Group #22559693 — subgroups

Subgroups — #22559693

Order 8576.

184 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 184 conjugacy classes of subgroups — too many to draw. Only its 57 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	#16863773	C₂	1	1	yes	maximal, characteristic
2	#16863596	C₂	1	2	yes	maximal, complemented
3	#16863147	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863139	C₂	1	2	yes	maximal, complemented
5	#446×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11761615	—	2	2	no	—
7	C₂×D₅₃₆	—	2	1	no	—
8	C₂×D₅₃₆	—	2	1	no	—
9	#14524040	C₂²	1	2	yes	—
10	C₄⋊(C₆₇⋊C₈)	C₂²	1	2	yes	—
11	#14523963	—	2	2	no	—
12	C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₄⋊C₈×C₆₇	—	2	1	no	—
14	C₄⋊C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#97×C₆₇	—	2	1	no	—
16	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×D₂₆₈	—	2	2	no	—
18	C₂×D₂₆₈	—	2	2	no	—
19	C₂×D₂₆₈	—	2	2	no	—
20	#11761615	—	4	4	no	—
21	D₅₃₆	—	4	2	no	—
22	D₅₃₆	—	4	2	no	—
23	C₂×C₆₇⋊C₈	—	2	2	no	—
24	C₂×Q₈×C₆₇	D₄	1	1	yes	characteristic
25	C₂×C₈×C₆₇	—	2	1	no	—
26	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
27	C₂×C₈×C₆₇	D₄	1	1	yes	characteristic
28	C₄⋊C₄×C₆₇	—	2	1	no	—
29	C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
30	C₄²×C₆₇	—	2	1	no	—
31	C₄²×C₆₇	C₂³	1	1	yes	characteristic
32	C₂²×D₆₇	—	4	2	no	—
33	Q₈×C₆₇	—	2	2	no	—
34	C₂×C₄×C₆₇	—	2	1	no	—
35	C₂×C₄×C₆₇	—	2	1	no	—
36	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
38	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
39	D₂₆₈	—	4	2	no	—
40	D₂₆₈	—	4	2	no	—
41	D₂₆₈	—	4	2	no	—
42	D₂₆₈	—	4	2	no	—
43	D₂₆₈	—	8	2	no	—
44	C₈×C₆₇	—	2	1	no	—
45	C₈×C₆₇	—	4	1	no	—
46	C₈×C₆₇	SD₁₆	1	2	yes	complemented
47	C₆₇⋊C₈	—	4	2	no	—
48	C₂²×C₆₇	#121	1	1	yes	characteristic
49	C₂×D₆₇	—	8	2	no	—
50	C₂×D₆₇	—	8	2	no	—
51	C₂×D₆₇	—	8	2	no	—
52	C₄×C₆₇	—	2	1	no	—
53	C₄×C₆₇	—	4	1	no	—
54	C₄×C₆₇	—	4	1	no	—
55	C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
56	C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
57	C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
58	C₂×C₆₇	#460	1	1	yes	characteristic
59	C₂×C₆₇	#469	1	1	yes	characteristic
60	C₂×C₆₇	#472	1	1	yes	characteristic
61	D₆₇	—	16	2	no	—
62	#1688	—	67	1	no	maximal, Sylow
63	C₆₇	#1688	1	1	yes	characteristic, complemented, Sylow
64	#493	—	67	1	no	—
65	#460	—	67	2	no	—
66	#446	D₆₇	1	1	yes	characteristic, complemented
67	#334	—	67	1	no	—
68	#331	—	67	2	no	—
69	C₂×SD₁₆	—	134	2	no	—
70	C₂×D₈	—	134	1	no	—
71	C₂×D₈	—	134	1	no	—
72	#127	—	67	2	no	—
73	C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
74	C₄⋊C₈	—	2	1	no	—
75	C₄⋊C₈	—	67	2	no	—
76	C₄⋊C₈	C₂×D₆₇	1	1	yes	characteristic
77	#102	—	134	2	no	—
78	#97	—	2	1	no	—
79	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
80	C₂×Q₈	D₂₆₈	1	1	yes	characteristic
81	C₂×D₄	—	134	2	no	—
82	C₂×D₄	—	134	2	no	—
83	C₂×D₄	—	134	2	no	—
84	SD₁₆	—	268	4	no	—
85	D₈	—	268	2	no	—
86	D₈	—	268	2	no	—
87	C₂×C₈	—	2	1	no	—
88	C₂×C₈	—	134	2	no	—
89	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
90	C₂×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
91	C₄⋊C₄	—	2	1	no	—
92	C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
93	C₄²	—	2	1	no	—
94	C₄²	C₂²×D₆₇	1	1	yes	characteristic
95	C₂³	—	268	2	no	—
96	Q₈	—	2	2	no	—
97	D₄	—	268	2	no	—
98	D₄	—	268	2	no	—
99	D₄	—	268	2	no	—
100	D₄	—	268	2	no	—
101	D₄	—	536	2	no	—
102	C₂×C₄	—	2	1	no	—
103	C₂×C₄	—	2	1	no	—
104	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
105	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
106	C₂×C₄	#11761628	1	1	yes	characteristic
107	C₈	—	2	1	no	—
108	C₈	—	4	1	no	—
109	C₈	—	268	2	no	—
110	C₈	#11761615	1	2	yes	complemented
111	C₂²	—	536	2	no	—
112	C₂²	—	536	2	no	—
113	C₂²	—	536	2	no	—
114	C₂²	#14524017	1	1	yes	characteristic
115	C₄	—	2	1	no	—
116	C₄	—	4	1	no	—
117	C₄	—	4	1	no	—
118	C₄	C₂×#11761615	1	1	yes	characteristic
119	C₄	#14524068	1	1	yes	characteristic
120	C₄	#14524071	1	2	yes	—
121	C₂	—	1072	2	no	—
122	C₂	#16863596	1	1	yes	characteristic
123	C₂	#16863649	1	1	yes	characteristic
124	C₂	#16863666	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863773

