Group #22561763 — subgroups

Subgroups — #22561763

Order 8576.

210 conjugacy classes of subgroups; only the 67 normal subgroups are shown.

normal subgroups only

This group has 210 conjugacy classes of subgroups — too many to draw. Only its 67 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	#16863445	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863423	C₂	1	2	yes	maximal, complemented
3	#16863414	C₂	1	2	yes	maximal
4	#16863364	C₂	1	1	yes	maximal, characteristic
5	H(3,ℤ/4)×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11761628	C₂²	1	1	yes	characteristic
7	C₂×#11761612	C₂²	1	2	yes	—
8	C₂²×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
9	#14523979	—	2	2	no	—
10	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
11	#14523971	—	2	4	no	—
12	(C₂×D₆₇)⋊C₈	—	2	2	no	—
13	C₂×M₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	#117×C₆₇	C₂²	1	2	yes	—
15	#11761628	—	2	2	no	—
16	#11761628	—	2	2	no	—
17	C₂×Dic₁₃₄	Q₈	1	1	yes	characteristic
18	C₂×D₂₆₈	Q₈	1	1	yes	characteristic
19	C₂×C₂²⋊D₆₇	—	2	1	no	—
20	C₂×C₄×D₆₇	—	2	1	no	—
21	#11761612	—	2	2	no	—
22	#11761612	—	2	2	no	—
23	C₂×C₆₇⋊C₈	—	2	1	no	—
24	C₂×C₆₇⋊C₈	—	2	2	no	—
25	C₂×C₆₇⋊C₈	—	2	2	no	—
26	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
27	M₁₆×C₆₇	—	2	2	no	—
28	C₂×C₈×C₆₇	D₄	1	2	yes	—
29	C₄⋊C₄×C₆₇	—	2	2	no	—
30	C₂²⋊C₄×C₆₇	—	2	2	no	—
31	C₄²×C₆₇	D₄	1	4	yes	—
32	C₂³×C₆₇	#41	1	1	yes	characteristic
33	C₂²×D₆₇	—	2	1	no	—
34	C₂×C₄×C₆₇	—	2	4	no	—
35	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
36	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
38	C₂×C₄×C₆₇	#41	1	2	yes	—
39	C₂²⋊D₆₇	—	4	2	no	—
40	C₂×Dic₆₇	—	2	1	no	—
41	D₂₆₈	—	2	2	no	—
42	C₄×D₆₇	—	4	1	no	—
43	C₄×D₆₇	—	4	1	no	—
44	Dic₁₃₄	—	2	2	no	—
45	C₈×C₆₇	—	2	2	no	—
46	C₆₇⋊C₈	—	4	1	no	—
47	C₆₇⋊C₈	—	4	2	no	—
48	C₂²×C₆₇	—	4	1	no	—
49	C₂²×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
50	C₂²×C₆₇	#123	1	2	yes	—
51	C₂×D₆₇	—	4	2	no	—
52	C₂×D₆₇	—	4	1	no	—
53	C₄×C₆₇	—	4	4	no	—
54	C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
55	C₄×C₆₇	#123	1	1	yes	characteristic
56	C₄×C₆₇	#127	1	1	yes	characteristic
57	Dic₆₇	—	4	1	no	—
58	C₂×C₆₇	—	2	2	no	—
59	C₂×C₆₇	—	2	1	no	—
60	C₂×C₆₇	#399	1	1	yes	characteristic
61	D₆₇	—	8	1	no	—
62	#2098	—	67	1	no	maximal, Sylow
63	C₆₇	#2098	1	1	yes	characteristic, complemented, Sylow
64	#430	—	67	1	no	—
65	#424	—	67	2	no	—
66	#421	—	67	2	no	—
67	#408	—	67	1	no	—
68	H(3,ℤ/4)	D₆₇	1	1	yes	characteristic, complemented
69	C₂×#41	—	67	1	no	—
70	C₂×M₁₆	—	67	2	no	—
71	C₂×M₁₆	C₂×D₆₇	1	1	yes	characteristic
72	C₂²×C₈	—	67	1	no	—
73	#117	C₂×D₆₇	1	2	yes	—
74	#108	—	134	2	no	—
75	C₄⋊C₈	—	134	4	no	—
76	#104	—	134	4	no	—
77	C₂²⋊C₈	—	134	2	no	—
78	#41	—	134	2	no	—
79	#41	—	134	2	no	—
80	C₂×Q₈	—	67	1	no	—
81	C₂×D₄	—	67	1	no	—
82	C₂×D₄	—	134	1	no	—
83	C₂²×C₄	—	134	1	no	—
84	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
85	M₁₆	—	2	2	no	—
86	M₁₆	—	134	2	no	—
87	M₁₆	—	134	2	no	—
88	C₂×C₈	—	134	1	no	—
89	C₂×C₈	—	134	2	no	—
90	C₂×C₈	—	134	2	no	—
91	C₂×C₈	D₂₆₈	1	2	yes	—
92	C₄⋊C₄	—	2	2	no	—
93	C₂²⋊C₄	—	2	2	no	—
94	C₄²	D₂₆₈	1	4	yes	—
95	C₂³	—	134	1	no	—
96	C₂³	#11761627	1	1	yes	characteristic
97	Q₈	—	134	2	no	—
98	D₄	—	268	2	no	—
99	D₄	—	134	2	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	268	1	no	—
102	C₂×C₄	—	268	1	no	—
103	C₂×C₄	—	134	1	no	—
104	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
105	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
106	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
107	C₂×C₄	#11761627	1	2	yes	—
108	C₈	—	2	2	no	—
109	C₈	—	268	1	no	—
110	C₈	—	268	2	no	—
111	C₂²	—	4	1	no	—
112	C₂²	—	268	2	no	—
113	C₂²	—	268	1	no	—
114	C₂²	#14524023	1	1	yes	characteristic
115	C₂²	#14524027	1	2	yes	—
116	C₄	—	4	4	no	—
117	C₄	—	268	1	no	—
118	C₄	#14524023	1	2	yes	—
119	C₄	#14524027	1	1	yes	characteristic
120	C₄	#14524040	1	1	yes	characteristic
121	C₂	—	2	2	no	—
122	C₂	—	2	1	no	—
123	C₂	—	536	1	no	—
124	C₂	#16863333	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863445

