Group #22609291 — subgroups

Subgroups — #22609291

Order 9344.

188 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 188 conjugacy classes of subgroups — too many to draw. Only its 61 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	#17988964	C₂	1	1	yes	maximal, characteristic
2	#17988787	C₂	1	2	yes	maximal, complemented
3	#17988340	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988330	C₂	1	2	yes	maximal, complemented
5	C₈×Q₈×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11924015	—	2	2	no	—
7	C₂×D₅₈₄	—	2	1	no	—
8	C₂×D₅₈₄	—	2	1	no	—
9	#14556588	C₂²	1	2	yes	—
10	C₄⋊#10596831	C₂²	1	2	yes	—
11	#14556511	—	2	2	no	—
12	C₄×Q₈×C₇₃	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₇₃	—	2	1	no	—
14	C₄⋊C₈×C₇₃	C₂²	1	1	yes	characteristic
15	C₄×C₈×C₇₃	—	2	1	no	—
16	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×D₂₉₂	—	2	2	no	—
18	C₂×D₂₉₂	—	2	2	no	—
19	C₂×D₂₉₂	—	2	2	no	—
20	#11924015	—	4	4	no	—
21	D₅₈₄	—	4	2	no	—
22	D₅₈₄	—	4	2	no	—
23	C₂×#10596831	—	2	2	no	—
24	C₂×Q₈×C₇₃	D₄	1	1	yes	characteristic
25	C₂×C₈×C₇₃	—	2	1	no	—
26	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
27	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
28	C₄⋊C₄×C₇₃	—	2	1	no	—
29	C₄⋊C₄×C₇₃	D₄	1	1	yes	characteristic
30	C₄²×C₇₃	—	2	1	no	—
31	C₄²×C₇₃	C₂³	1	1	yes	characteristic
32	C₂²×D₇₃	—	4	2	no	—
33	Q₈×C₇₃	—	2	1	no	—
34	Q₈×C₇₃	D₈	1	2	yes	complemented
35	C₂×C₄×C₇₃	—	2	1	no	—
36	C₂×C₄×C₇₃	—	2	1	no	—
37	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
40	D₂₉₂	—	4	2	no	—
41	D₂₉₂	—	4	2	no	—
42	D₂₉₂	—	4	2	no	—
43	D₂₉₂	—	4	2	no	—
44	D₂₉₂	—	8	2	no	—
45	C₈×C₇₃	—	2	1	no	—
46	C₈×C₇₃	—	4	1	no	—
47	C₈×C₇₃	SD₁₆	1	2	yes	complemented
48	#10596831	—	4	2	no	—
49	C₂²×C₇₃	#121	1	1	yes	characteristic
50	C₂×D₇₃	—	8	2	no	—
51	C₂×D₇₃	—	8	2	no	—
52	C₂×D₇₃	—	8	2	no	—
53	C₄×C₇₃	—	2	2	no	—
54	C₄×C₇₃	—	2	1	no	—
55	C₄×C₇₃	—	4	1	no	—
56	C₄×C₇₃	C₂×D₈	1	1	yes	characteristic
57	C₄×C₇₃	C₂×SD₁₆	1	1	yes	characteristic
58	C₄×C₇₃	#135	1	1	yes	characteristic
59	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
