Group #38742885 — subgroups

Subgroups — #38742885

Order 25216.

212 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 101 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×#20263262	C₂	1	1	yes	maximal, characteristic
2	C₂×#20263261	C₂	1	2	yes	maximal
3	#27161102	C₂	1	2	yes	maximal
4	C₂×#20263236	C₂	1	1	yes	maximal, characteristic
5	C₂×#108×C₁₉₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#15691342	C₂²	1	2	yes	—
7	C₂²×#13096802	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₉₇	C₂²	1	1	yes	characteristic
9	#20263262	—	2	2	no	—
10	#20263261	—	2	4	no	—
11	#20263255	—	2	4	no	—
12	#20263236	C₄	1	4	yes	—
13	C₂×M₁₆×C₁₉₇	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
15	#108×C₁₉₇	—	2	2	no	—
16	C₂²×Dic₁₉₇	Q₈	1	2	yes	—
17	#15691342	—	2	2	no	—
18	#15691342	—	2	2	no	—
19	C₂×#13096802	—	2	2	no	—
20	C₂×#13096802	C₂×C₄	1	2	yes	—
21	C₂×#13096802	D₄	1	2	yes	—
22	C₂×#13096802	Q₈	1	2	yes	—
23	C₄×Dic₁₉₇	C₂×C₄	1	2	yes	—
24	C₄×Dic₁₉₇	D₄	1	2	yes	—
25	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
26	M₁₆×C₁₉₇	—	2	2	no	—
27	M₁₆×C₁₉₇	—	2	2	no	—
28	C₂×C₈×C₁₉₇	—	2	2	no	—
29	C₂×C₈×C₁₉₇	C₂×C₄	1	2	yes	—
30	C₂×C₈×C₁₉₇	D₄	1	2	yes	—
31	C₂×C₈×C₁₉₇	Q₈	1	2	yes	—
32	C₂×Dic₁₉₇	—	4	2	no	—
33	C₂×Dic₁₉₇	C₄⋊C₄	1	4	yes	—
34	#13096802	—	4	2	no	—
35	#13096802	C₄⋊C₄	1	4	yes	—
36	C₂³×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
37	C₂×C₄×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
43	C₈×C₁₉₇	—	4	2	no	—
44	C₈×C₁₉₇	C₄⋊C₄	1	4	yes	—
45	C₂²×C₁₉₇	—	2	2	no	—
46	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₉₇	C₄⋊Q₈	1	1	yes	characteristic
48	C₂²×C₁₉₇	C₄⋊Q₈	1	1	yes	characteristic
49	C₄×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₄×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₄×C₁₉₇	#127	1	1	yes	characteristic
52	C₄×C₁₉₇	C₄⋊Q₈	1	1	yes	characteristic
53	Dic₁₉₇	—	2	4	no	—
54	C₂×C₁₉₇	—	2	1	no	—
