Group #38769630 — subgroups

Subgroups — #38769630

Order 25472.

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

normal subgroups only

This group has 196 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×#20263891	C₂	1	1	yes	maximal, characteristic
2	#27164061	C₂	1	2	yes	maximal
3	C₂×#20263887	C₂	1	1	yes	maximal, characteristic
4	#27163786	C₂	1	2	yes	maximal
5	#405×C₁₉₉	C₂	1	1	yes	maximal, characteristic
6	C₂×#15691870	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₁₉₉	C₂²	1	2	yes	—
8	#20263891	C₄	1	4	yes	complemented
9	#20263888	—	2	2	no	—
10	#20263887	—	2	1	no	—
11	#20263887	—	2	1	no	—
12	#20263886	—	2	2	no	—
13	C₂×M₁₆×C₁₉₉	C₂²	1	1	yes	characteristic
14	C₂²×C₈×C₁₉₉	C₂²	1	1	yes	characteristic
15	#117×C₁₉₉	C₂²	1	1	yes	characteristic
16	#97×C₁₉₉	—	2	1	no	—
17	C₄×C₈×C₁₉₉	—	2	1	no	—
18	C₂²×Dic₁₉₉	—	2	2	no	—
19	#15691870	—	2	2	no	—
20	#15691870	C₂×C₄	1	4	yes	—
21	C₂×C₁₉₉⋊C₈	—	2	2	no	—
22	C₄⋊Dic₁₉₉	—	2	2	no	—
23	C₄⋊Dic₁₉₉	—	2	2	no	—
24	C₂²×C₄×C₁₉₉	C₂³	1	1	yes	characteristic
25	M₁₆×C₁₉₉	—	2	1	no	—
26	M₁₆×C₁₉₉	Q₈	1	2	yes	—
27	C₂×C₈×C₁₉₉	—	2	1	no	—
28	C₂×C₈×C₁₉₉	C₂×C₄	1	2	yes	—
29	C₂×C₈×C₁₉₉	D₄	1	1	yes	characteristic
30	C₂×C₈×C₁₉₉	D₄	1	1	yes	characteristic
31	C₂×C₈×C₁₉₉	Q₈	1	2	yes	—
32	C₄⋊C₄×C₁₉₉	D₄	1	2	yes	—
33	C₂²⋊C₄×C₁₉₉	—	2	1	no	—
34	C₄²×C₁₉₉	—	2	1	no	—
35	C₂×Dic₁₉₉	—	4	2	no	—
36	C₂×Dic₁₉₉	—	8	2	no	—
37	C₁₉₉⋊C₈	—	2	4	no	—
38	C₂³×C₁₉₉	#41	1	1	yes	characteristic
39	C₂×C₄×C₁₉₉	—	2	2	no	—
40	C₂×C₄×C₁₉₉	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
42	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₁₉₉	C₂×Q₈	1	1	yes	characteristic
45	C₂×C₄×C₁₉₉	#41	1	1	yes	characteristic
46	C₈×C₁₉₉	—	2	2	no	—
47	C₈×C₁₉₉	C₄⋊C₄	1	2	yes	—
48	C₈×C₁₉₉	C₄⋊C₄	1	2	yes	—
49	C₂²×C₁₉₉	—	2	2	no	—
50	C₂²×C₁₉₉	C₄×Q₈	1	1	yes	characteristic
51	C₂²×C₁₉₉	#121	1	1	yes	characteristic
52	C₂²×C₁₉₉	#125	1	1	yes	characteristic
53	C₄×C₁₉₉	—	4	2	no	—
54	C₄×C₁₉₉	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₁₉₉	C₄×D₄	1	1	yes	characteristic
56	C₄×C₁₉₉	C₂²⋊Q₈	1	1	yes	characteristic
57	C₄×C₁₉₉	C₄⋊Q₈	1	1	yes	characteristic
58	Dic₁₉₉	—	8	2	no	—
59	C₂×C₁₉₉	—	2	1	no	—
60	C₂×C₁₉₉	—	2	1	no	—
61	C₂×C₁₉₉	#389	1	1	yes	characteristic
62	C₂×C₁₉₉	#471	1	2	yes	—
63	C₁₉₉	#1951	1	1	yes	characteristic, complemented, Sylow
64	#1951	—	199	1	no	maximal, Sylow
65	C₂×#108	—	199	1	no	—
66	#428	—	199	2	no	—
67	C₂×#106	—	199	1	no	—
68	#405	D₁₉₉	1	1	yes	characteristic
69	#343	—	199	2	no	—
70	C₂×M₁₆	—	199	2	no	—
71	C₂×M₁₆	C₂×D₁₉₉	1	1	yes	characteristic
72	C₂²×C₈	C₂×D₁₉₉	1	1	yes	characteristic
73	#117	C₂×D₁₉₉	1	1	yes	characteristic
74	C₂×C₄⋊C₄	—	199	2	no	—
75	#108	—	199	4	no	—
76	#107	—	398	2	no	—
77	#106	—	398	1	no	—
78	#106	—	398	1	no	—
79	#106	—	398	2	no	—
80	#97	—	2	1	no	—
81	C₄×C₈	—	2	1	no	—
82	C₂²×C₄	—	398	2	no	—
83	C₂²×C₄	C₂²×D₁₉₉	1	1	yes	characteristic
84	M₁₆	—	2	1	no	—
85	M₁₆	—	398	2	no	—
86	M₁₆	—	199	4	no	—
87	M₁₆	Dic₃₉₈	1	2	yes	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	398	2	no	—
90	C₂×C₈	C₄×D₁₉₉	1	2	yes	—
91	C₂×C₈	C₂²⋊D₁₉₉	1	1	yes	characteristic
92	C₂×C₈	C₂²⋊D₁₉₉	1	1	yes	characteristic
93	C₂×C₈	Dic₃₉₈	1	2	yes	—
94	C₄⋊C₄	—	398	2	no	—
95	C₄⋊C₄	—	398	2	no	—
96	C₄⋊C₄	D₇₉₆	1	2	yes	—
97	C₂²⋊C₄	—	2	1	no	—
98	C₄²	—	2	1	no	—
99	C₂³	#15691886	1	1	yes	characteristic
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	796	2	no	—
102	C₂×C₄	—	1592	2	no	—
103	C₂×C₄	C₂×C₄×D₁₉₉	1	1	yes	characteristic
104	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
105	C₂×C₄	C₂×D₇₉₆	1	1	yes	characteristic
106	C₂×C₄	C₂×Dic₃₉₈	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₃₉₈	1	1	yes	characteristic
108	C₂×C₄	#15691886	1	1	yes	characteristic
109	C₈	—	2	2	no	—
110	C₈	—	398	4	no	—
111	C₈	Dic₁₉₉⋊C₄	1	2	yes	—
112	C₈	Dic₁₉₉⋊C₄	1	2	yes	—
113	C₂²	—	2	2	no	—
114	C₂²	C₄×Dic₃₉₈	1	1	yes	characteristic
115	C₂²	#20263929	1	1	yes	characteristic
116	C₂²	#20263947	1	1	yes	characteristic
117	C₄	—	4	2	no	—
118	C₄	—	1592	2	no	—
119	C₄	C₂×Dic₁₉₉⋊C₄	1	1	yes	characteristic
120	C₄	C₄×D₇₉₆	1	1	yes	characteristic
121	C₄	#20263934	1	1	yes	characteristic
122	C₄	#20263955	1	1	yes	characteristic
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	#27163916	1	1	yes	characteristic
126	C₂	#27164281	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#20263891

