Group #30770490 — subgroups

Subgroups — #30770490

Order 17152.

262 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 262 conjugacy classes of subgroups — too many to draw. Only its 87 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	#22562462	C₂	1	1	yes	maximal, characteristic
2	#22562438	C₂	1	2	yes	maximal
3	#22559807	C₂	1	2	yes	maximal
4	C₁₆×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2252	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₃₆	—	2	2	no	—
7	#16863792	—	2	2	no	—
8	#16863776	C₂²	1	2	yes	—
9	#16863746	—	2	2	no	—
10	#16863199	—	2	4	no	—
11	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
12	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
13	#500×C₆₇	C₂²	1	2	yes	—
14	#366×C₆₇	—	2	2	no	—
15	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×Dic₂₆₈	—	2	2	no	—
17	C₂×Dic₂₆₈	—	2	2	no	—
18	#14524043	—	2	2	no	—
19	Dic₅₃₆	—	4	4	no	—
20	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
21	#14523974	—	4	4	no	—
22	#14523967	—	4	4	no	—
23	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
24	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
25	C₄⋊Q₈×C₆₇	—	2	2	no	—
26	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
27	#107×C₆₇	—	2	2	no	—
28	#107×C₆₇	—	2	2	no	—
29	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
30	C₂×Dic₁₃₄	—	4	2	no	—
31	Dic₂₆₈	—	4	4	no	—
32	Dic₂₆₈	—	4	4	no	—
33	C₂×C₆₇⋊C₈	D₈	1	2	yes	—
34	C₂×C₆₇⋊C₈	D₈	1	2	yes	—
35	C₄⋊Dic₆₇	—	4	2	no	—
36	C₄⋊Dic₆₇	—	4	2	no	—
37	C₆₇⋊C₁₆	—	2	2	no	—
38	C₂×Q₈×C₆₇	—	4	2	no	—
39	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₄⋊C₄×C₆₇	—	4	2	no	—
42	C₄⋊C₄×C₆₇	—	4	2	no	—
43	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₁₆×C₆₇	SD₁₆	1	4	yes	—
45	Q₈×C₆₇	—	8	4	no	—
46	C₂×C₄×C₆₇	—	8	2	no	—
47	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
48	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
50	C₂×Dic₆₇	—	8	2	no	—
51	Dic₁₃₄	—	8	2	no	—
52	Dic₁₃₄	—	8	2	no	—
53	C₈×C₆₇	C₂×SD₁₆	1	2	yes	—
54	C₈×C₆₇	#135	1	2	yes	—
55	C₆₇⋊C₈	—	2	2	no	—
56	C₆₇⋊C₈	Q₃₂	1	4	yes	complemented
57	C₂²×C₆₇	#493	1	1	yes	characteristic
58	C₄×C₆₇	—	16	2	no	—
59	C₄×C₆₇	#492	1	1	yes	characteristic
60	C₄×C₆₇	#495	1	1	yes	characteristic
61	C₄×C₆₇	C₂×Q₃₂	1	2	yes	—
62	C₄×C₆₇	#508	1	2	yes	—
63	Dic₆₇	—	16	2	no	—
64	#12504	—	67	1	no	maximal, Sylow
65	C₂×C₆₇	#1706	1	1	yes	characteristic
66	C₂×C₆₇	#2247	1	1	yes	characteristic
67	C₂×C₆₇	#2249	1	1	yes	characteristic
68	#2252	D₆₇	1	1	yes	characteristic
69	#2247	—	67	1	no	—
70	#2241	—	67	2	no	—
71	#1707	—	67	2	no	—
72	C₈×C₁₆	—	67	1	no	—
73	C₆₇	#12504	1	1	yes	characteristic, complemented, Sylow
74	C₂×Q₃₂	—	134	2	no	—
75	#500	C₂×D₆₇	1	2	yes	—
76	#498	—	134	2	no	—
77	#494	—	67	2	no	—
78	#487	—	134	2	no	—
79	#366	—	2	2	no	—
80	#358	—	134	4	no	—
81	C₄×C₁₆	—	67	1	no	—
82	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
83	C₈²	—	67	1	no	—
84	C₂×Q₁₆	—	134	2	no	—
85	C₂×Q₁₆	—	134	2	no	—
86	C₄⋊Q₈	—	2	2	no	—
87	C₄⋊Q₈	—	134	2	no	—
88	Q₃₂	—	268	4	no	—
89	C₂×C₁₆	—	67	2	no	—
90	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
91	#107	—	2	2	no	—
92	#107	—	2	2	no	—
93	#106	—	268	4	no	—
94	#103	—	268	4	no	—
95	C₄×C₈	—	67	1	no	—
96	C₄×C₈	—	67	1	no	—
97	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
98	C₂×Q₈	—	4	2	no	—
99	C₂×Q₈	—	268	2	no	—
100	Q₁₆	—	268	4	no	—
101	Q₁₆	—	268	4	no	—
102	C₂×C₈	—	67	2	no	—
103	C₂×C₈	—	67	2	no	—
104	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
105	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
106	C₄⋊C₄	—	4	2	no	—
107	C₄⋊C₄	—	4	2	no	—
108	C₄⋊C₄	—	268	2	no	—
109	C₄⋊C₄	—	268	2	no	—
110	C₄²	D₄×D₆₇	1	1	yes	characteristic
111	C₁₆	—	134	2	no	—
112	C₁₆	#11761615	1	4	yes	—
113	Q₈	—	8	4	no	—
114	Q₈	—	536	2	no	—
115	Q₈	—	536	2	no	—
116	C₂×C₄	—	8	2	no	—
117	C₂×C₄	—	536	2	no	—
118	C₂×C₄	#14524039	1	1	yes	characteristic
119	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
120	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
121	C₈	—	67	4	no	—
122	C₈	—	134	2	no	—
123	C₈	C₂×#11761615	1	2	yes	—
124	C₈	#14524066	1	2	yes	—
125	C₂²	#16863771	1	1	yes	characteristic
126	C₄	—	16	2	no	—
127	C₄	—	1072	2	no	—
128	C₄	#16863770	1	1	yes	characteristic
129	C₄	#16863781	1	1	yes	characteristic
130	C₄	Q₃₂×D₆₇	1	2	yes	—
131	C₄	#16863825	1	2	yes	—
132	C₂	#22559802	1	1	yes	characteristic
133	C₂	#22562460	1	1	yes	characteristic
134	C₂	#22562467	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562462

