Group #30743004 — subgroups

Subgroups — #30743004

Order 17152.

238 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 238 conjugacy classes of subgroups — too many to draw. Only its 111 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×#16863217	C₂	1	2	yes	maximal
2	C₆₇×#1845	C₂	1	1	yes	maximal, characteristic
3	C₂×#14523983	—	2	2	no	—
4	#16863217	—	2	2	no	—
5	#16863217	C₄	1	4	yes	complemented
6	C₂×#107×C₆₇	—	2	2	no	—
7	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
8	#384×C₆₇	C₄	1	2	yes	—
9	#14523983	—	2	4	no	—
10	#14523983	—	4	2	no	—
11	C₂×C₆₇⋊C₁₆	—	4	2	no	—
12	C₂²×C₈×C₆₇	D₄	1	2	yes	—
13	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
14	C₂×C₄⋊C₄×C₆₇	—	4	2	no	—
15	C₂×C₄²×C₆₇	C₂×C₄	1	1	yes	characteristic
16	#107×C₆₇	—	4	2	no	—
17	#107×C₆₇	—	4	2	no	—
18	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
19	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
20	C₄×C₈×C₆₇	Q₈	1	1	yes	characteristic
21	C₆₇⋊C₁₆	—	4	4	no	—
22	C₂²×C₄×C₆₇	—	2	1	no	—
23	C₂²×C₄×C₆₇	—	4	2	no	—
24	C₂²×C₄×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
25	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
26	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
27	C₂×C₈×C₆₇	D₈	1	2	yes	—
28	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
29	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
30	C₂×C₈×C₆₇	Q₁₆	1	2	yes	—
31	C₄⋊C₄×C₆₇	—	4	4	no	—
32	C₄⋊C₄×C₆₇	—	8	4	no	—
33	C₄²×C₆₇	—	2	1	no	—
34	C₄²×C₆₇	C₄²	1	1	yes	characteristic
35	C₄²×C₆₇	C₄⋊C₄	1	1	yes	characteristic
36	C₂³×C₆₇	#100	1	1	yes	characteristic
37	C₂×C₄×C₆₇	—	2	2	no	—
38	C₂×C₄×C₆₇	—	2	2	no	—
39	C₂×C₄×C₆₇	—	2	1	no	—
40	C₂×C₄×C₆₇	—	2	1	no	—
41	C₂×C₄×C₆₇	—	8	4	no	—
42	C₂×C₄×C₆₇	—	8	2	no	—
43	C₂×C₄×C₆₇	#95	1	1	yes	characteristic
44	C₂×C₄×C₆₇	#101	1	1	yes	characteristic
45	C₂×C₄×C₆₇	#102	1	2	yes	—
46	C₂×C₄×C₆₇	#103	1	2	yes	—
47	C₈×C₆₇	#102	1	2	yes	—
48	C₈×C₆₇	#103	1	2	yes	—
49	C₈×C₆₇	#106	1	2	yes	—
50	C₈×C₆₇	#107	1	2	yes	—
51	C₂²×C₆₇	—	2	1	no	—
52	C₂²×C₆₇	—	2	1	no	—
53	C₂²×C₆₇	#331	1	1	yes	characteristic
54	C₂²×C₆₇	#333	1	1	yes	characteristic
55	C₂²×C₆₇	#343	1	1	yes	characteristic
56	C₄×C₆₇	—	2	2	no	—
57	C₄×C₆₇	—	2	2	no	—
58	C₄×C₆₇	—	16	2	no	—
59	C₄×C₆₇	#332	1	2	yes	—
60	C₄×C₆₇	#340	1	2	yes	—
