Group #38767995 — subgroups

Subgroups — #38767995

Order 25472.

218 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 57 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	#27164397	C₂	1	1	yes	maximal, characteristic, complemented
2	#27164308	C₂	1	2	yes	maximal, complemented
3	#27163767	C₂	1	1	yes	maximal, characteristic, complemented
4	D₄⋊(C₁₉₉⋊C₈)	C₂	1	2	yes	maximal, complemented
5	#555×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15691876	—	2	2	no	—
7	#20263955	C₂²	1	1	yes	characteristic
8	#20263886	—	2	2	no	—
9	C₄⋊(C₁₉₉⋊C₈)	C₂²	1	2	yes	—
10	#20263876	—	2	2	no	—
11	C₄×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
12	C₄⋊Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
13	#124×C₁₉₉	—	2	2	no	—
14	#121×C₁₉₉	—	2	2	no	—
15	C₄×D₄×C₁₉₉	C₂²	1	2	yes	—
16	#117×C₁₉₉	—	2	2	no	—
17	C₂×Dic₃₉₈	—	2	1	no	—
18	#15691876	—	4	4	no	—
19	C₂×C₁₉₉⋊C₈	—	2	1	no	—
20	C₂×C₁₉₉⋊C₈	—	2	2	no	—
21	C₄⋊Dic₁₉₉	—	2	2	no	—
22	C₂×Q₈×C₁₉₉	D₄	1	2	yes	—
23	C₂×D₄×C₁₉₉	—	2	2	no	—
24	C₂×D₄×C₁₉₉	D₄	1	2	yes	—
25	C₂²×C₄×C₁₉₉	—	2	2	no	—
26	C₄⋊C₄×C₁₉₉	—	2	2	no	—
27	C₄⋊C₄×C₁₉₉	D₄	1	2	yes	—
28	C₂²⋊C₄×C₁₉₉	—	2	4	no	—
29	C₂²⋊C₄×C₁₉₉	—	2	2	no	—
30	C₄²×C₁₉₉	—	2	2	no	—
31	C₄²×C₁₉₉	C₂³	1	1	yes	characteristic
32	Dic₃₉₈	—	4	2	no	—
33	C₂×Dic₁₉₉	—	4	1	no	—
34	C₁₉₉⋊C₈	—	2	2	no	—
35	C₁₉₉⋊C₈	—	4	2	no	—
36	C₂³×C₁₉₉	—	2	2	no	—
37	Q₈×C₁₉₉	—	4	2	no	—
38	D₄×C₁₉₉	—	2	2	no	—
39	D₄×C₁₉₉	—	2	2	no	—
40	D₄×C₁₉₉	—	8	2	no	—
41	C₂×C₄×C₁₉₉	—	2	2	no	—
42	C₂×C₄×C₁₉₉	—	2	2	no	—
43	C₂×C₄×C₁₉₉	—	4	2	no	—
44	C₂×C₄×C₁₉₉	—	4	2	no	—
45	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₉₉	C₂×D₄	1	2	yes	—
47	C₂²×C₁₉₉	—	4	2	no	—
48	C₂²×C₁₉₉	—	4	2	no	—
49	C₂²×C₁₉₉	—	4	2	no	—
50	C₂²×C₁₉₉	#120	1	1	yes	characteristic
51	C₄×C₁₉₉	—	2	1	no	—
52	C₄×C₁₉₉	—	4	2	no	—
