Group #30772222 — subgroups

Subgroups — #30772222

Order 17152.

274 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 115 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×#16863171	C₂	1	1	yes	maximal, characteristic
2	#22560379	C₂	1	2	yes	maximal
3	#22558303	C₂	1	1	yes	maximal, characteristic
4	#22558302	C₂	1	2	yes	maximal
5	C₆₇×#1381	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14523979	C₂²	1	2	yes	—
8	C₂×#14523978	—	2	2	no	—
9	#16863419	—	2	2	no	—
10	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
11	#16863171	C₄	1	4	yes	—
12	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	C₂×#108×C₆₇	C₂²	1	2	yes	—
14	C₂×#11761612	—	2	2	no	—
15	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
16	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
17	C₂×C₆₇⋊C₁₆	—	2	2	no	—
18	C₂×C₆₇⋊C₁₆	C₂×C₄	1	2	yes	—
19	#14523979	—	2	2	no	—
20	#14523979	—	2	2	no	—
21	#14523978	—	4	4	no	—
22	Dic₆₇⋊C₈	—	4	4	no	—
23	C₈×Dic₆₇	C₂×C₄	1	2	yes	—
24	C₈×Dic₆₇	Q₈	1	2	yes	—
25	C₂×M₁₆×C₆₇	—	2	2	no	—
26	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
27	C₂×C₁₆×C₆₇	—	2	2	no	—
28	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
29	#108×C₆₇	—	2	2	no	—
30	#108×C₆₇	—	2	2	no	—
31	C₂²×Dic₆₇	Q₁₆	1	2	yes	—
32	#11761612	—	4	2	no	—
33	#11761612	—	4	2	no	—
34	C₂×C₆₇⋊C₈	—	4	2	no	—
35	C₂×C₆₇⋊C₈	C₄⋊C₄	1	2	yes	—
36	C₂×C₆₇⋊C₈	SD₁₆	1	4	yes	—
37	C₄×Dic₆₇	C₄⋊C₄	1	2	yes	—
38	C₄×Dic₆₇	D₈	1	2	yes	—
39	C₆₇⋊C₁₆	—	2	2	no	—
40	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
41	M₁₆×C₆₇	—	4	2	no	—
42	M₁₆×C₆₇	—	4	2	no	—
43	C₂×C₈×C₆₇	—	4	2	no	—
44	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
46	C₂×C₈×C₆₇	#41	1	2	yes	—
47	C₂×C₈×C₆₇	#41	1	2	yes	—
48	C₁₆×C₆₇	—	2	2	no	—
49	C₂³×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
51	C₂×C₄×C₆₇	#124	1	2	yes	—
52	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₂×SD₁₆	1	2	yes	—
54	C₂×Dic₆₇	—	4	2	no	—
55	C₂×Dic₆₇	#107	1	4	yes	—
56	C₈×C₆₇	—	8	2	no	—
57	C₈×C₆₇	#117	1	1	yes	characteristic
58	C₈×C₆₇	#117	1	1	yes	characteristic
59	C₈×C₆₇	#125	1	2	yes	—
60	C₆₇⋊C₈	—	8	2	no	—
61	C₆₇⋊C₈	#106	1	4	yes	—
62	C₂²×C₆₇	—	2	2	no	—
63	C₂²×C₆₇	C₂×#107	1	1	yes	characteristic
64	C₂²×C₆₇	#487	1	2	yes	—
65	C₄×C₆₇	#382	1	1	yes	characteristic
66	C₄×C₆₇	C₂×#106	1	1	yes	characteristic
67	C₄×C₆₇	#486	1	2	yes	—
