Group #30755332 — subgroups

Subgroups — #30755332

Order 17152.

454 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 127 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	#22560347	C₂	1	1	yes	maximal, characteristic
2	#22558881	C₂	1	4	yes	maximal
3	#22558501	C₂	1	1	yes	maximal, characteristic
4	C₂×#494×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#16863449	C₂²	1	2	yes	—
6	#16863420	C₂²	1	2	yes	—
7	#16863161	—	2	2	no	—
8	#16863142	—	2	8	no	—
9	#16863118	—	2	2	no	—
10	C₂²×Q₁₆×C₆₇	—	2	2	no	—
11	C₂×C₄⋊Q₈×C₆₇	C₂²	1	2	yes	—
12	#494×C₆₇	—	2	2	no	—
13	#494×C₆₇	C₄	1	4	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#11761612	—	2	2	no	—
16	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
17	C₄⋊(C₆₇⋊C₈)	D₄	1	4	yes	—
18	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
19	C₄×C₆₇⋊C₈	—	2	2	no	—
20	C₂²×Q₈×C₆₇	—	2	2	no	—
21	C₂×Q₁₆×C₆₇	—	2	8	no	—
22	C₂×Q₁₆×C₆₇	—	4	8	no	—
23	C₂²×C₈×C₆₇	D₄	1	2	yes	—
24	C₄⋊Q₈×C₆₇	—	2	4	no	—
25	C₄⋊Q₈×C₆₇	—	2	2	no	—
26	C₄⋊Q₈×C₆₇	C₂×C₄	1	4	yes	—
27	C₂×C₄⋊C₄×C₆₇	—	2	4	no	—
28	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
29	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
30	C₄×C₈×C₆₇	D₄	1	2	yes	—
31	#11761612	—	4	4	no	—
32	C₂×C₆₇⋊C₈	—	2	4	no	—
33	C₂×C₆₇⋊C₈	—	4	2	no	—
34	C₂×Q₈×C₆₇	—	2	4	no	—
35	C₂×Q₈×C₆₇	—	4	2	no	—
36	C₂×Q₈×C₆₇	—	4	8	no	—
37	C₂²×C₄×C₆₇	—	4	2	no	—
38	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₂²×C₄×C₆₇	#41	1	2	yes	—
40	Q₁₆×C₆₇	—	4	16	no	—
41	C₂×C₈×C₆₇	—	2	4	no	—
42	C₂×C₈×C₆₇	C₂²⋊C₄	1	4	yes	—
43	C₄⋊C₄×C₆₇	—	2	8	no	—
44	C₄⋊C₄×C₆₇	—	4	4	no	—
45	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
46	C₄²×C₆₇	C₂×D₄	1	2	yes	—
47	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
48	C₂³×C₆₇	#124	1	1	yes	characteristic
49	Q₈×C₆₇	—	4	8	no	—
50	Q₈×C₆₇	—	8	4	no	—
51	C₂×C₄×C₆₇	—	2	4	no	—
52	C₂×C₄×C₆₇	—	4	4	no	—
53	C₂×C₄×C₆₇	—	8	4	no	—
54	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
56	C₂×C₄×C₆₇	#121	1	2	yes	—
57	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
58	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
59	C₈×C₆₇	—	2	4	no	—
60	C₆₇⋊C₈	—	4	4	no	—
61	C₆₇⋊C₈	—	8	2	no	—
62	C₂²×C₆₇	—	2	2	no	—
63	C₂²×C₆₇	#390	1	1	yes	characteristic
64	C₂²×C₆₇	#490	1	2	yes	—
65	C₄×C₆₇	—	2	2	no	—
66	C₄×C₆₇	—	8	4	no	—
67	C₄×C₆₇	#418	1	2	yes	—
68	C₄×C₆₇	#473	1	4	yes	—
69	C₄×C₆₇	#496	1	2	yes	—
