Group #22561924 — subgroups

Subgroups — #22561924

Order 8576.

260 conjugacy classes of subgroups; only the 85 normal subgroups are shown.

normal subgroups only

This group has 260 conjugacy classes of subgroups — too many to draw. Only its 85 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	C₂²×D₅₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#14523983	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863196	C₂	1	4	yes	maximal, complemented
4	#427×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂²×D₂₆₈	—	2	1	no	—
6	C₂×D₅₃₆	—	2	2	no	—
7	C₂×D₅₃₆	—	2	1	no	—
8	C₂×D₅₃₆	C₄	1	4	yes	complemented
9	C₂×D₅₃₆	C₂²	1	2	yes	—
10	#14523983	—	2	2	no	—
11	C₂×C₆₇⋊C₁₆	C₂²	1	2	yes	complemented
12	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	#117×C₆₇	—	2	1	no	—
14	#107×C₆₇	C₂²	1	2	yes	complemented
15	#106×C₆₇	—	2	1	no	—
16	C₂³×D₆₇	—	4	1	no	—
17	C₂×D₂₆₈	—	2	4	no	—
18	C₂×D₂₆₈	—	2	2	no	—
19	C₂×D₂₆₈	—	4	1	no	—
20	C₂×D₂₆₈	—	4	2	no	—
21	D₅₃₆	—	2	4	no	—
22	D₅₃₆	—	2	2	no	—
23	D₅₃₆	C₂×C₄	1	4	yes	—
24	C₆₇⋊C₁₆	—	2	2	no	—
25	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
26	C₂×C₈×C₆₇	C₂×C₄	1	2	yes	—
27	C₂×C₈×C₆₇	D₄	1	2	yes	—
28	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
29	C₂×C₈×C₆₇	C₂³	1	1	yes	characteristic
30	C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₂²⋊C₄×C₆₇	—	4	1	no	—
32	C₄²×C₆₇	—	4	1	no	—
33	C₂³×C₆₇	D₈	1	1	yes	characteristic
34	C₂²×D₆₇	—	4	4	no	—
35	C₂²×D₆₇	—	4	2	no	—
36	C₂²×D₆₇	—	8	4	no	—
37	C₂×C₄×C₆₇	—	4	2	no	—
38	C₂×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
39	C₂×C₄×C₆₇	D₈	1	1	yes	characteristic
40	C₂×C₄×C₆₇	SD₁₆	1	2	yes	—
41	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
42	D₂₆₈	—	2	4	no	—
43	D₂₆₈	—	4	4	no	—
44	D₂₆₈	—	4	2	no	—
45	C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
46	C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
47	C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂²×C₆₇	—	2	2	no	—
49	C₂²×C₆₇	#102	1	2	yes	—
50	C₂²×C₆₇	C₂×D₈	1	1	yes	characteristic
51	C₂×D₆₇	—	4	4	no	—
52	C₂×D₆₇	—	8	8	no	—
53	C₂×D₆₇	—	8	4	no	—
54	C₄×C₆₇	—	8	2	no	—
55	C₄×C₆₇	#102	1	2	yes	—
56	C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
57	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₆₇	—	2	2	no	—
59	C₂×C₆₇	C₂×#102	1	1	yes	characteristic
60	C₂×C₆₇	#509	1	2	yes	—
61	D₆₇	—	8	4	no	—
62	#2140	—	67	1	no	maximal, Sylow
63	C₆₇	#2140	1	1	yes	characteristic, complemented, Sylow
64	C₂²×D₈	—	67	1	no	—
65	C₂×M₃₂	—	67	1	no	—
66	#427	D₆₇	1	1	yes	characteristic, complemented
67	#357	—	67	4	no	—
68	C₂²×D₄	—	134	1	no	—
69	C₂×D₈	—	67	4	no	—
70	C₂×D₈	—	67	2	no	—
71	C₂×D₈	—	134	2	no	—
72	C₂×D₈	—	134	1	no	—
73	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
74	#117	—	2	1	no	—
75	M₃₂	—	134	2	no	—
76	C₂×C₁₆	—	67	2	no	—
77	#107	C₂×D₆₇	1	2	yes	complemented
78	#106	—	2	1	no	—
79	C₂⁴	—	268	1	no	—
80	C₂×D₄	—	268	1	no	—
81	C₂×D₄	—	268	2	no	—
82	C₂×D₄	—	134	4	no	—
83	C₂×D₄	—	134	2	no	—
84	C₂²×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
85	D₈	—	67	4	no	—
86	D₈	—	134	4	no	—
87	D₈	—	134	2	no	—
88	C₂×C₈	C₄×D₆₇	1	2	yes	—
89	C₂×C₈	D₂₆₈	1	2	yes	—
90	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
91	C₂×C₈	C₂²×D₆₇	1	1	yes	characteristic
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	4	1	no	—
94	C₄²	—	4	1	no	—
95	C₁₆	—	134	2	no	—
96	C₂³	—	268	4	no	—
97	C₂³	—	268	2	no	—
98	C₂³	—	536	4	no	—
99	C₂³	#11761613	1	1	yes	characteristic
100	D₄	—	268	4	no	—
101	D₄	—	268	2	no	—
102	D₄	—	134	4	no	—
103	C₂×C₄	—	4	2	no	—
104	C₂×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
105	C₂×C₄	#11761613	1	1	yes	characteristic
106	C₂×C₄	#11761615	1	2	yes	—
107	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
108	C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
109	C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
110	C₈	C₂×D₂₆₈	1	1	yes	characteristic
111	C₂²	—	2	2	no	—
112	C₂²	—	268	4	no	—
113	C₂²	—	536	8	no	—
114	C₂²	—	536	4	no	—
115	C₂²	#14523963	1	2	yes	—
116	C₂²	C₂×#11761613	1	1	yes	characteristic
117	C₄	—	8	2	no	—
118	C₄	#14523963	1	2	yes	—
119	C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
120	C₄	C₂×#11761615	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	536	4	no	—
123	C₂	C₂×#14523963	1	1	yes	characteristic
124	C₂	#16863831	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₅₃₆

