Group #38768968 — subgroups

Subgroups — #38768968

Order 25472.

216 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 216 conjugacy classes of subgroups — too many to draw. Only its 93 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×Dic₁₉₉⋊C₈	C₂	1	2	yes	maximal
2	#27163864	C₂	1	1	yes	maximal, characteristic
3	#27163773	C₂	1	1	yes	maximal, characteristic
4	#27163772	C₂	1	2	yes	maximal
5	#336×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₁₉₉⋊C₈	C₂²	1	2	yes	—
7	C₂×C₄×Dic₁₉₉	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₁₉₉	C₂²	1	1	yes	characteristic
9	Dic₁₉₉⋊C₈	—	2	4	no	—
10	C₁₉₉⋊#95	C₄	1	2	yes	—
11	C₁₉₉⋊#95	C₄	1	2	yes	—
12	C₂²×C₈×C₁₉₉	C₂²	1	2	yes	—
13	C₂×C₄²×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₂²×Dic₁₉₉	C₂×C₄	1	1	yes	characteristic
15	C₂²×Dic₁₉₉	C₂×C₄	1	1	yes	characteristic
16	C₂²×Dic₁₉₉	C₂×C₄	1	2	yes	—
17	C₂×C₁₉₉⋊C₈	—	2	2	no	—
18	C₂×C₁₉₉⋊C₈	—	2	2	no	—
19	C₂×C₁₉₉⋊C₈	—	2	2	no	—
20	C₄×Dic₁₉₉	—	2	2	no	—
21	C₄×Dic₁₉₉	D₄	1	2	yes	—
22	C₄×Dic₁₉₉	Q₈	1	2	yes	—
23	C₂²×C₄×C₁₉₉	C₂×C₄	1	2	yes	—
24	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₁₉₉	—	2	2	no	—
26	C₂×C₈×C₁₉₉	—	2	2	no	—
27	C₂×C₈×C₁₉₉	—	2	2	no	—
28	C₄²×C₁₉₉	—	2	2	no	—
29	C₂×Dic₁₉₉	—	2	2	no	—
30	C₂×Dic₁₉₉	—	2	2	no	—
31	C₂×Dic₁₉₉	—	2	1	no	—
32	C₂×Dic₁₉₉	—	2	1	no	—
33	C₂×Dic₁₉₉	—	2	2	no	—
34	C₂×Dic₁₉₉	—	2	2	no	—
35	C₂×Dic₁₉₉	C₄⋊C₄	1	4	yes	—
36	C₁₉₉⋊C₈	—	4	2	no	—
37	C₂³×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₁₉₉	—	2	2	no	—
39	C₂×C₄×C₁₉₉	—	2	2	no	—
40	C₂×C₄×C₁₉₉	—	2	2	no	—
41	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
43	C₂×C₄×C₁₉₉	#41	1	2	yes	—
44	C₂×C₄×C₁₉₉	#41	1	2	yes	—
45	C₈×C₁₉₉	—	4	2	no	—
46	C₂²×C₁₉₉	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂²×C₁₉₉	#117	1	1	yes	characteristic
48	C₂²×C₁₉₉	#117	1	1	yes	characteristic
49	C₂²×C₁₉₉	#131	1	2	yes	—
50	C₂²×C₁₉₉	#131	1	2	yes	—
51	C₄×C₁₉₉	—	4	2	no	—
52	C₄×C₁₉₉	#124	1	2	yes	—
53	C₄×C₁₉₉	#125	1	2	yes	—
54	Dic₁₉₉	—	2	4	no	—
55	Dic₁₉₉	—	4	1	no	—
