Group #30776803 — subgroups

Subgroups — #30776803

Order 17152.

182 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 182 conjugacy classes of subgroups — too many to draw. Only its 75 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	#22562484	C₂	1	1	yes	maximal, characteristic
2	#22558390	C₂	1	4	yes	maximal
3	C₄×C₆₇⋊C₃₂	C₂	1	1	yes	maximal, characteristic
4	C₆₇×#2247	C₂	1	1	yes	maximal, characteristic
5	#16863801	—	2	1	no	—
6	C₂×C₆₇⋊C₃₂	C₂²	1	2	yes	—
7	#16863222	—	2	1	no	—
8	#16863222	C₂²	1	2	yes	—
9	C₂×Q₃₂×C₆₇	—	2	1	no	—
10	C₂×Q₃₂×C₆₇	C₂²	1	2	yes	—
11	#494×C₆₇	—	2	1	no	—
12	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
13	#14524043	—	4	1	no	—
14	#14523977	—	2	2	no	—
15	#14523977	—	4	1	no	—
16	C₆₇⋊C₃₂	—	2	2	no	—
17	C₂×Q₁₆×C₆₇	—	2	2	no	—
18	C₂×Q₁₆×C₆₇	—	4	1	no	—
19	C₄⋊Q₈×C₆₇	—	4	1	no	—
20	Q₃₂×C₆₇	—	2	4	no	—
21	Q₃₂×C₆₇	—	4	2	no	—
22	C₂×C₁₆×C₆₇	D₄	1	1	yes	characteristic
23	C₂×C₁₆×C₆₇	C₂³	1	1	yes	characteristic
24	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
25	C₂×Dic₁₃₄	—	8	1	no	—
26	C₄⋊Dic₆₇	—	4	2	no	—
27	C₄⋊Dic₆₇	—	8	1	no	—
28	C₂×Q₈×C₆₇	—	4	2	no	—
29	Q₁₆×C₆₇	—	4	2	no	—
30	Q₁₆×C₆₇	—	4	2	no	—
31	Q₁₆×C₆₇	—	8	2	no	—
32	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
33	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
34	C₄⋊C₄×C₆₇	—	8	1	no	—
35	C₄⋊C₄×C₆₇	—	8	1	no	—
36	C₄²×C₆₇	D₈	1	1	yes	characteristic
37	C₁₆×C₆₇	SD₁₆	1	2	yes	—
38	C₁₆×C₆₇	#41	1	2	yes	—
39	Q₈×C₆₇	—	8	2	no	—
40	Q₈×C₆₇	—	8	2	no	—
41	C₂×C₄×C₆₇	—	8	2	no	—
42	C₂×C₄×C₆₇	D₁₆	1	1	yes	characteristic
43	C₂×C₄×C₆₇	D₁₆	1	1	yes	characteristic
44	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
45	C₂×Dic₆₇	—	8	2	no	—
46	Dic₁₃₄	—	16	2	no	—
47	C₈×C₆₇	SD₃₂	1	2	yes	—
48	C₈×C₆₇	#124	1	1	yes	characteristic
49	C₈×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
50	C₂²×C₆₇	C₂×D₁₆	1	1	yes	characteristic
51	C₄×C₆₇	—	16	2	no	—
52	C₄×C₆₇	SD₆₄	1	2	yes	—
53	C₄×C₆₇	Q₆₄	1	2	yes	—
54	C₄×C₆₇	#486	1	1	yes	characteristic
55	C₄×C₆₇	C₂×SD₃₂	1	1	yes	characteristic
