Group #30767711 — subgroups

Subgroups — #30767711

Order 17152.

214 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 214 conjugacy classes of subgroups — too many to draw. Only its 93 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	#22559028	C₂	1	2	yes	maximal
2	#22558275	C₂	1	1	yes	maximal, characteristic
3	#22558179	C₂	1	2	yes	maximal
4	#22558129	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#2176	C₂	1	1	yes	maximal, characteristic
6	Q₈×C₆₇⋊C₈	—	2	2	no	—
7	#16863172	C₂²	1	1	yes	characteristic
8	#16863147	C₂²	1	2	yes	—
9	Q₈⋊(C₆₇⋊C₈)	—	2	2	no	—
10	#16863117	C₂²	1	1	yes	characteristic
11	C₂×Q₃₂×C₆₇	C₄	1	1	yes	characteristic
12	C₄×Q₁₆×C₆₇	C₂²	1	2	yes	—
13	#367×C₆₇	C₄	1	1	yes	characteristic
14	#358×C₆₇	C₄	1	2	yes	—
15	#346×C₆₇	C₂²	1	1	yes	characteristic
16	C₂×C₆₇⋊C₁₆	—	2	1	no	—
17	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
18	C₄⋊(C₆₇⋊C₈)	—	4	2	no	—
19	C₄×C₆₇⋊C₈	—	4	2	no	—
20	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
21	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
22	C₄×Q₈×C₆₇	—	2	2	no	—
23	Q₃₂×C₆₇	C₈	1	4	yes	complemented
24	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
25	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
26	#107×C₆₇	C₂×C₄	1	2	yes	—
27	#103×C₆₇	—	2	2	no	—
28	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×C₆₇⋊C₈	—	2	2	no	—
30	C₂×C₆₇⋊C₈	—	4	2	no	—
31	C₆₇⋊C₁₆	—	4	1	no	—
32	C₂×Q₈×C₆₇	—	2	2	no	—
33	Q₁₆×C₆₇	—	2	2	no	—
34	Q₁₆×C₆₇	C₂×C₈	1	4	yes	—
35	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
36	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
37	C₄⋊C₄×C₆₇	—	2	2	no	—
38	C₄⋊C₄×C₆₇	—	4	2	no	—
39	C₄²×C₆₇	—	4	2	no	—
40	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₆₇	—	2	1	no	—
42	C₁₆×C₆₇	C₂×C₈	1	2	yes	—
43	Q₈×C₆₇	—	2	4	no	—
44	Q₈×C₆₇	—	4	2	no	—
45	C₂×C₄×C₆₇	—	4	2	no	—
46	C₂×C₄×C₆₇	—	4	2	no	—
47	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
48	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
49	C₈×C₆₇	—	2	1	no	—
50	C₈×C₆₇	C₂²×C₈	1	1	yes	characteristic
51	C₈×C₆₇	#131	1	1	yes	characteristic
52	C₆₇⋊C₈	—	4	2	no	—
53	C₆₇⋊C₈	—	8	2	no	—
54	C₂²×C₆₇	#442	1	1	yes	characteristic
55	C₄×C₆₇	—	2	2	no	—
