Group #38741793 — subgroups

Subgroups — #38741793

Order 25216.

240 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 240 conjugacy classes of subgroups — too many to draw. Only its 53 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	#27161795	C₂	1	1	yes	maximal, characteristic, complemented
2	#27161271	C₂	1	2	yes	maximal, complemented
3	#27160828	C₂	1	1	yes	maximal, characteristic, complemented
4	#27160808	C₂	1	2	yes	maximal, complemented
5	#492×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15691343	—	2	2	no	—
7	#20263315	—	2	2	no	—
8	#20263301	—	2	2	no	—
9	C₄×D₇₈₈	C₂²	1	2	yes	—
10	#20263284	—	2	2	no	—
11	C₄⋊#13096802	C₂²	1	2	yes	—
12	#20263247	—	2	2	no	—
13	C₂×SD₁₆×C₁₉₇	—	2	2	no	—
14	C₄⋊Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
15	#127×C₁₉₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
17	C₂×D₇₈₈	—	2	2	no	—
18	C₂×D₇₈₈	D₄	1	2	yes	—
19	C₂×C₄×D₁₉₇	—	2	2	no	—
20	#15691343	—	4	4	no	—
21	C₂×#13096802	—	2	2	no	—
22	C₄⋊Dic₁₉₇	D₄	1	2	yes	—
23	#15691335	—	2	4	no	—
24	#15691335	—	2	2	no	—
25	C₄×Dic₁₉₇	—	2	2	no	—
26	C₂×Q₈×C₁₉₇	D₄	1	2	yes	—
27	C₂×D₄×C₁₉₇	—	2	1	no	—
28	C₂×D₄×C₁₉₇	—	2	2	no	—
29	SD₁₆×C₁₉₇	—	4	4	no	—
30	C₂×C₈×C₁₉₇	—	2	1	no	—
31	C₄⋊C₄×C₁₉₇	—	2	2	no	—
32	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
33	C₂²×D₁₉₇	—	2	2	no	—
34	D₇₈₈	—	2	2	no	—
35	D₇₈₈	—	2	2	no	—
36	D₇₈₈	—	8	2	no	—
37	C₂×Dic₁₉₇	—	2	2	no	—
38	C₄×D₁₉₇	—	4	2	no	—
39	C₄×D₁₉₇	—	4	2	no	—
40	#13096802	—	4	2	no	—
41	C₂³×C₁₉₇	—	4	1	no	—
42	Q₈×C₁₉₇	—	4	2	no	—
43	D₄×C₁₉₇	—	4	2	no	—
44	D₄×C₁₉₇	—	4	2	no	—
45	D₄×C₁₉₇	—	4	2	no	—
46	C₂×C₄×C₁₉₇	—	2	2	no	—
47	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
48	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
49	C₈×C₁₉₇	—	2	2	no	—
50	C₂²×C₁₉₇	—	8	2	no	—
51	C₂²×C₁₉₇	—	8	1	no	—
52	C₂²×C₁₉₇	#120	1	1	yes	characteristic
53	C₂×D₁₉₇	—	4	2	no	—
54	C₂×D₁₉₇	—	4	2	no	—
55	C₂×D₁₉₇	—	4	2	no	—
56	C₄×C₁₉₇	—	2	1	no	—
57	C₄×C₁₉₇	—	4	2	no	—
58	C₄×C₁₉₇	#135	1	2	yes	—
59	C₄×C₁₉₇	AGL(1,ℤ/8)	1	2	yes	—
60	Dic₁₉₇	—	4	2	no	—
61	C₂×C₁₉₇	—	16	1	no	—
62	C₂×C₁₉₇	#449	1	2	yes	—
63	C₂×C₁₉₇	#453	1	1	yes	characteristic
64	D₁₉₇	—	8	2	no	—
65	C₁₉₇	#1624	1	1	yes	characteristic, complemented, Sylow
66	#1624	—	197	1	no	maximal, Sylow
67	#555	—	197	1	no	—
68	#492	D₁₉₇	1	1	yes	characteristic, complemented
69	#459	—	197	2	no	—
70	#333	—	197	1	no	—
71	D₄⋊C₈	—	197	2	no	—
72	C₂×SD₁₆	—	2	2	no	—
73	C₂×D₈	—	394	2	no	—
74	C₄⋊Q₈	C₂×D₁₉₇	1	1	yes	characteristic
75	#127	C₂×D₁₉₇	1	1	yes	characteristic
76	#124	—	394	2	no	—
77	#121	—	394	2	no	—
78	C₄×D₄	—	197	2	no	—
79	#117	—	394	2	no	—
80	C₄⋊C₈	—	197	2	no	—
81	#102	—	394	2	no	—
82	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
83	C₂×Q₈	C₂²⋊D₁₉₇	1	2	yes	—
84	C₂×D₄	—	2	2	no	—
85	C₂×D₄	—	2	1	no	—
86	C₂×D₄	—	394	2	no	—
87	C₂×D₄	—	197	2	no	—
88	C₂²×C₄	—	394	2	no	—
89	SD₁₆	—	4	4	no	—
90	D₈	—	788	4	no	—
91	C₂×C₈	—	2	1	no	—
92	C₂×C₈	—	394	2	no	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	197	2	no	—
95	C₂²⋊C₄	—	394	4	no	—
96	C₂²⋊C₄	—	394	2	no	—
97	C₄²	—	394	2	no	—
98	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
99	C₂³	—	4	1	no	—
100	C₂³	—	394	2	no	—
101	Q₈	—	4	2	no	—
102	D₄	—	4	2	no	—
103	D₄	—	4	2	no	—
104	D₄	—	4	2	no	—
105	D₄	—	1576	2	no	—
106	D₄	—	394	2	no	—
107	D₄	—	394	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	788	2	no	—
110	C₂×C₄	—	788	2	no	—
111	C₂×C₄	—	394	2	no	—
112	C₂×C₄	D₄×D₁₉₇	1	2	yes	—
113	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	788	2	no	—
116	C₂²	—	8	2	no	—
117	C₂²	—	8	1	no	—
118	C₂²	—	788	2	no	—
119	C₂²	—	788	2	no	—
120	C₂²	—	788	2	no	—
121	C₂²	#20263294	1	1	yes	characteristic
122	C₄	—	2	1	no	—
123	C₄	—	4	2	no	—
124	C₄	—	788	2	no	—
125	C₄	#20263347	1	2	yes	—
126	C₄	#20263353	1	2	yes	—
127	C₂	—	16	1	no	—
128	C₂	—	1576	2	no	—
129	C₂	#27161223	1	2	yes	—
130	C₂	#27161246	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27161795