56	Dic₆₇	—	16	2	no	—
57	#13720	—	67	1	no	maximal, Sylow
58	C₂×C₆₇	#2240	1	1	yes	characteristic
59	C₂×C₆₇	C₂×SD₆₄	1	1	yes	characteristic
60	C₂×C₆₇	C₂×Q₆₄	1	1	yes	characteristic
61	#2252	—	67	1	no	—
62	#2247	D₆₇	1	1	yes	characteristic
63	#1416	—	67	4	no	—
64	C₄×C₃₂	—	67	1	no	—
65	C₆₇	#13720	1	1	yes	characteristic, complemented, Sylow
66	C₂×Q₃₂	—	2	1	no	—
67	C₂×Q₃₂	C₂×D₆₇	1	2	yes	—
68	#500	—	134	1	no	—
69	#494	—	2	1	no	—
70	C₂×C₃₂	—	67	2	no	—
71	#366	—	67	2	no	—
72	#366	—	134	1	no	—
73	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
74	C₂×Q₁₆	—	2	2	no	—
75	C₂×Q₁₆	—	4	1	no	—
76	C₄⋊Q₈	—	4	1	no	—
77	C₄⋊Q₈	—	268	1	no	—
78	Q₃₂	—	2	4	no	—
79	Q₃₂	—	4	2	no	—
80	C₂×C₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
81	C₂×C₁₆	C₂²×D₆₇	1	1	yes	characteristic
82	#107	—	268	1	no	—
83	#107	—	134	2	no	—
84	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
85	C₃₂	—	134	2	no	—
86	C₂×Q₈	—	4	2	no	—
87	C₂×Q₈	—	536	1	no	—
88	Q₁₆	—	4	2	no	—
89	Q₁₆	—	4	2	no	—
90	Q₁₆	—	8	2	no	—
91	C₂×C₈	#11761613	1	1	yes	characteristic
92	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
93	C₄⋊C₄	—	8	1	no	—
94	C₄⋊C₄	—	8	1	no	—
95	C₄⋊C₄	—	268	2	no	—
96	C₄⋊C₄	—	536	1	no	—
97	C₄²	#11761613	1	1	yes	characteristic
98	C₁₆	#11761616	1	2	yes	—
99	C₁₆	#11761627	1	2	yes	—
100	Q₈	—	8	2	no	—
101	Q₈	—	8	2	no	—
102	Q₈	—	1072	2	no	—
103	C₂×C₄	—	8	2	no	—
104	C₂×C₄	—	536	2	no	—
105	C₂×C₄	#14523984	1	1	yes	characteristic
106	C₂×C₄	#14523984	1	1	yes	characteristic
107	C₂×C₄	C₂×#11761613	1	1	yes	characteristic
108	C₈	#14523987	1	2	yes	—
109	C₈	#14524030	1	1	yes	characteristic
110	C₈	C₂×#11761616	1	1	yes	characteristic
111	C₂²	C₂×#14523984	1	1	yes	characteristic
112	C₄	—	16	2	no	—
113	C₄	—	1072	2	no	—
114	C₄	#16863233	1	2	yes	—
115	C₄	#16863236	1	2	yes	—
116	C₄	#16863742	1	1	yes	characteristic
117	C₄	C₂×#14523987	1	1	yes	characteristic
118	C₂	#22562432	1	1	yes	characteristic
119	C₂	C₂×#16863233	1	1	yes	characteristic
120	C₂	C₂×#16863236	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562484

