Group #30770560 — subgroups

Subgroups — #30770560

Order 17152.

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

normal subgroups only

This group has 360 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	#22562458	C₂	1	2	yes	maximal, complemented
3	#22559824	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558126	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#1709	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14523986	—	2	4	no	—
7	C₂×#14523984	—	2	4	no	—
8	#16863773	—	2	2	no	—
9	#16863773	C₂²	1	2	yes	—
10	C₄×C₆₇⋊C₁₆	C₂²	1	2	yes	—
11	#494×C₆₇	C₂²	1	1	yes	characteristic
12	#493×C₆₇	C₂²	1	1	yes	characteristic
13	#486×C₆₇	—	2	2	no	—
14	C₈²×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×D₅₃₆	—	2	2	no	—
16	C₂×D₅₃₆	—	2	2	no	—
17	C₂×D₅₃₆	—	4	4	no	—
18	#14524040	—	2	2	no	—
19	#14523986	—	4	8	no	—
20	#14523984	—	4	8	no	—
21	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	—
22	C₂×Q₁₆×C₆₇	—	2	2	no	—
23	C₂×D₈×C₆₇	—	2	2	no	—
24	C₄⋊Q₈×C₆₇	—	2	1	no	—
25	#127×C₆₇	—	2	1	no	—
26	#102×C₆₇	—	4	4	no	—
27	C₄×C₈×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	D₅₃₆	—	8	8	no	—
34	C₆₇⋊C₁₆	D₈	1	8	yes	complemented
35	C₂×Q₈×C₆₇	—	4	1	no	—
36	C₂×D₄×C₆₇	—	4	1	no	—
37	C₂×D₄×C₆₇	—	4	2	no	—
38	Q₁₆×C₆₇	—	4	4	no	—
39	D₈×C₆₇	—	4	4	no	—
40	C₂×C₈×C₆₇	D₈	1	4	yes	—
41	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
42	C₄⋊C₄×C₆₇	—	4	2	no	—
43	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
44	C₂³×C₆₇	—	8	1	no	—
45	C₂²×D₆₇	—	8	2	no	—
46	Q₈×C₆₇	—	8	2	no	—
47	D₄×C₆₇	—	8	2	no	—
48	D₄×C₆₇	—	8	4	no	—
49	C₂×C₄×C₆₇	—	8	1	no	—
50	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
51	C₂×C₄×C₆₇	C₂×D₈	1	2	yes	—
52	D₂₆₈	—	8	2	no	—
53	D₂₆₈	—	8	2	no	—
54	D₂₆₈	—	8	8	no	—
55	C₈×C₆₇	—	2	4	no	—
56	C₈×C₆₇	C₂×D₈	1	4	yes	—
57	C₂²×C₆₇	—	16	1	no	—
58	C₂²×C₆₇	—	16	2	no	—
59	C₂²×C₆₇	#493	1	1	yes	characteristic
60	C₂×D₆₇	—	16	2	no	—
61	C₂×D₆₇	—	16	4	no	—
62	C₄×C₆₇	—	16	1	no	—
63	C₄×C₆₇	#493	1	2	yes	—
64	C₄×C₆₇	#509	1	4	yes	—
