Group #30770570 — subgroups

Subgroups — #30770570

Order 17152.

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

normal subgroups only

This group has 238 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	#22562489	C₂	1	2	yes	maximal
2	#22562484	C₂	1	1	yes	maximal, characteristic
3	#22559818	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	#16863801	C₂²	1	2	yes	—
7	#16863748	—	2	2	no	—
8	#16863746	—	2	2	no	—
9	#16863223	—	2	4	no	—
10	#16863222	—	2	2	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	#14524043	—	2	2	no	—
17	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
18	#14523977	—	2	2	no	—
19	#14523977	—	2	2	no	—
20	#14523967	—	4	4	no	—
21	#14523966	—	4	4	no	—
22	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
23	C₄×C₆₇⋊C₈	D₄	1	1	yes	characteristic
24	C₄⋊Q₈×C₆₇	—	2	2	no	—
25	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
26	#107×C₆₇	—	2	2	no	—
27	#107×C₆₇	—	2	2	no	—
28	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₃₄	—	4	2	no	—
30	C₂×C₆₇⋊C₈	D₈	1	2	yes	—
31	C₂×C₆₇⋊C₈	D₈	1	2	yes	—
32	C₄⋊Dic₆₇	—	4	2	no	—
33	C₄⋊Dic₆₇	—	4	2	no	—
34	C₆₇⋊C₁₆	Q₁₆	1	4	yes	—
35	C₂×Q₈×C₆₇	—	4	2	no	—
36	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
37	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
38	C₄⋊C₄×C₆₇	—	4	2	no	—
39	C₄⋊C₄×C₆₇	—	4	2	no	—
40	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₆₇	Q₁₆	1	4	yes	—
42	Q₈×C₆₇	—	8	4	no	—
43	C₂×C₄×C₆₇	—	8	2	no	—
44	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
45	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
46	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
47	C₂×Dic₆₇	—	8	2	no	—
48	Dic₁₃₄	—	8	4	no	—
49	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
50	C₈×C₆₇	C₂×Q₁₆	1	2	yes	—
51	C₆₇⋊C₈	—	2	2	no	—
52	C₆₇⋊C₈	—	2	2	no	—
53	C₂²×C₆₇	#493	1	1	yes	characteristic
54	C₄×C₆₇	—	16	2	no	—
55	C₄×C₆₇	#494	1	1	yes	characteristic
56	C₄×C₆₇	#494	1	1	yes	characteristic
57	C₄×C₆₇	#508	1	2	yes	—
58	C₄×C₆₇	#508	1	2	yes	—
59	Dic₆₇	—	16	2	no	—
60	#12522	—	67	1	no	maximal, Sylow
61	C₂×C₆₇	#1713	1	1	yes	characteristic
62	C₂×C₆₇	#2249	1	1	yes	characteristic
63	C₂×C₆₇	#2249	1	1	yes	characteristic
64	#2254	—	67	2	no	—
65	#2252	—	67	1	no	—
66	#2252	D₆₇	1	1	yes	characteristic
67	#1710	—	67	2	no	—
68	C₈×C₁₆	—	67	1	no	—
69	C₆₇	#12522	1	1	yes	characteristic, complemented, Sylow
70	#500	—	67	2	no	—
71	#500	C₂×D₆₇	1	2	yes	—
72	#487	—	134	2	no	—
73	#487	—	134	2	no	—
74	#367	—	134	4	no	—
75	#366	—	2	2	no	—
76	#366	—	134	2	no	—
77	C₄×C₁₆	—	67	1	no	—
78	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
79	C₈²	—	67	1	no	—
80	C₄⋊Q₈	—	2	2	no	—
81	C₄⋊Q₈	—	134	2	no	—
82	C₂×C₁₆	—	67	2	no	—
83	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
84	#107	—	2	2	no	—
85	#107	—	2	2	no	—
86	#107	—	134	2	no	—
87	#107	—	134	2	no	—
88	#103	—	268	4	no	—
89	#103	—	268	4	no	—
90	C₄×C₈	—	67	1	no	—
91	C₄×C₈	—	67	1	no	—
92	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
93	C₂×Q₈	—	4	2	no	—
94	C₂×Q₈	—	268	2	no	—
95	C₂×C₈	—	67	2	no	—
96	C₂×C₈	—	67	2	no	—
97	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
98	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
99	C₄⋊C₄	—	4	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄⋊C₄	—	268	2	no	—
102	C₄⋊C₄	—	268	2	no	—
103	C₄²	D₄×D₆₇	1	1	yes	characteristic
104	C₁₆	—	67	4	no	—
105	C₁₆	#11761618	1	4	yes	—
106	Q₈	—	8	4	no	—
107	Q₈	—	536	4	no	—
108	C₂×C₄	—	8	2	no	—
109	C₂×C₄	—	536	2	no	—
110	C₂×C₄	#14524039	1	1	yes	characteristic
111	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
112	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
113	C₈	—	134	2	no	—
114	C₈	—	134	2	no	—
115	C₈	Q₁₆×D₆₇	1	2	yes	—
116	C₈	C₂×#11761618	1	2	yes	—
117	C₂²	#16863771	1	1	yes	characteristic
118	C₄	—	16	2	no	—
119	C₄	—	1072	2	no	—
120	C₄	#16863774	1	1	yes	characteristic
121	C₄	#16863775	1	1	yes	characteristic
122	C₄	#16863825	1	2	yes	—
123	C₄	#16863829	1	2	yes	—
124	C₂	#22559826	1	1	yes	characteristic
125	C₂	#22562467	1	1	yes	characteristic
126	C₂	#22562471	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562489

