Group #22626849 — subgroups

Subgroups — #22626849

Order 9344.

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

normal subgroups only

This group has 432 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	Q₈×#10596833	C₂	1	2	yes	maximal, complemented
2	#17989615	C₂	1	6	yes	maximal, complemented
3	#17989605	C₂	1	3	yes	maximal, complemented
4	#17989588	C₂	1	3	yes	maximal, complemented
5	C₄×#11924029	C₂	1	1	yes	maximal, characteristic
6	C₂×#11924036	C₂²	1	6	yes	—
7	#14556658	C₂²	1	3	yes	—
8	#14556658	C₂²	1	1	yes	characteristic
9	C₄⋊#10596833	C₂²	1	6	yes	—
10	#14556648	C₂²	1	6	yes	—
11	C₄×#10596833	C₂²	1	6	yes	—
12	C₂×#11924029	C₄	1	1	yes	characteristic
13	Q₈×Dic₇₃	C₂²	1	1	yes	characteristic
14	C₄×D₂₉₂	C₄	1	3	yes	—
15	#14556553	C₂²	1	3	yes	—
16	#14556549	C₄	1	3	yes	—
17	C₄²×D₇₃	C₂²	1	3	yes	—
18	C₄×Q₈×C₇₃	C₄	1	1	yes	characteristic
19	#11924036	—	4	6	no	—
20	C₂×#10596833	C₂³	1	6	yes	—
21	C₂×#10596833	C₂³	1	2	yes	—
22	#11924029	C₈	1	8	yes	complemented
23	C₂×D₂₉₂	C₂×C₄	1	3	yes	—
24	C₂×C₄×D₇₃	C₂×C₄	1	3	yes	—
25	C₂×C₄×D₇₃	C₂×C₄	1	3	yes	—
26	C₂×C₄×D₇₃	C₂³	1	3	yes	—
27	C₄⋊Dic₇₃	C₂×C₄	1	3	yes	—
28	#11924005	C₂×C₄	1	6	yes	—
29	C₄×Dic₇₃	C₂×C₄	1	3	yes	—
30	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
31	C₄×Dic₇₃	C₂³	1	3	yes	—
32	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₄⋊C₄×C₇₃	C₂×C₄	1	3	yes	—
34	C₄²×C₇₃	C₂×C₄	1	3	yes	—
35	C₂²×D₇₃	C₂²×C₄	1	3	yes	—
36	Q₈×C₇₃	C₂×C₈	1	4	yes	complemented
37	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
38	C₂×C₄×C₇₃	C₂²×C₄	1	3	yes	—
39	C₂×C₄×C₇₃	C₂²×C₄	1	3	yes	—
40	C₂×Dic₇₃	C₂²×C₄	1	3	yes	—
41	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
42	C₂×Dic₇₃	C₂⁴	1	1	yes	characteristic
43	D₂₉₂	C₂×C₈	1	12	yes	complemented
44	C₄×D₇₃	—	2	3	no	—
45	C₄×D₇₃	—	2	3	no	—
46	C₄×D₇₃	—	4	3	no	—
47	C₄×D₇₃	C₂×C₈	1	12	yes	complemented
48	#10596833	—	2	6	no	—
49	#10596833	—	2	2	no	—
50	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
51	C₂×D₇₃	—	2	3	no	—
52	C₂×D₇₃	—	2	3	no	—
53	C₂×D₇₃	C₂²×C₈	1	6	yes	—
54	C₄×C₇₃	—	2	3	no	—
