Group #22626777 — subgroups

Subgroups — #22626777

Order 9344.

402 conjugacy classes of subgroups; only the 168 normal subgroups are shown.

normal subgroups only

This group has 402 conjugacy classes of subgroups — too many to draw. Only its 168 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	#17989622	C₂	1	6	yes	maximal, complemented
3	#17989608	C₂	1	3	yes	maximal, complemented
4	C₄×#11924037	C₂	1	3	yes	maximal, complemented
5	#41×Dic₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11924037	C₂²	1	6	yes	—
7	C₄⋊#10596833	C₂²	1	6	yes	—
8	C₂²⋊#10596833	C₂²	1	6	yes	—
9	#14556646	C₂²	1	1	yes	characteristic
10	#14556646	C₂²	1	3	yes	—
11	C₄×#10596833	C₂²	1	6	yes	—
12	Q₈×Dic₇₃	C₄	1	1	yes	characteristic
13	Q₈×Dic₇₃	C₂²	1	1	yes	characteristic
14	D₄×Dic₇₃	C₄	1	3	yes	—
15	D₄×Dic₇₃	C₂²	1	3	yes	—
16	#14556550	C₄	1	3	yes	—
17	C₂×C₄×Dic₇₃	C₂²	1	3	yes	—
18	C₂×#41×C₇₃	C₄	1	1	yes	characteristic
19	#11924037	—	2	12	no	—
20	C₂×#10596833	C₂³	1	6	yes	—
21	C₂×#10596833	C₂³	1	2	yes	—
22	C₂²×Dic₇₃	C₂×C₄	1	3	yes	—
23	C₂²×Dic₇₃	C₂³	1	3	yes	—
24	C₄⋊Dic₇₃	C₂×C₄	1	3	yes	—
25	C₄⋊Dic₇₃	C₂×C₄	1	3	yes	—
26	C₂²⋊Dic₇₃	C₂×C₄	1	6	yes	—
27	C₄×Dic₇₃	C₂×C₄	1	3	yes	—
28	C₄×Dic₇₃	C₂×C₄	1	3	yes	—
29	C₄×Dic₇₃	C₂³	1	1	yes	characteristic
30	C₄×Dic₇₃	C₂³	1	3	yes	—
31	#41×C₇₃	—	2	4	no	—
32	C₂×Q₈×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₂×D₄×C₇₃	C₂×C₄	1	3	yes	—
34	C₂²×C₄×C₇₃	C₂×C₄	1	3	yes	—
35	C₂³×C₇₃	C₂²×C₄	1	3	yes	—
36	Q₈×C₇₃	M₁₆	1	4	yes	complemented
37	D₄×C₇₃	—	2	6	no	—
38	C₂×C₄×C₇₃	—	2	6	no	—
39	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₇₃	C₂²×C₄	1	3	yes	—
41	C₂×Dic₇₃	—	2	6	no	—
42	C₂×Dic₇₃	—	4	3	no	—
43	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×Dic₇₃	C₂²×C₄	1	3	yes	—
45	C₂×Dic₇₃	C₂²×C₄	1	3	yes	—
46	C₂×Dic₇₃	C₂⁴	1	1	yes	characteristic
47	#10596833	—	2	6	no	—
48	#10596833	#41	1	4	yes	complemented
49	C₂²×C₇₃	—	2	3	no	—
50	C₂²×C₇₃	—	2	3	no	—
51	C₂²×C₇₃	—	2	3	no	—
52	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