65	#12519	—	67	1	no	maximal, Sylow
66	C₂×C₆₇	—	32	1	no	—
67	C₂×C₆₇	#1712	1	1	yes	characteristic
68	C₂×C₆₇	#2250	1	2	yes	—
69	D₆₇	—	32	2	no	—
70	#2248	—	67	2	no	—
71	#2246	—	67	2	no	—
72	#1712	—	67	1	no	—
73	#1709	D₆₇	1	1	yes	characteristic, complemented
74	#1311	—	67	1	no	—
75	C₆₇	#12519	1	1	yes	characteristic, complemented, Sylow
76	C₂×SD₃₂	—	134	4	no	—
77	C₂×D₁₆	—	134	4	no	—
78	#494	C₂×D₆₇	1	1	yes	characteristic
79	#493	—	67	2	no	—
80	#493	—	134	2	no	—
81	#493	C₂×D₆₇	1	1	yes	characteristic
82	#486	—	2	2	no	—
83	C₄×C₁₆	—	67	2	no	—
84	C₈²	C₂×D₆₇	1	1	yes	characteristic
85	C₂×Q₁₆	—	2	2	no	—
86	C₂×D₈	—	2	2	no	—
87	C₂×D₈	—	268	4	no	—
88	C₂×D₈	—	134	2	no	—
89	C₂×D₈	—	134	2	no	—
90	C₄⋊Q₈	—	2	1	no	—
91	#127	—	2	1	no	—
92	#127	—	134	2	no	—
93	SD₃₂	—	268	8	no	—
94	D₁₆	—	268	8	no	—
95	C₂×C₁₆	—	67	4	no	—
96	#102	—	4	4	no	—
97	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
98	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
99	C₂×Q₈	—	4	1	no	—
100	C₂×D₄	—	4	1	no	—
101	C₂×D₄	—	4	2	no	—
102	C₂×D₄	—	268	2	no	—
103	C₂×D₄	—	268	4	no	—
104	Q₁₆	—	4	4	no	—
105	D₈	—	4	4	no	—
106	D₈	—	268	4	no	—
107	D₈	—	268	4	no	—
108	D₈	—	536	8	no	—
109	C₂×C₈	#11761613	1	4	yes	—
110	C₂×C₈	D₄×D₆₇	1	2	yes	—
111	C₄⋊C₄	—	4	2	no	—
112	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
113	C₁₆	—	67	8	no	—
114	C₂³	—	8	1	no	—
115	C₂³	—	536	2	no	—
116	Q₈	—	8	2	no	—
117	D₄	—	8	2	no	—
118	D₄	—	8	4	no	—
119	D₄	—	536	2	no	—
120	D₄	—	536	2	no	—
121	D₄	—	536	8	no	—
122	C₂×C₄	—	8	1	no	—
123	C₂×C₄	#14524039	1	1	yes	characteristic
124	C₂×C₄	C₂×#11761613	1	2	yes	—
125	C₈	—	2	4	no	—
126	C₈	D₈×D₆₇	1	4	yes	—
127	C₂²	—	16	1	no	—
128	C₂²	—	16	2	no	—
129	C₂²	—	1072	2	no	—
130	C₂²	—	1072	4	no	—
131	C₂²	#16863772	1	1	yes	characteristic
132	C₄	—	16	1	no	—
133	C₄	#16863771	1	2	yes	—
134	C₄	#16863831	1	4	yes	—
135	C₂	—	32	1	no	—
136	C₂	—	2144	2	no	—
137	C₂	#22559823	1	1	yes	characteristic
138	C₂	#22562476	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22562466

