Group #30746286 — subgroups

Subgroups — #30746286

Order 17152.

294 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 294 conjugacy classes of subgroups — too many to draw. Only its 131 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	#22559960	C₂	1	1	yes	maximal, characteristic
2	#22558501	C₂	1	1	yes	maximal, characteristic
3	#22558039	C₂	1	2	yes	maximal
4	#22558037	C₂	1	2	yes	maximal
5	C₆₇×#1450	C₂	1	1	yes	maximal, characteristic
6	#16863449	C₂²	1	2	yes	—
7	C₂×#14523954	C₂²	1	1	yes	characteristic
8	#16863242	C₂²	1	1	yes	characteristic
9	#16863150	C₄	1	4	yes	—
10	#16863118	C₄	1	4	yes	complemented
11	#431×C₆₇	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
13	#322×C₆₇	C₄	1	2	yes	complemented
14	C₈²×C₆₇	C₄	1	2	yes	complemented
15	C₂²×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
16	C₂×C₄×Dic₆₇	C₂×C₄	1	2	yes	—
17	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
18	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
19	#14523954	—	2	2	no	—
20	C₄×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
21	C₆₇⋊#95	—	2	2	no	—
22	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
23	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
24	C₄⋊C₈×C₆₇	—	2	2	no	—
25	C₂²⋊C₈×C₆₇	—	2	2	no	—
26	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
27	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
30	C₂²×Dic₆₇	—	2	2	no	—
31	C₂×C₆₇⋊C₈	—	2	4	no	—
32	C₂×C₆₇⋊C₈	—	2	2	no	—
33	C₂×C₆₇⋊C₈	—	2	4	no	—
34	C₄×Dic₆₇	—	2	4	no	—
35	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	#41	1	2	yes	—
37	C₂×C₈×C₆₇	—	2	4	no	—
38	C₂×C₈×C₆₇	—	2	4	no	—
39	C₂×C₈×C₆₇	C₄²	1	2	yes	—
40	C₂×C₈×C₆₇	C₄²	1	2	yes	—
41	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
42	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
43	C₄²×C₆₇	#41	1	2	yes	—
44	C₂³×C₆₇	#117	1	1	yes	characteristic
45	C₂×C₄×C₆₇	—	4	2	no	—
46	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
47	C₂×C₄×C₆₇	#117	1	2	yes	—
48	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
49	C₂×C₄×C₆₇	#117	1	2	yes	—
50	C₂×C₄×C₆₇	#131	1	4	yes	—
51	C₂×Dic₆₇	—	4	4	no	—
52	C₂×Dic₆₇	—	4	2	no	—
53	C₈×C₆₇	—	2	2	no	—
54	C₈×C₆₇	—	2	2	no	—
55	C₈×C₆₇	—	4	4	no	—
56	C₆₇⋊C₈	—	4	4	no	—
57	C₆₇⋊C₈	—	4	2	no	—
58	C₂²×C₆₇	—	2	2	no	—
59	C₂²×C₆₇	#376	1	1	yes	characteristic
60	C₂²×C₆₇	#433	1	2	yes	—
61	C₄×C₆₇	—	2	2	no	—