56	Dic₁₉₉	—	4	1	no	—
57	C₂×C₁₉₉	#382	1	1	yes	characteristic
58	C₂×C₁₉₉	#424	1	2	yes	—
59	C₂×C₁₉₉	#433	1	2	yes	—
60	C₂×C₁₉₉	#433	1	1	yes	characteristic
61	C₂×C₁₉₉	#433	1	1	yes	characteristic
62	C₁₉₉	#1832	1	1	yes	characteristic, complemented, Sylow
63	#1832	—	199	1	no	maximal, Sylow
64	C₂×C₄⋊C₈	—	199	2	no	—
65	#376	—	199	1	no	—
66	#336	—	199	2	no	—
67	#336	—	199	1	no	—
68	#336	D₁₉₉	1	1	yes	characteristic
69	C₂²×C₈	—	199	2	no	—
70	C₂²×C₈	C₂×D₁₉₉	1	2	yes	—
71	C₂×C₄²	—	199	1	no	—
72	C₂×C₄²	—	199	1	no	—
73	C₂×C₄²	C₂×D₁₉₉	1	1	yes	characteristic
74	C₄⋊C₈	—	398	4	no	—
75	#95	—	199	2	no	—
76	#95	—	199	2	no	—
77	C₂²×C₄	—	199	1	no	—
78	C₂²×C₄	—	199	1	no	—
79	C₂²×C₄	—	199	2	no	—
80	C₂²×C₄	C₄×D₁₉₉	1	2	yes	—
81	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
82	C₂×C₈	—	2	2	no	—
83	C₂×C₈	—	2	2	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	398	2	no	—
86	C₂×C₈	—	398	2	no	—
87	C₂×C₈	—	398	2	no	—
88	C₄²	—	2	2	no	—
89	C₄²	—	398	2	no	—
90	C₄²	—	199	2	no	—
91	C₄²	—	199	2	no	—
92	C₂³	C₂×C₄×D₁₉₉	1	1	yes	characteristic
93	C₂×C₄	—	2	2	no	—
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	398	2	no	—
97	C₂×C₄	—	398	2	no	—
98	C₂×C₄	—	398	1	no	—
99	C₂×C₄	—	398	1	no	—
100	C₂×C₄	—	398	2	no	—
101	C₂×C₄	—	398	2	no	—
102	C₂×C₄	—	199	4	no	—
103	C₂×C₄	D₄×D₁₉₉	1	1	yes	characteristic
104	C₂×C₄	Q₈×D₁₉₉	1	1	yes	characteristic
105	C₂×C₄	#15691885	1	2	yes	—
106	C₂×C₄	#15691886	1	2	yes	—
107	C₈	—	4	2	no	—
108	C₈	—	796	2	no	—
109	C₂²	C₄⋊C₄×D₁₉₉	1	1	yes	characteristic
110	C₂²	#20263911	1	1	yes	characteristic
111	C₂²	#20263912	1	1	yes	characteristic
112	C₂²	#20263961	1	2	yes	—
113	C₂²	#20263963	1	2	yes	—
114	C₄	—	4	2	no	—
115	C₄	—	796	1	no	—
116	C₄	—	796	1	no	—
117	C₄	—	398	4	no	—
118	C₄	#20263941	1	2	yes	—
119	C₄	#20263945	1	2	yes	—
120	C₂	#27163882	1	1	yes	characteristic
121	C₂	#27164043	1	2	yes	—
122	C₂	#27164076	1	2	yes	—
123	C₂	#27164077	1	1	yes	characteristic
124	C₂	#27164080	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Dic₁₉₉⋊C₈