55	C₂×C₁₉₇	—	2	1	no	—
56	C₂×C₁₉₇	#384	1	1	yes	characteristic
57	C₂×C₁₉₇	#430	1	2	yes	—
58	C₁₉₇	#1849	1	1	yes	characteristic, complemented, Sylow
59	#1849	—	197	1	no	maximal, Sylow
60	C₂×#108	—	197	2	no	—
61	C₂×#108	—	197	1	no	—
62	C₂×#108	D₁₉₇	1	1	yes	characteristic
63	#423	—	197	2	no	—
64	C₂×#97	—	197	1	no	—
65	C₂×M₁₆	—	197	2	no	—
66	C₂×M₁₆	C₂×D₁₉₇	1	2	yes	—
67	C₂²×C₈	—	197	1	no	—
68	C₂²×C₈	C₂×D₁₉₇	1	1	yes	characteristic
69	C₂×C₄²	—	197	1	no	—
70	#108	—	2	2	no	—
71	#108	—	394	4	no	—
72	#108	—	394	2	no	—
73	C₄⋊C₈	—	394	4	no	—
74	#97	—	197	4	no	—
75	C₂²×C₄	—	197	2	no	—
76	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
77	M₁₆	—	2	2	no	—
78	M₁₆	—	2	2	no	—
79	M₁₆	—	394	2	no	—
80	M₁₆	—	394	2	no	—
81	C₂×C₈	—	2	2	no	—
82	C₂×C₈	—	394	2	no	—
83	C₂×C₈	—	197	2	no	—
84	C₂×C₈	—	197	2	no	—
85	C₂×C₈	—	197	2	no	—
86	C₂×C₈	C₄×D₁₉₇	1	2	yes	—
87	C₂×C₈	C₂²⋊D₁₉₇	1	2	yes	—
88	C₂×C₈	Dic₃₉₄	1	2	yes	—
89	C₄²	—	197	2	no	—
90	C₄²	—	197	2	no	—
91	C₂³	Q₈×D₁₉₇	1	1	yes	characteristic
92	C₂×C₄	—	788	2	no	—
93	C₂×C₄	—	197	4	no	—
94	C₂×C₄	C₂×C₄×D₁₉₇	1	1	yes	characteristic
95	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
96	C₂×C₄	D₄×D₁₉₇	1	1	yes	characteristic
97	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
98	C₂×C₄	Q₈×D₁₉₇	1	1	yes	characteristic
99	C₂×C₄	C₂×Dic₃₉₄	1	1	yes	characteristic
100	C₈	—	4	2	no	—
101	C₈	—	788	2	no	—
102	C₈	—	197	4	no	—
103	C₈	#15691337	1	4	yes	—
104	C₂²	—	2	2	no	—
105	C₂²	C₄⋊C₄×D₁₉₇	1	1	yes	characteristic
106	C₂²	#20263324	1	1	yes	characteristic
107	C₂²	#20263325	1	1	yes	characteristic
108	C₄	—	394	4	no	—
109	C₄	C₄⋊C₄×D₁₉₇	1	1	yes	characteristic
110	C₄	C₂×#15691337	1	1	yes	characteristic
111	C₄	#20263322	1	1	yes	characteristic
112	C₄	#20263325	1	1	yes	characteristic
113	C₂	—	2	1	no	—
114	C₂	—	2	1	no	—
115	C₂	#27160948	1	1	yes	characteristic
116	C₂	#27161126	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263262

