Group #22614960 — subgroups

Subgroups — #22614960

Order 9344.

556 conjugacy classes of subgroups; only the 197 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 197 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×#14556590	C₂	1	4	yes	maximal
2	#17988507	C₂	1	4	yes	maximal
3	#17988483	C₂	1	4	yes	maximal
4	C₄×C₄⋊Dic₇₃	C₂	1	2	yes	maximal
5	#384×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₄₆	C₂²	1	4	yes	—
7	#14556590	—	2	16	no	—
8	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
10	C₂×#11924007	C₂²	1	8	yes	—
11	C₂×C₄×Dic₇₃	C₂²	1	4	yes	—
12	#14556498	C₂²	1	4	yes	—
13	#14556498	C₂²	1	4	yes	—
14	C₂×C₄⋊C₄×C₇₃	C₂²	1	4	yes	—
15	C₂×C₄⋊C₄×C₇₃	C₂²	1	2	yes	—
16	C₂×C₄²×C₇₃	C₂²	1	1	yes	characteristic
17	C₂×Dic₁₄₆	—	2	16	no	—
18	C₂×Dic₁₄₆	—	4	4	no	—
19	C₂²×Dic₇₃	C₂³	1	4	yes	—
20	C₂²×Dic₇₃	C₂³	1	4	yes	—
21	C₄⋊Dic₇₃	D₄	1	8	yes	—
22	C₄⋊Dic₇₃	Q₈	1	8	yes	complemented
23	#11924007	—	2	16	no	—
24	C₄×Dic₇₃	—	2	8	no	—
25	C₂²×C₄×C₇₃	C₂³	1	4	yes	—
26	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
27	C₂²×C₄×C₇₃	C₂³	1	2	yes	—
28	C₄⋊C₄×C₇₃	—	2	8	no	—
29	C₄⋊C₄×C₇₃	—	4	2	no	—
30	C₄²×C₇₃	Q₈	1	4	yes	complemented
31	C₂³×C₇₃	C₂⁴	1	1	yes	characteristic
32	C₂×C₄×C₇₃	—	2	1	no	—
33	C₂×C₄×C₇₃	—	2	1	no	—
34	C₂×C₄×C₇₃	—	2	8	no	—
35	C₂×C₄×C₇₃	—	2	4	no	—
36	C₂×C₄×C₇₃	C₂×D₄	1	4	yes	—
37	C₂×C₄×C₇₃	C₂×Q₈	1	4	yes	—
38	C₂×C₄×C₇₃	C₂×Q₈	1	2	yes	—
39	C₂×C₄×C₇₃	C₂×Q₈	1	4	yes	—
40	C₂×Dic₇₃	—	2	4	no	—
41	C₂×Dic₇₃	—	2	4	no	—
42	C₂×Dic₇₃	—	2	4	no	—
43	C₂×Dic₇₃	—	2	4	no	—
44	C₂×Dic₇₃	C₂×D₄	1	8	yes	—
45	C₂×Dic₇₃	C₂×Q₈	1	8	yes	—
46	Dic₁₄₆	—	4	16	no	—
47	C₂²×C₇₃	C₂²×D₄	1	2	yes	—
48	C₂²×C₇₃	C₂²×Q₈	1	2	yes	—
49	C₂²×C₇₃	C₂²×Q₈	1	1	yes	characteristic
50	C₂²×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
51	C₂²×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
52	C₄×C₇₃	—	2	2	no	—
53	C₄×C₇₃	—	4	4	no	—
54	C₄×C₇₃	C₄⋊Q₈	1	8	yes	complemented
55	Dic₇₃	—	2	8	no	—
56	Dic₇₃	—	2	8	no	—
57	C₂×C₇₃	C₂×C₄⋊Q₈	1	2	yes	—
