Group #38742865 — subgroups

Subgroups — #38742865

Order 25216.

256 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 117 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	1	yes	maximal, characteristic
2	C₂×#20263255	C₂	1	4	yes	maximal
3	#27160925	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄⋊C₈×C₁₉₇	C₂	1	1	yes	maximal, characteristic
5	C₂²×#13096802	C₂²	1	2	yes	—
6	C₂×C₄×Dic₁₉₇	C₂²	1	2	yes	—
7	C₄⋊#13096802	—	2	2	no	—
8	#20263255	—	2	8	no	—
9	#20263233	C₄	1	2	yes	—
10	#20263233	C₄	1	2	yes	—
11	C₂²×C₈×C₁₉₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₁₉₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₁₉₇	—	2	2	no	—
14	C₂²×Dic₁₉₇	C₂×C₄	1	4	yes	—
15	C₂×#13096802	—	2	4	no	—
16	C₂×#13096802	—	2	2	no	—
17	C₄×Dic₁₉₇	D₄	1	4	yes	—
18	C₄×Dic₁₉₇	Q₈	1	4	yes	—
19	C₂²×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
20	C₂²×C₄×C₁₉₇	C₂×C₄	1	1	yes	characteristic
21	C₂²×C₄×C₁₉₇	C₂³	1	1	yes	characteristic
22	C₂×C₈×C₁₉₇	—	2	2	no	—
23	C₂×C₈×C₁₉₇	—	2	4	no	—
24	C₄²×C₁₉₇	D₄	1	2	yes	—
25	C₄²×C₁₉₇	Q₈	1	2	yes	—
26	C₂×Dic₁₉₇	—	2	4	no	—
27	C₂×Dic₁₉₇	—	2	4	no	—
28	C₂×Dic₁₉₇	C₄⋊C₄	1	8	yes	—
29	#13096802	—	4	2	no	—
30	C₂³×C₁₉₇	C₂²×C₄	1	1	yes	characteristic
31	C₂×C₄×C₁₉₇	—	2	2	no	—
32	C₂×C₄×C₁₉₇	—	2	2	no	—
33	C₂×C₄×C₁₉₇	C₄⋊C₄	1	2	yes	—
34	C₂×C₄×C₁₉₇	C₄⋊C₄	1	2	yes	—
35	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₁₉₇	C₂×Q₈	1	2	yes	—
38	C₂×C₄×C₁₉₇	C₂×Q₈	1	1	yes	characteristic
39	C₈×C₁₉₇	—	4	2	no	—
40	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	2	yes	—
41	C₂²×C₁₉₇	C₂×C₄⋊C₄	1	1	yes	characteristic
42	C₂²×C₁₉₇	#131	1	2	yes	—
43	C₂²×C₁₉₇	#131	1	2	yes	—
44	C₄×C₁₉₇	—	2	2	no	—
45	C₄×C₁₉₇	—	2	2	no	—
46	C₄×C₁₉₇	#127	1	1	yes	characteristic
47	C₄×C₁₉₇	C₄⋊Q₈	1	1	yes	characteristic
48	C₄×C₁₉₇	C₄⋊Q₈	1	2	yes	—
49	Dic₁₉₇	—	2	8	no	—