56	C₄×C₆₇	—	4	4	no	—
57	C₄×C₆₇	—	8	2	no	—
58	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
59	C₄×C₆₇	#444	1	1	yes	characteristic
60	#12066	—	67	1	no	maximal, Sylow
61	C₂×C₆₇	#1580	1	1	yes	characteristic
62	C₂×C₆₇	#2180	1	2	yes	—
63	#2176	D₆₇	1	1	yes	characteristic
64	#1579	—	67	2	no	—
65	#1368	—	67	1	no	—
66	#1334	—	67	2	no	—
67	#1313	—	67	1	no	—
68	C₆₇	#12066	1	1	yes	characteristic, complemented, Sylow
69	C₂×Q₃₂	Dic₆₇	1	1	yes	characteristic
70	C₈×Q₈	—	134	2	no	—
71	C₄×Q₁₆	C₂×D₆₇	1	2	yes	—
72	#367	Dic₆₇	1	1	yes	characteristic
73	#358	Dic₆₇	1	2	yes	—
74	#346	—	67	1	no	—
75	#346	C₂×D₆₇	1	1	yes	characteristic
76	#334	—	67	2	no	—
77	Q₈⋊C₈	—	134	2	no	—
78	#322	—	67	1	no	—
79	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
80	C₄×Q₈	—	2	2	no	—
81	Q₃₂	C₆₇⋊C₈	1	4	yes	complemented
82	C₂×C₁₆	—	134	1	no	—
83	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
84	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
85	#107	C₂×Dic₆₇	1	2	yes	—
86	C₄⋊C₈	—	268	2	no	—
87	C₄⋊C₈	—	134	2	no	—
88	#103	—	2	2	no	—
89	C₄×C₈	—	268	2	no	—
90	C₄×C₈	—	67	2	no	—
91	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
92	C₂×Q₈	—	2	2	no	—
93	Q₁₆	—	2	2	no	—
94	Q₁₆	C₂×C₆₇⋊C₈	1	4	yes	—
95	C₂×C₈	—	268	2	no	—
96	C₂×C₈	—	134	2	no	—
97	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
98	C₂×C₈	#11761627	1	1	yes	characteristic
99	C₄⋊C₄	—	2	2	no	—
100	C₄⋊C₄	—	4	2	no	—
101	C₄²	—	4	2	no	—
102	C₄²	D₄×D₆₇	1	1	yes	characteristic
103	C₁₆	—	2	1	no	—
104	C₁₆	—	268	1	no	—
105	C₁₆	C₂×C₆₇⋊C₈	1	2	yes	—
106	Q₈	—	2	4	no	—
107	Q₈	—	4	2	no	—
108	C₂×C₄	—	4	2	no	—
109	C₂×C₄	—	4	2	no	—
110	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
111	C₂×C₄	#14524067	1	2	yes	—
112	C₈	—	2	1	no	—
113	C₈	—	268	2	no	—
114	C₈	—	536	2	no	—
115	C₈	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
116	C₈	#14524050	1	1	yes	characteristic
117	C₂²	#16863508	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	4	4	no	—
120	C₄	—	8	2	no	—
121	C₄	D₄×C₆₇⋊C₈	1	1	yes	characteristic
122	C₄	#16863518	1	1	yes	characteristic
123	C₂	#22559033	1	1	yes	characteristic
124	C₂	#22562101	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22559028

