Group #22568997 — subgroups

Subgroups — C₂×#16863504 #22568997

Order 8576.

480 conjugacy classes of subgroups; only the 205 normal subgroups are shown.

normal subgroups only

This group has 480 conjugacy classes of subgroups — too many to draw. Only its 205 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	C₂²×Dic₂₆₈	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×Dic₁₃₄	C₂	1	2	yes	maximal
3	#16863504	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14523975	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523968	C₂	1	2	yes	maximal
6	C₂×#97×C₆₇	C₂	1	1	yes	maximal, characteristic
7	C₂²×Dic₁₃₄	C₂²	1	2	yes	—
8	C₂×Dic₂₆₈	C₄	1	8	yes	complemented
9	C₂×Dic₂₆₈	C₂²	1	4	yes	—
10	C₄×Dic₁₃₄	—	2	4	no	—
11	C₄×Dic₁₃₄	C₂²	1	8	yes	—
12	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
13	C₂×Dic₆₇⋊C₄	—	2	2	no	—
14	C₂×C₄×Dic₆₇	—	2	2	no	—
15	#14523975	C₂²	1	4	yes	—
16	#14523968	C₂²	1	8	yes	—
17	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂²×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
20	#97×C₆₇	C₂²	1	4	yes	—
21	C₂×Dic₁₃₄	—	2	2	no	—
22	C₂×Dic₁₃₄	—	2	4	no	—
23	C₂×Dic₁₃₄	C₂×C₄	1	8	yes	—
24	C₂×Dic₁₃₄	C₂³	1	4	yes	—
25	C₂²×Dic₆₇	—	2	2	no	—
26	C₂²×Dic₆₇	—	2	2	no	—
27	Dic₂₆₈	C₂×C₄	1	16	yes	complemented
28	C₄⋊Dic₆₇	—	2	2	no	—
29	C₄⋊Dic₆₇	C₂³	1	4	yes	—
30	Dic₆₇⋊C₄	—	2	8	no	—
31	C₄×Dic₆₇	—	2	8	no	—
32	C₂²×C₄×C₆₇	D₄	1	2	yes	—
33	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
34	C₂×C₈×C₆₇	—	2	2	no	—
35	C₂×C₈×C₆₇	C₂×C₄	1	4	yes	—
36	C₂×C₈×C₆₇	C₂³	1	2	yes	—
37	C₂×C₈×C₆₇	C₂³	1	2	yes	—
38	C₄²×C₆₇	D₄	1	2	yes	—
39	C₄²×C₆₇	C₂³	1	2	yes	—
40	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₂×C₄×C₆₇	—	2	4	no	—
42	C₂×C₄×C₆₇	C₂²×C₄	1	2	yes	—
43	C₂×C₄×C₆₇	C₂×D₄	1	4	yes	—
44	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₂×C₄×C₆₇	#41	1	2	yes	—
46	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₆₇	—	2	8	no	—
48	C₂×Dic₆₇	—	2	4	no	—
49	C₂×Dic₆₇	—	2	4	no	—
50	C₂×Dic₆₇	—	4	4	no	—
51	Dic₁₃₄	—	2	8	no	—
52	Dic₁₃₄	—	2	4	no	—
53	Dic₁₃₄	C₂²×C₄	1	8	yes	—
54	C₈×C₆₇	—	2	2	no	—
55	C₈×C₆₇	C₂²×C₄	1	4	yes	—
56	C₂²×C₆₇	C₄×D₄	1	2	yes	—
57	C₂²×C₆₇	#137	1	4	yes	—
58	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₆₇	—	2	4	no	—
60	C₄×C₆₇	C₄×D₄	1	2	yes	—
61	C₄×C₆₇	C₂³×C₄	1	1	yes	characteristic