C₂

yes

maximal, characteristic

#27164061

C₂

yes

maximal

C₂×#20263887

C₂

yes

maximal, characteristic

#27163786

C₂

yes

maximal

#405×C₁₉₉

C₂

yes

maximal, characteristic

C₂×#15691870

C₂²

yes

—

C₂×C₄⋊Dic₁₉₉

C₂²

yes

—

#20263891

C₄

yes

—

—

—

—

C₂×M₁₆×C₁₉₉

C₂²

yes

characteristic

C₂²×C₈×C₁₉₉

C₂²

yes

characteristic

#117×C₁₉₉

C₂²

yes

characteristic

#97×C₁₉₉

—

C₄×C₈×C₁₉₉

—

C₂²×Dic₁₉₉

—

#15691870

—

#15691870

C₂×C₄

yes

—

C₂×C₁₉₉⋊C₈

—

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₂²×C₄×C₁₉₉

C₂³

yes

characteristic

M₁₆×C₁₉₉

—

M₁₆×C₁₉₉

Q₈

yes

—

C₂×C₈×C₁₉₉

—

C₂×C₈×C₁₉₉

C₂×C₄

yes

—

C₂×C₈×C₁₉₉

D₄

yes

characteristic

C₂×C₈×C₁₉₉

D₄

yes

characteristic

C₂×C₈×C₁₉₉

Q₈

yes

—

C₄⋊C₄×C₁₉₉

D₄

yes

—

C₂²⋊C₄×C₁₉₉

—

C₄²×C₁₉₉

—

C₂×Dic₁₉₉

—

C₂×Dic₁₉₉

—

C₁₉₉⋊C₈

—

C₂³×C₁₉₉

#41

yes

characteristic

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂²×C₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₉

