Group #22610712 — subgroups

Subgroups — #22610712

Order 9344.

196 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 196 conjugacy classes of subgroups — too many to draw. Only its 97 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	#17988629	C₂	1	2	yes	maximal
2	C₂×#14556523	C₂	1	1	yes	maximal, characteristic
3	#17988542	C₂	1	1	yes	maximal, characteristic
4	#17988355	C₂	1	2	yes	maximal
5	C₂×#108×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#11924012	C₂²	1	1	yes	characteristic
7	#14556550	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
9	#14556525	—	2	2	no	—
10	#14556523	—	2	2	no	—
11	#14556523	—	2	1	no	—
12	#14556523	—	2	1	no	—
13	#14556501	—	2	1	no	—
14	C₈×Dic₇₃	—	2	1	no	—
15	C₂×M₁₆×C₇₃	C₂²	1	2	yes	—
16	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
17	#108×C₇₃	C₄	1	4	yes	complemented
18	C₂²×Dic₇₃	—	2	2	no	—
19	#11924012	—	2	1	no	—
20	#11924012	Q₈	1	2	yes	—
21	C₂×#10596831	—	2	1	no	—
22	C₄⋊Dic₇₃	—	2	2	no	—
23	C₄⋊Dic₇₃	—	2	2	no	—
24	C₄⋊Dic₇₃	D₄	1	2	yes	—
25	C₂²⋊Dic₇₃	—	2	1	no	—
26	C₄×Dic₇₃	—	2	1	no	—
27	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
28	M₁₆×C₇₃	—	2	2	no	—
29	M₁₆×C₇₃	C₂×C₄	1	4	yes	—
30	C₂×C₈×C₇₃	—	2	2	no	—
31	C₂×C₈×C₇₃	C₂×C₄	1	2	yes	—
32	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
33	C₂×C₈×C₇₃	D₄	1	1	yes	characteristic
34	C₂×C₈×C₇₃	Q₈	1	2	yes	—
35	C₂³×C₇₃	#41	1	1	yes	characteristic
36	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
40	C₂×C₄×C₇₃	C₂×Q₈	1	1	yes	characteristic
41	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
42	C₂×Dic₇₃	—	2	2	no	—
43	C₂×Dic₇₃	—	4	2	no	—
44	C₂×Dic₇₃	—	8	2	no	—
45	C₈×C₇₃	—	2	4	no	—
46	C₈×C₇₃	C₄⋊C₄	1	2	yes	—
47	C₈×C₇₃	C₄⋊C₄	1	2	yes	—
48	#10596831	—	2	2	no	—
49	C₂²×C₇₃	—	2	2	no	—
50	C₂²×C₇₃	C₄×Q₈	1	1	yes	characteristic
51	C₂²×C₇₃	#121	1	1	yes	characteristic
52	C₂²×C₇₃	#125	1	1	yes	characteristic
53	C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₄×C₇₃	C₄×D₄	1	1	yes	characteristic
55	C₄×C₇₃	C₂²⋊Q₈	1	1	yes	characteristic
56	C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
57	Dic₇₃	—	4	2	no	—
58	Dic₇₃	—	8	2	no	—
59	C₂×C₇₃	—	2	1	no	—
60	C₂×C₇₃	—	2	1	no	—
61	C₂×C₇₃	#389	1	1	yes	characteristic
62	C₂×C₇₃	#471	1	2	yes	—
63	#1951	—	73	1	no	maximal, Sylow
64	C₇₃	#1951	1	1	yes	characteristic, complemented, Sylow
65	C₂×#108	D₇₃	1	1	yes	characteristic
66	#428	—	73	2	no	—
67	C₂×#106	—	73	1	no	—
68	#405	—	73	1	no	—
69	#343	—	73	2	no	—
70	C₂×M₁₆	—	73	1	no	—
71	C₂×M₁₆	C₂×D₇₃	1	2	yes	—
72	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
73	#117	—	73	1	no	—
74	C₂×C₄⋊C₄	—	73	2	no	—
75	#108	Dic₇₃	1	4	yes	complemented
76	#107	—	146	2	no	—
77	#106	—	146	2	no	—
78	#106	—	146	1	no	—
79	#106	—	146	1	no	—
80	#97	—	146	1	no	—
81	C₄×C₈	—	146	1	no	—
82	C₂²×C₄	—	146	2	no	—
83	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
84	M₁₆	—	2	2	no	—
85	M₁₆	—	73	2	no	—
86	M₁₆	—	146	1	no	—
87	M₁₆	C₂×Dic₇₃	1	4	yes	—
88	C₂×C₈	—	2	2	no	—
89	C₂×C₈	—	146	1	no	—
90	C₂×C₈	Dic₁₄₆	1	2	yes	—
91	C₂×C₈	D₂₉₂	1	1	yes	characteristic
92	C₂×C₈	D₂₉₂	1	1	yes	characteristic
93	C₂×C₈	C₂×Dic₇₃	1	2	yes	—
94	C₄⋊C₄	—	73	2	no	—
95	C₄⋊C₄	—	146	2	no	—
96	C₄⋊C₄	—	146	2	no	—
97	C₂²⋊C₄	—	146	1	no	—
98	C₄²	—	146	1	no	—
99	C₂³	#11924029	1	1	yes	characteristic
100	C₂×C₄	—	292	2	no	—
101	C₂×C₄	—	584	2	no	—
102	C₂×C₄	—	146	2	no	—
103	C₂×C₄	C₂²×Dic₇₃	1	1	yes	characteristic
104	C₂×C₄	D₄×D₇₃	1	1	yes	characteristic
105	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₇₃	1	1	yes	characteristic
107	C₂×C₄	C₂×Dic₁₄₆	1	1	yes	characteristic
108	C₂×C₄	#11924027	1	1	yes	characteristic
109	C₈	—	2	4	no	—
110	C₈	—	146	2	no	—
111	C₈	C₄⋊Dic₇₃	1	2	yes	—
112	C₈	C₄⋊Dic₇₃	1	2	yes	—
113	C₂²	—	2	2	no	—
114	C₂²	Q₈×Dic₇₃	1	1	yes	characteristic
115	C₂²	#14556566	1	1	yes	characteristic
116	C₂²	#14556582	1	1	yes	characteristic
117	C₄	—	292	2	no	—
118	C₄	—	584	2	no	—
119	C₄	C₂×C₄⋊Dic₇₃	1	1	yes	characteristic
120	C₄	D₄×Dic₇₃	1	1	yes	characteristic
121	C₄	#14556571	1	1	yes	characteristic
122	C₄	#14556590	1	1	yes	characteristic
123	C₂	—	2	1	no	—
124	C₂	—	2	1	no	—
125	C₂	#17988486	1	1	yes	characteristic
126	C₂	#17988851	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#17988629

