Group #22625868 — subgroups

Subgroups — #22625868

Order 9344.

248 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 248 conjugacy classes of subgroups — too many to draw. Only its 94 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×#14556660	C₂	1	2	yes	maximal
2	C₂×#14556659	C₂	1	2	yes	maximal
3	#17989531	C₂	1	2	yes	maximal
4	C₄×C₈×D₇₃	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄⋊#64408	C₂²	1	4	yes	—
6	#14556660	—	2	4	no	—
7	#14556659	—	2	4	no	—
8	(C₂×#64408)⋊C₄	—	2	4	no	—
9	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
10	C₂×C₈×D₇₃	C₂²	1	1	yes	characteristic
11	C₄²×D₇₃	C₂²	1	1	yes	characteristic
12	C₄×#10596831	C₄	1	1	yes	characteristic, complemented
13	C₈×Dic₇₃	C₄	1	1	yes	characteristic, complemented
14	C₈×Dic₇₃	C₄	1	1	yes	characteristic, complemented
15	C₄×C₈×C₇₃	C₄	1	1	yes	characteristic, complemented
16	C₂²×#64408	—	2	4	no	—
17	C₄⋊#64408	—	2	4	no	—
18	C₄⋊#64408	—	2	4	no	—
19	C₂×C₄×D₇₃	D₄	1	2	yes	—
20	C₂×C₄×D₇₃	C₂³	1	1	yes	characteristic
21	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
22	C₂×#10596831	C₂×C₄	1	1	yes	characteristic
23	C₈×D₇₃	D₄	1	2	yes	—
24	C₈×D₇₃	D₄	1	2	yes	—
25	C₈×D₇₃	Q₈	1	2	yes	—
26	C₈×D₇₃	Q₈	1	2	yes	—
27	C₄×Dic₇₃	C₂×C₄	1	1	yes	characteristic
28	C₄×Dic₇₃	Q₈	1	2	yes	—
29	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₈×C₇₃	C₂×C₄	1	1	yes	characteristic
31	C₄²×C₇₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×D₇₃	C₂×D₄	1	1	yes	characteristic
33	C₂×#64408	—	4	4	no	—
34	C₂×#64408	—	4	8	no	—
35	C₂×C₄×C₇₃	C₄⋊C₄	1	2	yes	—
36	C₂×C₄×C₇₃	C₂²×C₄	1	1	yes	characteristic
37	C₂×Dic₇₃	C₄⋊C₄	1	2	yes	—
38	C₂×Dic₇₃	C₂×Q₈	1	1	yes	characteristic
39	C₄×D₇₃	—	2	4	no	—
40	C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
41	C₄×D₇₃	C₂×D₄	1	1	yes	characteristic
42	C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
43	C₄×D₇₃	C₂×Q₈	1	1	yes	characteristic
44	C₈×C₇₃	C₄⋊C₄	1	2	yes	—
45	C₈×C₇₃	C₄⋊C₄	1	2	yes	—
46	#10596831	C₄⋊C₄	1	2	yes	—
47	#10596831	C₄⋊C₄	1	2	yes	—
48	C₂²×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×D₇₃	#127	1	1	yes	characteristic
50	C₂×D₇₃	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×D₇₃	#135	1	4	yes	—
