Group #38741855 — subgroups

Subgroups — #38741855

Order 25216.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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	#27161795	C₂	1	1	yes	maximal, characteristic
2	#27161366	C₂	1	2	yes	maximal, complemented
3	#27160828	C₂	1	1	yes	maximal, characteristic, complemented
4	#27160808	C₂	1	2	yes	maximal, complemented
5	#494×C₁₉₇	C₂	1	1	yes	maximal, characteristic, complemented
6	#20263315	—	2	2	no	—
7	#20263301	—	2	2	no	—
8	C₄×D₇₈₈	C₂²	1	2	yes	—
9	#20263284	—	2	2	no	—
10	#20263257	—	2	2	no	—
11	C₄⋊#13096802	C₂²	1	2	yes	—
12	#20263246	—	2	2	no	—
13	C₂×Q₁₆×C₁₉₇	—	2	2	no	—
14	C₄⋊Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₁₉₇	C₂²	1	1	yes	characteristic
16	C₄×C₈×C₁₉₇	C₂²	1	1	yes	characteristic
17	C₂×D₇₈₈	—	2	2	no	—
18	C₂×D₇₈₈	D₄	1	2	yes	—
19	C₂×C₄×D₁₉₇	—	2	2	no	—
20	C₂×#13096802	—	2	2	no	—
21	C₄⋊Dic₁₉₇	D₄	1	2	yes	—
22	#15691335	—	2	4	no	—
23	#15691335	—	2	2	no	—
24	C₄×Dic₁₉₇	—	2	2	no	—
25	C₂×Q₈×C₁₉₇	—	2	1	no	—
26	C₂×Q₈×C₁₉₇	D₄	1	2	yes	—
27	Q₁₆×C₁₉₇	—	4	4	no	—
28	C₂×C₈×C₁₉₇	—	2	1	no	—
29	C₄⋊C₄×C₁₉₇	—	2	2	no	—
30	C₄⋊C₄×C₁₉₇	—	2	2	no	—
31	C₄²×C₁₉₇	C₂³	1	1	yes	characteristic
32	C₂²×D₁₉₇	—	2	2	no	—
33	D₇₈₈	—	2	2	no	—
34	D₇₈₈	—	2	2	no	—
35	D₇₈₈	—	8	2	no	—
36	C₂×Dic₁₉₇	—	2	2	no	—
37	C₄×D₁₉₇	—	4	2	no	—
38	C₄×D₁₉₇	—	4	2	no	—
39	#13096802	—	4	2	no	—
40	Q₈×C₁₉₇	—	4	2	no	—
41	Q₈×C₁₉₇	—	4	2	no	—
42	C₂×C₄×C₁₉₇	—	2	2	no	—
43	C₂×C₄×C₁₉₇	—	4	1	no	—
44	C₂×C₄×C₁₉₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₁₉₇	C₂×D₄	1	1	yes	characteristic
46	C₈×C₁₉₇	—	2	2	no	—
47	C₂²×C₁₉₇	#120	1	1	yes	characteristic
48	C₂×D₁₉₇	—	4	2	no	—
49	C₂×D₁₉₇	—	4	2	no	—
50	C₂×D₁₉₇	—	4	2	no	—
51	C₄×C₁₉₇	—	2	1	no	—
52	C₄×C₁₉₇	—	4	2	no	—
53	C₄×C₁₉₇	—	8	1	no	—
54	C₄×C₁₉₇	#135	1	2	yes	—
