Group #38743557 — subgroups

Subgroups — #38743557

Order 25216.

292 conjugacy classes of subgroups; only the 105 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 105 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	#27161103	C₂	1	1	yes	maximal, characteristic
2	C₂×#20263245	C₂	1	4	yes	maximal
3	C₄×#15691342	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊Q₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂×#15691342	C₂²	1	2	yes	—
6	C₂×#15691342	C₂²	1	2	yes	—
7	C₄⋊#13096802	—	2	2	no	—
8	#20263245	—	2	8	no	—
9	#20263237	—	2	1	no	—
10	C₄×#13096802	—	2	1	no	—
11	C₂²×Q₈×C₁₉₇	C₂²	1	2	yes	—
12	C₄⋊Q₈×C₁₉₇	—	2	1	no	—
13	C₄⋊Q₈×C₁₉₇	—	2	1	no	—
14	C₄⋊Q₈×C₁₉₇	—	2	2	no	—
15	C₂×C₄⋊C₄×C₁₉₇	C₄	1	4	yes	—
16	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
17	#15691342	—	2	4	no	—
18	#15691342	—	2	2	no	—
19	#15691342	D₄	1	4	yes	—
20	C₂×#13096802	—	2	2	no	—
21	C₂×#13096802	—	2	2	no	—
22	C₂×Q₈×C₁₉₇	—	2	2	no	—
23	C₂×Q₈×C₁₉₇	—	2	2	no	—
24	C₂×Q₈×C₁₉₇	—	2	4	no	—
25	C₂×Q₈×C₁₉₇	—	2	4	no	—
26	C₂²×C₄×C₁₉₇	C₂×C₄	1	4	yes	—
27	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
28	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₁₉₇	—	2	4	no	—
30	C₄⋊C₄×C₁₉₇	—	2	4	no	—
31	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
32	C₄²×C₁₉₇	D₄	1	1	yes	characteristic
33	C₄²×C₁₉₇	D₄	1	2	yes	—
34	#13096802	—	2	4	no	—
35	#13096802	—	4	2	no	—
36	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
37	Q₈×C₁₉₇	—	4	4	no	—
38	Q₈×C₁₉₇	—	4	2	no	—
39	Q₈×C₁₉₇	—	4	2	no	—
40	C₂×C₄×C₁₉₇	—	2	4	no	—
41	C₂×C₄×C₁₉₇	—	2	4	no	—
42	C₂×C₄×C₁₉₇	—	2	8	no	—
43	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₉₇	C₂²⋊C₄	1	2	yes	—
45	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
46	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
47	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₁₉₇	#41	1	2	yes	—
49	C₂²×C₁₉₇	—	2	2	no	—
50	C₂²×C₁₉₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
51	C₂²×C₁₉₇	C₄×D₄	1	2	yes	—
52	C₄×C₁₉₇	—	2	2	no	—
53	C₄×C₁₉₇	—	4	4	no	—
54	C₄×C₁₉₇	#101	1	4	yes	complemented