68	Dic₆₇	—	2	4	no	—
69	#12892	—	67	1	no	maximal, Sylow
70	C₂×C₆₇	—	2	2	no	—
71	C₂×C₆₇	#1834	1	1	yes	characteristic
72	C₂×C₆₇	#2161	1	2	yes	—
73	C₂×#346	—	67	1	no	—
74	#1848	—	67	2	no	—
75	#1381	—	67	1	no	—
76	#1381	—	67	2	no	—
77	#1381	D₆₇	1	1	yes	characteristic
78	C₆₇	#12892	1	1	yes	characteristic, complemented, Sylow
79	C₂²×C₁₆	—	67	1	no	—
80	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
81	C₂×#108	—	67	2	no	—
82	C₂×#108	—	134	2	no	—
83	C₂×#108	C₂×D₆₇	1	2	yes	—
84	#423	—	134	2	no	—
85	C₂×C₄×C₈	—	67	1	no	—
86	#346	—	67	4	no	—
87	C₂×M₁₆	—	2	2	no	—
88	C₂×M₁₆	—	134	2	no	—
89	C₂²×C₈	—	67	1	no	—
90	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
91	C₂×C₄²	—	67	1	no	—
92	C₂×C₁₆	—	2	2	no	—
93	C₂×C₁₆	—	67	2	no	—
94	C₂×C₁₆	—	134	2	no	—
95	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
96	#108	—	2	2	no	—
97	#108	—	2	2	no	—
98	#108	—	268	4	no	—
99	#108	—	134	2	no	—
100	#108	—	134	2	no	—
101	C₄⋊C₈	—	268	4	no	—
102	C₄×C₈	—	67	2	no	—
103	C₄×C₈	—	67	2	no	—
104	C₂²×C₄	—	67	2	no	—
105	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
106	M₁₆	—	4	2	no	—
107	M₁₆	—	4	2	no	—
108	M₁₆	—	268	2	no	—
109	M₁₆	—	268	2	no	—
110	C₂×C₈	—	4	2	no	—
111	C₂×C₈	—	268	2	no	—
112	C₂×C₈	—	67	2	no	—
113	C₂×C₈	—	67	4	no	—
114	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
115	C₂×C₈	Q₈×D₆₇	1	1	yes	characteristic
116	C₂×C₈	#11761627	1	2	yes	—
117	C₂×C₈	#11761628	1	2	yes	—
118	C₄²	—	67	2	no	—
119	C₄²	—	67	2	no	—
120	C₁₆	—	2	2	no	—
121	C₁₆	—	134	2	no	—
122	C₂³	Q₁₆×D₆₇	1	1	yes	characteristic
123	C₂×C₄	—	268	2	no	—
124	C₂×C₄	—	67	4	no	—
125	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
126	C₂×C₄	#14524029	1	2	yes	—
127	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
128	C₂×C₄	SD₁₆×D₆₇	1	2	yes	—
129	C₈	—	8	2	no	—
130	C₈	—	536	2	no	—
131	C₈	—	67	4	no	—
132	C₈	#14523999	1	1	yes	characteristic
133	C₈	#14524000	1	1	yes	characteristic
134	C₈	#14524033	1	2	yes	—
135	C₂²	—	2	2	no	—
136	C₂²	#107×D₆₇	1	1	yes	characteristic
137	C₂²	#16863745	1	2	yes	—
138	C₄	—	134	4	no	—
139	C₄	#16863260	1	1	yes	characteristic
140	C₄	#106×D₆₇	1	1	yes	characteristic
141	C₄	#16863741	1	2	yes	—
142	C₂	—	2	2	no	—
143	C₂	#22560335	1	1	yes	characteristic
144	C₂	#22562002	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863171

