Group #22611419 — subgroups

Subgroups — #22611419

Order 9344.

224 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 224 conjugacy classes of subgroups — too many to draw. Only its 89 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×#14556525	C₂	1	1	yes	maximal, characteristic
2	#17988485	C₂	1	2	yes	maximal
3	#17988350	C₂	1	2	yes	maximal
4	#17988350	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊C₈×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
8	C₂×#11924007	—	2	2	no	—
9	#14556525	—	2	2	no	—
10	#14556498	—	2	2	no	—
11	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
12	C₂²×C₈×C₇₃	C₂²	1	1	yes	characteristic
13	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
14	C₄⋊C₈×C₇₃	—	2	2	no	—
15	C₂²×Dic₇₃	—	2	2	no	—
16	C₂²×Dic₇₃	—	2	2	no	—
17	C₄⋊Dic₇₃	—	2	2	no	—
18	C₄⋊Dic₇₃	—	2	2	no	—
19	C₄⋊Dic₇₃	—	2	2	no	—
20	C₄⋊Dic₇₃	Q₈	1	4	yes	—
21	#11924007	—	4	4	no	—
22	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
23	C₂²×C₄×C₇₃	D₄	1	1	yes	characteristic
24	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
25	C₂×C₈×C₇₃	—	2	2	no	—
26	C₂×C₈×C₇₃	—	2	1	no	—
27	C₂×C₈×C₇₃	—	2	2	no	—
28	C₂×C₈×C₇₃	D₄	1	2	yes	—
29	C₄²×C₇₃	—	2	1	no	—
30	C₄²×C₇₃	—	2	1	no	—
31	C₂³×C₇₃	C₂×D₄	1	1	yes	characteristic
32	C₂×C₄×C₇₃	—	2	1	no	—
33	C₂×C₄×C₇₃	—	2	1	no	—
34	C₂×C₄×C₇₃	—	2	1	no	—
35	C₂×C₄×C₇₃	—	2	2	no	—
36	C₂×C₄×C₇₃	D₈	1	2	yes	—
37	C₂×C₄×C₇₃	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₇₃	C₂×Q₈	1	2	yes	—
39	C₂×C₄×C₇₃	#41	1	2	yes	—
40	C₂×C₄×C₇₃	#41	1	1	yes	characteristic
41	C₂×Dic₇₃	—	2	4	no	—
42	C₂×Dic₇₃	—	4	2	no	—
43	C₂×Dic₇₃	—	4	2	no	—
44	C₂×Dic₇₃	—	4	2	no	—
45	C₂×Dic₇₃	—	4	2	no	—
46	C₂×Dic₇₃	—	4	2	no	—
47	C₈×C₇₃	—	2	2	no	—
48	C₈×C₇₃	—	4	1	no	—
49	C₂²×C₇₃	#121	1	1	yes	characteristic
50	C₂²×C₇₃	C₂²⋊Q₈	1	2	yes	—
51	C₂²×C₇₃	C₂×D₈	1	1	yes	characteristic
52	C₂²×C₇₃	#135	1	1	yes	characteristic
53	C₂²×C₇₃	#137	1	2	yes	—
54	C₄×C₇₃	—	2	2	no	—
55	C₄×C₇₃	—	4	1	no	—
56	C₄×C₇₃	#123	1	1	yes	characteristic
57	C₄×C₇₃	#125	1	2	yes	—
58	C₄×C₇₃	C₄⋊Q₈	1	1	yes	characteristic
59	Dic₇₃	—	4	4	no	—
60	Dic₇₃	—	8	2	no	—
61	C₂×C₇₃	#398	1	1	yes	characteristic
62	C₂×C₇₃	#466	1	1	yes	characteristic
