Group #22562929 — subgroups

Subgroups — #22562929

Order 8576.

516 conjugacy classes of subgroups; only the 237 normal subgroups are shown.

normal subgroups only

This group has 516 conjugacy classes of subgroups — too many to draw. Only its 237 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	C₂×Q₈×Dic₆₇	C₂	1	2	yes	maximal
2	#16863303	C₂	1	2	yes	maximal
3	#16863282	C₂	1	8	yes	maximal
4	#16863262	C₂	1	1	yes	maximal, characteristic
5	#16863242	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊Q₈×C₆₇	C₂	1	1	yes	maximal, characteristic
7	Q₈×Dic₆₇	—	2	8	no	—
8	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
9	C₂×C₄⋊Dic₆₇	C₂²	1	4	yes	—
10	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×Dic₆₇	C₂²	1	4	yes	—
12	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
13	C₆₇⋊#95	C₂²	1	4	yes	—
14	C₆₇⋊#95	C₂²	1	4	yes	—
15	C₆₇⋊#95	C₂²	1	8	yes	—
16	C₂²×Q₈×C₆₇	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₆₇	C₄	1	8	yes	complemented
18	C₂×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
19	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
20	C₂²×Dic₆₇	C₂³	1	2	yes	—
21	C₂²×Dic₆₇	C₂³	1	4	yes	—
22	C₂²×Dic₆₇	C₂³	1	2	yes	—
23	C₄⋊Dic₆₇	—	2	4	no	—
24	C₄⋊Dic₆₇	—	2	8	no	—
25	C₄×Dic₆₇	—	2	2	no	—
26	C₄×Dic₆₇	—	2	8	no	—
27	C₄×Dic₆₇	D₄	1	4	yes	—
28	C₂×Q₈×C₆₇	—	2	4	no	—
29	C₂×Q₈×C₆₇	—	2	4	no	—
30	C₂×Q₈×C₆₇	C₂×C₄	1	8	yes	—
31	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
32	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
33	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
34	C₄⋊C₄×C₆₇	C₂×C₄	1	16	yes	—
35	C₄²×C₆₇	C₂×C₄	1	4	yes	—
36	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
37	Q₈×C₆₇	—	2	16	no	—
38	C₂×C₄×C₆₇	—	2	4	no	—
39	C₂×C₄×C₆₇	—	2	4	no	—
40	C₂×C₄×C₆₇	—	2	4	no	—
41	C₂×C₄×C₆₇	C₂²×C₄	1	4	yes	—
42	C₂×C₄×C₆₇	C₂²×C₄	1	8	yes	—
43	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
44	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₆₇	#41	1	2	yes	—
46	C₂×Dic₆₇	—	2	4	no	—
47	C₂×Dic₆₇	—	2	4	no	—
48	C₂×Dic₆₇	—	2	2	no	—
49	C₂×Dic₆₇	—	2	8	no	—
50	C₂×Dic₆₇	—	2	4	no	—
51	C₂×Dic₆₇	C₂×D₄	1	4	yes	—
52	C₂²×C₆₇	C₂³×C₄	1	1	yes	characteristic
53	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
