Group #22611973 — subgroups

Subgroups — #22611973

Order 9344.

166 conjugacy classes of subgroups; only the 55 normal subgroups are shown.

normal subgroups only

This group has 166 conjugacy classes of subgroups — too many to draw. Only its 55 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×#14556534	C₂	1	2	yes	maximal
2	C₄×D₅₈₄	C₂	1	1	yes	maximal, characteristic
3	C₄⋊#11924003	C₂	1	1	yes	maximal, characteristic, complemented
4	#17988391	C₂	1	2	yes	maximal, complemented
5	#494×C₇₃	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×D₅₈₄	C₂²	1	1	yes	characteristic
7	C₄×D₂₉₂	—	2	1	no	—
8	#14556534	—	2	4	no	—
9	C₂×#11924003	C₂²	1	2	yes	—
10	#14556525	C₂²	1	1	yes	characteristic
11	#14556513	—	2	1	no	—
12	C₂×Q₁₆×C₇₃	—	2	1	no	—
13	C₂×Q₁₆×C₇₃	C₂²	1	2	yes	—
14	C₄⋊Q₈×C₇₃	—	2	1	no	—
15	C₄×C₈×C₇₃	C₂²	1	1	yes	characteristic
16	C₂×D₂₉₂	—	2	1	no	—
17	C₂×C₄×D₇₃	—	4	1	no	—
18	D₅₈₄	—	2	1	no	—
19	D₅₈₄	D₄	1	2	yes	—
20	C₄⋊Dic₇₃	—	2	1	no	—
21	#11924005	—	4	1	no	—
22	#11924003	—	2	2	no	—
23	C₂×Q₈×C₇₃	—	2	2	no	—
24	Q₁₆×C₇₃	—	2	2	no	—
25	Q₁₆×C₇₃	—	2	2	no	—
26	Q₁₆×C₇₃	—	4	2	no	—
27	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
28	C₂×C₈×C₇₃	C₂³	1	1	yes	characteristic
29	C₄⋊C₄×C₇₃	—	4	1	no	—
30	C₄⋊C₄×C₇₃	—	4	1	no	—
31	C₄²×C₇₃	D₄	1	1	yes	characteristic
32	C₂²×D₇₃	—	4	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	D₈	1	1	yes	characteristic
37	C₂×C₄×C₇₃	D₈	1	1	yes	characteristic
38	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
39	C₂×Dic₇₃	—	4	1	no	—
40	D₂₉₂	—	2	2	no	—
41	D₂₉₂	—	4	1	no	—
42	C₄×D₇₃	—	8	2	no	—
43	C₈×C₇₃	—	2	1	no	—
44	C₈×C₇₃	C₂×D₄	1	1	yes	characteristic
45	C₈×C₇₃	#41	1	1	yes	characteristic
46	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
47	C₂×D₇₃	—	4	2	no	—
48	C₂×D₇₃	—	8	1	no	—
49	C₂×D₇₃	—	8	1	no	—
50	C₄×C₇₃	—	2	1	no	—
51	C₄×C₇₃	—	8	2	no	—
52	C₄×C₇₃	SD₃₂	1	2	yes	complemented
53	C₄×C₇₃	#121	1	1	yes	characteristic
54	C₄×C₇₃	AGL(1,ℤ/8)	1	1	yes	characteristic