55	C₄×C₇₃	—	2	1	no	—
56	C₄×C₇₃	C₂²×C₈	1	6	yes	—
57	Dic₇₃	—	2	3	no	—
58	Dic₇₃	C₂²×C₈	1	2	yes	—
59	Dic₇₃	H(5,𝔽₂)	1	1	yes	characteristic
60	Dic₇₃	2₋¹⁺⁴	1	1	yes	characteristic
61	C₂×C₇₃	#520	1	1	yes	characteristic
62	C₂×C₇₃	C₂³×C₈	1	1	yes	characteristic
63	C₂×C₇₃	#568	1	1	yes	characteristic
64	D₇₃	—	2	6	no	—
65	#2988	—	73	1	no	maximal, Sylow
66	C₇₃	#2988	1	1	yes	characteristic, complemented, Sylow
67	C₄×#41	—	73	1	no	—
68	C₈×Q₈	—	73	2	no	—
69	C₈×D₄	—	73	6	no	—
70	#431	—	73	3	no	—
71	C₂×C₄⋊C₈	—	73	3	no	—
72	C₂×#41	—	73	1	no	—
73	C₂²×C₈	—	73	6	no	—
74	C₄×Q₈	—	73	1	no	—
75	C₄×Q₈	#64408	1	1	yes	characteristic
76	C₄×D₄	—	73	3	no	—
77	C₄×D₄	—	73	3	no	—
78	#117	—	73	3	no	—
79	C₂×C₄²	—	73	3	no	—
80	C₄⋊C₈	—	73	3	no	—
81	C₄⋊C₈	—	73	6	no	—
82	C₄⋊C₈	—	73	1	no	—
83	C₂²⋊C₈	—	73	6	no	—
84	C₄×C₈	—	73	6	no	—
85	#41	—	73	8	no	—
86	C₂×Q₈	C₂×#64408	1	1	yes	characteristic
87	C₂×D₄	—	73	3	no	—
88	C₂²×C₄	—	73	3	no	—
89	C₂²×C₄	—	73	3	no	—
90	C₂²×C₄	—	73	3	no	—
91	C₂×C₈	—	292	6	no	—
92	C₂×C₈	—	73	2	no	—
93	C₂×C₈	—	73	6	no	—
94	C₄⋊C₄	—	73	3	no	—
95	C₄⋊C₄	C₂×#64408	1	3	yes	—
96	C₂²⋊C₄	—	73	6	no	—
97	C₄²	—	73	3	no	—
98	C₄²	—	73	3	no	—
99	C₄²	—	73	1	no	—
100	C₄²	C₂×#64408	1	3	yes	—
101	C₂³	—	73	3	no	—
102	Q₈	#11924036	1	4	yes	complemented
103	D₄	—	73	12	no	—
104	C₂×C₄	—	292	3	no	—
105	C₂×C₄	—	73	1	no	—
106	C₂×C₄	—	73	1	no	—
107	C₂×C₄	—	73	12	no	—
108	C₂×C₄	—	73	3	no	—
109	C₂×C₄	—	146	3	no	—
110	C₂×C₄	—	146	3	no	—
111	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
112	C₂×C₄	C₂²×#64408	1	3	yes	—
113	C₂×C₄	C₂²×#64408	1	3	yes	—
114	C₈	—	146	6	no	—
115	C₈	—	146	2	no	—
116	C₂²	—	73	6	no	—
117	C₂²	—	146	3	no	—
118	C₂²	—	146	3	no	—
119	C₂²	C₂³×#64408	1	1	yes	characteristic
120	C₄	—	2	3	no	—
121	C₄	—	2	1	no	—
122	C₄	—	73	1	no	—
123	C₄	—	73	1	no	—
124	C₄	—	73	2	no	—
125	C₄	—	146	3	no	—
126	C₄	C₂×#11924036	1	6	yes	—
127	C₂	—	146	6	no	—
128	C₂	#17989656	1	1	yes	characteristic
129	C₂	C₂²×#11924036	1	1	yes	characteristic
130	C₂	#17989667	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#10596833

