Group #30770639 — subgroups

Subgroups — #30770639

Order 17152.

250 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 250 conjugacy classes of subgroups — too many to draw. Only its 91 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	#22562491	C₂	1	2	yes	maximal
2	#22562437	C₂	1	2	yes	maximal
3	#22559827	C₂	1	1	yes	maximal, characteristic
4	#22559806	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1312	C₂	1	1	yes	maximal, characteristic
6	#16863801	—	2	1	no	—
7	#16863801	C₂²	1	1	yes	characteristic
8	#16863801	C₂²	1	2	yes	—
9	#16863794	—	2	1	no	—
10	#16863776	—	2	1	no	—
11	#16863776	C₂²	1	1	yes	characteristic
12	#16863747	—	2	1	no	—
13	#16863224	—	2	4	no	—
14	#16863201	—	2	4	no	—
15	C₄×C₁₆×C₆₇	C₂²	1	2	yes	—
16	C₈²×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×Dic₂₆₈	—	2	2	no	—
18	C₂×Dic₂₆₈	—	4	2	no	—
19	#14524043	—	2	1	no	—
20	#14524043	—	2	1	no	—
21	#14524043	—	2	2	no	—
22	#14523977	—	2	2	no	—
23	#14523977	—	2	2	no	—
24	#14523977	—	2	2	no	—
25	#14523977	—	4	2	no	—
26	#14523975	—	4	2	no	—
27	#14523968	—	4	2	no	—
28	C₂×C₁₆×C₆₇	D₄	1	4	yes	—
29	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
30	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
31	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
32	C₂×Dic₁₃₄	—	4	1	no	—
33	C₂×Dic₁₃₄	—	4	1	no	—
34	C₂×Dic₁₃₄	—	4	2	no	—
35	Dic₂₆₈	—	4	4	no	—
36	Dic₂₆₈	—	8	4	no	—
37	C₄⋊Dic₆₇	—	4	2	no	—
38	C₄⋊Dic₆₇	—	4	1	no	—
39	C₄⋊Dic₆₇	—	4	1	no	—
40	C₄⋊Dic₆₇	—	4	1	no	—
41	C₄⋊Dic₆₇	—	4	1	no	—
42	C₄⋊Dic₆₇	—	4	2	no	—
43	C₂×C₈×C₆₇	D₈	1	2	yes	—
44	C₂×C₈×C₆₇	D₈	1	2	yes	—
45	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
46	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
47	C₁₆×C₆₇	—	2	4	no	—
48	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
51	C₂×Dic₆₇	—	8	1	no	—
52	C₂×Dic₆₇	—	8	2	no	—
53	C₂×Dic₆₇	—	8	1	no	—
54	Dic₁₃₄	—	8	2	no	—
55	Dic₁₃₄	—	8	4	no	—
56	Dic₁₃₄	—	8	2	no	—
57	C₈×C₆₇	—	2	2	no	—
58	C₈×C₆₇	SD₃₂	1	4	yes	—
59	C₈×C₆₇	#137	1	4	yes	—
60	C₂²×C₆₇	#493	1	1	yes	characteristic
61	C₄×C₆₇	#497	1	2	yes	—
62	C₄×C₆₇	C₂×SD₃₂	1	2	yes	—
63	C₄×C₆₇	#510	1	2	yes	—
64	Dic₆₇	—	16	1	no	—
65	Dic₆₇	—	16	2	no	—
