes	characteristic
72	C₁₉₇	#1546	1	1	yes	characteristic, direct product factor, Sylow
73	#1546	C₁₉₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂×C₄⋊Q₈	C₂×C₁₉₇	1	1	yes	characteristic
75	#431	C₂×C₁₉₇	1	1	yes	characteristic
76	#423	C₂×C₁₉₇	1	1	yes	characteristic
77	#332	C₂×C₁₉₇	1	4	yes	complemented
78	C₂²×Q₈	—	2	1	no	—
79	C₂×M₁₆	—	2	1	no	—
80	C₄⋊Q₈	—	2	1	no	—
81	C₄⋊Q₈	—	2	1	no	—
82	C₄⋊Q₈	C₄×C₁₉₇	1	2	yes	—
83	C₄⋊Q₈	C₂²×C₁₉₇	1	2	yes	—
84	C₂×C₄⋊C₄	—	2	1	no	—
85	C₂×C₄⋊C₄	C₄×C₁₉₇	1	2	yes	—
86	C₂×C₄²	C₂²×C₁₉₇	1	1	yes	characteristic
87	C₄⋊C₈	—	2	1	no	—
88	C₄⋊C₈	C₂²×C₁₉₇	1	2	yes	—
89	C₄⋊C₈	C₂²×C₁₉₇	1	2	yes	—
90	C₂²⋊C₈	—	2	1	no	—
91	C₄×C₈	—	2	1	no	—
92	C₂×Q₈	—	2	2	no	—
93	C₂×Q₈	—	2	2	no	—
94	C₂×Q₈	—	4	2	no	—
95	C₂×Q₈	—	4	2	no	—
96	C₂²×C₄	—	2	2	no	—
97	C₂²×C₄	C₂×C₄×C₁₉₇	1	1	yes	characteristic
98	C₂²×C₄	D₄×C₁₉₇	1	1	yes	characteristic
99	C₂²×C₄	D₄×C₁₉₇	1	1	yes	characteristic
100	M₁₆	—	4	2	no	—
101	C₂×C₈	—	2	2	no	—
102	C₂×C₈	—	2	2	no	—
103	C₄⋊C₄	—	2	2	no	—
104	C₄⋊C₄	—	2	2	no	—
105	C₄⋊C₄	—	2	2	no	—
106	C₄⋊C₄	C₂×C₄×C₁₉₇	1	4	yes	—
107	C₄²	C₂×C₄×C₁₉₇	1	1	yes	characteristic
108	C₄²	D₄×C₁₉₇	1	1	yes	characteristic
109	C₄²	D₄×C₁₉₇	1	1	yes	characteristic
110	C₄²	C₂³×C₁₉₇	1	1	yes	characteristic
111	C₂³	C₂²⋊C₄×C₁₉₇	1	1	yes	characteristic
112	Q₈	—	4	4	no	—
113	Q₈	—	4	2	no	—
114	Q₈	—	4	2	no	—
115	C₂×C₄	—	2	2	no	—
116	C₂×C₄	—	2	2	no	—
117	C₂×C₄	—	2	4	no	—
118	C₂×C₄	—	4	4	no	—
119	C₂×C₄	C₂²⋊C₄×C₁₉₇	1	1	yes	characteristic
120	C₂×C₄	C₂²⋊C₄×C₁₉₇	1	1	yes	characteristic
121	C₂×C₄	C₂²⋊C₄×C₁₉₇	1	1	yes	characteristic
122	C₂×C₄	D₈×C₁₉₇	1	2	yes	—
123	C₂×C₄	SD₁₆×C₁₉₇	1	2	yes	—
124	C₂×C₄	C₂²×C₄×C₁₉₇	1	1	yes	characteristic
125	C₂×C₄	C₂×D₄×C₁₉₇	1	1	yes	characteristic
126	C₂×C₄	C₂×D₄×C₁₉₇	1	1	yes	characteristic
127	C₈	—	4	2	no	—
128	C₈	—	4	2	no	—
129	C₂²	—	2	1	no	—
130	C₂²	—	2	1	no	—
131	C₂²	#102×C₁₉₇	1	2	yes	—
132	C₂²	C₂×C₂²⋊C₄×C₁₉₇	1	1	yes	characteristic
133	C₄	—	2	1	no	—
134	C₄	—	2	1	no	—
135	C₄	—	4	4	no	—
136	C₄	#101×C₁₉₇	1	2	yes	—
137	C₄	#102×C₁₉₇	1	2	yes	—
138	C₄	C₂×D₈×C₁₉₇	1	1	yes	characteristic
139	C₄	C₂×SD₁₆×C₁₉₇	1	1	yes	characteristic
140	C₄	#137×C₁₉₇	1	2	yes	—
141	C₂	—	2	2	no	—
142	C₂	C₂×#101×C₁₉₇	1	1	yes	characteristic
143	C₂	C₂×#102×C₁₉₇	1	1	yes	characteristic
144	C₂	#419×C₁₉₇	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊Q₈×C₁₉₇

