Group #38741791 — subgroups

Subgroups — #38741791

Order 25216.

228 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 57 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	#27161809	C₂	1	1	yes	maximal, characteristic, complemented
2	#27161453	C₂	1	1	yes	maximal, characteristic
3	#27161280	C₂	1	2	yes	maximal
4	#27160809	C₂	1	2	yes	maximal, complemented
5	#333×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15691347	—	2	2	no	—
7	C₂×#15691345	—	2	2	no	—
8	#20263325	—	2	2	no	—
9	#20263324	—	2	2	no	—
10	#20263323	C₂²	1	1	yes	characteristic
11	C₄×Dic₃₉₄	C₂²	1	2	yes	—
12	#20263291	—	2	2	no	—
13	#20263248	—	2	2	no	—
14	C₄×#13096802	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₁₉₇	C₂²	1	2	yes	—
17	C₂×Dic₃₉₄	—	2	2	no	—
18	C₂×Dic₃₉₄	D₄	1	2	yes	—
19	C₂×D₇₈₈	—	2	2	no	—
20	C₂×D₇₈₈	—	2	1	no	—
21	#15691347	—	4	4	no	—
22	#15691345	—	4	4	no	—
23	C₂×#13096802	—	2	1	no	—
24	C₄⋊Dic₁₉₇	D₄	1	2	yes	—
25	#15691337	—	2	4	no	—
26	#15691337	—	2	2	no	—
27	C₄×Dic₁₉₇	—	2	2	no	—
28	C₄×Dic₁₉₇	—	2	2	no	—
29	C₂×Q₈×C₁₉₇	D₄	1	2	yes	—
30	C₂×C₈×C₁₉₇	—	2	2	no	—
31	C₄⋊C₄×C₁₉₇	—	2	2	no	—
32	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
33	C₂²×D₁₉₇	—	4	1	no	—
34	Dic₃₉₄	—	2	2	no	—
35	Dic₃₉₄	—	8	2	no	—
36	Dic₃₉₄	SD₁₆	1	4	yes	complemented
37	D₇₈₈	—	4	2	no	—
38	D₇₈₈	—	4	2	no	—
39	D₇₈₈	—	4	2	no	—
40	C₂×Dic₁₉₇	—	2	2	no	—
41	C₂×Dic₁₉₇	—	2	2	no	—
42	#13096802	—	2	2	no	—
43	Q₈×C₁₉₇	—	4	2	no	—
44	C₂×C₄×C₁₉₇	—	2	2	no	—
45	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
47	C₈×C₁₉₇	—	4	2	no	—
48	C₂²×C₁₉₇	#120	1	1	yes	characteristic
49	C₂×D₁₉₇	—	8	2	no	—
50	C₂×D₁₉₇	—	8	1	no	—
51	C₄×C₁₉₇	—	2	1	no	—
52	C₄×C₁₉₇	—	4	2	no	—
53	C₄×C₁₉₇	C₂×SD₁₆	1	2	yes	—
54	C₄×C₁₉₇	#137	1	2	yes	—
55	Dic₁₉₇	—	2	4	no	—
