Group #38753218 — subgroups

Subgroups — C₂×#27161482 #38753218

Order 25216.

424 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 197 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×#20263326	C₂	1	1	yes	maximal, characteristic
2	#27161482	C₂	1	8	yes	maximal, direct product factor
3	C₂×#20263259	C₂	1	4	yes	maximal
4	C₂×C₄×#13096802	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊Q₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₃₉₄	—	2	1	no	—
7	C₂²×#13096802	C₂²	1	2	yes	—
8	#20263326	—	2	2	no	—
9	#20263326	C₂²	1	4	yes	—
10	C₂×C₄⋊Dic₁₉₇	—	2	1	no	—
11	C₂×C₄⋊Dic₁₉₇	C₂²	1	2	yes	—
12	#20263259	C₂²	1	16	yes	—
13	C₄×#13096802	C₂²	1	4	yes	—
14	C₂²×Q₈×C₁₉₇	—	2	1	no	—
15	C₄⋊Q₈×C₁₉₇	—	2	2	no	—
16	C₄⋊Q₈×C₁₉₇	C₂²	1	4	yes	—
17	C₂×C₄⋊C₄×C₁₉₇	—	2	1	no	—
18	C₂×C₄⋊C₄×C₁₉₇	C₂²	1	2	yes	—
19	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
20	C₂×Dic₃₉₄	—	2	4	no	—
21	C₂×Dic₃₉₄	—	4	4	no	—
22	C₂²×Dic₁₉₇	—	2	2	no	—
23	C₂×#13096802	Q₈	1	8	yes	complemented
24	C₂×#13096802	C₂³	1	4	yes	—
25	C₄⋊Dic₁₉₇	—	2	4	no	—
26	C₄⋊Dic₁₉₇	—	2	2	no	—
27	C₄⋊Dic₁₉₇	C₂³	1	4	yes	—
28	C₂×Q₈×C₁₉₇	—	2	4	no	—
29	C₂×Q₈×C₁₉₇	—	4	4	no	—
30	C₂²×C₄×C₁₉₇	—	2	2	no	—
31	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
32	C₂²×C₄×C₁₉₇	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₁₉₇	—	2	4	no	—
35	C₄⋊C₄×C₁₉₇	—	2	2	no	—
36	C₄⋊C₄×C₁₉₇	C₂³	1	4	yes	—
37	C₄²×C₁₉₇	D₄	1	2	yes	—
38	C₄²×C₁₉₇	C₂³	1	2	yes	—
39	Dic₃₉₄	—	4	8	no	—
40	C₂×Dic₁₉₇	—	2	4	no	—
41	C₂×Dic₁₉₇	—	4	4	no	—
42	#13096802	C₂×Q₈	1	8	yes	complemented
43	C₂³×C₁₉₇	C₂×D₄	1	1	yes	characteristic
44	Q₈×C₁₉₇	—	4	8	no	—
45	C₂×C₄×C₁₉₇	—	2	4	no	—
46	C₂×C₄×C₁₉₇	—	4	4	no	—
47	C₂×C₄×C₁₉₇	D₈	1	4	yes	—
48	C₂×C₄×C₁₉₇	Q₁₆	1	4	yes	—
49	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
50	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
51	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
52	C₂×C₄×C₁₉₇	C₂×Q₈	1	4	yes	—
53	C₂×C₄×C₁₉₇	C₂⁴	1	1	yes	characteristic
54	C₂²×C₁₉₇	C₄⋊Q₈	1	2	yes	—
55	C₂²×C₁₉₇	C₂×D₈	1	2	yes	—
56	C₂²×C₁₉₇	C₂×Q₁₆	1	2	yes	—
57	C₂²×C₁₉₇	C₂²×D₄	1	1	yes	characteristic
58	C₄×C₁₉₇	—	4	4	no	—
59	C₄×C₁₉₇	C₄⋊Q₈	1	2	yes	—
60	C₄×C₁₉₇	C₂×D₈	1	4	yes	—
61	C₄×C₁₉₇	C₂×Q₁₆	1	4	yes	—
62	C₄×C₁₉₇	C₂²×Q₈	1	2	yes	—
63	Dic₁₉₇	—	4	4	no	—
64	C₂×C₁₉₇	#500	1	4	yes	complemented