55	C₄×C₁₉₇	#137	1	2	yes	—
56	Dic₁₉₇	—	4	2	no	—
57	C₂×C₁₉₇	#452	1	2	yes	—
58	C₂×C₁₉₇	#453	1	1	yes	characteristic
59	D₁₉₇	—	8	2	no	—
60	C₁₉₇	#1634	1	1	yes	characteristic, complemented, Sylow
61	#1634	—	197	1	no	maximal, Sylow
62	#555	—	197	1	no	—
63	#494	D₁₉₇	1	1	yes	characteristic, complemented
64	#476	—	197	2	no	—
65	#333	—	197	1	no	—
66	D₄⋊C₈	—	197	2	no	—
67	C₂×Q₁₆	—	2	2	no	—
68	C₄⋊Q₈	C₂×D₁₉₇	1	1	yes	characteristic
69	C₄⋊Q₈	C₂×D₁₉₇	1	1	yes	characteristic
70	#124	—	394	2	no	—
71	#121	—	394	2	no	—
72	C₄×D₄	—	197	2	no	—
73	#117	—	394	2	no	—
74	#106	—	394	2	no	—
75	C₄⋊C₈	—	197	2	no	—
76	#102	—	394	2	no	—
77	C₄×C₈	C₂×D₁₉₇	1	1	yes	characteristic
78	C₂×Q₈	—	2	1	no	—
79	C₂×Q₈	C₂²⋊D₁₉₇	1	2	yes	—
80	C₂×D₄	—	394	2	no	—
81	C₂×D₄	—	197	2	no	—
82	C₂²×C₄	—	394	2	no	—
83	Q₁₆	—	4	4	no	—
84	C₂×C₈	—	2	1	no	—
85	C₂×C₈	—	394	2	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	2	2	no	—
88	C₄⋊C₄	—	197	2	no	—
89	C₂²⋊C₄	—	394	4	no	—
90	C₂²⋊C₄	—	394	2	no	—
91	C₄²	—	394	2	no	—
92	C₄²	C₂²×D₁₉₇	1	1	yes	characteristic
93	C₂³	—	394	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	4	2	no	—
96	D₄	—	1576	2	no	—
97	D₄	—	394	2	no	—
98	D₄	—	394	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	4	1	no	—
101	C₂×C₄	—	788	2	no	—
102	C₂×C₄	—	788	2	no	—
103	C₂×C₄	—	394	2	no	—
104	C₂×C₄	D₄×D₁₉₇	1	2	yes	—
105	C₂×C₄	C₂×C₂²⋊D₁₉₇	1	1	yes	characteristic
106	C₈	—	2	2	no	—
107	C₈	—	788	2	no	—
108	C₂²	—	788	2	no	—
109	C₂²	—	788	2	no	—
110	C₂²	—	788	2	no	—
111	C₂²	#20263294	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	8	1	no	—
115	C₄	—	788	2	no	—
116	C₄	#20263349	1	2	yes	—
117	C₄	#20263358	1	2	yes	—
118	C₂	—	1576	2	no	—
119	C₂	#27161240	1	2	yes	—
120	C₂	#27161246	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27161795

