Group #22559355 — subgroups

Subgroups — #22559355

Order 8576.

218 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 218 conjugacy classes of subgroups — too many to draw. Only its 53 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	#16864112	C₂	1	1	yes	maximal, characteristic
2	#16863775	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863687	C₂	1	2	yes	maximal, complemented
4	#16863123	C₂	1	2	yes	maximal, complemented
5	#333×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11761618	—	2	2	no	—
7	#14524043	C₂²	1	1	yes	characteristic
8	#14524032	—	2	2	no	—
9	#14524018	—	2	2	no	—
10	C₄×D₂₆₈	C₂²	1	2	yes	—
11	#14524001	—	2	2	no	—
12	#14523975	—	2	2	no	—
13	#14523965	—	2	2	no	—
14	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
16	C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
17	C₂×Dic₁₃₄	—	2	1	no	—
18	C₂×D₂₆₈	—	2	2	no	—
19	C₂×D₂₆₈	D₄	1	2	yes	—
20	C₂×C₄×D₆₇	—	2	2	no	—
21	#11761618	—	4	4	no	—
22	C₂×C₆₇⋊C₈	—	2	1	no	—
23	C₄⋊Dic₆₇	—	2	2	no	—
24	C₄⋊Dic₆₇	D₄	1	2	yes	—
25	(C₂×D₆₇)⋊C₄	—	2	4	no	—
26	(C₂×D₆₇)⋊C₄	—	2	2	no	—
27	C₄×Dic₆₇	—	2	2	no	—
28	C₂×Q₈×C₆₇	D₄	1	2	yes	—
29	C₂×C₈×C₆₇	—	2	2	no	—
30	C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₄²×C₆₇	C₂³	1	1	yes	characteristic
32	C₂²×D₆₇	—	2	2	no	—
33	Q₈×C₆₇	—	4	2	no	—
34	C₂×C₄×C₆₇	—	2	2	no	—
35	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
37	C₂×Dic₆₇	—	2	2	no	—
38	C₂×Dic₆₇	—	4	1	no	—
39	D₂₆₈	—	2	2	no	—
40	D₂₆₈	—	2	2	no	—
41	D₂₆₈	—	8	2	no	—
42	C₄×D₆₇	—	4	2	no	—
43	C₄×D₆₇	—	4	2	no	—
44	Dic₁₃₄	—	4	2	no	—
45	C₈×C₆₇	—	4	2	no	—
46	C₆₇⋊C₈	—	2	2	no	—
47	C₂²×C₆₇	#120	1	1	yes	characteristic
48	C₂×D₆₇	—	4	2	no	—
49	C₂×D₆₇	—	4	2	no	—
50	C₂×D₆₇	—	4	2	no	—
51	C₄×C₆₇	—	2	1	no	—
52	C₄×C₆₇	—	4	2	no	—
53	C₄×C₆₇	#135	1	2	yes	—
54	C₄×C₆₇	#137	1	2	yes	—
55	Dic₆₇	—	4	2	no	—
56	Dic₆₇	—	8	1	no	—
57	C₂×C₆₇	#452	1	2	yes	—
58	C₂×C₆₇	#453	1	1	yes	characteristic
59	D₆₇	—	8	2	no	—
60	#1634	—	67	1	no	maximal, Sylow
61	C₆₇	#1634	1	1	yes	characteristic, complemented, Sylow
62	#555	—	67	1	no	—
63	#494	—	67	1	no	—
64	#476	—	67	2	no	—
65	#333	D₆₇	1	1	yes	characteristic, complemented
66	D₄⋊C₈	—	67	2	no	—
67	C₂×Q₁₆	—	134	2	no	—
68	C₄⋊Q₈	—	67	1	no	—
69	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
70	#124	—	134	2	no	—
71	#121	—	134	2	no	—
72	C₄×D₄	—	67	2	no	—
73	#117	—	134	2	no	—
74	#106	—	134	2	no	—
75	C₄⋊C₈	C₂×D₆₇	1	2	yes	—
76	#102	—	134	2	no	—
77	C₄×C₈	—	67	1	no	—
78	C₂×Q₈	—	134	1	no	—
79	C₂×Q₈	D₂₆₈	1	2	yes	—
80	C₂×D₄	—	67	2	no	—
81	C₂×D₄	—	134	2	no	—
82	C₂²×C₄	—	134	2	no	—
83	Q₁₆	—	268	4	no	—
84	C₂×C₈	—	2	2	no	—
85	C₂×C₈	—	134	1	no	—
86	C₄⋊C₄	—	2	2	no	—
87	C₄⋊C₄	—	67	2	no	—
88	C₄⋊C₄	—	134	2	no	—
89	C₂²⋊C₄	—	134	4	no	—
90	C₂²⋊C₄	—	134	2	no	—
91	C₄²	—	134	2	no	—
92	C₄²	C₂²×D₆₇	1	1	yes	characteristic
93	C₂³	—	134	2	no	—
94	Q₈	—	4	2	no	—
95	Q₈	—	268	2	no	—
96	D₄	—	536	2	no	—
97	D₄	—	134	2	no	—
98	D₄	—	134	2	no	—
99	C₂×C₄	—	2	2	no	—
100	C₂×C₄	—	268	2	no	—
101	C₂×C₄	—	268	2	no	—
102	C₂×C₄	—	268	1	no	—
103	C₂×C₄	—	134	2	no	—
104	C₂×C₄	D₄×D₆₇	1	2	yes	—
105	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
106	C₈	—	4	2	no	—
107	C₈	—	134	2	no	—
108	C₂²	—	268	2	no	—
109	C₂²	—	268	2	no	—
110	C₂²	—	268	2	no	—
111	C₂²	#14524013	1	1	yes	characteristic
112	C₄	—	2	1	no	—
113	C₄	—	4	2	no	—
114	C₄	—	268	2	no	—
115	C₄	—	536	1	no	—
116	C₄	#14524064	1	2	yes	—
117	C₄	#14524075	1	2	yes	—
118	C₂	—	536	2	no	—
119	C₂	#16863561	1	2	yes	—
120	C₂	#16863566	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16864112

