Group #22626925 — subgroups

Subgroups — C₂×#17989667 #22626925

Order 9344.

1460 conjugacy classes of subgroups; only the 754 normal subgroups are shown.

normal subgroups only

This group has 1460 conjugacy classes of subgroups — too many to draw. Only its 754 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	#17989667	C₂	1	16	yes	maximal, direct product factor
2	C₂×#14556679	C₂	1	8	yes	maximal, complemented
3	C₂×D₇₃⋊M₁₆	C₂	1	6	yes	maximal, complemented
4	C₂²×#11924029	C₂	1	1	yes	maximal, characteristic
5	#14556679	C₂²	1	64	yes	complemented
6	C₂×#11924038	C₂²	1	12	yes	complemented
7	D₇₃⋊M₁₆	C₂²	1	48	yes	complemented
8	C₂×#11924037	C₂²	1	4	yes	—
9	C₂×#11924036	C₂²	1	12	yes	complemented
10	C₂×#11924029	C₄	1	4	yes	—
11	C₂×#11924029	C₄	1	8	yes	—
12	C₂×#11924029	C₂²	1	4	yes	—
13	C₂×#11924029	C₂²	1	8	yes	—
14	C₂²×D₂₉₂	C₄	1	3	yes	—
15	C₂²×C₄×D₇₃	C₂²	1	3	yes	—
16	C₂²×Q₈×C₇₃	C₄	1	1	yes	characteristic
17	#11924038	C₂³	1	48	yes	complemented
18	#11924037	C₂³	1	16	yes	—
19	#11924036	C₂³	1	48	yes	complemented
20	C₂×#10596833	C₂³	1	8	yes	—
21	C₂³×D₇₃	C₂×C₄	1	3	yes	—
22	#11924029	C₂×C₄	1	16	yes	—
23	#11924029	C₂×C₄	1	32	yes	—
24	#11924029	C₂³	1	16	yes	—
25	C₂×D₂₉₂	C₂×C₄	1	12	yes	—
26	C₂×D₂₉₂	C₂×C₄	1	24	yes	—
27	C₂²×Dic₇₃	C₂³	1	1	yes	characteristic
28	C₂×C₄×D₇₃	C₂×C₄	1	6	yes	—
29	C₂×C₄×D₇₃	C₂×C₄	1	12	yes	—
30	C₂×C₄×D₇₃	C₂³	1	6	yes	—
31	C₂×C₄×D₇₃	C₂³	1	12	yes	—
32	C₂×Q₈×C₇₃	C₂×C₄	1	4	yes	—
33	C₂×Q₈×C₇₃	C₂×C₄	1	8	yes	—
34	C₂²×C₄×C₇₃	C₂×C₄	1	3	yes	—
35	C₂³×C₇₃	C₂²×C₄	1	1	yes	characteristic
36	C₂²×D₇₃	—	4	6	no	—
37	C₂²×D₇₃	C₂²×C₄	1	6	yes	—
38	C₂²×D₇₃	C₂²×C₄	1	12	yes	—
39	Q₈×C₇₃	C₂²×C₄	1	16	yes	—
40	C₂×C₄×C₇₃	C₂²×C₄	1	12	yes	—
41	C₂×C₄×C₇₃	C₂²×C₄	1	6	yes	—
42	C₂×Dic₇₃	C₂²×C₄	1	2	yes	—
43	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×Dic₇₃	C₂⁴	1	1	yes	characteristic
45	C₂×Dic₇₃	C₂⁴	1	2	yes	—
46	D₂₉₂	C₂²×C₄	1	48	yes	—
47	C₄×D₇₃	C₂²×C₄	1	12	yes	—
48	C₄×D₇₃	C₂²×C₄	1	24	yes	—
49	C₄×D₇₃	C₂⁴	1	12	yes	—
50	#10596833	C₂⁴	1	16	yes	—
51	C₂²×C₇₃	—	2	2	no	—
52	C₂²×C₇₃	C₂³×C₄	1	2	yes	—