63	C₂×C₇₃	#470	1	1	yes	characteristic
64	C₂×C₇₃	#474	1	2	yes	—
65	C₂×C₇₃	#479	1	2	yes	—
66	#2086	—	73	1	no	maximal, Sylow
67	C₇₃	#2086	1	1	yes	characteristic, complemented, Sylow
68	C₂×#107	—	73	1	no	—
69	C₂×C₄⋊C₈	D₇₃	1	1	yes	characteristic
70	#389	—	73	2	no	—
71	#340	—	73	1	no	—
72	#340	—	73	2	no	—
73	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
74	C₂²×C₈	C₂×D₇₃	1	1	yes	characteristic
75	C₂×C₄⋊C₄	—	73	2	no	—
76	C₂×C₄⋊C₄	—	73	2	no	—
77	C₂×C₄⋊C₄	—	146	2	no	—
78	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
79	#107	—	146	2	no	—
80	C₄⋊C₈	—	2	2	no	—
81	#95	—	146	2	no	—
82	C₂²×C₄	—	146	2	no	—
83	C₂²×C₄	—	146	2	no	—
84	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
85	C₂²×C₄	D₂₉₂	1	1	yes	characteristic
86	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
87	C₂×C₈	—	2	2	no	—
88	C₂×C₈	—	2	1	no	—
89	C₂×C₈	—	2	2	no	—
90	C₂×C₈	C₂²⋊D₇₃	1	2	yes	—
91	C₄⋊C₄	—	292	4	no	—
92	C₄⋊C₄	—	73	4	no	—
93	C₄⋊C₄	—	146	2	no	—
94	C₄⋊C₄	—	146	2	no	—
95	C₄⋊C₄	—	146	2	no	—
96	C₄²	—	2	1	no	—
97	C₄²	—	2	1	no	—
98	C₂³	C₂×D₂₉₂	1	1	yes	characteristic
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	1	no	—
101	C₂×C₄	—	2	1	no	—
102	C₂×C₄	—	2	2	no	—
103	C₂×C₄	—	292	2	no	—
104	C₂×C₄	—	292	2	no	—
105	C₂×C₄	—	292	2	no	—
106	C₂×C₄	—	292	2	no	—
107	C₂×C₄	—	292	2	no	—
108	C₂×C₄	—	146	4	no	—
109	C₂×C₄	D₅₈₄	1	2	yes	—
110	C₂×C₄	C₂×C₂²⋊D₇₃	1	1	yes	characteristic
111	C₂×C₄	Q₈×D₇₃	1	2	yes	—
112	C₂×C₄	#11924027	1	2	yes	—
113	C₂×C₄	#11924028	1	1	yes	characteristic
114	C₈	—	2	2	no	—
115	C₈	—	4	1	no	—
116	C₂²	#14556565	1	1	yes	characteristic
117	C₂²	#14556571	1	2	yes	—
118	C₂²	C₂×D₅₈₄	1	1	yes	characteristic
119	C₂²	#14556613	1	1	yes	characteristic
120	C₂²	#14556624	1	2	yes	—
121	C₄	—	2	2	no	—
122	C₄	—	4	1	no	—
123	C₄	—	292	4	no	—
124	C₄	—	584	2	no	—
125	C₄	#14556572	1	1	yes	characteristic
126	C₄	#14556581	1	2	yes	—
127	C₄	#14556589	1	1	yes	characteristic
128	C₂	#17988517	1	1	yes	characteristic
129	C₂	#17988823	1	1	yes	characteristic
130	C₂	#17988845	1	1	yes	characteristic
131	C₂	#17988868	1	2	yes	—
132	C₂	#17988895	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556525

