Group #30746266 — subgroups

Subgroups — #30746266

Order 17152.

382 conjugacy classes of subgroups; only the 179 normal subgroups are shown.

normal subgroups only

This group has 382 conjugacy classes of subgroups — too many to draw. Only its 179 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	#22559955	C₂	1	1	yes	maximal, characteristic
2	#22559955	C₂	1	1	yes	maximal, characteristic
3	C₆₇⋊#1273	C₂	1	4	yes	maximal
4	C₆₇×#1744	C₂	1	1	yes	maximal, characteristic
5	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
6	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
7	#16863151	C₄	1	4	yes	—
8	#16863151	C₄	1	4	yes	—
9	C₂×C₄×C₈×C₆₇	C₂²	1	2	yes	—
10	C₄³×C₆₇	C₂²	1	1	yes	characteristic
11	#336×C₆₇	C₄	1	4	yes	—
12	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
13	C₂²×C₆₇⋊C₈	C₂×C₄	1	4	yes	—
14	C₄×C₆₇⋊C₈	—	2	4	no	—
15	C₄×C₆₇⋊C₈	—	2	4	no	—
16	C₂²×C₈×C₆₇	C₂×C₄	1	4	yes	—
17	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
20	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
21	C₄×C₈×C₆₇	—	2	4	no	—
22	C₂×C₆₇⋊C₈	—	2	4	no	—
23	C₂×C₆₇⋊C₈	—	2	4	no	—
24	C₂×C₆₇⋊C₈	—	2	8	no	—
25	C₂×C₆₇⋊C₈	—	2	8	no	—
26	C₂²×C₄×C₆₇	C₄²	1	4	yes	—
27	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
29	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
30	C₂×C₈×C₆₇	—	2	4	no	—
31	C₂×C₈×C₆₇	—	2	8	no	—
32	C₄²×C₆₇	—	2	4	no	—
33	C₄²×C₆₇	—	2	4	no	—
34	C₄²×C₆₇	—	2	4	no	—
35	C₄²×C₆₇	#41	1	4	yes	—
36	C₂³×C₆₇	C₂×C₄²	1	1	yes	characteristic
37	C₂×C₄×C₆₇	—	2	4	no	—
38	C₂×C₄×C₆₇	—	2	4	no	—
39	C₂×C₄×C₆₇	—	2	4	no	—
40	C₂×C₄×C₆₇	#117	1	2	yes	—
41	C₂×C₄×C₆₇	#117	1	2	yes	—
42	C₂×C₄×C₆₇	#117	1	2	yes	—
43	C₂×C₄×C₆₇	#131	1	4	yes	—
44	C₂×C₄×C₆₇	#131	1	4	yes	—
45	C₂×C₄×C₆₇	#131	1	4	yes	—
46	C₈×C₆₇	—	4	4	no	—
47	C₆₇⋊C₈	—	4	4	no	—
48	C₆₇⋊C₈	—	4	4	no	—
49	C₂²×C₆₇	#376	1	1	yes	characteristic
50	C₂²×C₆₇	#405	1	2	yes	—
51	C₂²×C₆₇	#405	1	2	yes	—
52	C₂²×C₆₇	#405	1	2	yes	—
53	C₄×C₆₇	—	4	4	no	—
54	C₄×C₆₇	#433	1	4	yes	—
55	C₄×C₆₇	#433	1	4	yes	—
56	C₄×C₆₇	#433	1	4	yes	—
57	#8135	—	67	1	no	maximal, Sylow
58	C₂×C₆₇	#1454	1	4	yes	—
59	C₂×C₆₇	#1746	1	1	yes	characteristic
60	C₂×C₆₇	#1746	1	1	yes	characteristic
61	C₂×C₆₇	#1746	1	1	yes	characteristic
62	#1744	—	67	1	no	—
63	#1744	—	67	1	no	—
64	#1744	D₆₇	1	1	yes	characteristic
65	#1273	—	67	4	no	—
66	C₆₇	#8135	1	1	yes	characteristic, complemented, Sylow
67	C₂×C₄×C₈	—	67	2	no	—
68	C₂×C₄×C₈	—	67	2	no	—
69	C₂×C₄×C₈	C₂×D₆₇	1	2	yes	—
70	C₄³	C₂×D₆₇	1	1	yes	characteristic
71	#336	—	67	4	no	—
72	#336	—	67	4	no	—
73	#336	Dic₆₇	1	4	yes	—
74	C₂²×C₈	—	67	4	no	—
75	C₂²×C₈	—	67	4	no	—
76	C₂²×C₈	C₂×Dic₆₇	1	4	yes	—
77	C₂×C₄²	C₄×D₆₇	1	2	yes	—
78	C₂×C₄²	C₄×D₆₇	1	2	yes	—
79	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
80	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
81	C₄×C₈	—	2	4	no	—
82	C₄×C₈	—	134	4	no	—
83	C₄×C₈	—	134	4	no	—
84	C₂²×C₄	C₄×Dic₆₇	1	4	yes	—
85	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
86	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
87	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
88	C₂×C₈	—	2	4	no	—
89	C₂×C₈	—	2	8	no	—
90	C₂×C₈	—	134	4	no	—
91	C₂×C₈	—	134	4	no	—
92	C₂×C₈	—	134	8	no	—
93	C₂×C₈	—	134	8	no	—
94	C₄²	—	2	4	no	—
95	C₄²	—	2	4	no	—
96	C₄²	—	2	4	no	—
97	C₄²	#11761628	1	4	yes	—
98	C₂³	C₂×C₄×Dic₆₇	1	1	yes	characteristic
99	C₂×C₄	—	2	4	no	—
100	C₂×C₄	—	2	4	no	—
101	C₂×C₄	—	2	4	no	—
102	C₂×C₄	#14524000	1	2	yes	—
103	C₂×C₄	#14524000	1	2	yes	—
104	C₂×C₄	#14524002	1	2	yes	—
105	C₂×C₄	#14524049	1	4	yes	—
106	C₂×C₄	#14524050	1	4	yes	—
107	C₂×C₄	#14524051	1	4	yes	—
108	C₈	—	4	4	no	—
109	C₈	—	268	4	no	—
110	C₈	—	268	4	no	—
111	C₂²	#16863242	1	1	yes	characteristic
112	C₂²	#16863351	1	2	yes	—
113	C₂²	#16863352	1	2	yes	—
114	C₂²	#16863354	1	2	yes	—
115	C₄	—	4	4	no	—
116	C₄	#16863455	1	4	yes	—
117	C₄	#16863457	1	4	yes	—
118	C₄	#16863458	1	4	yes	—
119	C₂	#22558509	1	4	yes	—
120	C₂	#22559960	1	1	yes	characteristic
121	C₂	#22559961	1	1	yes	characteristic
122	C₂	#22559962	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22559955