54	C₂²×C₆₇	C₂×#41	1	1	yes	characteristic
55	C₂²×C₆₇	2₋¹⁺⁴	1	4	yes	—
56	C₄×C₆₇	—	2	8	no	—
57	C₄×C₆₇	—	2	4	no	—
58	C₄×C₆₇	C₄×D₄	1	4	yes	—
59	Dic₆₇	—	2	4	no	—
60	Dic₆₇	—	4	4	no	—
61	Dic₆₇	—	4	2	no	—
62	C₂×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
63	C₂×C₆₇	#519	1	2	yes	—
64	C₂×C₆₇	#536	1	2	yes	—
65	C₂×C₆₇	#541	1	2	yes	—
66	#2327	—	67	1	no	maximal, Sylow
67	C₆₇	#2327	1	1	yes	characteristic, complemented, Sylow
68	C₂×C₄⋊Q₈	D₆₇	1	1	yes	characteristic
69	C₂×C₄×Q₈	—	67	2	no	—
70	#391	—	67	2	no	—
71	#387	—	67	8	no	—
72	#382	—	67	1	no	—
73	#376	—	67	1	no	—
74	C₂²×Q₈	C₂×D₆₇	1	2	yes	—
75	C₄⋊Q₈	Dic₆₇	1	8	yes	complemented
76	C₄×Q₈	—	134	8	no	—
77	C₂×C₄⋊C₄	—	67	2	no	—
78	C₂×C₄⋊C₄	—	67	4	no	—
79	C₂×C₄⋊C₄	C₂×D₆₇	1	4	yes	—
80	C₂×C₄²	—	67	1	no	—
81	C₂×C₄²	—	67	4	no	—
82	C₂×C₄²	—	67	1	no	—
83	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
84	#95	—	67	4	no	—
85	#95	—	67	4	no	—
86	#95	—	67	8	no	—
87	C₂×Q₈	—	2	4	no	—
88	C₂×Q₈	—	2	4	no	—
89	C₂×Q₈	C₂×Dic₆₇	1	8	yes	—
90	C₂²×C₄	—	67	2	no	—
91	C₂²×C₄	—	67	2	no	—
92	C₂²×C₄	—	67	4	no	—
93	C₂²×C₄	C₂²×D₆₇	1	2	yes	—
94	C₂²×C₄	C₂²×D₆₇	1	4	yes	—
95	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
96	C₄⋊C₄	—	134	8	no	—
97	C₄⋊C₄	—	134	4	no	—
98	C₄⋊C₄	C₂×Dic₆₇	1	16	yes	—
99	C₄²	—	67	4	no	—
100	C₄²	—	134	8	no	—
101	C₄²	—	134	2	no	—
102	C₄²	C₂×Dic₆₇	1	4	yes	—
103	C₂³	C₂³×D₆₇	1	1	yes	characteristic
104	Q₈	—	2	16	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	67	4	no	—
109	C₂×C₄	—	134	4	no	—
110	C₂×C₄	—	134	2	no	—
111	C₂×C₄	—	134	4	no	—
112	C₂×C₄	—	134	8	no	—
113	C₂×C₄	—	134	4	no	—
114	C₂×C₄	C₂²×Dic₆₇	1	4	yes	—
115	C₂×C₄	C₂²×Dic₆₇	1	8	yes	—
116	C₂×C₄	C₂²×Dic₆₇	1	2	yes	—
117	C₂×C₄	D₄×D₆₇	1	2	yes	—
118	C₂×C₄	#11761627	1	2	yes	—
119	C₂²	C₂³×Dic₆₇	1	1	yes	characteristic
120	C₂²	C₂×D₄×D₆₇	1	1	yes	characteristic
121	C₂²	C₂×#11761627	1	1	yes	characteristic
122	C₂²	#14524091	1	4	yes	—
123	C₄	—	2	8	no	—
124	C₄	—	2	4	no	—
125	C₄	—	268	2	no	—
126	C₄	—	268	4	no	—
127	C₄	—	134	4	no	—
128	C₄	D₄×Dic₆₇	1	4	yes	—
129	C₂	C₂×D₄×Dic₆₇	1	1	yes	characteristic
130	C₂	#16863874	1	2	yes	—
131	C₂	#16863971	1	2	yes	—
132	C₂	#16864007	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×Q₈×Dic₆₇

