Group #30749633 — subgroups

Subgroups — #30749633

Order 17152.

354 conjugacy classes of subgroups; only the 171 normal subgroups are shown.

normal subgroups only

This group has 354 conjugacy classes of subgroups — too many to draw. Only its 171 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	#22560381	C₂	1	1	yes	maximal, characteristic
2	#22558967	C₂	1	4	yes	maximal
3	C₂×C₈×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂×#14523979	—	2	2	no	—
6	#16863420	C₂²	1	2	yes	—
7	C₂×C₄×C₆₇⋊C₈	C₂²	1	2	yes	—
8	#16863148	—	2	8	no	—
9	C₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
10	C₈×C₆₇⋊C₈	C₄	1	2	yes	—
11	C₈×C₆₇⋊C₈	C₄	1	1	yes	characteristic
12	C₂×#108×C₆₇	—	2	2	no	—
13	#423×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#11761612	—	2	2	no	—
16	C₂²×C₆₇⋊C₈	Q₈	1	4	yes	—
17	#14523979	—	4	4	no	—
18	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
19	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
20	C₄×C₆₇⋊C₈	C₂×C₄	1	2	yes	—
21	C₄×C₆₇⋊C₈	D₄	1	4	yes	—
22	C₂×M₁₆×C₆₇	—	2	2	no	—
23	C₂²×C₈×C₆₇	D₄	1	2	yes	—
24	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
25	#108×C₆₇	—	4	4	no	—
26	C₄⋊C₈×C₆₇	—	2	4	no	—
27	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
29	C₄×C₈×C₆₇	Q₈	1	2	yes	—
30	#11761612	—	4	4	no	—
31	C₂×C₆₇⋊C₈	—	2	8	no	—
32	C₂×C₆₇⋊C₈	—	4	2	no	—
33	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
34	C₂×C₆₇⋊C₈	C₄⋊C₄	1	4	yes	—
35	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₇	C₂×Q₈	1	2	yes	—
37	M₁₆×C₆₇	—	4	4	no	—
38	C₂×C₈×C₆₇	—	4	2	no	—
39	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
41	C₂×C₈×C₆₇	D₈	1	4	yes	—
42	C₂×C₈×C₆₇	Q₁₆	1	4	yes	—
43	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
44	C₄²×C₆₇	C₂×D₄	1	2	yes	—
45	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
46	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
47	C₂×C₄×C₆₇	—	2	4	no	—
48	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	2	yes	—
49	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₄⋊Q₈	1	2	yes	—
52	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
53	C₂×C₄×C₆₇	C₂×Q₁₆	1	2	yes	—
54	C₈×C₆₇	—	8	2	no	—
55	C₈×C₆₇	#107	1	4	yes	—
56	C₈×C₆₇	#107	1	4	yes	—
57	C₆₇⋊C₈	—	2	4	no	—
58	C₆₇⋊C₈	—	8	2	no	—
59	C₆₇⋊C₈	#108	1	8	yes	complemented
60	C₂²×C₆₇	—	2	2	no	—
61	C₂²×C₆₇	#384	1	1	yes	characteristic
62	C₂²×C₆₇	#500	1	2	yes	—
63	C₄×C₆₇	—	2	2	no	—
64	C₄×C₆₇	C₂×#107	1	2	yes	—
65	C₄×C₆₇	C₂×#108	1	4	yes	—
66	C₄×C₆₇	#493	1	1	yes	characteristic