60	C₂×C₇₃	#460	1	1	yes	characteristic
61	C₂×C₇₃	#466	1	1	yes	characteristic
62	C₂×C₇₃	#472	1	1	yes	characteristic
63	D₇₃	—	16	2	no	—
64	#1668	—	73	1	no	maximal, Sylow
65	C₇₃	#1668	1	1	yes	characteristic, complemented, Sylow
66	#493	—	73	1	no	—
67	#460	—	73	2	no	—
68	C₈×Q₈	D₇₃	1	1	yes	characteristic, complemented
69	#335	—	73	1	no	—
70	#331	—	73	2	no	—
71	C₂×SD₁₆	—	146	2	no	—
72	C₂×D₈	—	146	1	no	—
73	C₂×D₈	—	146	1	no	—
74	#127	—	73	2	no	—
75	C₄×Q₈	C₂×D₇₃	1	1	yes	characteristic
76	C₄⋊C₈	—	2	1	no	—
77	C₄⋊C₈	—	73	2	no	—
78	C₄⋊C₈	C₂×D₇₃	1	1	yes	characteristic
79	#102	—	146	2	no	—
80	C₄×C₈	—	2	1	no	—
81	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
82	C₂×Q₈	D₂₉₂	1	1	yes	characteristic
83	C₂×D₄	—	146	2	no	—
84	C₂×D₄	—	146	2	no	—
85	C₂×D₄	—	146	2	no	—
86	SD₁₆	—	292	4	no	—
87	D₈	—	292	2	no	—
88	D₈	—	292	2	no	—
89	C₂×C₈	—	2	1	no	—
90	C₂×C₈	—	146	2	no	—
91	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
92	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
93	C₄⋊C₄	—	2	1	no	—
94	C₄⋊C₄	D₂₉₂	1	1	yes	characteristic
95	C₄²	—	2	1	no	—
96	C₄²	C₂²×D₇₃	1	1	yes	characteristic
97	C₂³	—	292	2	no	—
98	Q₈	—	2	1	no	—
99	Q₈	D₅₈₄	1	2	yes	complemented
100	D₄	—	292	2	no	—
101	D₄	—	292	2	no	—
102	D₄	—	292	2	no	—
103	D₄	—	292	2	no	—
104	D₄	—	584	2	no	—
105	C₂×C₄	—	2	1	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
108	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
109	C₂×C₄	#11924028	1	1	yes	characteristic
110	C₈	—	2	1	no	—
111	C₈	—	4	1	no	—
112	C₈	—	292	2	no	—
113	C₈	#11924015	1	2	yes	complemented
114	C₂²	—	584	2	no	—
115	C₂²	—	584	2	no	—
116	C₂²	—	584	2	no	—
117	C₂²	#14556565	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	2	1	no	—
120	C₄	—	4	1	no	—
121	C₄	C₂×D₅₈₄	1	1	yes	characteristic
122	C₄	C₂×#11924015	1	1	yes	characteristic
123	C₄	#14556613	1	1	yes	characteristic
124	C₄	#14556616	1	1	yes	characteristic
125	C₂	—	1168	2	no	—
126	C₂	#17988787	1	1	yes	characteristic
127	C₂	#17988823	1	1	yes	characteristic
128	C₂	#17988857	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#17988964