61	#7269	—	67	1	no	maximal, Sylow
62	C₂×C₆₇	—	2	1	no	—
63	C₂×C₆₇	—	2	1	no	—
64	C₂×C₆₇	#1296	1	1	yes	characteristic
65	C₂×C₆₇	#1357	1	1	yes	characteristic
66	C₂×C₆₇	#1358	1	1	yes	characteristic
67	C₂×#364	—	67	2	no	—
68	#1845	D₆₇	1	1	yes	characteristic
69	C₆₇	#7269	1	1	yes	characteristic, complemented, Sylow
70	C₂×M₃₂	—	134	2	no	—
71	C₂×#107	—	2	2	no	—
72	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
73	#384	Dic₆₇	1	2	yes	—
74	#364	—	67	4	no	—
75	#364	—	134	2	no	—
76	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
77	C₂×C₄⋊C₄	—	2	2	no	—
78	C₂×C₄⋊C₄	—	4	2	no	—
79	C₂×C₄²	C₂×Dic₆₇	1	1	yes	characteristic
80	M₃₂	—	268	2	no	—
81	M₃₂	—	134	4	no	—
82	C₂×C₁₆	—	268	2	no	—
83	#107	—	4	2	no	—
84	#107	—	4	2	no	—
85	C₄×C₈	Dic₁₃₄	1	1	yes	characteristic
86	C₄×C₈	C₄×D₆₇	1	2	yes	—
87	C₄×C₈	D₂₆₈	1	1	yes	characteristic
88	C₂²×C₄	—	2	1	no	—
89	C₂²×C₄	—	4	2	no	—
90	C₂²×C₄	C₂²⋊Dic₆₇	1	1	yes	characteristic
91	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
92	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
93	C₂×C₈	#11761613	1	2	yes	—
94	C₂×C₈	#11761615	1	2	yes	—
95	C₂×C₈	#11761616	1	2	yes	—
96	C₂×C₈	#11761618	1	2	yes	—
97	C₄⋊C₄	—	4	4	no	—
98	C₄⋊C₄	—	8	4	no	—
99	C₄²	—	2	1	no	—
100	C₄²	C₄×Dic₆₇	1	1	yes	characteristic
101	C₄²	C₄⋊Dic₆₇	1	1	yes	characteristic
102	C₁₆	—	268	4	no	—
103	C₂³	#14523960	1	1	yes	characteristic
104	C₂×C₄	—	2	2	no	—
105	C₂×C₄	—	2	2	no	—
106	C₂×C₄	—	2	1	no	—
107	C₂×C₄	—	2	1	no	—
108	C₂×C₄	—	8	4	no	—
109	C₂×C₄	—	8	2	no	—
110	C₂×C₄	C₆₇⋊#95	1	1	yes	characteristic
111	C₂×C₄	#14523962	1	1	yes	characteristic
112	C₂×C₄	#14523964	1	2	yes	—
113	C₂×C₄	#14523967	1	2	yes	—
114	C₈	#14523963	1	2	yes	—
115	C₈	#14523966	1	2	yes	—
116	C₈	#14523974	1	2	yes	—
117	C₈	#14523976	1	2	yes	—
118	C₂²	—	2	1	no	—
119	C₂²	—	2	1	no	—
120	C₂²	#16863140	1	1	yes	characteristic
121	C₂²	#16863145	1	1	yes	characteristic
122	C₂²	#16863165	1	1	yes	characteristic
123	C₄	—	2	2	no	—
124	C₄	—	2	2	no	—
125	C₄	—	16	2	no	—
126	C₄	#16863142	1	2	yes	—
127	C₄	#16863158	1	2	yes	—
128	C₂	—	2	1	no	—
129	C₂	—	2	1	no	—
130	C₂	#22558092	1	1	yes	characteristic
131	C₂	#22558246	1	1	yes	characteristic
132	C₂	#22558248	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863217