C₂

yes

maximal, characteristic, complemented

#16863423

C₂

yes

maximal, complemented

#16863414

C₂

yes

maximal

#16863364

C₂

yes

maximal, characteristic

H(3,ℤ/4)×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761628

C₂²

yes

characteristic

C₂×#11761612

C₂²

yes

—

C₂²×C₆₇⋊C₈

C₂²

yes

characteristic

#14523979

—

C₄⋊(C₆₇⋊C₈)

—

#14523971

—

(C₂×D₆₇)⋊C₈

—

C₂×M₁₆×C₆₇

C₂²

yes

characteristic

#117×C₆₇

C₂²

yes

—

#11761628

—

#11761628

—

C₂×Dic₁₃₄

Q₈

yes

characteristic

C₂×D₂₆₈

Q₈

yes

characteristic

C₂×C₂²⋊D₆₇

—

C₂×C₄×D₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

M₁₆×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

D₄

yes

—

C₂³×C₆₇

#41

yes

characteristic

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

#41

yes

—

C₂²⋊D₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

C₄×D₆₇

—

C₄×D₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₆₇

#123

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#399

yes

characteristic

D₆₇

—

#2098

—

maximal, Sylow

C₆₇

#2098

yes

characteristic, complemented, Sylow

—

—

—

—

D₆₇

yes

characteristic, complemented

C₂×#41

—

C₂×M₁₆

—

C₂×M₁₆

C₂×D₆₇

yes

characteristic

C₂²×C₈

—

#117

C₂×D₆₇

yes

—

#108

—

134

—

C₄⋊C₈

—

134

—

#104

—

134

—

C₂²⋊C₈

—

134

—

#41

—

134

—

#41

—

134

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

M₁₆

—

M₁₆

—

134

—

M₁₆

—

134

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₂×C₈

D₂₆₈

yes

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₄²

D₂₆₈

yes

—

C₂³

—

134

—

C₂³

#11761627

yes

characteristic

Q₈

—

134

—

D₄

—

268

—

D₄

—

134

—

100

C₂×C₄

—

101

C₂×C₄

—

268

—

102

C₂×C₄

—

268

—

103

C₂×C₄

—

134

—

104

C₂×C₄

C₂×D₂₆₈

yes

—

105

C₂×C₄

C₂×D₂₆₈

yes

characteristic

106

C₂×C₄

Q₈×D₆₇

yes

characteristic

107

C₂×C₄

#11761627

yes

—

108

C₈

—

109

C₈

—

268

—

110

C₈

—

268

—

111

C₂²

—

112

C₂²

—

268

—

113

C₂²

—

268

—

114

C₂²

#14524023

yes

characteristic

115

C₂²

#14524027

yes

—

116

C₄

—

117

C₄

—

268

—

118

C₄

#14524023

yes

—

119

C₄

#14524027

yes

characteristic

120

C₄

#14524040

yes

characteristic

121

C₂

—

122

C₂

—

123

C₂

—

536

—

124

C₂

#16863333

yes

characteristic

	subgroup	G/H
center	C₄	#14524040
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	H(3,ℤ/4)×C₆₇	C₂
socle	C₂×C₆₇	#399
hypercenter	H(3,ℤ/4)	D₆₇
hypocenter	C₆₇	#2098
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

subgroup

G/H

center

C₄

#14524040

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

H(3,ℤ/4)×C₆₇

C₂

C₂×C₆₇

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂²×C₄ ◁ H(3,ℤ/4)

N	H ≅ G/N	type
1. #16863445	C₂	semidirect
2. #16863423	C₂	semidirect
5. H(3,ℤ/4)×C₆₇	C₂	semidirect
63. C₆₇	#2098	semidirect
68. H(3,ℤ/4)	D₆₇	semidirect

H ≅ G/N

type

1. #16863445

C₂

semidirect

2. #16863423

C₂

semidirect

5. H(3,ℤ/4)×C₆₇

C₂

semidirect

63. C₆₇

#2098

semidirect

68. H(3,ℤ/4)

D₆₇

semidirect

Subgroups — #22561763

Conjugacy classes of subgroups

Complements