Group #30767850 — subgroups

Subgroups — #30767850

Order 17152.

198 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 198 conjugacy classes of subgroups — too many to draw. Only its 69 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	#22562081	C₂	1	1	yes	maximal, characteristic
2	#22559117	C₂	1	2	yes	maximal
3	#22558178	C₂	1	2	yes	maximal
4	#22558129	C₂	1	1	yes	maximal, characteristic
5	C₆₇×#1368	C₂	1	1	yes	maximal, characteristic
6	C₂×Dic₅₃₆	C₄	1	1	yes	characteristic
7	#16863523	—	2	2	no	—
8	C₄×Dic₂₆₈	C₂²	1	2	yes	—
9	#16863224	C₄	1	1	yes	characteristic
10	#16863201	C₄	1	2	yes	—
11	#16863172	C₂²	1	1	yes	characteristic
12	#16863126	—	2	2	no	—
13	#16863117	C₂²	1	1	yes	characteristic
14	#346×C₆₇	C₂²	1	1	yes	characteristic
15	#334×C₆₇	C₂²	1	2	yes	—
16	C₂×Dic₂₆₈	C₂×C₄	1	2	yes	—
17	C₄×Dic₁₃₄	—	2	2	no	—
18	Dic₅₃₆	—	2	2	no	—
19	C₂×C₆₇⋊C₁₆	—	2	1	no	—
20	#14523977	C₂×C₄	1	2	yes	—
21	Dic₆₇⋊C₈	—	4	2	no	—
22	#14523968	—	2	2	no	—
23	#14523953	—	4	2	no	—
24	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
25	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
26	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₄⋊C₈×C₆₇	—	2	2	no	—
28	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×Dic₁₃₄	—	2	2	no	—
30	Dic₂₆₈	—	2	2	no	—
31	Dic₂₆₈	—	2	2	no	—
32	C₂×C₆₇⋊C₈	—	2	2	no	—
33	C₄⋊Dic₆₇	—	2	2	no	—
34	Dic₆₇⋊C₄	—	4	2	no	—
35	C₄×Dic₆₇	—	4	2	no	—
36	C₆₇⋊C₁₆	—	4	1	no	—
37	C₂×C₈×C₆₇	—	4	2	no	—
38	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
39	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
40	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
41	C₁₆×C₆₇	—	2	1	no	—
42	C₁₆×C₆₇	M₁₆	1	2	yes	—
43	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
44	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
45	C₂×Dic₆₇	—	4	2	no	—
46	C₂×Dic₆₇	—	4	2	no	—
47	Dic₁₃₄	—	4	2	no	—
48	Dic₁₃₄	—	4	2	no	—
49	C₈×C₆₇	—	2	1	no	—
50	C₈×C₆₇	—	8	2	no	—
51	C₈×C₆₇	C₂×M₁₆	1	1	yes	characteristic
52	C₈×C₆₇	#131	1	1	yes	characteristic
53	C₆₇⋊C₈	—	4	2	no	—
54	C₂²×C₆₇	#442	1	1	yes	characteristic
55	C₄×C₆₇	—	2	2	no	—
56	C₄×C₆₇	#436	1	1	yes	characteristic
57	C₄×C₆₇	#444	1	1	yes	characteristic
58	Dic₆₇	—	8	2	no	—
59	Dic₆₇	—	8	2	no	—
60	#12088	—	67	1	no	maximal, Sylow
61	C₂×C₆₇	#1594	1	1	yes	characteristic
62	C₂×C₆₇	#2180	1	2	yes	—
63	#2176	—	67	1	no	—
64	#1593	—	67	2	no	—
65	#1368	D₆₇	1	1	yes	characteristic
66	#1334	—	67	2	no	—
67	#1313	—	67	1	no	—
68	C₆₇	#12088	1	1	yes	characteristic, complemented, Sylow
69	C₂×Q₃₂	—	67	1	no	—
70	#446	—	134	2	no	—
71	C₄×Q₁₆	—	67	2	no	—
72	#367	—	67	1	no	—
73	#358	—	67	2	no	—
74	#346	—	67	1	no	—
75	#346	C₂×D₆₇	1	1	yes	characteristic
76	#334	C₂×D₆₇	1	2	yes	—
77	Q₈⋊C₈	—	134	2	no	—
78	#322	—	67	1	no	—
79	C₂×Q₁₆	—	67	2	no	—
80	C₄×Q₈	—	134	2	no	—
81	Q₃₂	—	134	2	no	—
82	C₂×C₁₆	—	134	1	no	—
83	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
84	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
85	#107	—	67	2	no	—
86	C₄⋊C₈	—	2	2	no	—
87	C₄⋊C₈	—	268	2	no	—
88	#103	—	134	2	no	—
89	#97	—	268	2	no	—
90	C₄×C₈	—	67	2	no	—
91	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
92	C₂×Q₈	—	134	2	no	—
93	Q₁₆	—	134	2	no	—
94	Q₁₆	—	134	2	no	—
95	C₂×C₈	—	4	2	no	—
96	C₂×C₈	—	134	2	no	—
97	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
98	C₂×C₈	#11761629	1	1	yes	characteristic
99	C₄⋊C₄	—	268	2	no	—
100	C₄⋊C₄	—	134	2	no	—
101	C₄²	—	268	2	no	—
102	C₄²	D₄×D₆₇	1	1	yes	characteristic
103	C₁₆	—	2	1	no	—
104	C₁₆	—	268	1	no	—
105	C₁₆	#11761611	1	2	yes	—
106	Q₈	—	268	2	no	—
107	Q₈	—	268	2	no	—
108	C₂×C₄	—	268	2	no	—
109	C₂×C₄	—	268	2	no	—
110	C₂×C₄	#14524005	1	1	yes	characteristic
111	C₂×C₄	#14524070	1	2	yes	—
112	C₈	—	2	1	no	—
113	C₈	—	8	2	no	—
114	C₈	—	268	2	no	—
115	C₈	C₂×#11761611	1	1	yes	characteristic
116	C₈	#14524051	1	1	yes	characteristic
117	C₂²	#16863507	1	1	yes	characteristic
118	C₄	—	2	2	no	—
119	C₄	—	536	2	no	—
120	C₄	—	536	2	no	—
121	C₄	#16863473	1	1	yes	characteristic
122	C₄	#16863517	1	1	yes	characteristic
123	C₂	#22559122	1	1	yes	characteristic
124	C₂	#22562099	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22562081

