Group #22557039 — subgroups

Subgroups — C₆₇×#2608 #22557039

Order 8576.

428 conjugacy classes of subgroups; only the 200 normal subgroups are shown.

normal subgroups only

This group has 428 conjugacy classes of subgroups — too many to draw. Only its 200 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	#396×C₆₇	C₂	1	4	yes	maximal
2	#389×C₆₇	C₂	1	4	yes	maximal, complemented
3	#387×C₆₇	C₂	1	4	yes	maximal, complemented
4	#381×C₆₇	C₂	1	1	yes	maximal, characteristic
5	C₄×C₂²⋊C₄×C₆₇	C₂	1	2	yes	maximal
6	C₂³×C₄×C₆₇	C₂²	1	1	yes	characteristic
7	C₂×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
8	C₂×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
9	C₂×C₄⋊C₄×C₆₇	C₂²	1	4	yes	—
10	C₂×C₂²⋊C₄×C₆₇	C₂²	1	4	yes	—
11	C₂×C₂²⋊C₄×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄²×C₆₇	C₂²	1	2	yes	—
13	C₂×C₄²×C₆₇	C₂²	1	2	yes	—
14	#95×C₆₇	C₂²	1	4	yes	—
15	#95×C₆₇	C₂²	1	4	yes	—
16	#95×C₆₇	C₂²	1	4	yes	—
17	C₂⁴×C₆₇	C₂³	1	1	yes	characteristic
18	C₂²×C₄×C₆₇	—	4	2	no	—
19	C₂²×C₄×C₆₇	Q₈	1	4	yes	—
20	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
21	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
22	C₂²×C₄×C₆₇	C₂³	1	2	yes	—
23	C₂²×C₄×C₆₇	C₂³	1	4	yes	—
24	C₄⋊C₄×C₆₇	—	2	8	no	—
25	C₄⋊C₄×C₆₇	—	2	8	no	—
26	C₄⋊C₄×C₆₇	—	4	4	no	—
27	C₂²⋊C₄×C₆₇	—	2	8	no	—
28	C₂²⋊C₄×C₆₇	—	4	2	no	—
29	C₄²×C₆₇	—	4	2	no	—
30	C₄²×C₆₇	—	4	2	no	—
31	C₂³×C₆₇	—	2	4	no	—
32	C₂³×C₆₇	—	2	2	no	—
33	C₂³×C₆₇	C₂×Q₈	1	2	yes	—
34	C₂³×C₆₇	C₂⁴	1	1	yes	characteristic
35	C₂×C₄×C₆₇	—	2	4	no	—
36	C₂×C₄×C₆₇	—	2	4	no	—
37	C₂×C₄×C₆₇	—	2	2	no	—
38	C₂×C₄×C₆₇	—	2	2	no	—
39	C₂×C₄×C₆₇	—	2	4	no	—
40	C₂×C₄×C₆₇	—	2	4	no	—
41	C₂×C₄×C₆₇	—	2	4	no	—
42	C₂×C₄×C₆₇	—	2	4	no	—
43	C₂×C₄×C₆₇	—	2	4	no	—
44	C₂×C₄×C₆₇	—	2	4	no	—
45	C₂×C₄×C₆₇	—	4	4	no	—
46	C₂×C₄×C₆₇	C₂×Q₈	1	4	yes	—
47	C₂×C₄×C₆₇	#41	1	8	yes	—
48	C₂²×C₆₇	—	2	4	no	—
49	C₂²×C₆₇	—	2	2	no	—
50	C₂²×C₆₇	—	4	4	no	—
51	C₂²×C₆₇	C₂²×Q₈	1	1	yes	characteristic
52	C₂²×C₆₇	C₂×#41	1	4	yes	—
53	C₂²×C₆₇	H(5,𝔽₂)	1	2	yes	—
54	C₄×C₆₇	—	2	4	no	—
55	C₄×C₆₇	—	2	8	no	—