C₂

yes

maximal, characteristic

#431×C₁₉₇

C₂

yes

maximal, characteristic

#423×C₁₉₇

C₂

yes

maximal, characteristic

#332×C₁₉₇

C₂

yes

maximal, complemented

C₂²×Q₈×C₁₉₇

—

C₂×M₁₆×C₁₉₇

—

C₄⋊Q₈×C₁₉₇

—

C₄⋊Q₈×C₁₉₇

—

C₄⋊Q₈×C₁₉₇

C₄

yes

—

C₄⋊Q₈×C₁₉₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₉₇

—

C₂×C₄⋊C₄×C₁₉₇

C₄

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₇

—

C₄⋊C₈×C₁₉₇

C₂²

yes

—

C₄⋊C₈×C₁₉₇

C₂²

yes

—

C₂²⋊C₈×C₁₉₇

—

C₄×C₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₇

D₄

yes

characteristic

C₂²×C₄×C₁₉₇

D₄

yes

characteristic

M₁₆×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

C₂×C₄

yes

—

C₄²×C₁₉₇

C₂×C₄

yes

characteristic

C₄²×C₁₉₇

D₄

yes

characteristic

C₄²×C₁₉₇

D₄

yes

characteristic

C₄²×C₁₉₇

C₂³

yes

characteristic

C₂³×C₁₉₇

C₂²⋊C₄

yes

characteristic

Q₈×C₁₉₇

—

Q₈×C₁₉₇

—

Q₈×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₁₉₇

D₈

yes

—

C₂×C₄×C₁₉₇

SD₁₆

yes

—

C₂×C₄×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₈×C₁₉₇

—

C₈×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

#102

yes

—

C₂²×C₁₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

#101

yes

—

C₄×C₁₉₇

#102

yes

—

C₄×C₁₉₇

C₂×D₈

yes

characteristic

C₄×C₁₉₇

C₂×SD₁₆

yes

characteristic

C₄×C₁₉₇

#137

yes

—

C₂×C₁₉₇

—

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₁₉₇

yes

characteristic, direct product factor, Sylow

#1546

C₁₉₇

yes

maximal, characteristic, direct product factor, Sylow

C₂×C₄⋊Q₈

C₂×C₁₉₇

yes

characteristic

#431

C₂×C₁₉₇

yes

characteristic

#423

C₂×C₁₉₇

yes

characteristic

#332

C₂×C₁₉₇

yes

complemented

C₂²×Q₈

—

C₂×M₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₄×C₁₉₇

yes

—

C₄⋊Q₈

C₂²×C₁₉₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₄×C₁₉₇

yes

—

C₂×C₄²

C₂²×C₁₉₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂²×C₁₉₇

yes

—

C₄⋊C₈

C₂²×C₁₉₇

yes

—

C₂²⋊C₈

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

C₂×C₄×C₁₉₇

yes

characteristic

C₂²×C₄

D₄×C₁₉₇

yes

characteristic

C₂²×C₄

D₄×C₁₉₇

yes

characteristic

100

M₁₆

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

C₂×C₄×C₁₉₇

yes

—

107

C₄²