65	C₂×C₁₉₇	C₂×C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₁₉₇	C₂²×D₈	1	1	yes	characteristic
67	C₂×C₁₉₇	C₂²×Q₁₆	1	1	yes	characteristic
68	C₁₉₇	C₂×#500	1	1	yes	characteristic, complemented, Sylow
69	C₂×#500	—	197	1	no	maximal, Sylow
70	C₂×C₄⋊Q₈	—	197	1	no	—
71	C₂×C₄⋊Q₈	D₁₉₇	1	1	yes	characteristic
72	#500	—	197	8	no	—
73	C₂×#107	—	197	4	no	—
74	C₂×C₄×C₈	—	197	1	no	—
75	C₂²×Q₈	—	2	1	no	—
76	C₂²×Q₈	—	394	1	no	—
77	C₂²×C₈	—	197	2	no	—
78	C₄⋊Q₈	—	2	2	no	—
79	C₄⋊Q₈	—	394	2	no	—
80	C₄⋊Q₈	—	197	4	no	—
81	C₄⋊Q₈	C₂×D₁₉₇	1	4	yes	—
82	C₂×C₄⋊C₄	—	2	1	no	—
83	C₂×C₄⋊C₄	—	394	1	no	—
84	C₂×C₄⋊C₄	—	197	2	no	—
85	C₂×C₄⋊C₄	C₂×D₁₉₇	1	2	yes	—
86	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
87	#107	—	197	16	no	—
88	C₄×C₈	—	197	4	no	—
89	C₂×Q₈	—	2	4	no	—
90	C₂×Q₈	—	4	4	no	—
91	C₂×Q₈	—	788	4	no	—
92	C₂×Q₈	—	394	4	no	—
93	C₂²×C₄	—	2	2	no	—
94	C₂²×C₄	—	394	2	no	—
95	C₂²×C₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
96	C₂²×C₄	C₂²⋊D₁₉₇	1	1	yes	characteristic
97	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
98	C₂×C₈	—	197	8	no	—
99	C₂×C₈	—	197	4	no	—
100	C₄⋊C₄	—	2	4	no	—
101	C₄⋊C₄	—	2	2	no	—
102	C₄⋊C₄	—	394	4	no	—
103	C₄⋊C₄	—	394	2	no	—
104	C₄⋊C₄	—	197	4	no	—
105	C₄⋊C₄	C₂²×D₁₉₇	1	4	yes	—
106	C₄²	C₂²⋊D₁₉₇	1	2	yes	—
107	C₄²	C₂²×D₁₉₇	1	2	yes	—
108	C₂³	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
109	Q₈	—	4	8	no	—
110	Q₈	—	788	8	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	4	4	no	—
113	C₂×C₄	—	788	4	no	—
114	C₂×C₄	—	394	4	no	—
115	C₂×C₄	#15691343	1	4	yes	—
116	C₂×C₄	#15691348	1	4	yes	—
117	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
118	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
119	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	2	yes	—
120	C₂×C₄	Q₈×D₁₉₇	1	4	yes	—
121	C₂×C₄	C₂³×D₁₉₇	1	1	yes	characteristic
122	C₈	—	197	8	no	—
123	C₂²	#20263324	1	2	yes	—
124	C₂²	C₂×#15691343	1	2	yes	—
125	C₂²	C₂×#15691348	1	2	yes	—
126	C₂²	C₂²×C₂²⋊D₁₉₇	1	1	yes	characteristic
127	C₄	—	4	4	no	—
128	C₄	—	788	4	no	—
129	C₄	#20263324	1	2	yes	—
130	C₄	C₂×#15691343	1	4	yes	—
131	C₄	C₂×#15691348	1	4	yes	—
132	C₄	C₂×Q₈×D₁₉₇	1	2	yes	—
133	C₂	#27161482	1	4	yes	direct product factor
134	C₂	C₂×#20263324	1	1	yes	characteristic
135	C₂	C₂²×#15691343	1	1	yes	characteristic
136	C₂	C₂²×#15691348	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263326