C₂

yes

maximal

#22562484

C₂

yes

maximal, characteristic

#22559818

C₂

yes

maximal

C₁₆×C₆₇⋊C₈

C₂

yes

maximal, characteristic

C₆₇×#2252

C₂

yes

maximal, characteristic

#16863801

C₂²

yes

—

—

—

—

—

C₄×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₈×C₆₇⋊C₈

C₂²

yes

characteristic

#500×C₆₇

C₂²

yes

—

#366×C₆₇

—

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

#14524043

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

—

—

—

—

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

—

C₂×C₆₇⋊C₈

D₈

yes

—

C₂×C₆₇⋊C₈

D₈

yes

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₆₇⋊C₁₆

Q₁₆

yes

—

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

Q₁₆

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

—

C₈×C₆₇

C₂×Q₁₆

yes

—

C₈×C₆₇

C₂×Q₁₆

yes

—

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

#493

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

C₄×C₆₇

#508

yes

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

—

D₆₇

yes

characteristic

#1710

—

C₈×C₁₆

—

C₆₇

#12522

yes

characteristic, complemented, Sylow

#500

—

#500

C₂×D₆₇

yes

—

#487

—

134

—

#487

—

134

—

#367

—

134

—

#366

—

#366

—

134

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₇

yes

characteristic

C₈²

—

C₄⋊Q₈

—

C₄⋊Q₈

—

134

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₆₇

yes

—

#107

—

#107

—

#107

—

134

—

#107

—

134

—

#103

—

268

—

#103

—

268

—

C₄×C₈

—

C₄×C₈

—

C₄×C₈

C₂²×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×Q₈

—

268

—

C₂×C₈

—

C₂×C₈

—

C₂×C₈

D₄×D₆₇

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄⋊C₄

—

268

—

102

C₄⋊C₄

—

268

—

103

C₄²

D₄×D₆₇

yes

characteristic

104

C₁₆

—

105

C₁₆

#11761618

yes

—

106

Q₈

—

107

Q₈

—

536

—

108

C₂×C₄

—

109

C₂×C₄

—

536

—

110

C₂×C₄

#14524039

yes

characteristic

111

C₂×C₄

D₈×D₆₇

yes

characteristic

112

C₂×C₄

D₈×D₆₇

yes

characteristic

113

C₈

—

134

—

114

C₈

—

134

—

115

C₈

Q₁₆×D₆₇

yes

—

116

C₈

C₂×#11761618

yes

—

117

C₂²

#16863771

yes

characteristic

118

C₄

—

119

C₄

—

1072

—

120

C₄

#16863774

yes

characteristic

121

C₄

#16863775

yes

characteristic

122

C₄

#16863825

yes

—

123

C₄

#16863829

yes

—

124

C₂

#22559826

yes

characteristic

125

C₂

#22562467

yes

characteristic

126

C₂

#22562471

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	#12522
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
69. C₆₇	#12522	semidirect

H ≅ G/N

type

69. C₆₇

#12522

semidirect

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

subgroup	index
#22562489	2
#22562484	2
#22559818	2
C₁₆×C₆₇⋊C₈	2
C₆₇×#2252	2
#12522	67

subgroup

index

#22562489

#22562484

#22559818

C₁₆×C₆₇⋊C₈

C₆₇×#2252

#12522

Subgroups — #30770570

Conjugacy classes of subgroups

Complements

Maximal subgroups