53	C₄×C₇₃	—	2	1	no	—
54	C₄×C₇₃	C₂×M₁₆	1	6	yes	—
55	Dic₇₃	—	2	3	no	—
56	Dic₇₃	—	2	1	no	—
57	Dic₇₃	—	2	3	no	—
58	Dic₇₃	C₂×#41	1	2	yes	—
59	C₂×C₇₃	—	4	3	no	—
60	C₂×C₇₃	C₄×#41	1	1	yes	characteristic
61	C₂×C₇₃	C₂²×M₁₆	1	1	yes	characteristic
62	C₂×C₇₃	#568	1	1	yes	characteristic
63	#2971	—	73	1	no	maximal, Sylow
64	C₇₃	#2971	1	1	yes	characteristic, complemented, Sylow
65	C₄×#41	—	73	1	no	—
66	C₈×Q₈	—	73	2	no	—
67	#436	—	73	6	no	—
68	#432	—	73	3	no	—
69	C₄×M₁₆	—	73	3	no	—
70	C₂×#41	#64408	1	1	yes	characteristic
71	C₂×M₁₆	—	73	6	no	—
72	C₄×Q₈	—	73	1	no	—
73	C₄×Q₈	—	73	1	no	—
74	C₄×D₄	—	73	3	no	—
75	C₄×D₄	—	73	3	no	—
76	#117	—	73	3	no	—
77	C₂×C₄²	—	73	3	no	—
78	C₄⋊C₈	—	73	6	no	—
79	C₂²⋊C₈	—	73	6	no	—
80	#97	—	73	1	no	—
81	#97	—	73	3	no	—
82	C₄×C₈	—	73	6	no	—
83	#41	—	2	4	no	—
84	C₂×Q₈	C₂×#64408	1	1	yes	characteristic
85	C₂×D₄	C₂×#64408	1	3	yes	—
86	C₂²×C₄	—	73	3	no	—
87	C₂²×C₄	—	73	3	no	—
88	C₂²×C₄	C₂×#64408	1	3	yes	—
89	M₁₆	—	146	12	no	—
90	C₂×C₈	—	73	2	no	—
91	C₂×C₈	—	73	6	no	—
92	C₄⋊C₄	—	73	3	no	—
93	C₄⋊C₄	—	73	3	no	—
94	C₂²⋊C₄	—	73	6	no	—
95	C₄²	—	73	3	no	—
96	C₄²	—	73	3	no	—
97	C₄²	—	73	3	no	—
98	C₄²	—	73	1	no	—
99	C₂³	C₂²×#64408	1	3	yes	—
100	Q₈	#11924037	1	4	yes	complemented
101	D₄	—	2	6	no	—
102	C₂×C₄	—	2	6	no	—
103	C₂×C₄	—	292	3	no	—
104	C₂×C₄	—	73	1	no	—
105	C₂×C₄	—	73	1	no	—
106	C₂×C₄	—	73	3	no	—
107	C₂×C₄	—	73	3	no	—
108	C₂×C₄	—	146	6	no	—
109	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
110	C₂×C₄	C₂²×#64408	1	3	yes	—
111	C₈	—	73	4	no	—
112	C₈	—	146	6	no	—
113	C₂²	—	2	3	no	—
114	C₂²	—	2	3	no	—
115	C₂²	—	2	3	no	—
116	C₂²	C₂³×#64408	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	73	2	no	—
119	C₄	—	146	3	no	—
120	C₄	—	146	1	no	—
121	C₄	—	146	3	no	—
122	C₄	C₂×#11924037	1	6	yes	—
123	C₂	—	4	3	no	—
124	C₂	#41×#64408	1	1	yes	characteristic
125	C₂	C₂²×#11924037	1	1	yes	characteristic
126	C₂	#17989666	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