53	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
54	C₂×D₇₃	—	2	12	no	—
55	C₂×D₇₃	—	4	12	no	—
56	C₂×D₇₃	C₂³×C₄	1	12	yes	—
57	C₄×C₇₃	C₂³×C₄	1	12	yes	—
58	Dic₇₃	C₂³×C₄	1	1	yes	characteristic
59	Dic₇₃	C₂³×C₄	1	2	yes	—
60	Dic₇₃	C₂⁵	1	1	yes	characteristic
61	C₂×C₇₃	—	2	2	no	—
62	C₂×C₇₃	#568	1	2	yes	complemented
63	C₂×C₇₃	C₂⁴×C₄	1	1	yes	characteristic
64	D₇₃	—	2	12	no	—
65	C₂×#568	—	73	1	no	maximal, Sylow
66	C₇₃	C₂×#568	1	1	yes	characteristic, complemented, Sylow
67	C₂²×#41	—	73	1	no	—
68	#568	—	73	16	no	—
69	C₂×#131	—	73	8	no	—
70	C₂²×M₁₆	—	73	6	no	—
71	C₂×#41	—	73	4	no	—
72	C₂×#41	—	73	8	no	—
73	C₂×#41	—	73	8	no	—
74	C₂×#41	—	73	4	no	—
75	C₂²×Q₈	#64408	1	1	yes	characteristic
76	C₂²×D₄	—	73	3	no	—
77	C₂³×C₄	—	73	3	no	—
78	#131	—	73	64	no	—
79	C₂×M₁₆	—	73	4	no	—
80	C₂×M₁₆	—	73	48	no	—
81	C₂×M₁₆	—	73	12	no	—
82	C₂²×C₈	—	73	12	no	—
83	C₂⁴	—	73	3	no	—
84	#41	—	73	16	no	—
85	#41	—	73	32	no	—
86	#41	—	73	16	no	—
87	C₂×Q₈	C₂×#64408	1	8	yes	—
88	C₂×Q₈	C₂×#64408	1	4	yes	—
89	C₂×D₄	—	73	24	no	—
90	C₂×D₄	—	73	12	no	—
91	C₂²×C₄	—	73	1	no	—
92	C₂²×C₄	—	73	12	no	—
93	C₂²×C₄	—	73	6	no	—
94	C₂²×C₄	—	73	6	no	—
95	C₂²×C₄	—	73	12	no	—
96	C₂²×C₄	C₂×#64408	1	3	yes	—
97	M₁₆	—	73	16	no	—
98	M₁₆	—	73	48	no	—
99	C₂×C₈	—	73	8	no	—
100	C₂×C₈	—	73	48	no	—
101	C₂³	—	292	6	no	—
102	C₂³	—	73	12	no	—
103	C₂³	—	73	6	no	—
104	C₂³	C₂²×#64408	1	1	yes	characteristic
105	Q₈	C₂²×#64408	1	16	yes	—
106	D₄	—	73	48	no	—
107	C₂×C₄	—	73	2	no	—
108	C₂×C₄	—	73	2	no	—
109	C₂×C₄	—	73	1	no	—
110	C₂×C₄	—	73	1	no	—
111	C₂×C₄	—	73	12	no	—
112	C₂×C₄	—	73	12	no	—
113	C₂×C₄	—	73	24	no	—
114	C₂×C₄	C₂²×#64408	1	12	yes	—
115	C₂×C₄	C₂²×#64408	1	6	yes	—
116	C₈	—	73	16	no	—
117	C₂²	—	2	2	no	—
118	C₂²	—	292	12	no	—
119	C₂²	—	73	12	no	—
120	C₂²	—	146	12	no	—
121	C₂²	C₂³×#64408	1	2	yes	—
122	C₂²	C₂³×#64408	1	1	yes	characteristic
123	C₄	—	73	1	no	—
124	C₄	—	73	2	no	—
125	C₄	—	73	1	no	—
126	C₄	C₂³×#64408	1	12	yes	—
127	C₂	—	2	2	no	—
128	C₂	—	146	12	no	—
129	C₂	#17989667	1	2	yes	direct product factor
130	C₂	C₂⁴×#64408	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989667