C₂

yes

maximal, characteristic

#27161482

C₂

yes

maximal, direct product factor

C₂×#20263259

C₂

yes

maximal

C₂×C₄×#13096802

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₉₇

C₂

yes

maximal, characteristic

C₂²×Dic₃₉₄

—

C₂²×#13096802

C₂²

yes

—

#20263326

—

#20263326

C₂²

yes

—

C₂×C₄⋊Dic₁₉₇

—

C₂×C₄⋊Dic₁₉₇

C₂²

yes

—

#20263259

C₂²

yes

—

C₄×#13096802

C₂²

yes

—

C₂²×Q₈×C₁₉₇

—

C₄⋊Q₈×C₁₉₇

—

C₄⋊Q₈×C₁₉₇

C₂²

yes

—

C₂×C₄⋊C₄×C₁₉₇

—

C₂×C₄⋊C₄×C₁₉₇

C₂²

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₂×Dic₃₉₄

—

C₂×Dic₃₉₄

—

C₂²×Dic₁₉₇

—

C₂×#13096802

Q₈

yes

complemented

C₂×#13096802

C₂³

yes

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

—

C₄⋊Dic₁₉₇

C₂³

yes

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂²×C₄×C₁₉₇

—

C₂²×C₄×C₁₉₇

D₄

yes

characteristic

C₂²×C₄×C₁₉₇

D₄

yes

characteristic

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

C₂³

yes

—

C₄²×C₁₉₇

D₄

yes

—

C₄²×C₁₉₇

C₂³

yes

—

Dic₃₉₄

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

#13096802

C₂×Q₈

yes

complemented

C₂³×C₁₉₇

C₂×D₄

yes

characteristic

Q₈×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

D₈

yes

—

C₂×C₄×C₁₉₇

Q₁₆

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂×Q₈

yes

—

C₂×C₄×C₁₉₇

C₂⁴

yes

characteristic

C₂²×C₁₉₇

C₄⋊Q₈

yes

—

C₂²×C₁₉₇

C₂×D₈

yes

—

C₂²×C₁₉₇

C₂×Q₁₆

yes

—

C₂²×C₁₉₇

C₂²×D₄

yes

characteristic

C₄×C₁₉₇

—

C₄×C₁₉₇

C₄⋊Q₈

yes

—

C₄×C₁₉₇

C₂×D₈

yes

—

C₄×C₁₉₇

C₂×Q₁₆

yes

—

C₄×C₁₉₇

C₂²×Q₈

yes

—

Dic₁₉₇

—

C₂×C₁₉₇

#500

yes

complemented

C₂×C₁₉₇

C₂×C₄⋊Q₈

yes

characteristic

C₂×C₁₉₇

C₂²×D₈

yes

characteristic

C₂×C₁₉₇

C₂²×Q₁₆

yes

characteristic

C₁₉₇

C₂×#500

yes

characteristic, complemented, Sylow

C₂×#500

—

197

maximal, Sylow

C₂×C₄⋊Q₈

—

197

—

C₂×C₄⋊Q₈

D₁₉₇

yes

characteristic

#500

—

197

—

C₂×#107

—

197

—

C₂×C₄×C₈

—

197

—

C₂²×Q₈

—

C₂²×Q₈

—

394

—

C₂²×C₈

—

197

—

C₄⋊Q₈

—

C₄⋊Q₈

—

394

—

C₄⋊Q₈

—

197

—

C₄⋊Q₈

C₂×D₁₉₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

394

—

C₂×C₄⋊C₄

—

197

—

C₂×C₄⋊C₄

C₂×D₁₉₇

yes

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

#107

—

197

—

C₄×C₈

—

197

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

788

—

C₂×Q₈

—

394

—

C₂²×C₄

—

C₂²×C₄

—

394

—

C₂²×C₄

C₂²⋊D₁₉₇

yes

characteristic

C₂²×C₄

C₂²⋊D₁₉₇

yes

characteristic

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

C₂×C₈

—

197

—

C₂×C₈

—

197

