Group #30770555 — subgroups

Subgroups — #30770555

Order 17152.

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

normal subgroups only

This group has 330 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	#22562466	C₂	1	2	yes	maximal, complemented
2	#22562459	C₂	1	1	yes	maximal, characteristic
3	#22559816	C₂	1	2	yes	maximal, complemented
4	C₁₆×C₆₇⋊C₈	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2247	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523986	—	2	4	no	—
7	C₂×D₁₀₇₂	—	2	2	no	—
8	#16863773	C₂²	1	2	yes	—
9	#16863744	—	2	4	no	—
10	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₈×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
12	C₂×Q₃₂×C₆₇	—	2	2	no	—
13	#494×C₆₇	C₂²	1	2	yes	—
14	C₄×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×D₅₃₆	—	2	2	no	—
16	C₂×D₅₃₆	—	2	2	no	—
17	#14524040	—	2	2	no	—
18	#14523986	—	4	8	no	—
19	D₁₀₇₂	—	4	4	no	—
20	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
21	#14523963	—	4	8	no	—
22	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
23	C₂×Q₁₆×C₆₇	—	2	2	no	—
24	C₂×Q₁₆×C₆₇	—	2	2	no	—
25	C₄⋊Q₈×C₆₇	—	2	2	no	—
26	Q₃₂×C₆₇	—	4	4	no	—
27	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
28	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
29	C₂×D₂₆₈	—	4	2	no	—
30	C₂×D₂₆₈	—	4	4	no	—
31	D₅₃₆	—	4	4	no	—
32	D₅₃₆	—	4	4	no	—
33	C₂×C₆₇⋊C₈	D₈	1	4	yes	—
34	C₆₇⋊C₁₆	D₈	1	4	yes	—
35	C₂×Q₈×C₆₇	—	4	2	no	—
36	Q₁₆×C₆₇	—	4	4	no	—
37	Q₁₆×C₆₇	—	4	4	no	—
38	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₄⋊C₄×C₆₇	—	4	4	no	—
41	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
42	C₁₆×C₆₇	D₈	1	4	yes	—
43	C₂²×D₆₇	—	8	2	no	—
44	Q₈×C₆₇	—	8	2	no	—
45	Q₈×C₆₇	—	8	2	no	—
46	C₂×C₄×C₆₇	—	8	2	no	—
47	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
48	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
49	D₂₆₈	—	8	2	no	—
50	D₂₆₈	—	8	2	no	—
51	D₂₆₈	—	8	8	no	—
52	C₈×C₆₇	C₂×D₈	1	2	yes	—
53	C₈×C₆₇	C₂×D₈	1	2	yes	—
54	C₆₇⋊C₈	—	2	4	no	—
55	C₂²×C₆₇	#493	1	1	yes	characteristic
56	C₂×D₆₇	—	16	2	no	—
57	C₂×D₆₇	—	16	4	no	—
58	C₄×C₆₇	—	16	2	no	—
