Group #38769824 — subgroups

Subgroups — #38769824

Order 25472.

276 conjugacy classes of subgroups; only the 113 normal subgroups are shown.

normal subgroups only

This group has 276 conjugacy classes of subgroups — too many to draw. Only its 113 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×#20263866	C₂	1	1	yes	maximal, characteristic
2	#27163884	C₂	1	1	yes	maximal, characteristic
3	#27163780	C₂	1	4	yes	maximal
4	C₂×C₄⋊Q₈×C₁₉₉	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₁₉₉⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄⋊Dic₁₉₉	C₂²	1	2	yes	—
7	#20263866	—	2	2	no	—
8	C₁₉₉⋊#95	—	2	1	no	—
9	C₁₉₉⋊#95	—	2	1	no	—
10	C₂²×Q₈×C₁₉₉	—	2	1	no	—
11	C₄⋊Q₈×C₁₉₉	—	2	2	no	—
12	C₄⋊Q₈×C₁₉₉	C₄	1	4	yes	complemented
13	C₂×C₄⋊C₄×C₁₉₉	—	2	1	no	—
14	C₂×C₄⋊C₄×C₁₉₉	C₂²	1	2	yes	—
15	C₂×C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
16	C₂²×Dic₁₉₉	—	2	2	no	—
17	C₂×C₁₉₉⋊C₈	—	2	2	no	—
18	C₂×C₁₉₉⋊C₈	—	2	2	no	—
19	C₂×C₁₉₉⋊C₈	Q₈	1	4	yes	complemented
20	C₄⋊Dic₁₉₉	—	2	4	no	—
21	C₂×Q₈×C₁₉₉	—	2	4	no	—
22	C₂×Q₈×C₁₉₉	—	4	4	no	—
23	C₂²×C₄×C₁₉₉	—	2	2	no	—
24	C₂²×C₄×C₁₉₉	D₄	1	1	yes	characteristic
25	C₂²×C₄×C₁₉₉	D₄	1	1	yes	characteristic
26	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
27	C₄⋊C₄×C₁₉₉	—	2	4	no	—
28	C₄⋊C₄×C₁₉₉	—	2	2	no	—
29	C₄⋊C₄×C₁₉₉	C₂×C₄	1	4	yes	—
30	C₄²×C₁₉₉	C₂×C₄	1	2	yes	—
31	C₄²×C₁₉₉	D₄	1	2	yes	—
32	C₂×Dic₁₉₉	—	4	2	no	—
33	C₂×Dic₁₉₉	—	4	2	no	—
34	C₂×Dic₁₉₉	—	4	2	no	—
35	C₁₉₉⋊C₈	—	2	4	no	—
36	C₂³×C₁₉₉	C₂×D₄	1	1	yes	characteristic
37	Q₈×C₁₉₉	—	4	8	no	—
38	C₂×C₄×C₁₉₉	—	2	2	no	—
39	C₂×C₄×C₁₉₉	—	2	2	no	—
40	C₂×C₄×C₁₉₉	—	2	4	no	—
41	C₂×C₄×C₁₉₉	—	4	4	no	—
42	C₂×C₄×C₁₉₉	C₂²⋊C₄	1	2	yes	—
43	C₂×C₄×C₁₉₉	C₂²⋊C₄	1	2	yes	—
44	C₂×C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
46	C₂×C₄×C₁₉₉	C₂×Q₈	1	2	yes	—
47	C₂×C₄×C₁₉₉	#41	1	2	yes	—
48	C₂²×C₁₉₉	C₂×C₂²⋊C₄	1	1	yes	characteristic
49	C₂²×C₁₉₉	#124	1	1	yes	characteristic
50	C₂²×C₁₉₉	C₄⋊Q₈	1	1	yes	characteristic
51	C₂²×C₁₉₉	AGL(1,ℤ/8)	1	2	yes	—
52	C₂²×C₁₉₉	#137	1	2	yes	—
53	C₄×C₁₉₉	—	2	2	no	—
54	C₄×C₁₉₉	—	2	2	no	—
55	C₄×C₁₉₉	—	4	4	no	—
56	C₄×C₁₉₉	C₄×Q₈	1	2	yes	—
57	C₄×C₁₉₉	C₂²⋊Q₈	1	2	yes	—
58	Dic₁₉₉	—	8	2	no	—
59	C₂×C₁₉₉	#391	1	1	yes	characteristic
60	C₂×C₁₉₉	#418	1	1	yes	characteristic
61	C₂×C₁₉₉	#419	1	1	yes	characteristic
62	C₂×C₁₉₉	#489	1	2	yes	—
63	C₂×C₁₉₉	#501	1	2	yes	—
64	C₁₉₉	#1993	1	1	yes	characteristic, complemented, Sylow
65	#1993	—	199	1	no	maximal, Sylow
66	C₂×C₄⋊Q₈	D₁₉₉	1	1	yes	characteristic
67	C₂×#97	—	199	1	no	—
68	#382	—	199	1	no	—
69	#340	—	199	4	no	—
70	C₂²×Q₈	—	2	1	no	—
71	C₂²×C₈	—	199	2	no	—
72	C₄⋊Q₈	—	2	2	no	—
73	C₄⋊Q₈	Dic₁₉₉	1	4	yes	complemented
74	C₂×C₄⋊C₄	—	2	1	no	—
75	C₂×C₄⋊C₄	—	199	2	no	—
76	C₂×C₄⋊C₄	C₂×D₁₉₉	1	2	yes	—
77	C₂×C₄²	C₂×D₁₉₉	1	1	yes	characteristic
78	#97	—	398	2	no	—
79	#95	—	398	1	no	—
80	#95	—	398	1	no	—
81	C₂×Q₈	—	2	4	no	—
82	C₂×Q₈	—	4	4	no	—
83	C₂²×C₄	—	2	2	no	—
84	C₂²×C₄	—	398	2	no	—
85	C₂²×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
86	C₂²×C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
87	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
88	C₂×C₈	—	398	2	no	—
89	C₂×C₈	—	398	2	no	—
90	C₂×C₈	—	199	4	no	—
91	C₄⋊C₄	—	2	4	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	398	4	no	—
94	C₄⋊C₄	C₂×Dic₁₉₉	1	4	yes	—
95	C₄²	C₂×Dic₁₉₉	1	2	yes	—
96	C₄²	C₂²⋊D₁₉₉	1	2	yes	—
97	C₂³	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
98	Q₈	—	4	8	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	4	4	no	—
103	C₂×C₄	—	796	2	no	—
104	C₂×C₄	—	796	2	no	—
105	C₂×C₄	—	796	2	no	—
106	C₂×C₄	C₂²⋊Dic₁₉₉	1	2	yes	—
107	C₂×C₄	C₂²⋊Dic₁₉₉	1	2	yes	—
108	C₂×C₄	C₂²×Dic₁₉₉	1	1	yes	characteristic
109	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
110	C₂×C₄	Q₈×D₁₉₉	1	2	yes	—
111	C₂×C₄	#15691887	1	2	yes	—
112	C₈	—	398	4	no	—
113	C₂²	C₂×C₂²⋊Dic₁₉₉	1	1	yes	characteristic
114	C₂²	#20263944	1	1	yes	characteristic
115	C₂²	#20263953	1	1	yes	characteristic
116	C₂²	#20263981	1	2	yes	—
117	C₂²	#20263986	1	2	yes	—
118	C₄	—	2	2	no	—
119	C₄	—	2	2	no	—
120	C₄	—	4	4	no	—
121	C₄	—	1592	2	no	—
122	C₄	Q₈×Dic₁₉₉	1	2	yes	—
123	C₄	#20263931	1	2	yes	—
124	C₂	#27163925	1	1	yes	characteristic
125	C₂	#27164023	1	1	yes	characteristic
126	C₂	#27164027	1	1	yes	characteristic
127	C₂	#27164380	1	2	yes	—
128	C₂	#27164424	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263866