C₂

yes

maximal, characteristic

C₂×#20263261

C₂

yes

maximal

#27161102

C₂

yes

maximal

C₂×#20263236

C₂

yes

maximal, characteristic

C₂×#108×C₁₉₇

C₂

yes

maximal, characteristic

C₂×#15691342

C₂²

yes

—

C₂²×#13096802

C₂²

yes

characteristic

C₂×C₄×Dic₁₉₇

C₂²

yes

—

—

—

C₄

yes

—

C₂×M₁₆×C₁₉₇

C₂²

yes

—

C₂²×C₈×C₁₉₇

C₂²

yes

characteristic

#108×C₁₉₇

—

C₂²×Dic₁₉₇

Q₈

yes

—

—

—

—

C₂×C₄

yes

—

C₂×#13096802

D₄

yes

—

C₂×#13096802

Q₈

yes

—

C₄×Dic₁₉₇

C₂×C₄

yes

—

C₄×Dic₁₉₇

D₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

M₁₆×C₁₉₇

—

M₁₆×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

C₂×C₄

yes

—

C₂×C₈×C₁₉₇

D₄

yes

—

C₂×C₈×C₁₉₇

Q₈

yes

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

C₄⋊C₄

yes

—

#13096802

—

#13096802

C₄⋊C₄

yes

—

C₂³×C₁₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₇

C₂×Q₈

yes

characteristic

C₈×C₁₉₇

—

C₈×C₁₉₇

C₄⋊C₄

yes

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₉₇

C₄⋊Q₈

yes

characteristic

C₂²×C₁₉₇

C₄⋊Q₈

yes

characteristic

C₄×C₁₉₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₉₇

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₉₇

#127

yes

characteristic

C₄×C₁₉₇

C₄⋊Q₈

yes

characteristic

Dic₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

—

C₂×C₁₉₇

#384

yes

characteristic

C₂×C₁₉₇

#430

yes

—

C₁₉₇

#1849

yes

characteristic, complemented, Sylow

#1849

—

197

maximal, Sylow

C₂×#108

—

197

—

C₂×#108

—

197

—

C₂×#108

D₁₉₇

yes

characteristic

#423

—

197

—

C₂×#97

—

197

—

C₂×M₁₆

—

197

—

C₂×M₁₆

C₂×D₁₉₇

yes

—

C₂²×C₈

—

197

—

C₂²×C₈

C₂×D₁₉₇

yes

characteristic

C₂×C₄²

—

197

—

#108

—

#108

—

394

—

#108

—

394

—

C₄⋊C₈

—

394

—

#97

—

197

—

C₂²×C₄

—

197

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

394

—

M₁₆

—

394

—

C₂×C₈

—

C₂×C₈

—

394

—

C₂×C₈

—

197

—

C₂×C₈

—

197

—

C₂×C₈

—

197

—

C₂×C₈

C₄×D₁₉₇

yes

—

C₂×C₈

C₂²⋊D₁₉₇

yes

—

C₂×C₈

Dic₃₉₄

yes

—

C₄²

—

197

—

C₄²

—

197

—

C₂³

Q₈×D₁₉₇

yes

characteristic

C₂×C₄

—

788

—

C₂×C₄

—

197

—

C₂×C₄

C₂×C₄×D₁₉₇

yes

characteristic

C₂×C₄

D₄×D₁₉₇

yes

characteristic

C₂×C₄

D₄×D₁₉₇

yes

characteristic

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

C₂×C₄

Q₈×D₁₉₇

yes

characteristic

C₂×C₄

C₂×Dic₃₉₄

yes

characteristic

100

C₈

—

101

C₈

—

788

—

102

C₈

—

197

—

103

C₈

#15691337

yes

—

104

C₂²

—

105

C₂²

C₄⋊C₄×D₁₉₇

yes

characteristic

106

C₂²

#20263324

yes

characteristic

107

C₂²

#20263325

yes

characteristic

108

C₄

—

394

—

109

C₄

C₄⋊C₄×D₁₉₇

yes

characteristic

110

C₄

C₂×#15691337

yes

characteristic

111

C₄

#20263322

yes

characteristic

112

C₄

#20263325

yes

characteristic

113

C₂

—

114

C₂

—

115

C₂

#27160948

yes

characteristic

116

C₂

#27161126

yes

—

	subgroup	G/H
center	C₂×C₄	D₄×D₁₉₇
commutator	C₂×C₄×C₁₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	C₂×#108×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂×C₄⋊C₄
prime radical	C₂³×C₁₉₇	C₂×Q₈
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	C₂×#108	D₁₉₇
hypocenter	C₁₉₇	#1849
LCS	G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂×C₄

D₄×D₁₉₇

commutator

C₂×C₄×C₁₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

C₂×#108×C₁₉₇

C₂

socle

C₂²×C₁₉₇

C₂×C₄⋊C₄

prime radical

C₂³×C₁₉₇

C₂×Q₈

cosocle

C₂²×C₄×C₁₉₇

C₂³

D₁₉₇

C₁₉₇

G ▷ C₂×C₄×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
58. C₁₉₇	#1849	semidirect

H ≅ G/N

type

58. C₁₉₇

#1849

semidirect

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

subgroup	index
C₂×#20263262	2
C₂×#20263261	2
#27161102	2
C₂×#20263236	2
C₂×#108×C₁₉₇	2
#1849	197

subgroup

index

C₂×#108×C₁₉₇

197

Subgroups — #38742885

Conjugacy classes of subgroups

Complements

Maximal subgroups