62	C₄×C₆₇	C₂×#41	1	1	yes	characteristic
63	Dic₆₇	—	2	8	no	—
64	Dic₆₇	—	4	4	no	—
65	C₂×C₆₇	#441	1	4	yes	complemented
66	C₂×C₆₇	C₂×C₄×D₄	1	1	yes	characteristic
67	C₂×C₆₇	C₂×#137	1	2	yes	—
68	C₂×#441	—	67	1	no	maximal, Sylow
69	C₆₇	C₂×#441	1	1	yes	characteristic, complemented, Sylow
70	C₂²×Q₁₆	—	67	1	no	—
71	C₂×C₄×Q₈	—	67	2	no	—
72	#441	—	67	8	no	—
73	C₂×#106	—	67	1	no	—
74	C₂×#103	—	67	2	no	—
75	C₂×#97	D₆₇	1	1	yes	characteristic
76	C₂²×Q₈	—	67	2	no	—
77	C₂×Q₁₆	—	67	8	no	—
78	C₂×Q₁₆	—	67	4	no	—
79	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
80	C₂²×C₈	C₂×D₆₇	1	1	yes	characteristic
81	C₄×Q₈	—	67	8	no	—
82	C₄×Q₈	—	134	4	no	—
83	C₂×C₄⋊C₄	—	67	2	no	—
84	C₂×C₄⋊C₄	—	134	2	no	—
85	C₂×C₄²	—	134	2	no	—
86	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
87	#106	—	67	4	no	—
88	#103	—	67	8	no	—
89	#97	C₂×D₆₇	1	4	yes	—
90	C₂×Q₈	—	67	8	no	—
91	C₂×Q₈	—	67	4	no	—
92	C₂×Q₈	—	134	2	no	—
93	C₂×Q₈	—	134	4	no	—
94	C₂²×C₄	—	134	2	no	—
95	C₂²×C₄	—	134	2	no	—
96	C₂²×C₄	D₂₆₈	1	2	yes	—
97	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
98	Q₁₆	—	67	16	no	—
99	C₂×C₈	—	2	2	no	—
100	C₂×C₈	C₄×D₆₇	1	4	yes	—
101	C₂×C₈	C₂²×D₆₇	1	2	yes	—
102	C₂×C₈	C₂²×D₆₇	1	2	yes	—
103	C₄⋊C₄	—	67	4	no	—
104	C₄⋊C₄	—	134	2	no	—
105	C₄⋊C₄	—	134	8	no	—
106	C₄²	—	134	8	no	—
107	C₄²	D₂₆₈	1	2	yes	—
108	C₄²	C₂²×D₆₇	1	2	yes	—
109	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
110	Q₈	—	67	8	no	—
111	Q₈	—	134	8	no	—
112	Q₈	—	134	4	no	—
113	C₂×C₄	—	2	4	no	—
114	C₂×C₄	—	268	4	no	—
115	C₂×C₄	—	134	4	no	—
116	C₂×C₄	—	134	8	no	—
117	C₂×C₄	—	134	4	no	—
118	C₂×C₄	C₂×C₄×D₆₇	1	2	yes	—
119	C₂×C₄	C₂×D₂₆₈	1	4	yes	—
120	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
121	C₂×C₄	#11761628	1	2	yes	—
122	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
123	C₈	—	2	2	no	—
124	C₈	C₂×C₄×D₆₇	1	4	yes	—
125	C₂²	C₄×D₂₆₈	1	2	yes	—
126	C₂²	#14524076	1	4	yes	—
127	C₂²	C₂²×D₂₆₈	1	1	yes	characteristic
128	C₄	—	2	4	no	—
129	C₄	—	268	4	no	—
130	C₄	—	134	8	no	—
131	C₄	C₄×D₂₆₈	1	2	yes	—
132	C₄	C₂²×C₄×D₆₇	1	1	yes	characteristic
133	C₄	C₂×#11761628	1	1	yes	characteristic
134	C₂	#16863504	1	4	yes	direct product factor
135	C₂	C₂×C₄×D₂₆₈	1	1	yes	characteristic
136	C₂	C₂×#14524076	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₂₆₈

