Group #38770404 — subgroups

Subgroups — #38770404

Order 25472.

210 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 67 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	#27164067	C₂	1	1	yes	maximal, characteristic, complemented
2	#27164045	C₂	1	2	yes	maximal, complemented
3	#27164036	C₂	1	2	yes	maximal
4	#27163986	C₂	1	1	yes	maximal, characteristic
5	H(3,ℤ/4)×C₁₉₉	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#15691886	C₂²	1	1	yes	characteristic
7	C₂×#15691870	C₂²	1	2	yes	—
8	C₂²×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
9	#20263891	—	2	2	no	—
10	C₄⋊(C₁₉₉⋊C₈)	—	2	4	no	—
11	#20263883	—	2	4	no	—
12	(C₂×D₁₉₉)⋊C₈	—	2	2	no	—
13	C₂×M₁₆×C₁₉₉	C₂²	1	1	yes	characteristic
14	#117×C₁₉₉	C₂²	1	2	yes	—
15	#15691886	—	2	2	no	—
16	#15691886	—	2	2	no	—
17	C₂×Dic₃₉₈	Q₈	1	1	yes	characteristic
18	C₂×D₇₉₆	Q₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₁₉₉	—	2	1	no	—
20	C₂×C₄×D₁₉₉	—	2	1	no	—
21	#15691870	—	2	2	no	—
22	#15691870	—	2	2	no	—
23	C₂×C₁₉₉⋊C₈	—	2	1	no	—
24	C₂×C₁₉₉⋊C₈	—	2	2	no	—
25	C₂×C₁₉₉⋊C₈	—	2	2	no	—
26	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
27	M₁₆×C₁₉₉	—	2	2	no	—
28	C₂×C₈×C₁₉₉	D₄	1	2	yes	—
29	C₄⋊C₄×C₁₉₉	—	2	2	no	—
30	C₂²⋊C₄×C₁₉₉	—	2	2	no	—
31	C₄²×C₁₉₉	D₄	1	4	yes	—
32	C₂²×D₁₉₉	—	2	1	no	—
33	Dic₃₉₈	—	2	2	no	—
34	D₇₉₆	—	2	2	no	—
35	C₂²⋊D₁₉₉	—	4	2	no	—
36	C₂×Dic₁₉₉	—	2	1	no	—
37	C₄×D₁₉₉	—	4	1	no	—
38	C₄×D₁₉₉	—	4	1	no	—
39	C₁₉₉⋊C₈	—	4	1	no	—
40	C₁₉₉⋊C₈	—	4	2	no	—
41	C₂³×C₁₉₉	#41	1	1	yes	characteristic
42	C₂×C₄×C₁₉₉	—	2	4	no	—
43	C₂×C₄×C₁₉₉	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₄×C₁₉₉	#41	1	2	yes	—
47	C₈×C₁₉₉	—	2	2	no	—
48	C₂²×C₁₉₉	—	4	1	no	—
49	C₂²×C₁₉₉	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₁₉₉	#123	1	2	yes	—
51	C₂×D₁₉₉	—	4	2	no	—
52	C₂×D₁₉₉	—	4	1	no	—
53	C₄×C₁₉₉	—	4	4	no	—
54	C₄×C₁₉₉	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₁₉₉	#123	1	1	yes	characteristic
56	C₄×C₁₉₉	#127	1	1	yes	characteristic
57	Dic₁₉₉	—	4	1	no	—
58	C₂×C₁₉₉	—	2	2	no	—
59	C₂×C₁₉₉	—	2	1	no	—
60	C₂×C₁₉₉	#399	1	1	yes	characteristic
61	D₁₉₉	—	8	1	no	—
62	C₁₉₉	#2098	1	1	yes	characteristic, complemented, Sylow
63	#2098	—	199	1	no	maximal, Sylow
64	#430	—	199	1	no	—
65	#424	—	199	2	no	—
66	#421	—	199	2	no	—
67	#408	—	199	1	no	—
68	H(3,ℤ/4)	D₁₉₉	1	1	yes	characteristic, complemented
69	C₂×#41	—	199	1	no	—
70	C₂×M₁₆	—	199	2	no	—
71	C₂×M₁₆	C₂×D₁₉₉	1	1	yes	characteristic
72	C₂²×C₈	—	199	1	no	—
73	#117	C₂×D₁₉₉	1	2	yes	—
74	#108	—	398	2	no	—
75	C₄⋊C₈	—	398	4	no	—
76	#104	—	398	4	no	—
77	C₂²⋊C₈	—	398	2	no	—
78	#41	—	398	2	no	—
79	#41	—	398	2	no	—
80	C₂×Q₈	—	199	1	no	—
81	C₂×D₄	—	398	1	no	—
82	C₂×D₄	—	199	1	no	—
83	C₂²×C₄	—	398	1	no	—
84	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	398	2	no	—
87	M₁₆	—	398	2	no	—
88	C₂×C₈	—	398	1	no	—
89	C₂×C₈	—	398	2	no	—
90	C₂×C₈	—	398	2	no	—
91	C₂×C₈	D₇₉₆	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₇₉₆	1	4	yes	—
95	C₂³	—	398	1	no	—
96	C₂³	#15691885	1	1	yes	characteristic
97	Q₈	—	398	2	no	—
98	D₄	—	796	2	no	—
99	D₄	—	398	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	796	1	no	—
102	C₂×C₄	—	796	1	no	—
103	C₂×C₄	—	398	1	no	—
104	C₂×C₄	C₂×D₇₉₆	1	1	yes	characteristic
105	C₂×C₄	C₂×D₇₉₆	1	2	yes	—
106	C₂×C₄	Q₈×D₁₉₉	1	1	yes	characteristic
107	C₂×C₄	#15691885	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	796	1	no	—
110	C₈	—	796	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	796	2	no	—
113	C₂²	—	796	1	no	—
114	C₂²	#20263935	1	1	yes	characteristic
115	C₂²	#20263939	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	796	1	no	—
118	C₄	#20263935	1	2	yes	—
119	C₄	#20263939	1	1	yes	characteristic
120	C₄	#20263952	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	1592	1	no	—
124	C₂	#27163955	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27164067

