Group #69911 — subgroups

Subgroups — #69911

Order 384.

292 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 101 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×#926	C₂	1	1	yes	maximal, characteristic
2	C₂×C₃×C₂²⋊C₈	C₂	1	1	yes	maximal, characteristic
3	C₂×#830	C₂	1	1	yes	maximal, characteristic
4	#4550	C₂	1	4	yes	maximal, complemented
5	#1818	—	3	1	no	maximal, Sylow
6	C₂³×C₃×C₄	C₂²	1	1	yes	characteristic
7	C₂²×C₃×C₈	C₂²	1	2	yes	—
8	C₂×#217	C₄	1	2	yes	—
9	#926	—	2	2	no	—
10	#926	—	2	2	no	—
11	C₂×C₄⋊Dic₃	C₄	1	1	yes	characteristic
12	C₂×C₄⋊Dic₃	C₄	1	1	yes	characteristic
13	C₂×C₄×Dic₃	C₂²	1	2	yes	—
14	C₂²×C₃⋊C₈	C₂²	1	2	yes	—
15	C₃×C₂²⋊C₈	—	2	2	no	—
16	#830	—	2	2	no	—
17	C₂×#117	—	3	1	no	—
18	C₂×C₂²⋊C₈	—	3	1	no	—
19	C₂×C₂²⋊C₈	D₃	1	1	yes	characteristic
20	#336	—	3	4	no	—
21	C₂⁴×C₃	C₂×C₄	1	1	yes	characteristic
22	C₂²×C₃×C₄	—	2	4	no	—
23	C₂²×C₃×C₄	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₃×C₄	D₄	1	2	yes	—
25	C₂²×C₃×C₄	D₄	1	2	yes	—
26	C₂²×C₃×C₄	C₂³	1	1	yes	characteristic
27	C₂²×Dic₃	C₂×C₄	1	2	yes	—
28	C₂²×Dic₃	C₂×C₄	1	2	yes	—
29	C₂×C₃×C₈	—	2	2	no	—
30	C₂×C₃×C₈	—	2	4	no	—
31	#217	—	2	4	no	—
32	C₄⋊Dic₃	—	2	2	no	—
33	C₄⋊Dic₃	—	2	2	no	—
34	C₄×Dic₃	—	2	2	no	—
35	C₄×Dic₃	—	2	2	no	—
36	C₂×C₃⋊C₈	—	2	2	no	—
37	C₂×C₃⋊C₈	—	2	4	no	—
38	C₂³×C₄	C₂×D₃	1	1	yes	characteristic
39	C₂²×C₈	—	3	2	no	—
40	C₂²×C₈	C₂×D₃	1	2	yes	—
41	#117	—	6	2	no	—
42	#117	—	6	2	no	—
43	C₂×C₄⋊C₄	—	3	1	no	—
44	C₂×C₄⋊C₄	—	3	1	no	—
45	C₂×C₂²⋊C₄	—	3	2	no	—
46	C₂×C₄²	—	3	2	no	—
47	C₂²⋊C₈	—	2	2	no	—
48	C₂²⋊C₈	—	6	2	no	—
49	C₂³×C₃	—	2	2	no	—
50	C₂³×C₃	—	2	2	no	—
51	C₂³×C₃	—	2	2	no	—
52	C₂³×C₃	C₂²⋊C₄	1	2	yes	—
53	C₂³×C₃	C₂²×C₄	1	1	yes	characteristic
54	C₂×C₃×C₄	—	2	2	no	—
55	C₂×C₃×C₄	—	2	4	no	—
56	C₂×C₃×C₄	—	2	4	no	—
57	C₂×C₃×C₄	C₂²⋊C₄	1	2	yes	—
58	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
59	C₂×C₃×C₄	C₂×D₄	1	1	yes	characteristic
60	C₂×C₃×C₄	#41	1	4	yes	—
61	C₂×Dic₃	—	2	2	no	—
62	C₂×Dic₃	—	2	2	no	—