70	#9779	—	67	1	no	maximal, Sylow
71	C₂×C₆₇	—	2	2	no	—
72	C₂×C₆₇	#1961	1	1	yes	characteristic
73	C₂×C₆₇	#1979	1	2	yes	—
74	C₂×#494	D₆₇	1	1	yes	characteristic
75	#1837	—	67	1	no	—
76	#1546	—	67	4	no	—
77	#1452	—	67	1	no	—
78	C₆₇	#9779	1	1	yes	characteristic, complemented, Sylow
79	C₂²×Q₁₆	—	2	2	no	—
80	C₂×C₄⋊Q₈	C₂×D₆₇	1	2	yes	—
81	#494	—	2	2	no	—
82	#494	Dic₆₇	1	4	yes	—
83	#431	—	67	2	no	—
84	#423	—	67	2	no	—
85	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
86	#341	—	134	2	no	—
87	#332	—	134	8	no	—
88	#322	—	134	2	no	—
89	C₂²×Q₈	—	2	2	no	—
90	C₂×Q₁₆	—	2	8	no	—
91	C₂×Q₁₆	—	4	8	no	—
92	C₂×M₁₆	—	134	2	no	—
93	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
94	C₄⋊Q₈	—	2	4	no	—
95	C₄⋊Q₈	—	2	2	no	—
96	C₄⋊Q₈	C₂×Dic₆₇	1	4	yes	—
97	C₂×C₄⋊C₄	—	2	4	no	—
98	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
99	C₄⋊C₈	—	67	4	no	—
100	C₄⋊C₈	—	134	4	no	—
101	C₂²⋊C₈	—	134	2	no	—
102	C₄×C₈	—	134	2	no	—
103	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
104	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
105	C₂×Q₈	—	2	4	no	—
106	C₂×Q₈	—	4	2	no	—
107	C₂×Q₈	—	4	8	no	—
108	C₂²×C₄	—	4	2	no	—
109	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
110	C₂²×C₄	#11761627	1	2	yes	—
111	Q₁₆	—	4	16	no	—
112	M₁₆	—	268	4	no	—
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	—	268	2	no	—
115	C₂×C₈	—	134	4	no	—
116	C₂×C₈	C₂²⋊Dic₆₇	1	4	yes	—
117	C₄⋊C₄	—	2	8	no	—
118	C₄⋊C₄	—	4	4	no	—
119	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
120	C₄²	D₄×D₆₇	1	2	yes	—
121	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
122	C₂³	#14524030	1	1	yes	characteristic
123	Q₈	—	4	8	no	—
124	Q₈	—	8	4	no	—
125	C₂×C₄	—	2	4	no	—
126	C₂×C₄	—	4	4	no	—
127	C₂×C₄	—	8	4	no	—
128	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
129	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
130	C₂×C₄	#14524014	1	2	yes	—
131	C₂×C₄	#14524039	1	1	yes	characteristic
132	C₂×C₄	#14524069	1	4	yes	—
133	C₈	—	2	4	no	—
134	C₈	—	268	4	no	—
135	C₈	—	536	2	no	—
136	C₂²	—	2	2	no	—
137	C₂²	#16863299	1	1	yes	characteristic
138	C₂²	#16863760	1	2	yes	—
139	C₄	—	2	2	no	—
140	C₄	—	8	4	no	—
141	C₄	#16863401	1	2	yes	—
142	C₄	#16863668	1	4	yes	—
143	C₄	#16863783	1	2	yes	—
144	C₂	—	2	2	no	—
145	C₂	#22561034	1	1	yes	characteristic
146	C₂	#22561125	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22560347

