Group #22626155 — subgroups

Subgroups — C₂×C₂²⋊#11924031 #22626155

Order 9344.

272 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 122 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	C₂²×#11924031	C₂	1	2	yes	maximal, complemented
2	C₂²⋊#11924031	C₂	1	4	yes	maximal, direct product factor
3	C₂³×#10596831	C₂	1	1	yes	maximal, characteristic
4	C₂×#11924031	—	2	4	no	—
5	C₂×#11924031	C₂²	1	4	yes	complemented
6	C₂²×#10596831	—	2	4	no	—
7	C₂²×#10596831	C₄	1	2	yes	—
8	C₂²×#10596831	C₄	1	1	yes	characteristic
9	C₂²×#10596831	C₂²	1	2	yes	—
10	C₂²×#10596831	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₇₃	C₄	1	1	yes	characteristic
12	#11924031	—	2	4	no	—
13	C₂×#10596831	—	2	2	no	—
14	C₂×#10596831	—	2	4	no	—
15	C₂×#10596831	—	2	4	no	—
16	C₂×#10596831	C₂×C₄	1	2	yes	—
17	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
18	C₂×#10596831	D₄	1	4	yes	—
19	C₂×#10596831	C₂³	1	1	yes	characteristic
20	C₂⁴×C₇₃	C₈	1	1	yes	characteristic
21	C₂²×C₄×C₇₃	—	2	2	no	—
22	C₂²×C₄×C₇₃	—	2	2	no	—
23	C₂²×C₄×C₇₃	C₈	1	1	yes	characteristic
24	C₂²×C₄×C₇₃	C₈	1	2	yes	—
25	C₂²×C₄×C₇₃	C₂×C₄	1	2	yes	—
26	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
27	C₂³×C₇₃	—	2	2	no	—
28	C₂³×C₇₃	—	2	2	no	—
29	C₂³×C₇₃	C₁₆	1	4	yes	complemented
30	C₂³×C₇₃	C₂×C₈	1	2	yes	—
31	C₂³×C₇₃	C₂×C₈	1	1	yes	characteristic
32	C₂×C₄×C₇₃	—	2	2	no	—
33	C₂×C₄×C₇₃	—	2	4	no	—
34	C₂×C₄×C₇₃	—	2	2	no	—
35	C₂×C₄×C₇₃	—	2	2	no	—
36	C₂×C₄×C₇₃	C₂²⋊C₄	1	2	yes	—
37	C₂×C₄×C₇₃	C₂×C₈	1	2	yes	—
38	C₂×C₄×C₇₃	C₂×C₈	1	1	yes	characteristic
39	C₂×C₄×C₇₃	M₁₆	1	2	yes	—
40	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
41	#10596831	—	2	2	no	—
42	#10596831	C₂²⋊C₄	1	2	yes	—
43	#10596831	C₂×D₄	1	2	yes	—
44	C₂²×C₇₃	—	2	8	no	—
45	C₂²×C₇₃	—	2	2	no	—
46	C₂²×C₇₃	—	2	2	no	—
47	C₂²×C₇₃	C₂²⋊C₈	1	2	yes	—
48	C₂²×C₇₃	C₂×C₁₆	1	2	yes	—
49	C₂²×C₇₃	C₂×C₁₆	1	4	yes	complemented
50	C₂²×C₇₃	M₃₂	1	2	yes	—
51	C₂²×C₇₃	C₂²×C₈	1	1	yes	characteristic
52	C₄×C₇₃	—	2	2	no	—
53	C₄×C₇₃	C₂²⋊C₈	1	2	yes	—
54	C₄×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₄×C₇₃	C₂×M₁₆	1	1	yes	characteristic
56	C₂×C₇₃	—	2	4	no	—
