Group #30746280 — subgroups

Subgroups — #30746280

Order 17152.

330 conjugacy classes of subgroups; only the 147 normal subgroups are shown.

normal subgroups only

This group has 330 conjugacy classes of subgroups — too many to draw. Only its 147 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	#22558501	C₂	1	2	yes	maximal
2	#22558040	C₂	1	2	yes	maximal
3	#22558038	C₂	1	2	yes	maximal
4	C₆₇×#1726	C₂	1	1	yes	maximal, characteristic
5	#16863449	C₂²	1	4	yes	—
6	#16863118	C₄	1	8	yes	complemented
7	C₂×#97×C₆₇	C₄	1	2	yes	—
8	C₂×#97×C₆₇	C₂²	1	1	yes	characteristic
9	C₂×C₄×C₈×C₆₇	C₄	1	2	yes	—
10	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	#376×C₆₇	C₂²	1	1	yes	characteristic
12	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
13	C₂²⋊(C₆₇⋊C₈)	—	2	4	no	—
14	C₄×C₆₇⋊C₈	C₂×C₄	1	8	yes	—
15	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
16	C₂²×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
17	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
18	C₂×C₄²×C₆₇	C₂×C₄	1	2	yes	—
19	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
20	#97×C₆₇	—	2	4	no	—
21	#97×C₆₇	—	2	2	no	—
22	C₄×C₈×C₆₇	—	2	4	no	—
23	C₄×C₈×C₆₇	C₂×C₄	1	4	yes	—
24	#95×C₆₇	—	2	2	no	—
25	C₂×C₆₇⋊C₈	—	2	8	no	—
26	C₂²×C₄×C₆₇	—	2	2	no	—
27	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
28	C₂²×C₄×C₆₇	#41	1	2	yes	—
29	C₂×C₈×C₆₇	—	2	2	no	—
30	C₂×C₈×C₆₇	—	2	2	no	—
31	C₂×C₈×C₆₇	—	2	2	no	—
32	C₂×C₈×C₆₇	—	2	2	no	—
33	C₂×C₈×C₆₇	C₄²	1	2	yes	—
34	C₂×C₈×C₆₇	C₄²	1	2	yes	—
35	C₂×C₈×C₆₇	M₁₆	1	4	yes	—
36	C₄²×C₆₇	—	2	2	no	—
37	C₄²×C₆₇	M₁₆	1	4	yes	—
38	C₄²×C₆₇	C₂²×C₄	1	2	yes	—
39	C₄²×C₆₇	#41	1	2	yes	—
40	C₂³×C₆₇	#117	1	1	yes	characteristic
41	C₂×C₄×C₆₇	—	2	4	no	—
42	C₂×C₄×C₆₇	—	4	2	no	—
43	C₂×C₄×C₆₇	—	4	4	no	—
44	C₂×C₄×C₆₇	C₂×C₄²	1	1	yes	characteristic
45	C₂×C₄×C₆₇	#117	1	4	yes	—
46	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
47	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
48	C₂×C₄×C₆₇	C₂×M₁₆	1	2	yes	—
49	C₈×C₆₇	—	2	2	no	—
50	C₈×C₆₇	—	2	2	no	—
51	C₈×C₆₇	—	2	4	no	—
52	C₆₇⋊C₈	—	4	8	no	—
53	C₂²×C₆₇	—	2	2	no	—
54	C₂²×C₆₇	#376	1	1	yes	characteristic
55	C₂²×C₆₇	#432	1	2	yes	—
56	C₄×C₆₇	—	2	4	no	—
57	C₄×C₆₇	—	4	4	no	—