53	C₄×C₁₉₉	—	4	2	no	—
54	C₄×C₁₉₉	#135	1	2	yes	—
55	C₄×C₁₉₉	#137	1	2	yes	—
56	Dic₁₉₉	—	8	1	no	—
57	C₂×C₁₉₉	—	8	2	no	—
58	C₂×C₁₉₉	#452	1	2	yes	—
59	C₂×C₁₉₉	#453	1	1	yes	characteristic
60	C₁₉₉	#1634	1	1	yes	characteristic, complemented, Sylow
61	#1634	—	199	1	no	maximal, Sylow
62	#555	D₁₉₉	1	1	yes	characteristic
63	#494	—	199	1	no	—
64	#476	—	199	2	no	—
65	#333	—	199	1	no	—
66	D₄⋊C₈	—	199	2	no	—
67	C₂×Q₁₆	—	398	2	no	—
68	C₄⋊Q₈	—	199	1	no	—
69	C₄⋊Q₈	C₂×D₁₉₉	1	1	yes	characteristic
70	#124	—	2	2	no	—
71	#121	—	2	2	no	—
72	C₄×D₄	C₂×D₁₉₉	1	2	yes	—
73	#117	—	2	2	no	—
74	#106	—	398	2	no	—
75	C₄⋊C₈	—	199	2	no	—
76	#102	—	398	2	no	—
77	C₄×C₈	—	199	1	no	—
78	C₂×Q₈	—	398	1	no	—
79	C₂×Q₈	C₂²⋊D₁₉₉	1	2	yes	—
80	C₂×D₄	—	2	2	no	—
81	C₂×D₄	C₂²⋊D₁₉₉	1	2	yes	—
82	C₂²×C₄	—	2	2	no	—
83	Q₁₆	—	796	4	no	—
84	C₂×C₈	—	398	2	no	—
85	C₂×C₈	—	398	1	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	398	2	no	—
88	C₄⋊C₄	C₂²⋊D₁₉₉	1	2	yes	—
89	C₂²⋊C₄	—	2	4	no	—
90	C₂²⋊C₄	—	2	2	no	—
91	C₄²	—	2	2	no	—
92	C₄²	C₂²×D₁₉₉	1	1	yes	characteristic
93	C₂³	—	2	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	796	2	no	—
96	D₄	—	2	2	no	—
97	D₄	—	2	2	no	—
98	D₄	—	8	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	4	2	no	—
103	C₂×C₄	—	796	1	no	—
104	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
105	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	2	yes	—
106	C₈	—	796	2	no	—
107	C₈	—	398	2	no	—
108	C₂²	—	4	2	no	—
109	C₂²	—	4	2	no	—
110	C₂²	—	4	2	no	—
111	C₂²	#20263924	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	4	2	no	—
115	C₄	—	1592	1	no	—
116	C₄	#20263975	1	2	yes	—
117	C₄	#20263985	1	2	yes	—
118	C₂	—	8	2	no	—
119	C₂	#27164180	1	2	yes	—
120	C₂	#27164186	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27164397