67	C₄×C₆₇	#494	1	1	yes	characteristic
68	#8823	—	67	1	no	maximal, Sylow
69	C₂×C₆₇	—	2	2	no	—
70	C₂×C₆₇	#1845	1	1	yes	characteristic
71	C₂×C₆₇	#1848	1	2	yes	—
72	#1848	—	67	1	no	—
73	#1848	D₆₇	1	1	yes	characteristic
74	#1567	—	67	4	no	—
75	C₂×C₈²	—	67	1	no	—
76	C₆₇	#8823	1	1	yes	characteristic, complemented, Sylow
77	C₂×#108	—	2	2	no	—
78	C₂×#108	—	134	2	no	—
79	#423	—	67	2	no	—
80	#423	C₂×D₆₇	1	2	yes	—
81	C₂×C₄×C₈	—	67	2	no	—
82	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
83	#335	—	134	8	no	—
84	C₈²	—	67	1	no	—
85	C₈²	—	67	2	no	—
86	C₈²	—	67	1	no	—
87	C₂×M₁₆	—	2	2	no	—
88	C₂×M₁₆	—	134	2	no	—
89	C₂²×C₈	—	67	4	no	—
90	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
91	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
92	#108	—	4	4	no	—
93	#108	—	268	4	no	—
94	C₄⋊C₈	—	2	4	no	—
95	C₄⋊C₈	—	134	4	no	—
96	C₄×C₈	—	67	2	no	—
97	C₄×C₈	—	67	2	no	—
98	C₄×C₈	—	67	4	no	—
99	C₄×C₈	Dic₁₃₄	1	2	yes	—
100	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
101	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
102	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
103	C₂²×C₄	Q₈×D₆₇	1	2	yes	—
104	M₁₆	—	4	4	no	—
105	M₁₆	—	268	4	no	—
106	C₂×C₈	—	4	2	no	—
107	C₂×C₈	—	268	2	no	—
108	C₂×C₈	—	67	4	no	—
109	C₂×C₈	—	67	4	no	—
110	C₂×C₈	—	134	8	no	—
111	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
112	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
113	C₂×C₈	#11761613	1	4	yes	—
114	C₂×C₈	#11761618	1	4	yes	—
115	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
116	C₄²	D₄×D₆₇	1	2	yes	—
117	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
118	C₂³	#14524041	1	1	yes	characteristic
119	C₂×C₄	—	2	4	no	—
120	C₂×C₄	C₄⋊C₄×D₆₇	1	2	yes	—
121	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
122	C₂×C₄	#14524039	1	1	yes	characteristic
123	C₂×C₄	#14524042	1	2	yes	—
124	C₂×C₄	C₂×#11761613	1	2	yes	—
125	C₂×C₄	C₂×#11761618	1	2	yes	—
126	C₈	—	8	2	no	—
127	C₈	—	536	2	no	—
128	C₈	—	67	8	no	—
129	C₈	—	134	4	no	—
130	C₈	#14523976	1	4	yes	—
131	C₈	#14523976	1	4	yes	—
132	C₂²	—	2	2	no	—
133	C₂²	#16863265	1	1	yes	characteristic
134	C₂²	#16863799	1	2	yes	—
135	C₄	—	2	2	no	—
136	C₄	C₂×#14523976	1	2	yes	—
137	C₄	#108×D₆₇	1	4	yes	—
138	C₄	#16863772	1	1	yes	characteristic
139	C₄	#16863775	1	1	yes	characteristic
140	C₂	—	2	2	no	—
141	C₂	#22560368	1	1	yes	characteristic
142	C₂	#22560379	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22560381

