Group #38742565 — subgroups

Subgroups — #38742565

Order 25216.

292 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 165 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×C₄⋊#13096802	C₂	1	2	yes	maximal
2	C₂×#20263237	C₂	1	2	yes	maximal
3	#27160834	C₂	1	2	yes	maximal
4	C₄³×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13096802	C₂²	1	4	yes	—
6	C₄⋊#13096802	C₄	1	8	yes	complemented
7	#20263237	—	2	4	no	—
8	C₂×C₄²×C₁₉₇	C₄	1	2	yes	—
9	C₂×C₄²×C₁₉₇	C₄	1	2	yes	—
10	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
11	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
12	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
13	C₂×#13096802	—	2	4	no	—
14	C₂×#13096802	—	2	4	no	—
15	C₂×#13096802	C₂×C₄	1	8	yes	—
16	C₂²×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
18	C₂²×C₄×C₁₉₇	C₂×C₄	1	2	yes	—
19	C₂²×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
21	C₄²×C₁₉₇	—	2	4	no	—
22	C₄²×C₁₉₇	—	2	4	no	—
23	C₄²×C₁₉₇	—	2	2	no	—
24	C₄²×C₁₉₇	C₂×C₄	1	4	yes	—
25	C₄²×C₁₉₇	D₄	1	2	yes	—
26	C₄²×C₁₉₇	Q₈	1	2	yes	—
27	#13096802	—	2	8	no	—
28	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
29	C₂×C₄×C₁₉₇	—	2	2	no	—
30	C₂×C₄×C₁₉₇	—	2	2	no	—
31	C₂×C₄×C₁₉₇	—	2	2	no	—
32	C₂×C₄×C₁₉₇	—	2	2	no	—
33	C₂×C₄×C₁₉₇	—	2	2	no	—
34	C₂×C₄×C₁₉₇	C₄²	1	2	yes	—
35	C₂×C₄×C₁₉₇	C₄²	1	2	yes	—
36	C₂×C₄×C₁₉₇	C₄⋊C₄	1	4	yes	—
37	C₂×C₄×C₁₉₇	M₁₆	1	4	yes	—
38	C₂×C₄×C₁₉₇	M₁₆	1	4	yes	—
39	C₂×C₄×C₁₉₇	C₂²×C₄	1	2	yes	—
40	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
42	C₂×C₄×C₁₉₇	#41	1	2	yes	—
43	C₂²×C₁₉₇	C₂×C₄²	1	1	yes	characteristic
44	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂²×C₁₉₇	#117	1	1	yes	characteristic
46	C₂²×C₁₉₇	C₂×M₁₆	1	2	yes	—
47	C₂²×C₁₉₇	C₂×M₁₆	1	2	yes	—
48	C₄×C₁₉₇	—	2	2	no	—
49	C₄×C₁₉₇	—	2	4	no	—
50	C₄×C₁₉₇	—	2	4	no	—
51	C₄×C₁₉₇	#97	1	4	yes	complemented
52	C₄×C₁₉₇	C₄×D₄	1	2	yes	—
53	C₄×C₁₉₇	C₄×Q₈	1	2	yes	—
54	C₂×C₁₉₇	C₄×C₄⋊C₄	1	1	yes	characteristic
55	C₂×C₁₉₇	C₂×#97	1	1	yes	characteristic
56	C₂×C₁₉₇	C₄×M₁₆	1	1	yes	characteristic
