Group #30745221 — subgroups

Subgroups — #30745221

Order 17152.

610 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 207 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×#16863447	C₂	1	3	yes	maximal
2	C₆₇⋊#1273	C₂	1	3	yes	maximal, complemented
3	C₆₇×#1273	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₄²×D₆₇	C₂²	1	1	yes	characteristic
5	#16863447	—	2	12	no	—
6	C₂×Dic₆₇⋊C₈	C₄	1	6	yes	—
7	C₂×(C₂×D₆₇)⋊C₈	C₄	1	6	yes	—
8	C₂×C₄×C₆₇⋊C₈	C₂²	1	3	yes	—
9	C₂×C₄×C₈×C₆₇	C₂²	1	3	yes	—
10	C₂²×C₄×D₆₇	C₂×C₄	1	3	yes	—
11	C₂²×C₆₇⋊C₈	C₂×C₄	1	6	yes	—
12	C₂×C₄×Dic₆₇	C₂×C₄	1	3	yes	—
13	C₄²×D₆₇	D₄	1	6	yes	—
14	C₄²×D₆₇	Q₈	1	2	yes	—
15	Dic₆₇⋊C₈	—	2	12	no	—
16	(C₂×D₆₇)⋊C₈	—	2	12	no	—
17	C₄×C₆₇⋊C₈	—	2	6	no	—
18	C₂²×C₈×C₆₇	C₂×C₄	1	6	yes	—
19	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
20	C₄×C₈×C₆₇	—	2	6	no	—
21	C₂³×D₆₇	C₄²	1	1	yes	characteristic
22	C₂²×Dic₆₇	C₄²	1	3	yes	—
23	C₂×C₄×D₆₇	—	2	6	no	—
24	C₂×C₄×D₆₇	—	2	6	no	—
25	C₂×C₄×D₆₇	C₂²⋊C₄	1	6	yes	—
26	C₂×C₄×D₆₇	C₄⋊C₄	1	6	yes	—
27	C₂×C₆₇⋊C₈	—	2	6	no	—
28	C₂×C₆₇⋊C₈	—	2	6	no	—
29	C₂×C₆₇⋊C₈	—	2	6	no	—
30	C₄×Dic₆₇	C₂²⋊C₄	1	6	yes	—
31	C₄×Dic₆₇	C₄⋊C₄	1	6	yes	—
32	C₂²×C₄×C₆₇	C₂²×C₄	1	3	yes	—
33	C₂×C₈×C₆₇	—	2	6	no	—
34	C₂×C₈×C₆₇	—	2	6	no	—
35	C₂×C₈×C₆₇	—	2	6	no	—
36	C₄²×C₆₇	C₂×D₄	1	3	yes	—
37	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
38	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
39	C₂²×D₆₇	—	2	3	no	—
40	C₂²×D₆₇	—	2	3	no	—
41	C₂²×D₆₇	#95	1	2	yes	—
42	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	3	yes	—
43	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	3	yes	—
44	C₂×C₄×C₆₇	#131	1	6	yes	—
45	C₂×C₄×C₆₇	#131	1	6	yes	—
46	C₂×Dic₆₇	—	2	3	no	—
47	C₂×Dic₆₇	—	2	3	no	—
48	C₂×Dic₆₇	#95	1	6	yes	—
49	C₄×D₆₇	—	2	12	no	—
50	C₄×D₆₇	—	2	12	no	—
51	C₈×C₆₇	—	4	6	no	—
52	C₆₇⋊C₈	—	4	6	no	—
53	C₂²×C₆₇	C₂×#95	1	1	yes	characteristic
54	C₂²×C₆₇	#405	1	3	yes	—
55	C₂²×C₆₇	#405	1	3	yes	—
56	C₂×D₆₇	—	2	6	no	—
57	C₂×D₆₇	—	4	1	no	—
58	C₂×D₆₇	—	4	3	no	—