C₂

yes

maximal, characteristic

#17988485

C₂

yes

maximal

#17988350

C₂

yes

maximal

#17988350

C₂

yes

maximal, characteristic

C₂×C₄⋊C₈×C₇₃

C₂

yes

maximal, characteristic

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₂×#11924007

—

#14556525

—

#14556498

—

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂²×C₈×C₇₃

C₂²

yes

characteristic

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₄⋊C₈×C₇₃

—

C₂²×Dic₇₃

—

C₂²×Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

Q₈

yes

—

#11924007

—

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

D₄

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

—

C₂×C₈×C₇₃

D₄

yes

—

C₄²×C₇₃

—

C₄²×C₇₃

—

C₂³×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

D₈

yes

—

C₂×C₄×C₇₃

C₂×D₄

yes

characteristic

C₂×C₄×C₇₃

C₂×Q₈

yes

—

C₂×C₄×C₇₃

#41

yes

—

C₂×C₄×C₇₃

#41

yes

characteristic

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₈×C₇₃

—

C₈×C₇₃

—

C₂²×C₇₃

#121

yes

characteristic

C₂²×C₇₃

C₂²⋊Q₈

yes

—

C₂²×C₇₃

C₂×D₈

yes

C₂²×C₇₃

yes

C₂²×C₇₃

yes

—

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

#123

yes

characteristic

C₄×C₇₃

#125

yes

—

C₄×C₇₃

C₄⋊Q₈

yes

characteristic

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

C₂×C₇₃

#479

yes

—

#2086

—

maximal, Sylow

C₇₃

#2086

yes

characteristic, complemented, Sylow

C₂×#107

—

C₂×C₄⋊C₈

D₇₃

yes

—

—

—

C₂²×C₈

C₂×D₇₃

yes

characteristic

C₂²×C₈

C₂×D₇₃

yes

characteristic

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

146

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

#107

—

146

—

C₄⋊C₈

—

#95

—

146

—

C₂²×C₄

—

146

—

C₂²×C₄

—

146

—

C₂²×C₄

D₂₉₂

yes

characteristic

C₂²×C₄

D₂₉₂

yes

characteristic

C₂²×C₄

C₂²×D₇₃

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₇₃

yes

—

C₄⋊C₄

—

292

—

C₄⋊C₄

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

C₄²

—

C₄²

—

C₂³

C₂×D₂₉₂

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

292

—

104

C₂×C₄

—

292

—

105

C₂×C₄

—

292

—

106

C₂×C₄

—

292

—

107

C₂×C₄

—

292

—

108

C₂×C₄

—

146

—

109

C₂×C₄

D₅₈₄

yes

—

110

C₂×C₄

C₂×C₂²⋊D₇₃

yes

characteristic

111

C₂×C₄

Q₈×D₇₃

yes

—

112

C₂×C₄

#11924027

yes

—

113

C₂×C₄

#11924028

yes

characteristic

114

C₈

—

115

C₈

—

116

C₂²

#14556565

yes

characteristic

117

C₂²

#14556571

yes

—

118

C₂²

C₂×D₅₈₄

yes

characteristic

119

C₂²

#14556613

yes

characteristic

120

C₂²

#14556624

yes

—

121

C₄

—

122

C₄

—

123

C₄

—

292

—

124

C₄

—

584

—

125

C₄

#14556572

yes

characteristic

126

C₄

#14556581

yes

—

127

C₄

#14556589

yes

characteristic

128

C₂

#17988517

yes

characteristic

129

C₂

#17988823

yes

characteristic

130

C₂

#17988845

yes

characteristic

131

C₂

#17988868

yes

—

132

C₂

#17988895

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₉₂
commutator	C₂²×C₄×C₇₃	C₂³
Frattini	C₂²×C₄	C₂²×D₇₃
Fitting	C₂×C₄⋊C₈×C₇₃	C₂
socle	C₂³×C₇₃	C₂×D₄
hypercenter	C₂×C₄⋊C₈	D₇₃
hypocenter	C₇₃	#2086
LCS	G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

subgroup

G/H

center

C₂³

C₂×D₂₉₂

commutator

C₂²×C₄×C₇₃

C₂³

Frattini

C₂²×C₄

C₂²×D₇₃

Fitting

C₂×C₄⋊C₈×C₇₃

C₂

socle

C₂³×C₇₃

C₂×D₄

hypercenter

C₂×C₄⋊C₈

D₇₃

hypocenter

C₇₃

#2086

LCS

G ▷ C₂²×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄⋊C₈

N	H ≅ G/N	type
67. C₇₃	#2086	semidirect

H ≅ G/N

type

67. C₇₃

#2086

semidirect

Subgroups — #22611419

Conjugacy classes of subgroups

Complements