C₂

yes

maximal, characteristic

#27163884

C₂

yes

maximal, characteristic

#27163780

C₂

yes

maximal

C₂×C₄⋊Q₈×C₁₉₉

C₂

yes

maximal, characteristic

C₂²×C₁₉₉⋊C₈

C₂²

yes

—

C₂×C₄⋊Dic₁₉₉

C₂²

yes

—

#20263866

—

C₁₉₉⋊#95

—

C₁₉₉⋊#95

—

C₂²×Q₈×C₁₉₉

—

C₄⋊Q₈×C₁₉₉

—

C₄⋊Q₈×C₁₉₉

C₄

yes

complemented

C₂×C₄⋊C₄×C₁₉₉

—

C₂×C₄⋊C₄×C₁₉₉

C₂²

yes

—

C₂×C₄²×C₁₉₉

C₂²

yes

characteristic

C₂²×Dic₁₉₉

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

Q₈

yes

complemented

C₄⋊Dic₁₉₉

—

C₂×Q₈×C₁₉₉

—

C₂×Q₈×C₁₉₉

—

C₂²×C₄×C₁₉₉

—

C₂²×C₄×C₁₉₉

D₄

yes

characteristic

C₂²×C₄×C₁₉₉

D₄

yes

characteristic

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

C₂×C₄

yes

—

C₄²×C₁₉₉

C₂×C₄

yes

—

C₄²×C₁₉₉

D₄

yes

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

C₂×D₄

yes

characteristic

Q₈×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₉

C₂²⋊C₄

yes

—

C₂×C₄×C₁₉₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×Q₈

yes

—

C₂×C₄×C₁₉₉

#41

yes

—

C₂²×C₁₉₉

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₁₉₉

#124

yes

characteristic

C₂²×C₁₉₉

C₄⋊Q₈

yes

characteristic

C₂²×C₁₉₉

AGL(1,ℤ/8)