C₂

yes

maximal, characteristic

#22559955

C₂

yes

maximal, characteristic

C₆₇⋊#1273

C₂

yes

maximal

C₆₇×#1744

C₂

yes

maximal, characteristic

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

#16863151

C₄

yes

—

#16863151

C₄

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

—

C₄³×C₆₇

C₂²

yes

characteristic

#336×C₆₇

C₄

yes

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

—

C₂²×C₆₇⋊C₈

C₂×C₄

yes

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄×C₈×C₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

C₄²

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₂×C₄×C₆₇

#131

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

#376

yes

characteristic

C₂²×C₆₇

#405

yes

—

C₂²×C₆₇

#405

yes

—

C₂²×C₆₇

#405

yes

—

C₄×C₆₇

—

C₄×C₆₇

#433

yes

—

C₄×C₆₇

#433

yes

—

C₄×C₆₇

#433

yes

—

#8135

—

maximal, Sylow

C₂×C₆₇

#1454

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

—

D₆₇

yes

characteristic

#1273

—

C₆₇

#8135

yes

characteristic, complemented, Sylow

C₂×C₄×C₈

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₇

yes

—

C₄³

C₂×D₆₇

yes

—

—

Dic₆₇

yes

—

C₂²×C₈

—

C₂²×C₈

—

C₂²×C₈

C₂×Dic₆₇

yes

—

C₂×C₄²

C₄×D₆₇

yes

—

C₂×C₄²

C₄×D₆₇

yes

—

C₂×C₄²

C₂×Dic₆₇

yes

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

C₄×C₈

—

C₄×C₈

—

134

—

C₄×C₈

—

134

—

C₂²×C₄

C₄×Dic₆₇

yes

—

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₂×C₈

—

134

—

C₄²

—

C₄²

—

C₄²

—

C₄²

#11761628

yes

—

C₂³

C₂×C₄×Dic₆₇

yes

characteristic

C₂×C₄

—

100

C₂×C₄

—

101

C₂×C₄

—

102

C₂×C₄

#14524000

yes

—

103

C₂×C₄

#14524000

yes

—

104

C₂×C₄

#14524002

yes

—

105

C₂×C₄

#14524049

yes

—

106

C₂×C₄

#14524050

yes

—

107

C₂×C₄

#14524051

yes

—

108

C₈

—

109

C₈

—

268

—

110

C₈

—

268

—

111

C₂²

#16863242

yes

characteristic

112

C₂²

#16863351

yes

—

113

C₂²

#16863352

yes

—

114

C₂²

#16863354

yes

—

115

C₄

—

116

C₄

#16863455

yes

—

117

C₄

#16863457

yes

—

118

C₄

#16863458

yes

—

119

C₂

#22558509

yes

—

120

C₂

#22559960

yes

characteristic

121

C₂

#22559961

yes

characteristic

122

C₂

#22559962

yes

characteristic

	subgroup	G/H
center, Frattini	C₂×C₄²	C₂²×D₆₇
commutator, socle, prime radical	C₂³×C₆₇	C₂×C₄²
Fitting	C₆₇×#1744	C₂
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1744	D₆₇
hypocenter	C₆₇	#8135
LCS	G ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄² ◁ #1744

subgroup

G/H

center, Frattini

C₂×C₄²

C₂²×D₆₇

commutator, socle, prime radical

C₂³×C₆₇

C₂×C₄²

Fitting

C₆₇×#1744

C₂

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂³×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₄² ◁ #1744

N	H ≅ G/N	type
66. C₆₇	#8135	semidirect

H ≅ G/N

type

66. C₆₇

#8135

semidirect

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

subgroup	index
#22559955	2
C₆₇⋊#1273	2
C₆₇×#1744	2
#8135	67

subgroup

index

#22559955

C₆₇⋊#1273

C₆₇×#1744

#8135

Subgroups — #30746266

Conjugacy classes of subgroups

Complements

Maximal subgroups