Group #22626715 — subgroups

Subgroups — C₂×#17989637 #22626715

Order 9344.

400 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 400 conjugacy classes of subgroups — too many to draw. Only its 194 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	#17989637	C₂	1	8	yes	maximal, direct product factor
2	C₂×C₄⋊#10596833	C₂	1	2	yes	maximal
3	C₂×C₄⋊#10596833	C₂	1	1	yes	maximal, characteristic
4	C₂×#14556646	C₂	1	2	yes	maximal
5	C₂×C₄×#10596833	C₂	1	1	yes	maximal, characteristic
6	C₂×#14556556	C₂	1	1	yes	maximal, characteristic
7	C₂²×#10596833	C₂²	1	1	yes	characteristic
8	C₂²×#10596833	C₂²	1	2	yes	—
9	C₂²×#10596833	C₂²	1	1	yes	characteristic
10	C₄⋊#10596833	C₂²	1	4	yes	—
11	C₄⋊#10596833	C₂²	1	8	yes	—
12	#14556646	C₂²	1	8	yes	—
13	C₄×#10596833	C₂²	1	4	yes	—
14	C₂²×Dic₁₄₆	C₄	1	1	yes	characteristic
15	#14556556	C₄	1	4	yes	—
16	#14556556	C₂²	1	4	yes	—
17	C₂×#11924007	C₄	1	2	yes	—
18	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×Dic₇₃	C₂²	1	2	yes	—
20	C₂×C₄⋊C₄×C₇₃	C₄	1	1	yes	characteristic
21	C₂×#10596833	—	2	2	no	—
22	C₂×#10596833	—	2	4	no	—
23	C₂×#10596833	Q₈	1	4	yes	complemented
24	C₂×#10596833	C₂³	1	2	yes	—
25	C₂×#10596833	C₂³	1	4	yes	—
26	C₂×#10596833	C₂³	1	2	yes	—
27	C₂×Dic₁₄₆	—	2	4	no	—
28	C₂×Dic₁₄₆	C₂×C₄	1	4	yes	—
29	C₂²×Dic₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×Dic₇₃	C₂×C₄	1	2	yes	—
31	C₂²×Dic₇₃	C₂³	1	1	yes	characteristic
32	#11924007	C₂×C₄	1	8	yes	—
33	C₄×Dic₇₃	C₂×C₄	1	2	yes	—
34	C₄×Dic₇₃	C₂×C₄	1	4	yes	—
35	C₄×Dic₇₃	C₂³	1	2	yes	—
36	C₄×Dic₇₃	C₂³	1	4	yes	—
37	C₂²×C₄×C₇₃	C₂×C₄	1	2	yes	—
38	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
39	C₄⋊C₄×C₇₃	C₂×C₄	1	4	yes	—
40	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₄×C₇₃	—	2	4	no	—
42	C₂×C₄×C₇₃	M₁₆	1	4	yes	—
43	C₂×C₄×C₇₃	C₂²×C₄	1	4	yes	—
44	C₂×C₄×C₇₃	C₂²×C₄	1	2	yes	—
45	C₂×Dic₇₃	—	2	4	no	—
46	C₂×Dic₇₃	—	2	2	no	—
47	C₂×Dic₇₃	C₂²×C₄	1	2	yes	—
48	C₂×Dic₇₃	C₂²×C₄	1	4	yes	—
49	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×Dic₇₃	C₂×Q₈	1	2	yes	—
51	C₂×Dic₇₃	#41	1	2	yes	—
52	C₂×Dic₇₃	C₂⁴	1	1	yes	characteristic
53	Dic₁₄₆	—	2	8	no	—
54	#10596833	—	2	2	no	—
55	#10596833	—	2	4	no	—
56	#10596833	C₂×Q₈	1	4	yes	complemented
57	C₂²×C₇₃	C₄×Q₈	1	2	yes	—
58	C₂²×C₇₃	C₂×M₁₆	1	2	yes	—
59	C₂²×C₇₃	#131	1	2	yes	—
60	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
61	C₄×C₇₃	—	2	4	no	—
62	C₄×C₇₃	C₂×M₁₆	1	4	yes	—
63	Dic₇₃	—	2	4	no	—
64	Dic₇₃	—	2	2	no	—
65	Dic₇₃	C₄×Q₈	1	2	yes	—
66	Dic₇₃	C₂²×Q₈	1	1	yes	characteristic
67	Dic₇₃	C₂×#41	1	1	yes	characteristic
68	C₂×C₇₃	#446	1	4	yes	complemented
69	C₂×C₇₃	C₂×C₄×Q₈	1	1	yes	characteristic
70	C₂×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
71	C₂×C₇₃	C₂×#131	1	1	yes	characteristic
72	C₂×#446	—	73	1	no	maximal, Sylow
73	C₇₃	C₂×#446	1	1	yes	characteristic, complemented, Sylow
74	C₂×C₄×Q₈	—	73	1	no	—
75	#446	—	73	8	no	—
76	C₂×C₄⋊C₈	—	73	1	no	—
77	C₂×C₄⋊C₈	—	73	2	no	—
78	C₂×#97	—	73	2	no	—
79	C₂×C₄×C₈	—	73	1	no	—
80	C₂²×Q₈	—	73	1	no	—
81	C₂²×C₈	—	73	1	no	—
82	C₂²×C₈	—	73	1	no	—
83	C₂²×C₈	—	73	2	no	—
84	C₄×Q₈	—	73	4	no	—
85	C₄×Q₈	—	73	4	no	—
86	C₂×C₄⋊C₄	—	73	2	no	—
87	C₂×C₄⋊C₄	#64408	1	1	yes	characteristic
88	C₂×C₄²	—	73	1	no	—
89	C₂×C₄²	—	73	2	no	—
90	C₄⋊C₈	—	73	4	no	—
91	C₄⋊C₈	—	73	8	no	—
92	#97	—	73	8	no	—
93	C₄×C₈	—	73	4	no	—
94	C₂×Q₈	—	73	4	no	—
95	C₂×Q₈	—	146	4	no	—
96	C₂²×C₄	—	73	1	no	—
97	C₂²×C₄	—	73	1	no	—
98	C₂²×C₄	—	73	2	no	—
99	C₂²×C₄	C₂×#64408	1	2	yes	—
100	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
101	C₂×C₈	—	73	4	no	—
102	C₂×C₈	—	73	2	no	—
103	C₂×C₈	—	73	2	no	—
104	C₂×C₈	—	73	4	no	—
105	C₂×C₈	—	146	2	no	—
106	C₂×C₈	—	146	4	no	—
107	C₄⋊C₄	—	73	8	no	—
108	C₄⋊C₄	C₂×#64408	1	4	yes	—
109	C₄²	—	73	2	no	—
110	C₄²	—	73	4	no	—
111	C₄²	—	73	2	no	—
112	C₄²	—	73	4	no	—
113	C₂³	C₂²×#64408	1	1	yes	characteristic
114	Q₈	—	146	8	no	—
115	C₂×C₄	—	2	4	no	—
116	C₂×C₄	—	73	2	no	—
117	C₂×C₄	—	73	2	no	—
118	C₂×C₄	—	73	1	no	—
119	C₂×C₄	—	73	1	no	—
120	C₂×C₄	—	73	2	no	—
121	C₂×C₄	—	73	4	no	—
122	C₂×C₄	—	146	4	no	—
123	C₂×C₄	—	146	2	no	—
124	C₂×C₄	#11924038	1	4	yes	—
125	C₂×C₄	C₂²×#64408	1	4	yes	—
126	C₂×C₄	C₂²×#64408	1	2	yes	—
127	C₈	—	73	4	no	—
128	C₈	—	146	2	no	—
129	C₈	—	146	4	no	—
130	C₂²	Q₈×#64408	1	2	yes	—
131	C₂²	C₂×#11924038	1	2	yes	—
132	C₂²	#14556678	1	2	yes	—
133	C₂²	C₂³×#64408	1	1	yes	characteristic
134	C₄	—	2	4	no	—
135	C₄	—	73	1	no	—
136	C₄	—	73	1	no	—
137	C₄	—	73	2	no	—
138	C₄	—	146	4	no	—
139	C₄	—	146	2	no	—
140	C₄	C₂×#11924038	1	4	yes	—
141	C₂	#17989637	1	4	yes	direct product factor
142	C₂	C₂×Q₈×#64408	1	1	yes	characteristic
143	C₂	C₂²×#11924038	1	1	yes	characteristic
144	C₂	C₂×#14556678	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989637