C₂

yes

maximal, complemented

#22562458

C₂

yes

maximal, complemented

#22559824

C₂

yes

maximal, characteristic, complemented

#22558126

C₂

yes

maximal, characteristic, complemented

C₆₇×#1709

C₂

yes

maximal, characteristic, complemented

—

—

—

C₂²

yes

—

C₄×C₆₇⋊C₁₆

C₂²

yes

—

#494×C₆₇

C₂²

yes

characteristic

#493×C₆₇

C₂²

yes

characteristic

#486×C₆₇

—

C₈²×C₆₇

C₂²

yes

characteristic

C₂×D₅₃₆

—

C₂×D₅₃₆

—

C₂×D₅₃₆

—

#14524040

—

#14523986

—

#14523984

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

C₂×Q₁₆×C₆₇

—

C₂×D₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

#127×C₆₇

—

#102×C₆₇

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×D₂₆₈

—

C₂×D₂₆₈

—

D₅₃₆

—

D₅₃₆

—

D₅₃₆

—

C₆₇⋊C₁₆

D₈

yes

complemented

C₂×Q₈×C₆₇

—

C₂×D₄×C₆₇

—

C₂×D₄×C₆₇

—

Q₁₆×C₆₇

—

D₈×C₆₇

—

C₂×C₈×C₆₇

D₈

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₂³×C₆₇

—

C₂²×D₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₂×D₈

yes

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₈×C₆₇

—

C₈×C₆₇

C₂×D₈

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#493

yes

characteristic

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#493

yes

—

C₄×C₆₇

#509

yes

—

#12519

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1712

yes

characteristic

C₂×C₆₇

#2250

yes

—

D₆₇

—

—

—

—

D₆₇

yes

characteristic, complemented

#1311

—

C₆₇

#12519

yes

characteristic, complemented, Sylow

C₂×SD₃₂

—

134

—

C₂×D₁₆

—

134

—

#494

C₂×D₆₇

yes

characteristic

#493

—

#493

—

134

—

#493

C₂×D₆₇

yes

characteristic

#486

—

C₄×C₁₆

—

C₈²

C₂×D₆₇

yes

characteristic

C₂×Q₁₆

—

C₂×D₈

—

C₂×D₈

—

268

—

C₂×D₈

—

134

—

C₂×D₈

—

134

—

C₄⋊Q₈

—

#127

—

#127

—

134

—

SD₃₂

—

268

—

D₁₆

—

268

—

C₂×C₁₆

—

#102

—

C₄×C₈

C₂²⋊D₆₇

yes

—

C₄×C₈

C₂²×D₆₇

yes

characteristic

C₂×Q₈

—

100

C₂×D₄

—

101

C₂×D₄

—

102

C₂×D₄

—

268

—

103

C₂×D₄

—

268

—

104

Q₁₆

—

105

D₈

—

106

D₈

—

268

—

107

D₈

—

268

—

108

D₈

—

536

—

109

C₂×C₈

#11761613

yes

—

110

C₂×C₈

D₄×D₆₇

yes

—

111

C₄⋊C₄

—

112

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

113

C₁₆

—

114

C₂³

—

115

C₂³

—

536

—

116

Q₈

—

117

D₄

—

118

D₄

—

119

D₄

—

536

—

120

D₄

—

536

—

121

D₄

—

536

—

122

C₂×C₄

—

123

C₂×C₄

#14524039

yes

characteristic

124

C₂×C₄

C₂×#11761613

yes

—

125

C₈

—

126

C₈

D₈×D₆₇

yes

—

127

C₂²

—

128

C₂²

—

129

C₂²

—

1072

—

130

C₂²

—

1072

—

131

C₂²

#16863772

yes

characteristic

132

C₄

—

133

C₄

#16863771

yes

—

134

C₄

#16863831

yes

—

135

C₂

—

136

C₂

—

2144

—

137

C₂

#22559823

yes

characteristic

138

C₂

#22562476

yes

—

	subgroup	G/H
center	C₂²	#16863772
commutator, cosocle	C₄×C₈×C₆₇	C₂³
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1709	C₂
socle	C₂²×C₆₇	#493
hypercenter	#1709	D₆₇
hypocenter	C₆₇	#12519
LCS	G ▷ C₄×C₈×C₆₇ ▷ C₂×C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

subgroup

G/H

center

C₂²

#16863772

commutator, cosocle

C₄×C₈×C₆₇

C₂³

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#1709

C₂

C₂²×C₆₇

D₆₇

C₆₇

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

UCS

C₁ ◁ C₂² ◁ C₄² ◁ C₈² ◁ #1709

N	H ≅ G/N	type
1. #22562466	C₂	semidirect
2. #22562458	C₂	semidirect
3. #22559824	C₂	semidirect
4. #22558126	C₂	semidirect
5. C₆₇×#1709	C₂	semidirect
34. C₆₇⋊C₁₆	D₈	semidirect
73. #1709	D₆₇	semidirect
75. C₆₇	#12519	semidirect

H ≅ G/N

type

1. #22562466

C₂

semidirect

2. #22562458

C₂

semidirect

3. #22559824

C₂

semidirect

4. #22558126

C₂

semidirect

5. C₆₇×#1709

C₂

semidirect

34. C₆₇⋊C₁₆

D₈

semidirect

73. #1709

D₆₇

semidirect

75. C₆₇

#12519

semidirect

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

subgroup	index
#22562466	2
#22562458	2
#22559824	2
#22558126	2
C₆₇×#1709	2
#12519	67

subgroup

index

C₆₇×#1709

Subgroups — #30770560

Conjugacy classes of subgroups

Complements

Maximal subgroups