C₂

yes

maximal

#27163864

C₂

yes

maximal, characteristic

#27163773

C₂

yes

maximal, characteristic

#27163772

C₂

yes

maximal

#336×C₁₉₉

C₂

yes

maximal, characteristic

C₂²×C₁₉₉⋊C₈

C₂²

yes

—

C₂×C₄×Dic₁₉₉

C₂²

yes

characteristic

C₂×C₄×Dic₁₉₉

C₂²

yes

characteristic

Dic₁₉₉⋊C₈

—

C₁₉₉⋊#95

C₄

yes

—

C₁₉₉⋊#95

C₄

yes

—

C₂²×C₈×C₁₉₉

C₂²

yes

—

C₂×C₄²×C₁₉₉

C₂²

yes

characteristic

C₂²×Dic₁₉₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₉₉

C₂×C₄

yes

characteristic

C₂²×Dic₁₉₉

C₂×C₄

yes

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₄×Dic₁₉₉

—

C₄×Dic₁₉₉

D₄

yes

—

C₄×Dic₁₉₉

Q₈

yes

—

C₂²×C₄×C₁₉₉

C₂×C₄

yes

—

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

C₂×C₈×C₁₉₉

—

C₂×C₈×C₁₉₉

—

C₂×C₈×C₁₉₉

—

C₄²×C₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

C₄⋊C₄

yes

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₉

#41

yes

—

C₂×C₄×C₁₉₉

#41

yes

—

C₈×C₁₉₉

—

C₂²×C₁₉₉

C₂×C₄⋊C₄

yes

C₂²×C₁₉₉

yes

C₂²×C₁₉₉

yes

C₂²×C₁₉₉

yes

—

C₂²×C₁₉₉

#131

yes

—

C₄×C₁₉₉

—

C₄×C₁₉₉

#124

yes

—

C₄×C₁₉₉

#125

yes

—

Dic₁₉₉

—

Dic₁₉₉

—

Dic₁₉₉

—

C₂×C₁₉₉

#382

yes

characteristic

C₂×C₁₉₉

#424

yes

—

C₂×C₁₉₉

#433

yes

—

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₁₉₉

yes

characteristic, complemented, Sylow

#1832

—

199

maximal, Sylow

C₂×C₄⋊C₈

—

199

—

#376

—

199

—

#336

—

199

—

#336

—

199

—

#336

D₁₉₉

yes

characteristic

C₂²×C₈

—

199

—

C₂²×C₈

C₂×D₁₉₉

yes

—

C₂×C₄²

—

199

—

C₂×C₄²

—

199

—

C₂×C₄²

C₂×D₁₉₉

yes

characteristic

C₄⋊C₈

—

398

—

#95

—

199

—

#95

—

199

—

C₂²×C₄

—

199

—

C₂²×C₄

—

199

—

C₂²×C₄

—

199

—

C₂²×C₄

C₄×D₁₉₉

yes

—

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

398

—

C₂×C₈

—

398

—

C₂×C₈

—

398

—

C₄²

—

C₄²

—

398

—

C₄²

—

199

—

C₄²

—

199

—

C₂³

C₂×C₄×D₁₉₉

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

398

—

C₂×C₄

—

398

—

C₂×C₄

—

398

—

C₂×C₄

—

398

—

100

C₂×C₄

—

398

—

101

C₂×C₄

—

398

—

102

C₂×C₄

—

199

—

103

C₂×C₄

D₄×D₁₉₉

yes

characteristic

104

C₂×C₄

Q₈×D₁₉₉

yes

characteristic

105

C₂×C₄

#15691885

yes

—

106

C₂×C₄

#15691886

yes

—

107

C₈

—

108

C₈

—

796

—

109

C₂²

C₄⋊C₄×D₁₉₉

yes

characteristic

110

C₂²

#20263911

yes

characteristic

111

C₂²

#20263912

yes

characteristic

112

C₂²

#20263961

yes

—

113

C₂²

#20263963

yes

—

114

C₄

—

115

C₄

—

796

—

116

C₄

—

796

—

117

C₄

—

398

—

118

C₄

#20263941

yes

—

119

C₄

#20263945

yes

—

120

C₂

#27163882

yes

characteristic

121

C₂

#27164043

yes

—

122

C₂

#27164076

yes

—

123

C₂

#27164077

yes

characteristic

124

C₂

#27164080

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₁₉₉
commutator, socle, prime radical	C₂³×C₁₉₉	C₂²×C₄
Fitting	#336×C₁₉₉	C₂
cosocle	C₂²×C₄×C₁₉₉	C₂³
hypercenter	#336	D₁₉₉
hypocenter	C₁₉₉	#1832
LCS	G ▷ C₂³×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂²×C₄ ◁ #336

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₁₉₉

commutator, socle, prime radical

C₂³×C₁₉₉

C₂²×C₄

Fitting

#336×C₁₉₉

C₂

cosocle

C₂²×C₄×C₁₉₉

C₂³

D₁₉₉

C₁₉₉

G ▷ C₂³×C₁₉₉ ▷ C₁₉₉

C₁ ◁ C₂²×C₄ ◁ #336

N	H ≅ G/N	type
62. C₁₉₉	#1832	semidirect

H ≅ G/N

type

62. C₁₉₉

#1832

semidirect

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

subgroup	index
C₂×Dic₁₉₉⋊C₈	2
#27163864	2
#27163773	2
#27163772	2
#336×C₁₉₉	2
#1832	199

subgroup

index

C₂×Dic₁₉₉⋊C₈

#336×C₁₉₉

199

Subgroups — #38768968

Conjugacy classes of subgroups

Complements

Maximal subgroups