55	C₄×C₁₉₇	#121	1	2	yes	—
56	C₄×C₁₉₇	#124	1	1	yes	characteristic
57	C₄×C₁₉₇	#127	1	1	yes	characteristic
58	C₂×C₁₉₇	—	2	2	no	—
59	C₂×C₁₉₇	#390	1	1	yes	characteristic
60	C₂×C₁₉₇	C₂×#101	1	2	yes	—
61	C₁₉₇	#1966	1	1	yes	characteristic, complemented, Sylow
62	#1966	—	197	1	no	maximal, Sylow
63	C₂×C₄⋊Q₈	D₁₉₇	1	1	yes	characteristic
64	#423	—	197	1	no	—
65	C₂×#101	—	197	4	no	—
66	C₄×M₁₆	—	197	1	no	—
67	C₂²×Q₈	C₂×D₁₉₇	1	2	yes	—
68	C₂×M₁₆	—	197	2	no	—
69	C₂×M₁₆	—	197	2	no	—
70	C₄⋊Q₈	—	2	1	no	—
71	C₄⋊Q₈	—	2	1	no	—
72	C₄⋊Q₈	—	2	2	no	—
73	C₂×C₄⋊C₄	Dic₁₉₇	1	4	yes	—
74	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
75	C₄⋊C₈	—	394	2	no	—
76	#101	—	394	8	no	—
77	#97	—	394	1	no	—
78	C₄×C₈	—	394	1	no	—
79	C₂×Q₈	—	2	4	no	—
80	C₂×Q₈	—	2	2	no	—
81	C₂×Q₈	—	2	2	no	—
82	C₂×Q₈	—	2	4	no	—
83	C₂²×C₄	C₂×Dic₁₉₇	1	4	yes	—
84	C₂²×C₄	C₂×Dic₁₉₇	1	2	yes	—
85	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
86	M₁₆	—	394	4	no	—
87	M₁₆	—	394	2	no	—
88	M₁₆	—	197	4	no	—
89	C₂×C₈	—	394	2	no	—
90	C₂×C₈	—	394	2	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	2	4	no	—
93	C₄²	C₂²⋊D₁₉₇	1	1	yes	characteristic
94	C₄²	C₂²⋊D₁₉₇	1	1	yes	characteristic
95	C₄²	C₂²⋊D₁₉₇	1	2	yes	—
96	C₂³	C₂²×Dic₁₉₇	1	1	yes	characteristic
97	Q₈	—	4	4	no	—
98	Q₈	—	4	2	no	—
99	Q₈	—	4	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	2	8	no	—
103	C₂×C₄	C₂²⋊Dic₁₉₇	1	2	yes	—
104	C₂×C₄	C₂²⋊Dic₁₉₇	1	2	yes	—
105	C₂×C₄	D₄×D₁₉₇	1	2	yes	—
106	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
107	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
108	C₂×C₄	#15691357	1	2	yes	—
109	C₈	—	788	2	no	—
110	C₈	—	394	4	no	—
111	C₂²	—	2	2	no	—
112	C₂²	C₂×C₂²⋊Dic₁₉₇	1	1	yes	characteristic
113	C₂²	D₄×Dic₁₉₇	1	2	yes	—
114	C₄	—	2	2	no	—
115	C₄	—	4	4	no	—
116	C₄	#20263245	1	4	yes	complemented
117	C₄	#20263297	1	2	yes	—
118	C₄	#20263313	1	1	yes	characteristic
119	C₄	#20263322	1	1	yes	characteristic
120	C₂	—	2	2	no	—
121	C₂	#27160982	1	1	yes	characteristic
122	C₂	C₂×#20263245	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#27161103