C₂

yes

maximal, characteristic

#22558390

C₂

yes

maximal

C₄×C₆₇⋊C₃₂

C₂

yes

maximal, characteristic

C₆₇×#2247

C₂

yes

maximal, characteristic

#16863801

—

C₂×C₆₇⋊C₃₂

C₂²

yes

—

#16863222

—

#16863222

C₂²

yes

—

C₂×Q₃₂×C₆₇

—

C₂×Q₃₂×C₆₇

C₂²

yes

—

#494×C₆₇

—

C₄×C₁₆×C₆₇

C₂²

yes

—

—

—

C₆₇⋊C₃₂

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₄⋊Q₈×C₆₇

—

Q₃₂×C₆₇

—

Q₃₂×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

characteristic

C₂×C₁₆×C₆₇

C₂³

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

characteristic

C₂×Dic₁₃₄

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂×Q₈×C₆₇

—

Q₁₆×C₆₇

—

Q₁₆×C₆₇

—

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

D₈

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

D₈

yes

characteristic

C₁₆×C₆₇

SD₁₆

yes

—

C₁₆×C₆₇

#41

yes

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

D₁₆

yes

characteristic

C₂×C₄×C₆₇

D₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

SD₃₂

yes

—

C₈×C₆₇

#124

yes

characteristic

C₈×C₆₇

C₂×SD₁₆

yes

characteristic

C₂²×C₆₇

C₂×D₁₆

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

SD₆₄

yes

—

C₄×C₆₇

Q₆₄

yes

—

C₄×C₆₇

#486

yes

characteristic

C₄×C₆₇

C₂×SD₃₂

yes

characteristic

Dic₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

C₂×SD₆₄

yes

characteristic

C₂×C₆₇

C₂×Q₆₄

yes

characteristic

#2252

—

#2247

D₆₇

yes

characteristic

#1416

—

C₄×C₃₂

—

C₆₇

#13720

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×Q₃₂

C₂×D₆₇

yes

—

#500

—

134

—

#494

—

C₂×C₃₂

—

#366

—

#366

—

134

—

C₄×C₁₆

C₂×D₆₇

yes

characteristic

C₂×Q₁₆

—

C₂×Q₁₆

—

C₄⋊Q₈

—

C₄⋊Q₈

—

268

—

Q₃₂

—

Q₃₂

—

C₂×C₁₆

C₂²⋊D₆₇

yes

characteristic

C₂×C₁₆

C₂²×D₆₇

yes

characteristic

#107

—

268

—

#107

—

134

—

C₄×C₈

C₂²⋊D₆₇

yes

characteristic

C₃₂

—

134

—

C₂×Q₈

—

C₂×Q₈

—

536

—

Q₁₆

—

Q₁₆

—

Q₁₆

—

C₂×C₈

#11761613

yes

characteristic

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

C₄⋊C₄

—

C₄⋊C₄

—

C₄⋊C₄

—

268

—

C₄⋊C₄

—

536

—

C₄²

#11761613

yes

characteristic

C₁₆

#11761616

yes

—

C₁₆

#11761627

yes

—

100

Q₈

—

101

Q₈

—

102

Q₈

—

1072

—

103

C₂×C₄

—

104

C₂×C₄

—

536

—

105

C₂×C₄

#14523984

yes

characteristic

106

C₂×C₄

#14523984

yes

characteristic

107

C₂×C₄

C₂×#11761613

yes

characteristic

108

C₈

#14523987

yes

—

109

C₈

#14524030

yes

characteristic

110

C₈

C₂×#11761616

yes

characteristic

111

C₂²

C₂×#14523984

yes

characteristic

112

C₄

—

113

C₄

—

1072

—

114

C₄

#16863233

yes

—

115

C₄

#16863236

yes

—

116

C₄

#16863742

yes

characteristic

117

C₄

C₂×#14523987

yes

characteristic

118

C₂

#22562432

yes

characteristic

119

C₂

C₂×#16863233

yes

characteristic

120

C₂

C₂×#16863236

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×#14523984
commutator, cosocle	C₂×C₁₆×C₆₇	C₂³
Frattini	C₂×C₁₆	C₂²×D₆₇
Fitting	C₆₇×#2247	C₂
socle, prime radical	C₂²×C₆₇	C₂×D₁₆
hypercenter	#2247	D₆₇
hypocenter	C₆₇	#13720
LCS	G ▷ C₂×C₁₆×C₆₇ ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2247

subgroup

G/H

center

C₂²

C₂×#14523984

commutator, cosocle

C₂×C₁₆×C₆₇

C₂³

Frattini

C₂×C₁₆

C₂²×D₆₇

Fitting

C₆₇×#2247

C₂

socle, prime radical

C₂²×C₆₇

C₂×D₁₆

D₆₇

C₆₇

G ▷ C₂×C₁₆×C₆₇ ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ C₄×C₁₆ ◁ #2247

N	H ≅ G/N	type
65. C₆₇	#13720	semidirect

H ≅ G/N

type

65. C₆₇

#13720

semidirect

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

subgroup	index
#22562484	2
#22558390	2
C₄×C₆₇⋊C₃₂	2
C₆₇×#2247	2
#13720	67

subgroup

index

#22562484

#22558390

C₄×C₆₇⋊C₃₂

C₆₇×#2247

#13720

Subgroups — #30776803

Conjugacy classes of subgroups

Complements

Maximal subgroups