Group #22626476 — subgroups

Subgroups — #22626476

Order 9344.

424 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 152 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	(C₂×#64408)⋊Q₈	C₂	1	4	yes	maximal
2	#17989536	C₂	1	8	yes	maximal
3	#17989532	C₂	1	2	yes	maximal
4	C₄⋊Q₈×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#64408	C₂²	1	4	yes	—
6	C₂×C₄×#64408	C₂²	1	4	yes	—
7	(C₂×#64408)⋊C₄	C₂²	1	8	yes	—
8	(C₂×#64408)⋊C₄	C₂²	1	4	yes	—
9	(C₂×#64408)⋊C₄	C₂²	1	8	yes	—
10	C₂×Q₈×D₇₃	C₂²	1	2	yes	—
11	#14556591	C₄	1	1	yes	characteristic, complemented
12	#14556590	C₄	1	4	yes	complemented
13	#14556589	C₄	1	2	yes	complemented
14	C₄⋊C₄×D₇₃	C₂²	1	4	yes	—
15	C₄²×D₇₃	C₂²	1	1	yes	characteristic
16	C₄⋊Q₈×C₇₃	C₄	1	1	yes	characteristic, complemented
17	C₂²×#64408	C₂³	1	8	yes	—
18	C₄⋊#64408	—	4	4	no	—
19	C₄×#64408	—	4	4	no	—
20	C₂×Dic₁₄₆	C₂×C₄	1	2	yes	—
21	C₂×Dic₁₄₆	C₂×C₄	1	4	yes	—
22	Q₈×D₇₃	—	4	2	no	—
23	Q₈×D₇₃	—	4	2	no	—
24	C₂×C₄×D₇₃	C₂³	1	2	yes	—
25	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
26	C₂×C₄×D₇₃	C₂³	1	4	yes	—
27	C₄⋊Dic₇₃	C₂×C₄	1	4	yes	—
28	#11924007	C₂×C₄	1	8	yes	—
29	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
30	C₄×Dic₇₃	C₂×C₄	1	2	yes	—
31	C₂×Q₈×C₇₃	C₂×C₄	1	2	yes	—
32	C₄⋊C₄×C₇₃	C₂×C₄	1	4	yes	—
33	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×D₇₃	C₂⁴	1	1	yes	characteristic
35	C₂×#64408	—	2	8	no	—
36	C₂×#64408	—	2	16	no	—
37	Q₈×C₇₃	—	4	2	no	—
38	C₂×C₄×C₇₃	C₂²×C₄	1	2	yes	—
39	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
40	C₂×C₄×C₇₃	C₂²×C₄	1	4	yes	—
41	C₂×Dic₇₃	C₂²×C₄	1	2	yes	—
42	C₂×Dic₇₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×Dic₇₃	C₂²×C₄	1	4	yes	—
44	C₄×D₇₃	—	2	2	no	—
45	C₄×D₇₃	—	2	4	no	—
46	C₄×D₇₃	—	2	4	no	—
47	C₄×D₇₃	—	2	4	no	—
48	Dic₁₄₆	—	4	4	no	—
49	Dic₁₄₆	—	4	2	no	—
50	C₂²×C₇₃	C₂³×C₄	1	1	yes	characteristic
51	C₂×D₇₃	H(5,𝔽₂)	1	2	yes	—
52	C₂×D₇₃	2₋¹⁺⁴	1	4	yes	—