56	Dic₁₉₇	—	4	2	no	—
57	C₂×C₁₉₇	#448	1	2	yes	—
58	C₂×C₁₉₇	#453	1	1	yes	characteristic
59	D₁₉₇	—	16	1	no	—
60	C₁₉₇	#1623	1	1	yes	characteristic, complemented, Sylow
61	#1623	—	197	1	no	maximal, Sylow
62	Q₈²	—	197	1	no	—
63	#492	—	197	1	no	—
64	#460	—	197	2	no	—
65	#333	D₁₉₇	1	1	yes	characteristic, complemented
66	Q₈⋊C₈	—	197	2	no	—
67	C₂×SD₁₆	—	394	2	no	—
68	C₂×SD₁₆	—	394	2	no	—
69	C₄⋊Q₈	—	394	2	no	—
70	C₄⋊Q₈	—	394	2	no	—
71	C₄⋊Q₈	C₂×D₁₉₇	1	1	yes	characteristic
72	#127	—	197	1	no	—
73	C₄×Q₈	—	394	2	no	—
74	C₄×Q₈	—	197	2	no	—
75	C₄⋊C₈	C₂×D₁₉₇	1	2	yes	—
76	#102	—	394	2	no	—
77	C₄×C₈	—	197	1	no	—
78	C₂×Q₈	—	394	2	no	—
79	C₂×Q₈	—	197	2	no	—
80	C₂×Q₈	D₇₈₈	1	2	yes	—
81	C₂×D₄	—	394	2	no	—
82	C₂×D₄	—	394	1	no	—
83	SD₁₆	—	788	4	no	—
84	SD₁₆	—	788	4	no	—
85	C₂×C₈	—	2	2	no	—
86	C₂×C₈	—	394	1	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	394	4	no	—
89	C₄⋊C₄	—	394	2	no	—
90	C₄⋊C₄	—	197	2	no	—
91	C₄²	—	394	2	no	—
92	C₄²	—	394	2	no	—
93	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
94	C₂³	—	788	1	no	—
95	Q₈	—	4	2	no	—
96	Q₈	—	1576	2	no	—
97	Q₈	—	394	2	no	—
98	Q₈	—	197	4	no	—
99	D₄	—	788	2	no	—
100	D₄	—	788	2	no	—
101	D₄	—	788	2	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	394	2	no	—
104	C₂×C₄	—	394	2	no	—
105	C₂×C₄	D₄×D₁₉₇	1	2	yes	—
106	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
107	C₈	—	4	2	no	—
108	C₈	—	394	2	no	—
109	C₂²	—	1576	2	no	—
110	C₂²	—	1576	1	no	—
111	C₂²	#20263296	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	788	2	no	—
115	C₄	—	394	4	no	—
116	C₄	SD₁₆×D₁₉₇	1	2	yes	—
117	C₄	#20263358	1	2	yes	—
118	C₂	—	3152	1	no	—
119	C₂	#27161219	1	2	yes	—
120	C₂	#27161249	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27161809