58	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
59	C₄×C₆₇	C₄×M₁₆	1	1	yes	characteristic
60	C₄×C₆₇	#432	1	2	yes	—
61	#8139	—	67	1	no	maximal, Sylow
62	C₂×C₆₇	—	2	2	no	—
63	C₂×C₆₇	#1745	1	1	yes	characteristic
64	C₂×C₆₇	#1747	1	2	yes	—
65	#1726	D₆₇	1	1	yes	characteristic
66	#1452	—	67	2	no	—
67	#1275	—	67	2	no	—
68	#1274	—	67	2	no	—
69	C₆₇	#8139	1	1	yes	characteristic, complemented, Sylow
70	#431	—	67	4	no	—
71	C₂×#97	Dic₆₇	1	2	yes	—
72	C₂×#97	C₂×D₆₇	1	1	yes	characteristic
73	C₂×C₄×C₈	Dic₆₇	1	2	yes	—
74	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
75	#376	C₂×D₆₇	1	1	yes	characteristic
76	#322	—	67	8	no	—
77	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
78	C₂²×C₈	C₂×Dic₆₇	1	1	yes	characteristic
79	C₂²×C₈	C₂×Dic₆₇	1	2	yes	—
80	C₂×C₄²	C₂×Dic₆₇	1	2	yes	—
81	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
82	C₄⋊C₈	—	134	4	no	—
83	C₂²⋊C₈	—	134	4	no	—
84	#97	—	2	4	no	—
85	#97	—	2	2	no	—
86	C₄×C₈	—	2	4	no	—
87	C₄×C₈	—	67	8	no	—
88	C₄×C₈	C₄×D₆₇	1	4	yes	—
89	#95	—	2	2	no	—
90	C₂²×C₄	—	2	2	no	—
91	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
92	C₂²×C₄	#11761628	1	2	yes	—
93	C₂×C₈	—	2	2	no	—
94	C₂×C₈	—	2	2	no	—
95	C₂×C₈	—	2	2	no	—
96	C₂×C₈	—	2	2	no	—
97	C₂×C₈	—	134	8	no	—
98	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
99	C₂×C₈	C₄×Dic₆₇	1	2	yes	—
100	C₂×C₈	#11761612	1	4	yes	—
101	C₄²	—	2	2	no	—
102	C₄²	#11761612	1	4	yes	—
103	C₄²	C₂×C₄×D₆₇	1	2	yes	—
104	C₄²	#11761628	1	2	yes	—
105	C₂³	#14524002	1	1	yes	characteristic
106	C₂×C₄	—	2	4	no	—
107	C₂×C₄	—	4	2	no	—
108	C₂×C₄	—	4	4	no	—
109	C₂×C₄	C₂×C₄×Dic₆₇	1	1	yes	characteristic
110	C₂×C₄	#14524000	1	4	yes	—
111	C₂×C₄	#14524002	1	1	yes	characteristic
112	C₂×C₄	C₂×#11761612	1	2	yes	—
113	C₂×C₄	C₂×#11761612	1	2	yes	—
114	C₈	—	2	2	no	—
115	C₈	—	2	2	no	—
116	C₈	—	2	4	no	—
117	C₈	—	268	8	no	—
118	C₂²	—	2	2	no	—
119	C₂²	#16863242	1	1	yes	characteristic
120	C₂²	#16863453	1	2	yes	—
121	C₄	—	2	4	no	—
122	C₄	—	4	4	no	—
123	C₄	C₄×#11761612	1	1	yes	characteristic
124	C₄	C₄×#11761612	1	1	yes	characteristic
125	C₄	#16863453	1	2	yes	—
126	C₂	—	2	2	no	—
127	C₂	#22559958	1	1	yes	characteristic
128	C₂	#22559964	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22558501