C₂

yes

maximal, characteristic

C₂×#14523983

C₂

yes

maximal, characteristic, complemented

#16863196

C₂

yes

maximal, complemented

#427×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂²×D₂₆₈

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

C₂×D₅₃₆

C₄

yes

complemented

C₂×D₅₃₆

C₂²

yes

—

#14523983

—

C₂×C₆₇⋊C₁₆

C₂²

yes

complemented

C₂²×C₈×C₆₇

C₂²

yes

characteristic

#117×C₆₇

—

#107×C₆₇

C₂²

yes

complemented

#106×C₆₇

—

C₂³×D₆₇

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

D₅₃₆

—

D₅₃₆

—

D₅₃₆

C₂×C₄

yes

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

D₈

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂×C₄×C₆₇

D₈

yes

characteristic

C₂×C₄×C₆₇

SD₁₆

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

C₂²⋊C₄

yes

—

C₈×C₆₇

C₂²×C₄

yes

characteristic

C₈×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₆₇

—

C₂²×C₆₇

#102

yes

—

C₂²×C₆₇

C₂×D₈

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#102

yes

—

C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₆₇

—

C₂×C₆₇

C₂×#102

yes

characteristic

C₂×C₆₇

#509

yes

—

D₆₇

—

#2140

—

maximal, Sylow

C₆₇

#2140

yes

characteristic, complemented, Sylow

C₂²×D₈

—

C₂×M₃₂

—

#427

D₆₇

yes

characteristic, complemented

#357

—

C₂²×D₄

—

134

—

C₂×D₈

—

C₂×D₈

—

C₂×D₈

—

134

—

C₂×D₈

—

134

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

#117

—

M₃₂

—

134

—

C₂×C₁₆

—

#107

C₂×D₆₇

yes

complemented

#106

—

C₂⁴

—

268

—

C₂×D₄

—

268

—

C₂×D₄

—

268

—

C₂×D₄

—

134

—

C₂×D₄

—

134

—

C₂²×C₄

C₂²⋊D₆₇

yes

characteristic

D₈

—

D₈

—

134

—

D₈

—

134

—

C₂×C₈

C₄×D₆₇

yes

—

C₂×C₈

D₂₆₈

yes

—

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₂×C₈

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₁₆

—

134

—

C₂³

—

268

—

C₂³

—

268

—

C₂³

—

536

—

C₂³

#11761613

yes

characteristic

100

D₄

—

268

—

101

D₄

—

268

—

102

D₄

—

134

—

103

C₂×C₄

—

104

C₂×C₄

(C₂×D₆₇)⋊C₄

yes

—

105

C₂×C₄

#11761613

yes

characteristic

106

C₂×C₄

#11761615

yes

—

107

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

108

C₈

(C₂×D₆₇)⋊C₄

yes

—

109

C₈

C₂×C₄×D₆₇

yes

characteristic

110

C₈

C₂×D₂₆₈

yes

characteristic

111

C₂²

—

112

C₂²

—

268

—

113

C₂²

—

536

—

114

C₂²

—

536

—

115

C₂²

#14523963

yes

—

116

C₂²

C₂×#11761613

yes

characteristic

117

C₄

—

118

C₄

#14523963

yes

—

119

C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

120

C₄

C₂×#11761615

yes

characteristic

121

C₂

—

122

C₂

—

536

—

123

C₂

C₂×#14523963

yes

characteristic

124

C₂

#16863831

yes

—

	subgroup	G/H
center	C₂²	C₂×#11761613
commutator	C₈×C₆₇	C₂²×C₄
Frattini	C₂×C₈	C₂²×D₆₇
Fitting	#427×C₆₇	C₂
socle	C₂²×C₆₇	C₂×D₈
hypercenter	#427	D₆₇
hypocenter	C₆₇	#2140
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

subgroup

G/H

center

C₂²

C₂×#11761613

commutator

C₈×C₆₇

C₂²×C₄

Frattini

C₂×C₈

C₂²×D₆₇

Fitting

#427×C₆₇

C₂

socle

C₂²×C₆₇

C₂×D₈

D₆₇

C₆₇

G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #427

N	H ≅ G/N	type
2. C₂×#14523983	C₂	semidirect
3. #16863196	C₂	semidirect
4. #427×C₆₇	C₂	semidirect
8. C₂×D₅₃₆	C₄	semidirect
11. C₂×C₆₇⋊C₁₆	C₂²	semidirect
14. #107×C₆₇	C₂²	semidirect
63. C₆₇	#2140	semidirect
66. #427	D₆₇	semidirect
77. #107	C₂×D₆₇	semidirect

H ≅ G/N

type

2. C₂×#14523983

C₂

semidirect

3. #16863196

C₂

semidirect

4. #427×C₆₇

C₂

semidirect

8. C₂×D₅₃₆

C₄

semidirect

11. C₂×C₆₇⋊C₁₆

C₂²

semidirect

14. #107×C₆₇

C₂²

semidirect

63. C₆₇

#2140

semidirect

66. #427

D₆₇

semidirect

77. #107

C₂×D₆₇

semidirect

Subgroups — #22561924

Conjugacy classes of subgroups

Complements