C₂×C₄×C₁₉₇

yes

characteristic

108

C₄²

D₄×C₁₉₇

yes

characteristic

109

C₄²

D₄×C₁₉₇

yes

characteristic

110

C₄²

C₂³×C₁₉₇

yes

characteristic

111

C₂³

C₂²⋊C₄×C₁₉₇

yes

characteristic

112

Q₈

—

113

Q₈

—

114

Q₈

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

C₂²⋊C₄×C₁₉₇

yes

characteristic

120

C₂×C₄

C₂²⋊C₄×C₁₉₇

yes

characteristic

121

C₂×C₄

C₂²⋊C₄×C₁₉₇

yes

characteristic

122

C₂×C₄

D₈×C₁₉₇

yes

—

123

C₂×C₄

SD₁₆×C₁₉₇

yes

—

124

C₂×C₄

C₂²×C₄×C₁₉₇

yes

characteristic

125

C₂×C₄

C₂×D₄×C₁₉₇

yes

characteristic

126

C₂×C₄

C₂×D₄×C₁₉₇

yes

characteristic

127

C₈

—

128

C₈

—

129

C₂²

—

130

C₂²

—

131

C₂²

#102×C₁₉₇

yes

—

132

C₂²

C₂×C₂²⋊C₄×C₁₉₇

yes

characteristic

133

C₄

—

134

C₄

—

135

C₄

—

136

C₄

#101×C₁₉₇

yes

—

137

C₄

#102×C₁₉₇

yes

—

138

C₄

C₂×D₈×C₁₉₇

yes

characteristic

139

C₄

C₂×SD₁₆×C₁₉₇

yes

characteristic

140

C₄

#137×C₁₉₇

yes

—

141

C₂

—

142

C₂

C₂×#101×C₁₉₇

yes

characteristic

143

C₂

C₂×#102×C₁₉₇

yes

characteristic

144

C₂

#419×C₁₉₇

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₁₉₇	C₂×C₂²⋊C₄
commutator	C₂×C₄	C₂²×C₄×C₁₉₇
Frattini, cosocle	C₄²	C₂³×C₁₉₇
prime radical	C₂³×C₁₉₇	C₂²⋊C₄
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₉₇ ◁ C₂×C₄²×C₁₉₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₁₉₇

C₂×C₂²⋊C₄

commutator

C₂×C₄

C₂²×C₄×C₁₉₇

Frattini, cosocle

C₄²

C₂³×C₁₉₇

prime radical

C₂³×C₁₉₇

C₂²⋊C₄

LCS

G ▷ C₂×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₁₉₇ ◁ C₂×C₄²×C₁₉₇ ◁ G

N	H ≅ G/N	type
4. #332×C₁₉₇	C₂	semidirect
72. C₁₉₇	#1546	direct (N × H)
73. #1546	C₁₉₇	direct (N × H)
77. #332	C₂×C₁₉₇	semidirect

H ≅ G/N

type

4. #332×C₁₉₇

C₂

semidirect

72. C₁₉₇

#1546

direct (N × H)

73. #1546

C₁₉₇

direct (N × H)

77. #332

C₂×C₁₉₇

semidirect

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

subgroup	index
C₂×C₄⋊Q₈×C₁₉₇	2
#431×C₁₉₇	2
#423×C₁₉₇	2
#332×C₁₉₇	2
#1546	197

subgroup

index

C₂×C₄⋊Q₈×C₁₉₇

#431×C₁₉₇

#423×C₁₉₇

#332×C₁₉₇

#1546

197

Subgroups — #1546×C₁₉₇ #38738480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1546×C197 #38738480

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #1546×C₁₉₇ #38738480