Group #30745145 — subgroups

Subgroups — #30745145

Order 17152.

322 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 83 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	#22561772	C₂	1	3	yes	maximal
2	#22558032	C₂	1	3	yes	maximal
3	C₆₇×#2810	C₂	1	1	yes	maximal, characteristic
4	#16863266	C₂²	1	1	yes	characteristic
5	#16863158	—	2	3	no	—
6	#16863158	—	2	3	no	—
7	#16863151	C₂²	1	3	yes	—
8	#400×C₆₇	—	2	3	no	—
9	#389×C₆₇	C₂²	1	3	yes	—
10	#387×C₆₇	—	2	3	no	—
11	C₂²×C₆₇⋊C₈	—	2	3	no	—
12	C₂×C₄⋊Dic₆₇	—	2	3	no	—
13	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
14	C₂×C₄⋊C₄×C₆₇	D₄	1	3	yes	—
15	C₂×C₄⋊C₄×C₆₇	D₄	1	3	yes	—
16	C₂×C₄²×C₆₇	—	2	1	no	—
17	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
18	#95×C₆₇	—	2	3	no	—
19	#95×C₆₇	—	2	3	no	—
20	#95×C₆₇	—	2	1	no	—
21	#95×C₆₇	—	2	3	no	—
22	C₂²×Dic₆₇	—	4	1	no	—
23	C₂×C₆₇⋊C₈	—	4	3	no	—
24	C₂×C₆₇⋊C₈	—	4	3	no	—
25	C₂×C₆₇⋊C₈	—	4	3	no	—
26	C₄⋊Dic₆₇	—	4	6	no	—
27	C₂²×C₄×C₆₇	—	2	3	no	—
28	C₂²×C₄×C₆₇	—	2	3	no	—
29	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
30	C₄⋊C₄×C₆₇	—	2	3	no	—
31	C₄⋊C₄×C₆₇	—	2	3	no	—
32	C₄⋊C₄×C₆₇	—	4	3	no	—
33	C₄⋊C₄×C₆₇	—	4	3	no	—
34	C₄⋊C₄×C₆₇	—	4	3	no	—
35	C₄²×C₆₇	—	4	1	no	—
36	C₄²×C₆₇	—	8	1	no	—
37	C₂³×C₆₇	#120	1	1	yes	characteristic
38	C₂×C₄×C₆₇	—	2	3	no	—
39	C₂×C₄×C₆₇	—	2	3	no	—
40	C₂×C₄×C₆₇	—	4	3	no	—
41	C₂×C₄×C₆₇	—	4	3	no	—
42	C₂×C₄×C₆₇	—	4	3	no	—
43	C₂×C₄×C₆₇	—	4	3	no	—
44	C₂×C₄×C₆₇	—	4	3	no	—
45	C₂×C₄×C₆₇	—	4	3	no	—
46	C₂×C₄×C₆₇	#135	1	3	yes	—
47	C₂×C₄×C₆₇	#137	1	3	yes	—
48	C₂×Dic₆₇	—	8	3	no	—
49	C₆₇⋊C₈	—	8	3	no	—
50	C₂²×C₆₇	#452	1	3	yes	—
51	C₂²×C₆₇	#454	1	3	yes	—
52	C₂²×C₆₇	#458	1	1	yes	characteristic
53	C₄×C₆₇	—	2	3	no	—
54	C₄×C₆₇	—	2	3	no	—
55	C₄×C₆₇	—	8	3	no	—
56	C₄×C₆₇	—	8	3	no	—
57	Dic₆₇	—	16	1	no	—
58	#7832	—	67	1	no	maximal, Sylow
59	C₂×C₆₇	#1615	1	3	yes	—
60	C₂×C₆₇	#1638	1	3	yes	—
61	C₂×C₆₇	#1658	1	1	yes	characteristic
62	#2810	D₆₇	1	1	yes	characteristic
63	#2100	—	67	3	no	—
64	#1271	—	67	3	no	—
65	C₆₇	#7832	1	1	yes	characteristic, complemented, Sylow
66	#400	—	2	3	no	—
67	#389	C₂×D₆₇	1	3	yes	—
68	#387	—	2	3	no	—
69	#384	—	67	1	no	—
70	#340	—	134	3	no	—
71	#340	—	134	3	no	—
72	#336	—	67	3	no	—
73	C₂²×C₈	—	134	3	no	—
74	C₂×C₄⋊C₄	—	2	3	no	—
75	C₂×C₄⋊C₄	—	134	3	no	—
76	C₂×C₄⋊C₄	C₂²⋊D₆₇	1	3	yes	—
77	C₂×C₄⋊C₄	C₂²⋊D₆₇	1	3	yes	—
78	C₂×C₄²	—	2	1	no	—
79	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
80	#95	—	2	3	no	—
81	#95	—	2	3	no	—
82	#95	—	2	1	no	—
83	#95	—	2	3	no	—
84	C₂²×C₄	—	2	3	no	—
85	C₂²×C₄	—	2	3	no	—
86	C₂²×C₄	—	268	1	no	—
87	C₂²×C₄	C₂×C₂²⋊D₆₇	1	3	yes	—
88	C₂×C₈	—	268	3	no	—
89	C₂×C₈	—	268	3	no	—
90	C₂×C₈	—	268	3	no	—
91	C₄⋊C₄	—	2	3	no	—
92	C₄⋊C₄	—	2	3	no	—
93	C₄⋊C₄	—	4	3	no	—
94	C₄⋊C₄	—	4	3	no	—
95	C₄⋊C₄	—	4	3	no	—
96	C₄⋊C₄	—	268	6	no	—
97	C₄²	—	4	1	no	—
98	C₄²	—	8	1	no	—
99	C₂³	#14524012	1	1	yes	characteristic
100	C₂×C₄	—	2	3	no	—
101	C₂×C₄	—	2	3	no	—
102	C₂×C₄	—	4	3	no	—
103	C₂×C₄	—	4	3	no	—
104	C₂×C₄	—	4	3	no	—
105	C₂×C₄	—	4	3	no	—
106	C₂×C₄	—	4	3	no	—
107	C₂×C₄	—	4	3	no	—
108	C₂×C₄	—	536	3	no	—
109	C₂×C₄	#14524063	1	3	yes	—
110	C₂×C₄	#14524073	1	3	yes	—
111	C₈	—	536	3	no	—
112	C₂²	#16863558	1	3	yes	—
113	C₂²	#16863569	1	3	yes	—
114	C₂²	#16863586	1	1	yes	characteristic
115	C₄	—	2	3	no	—
116	C₄	—	2	3	no	—
117	C₄	—	8	3	no	—
118	C₄	—	8	3	no	—
119	C₄	—	1072	1	no	—
120	C₂	#22559240	1	3	yes	—
121	C₂	#22559381	1	3	yes	—
122	C₂	#22559511	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22561772