C₂

yes

maximal, characteristic

#22562438

C₂

yes

maximal

#22559807

C₂

yes

maximal

C₁₆×C₆₇⋊C₈

C₂

yes

maximal, characteristic

C₆₇×#2252

C₂

yes

maximal, characteristic

C₂×Dic₅₃₆

—

#16863792

—

#16863776

C₂²

yes

—

#16863746

—

#16863199

—

C₄×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₈×C₆₇⋊C₈

C₂²

yes

characteristic

#500×C₆₇

C₂²

yes

—

#366×C₆₇

—

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

#14524043

—

Dic₅₃₆

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

#14523974

—

#14523967

—

C₄×C₆₇⋊C₈

D₄

yes

characteristic

C₄×C₆₇⋊C₈

D₄

yes

characteristic

C₄⋊Q₈×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

—

#107×C₆₇

—

#107×C₆₇

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

—

Dic₂₆₈

—

Dic₂₆₈

—

C₂×C₆₇⋊C₈

D₈

yes

—

C₂×C₆₇⋊C₈

D₈

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₆₇⋊C₁₆

—

C₂×Q₈×C₆₇

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₁₆×C₆₇

SD₁₆

yes

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

C₂×SD₁₆

yes

—

C₈×C₆₇

#135

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

Q₃₂

yes

complemented

C₂²×C₆₇

#493

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

C₂×Q₃₂

yes

—

C₄×C₆₇

#508

yes

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

yes

—

—

—

C₈×C₁₆

—

C₆₇

#12504

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

134

—

#500

C₂×D₆₇

yes

—

#498

—

134

—

#494

—

#487

—

134

—

#366

—

#358

—

134

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₇

yes

characteristic

C₈²

—

C₂×Q₁₆

—

134

—

C₂×Q₁₆

—

134

—

C₄⋊Q₈

—

C₄⋊Q₈

—

134

—

Q₃₂

—

268

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₆₇

yes

—

#107

—

#107

—

#106

—

268

—

#103

—

268

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

268

—

100

Q₁₆

—

268

—

101

Q₁₆

—

268

—

102

C₂×C₈

—

103

C₂×C₈

—

104

C₂×C₈

D₄×D₆₇

yes

characteristic

105

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

106

C₄⋊C₄

—

107

C₄⋊C₄

—

108

C₄⋊C₄

—

268

—

109

C₄⋊C₄

—

268

—

110

C₄²

D₄×D₆₇

yes

characteristic

111

C₁₆

—

134

—

112

C₁₆

#11761615

yes

—

113

Q₈

—

114

Q₈

—

536

—

115

Q₈

—

536

—

116

C₂×C₄

—

117

C₂×C₄

—

536

—

118

C₂×C₄

#14524039

yes

characteristic

119

C₂×C₄

D₈×D₆₇

yes

characteristic

120

C₂×C₄

D₈×D₆₇

yes

characteristic

121

C₈

—

122

C₈

—

134

—

123

C₈

C₂×#11761615

yes

—

124

C₈

#14524066

yes

—

125

C₂²

#16863771

yes

characteristic

126

C₄

—

127

C₄

—

1072

—

128

C₄

#16863770

yes

characteristic

129

C₄

#16863781

yes

characteristic

130

C₄

Q₃₂×D₆₇

yes

—

131

C₄

#16863825

yes

—

132

C₂

#22559802

yes

characteristic

133

C₂

#22562460

yes

characteristic

134

C₂

#22562467

yes

characteristic

	subgroup	G/H
center	C₂²	#16863771
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#2252	C₂
socle, prime radical	C₂²×C₆₇	#493
hypercenter	#2252	D₆₇
hypocenter	C₆₇	#12504
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

subgroup

G/H

center

C₂²

#16863771

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#2252

C₂

C₂²×C₆₇

D₆₇

C₆₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2252

N	H ≅ G/N	type
56. C₆₇⋊C₈	Q₃₂	semidirect
73. C₆₇	#12504	semidirect

H ≅ G/N

type

56. C₆₇⋊C₈

Q₃₂

semidirect

73. C₆₇

#12504

semidirect

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

subgroup	index
#22562462	2
#22562438	2
#22559807	2
C₁₆×C₆₇⋊C₈	2
C₆₇×#2252	2
#12504	67

subgroup

index

#22562462

#22562438

#22559807

C₁₆×C₆₇⋊C₈

C₆₇×#2252

#12504

Subgroups — #30770490

Conjugacy classes of subgroups

Complements

Maximal subgroups