C₂

yes

maximal, characteristic

#22558881

C₂

yes

maximal

#22558501

C₂

yes

maximal, characteristic

C₂×#494×C₆₇

C₂

yes

maximal, characteristic

#16863449

C₂²

yes

—

#16863420

C₂²

yes

—

#16863161

—

#16863142

—

#16863118

—

C₂²×Q₁₆×C₆₇

—

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

—

#494×C₆₇

—

#494×C₆₇

C₄

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×#11761612

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

C₂²⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

—

C₂²×Q₈×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

—

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₂³×C₆₇

#124

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#390

yes

characteristic

C₂²×C₆₇

#490

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#418

yes

—

C₄×C₆₇

#473

yes

—

C₄×C₆₇

#496

yes

—

#9779

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1961

yes

characteristic

C₂×C₆₇

#1979

yes

—

C₂×#494

D₆₇

yes

—

—

—

C₆₇

#9779

yes

characteristic, complemented, Sylow

C₂²×Q₁₆

—

C₂×C₄⋊Q₈

C₂×D₆₇

yes

—

#494

—

#494

Dic₆₇

yes

—

#431

—

#423

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#341

—

134

—

#332

—

134

—

#322

—

134

—

C₂²×Q₈

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×M₁₆

—

134

—

C₂²×C₈

C₂²⋊D₆₇

yes

—

C₄⋊Q₈

—

C₄⋊Q₈

—

C₄⋊Q₈

C₂×Dic₆₇

yes

—

C₂×C₄⋊C₄

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

C₄⋊C₈

—

100

C₄⋊C₈

—

134

—

101

C₂²⋊C₈

—

134

—

102

C₄×C₈

—

134

—

103

C₄×C₈

C₂×Dic₆₇

yes

—

104

C₄×C₈

C₂²⋊D₆₇

yes

—

105

C₂×Q₈

—

106

C₂×Q₈

—

107

C₂×Q₈

—

108

C₂²×C₄

—

109

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

110

C₂²×C₄

#11761627

yes

—

111

Q₁₆

—

112

M₁₆

—

268

—

113

C₂×C₈

—

114

C₂×C₈

—

268

—

115

C₂×C₈

—

134

—

116

C₂×C₈

C₂²⋊Dic₆₇

yes

—

117

C₄⋊C₄

—

118

C₄⋊C₄

—

119

C₄²

C₂²×Dic₆₇

yes

characteristic

120

C₄²

D₄×D₆₇

yes

—

121

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

122

C₂³

#14524030

yes

characteristic

123

Q₈

—

124

Q₈

—

125

C₂×C₄

—

126

C₂×C₄

—

127

C₂×C₄

—

128

C₂×C₄

C₂×C₂²⋊Dic₆₇

yes

characteristic

129

C₂×C₄

D₄×Dic₆₇

yes

—

130

C₂×C₄

#14524014

yes

—

131

C₂×C₄

#14524039

yes

characteristic

132

C₂×C₄

#14524069

yes

—

133

C₈

—

134

C₈

—

268

—

135

C₈

—

536

—

136

C₂²

—

137

C₂²

#16863299

yes

characteristic

138

C₂²

#16863760

yes

—

139

C₄

—

140

C₄

—

141

C₄

#16863401

yes

—

142

C₄

#16863668

yes

—

143

C₄

#16863783

yes

—

144

C₂

—

145

C₂

#22561034

yes

characteristic

146

C₂

#22561125

yes

—

	subgroup	G/H
center	C₂²	#16863299
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#494×C₆₇	C₂
socle	C₂²×C₆₇	#390
prime radical	C₂³×C₆₇	#124
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#494	D₆₇
hypocenter	C₆₇	#9779
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

subgroup

G/H

center

C₂²

#16863299

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₂×#494×C₆₇

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#494

N	H ≅ G/N	type
78. C₆₇	#9779	semidirect

H ≅ G/N

type

78. C₆₇

#9779

semidirect

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

subgroup	index
#22560347	2
#22558881	2
#22558501	2
C₂×#494×C₆₇	2
#9779	67

subgroup

index

C₂×#494×C₆₇

Subgroups — #30755332

Conjugacy classes of subgroups

Complements

Maximal subgroups