Group #22557686 — subgroups

Subgroups — C₆₇×#3255 #22557686

Order 8576.

342 conjugacy classes of subgroups; only the 176 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 176 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	#533×C₆₇	C₂	1	2	yes	maximal
2	#475×C₆₇	C₂	1	4	yes	maximal, complemented
3	#470×C₆₇	C₂	1	4	yes	maximal
4	#462×C₆₇	C₂	1	4	yes	maximal, complemented
5	#423×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂×Q₁₆×C₆₇	C₂²	1	4	yes	—
7	C₂×M₁₆×C₆₇	C₂²	1	2	yes	—
8	C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
9	C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
10	#125×C₆₇	—	2	2	no	—
11	C₂²⋊Q₈×C₆₇	—	2	2	no	—
12	C₂²⋊Q₈×C₆₇	C₂²	1	4	yes	—
13	C₄×Q₈×C₆₇	—	2	2	no	—
14	C₄×Q₈×C₆₇	C₂²	1	4	yes	—
15	#117×C₆₇	—	2	2	no	—
16	C₂×C₄²×C₆₇	C₂²	1	1	yes	characteristic
17	#106×C₆₇	C₂²	1	4	yes	—
18	C₄⋊C₈×C₆₇	C₂²	1	4	yes	—
19	#103×C₆₇	C₂²	1	8	yes	—
20	C₂×Q₈×C₆₇	—	2	2	no	—
21	C₂×Q₈×C₆₇	C₂³	1	4	yes	—
22	C₂²×C₄×C₆₇	D₄	1	2	yes	—
23	C₂²×C₄×C₆₇	C₂³	1	1	yes	characteristic
24	Q₁₆×C₆₇	—	4	4	no	—
25	M₁₆×C₆₇	—	4	2	no	—
26	C₂×C₈×C₆₇	C₂³	1	4	yes	—
27	C₄⋊C₄×C₆₇	—	2	2	no	—
28	C₄⋊C₄×C₆₇	—	2	2	no	—
29	C₄⋊C₄×C₆₇	—	2	2	no	—
30	C₄⋊C₄×C₆₇	—	2	4	no	—
31	C₄⋊C₄×C₆₇	—	2	4	no	—
32	C₄⋊C₄×C₆₇	C₂³	1	4	yes	—
33	C₂²⋊C₄×C₆₇	—	2	4	no	—
34	C₄²×C₆₇	—	2	4	no	—
35	C₄²×C₆₇	D₄	1	1	yes	characteristic
36	C₄²×C₆₇	D₄	1	1	yes	characteristic
37	C₄²×C₆₇	C₂³	1	2	yes	—
38	C₂³×C₆₇	C₂×D₄	1	1	yes	characteristic
39	Q₈×C₆₇	—	2	4	no	—
40	Q₈×C₆₇	—	2	4	no	—
41	Q₈×C₆₇	—	4	4	no	—
42	C₂×C₄×C₆₇	—	2	4	no	—
43	C₂×C₄×C₆₇	—	2	1	no	—
44	C₂×C₄×C₆₇	—	2	1	no	—
45	C₂×C₄×C₆₇	—	2	4	no	—
46	C₂×C₄×C₆₇	—	2	4	no	—
47	C₂×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
49	C₂×C₄×C₆₇	C₂×D₄	1	2	yes	—
50	C₂×C₄×C₆₇	C₂⁴	1	1	yes	characteristic
51	C₈×C₆₇	—	2	4	no	—
52	C₂²×C₆₇	—	2	1	no	—
53	C₂²×C₆₇	—	2	2	no	—
54	C₂²×C₆₇	C₂²×D₄	1	1	yes	characteristic
55	C₄×C₆₇	—	2	1	no	—
56	C₄×C₆₇	—	2	2	no	—
57	C₄×C₆₇	—	4	4	no	—
58	C₄×C₆₇	—	4	4	no	—
59	C₄×C₆₇	#137	1	4	yes	—
60	C₄×C₆₇	H(5,𝔽₂)	1	1	yes	characteristic
61	C₄×C₆₇	2₋¹⁺⁴	1	1	yes	characteristic
62	C₂×C₆₇	—	4	1	no	—
63	C₂×C₆₇	#537	1	1	yes	characteristic
64	C₂×C₆₇	C₂×#137	1	2	yes	—
65	#3255	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₆₇	#3255	1	1	yes	characteristic, direct product factor, Sylow
67	#533	C₂×C₆₇	1	2	yes	—
68	#475	C₂×C₆₇	1	4	yes	complemented
69	#470	C₂×C₆₇	1	4	yes	—
70	#462	C₂×C₆₇	1	4	yes	complemented
71	#423	C₂×C₆₇	1	1	yes	characteristic
72	C₂×Q₁₆	C₂²×C₆₇	1	4	yes	—
73	C₂×M₁₆	C₂²×C₆₇	1	2	yes	—
74	C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
75	C₄⋊Q₈	C₂²×C₆₇	1	2	yes	—
76	#125	—	2	2	no	—
77	C₂²⋊Q₈	—	2	2	no	—
78	C₂²⋊Q₈	C₂²×C₆₇	1	4	yes	—
79	C₄×Q₈	—	2	2	no	—
80	C₄×Q₈	C₂²×C₆₇	1	4	yes	—
81	#117	—	2	2	no	—
82	C₂×C₄²	C₂²×C₆₇	1	1	yes	characteristic
83	#106	C₂²×C₆₇	1	4	yes	—
84	C₄⋊C₈	C₂²×C₆₇	1	4	yes	—
85	#103	C₂²×C₆₇	1	8	yes	—
86	C₂×Q₈	—	2	2	no	—
87	C₂×Q₈	C₂³×C₆₇	1	4	yes	—
88	C₂²×C₄	D₄×C₆₇	1	2	yes	—
89	C₂²×C₄	C₂³×C₆₇	1	1	yes	characteristic
90	Q₁₆	—	4	4	no	—
91	M₁₆	—	4	2	no	—
92	C₂×C₈	C₂³×C₆₇	1	4	yes	—
93	C₄⋊C₄	—	2	2	no	—
94	C₄⋊C₄	—	2	2	no	—
95	C₄⋊C₄	—	2	2	no	—
96	C₄⋊C₄	—	2	4	no	—
97	C₄⋊C₄	—	2	4	no	—
98	C₄⋊C₄	C₂³×C₆₇	1	4	yes	—
99	C₂²⋊C₄	—	2	4	no	—
100	C₄²	—	2	4	no	—
101	C₄²	D₄×C₆₇	1	1	yes	characteristic
102	C₄²	D₄×C₆₇	1	1	yes	characteristic
103	C₄²	C₂³×C₆₇	1	2	yes	—
104	C₂³	C₂×D₄×C₆₇	1	1	yes	characteristic
105	Q₈	—	2	4	no	—
106	Q₈	—	2	4	no	—
107	Q₈	—	4	4	no	—
108	C₂×C₄	—	2	4	no	—
109	C₂×C₄	—	2	1	no	—
110	C₂×C₄	—	2	1	no	—
111	C₂×C₄	—	2	4	no	—
112	C₂×C₄	—	2	4	no	—
113	C₂×C₄	C₂×D₄×C₆₇	1	1	yes	characteristic
114	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
115	C₂×C₄	C₂×D₄×C₆₇	1	2	yes	—
116	C₂×C₄	C₂⁴×C₆₇	1	1	yes	characteristic
117	C₈	—	2	4	no	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	2	no	—
120	C₂²	C₂²×D₄×C₆₇	1	1	yes	characteristic
121	C₄	—	2	1	no	—
122	C₄	—	2	2	no	—
123	C₄	—	4	4	no	—
124	C₄	—	4	4	no	—
125	C₄	#137×C₆₇	1	4	yes	—
126	C₄	H(5,𝔽₂)×C₆₇	1	1	yes	characteristic
127	C₄	2₋¹⁺⁴×C₆₇	1	1	yes	characteristic
128	C₂	—	4	1	no	—
129	C₂	#537×C₆₇	1	1	yes	characteristic
130	C₂	C₂×#137×C₆₇	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#533×C₆₇

