Group #22570520 — subgroups

Subgroups — C₂×#16863747 #22570520

Order 8576.

424 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 181 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×#14524043	C₂	1	1	yes	maximal, characteristic
2	C₂×#14524043	C₂	1	1	yes	maximal, characteristic
3	#16863747	C₂	1	8	yes	maximal, direct product factor
4	C₂×#14523968	C₂	1	4	yes	maximal
5	C₂×C₄×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₁₃₄	—	2	1	no	—
7	C₂²×Dic₁₃₄	C₂²	1	2	yes	—
8	#14524043	—	2	2	no	—
9	#14524043	—	2	2	no	—
10	#14524043	C₂²	1	4	yes	—
11	#14524043	C₂²	1	4	yes	—
12	C₂×C₄⋊Dic₆₇	—	2	1	no	—
13	C₂×C₄⋊Dic₆₇	—	2	1	no	—
14	C₂×C₄⋊Dic₆₇	—	2	1	no	—
15	C₂×C₄⋊Dic₆₇	C₂²	1	2	yes	—
16	#14523968	C₂²	1	16	yes	—
17	C₂²×C₈×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
19	C₄×C₈×C₆₇	C₂²	1	4	yes	—
20	C₂×Dic₁₃₄	—	2	8	no	—
21	C₂×Dic₁₃₄	—	2	2	no	—
22	C₂×Dic₁₃₄	—	2	4	no	—
23	C₂×Dic₁₃₄	—	4	4	no	—
24	C₂×Dic₁₃₄	C₂³	1	4	yes	—
25	C₂²×Dic₆₇	—	2	2	no	—
26	C₂²×Dic₆₇	—	2	2	no	—
27	C₄⋊Dic₆₇	—	2	4	no	—
28	C₄⋊Dic₆₇	—	2	4	no	—
29	C₄⋊Dic₆₇	—	2	4	no	—
30	C₄⋊Dic₆₇	—	2	2	no	—
31	C₄⋊Dic₆₇	C₂³	1	4	yes	—
32	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
33	C₂²×C₄×C₆₇	D₄	1	1	yes	characteristic
34	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
35	C₂×C₈×C₆₇	—	2	4	no	—
36	C₂×C₈×C₆₇	C₂³	1	4	yes	—
37	C₄²×C₆₇	D₄	1	2	yes	—
38	C₄²×C₆₇	C₂³	1	2	yes	—
39	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₄×C₆₇	SD₁₆	1	4	yes	—
41	C₂×C₄×C₆₇	Q₁₆	1	4	yes	—
42	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
44	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
45	C₂×C₄×C₆₇	#41	1	4	yes	—
46	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
47	C₂×Dic₆₇	—	2	4	no	—
48	C₂×Dic₆₇	—	2	4	no	—
49	C₂×Dic₆₇	—	4	4	no	—
50	C₂×Dic₆₇	—	4	4	no	—
51	Dic₁₃₄	—	2	8	no	—
52	Dic₁₃₄	—	4	4	no	—
53	Dic₁₃₄	—	4	8	no	—
54	C₈×C₆₇	—	2	4	no	—
55	C₂²×C₆₇	#124	1	2	yes	—
56	C₂²×C₆₇	C₂×SD₁₆	1	2	yes	—
57	C₂²×C₆₇	C₂×Q₁₆	1	2	yes	—
58	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
59	C₄×C₆₇	#124	1	2	yes	—
60	C₄×C₆₇	C₂×SD₁₆	1	4	yes	—
61	C₄×C₆₇	C₂×Q₁₆	1	4	yes	—
62	C₄×C₆₇	C₂×#41	1	2	yes	—
63	Dic₆₇	—	4	4	no	—
64	Dic₆₇	—	4	4	no	—
65	C₂×C₆₇	#487	1	4	yes	complemented
