Group #22559820 — subgroups

Subgroups — #22559820

Order 8576.

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

normal subgroups only

This group has 178 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	#16863748	C₂	1	1	yes	maximal, characteristic, complemented
2	#16863747	C₂	1	1	yes	maximal, characteristic, complemented
3	#16863746	C₂	1	1	yes	maximal, characteristic, complemented
4	#16863742	C₂	1	1	yes	maximal, characteristic
5	#16863742	C₂	1	1	yes	maximal, characteristic
6	C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
7	#486×C₆₇	C₂	1	1	yes	maximal, characteristic
8	#14524043	C₂²	1	1	yes	characteristic
9	#14524043	C₂²	1	1	yes	characteristic
10	#14523968	—	2	2	no	—
11	#14523967	—	2	2	no	—
12	#14523966	—	2	2	no	—
13	#14523964	—	2	2	no	—
14	#14523964	—	2	2	no	—
15	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
16	C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
17	C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
18	#127×C₆₇	C₂²	1	1	yes	characteristic
19	#102×C₆₇	—	2	2	no	—
20	C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	C₂×Dic₁₃₄	—	2	1	no	—
22	C₂×Dic₁₃₄	—	2	1	no	—
23	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
24	C₂×C₆₇⋊C₈	D₄	1	2	yes	—
25	C₄⋊Dic₆₇	—	2	1	no	—
26	C₄⋊Dic₆₇	—	2	1	no	—
27	C₄⋊Dic₆₇	—	2	1	no	—
28	C₄⋊Dic₆₇	—	2	1	no	—
29	C₂×Q₈×C₆₇	—	2	1	no	—
30	C₂×D₄×C₆₇	—	2	1	no	—
31	C₂×D₄×C₆₇	—	2	1	no	—
32	C₂×D₄×C₆₇	—	2	1	no	—
33	C₂×C₈×C₆₇	D₄	1	2	yes	—
34	C₄⋊C₄×C₆₇	—	2	1	no	—
35	C₄⋊C₄×C₆₇	—	2	1	no	—
36	C₄²×C₆₇	C₂³	1	1	yes	characteristic
37	C₂³×C₆₇	—	4	1	no	—
38	Q₈×C₆₇	—	4	2	no	—
39	D₄×C₆₇	—	4	2	no	—
40	D₄×C₆₇	—	4	2	no	—
41	D₄×C₆₇	—	4	2	no	—
42	C₂×C₄×C₆₇	—	4	1	no	—
43	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
46	C₂×Dic₆₇	—	4	1	no	—
47	C₂×Dic₆₇	—	4	1	no	—
48	Dic₁₃₄	—	4	2	no	—
49	Dic₁₃₄	—	4	2	no	—
50	C₈×C₆₇	—	2	2	no	—
51	C₆₇⋊C₈	—	2	2	no	—
52	C₆₇⋊C₈	—	2	2	no	—
53	C₂²×C₆₇	—	8	1	no	—
54	C₂²×C₆₇	—	8	1	no	—
55	C₂²×C₆₇	—	8	1	no	—
56	C₂²×C₆₇	#127	1	1	yes	characteristic
57	C₄×C₆₇	—	8	1	no	—
58	C₄×C₆₇	#135	1	2	yes	—
59	C₄×C₆₇	#135	1	2	yes	—
60	C₄×C₆₇	#135	1	2	yes	—
61	Dic₆₇	—	8	1	no	—
62	Dic₆₇	—	8	1	no	—
63	C₂×C₆₇	—	16	1	no	—
64	C₂×C₆₇	#495	1	1	yes	characteristic
65	C₂×C₆₇	#495	1	1	yes	characteristic
66	C₂×C₆₇	#495	1	1	yes	characteristic
67	#1711	—	67	1	no	maximal, Sylow
68	C₆₇	#1711	1	1	yes	characteristic, complemented, Sylow
69	#487	—	67	1	no	—
70	#487	—	67	1	no	—
71	#487	—	67	1	no	—
72	#486	—	67	1	no	—
73	#486	—	67	1	no	—
74	#486	D₆₇	1	1	yes	characteristic
75	C₈²	—	67	1	no	—
76	C₄⋊Q₈	—	67	1	no	—
77	C₄⋊Q₈	—	67	1	no	—
78	C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
79	#127	C₂×D₆₇	1	1	yes	characteristic
80	#103	—	134	2	no	—
81	#103	—	134	2	no	—
82	#103	—	134	2	no	—
83	#102	—	2	2	no	—
84	#102	—	134	2	no	—
85	#102	—	134	2	no	—
86	C₄×C₈	—	67	1	no	—
87	C₄×C₈	—	67	1	no	—
88	C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
89	C₂×Q₈	—	2	1	no	—
90	C₂×Q₈	—	134	1	no	—
91	C₂×Q₈	—	134	1	no	—
92	C₂×D₄	—	2	1	no	—
93	C₂×D₄	—	2	1	no	—
94	C₂×D₄	—	2	1	no	—
95	C₂×C₈	—	67	2	no	—
96	C₂×C₈	—	67	2	no	—
97	C₂×C₈	C₂²⋊D₆₇	1	2	yes	—
98	C₄⋊C₄	—	2	1	no	—
99	C₄⋊C₄	—	2	1	no	—
100	C₄⋊C₄	—	134	1	no	—
101	C₄⋊C₄	—	134	1	no	—
102	C₄⋊C₄	—	134	1	no	—
103	C₄⋊C₄	—	134	1	no	—
104	C₄²	C₂²×D₆₇	1	1	yes	characteristic
105	C₂³	—	4	1	no	—
106	Q₈	—	4	2	no	—
107	Q₈	—	268	2	no	—
108	Q₈	—	268	2	no	—
109	D₄	—	4	2	no	—
110	D₄	—	4	2	no	—
111	D₄	—	4	2	no	—
112	C₂×C₄	—	4	1	no	—
113	C₂×C₄	—	268	1	no	—
114	C₂×C₄	—	268	1	no	—
115	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
116	C₂×C₄	D₄×D₆₇	1	1	yes	characteristic
117	C₂×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
118	C₈	—	2	2	no	—
119	C₈	—	134	2	no	—
120	C₈	—	134	2	no	—
121	C₂²	—	8	1	no	—
122	C₂²	—	8	1	no	—
123	C₂²	—	8	1	no	—
124	C₂²	#14524039	1	1	yes	characteristic
125	C₄	—	8	1	no	—
126	C₄	—	536	1	no	—
127	C₄	—	536	1	no	—
128	C₄	#14524062	1	2	yes	—
129	C₄	#14524063	1	2	yes	—
130	C₄	#14524064	1	2	yes	—
131	C₂	—	16	1	no	—
132	C₂	#16863777	1	1	yes	characteristic
133	C₂	#16863778	1	1	yes	characteristic
134	C₂	#16863779	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863748