57	C₂×C₇₃	C₂²⋊C₁₆	1	4	yes	complemented
58	C₂×C₇₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
59	C₂×C₇₃	C₂²×C₁₆	1	1	yes	characteristic
60	C₂×C₇₃	C₂×M₃₂	1	1	yes	characteristic
61	C₂×C₂²⋊C₁₆	—	73	1	no	maximal, Sylow
62	C₇₃	C₂×C₂²⋊C₁₆	1	1	yes	characteristic, complemented, Sylow
63	C₂³×C₈	—	73	1	no	—
64	C₂²×C₁₆	—	73	2	no	—
65	C₂²⋊C₁₆	—	73	4	no	—
66	C₂³×C₄	#64408	1	1	yes	characteristic
67	C₂²×C₈	—	73	1	no	—
68	C₂²×C₈	—	73	2	no	—
69	C₂²×C₈	—	73	2	no	—
70	C₂²×C₈	—	73	1	no	—
71	C₂²×C₈	—	146	4	no	—
72	C₂×C₁₆	—	73	4	no	—
73	C₂×C₁₆	—	146	4	no	—
74	C₂⁴	#10596833	1	1	yes	characteristic
75	C₂²×C₄	—	2	2	no	—
76	C₂²×C₄	—	2	2	no	—
77	C₂²×C₄	#10596833	1	1	yes	characteristic
78	C₂²×C₄	#10596833	1	2	yes	—
79	C₂²×C₄	C₂×#64408	1	2	yes	—
80	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
81	C₂×C₈	—	73	4	no	—
82	C₂×C₈	—	73	1	no	—
83	C₂×C₈	—	73	1	no	—
84	C₂×C₈	—	73	2	no	—
85	C₂×C₈	—	146	4	no	—
86	C₂×C₈	—	146	4	no	—
87	C₂×C₈	—	146	2	no	—
88	C₁₆	—	146	4	no	—
89	C₂³	—	2	2	no	—
90	C₂³	—	2	2	no	—
91	C₂³	#11924031	1	4	yes	complemented
92	C₂³	C₂×#10596833	1	2	yes	—
93	C₂³	C₂×#10596833	1	1	yes	characteristic
94	C₂×C₄	—	2	2	no	—
95	C₂×C₄	—	2	2	no	—
96	C₂×C₄	—	2	4	no	—
97	C₂×C₄	—	2	2	no	—
98	C₂×C₄	C₂²⋊#64408	1	2	yes	—
99	C₂×C₄	C₂×#10596833	1	2	yes	—
100	C₂×C₄	C₂×#10596833	1	1	yes	characteristic
101	C₂×C₄	#11924037	1	2	yes	—
102	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
103	C₈	—	73	2	no	—
104	C₈	—	73	2	no	—
105	C₈	—	146	2	no	—
106	C₂²	—	2	2	no	—
107	C₂²	—	2	8	no	—
108	C₂²	—	2	2	no	—
109	C₂²	C₂²⋊#10596833	1	2	yes	—
110	C₂²	C₂×#11924031	1	2	yes	—
111	C₂²	C₂×#11924031	1	4	yes	complemented
112	C₂²	#14556665	1	2	yes	—
113	C₂²	C₂²×#10596833	1	1	yes	characteristic
114	C₄	—	2	2	no	—
115	C₄	C₂²⋊#10596833	1	2	yes	—
116	C₄	C₂×C₂²⋊#64408	1	1	yes	characteristic
117	C₄	C₂×#11924037	1	1	yes	characteristic
118	C₂	—	2	4	no	—
119	C₂	C₂²⋊#11924031	1	4	yes	direct product factor
120	C₂	C₂×C₂²⋊#10596833	1	1	yes	characteristic
121	C₂	C₂²×#11924031	1	1	yes	characteristic
122	C₂	C₂×#14556665	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×#11924031