59	C₄×C₆₇	#408	1	6	yes	—
60	C₄×C₆₇	#424	1	6	yes	—
61	Dic₆₇	—	2	6	no	—
62	#7851	—	67	1	no	maximal, Sylow
63	C₂×C₆₇	#1454	1	1	yes	characteristic
64	C₂×C₆₇	#1454	1	3	yes	—
65	C₂×C₆₇	#1729	1	3	yes	—
66	D₆₇	—	4	2	no	—
67	C₂×#431	—	67	3	no	—
68	#1273	—	67	3	no	—
69	#1273	D₆₇	1	1	yes	characteristic, complemented
70	C₆₇	#7851	1	1	yes	characteristic, complemented, Sylow
71	C₂²×C₄²	—	67	1	no	—
72	#431	—	134	12	no	—
73	C₂×C₄⋊C₈	—	67	6	no	—
74	C₂×C₂²⋊C₈	—	67	6	no	—
75	C₂×C₄×C₈	—	67	3	no	—
76	C₂×C₄×C₈	C₂×D₆₇	1	3	yes	—
77	C₂³×C₄	—	67	3	no	—
78	C₂²×C₈	—	67	6	no	—
79	C₂²×C₈	C₄×D₆₇	1	6	yes	—
80	C₂×C₄²	—	67	3	no	—
81	C₂×C₄²	—	67	6	no	—
82	C₂×C₄²	—	67	2	no	—
83	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
84	C₄⋊C₈	—	134	12	no	—
85	C₂²⋊C₈	—	134	12	no	—
86	C₄×C₈	—	2	6	no	—
87	C₄×C₈	—	134	6	no	—
88	C₂⁴	—	67	1	no	—
89	C₂²×C₄	—	67	3	no	—
90	C₂²×C₄	—	67	6	no	—
91	C₂²×C₄	—	67	6	no	—
92	C₂²×C₄	—	134	6	no	—
93	C₂²×C₄	—	134	6	no	—
94	C₂²×C₄	C₂×C₄×D₆₇	1	3	yes	—
95	C₂×C₈	—	2	6	no	—
96	C₂×C₈	—	2	6	no	—
97	C₂×C₈	—	2	6	no	—
98	C₂×C₈	—	134	6	no	—
99	C₂×C₈	—	134	6	no	—
100	C₂×C₈	—	134	6	no	—
101	C₄²	—	67	6	no	—
102	C₄²	—	67	6	no	—
103	C₄²	D₄×D₆₇	1	3	yes	—
104	C₄²	Q₈×D₆₇	1	1	yes	characteristic
105	C₂³	—	67	2	no	—
106	C₂³	—	134	3	no	—
107	C₂³	—	134	3	no	—
108	C₂³	C₄²×D₆₇	1	1	yes	characteristic
109	C₂×C₄	—	67	6	no	—
110	C₂×C₄	—	134	3	no	—
111	C₂×C₄	—	134	3	no	—
112	C₂×C₄	—	134	12	no	—
113	C₂×C₄	—	134	12	no	—
114	C₂×C₄	C₂²⋊C₄×D₆₇	1	3	yes	—
115	C₂×C₄	C₄⋊C₄×D₆₇	1	3	yes	—
116	C₂×C₄	#14524049	1	6	yes	—
117	C₂×C₄	#14524051	1	6	yes	—
118	C₈	—	4	6	no	—
119	C₈	—	268	6	no	—
120	C₂²	—	268	1	no	—
121	C₂²	—	268	3	no	—
122	C₂²	—	134	6	no	—
123	C₂²	#95×D₆₇	1	1	yes	characteristic
124	C₂²	#16863350	1	3	yes	—
125	C₂²	#16863353	1	3	yes	—
126	C₄	—	134	6	no	—
127	C₄	#16863361	1	6	yes	—
128	C₄	#16863421	1	6	yes	—
129	C₂	—	268	2	no	—
130	C₂	#22558508	1	1	yes	characteristic
131	C₂	#22558510	1	3	yes	—
132	C₂	#22559904	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863447

