Group #22626781 — subgroups

Subgroups — #22626781

Order 9344.

500 conjugacy classes of subgroups; only the 204 normal subgroups are shown.

normal subgroups only

This group has 500 conjugacy classes of subgroups — too many to draw. Only its 204 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	#17989615	C₂	1	8	yes	maximal, complemented
2	#17989603	C₂	1	2	yes	maximal, complemented
3	#17989555	C₂	1	4	yes	maximal, complemented
4	C₂×#14556553	C₂	1	1	yes	maximal, characteristic
5	C₂×#11924036	C₂²	1	8	yes	complemented
6	C₄⋊#10596833	C₂²	1	4	yes	complemented
7	#14556648	C₂²	1	8	yes	complemented
8	C₂²⋊#10596833	C₂²	1	4	yes	—
9	C₄×#10596833	C₂²	1	4	yes	complemented
10	C₂²×D₂₉₂	C₄	1	1	yes	characteristic
11	C₂²×C₄×D₇₃	C₂²	1	2	yes	—
12	#14556553	C₄	1	4	yes	—
13	#14556553	C₂²	1	4	yes	—
14	C₂×#11924005	C₄	1	2	yes	—
15	C₂×C₄×Dic₇₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₄×C₇₃	C₄	1	1	yes	characteristic
17	#11924036	—	4	8	no	—
18	C₂×#10596833	C₂³	1	8	yes	—
19	C₂³×D₇₃	C₂×C₄	1	2	yes	—
20	C₂×D₂₉₂	C₈	1	8	yes	complemented
21	C₂×D₂₉₂	C₂×C₄	1	4	yes	—
22	C₂²×Dic₇₃	C₂×C₄	1	1	yes	characteristic
23	C₂²×Dic₇₃	C₂³	1	1	yes	characteristic
24	C₂×C₄×D₇₃	—	4	4	no	—
25	C₂×C₄×D₇₃	C₂×C₄	1	4	yes	—
26	C₂×C₄×D₇₃	C₂³	1	4	yes	—
27	#11924005	C₂×C₄	1	8	yes	—
28	C₄×Dic₇₃	C₂×C₄	1	2	yes	—
29	C₄×Dic₇₃	C₂³	1	2	yes	—
30	C₂²×C₄×C₇₃	C₂×C₄	1	2	yes	—
31	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₄⋊C₄×C₇₃	C₂×C₄	1	4	yes	—
33	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
34	C₂²×D₇₃	—	4	2	no	—
35	C₂²×D₇₃	—	4	2	no	—
36	C₂²×D₇₃	C₂×C₈	1	8	yes	complemented
37	C₂²×D₇₃	C₂²×C₄	1	4	yes	—
38	C₂×C₄×C₇₃	—	4	2	no	—
39	C₂×C₄×C₇₃	C₂×C₈	1	4	yes	complemented
40	C₂×C₄×C₇₃	C₂²×C₄	1	4	yes	—
41	C₂×C₄×C₇₃	C₂²×C₄	1	2	yes	—
42	C₂×Dic₇₃	—	2	2	no	—
43	C₂×Dic₇₃	—	4	1	no	—
44	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
45	C₂×Dic₇₃	C₂²×C₄	1	2	yes	—
46	C₂×Dic₇₃	C₂⁴	1	1	yes	characteristic
47	D₂₉₂	C₂×C₈	1	16	yes	complemented
48	C₄×D₇₃	—	2	8	no	—
49	C₄×D₇₃	—	4	4	no	—
50	#10596833	—	2	8	no	—
51	C₂²×C₇₃	—	2	1	no	—
52	C₂²×C₇₃	—	2	1	no	—
53	C₂²×C₇₃	C₂²×C₈	1	2	yes	—
54	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
55	C₂×D₇₃	—	2	4	no	—
56	C₂×D₇₃	—	2	4	no	—
57	C₂×D₇₃	—	4	8	no	—
58	C₂×D₇₃	C₂²×C₈	1	8	yes	—
59	C₄×C₇₃	—	2	4	no	—
60	C₄×C₇₃	C₂²×C₈	1	4	yes	—
61	Dic₇₃	—	2	1	no	—
62	Dic₇₃	—	2	2	no	—
63	Dic₇₃	H(5,𝔽₂)	1	2	yes	—
64	C₂×C₇₃	—	2	2	no	—
65	C₂×C₇₃	#518	1	1	yes	characteristic
66	C₂×C₇₃	C₂³×C₈	1	1	yes	characteristic
67	C₂×C₇₃	#568	1	1	yes	characteristic
68	D₇₃	—	2	8	no	—
69	#2972	—	73	1	no	maximal, Sylow
70	C₇₃	#2972	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₄×D₄	—	73	1	no	—
72	C₈×D₄	—	73	8	no	—
73	#431	—	73	2	no	—
74	C₂×C₂²⋊C₈	—	73	4	no	—
75	C₂²×D₄	—	73	1	no	—
76	C₂³×C₄	—	73	2	no	—
77	C₂²×C₈	—	73	8	no	—
78	C₄×D₄	—	73	4	no	—
79	C₄×D₄	—	73	4	no	—
80	C₂×C₄⋊C₄	#64408	1	1	yes	characteristic
81	C₂×C₂²⋊C₄	—	73	2	no	—
82	C₂×C₄²	—	73	1	no	—
83	C₄⋊C₈	—	73	4	no	—
84	C₂²⋊C₈	—	73	4	no	—
85	C₂²⋊C₈	—	73	8	no	—
86	C₄×C₈	—	73	4	no	—
87	C₂⁴	—	73	2	no	—
88	C₂×D₄	—	73	8	no	—
89	C₂×D₄	—	73	4	no	—
90	C₂²×C₄	—	292	4	no	—
91	C₂²×C₄	—	73	1	no	—
92	C₂²×C₄	—	73	4	no	—
93	C₂²×C₄	—	73	1	no	—
94	C₂²×C₄	—	73	4	no	—
95	C₂²×C₄	C₂×#64408	1	2	yes	—
96	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
97	C₂×C₈	—	292	8	no	—
98	C₂×C₈	—	73	8	no	—
99	C₄⋊C₄	C₂×#64408	1	4	yes	—
100	C₂²⋊C₄	—	73	8	no	—
101	C₄²	—	73	2	no	—
102	C₄²	—	73	2	no	—
103	C₂³	—	292	2	no	—
104	C₂³	—	292	2	no	—
105	C₂³	—	73	8	no	—
106	C₂³	—	73	4	no	—
107	C₂³	C₂²×#64408	1	1	yes	characteristic
108	D₄	—	73	16	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	292	4	no	—
111	C₂×C₄	—	292	1	no	—
112	C₂×C₄	—	73	1	no	—
113	C₂×C₄	—	73	2	no	—
114	C₂×C₄	—	73	1	no	—
115	C₂×C₄	—	146	8	no	—
116	C₂×C₄	—	146	2	no	—
117	C₂×C₄	#11924036	1	4	yes	complemented
118	C₂×C₄	C₂²×#64408	1	4	yes	—
119	C₂×C₄	C₂²×#64408	1	2	yes	—
120	C₈	—	146	8	no	—
121	C₂²	—	2	1	no	—
122	C₂²	—	2	1	no	—
123	C₂²	—	292	8	no	—
124	C₂²	—	73	8	no	—
125	C₂²	—	146	4	no	—
126	C₂²	—	146	4	no	—
127	C₂²	C₂×#11924036	1	2	yes	—
128	C₂²	C₂³×#64408	1	1	yes	characteristic
129	C₄	—	2	4	no	—
130	C₄	—	73	2	no	—
131	C₄	—	146	1	no	—
132	C₄	—	146	2	no	—
133	C₄	C₂×#11924036	1	4	yes	—
134	C₂	—	2	2	no	—
135	C₂	—	146	8	no	—
136	C₂	#17989654	1	1	yes	characteristic
137	C₂	C₂²×#11924036	1	1	yes	characteristic
138	C₂	#17989667	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989615