C₂

yes

maximal

#475×C₆₇

C₂

yes

maximal, complemented

#470×C₆₇

C₂

yes

maximal

#462×C₆₇

C₂

yes

maximal, complemented

#423×C₆₇

C₂

yes

maximal, characteristic

C₂×Q₁₆×C₆₇

C₂²

yes

—

C₂×M₁₆×C₆₇

C₂²

yes

—

C₄⋊Q₈×C₆₇

C₂²

yes

—

C₄⋊Q₈×C₆₇

C₂²

yes

—

#125×C₆₇

—

C₂²⋊Q₈×C₆₇

—

C₂²⋊Q₈×C₆₇

C₂²

yes

—

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

C₂²

yes

—

#117×C₆₇

—

C₂×C₄²×C₆₇

C₂²

yes

characteristic

#106×C₆₇

C₂²

yes

—

C₄⋊C₈×C₆₇

C₂²

yes

—

#103×C₆₇

C₂²

yes

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

D₄

yes

—

C₂²×C₄×C₆₇

C₂³

yes

characteristic

Q₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

C₂³

yes

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

D₄

yes

characteristic

C₄²×C₆₇

C₂³

yes

—

C₂³×C₆₇

C₂×D₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂×D₄

yes

—

C₂×C₄×C₆₇

C₂⁴

yes

characteristic

C₈×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²×D₄

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#137

yes

—

C₄×C₆₇

H(5,𝔽₂)