C₂

yes

maximal, direct product factor

C₂×#14556679

C₂

yes

maximal, complemented

C₂×D₇₃⋊M₁₆

C₂

yes

maximal, complemented

C₂²×#11924029

C₂

yes

maximal, characteristic

#14556679

C₂²

yes

complemented

C₂×#11924038

C₂²

yes

complemented

D₇₃⋊M₁₆

C₂²

yes

complemented

C₂×#11924037

C₂²

yes

—

C₂×#11924036

C₂²

yes

complemented

C₂×#11924029

C₄

yes

—

C₂×#11924029

C₄

yes

—

C₂×#11924029

C₂²

yes

—

C₂×#11924029

C₂²

yes

—

C₂²×D₂₉₂

C₄

yes

—

C₂²×C₄×D₇₃

C₂²

yes

—

C₂²×Q₈×C₇₃

C₄

yes

characteristic

#11924038

C₂³

yes

complemented

#11924037

C₂³

yes

—

#11924036

C₂³

yes

complemented

C₂×#10596833

C₂³

yes

—

C₂³×D₇₃

C₂×C₄

yes

—

#11924029

C₂×C₄

yes

—

#11924029

C₂×C₄

yes

—

#11924029

C₂³

yes

—

C₂×D₂₉₂

C₂×C₄

yes

—

C₂×D₂₉₂

C₂×C₄

yes

—

C₂²×Dic₇₃

C₂³

yes

characteristic

C₂×C₄×D₇₃

C₂×C₄

yes

—

C₂×C₄×D₇₃

C₂×C₄

yes

—

C₂×C₄×D₇₃

C₂³

yes

—

C₂×C₄×D₇₃

C₂³

yes

—

C₂×Q₈×C₇₃

C₂×C₄

yes

—

C₂×Q₈×C₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

C₂×C₄

yes

—

C₂³×C₇₃

C₂²×C₄

yes

characteristic

C₂²×D₇₃

—

C₂²×D₇₃

C₂²×C₄

yes

—

C₂²×D₇₃

C₂²×C₄

yes

—

Q₈×C₇₃

C₂²×C₄

yes

—

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

C₂×Dic₇₃

C₂⁴

yes

—

D₂₉₂

C₂²×C₄

yes

—

C₄×D₇₃

C₂²×C₄

yes

—

C₄×D₇₃

C₂²×C₄

yes

—

C₄×D₇₃

C₂⁴

yes

—

#10596833

C₂⁴

yes

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂³×C₄

yes

—

C₂²×C₇₃

C₂³×C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

C₂³×C₄

yes

—

C₄×C₇₃

C₂³×C₄

yes

—

Dic₇₃

C₂³×C₄

yes

characteristic

Dic₇₃

C₂³×C₄

yes

—

Dic₇₃

C₂⁵

yes

characteristic

C₂×C₇₃

—

C₂×C₇₃

#568

yes

complemented

C₂×C₇₃

C₂⁴×C₄

yes

characteristic

D₇₃

—

C₂×#568

—

maximal, Sylow

C₇₃

C₂×#568

yes

characteristic, complemented, Sylow

C₂²×#41

—

#568

—

C₂×#131

—

C₂²×M₁₆

—

C₂×#41

—

C₂×#41

—

C₂×#41

—

C₂×#41

—

C₂²×Q₈

#64408

yes

characteristic

C₂²×D₄

—

C₂³×C₄

—

#131

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

—

C₂⁴

—

#41

—

#41

—

#41

—

C₂×Q₈

C₂×#64408

yes

—

C₂×Q₈

C₂×#64408

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#64408

yes

—

M₁₆

—

M₁₆

—

C₂×C₈

—

100

C₂×C₈

—

101

C₂³

—

292

—

102

C₂³

—

103

C₂³

—

104

C₂³

C₂²×#64408

yes

characteristic

105

Q₈

C₂²×#64408

yes

—

106

D₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂²×#64408

yes

—

115

C₂×C₄

C₂²×#64408

yes

—

116

C₈

—

117

C₂²

—

118

C₂²

—

292

—

119

C₂²

—

120

C₂²

—

146

—

121

C₂²

C₂³×#64408

yes

—

122

C₂²

C₂³×#64408

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

C₂³×#64408

yes

—

127

C₂

—

128

C₂

—

146

—

129

C₂

#17989667

yes

direct product factor

130

C₂

C₂⁴×#64408

yes

characteristic

	subgroup	G/H
center	C₂²	C₂³×#64408
commutator	C₂×C₇₃	C₂⁴×C₄
Frattini	C₂	C₂⁴×#64408
Fitting	C₂²×Q₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂³×C₄
hypercenter	C₂²×Q₈	#64408
hypocenter	C₇₃	C₂×#568
LCS	G ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×Q₈

subgroup

G/H

center

C₂²

C₂³×#64408

commutator

C₂×C₇₃

C₂⁴×C₄

Frattini

C₂

C₂⁴×#64408

Fitting

C₂²×Q₈×C₇₃

C₄

socle

C₂²×C₇₃

C₂³×C₄

C₂²×Q₈

C₇₃

G ▷ C₂×C₇₃ ▷ C₇₃

C₁ ◁ C₂² ◁ C₂²×Q₈

N	H ≅ G/N	type
1. #17989667	C₂	direct (N × H)
2. C₂×#14556679	C₂	semidirect
3. C₂×D₇₃⋊M₁₆	C₂	semidirect
5. #14556679	C₂²	semidirect
6. C₂×#11924038	C₂²	semidirect
7. D₇₃⋊M₁₆	C₂²	semidirect
9. C₂×#11924036	C₂²	semidirect
17. #11924038	C₂³	semidirect
19. #11924036	C₂³	semidirect
62. C₂×C₇₃	#568	semidirect
66. C₇₃	C₂×#568	semidirect
129. C₂	#17989667	direct (N × H)

H ≅ G/N

type

1. #17989667

C₂

direct (N × H)

2. C₂×#14556679

C₂

semidirect

3. C₂×D₇₃⋊M₁₆

C₂

semidirect

5. #14556679

C₂²

semidirect

6. C₂×#11924038

C₂²

semidirect

7. D₇₃⋊M₁₆

C₂²

semidirect

9. C₂×#11924036

C₂²

semidirect

17. #11924038

C₂³

semidirect

19. #11924036

C₂³

semidirect

62. C₂×C₇₃

#568

semidirect

66. C₇₃

C₂×#568

semidirect

129. C₂

#17989667

direct (N × H)

Subgroups — C₂×#17989667 #22626925

Conjugacy classes of subgroups

Complements

Subgroups — C2×#17989667 #22626925

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#17989667 #22626925