66	C₂×C₆₇	C₂×#124	1	1	yes	characteristic
67	C₂×C₆₇	C₂²×SD₁₆	1	1	yes	characteristic
68	C₂×C₆₇	C₂²×Q₁₆	1	1	yes	characteristic
69	C₂×#487	—	67	1	no	maximal, Sylow
70	C₆₇	C₂×#487	1	1	yes	characteristic, complemented, Sylow
71	C₂×C₄⋊Q₈	—	67	1	no	—
72	C₂×C₄⋊Q₈	—	67	1	no	—
73	#487	—	67	8	no	—
74	C₂×#103	—	67	4	no	—
75	C₂×C₄×C₈	D₆₇	1	1	yes	characteristic
76	C₂²×Q₈	—	67	2	no	—
77	C₂²×Q₈	—	134	1	no	—
78	C₂²×C₈	C₂×D₆₇	1	2	yes	—
79	C₄⋊Q₈	—	67	4	no	—
80	C₄⋊Q₈	—	67	4	no	—
81	C₄⋊Q₈	—	134	2	no	—
82	C₄⋊Q₈	—	134	2	no	—
83	C₂×C₄⋊C₄	—	67	2	no	—
84	C₂×C₄⋊C₄	—	134	1	no	—
85	C₂×C₄⋊C₄	—	134	1	no	—
86	C₂×C₄⋊C₄	—	134	1	no	—
87	C₂×C₄²	C₂×D₆₇	1	1	yes	characteristic
88	#103	—	67	16	no	—
89	C₄×C₈	C₂×D₆₇	1	4	yes	—
90	C₂×Q₈	—	268	4	no	—
91	C₂×Q₈	—	67	4	no	—
92	C₂×Q₈	—	134	8	no	—
93	C₂×Q₈	—	134	4	no	—
94	C₂×Q₈	—	134	2	no	—
95	C₂²×C₄	—	134	2	no	—
96	C₂²×C₄	—	134	2	no	—
97	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
98	C₂²×C₄	D₂₆₈	1	1	yes	characteristic
99	C₂²×C₄	C₂²×D₆₇	1	1	yes	characteristic
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	C₂²×D₆₇	1	4	yes	—
102	C₄⋊C₄	—	67	4	no	—
103	C₄⋊C₄	—	134	2	no	—
104	C₄⋊C₄	—	134	4	no	—
105	C₄⋊C₄	—	134	4	no	—
106	C₄⋊C₄	—	134	4	no	—
107	C₄²	D₂₆₈	1	2	yes	—
108	C₄²	C₂²×D₆₇	1	2	yes	—
109	C₂³	C₂×D₂₆₈	1	1	yes	characteristic
110	Q₈	—	268	4	no	—
111	Q₈	—	268	8	no	—
112	Q₈	—	134	8	no	—
113	C₂×C₄	—	268	4	no	—
114	C₂×C₄	—	268	4	no	—
115	C₂×C₄	—	134	4	no	—
116	C₂×C₄	—	134	4	no	—
117	C₂×C₄	#11761617	1	4	yes	—
118	C₂×C₄	Dic₂₆₈	1	4	yes	—
119	C₂×C₄	C₂×D₂₆₈	1	1	yes	characteristic
120	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
121	C₂×C₄	C₂×D₂₆₈	1	2	yes	—
122	C₂×C₄	#11761628	1	4	yes	—
123	C₂×C₄	C₂³×D₆₇	1	1	yes	characteristic
124	C₈	—	2	4	no	—
125	C₂²	#14524031	1	2	yes	—
126	C₂²	C₂×#11761617	1	2	yes	—
127	C₂²	C₂×Dic₂₆₈	1	2	yes	—
128	C₂²	C₂²×D₂₆₈	1	1	yes	characteristic
129	C₄	—	268	4	no	—
130	C₄	—	268	4	no	—
131	C₄	#14524031	1	2	yes	—
132	C₄	C₂×#11761617	1	4	yes	—
133	C₄	C₂×Dic₂₆₈	1	4	yes	—
134	C₄	C₂×#11761628	1	2	yes	—
135	C₂	#16863747	1	4	yes	direct product factor
136	C₂	C₂×#14524031	1	1	yes	characteristic
137	C₂	C₂²×#11761617	1	1	yes	characteristic
138	C₂	C₂²×Dic₂₆₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#14524043