66	Dic₆₇	—	16	1	no	—
67	#12535	—	67	1	no	maximal, Sylow
68	C₂×C₆₇	#1722	1	1	yes	characteristic
69	C₂×C₆₇	#2248	1	1	yes	characteristic
70	C₂×C₆₇	#2251	1	1	yes	characteristic
71	#2254	—	67	2	no	—
72	#2241	—	67	2	no	—
73	#1713	—	67	1	no	—
74	#1707	—	67	1	no	—
75	#1312	D₆₇	1	1	yes	characteristic
76	C₆₇	#12535	1	1	yes	characteristic, complemented, Sylow
77	#500	—	67	1	no	—
78	#500	—	67	2	no	—
79	#500	—	134	1	no	—
80	#498	—	134	1	no	—
81	#494	—	67	1	no	—
82	#494	—	134	1	no	—
83	#487	—	134	1	no	—
84	#367	—	134	4	no	—
85	#358	—	134	4	no	—
86	C₄×C₁₆	C₂×D₆₇	1	2	yes	—
87	C₈²	C₂×D₆₇	1	1	yes	characteristic
88	C₂×Q₁₆	—	268	2	no	—
89	C₂×Q₁₆	—	134	2	no	—
90	C₄⋊Q₈	—	134	1	no	—
91	C₄⋊Q₈	—	134	1	no	—
92	C₄⋊Q₈	—	134	2	no	—
93	C₂×C₁₆	D₂₆₈	1	4	yes	—
94	#107	—	268	2	no	—
95	#107	—	134	2	no	—
96	#107	—	134	2	no	—
97	#107	—	134	2	no	—
98	#106	—	268	2	no	—
99	#103	—	268	2	no	—
100	C₄×C₈	D₂₆₈	1	1	yes	characteristic
101	C₄×C₈	D₂₆₈	1	1	yes	characteristic
102	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
103	C₂×Q₈	—	268	1	no	—
104	C₂×Q₈	—	268	1	no	—
105	C₂×Q₈	—	268	2	no	—
106	Q₁₆	—	268	4	no	—
107	Q₁₆	—	536	4	no	—
108	C₂×C₈	D₅₃₆	1	2	yes	—
109	C₂×C₈	D₅₃₆	1	2	yes	—
110	C₂×C₈	C₂×D₂₆₈	1	2	yes	—
111	C₄⋊C₄	—	268	2	no	—
112	C₄⋊C₄	—	268	1	no	—
113	C₄⋊C₄	—	268	1	no	—
114	C₄⋊C₄	—	268	1	no	—
115	C₄⋊C₄	—	268	1	no	—
116	C₄⋊C₄	—	268	2	no	—
117	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
118	C₁₆	—	2	4	no	—
119	Q₈	—	536	2	no	—
120	Q₈	—	536	4	no	—
121	Q₈	—	536	2	no	—
122	C₂×C₄	—	536	1	no	—
123	C₂×C₄	—	536	2	no	—
124	C₂×C₄	—	536	1	no	—
125	C₂×C₄	#14524040	1	1	yes	characteristic
126	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
127	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
128	C₈	—	2	2	no	—
129	C₈	#14523988	1	4	yes	—
130	C₈	#14524076	1	4	yes	—
131	C₂²	#16863773	1	1	yes	characteristic
132	C₄	—	1072	1	no	—
133	C₄	—	1072	2	no	—
134	C₄	—	1072	1	no	—
135	C₄	#16863790	1	2	yes	—
136	C₄	C₂×#14523988	1	2	yes	—
137	C₄	#16863839	1	2	yes	—
138	C₂	#22559880	1	1	yes	characteristic
139	C₂	#22562465	1	1	yes	characteristic
140	C₂	#22562480	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562491