C₂

yes

maximal, direct product factor

C₂×C₄⋊#10596833

C₂

yes

maximal

C₂×C₄⋊#10596833

C₂

yes

maximal, characteristic

C₂×#14556646

C₂

yes

maximal

C₂×C₄×#10596833

C₂

yes

maximal, characteristic

C₂×#14556556

C₂

yes

maximal, characteristic

C₂²×#10596833

C₂²

yes

characteristic

C₂²×#10596833

C₂²

yes

—

C₂²×#10596833

C₂²

yes

characteristic

C₄⋊#10596833

C₂²

yes

—

C₄⋊#10596833

C₂²

yes

—

#14556646

C₂²

yes

—

C₄×#10596833

C₂²

yes

—

C₂²×Dic₁₄₆

C₄

yes

characteristic

#14556556

C₄

yes

—

#14556556

C₂²

yes

—

C₂×#11924007

C₄

yes

—

C₂×C₄×Dic₇₃

C₂²

yes

characteristic

C₂×C₄×Dic₇₃

C₂²

yes

—

C₂×C₄⋊C₄×C₇₃

C₄

yes

—

—

Q₈

yes

complemented

C₂×#10596833

C₂³

yes

—

C₂×#10596833

C₂³

yes

—

C₂×#10596833

C₂³

yes

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂×C₄

yes

—

C₂²×Dic₇₃

C₂×C₄

yes

characteristic

C₂²×Dic₇₃

C₂×C₄

yes

—

C₂²×Dic₇₃

C₂³

yes

characteristic

#11924007

C₂×C₄

yes

—

C₄×Dic₇₃

C₂×C₄

yes

—

C₄×Dic₇₃

C₂×C₄

yes

—

C₄×Dic₇₃

C₂³

yes

—

C₄×Dic₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₂³×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