62	C₄×C₆₇	—	2	2	no	—
63	C₄×C₆₇	#405	1	2	yes	—
64	C₄×C₆₇	#433	1	2	yes	—
65	Dic₆₇	—	8	2	no	—
66	#8141	—	67	1	no	maximal, Sylow
67	C₂×C₆₇	—	2	2	no	—
68	C₂×C₆₇	#1746	1	1	yes	characteristic
69	C₂×C₆₇	#1747	1	2	yes	—
70	#1746	—	67	1	no	—
71	#1452	—	67	1	no	—
72	#1450	D₆₇	1	1	yes	characteristic
73	#1275	—	67	2	no	—
74	#1274	—	67	2	no	—
75	C₆₇	#8141	1	1	yes	characteristic, complemented, Sylow
76	#431	—	67	2	no	—
77	#431	C₂×D₆₇	1	2	yes	—
78	C₂×#97	—	67	1	no	—
79	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
80	#376	—	67	1	no	—
81	#336	—	67	4	no	—
82	#322	—	67	4	no	—
83	#322	Dic₆₇	1	2	yes	complemented
84	C₈²	Dic₆₇	1	2	yes	complemented
85	C₂²×C₈	—	67	2	no	—
86	C₂²×C₈	C₄×D₆₇	1	2	yes	—
87	C₂×C₄²	—	67	2	no	—
88	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
89	C₄⋊C₈	—	2	2	no	—
90	C₄⋊C₈	—	134	2	no	—
91	C₂²⋊C₈	—	2	2	no	—
92	C₂²⋊C₈	—	134	2	no	—
93	#97	—	134	2	no	—
94	C₄×C₈	—	67	4	no	—
95	C₄×C₈	C₄×D₆₇	1	2	yes	—
96	C₄×C₈	C₂×Dic₆₇	1	4	yes	—
97	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
98	C₄×C₈	C₂×Dic₆₇	1	1	yes	characteristic
99	#95	—	134	2	no	—
100	C₂²×C₄	—	134	2	no	—
101	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
102	C₂²×C₄	#11761628	1	2	yes	—
103	C₂×C₈	—	2	4	no	—
104	C₂×C₈	—	2	4	no	—
105	C₂×C₈	—	134	4	no	—
106	C₂×C₈	—	134	2	no	—
107	C₂×C₈	—	134	4	no	—
108	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
109	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
110	C₄²	—	134	4	no	—
111	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
112	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
113	C₄²	#11761628	1	2	yes	—
114	C₂³	#14524000	1	1	yes	characteristic
115	C₂×C₄	—	4	2	no	—
116	C₂×C₄	—	268	4	no	—
117	C₂×C₄	—	268	2	no	—
118	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
119	C₂×C₄	#14524000	1	2	yes	—
120	C₂×C₄	#14524000	1	1	yes	characteristic
121	C₂×C₄	#14524002	1	2	yes	—
122	C₂×C₄	#14524049	1	4	yes	—
123	C₈	—	2	2	no	—
124	C₈	—	2	2	no	—
125	C₈	—	4	4	no	—
126	C₈	—	268	4	no	—
127	C₈	—	268	2	no	—
128	C₂²	—	2	2	no	—
129	C₂²	#16863242	1	1	yes	characteristic
130	C₂²	#16863455	1	2	yes	—
131	C₄	—	2	2	no	—
132	C₄	—	2	2	no	—
133	C₄	—	536	2	no	—
134	C₄	#16863351	1	2	yes	—
135	C₄	#16863455	1	2	yes	—
136	C₂	—	2	2	no	—
137	C₂	#22559960	1	1	yes	characteristic
138	C₂	#22559964	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22559960