52	#64408	—	8	4	no	—
53	C₄×C₇₃	—	2	2	no	—
54	C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
55	C₄×C₇₃	C₂×C₄⋊C₄	1	1	yes	characteristic
56	Dic₇₃	—	2	2	no	—
57	Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
58	Dic₇₃	C₄⋊Q₈	1	1	yes	characteristic
59	C₂×C₇₃	#384	1	1	yes	characteristic
60	C₂×C₇₃	#427	1	2	yes	—
61	D₇₃	#495	1	2	yes	—
62	D₇₃	#499	1	2	yes	—
63	#1846	—	73	1	no	maximal, Sylow
64	C₇₃	#1846	1	1	yes	characteristic, complemented, Sylow
65	C₂×#107	—	73	2	no	—
66	C₂×#106	—	73	2	no	—
67	C₂×C₄×C₈	—	73	1	no	—
68	#382	—	73	2	no	—
69	C₂²×C₈	—	73	1	no	—
70	C₂²×C₈	—	73	1	no	—
71	C₂×C₄⋊C₄	—	73	4	no	—
72	C₂×C₄²	—	73	1	no	—
73	#107	—	146	4	no	—
74	#106	—	146	4	no	—
75	C₄×C₈	—	73	1	no	—
76	C₄×C₈	—	73	1	no	—
77	C₄×C₈	—	73	1	no	—
78	C₄×C₈	#64408	1	1	yes	characteristic, complemented
79	#95	—	146	4	no	—
80	C₂²×C₄	—	73	1	no	—
81	C₂²×C₄	—	73	2	no	—
82	C₂²×C₄	—	146	4	no	—
83	C₂×C₈	—	73	1	no	—
84	C₂×C₈	—	73	1	no	—
85	C₂×C₈	—	73	2	no	—
86	C₂×C₈	—	73	2	no	—
87	C₂×C₈	—	73	2	no	—
88	C₂×C₈	—	73	2	no	—
89	C₂×C₈	C₂×#64408	1	1	yes	characteristic
90	C₂×C₈	C₂×#64408	1	1	yes	characteristic
91	C₄⋊C₄	—	146	4	no	—
92	C₄⋊C₄	—	146	4	no	—
93	C₄²	—	73	2	no	—
94	C₄²	—	73	1	no	—
95	C₄²	C₂×#64408	1	1	yes	characteristic
96	C₂³	—	73	1	no	—
97	C₂×C₄	—	292	4	no	—
98	C₂×C₄	—	292	8	no	—
99	C₂×C₄	—	73	1	no	—
100	C₂×C₄	—	73	1	no	—
101	C₂×C₄	—	73	1	no	—
102	C₂×C₄	—	73	1	no	—
103	C₂×C₄	—	73	1	no	—
104	C₂×C₄	—	73	2	no	—
105	C₂×C₄	—	146	4	no	—
106	C₂×C₄	C₄⋊#64408	1	2	yes	—
107	C₂×C₄	C₂²×#64408	1	1	yes	characteristic
108	C₈	—	73	2	no	—
109	C₈	—	73	2	no	—
110	C₈	C₄⋊#64408	1	2	yes	—
111	C₈	C₄⋊#64408	1	2	yes	—
112	C₂²	—	73	1	no	—
113	C₂²	—	73	1	no	—
114	C₂²	—	73	4	no	—
115	C₂²	C₂×C₄⋊#64408	1	1	yes	characteristic
116	C₄	—	2	2	no	—
117	C₄	—	584	4	no	—
118	C₄	—	73	1	no	—
119	C₄	—	73	1	no	—
120	C₄	—	146	2	no	—
121	C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
122	C₄	C₂×C₄⋊#64408	1	1	yes	characteristic
123	C₂	—	73	2	no	—
124	C₂	—	73	2	no	—
125	C₂	#17989533	1	1	yes	characteristic
126	C₂	#17989597	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14556660