59	C₄×C₆₇	#493	1	1	yes	characteristic
60	C₄×C₆₇	#493	1	1	yes	characteristic
61	C₄×C₆₇	#508	1	4	yes	—
62	#12518	—	67	1	no	maximal, Sylow
63	C₂×C₆₇	#1712	1	1	yes	characteristic
64	C₂×C₆₇	#2249	1	2	yes	—
65	D₆₇	—	32	2	no	—
66	#2248	—	67	2	no	—
67	#2247	D₆₇	1	1	yes	characteristic, complemented
68	#2246	—	67	1	no	—
69	#1709	—	67	2	no	—
70	C₈×C₁₆	—	67	1	no	—
71	C₆₇	#12518	1	1	yes	characteristic, complemented, Sylow
72	C₂×Q₃₂	—	2	2	no	—
73	C₂×SD₃₂	—	134	4	no	—
74	C₂×D₁₆	—	134	2	no	—
75	#494	C₂×D₆₇	1	2	yes	—
76	#493	—	67	2	no	—
77	#486	—	134	4	no	—
78	C₄×C₁₆	—	67	1	no	—
79	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
80	C₈²	—	67	1	no	—
81	C₂×Q₁₆	—	2	2	no	—
82	C₂×Q₁₆	—	2	2	no	—
83	C₂×D₈	—	134	2	no	—
84	C₂×D₈	—	134	2	no	—
85	C₄⋊Q₈	—	2	2	no	—
86	#127	—	134	2	no	—
87	Q₃₂	—	4	4	no	—
88	SD₃₂	—	268	8	no	—
89	D₁₆	—	268	4	no	—
90	C₂×C₁₆	—	67	2	no	—
91	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
92	#102	—	268	8	no	—
93	C₄×C₈	—	67	2	no	—
94	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
95	C₂×Q₈	—	4	2	no	—
96	C₂×D₄	—	268	2	no	—
97	C₂×D₄	—	268	4	no	—
98	Q₁₆	—	4	4	no	—
99	Q₁₆	—	4	4	no	—
100	D₈	—	268	4	no	—
101	D₈	—	268	4	no	—
102	C₂×C₈	—	67	4	no	—
103	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
104	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
105	C₄⋊C₄	—	4	4	no	—
106	C₄²	D₄×D₆₇	1	1	yes	characteristic
107	C₁₆	—	67	4	no	—
108	C₁₆	#11761613	1	4	yes	—
109	C₂³	—	536	2	no	—
110	Q₈	—	8	2	no	—
111	Q₈	—	8	2	no	—
112	D₄	—	536	2	no	—
113	D₄	—	536	2	no	—
114	D₄	—	536	8	no	—
115	C₂×C₄	—	8	2	no	—
116	C₂×C₄	#14524039	1	1	yes	characteristic
117	C₂×C₄	D₈×D₆₇	1	2	yes	—
118	C₈	—	134	4	no	—
119	C₈	D₈×D₆₇	1	2	yes	—
120	C₈	C₂×#11761613	1	2	yes	—
121	C₂²	—	1072	2	no	—
122	C₂²	—	1072	4	no	—
123	C₂²	#16863771	1	1	yes	characteristic
124	C₄	—	16	2	no	—
125	C₄	#16863771	1	1	yes	characteristic
126	C₄	#16863772	1	1	yes	characteristic
127	C₄	#16863829	1	4	yes	—
128	C₂	—	2144	2	no	—
129	C₂	#22559823	1	1	yes	characteristic
130	C₂	#22562471	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22562466

