Group #22561655 — subgroups

Subgroups — #22561655

Order 8576.

198 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 63 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	#16863395	C₂	1	2	yes	maximal
2	#16863362	C₂	1	1	yes	maximal, characteristic
3	#16863165	C₂	1	2	yes	maximal, complemented
4	#16863161	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#106×C₆₇	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#11761628	C₂²	1	1	yes	characteristic
7	C₂×#11761612	C₂²	1	1	yes	characteristic
8	C₂×#11761612	C₂²	1	2	yes	—
9	#14523966	—	2	2	no	—
10	#14523963	—	2	2	no	—
11	(C₂×D₆₇)⋊C₈	—	2	2	no	—
12	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×C₄⋊C₄×C₆₇	C₂²	1	2	yes	—
14	#106×C₆₇	—	2	2	no	—
15	#11761628	—	2	2	no	—
16	#11761628	—	2	2	no	—
17	C₂×Dic₁₃₄	D₄	1	1	yes	characteristic
18	C₂×D₂₆₈	D₄	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	#11761612	—	2	2	no	—
24	C₂×C₆₇⋊C₈	—	2	1	no	—
25	C₂×C₆₇⋊C₈	—	2	2	no	—
26	C₂²×C₄×C₆₇	—	2	2	no	—
27	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
28	C₂×C₈×C₆₇	—	2	2	no	—
29	C₂×C₈×C₆₇	D₄	1	2	yes	—
30	C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₄⋊C₄×C₆₇	—	2	2	no	—
32	C₂³×C₆₇	#41	1	1	yes	characteristic
33	C₂²×D₆₇	—	2	1	no	—
34	C₂×C₄×C₆₇	—	4	2	no	—
35	C₂×C₄×C₆₇	—	8	2	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	2	yes	—
39	C₂×C₄×C₆₇	#41	1	2	yes	—
40	C₂²⋊D₆₇	—	4	2	no	—
41	C₂×Dic₆₇	—	2	1	no	—
42	D₂₆₈	—	2	2	no	—
43	C₄×D₆₇	—	4	1	no	—
44	C₄×D₆₇	—	4	1	no	—
45	Dic₁₃₄	—	2	2	no	—
46	C₈×C₆₇	—	2	2	no	—
47	C₆₇⋊C₈	—	4	2	no	—
48	C₆₇⋊C₈	—	4	1	no	—
49	C₂²×C₆₇	—	2	2	no	—
50	C₂²×C₆₇	#121	1	1	yes	characteristic
51	C₂²×C₆₇	#126	1	2	yes	—
52	C₂×D₆₇	—	4	1	no	—
53	C₂×D₆₇	—	4	2	no	—
54	C₄×C₆₇	—	8	2	no	—
55	C₄×C₆₇	#123	1	1	yes	characteristic
56	C₄×C₆₇	#123	1	2	yes	—
57	C₄×C₆₇	#124	1	1	yes	characteristic
58	Dic₆₇	—	4	1	no	—
59	C₂×C₆₇	—	2	2	no	—
60	C₂×C₆₇	#397	1	1	yes	characteristic
61	C₂×C₆₇	#471	1	2	yes	—
62	D₆₇	—	8	1	no	—
63	#2079	—	67	1	no	maximal, Sylow
64	C₆₇	#2079	1	1	yes	characteristic, complemented, Sylow
65	C₂×#106	D₆₇	1	1	yes	characteristic, complemented
66	#417	—	67	2	no	—
67	#408	—	67	1	no	—
68	#343	—	67	2	no	—
69	#341	—	67	1	no	—
70	C₂×#41	—	67	1	no	—
71	C₂×M₁₆	—	67	2	no	—
72	C₂×M₁₆	—	67	1	no	—
73	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
74	C₂×C₄⋊C₄	C₂×D₆₇	1	2	yes	—
75	#106	—	2	2	no	—
76	#103	—	134	2	no	—
77	#102	—	134	2	no	—
78	C₂²⋊C₈	—	134	2	no	—
79	#41	—	134	2	no	—
80	#41	—	134	2	no	—
81	C₂×Q₈	—	67	1	no	—
82	C₂×D₄	—	67	1	no	—
83	C₂×D₄	—	134	1	no	—
84	C₂²×C₄	—	2	2	no	—
85	C₂²×C₄	—	134	1	no	—
86	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
87	M₁₆	—	134	2	no	—
88	M₁₆	—	134	2	no	—
89	M₁₆	—	134	2	no	—
90	C₂×C₈	—	2	2	no	—
91	C₂×C₈	—	134	2	no	—
92	C₂×C₈	—	134	1	no	—
93	C₂×C₈	C₂²⋊D₆₇	1	2	yes	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₂³	—	134	1	no	—
97	C₂³	#11761627	1	1	yes	characteristic
98	Q₈	—	134	2	no	—
99	D₄	—	268	2	no	—
100	D₄	—	134	2	no	—
101	C₂×C₄	—	4	2	no	—
102	C₂×C₄	—	8	2	no	—
103	C₂×C₄	—	268	1	no	—
104	C₂×C₄	—	268	1	no	—
105	C₂×C₄	—	134	1	no	—
106	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
107	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
108	C₂×C₄	#11761628	1	2	yes	—
109	C₂×C₄	#11761629	1	2	yes	—
110	C₈	—	2	2	no	—
111	C₈	—	268	2	no	—
112	C₈	—	268	1	no	—
113	C₂²	—	2	2	no	—
114	C₂²	—	268	1	no	—
115	C₂²	—	268	2	no	—
116	C₂²	#14524014	1	1	yes	characteristic
117	C₂²	#14524038	1	2	yes	—
118	C₄	—	8	2	no	—
119	C₄	—	268	1	no	—
120	C₄	#14524026	1	1	yes	characteristic
121	C₄	#14524028	1	2	yes	—
122	C₄	#14524032	1	1	yes	characteristic
123	C₂	—	2	2	no	—
124	C₂	—	536	1	no	—
125	C₂	#16863322	1	1	yes	characteristic
126	C₂	#16863656	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#16863395