53	#64408	—	4	8	no	—
54	C₄×C₇₃	—	2	1	no	—
55	C₄×C₇₃	—	2	4	no	—
56	C₄×C₇₃	—	2	2	no	—
57	Dic₇₃	—	2	1	no	—
58	Dic₇₃	—	2	4	no	—
59	Dic₇₃	—	2	2	no	—
60	C₂×C₇₃	#518	1	1	yes	characteristic
61	C₂×C₇₃	#519	1	2	yes	—
62	D₇₃	#563	1	4	yes	—
63	#2381	—	73	1	no	maximal, Sylow
64	C₇₃	#2381	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₄⋊Q₈	—	73	1	no	—
66	#391	—	73	4	no	—
67	#387	—	73	8	no	—
68	#383	—	73	2	no	—
69	C₂²×Q₈	—	73	2	no	—
70	C₄⋊Q₈	—	73	2	no	—
71	C₄⋊Q₈	—	73	4	no	—
72	C₄⋊Q₈	—	73	1	no	—
73	C₄⋊Q₈	#64408	1	1	yes	characteristic, complemented
74	C₂×C₄⋊C₄	—	73	4	no	—
75	C₂×C₄⋊C₄	—	73	4	no	—
76	C₂×C₄²	—	73	4	no	—
77	C₂×C₄²	—	73	1	no	—
78	#95	—	73	8	no	—
79	#95	—	73	4	no	—
80	#95	—	73	8	no	—
81	C₂×Q₈	—	292	2	no	—
82	C₂×Q₈	—	292	2	no	—
83	C₂×Q₈	—	73	4	no	—
84	C₂×Q₈	—	73	2	no	—
85	C₂×Q₈	C₂×#64408	1	2	yes	—
86	C₂²×C₄	—	73	2	no	—
87	C₂²×C₄	—	73	1	no	—
88	C₂²×C₄	—	73	4	no	—
89	C₂²×C₄	—	73	8	no	—
90	C₄⋊C₄	—	292	4	no	—
91	C₄⋊C₄	—	73	8	no	—
92	C₄⋊C₄	—	73	4	no	—
93	C₄⋊C₄	C₂×#64408	1	4	yes	—
94	C₄²	—	292	4	no	—
95	C₄²	—	73	1	no	—
96	C₄²	—	73	2	no	—
97	C₄²	C₂×#64408	1	1	yes	characteristic
98	C₂³	—	73	1	no	—
99	Q₈	—	4	2	no	—
100	Q₈	—	292	2	no	—
101	Q₈	—	292	4	no	—
102	C₂×C₄	—	73	2	no	—
103	C₂×C₄	—	73	1	no	—
104	C₂×C₄	—	73	4	no	—
105	C₂×C₄	—	146	4	no	—
106	C₂×C₄	—	146	4	no	—
107	C₂×C₄	—	146	2	no	—
108	C₂×C₄	—	146	4	no	—
109	C₂×C₄	—	146	8	no	—
110	C₂×C₄	—	146	16	no	—
111	C₂×C₄	C₂²×#64408	1	2	yes	—
112	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
113	C₂×C₄	C₂²×#64408	1	4	yes	—
114	C₂²	—	73	2	no	—
115	C₂²	—	73	4	no	—
116	C₂²	C₂³×#64408	1	1	yes	characteristic
117	C₄	—	2	1	no	—
118	C₄	—	2	4	no	—
119	C₄	—	2	2	no	—
120	C₄	—	292	8	no	—
121	C₄	—	146	1	no	—
122	C₄	—	146	4	no	—
123	C₄	—	146	2	no	—
124	C₂	—	73	4	no	—
125	C₂	#17989654	1	1	yes	characteristic
126	C₂	#17989655	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