M₁₆

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²×C₄

yes

—

C₂×Dic₇₃

C₂²×C₄

yes

—

C₂×Dic₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

C₂×Q₈

yes

—

C₂×Dic₇₃

#41

yes

—

C₂×Dic₇₃

C₂⁴

yes

characteristic

Dic₁₄₆

—

#10596833

—

#10596833

—

#10596833

C₂×Q₈

yes

complemented

C₂²×C₇₃

C₄×Q₈

yes

—

C₂²×C₇₃

C₂×M₁₆

yes

—

C₂²×C₇₃

#131

yes

—

C₂²×C₇₃

C₂³×C₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₂×M₁₆

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

C₄×Q₈

yes

—

Dic₇₃

C₂²×Q₈

yes

characteristic

Dic₇₃

C₂×#41

yes

characteristic

C₂×C₇₃

#446

yes

complemented

C₂×C₇₃

C₂×C₄×Q₈

yes

characteristic

C₂×C₇₃

C₂²×M₁₆

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

#446

—

C₂×C₄⋊C₈

—

C₂×C₄⋊C₈

—

C₂×#97

—

C₂×C₄×C₈

—

C₂²×Q₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

#64408

yes

characteristic

C₂×C₄²

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

#97

—

C₄×C₈

—

C₂×Q₈

—

C₂×Q₈

—

146

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#64408

yes

—

100

C₂²×C₄

C₂×#64408

yes

characteristic

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

146

—

106

C₂×C₈

—

146

—

107

C₄⋊C₄

—

108

C₄⋊C₄

C₂×#64408

yes

—

109

C₄²

—

110

C₄²

—

111

C₄²

—

112

C₄²

—

113

C₂³

C₂²×#64408

yes

characteristic

114

Q₈

—

146

—

115

C₂×C₄

—

116

C₂×C₄

—

117

C₂×C₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

—

121

C₂×C₄

—

122

C₂×C₄

—

146

—

123

C₂×C₄

—

146

—

124

C₂×C₄

#11924038

yes

—

125

C₂×C₄

C₂²×#64408

yes

—

126

C₂×C₄

C₂²×#64408

yes

—

127

C₈

—

128

C₈

—

146

—

129

C₈

—

146

—

130

C₂²

Q₈×#64408

yes

—

131

C₂²

C₂×#11924038

yes

—

132

C₂²

#14556678

yes

—

133

C₂²

C₂³×#64408

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

—

137

C₄

—

138

C₄

—

146

—

139

C₄

—

146

—

140

C₄

C₂×#11924038

yes

—

141

C₂

#17989637

yes

direct product factor

142

C₂

C₂×Q₈×#64408

yes

characteristic

143

C₂

C₂²×#11924038

yes

characteristic

144

C₂

C₂×#14556678

yes

characteristic

	subgroup	G/H
center	C₂³	C₂²×#64408
commutator	C₂²×C₇₃	C₂³×C₄
Frattini	C₂²	C₂³×#64408
Fitting	C₂×C₄⋊C₄×C₇₃	C₄
socle	C₂³×C₇₃	C₂²×C₄
hypercenter	C₂×C₄⋊C₄	#64408
hypocenter	C₇₃	C₂×#446
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

subgroup

G/H

center

C₂³

C₂²×#64408

commutator

C₂²×C₇₃

C₂³×C₄

Frattini

C₂²

C₂³×#64408

Fitting

C₂×C₄⋊C₄×C₇₃

C₄

socle

C₂³×C₇₃

C₂²×C₄

C₂×C₄⋊C₄

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂³ ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
1. #17989637	C₂	direct (N × H)
23. C₂×#10596833	Q₈	semidirect
56. #10596833	C₂×Q₈	semidirect
68. C₂×C₇₃	#446	semidirect
73. C₇₃	C₂×#446	semidirect
141. C₂	#17989637	direct (N × H)

H ≅ G/N

type

1. #17989637

C₂

direct (N × H)

23. C₂×#10596833

Q₈

semidirect

56. #10596833

C₂×Q₈

semidirect

68. C₂×C₇₃

#446

semidirect

73. C₇₃

C₂×#446

semidirect

141. C₂

#17989637

direct (N × H)

Subgroups — C₂×#17989637 #22626715

Conjugacy classes of subgroups

Complements

Subgroups — C2×#17989637 #22626715

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#17989637 #22626715