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

102

C₄⋊C₄

—

394

—

103

C₄⋊C₄

—

394

—

104

C₄⋊C₄

—

197

—

105

C₄⋊C₄

C₂²×D₁₉₇

yes

—

106

C₄²

C₂²⋊D₁₉₇

yes

—

107

C₄²

C₂²×D₁₉₇

yes

—

108

C₂³

C₂×C₂²⋊D₁₉₇

yes

characteristic

109

Q₈

—

110

Q₈

—

788

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

788

—

114

C₂×C₄

—

394

—

115

C₂×C₄

#15691343

yes

—

116

C₂×C₄

#15691348

yes

—

117

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

118

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

—

119

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

—

120

C₂×C₄

Q₈×D₁₉₇

yes

—

121

C₂×C₄

C₂³×D₁₉₇

yes

characteristic

122

C₈

—

197

—

123

C₂²

#20263324

yes

—

124

C₂²

C₂×#15691343

yes

—

125

C₂²

C₂×#15691348

yes

—

126

C₂²

C₂²×C₂²⋊D₁₉₇

yes

characteristic

127

C₄

—

128

C₄

—

788

—

129

C₄

#20263324

yes

—

130

C₄

C₂×#15691343

yes

—

131

C₄

C₂×#15691348

yes

—

132

C₄

C₂×Q₈×D₁₉₇

yes

—

133

C₂

#27161482

yes

direct product factor

134

C₂

C₂×#20263324

yes

characteristic

135

C₂

C₂²×#15691343

yes

characteristic

136

C₂

C₂²×#15691348

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₇
commutator, cosocle	C₂×C₄×C₁₉₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₉₇
Fitting	C₂×C₄⋊Q₈×C₁₉₇	C₂
socle, prime radical	C₂³×C₁₉₇	C₂×D₄
hypercenter	C₂×C₄⋊Q₈	D₁₉₇
hypocenter	C₁₉₇	C₂×#500
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₇

commutator, cosocle

C₂×C₄×C₁₉₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₁₉₇

Fitting

C₂×C₄⋊Q₈×C₁₉₇

C₂

socle, prime radical

C₂³×C₁₉₇

C₂×D₄

hypercenter

C₂×C₄⋊Q₈

D₁₉₇

hypocenter

C₁₉₇

C₂×#500

LCS

G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
2. #27161482	C₂	direct (N × H)
23. C₂×#13096802	Q₈	semidirect
42. #13096802	C₂×Q₈	semidirect
64. C₂×C₁₉₇	#500	semidirect
68. C₁₉₇	C₂×#500	semidirect
133. C₂	#27161482	direct (N × H)

H ≅ G/N

type

2. #27161482

C₂

direct (N × H)

23. C₂×#13096802

Q₈

semidirect

42. #13096802

C₂×Q₈

semidirect

64. C₂×C₁₉₇

#500

semidirect

68. C₁₉₇

C₂×#500

semidirect

133. C₂

#27161482

direct (N × H)

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

subgroup	index
C₂×#20263326	2
#27161482	2
C₂×#20263259	2
C₂×C₄×#13096802	2
C₂×C₄⋊Q₈×C₁₉₇	2
C₂×#500	197

subgroup

index

C₂×C₄⋊Q₈×C₁₉₇

197

Subgroups — C₂×#27161482 #38753218

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C2×#27161482 #38753218

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#27161482 #38753218