Group #22626278 — subgroups

Subgroups — #22626278

Order 9344.

144 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 144 conjugacy classes of subgroups — too many to draw. Only its 62 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	D₇₃⋊#108	C₂	1	1	yes	maximal, characteristic
2	#17989597	C₂	1	1	yes	maximal, characteristic
3	#17989512	C₂	1	2	yes	maximal, complemented
4	#17989511	C₂	1	2	yes	maximal, complemented
5	C₂×#14556530	C₂	1	1	yes	maximal, characteristic
6	D₇₃⋊M₁₆	—	2	1	no	—
7	#14556670	—	2	1	no	—
8	#14556662	—	2	1	no	—
9	#14556661	C₂²	1	2	yes	—
10	#14556660	—	2	1	no	—
11	#14556659	C₂²	1	2	yes	—
12	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
13	#14556530	C₄	1	1	yes	characteristic, complemented
14	#14556530	C₄	1	1	yes	characteristic, complemented
15	#14556530	C₂²	1	1	yes	characteristic
16	#14556530	C₂²	1	1	yes	characteristic
17	C₂×#11924003	C₄	1	1	yes	characteristic, complemented
18	C₂×C₁₆×C₇₃	C₄	1	1	yes	characteristic, complemented
19	#11924038	—	2	2	no	—
20	#11924037	—	4	1	no	—
21	#11924036	—	4	1	no	—
22	C₄⋊#64408	—	2	2	no	—
23	C₂²⋊#64408	—	4	1	no	—
24	C₄×#64408	—	4	1	no	—
25	C₂×C₄×D₇₃	D₄	1	1	yes	characteristic
26	C₂×#10596831	Q₈	1	1	yes	characteristic
27	C₈×D₇₃	C₂×C₄	1	1	yes	characteristic
28	C₈×D₇₃	D₄	1	1	yes	characteristic
29	C₈×D₇₃	Q₈	1	1	yes	characteristic
30	C₈×D₇₃	C₂³	1	1	yes	characteristic
31	#11924003	C₂×C₄	1	1	yes	characteristic
32	#11924003	C₂×C₄	1	1	yes	characteristic
33	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
34	C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
35	C₁₆×C₇₃	C₂×C₄	1	1	yes	characteristic
36	C₂²×D₇₃	SD₁₆	1	1	yes	characteristic
37	C₂×#64408	—	4	2	no	—
38	C₂×C₄×C₇₃	C₄⋊C₄	1	1	yes	characteristic
39	C₂×Dic₇₃	SD₁₆	1	1	yes	characteristic
40	C₄×D₇₃	C₄⋊C₄	1	1	yes	characteristic
41	C₄×D₇₃	SD₁₆	1	1	yes	characteristic
42	C₄×D₇₃	SD₁₆	1	1	yes	characteristic
43	C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
44	#10596833	—	4	2	no	—
45	C₈×C₇₃	C₄⋊C₄	1	1	yes	characteristic
46	C₈×C₇₃	C₂²×C₄	1	1	yes	characteristic
47	#10596831	C₄⋊C₄	1	1	yes	characteristic
48	#10596831	C₂×Q₈	1	1	yes	characteristic