C₂

yes

maximal

#22558275

C₂

yes

maximal, characteristic

#22558179

C₂

yes

maximal

#22558129

C₂

yes

maximal, characteristic

C₆₇×#2176

C₂

yes

maximal, characteristic

Q₈×C₆₇⋊C₈

—

#16863172

C₂²

yes

characteristic

#16863147

C₂²

yes

—

Q₈⋊(C₆₇⋊C₈)

—

#16863117

C₂²

yes

characteristic

C₂×Q₃₂×C₆₇

C₄

yes

characteristic

C₄×Q₁₆×C₆₇

C₂²

yes

—

#367×C₆₇

C₄

yes

characteristic

#358×C₆₇

C₄

yes

—

#346×C₆₇

C₂²

yes

characteristic

C₂×C₆₇⋊C₁₆

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₂×Q₁₆×C₆₇

C₂×C₄

yes

—

C₄×Q₈×C₆₇

—

Q₃₂×C₆₇

C₈

yes

complemented

C₂×C₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₇

C₂×C₄

yes

characteristic

#107×C₆₇

C₂×C₄

yes

—

#103×C₆₇

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₆₇⋊C₁₆

—

C₂×Q₈×C₆₇

—

Q₁₆×C₆₇

—

Q₁₆×C₆₇

C₂×C₈

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

#41

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₁₆×C₆₇

—

C₁₆×C₆₇

C₂×C₈

yes

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄×D₄

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₈×C₆₇

—

C₈×C₆₇

C₂²×C₈

yes

characteristic

C₈×C₆₇

#131

yes

characteristic

C₆₇⋊C₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

#442

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₈×D₄

yes

C₄×C₆₇

yes

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

#2176

D₆₇

yes

—

—

—

—

C₆₇

#12066

yes

characteristic, complemented, Sylow

C₂×Q₃₂

Dic₆₇

yes

characteristic

C₈×Q₈

—

134

—

C₄×Q₁₆

C₂×D₆₇

yes

—

#367

Dic₆₇

yes

characteristic

#358

Dic₆₇

yes

—

#346

—

#346

C₂×D₆₇

yes

characteristic

#334

—

Q₈⋊C₈

—

134

—

#322

—

C₂×Q₁₆

C₂×Dic₆₇

yes

—

C₄×Q₈

—

Q₃₂

C₆₇⋊C₈

yes

complemented

C₂×C₁₆

—

134

—

C₂×C₁₆

C₂×Dic₆₇

yes

characteristic

C₂×C₁₆

C₂×Dic₆₇

yes

characteristic

#107

C₂×Dic₆₇

yes

—

C₄⋊C₈

—

268

—

C₄⋊C₈

—

134

—

#103

—

C₄×C₈

—

268

—

C₄×C₈

—

C₄×C₈

C₂²×D₆₇

yes

characteristic

C₂×Q₈

—

Q₁₆

—

Q₁₆

C₂×C₆₇⋊C₈

yes

—

C₂×C₈

—

268

—

C₂×C₈

—

134

—

C₂×C₈

C₂²×Dic₆₇

yes

characteristic

C₂×C₈

#11761627

yes

characteristic

C₄⋊C₄

—

100

C₄⋊C₄

—

101

C₄²

—

102

C₄²

D₄×D₆₇

yes

characteristic

103

C₁₆

—

104

C₁₆

—

268

—

105

C₁₆

C₂×C₆₇⋊C₈

yes

—

106

Q₈

—

107

Q₈

—

108

C₂×C₄

—

109

C₂×C₄

—

110

C₂×C₄

D₄×Dic₆₇

yes

characteristic

111

C₂×C₄

#14524067

yes

—

112

C₈

—

113

C₈

—

268

—

114

C₈

—

536

—

115

C₈

C₂²×C₆₇⋊C₈

yes

characteristic

116

C₈

#14524050

yes

characteristic

117

C₂²

#16863508

yes

characteristic

118

C₄

—

119

C₄

—

120

C₄

—

121

C₄

D₄×C₆₇⋊C₈

yes

characteristic

122

C₄

#16863518

yes

characteristic

123

C₂

#22559033

yes

characteristic

124

C₂

#22562101

yes

—

	subgroup	G/H
center	C₂²	#16863508
commutator	C₈×C₆₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#2176	C₂
socle, prime radical	C₂²×C₆₇	#442
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#2176	D₆₇
hypocenter	C₆₇	#12066
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

subgroup

G/H

center

C₂²

#16863508

commutator

C₈×C₆₇

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#2176

C₂

socle, prime radical

C₂²×C₆₇

#442

cosocle

C₄×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2176

N	H ≅ G/N	type
23. Q₃₂×C₆₇	C₈	semidirect
68. C₆₇	#12066	semidirect
81. Q₃₂	C₆₇⋊C₈	semidirect

H ≅ G/N

type

23. Q₃₂×C₆₇

C₈

semidirect

68. C₆₇

#12066

semidirect

81. Q₃₂

C₆₇⋊C₈

semidirect

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

subgroup	index
#22559028	2
#22558275	2
#22558179	2
#22558129	2
C₆₇×#2176	2
#12066	67

subgroup

index

C₆₇×#2176

Subgroups — #30767711

Conjugacy classes of subgroups

Complements

Maximal subgroups