C₂

yes

maximal, complemented

#17989603

C₂

yes

maximal, complemented

#17989555

C₂

yes

maximal, complemented

C₂×#14556553

C₂

yes

maximal, characteristic

C₂×#11924036

C₂²

yes

complemented

C₄⋊#10596833

C₂²

yes

complemented

#14556648

C₂²

yes

complemented

C₂²⋊#10596833

C₂²

yes

—

C₄×#10596833

C₂²

yes

complemented

C₂²×D₂₉₂

C₄

yes

characteristic

C₂²×C₄×D₇₃

C₂²

yes

—

#14556553

C₄

yes

—

#14556553

C₂²

yes

—

C₂×#11924005

C₄

yes

—

C₂×C₄×Dic₇₃

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₇₃

C₄

yes

characteristic

#11924036

—

C₂×#10596833

C₂³

yes

—

C₂³×D₇₃

C₂×C₄

yes

—

C₂×D₂₉₂

C₈

yes

complemented

C₂×D₂₉₂

C₂×C₄

yes

—

C₂²×Dic₇₃

C₂×C₄

yes

characteristic

C₂²×Dic₇₃

C₂³

yes

characteristic

C₂×C₄×D₇₃

—

C₂×C₄×D₇₃

C₂×C₄

yes

—

C₂×C₄×D₇₃

C₂³

yes

—

#11924005

C₂×C₄

yes

—

C₄×Dic₇₃

C₂×C₄

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

C₂²×D₇₃

—

C₂²×D₇₃

—

C₂²×D₇₃

C₂×C₈

yes

complemented

C₂²×D₇₃

C₂²×C₄

yes

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×C₈

yes

complemented

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

C₂²×C₄

yes

—

C₂×Dic₇₃

C₂⁴

yes

characteristic

D₂₉₂

C₂×C₈

yes

complemented

C₄×D₇₃

—

C₄×D₇₃

—

#10596833

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²×C₈

yes

—

C₂²×C₇₃

C₂³×C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₂²×C₈

yes

—

C₄×C₇₃

—

C₄×C₇₃

C₂²×C₈

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

H(5,𝔽₂)