C₂

yes

maximal

#22562437

C₂

yes

maximal

#22559827

C₂

yes

maximal, characteristic

#22559806

C₂

yes

maximal, characteristic

C₆₇×#1312

C₂

yes

maximal, characteristic

#16863801

—

#16863801

C₂²

yes

characteristic

#16863801

C₂²

yes

—

#16863794

—

#16863776

—

#16863776

C₂²

yes

—

—

—

C₄×C₁₆×C₆₇

C₂²

yes

—

C₈²×C₆₇

C₂²

yes

characteristic

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

—

—

—

—

—

—

—

—

—

C₂×C₁₆×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

characteristic

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

Dic₂₆₈

—

Dic₂₆₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₁₆×C₆₇

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

SD₃₂

yes

—

C₈×C₆₇

#137

yes

—

C₂²×C₆₇

#493

yes

characteristic

C₄×C₆₇

#497

yes

—

C₄×C₆₇

C₂×SD₃₂

yes

—

C₄×C₆₇

#510

yes

—

Dic₆₇

—

Dic₆₇

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

—

—

—

D₆₇

yes

characteristic

C₆₇

#12535

yes

characteristic, complemented, Sylow

#500

—

#500

—

#500

—

134

—

#498

—

134

—

#494

—

#494

—

134

—

#487

—

134

—

#367

—

134

—

#358

—

134

—

C₄×C₁₆

C₂×D₆₇

yes

—

C₈²

C₂×D₆₇

yes

characteristic

C₂×Q₁₆

—

268

—

C₂×Q₁₆

—

134

—

C₄⋊Q₈

—

134

—

C₄⋊Q₈

—

134

—

C₄⋊Q₈

—

134

—

C₂×C₁₆

D₂₆₈

yes

—

#107

—

268

—

#107

—

134

—

#107

—

134

—

#107

—

134

—

#106

—

268

—

#103

—

268

—

100

C₄×C₈

D₂₆₈

yes

characteristic

101

C₄×C₈

D₂₆₈

yes

characteristic

102

C₄×C₈

C₂²×D₆₇

yes

characteristic

103

C₂×Q₈

—

268

—

104

C₂×Q₈

—

268

—

105

C₂×Q₈

—

268

—

106

Q₁₆

—

268

—

107

Q₁₆

—

536

—

108

C₂×C₈

D₅₃₆

yes

—

109

C₂×C₈

D₅₃₆

yes

—

110

C₂×C₈

C₂×D₂₆₈

yes

—

111

C₄⋊C₄

—

268

—

112

C₄⋊C₄

—

268

—

113

C₄⋊C₄

—

268

—

114

C₄⋊C₄

—

268

—

115

C₄⋊C₄

—

268

—

116

C₄⋊C₄

—

268

—

117

C₄²

C₂×D₂₆₈

yes

characteristic

118

C₁₆

—

119

Q₈

—

536

—

120

Q₈

—

536

—

121

Q₈

—

536

—

122

C₂×C₄

—

536

—

123

C₂×C₄

—

536

—

124

C₂×C₄

—

536

—

125

C₂×C₄

#14524040

yes

characteristic

126

C₂×C₄

C₂×D₅₃₆

yes

characteristic

127

C₂×C₄

C₂×D₅₃₆

yes

characteristic

128

C₈

—

129

C₈

#14523988

yes

—

130

C₈

#14524076

yes

—

131

C₂²

#16863773

yes

characteristic

132

C₄

—

1072

—

133

C₄

—

1072

—

134

C₄

—

1072

—

135

C₄

#16863790

yes

—

136

C₄

C₂×#14523988

yes

—

137

C₄

#16863839

yes

—

138

C₂

#22559880

yes

characteristic

139

C₂

#22562465

yes

characteristic

140

C₂

#22562480

yes

characteristic

	subgroup	G/H
center	C₂²	#16863773
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1312	C₂
socle, prime radical	C₂²×C₆₇	#493
hypercenter	#1312	D₆₇
hypocenter	C₆₇	#12535
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

subgroup

G/H

center

C₂²

#16863773

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#1312

C₂

C₂²×C₆₇

D₆₇

C₆₇

G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1312

N	H ≅ G/N	type
76. C₆₇	#12535	semidirect

H ≅ G/N

type

76. C₆₇

#12535

semidirect

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

subgroup	index
#22562491	2
#22562437	2
#22559827	2
#22559806	2
C₆₇×#1312	2
#12535	67

subgroup

index

C₆₇×#1312

Subgroups — #30770639

Conjugacy classes of subgroups

Complements

Maximal subgroups