50	C₂×C₁₉₇	#384	1	1	yes	characteristic
51	C₂×C₁₉₇	#424	1	4	yes	—
52	C₂×C₁₉₇	#424	1	1	yes	characteristic
53	C₂×C₁₉₇	#424	1	1	yes	characteristic
54	C₁₉₇	#1843	1	1	yes	characteristic, complemented, Sylow
55	#1843	—	197	1	no	maximal, Sylow
56	C₂×C₄⋊C₈	—	197	4	no	—
57	C₂×C₄⋊C₈	—	197	1	no	—
58	C₂×C₄⋊C₈	D₁₉₇	1	1	yes	characteristic
59	#376	—	197	1	no	—
60	C₂²×C₈	—	197	2	no	—
61	C₂²×C₈	C₂×D₁₉₇	1	2	yes	—
62	C₂×C₄²	—	197	2	no	—
63	C₂×C₄²	C₂×D₁₉₇	1	1	yes	characteristic
64	C₄⋊C₈	—	2	2	no	—
65	C₄⋊C₈	—	394	8	no	—
66	C₄⋊C₈	—	394	2	no	—
67	#95	—	197	2	no	—
68	#95	—	197	2	no	—
69	C₂²×C₄	—	197	4	no	—
70	C₂²×C₄	C₄×D₁₉₇	1	1	yes	characteristic
71	C₂²×C₄	C₄×D₁₉₇	1	1	yes	characteristic
72	C₂²×C₄	C₂²×D₁₉₇	1	1	yes	characteristic
73	C₂×C₈	—	2	4	no	—
74	C₂×C₈	—	2	2	no	—
75	C₂×C₈	—	394	4	no	—
76	C₂×C₈	—	394	2	no	—
77	C₄²	—	197	4	no	—
78	C₄²	—	197	4	no	—
79	C₄²	C₂²⋊D₁₉₇	1	2	yes	—
80	C₄²	Dic₃₉₄	1	2	yes	—
81	C₂³	C₂×C₄×D₁₉₇	1	1	yes	characteristic
82	C₂×C₄	—	2	2	no	—
83	C₂×C₄	—	2	2	no	—
84	C₂×C₄	—	394	4	no	—
85	C₂×C₄	—	394	4	no	—
86	C₂×C₄	—	197	8	no	—
87	C₂×C₄	#15691337	1	2	yes	—
88	C₂×C₄	#15691337	1	2	yes	—
89	C₂×C₄	D₄×D₁₉₇	1	2	yes	—
90	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
91	C₂×C₄	Q₈×D₁₉₇	1	2	yes	—
92	C₂×C₄	C₂×Dic₃₉₄	1	1	yes	characteristic
93	C₈	—	4	2	no	—
94	C₈	—	788	2	no	—
95	C₂²	C₄⋊C₄×D₁₉₇	1	2	yes	—
96	C₂²	C₂×#15691337	1	1	yes	characteristic
97	C₂²	#20263332	1	2	yes	—
98	C₂²	#20263334	1	2	yes	—
99	C₄	—	2	2	no	—
100	C₄	—	2	2	no	—
101	C₄	—	394	8	no	—
102	C₄	#20263322	1	1	yes	characteristic
103	C₄	#20263324	1	1	yes	characteristic
104	C₄	#20263325	1	2	yes	—
105	C₂	#27160948	1	1	yes	characteristic
106	C₂	#27161104	1	4	yes	—
107	C₂	#27161105	1	1	yes	characteristic
108	C₂	#27161107	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄⋊#13096802

