Group #22561556 — subgroups

Subgroups — #22561556

Order 8576.

230 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 75 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	#16863439	C₂	1	2	yes	maximal
2	#16863261	C₂	1	1	yes	maximal, characteristic
3	#16863165	C₂	1	1	yes	maximal, characteristic
4	#16863155	C₂	1	2	yes	maximal
5	#544×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×#11761612	C₂²	1	2	yes	—
7	C₂×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
8	C₂×C₄×Dic₆₇	C₂²	1	1	yes	characteristic
9	#14523976	—	2	2	no	—
10	#14523974	—	2	2	no	—
11	C₆₇⋊#95	—	2	2	no	—
12	C₄⋊Q₈×C₆₇	—	2	2	no	—
13	#125×C₆₇	—	2	2	no	—
14	C₂²⋊Q₈×C₆₇	—	2	2	no	—
15	#117×C₆₇	C₂²	1	2	yes	—
16	C₂×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
17	C₂²×Dic₆₇	—	2	1	no	—
18	C₂²×Dic₆₇	—	2	1	no	—
19	#11761612	—	2	2	no	—
20	#11761612	Q₈	1	4	yes	complemented
21	C₂×C₆₇⋊C₈	—	2	2	no	—
22	C₄⋊Dic₆₇	—	2	2	no	—
23	C₄×Dic₆₇	—	2	1	no	—
24	C₄×Dic₆₇	Q₈	1	2	yes	complemented
25	C₂×Q₈×C₆₇	—	2	2	no	—
26	C₂²×C₄×C₆₇	D₄	1	2	yes	—
27	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
28	C₄⋊C₄×C₆₇	—	2	2	no	—
29	C₄⋊C₄×C₆₇	—	2	4	no	—
30	C₄⋊C₄×C₆₇	—	2	2	no	—
31	C₄⋊C₄×C₆₇	D₄	1	2	yes	—
32	C₄⋊C₄×C₆₇	D₄	1	2	yes	—
33	C₂²⋊C₄×C₆₇	—	2	2	no	—
34	C₄²×C₆₇	—	2	2	no	—
35	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
36	Q₈×C₆₇	—	8	2	no	—
37	C₂×C₄×C₆₇	—	2	2	no	—
38	C₂×C₄×C₆₇	—	2	2	no	—
39	C₂×C₄×C₆₇	—	2	2	no	—
40	C₂×C₄×C₆₇	—	4	2	no	—
41	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
42	C₂×C₄×C₆₇	C₂×Q₈	1	2	yes	—
43	C₂×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₄×C₆₇	#41	1	1	yes	characteristic
45	C₂×Dic₆₇	—	2	2	no	—
46	C₂×Dic₆₇	—	4	1	no	—
47	C₂×Dic₆₇	—	4	1	no	—
48	C₂×Dic₆₇	—	4	2	no	—
49	C₂×Dic₆₇	—	4	1	no	—
50	C₆₇⋊C₈	—	2	4	no	—
51	C₂²×C₆₇	—	2	1	no	—
52	C₂²×C₆₇	—	2	1	no	—
53	C₂²×C₆₇	#120	1	1	yes	characteristic
54	C₂²×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
55	C₂²×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
56	C₄×C₆₇	—	4	2	no	—
57	C₄×C₆₇	—	4	2	no	—
58	C₄×C₆₇	—	4	2	no	—
59	C₄×C₆₇	C₂²⋊Q₈	1	2	yes	—
60	C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
61	Dic₆₇	—	4	2	no	—
62	Dic₆₇	—	8	1	no	—
63	C₂×C₆₇	—	2	1	no	—
64	C₂×C₆₇	—	2	1	no	—
65	C₂×C₆₇	#395	1	1	yes	characteristic
66	C₂×C₆₇	#454	1	1	yes	characteristic
67	C₂×C₆₇	#455	1	1	yes	characteristic
68	#2061	—	67	1	no	maximal, Sylow
69	C₆₇	#2061	1	1	yes	characteristic, complemented, Sylow
70	#544	D₆₇	1	1	yes	characteristic
71	#428	—	67	2	no	—
72	#382	—	67	1	no	—
73	#343	—	67	1	no	—
74	#339	—	67	2	no	—
75	C₂×M₁₆	—	67	2	no	—
76	C₄⋊Q₈	—	2	2	no	—
77	#125	—	2	2	no	—
78	C₂²⋊Q₈	—	2	2	no	—
79	#117	C₂×D₆₇	1	2	yes	—
80	C₂×C₄⋊C₄	—	67	1	no	—
81	C₂×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
82	C₂×C₄²	—	67	1	no	—
83	#107	—	134	2	no	—
84	#106	—	134	2	no	—
85	#95	—	134	2	no	—
86	C₂×Q₈	—	2	2	no	—
87	C₂²×C₄	—	134	1	no	—
88	C₂²×C₄	—	134	1	no	—
89	C₂²×C₄	C₂²⋊D₆₇	1	2	yes	—
90	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
91	M₁₆	—	67	4	no	—
92	M₁₆	—	134	2	no	—
93	C₂×C₈	—	134	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	4	no	—
96	C₄⋊C₄	—	2	2	no	—
97	C₄⋊C₄	—	134	2	no	—
98	C₄⋊C₄	C₂²⋊D₆₇	1	2	yes	—
99	C₄⋊C₄	C₂²⋊D₆₇	1	2	yes	—
100	C₂²⋊C₄	—	2	2	no	—
101	C₄²	—	2	2	no	—
102	C₄²	—	67	2	no	—
103	C₄²	—	134	1	no	—
104	C₂³	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
105	Q₈	—	8	2	no	—
106	C₂×C₄	—	2	2	no	—
107	C₂×C₄	—	2	2	no	—
108	C₂×C₄	—	2	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	—	268	1	no	—
111	C₂×C₄	—	268	1	no	—
112	C₂×C₄	—	268	2	no	—
113	C₂×C₄	—	268	1	no	—
114	C₂×C₄	—	134	2	no	—
115	C₂×C₄	C₂×C₂²⋊D₆₇	1	2	yes	—
116	C₂×C₄	Q₈×D₆₇	1	2	yes	—
117	C₂×C₄	Q₈×D₆₇	1	1	yes	characteristic
118	C₂×C₄	#11761629	1	1	yes	characteristic
119	C₈	—	134	4	no	—
120	C₂²	—	2	1	no	—
121	C₂²	—	2	1	no	—
122	C₂²	#14524012	1	1	yes	characteristic
123	C₂²	#14524019	1	1	yes	characteristic
124	C₂²	#14524041	1	1	yes	characteristic
125	C₄	—	4	2	no	—
126	C₄	—	4	2	no	—
127	C₄	—	4	2	no	—
128	C₄	—	268	2	no	—
129	C₄	—	536	1	no	—
130	C₄	#14524019	1	2	yes	—
131	C₄	#14524041	1	2	yes	—
132	C₂	—	2	1	no	—
133	C₂	—	2	1	no	—
134	C₂	#16863315	1	1	yes	characteristic
135	C₂	#16863569	1	1	yes	characteristic
136	C₂	#16863574	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16863439

