Group #38768230 — subgroups

Subgroups — #38768230

Order 25472.

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

normal subgroups only

This group has 150 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	#27164414	C₂	1	1	yes	maximal, characteristic
2	#27164313	C₂	1	1	yes	maximal, characteristic
3	Q₈×C₁₉₉⋊C₈	C₂	1	1	yes	maximal, characteristic
4	#27163768	C₂	1	1	yes	maximal, characteristic
5	#27163766	C₂	1	1	yes	maximal, characteristic
6	#27163764	C₂	1	1	yes	maximal, characteristic
7	#462×C₁₉₉	C₂	1	1	yes	maximal, characteristic
8	#20263955	C₂²	1	1	yes	characteristic
9	#20263888	—	2	1	no	—
10	#20263886	—	2	1	no	—
11	#20263886	—	2	1	no	—
12	C₄⋊(C₁₉₉⋊C₈)	—	2	1	no	—
13	C₄⋊(C₁₉₉⋊C₈)	C₂²	1	1	yes	characteristic
14	C₄⋊(C₁₉₉⋊C₈)	C₂²	1	1	yes	characteristic
15	#20263879	—	2	1	no	—
16	C₄×C₁₉₉⋊C₈	—	2	1	no	—
17	C₄×C₁₉₉⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×Q₁₆×C₁₉₉	—	2	1	no	—
19	C₄⋊Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
20	C₄×Q₈×C₁₉₉	C₂²	1	1	yes	characteristic
21	C₄⋊C₈×C₁₉₉	C₂²	1	1	yes	characteristic
22	#103×C₁₉₉	—	2	1	no	—
23	C₂×Dic₃₉₈	—	2	1	no	—
24	C₂×C₁₉₉⋊C₈	—	2	1	no	—
25	C₂×C₁₉₉⋊C₈	—	2	1	no	—
26	C₂×C₁₉₉⋊C₈	D₄	1	1	yes	characteristic
27	C₂×C₁₉₉⋊C₈	D₄	1	1	yes	characteristic
28	C₄⋊Dic₁₉₉	—	2	1	no	—
29	C₄⋊Dic₁₉₉	—	2	1	no	—
30	C₂×Q₈×C₁₉₉	—	2	1	no	—
31	C₂×Q₈×C₁₉₉	D₄	1	1	yes	characteristic
32	Q₁₆×C₁₉₉	—	4	2	no	—
33	C₂×C₈×C₁₉₉	—	2	1	no	—
34	C₄⋊C₄×C₁₉₉	—	2	1	no	—
35	C₄⋊C₄×C₁₉₉	—	2	1	no	—
36	C₄⋊C₄×C₁₉₉	—	2	1	no	—
37	C₄⋊C₄×C₁₉₉	D₄	1	1	yes	characteristic
38	C₄²×C₁₉₉	—	2	1	no	—
39	C₄²×C₁₉₉	C₂³	1	1	yes	characteristic
40	Dic₃₉₈	—	8	1	no	—
41	C₂×Dic₁₉₉	—	4	1	no	—
42	C₁₉₉⋊C₈	—	2	1	no	—
43	C₁₉₉⋊C₈	—	4	1	no	—
44	C₁₉₉⋊C₈	—	4	1	no	—
45	C₁₉₉⋊C₈	Q₁₆	1	2	yes	complemented
46	Q₈×C₁₉₉	—	2	1	no	—
47	Q₈×C₁₉₉	—	4	1	no	—
48	Q₈×C₁₉₉	—	4	1	no	—
49	Q₈×C₁₉₉	SD₁₆	1	2	yes	—
50	C₂×C₄×C₁₉₉	—	2	1	no	—
51	C₂×C₄×C₁₉₉	—	2	1	no	—
52	C₂×C₄×C₁₉₉	—	4	1	no	—
53	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
54	C₂×C₄×C₁₉₉	C₂×D₄	1	1	yes	characteristic
55	C₂×C₄×C₁₉₉	#41	1	1	yes	characteristic
56	C₈×C₁₉₉	—	4	1	no	—
57	C₂²×C₁₉₉	#121	1	1	yes	characteristic
58	C₄×C₁₉₉	—	2	2	no	—
59	C₄×C₁₉₉	—	2	1	no	—
60	C₄×C₁₉₉	—	4	1	no	—
61	C₄×C₁₉₉	—	8	1	no	—
62	C₄×C₁₉₉	C₂×SD₁₆	1	1	yes	characteristic
