Group #22608656 — subgroups

Subgroups — #22608656

Order 9344.

228 conjugacy classes of subgroups; only the 77 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 77 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×#14556591	C₂	1	1	yes	maximal, characteristic
2	#17988644	C₂	1	1	yes	maximal, characteristic
3	#17988335	C₂	1	4	yes	maximal, complemented
4	#432×C₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×Dic₁₄₆	C₄	1	2	yes	—
6	#14556591	—	2	1	no	—
7	#14556591	—	2	1	no	—
8	#14556591	C₄	1	2	yes	—
9	#14556591	C₂²	1	2	yes	—
10	C₂×C₄⋊Dic₇₃	—	2	1	no	—
11	C₂×C₄⋊Dic₇₃	—	2	1	no	—
12	C₄⋊#10596831	C₂²	1	2	yes	—
13	C₂²⋊#10596831	—	2	1	no	—
14	#14556502	—	2	1	no	—
15	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₇₃	C₂²	1	2	yes	—
17	C₂²⋊C₈×C₇₃	—	2	1	no	—
18	#97×C₇₃	—	2	1	no	—
19	C₂×Dic₁₄₆	—	2	2	no	—
20	C₂×Dic₁₄₆	—	4	4	no	—
21	C₂×Dic₁₄₆	C₂×C₄	1	4	yes	—
22	C₂²×Dic₇₃	—	2	2	no	—
23	C₂×#10596831	—	2	2	no	—
24	C₄⋊Dic₇₃	—	2	2	no	—
25	C₄⋊Dic₇₃	—	2	2	no	—
26	C₄⋊Dic₇₃	—	2	2	no	—
27	C₄⋊Dic₇₃	—	2	2	no	—
28	C₂²×C₄×C₇₃	C₂×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
30	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
31	C₂×C₈×C₇₃	—	2	2	no	—
32	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
33	C₄²×C₇₃	D₄	1	1	yes	characteristic
34	C₄²×C₇₃	D₄	1	1	yes	characteristic
35	C₄²×C₇₃	C₂³	1	1	yes	characteristic
36	C₂³×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
37	C₂×C₄×C₇₃	—	2	2	no	—
38	C₂×C₄×C₇₃	—	2	2	no	—
39	C₂×C₄×C₇₃	—	2	2	no	—
40	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
41	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
42	C₂×C₄×C₇₃	C₂²⋊C₄	1	1	yes	characteristic
43	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
46	C₂×Dic₇₃	—	2	4	no	—
47	C₂×Dic₇₃	—	4	4	no	—
48	Dic₁₄₆	—	4	4	no	—
49	Dic₁₄₆	—	4	4	no	—
50	C₈×C₇₃	—	4	2	no	—
51	#10596831	—	4	2	no	—
52	C₂²×C₇₃	—	2	1	no	—
53	C₂²×C₇₃	—	2	1	no	—
54	C₂²×C₇₃	#100	1	2	yes	—
55	C₂²×C₇₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
56	C₄×C₇₃	—	2	1	no	—
57	C₄×C₇₃	—	2	1	no	—
58	C₄×C₇₃	—	2	1	no	—
59	C₄×C₇₃	—	2	1	no	—
60	C₄×C₇₃	#137	1	2	yes	—
61	C₄×C₇₃	#137	1	2	yes	—
62	Dic₇₃	—	4	4	no	—
63	C₂×C₇₃	—	2	2	no	—
64	C₂×C₇₃	C₂×#100	1	1	yes	characteristic
65	C₂×C₇₃	#419	1	1	yes	characteristic
66	C₂×C₇₃	#419	1	1	yes	characteristic
67	#1558	—	73	1	no	maximal, Sylow
68	C₇₃	#1558	1	1	yes	characteristic, complemented, Sylow
69	C₂×C₄⋊Q₈	—	73	1	no	—
70	#432	—	73	1	no	—
71	#432	D₇₃	1	1	yes	characteristic
72	#333	—	73	4	no	—
73	C₂²×Q₈	—	73	2	no	—
74	C₄⋊Q₈	—	73	2	no	—
75	C₄⋊Q₈	—	73	2	no	—
76	C₄⋊Q₈	—	146	1	no	—
77	C₄⋊Q₈	—	146	1	no	—
78	C₂×C₄⋊C₄	—	146	1	no	—
79	C₂×C₄⋊C₄	—	146	1	no	—
80	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
81	C₄⋊C₈	—	73	2	no	—
82	C₄⋊C₈	C₂×D₇₃	1	2	yes	—
83	C₂²⋊C₈	—	2	1	no	—
84	C₂²⋊C₈	—	146	1	no	—
85	#97	—	2	1	no	—
86	#97	—	146	1	no	—
87	C₂×Q₈	—	292	4	no	—
88	C₂×Q₈	—	73	4	no	—
89	C₂×Q₈	—	146	2	no	—
90	C₂²×C₄	—	146	2	no	—
91	C₂²×C₄	C₄×D₇₃	1	1	yes	characteristic
92	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
93	C₂²×C₄	C₂²⋊D₇₃	1	1	yes	characteristic
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	146	2	no	—
96	C₄⋊C₄	—	146	2	no	—
97	C₄⋊C₄	—	146	2	no	—
98	C₄⋊C₄	—	146	2	no	—
99	C₄⋊C₄	—	146	2	no	—
100	C₄²	C₄×D₇₃	1	1	yes	characteristic
101	C₄²	D₂₉₂	1	1	yes	characteristic
102	C₄²	C₂²⋊D₇₃	1	1	yes	characteristic
103	C₄²	C₂²×D₇₃	1	1	yes	characteristic
104	C₂³	#11924005	1	1	yes	characteristic
105	Q₈	—	292	4	no	—
106	Q₈	—	292	4	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	2	2	no	—
110	C₂×C₄	—	292	4	no	—
111	C₂×C₄	—	146	4	no	—
112	C₂×C₄	#11924005	1	1	yes	characteristic
113	C₂×C₄	#11924005	1	1	yes	characteristic
114	C₂×C₄	#11924005	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄×D₇₃	1	1	yes	characteristic
116	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
117	C₂×C₄	C₂×D₂₉₂	1	1	yes	characteristic
118	C₈	—	4	2	no	—
119	C₈	—	292	2	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	#14556507	1	2	yes	—
123	C₂²	C₂×#11924005	1	1	yes	characteristic
124	C₄	—	2	1	no	—
125	C₄	—	2	1	no	—
126	C₄	—	2	1	no	—
127	C₄	—	2	1	no	—
128	C₄	—	292	4	no	—
129	C₄	#14556621	1	2	yes	—
130	C₄	#14556624	1	2	yes	—
131	C₂	—	2	2	no	—
132	C₂	C₂×#14556507	1	1	yes	characteristic
133	C₂	#17988595	1	1	yes	characteristic
134	C₂	#17988597	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556591