C₂

yes

maximal, characteristic

C₂×#14524043

C₂

yes

maximal, characteristic

#16863747

C₂

yes

maximal, direct product factor

C₂×#14523968

C₂

yes

maximal

C₂×C₄×C₈×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₁₃₄

—

C₂²×Dic₁₃₄

C₂²

yes

—

#14524043

—

#14524043

—

#14524043

C₂²

yes

—

#14524043

C₂²

yes

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

C₂²

yes

—

#14523968

C₂²

yes

—

C₂²×C₈×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

C₄×C₈×C₆₇

C₂²

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₂³

yes

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂³

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

—

C₄²×C₆₇

D₄

yes

—

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

SD₁₆

yes

—

C₂×C₄×C₆₇

Q₁₆

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₂²×C₆₇

#124

yes

—

C₂²×C₆₇

C₂×SD₁₆

yes

—

C₂²×C₆₇

C₂×Q₁₆

yes

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

#124

yes

—

C₄×C₆₇

C₂×SD₁₆

yes

—

C₄×C₆₇

C₂×Q₁₆

yes

—

C₄×C₆₇

C₂×#41

yes

—

Dic₆₇

—

Dic₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

C₂²×SD₁₆

yes

characteristic

C₂×C₆₇

C₂²×Q₁₆

yes

—

C₆₇

yes

characteristic, complemented, Sylow

C₂×C₄⋊Q₈

—

C₂×C₄⋊Q₈

—

#487

—

C₂×#103

—

C₂×C₄×C₈

D₆₇

yes

characteristic

C₂²×Q₈

—

C₂²×Q₈

—

134

—

C₂²×C₈

C₂×D₆₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

—

134

—

C₄⋊Q₈

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄²

C₂×D₆₇

yes

characteristic

#103

—

C₄×C₈

C₂×D₆₇

yes

—

C₂×Q₈

—

268

—

C₂×Q₈

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂×Q₈

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

D₂₆₈

yes

characteristic

C₂²×C₄

D₂₆₈

yes

characteristic

C₂²×C₄

C₂²×D₆₇

yes

characteristic

100

C₂×C₈

—

101

C₂×C₈

C₂²×D₆₇

yes

—

102

C₄⋊C₄

—

103

C₄⋊C₄

—

134

—

104

C₄⋊C₄

—

134

—

105

C₄⋊C₄

—

134

—

106

C₄⋊C₄

—

134

—

107

C₄²

D₂₆₈

yes

—

108

C₄²

C₂²×D₆₇

yes

—

109

C₂³

C₂×D₂₆₈

yes

characteristic

110

Q₈

—

268

—

111

Q₈

—

268

—

112

Q₈

—

134

—

113

C₂×C₄

—

268

—

114

C₂×C₄

—

268

—

115

C₂×C₄

—

134

—

116

C₂×C₄

—

134

—

117

C₂×C₄

#11761617

yes

—

118

C₂×C₄

Dic₂₆₈

yes

—

119

C₂×C₄

C₂×D₂₆₈

yes

characteristic

120

C₂×C₄

C₂×D₂₆₈

yes

—

121

C₂×C₄

C₂×D₂₆₈

yes

—

122

C₂×C₄

#11761628

yes

—

123

C₂×C₄

C₂³×D₆₇

yes

characteristic

124

C₈

—

125

C₂²

#14524031

yes

—

126

C₂²

C₂×#11761617

yes

—

127

C₂²

C₂×Dic₂₆₈

yes

—

128

C₂²

C₂²×D₂₆₈

yes

characteristic

129

C₄

—

268

—

130

C₄

—

268

—

131

C₄

#14524031

yes

—

132

C₄

C₂×#11761617

yes

—

133

C₄

C₂×Dic₂₆₈

yes

—

134

C₄

C₂×#11761628

yes

—

135

C₂

#16863747

yes

direct product factor

136

C₂

C₂×#14524031

yes

characteristic

137

C₂

C₂²×#11761617

yes

characteristic

138

C₂

C₂²×Dic₂₆₈

yes

characteristic

	subgroup	G/H
center	C₂³	C₂×D₂₆₈
commutator	C₂×C₄×C₆₇	C₂⁴
Frattini	C₂×C₄	C₂³×D₆₇
Fitting	C₂×C₄×C₈×C₆₇	C₂
socle	C₂³×C₆₇	C₂×D₄
hypercenter	C₂×C₄×C₈	D₆₇
hypocenter	C₆₇	C₂×#487
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

subgroup

G/H

center

C₂³

C₂×D₂₆₈

commutator

C₂×C₄×C₆₇

C₂⁴

Frattini

C₂×C₄

C₂³×D₆₇

Fitting

C₂×C₄×C₈×C₆₇

C₂

socle

C₂³×C₆₇

C₂×D₄

hypercenter

C₂×C₄×C₈

D₆₇

hypocenter

C₆₇

C₂×#487

LCS

G ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈

N	H ≅ G/N	type
3. #16863747	C₂	direct (N × H)
65. C₂×C₆₇	#487	semidirect
70. C₆₇	C₂×#487	semidirect
135. C₂	#16863747	direct (N × H)

H ≅ G/N

type

3. #16863747

C₂

direct (N × H)

65. C₂×C₆₇

#487

semidirect

70. C₆₇

C₂×#487

semidirect

135. C₂

#16863747

direct (N × H)

Subgroups — C₂×#16863747 #22570520

Conjugacy classes of subgroups

Complements

Subgroups — C2×#16863747 #22570520

Conjugacy classes of subgroups

Complements

Subgroups — C₂×#16863747 #22570520