Group #30767721 — subgroups

Subgroups — #30767721

Order 17152.

204 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 204 conjugacy classes of subgroups — too many to draw. Only its 89 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	#22562051	C₂	1	2	yes	maximal, complemented
2	#22558277	C₂	1	1	yes	maximal, characteristic, complemented
3	#22558185	C₂	1	2	yes	maximal, complemented
4	#22558129	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2179	C₂	1	1	yes	maximal, characteristic
6	#16863813	—	2	2	no	—
7	#16863217	C₂²	1	2	yes	—
8	#16863180	—	2	2	no	—
9	#16863172	C₂²	1	1	yes	characteristic
10	#16863117	C₂²	1	1	yes	characteristic
11	#508×C₆₇	C₄	1	1	yes	characteristic
12	#443×C₆₇	C₂²	1	2	yes	—
13	#368×C₆₇	C₄	1	1	yes	characteristic
14	#359×C₆₇	C₄	1	2	yes	—
15	#346×C₆₇	C₂²	1	1	yes	characteristic
16	#14523983	—	2	2	no	—
17	#14523983	—	4	2	no	—
18	C₂×C₆₇⋊C₁₆	—	4	2	no	—
19	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
20	C₄×C₆₇⋊C₈	—	2	1	no	—
21	#135×C₆₇	C₂×C₄	1	2	yes	—
22	#131×C₆₇	—	2	2	no	—
23	Q₃₂×C₆₇	C₈	1	1	yes	characteristic, complemented
24	SD₃₂×C₆₇	C₈	1	2	yes	complemented
25	D₁₆×C₆₇	C₈	1	1	yes	characteristic, complemented
26	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
28	#108×C₆₇	C₂×C₄	1	2	yes	—
29	#104×C₆₇	—	2	2	no	—
30	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
31	C₂×C₆₇⋊C₈	—	4	1	no	—
32	C₆₇⋊C₁₆	—	2	2	no	—
33	C₆₇⋊C₁₆	—	4	2	no	—
34	#41×C₆₇	—	2	2	no	—
35	Q₁₆×C₆₇	C₂×C₈	1	2	yes	—
36	SD₁₆×C₆₇	—	2	2	no	—
37	D₈×C₆₇	C₂×C₈	1	2	yes	—
38	M₁₆×C₆₇	—	2	2	no	—
39	M₁₆×C₆₇	—	4	2	no	—
40	C₂×C₈×C₆₇	—	4	2	no	—
41	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
42	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₄²×C₆₇	#41	1	1	yes	characteristic
44	C₁₆×C₆₇	—	2	1	no	—
45	C₁₆×C₆₇	C₂×C₈	1	1	yes	characteristic
46	C₁₆×C₆₇	C₂×C₈	1	1	yes	characteristic
47	Q₈×C₆₇	—	2	2	no	—
48	D₄×C₆₇	—	2	2	no	—
49	D₄×C₆₇	—	4	2	no	—
50	C₂×C₄×C₆₇	—	2	1	no	—
51	C₂×C₄×C₆₇	—	4	2	no	—
52	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
53	C₈×C₆₇	—	4	2	no	—
54	C₈×C₆₇	C₂²×C₈	1	1	yes	characteristic
55	C₈×C₆₇	#131	1	1	yes	characteristic
56	C₈×C₆₇	AGL(1,ℤ/8)	1	2	yes	—
57	C₆₇⋊C₈	—	8	1	no	—
58	C₂²×C₆₇	—	4	2	no	—
59	C₂²×C₆₇	#444	1	1	yes	characteristic
60	C₄×C₆₇	—	4	1	no	—
61	C₄×C₆₇	—	4	2	no	—