C₂

yes

maximal, characteristic

#17988644

C₂

yes

maximal, characteristic

#17988335

C₂

yes

maximal, complemented

#432×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

C₄

yes

—

#14556591

—

#14556591

—

#14556591

C₄

yes

—

#14556591

C₂²

yes

—

C₂×C₄⋊Dic₇₃

—

C₂×C₄⋊Dic₇₃

—

C₄⋊#10596831

C₂²

yes

—

C₂²⋊#10596831

—

#14556502

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

C₂²

yes

—

C₂²⋊C₈×C₇₃

—

#97×C₇₃

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

C₂×C₄

yes

—

C₂²×Dic₇₃

—

C₂×#10596831

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₂²×C₄×C₇₃

C₂×C₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂×C₈×C₇₃

—

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

D₄

yes

characteristic

C₄²×C₇₃

C₂³

yes

characteristic

C₂³×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²⋊C₄

yes

characteristic

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

Dic₁₄₆

—

Dic₁₄₆

—

C₈×C₇₃

—

#10596831

—

C₂²×C₇₃

—

C₂²×C₇₃

—

C₂²×C₇₃

#100

yes

—

C₂²×C₇₃

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#137

yes

—

C₄×C₇₃

#137

yes

—

Dic₇₃

—

C₂×C₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#432

—

#432

D₇₃

yes

characteristic

#333

—

C₂²×Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

146

—

C₄⋊Q₈

—

146

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

C₂×D₇₃

yes

—

C₂²⋊C₈

—

C₂²⋊C₈

—

146

—

#97

—

#97

—

146

—

C₂×Q₈

—

292

—

C₂×Q₈

—

C₂×Q₈

—

146

—

C₂²×C₄

—

146

—

C₂²×C₄

C₄×D₇₃

yes

characteristic

C₂²×C₄

D₂₉₂

yes

characteristic

C₂²×C₄

C₂²⋊D₇₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

146

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

100

C₄²

C₄×D₇₃

yes

characteristic

101

C₄²

D₂₉₂

yes

characteristic

102

C₄²

C₂²⋊D₇₃

yes

characteristic

103

C₄²

C₂²×D₇₃

yes

characteristic

104

C₂³

#11924005

yes

characteristic

105

Q₈

—

292

—

106

Q₈

—

292

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

292

—

111

C₂×C₄

—

146

—

112

C₂×C₄

#11924005

yes

characteristic

113

C₂×C₄

#11924005

yes

characteristic

114

C₂×C₄

#11924005

yes

characteristic

115

C₂×C₄

C₂×C₄×D₇₃

yes

characteristic

116

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

117

C₂×C₄

C₂×D₂₉₂

yes

characteristic

118

C₈

—

119

C₈

—

292

—

120

C₂²

—

121

C₂²

—

122

C₂²

#14556507

yes

—

123

C₂²

C₂×#11924005

yes

characteristic

124

C₄

—

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

292

—

129

C₄

#14556621

yes

—

130

C₄

#14556624

yes

—

131

C₂

—

132

C₂

C₂×#14556507

yes

characteristic

133

C₂

#17988595

yes

characteristic

134

C₂

#17988597

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#11924005
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₄²	C₂²×D₇₃
Fitting	#432×C₇₃	C₂
socle	C₂²×C₇₃	C₂×C₂²⋊C₄
hypercenter	#432	D₇₃
hypocenter	C₇₃	#1558
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

subgroup

G/H

center

C₂²

C₂×#11924005

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₄²

C₂²×D₇₃

Fitting

#432×C₇₃

C₂

socle

C₂²×C₇₃

C₂×C₂²⋊C₄

D₇₃

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂²×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #432

N	H ≅ G/N	type
3. #17988335	C₂	semidirect
68. C₇₃	#1558	semidirect

H ≅ G/N

type

3. #17988335

C₂

semidirect

68. C₇₃

#1558

semidirect

Subgroups — #22608656

Conjugacy classes of subgroups

Complements