C₂

yes

maximal, characteristic, complemented

#27161453

C₂

yes

maximal, characteristic

#27161280

C₂

yes

maximal

#27160809

C₂

yes

maximal, complemented

#333×C₁₉₇

C₂

yes

maximal, characteristic, complemented

—

—

—

—

C₂²

yes

characteristic

C₄×Dic₃₉₄

C₂²

yes

—

#20263291

—

#20263248

—

C₄×#13096802

C₂²

yes

characteristic

C₄⋊Q₈×C₁₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₇

C₂²

yes

—

C₂×Dic₃₉₄

—

C₂×Dic₃₉₄

D₄

yes

—

C₂×D₇₈₈

—

C₂×D₇₈₈

—

#15691347

—

#15691345

—

C₂×#13096802

—

C₄⋊Dic₁₉₇

D₄

yes

—

#15691337

—

#15691337

—

C₄×Dic₁₉₇

—

C₄×Dic₁₉₇

—

C₂×Q₈×C₁₉₇

D₄

yes

—

C₂×C₈×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄²×C₁₉₇

C₂³

yes

characteristic

C₂²×D₁₉₇

—

Dic₃₉₄

—

Dic₃₉₄

—

Dic₃₉₄

SD₁₆

yes

complemented

D₇₈₈

—

D₇₈₈

—

D₇₈₈

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

#13096802

—

Q₈×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₈×C₁₉₇

—

C₂²×C₁₉₇

#120

yes

characteristic

C₂×D₁₉₇

—

C₂×D₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

C₂×SD₁₆

yes

—

C₄×C₁₉₇

#137

yes

—

Dic₁₉₇

—

Dic₁₉₇

—

C₂×C₁₉₇

#448

yes

—

C₂×C₁₉₇

#453

yes

characteristic

D₁₉₇

—

C₁₉₇

#1623

yes

characteristic, complemented, Sylow

#1623

—

197

maximal, Sylow

Q₈²

—

197

—

#492

—

197

—

#460

—

197

—

#333

D₁₉₇

yes

characteristic, complemented

Q₈⋊C₈

—

197

—

C₂×SD₁₆

—

394

—

C₂×SD₁₆

—

394

—

C₄⋊Q₈

—

394

—

C₄⋊Q₈

—

394

—

C₄⋊Q₈

C₂×D₁₉₇

yes

characteristic

#127

—

197

—

C₄×Q₈

—

394

—

C₄×Q₈

—

197

—

C₄⋊C₈

C₂×D₁₉₇

yes

—

#102

—

394

—

C₄×C₈

—

197

—

C₂×Q₈

—

394

—

C₂×Q₈

—

197

—

C₂×Q₈

D₇₈₈

yes

—

C₂×D₄

—

394

—

C₂×D₄

—

394

—

SD₁₆

—

788

—

SD₁₆

—

788

—

C₂×C₈

—

C₂×C₈

—

394

—

C₄⋊C₄

—

C₄⋊C₄

—

394

—

C₄⋊C₄

—

394

—

C₄⋊C₄

—

197

—

C₄²

—

394

—

C₄²

—

394

—

C₄²

C₂²×D₁₉₇

yes

characteristic

C₂³

—

788

—

Q₈

—

Q₈

—

1576

—

Q₈

—

394

—

Q₈

—

197

—

D₄

—

788

—

100

D₄

—

788

—

101

D₄

—

788

—

102

C₂×C₄

—

103

C₂×C₄

—

394

—

104

C₂×C₄

—

394

—

105

C₂×C₄

D₄×D₁₉₇

yes

—

106

C₂×C₄

C₂×D₇₈₈

yes

characteristic

107

C₈

—

108

C₈

—

394

—

109

C₂²

—

1576

—

110

C₂²

—

1576

—

111

C₂²

#20263296

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

788

—

115

C₄

—

394

—

116

C₄

SD₁₆×D₁₉₇

yes

—

117

C₄

#20263358

yes

—

118

C₂

—

3152

—

119

C₂

#27161219

yes

—

120

C₂

#27161249

yes

characteristic

	subgroup	G/H
center	C₂²	#20263296
commutator, cosocle	C₄²×C₁₉₇	C₂³
Frattini	C₄²	C₂²×D₁₉₇
Fitting	#333×C₁₉₇	C₂
socle	C₂²×C₁₉₇	#120
prime radical	#20263323	C₂²
hypercenter	#333	D₁₉₇
hypocenter	C₁₉₇	#1623
LCS	G ▷ C₄²×C₁₉₇ ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#20263296

commutator, cosocle

C₄²×C₁₉₇

C₂³

Frattini

C₄²

C₂²×D₁₉₇

Fitting

#333×C₁₉₇

C₂

C₂²×C₁₉₇

C₂²

D₁₉₇

C₁₉₇

G ▷ C₄²×C₁₉₇ ▷ C₂²×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
1. #27161809	C₂	semidirect
4. #27160809	C₂	semidirect
5. #333×C₁₉₇	C₂	semidirect
36. Dic₃₉₄	SD₁₆	semidirect
60. C₁₉₇	#1623	semidirect
65. #333	D₁₉₇	semidirect

H ≅ G/N

type

1. #27161809

C₂

semidirect

4. #27160809

C₂

semidirect

5. #333×C₁₉₇

C₂

semidirect

36. Dic₃₉₄

SD₁₆

semidirect

60. C₁₉₇

#1623

semidirect

65. #333

D₁₉₇

semidirect

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

subgroup	index
#27161809	2
#27161453	2
#27161280	2
#27160809	2
#333×C₁₉₇	2
#1623	197

subgroup

index

#333×C₁₉₇

197

Subgroups — #38741791

Conjugacy classes of subgroups

Complements

Maximal subgroups