C₂

yes

maximal, characteristic

#22560379

C₂

yes

maximal

#22558303

C₂

yes

maximal, characteristic

#22558302

C₂

yes

maximal

C₆₇×#1381

C₂

yes

maximal, characteristic

C₂²×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₂×#14523979

C₂²

yes

—

C₂×#14523978

—

#16863419

—

C₂×C₈×Dic₆₇

C₂²

yes

characteristic

#16863171

C₄

yes

—

C₂²×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×#108×C₆₇

C₂²

yes

—

C₂×#11761612

—

C₂²×C₆₇⋊C₈

D₄

yes

characteristic

C₂×C₄×Dic₆₇

D₄

yes

characteristic

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

—

#14523979

—

#14523979

—

#14523978

—

Dic₆₇⋊C₈

—

C₈×Dic₆₇

C₂×C₄

yes

—

C₈×Dic₆₇

Q₈

yes

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₁₆×C₆₇

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

—

#108×C₆₇

—

#108×C₆₇

—

C₂²×Dic₆₇

Q₁₆

yes

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

C₄⋊C₄

yes

—

C₂×C₆₇⋊C₈

SD₁₆

yes

—

C₄×Dic₆₇

C₄⋊C₄

yes

—

C₄×Dic₆₇

D₈

yes

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₂×C₈×C₆₇

#41

yes

—

C₁₆×C₆₇

—

C₂³×C₆₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#124

yes

—

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

#107

yes

—

C₈×C₆₇

—

C₈×C₆₇

yes

C₈×C₆₇

yes

C₈×C₆₇

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

#106

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

C₂×#107

yes

characteristic

C₂²×C₆₇

#487

yes

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

Dic₆₇

—

#12892

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1834

yes

characteristic

C₂×C₆₇

#2161

yes

—

—

—

—

—

D₆₇

yes

characteristic

C₆₇

#12892

yes

characteristic, complemented, Sylow

C₂²×C₁₆

—

C₂²×C₁₆

C₂×D₆₇

yes

characteristic

C₂×#108

—

C₂×#108

—

134

—

C₂×#108

C₂×D₆₇

yes

—

#423

—

134

—

C₂×C₄×C₈

—

#346

—

C₂×M₁₆

—

C₂×M₁₆

—

134

—

C₂²×C₈

—

C₂²×C₈

C₂²×D₆₇

yes

characteristic

C₂×C₄²

—

C₂×C₁₆

—

C₂×C₁₆

—

C₂×C₁₆

—

134

—

C₂×C₁₆

C₄×D₆₇

yes

—

#108

—

#108

—

#108

—

268

—

#108

—

134

—

100

#108

—

134

—

101

C₄⋊C₈

—

268

—

102

C₄×C₈

—

103

C₄×C₈

—

104

C₂²×C₄

—

105

C₂²×C₄

D₄×D₆₇

yes

characteristic

106

M₁₆

—

107

M₁₆

—

108

M₁₆

—

268

—

109

M₁₆

—

268

—

110

C₂×C₈

—

111

C₂×C₈

—

268

—

112

C₂×C₈

—

113

C₂×C₈

—

114

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

115

C₂×C₈

Q₈×D₆₇

yes

characteristic

116

C₂×C₈

#11761627

yes

—

117

C₂×C₈

#11761628

yes

—

118

C₄²

—

119

C₄²

—

120

C₁₆

—

121

C₁₆

—

134

—

122

C₂³

Q₁₆×D₆₇

yes

characteristic

123

C₂×C₄

—

268

—

124

C₂×C₄

—

125

C₂×C₄

C₄⋊C₄×D₆₇

yes

characteristic

126

C₂×C₄

#14524029

yes

—

127

C₂×C₄

D₈×D₆₇

yes

characteristic

128

C₂×C₄

SD₁₆×D₆₇

yes

—

129

C₈

—

130

C₈

—

536

—

131

C₈

—

132

C₈

#14523999

yes

characteristic

133

C₈

#14524000

yes

characteristic

134

C₈

#14524033

yes

—

135

C₂²

—

136

C₂²

#107×D₆₇

yes

characteristic

137

C₂²

#16863745

yes

—

138

C₄

—

134

—

139

C₄

#16863260

yes

characteristic

140

C₄

#106×D₆₇

yes

characteristic

141

C₄

#16863741

yes

—

142

C₂

—

143

C₂

#22560335

yes

characteristic

144

C₂

#22562002

yes

—

	subgroup	G/H
center	C₂×C₄	D₈×D₆₇
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1381	C₂
socle	C₂²×C₆₇	C₂×#107
prime radical	C₂³×C₆₇	C₂×Q₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1381	D₆₇
hypocenter	C₆₇	#12892
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

subgroup

G/H

center

C₂×C₄

D₈×D₆₇

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1381

C₂

socle

C₂²×C₆₇

C₂×#107

prime radical

C₂³×C₆₇

C₂×Q₁₆

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ #1381

N	H ≅ G/N	type
78. C₆₇	#12892	semidirect

H ≅ G/N

type

78. C₆₇

#12892

semidirect

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

subgroup	index
C₂×#16863171	2
#22560379	2
#22558303	2
#22558302	2
C₆₇×#1381	2
#12892	67

subgroup

index

C₆₇×#1381

Subgroups — #30772222

Conjugacy classes of subgroups

Complements

Maximal subgroups