49	C₂²×C₇₃	#106	1	1	yes	characteristic
50	C₂×D₇₃	—	4	1	no	—
51	C₂×D₇₃	#106	1	1	yes	characteristic
52	C₂×D₇₃	C₂×SD₁₆	1	1	yes	characteristic
53	#64408	—	8	2	no	—
54	C₄×C₇₃	#106	1	1	yes	characteristic
55	C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	Dic₇₃	#106	1	1	yes	characteristic
57	Dic₇₃	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₇₃	—	2	1	no	—
59	C₂×C₇₃	C₂×#106	1	1	yes	characteristic
60	D₇₃	—	2	1	no	—
61	D₇₃	—	2	1	no	—
62	#2155	—	73	1	no	maximal, Sylow
63	C₇₃	#2155	1	1	yes	characteristic, complemented, Sylow
64	C₂×M₃₂	—	73	1	no	—
65	C₂×#108	—	73	1	no	—
66	#427	—	73	1	no	—
67	#365	—	73	2	no	—
68	#365	—	73	2	no	—
69	C₂×M₁₆	—	146	1	no	—
70	C₂²×C₈	—	73	1	no	—
71	#117	—	146	1	no	—
72	M₃₂	—	73	1	no	—
73	M₃₂	—	73	1	no	—
74	M₃₂	—	73	1	no	—
75	M₃₂	—	73	1	no	—
76	C₂×C₁₆	—	73	1	no	—
77	C₂×C₁₆	#64408	1	1	yes	characteristic, complemented
78	#108	—	73	2	no	—
79	#108	—	146	1	no	—
80	#107	—	146	1	no	—
81	#106	—	73	2	no	—
82	C₂²×C₄	—	73	1	no	—
83	M₁₆	—	292	1	no	—
84	M₁₆	—	146	2	no	—
85	C₂×C₈	—	292	1	no	—
86	C₂×C₈	—	73	1	no	—
87	C₂×C₈	—	73	1	no	—
88	C₂×C₈	—	73	1	no	—
89	C₂×C₈	—	73	1	no	—
90	C₂×C₈	—	73	1	no	—
91	C₂×C₈	C₂×#64408	1	1	yes	characteristic
92	C₄⋊C₄	—	146	2	no	—
93	C₂²⋊C₄	—	292	1	no	—
94	C₄²	—	292	1	no	—
95	C₁₆	—	73	1	no	—
96	C₁₆	—	73	1	no	—
97	C₁₆	C₂×#64408	1	1	yes	characteristic
98	C₁₆	C₂×#64408	1	1	yes	characteristic
99	C₂³	—	73	1	no	—
100	C₂×C₄	—	292	2	no	—
101	C₂×C₄	—	73	1	no	—
102	C₂×C₄	—	73	1	no	—
103	C₂×C₄	—	73	1	no	—
104	C₂×C₄	—	73	1	no	—
105	C₂×C₄	—	73	1	no	—
106	C₂×C₄	C₄⋊#64408	1	1	yes	characteristic
107	C₈	—	292	2	no	—
108	C₈	—	73	1	no	—
109	C₈	—	73	1	no	—
110	C₈	C₄⋊#64408	1	1	yes	characteristic
111	C₈	C₂²×#64408	1	1	yes	characteristic
112	C₂²	—	292	1	no	—
113	C₂²	—	73	1	no	—
114	C₂²	—	73	1	no	—
115	C₂²	#14556659	1	1	yes	characteristic
116	C₄	—	584	2	no	—
117	C₄	—	73	1	no	—
118	C₄	—	73	1	no	—
119	C₄	#14556659	1	1	yes	characteristic
120	C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
121	C₂	—	2	1	no	—
122	C₂	—	146	1	no	—
123	C₂	—	146	1	no	—
124	C₂	C₂×#14556659	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