yes

—

C₂×C₇₃

—

C₂×C₇₃

#518

yes

characteristic

C₂×C₇₃

C₂³×C₈

yes

characteristic

C₂×C₇₃

#568

yes

characteristic

D₇₃

—

#2972

—

maximal, Sylow

C₇₃

#2972

yes

characteristic, complemented, Sylow

C₂×C₄×D₄

—

C₈×D₄

—

#431

—

C₂×C₂²⋊C₈

—

C₂²×D₄

—

C₂³×C₄

—

C₂²×C₈

—

C₄×D₄

—

C₄×D₄

—

C₂×C₄⋊C₄

#64408

yes

characteristic

C₂×C₂²⋊C₄

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

C₂⁴

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

292

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#64408

yes

—

C₂²×C₄

C₂×#64408

yes

characteristic

C₂×C₈

—

292

—

C₂×C₈

—

C₄⋊C₄

C₂×#64408

yes

—

100

C₂²⋊C₄

—

101

C₄²

—

102

C₄²

—

103

C₂³

—

292

—

104

C₂³

—

292

—

105

C₂³

—

106

C₂³

—

107

C₂³

C₂²×#64408

yes

characteristic

108

D₄

—

109

C₂×C₄

—

110

C₂×C₄

—

292

—

111

C₂×C₄

—

292

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

—

115

C₂×C₄

—

146

—

116

C₂×C₄

—

146

—

117

C₂×C₄

#11924036

yes

complemented

118

C₂×C₄

C₂²×#64408

yes

—

119

C₂×C₄

C₂²×#64408

yes

—

120

C₈

—

146

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

292

—

124

C₂²

—

125

C₂²

—

146

—

126

C₂²

—

146

—

127

C₂²

C₂×#11924036

yes

—

128

C₂²

C₂³×#64408

yes

characteristic

129

C₄

—

130

C₄

—

131

C₄

—

146

—

132

C₄

—

146

—

133

C₄

C₂×#11924036

yes

—

134

C₂

—

135

C₂

—

146

—

136

C₂

#17989654

yes

characteristic

137

C₂

C₂²×#11924036

yes

characteristic

138

C₂

#17989667

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#64408
commutator	C₂×C₇₃	C₂³×C₈
Fitting	C₂×C₄⋊C₄×C₇₃	C₄
socle	C₂²×C₇₃	C₂³×C₄
hypercenter	C₂×C₄⋊C₄	#64408
hypocenter	C₇₃	#2972
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

subgroup

G/H

center, Frattini

C₂²

C₂³×#64408

commutator

C₂×C₇₃

C₂³×C₈

Fitting

C₂×C₄⋊C₄×C₇₃

C₄

socle

C₂²×C₇₃

C₂³×C₄

C₂×C₄⋊C₄

C₇₃

G ▷ C₂×C₇₃ ▷ C₇₃

C₁ ◁ C₂² ◁ C₂×C₄⋊C₄

N	H ≅ G/N	type
1. #17989615	C₂	semidirect
2. #17989603	C₂	semidirect
3. #17989555	C₂	semidirect
5. C₂×#11924036	C₂²	semidirect
6. C₄⋊#10596833	C₂²	semidirect
7. #14556648	C₂²	semidirect
9. C₄×#10596833	C₂²	semidirect
20. C₂×D₂₉₂	C₈	semidirect
36. C₂²×D₇₃	C₂×C₈	semidirect
39. C₂×C₄×C₇₃	C₂×C₈	semidirect
47. D₂₉₂	C₂×C₈	semidirect
70. C₇₃	#2972	semidirect
117. C₂×C₄	#11924036	semidirect

H ≅ G/N

type

1. #17989615

C₂

semidirect

2. #17989603

C₂

semidirect

3. #17989555

C₂

semidirect

5. C₂×#11924036

C₂²

semidirect

6. C₄⋊#10596833

C₂²

semidirect

7. #14556648

C₂²

semidirect

9. C₄×#10596833

C₂²

semidirect

20. C₂×D₂₉₂

C₈

semidirect

36. C₂²×D₇₃

C₂×C₈

semidirect

39. C₂×C₄×C₇₃

C₂×C₈

semidirect

47. D₂₉₂

C₂×C₈

semidirect

70. C₇₃

#2972

semidirect

117. C₂×C₄

#11924036

semidirect

Subgroups — #22626781

Conjugacy classes of subgroups

Complements