57	C₂×C₁₉₇	#423	1	2	yes	—
58	C₂×C₁₉₇	#432	1	2	yes	—
59	C₁₉₇	#1767	1	1	yes	characteristic, complemented, Sylow
60	#1767	—	197	1	no	maximal, Sylow
61	C₂×C₄⋊C₈	—	197	2	no	—
62	C₂×#97	—	197	2	no	—
63	C₄³	D₁₉₇	1	1	yes	characteristic
64	#336	—	197	2	no	—
65	C₂²×C₈	—	197	4	no	—
66	C₂×C₄²	Dic₁₉₇	1	2	yes	—
67	C₂×C₄²	Dic₁₉₇	1	2	yes	—
68	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
69	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
70	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
71	C₄⋊C₈	—	197	8	no	—
72	#97	—	394	4	no	—
73	C₂²×C₄	C₂×Dic₁₉₇	1	1	yes	characteristic
74	C₂²×C₄	C₂×Dic₁₉₇	1	2	yes	—
75	C₂²×C₄	C₂×Dic₁₉₇	1	2	yes	—
76	C₂²×C₄	C₂×Dic₁₉₇	1	1	yes	characteristic
77	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
78	C₂×C₈	—	394	4	no	—
79	C₂×C₈	—	394	4	no	—
80	C₂×C₈	—	197	8	no	—
81	C₄²	—	2	4	no	—
82	C₄²	—	2	4	no	—
83	C₄²	—	2	2	no	—
84	C₄²	C₄×D₁₉₇	1	4	yes	—
85	C₄²	D₇₈₈	1	2	yes	—
86	C₄²	Dic₃₉₄	1	2	yes	—
87	C₂³	C₂²×Dic₁₉₇	1	1	yes	characteristic
88	C₂×C₄	—	2	2	no	—
89	C₂×C₄	—	2	2	no	—
90	C₂×C₄	—	2	2	no	—
91	C₂×C₄	—	2	2	no	—
92	C₂×C₄	—	2	2	no	—
93	C₂×C₄	C₄×Dic₁₉₇	1	2	yes	—
94	C₂×C₄	C₄×Dic₁₉₇	1	2	yes	—
95	C₂×C₄	C₄⋊Dic₁₉₇	1	4	yes	—
96	C₂×C₄	#15691342	1	4	yes	—
97	C₂×C₄	#15691342	1	4	yes	—
98	C₂×C₄	C₂×C₄×D₁₉₇	1	2	yes	—
99	C₂×C₄	C₂×D₇₈₈	1	1	yes	characteristic
100	C₂×C₄	C₂×Dic₃₉₄	1	1	yes	characteristic
101	C₂×C₄	#15691358	1	2	yes	—
102	C₈	—	394	8	no	—
103	C₂²	C₂×C₄×Dic₁₉₇	1	1	yes	characteristic
104	C₂²	C₂×C₄⋊Dic₁₉₇	1	1	yes	characteristic
105	C₂²	#20263285	1	1	yes	characteristic
106	C₂²	C₂×#15691342	1	2	yes	—
107	C₂²	C₂×#15691342	1	2	yes	—
108	C₄	—	2	4	no	—
109	C₄	—	2	4	no	—
110	C₄	—	2	2	no	—
111	C₄	#20263237	1	4	yes	complemented
112	C₄	C₄×D₇₈₈	1	2	yes	—
113	C₄	C₄×Dic₃₉₄	1	2	yes	—
114	C₂	C₄×C₄⋊Dic₁₉₇	1	1	yes	characteristic
115	C₂	C₂×#20263237	1	1	yes	characteristic
116	C₂	C₄×#15691342	1	1	yes	characteristic
117	C₂	#27161103	1	2	yes	—
118	C₂	#27161136	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#13096802