63	C₄×C₁₉₉	C₂×Q₁₆	1	1	yes	characteristic
64	C₄×C₁₉₉	#135	1	1	yes	characteristic
65	C₄×C₁₉₉	#137	1	1	yes	characteristic
66	Dic₁₉₉	—	8	1	no	—
67	C₂×C₁₉₉	#462	1	1	yes	characteristic
68	C₂×C₁₉₉	#465	1	1	yes	characteristic
69	C₂×C₁₉₉	#471	1	1	yes	characteristic
70	C₁₉₉	#1672	1	1	yes	characteristic, complemented, Sylow
71	#1672	—	199	1	no	maximal, Sylow
72	#498	—	199	1	no	—
73	#477	—	199	1	no	—
74	#462	D₁₉₉	1	1	yes	characteristic
75	C₈×Q₈	—	199	1	no	—
76	#334	—	199	1	no	—
77	#333	—	199	1	no	—
78	#332	—	199	1	no	—
79	C₂×Q₁₆	—	2	1	no	—
80	C₄⋊Q₈	—	199	1	no	—
81	C₄⋊Q₈	C₂×D₁₉₉	1	1	yes	characteristic
82	C₄×Q₈	C₂×D₁₉₉	1	1	yes	characteristic
83	#107	—	398	1	no	—
84	#106	—	398	1	no	—
85	#106	—	398	1	no	—
86	C₄⋊C₈	—	398	1	no	—
87	C₄⋊C₈	—	199	1	no	—
88	C₄⋊C₈	—	199	1	no	—
89	C₄⋊C₈	C₂×D₁₉₉	1	1	yes	characteristic
90	#103	—	2	1	no	—
91	#103	—	398	1	no	—
92	C₄×C₈	—	398	1	no	—
93	C₄×C₈	—	199	1	no	—
94	C₂×Q₈	—	2	1	no	—
95	C₂×Q₈	—	398	1	no	—
96	C₂×Q₈	C₂²⋊D₁₉₉	1	1	yes	characteristic
97	Q₁₆	—	4	2	no	—
98	C₂×C₈	—	2	1	no	—
99	C₂×C₈	—	398	1	no	—
100	C₂×C₈	—	398	1	no	—
101	C₂×C₈	—	199	1	no	—
102	C₂×C₈	—	199	1	no	—
103	C₄⋊C₄	—	2	1	no	—
104	C₄⋊C₄	—	2	1	no	—
105	C₄⋊C₄	—	2	1	no	—
106	C₄⋊C₄	—	398	1	no	—
107	C₄⋊C₄	—	398	1	no	—
108	C₄⋊C₄	C₂²⋊D₁₉₉	1	1	yes	characteristic
109	C₄²	—	2	1	no	—
110	C₄²	C₂²×D₁₉₉	1	1	yes	characteristic
111	Q₈	—	2	1	no	—
112	Q₈	—	4	1	no	—
113	Q₈	—	4	1	no	—
114	Q₈	—	1592	1	no	—
115	Q₈	#15691874	1	2	yes	—
116	C₂×C₄	—	2	1	no	—
117	C₂×C₄	—	2	1	no	—
118	C₂×C₄	—	4	1	no	—
119	C₂×C₄	—	796	1	no	—
120	C₂×C₄	D₄×D₁₉₉	1	1	yes	characteristic
121	C₂×C₄	C₂×C₂²⋊D₁₉₉	1	1	yes	characteristic
122	C₂×C₄	#15691885	1	1	yes	characteristic
123	C₈	—	4	1	no	—
124	C₈	—	796	1	no	—
125	C₈	—	796	1	no	—
126	C₈	—	398	1	no	—
127	C₈	—	199	2	no	—
128	C₂²	#20263927	1	1	yes	characteristic
129	C₄	—	2	2	no	—
130	C₄	—	2	1	no	—
131	C₄	—	4	1	no	—
132	C₄	—	8	1	no	—
133	C₄	—	1592	1	no	—
134	C₄	C₂×#15691874	1	1	yes	characteristic
135	C₄	Q₁₆×D₁₉₉	1	1	yes	characteristic
136	C₄	#20263975	1	1	yes	characteristic
137	C₄	#20263987	1	1	yes	characteristic
138	C₂	#27164226	1	1	yes	characteristic
139	C₂	#27164245	1	1	yes	characteristic
140	C₂	#27164283	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#27164414