C₂

yes

maximal, characteristic

C₂×#20263245

C₂

yes

maximal

C₄×#15691342

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₁₉₇

C₂

yes

maximal, characteristic

C₂×#15691342

C₂²

yes

—

C₂×#15691342

C₂²

yes

—

—

—

—

—

C₂²×Q₈×C₁₉₇

C₂²

yes

—

C₄⋊Q₈×C₁₉₇

—

C₄⋊Q₈×C₁₉₇

—

C₄⋊Q₈×C₁₉₇

—

C₂×C₄⋊C₄×C₁₉₇

C₄

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

—

—

D₄

yes

—

C₂×#13096802

—

C₂×#13096802

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

—

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄²×C₁₉₇

D₄

yes

characteristic

C₄²×C₁₉₇

D₄

yes

characteristic

C₄²×C₁₉₇

D₄

yes

—

#13096802

—

#13096802

—

C₂³×C₁₉₇

C₂²×C₄

yes

characteristic

Q₈×C₁₉₇

—

Q₈×C₁₉₇

—

Q₈×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₇

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₂²×C₁₉₇

—

C₂²×C₁₉₇

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₇

C₄×D₄

yes

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

#101

yes

complemented

C₄×C₁₉₇

#121

yes

—

C₄×C₁₉₇

yes

C₄×C₁₉₇

yes

C₂×C₁₉₇

—

C₂×C₁₉₇

#390

yes

characteristic

C₂×C₁₉₇

C₂×#101

yes

—

C₁₉₇

#1966

yes

characteristic, complemented, Sylow

#1966

—

197

maximal, Sylow

C₂×C₄⋊Q₈

D₁₉₇

yes

characteristic

#423

—

197

—

C₂×#101

—

197

—

C₄×M₁₆

—

197

—

C₂²×Q₈

C₂×D₁₉₇

yes

—

C₂×M₁₆

—

197

—

C₂×M₁₆

—

197

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₂×C₄⋊C₄

Dic₁₉₇

yes

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

C₄⋊C₈

—

394

—

#101

—

394

—

#97

—

394

—

C₄×C₈

—

394

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

C₂×Dic₁₉₇

yes

—

C₂²×C₄

C₂×Dic₁₉₇

yes

—

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

M₁₆

—

394

—

M₁₆

—

394

—

M₁₆

—

197

—

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

C₂²⋊D₁₉₇

yes

characteristic

C₄²

C₂²⋊D₁₉₇

yes

characteristic

C₄²

C₂²⋊D₁₉₇

yes

—

C₂³

C₂²×Dic₁₉₇

yes

characteristic

Q₈

—

Q₈

—

Q₈

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

C₂²⋊Dic₁₉₇

yes

—

104

C₂×C₄

C₂²⋊Dic₁₉₇

yes

—

105

C₂×C₄

D₄×D₁₉₇

yes

—

106

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

107

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

108

C₂×C₄

#15691357

yes

—

109

C₈

—

788

—

110

C₈

—

394

—

111

C₂²

—

112

C₂²

C₂×C₂²⋊Dic₁₉₇

yes

characteristic

113

C₂²

D₄×Dic₁₉₇

yes

—

114

C₄

—

115

C₄

—

116

C₄

#20263245

yes

complemented

117

C₄

#20263297

yes

—

118

C₄

#20263313

yes

characteristic

119

C₄

#20263322

yes

characteristic

120

C₂

—

121

C₂

#27160982

yes

characteristic

122

C₂

C₂×#20263245

yes

—

	subgroup	G/H
center	C₂²	C₂×C₂²⋊Dic₁₉₇
commutator, prime radical	C₂³×C₁₉₇	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₇
Fitting	C₂×C₄⋊Q₈×C₁₉₇	C₂
socle	C₂²×C₁₉₇	C₂×C₂²⋊C₄
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₉₇
hypocenter	C₁₉₇	#1966
LCS	G ▷ C₂³×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂²

C₂×C₂²⋊Dic₁₉₇

commutator, prime radical

C₂³×C₁₉₇

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₇

Fitting

C₂×C₄⋊Q₈×C₁₉₇

C₂

socle

C₂²×C₁₉₇

C₂×C₂²⋊C₄

cosocle

C₂²×C₄×C₁₉₇

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₉₇

hypocenter

C₁₉₇

#1966

LCS

G ▷ C₂³×C₁₉₇ ▷ C₂×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
54. C₄×C₁₉₇	#101	semidirect
61. C₁₉₇	#1966	semidirect
116. C₄	#20263245	semidirect

H ≅ G/N

type

54. C₄×C₁₉₇

#101

semidirect

61. C₁₉₇

#1966

semidirect

116. C₄

#20263245

semidirect

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

subgroup	index
#27161103	2
C₂×#20263245	2
C₄×#15691342	2
C₂×C₄⋊Q₈×C₁₉₇	2
#1966	197

subgroup

index

C₂×C₄⋊Q₈×C₁₉₇

197

Subgroups — #38743557

Conjugacy classes of subgroups

Complements

Maximal subgroups