63	C₂×Dic₃	—	2	4	no	—
64	C₂×Dic₃	—	2	4	no	—
65	C₃×C₈	—	4	2	no	—
66	C₃⋊C₈	—	4	2	no	—
67	C₂⁴	C₄×D₃	1	1	yes	characteristic
68	C₂²×C₄	—	2	4	no	—
69	C₂²×C₄	—	3	2	no	—
70	C₂²×C₄	—	3	2	no	—
71	C₂²×C₄	C₄×D₃	1	1	yes	characteristic
72	C₂²×C₄	D₁₂	1	2	yes	—
73	C₂²×C₄	#67	1	2	yes	—
74	C₂²×C₄	C₂²×D₃	1	1	yes	characteristic
75	C₂×C₈	—	2	4	no	—
76	C₂×C₈	—	2	2	no	—
77	C₂×C₈	—	6	2	no	—
78	C₂×C₈	—	6	4	no	—
79	C₄⋊C₄	—	6	2	no	—
80	C₄⋊C₄	—	6	2	no	—
81	C₂²⋊C₄	—	6	4	no	—
82	C₄²	—	6	2	no	—
83	C₄²	—	6	2	no	—
84	C₂²×C₃	—	2	2	no	—
85	C₂²×C₃	—	2	2	no	—
86	C₂²×C₃	—	2	2	no	—
87	C₂²×C₃	—	4	4	no	—
88	C₂²×C₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂²×C₃	#117	1	2	yes	—
90	C₂²×C₃	#131	1	2	yes	—
91	C₂²×C₃	#131	1	2	yes	—
92	C₃×C₄	—	2	2	no	—
93	C₃×C₄	#123	1	2	yes	—
94	C₃×C₄	#123	1	2	yes	—
95	Dic₃	—	4	2	no	—
96	Dic₃	—	4	2	no	—
97	C₂³	—	2	2	no	—
98	C₂³	—	2	2	no	—
99	C₂³	—	2	2	no	—
100	C₂³	(C₂×D₃)⋊C₄	1	2	yes	—
101	C₂³	C₂×C₄×D₃	1	1	yes	characteristic
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	6	4	no	—
106	C₂×C₄	—	6	4	no	—
107	C₂×C₄	—	6	2	no	—
108	C₂×C₄	—	6	2	no	—
109	C₂×C₄	(C₂×D₃)⋊C₄	1	2	yes	—
110	C₂×C₄	C₂×D₁₂	1	1	yes	characteristic
111	C₂×C₄	#237	1	4	yes	—
112	C₂×C₄	C₂×#67	1	1	yes	characteristic
113	C₈	—	4	2	no	—
114	C₈	—	12	2	no	—
115	C₂×C₃	—	4	2	no	—
116	C₂×C₃	#381	1	1	yes	characteristic
117	C₂×C₃	#408	1	1	yes	characteristic
118	C₂×C₃	#408	1	1	yes	characteristic
119	C₂×C₃	#433	1	4	yes	—
120	C₂²	—	2	2	no	—
121	C₂²	—	2	2	no	—
122	C₂²	—	2	2	no	—
123	C₂²	—	4	4	no	—
124	C₂²	#877	1	2	yes	—
125	C₂²	#901	1	2	yes	—
126	C₂²	#907	1	2	yes	—
127	C₂²	C₂×(C₂×D₃)⋊C₄	1	1	yes	characteristic
128	C₄	—	2	2	no	—
129	C₄	—	12	2	no	—
130	C₄	—	12	2	no	—
131	C₄	#886	1	2	yes	—
132	C₄	#935	1	2	yes	—
133	C₃	#1818	1	1	yes	characteristic, complemented, Sylow
134	C₂	—	4	2	no	—
135	C₂	#4720	1	4	yes	—
136	C₂	#4943	1	1	yes	characteristic
137	C₂	#5110	1	1	yes	characteristic
138	C₂	#5129	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#926

