Group #22615304 — subgroups

Subgroups — #22615304

Order 9344.

484 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 181 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×C₄×Dic₁₄₆	C₂	1	2	yes	maximal
2	#17988517	C₂	1	4	yes	maximal
3	#17988492	C₂	1	4	yes	maximal
4	#17988486	C₂	1	4	yes	maximal
5	#384×C₇₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₄₆	C₂²	1	2	yes	—
7	C₄×Dic₁₄₆	—	2	8	no	—
8	C₂×C₄⋊Dic₇₃	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₇₃	C₂²	1	4	yes	—
10	C₂×#11924007	C₂²	1	4	yes	—
11	C₂×C₄×Dic₇₃	C₂²	1	4	yes	—
12	#14556498	C₂²	1	8	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	4	no	—
18	C₂×Dic₁₄₆	—	2	4	no	—
19	C₂×Dic₁₄₆	Q₈	1	8	yes	complemented
20	C₂²×Dic₇₃	C₂³	1	4	yes	—
21	C₂²×Dic₇₃	C₂³	1	4	yes	—
22	C₄⋊Dic₇₃	—	2	4	no	—
23	C₄⋊Dic₇₃	—	4	4	no	—
24	#11924007	—	2	8	no	—
25	C₄×Dic₇₃	—	2	8	no	—
26	C₂²×C₄×C₇₃	C₂³	1	4	yes	—
27	C₂²×C₄×C₇₃	C₂³	1	1	yes	characteristic
28	C₂²×C₄×C₇₃	C₂³	1	2	yes	—
29	C₄⋊C₄×C₇₃	—	2	8	no	—
30	C₄⋊C₄×C₇₃	—	4	2	no	—
31	C₄²×C₇₃	D₄	1	4	yes	—
32	C₂³×C₇₃	C₂⁴	1	1	yes	characteristic
33	C₂×C₄×C₇₃	—	2	1	no	—
34	C₂×C₄×C₇₃	—	2	1	no	—
35	C₂×C₄×C₇₃	—	2	8	no	—
36	C₂×C₄×C₇₃	—	2	4	no	—
37	C₂×C₄×C₇₃	C₂×D₄	1	2	yes	—
38	C₂×C₄×C₇₃	C₂×D₄	1	4	yes	—
39	C₂×C₄×C₇₃	C₂×Q₈	1	4	yes	—
40	C₂×C₄×C₇₃	#41	1	4	yes	—
41	C₂×Dic₇₃	—	2	4	no	—
42	C₂×Dic₇₃	—	2	4	no	—
43	C₂×Dic₇₃	—	2	4	no	—
44	C₂×Dic₇₃	—	2	4	no	—
45	C₂×Dic₇₃	—	2	4	no	—
46	C₂×Dic₇₃	C₂×Q₈	1	8	yes	—
47	Dic₁₄₆	—	2	16	no	—
48	C₂²×C₇₃	C₂²×D₄	1	1	yes	characteristic
49	C₂²×C₇₃	C₂²×Q₈	1	2	yes	—
50	C₂²×C₇₃	C₂×#41	1	2	yes	—
51	C₂²×C₇₃	H(5,𝔽₂)	1	1	yes	characteristic
52	C₂²×C₇₃	2₋¹⁺⁴	1	1	yes	characteristic
53	C₄×C₇₃	—	2	2	no	—
54	C₄×C₇₃	—	4	4	no	—
55	C₄×C₇₃	C₂²⋊Q₈	1	8	yes	—
56	Dic₇₃	—	2	8	no	—
57	Dic₇₃	—	4	4	no	—
58	C₂×C₇₃	C₂×C₂²⋊Q₈	1	2	yes	—
59	C₂×C₇₃	#537	1	1	yes	characteristic
60	C₂×C₇₃	#557	1	2	yes	—
61	C₂×C₇₃	Q₈²	1	1	yes	characteristic
62	C₂×C₇₃	#559	1	1	yes	characteristic
63	#2582	—	73	1	no	maximal, Sylow
64	C₇₃	#2582	1	1	yes	characteristic, complemented, Sylow
65	C₂×C₄×Q₈	—	73	2	no	—
66	#398	—	73	4	no	—
67	#391	—	73	4	no	—
68	#389	—	73	4	no	—
69	#384	D₇₃	1	1	yes	characteristic
70	C₂²×Q₈	—	73	2	no	—
71	C₄×Q₈	—	146	8	no	—
72	C₂×C₄⋊C₄	—	73	4	no	—
73	C₂×C₄⋊C₄	—	73	4	no	—
74	C₂×C₄⋊C₄	—	73	2	no	—
75	C₂×C₄⋊C₄	C₂×D₇₃	1	4	yes	—
76	C₂×C₄⋊C₄	C₂×D₇₃	1	2	yes	—
77	C₂×C₄²	—	73	4	no	—
78	C₂×C₄²	C₂×D₇₃	1	1	yes	characteristic
79	#95	—	73	8	no	—
80	#95	—	73	4	no	—
81	C₂×Q₈	—	73	8	no	—
82	C₂×Q₈	—	146	4	no	—
83	C₂×Q₈	—	146	4	no	—
84	C₂²×C₄	—	73	4	no	—
85	C₂²×C₄	—	73	4	no	—
86	C₂²×C₄	C₂²×D₇₃	1	4	yes	—
87	C₂²×C₄	C₂²×D₇₃	1	1	yes	characteristic
88	C₂²×C₄	C₂²×D₇₃	1	2	yes	—
89	C₄⋊C₄	—	2	8	no	—
90	C₄⋊C₄	—	4	2	no	—
91	C₄⋊C₄	—	292	4	no	—
92	C₄⋊C₄	—	146	8	no	—
93	C₄⋊C₄	—	146	4	no	—
94	C₄²	—	146	8	no	—
95	C₄²	D₂₉₂	1	4	yes	—
96	C₂³	C₂³×D₇₃	1	1	yes	characteristic
97	Q₈	—	146	16	no	—
98	C₂×C₄	—	2	1	no	—
99	C₂×C₄	—	2	1	no	—
100	C₂×C₄	—	2	8	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	—	73	8	no	—
103	C₂×C₄	—	146	4	no	—
104	C₂×C₄	—	146	4	no	—
105	C₂×C₄	—	146	4	no	—
106	C₂×C₄	—	146	4	no	—
107	C₂×C₄	—	146	4	no	—
108	C₂×C₄	C₂×D₂₉₂	1	2	yes	—
109	C₂×C₄	C₂×D₂₉₂	1	4	yes	—
110	C₂×C₄	Q₈×D₇₃	1	4	yes	—
111	C₂×C₄	#11924027	1	4	yes	—
112	C₂²	C₂²×D₂₉₂	1	1	yes	characteristic
113	C₂²	C₂×Q₈×D₇₃	1	2	yes	—
114	C₂²	C₂×#11924027	1	2	yes	—
115	C₂²	#14556636	1	1	yes	characteristic
116	C₂²	#14556639	1	1	yes	characteristic
117	C₄	—	2	2	no	—
118	C₄	—	4	4	no	—
119	C₄	—	292	4	no	—
120	C₄	—	146	8	no	—
121	C₄	#14556571	1	8	yes	—
122	C₂	C₂×#14556571	1	2	yes	—
123	C₂	#17989168	1	1	yes	characteristic
124	C₂	#17989312	1	2	yes	—
125	C₂	#17989317	1	1	yes	characteristic
126	C₂	#17989323	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₄×Dic₁₄₆