C₂

yes

maximal

C₆₇⋊#1273

C₂

yes

maximal, complemented

C₆₇×#1273

C₂

yes

maximal, characteristic, complemented

C₂×C₄²×D₆₇

C₂²

yes

characteristic

#16863447

—

C₂×Dic₆₇⋊C₈

C₄

yes

—

C₂×(C₂×D₆₇)⋊C₈

C₄

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

—

C₂²×C₄×D₆₇

C₂×C₄

yes

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₆₇

C₂×C₄

yes

—

C₄²×D₆₇

D₄

yes

—

C₄²×D₆₇

Q₈

yes

—

Dic₆₇⋊C₈

—

(C₂×D₆₇)⋊C₈

—

C₄×C₆₇⋊C₈

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄×C₈×C₆₇

—

C₂³×D₆₇

C₄²

yes

characteristic

C₂²×Dic₆₇

C₄²

yes

—

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

—

C₂×C₄×D₆₇

C₂²⋊C₄

yes

—

C₂×C₄×D₆₇

C₄⋊C₄

yes

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄×Dic₆₇

C₂²⋊C₄

yes

—

C₄×Dic₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₂³×C₆₇

C₂×C₄²

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

#95

yes

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

#95

yes

—

C₄×D₆₇

—

C₄×D₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

C₂×#95

yes

characteristic

C₂²×C₆₇

#405

yes

—

C₂²×C₆₇

#405

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

#408

yes

—

C₄×C₆₇

#424

yes

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₆₇

#1729

yes

—

D₆₇

—

C₂×#431

—

#1273

—

#1273

D₆₇

yes

characteristic, complemented

C₆₇

#7851

yes

characteristic, complemented, Sylow

C₂²×C₄²

—

#431

—

134

—

C₂×C₄⋊C₈

—

C₂×C₂²⋊C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₇

yes

—

C₂³×C₄

—

C₂²×C₈

—

C₂²×C₈

C₄×D₆₇

yes

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

C₄⋊C₈

—

134

—

C₂²⋊C₈

—

134

—

C₄×C₈

—

C₄×C₈

—

134

—

C₂⁴

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

134

—

C₂²×C₄

—

134

—

C₂²×C₄

C₂×C₄×D₆₇

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

100

C₂×C₈

—

134

—

101

C₄²

—

102

C₄²

—

103

C₄²

D₄×D₆₇

yes

—

104

C₄²

Q₈×D₆₇

yes

characteristic

105

C₂³

—

106

C₂³

—

134

—

107

C₂³

—

134

—

108

C₂³

C₄²×D₆₇

yes

characteristic

109

C₂×C₄

—

110

C₂×C₄

—

134

—

111

C₂×C₄

—

134

—

112

C₂×C₄

—

134

—

113

C₂×C₄

—

134

—

114

C₂×C₄

C₂²⋊C₄×D₆₇

yes

—

115

C₂×C₄

C₄⋊C₄×D₆₇

yes

—

116

C₂×C₄

#14524049

yes

—

117

C₂×C₄

#14524051

yes

—

118

C₈

—

119

C₈

—

268

—

120

C₂²

—

268

—

121

C₂²

—

268

—

122

C₂²

—

134

—

123

C₂²

#95×D₆₇

yes

characteristic

124

C₂²

#16863350

yes

—

125

C₂²

#16863353

yes

—

126

C₄

—

134

—

127

C₄

#16863361

yes

—

128

C₄

#16863421

yes

—

129

C₂

—

268

—

130

C₂

#22558508

yes

characteristic

131

C₂

#22558510

yes

—

132

C₂

#22559904

yes

—

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₆₇
commutator, socle	C₂³×C₆₇	C₂×C₄²
Fitting	C₆₇×#1273	C₂
prime radical	C₂³×D₆₇	C₄²
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1273	D₆₇
hypocenter	C₆₇	#7851
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ #1273

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₆₇

commutator, socle

C₂³×C₆₇

C₂×C₄²

Fitting

C₆₇×#1273

C₂

prime radical

C₂³×D₆₇

C₄²

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₄² ◁ #1273

N	H ≅ G/N	type
2. C₆₇⋊#1273	C₂	semidirect
3. C₆₇×#1273	C₂	semidirect
69. #1273	D₆₇	semidirect
70. C₆₇	#7851	semidirect

H ≅ G/N

type

2. C₆₇⋊#1273

C₂

semidirect

3. C₆₇×#1273

C₂

semidirect

69. #1273

D₆₇

semidirect

70. C₆₇

#7851

semidirect

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

subgroup	index
C₂×#16863447	2
C₆₇⋊#1273	2
C₆₇×#1273	2
#7851	67

subgroup

index

C₂×#16863447

C₆₇⋊#1273

C₆₇×#1273

#7851

Subgroups — #30745221

Conjugacy classes of subgroups

Complements

Maximal subgroups