C₂

yes

maximal, characteristic

#22558967

C₂

yes

maximal

C₂×C₈×C₆₇⋊C₈

C₂

yes

maximal, characteristic

C₆₇×#1848

C₂

yes

maximal, characteristic

C₂×#14523979

—

#16863420

C₂²

yes

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

—

#16863148

—

C₈×C₆₇⋊C₈

C₄

yes

characteristic

C₈×C₆₇⋊C₈

C₄

yes

—

C₈×C₆₇⋊C₈

C₄

yes

characteristic

C₂×#108×C₆₇

—

#423×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×#11761612

—

C₂²×C₆₇⋊C₈

Q₈

yes

—

#14523979

—

C₄⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

—

C₄×C₆₇⋊C₈

C₂×C₄

yes

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#108×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

Q₈

yes

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

Q₁₆

yes

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₂³×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

—

C₂×C₄×C₆₇

C₂×Q₁₆

yes

—

C₈×C₆₇

—

C₈×C₆₇

#107

yes

—

C₈×C₆₇

#107

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₆₇⋊C₈

#108

yes

complemented

C₂²×C₆₇

—

C₂²×C₆₇

#384

yes

characteristic

C₂²×C₆₇

#500

yes

—

C₄×C₆₇

—

C₄×C₆₇

C₂×#107

yes

—

C₄×C₆₇

C₂×#108

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

#1845

yes

characteristic

C₂×C₆₇

#1848

yes

—

#1848

—

#1848

D₆₇

yes

characteristic

#1567

—

C₂×C₈²

—

C₆₇

#8823

yes

characteristic, complemented, Sylow

C₂×#108

—

C₂×#108

—

134

—

#423

—

#423

C₂×D₆₇

yes

—

C₂×C₄×C₈

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#335

—

134

—

C₈²

—

C₈²

—

C₈²

—

C₂×M₁₆

—

C₂×M₁₆

—

134

—

C₂²×C₈

—

C₂²×C₈

C₂²⋊D₆₇

yes

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

#108

—

#108

—

268

—

C₄⋊C₈

—

C₄⋊C₈

—

134

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

Dic₁₃₄

yes

—

100

C₄×C₈

C₄×D₆₇

yes

characteristic

101

C₄×C₈

C₄×D₆₇

yes

characteristic

102

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

103

C₂²×C₄

Q₈×D₆₇

yes

—

104

M₁₆

—

105

M₁₆

—

268

—

106

C₂×C₈

—

107

C₂×C₈

—

268

—

108

C₂×C₈

—

109

C₂×C₈

—

110

C₂×C₈

—

134

—

111

C₂×C₈

Dic₆₇⋊C₄

yes

—

112

C₂×C₈

Dic₆₇⋊C₄

yes

—

113

C₂×C₈

#11761613

yes

—

114

C₂×C₈

#11761618

yes

—

115

C₄²

C₂×C₄×D₆₇

yes

characteristic

116

C₄²

D₄×D₆₇

yes

—

117

C₄²

C₂×Dic₁₃₄

yes

characteristic

118

C₂³

#14524041

yes

characteristic

119

C₂×C₄

—

120

C₂×C₄

C₄⋊C₄×D₆₇

yes

—

121

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

122

C₂×C₄

#14524039

yes

characteristic

123

C₂×C₄

#14524042

yes

—

124

C₂×C₄

C₂×#11761613

yes

—

125

C₂×C₄

C₂×#11761618

yes

—

126

C₈

—

127

C₈

—

536

—

128

C₈

—

129

C₈

—

134

—

130

C₈

#14523976

yes

—

131

C₈

#14523976

yes

—

132

C₂²

—

133

C₂²

#16863265

yes

characteristic

134

C₂²

#16863799

yes

—

135

C₄

—

136

C₄

C₂×#14523976

yes

—

137

C₄

#108×D₆₇

yes

—

138

C₄

#16863772

yes

characteristic

139

C₄

#16863775

yes

characteristic

140

C₂

—

141

C₂

#22560368

yes

characteristic

142

C₂

#22560379

yes

—

	subgroup	G/H
center	C₂×C₄	#14524039
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1848	C₂
socle	C₂²×C₆₇	#384
prime radical	C₂³×C₆₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1848	D₆₇
hypocenter	C₆₇	#8823
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

subgroup

G/H

center

C₂×C₄

#14524039

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1848

C₂

socle

C₂²×C₆₇

#384

prime radical

C₂³×C₆₇

C₄⋊Q₈

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1848

N	H ≅ G/N	type
59. C₆₇⋊C₈	#108	semidirect
76. C₆₇	#8823	semidirect

H ≅ G/N

type

59. C₆₇⋊C₈

#108

semidirect

76. C₆₇

#8823

semidirect

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

subgroup	index
#22560381	2
#22558967	2
C₂×C₈×C₆₇⋊C₈	2
C₆₇×#1848	2
#8823	67

subgroup

index

#22560381

#22558967

C₂×C₈×C₆₇⋊C₈

C₆₇×#1848

#8823

Subgroups — #30749633

Conjugacy classes of subgroups

Complements

Maximal subgroups