D₇₃⋊#108

C₂

yes

maximal, characteristic

#17989597

C₂

yes

maximal, characteristic

#17989512

C₂

yes

maximal, complemented

#17989511

C₂

yes

maximal, complemented

C₂×#14556530

C₂

yes

maximal, characteristic

D₇₃⋊M₁₆

—

#14556670

—

#14556662

—

#14556661

C₂²

yes

—

#14556660

—

#14556659

C₂²

yes

—

C₂×C₈×D₇₃

C₂²

yes

characteristic

#14556530

C₄

yes

characteristic, complemented

#14556530

C₄

yes

characteristic, complemented

#14556530

C₂²

yes

characteristic

#14556530

C₂²

yes

characteristic

C₂×#11924003

C₄

yes

characteristic, complemented

C₂×C₁₆×C₇₃

C₄

yes

characteristic, complemented

—

—

—

—

—

—

C₂×C₄×D₇₃

D₄

yes

characteristic

C₂×#10596831

Q₈

yes

characteristic

C₈×D₇₃

C₂×C₄

yes

characteristic

C₈×D₇₃

D₄

yes

characteristic

C₈×D₇₃

Q₈

yes

characteristic

C₈×D₇₃

C₂³

yes

characteristic

#11924003

C₂×C₄

yes

characteristic

#11924003

C₂×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₁₆×C₇₃

C₂×C₄

yes

characteristic

C₂²×D₇₃

SD₁₆

yes

characteristic

C₂×#64408

—

C₂×C₄×C₇₃

C₄⋊C₄

yes

characteristic

C₂×Dic₇₃

SD₁₆

yes

characteristic

C₄×D₇₃

C₄⋊C₄

yes

characteristic

C₄×D₇₃

SD₁₆

yes

characteristic

C₄×D₇₃

SD₁₆

yes

characteristic

C₄×D₇₃

C₂×D₄

yes

characteristic

#10596833

—

C₈×C₇₃

C₄⋊C₄

yes

characteristic

C₈×C₇₃

C₂²×C₄

yes

characteristic

#10596831

C₄⋊C₄

yes

characteristic

#10596831

C₂×Q₈

yes

characteristic

C₂²×C₇₃

#106

yes

characteristic

C₂×D₇₃

—

C₂×D₇₃

#106

yes

characteristic

C₂×D₇₃

C₂×SD₁₆

yes

characteristic

#64408

—

C₄×C₇₃

#106

yes

characteristic

C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

Dic₇₃

#106

yes

characteristic

Dic₇₃

C₂×SD₁₆

yes

characteristic

C₂×C₇₃

—

C₂×C₇₃

C₂×#106

yes

characteristic

D₇₃

—

D₇₃

—

#2155

—

maximal, Sylow

C₇₃

#2155

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

—

—

—

—

C₂×M₁₆

—

146

—

C₂²×C₈

—

#117

—

146

—

M₃₂

—

M₃₂

—

M₃₂

—

M₃₂

—

C₂×C₁₆

—

C₂×C₁₆

#64408

yes

characteristic, complemented

#108

—

#108

—

146

—

#107

—

146

—

#106

—

C₂²×C₄

—

M₁₆

—

292

—

M₁₆

—

146

—

C₂×C₈

—

292

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂×#64408

yes

characteristic

C₄⋊C₄

—

146

—

C₂²⋊C₄

—

292

—

C₄²

—

292

—

C₁₆

—

C₁₆

—

C₁₆

yes

C₁₆

yes

C₂³

—

100

C₂×C₄

—

292

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

C₄⋊#64408

yes

characteristic

107

C₈

—

292

—

108

C₈

—

109

C₈

—

110

C₈

C₄⋊#64408

yes

characteristic

111

C₈

C₂²×#64408

yes

characteristic

112

C₂²

—

292

—

113

C₂²

—

114

C₂²

—

115

C₂²

#14556659

yes

characteristic

116

C₄

—

584

—

117

C₄

—

118

C₄

—

119

C₄

#14556659

yes

characteristic

120

C₄

C₂×C₄⋊#64408

yes

characteristic

121

C₂

—

122

C₂

—

146

—

123

C₂

—

146

—

124

C₂

C₂×#14556659

yes

characteristic

	subgroup	G/H
center	C₄	#14556659
commutator	C₈×C₇₃	C₂²×C₄
Frattini	C₈	C₂²×#64408
Fitting	C₂×C₁₆×C₇₃	C₄
socle	C₂×C₇₃	C₂×#106
hypercenter	C₂×C₁₆	#64408
hypocenter	C₇₃	#2155
LCS	G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

subgroup

G/H

center

C₄

#14556659

commutator

C₈×C₇₃

C₂²×C₄

Frattini

C₈

C₂²×#64408

Fitting

C₂×C₁₆×C₇₃

C₄

C₂×C₇₃

C₂×C₁₆

C₇₃

G ▷ C₈×C₇₃ ▷ C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₄ ◁ C₂×C₄ ◁ C₂×C₈ ◁ C₂×C₁₆

N	H ≅ G/N	type
3. #17989512	C₂	semidirect
4. #17989511	C₂	semidirect
13. #14556530	C₄	semidirect
14. #14556530	C₄	semidirect
17. C₂×#11924003	C₄	semidirect
18. C₂×C₁₆×C₇₃	C₄	semidirect
63. C₇₃	#2155	semidirect
77. C₂×C₁₆	#64408	semidirect

H ≅ G/N

type

3. #17989512

C₂

semidirect

4. #17989511

C₂

semidirect

13. #14556530

C₄

semidirect

14. #14556530

C₄

semidirect

17. C₂×#11924003

C₄

semidirect

18. C₂×C₁₆×C₇₃

C₄

semidirect

63. C₇₃

#2155

semidirect

77. C₂×C₁₆

#64408

semidirect

Subgroups — #22626278

Conjugacy classes of subgroups

Complements