55	Dic₇₃	—	8	1	no	—
56	C₂×C₇₃	#459	1	1	yes	characteristic
57	C₂×C₇₃	C₂×SD₃₂	1	1	yes	characteristic
58	C₂×C₇₃	#510	1	1	yes	characteristic
59	D₇₃	—	8	2	no	—
60	#2208	—	73	1	no	maximal, Sylow
61	C₇₃	#2208	1	1	yes	characteristic, complemented, Sylow
62	C₂×SD₃₂	—	73	2	no	—
63	#494	D₇₃	1	1	yes	characteristic, complemented
64	C₄×D₈	—	73	1	no	—
65	C₄⋊C₁₆	—	73	1	no	—
66	#358	—	73	2	no	—
67	C₂×Q₁₆	—	2	1	no	—
68	C₂×Q₁₆	C₂×D₇₃	1	2	yes	—
69	C₂×D₈	—	73	1	no	—
70	C₄⋊Q₈	—	2	1	no	—
71	C₄×D₄	—	146	1	no	—
72	SD₃₂	—	146	4	no	—
73	C₂×C₁₆	—	73	2	no	—
74	#107	—	73	1	no	—
75	#102	—	146	1	no	—
76	C₄×C₈	C₂×D₇₃	1	1	yes	characteristic
77	C₂×Q₈	—	2	2	no	—
78	C₂×D₄	—	146	1	no	—
79	C₂²×C₄	—	292	1	no	—
80	Q₁₆	—	2	2	no	—
81	Q₁₆	—	2	2	no	—
82	Q₁₆	—	4	2	no	—
83	D₈	—	73	2	no	—
84	D₈	—	146	1	no	—
85	C₂×C₈	C₂²⋊D₇₃	1	1	yes	characteristic
86	C₂×C₈	C₂²×D₇₃	1	1	yes	characteristic
87	C₄⋊C₄	—	4	1	no	—
88	C₄⋊C₄	—	4	1	no	—
89	C₄⋊C₄	—	146	1	no	—
90	C₂²⋊C₄	—	292	1	no	—
91	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
92	C₁₆	—	146	2	no	—
93	C₂³	—	292	1	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	4	2	no	—
96	D₄	—	292	1	no	—
97	D₄	—	146	2	no	—
98	C₂×C₄	—	4	2	no	—
99	C₂×C₄	—	292	1	no	—
100	C₂×C₄	—	584	2	no	—
101	C₂×C₄	#11924013	1	1	yes	characteristic
102	C₂×C₄	#11924013	1	1	yes	characteristic
103	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
104	C₈	—	2	1	no	—
105	C₈	D₄×D₇₃	1	1	yes	characteristic
106	C₈	#11924027	1	1	yes	characteristic
107	C₂²	—	292	2	no	—
108	C₂²	—	584	1	no	—
109	C₂²	—	584	1	no	—
110	C₂²	C₂×#11924013	1	1	yes	characteristic
111	C₄	—	2	1	no	—
112	C₄	—	8	2	no	—
113	C₄	—	584	1	no	—
114	C₄	#14556534	1	2	yes	complemented
115	C₄	#14556562	1	1	yes	characteristic
116	C₄	#14556617	1	1	yes	characteristic
117	C₂	—	584	2	no	—
118	C₂	#17988779	1	1	yes	characteristic
119	C₂	C₂×#14556534	1	1	yes	characteristic
120	C₂	#17989028	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556534