C₂

yes

maximal, characteristic

C₂×C₃×C₂²⋊C₈

C₂

yes

maximal, characteristic

C₂×#830

C₂

yes

maximal, characteristic

#4550

C₂

yes

maximal, complemented

#1818

—

maximal, Sylow

C₂³×C₃×C₄

C₂²

yes

characteristic

C₂²×C₃×C₈

C₂²

yes

—

C₂×#217

C₄

yes

—

#926

—

#926

—

C₂×C₄⋊Dic₃

C₄

yes

characteristic

C₂×C₄⋊Dic₃

C₄

yes

characteristic

C₂×C₄×Dic₃

C₂²

yes

—

C₂²×C₃⋊C₈

C₂²

yes

—

C₃×C₂²⋊C₈

—

#830

—

C₂×#117

—

C₂×C₂²⋊C₈

—

C₂×C₂²⋊C₈

D₃

yes

characteristic

#336

—

C₂⁴×C₃

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

—

C₂²×C₃×C₄

C₂×C₄

yes

characteristic

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

D₄

yes

—

C₂²×C₃×C₄

C₂³

yes

characteristic

C₂²×Dic₃

C₂×C₄

yes

—

C₂²×Dic₃

C₂×C₄

yes

—

C₂×C₃×C₈

—

C₂×C₃×C₈

—

#217

—

C₄⋊Dic₃

—

C₄⋊Dic₃

—

C₄×Dic₃

—

C₄×Dic₃

—

C₂×C₃⋊C₈

—

C₂×C₃⋊C₈

—

C₂³×C₄

C₂×D₃

yes

characteristic

C₂²×C₈

—

C₂²×C₈

C₂×D₃

yes

—

#117

—

#117

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

—

C₂³×C₃

C₂²⋊C₄

yes

—

C₂³×C₃

C₂²×C₄

yes

characteristic

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

—

C₂×C₃×C₄

C₂²⋊C₄

yes

—

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

C₂×D₄

yes

characteristic

C₂×C₃×C₄

#41

yes

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₂×Dic₃

—

C₃×C₈

—

C₃⋊C₈

—

C₂⁴

C₄×D₃

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₄×D₃

yes

characteristic

C₂²×C₄

D₁₂

yes

—

C₂²×C₄

#67

yes

—

C₂²×C₄

C₂²×D₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

—

C₂²×C₃

C₂×C₂²⋊C₄

yes

characteristic

C₂²×C₃

#117

yes

—

C₂²×C₃

#131

yes

—

C₂²×C₃

#131

yes

—

C₃×C₄

—

C₃×C₄

#123

yes

—

C₃×C₄

#123

yes

—

Dic₃

—

Dic₃

—

C₂³

—

C₂³

—

C₂³

—

100

C₂³

(C₂×D₃)⋊C₄

yes

—

101

C₂³

C₂×C₄×D₃

yes

characteristic

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

(C₂×D₃)⋊C₄

yes

—

110

C₂×C₄

C₂×D₁₂

yes

characteristic

111

C₂×C₄

#237

yes

—

112

C₂×C₄

C₂×#67

yes

characteristic

113

C₈

—

114

C₈

—

115

C₂×C₃

—

116

C₂×C₃

#381

yes

characteristic

117

C₂×C₃

#408

yes

characteristic

118

C₂×C₃

#408

yes

characteristic

119

C₂×C₃

#433

yes

—

120

C₂²

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

#877

yes

—

125

C₂²

#901

yes

—

126

C₂²

#907

yes

—

127

C₂²

C₂×(C₂×D₃)⋊C₄

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

—

131

C₄

#886

yes

—

132

C₄

#935

yes

—

133

C₃

#1818

yes

characteristic, complemented, Sylow

134

C₂

—

135

C₂

#4720

yes

—

136

C₂

#4943

yes

characteristic

137

C₂

#5110

yes

characteristic

138

C₂

#5129

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₃
commutator, socle	C₂³×C₃	C₂²×C₄
Fitting	C₂×C₃×C₂²⋊C₈	C₂
prime radical	C₂⁴×C₃	C₂×C₄
cosocle	C₂²×C₃×C₄	C₂³
hypercenter	C₂×C₂²⋊C₈	D₃
hypocenter	C₃	#1818
LCS	G ▷ C₂³×C₃ ▷ C₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

subgroup

G/H

center, Frattini

C₂²×C₄

C₂²×D₃

commutator, socle

C₂³×C₃

C₂²×C₄

Fitting

C₂×C₃×C₂²⋊C₈

C₂

prime radical

C₂⁴×C₃

C₂×C₄

cosocle

C₂²×C₃×C₄

C₂³

hypercenter

C₂×C₂²⋊C₈

D₃

C₃

G ▷ C₂³×C₃ ▷ C₃

C₁ ◁ C₂²×C₄ ◁ C₂×C₂²⋊C₈

N	H ≅ G/N	type
4. #4550	C₂	semidirect
133. C₃	#1818	semidirect

H ≅ G/N

type

4. #4550

C₂

semidirect

133. C₃

#1818

semidirect

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

subgroup	index
C₂×#926	2
C₂×C₃×C₂²⋊C₈	2
C₂×#830	2
#4550	2
#1818	3

subgroup

index

C₂×C₃×C₂²⋊C₈

Subgroups — #69911

Conjugacy classes of subgroups

Complements

Maximal subgroups