C₂

yes

maximal, characteristic, complemented

#27161271

C₂

yes

maximal, complemented

#27160828

C₂

yes

maximal, characteristic, complemented

#27160808

C₂

yes

maximal, complemented

#492×C₁₉₇

C₂

yes

maximal, characteristic, complemented

C₂×#15691343

—

#20263315

—

#20263301

—

C₄×D₇₈₈

C₂²

yes

—

#20263284

—

C₄⋊#13096802

C₂²

yes

—

#20263247

—

C₂×SD₁₆×C₁₉₇

—

C₄⋊Q₈×C₁₉₇

C₂²

yes

characteristic

#127×C₁₉₇

C₂²

yes

characteristic

C₄×C₈×C₁₉₇

C₂²

yes

characteristic

C₂×D₇₈₈

—

C₂×D₇₈₈

D₄

yes

—

C₂×C₄×D₁₉₇

—

#15691343

—

C₂×#13096802

—

C₄⋊Dic₁₉₇

D₄

yes

—

#15691335

—

#15691335

—

C₄×Dic₁₉₇

—

C₂×Q₈×C₁₉₇

D₄

yes

—

C₂×D₄×C₁₉₇

—

C₂×D₄×C₁₉₇

—

SD₁₆×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄²×C₁₉₇

C₂³

yes

characteristic

C₂²×D₁₉₇

—

D₇₈₈

—

D₇₈₈

—

D₇₈₈

—

C₂×Dic₁₉₇

—

C₄×D₁₉₇

—

C₄×D₁₉₇

—

#13096802

—

C₂³×C₁₉₇

—

Q₈×C₁₉₇

—

D₄×C₁₉₇

—

D₄×C₁₉₇

—

D₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₈×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