Q₈×#10596833

C₂

yes

maximal, complemented

#17989622

C₂

yes

maximal, complemented

#17989608

C₂

yes

maximal, complemented

C₄×#11924037

C₂

yes

maximal, complemented

#41×Dic₇₃

C₂

yes

maximal, characteristic

C₂×#11924037

C₂²

yes

—

C₄⋊#10596833

C₂²

yes

—

C₂²⋊#10596833

C₂²

yes

—

#14556646

C₂²

yes

characteristic

#14556646

C₂²

yes

—

C₄×#10596833

C₂²

yes

—

Q₈×Dic₇₃

C₄

yes

characteristic

Q₈×Dic₇₃

C₂²

yes

characteristic

D₄×Dic₇₃

C₄

yes

—

D₄×Dic₇₃

C₂²

yes

—

#14556550

C₄

yes

—

C₂×C₄×Dic₇₃

C₂²

yes

—

C₂×#41×C₇₃

C₄

yes

characteristic

#11924037

—

C₂×#10596833

C₂³

yes

—

C₂×#10596833

C₂³

yes

—

C₂²×Dic₇₃

C₂×C₄

yes

—

C₂²×Dic₇₃

C₂³

yes

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

C₂²⋊Dic₇₃

C₂×C₄

yes

—

C₄×Dic₇₃

C₂×C₄

yes

—

C₄×Dic₇₃

C₂×C₄

yes

—

C₄×Dic₇₃

C₂³

yes

characteristic

C₄×Dic₇₃

C₂³

yes

—

#41×C₇₃

—

C₂×Q₈×C₇₃

C₂×C₄

yes

characteristic

C₂×D₄×C₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

C₂×C₄

yes

—

C₂³×C₇₃

C₂²×C₄

yes

—

Q₈×C₇₃

M₁₆

yes

complemented

D₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

C₂²×C₄

yes

—

C₂×Dic₇₃

C₂²×C₄

yes

—

C₂×Dic₇₃

C₂⁴

yes

—

yes

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂³×C₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

C₂×M₁₆

yes

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

C₂×#41

yes

—

C₂×C₇₃

—

C₂×C₇₃

C₄×#41

yes

characteristic

C₂×C₇₃

C₂²×M₁₆

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₄×#41

—

C₈×Q₈

—

#436

—

#432

—

C₄×M₁₆

—

C₂×#41

#64408

yes

characteristic

C₂×M₁₆

—

C₄×Q₈

—

C₄×Q₈

—

C₄×D₄

—

C₄×D₄

—

#117

—

C₂×C₄²

—

C₄⋊C₈

—

C₂²⋊C₈

—

#97

—

#97

—

C₄×C₈

—

#41

—

C₂×Q₈

C₂×#64408

yes

characteristic

C₂×D₄

C₂×#64408

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#64408

yes

—

M₁₆

—

146

—

C₂×C₈

—

C₂×C₈

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₄²

—

C₄²

—

C₂³

C₂²×#64408

yes

—

100

Q₈

#11924037

yes

complemented

101

D₄

—

102

C₂×C₄

—

103

C₂×C₄

—

292

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

146

—

109

C₂×C₄

C₂²×#64408

yes

characteristic

110

C₂×C₄

C₂²×#64408

yes

—

111

C₈

—

112

C₈

—

146

—

113

C₂²

—

114

C₂²

—

115

C₂²

—

116

C₂²

C₂³×#64408

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

146

—

120

C₄

—

146

—

121

C₄

—

146

—

122

C₄

C₂×#11924037

yes

—

123

C₂

—

124

C₂

#41×#64408

yes

characteristic

125

C₂

C₂²×#11924037

yes

characteristic

126

C₂

#17989666

yes

characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₄
Fitting	C₂×#41×C₇₃	C₄
hypercenter	C₂×#41	#64408
hypocenter	C₇₃	#2971
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×#41

subgroup

G/H

center, Frattini

C₂²

C₂³×#64408

commutator, socle

C₂²×C₇₃

C₂³×C₄

Fitting

C₂×#41×C₇₃

C₄

C₂×#41

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂² ◁ C₂×#41

N	H ≅ G/N	type
1. Q₈×#10596833	C₂	semidirect
2. #17989622	C₂	semidirect
3. #17989608	C₂	semidirect
4. C₄×#11924037	C₂	semidirect
36. Q₈×C₇₃	M₁₆	semidirect
48. #10596833	#41	semidirect
64. C₇₃	#2971	semidirect
100. Q₈	#11924037	semidirect

H ≅ G/N

type

1. Q₈×#10596833

C₂

semidirect

2. #17989622

C₂

semidirect

3. #17989608

C₂

semidirect

4. C₄×#11924037

C₂

semidirect

36. Q₈×C₇₃

M₁₆

semidirect

48. #10596833

#41

semidirect

64. C₇₃

#2971

semidirect

100. Q₈

#11924037

semidirect

Subgroups — #22626777

Conjugacy classes of subgroups

Complements