C₂

yes

maximal, complemented

#17989615

C₂

yes

maximal, complemented

#17989605

C₂

yes

maximal, complemented

#17989588

C₂

yes

maximal, complemented

C₄×#11924029

C₂

yes

maximal, characteristic

C₂×#11924036

C₂²

yes

—

#14556658

C₂²

yes

—

#14556658

C₂²

yes

characteristic

C₄⋊#10596833

C₂²

yes

—

#14556648

C₂²

yes

—

C₄×#10596833

C₂²

yes

—

C₂×#11924029

C₄

yes

characteristic

Q₈×Dic₇₃

C₂²

yes

characteristic

C₄×D₂₉₂

C₄

yes

—

#14556553

C₂²

yes

—

#14556549

C₄

yes

—

C₄²×D₇₃

C₂²

yes

—

C₄×Q₈×C₇₃

C₄

yes

characteristic

#11924036

—

C₂×#10596833

C₂³

yes

—

C₂×#10596833

C₂³

yes

—

#11924029

C₈

yes

complemented

C₂×D₂₉₂

C₂×C₄

yes

—

C₂×C₄×D₇₃

C₂×C₄

yes

—

C₂×C₄×D₇₃

C₂×C₄

yes

—

C₂×C₄×D₇₃

C₂³

yes

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

#11924005

C₂×C₄

yes

—

C₄×Dic₇₃

C₂×C₄

yes

—

C₄×Dic₇₃

C₂³

yes

characteristic

C₄×Dic₇₃

C₂³

yes

—

C₂×Q₈×C₇₃

C₂×C₄

yes

characteristic

C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₄²×C₇₃

C₂×C₄

yes

—

C₂²×D₇₃

C₂²×C₄

yes

—

Q₈×C₇₃

C₂×C₈

yes

complemented

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂×Dic₇₃

C₂²×C₄

yes

—

C₂×Dic₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

C₂⁴

yes

characteristic

D₂₉₂

C₂×C₈

yes

complemented

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

C₂×C₈

yes

complemented

#10596833

—

#10596833

—

C₂²×C₇₃

C₂³×C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₂²×C₈

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂²×C₈

yes

—

Dic₇₃

—

Dic₇₃

C₂²×C₈

yes

—

Dic₇₃

H(5,𝔽₂)

yes

characteristic

Dic₇₃

2₋¹⁺⁴

yes

characteristic

C₂×C₇₃

#520

yes

characteristic

C₂×C₇₃

C₂³×C₈

yes

characteristic

C₂×C₇₃

#568

yes

characteristic

D₇₃

—

#2988

—

maximal, Sylow

C₇₃

#2988

yes

characteristic, complemented, Sylow

C₄×#41

—

C₈×Q₈

—

C₈×D₄

—

#431

—

C₂×C₄⋊C₈

—

C₂×#41

—

C₂²×C₈

—

C₄×Q₈

—

C₄×Q₈

#64408

yes

characteristic

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

—

C₂²⋊C₈

—

C₄×C₈

—

#41

—

C₂×Q₈

C₂×#64408

yes

characteristic

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂×C₈

—

292

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

C₂×#64408

yes

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

100

C₄²

C₂×#64408

yes

—

101

C₂³

—

102

Q₈

#11924036

yes

complemented

103

D₄

—

104

C₂×C₄

—

292

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

146

—

110

C₂×C₄

—

146

—

111

C₂×C₄

C₂²×#64408

yes

characteristic

112

C₂×C₄

C₂²×#64408

yes

—

113

C₂×C₄

C₂²×#64408

yes

—

114

C₈

—

146

—

115

C₈

—

146

—

116

C₂²

—

117

C₂²

—

146

—

118

C₂²

—

146

—

119

C₂²

C₂³×#64408

yes

characteristic

120

C₄

—

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

—

146

—

126

C₄

C₂×#11924036

yes

—

127

C₂

—

146

—

128

C₂

#17989656

yes

characteristic

129

C₂

C₂²×#11924036

yes

characteristic

130

C₂

#17989667

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#64408
commutator	C₂×C₇₃	C₂³×C₈
Fitting	C₄×Q₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂³×C₄
hypercenter	C₄×Q₈	#64408
hypocenter	C₇₃	#2988
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄×Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#64408

commutator

C₂×C₇₃

C₂³×C₈

Fitting

C₄×Q₈×C₇₃

C₄

socle

C₂²×C₇₃

C₂³×C₄

C₄×Q₈

C₇₃

G ▷ C₂×C₇₃ ▷ C₇₃

C₁ ◁ C₂² ◁ C₄×Q₈

N	H ≅ G/N	type
1. Q₈×#10596833	C₂	semidirect
2. #17989615	C₂	semidirect
3. #17989605	C₂	semidirect
4. #17989588	C₂	semidirect
22. #11924029	C₈	semidirect
36. Q₈×C₇₃	C₂×C₈	semidirect
43. D₂₉₂	C₂×C₈	semidirect
47. C₄×D₇₃	C₂×C₈	semidirect
66. C₇₃	#2988	semidirect
102. Q₈	#11924036	semidirect

H ≅ G/N

type

1. Q₈×#10596833

C₂

semidirect

2. #17989615

C₂

semidirect

3. #17989605

C₂

semidirect

4. #17989588

C₂

semidirect

22. #11924029

C₈

semidirect

36. Q₈×C₇₃

C₂×C₈

semidirect

43. D₂₉₂

C₂×C₈

semidirect

47. C₄×D₇₃

C₂×C₈

semidirect

66. C₇₃

#2988

semidirect

102. Q₈

#11924036

semidirect

Subgroups — #22626849

Conjugacy classes of subgroups

Complements