C₂

yes

maximal, characteristic, complemented

#27164308

C₂

yes

maximal, complemented

#27163767

C₂

yes

maximal, characteristic, complemented

D₄⋊(C₁₉₉⋊C₈)

C₂

yes

maximal, complemented

#555×C₁₉₉

C₂

yes

maximal, characteristic

C₂×#15691876

—

#20263955

C₂²

yes

characteristic

#20263886

—

C₄⋊(C₁₉₉⋊C₈)

C₂²

yes

—

#20263876

—

C₄×C₁₉₉⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₁₉₉

C₂²

yes

characteristic

#124×C₁₉₉

—

#121×C₁₉₉

—

C₄×D₄×C₁₉₉

C₂²

yes

—

#117×C₁₉₉

—

C₂×Dic₃₉₈

—

#15691876

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₄⋊Dic₁₉₉

—

C₂×Q₈×C₁₉₉

D₄

yes

—

C₂×D₄×C₁₉₉

—

C₂×D₄×C₁₉₉

D₄

yes

—

C₂²×C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

D₄

yes

—

C₂²⋊C₄×C₁₉₉

—

C₂²⋊C₄×C₁₉₉

—

C₄²×C₁₉₉

—

C₄²×C₁₉₉

C₂³

yes

characteristic

Dic₃₉₈

—

C₂×Dic₁₉₉

—

C₁₉₉⋊C₈

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

—

Q₈×C₁₉₉

—

D₄×C₁₉₉

—

D₄×C₁₉₉

—

D₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

#120

yes

characteristic

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

#135

yes

—

C₄×C₁₉₉

#137

yes

—

Dic₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

#452

yes

—

C₂×C₁₉₉

#453

yes

characteristic

C₁₉₉

#1634

yes

characteristic, complemented, Sylow

#1634

—

199

maximal, Sylow

#555

D₁₉₉

yes

characteristic

#494

—

199

—

#476

—

199

—

#333

—

199

—

D₄⋊C₈

—

199

—

C₂×Q₁₆

—

398

—

C₄⋊Q₈

—

199

—

C₄⋊Q₈

C₂×D₁₉₉

yes

characteristic

#124

—

#121

—

C₄×D₄

C₂×D₁₉₉

yes

—

#117

—

#106

—

398

—

C₄⋊C₈

—

199

—

#102

—

398

—

C₄×C₈

—

199

—

C₂×Q₈

—

398

—

C₂×Q₈

C₂²⋊D₁₉₉

yes

—

C₂×D₄

—

C₂×D₄

C₂²⋊D₁₉₉

yes

—

C₂²×C₄

—

Q₁₆

—

796

—

C₂×C₈

—

398

—

C₂×C₈

—

398

—

C₄⋊C₄

—

C₄⋊C₄

—

398

—

C₄⋊C₄

C₂²⋊D₁₉₉

yes

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

C₂²×D₁₉₉

yes

characteristic

C₂³

—

Q₈

—

Q₈

—

796

—

D₄

—

D₄

—

D₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

796

—

104

C₂×C₄

C₂×C₂²⋊D₁₉₉

yes

characteristic

105

C₂×C₄

C₂×C₂²⋊D₁₉₉

yes

—

106

C₈

—

796

—

107

C₈

—

398

—

108

C₂²

—

109

C₂²

—

110

C₂²

—

111

C₂²

#20263924

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

115

C₄

—

1592

—

116

C₄

#20263975

yes

—

117

C₄

#20263985

yes

—

118

C₂

—

119

C₂

#27164180

yes

—

120

C₂

#27164186

yes

characteristic

	subgroup	G/H
center	C₂²	#20263924
commutator, cosocle	C₄²×C₁₉₉	C₂³
Frattini	C₄²	C₂²×D₁₉₉
Fitting, prime radical	#555×C₁₉₉	C₂
socle	C₂²×C₁₉₉	#120
hypercenter	#555	D₁₉₉
hypocenter	C₁₉₉	#1634
LCS	G ▷ C₄²×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #555

subgroup

G/H

center

C₂²

#20263924

commutator, cosocle

C₄²×C₁₉₉

C₂³

Frattini

C₄²

C₂²×D₁₉₉

Fitting, prime radical

#555×C₁₉₉

C₂

C₂²×C₁₉₉

D₁₉₉

C₁₉₉

G ▷ C₄²×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #555

N	H ≅ G/N	type
1. #27164397	C₂	semidirect
2. #27164308	C₂	semidirect
3. #27163767	C₂	semidirect
4. D₄⋊(C₁₉₉⋊C₈)	C₂	semidirect
60. C₁₉₉	#1634	semidirect

H ≅ G/N

type

1. #27164397

C₂

semidirect

2. #27164308

C₂

semidirect

3. #27163767

C₂

semidirect

4. D₄⋊(C₁₉₉⋊C₈)

C₂

semidirect

60. C₁₉₉

#1634

semidirect

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

subgroup	index
#27164397	2
#27164308	2
#27163767	2
D₄⋊(C₁₉₉⋊C₈)	2
#555×C₁₉₉	2
#1634	199

subgroup

index

#27164397

#27164308

#27163767

D₄⋊(C₁₉₉⋊C₈)

#555×C₁₉₉

#1634

199

Subgroups — #38767995

Conjugacy classes of subgroups

Complements

Maximal subgroups