C₂

yes

maximal, complemented

#22562459

C₂

yes

maximal, characteristic

#22559816

C₂

yes

maximal, complemented

C₁₆×C₆₇⋊C₈

C₂

yes

maximal, characteristic, complemented

C₆₇×#2247

C₂

yes

maximal, characteristic, complemented

C₂×#14523986

—

C₂×D₁₀₇₂

—

#16863773

C₂²

yes

—

#16863744

—

C₄×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₈×C₆₇⋊C₈

C₂²

yes

characteristic

C₂×Q₃₂×C₆₇

—

#494×C₆₇

C₂²

yes

—

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×D₅₃₆

—

C₂×D₅₃₆

—

#14524040

—

#14523986

—

D₁₀₇₂

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

#14523963

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₄⋊Q₈×C₆₇

—

Q₃₂×C₆₇

—

C₂×C₁₆×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×D₂₆₈

—

C₂×D₂₆₈

—

D₅₃₆

—

D₅₃₆

—

C₂×C₆₇⋊C₈

D₈

yes

—

C₆₇⋊C₁₆

D₈

yes

—

C₂×Q₈×C₆₇

—

Q₁₆×C₆₇

—

Q₁₆×C₆₇

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₁₆×C₆₇

D₈

yes

—

C₂²×D₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

C₂×D₈

yes

—

C₈×C₆₇

C₂×D₈

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

#493

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

yes

C₄×C₆₇

yes

C₄×C₆₇

yes

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

D₆₇

—

#2248

—

#2247

D₆₇

yes

characteristic, complemented

#2246

—

#1709

—

C₈×C₁₆

—

C₆₇

#12518

yes

characteristic, complemented, Sylow

C₂×Q₃₂

—

C₂×SD₃₂

—

134

—

C₂×D₁₆

—

134

—

#494

C₂×D₆₇

yes

—

#493

—

#486

—

134

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₇

yes

characteristic

C₈²

—

C₂×Q₁₆

—

C₂×Q₁₆

—

C₂×D₈

—

134

—

C₂×D₈

—

134

—

C₄⋊Q₈

—

#127

—

134

—

Q₃₂

—

SD₃₂

—

268

—

D₁₆

—

268

—

C₂×C₁₆

—

C₂×C₁₆

C₂²⋊D₆₇

yes

—

#102

—

268

—

C₄×C₈

—

C₄×C₈

C₂²×D₆₇

yes

characteristic

C₂×Q₈

—

C₂×D₄

—

268

—

C₂×D₄

—

268

—

Q₁₆

—

Q₁₆

—

100

D₈

—

268

—

101

D₈

—

268

—

102

C₂×C₈

—

103

C₂×C₈

D₄×D₆₇

yes

characteristic

104

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

105

C₄⋊C₄

—

106

C₄²

D₄×D₆₇

yes

characteristic

107

C₁₆

—

108

C₁₆

#11761613

yes

—

109

C₂³

—

536

—

110

Q₈

—

111

Q₈

—

112

D₄

—

536

—

113

D₄

—

536

—

114

D₄

—

536

—

115

C₂×C₄

—

116

C₂×C₄

#14524039

yes

characteristic

117

C₂×C₄

D₈×D₆₇

yes

—

118

C₈

—

134

—

119

C₈

D₈×D₆₇

yes

—

120

C₈

C₂×#11761613

yes

—

121

C₂²

—

1072

—

122

C₂²

—

1072

—

123

C₂²

#16863771

yes

characteristic

124

C₄

—

125

C₄

#16863771

yes

characteristic

126

C₄

#16863772

yes

characteristic

127

C₄

#16863829

yes

—

128

C₂

—

2144

—

129

C₂

#22559823

yes

characteristic

130

C₂

#22562471

yes

—

	subgroup	G/H
center	C₂²	#16863771
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#2247	C₂
socle	C₂²×C₆₇	#493
prime radical	#22562459	C₂
hypercenter	#2247	D₆₇
hypocenter	C₆₇	#12518
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

subgroup

G/H

center

C₂²

#16863771

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#2247

C₂

C₂²×C₆₇

C₂

D₆₇

C₆₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2247

N	H ≅ G/N	type
1. #22562466	C₂	semidirect
3. #22559816	C₂	semidirect
4. C₁₆×C₆₇⋊C₈	C₂	semidirect
5. C₆₇×#2247	C₂	semidirect
67. #2247	D₆₇	semidirect
71. C₆₇	#12518	semidirect

H ≅ G/N

type

1. #22562466

C₂

semidirect

3. #22559816

C₂

semidirect

4. C₁₆×C₆₇⋊C₈

C₂

semidirect

5. C₆₇×#2247

C₂

semidirect

67. #2247

D₆₇

semidirect

71. C₆₇

#12518

semidirect

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

subgroup	index
#22562466	2
#22562459	2
#22559816	2
C₁₆×C₆₇⋊C₈	2
C₆₇×#2247	2
#12518	67

subgroup

index

#22562466

#22562459

#22559816

C₁₆×C₆₇⋊C₈

C₆₇×#2247

#12518

Subgroups — #30770555

Conjugacy classes of subgroups

Complements

Maximal subgroups