C₂

yes

maximal

C₂×#14556659

C₂

yes

maximal

#17989531

C₂

yes

maximal

C₄×C₈×D₇₃

C₂

yes

maximal, characteristic

C₂×C₄⋊#64408

C₂²

yes

—

#14556660

—

#14556659

—

(C₂×#64408)⋊C₄

—

C₂×C₈×D₇₃

C₂²

yes

characteristic

C₂×C₈×D₇₃

C₂²

yes

characteristic

C₄²×D₇₃

C₂²

yes

characteristic

C₄×#10596831

C₄

yes

characteristic, complemented

C₈×Dic₇₃

C₄

yes

characteristic, complemented

C₈×Dic₇₃

C₄

yes

characteristic, complemented

C₄×C₈×C₇₃

C₄

yes

characteristic, complemented

C₂²×#64408

—

C₄⋊#64408

—

C₄⋊#64408

—

C₂×C₄×D₇₃

D₄

yes

—

C₂×C₄×D₇₃

C₂³

yes

characteristic

C₂×#10596831

C₂×C₄

yes

characteristic

C₂×#10596831

C₂×C₄

yes

characteristic

C₈×D₇₃

D₄

yes

—

C₈×D₇₃

D₄

yes

—

C₈×D₇₃

Q₈

yes

—

C₈×D₇₃

Q₈

yes

—

C₄×Dic₇₃

C₂×C₄

yes

characteristic

C₄×Dic₇₃

Q₈

yes

—

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₂×C₈×C₇₃

C₂×C₄

yes

characteristic

C₄²×C₇₃

C₂×C₄

yes

characteristic

C₂²×D₇₃

C₂×D₄

yes

characteristic

C₂×#64408

—

C₂×#64408

—

C₂×C₄×C₇₃

C₄⋊C₄

yes

—

C₂×C₄×C₇₃

C₂²×C₄

yes

characteristic

C₂×Dic₇₃

C₄⋊C₄

yes

—

C₂×Dic₇₃

C₂×Q₈

yes

characteristic

C₄×D₇₃

—

C₄×D₇₃

C₂×D₄

yes

characteristic

C₄×D₇₃

C₂×D₄

yes

characteristic

C₄×D₇₃

C₂×Q₈

yes

characteristic

C₄×D₇₃

C₂×Q₈

yes

characteristic

C₈×C₇₃

C₄⋊C₄

yes

—

C₈×C₇₃

C₄⋊C₄

yes

—

#10596831

C₄⋊C₄

yes

—

#10596831

C₄⋊C₄

yes

—

C₂²×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₂×D₇₃

#127

yes

characteristic

C₂×D₇₃

C₄⋊Q₈

yes

characteristic

C₂×D₇₃

#135

yes

—

#64408

—

C₄×C₇₃

—

C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

C₄×C₇₃

C₂×C₄⋊C₄

yes

characteristic

Dic₇₃

—

Dic₇₃

C₄⋊Q₈

yes

characteristic

Dic₇₃

C₄⋊Q₈

yes

C₂×C₇₃

yes

C₂×C₇₃

yes

—

D₇₃

#495

yes

—

D₇₃

#499

yes

—

#1846

—

maximal, Sylow

C₇₃

#1846

yes

characteristic, complemented, Sylow

C₂×#107

—

C₂×#106

—

C₂×C₄×C₈

—

#382

—

C₂²×C₈

—

C₂²×C₈

—

C₂×C₄⋊C₄

—

C₂×C₄²

—

#107

—

146

—

#106

—

146

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

#64408

yes

characteristic, complemented

#95

—

146

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

146

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

yes

C₂×C₈

yes

C₄⋊C₄

—

146

—

C₄⋊C₄

—

146

—

C₄²

—

C₄²

—

C₄²

C₂×#64408

yes

characteristic

C₂³

—

C₂×C₄

—

292

—

C₂×C₄

—

292

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

146

—

106

C₂×C₄

C₄⋊#64408

yes

—

107

C₂×C₄

C₂²×#64408

yes

characteristic

108

C₈

—

109

C₈

—

110

C₈

C₄⋊#64408

yes

—

111

C₈

C₄⋊#64408

yes

—

112

C₂²

—

113

C₂²

—

114

C₂²

—

115

C₂²

C₂×C₄⋊#64408

yes

characteristic

116

C₄

—

117

C₄

—

584

—

118

C₄

—

119

C₄

—

120

C₄

—

146

—

121

C₄

C₂×C₄⋊#64408

yes

characteristic

122

C₄

C₂×C₄⋊#64408

yes

characteristic

123

C₂

—

124

C₂

—

125

C₂

#17989533

yes

characteristic

126

C₂

#17989597

yes

—

	subgroup	G/H
center	C₂²	C₂×C₄⋊#64408
commutator	C₂×C₄×C₇₃	C₂²×C₄
Frattini	C₂×C₄	C₂²×#64408
Fitting	C₄×C₈×C₇₃	C₄
socle	C₂²×C₇₃	C₂×C₄⋊C₄
hypercenter	C₄×C₈	#64408
hypocenter	C₇₃	#1846
LCS	G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

subgroup

G/H

center

C₂²

C₂×C₄⋊#64408

commutator

C₂×C₄×C₇₃

C₂²×C₄

Frattini

C₂×C₄

C₂²×#64408

Fitting

C₄×C₈×C₇₃

C₄

socle

C₂²×C₇₃

C₂×C₄⋊C₄

C₄×C₈

C₇₃

G ▷ C₂×C₄×C₇₃ ▷ C₂×C₇₃ ▷ C₇₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈

N	H ≅ G/N	type
12. C₄×#10596831	C₄	semidirect
13. C₈×Dic₇₃	C₄	semidirect
14. C₈×Dic₇₃	C₄	semidirect
15. C₄×C₈×C₇₃	C₄	semidirect
64. C₇₃	#1846	semidirect
78. C₄×C₈	#64408	semidirect

H ≅ G/N

type

12. C₄×#10596831

C₄

semidirect

13. C₈×Dic₇₃

C₄

semidirect

14. C₈×Dic₇₃

C₄

semidirect

15. C₄×C₈×C₇₃

C₄

semidirect

64. C₇₃

#1846

semidirect

78. C₄×C₈

#64408

semidirect

Subgroups — #22625868

Conjugacy classes of subgroups

Complements