Group #22609293 — subgroups

Subgroups — #22609293

Order 9344.

172 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 172 conjugacy classes of subgroups — too many to draw. Only its 57 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	#17988864	C₂	1	2	yes	maximal
2	C₈×D₂₉₂	C₂	1	1	yes	maximal, characteristic
3	#17988340	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988333	C₂	1	2	yes	maximal, complemented
5	#494×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₈×D₇₃	—	2	1	no	—
7	C₄×D₂₉₂	C₂²	1	1	yes	characteristic
8	#14556524	—	2	2	no	—
9	C₄⋊#10596831	C₂²	1	2	yes	—
10	C₄⋊#10596831	C₂²	1	1	yes	characteristic
11	#14556511	—	2	2	no	—
12	#14556503	—	2	1	no	—
13	C₂×Q₁₆×C₇₃	—	2	1	no	—
14	C₂×Q₁₆×C₇₃	—	2	1	no	—
15	C₄⋊Q₈×C₇₃	C₂²	1	2	yes	—
16	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×D₂₉₂	D₄	1	1	yes	characteristic
18	C₂×C₄×D₇₃	—	2	1	no	—
19	C₂×#10596831	—	2	1	no	—
20	C₂×#10596831	—	2	2	no	—
21	C₈×D₇₃	—	4	2	no	—
22	C₄⋊Dic₇₃	D₄	1	1	yes	characteristic
23	#11924005	—	2	1	no	—
24	C₂×Q₈×C₇₃	—	2	2	no	—
25	Q₁₆×C₇₃	—	4	2	no	—
26	Q₁₆×C₇₃	—	4	2	no	—
27	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
28	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
29	C₄⋊C₄×C₇₃	—	2	2	no	—
30	C₄⋊C₄×C₇₃	—	2	2	no	—
31	C₄²×C₇₃	C₂³	1	1	yes	characteristic
32	C₂²×D₇₃	—	2	1	no	—
33	Q₈×C₇₃	—	4	2	no	—
34	Q₈×C₇₃	—	4	2	no	—
35	C₂×C₄×C₇₃	—	4	2	no	—
36	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
39	C₂×Dic₇₃	—	2	1	no	—
40	D₂₉₂	—	2	1	no	—
41	D₂₉₂	Q₁₆	1	2	yes	complemented
42	C₄×D₇₃	—	4	1	no	—
43	C₄×D₇₃	—	4	1	no	—
44	C₈×C₇₃	—	2	1	no	—
45	C₈×C₇₃	D₈	1	2	yes	—
46	#10596831	—	4	1	no	—
47	#10596831	—	4	2	no	—
48	C₂²×C₇₃	#121	1	1	yes	characteristic
49	C₂×D₇₃	—	2	2	no	—
50	C₂×D₇₃	—	4	1	no	—
51	C₂×D₇₃	—	4	1	no	—
52	C₄×C₇₃	—	2	1	no	—
53	C₄×C₇₃	—	8	2	no	—
54	C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
55	C₄×C₇₃	C₂×Q₁₆	1	1	yes	characteristic
56	C₄×C₇₃	#135	1	1	yes	characteristic
57	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
58	Dic₇₃	—	4	1	no	—
59	C₂×C₇₃	#459	1	1	yes	characteristic
60	C₂×C₇₃	#467	1	1	yes	characteristic
61	C₂×C₇₃	#473	1	1	yes	characteristic
62	D₇₃	—	4	2	no	—
63	#1669	—	73	1	no	maximal, Sylow
64	C₇₃	#1669	1	1	yes	characteristic, complemented, Sylow
65	#494	D₇₃	1	1	yes	characteristic, complemented
66	#474	—	73	2	no	—
67	C₈×D₄	—	73	1	no	—
68	#335	—	73	1	no	—
69	#332	—	73	2	no	—
70	C₂×Q₁₆	—	2	1	no	—
71	C₂×Q₁₆	—	2	1	no	—
72	C₂²×C₈	—	146	1	no	—
73	C₄⋊Q₈	C₂×D₇₃	1	2	yes	—
74	C₄×D₄	—	73	1	no	—
75	#107	—	146	2	no	—
76	C₄⋊C₈	—	73	1	no	—
77	C₄⋊C₈	—	73	2	no	—
78	#102	—	146	2	no	—
79	C₂²⋊C₈	—	146	1	no	—
80	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
81	C₂×Q₈	—	2	2	no	—
82	C₂×D₄	—	73	1	no	—
83	C₂²×C₄	—	146	1	no	—
84	Q₁₆	—	4	2	no	—
85	Q₁₆	—	4	2	no	—
86	C₂×C₈	—	292	2	no	—
87	C₂×C₈	—	146	2	no	—
88	C₂×C₈	—	146	1	no	—
89	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
91	C₄⋊C₄	—	2	2	no	—
92	C₄⋊C₄	—	2	2	no	—
93	C₄⋊C₄	—	73	1	no	—
94	C₂²⋊C₄	—	146	1	no	—
95	C₄²	C₂²×D₇₃	1	1	yes	characteristic
96	C₂³	—	146	1	no	—
97	Q₈	—	4	2	no	—
98	Q₈	—	4	2	no	—
99	D₄	—	73	2	no	—
100	D₄	—	146	1	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	292	1	no	—
103	C₂×C₄	—	292	1	no	—
104	C₂×C₄	—	146	1	no	—
105	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
106	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
107	C₂×C₄	#11924027	1	1	yes	characteristic
108	C₈	—	2	1	no	—
109	C₈	—	292	1	no	—
110	C₈	—	292	2	no	—
111	C₈	#11924013	1	2	yes	—
112	C₂²	—	292	1	no	—
113	C₂²	—	292	1	no	—
114	C₂²	—	146	2	no	—
115	C₂²	#14556562	1	1	yes	characteristic
116	C₄	—	2	1	no	—
117	C₄	—	8	2	no	—
118	C₄	—	292	1	no	—
119	C₄	C₂×#11924013	1	1	yes	characteristic
120	C₄	Q₁₆×D₇₃	1	1	yes	characteristic
121	C₄	#14556614	1	1	yes	characteristic
122	C₄	#14556617	1	1	yes	characteristic
123	C₂	—	292	2	no	—
124	C₂	#17988779	1	1	yes	characteristic
125	C₂	#17988825	1	1	yes	characteristic
126	C₂	#17988859	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988864