C₂

yes

maximal, characteristic

C₂×C₄×Dic₁₃₄

C₂

yes

maximal

#16863504

C₂

yes

maximal, direct product factor

C₂×#14523975

C₂

yes

maximal, characteristic

C₂×#14523968

C₂

yes

maximal

C₂×#97×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

C₂²

yes

—

C₂×Dic₂₆₈

C₄

yes

complemented

C₂×Dic₂₆₈

C₂²

yes

—

C₄×Dic₁₃₄

—

C₄×Dic₁₃₄

C₂²

yes

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

C₂×Dic₆₇⋊C₄

—

C₂×C₄×Dic₆₇

—

#14523975

C₂²

yes

—

#14523968

C₂²

yes

—

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂²×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄²×C₆₇

C₂²

yes

characteristic

#97×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₂×C₄

yes

—

C₂×Dic₁₃₄

C₂³

yes

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

Dic₂₆₈

C₂×C₄

yes

complemented

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₈×C₆₇

C₂³

yes

—

C₂×C₈×C₆₇

C₂³

yes

—

C₄²×C₆₇

D₄

yes

—

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

C₂²×C₄

yes

—

C₈×C₆₇

—

C₈×C₆₇

C₂²×C₄

yes

—

C₂²×C₆₇

C₄×D₄

yes

—

C₂²×C₆₇

#137

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

C₄×D₄

yes

—

C₄×C₆₇

C₂³×C₄

yes

characteristic

C₄×C₆₇

C₂×#41

yes

characteristic

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

#441

yes

complemented

C₂×C₆₇

C₂×C₄×D₄

yes

characteristic

C₂×C₆₇

C₂×#137

yes

—

C₂×#441

—

maximal, Sylow

C₆₇

C₂×#441

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×C₄×Q₈

—

#441

—

C₂×#106

—

C₂×#103

—

C₂×#97

D₆₇

yes

characteristic

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₂²×C₈

C₂×D₆₇

yes

characteristic

C₄×Q₈

—

C₄×Q₈

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄²

—

134

—

C₂×C₄²

C₂×D₆₇

yes

—

—

C₂×D₆₇

yes

—

C₂×Q₈

—

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

D₂₆₈

yes

—

C₂²×C₄

C₂²×D₆₇

yes

characteristic

Q₁₆

—

C₂×C₈

—

100

C₂×C₈

C₄×D₆₇

yes

—

101

C₂×C₈

C₂²×D₆₇

yes

—

102

C₂×C₈

C₂²×D₆₇

yes

—

103

C₄⋊C₄

—

104

C₄⋊C₄

—

134

—

105

C₄⋊C₄

—

134

—

106

C₄²

—

134

—

107

C₄²

D₂₆₈

yes

—

108

C₄²

C₂²×D₆₇

yes

—

109

C₂³

C₂×D₂₆₈

yes

characteristic

110

Q₈

—

111

Q₈

—

134

—

112

Q₈

—

134

—

113

C₂×C₄

—

114

C₂×C₄

—

268

—

115

C₂×C₄

—

134

—

116

C₂×C₄

—

134

—

117

C₂×C₄

—

134

—

118

C₂×C₄

C₂×C₄×D₆₇

yes

—

119

C₂×C₄

C₂×D₂₆₈

yes

—

120

C₂×C₄

C₂×D₂₆₈

yes

characteristic

121

C₂×C₄

#11761628

yes

—

122

C₂×C₄

C₂³×D₆₇

yes

characteristic

123

C₈

—

124

C₈

C₂×C₄×D₆₇

yes

—

125

C₂²

C₄×D₂₆₈

yes

—

126

C₂²

#14524076

yes

—

127

C₂²

C₂²×D₂₆₈

yes

characteristic

128

C₄

—

129

C₄

—

268

—

130

C₄

—

134

—

131

C₄

C₄×D₂₆₈

yes

—

132

C₄

C₂²×C₄×D₆₇

yes

characteristic

133

C₄

C₂×#11761628

yes

characteristic

134

C₂

#16863504

yes

direct product factor

135

C₂

C₂×C₄×D₂₆₈

yes

characteristic

136

C₂

C₂×#14524076

yes

—

	subgroup	G/H
center	C₂³	C₂×D₂₆₈
commutator	C₄×C₆₇	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂×#97×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂×#97	D₆₇
hypocenter	C₆₇	C₂×#441
LCS	G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

subgroup

G/H

center

C₂³

C₂×D₂₆₈

commutator

C₄×C₆₇

C₂³×C₄

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₂×#97×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

hypercenter

C₂×#97

D₆₇

hypocenter

C₆₇

C₂×#441

LCS

G ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#97

N	H ≅ G/N	type
3. #16863504	C₂	direct (N × H)
8. C₂×Dic₂₆₈	C₄	semidirect
27. Dic₂₆₈	C₂×C₄	semidirect
65. C₂×C₆₇	#441	semidirect
69. C₆₇	C₂×#441	semidirect
134. C₂	#16863504	direct (N × H)

H ≅ G/N

type

3. #16863504

C₂

direct (N × H)

8. C₂×Dic₂₆₈

C₄

semidirect

27. Dic₂₆₈

C₂×C₄

semidirect

65. C₂×C₆₇

#441

semidirect

69. C₆₇

C₂×#441

semidirect

134. C₂

#16863504

direct (N × H)

Subgroups — C₂×#16863504 #22568997

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16863504 #22568997

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16863504 #22568997