C₂

yes

maximal, characteristic

#22558501

C₂

yes

maximal, characteristic

#22558039

C₂

yes

maximal

#22558037

C₂

yes

maximal

C₆₇×#1450

C₂

yes

maximal, characteristic

#16863449

C₂²

yes

—

C₂×#14523954

C₂²

yes

characteristic

#16863242

C₂²

yes

characteristic

#16863150

C₄

yes

—

#16863118

C₄

yes

complemented

#431×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#322×C₆₇

C₄

yes

complemented

C₈²×C₆₇

C₄

yes

complemented

C₂²×C₆₇⋊C₈

C₂×C₄

yes

—

C₂×C₄×Dic₆₇

C₂×C₄

yes

—

C₄⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

#14523954

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄²

yes

—

C₂×C₈×C₆₇

C₄²

yes

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

#117

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

#117

yes

characteristic

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#376

yes

characteristic

C₂²×C₆₇

#433

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#405

yes

—

C₄×C₆₇

#433

yes

—

Dic₆₇

—

#8141

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1746

yes

characteristic

C₂×C₆₇

#1747

yes

—

#1746

—

#1452

—

#1450

D₆₇

yes

characteristic

#1275

—

#1274

—

C₆₇

#8141

yes

characteristic, complemented, Sylow

#431

—

#431

C₂×D₆₇

yes

—

C₂×#97

—

C₂×C₄×C₈

C₂×D₆₇

yes

—

—

—

Dic₆₇

yes

complemented

C₈²

Dic₆₇

yes

complemented

C₂²×C₈

—

C₂²×C₈

C₄×D₆₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₂²⋊C₈

—

C₂²⋊C₈

—

134

—

#97

—

134

—

C₄×C₈

—

C₄×C₈

C₄×D₆₇

yes

—

C₄×C₈

C₂×Dic₆₇

yes

—

C₄×C₈

C₂×Dic₆₇

yes

characteristic

C₄×C₈

C₂×Dic₆₇

yes

characteristic

#95

—

134

—

100

C₂²×C₄

—

134

—

101

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

102

C₂²×C₄

#11761628

yes

—

103

C₂×C₈

—

104

C₂×C₈

—

105

C₂×C₈

—

134

—

106

C₂×C₈

—

134

—

107

C₂×C₈

—

134

—

108

C₂×C₈

C₄×Dic₆₇

yes

—

109

C₂×C₈

C₄×Dic₆₇

yes

—

110

C₄²

—

134

—

111

C₄²

C₂×C₄×D₆₇

yes

characteristic

112

C₄²

C₂²×Dic₆₇

yes

characteristic

113

C₄²

#11761628

yes

—

114

C₂³

#14524000

yes

characteristic

115

C₂×C₄

—

116

C₂×C₄

—

268

—

117

C₂×C₄

—

268

—

118

C₂×C₄

C₂×C₄×Dic₆₇

yes

characteristic

119

C₂×C₄

#14524000

yes

—

120

C₂×C₄

#14524000

yes

characteristic

121

C₂×C₄

#14524002

yes

—

122

C₂×C₄

#14524049

yes

—

123

C₈

—

124

C₈

—

125

C₈

—

126

C₈

—

268

—

127

C₈

—

268

—

128

C₂²

—

129

C₂²

#16863242

yes

characteristic

130

C₂²

#16863455

yes

—

131

C₄

—

132

C₄

—

133

C₄

—

536

—

134

C₄

#16863351

yes

—

135

C₄

#16863455

yes

—

136

C₂

—

137

C₂

#22559960

yes

characteristic

138

C₂

#22559964

yes

—

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₆₇
commutator	C₂×C₄×C₆₇	C₂×C₄²
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1450	C₂
socle	C₂²×C₆₇	#376
prime radical	C₂³×C₆₇	#117
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1450	D₆₇
hypocenter	C₆₇	#8141
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #1450

subgroup

G/H

center

C₂×C₄

C₂×C₄×Dic₆₇

commutator

C₂×C₄×C₆₇

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1450

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₄ ◁ #1450

N	H ≅ G/N	type
10. #16863118	C₄	semidirect
13. #322×C₆₇	C₄	semidirect
14. C₈²×C₆₇	C₄	semidirect
75. C₆₇	#8141	semidirect
83. #322	Dic₆₇	semidirect
84. C₈²	Dic₆₇	semidirect

H ≅ G/N

type

10. #16863118

C₄

semidirect

13. #322×C₆₇

C₄

semidirect

14. C₈²×C₆₇

C₄

semidirect

75. C₆₇

#8141

semidirect

83. #322

Dic₆₇

semidirect

84. C₈²

Dic₆₇

semidirect

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

subgroup	index
#22559960	2
#22558501	2
#22558039	2
#22558037	2
C₆₇×#1450	2
#8141	67

subgroup

index

C₆₇×#1450

Subgroups — #30746286

Conjugacy classes of subgroups

Complements

Maximal subgroups