C₂

yes

maximal, characteristic, complemented

#16863747

C₂

yes

maximal, characteristic, complemented

#16863746

C₂

yes

maximal, characteristic, complemented

#16863742

C₂

yes

maximal, characteristic

#16863742

C₂

yes

maximal, characteristic

C₈×C₆₇⋊C₈

C₂

yes

maximal, characteristic, complemented

#486×C₆₇

C₂

yes

maximal, characteristic

#14524043

C₂²

yes

characteristic

#14524043

C₂²

yes

—

—

—

—

—

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

#127×C₆₇

C₂²

yes

characteristic

#102×C₆₇

—

C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₂×C₆₇⋊C₈

D₄

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

C₂×C₈×C₆₇

D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂³

yes

characteristic

C₂³×C₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#127

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

#135

yes

—

C₄×C₆₇

#135

yes

—

C₄×C₆₇

#135

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

—

—

—

—

—

D₆₇

yes

characteristic

C₈²

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×D₆₇

yes

characteristic

#127

C₂×D₆₇

yes

characteristic

#103

—

134

—

#103

—

134

—

#103

—

134

—

#102

—

#102

—

134

—

#102

—

134

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂×D₄

—

C₂×D₄

—

C₂×D₄

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

C₂²⋊D₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

100

C₄⋊C₄

—

134

—

101

C₄⋊C₄

—

134

—

102

C₄⋊C₄

—

134

—

103

C₄⋊C₄

—

134

—

104

C₄²

C₂²×D₆₇

yes

characteristic

105

C₂³

—

106

Q₈

—

107

Q₈

—

268

—

108

Q₈

—

268

—

109

D₄

—

110

D₄

—

111

D₄

—

112

C₂×C₄

—

113

C₂×C₄

—

268

—

114

C₂×C₄

—

268

—

115

C₂×C₄

D₄×D₆₇

yes

characteristic

116

C₂×C₄

D₄×D₆₇

yes

characteristic

117

C₂×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

118

C₈

—

119

C₈

—

134

—

120

C₈

—

134

—

121

C₂²

—

122

C₂²

—

123

C₂²

—

124

C₂²

#14524039

yes

characteristic

125

C₄

—

126

C₄

—

536

—

127

C₄

—

536

—

128

C₄

#14524062

yes

—

129

C₄

#14524063

yes

—

130

C₄

#14524064

yes

—

131

C₂

—

132

C₂

#16863777

yes

characteristic

133

C₂

#16863778

yes

characteristic

134

C₂

#16863779

yes

characteristic

	subgroup	G/H
center	C₂²	#14524039
commutator	C₄²×C₆₇	C₂³
Frattini	C₄²	C₂²×D₆₇
Fitting	#486×C₆₇	C₂
socle	C₂²×C₆₇	#127
hypercenter	#486	D₆₇
hypocenter	C₆₇	#1711
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #486

subgroup

G/H

center

C₂²

#14524039

commutator

C₄²×C₆₇

C₂³

Frattini

C₄²

C₂²×D₆₇

Fitting

#486×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #486

N	H ≅ G/N	type
1. #16863748	C₂	semidirect
2. #16863747	C₂	semidirect
3. #16863746	C₂	semidirect
6. C₈×C₆₇⋊C₈	C₂	semidirect
68. C₆₇	#1711	semidirect

H ≅ G/N

type

1. #16863748

C₂

semidirect

2. #16863747

C₂

semidirect

3. #16863746

C₂

semidirect

6. C₈×C₆₇⋊C₈

C₂

semidirect

68. C₆₇

#1711

semidirect

Subgroups — #22559820

Conjugacy classes of subgroups

Complements