yes

—

C₂²×C₁₉₉

#137

yes

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

C₄×Q₈

yes

—

C₄×C₁₉₉

C₂²⋊Q₈

yes

—

Dic₁₉₉

—

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

—

C₂×C₁₉₉

#501

yes

—

C₁₉₉

#1993

yes

characteristic, complemented, Sylow

#1993

—

199

maximal, Sylow

C₂×C₄⋊Q₈

D₁₉₉

yes

characteristic

C₂×#97

—

199

—

#382

—

199

—

#340

—

199

—

C₂²×Q₈

—

C₂²×C₈

—

199

—

C₄⋊Q₈

—

C₄⋊Q₈

Dic₁₉₉

yes

complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

199

—

C₂×C₄⋊C₄

C₂×D₁₉₉

yes

—

C₂×C₄²

C₂×D₁₉₉

yes

characteristic

#97

—

398

—

#95

—

398

—

#95

—

398

—

C₂×Q₈

—

C₂×Q₈

—

C₂²×C₄

—

C₂²×C₄

—

398

—

C₂²×C₄

C₂²⋊D₁₉₉

yes

characteristic

C₂²×C₄

C₂²⋊D₁₉₉

yes

characteristic

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

C₂×C₈

—

398

—

C₂×C₈

—

398

—

C₂×C₈

—

199

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

398

—

C₄⋊C₄

C₂×Dic₁₉₉

yes

—

C₄²

C₂×Dic₁₉₉

yes

—

C₄²

C₂²⋊D₁₉₉

yes

—

C₂³

C₂×C₂²⋊D₁₉₉

yes

characteristic

Q₈

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

796

—

104

C₂×C₄

—

796

—

105

C₂×C₄

—

796

—

106

C₂×C₄

C₂²⋊Dic₁₉₉

yes

—

107

C₂×C₄

C₂²⋊Dic₁₉₉

yes

—

108

C₂×C₄

C₂²×Dic₁₉₉

yes

characteristic

109

C₂×C₄

C₂×C₂²⋊D₁₉₉

yes

characteristic

110

C₂×C₄

Q₈×D₁₉₉

yes

—

111

C₂×C₄

#15691887

yes

—

112

C₈

—

398

—

113

C₂²

C₂×C₂²⋊Dic₁₉₉

yes

characteristic

114

C₂²

#20263944

yes

characteristic

115

C₂²

#20263953

yes

characteristic

116

C₂²

#20263981

yes

—

117

C₂²

#20263986

yes

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

—

1592

—

122

C₄

Q₈×Dic₁₉₉

yes

—

123

C₄

#20263931

yes

—

124

C₂

#27163925

yes

characteristic

125

C₂

#27164023

yes

characteristic

126

C₂

#27164027

yes

characteristic

127

C₂

#27164380

yes

—

128

C₂

#27164424

yes

—

	subgroup	G/H
center	C₂³	C₂×C₂²⋊D₁₉₉
commutator	C₂×C₄×C₁₉₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₉
Fitting	C₂×C₄⋊Q₈×C₁₉₉	C₂
socle, prime radical	C₂³×C₁₉₉	C₂×D₄
cosocle	C₂²×C₄×C₁₉₉	C₂³
hypercenter	C₂×C₄⋊Q₈	D₁₉₉
hypocenter	C₁₉₉	#1993
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

subgroup

G/H

center

C₂³

C₂×C₂²⋊D₁₉₉

commutator

C₂×C₄×C₁₉₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₉

Fitting

C₂×C₄⋊Q₈×C₁₉₉

C₂

socle, prime radical

C₂³×C₁₉₉

C₂×D₄

cosocle

C₂²×C₄×C₁₉₉

C₂³

hypercenter

C₂×C₄⋊Q₈

D₁₉₉

hypocenter

C₁₉₉

#1993

LCS

G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
12. C₄⋊Q₈×C₁₉₉	C₄	semidirect
19. C₂×C₁₉₉⋊C₈	Q₈	semidirect
64. C₁₉₉	#1993	semidirect
73. C₄⋊Q₈	Dic₁₉₉	semidirect

H ≅ G/N

type

12. C₄⋊Q₈×C₁₉₉

C₄

semidirect

19. C₂×C₁₉₉⋊C₈

Q₈

semidirect

64. C₁₉₉

#1993

semidirect

73. C₄⋊Q₈

Dic₁₉₉

semidirect

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

subgroup	index
C₂×#20263866	2
#27163884	2
#27163780	2
C₂×C₄⋊Q₈×C₁₉₉	2
#1993	199

subgroup

index

C₂×C₄⋊Q₈×C₁₉₉

199

Subgroups — #38769824

Conjugacy classes of subgroups

Complements

Maximal subgroups