C₂

yes

maximal

C₂×#14556523

C₂

yes

maximal, characteristic

#17988542

C₂

yes

maximal, characteristic

#17988355

C₂

yes

maximal

C₂×#108×C₇₃

C₂

yes

maximal, characteristic

C₂×#11924012

C₂²

yes

characteristic

#14556550

C₂²

yes

characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

—

—

—

—

—

—

C₈×Dic₇₃

—

C₂×M₁₆×C₇₃

C₂²

yes

—

C₂²×C₈×C₇₃

C₂²

yes

characteristic

#108×C₇₃

C₄

yes

complemented

C₂²×Dic₇₃

—

#11924012

—

#11924012

Q₈

yes

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

D₄

yes

—

C₂²⋊Dic₇₃

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

M₁₆×C₇₃

—

M₁₆×C₇₃

C₂×C₄

yes

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

C₂×C₄

yes

—

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

Q₈

yes

—

C₂³×C₇₃

#41

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

characteristic

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

C₄⋊C₄

yes

—

C₈×C₇₃

C₄⋊C₄

yes

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

C₄×Q₈

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₃

C₄×D₄

yes

characteristic

C₄×C₇₃

C₂²⋊Q₈

yes

characteristic

C₄×C₇₃

C₄⋊Q₈

yes

characteristic

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

—

C₂×C₇₃

#389

yes

characteristic

C₂×C₇₃

#471

yes

—

#1951

—

maximal, Sylow

C₇₃

#1951

yes

characteristic, complemented, Sylow

C₂×#108

D₇₃

yes

—

—

—

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₇₃

yes

—

C₂²×C₈

C₂×D₇₃

yes

characteristic

#117

—

C₂×C₄⋊C₄

—

#108

Dic₇₃

yes

complemented

#107

—

146

—

#106

—

146

—

#106

—

146

—

#106

—

146

—

#97

—

146

—

C₄×C₈

—

146

—

C₂²×C₄

—

146

—

C₂²×C₄

C₂²×D₇₃

yes

characteristic

M₁₆

—

M₁₆

—

M₁₆

—

146

—

M₁₆

C₂×Dic₇₃

yes

—

C₂×C₈

—

C₂×C₈

—

146

—

C₂×C₈

Dic₁₄₆

yes

—

C₂×C₈

D₂₉₂

yes

characteristic

C₂×C₈

D₂₉₂

yes

characteristic

C₂×C₈

C₂×Dic₇₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

C₂²⋊C₄

—

146

—

C₄²

—

146

—

C₂³

#11924029

yes

characteristic

100

C₂×C₄

—

292

—

101

C₂×C₄

—

584

—

102

C₂×C₄

—

146

—

103

C₂×C₄

C₂²×Dic₇₃

yes

characteristic

104

C₂×C₄

D₄×D₇₃

yes

characteristic

105

C₂×C₄

C₂×D₂₉₂

yes

characteristic

106

C₂×C₄

Q₈×D₇₃

yes

characteristic

107

C₂×C₄

C₂×Dic₁₄₆

yes

characteristic

108

C₂×C₄

#11924027

yes

characteristic

109

C₈

—

110

C₈

—

146

—

111

C₈

C₄⋊Dic₇₃

yes

—

112

C₈

C₄⋊Dic₇₃

yes

—

113

C₂²

—

114

C₂²

Q₈×Dic₇₃

yes

characteristic

115

C₂²

#14556566

yes

characteristic

116

C₂²

#14556582

yes

characteristic

117

C₄

—

292

—

118

C₄

—

584

—

119

C₄

C₂×C₄⋊Dic₇₃

yes

characteristic

120

C₄

D₄×Dic₇₃

yes

characteristic

121

C₄

#14556571

yes

characteristic

122

C₄

#14556590

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#17988486

yes

characteristic

126

C₂

#17988851

yes

—

	subgroup	G/H
center	C₂²	#14556566
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×#108×C₇₃	C₂
socle	C₂²×C₇₃	#121
hypercenter	C₂×#108	D₇₃
hypocenter	C₇₃	#1951
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

subgroup

G/H

center

C₂²

#14556566

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂×#108×C₇₃

C₂

C₂²×C₇₃

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#108

N	H ≅ G/N	type
17. #108×C₇₃	C₄	semidirect
64. C₇₃	#1951	semidirect
75. #108	Dic₇₃	semidirect

H ≅ G/N

type

17. #108×C₇₃

C₄

semidirect

64. C₇₃

#1951

semidirect

75. #108

Dic₇₃

semidirect

Subgroups — #22610712

Conjugacy classes of subgroups

Complements