C₂

yes

maximal, characteristic, complemented

#27164045

C₂

yes

maximal, complemented

#27164036

C₂

yes

maximal

#27163986

C₂

yes

maximal, characteristic

H(3,ℤ/4)×C₁₉₉

C₂

yes

maximal, characteristic, complemented

C₂×#15691886

C₂²

yes

characteristic

C₂×#15691870

C₂²

yes

—

C₂²×C₁₉₉⋊C₈

C₂²

yes

characteristic

#20263891

—

C₄⋊(C₁₉₉⋊C₈)

—

#20263883

—

(C₂×D₁₉₉)⋊C₈

—

C₂×M₁₆×C₁₉₉

C₂²

yes

characteristic

#117×C₁₉₉

C₂²

yes

—

#15691886

—

#15691886

—

C₂×Dic₃₉₈

Q₈

yes

characteristic

C₂×D₇₉₆

Q₈

yes

characteristic

C₂×C₂²⋊D₁₉₉

—

C₂×C₄×D₁₉₉

—

#15691870

—

#15691870

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

M₁₆×C₁₉₉

—

C₂×C₈×C₁₉₉

D₄

yes

—

C₄⋊C₄×C₁₉₉

—

C₂²⋊C₄×C₁₉₉

—

C₄²×C₁₉₉

D₄

yes

—

C₂²×D₁₉₉

—

Dic₃₉₈

—

D₇₉₆

—

C₂²⋊D₁₉₉

—

C₂×Dic₁₉₉

—

C₄×D₁₉₉

—

C₄×D₁₉₉

—

C₁₉₉⋊C₈

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

#41

yes

characteristic

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₉

#41

yes

—

C₈×C₁₉₉

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

C₂²⋊Q₈

yes

characteristic

C₂²×C₁₉₉

#123

yes

—

C₂×D₁₉₉

—

C₂×D₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

C₂²⋊Q₈

yes

—

C₄×C₁₉₉

yes

C₄×C₁₉₉

yes

Dic₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

#399

yes

characteristic

D₁₉₉

—

C₁₉₉

#2098

yes

characteristic, complemented, Sylow

#2098

—

199

maximal, Sylow

#430

—

199

—

#424

—

199

—

#421

—

199

—

#408

—

199

—

H(3,ℤ/4)