C₂

yes

maximal

#17988517

C₂

yes

maximal

#17988492

C₂

yes

maximal

#17988486

C₂

yes

maximal

#384×C₇₃

C₂

yes

maximal, characteristic

C₂²×Dic₁₄₆

C₂²

yes

—

C₄×Dic₁₄₆

—

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

—

C₂×Dic₁₄₆

Q₈

yes

complemented

C₂²×Dic₇₃

C₂³

yes

—

C₂²×Dic₇₃

C₂³

yes

—

C₄⋊Dic₇₃

—

C₄⋊Dic₇₃

—

#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

D₄

yes

—

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×D₄

yes

—

C₂×C₄×C₇₃

C₂×Q₈

yes

—

C₂×C₄×C₇₃

#41

yes

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

—

C₂×Dic₇₃

C₂×Q₈

yes

—

Dic₁₄₆

—

C₂²×C₇₃

C₂²×D₄

yes

characteristic

C₂²×C₇₃

C₂²×Q₈

yes

—

C₂²×C₇₃

C₂×#41

yes

—

C₂²×C₇₃

H(5,𝔽₂)

yes

characteristic

C₂²×C₇₃

2₋¹⁺⁴

yes

characteristic

C₄×C₇₃

—

C₄×C₇₃

—

C₄×C₇₃

C₂²⋊Q₈

yes

—

Dic₇₃

—

Dic₇₃

—

C₂×C₇₃

C₂×C₂²⋊Q₈

yes

—

C₂×C₇₃

#537

yes

characteristic

C₂×C₇₃

#557

yes

—

C₂×C₇₃

Q₈²

yes

C₂×C₇₃

yes

—

C₇₃

yes

characteristic, complemented, Sylow

C₂×C₄×Q₈

—

—

—

—

D₇₃

yes

characteristic

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

—

C₂×Q₈

—

146

—

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

—

292

—

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

C₄²

—

146

—

C₄²

D₂₉₂

yes

—

C₂³

C₂³×D₇₃

yes

characteristic

Q₈

—

146

—

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

146

—

104

C₂×C₄

—

146

—

105

C₂×C₄

—

146

—

106

C₂×C₄

—

146

—

107

C₂×C₄

—

146

—

108

C₂×C₄

C₂×D₂₉₂

yes

—

109

C₂×C₄

C₂×D₂₉₂

yes

—

110

C₂×C₄

Q₈×D₇₃

yes

—

111

C₂×C₄

#11924027

yes

—

112

C₂²

C₂²×D₂₉₂

yes

characteristic

113

C₂²

C₂×Q₈×D₇₃

yes

—

114

C₂²

C₂×#11924027

yes

—

115

C₂²

#14556636

yes

characteristic

116

C₂²

#14556639

yes

characteristic

117

C₄

—

118

C₄

—

119

C₄

—

292

—

120

C₄

—

146

—

121

C₄

#14556571

yes

—

122

C₂

C₂×#14556571

yes

—

123

C₂

#17989168

yes

characteristic

124

C₂

#17989312

yes

—

125

C₂

#17989317

yes

characteristic

126

C₂

#17989323

yes

characteristic

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₇₃
commutator, socle	C₂³×C₇₃	C₂⁴
Fitting	#384×C₇₃	C₂
hypercenter	#384	D₇₃
hypocenter	C₇₃	#2582
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
19. C₂×Dic₁₄₆	Q₈	semidirect
64. C₇₃	#2582	semidirect

H ≅ G/N

type

19. C₂×Dic₁₄₆

Q₈

semidirect

64. C₇₃

#2582

semidirect

Subgroups — #22615304

Conjugacy classes of subgroups

Complements