56	C₄×C₆₇	—	4	4	no	—
57	C₄×C₆₇	—	4	4	no	—
58	C₂×C₆₇	—	4	2	no	—
59	C₂×C₆₇	#533	1	2	yes	—
60	C₂×C₆₇	#543	1	1	yes	characteristic
61	C₂×C₆₇	#553	1	4	yes	—
62	#2608	C₆₇	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₆₇	#2608	1	1	yes	characteristic, direct product factor, Sylow
64	#396	C₂×C₆₇	1	4	yes	—
65	#389	C₂×C₆₇	1	4	yes	complemented
66	#387	C₂×C₆₇	1	4	yes	complemented
67	#381	C₂×C₆₇	1	1	yes	characteristic
68	C₄×C₂²⋊C₄	C₂×C₆₇	1	2	yes	—
69	C₂³×C₄	C₂²×C₆₇	1	1	yes	characteristic
70	C₂×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
71	C₂×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
72	C₂×C₄⋊C₄	C₂²×C₆₇	1	4	yes	—
73	C₂×C₂²⋊C₄	C₂²×C₆₇	1	4	yes	—
74	C₂×C₂²⋊C₄	C₂²×C₆₇	1	2	yes	—
75	C₂×C₄²	C₂²×C₆₇	1	2	yes	—
76	C₂×C₄²	C₂²×C₆₇	1	2	yes	—
77	#95	C₂²×C₆₇	1	4	yes	—
78	#95	C₂²×C₆₇	1	4	yes	—
79	#95	C₂²×C₆₇	1	4	yes	—
80	C₂⁴	C₂³×C₆₇	1	1	yes	characteristic
81	C₂²×C₄	—	4	2	no	—
82	C₂²×C₄	Q₈×C₆₇	1	4	yes	—
83	C₂²×C₄	C₂³×C₆₇	1	4	yes	—
84	C₂²×C₄	C₂³×C₆₇	1	4	yes	—
85	C₂²×C₄	C₂³×C₆₇	1	2	yes	—
86	C₂²×C₄	C₂³×C₆₇	1	4	yes	—
87	C₄⋊C₄	—	2	8	no	—
88	C₄⋊C₄	—	2	8	no	—
89	C₄⋊C₄	—	4	4	no	—
90	C₂²⋊C₄	—	2	8	no	—
91	C₂²⋊C₄	—	4	2	no	—
92	C₄²	—	4	2	no	—
93	C₄²	—	4	2	no	—
94	C₂³	—	2	4	no	—
95	C₂³	—	2	2	no	—
96	C₂³	C₂×Q₈×C₆₇	1	2	yes	—
97	C₂³	C₂⁴×C₆₇	1	1	yes	characteristic
98	C₂×C₄	—	2	4	no	—
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	2	no	—
101	C₂×C₄	—	2	2	no	—
102	C₂×C₄	—	2	4	no	—
103	C₂×C₄	—	2	4	no	—
104	C₂×C₄	—	2	4	no	—
105	C₂×C₄	—	2	4	no	—
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	2	4	no	—
108	C₂×C₄	—	4	4	no	—
109	C₂×C₄	C₂×Q₈×C₆₇	1	4	yes	—
110	C₂×C₄	#41×C₆₇	1	8	yes	—
111	C₂²	—	2	4	no	—
112	C₂²	—	2	2	no	—
113	C₂²	—	4	4	no	—
114	C₂²	C₂²×Q₈×C₆₇	1	1	yes	characteristic
115	C₂²	C₂×#41×C₆₇	1	4	yes	—
116	C₂²	H(5,𝔽₂)×C₆₇	1	2	yes	—
117	C₄	—	2	4	no	—
118	C₄	—	2	8	no	—
119	C₄	—	4	4	no	—
120	C₄	—	4	4	no	—
121	C₂	—	4	2	no	—
122	C₂	#533×C₆₇	1	2	yes	—
123	C₂	#543×C₆₇	1	1	yes	characteristic
124	C₂	#553×C₆₇	1	4	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#396×C₆₇