C₂

yes

maximal, characteristic

#16863775

C₂

yes

maximal, characteristic, complemented

#16863687

C₂

yes

maximal, complemented

#16863123

C₂

yes

maximal, complemented

#333×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761618

—

#14524043

C₂²

yes

characteristic

#14524032

—

#14524018

—

C₄×D₂₆₈

C₂²

yes

—

#14524001

—

#14523975

—

#14523965

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

—

C₂×D₂₆₈

—

C₂×D₂₆₈

D₄

yes

—

C₂×C₄×D₆₇

—

#11761618

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

D₄

yes

—

(C₂×D₆₇)⋊C₄

—

(C₂×D₆₇)⋊C₄

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

D₄

yes

—

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂²×D₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₄×D₆₇

—

C₄×D₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

#120

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#135

yes

—

C₄×C₆₇

#137

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

#452

yes

—

C₂×C₆₇

#453

yes

characteristic

D₆₇

—

#1634

—

maximal, Sylow

C₆₇

#1634

yes

characteristic, complemented, Sylow

—

—

—

D₆₇

yes

characteristic, complemented

D₄⋊C₈

—

C₂×Q₁₆

—

134

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₇

yes

characteristic

#124

—

134

—

#121

—

134

—

C₄×D₄

—

#117

—

134

—

#106

—

134

—

C₄⋊C₈

C₂×D₆₇

yes

—

#102

—

134

—

C₄×C₈

—

C₂×Q₈

—

134

—

C₂×Q₈

D₂₆₈

yes

—

C₂×D₄

—

C₂×D₄

—

134

—

C₂²×C₄

—

134

—

Q₁₆

—

268

—

C₂×C₈

—

C₂×C₈

—

134

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

134

—

C₂²⋊C₄

—

134

—

C₂²⋊C₄

—

134

—

C₄²

—

134

—

C₄²

C₂²×D₆₇

yes

characteristic

C₂³

—

134

—

Q₈

—

Q₈

—

268

—

D₄

—

536

—

D₄

—

134

—

D₄

—

134

—

C₂×C₄

—

100

C₂×C₄

—

268

—

101

C₂×C₄

—

268

—

102

C₂×C₄

—

268

—

103

C₂×C₄

—

134

—

104

C₂×C₄

D₄×D₆₇

yes

—

105

C₂×C₄

C₂×D₂₆₈

yes

characteristic

106

C₈

—

107

C₈

—

134

—

108

C₂²

—

268

—

109

C₂²

—

268

—

110

C₂²

—

268

—

111

C₂²

#14524013

yes

characteristic

112

C₄

—

113

C₄

—

114

C₄

—

268

—

115

C₄

—

536

—

116

C₄

#14524064

yes

—

117

C₄

#14524075

yes

—

118

C₂

—

536

—

119

C₂

#16863561

yes

—

120

C₂

#16863566

yes

characteristic

	subgroup	G/H
center	C₂²	#14524013
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#333×C₆₇	C₂
socle	C₂²×C₆₇	#120
hypercenter	#333	D₆₇
hypocenter	C₆₇	#1634
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #333

subgroup

G/H

center

C₂²

#14524013

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#333×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #333

N	H ≅ G/N	type
2. #16863775	C₂	semidirect
3. #16863687	C₂	semidirect
4. #16863123	C₂	semidirect
5. #333×C₆₇	C₂	semidirect
61. C₆₇	#1634	semidirect
65. #333	D₆₇	semidirect

H ≅ G/N

type

2. #16863775

C₂

semidirect

3. #16863687

C₂

semidirect

4. #16863123

C₂

semidirect

5. #333×C₆₇

C₂

semidirect

61. C₆₇

#1634

semidirect

65. #333

D₆₇

semidirect

Subgroups — #22559355

Conjugacy classes of subgroups

Complements