C₂

yes

maximal

C₄×D₅₈₄

C₂

yes

maximal, characteristic

C₄⋊#11924003

C₂

yes

maximal, characteristic, complemented

#17988391

C₂

yes

maximal, complemented

#494×C₇₃

C₂

yes

maximal, characteristic, complemented

C₂×D₅₈₄

C₂²

yes

characteristic

C₄×D₂₉₂

—

#14556534

—

C₂×#11924003

C₂²

yes

—

#14556525

C₂²

yes

characteristic

#14556513

—

C₂×Q₁₆×C₇₃

—

C₂×Q₁₆×C₇₃

C₂²

yes

—

C₄⋊Q₈×C₇₃

—

C₄×C₈×C₇₃

C₂²

yes

characteristic

C₂×D₂₉₂

—

C₂×C₄×D₇₃

—

D₅₈₄

—

D₅₈₄

D₄

yes

—

C₄⋊Dic₇₃

—

#11924005

—

#11924003

—

C₂×Q₈×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

Q₁₆×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

C₂³

yes

characteristic

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

D₄

yes

characteristic

C₂²×D₇₃

—

Q₈×C₇₃

—

Q₈×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₈

yes

characteristic

C₂×C₄×C₇₃

D₈

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

D₂₉₂

—

D₂₉₂

—

C₄×D₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₂×D₄

yes

characteristic

C₈×C₇₃

#41

yes

characteristic

C₂²×C₇₃

C₂×D₈

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

—

C₂×D₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

SD₃₂

yes

C₄×C₇₃

yes

C₄×C₇₃

yes

Dic₇₃

—

C₂×C₇₃

#459

yes

characteristic

C₂×C₇₃

C₂×SD₃₂

yes

characteristic

C₂×C₇₃

#510

yes

characteristic

D₇₃

—

#2208

—

maximal, Sylow

C₇₃

#2208

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

#494

D₇₃

yes

characteristic, complemented

C₄×D₈

—

C₄⋊C₁₆

—

#358

—

C₂×Q₁₆

—

C₂×Q₁₆

C₂×D₇₃

yes

—

C₂×D₈

—

C₄⋊Q₈

—

C₄×D₄

—

146

—

SD₃₂

—

146

—

C₂×C₁₆

—

#107

—

#102

—

146

—

C₄×C₈

C₂×D₇₃

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

146

—

C₂²×C₄

—

292

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

D₈

—

D₈

—

146

—

C₂×C₈

C₂²⋊D₇₃

yes

characteristic

C₂×C₈

C₂²×D₇₃

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

146

—

C₂²⋊C₄

—

292

—

C₄²

C₂²⋊D₇₃

yes

characteristic

C₁₆

—

146

—

C₂³

—

292

—

Q₈

—

Q₈

—

D₄

—

292

—

D₄

—

146

—

C₂×C₄

—

C₂×C₄

—

292

—

100

C₂×C₄

—

584

—

101

C₂×C₄

#11924013

yes

characteristic

102

C₂×C₄

#11924013

yes

characteristic

103

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

104

C₈

—

105

C₈

D₄×D₇₃

yes

characteristic

106

C₈

#11924027

yes

characteristic

107

C₂²

—

292

—

108

C₂²

—

584

—

109

C₂²

—

584

—

110

C₂²

C₂×#11924013

yes

characteristic

111

C₄

—

112

C₄

—

113

C₄

—

584

—

114

C₄

#14556534

yes

complemented

115

C₄

#14556562

yes

characteristic

116

C₄

#14556617

yes

characteristic

117

C₂

—

584

—

118

C₂

#17988779

yes

characteristic

119

C₂

C₂×#14556534

yes

characteristic

120

C₂

#17989028

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11924013
commutator	C₂×C₈×C₇₃	C₂³
Frattini	C₂×C₈	C₂²×D₇₃
Fitting	#494×C₇₃	C₂
socle	C₂²×C₇₃	C₂×D₈
hypercenter	#494	D₇₃
hypocenter	C₇₃	#2208
LCS	G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

subgroup

G/H

center

C₂²

C₂×#11924013

commutator

C₂×C₈×C₇₃

C₂³

Frattini

C₂×C₈

C₂²×D₇₃

Fitting

#494×C₇₃

C₂

socle

C₂²×C₇₃

C₂×D₈

D₇₃

C₇₃

G ▷ C₂×C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #494

N	H ≅ G/N	type
3. C₄⋊#11924003	C₂	semidirect
4. #17988391	C₂	semidirect
5. #494×C₇₃	C₂	semidirect
52. C₄×C₇₃	SD₃₂	semidirect
61. C₇₃	#2208	semidirect
63. #494	D₇₃	semidirect
114. C₄	#14556534	semidirect

H ≅ G/N

type

3. C₄⋊#11924003

C₂

semidirect

4. #17988391

C₂

semidirect

5. #494×C₇₃

C₂

semidirect

52. C₄×C₇₃

SD₃₂

semidirect

61. C₇₃

#2208

semidirect

63. #494

D₇₃

semidirect

114. C₄

#14556534

semidirect

Subgroups — #22611973

Conjugacy classes of subgroups

Complements