C₂

yes

maximal, characteristic

#17988787

C₂

yes

maximal, complemented

#17988340

C₂

yes

maximal, characteristic, complemented

#17988330

C₂

yes

maximal, complemented

C₈×Q₈×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×#11924015

—

C₂×D₅₈₄

—

C₂×D₅₈₄

—

#14556588

C₂²

yes

—

C₄⋊#10596831

C₂²

yes

—

#14556511

—

C₄×Q₈×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

—

C₄⋊C₈×C₇₃

C₂²

yes

characteristic

C₄×C₈×C₇₃

—

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×D₂₉₂

—

C₂×D₂₉₂

—

C₂×D₂₉₂

—

#11924015

—

D₅₈₄

—

D₅₈₄

—

C₂×#10596831

—

C₂×Q₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

D₄

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

—

C₄²×C₇₃

C₂³

yes

characteristic

C₂²×D₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

D₈

yes

complemented

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

#41

yes

characteristic

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

D₂₉₂

—

C₈×C₇₃

—

C₈×C₇₃

—

C₈×C₇₃

SD₁₆

yes

complemented

#10596831

—

C₂²×C₇₃

#121

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂×D₈

yes

characteristic

C₄×C₇₃

C₂×SD₁₆

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

D₇₃

—

#1668

—

maximal, Sylow

C₇₃

#1668

yes

characteristic, complemented, Sylow

#493

—

#460

—

C₈×Q₈

D₇₃

yes

characteristic, complemented

#335

—

#331

—

C₂×SD₁₆

—

146

—

C₂×D₈

—

146

—

C₂×D₈

—

146

—

#127

—

C₄×Q₈

C₂×D₇₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇₃

yes

characteristic

#102

—

146

—

C₄×C₈

—

C₄×C₈

C₂×D₇₃

yes

characteristic

C₂×Q₈

D₂₉₂

yes

characteristic

C₂×D₄

—

146

—

C₂×D₄

—

146

—

C₂×D₄

—

146

—

SD₁₆

—

292

—

D₈

—

292

—

D₈

—

292

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

C₂²⋊D₇₃

yes

characteristic

C₂×C₈

C₂²⋊D₇₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

D₂₉₂

yes

characteristic

C₄²

—

C₄²

C₂²×D₇₃

yes

characteristic

C₂³

—

292

—

Q₈

—

Q₈

D₅₈₄

yes

complemented

100

D₄

—

292

—

101

D₄

—

292

—

102

D₄

—

292

—

103

D₄

—

292

—

104

D₄

—

584

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

108

C₂×C₄

C₂×D₂₉₂

yes

characteristic

109

C₂×C₄

#11924028

yes

characteristic

110

C₈

—

111

C₈

—

112

C₈

—

292

—

113

C₈

#11924015

yes

complemented

114

C₂²

—

584

—

115

C₂²

—

584

—

116

C₂²

—

584

—

117

C₂²

#14556565

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

C₂×D₅₈₄

yes

characteristic

122

C₄

C₂×#11924015

yes

characteristic

123

C₄

#14556613

yes

characteristic

124

C₄

#14556616

yes

characteristic

125

C₂

—

1168

—

126

C₂

#17988787

yes

characteristic

127

C₂

#17988823

yes

characteristic

128

C₂

#17988857

yes

characteristic

	subgroup	G/H
center	C₂²	#14556565
commutator	C₄²×C₇₃	C₂³
Frattini	C₄²	C₂²×D₇₃
Fitting	C₈×Q₈×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	C₈×Q₈	D₇₃
hypocenter	C₇₃	#1668
LCS	G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

subgroup

G/H

center

C₂²

#14556565

commutator

C₄²×C₇₃

C₂³

Frattini

C₄²

C₂²×D₇₃

Fitting

C₈×Q₈×C₇₃

C₂

socle

C₂²×C₇₃

#121

hypercenter

C₈×Q₈

D₇₃

hypocenter

C₇₃

#1668

LCS

G ▷ C₄²×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈×Q₈

N	H ≅ G/N	type
2. #17988787	C₂	semidirect
3. #17988340	C₂	semidirect
4. #17988330	C₂	semidirect
5. C₈×Q₈×C₇₃	C₂	semidirect
34. Q₈×C₇₃	D₈	semidirect
47. C₈×C₇₃	SD₁₆	semidirect
65. C₇₃	#1668	semidirect
68. C₈×Q₈	D₇₃	semidirect
99. Q₈	D₅₈₄	semidirect
113. C₈	#11924015	semidirect

H ≅ G/N

type

2. #17988787

C₂

semidirect

3. #17988340

C₂

semidirect

4. #17988330

C₂

semidirect

5. C₈×Q₈×C₇₃

C₂

semidirect

34. Q₈×C₇₃

D₈

semidirect

47. C₈×C₇₃

SD₁₆

semidirect

65. C₇₃

#1668

semidirect

68. C₈×Q₈

D₇₃

semidirect

99. Q₈

D₅₈₄

semidirect

113. C₈

#11924015

semidirect

Subgroups — #22609291

Conjugacy classes of subgroups

Complements