62	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
63	C₄×C₆₇	#442	1	1	yes	characteristic
64	#12068	—	67	1	no	maximal, Sylow
65	C₂×C₆₇	—	2	1	no	—
66	C₂×C₆₇	—	8	2	no	—
67	C₂×C₆₇	#1580	1	1	yes	characteristic
68	#2179	D₆₇	1	1	yes	characteristic
69	#2171	—	67	2	no	—
70	#1369	—	67	1	no	—
71	#1336	—	67	2	no	—
72	#1313	—	67	1	no	—
73	C₆₇	#12068	1	1	yes	characteristic, complemented, Sylow
74	#508	Dic₆₇	1	1	yes	characteristic
75	#504	—	134	2	no	—
76	#443	C₂×D₆₇	1	2	yes	—
77	#368	Dic₆₇	1	1	yes	characteristic
78	#364	—	67	2	no	—
79	#359	Dic₆₇	1	2	yes	—
80	#350	—	134	2	no	—
81	#346	—	67	1	no	—
82	#346	C₂×D₆₇	1	1	yes	characteristic
83	#322	—	67	1	no	—
84	#135	C₂×Dic₆₇	1	2	yes	—
85	#131	—	2	2	no	—
86	Q₃₂	C₆₇⋊C₈	1	1	yes	characteristic, complemented
87	SD₃₂	C₆₇⋊C₈	1	2	yes	complemented
88	D₁₆	C₆₇⋊C₈	1	1	yes	characteristic, complemented
89	M₃₂	—	268	2	no	—
90	M₃₂	—	134	2	no	—
91	C₂×C₁₆	—	268	2	no	—
92	C₂×C₁₆	—	67	2	no	—
93	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
94	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
95	#108	C₂×Dic₆₇	1	2	yes	—
96	#104	—	2	2	no	—
97	C₄×C₈	—	134	1	no	—
98	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
99	#41	—	2	2	no	—
100	Q₁₆	C₂×C₆₇⋊C₈	1	2	yes	—
101	SD₁₆	—	2	2	no	—
102	D₈	C₂×C₆₇⋊C₈	1	2	yes	—
103	M₁₆	—	2	2	no	—
104	M₁₆	—	4	2	no	—
105	C₂×C₈	—	4	2	no	—
106	C₂×C₈	—	268	1	no	—
107	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
108	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
109	C₄²	#11761627	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	268	2	no	—
112	C₁₆	—	134	2	no	—
113	C₁₆	C₂×C₆₇⋊C₈	1	1	yes	characteristic
114	C₁₆	C₂×C₆₇⋊C₈	1	1	yes	characteristic
115	Q₈	—	2	2	no	—
116	D₄	—	2	2	no	—
117	D₄	—	4	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	4	2	no	—
120	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
121	C₈	—	4	2	no	—
122	C₈	—	536	1	no	—
123	C₈	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
124	C₈	#14524050	1	1	yes	characteristic
125	C₈	#14524067	1	2	yes	—
126	C₂²	—	4	2	no	—
127	C₂²	#16863518	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	4	2	no	—
130	C₄	D₄×C₆₇⋊C₈	1	1	yes	characteristic
131	C₄	#16863508	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	8	2	no	—
134	C₂	#22559033	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562051