C₂

yes

maximal

#389×C₆₇

C₂

yes

maximal, complemented

#387×C₆₇

C₂

yes

maximal, complemented

#381×C₆₇

C₂

yes

maximal, characteristic

C₄×C₂²⋊C₄×C₆₇

C₂

yes

maximal

C₂³×C₄×C₆₇

C₂²

yes

characteristic

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄⋊C₄×C₆₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₆₇

C₂²

yes

—

C₂×C₂²⋊C₄×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

—

C₂×C₄²×C₆₇

C₂²

yes

—

#95×C₆₇

C₂²

yes

—

#95×C₆₇

C₂²

yes

—

#95×C₆₇

C₂²

yes

—

C₂⁴×C₆₇

C₂³

yes

characteristic

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

Q₈

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₂²×C₄×C₆₇

C₂³

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

C₂×Q₈

yes

—

C₂³×C₆₇

C₂⁴

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×Q₈

yes

—

C₂×C₄×C₆₇

#41

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂²×Q₈

yes

characteristic

C₂²×C₆₇

C₂×#41

yes

—

C₂²×C₆₇

H(5,𝔽₂)

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#533

yes

—

C₂×C₆₇

#543

yes

characteristic

C₂×C₆₇

#553

yes

—

#2608

C₆₇

yes

maximal, characteristic, direct product factor, Sylow

C₆₇

#2608

yes

characteristic, direct product factor, Sylow

#396

C₂×C₆₇

yes

—

#389

C₂×C₆₇

yes

complemented

#387

C₂×C₆₇

yes

complemented

#381

C₂×C₆₇

yes

characteristic

C₄×C₂²⋊C₄

C₂×C₆₇

yes

—

C₂³×C₄

C₂²×C₆₇

yes

characteristic

C₂×C₄⋊C₄

C₂²×C₆₇

yes

—

C₂×C₄⋊C₄

C₂²×C₆₇

yes

—

C₂×C₄⋊C₄

C₂²×C₆₇

yes

—

C₂×C₂²⋊C₄

C₂²×C₆₇

yes

—

C₂×C₂²⋊C₄

C₂²×C₆₇

yes

—

C₂×C₄²

C₂²×C₆₇

yes

—

C₂×C₄²

C₂²×C₆₇

yes

—

#95

C₂²×C₆₇

yes

—

#95

C₂²×C₆₇

yes

—

#95

C₂²×C₆₇

yes

—

C₂⁴

C₂³×C₆₇

yes

characteristic

C₂²×C₄

—

C₂²×C₄

Q₈×C₆₇

yes

—

C₂²×C₄

C₂³×C₆₇

yes

—

C₂²×C₄

C₂³×C₆₇

yes

—

C₂²×C₄

C₂³×C₆₇

yes

—

C₂²×C₄

C₂³×C₆₇

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₂²⋊C₄

—

C₂²⋊C₄

—

C₄²

—

C₄²

—

C₂³

—

C₂³

—

C₂³

C₂×Q₈×C₆₇

yes

—

C₂³

C₂⁴×C₆₇

yes

characteristic

C₂×C₄

—

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

—

103

C₂×C₄

—

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂×Q₈×C₆₇

yes

—

110

C₂×C₄

#41×C₆₇

yes

—

111

C₂²

—

112

C₂²

—

113

C₂²

—

114

C₂²

C₂²×Q₈×C₆₇

yes

characteristic

115

C₂²

C₂×#41×C₆₇

yes

—

116

C₂²

H(5,𝔽₂)×C₆₇

yes

—

117

C₄

—

118

C₄

—

119

C₄

—

120

C₄

—

121

C₂

—

122

C₂

#533×C₆₇

yes

—

123

C₂

#543×C₆₇

yes

characteristic

124

C₂

#553×C₆₇

yes

—

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

subgroup

G/H

center, socle

C₂³×C₆₇

C₂⁴

commutator, Frattini, cosocle

C₂³

C₂⁴×C₆₇

prime radical

C₂⁴×C₆₇

C₂³

LCS

G ▷ C₂³ ▷ C₁

UCS

C₁ ◁ C₂³×C₆₇ ◁ G

N	H ≅ G/N	type
2. #389×C₆₇	C₂	semidirect
3. #387×C₆₇	C₂	semidirect
62. #2608	C₆₇	direct (N × H)
63. C₆₇	#2608	direct (N × H)
65. #389	C₂×C₆₇	semidirect
66. #387	C₂×C₆₇	semidirect

H ≅ G/N

type

2. #389×C₆₇

C₂

semidirect

3. #387×C₆₇

C₂

semidirect

62. #2608

C₆₇

direct (N × H)

63. C₆₇

#2608

direct (N × H)

65. #389

C₂×C₆₇

semidirect

66. #387

C₂×C₆₇

semidirect

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

subgroup	index
#396×C₆₇	2
#389×C₆₇	2
#387×C₆₇	2
#381×C₆₇	2
C₄×C₂²⋊C₄×C₆₇	2
#2608	67

subgroup

index

#396×C₆₇

#389×C₆₇

#387×C₆₇

#381×C₆₇

C₄×C₂²⋊C₄×C₆₇

#2608

Subgroups — C₆₇×#2608 #22557039

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C67×#2608 #22557039

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₆₇×#2608 #22557039