C₂×Q₈

yes

characteristic

C₂×C₄×C₁₉₉

#41

yes

characteristic

C₈×C₁₉₉

—

C₈×C₁₉₉

C₄⋊C₄

yes

—

C₈×C₁₉₉

C₄⋊C₄

yes

—

C₂²×C₁₉₉

—

C₂²×C₁₉₉

C₄×Q₈

yes

C₂²×C₁₉₉

yes

C₂²×C₁₉₉

yes

C₄×C₁₉₉

—

C₄×C₁₉₉

C₂×C₄⋊C₄

yes

characteristic

C₄×C₁₉₉

C₄×D₄

yes

characteristic

C₄×C₁₉₉

C₂²⋊Q₈

yes

characteristic

C₄×C₁₉₉

C₄⋊Q₈

yes

characteristic

Dic₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

—

C₂×C₁₉₉

#389

yes

characteristic

C₂×C₁₉₉

#471

yes

—

C₁₉₉

#1951

yes

characteristic, complemented, Sylow

#1951

—

199

maximal, Sylow

C₂×#108

—

199

—

#428

—

199

—

C₂×#106

—

199

—

#405

D₁₉₉

yes

characteristic

#343

—

199

—

C₂×M₁₆

—

199

—

C₂×M₁₆

C₂×D₁₉₉

yes

characteristic

C₂²×C₈

C₂×D₁₉₉

yes

characteristic

#117

C₂×D₁₉₉

yes

characteristic

C₂×C₄⋊C₄

—

199

—

#108

—

199

—

#107

—

398

—

#106

—

398

—

#106

—

398

—

#106

—

398

—

#97

—

C₄×C₈

—

C₂²×C₄

—

398

—

C₂²×C₄

C₂²×D₁₉₉

yes

characteristic

M₁₆

—

M₁₆

—

398

—

M₁₆

—

199

—

M₁₆

Dic₃₉₈

yes

—

C₂×C₈

—

C₂×C₈

—

398

—

C₂×C₈

C₄×D₁₉₉

yes

—

C₂×C₈

C₂²⋊D₁₉₉

yes

characteristic

C₂×C₈

C₂²⋊D₁₉₉

yes

characteristic

C₂×C₈

Dic₃₉₈

yes

—

C₄⋊C₄

—

398

—

C₄⋊C₄

—

398

—

C₄⋊C₄

D₇₉₆

yes

—

C₂²⋊C₄

—

C₄²

—

C₂³

#15691886

yes

characteristic

100

C₂×C₄

—

101

C₂×C₄

—

796

—

102

C₂×C₄

—

1592

—

103

C₂×C₄

C₂×C₄×D₁₉₉

yes

characteristic

104

C₂×C₄

C₂×C₂²⋊D₁₉₉

yes

characteristic

105

C₂×C₄

C₂×D₇₉₆

yes

characteristic

106

C₂×C₄

C₂×Dic₃₉₈

yes

characteristic

107

C₂×C₄

C₂×Dic₃₉₈

yes

characteristic

108

C₂×C₄

#15691886

yes

characteristic

109

C₈

—

110

C₈

—

398

—

111

C₈

Dic₁₉₉⋊C₄

yes

—

112

C₈

Dic₁₉₉⋊C₄

yes

—

113

C₂²

—

114

C₂²

C₄×Dic₃₉₈

yes

characteristic

115

C₂²

#20263929

yes

characteristic

116

C₂²

#20263947

yes

characteristic

117

C₄

—

118

C₄

—

1592

—

119

C₄

C₂×Dic₁₉₉⋊C₄

yes

characteristic

120

C₄

C₄×D₇₉₆

yes

characteristic

121

C₄

#20263934

yes

characteristic

122

C₄

#20263955

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#27163916

yes

characteristic

126

C₂

#27164281

yes

—

	subgroup	G/H
center	C₂²	#20263929
commutator	C₂×C₄×C₁₉₉	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₁₉₉
Fitting	#405×C₁₉₉	C₂
socle	C₂²×C₁₉₉	#121
prime radical	C₂³×C₁₉₉	#41
cosocle	C₂²×C₄×C₁₉₉	C₂³
hypercenter	#405	D₁₉₉
hypocenter	C₁₉₉	#1951
LCS	G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

subgroup

G/H

center

C₂²

#20263929

commutator

C₂×C₄×C₁₉₉

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₁₉₉

Fitting

#405×C₁₉₉

C₂

C₂²×C₁₉₉

C₂³×C₁₉₉

C₂²×C₄×C₁₉₉

C₂³

D₁₉₉

C₁₉₉

G ▷ C₂×C₄×C₁₉₉ ▷ C₂×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #405

N	H ≅ G/N	type
8. #20263891	C₄	semidirect
63. C₁₉₉	#1951	semidirect

H ≅ G/N

type

8. #20263891

C₄

semidirect

63. C₁₉₉

#1951

semidirect

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

subgroup	index
C₂×#20263891	2
#27164061	2
C₂×#20263887	2
#27163786	2
#405×C₁₉₉	2
#1951	199

subgroup

index

#405×C₁₉₉

199

Subgroups — #38769630

Conjugacy classes of subgroups

Complements

Maximal subgroups