yes

characteristic

C₄×C₆₇

2₋¹⁺⁴

yes

characteristic

C₂×C₆₇

—

C₂×C₆₇

#537

yes

characteristic

C₂×C₆₇

C₂×#137

yes

—

#3255

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₆₇

#3255

yes

characteristic, direct product factor, Sylow

#533

C₂×C₆₇

yes

—

#475

C₂×C₆₇

yes

complemented

#470

C₂×C₆₇

yes

—

#462

C₂×C₆₇

yes

complemented

#423

C₂×C₆₇

yes

characteristic

C₂×Q₁₆

C₂²×C₆₇

yes

—

C₂×M₁₆

C₂²×C₆₇

yes

—

C₄⋊Q₈

C₂²×C₆₇

yes

—

C₄⋊Q₈

C₂²×C₆₇

yes

—

#125

—

C₂²⋊Q₈

—

C₂²⋊Q₈

C₂²×C₆₇

yes

—

C₄×Q₈

—

C₄×Q₈

C₂²×C₆₇

yes

—

#117

—

C₂×C₄²

C₂²×C₆₇

yes

characteristic

#106

C₂²×C₆₇

yes

—

C₄⋊C₈

C₂²×C₆₇

yes

—

#103

C₂²×C₆₇

yes

—

C₂×Q₈

—

C₂×Q₈

C₂³×C₆₇

yes

—

C₂²×C₄

D₄×C₆₇

yes

—

C₂²×C₄

C₂³×C₆₇

yes

characteristic

Q₁₆

—

M₁₆

—

C₂×C₈

C₂³×C₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

C₂³×C₆₇

yes

—

C₂²⋊C₄

—

100

C₄²

—

101

C₄²

D₄×C₆₇

yes

characteristic

102

C₄²

D₄×C₆₇

yes

characteristic

103

C₄²

C₂³×C₆₇

yes

—

104

C₂³

C₂×D₄×C₆₇

yes

characteristic

105

Q₈

—

106

Q₈

—

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

—

111

C₂×C₄

—

112

C₂×C₄

—

113

C₂×C₄

C₂×D₄×C₆₇

yes

characteristic

114

C₂×C₄

C₂×D₄×C₆₇

yes

—

115

C₂×C₄

C₂×D₄×C₆₇

yes

—

116

C₂×C₄

C₂⁴×C₆₇

yes

characteristic

117

C₈

—

118

C₂²

—

119

C₂²

—

120

C₂²

C₂²×D₄×C₆₇

yes

characteristic

121

C₄

—

122

C₄

—

123

C₄

—

124

C₄

—

125

C₄

#137×C₆₇

yes

—

126

C₄

H(5,𝔽₂)×C₆₇

yes

characteristic

127

C₄

2₋¹⁺⁴×C₆₇

yes

characteristic

128

C₂

—

129

C₂

#537×C₆₇

yes

characteristic

130

C₂

C₂×#137×C₆₇

yes

—

	subgroup	G/H
center, socle	C₂²×C₆₇	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₆₇
prime radical	C₂³×C₆₇	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

subgroup

G/H

center, socle

C₂²×C₆₇

C₂²×D₄

commutator, Frattini, cosocle

C₂×C₄

C₂⁴×C₆₇

prime radical

C₂³×C₆₇

C₂×D₄

LCS

G ▷ C₂×C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₆₇ ◁ C₂×C₄²×C₆₇ ◁ G

N	H ≅ G/N	type
2. #475×C₆₇	C₂	semidirect
4. #462×C₆₇	C₂	semidirect
65. #3255	C₆₇	direct (N × H)
66. C₆₇	#3255	direct (N × H)
68. #475	C₂×C₆₇	semidirect
70. #462	C₂×C₆₇	semidirect

H ≅ G/N

type

2. #475×C₆₇

C₂

semidirect

4. #462×C₆₇

C₂

semidirect

65. #3255

C₆₇

direct (N × H)

66. C₆₇

#3255

direct (N × H)

68. #475

C₂×C₆₇

semidirect

70. #462

C₂×C₆₇

semidirect

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#533×C₆₇	2
#475×C₆₇	2
#470×C₆₇	2
#462×C₆₇	2
#423×C₆₇	2
#3255	67

subgroup

index

#533×C₆₇

#475×C₆₇

#470×C₆₇

#462×C₆₇

#423×C₆₇

#3255

Subgroups — C₆₇×#3255 #22557686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#3255 #22557686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#3255 #22557686