C₂

yes

maximal, characteristic

#27164313

C₂

yes

maximal, characteristic

Q₈×C₁₉₉⋊C₈

C₂

yes

maximal, characteristic

#27163768

C₂

yes

maximal, characteristic

#27163766

C₂

yes

maximal, characteristic

#27163764

C₂

yes

maximal, characteristic

#462×C₁₉₉

C₂

yes

maximal, characteristic

#20263955

C₂²

yes

—

—

—

C₄⋊(C₁₉₉⋊C₈)

—

C₄⋊(C₁₉₉⋊C₈)

C₂²

yes

characteristic

C₄⋊(C₁₉₉⋊C₈)

C₂²

yes

characteristic

#20263879

—

C₄×C₁₉₉⋊C₈

—

C₄×C₁₉₉⋊C₈

C₂²

yes

characteristic

C₂×Q₁₆×C₁₉₉

—

C₄⋊Q₈×C₁₉₉

C₂²

yes

characteristic

C₄×Q₈×C₁₉₉

C₂²

yes

characteristic

C₄⋊C₈×C₁₉₉

C₂²

yes

characteristic

#103×C₁₉₉

—

C₂×Dic₃₉₈

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

—

C₂×C₁₉₉⋊C₈

D₄

yes

characteristic

C₂×C₁₉₉⋊C₈

D₄

yes

characteristic

C₄⋊Dic₁₉₉

—

C₄⋊Dic₁₉₉

—

C₂×Q₈×C₁₉₉

—

C₂×Q₈×C₁₉₉

D₄

yes

characteristic

Q₁₆×C₁₉₉

—

C₂×C₈×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

—

C₄⋊C₄×C₁₉₉

D₄

yes

characteristic

C₄²×C₁₉₉

—

C₄²×C₁₉₉

C₂³

yes

characteristic

Dic₃₉₈

—

C₂×Dic₁₉₉

—

C₁₉₉⋊C₈

—

C₁₉₉⋊C₈

—

C₁₉₉⋊C₈

—

C₁₉₉⋊C₈

Q₁₆

yes

complemented

Q₈×C₁₉₉

—

Q₈×C₁₉₉

—

Q₈×C₁₉₉

—

Q₈×C₁₉₉

SD₁₆

yes

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

—

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

C₂×D₄

yes

characteristic

C₂×C₄×C₁₉₉

#41

yes

characteristic

C₈×C₁₉₉

—

C₂²×C₁₉₉

#121

yes

characteristic

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

—

C₄×C₁₉₉

C₂×SD₁₆

yes

characteristic

C₄×C₁₉₉

C₂×Q₁₆

yes

C₄×C₁₉₉

yes

C₄×C₁₉₉

yes

Dic₁₉₉

—

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₂×C₁₉₉

yes

C₁₉₉

yes

characteristic, complemented, Sylow

#1672

—

199

maximal, Sylow

#498

—

199

—

#477

—

199

—

#462

D₁₉₉

yes

characteristic

C₈×Q₈

—

199

—

#334

—

199

—

#333

—

199

—

#332

—

199

—

C₂×Q₁₆

—

C₄⋊Q₈

—

199

—

C₄⋊Q₈

C₂×D₁₉₉

yes

characteristic

C₄×Q₈

C₂×D₁₉₉

yes

characteristic

#107

—

398

—

#106

—

398

—

#106

—

398

—

C₄⋊C₈

—

398

—

C₄⋊C₈

—

199

—

C₄⋊C₈

—

199

—

C₄⋊C₈

C₂×D₁₉₉

yes

characteristic

#103

—

#103

—

398

—