58	C₂×C₇₃	#544	1	1	yes	characteristic
59	C₂×C₇₃	D₄²	1	1	yes	characteristic
60	C₂×C₇₃	D₄×Q₈	1	2	yes	—
61	C₂×C₇₃	Q₈²	1	1	yes	characteristic
62	#2551	—	73	1	no	maximal, Sylow
63	C₇₃	#2551	1	1	yes	characteristic, complemented, Sylow
64	C₂×C₄⋊Q₈	—	73	4	no	—
65	#395	—	73	4	no	—
66	#389	—	73	4	no	—
67	#384	D₇₃	1	1	yes	characteristic
68	C₄×C₄⋊C₄	—	73	2	no	—
69	C₂²×Q₈	—	73	4	no	—
70	C₄⋊Q₈	—	146	16	no	—
71	C₂×C₄⋊C₄	—	73	2	no	—
72	C₂×C₄⋊C₄	—	73	2	no	—
73	C₂×C₄⋊C₄	—	73	8	no	—
74	C₂×C₄⋊C₄	C₂×D₇₃	1	4	yes	—
75	C₂×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
76	C₂×C₄²	—	73	4	no	—
77	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
78	#95	—	73	4	no	—
79	#95	—	73	4	no	—
80	C₂×Q₈	—	292	4	no	—
81	C₂×Q₈	—	146	16	no	—
82	C₂²×C₄	—	73	4	no	—
83	C₂²×C₄	—	73	4	no	—
84	C₂²×C₄	C₂²×D₇₃	1	4	yes	—
85	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
86	C₂²×C₄	C₂²×D₇₃	1	2	yes	—
87	C₄⋊C₄	—	2	8	no	—
88	C₄⋊C₄	—	4	2	no	—
89	C₄⋊C₄	—	73	8	no	—
90	C₄⋊C₄	—	73	8	no	—
91	C₄⋊C₄	—	146	16	no	—
92	C₄²	—	146	8	no	—
93	C₄²	Dic₁₄₆	1	4	yes	complemented
94	C₂³	C₂³×D₇₃	1	1	yes	characteristic
95	Q₈	—	292	16	no	—
96	C₂×C₄	—	2	1	no	—
97	C₂×C₄	—	2	1	no	—
98	C₂×C₄	—	2	8	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	73	8	no	—
101	C₂×C₄	—	73	8	no	—
102	C₂×C₄	—	146	4	no	—
103	C₂×C₄	—	146	4	no	—
104	C₂×C₄	—	146	4	no	—
105	C₂×C₄	—	146	4	no	—
106	C₂×C₄	D₄×D₇₃	1	4	yes	—
107	C₂×C₄	Q₈×D₇₃	1	4	yes	—
108	C₂×C₄	C₂×Dic₁₄₆	1	2	yes	—
109	C₂×C₄	C₂×Dic₁₄₆	1	4	yes	—
110	C₂²	C₂×D₄×D₇₃	1	2	yes	—
111	C₂²	C₂×Q₈×D₇₃	1	2	yes	—
112	C₂²	C₂²×Dic₁₄₆	1	1	yes	characteristic
113	C₂²	#14556636	1	1	yes	characteristic
114	C₂²	#14556639	1	1	yes	characteristic
115	C₄	—	2	2	no	—
116	C₄	—	4	4	no	—
117	C₄	—	146	8	no	—
118	C₄	—	146	8	no	—
119	C₄	#14556590	1	8	yes	complemented
120	C₂	C₂×#14556590	1	2	yes	—
121	C₂	#17989218	1	1	yes	characteristic
122	C₂	#17989222	1	1	yes	characteristic
123	C₂	#17989252	1	2	yes	—
124	C₂	#17989317	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556590