(C₂×#64408)⋊Q₈

C₂

yes

maximal

#17989536

C₂

yes

maximal

#17989532

C₂

yes

maximal

C₄⋊Q₈×D₇₃

C₂

yes

maximal, characteristic

C₂×C₄⋊#64408

C₂²

yes

—

C₂×C₄×#64408

C₂²

yes

—

(C₂×#64408)⋊C₄

C₂²

yes

—

(C₂×#64408)⋊C₄

C₂²

yes

—

(C₂×#64408)⋊C₄

C₂²

yes

—

C₂×Q₈×D₇₃

C₂²

yes

—

#14556591

C₄

yes

characteristic, complemented

#14556590

C₄

yes

complemented

#14556589

C₄

yes

complemented

C₄⋊C₄×D₇₃

C₂²

yes

—

C₄²×D₇₃

C₂²

yes

characteristic

C₄⋊Q₈×C₇₃

C₄

yes

characteristic, complemented

C₂²×#64408

C₂³

yes

—

C₄⋊#64408

—

C₄×#64408

—

C₂×Dic₁₄₆

C₂×C₄

yes

—

C₂×Dic₁₄₆

C₂×C₄

yes

—

Q₈×D₇₃

—

Q₈×D₇₃

—

C₂×C₄×D₇₃

C₂³

yes

—

C₂×C₄×D₇₃

C₂³

yes

characteristic

C₂×C₄×D₇₃

C₂³

yes

—

C₄⋊Dic₇₃

C₂×C₄

yes

—

#11924007

C₂×C₄

yes

—

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₄×Dic₇₃

C₂×C₄

yes

—

C₂×Q₈×C₇₃

C₂×C₄

yes

—

C₄⋊C₄×C₇₃

C₂×C₄

yes

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂²×D₇₃

C₂⁴

yes

characteristic

C₂×#64408

—

C₂×#64408

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

—

C₂×Dic₇₃

C₂²×C₄

yes

—

C₂×Dic₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

C₂²×C₄

yes

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₂²×C₇₃

C₂³×C₄

yes

characteristic

C₂×D₇₃

H(5,𝔽₂)

yes

—

C₂×D₇₃

2₋¹⁺⁴

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

Dic₇₃

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

#518

yes

characteristic

C₂×C₇₃

#519

yes

—

D₇₃

#563

yes

—

#2381

—

maximal, Sylow

C₇₃

#2381

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#391

—

#387

—

#383

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

#64408

yes

characteristic, complemented

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

C₂×C₄²

—

#95

—

#95

—

#95

—

C₂×Q₈

—

292

—

C₂×Q₈

—

292

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×#64408

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂×#64408

yes

—

C₄²

—

292

—

C₄²

—

C₄²

—

C₄²

C₂×#64408

yes

characteristic

C₂³

—

Q₈

—

100

Q₈

—

292

—

101

Q₈

—

292

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

146

—

106

C₂×C₄

—

146

—

107

C₂×C₄

—

146

—

108

C₂×C₄

—

146

—

109

C₂×C₄

—

146

—

110

C₂×C₄

—

146

—

111

C₂×C₄

C₂²×#64408

yes

—

112

C₂×C₄

C₂²×#64408

yes

characteristic

113

C₂×C₄

C₂²×#64408

yes

—

114

C₂²

—

115

C₂²

—

116

C₂²

C₂³×#64408

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

292

—

121

C₄

—

146

—

122

C₄

—

146

—

123

C₄

—

146

—

124

C₂

—

125

C₂

#17989654

yes

characteristic

126

C₂

#17989655

yes

—

	subgroup	G/H
center, Frattini	C₂²	C₂³×#64408
commutator, socle	C₂²×C₇₃	C₂³×C₄
Fitting	C₄⋊Q₈×C₇₃	C₄
hypercenter	C₄⋊Q₈	#64408
hypocenter	C₇₃	#2381
LCS	G ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂²

C₂³×#64408

commutator, socle

C₂²×C₇₃

C₂³×C₄

Fitting

C₄⋊Q₈×C₇₃

C₄

C₄⋊Q₈

C₇₃

G ▷ C₂²×C₇₃ ▷ C₇₃

C₁ ◁ C₂² ◁ C₄⋊Q₈

N	H ≅ G/N	type
11. #14556591	C₄	semidirect
12. #14556590	C₄	semidirect
13. #14556589	C₄	semidirect
16. C₄⋊Q₈×C₇₃	C₄	semidirect
64. C₇₃	#2381	semidirect
73. C₄⋊Q₈	#64408	semidirect

H ≅ G/N

type

11. #14556591

C₄

semidirect

12. #14556590

C₄

semidirect

13. #14556589

C₄

semidirect

16. C₄⋊Q₈×C₇₃

C₄

semidirect

64. C₇₃

#2381

semidirect

73. C₄⋊Q₈

#64408

semidirect

Subgroups — #22626476

Conjugacy classes of subgroups

Complements