C₂

yes

maximal, characteristic

C₂×#20263255

C₂

yes

maximal

#27160925

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₁₉₇

C₂

yes

maximal, characteristic

C₂²×#13096802

C₂²

yes

—

C₂×C₄×Dic₁₉₇

C₂²

yes

—

C₄⋊#13096802

—

#20263255

—

#20263233

C₄

yes

—

#20263233

C₄

yes

—

C₂²×C₈×C₁₉₇

C₂²

yes

—

C₂×C₄²×C₁₉₇

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₇

—

C₂²×Dic₁₉₇

C₂×C₄

yes

—

C₂×#13096802

—

C₂×#13096802

—

C₄×Dic₁₉₇

D₄

yes

—

C₄×Dic₁₉₇

Q₈

yes

—

C₂²×C₄×C₁₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₇

C₂×C₄

yes

characteristic

C₂²×C₄×C₁₉₇

C₂³

yes

characteristic

C₂×C₈×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₄²×C₁₉₇

D₄

yes

—

C₄²×C₁₉₇

Q₈

yes

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

—

C₂×Dic₁₉₇

C₄⋊C₄

yes

—

#13096802

—

C₂³×C₁₉₇

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

—

C₂×C₄×C₁₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₇

C₄⋊C₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

—

C₂×C₄×C₁₉₇

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₇

C₂×Q₈

yes

—

C₂×C₄×C₁₉₇

C₂×Q₈

yes

characteristic

C₈×C₁₉₇

—

C₂²×C₁₉₇

C₂×C₄⋊C₄

yes

—

C₂²×C₁₉₇

C₂×C₄⋊C₄

yes

characteristic

C₂²×C₁₉₇

#131

yes

—

C₂²×C₁₉₇

#131

yes

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

#127

yes

characteristic

C₄×C₁₉₇

C₄⋊Q₈

yes

characteristic

C₄×C₁₉₇

C₄⋊Q₈

yes

—

Dic₁₉₇

—

C₂×C₁₉₇

#384

yes

characteristic

C₂×C₁₉₇

#424

yes

—

C₂×C₁₉₇

yes

C₂×C₁₉₇

yes

C₁₉₇

yes

characteristic, complemented, Sylow

#1843

—

197

maximal, Sylow

C₂×C₄⋊C₈

—

197

—

C₂×C₄⋊C₈

—

197

—

C₂×C₄⋊C₈

D₁₉₇

yes

characteristic

#376

—

197

—

C₂²×C₈

—

197

—

C₂²×C₈

C₂×D₁₉₇

yes

—

C₂×C₄²

—

197

—

C₂×C₄²

C₂×D₁₉₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

394

—

C₄⋊C₈

—

394

—

#95

—

197

—

#95

—

197

—

C₂²×C₄

—

197

—

C₂²×C₄

C₄×D₁₉₇

yes

characteristic

C₂²×C₄

C₄×D₁₉₇

yes

characteristic

C₂²×C₄

C₂²×D₁₉₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

394

—

C₂×C₈

—

394

—

C₄²

—

197

—

C₄²

—

197

—

C₄²

C₂²⋊D₁₉₇

yes

—

C₄²

Dic₃₉₄

yes

—

C₂³

C₂×C₄×D₁₉₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

394

—

C₂×C₄

—

394

—

C₂×C₄

—

197

—

C₂×C₄

#15691337

yes

—

C₂×C₄

#15691337

yes

—

C₂×C₄

D₄×D₁₉₇

yes

—

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

C₂×C₄

Q₈×D₁₉₇

yes

—

C₂×C₄

C₂×Dic₃₉₄

yes

characteristic

C₈

—

C₈

—

788

—

C₂²

C₄⋊C₄×D₁₉₇

yes

—

C₂²

C₂×#15691337

yes

characteristic

C₂²

#20263332

yes

—

C₂²

#20263334

yes

—

C₄

—

100

C₄

—

101

C₄

—

394

—

102

C₄

#20263322

yes

characteristic

103

C₄

#20263324

yes

characteristic

104

C₄

#20263325

yes

—

105

C₂

#27160948

yes

characteristic

106

C₂

#27161104

yes

—

107

C₂

#27161105

yes

characteristic

108

C₂

#27161107

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₉₇
commutator, socle, prime radical	C₂³×C₁₉₇	C₂²×C₄
Fitting	C₂×C₄⋊C₈×C₁₉₇	C₂
cosocle	C₂²×C₄×C₁₉₇	C₂³
hypercenter	C₂×C₄⋊C₈	D₁₉₇
hypocenter	C₁₉₇	#1843
LCS	G ▷ C₂³×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₉₇

commutator, socle, prime radical

C₂³×C₁₉₇

C₂²×C₄

Fitting

C₂×C₄⋊C₈×C₁₉₇

C₂

cosocle

C₂²×C₄×C₁₉₇

C₂³

hypercenter

C₂×C₄⋊C₈

D₁₉₇

C₁₉₇

G ▷ C₂³×C₁₉₇ ▷ C₁₉₇

C₁ ◁ C₂²×C₄ ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
54. C₁₉₇	#1843	semidirect

H ≅ G/N

type

54. C₁₉₇

#1843

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×#20263255	2
#27160925	2
C₂×C₄⋊C₈×C₁₉₇	2
#1843	197

subgroup

index

C₂×C₄⋊C₈×C₁₉₇

197

Subgroups — #38742865

Conjugacy classes of subgroups

Complements

Maximal subgroups