C₂

yes

maximal

C₈×D₂₉₂

C₂

yes

maximal, characteristic

#17988340

C₂

yes

maximal, characteristic, complemented

#17988333

C₂

yes

maximal, complemented

#494×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×C₈×D₇₃

—

C₄×D₂₉₂

C₂²

yes

characteristic

#14556524

—

C₄⋊#10596831

C₂²

yes

—

C₄⋊#10596831

C₂²

yes

characteristic

#14556511

—

#14556503

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

—

C₄⋊Q₈×C₇₃

C₂²

yes

—

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×D₂₉₂

D₄

yes

characteristic

C₂×C₄×D₇₃

—

C₂×#10596831

—

C₂×#10596831

—

C₈×D₇₃

—

C₄⋊Dic₇₃

D₄

yes

characteristic

#11924005

—

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂²×D₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

Q₁₆

yes

complemented

C₄×D₇₃

—

C₄×D₇₃

—

C₈×C₇₃

—

C₈×C₇₃

D₈

yes

—

#10596831

—

#10596831

—

C₂²×C₇₃

#121

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

C₂×Q₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1669

—

maximal, Sylow

C₇₃

#1669

yes

characteristic, complemented, Sylow

#494

D₇₃

yes

characteristic, complemented

#474

—

C₈×D₄

—

#335

—

#332

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

—

146

—

C₄⋊Q₈

C₂×D₇₃

yes

—

C₄×D₄

—

#107

—

146

—

C₄⋊C₈

—

C₄⋊C₈

—

#102

—

146

—

C₂²⋊C₈

—

146

—

C₄×C₈

C₂×D₇₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

C₂²×C₄

—

146

—

Q₁₆

—

Q₁₆

—

C₂×C₈

—

292

—

C₂×C₈

—

146

—

C₂×C₈

—

146

—

C₂×C₈

C₂²⋊D₇₃

yes

characteristic

C₂×C₈

C₂²⋊D₇₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

146

—

C₄²

C₂²×D₇₃

yes

characteristic

C₂³

—

146

—

Q₈

—

Q₈

—

D₄

—

100

D₄

—

146

—

101

C₂×C₄

—

102

C₂×C₄

—

292

—

103

C₂×C₄

—

292

—

104

C₂×C₄

—

146

—

105

C₂×C₄

D₄×D₇₃

yes

characteristic

106

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

107

C₂×C₄

#11924027

yes

characteristic

108

C₈

—

109

C₈

—

292

—

110

C₈

—

292

—

111

C₈

#11924013

yes

—

112

C₂²

—

292

—

113

C₂²

—

292

—

114

C₂²

—

146

—

115

C₂²

#14556562

yes

characteristic

116

C₄

—

117

C₄

—

118

C₄

—

292

—

119

C₄

C₂×#11924013

yes

characteristic

120

C₄

Q₁₆×D₇₃

yes

characteristic

121

C₄

#14556614

yes

characteristic

122

C₄

#14556617

yes

characteristic

123

C₂

—

292

—

124

C₂

#17988779

yes

characteristic

125

C₂

#17988825

yes

characteristic

126

C₂

#17988859

yes

characteristic

	subgroup	G/H
center	C₂²	#14556562
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	#494×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	#494	D₇₃
hypocenter	C₇₃	#1669
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #494

subgroup

G/H

center

C₂²

#14556562

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

#494×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #494

N	H ≅ G/N	type
3. #17988340	C₂	semidirect
4. #17988333	C₂	semidirect
5. #494×C₇₃	C₂	semidirect
41. D₂₉₂	Q₁₆	semidirect
64. C₇₃	#1669	semidirect
65. #494	D₇₃	semidirect

H ≅ G/N

type

3. #17988340

C₂

semidirect

4. #17988333

C₂

semidirect

5. #494×C₇₃

C₂

semidirect

41. D₂₉₂

Q₁₆

semidirect

64. C₇₃

#1669

semidirect

65. #494

D₇₃

semidirect

Subgroups — #22609293

Conjugacy classes of subgroups

Complements