Group #22626213 — subgroups

Subgroups — #22626213

Order 9344.

238 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 54 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	#17989566	C₂	1	2	yes	maximal
2	#17989493	C₂	1	4	yes	maximal, complemented
3	#17989161	C₂	1	1	yes	maximal, characteristic
4	#14556670	—	2	2	no	—
5	#14556650	C₂²	1	4	yes	—
6	#14556649	—	2	2	no	—
7	C₂×#11924028	C₂²	1	1	yes	characteristic
8	#14556589	C₄	1	2	yes	complemented
9	#14556580	C₂²	1	2	yes	—
10	#14556574	—	2	2	no	—
11	#14556567	—	2	2	no	—
12	#14556550	C₄	1	1	yes	characteristic, complemented
13	C₂²×Q₈×C₇₃	C₄	1	1	yes	characteristic, complemented
14	C₄⋊#64408	—	4	2	no	—
15	C₂²⋊#64408	—	2	4	no	—
16	C₄×#64408	—	4	2	no	—
17	#11924028	—	4	2	no	—
18	C₂×Dic₁₄₆	C₂×C₄	1	1	yes	characteristic
19	C₂×D₂₉₂	C₂³	1	1	yes	characteristic
20	C₂×C₂²⋊D₇₃	D₄	1	2	yes	—
21	C₂×C₄×D₇₃	D₄	1	2	yes	—
22	C₄⋊Dic₇₃	—	2	1	no	—
23	#11924007	—	2	4	no	—
24	C₂²⋊Dic₇₃	—	2	1	no	—
25	#11924005	—	2	4	no	—
26	C₄×Dic₇₃	C₂×C₄	1	2	yes	—
27	C₂×Q₈×C₇₃	—	2	1	no	—
28	C₂×Q₈×C₇₃	—	4	2	no	—
29	C₂×Q₈×C₇₃	C₂×C₄	1	2	yes	—
30	C₂²×C₄×C₇₃	—	2	1	no	—
31	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂³×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂²×D₇₃	C₂×D₄	1	2	yes	—
34	C₂×#64408	—	4	4	no	—
35	Q₈×C₇₃	—	4	2	no	—
36	Q₈×C₇₃	—	4	2	no	—
37	C₂×C₄×C₇₃	—	2	2	no	—
38	C₂×C₄×C₇₃	—	2	2	no	—
39	C₂×C₄×C₇₃	—	4	2	no	—
40	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
42	C₂²⋊D₇₃	—	4	2	no	—
43	C₂×Dic₇₃	—	2	2	no	—
44	C₂×Dic₇₃	C₂²⋊C₄	1	2	yes	—
45	D₂₉₂	—	4	1	no	—
46	C₄×D₇₃	—	2	2	no	—
47	C₄×D₇₃	—	2	2	no	—
48	Dic₁₄₆	—	4	1	no	—
49	C₂²×C₇₃	—	2	1	no	—
50	C₂²×C₇₃	—	4	1	no	—
51	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
52	C₂×D₇₃	—	2	2	no	—
53	C₂×D₇₃	—	4	2	no	—
54	#64408	—	8	4	no	—
55	C₄×C₇₃	—	2	1	no	—
56	C₄×C₇₃	—	4	2	no	—
57	C₄×C₇₃	#99	1	2	yes	—
58	Dic₇₃	—	2	2	no	—
59	Dic₇₃	—	4	2	no	—
60	C₂×C₇₃	—	2	1	no	—
61	C₂×C₇₃	—	4	1	no	—
62	C₂×C₇₃	C₂×#99	1	1	yes	characteristic
63	D₇₃	—	4	2	no	—
64	#2129	—	73	1	no	maximal, Sylow
65	C₇₃	#2129	1	1	yes	characteristic, complemented, Sylow
66	#536	—	73	1	no	—
67	#410	—	73	2	no	—
68	#354	—	73	4	no	—
69	C₂×#41	—	73	1	no	—
70	C₂²×Q₈	#64408	1	1	yes	characteristic, complemented
71	C₄⋊Q₈	—	73	2	no	—
72	#124	—	73	2	no	—
73	#123	—	146	2	no	—
74	C₂²⋊Q₈	—	146	2	no	—
75	#117	—	73	1	no	—
76	#117	—	146	2	no	—
77	#99	—	73	4	no	—
78	#99	—	146	2	no	—
79	#41	—	292	2	no	—
80	C₂×Q₈	—	2	1	no	—
81	C₂×Q₈	—	4	2	no	—
82	C₂×Q₈	—	73	1	no	—
83	C₂×Q₈	C₂×#64408	1	2	yes	—
84	C₂×D₄	—	73	2	no	—
85	C₂×D₄	—	73	1	no	—
86	C₂²×C₄	—	2	1	no	—
87	C₂²×C₄	—	73	2	no	—
88	C₂²×C₄	C₂×#64408	1	1	yes	characteristic
89	C₄⋊C₄	—	292	2	no	—
90	C₄⋊C₄	—	146	4	no	—
91	C₄⋊C₄	—	146	1	no	—
92	C₂²⋊C₄	—	146	4	no	—
93	C₂²⋊C₄	—	146	4	no	—
94	C₂²⋊C₄	—	146	1	no	—
95	C₄²	—	292	2	no	—
96	C₄²	—	73	2	no	—
97	C₂³	—	73	2	no	—
98	C₂³	C₂²⋊#64408	1	1	yes	characteristic
99	Q₈	—	4	2	no	—
100	Q₈	—	4	2	no	—
101	Q₈	—	292	1	no	—
102	D₄	—	292	1	no	—
103	D₄	—	292	2	no	—
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	4	2	no	—
107	C₂×C₄	—	292	4	no	—
108	C₂×C₄	—	73	2	no	—
109	C₂×C₄	—	146	2	no	—
110	C₂×C₄	—	146	2	no	—
111	C₂×C₄	—	146	2	no	—
112	C₂×C₄	C₂²⋊#64408	1	1	yes	characteristic
113	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
114	C₂²	—	2	1	no	—
115	C₂²	—	4	1	no	—
116	C₂²	—	292	2	no	—
117	C₂²	—	146	2	no	—
118	C₂²	C₂×C₂²⋊#64408	1	1	yes	characteristic
119	C₄	—	2	1	no	—
120	C₄	—	4	2	no	—
121	C₄	—	292	2	no	—
122	C₄	—	584	4	no	—
123	C₄	—	146	2	no	—
124	C₄	#14556649	1	2	yes	—
125	C₂	—	2	1	no	—
126	C₂	—	4	1	no	—
127	C₂	—	292	2	no	—
128	C₂	C₂×#14556649	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17989566