C₂

yes

maximal

#16863303

C₂

yes

maximal

#16863282

C₂

yes

maximal

#16863262

C₂

yes

maximal, characteristic

#16863242

C₂

yes

maximal, characteristic

C₂×C₄⋊Q₈×C₆₇

C₂

yes

maximal, characteristic

Q₈×Dic₆₇

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

C₂×C₄×Dic₆₇

C₂²

yes

—

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

C₆₇⋊#95

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₆₇⋊#95

C₂²

yes

—

C₂²×Q₈×C₆₇

C₂²

yes

—

C₄⋊Q₈×C₆₇

C₄

yes

complemented

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₂²×Dic₆₇

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₂²×Dic₆₇

C₂³

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

D₄

yes

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₂×C₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₄²×C₆₇

C₂×C₄

yes

—

C₂³×C₆₇

C₂⁴

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

C₂×D₄

yes

—

C₂²×C₆₇

C₂³×C₄

yes

characteristic

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₂²×C₆₇

C₂×#41

yes

characteristic

C₂²×C₆₇

2₋¹⁺⁴

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₄×D₄

yes

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

C₂×C₄×D₄

yes

characteristic

C₂×C₆₇

#519

yes

—

C₂×C₆₇

#536

yes

—

C₂×C₆₇

#541

yes

—

#2327

—

maximal, Sylow

C₆₇

#2327

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

D₆₇

yes

characteristic

C₂×C₄×Q₈

—

—

—

—

—

C₂²×Q₈

C₂×D₆₇

yes

—

C₄⋊Q₈

Dic₆₇

yes

complemented

C₄×Q₈

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂×D₆₇

yes

—

—

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

C₂×Dic₆₇

yes

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²×D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

C₄⋊C₄

—

134

—

C₄⋊C₄

—

134

—

C₄⋊C₄

C₂×Dic₆₇

yes

—

C₄²

—

100

C₄²

—

134

—

101

C₄²

—

134

—

102

C₄²

C₂×Dic₆₇

yes

—

103

C₂³

C₂³×D₆₇

yes

characteristic

104

Q₈

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

134

—

110

C₂×C₄

—

134

—

111

C₂×C₄

—

134

—

112

C₂×C₄

—

134

—

113

C₂×C₄

—

134

—

114

C₂×C₄

C₂²×Dic₆₇

yes

—

115

C₂×C₄

C₂²×Dic₆₇

yes

—

116

C₂×C₄

C₂²×Dic₆₇

yes

—

117

C₂×C₄

D₄×D₆₇

yes

—

118

C₂×C₄

#11761627

yes

—

119

C₂²

C₂³×Dic₆₇

yes

characteristic

120

C₂²

C₂×D₄×D₆₇

yes

characteristic

121

C₂²

C₂×#11761627

yes

characteristic

122

C₂²

#14524091

yes

—

123

C₄

—

124

C₄

—

125

C₄

—

268

—

126

C₄

—

268

—

127

C₄

—

134

—

128

C₄

D₄×Dic₆₇

yes

—

129

C₂

C₂×D₄×Dic₆₇

yes

characteristic

130

C₂

#16863874

yes

—

131

C₂

#16863971

yes

—

132

C₂

#16864007

yes

—

	subgroup	G/H
center, Frattini	C₂³	C₂³×D₆₇
commutator	C₂²×C₆₇	C₂³×C₄
Fitting	C₂×C₄⋊Q₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂⁴
hypercenter	C₂×C₄⋊Q₈	D₆₇
hypocenter	C₆₇	#2327
LCS	G ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

subgroup

G/H

center, Frattini

C₂³

C₂³×D₆₇

commutator

C₂²×C₆₇

C₂³×C₄

Fitting

C₂×C₄⋊Q₈×C₆₇

C₂

socle

C₂³×C₆₇

C₂⁴

hypercenter

C₂×C₄⋊Q₈

D₆₇

C₆₇

G ▷ C₂²×C₆₇ ▷ C₆₇

C₁ ◁ C₂³ ◁ C₂×C₄⋊Q₈

N	H ≅ G/N	type
17. C₄⋊Q₈×C₆₇	C₄	semidirect
67. C₆₇	#2327	semidirect
75. C₄⋊Q₈	Dic₆₇	semidirect

H ≅ G/N

type

17. C₄⋊Q₈×C₆₇

C₄

semidirect

67. C₆₇

#2327

semidirect

75. C₄⋊Q₈

Dic₆₇

semidirect

Subgroups — #22562929

Conjugacy classes of subgroups

Complements