D₁₉₉

yes

characteristic, complemented

C₂×#41

—

199

—

C₂×M₁₆

—

199

—

C₂×M₁₆

C₂×D₁₉₉

yes

characteristic

C₂²×C₈

—

199

—

#117

C₂×D₁₉₉

yes

—

#108

—

398

—

C₄⋊C₈

—

398

—

#104

—

398

—

C₂²⋊C₈

—

398

—

#41

—

398

—

#41

—

398

—

C₂×Q₈

—

199

—

C₂×D₄

—

398

—

C₂×D₄

—

199

—

C₂²×C₄

—

398

—

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

M₁₆

—

M₁₆

—

398

—

M₁₆

—

398

—

C₂×C₈

—

398

—

C₂×C₈

—

398

—

C₂×C₈

—

398

—

C₂×C₈

D₇₉₆

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₇₉₆

yes

—

C₂³

—

398

—

C₂³

#15691885

yes

characteristic

Q₈

—

398

—

D₄

—

796

—

D₄

—

398

—

100

C₂×C₄

—

101

C₂×C₄

—

796

—

102

C₂×C₄

—

796

—

103

C₂×C₄

—

398

—

104

C₂×C₄

C₂×D₇₉₆

yes

characteristic

105

C₂×C₄

C₂×D₇₉₆

yes

—

106

C₂×C₄

Q₈×D₁₉₉

yes

characteristic

107

C₂×C₄

#15691885

yes

—

108

C₈

—

109

C₈

—

796

—

110

C₈

—

796

—

111

C₂²

—

112

C₂²

—

796

—

113

C₂²

—

796

—

114

C₂²

#20263935

yes

characteristic

115

C₂²

#20263939

yes

—

116

C₄

—

117

C₄

—

796

—

118

C₄

#20263935

yes

—

119

C₄

#20263939

yes

characteristic

120

C₄

#20263952

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

1592

—

124

C₂

#27163955

yes

characteristic

	subgroup	G/H
center	C₄	#20263952
commutator, cosocle	C₂²×C₄×C₁₉₉	C₂³
Frattini	C₂²×C₄	C₂²×D₁₉₉
Fitting	H(3,ℤ/4)×C₁₉₉	C₂
socle	C₂×C₁₉₉	#399
prime radical	C₂×#15691886	C₂²
hypercenter	H(3,ℤ/4)	D₁₉₉
hypocenter	C₁₉₉	#2098
LCS	G ▷ C₂²×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#20263952

commutator, cosocle

C₂²×C₄×C₁₉₉

C₂³

Frattini

C₂²×C₄

C₂²×D₁₉₉

Fitting

H(3,ℤ/4)×C₁₉₉

C₂

C₂×C₁₉₉

C₂²

D₁₉₉

C₁₉₉

G ▷ C₂²×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #27164067	C₂	semidirect
2. #27164045	C₂	semidirect
5. H(3,ℤ/4)×C₁₉₉	C₂	semidirect
62. C₁₉₉	#2098	semidirect
68. H(3,ℤ/4)	D₁₉₉	semidirect

H ≅ G/N

type

1. #27164067

C₂

semidirect

2. #27164045

C₂

semidirect

5. H(3,ℤ/4)×C₁₉₉

C₂

semidirect

62. C₁₉₉

#2098

semidirect

68. H(3,ℤ/4)

D₁₉₉

semidirect

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

subgroup	index
#27164067	2
#27164045	2
#27164036	2
#27163986	2
H(3,ℤ/4)×C₁₉₉	2
#2098	199

subgroup

index

H(3,ℤ/4)×C₁₉₉

199

Subgroups — #38770404

Conjugacy classes of subgroups

Complements

Maximal subgroups