#120

yes

characteristic

C₂×D₁₉₇

—

C₂×D₁₉₇

—

C₂×D₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

#135

yes

—

C₄×C₁₉₇

AGL(1,ℤ/8)

yes

—

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

#449

yes

—

C₂×C₁₉₇

#453

yes

characteristic

D₁₉₇

—

C₁₉₇

#1624

yes

characteristic, complemented, Sylow

#1624

—

197

maximal, Sylow

#555

—

197

—

#492

D₁₉₇

yes

characteristic, complemented

#459

—

197

—

#333

—

197

—

D₄⋊C₈

—

197

—

C₂×SD₁₆

—

C₂×D₈

—

394

—

C₄⋊Q₈

C₂×D₁₉₇

yes

characteristic

#127

C₂×D₁₉₇

yes

characteristic

#124

—

394

—

#121

—

394

—

C₄×D₄

—

197

—

#117

—

394

—

C₄⋊C₈

—

197

—

#102

—

394

—

C₄×C₈

C₂×D₁₉₇

yes

characteristic

C₂×Q₈

C₂²⋊D₁₉₇

yes

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

394

—

C₂×D₄

—

197

—

C₂²×C₄

—

394

—

SD₁₆

—

D₈

—

788

—

C₂×C₈

—

C₂×C₈

—

394

—

C₄⋊C₄

—

C₄⋊C₄

—

197

—

C₂²⋊C₄

—

394

—

C₂²⋊C₄

—

394

—

C₄²

—

394

—

C₄²

C₂²×D₁₉₇

yes

characteristic

C₂³

—

100

C₂³

—

394

—

101

Q₈

—

102

D₄

—

103

D₄

—

104

D₄

—

105

D₄

—

1576

—

106

D₄

—

394

—

107

D₄

—

394

—

108

C₂×C₄

—

109

C₂×C₄

—

788

—

110

C₂×C₄

—

788

—

111

C₂×C₄

—

394

—

112

C₂×C₄

D₄×D₁₉₇

yes

—

113

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

114

C₈

—

115

C₈

—

788

—

116

C₂²

—

117

C₂²

—

118

C₂²

—

788

—

119

C₂²

—

788

—

120

C₂²

—

788

—

121

C₂²

#20263294

yes

characteristic

122

C₄

—

123

C₄

—

124

C₄

—

788

—

125

C₄

#20263347

yes

—

126

C₄

#20263353

yes

—

127

C₂

—

128

C₂

—

1576

—

129

C₂

#27161223

yes

—

130

C₂

#27161246

yes

characteristic

	subgroup	G/H
center	C₂²	#20263294
commutator, cosocle	C₄²×C₁₉₇	C₂³
Frattini	C₄²	C₂²×D₁₉₇
Fitting	#492×C₁₉₇	C₂
socle	C₂²×C₁₉₇	#120
hypercenter	#492	D₁₉₇
hypocenter	C₁₉₇	#1624
LCS	G ▷ C₄²×C₁₉₇ ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #492

subgroup

G/H

center

C₂²

#20263294

commutator, cosocle

C₄²×C₁₉₇

C₂³

Frattini

C₄²

C₂²×D₁₉₇

Fitting

#492×C₁₉₇

C₂

C₂²×C₁₉₇

D₁₉₇

C₁₉₇

G ▷ C₄²×C₁₉₇ ▷ C₂²×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #492

N	H ≅ G/N	type
1. #27161795	C₂	semidirect
2. #27161271	C₂	semidirect
3. #27160828	C₂	semidirect
4. #27160808	C₂	semidirect
5. #492×C₁₉₇	C₂	semidirect
65. C₁₉₇	#1624	semidirect
68. #492	D₁₉₇	semidirect

H ≅ G/N

type

1. #27161795

C₂

semidirect

2. #27161271

C₂

semidirect

3. #27160828

C₂

semidirect

4. #27160808

C₂

semidirect

5. #492×C₁₉₇

C₂

semidirect

65. C₁₉₇

#1624

semidirect

68. #492

D₁₉₇

semidirect

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

subgroup	index
#27161795	2
#27161271	2
#27160828	2
#27160808	2
#492×C₁₉₇	2
#1624	197

subgroup

index

#492×C₁₉₇

197

Subgroups — #38741793

Conjugacy classes of subgroups

Complements

Maximal subgroups