C₂

yes

maximal

C₆₇×#1845

C₂

yes

maximal, characteristic

C₂×#14523983

—

#16863217

—

#16863217

C₄

yes

complemented

C₂×#107×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#384×C₆₇

C₄

yes

—

#14523983

—

#14523983

—

C₂×C₆₇⋊C₁₆

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂×C₄

yes

characteristic

#107×C₆₇

—

#107×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

Q₈

yes

characteristic

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₂×C₈×C₆₇

SD₁₆

yes

—

C₂×C₈×C₆₇

Q₁₆

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₄²

yes

characteristic

C₄²×C₆₇

C₄⋊C₄

yes

characteristic

C₂³×C₆₇

#100

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂×C₄×C₆₇

#103

yes

—

C₈×C₆₇

#102

yes

—

C₈×C₆₇

#103

yes

—

C₈×C₆₇

#106

yes

—

C₈×C₆₇

#107

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#332

yes

—

C₄×C₆₇

#340

yes

—

#7269

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

D₆₇

yes

characteristic

C₆₇

#7269

yes

characteristic, complemented, Sylow

C₂×M₃₂

—

134

—

C₂×#107

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#384

Dic₆₇

yes

—

#364

—

#364

—

134

—

C₂²×C₈

C₂²⋊D₆₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂×Dic₆₇

yes

characteristic

M₃₂

—

268

—

M₃₂

—

134

—

C₂×C₁₆

—

268

—

#107

—

#107

—

C₄×C₈

Dic₁₃₄

yes

characteristic

C₄×C₈

C₄×D₆₇

yes

—

C₄×C₈

D₂₆₈

yes

characteristic

C₂²×C₄

—

C₂²×C₄

—

C₂²×C₄

C₂²⋊Dic₆₇

yes

characteristic

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

C₂×C₈

Dic₆₇⋊C₄

yes

—

C₂×C₈

#11761613

yes

—

C₂×C₈

#11761615

yes

—

C₂×C₈

#11761616

yes

—

C₂×C₈

#11761618

yes

—

C₄⋊C₄

—

C₄⋊C₄

—

C₄²

—

100

C₄²

C₄×Dic₆₇

yes

characteristic

101

C₄²

C₄⋊Dic₆₇

yes

characteristic

102

C₁₆

—

268

—

103

C₂³

#14523960

yes

characteristic

104

C₂×C₄

—

105

C₂×C₄

—

106

C₂×C₄

—

107

C₂×C₄

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

C₆₇⋊#95

yes

characteristic

111

C₂×C₄

#14523962

yes

characteristic

112

C₂×C₄

#14523964

yes

—

113

C₂×C₄

#14523967

yes

—

114

C₈

#14523963

yes

—

115

C₈

#14523966

yes

—

116

C₈

#14523974

yes

—

117

C₈

#14523976

yes

—

118

C₂²

—

119

C₂²

—

120

C₂²

#16863140

yes

characteristic

121

C₂²

#16863145

yes

characteristic

122

C₂²

#16863165

yes

characteristic

123

C₄

—

124

C₄

—

125

C₄

—

126

C₄

#16863142

yes

—

127

C₄

#16863158

yes

—

128

C₂

—

129

C₂

—

130

C₂

#22558092

yes

characteristic

131

C₂

#22558246

yes

characteristic

132

C₂

#22558248

yes

characteristic

	subgroup	G/H
center	C₂²	#16863140
commutator	C₄²×C₆₇	C₄²
Frattini	C₂×C₄×C₈	C₂×D₆₇
Fitting	C₆₇×#1845	C₂
socle	C₂²×C₆₇	#331
prime radical	C₂³×C₆₇	#100
cosocle	C₂×C₄×C₈×C₆₇	C₂²
hypercenter	#1845	D₆₇
hypocenter	C₆₇	#7269
LCS	G ▷ C₄²×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1845

subgroup

G/H

center

C₂²

#16863140

commutator

C₄²×C₆₇

C₄²

Frattini

C₂×C₄×C₈

C₂×D₆₇

Fitting

C₆₇×#1845

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄×C₈×C₆₇

C₂²

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1845

N	H ≅ G/N	type
5. #16863217	C₄	semidirect
69. C₆₇	#7269	semidirect

H ≅ G/N

type

5. #16863217

C₄

semidirect

69. C₆₇

#7269

semidirect

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

subgroup	index
C₂×#16863217	2
C₆₇×#1845	2
#7269	67

subgroup

index

C₂×#16863217

C₆₇×#1845

#7269

Subgroups — #30743004

Conjugacy classes of subgroups

Complements

Maximal subgroups