C₂

yes

maximal

#22558032

C₂

yes

maximal

C₆₇×#2810

C₂

yes

maximal, characteristic

#16863266

C₂²

yes

—

—

C₂²

yes

—

#400×C₆₇

—

#389×C₆₇

C₂²

yes

—

#387×C₆₇

—

C₂²×C₆₇⋊C₈

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

D₄

yes

—

C₂×C₄⋊C₄×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#95×C₆₇

—

#95×C₆₇

—

#95×C₆₇

—

#95×C₆₇

—

C₂²×Dic₆₇

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

#120

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

#135

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₂×Dic₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

#452

yes

—

C₂²×C₆₇

#454

yes

—

C₂²×C₆₇

#458

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

Dic₆₇

—

#7832

—

maximal, Sylow

C₂×C₆₇

#1615

yes

—

C₂×C₆₇

#1638

yes

—

C₂×C₆₇

#1658

yes

characteristic

#2810

D₆₇

yes

characteristic

#2100

—

#1271

—

C₆₇

#7832

yes

characteristic, complemented, Sylow

#400

—

#389

C₂×D₆₇

yes

—

#387

—

#384

—

#340

—

134

—

#340

—

134

—

#336

—

C₂²×C₈

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

134

—

C₂×C₄⋊C₄

C₂²⋊D₆₇

yes

—

C₂×C₄⋊C₄

C₂²⋊D₆₇

yes

—

C₂×C₄²

—

C₂×C₄²

C₂²×D₆₇

yes

—

—

—

—

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

—

268

—

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

—

C₂×C₈

—

268

—

C₂×C₈

—

268

—

C₂×C₈

—

268

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

268

—

C₄²

—

C₄²

—

C₂³

#14524012

yes

characteristic

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

—

536

—

109

C₂×C₄

#14524063

yes

—

110

C₂×C₄

#14524073

yes

—

111

C₈

—

536

—

112

C₂²

#16863558

yes

—

113

C₂²

#16863569

yes

—

114

C₂²

#16863586

yes

characteristic

115

C₄

—

116

C₄

—

117

C₄

—

118

C₄

—

119

C₄

—

1072

—

120

C₂

#22559240

yes

—

121

C₂

#22559381

yes

—

122

C₂

#22559511

yes

characteristic

	subgroup	G/H
center	C₂³	#14524012
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2810	C₂
socle, prime radical	C₂³×C₆₇	#120
hypercenter	#2810	D₆₇
hypocenter	C₆₇	#7832
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂³×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2810

subgroup

G/H

center

C₂³

#14524012

commutator, cosocle

C₂×C₄²×C₆₇

C₂³

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#2810

C₂

C₂³×C₆₇

D₆₇

C₆₇

G ▷ C₂×C₄²×C₆₇ ▷ C₂³×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2810

N	H ≅ G/N	type
65. C₆₇	#7832	semidirect

H ≅ G/N

type

65. C₆₇

#7832

semidirect

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

subgroup	index
#22561772	2
#22558032	2
C₆₇×#2810	2
#7832	67

subgroup

index

#22561772

#22558032

C₆₇×#2810

#7832

Subgroups — #30745145

Conjugacy classes of subgroups

Complements

Maximal subgroups