C₄×C₈

—

398

—

C₄×C₈

—

199

—

C₂×Q₈

—

C₂×Q₈

—

398

—

C₂×Q₈

C₂²⋊D₁₉₉

yes

characteristic

Q₁₆

—

C₂×C₈

—

C₂×C₈

—

398

—

100

C₂×C₈

—

398

—

101

C₂×C₈

—

199

—

102

C₂×C₈

—

199

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

105

C₄⋊C₄

—

106

C₄⋊C₄

—

398

—

107

C₄⋊C₄

—

398

—

108

C₄⋊C₄

C₂²⋊D₁₉₉

yes

characteristic

109

C₄²

—

110

C₄²

C₂²×D₁₉₉

yes

characteristic

111

Q₈

—

112

Q₈

—

113

Q₈

—

114

Q₈

—

1592

—

115

Q₈

#15691874

yes

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

796

—

120

C₂×C₄

D₄×D₁₉₉

yes

characteristic

121

C₂×C₄

C₂×C₂²⋊D₁₉₉

yes

characteristic

122

C₂×C₄

#15691885

yes

characteristic

123

C₈

—

124

C₈

—

796

—

125

C₈

—

796

—

126

C₈

—

398

—

127

C₈

—

199

—

128

C₂²

#20263927

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

132

C₄

—

133

C₄

—

1592

—

134

C₄

C₂×#15691874

yes

characteristic

135

C₄

Q₁₆×D₁₉₉

yes

characteristic

136

C₄

#20263975

yes

characteristic

137

C₄

#20263987

yes

characteristic

138

C₂

#27164226

yes

characteristic

139

C₂

#27164245

yes

characteristic

140

C₂

#27164283

yes

characteristic

	subgroup	G/H
center	C₂²	#20263927
commutator, cosocle	C₄²×C₁₉₉	C₂³
Frattini	C₄²	C₂²×D₁₉₉
Fitting	#462×C₁₉₉	C₂
socle, prime radical	C₂²×C₁₉₉	#121
hypercenter	#462	D₁₉₉
hypocenter	C₁₉₉	#1672
LCS	G ▷ C₄²×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #462

subgroup

G/H

center

C₂²

#20263927

commutator, cosocle

C₄²×C₁₉₉

C₂³

Frattini

C₄²

C₂²×D₁₉₉

Fitting

#462×C₁₉₉

C₂

C₂²×C₁₉₉

D₁₉₉

C₁₉₉

G ▷ C₄²×C₁₉₉ ▷ C₂²×C₁₉₉ ▷ C₁₉₉

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #462

N	H ≅ G/N	type
45. C₁₉₉⋊C₈	Q₁₆	semidirect
70. C₁₉₉	#1672	semidirect

H ≅ G/N

type

45. C₁₉₉⋊C₈

Q₁₆

semidirect

70. C₁₉₉

#1672

semidirect

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

subgroup	index
#27164414	2
#27164313	2
Q₈×C₁₉₉⋊C₈	2
#27163768	2
#27163766	2
#27163764	2
#462×C₁₉₉	2
#1672	199

subgroup

index

Q₈×C₁₉₉⋊C₈

#462×C₁₉₉

199

Subgroups — #38768230

Conjugacy classes of subgroups

Complements

Maximal subgroups