C₂

yes

maximal

#16863261

C₂

yes

maximal, characteristic

#16863165

C₂

yes

maximal, characteristic

#16863155

C₂

yes

maximal

#544×C₆₇

C₂

yes

maximal, characteristic

C₂×#11761612

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

characteristic

C₂×C₄×Dic₆₇

C₂²

yes

characteristic

#14523976

—

#14523974

—

C₆₇⋊#95

—

C₄⋊Q₈×C₆₇

—

#125×C₆₇

—

C₂²⋊Q₈×C₆₇

—

#117×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

#11761612

—

#11761612

Q₈

yes

complemented

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

Q₈

yes

complemented

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

D₄

yes

—

C₄⋊C₄×C₆₇

D₄

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₄×C₆₇

#41

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#120

yes

characteristic

C₂²×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂²×C₆₇

C₄⋊Q₈

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₂²⋊Q₈

yes

—

C₄×C₆₇

C₄⋊Q₈

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₆₇

yes

characteristic, complemented, Sylow

#544

D₆₇

yes

—

—

—

—

C₂×M₁₆

—

C₄⋊Q₈

—

#125

—

C₂²⋊Q₈

—

#117

C₂×D₆₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂×D₆₇

yes

characteristic

C₂×C₄²

—

#107

—

134

—

#106

—

134

—

#95

—

134

—

C₂×Q₈

—

C₂²×C₄

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

C₂²⋊D₆₇

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

M₁₆

—

M₁₆

—

134

—

C₂×C₈

—

134

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

134

—

C₄⋊C₄

C₂²⋊D₆₇

yes

—

C₄⋊C₄

C₂²⋊D₆₇

yes

—

100

C₂²⋊C₄

—

101

C₄²

—

102

C₄²

—

103

C₄²

—

134

—

104

C₂³

C₂×C₂²⋊D₆₇

yes

characteristic

105

Q₈

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

268

—

111

C₂×C₄

—

268

—

112

C₂×C₄

—

268

—

113

C₂×C₄

—

268

—

114

C₂×C₄

—

134

—

115

C₂×C₄

C₂×C₂²⋊D₆₇

yes

—

116

C₂×C₄

Q₈×D₆₇

yes

—

117

C₂×C₄

Q₈×D₆₇

yes

characteristic

118

C₂×C₄

#11761629

yes

characteristic

119

C₈

—

134

—

120

C₂²

—

121

C₂²

—

122

C₂²

#14524012

yes

characteristic

123

C₂²

#14524019

yes

characteristic

124

C₂²

#14524041

yes

characteristic

125

C₄

—

126

C₄

—

127

C₄

—

128

C₄

—

268

—

129

C₄

—

536

—

130

C₄

#14524019

yes

—

131

C₄

#14524041

yes

—

132

C₂

—

133

C₂

—

134

C₂

#16863315

yes

characteristic

135

C₂

#16863569

yes

characteristic

136

C₂

#16863574

yes

characteristic

	subgroup	G/H
center	C₂²	#14524012
commutator	C₂²×C₄×C₆₇	C₂³
Frattini	C₂²×C₄	C₂²×D₆₇
Fitting	#544×C₆₇	C₂
socle	C₂²×C₆₇	#120
hypercenter	#544	D₆₇
hypocenter	C₆₇	#2061
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

subgroup

G/H

center

C₂²

#14524012

commutator

C₂²×C₄×C₆₇

C₂³

Frattini

C₂²×C₄

C₂²×D₆₇

Fitting

#544×C₆₇

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #544

N	H ≅ G/N	type
20. #11761612	Q₈	semidirect
24. C₄×Dic₆₇	Q₈	semidirect
69. C₆₇	#2061	semidirect

H ≅ G/N

type

20. #11761612

Q₈

semidirect

24. C₄×Dic₆₇

Q₈

semidirect

69. C₆₇

#2061

semidirect

Subgroups — #22561556

Conjugacy classes of subgroups

Complements