C₂

yes

maximal, characteristic

#22559117

C₂

yes

maximal

#22558178

C₂

yes

maximal

#22558129

C₂

yes

maximal, characteristic

C₆₇×#1368

C₂

yes

maximal, characteristic

C₂×Dic₅₃₆

C₄

yes

characteristic

#16863523

—

C₄×Dic₂₆₈

C₂²

yes

—

#16863224

C₄

yes

characteristic

#16863201

C₄

yes

—

#16863172

C₂²

yes

characteristic

#16863126

—

#16863117

C₂²

yes

characteristic

#346×C₆₇

C₂²

yes

characteristic

#334×C₆₇

C₂²

yes

—

C₂×Dic₂₆₈

C₂×C₄

yes

—

C₄×Dic₁₃₄

—

Dic₅₃₆

—

C₂×C₆₇⋊C₁₆

—

#14523977

C₂×C₄

yes

—

Dic₆₇⋊C₈

—

#14523968

—

#14523953

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₇

C₂×C₄

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×Dic₁₃₄

—

Dic₂₆₈

—

Dic₂₆₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

#41

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₁₆×C₆₇

—

C₁₆×C₆₇

M₁₆

yes

—

C₂×C₄×C₆₇

C₄×D₄

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₂×M₁₆

yes

characteristic

C₈×C₆₇

#131

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

#442

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

Dic₆₇

—

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

#2176

—

#1593

—

#1368

D₆₇

yes

characteristic

#1334

—

#1313

—

C₆₇

#12088

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

#446

—

134

—

C₄×Q₁₆

—

—

—

—

C₂×D₆₇

yes

characteristic

#334

C₂×D₆₇

yes

—

Q₈⋊C₈

—

134

—

#322

—

C₂×Q₁₆

—

C₄×Q₈

—

134

—

Q₃₂

—

134

—

C₂×C₁₆

—

134

—

C₂×C₁₆

C₄×D₆₇

yes

characteristic

C₂×C₁₆

C₄×D₆₇

yes

characteristic

#107

—

C₄⋊C₈

—

C₄⋊C₈

—

268

—

#103

—

134

—

#97

—

268

—

C₄×C₈

—

C₄×C₈

C₂²×D₆₇

yes

characteristic

C₂×Q₈

—

134

—

Q₁₆

—

134

—

Q₁₆

—

134

—

C₂×C₈

—

C₂×C₈

—

134

—

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

C₂×C₈

#11761629

yes

characteristic

C₄⋊C₄

—

268

—

100

C₄⋊C₄

—

134

—

101

C₄²

—

268

—

102

C₄²

D₄×D₆₇

yes

characteristic

103

C₁₆

—

104

C₁₆

—

268

—

105

C₁₆

#11761611

yes

—

106

Q₈

—

268

—

107

Q₈

—

268

—

108

C₂×C₄

—

268

—

109

C₂×C₄

—

268

—

110

C₂×C₄

#14524005

yes

characteristic

111

C₂×C₄

#14524070

yes

—

112

C₈

—

113

C₈

—

114

C₈

—

268

—

115

C₈

C₂×#11761611

yes

characteristic

116

C₈

#14524051

yes

characteristic

117

C₂²

#16863507

yes

characteristic

118

C₄

—

119

C₄

—

536

—

120

C₄

—

536

—

121

C₄

#16863473

yes

characteristic

122

C₄

#16863517

yes

characteristic

123

C₂

#22559122

yes

characteristic

124

C₂

#22562099

yes

—

	subgroup	G/H
center	C₂²	#16863507
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1368	C₂
socle, prime radical	C₂²×C₆₇	#442
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#1368	D₆₇
hypocenter	C₆₇	#12088
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

subgroup

G/H

center

C₂²

#16863507

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#1368

C₂

socle, prime radical

C₂²×C₆₇

#442

cosocle

C₄×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #1368

N	H ≅ G/N	type
68. C₆₇	#12088	semidirect

H ≅ G/N

type

68. C₆₇

#12088

semidirect

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

subgroup	index
#22562081	2
#22559117	2
#22558178	2
#22558129	2
C₆₇×#1368	2
#12088	67

subgroup

index

C₆₇×#1368

Subgroups — #30767850

Conjugacy classes of subgroups

Complements

Maximal subgroups