C₂

yes

maximal, complemented

#22558277

C₂

yes

maximal, characteristic, complemented

#22558185

C₂

yes

maximal, complemented

#22558129

C₂

yes

maximal, characteristic, complemented

C₆₇×#2179

C₂

yes

maximal, characteristic

#16863813

—

#16863217

C₂²

yes

—

#16863180

—

#16863172

C₂²

yes

characteristic

#16863117

C₂²

yes

characteristic

#508×C₆₇

C₄

yes

characteristic

#443×C₆₇

C₂²

yes

—

#368×C₆₇

C₄

yes

characteristic

#359×C₆₇

C₄

yes

—

#346×C₆₇

C₂²

yes

characteristic

#14523983

—

#14523983

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

C₄×C₆₇⋊C₈

—

#135×C₆₇

C₂×C₄

yes

—

#131×C₆₇

—

Q₃₂×C₆₇

C₈

yes

characteristic, complemented

SD₃₂×C₆₇

C₈

yes

complemented

D₁₆×C₆₇

C₈

yes

characteristic, complemented

C₂×C₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₇

C₂×C₄

yes

characteristic

#108×C₆₇

C₂×C₄

yes

—

#104×C₆₇

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₆₇⋊C₈

—

C₆₇⋊C₁₆

—

C₆₇⋊C₁₆

—

#41×C₆₇

—

Q₁₆×C₆₇

C₂×C₈

yes

—

SD₁₆×C₆₇

—

D₈×C₆₇

C₂×C₈

yes

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

#41

yes

characteristic

C₁₆×C₆₇

—

C₁₆×C₆₇

C₂×C₈

yes

characteristic

C₁₆×C₆₇

C₂×C₈

yes

characteristic

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄×D₄

yes

characteristic

C₈×C₆₇

—

C₈×C₆₇

C₂²×C₈

yes

C₈×C₆₇

yes

C₈×C₆₇

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#444

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₈×D₄

yes

C₄×C₆₇

yes

—

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

#1580

yes

characteristic

#2179

D₆₇

yes

—

—

—

—

C₆₇

#12068

yes

characteristic, complemented, Sylow

#508

Dic₆₇

yes

characteristic

#504

—

134

—

#443

C₂×D₆₇

yes

—

#368

Dic₆₇

yes

characteristic

#364

—

#359

Dic₆₇

yes

—

#350

—

134

—

#346

—

#346

C₂×D₆₇

yes

characteristic

#322

—

#135

C₂×Dic₆₇

yes

—

#131

—

Q₃₂

C₆₇⋊C₈

yes

characteristic, complemented

SD₃₂

C₆₇⋊C₈

yes

complemented

D₁₆

C₆₇⋊C₈

yes

characteristic, complemented

M₃₂

—

268

—

M₃₂

—

134

—

C₂×C₁₆

—

268

—

C₂×C₁₆

—

C₂×C₁₆

C₂×Dic₆₇

yes

characteristic

C₂×C₁₆

C₂×Dic₆₇

yes

characteristic

#108

C₂×Dic₆₇

yes

—

#104

—

C₄×C₈

—

134

—

C₄×C₈

C₂²×D₆₇

yes

characteristic

#41

—

100

Q₁₆

C₂×C₆₇⋊C₈

yes

—

101

SD₁₆

—

102

D₈

C₂×C₆₇⋊C₈

yes

—

103

M₁₆

—

104

M₁₆

—

105

C₂×C₈

—

106

C₂×C₈

—

268

—

107

C₂×C₈

C₂²×Dic₆₇

yes

characteristic

108

C₂×C₈

D₄×D₆₇

yes

characteristic

109

C₄²

#11761627

yes

characteristic

110

C₁₆

—

111

C₁₆

—

268

—

112

C₁₆

—

134

—

113

C₁₆

C₂×C₆₇⋊C₈

yes

characteristic

114

C₁₆

C₂×C₆₇⋊C₈

yes

characteristic

115

Q₈

—

116

D₄

—

117

D₄

—

118

C₂×C₄

—

119

C₂×C₄

—

120

C₂×C₄

D₄×Dic₆₇

yes

characteristic

121

C₈

—

122

C₈

—

536

—

123

C₈

C₂²×C₆₇⋊C₈

yes

characteristic

124

C₈

#14524050

yes

characteristic

125

C₈

#14524067

yes

—

126

C₂²

—

127

C₂²

#16863518

yes

characteristic

128

C₄

—

129

C₄

—

130

C₄

D₄×C₆₇⋊C₈

yes

characteristic

131

C₄

#16863508

yes

characteristic

132

C₂

—

133

C₂

—

134

C₂

#22559033

yes

characteristic

	subgroup	G/H
center	C₄	#16863508
commutator	C₈×C₆₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#2179	C₂
socle	C₂×C₆₇	#1580
prime radical	#508×C₆₇	C₄
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#2179	D₆₇
hypocenter	C₆₇	#12068
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2179

subgroup

G/H

center

C₄

#16863508

commutator

C₈×C₆₇

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#2179

C₂

socle

C₂×C₆₇

#1580

prime radical

#508×C₆₇

C₄

cosocle

C₄×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2179

N	H ≅ G/N	type
1. #22562051	C₂	semidirect
2. #22558277	C₂	semidirect
3. #22558185	C₂	semidirect
4. #22558129	C₂	semidirect
23. Q₃₂×C₆₇	C₈	semidirect
24. SD₃₂×C₆₇	C₈	semidirect
25. D₁₆×C₆₇	C₈	semidirect
73. C₆₇	#12068	semidirect
86. Q₃₂	C₆₇⋊C₈	semidirect
87. SD₃₂	C₆₇⋊C₈	semidirect
88. D₁₆	C₆₇⋊C₈	semidirect

H ≅ G/N

type

1. #22562051

C₂

semidirect

2. #22558277

C₂

semidirect

3. #22558185

C₂

semidirect

4. #22558129

C₂

semidirect

23. Q₃₂×C₆₇

C₈

semidirect

24. SD₃₂×C₆₇

C₈

semidirect

25. D₁₆×C₆₇

C₈

semidirect

73. C₆₇

#12068

semidirect

86. Q₃₂

C₆₇⋊C₈

semidirect

87. SD₃₂

C₆₇⋊C₈

semidirect

88. D₁₆

C₆₇⋊C₈

semidirect

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

subgroup	index
#22562051	2
#22558277	2
#22558185	2
#22558129	2
C₆₇×#2179	2
#12068	67

subgroup

index

C₆₇×#2179

Subgroups — #30767721

Conjugacy classes of subgroups

Complements

Maximal subgroups