C₂

yes

maximal, complemented

C₂²⋊#11924031

C₂

yes

maximal, direct product factor

C₂³×#10596831

C₂

yes

maximal, characteristic

C₂×#11924031

—

C₂×#11924031

C₂²

yes

complemented

C₂²×#10596831

—

C₂²×#10596831

C₄

yes

—

C₂²×#10596831

C₄

yes

characteristic

C₂²×#10596831

C₂²

yes

—

C₂²×#10596831

C₂²

yes

characteristic

C₂³×C₄×C₇₃

C₄

yes

—

—

—

—

C₂×C₄

yes

—

C₂×#10596831

C₂×C₄

yes

characteristic

C₂×#10596831

D₄

yes

—

C₂×#10596831

C₂³

yes

characteristic

C₂⁴×C₇₃

C₈

yes

characteristic

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₈

yes

characteristic

C₂²×C₄×C₇₃

C₈

yes

—

C₂²×C₄×C₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂³×C₇₃

—

C₂³×C₇₃

—

C₂³×C₇₃

C₁₆

yes

complemented

C₂³×C₇₃

C₂×C₈

yes

—

C₂³×C₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

—

C₂×C₄×C₇₃

C₂×C₈

yes

—

C₂×C₄×C₇₃

C₂×C₈

yes

characteristic

C₂×C₄×C₇₃

M₁₆

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

#10596831

—

#10596831

C₂²⋊C₄

yes

—

#10596831

C₂×D₄

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂²⋊C₈

yes

—

C₂²×C₇₃

C₂×C₁₆

yes

—

C₂²×C₇₃

C₂×C₁₆

yes

complemented

C₂²×C₇₃

M₃₂

yes

—

C₂²×C₇₃

C₂²×C₈

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₂²⋊C₈

yes

—

C₄×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₃

C₂×M₁₆

yes

characteristic

C₂×C₇₃

—

C₂×C₇₃

C₂²⋊C₁₆

yes

complemented

C₂×C₇₃

C₂×C₂²⋊C₈

yes

characteristic

C₂×C₇₃

C₂²×C₁₆

yes

characteristic

C₂×C₇₃

C₂×M₃₂

yes

characteristic

C₂×C₂²⋊C₁₆

—

maximal, Sylow

C₇₃

C₂×C₂²⋊C₁₆

yes

characteristic, complemented, Sylow

C₂³×C₈

—

C₂²×C₁₆

—

C₂²⋊C₁₆

—

C₂³×C₄

#64408

yes

characteristic

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

—

146

—

C₂×C₁₆

—

C₂×C₁₆

—

146

—

C₂⁴

#10596833

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

#10596833

yes

characteristic

C₂²×C₄

#10596833

yes

—

C₂²×C₄

C₂×#64408

yes

—

C₂²×C₄

C₂×#64408

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₁₆

—

146

—

C₂³

—

C₂³

—

C₂³

#11924031

yes

complemented

C₂³

C₂×#10596833

yes

—

C₂³

C₂×#10596833

yes

characteristic

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

C₂²⋊#64408

yes

—

C₂×C₄

C₂×#10596833

yes

—

100

C₂×C₄

C₂×#10596833

yes

characteristic

101

C₂×C₄

#11924037

yes

—

102

C₂×C₄

C₂²×#64408

yes

characteristic

103

C₈

—

104

C₈

—

105

C₈

—

146

—

106

C₂²

—

107

C₂²

—

108

C₂²

—

109

C₂²

C₂²⋊#10596833

yes

—

110

C₂²

C₂×#11924031

yes

—

111

C₂²

C₂×#11924031

yes

complemented

112

C₂²

#14556665

yes

—

113

C₂²

C₂²×#10596833

yes

characteristic

114

C₄

—

115

C₄

C₂²⋊#10596833

yes

—

116

C₄

C₂×C₂²⋊#64408

yes

characteristic

117

C₄

C₂×#11924037

yes

characteristic

118

C₂

—

119

C₂

C₂²⋊#11924031

yes

direct product factor

120

C₂

C₂×C₂²⋊#10596833

yes

characteristic

121

C₂

C₂²×#11924031

yes

characteristic

122

C₂

C₂×#14556665

yes

characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×#64408
commutator	C₂×C₇₃	C₂²×C₁₆
Frattini	C₂×C₄	C₂²×#64408
Fitting	C₂³×C₄×C₇₃	C₄
socle	C₂³×C₇₃	C₂×C₈
hypercenter	C₂³×C₄	#64408
hypocenter	C₇₃	C₂×C₂²⋊C₁₆
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

subgroup

G/H

center

C₂²×C₄

C₂×#64408

commutator

C₂×C₇₃

C₂²×C₁₆

Frattini

C₂×C₄

C₂²×#64408

Fitting

C₂³×C₄×C₇₃

C₄

socle

C₂³×C₇₃

C₂×C₈

hypercenter

C₂³×C₄

#64408

hypocenter

C₇₃

C₂×C₂²⋊C₁₆

LCS

G ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂²×C₄ ◁ C₂³×C₄

N	H ≅ G/N	type
1. C₂²×#11924031	C₂	semidirect
2. C₂²⋊#11924031	C₂	direct (N × H)
5. C₂×#11924031	C₂²	semidirect
29. C₂³×C₇₃	C₁₆	semidirect
49. C₂²×C₇₃	C₂×C₁₆	semidirect
57. C₂×C₇₃	C₂²⋊C₁₆	semidirect
62. C₇₃	C₂×C₂²⋊C₁₆	semidirect
91. C₂³	#11924031	semidirect
111. C₂²	C₂×#11924031	semidirect
119. C₂	C₂²⋊#11924031	direct (N × H)

H ≅ G/N

type

1. C₂²×#11924031

C₂

semidirect

2. C₂²⋊#11924031

C₂

direct (N × H)

5. C₂×#11924031

C₂²

semidirect

29. C₂³×C₇₃

C₁₆

semidirect

49. C₂²×C₇₃

C₂×C₁₆

semidirect

57. C₂×C₇₃

C₂²⋊C₁₆

semidirect

62. C₇₃

C₂×C₂²⋊C₁₆

semidirect

91. C₂³

#11924031

semidirect

111. C₂²

C₂×#11924031

semidirect

119. C₂

C₂²⋊#11924031

direct (N × H)

Subgroups — C₂×C₂²⋊#11924031 #22626155

Conjugacy classes of subgroups

Complements

Subgroups — C2×C22⋊#11924031 #22626155

Conjugacy classes of subgroups

Complements

Subgroups — C₂×C₂²⋊#11924031 #22626155