C₂

yes

maximal

#17989493

C₂

yes

maximal, complemented

#17989161

C₂

yes

maximal, characteristic

#14556670

—

#14556650

C₂²

yes

—

#14556649

—

C₂×#11924028

C₂²

yes

characteristic

#14556589

C₄

yes

complemented

#14556580

C₂²

yes

—

#14556574

—

#14556567

—

#14556550

C₄

yes

characteristic, complemented

C₂²×Q₈×C₇₃

C₄

yes

characteristic, complemented

—

—

—

—

C₂×Dic₁₄₆

C₂×C₄

yes

characteristic

C₂×D₂₉₂

C₂³

yes

characteristic

C₂×C₂²⋊D₇₃

D₄

yes

—

C₂×C₄×D₇₃

D₄

yes

—

C₄⋊Dic₇₃

—

#11924007

—

C₂²⋊Dic₇₃

—

#11924005

—

C₄×Dic₇₃

C₂×C₄

yes

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

—

C₂×Q₈×C₇₃

C₂×C₄

yes

—

C₂²×C₄×C₇₃

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂³×C₇₃

C₂²⋊C₄

yes

characteristic

C₂²×D₇₃

C₂×D₄

yes

—

C₂×#64408

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂²⋊D₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂²⋊C₄

yes

—

D₂₉₂

—

C₄×D₇₃

—

C₄×D₇₃

—

Dic₁₄₆

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#99

yes

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

C₂×#99

yes

characteristic

D₇₃

—

#2129

—

maximal, Sylow

C₇₃

#2129

yes

characteristic, complemented, Sylow

#536

—

#410

—

#354

—

C₂×#41

—

C₂²×Q₈

#64408

yes

characteristic, complemented

C₄⋊Q₈

—

#124

—

#123

—

146

—

C₂²⋊Q₈

—

146

—

#117

—

#117

—

146

—

#99

—

#99

—

146

—

#41

—

292

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×#64408

yes

—

C₂×D₄

—

C₂×D₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂×#64408

yes

characteristic

C₄⋊C₄

—

292

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

C₂²⋊C₄

—

146

—

C₂²⋊C₄

—

146

—

C₂²⋊C₄

—

146

—

C₄²

—

292

—

C₄²

—

C₂³

—

C₂³

C₂²⋊#64408

yes

characteristic

Q₈

—

100

Q₈

—

101

Q₈

—

292

—

102

D₄

—

292

—

103

D₄

—

292

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

292

—

108

C₂×C₄

—

109

C₂×C₄

—

146

—

110

C₂×C₄

—

146

—

111

C₂×C₄

—

146

—

112

C₂×C₄

C₂²⋊#64408

yes

characteristic

113

C₂×C₄

C₂²×#64408

yes

characteristic

114

C₂²

—

115

C₂²

—

116

C₂²

—

292

—

117

C₂²

—

146

—

118

C₂²

C₂×C₂²⋊#64408

yes

characteristic

119

C₄

—

120

C₄

—

121

C₄

—

292

—

122

C₄

—

584

—

123

C₄

—

146

—

124

C₄

#14556649

yes

—

125

C₂

—

126

C₂

—

127

C₂

—

292

—

128

C₂

C₂×#14556649

yes

characteristic

	subgroup	G/H
center	C₂	C₂×#14556649
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#64408
Fitting	C₂²×Q₈×C₇₃	C₄
socle	C₂×C₇₃	C₂×#99
hypercenter	C₂²×Q₈	#64408
hypocenter	C₇₃	#2129
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×Q₈

subgroup

G/H

center

C₂

C₂×#14556649

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#64408

Fitting

C₂²×Q₈×C₇₃

C₄

socle

C₂×C₇₃

C₂×#99

C₂²×Q₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×Q₈

N	H ≅ G/N	type
2. #17989493	C₂	semidirect
8. #14556589	C₄	semidirect
12. #14556550	C₄	semidirect
13. C₂²×Q₈×C₇₃	C₄	semidirect
65. C₇₃	#2129	semidirect
70. C₂²×Q₈	#64408	semidirect

H ≅ G/N

type

2. #17989493

C₂

semidirect

8. #14556589

C₄

semidirect

12. #14556550

C₄

semidirect

13. C₂²×Q₈×C₇₃

C₄

semidirect

65. C₇₃

#2129

semidirect

70. C₂²×Q₈

#64408

semidirect

Subgroups — #22626213

Conjugacy classes of subgroups

Complements