C₂

yes

maximal

#17988507

C₂

yes

maximal

#17988483

C₂

yes

maximal

C₄×C₄⋊Dic₇₃

C₂

yes

maximal

#384×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

C₂²

yes

—

#14556590

—

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₂×C₄⋊Dic₇₃

C₂²

yes

—

C₂×#11924007

C₂²

yes

—

C₂×C₄×Dic₇₃

C₂²

yes

—

#14556498

C₂²

yes

—

#14556498

C₂²

yes

—

C₂×C₄⋊C₄×C₇₃

C₂²

yes

—

C₂×C₄⋊C₄×C₇₃

C₂²

yes

—

C₂×C₄²×C₇₃

C₂²

yes

characteristic

C₂×Dic₁₄₆

—

C₂×Dic₁₄₆

—

C₂²×Dic₇₃

C₂³

yes

—

C₂²×Dic₇₃

C₂³

yes

—

C₄⋊Dic₇₃

D₄

yes

—

C₄⋊Dic₇₃

Q₈

yes

complemented

#11924007

—

C₄×Dic₇₃

—

C₂²×C₄×C₇₃

C₂³

yes

—

C₂²×C₄×C₇₃

C₂³

yes

characteristic

C₂²×C₄×C₇₃

C₂³

yes

—

C₄⋊C₄×C₇₃

—

C₄⋊C₄×C₇₃

—

C₄²×C₇₃

Q₈

yes

complemented

C₂³×C₇₃

C₂⁴

yes

characteristic

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

—

C₂×C₄×C₇₃

C₂×D₄

yes

—

C₂×C₄×C₇₃

C₂×Q₈

yes

—

C₂×C₄×C₇₃

C₂×Q₈

yes

—

C₂×C₄×C₇₃

C₂×Q₈

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×D₄

yes

—

C₂×Dic₇₃

C₂×Q₈

yes

—

Dic₁₄₆

—

C₂²×C₇₃

C₂²×D₄

yes

—

C₂²×C₇₃

C₂²×Q₈

yes

—

C₂²×C₇₃

C₂²×Q₈

yes

characteristic

C₂²×C₇₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₇₃

2₋¹⁺⁴

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₄⋊Q₈

yes

complemented

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

C₂×C₄⋊Q₈

yes

—

C₂×C₇₃

#544

yes

characteristic

C₂×C₇₃

D₄²

yes

characteristic

C₂×C₇₃

D₄×Q₈

yes

—

C₂×C₇₃

Q₈²

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

#395

—

#389

—

#384

D₇₃

yes

characteristic

C₄×C₄⋊C₄

—

C₂²×Q₈

—

C₄⋊Q₈

—

146

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₇₃

yes

—

C₂×C₄⋊C₄

C₂×D₇₃

yes

—

C₂×C₄²

—

C₂×C₄²

C₂×D₇₃

yes

characteristic

#95

—

#95

—

C₂×Q₈

—

292

—

C₂×Q₈

—

146

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₇₃

yes

—

C₂²×C₄

C₂²×D₇₃

yes

characteristic

C₂²×C₄

C₂²×D₇₃

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

146

—

C₄²

—

146

—

C₄²

Dic₁₄₆

yes

complemented

C₂³

C₂³×D₇₃

yes

characteristic

Q₈

—

292

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

146

—

103

C₂×C₄

—

146

—

104

C₂×C₄

—

146

—

105

C₂×C₄

—

146

—

106

C₂×C₄

D₄×D₇₃

yes

—

107

C₂×C₄

Q₈×D₇₃

yes

—

108

C₂×C₄

C₂×Dic₁₄₆

yes

—

109

C₂×C₄

C₂×Dic₁₄₆

yes

—

110

C₂²

C₂×D₄×D₇₃

yes

—

111

C₂²

C₂×Q₈×D₇₃

yes

—

112

C₂²

C₂²×Dic₁₄₆

yes

characteristic

113

C₂²

#14556636

yes

characteristic

114

C₂²

#14556639

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

146

—

118

C₄

—

146

—

119

C₄

#14556590

yes

complemented

120

C₂

C₂×#14556590

yes

—

121

C₂

#17989218

yes

characteristic

122

C₂

#17989222

yes

characteristic

123

C₂

#17989252

yes

—

124

C₂

#17989317

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₃
commutator, socle	C₂³×C₇₃	C₂⁴
Fitting	#384×C₇₃	C₂
hypercenter	#384	D₇₃
hypocenter	C₇₃	#2551
LCS	G ▷ C₂³×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂³ ◁ #384

subgroup

G/H

center, Frattini

C₂³

C₂³×D₇₃

commutator, socle

C₂³×C₇₃

C₂⁴

Fitting

#384×C₇₃

C₂

D₇₃

C₇₃

G ▷ C₂³×C₇₃ ▷ C₇₃

C₁ ◁ C₂³ ◁ #384

N	H ≅ G/N	type
22. C₄⋊Dic₇₃	Q₈	semidirect
30. C₄²×C₇₃	Q₈	semidirect
54. C₄×C₇₃	C₄⋊Q₈	semidirect
63. C₇₃	#2551	semidirect
93. C₄²	Dic₁₄₆	semidirect
119. C₄	#14556590	semidirect

H ≅ G/N

type

22. C₄⋊Dic₇₃

Q₈

semidirect

30. C₄²×C₇₃

Q₈

semidirect

54. C₄×C₇₃

C₄⋊Q₈

semidirect

63. C₇₃

#2551

semidirect

93. C₄²

Dic₁₄₆

semidirect

119. C₄

#14556590

semidirect

Subgroups — #22614960

Conjugacy classes of subgroups

Complements