C₂

yes

maximal

#16863362

C₂

yes

maximal, characteristic

#16863165

C₂

yes

maximal, complemented

#16863161

C₂

yes

maximal, characteristic, complemented

C₂×#106×C₆₇

C₂

yes

maximal, characteristic, complemented

C₂×#11761628

C₂²

yes

characteristic

C₂×#11761612

C₂²

yes

characteristic

C₂×#11761612

C₂²

yes

—

#14523966

—

#14523963

—

(C₂×D₆₇)⋊C₈

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

#106×C₆₇

—

#11761628

—

#11761628

—

C₂×Dic₁₃₄

D₄

yes

characteristic

C₂×D₂₆₈

D₄

yes

characteristic

C₂×C₂²⋊D₆₇

—

C₂×C₄×D₆₇

—

#11761612

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂³×C₆₇

#41

yes

characteristic

C₂²×D₆₇

—

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

—

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

#121

yes

characteristic

C₂²×C₆₇

#126

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#123

yes

characteristic

C₄×C₆₇

#123

yes

—

C₄×C₆₇

#124

yes

characteristic

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#397

yes

characteristic

C₂×C₆₇

#471

yes

—

D₆₇

—

#2079

—

maximal, Sylow

C₆₇

#2079

yes

characteristic, complemented, Sylow

C₂×#106

D₆₇

yes

characteristic, complemented

—

—

—

—

C₂×#41

—

C₂×M₁₆

—

C₂×M₁₆

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

#106

—

#103

—

134

—

#102

—

134

—

C₂²⋊C₈

—

134

—

#41

—

134

—

#41

—

134

—

C₂×Q₈

—

C₂×D₄

—

C₂×D₄

—

134

—

C₂²×C₄

—

C₂²×C₄

—

134

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

M₁₆

—

134

—

M₁₆

—

134

—

M₁₆

—

134

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₂×C₈

C₂²⋊D₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂³

—

134

—

C₂³

#11761627

yes

characteristic

Q₈

—

134

—

D₄

—

268

—

100

D₄

—

134

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

268

—

104

C₂×C₄

—

268

—

105

C₂×C₄

—

134

—

106

C₂×C₄

D₄×D₆₇

yes

characteristic

107

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

108

C₂×C₄

#11761628

yes

—

109

C₂×C₄

#11761629

yes

—

110

C₈

—

111

C₈

—

268

—

112

C₈

—

268

—

113

C₂²

—

114

C₂²

—

268

—

115

C₂²

—

268

—

116

C₂²

#14524014

yes

characteristic

117

C₂²

#14524038

yes

—

118

C₄

—

119

C₄

—

268

—

120

C₄

#14524026

yes

characteristic

121

C₄

#14524028

yes

—

122

C₄

#14524032

yes

characteristic

123

C₂

—

124

C₂

—

536

—

125

C₂

#16863322

yes

characteristic

126

C₂

#16863656

yes

—

	subgroup	G/H
center	C₂²	#14524014
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	C₂×#106×C₆₇	C₂
socle	C₂²×C₆₇	#121
hypercenter	C₂×#106	D₆₇
hypocenter	C₆₇	#2079
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

subgroup

G/H

center

C₂²

#14524014

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

C₂×#106×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#106

N	H ≅ G/N	type
3. #16863165	C₂	semidirect
4. #16863161	C₂	semidirect
5. C₂×#106×C₆₇	C₂	semidirect
64. C₆₇	#2079	semidirect
65. C₂×#106	D₆₇	semidirect

H ≅ G/N

type

3. #16863165

C₂

semidirect

4. #16863161

C₂

semidirect

5. C₂×#106×C₆₇

C₂

semidirect

64. C₆₇

#2079

semidirect

65. C₂×#106

D₆₇

semidirect

Subgroups — #22561655

Conjugacy classes of subgroups

Complements