C₂

yes

maximal, characteristic

#27161366

C₂

yes

maximal, complemented

#27160828

C₂

yes

maximal, characteristic, complemented

#27160808

C₂

yes

maximal, complemented

#494×C₁₉₇

C₂

yes

maximal, characteristic, complemented

#20263315

—

#20263301

—

C₄×D₇₈₈

C₂²

yes

—

#20263284

—

#20263257

—

C₄⋊#13096802

C₂²

yes

—

#20263246

—

C₂×Q₁₆×C₁₉₇

—

C₄⋊Q₈×C₁₉₇

C₂²

yes

characteristic

C₄⋊Q₈×C₁₉₇

C₂²

yes

characteristic

C₄×C₈×C₁₉₇

C₂²

yes

characteristic

C₂×D₇₈₈

—

C₂×D₇₈₈

D₄

yes

—

C₂×C₄×D₁₉₇

—

C₂×#13096802

—

C₄⋊Dic₁₉₇

D₄

yes

—

#15691335

—

#15691335

—

C₄×Dic₁₉₇

—

C₂×Q₈×C₁₉₇

—

C₂×Q₈×C₁₉₇

D₄

yes

—

Q₁₆×C₁₉₇

—

C₂×C₈×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄⋊C₄×C₁₉₇

—

C₄²×C₁₉₇

C₂³

yes

characteristic

C₂²×D₁₉₇

—

D₇₈₈

—

D₇₈₈

—

D₇₈₈

—

C₂×Dic₁₉₇

—

C₄×D₁₉₇

—

C₄×D₁₉₇

—

#13096802

—

Q₈×C₁₉₇

—

Q₈×C₁₉₇

—

C₂×C₄×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

—

C₂×D₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

—

C₄×C₁₉₇

#135

yes

—

C₄×C₁₉₇

#137

yes

—

Dic₁₉₇

—

C₂×C₁₉₇

#452

yes

—

C₂×C₁₉₇

#453

yes

characteristic

D₁₉₇

—

C₁₉₇

#1634

yes

characteristic, complemented, Sylow

#1634

—

197

maximal, Sylow

#555

—

197

—

#494

D₁₉₇

yes

characteristic, complemented

#476

—

197

—

#333

—

197

—

D₄⋊C₈

—

197

—

C₂×Q₁₆

—

C₄⋊Q₈

C₂×D₁₉₇

yes

characteristic

C₄⋊Q₈

C₂×D₁₉₇

yes

characteristic

#124

—

394

—

#121

—

394

—

C₄×D₄

—

197

—

#117

—

394

—

#106

—

394

—

C₄⋊C₈

—

197

—

#102

—

394

—

C₄×C₈

C₂×D₁₉₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

C₂²⋊D₁₉₇

yes

—

C₂×D₄

—

394

—

C₂×D₄

—

197

—

C₂²×C₄

—

394

—

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

394

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

197

—

C₂²⋊C₄

—

394

—

C₂²⋊C₄

—

394

—

C₄²

—

394

—

C₄²

C₂²×D₁₉₇

yes

characteristic

C₂³

—

394

—

Q₈

—

Q₈

—

D₄

—

1576

—

D₄

—

394

—

D₄

—

394

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

788

—

102

C₂×C₄

—

788

—

103

C₂×C₄

—

394

—

104

C₂×C₄

D₄×D₁₉₇

yes

—

105

C₂×C₄

C₂×C₂²⋊D₁₉₇

yes

characteristic

106

C₈

—

107

C₈

—

788

—

108

C₂²

—

788

—

109

C₂²

—

788

—

110

C₂²

—

788

—

111

C₂²

#20263294

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

788

—

116

C₄

#20263349

yes

—

117

C₄

#20263358

yes

—

118

C₂

—

1576

—

119

C₂

#27161240

yes

—

120

C₂

#27161246

yes

characteristic

	subgroup	G/H
center	C₂²	#20263294
commutator, cosocle	C₄²×C₁₉₇	C₂³
Frattini	C₄²	C₂²×D₁₉₇
Fitting	#494×C₁₉₇	C₂
socle	C₂²×C₁₉₇	#120
prime radical	#27161795	C₂
hypercenter	#494	D₁₉₇
hypocenter	C₁₉₇	#1634
LCS	G ▷ C₄²×C₁₉₇ ▷ C₂²×C₁₉₇ ▷ C₁₉₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#20263294

commutator, cosocle

C₄²×C₁₉₇

C₂³

Frattini

C₄²

C₂²×D₁₉₇

Fitting

#494×C₁₉₇

C₂

C₂²×C₁₉₇

C₂

D₁₉₇

C₁₉₇

G ▷ C₄²×C₁₉₇ ▷ C₂²×C₁₉₇ ▷ C₁₉₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
2. #27161366	C₂	semidirect
3. #27160828	C₂	semidirect
4. #27160808	C₂	semidirect
5. #494×C₁₉₇	C₂	semidirect
60. C₁₉₇	#1634	semidirect
63. #494	D₁₉₇	semidirect

H ≅ G/N

type

2. #27161366

C₂

semidirect

3. #27160828

C₂

semidirect

4. #27160808

C₂

semidirect

5. #494×C₁₉₇

C₂

semidirect

60. C₁₉₇

#1634

semidirect

63. #494

D₁₉₇

semidirect

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

subgroup	index
#27161795	2
#27161366	2
#27160828	2
#27160808	2
#494×C₁₉₇	2
#1634	197

subgroup

index

#494×C₁₉₇

197

Subgroups — #38741855

Conjugacy classes of subgroups

Complements

Maximal subgroups