C₂

yes

maximal

#22558040

C₂

yes

maximal

#22558038

C₂

yes

maximal

C₆₇×#1726

C₂

yes

maximal, characteristic

#16863449

C₂²

yes

—

#16863118

C₄

yes

complemented

C₂×#97×C₆₇

C₄

yes

—

C₂×#97×C₆₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₆₇

C₄

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#376×C₆₇

C₂²

yes

characteristic

C₄⋊(C₆₇⋊C₈)

—

C₂²⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

—

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂×C₄

yes

characteristic

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂×C₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#97×C₆₇

—

#97×C₆₇

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

#95×C₆₇

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄²

yes

—

C₂×C₈×C₆₇

C₄²

yes

—

C₂×C₈×C₆₇

M₁₆

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

M₁₆

yes

—

C₄²×C₆₇

C₂²×C₄

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

#117

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄²

yes

characteristic

C₂×C₄×C₆₇

#117

yes

—

C₂×C₄×C₆₇

#117

yes

characteristic

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₂×C₄×C₆₇

C₂×M₁₆

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#376

yes

characteristic

C₂²×C₆₇

#432

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₄×M₁₆

yes

characteristic

C₄×C₆₇

C₄×M₁₆

yes

characteristic

C₄×C₆₇

#432

yes

—

#8139

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1745

yes

characteristic

C₂×C₆₇

#1747

yes

—

#1726

D₆₇

yes

—

—

—

C₆₇

#8139

yes

characteristic, complemented, Sylow

#431

—

C₂×#97

Dic₆₇

yes

—

C₂×#97

C₂×D₆₇

yes

characteristic

C₂×C₄×C₈

Dic₆₇

yes

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#376

C₂×D₆₇

yes

characteristic

#322

—

C₂²×C₈

C₂×Dic₆₇

yes

characteristic

C₂²×C₈

C₂×Dic₆₇

yes

characteristic

C₂²×C₈

C₂×Dic₆₇

yes

—

C₂×C₄²

C₂×Dic₆₇

yes

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

C₄⋊C₈

—

134

—

C₂²⋊C₈

—

134

—

#97

—

#97

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₄×D₆₇

yes

—

#95

—

C₂²×C₄

—

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

C₂²×C₄

#11761628

yes

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

C₄×Dic₆₇

yes

—

C₂×C₈

C₄×Dic₆₇

yes

—

100

C₂×C₈

#11761612

yes

—

101

C₄²

—

102

C₄²

#11761612

yes

—

103

C₄²

C₂×C₄×D₆₇

yes

—

104

C₄²

#11761628

yes

—

105

C₂³

#14524002

yes

characteristic

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

C₂×C₄×Dic₆₇

yes

characteristic

110

C₂×C₄

#14524000

yes

—

111

C₂×C₄

#14524002

yes

characteristic

112

C₂×C₄

C₂×#11761612

yes

—

113

C₂×C₄

C₂×#11761612

yes

—

114

C₈

—

115

C₈

—

116

C₈

—

117

C₈

—

268

—

118

C₂²

—

119

C₂²

#16863242

yes

characteristic

120

C₂²

#16863453

yes

—

121

C₄

—

122

C₄

—

123

C₄

C₄×#11761612

yes

characteristic

124

C₄

C₄×#11761612

yes

characteristic

125

C₄

#16863453

yes

—

126

C₂

—

127

C₂

#22559958

yes

characteristic

128

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×#1726	C₂
socle	C₂²×C₆₇	#376
prime radical	C₂³×C₆₇	#117
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1726	D₆₇
hypocenter	C₆₇	#8139
LCS	G ▷ C₂×C₄×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ #1726

subgroup

G/H

center

C₂×C₄

C₂×C₄×Dic₆₇

commutator

C₂×C₄×C₆₇

C₂×C₄²

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1726

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₄×C₆₇ ▷ C₆₇

C₁ ◁ C₂×C₄ ◁ #1726

N	H ≅ G/N	type
6. #16863118	C₄	semidirect
69. C₆₇	#8139	semidirect

H ≅ G/N

type

6. #16863118

C₄

semidirect

69. C₆₇

#8139

semidirect

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

subgroup	index
#22558501	2
#22558040	2
#22558038	2
C₆₇×#1726	2
#8139	67

subgroup

index

C₆₇×#1726

Subgroups — #30746280

Conjugacy classes of subgroups

Complements

Maximal subgroups