C₂

yes

maximal, characteristic

#16863596

C₂

yes

maximal, complemented

#16863147

C₂

yes

maximal, characteristic, complemented

#16863139

C₂

yes

maximal, complemented

#446×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761615

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

#14524040

C₂²

yes

—

C₄⋊(C₆₇⋊C₈)

C₂²

yes

—

#14523963

—

C₄×Q₈×C₆₇

C₂²

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

C₂²

yes

characteristic

#97×C₆₇

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

#11761615

—

D₅₃₆

—

D₅₃₆

—

C₂×C₆₇⋊C₈

—

C₂×Q₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

characteristic

C₂×C₈×C₆₇

D₄

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂²×D₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

SD₁₆

yes

complemented

C₆₇⋊C₈

—

C₂²×C₆₇

#121

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂×SD₁₆

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

—

#1688

—

maximal, Sylow

C₆₇

#1688

yes

characteristic, complemented, Sylow

#493

—

#460

—

#446

D₆₇

yes

characteristic, complemented

#334

—

#331

—

C₂×SD₁₆

—

134

—

C₂×D₈

—

134

—

C₂×D₈

—

134

—

#127

—

C₄×Q₈

C₂×D₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

C₄⋊C₈

C₂×D₆₇

yes

characteristic

#102

—

134

—

#97

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×Q₈

D₂₆₈

yes

characteristic

C₂×D₄

—

134

—

C₂×D₄

—

134

—

C₂×D₄

—

134

—

SD₁₆

—

268

—

D₈

—

268

—

D₈

—

268

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₂×C₈

C₂²⋊D₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

D₂₆₈

yes

characteristic

C₄²

—

C₄²

C₂²×D₆₇

yes

characteristic

C₂³

—

268

—

Q₈

—

D₄

—

268

—

D₄

—

268

—

D₄

—

268

—

100

D₄

—

268

—

101

D₄

—

536

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

105

C₂×C₄

C₂×D₂₆₈

yes

characteristic

106

C₂×C₄

#11761628

yes

characteristic

107

C₈

—

108

C₈

—

109

C₈

—

268

—

110

C₈

#11761615

yes

complemented

111

C₂²

—

536

—

112

C₂²

—

536

—

113

C₂²

—

536

—

114

C₂²

#14524017

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

C₂×#11761615

yes

characteristic

119

C₄

#14524068

yes

characteristic

120

C₄

#14524071

yes

—

121

C₂

—

1072

—

122

C₂

#16863596

yes

characteristic

123

C₂

#16863649

yes

characteristic

124

C₂

#16863666

yes

characteristic

	subgroup	G/H
center	C₂²	#14524017
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#446×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	#446	D₆₇
hypocenter	C₆₇	#1688
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #446

subgroup

G/H

center

C₂²

#14524017

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#446×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #446

N	H ≅ G/N	type
2. #16863596	C₂	semidirect
3. #16863147	C₂	semidirect
4. #16863139	C₂	semidirect
5. #446×C₆₇	C₂	semidirect
46. C₈×C₆₇	SD₁₆	semidirect
63. C₆₇	#1688	semidirect
66. #446	D₆₇	semidirect
110. C₈	#11761615	semidirect

H ≅ G/N

type

2. #16863596

C₂

semidirect

3. #16863147

C₂

semidirect

4. #16863139

C₂

semidirect

5. #446×C₆₇

C₂

semidirect

46. C₈×C₆₇

SD₁₆

semidirect

63. C₆₇

#1688

semidirect

66. #446

D₆₇

semidirect

110. C₈

#11761615

semidirect

Subgroups — #22559693

Conjugacy classes of subgroups

Complements