C₂

yes

maximal

C₂×#20263237

C₂

yes

maximal

#27160834

C₂

yes

maximal

C₄³×C₁₉₇

C₂

yes

maximal, characteristic

C₂²×#13096802

C₂²

yes

—

C₄⋊#13096802

C₄

yes

complemented

#20263237

—

C₂×C₄²×C₁₉₇

C₄

yes

—

C₂×C₄²×C₁₉₇

C₄

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₂×C₄²×C₁₉₇

C₂²

yes

—

—

C₂×C₄

yes

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₇

C₂×C₄

yes

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₄²×C₁₉₇

—

C₄²×C₁₉₇

—

C₄²×C₁₉₇

—

C₄²×C₁₉₇

C₂×C₄

yes

—

C₄²×C₁₉₇

D₄

yes

—

C₄²×C₁₉₇

Q₈

yes

—

#13096802

—

C₂³×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₄²

yes

—

C₂×C₄×C₁₉₇

C₄²

yes

—

C₂×C₄×C₁₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₇

M₁₆

yes

—

C₂×C₄×C₁₉₇

M₁₆

yes

—

C₂×C₄×C₁₉₇

C₂²×C₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₇

#41

yes

—

C₂²×C₁₉₇

C₂×C₄²

yes

characteristic

C₂²×C₁₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₉₇

#117

yes

characteristic

C₂²×C₁₉₇

C₂×M₁₆

yes

—

C₂²×C₁₉₇

C₂×M₁₆

yes

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

#97

yes

complemented

C₄×C₁₉₇

C₄×D₄

yes

—

C₄×C₁₉₇

C₄×Q₈

yes

—

C₂×C₁₉₇

C₄×C₄⋊C₄

yes

characteristic

C₂×C₁₉₇

C₂×#97

yes

characteristic

C₂×C₁₉₇

C₄×M₁₆

yes

characteristic

C₂×C₁₉₇

#423

yes

—

C₂×C₁₉₇

#432

yes

—

C₁₉₇

#1767

yes

characteristic, complemented, Sylow

#1767

—

197

maximal, Sylow

C₂×C₄⋊C₈

—

197

—

C₂×#97

—

197

—

C₄³

D₁₉₇

yes

characteristic

#336

—

197

—

C₂²×C₈

—

197

—

C₂×C₄²

Dic₁₉₇

yes

—

C₂×C₄²

Dic₁₉₇

yes

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

C₄⋊C₈

—

197

—

#97

—

394

—

C₂²×C₄

C₂×Dic₁₉₇

yes

characteristic

C₂²×C₄

C₂×Dic₁₉₇

yes

—

C₂²×C₄

C₂×Dic₁₉₇

yes

—

C₂²×C₄

C₂×Dic₁₉₇

yes

characteristic

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₂×C₈

—

197

—

C₄²

—

C₄²

—

C₄²

—

C₄²

C₄×D₁₉₇

yes

—

C₄²

D₇₈₈

yes

—

C₄²

Dic₃₉₄

yes

—

C₂³

C₂²×Dic₁₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₄×Dic₁₉₇

yes

—

C₂×C₄

C₄×Dic₁₉₇

yes

—

C₂×C₄

C₄⋊Dic₁₉₇

yes

—

C₂×C₄

#15691342

yes

—

C₂×C₄

#15691342

yes

—

C₂×C₄

C₂×C₄×D₁₉₇

yes

—

C₂×C₄

C₂×D₇₈₈

yes

characteristic

100

C₂×C₄

C₂×Dic₃₉₄

yes

characteristic

101

C₂×C₄

#15691358

yes

—

102

C₈

—

394

—

103

C₂²

C₂×C₄×Dic₁₉₇

yes

characteristic

104

C₂²

C₂×C₄⋊Dic₁₉₇

yes

characteristic

105

C₂²

#20263285

yes

characteristic

106

C₂²

C₂×#15691342

yes

—

107

C₂²

C₂×#15691342

yes

—

108

C₄

—

109

C₄

—

110

C₄

—

111

C₄

#20263237

yes

complemented

112

C₄

C₄×D₇₈₈

yes

—

113

C₄

C₄×Dic₃₉₄

yes

—

114

C₂

C₄×C₄⋊Dic₁₉₇

yes

characteristic

115

C₂

C₂×#20263237

yes

characteristic

116

C₂

C₄×#15691342

yes

characteristic

117

C₂

#27161103

yes

—

118

C₂

#27161136

yes

—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₉₇
commutator	C₂²×C₁₉₇	C₂×C₄²
Fitting	C₄³×C₁₉₇	C₂
socle, prime radical	C₂³×C₁₉₇	C₂²×C₄
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	C₄³	D₁₉₇
hypocenter	C₁₉₇	#1767
LCS	G ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₄³

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₉₇

commutator

C₂²×C₁₉₇

C₂×C₄²

Fitting

C₄³×C₁₉₇

C₂

socle, prime radical

C₂³×C₁₉₇

C₂²×C₄

cosocle

C₂²×C₄×C₁₉₇

C₂³

hypercenter

C₄³

D₁₉₇

C₁₉₇

G ▷ C₂²×C₁₉₇ ▷ C₁₉₇

C₁ ◁ C₂²×C₄ ◁ C₄³

N	H ≅ G/N	type
6. C₄⋊#13096802	C₄	semidirect
51. C₄×C₁₉₇	#97	semidirect
59. C₁₉₇	#1767	semidirect
111. C₄	#20263237	semidirect

H ≅ G/N

type

6. C₄⋊#13096802

C₄

semidirect

51. C₄×C₁₉₇

#97

semidirect

59. C₁₉₇

#1767

semidirect

111. C₄

#20263237

semidirect

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

subgroup	index
C₂×C₄⋊#13096802	2
C₂×#20263237	2
#27160834	2
C₄³×C₁₉₇	2
#1767	197

subgroup

index

C₄³×C₁₉₇

197

Subgroups — #38742565

Conjugacy classes of subgroups

Complements

Maximal subgroups