Group #30767592 — subgroups

Subgroups — #30767592

Order 17152.

212 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 91 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	#22562046	C₂	1	2	yes	maximal, complemented
2	#22558285	C₂	1	1	yes	maximal, characteristic, complemented
3	#22558182	C₂	1	2	yes	maximal, complemented
4	#22558125	C₂	1	1	yes	maximal, characteristic, complemented
5	C₆₇×#2178	C₂	1	1	yes	maximal, characteristic
6	#16863813	—	2	2	no	—
7	#16863217	C₂²	1	2	yes	—
8	#16863180	—	2	2	no	—
9	C₄×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
10	C₈×C₆₇⋊C₈	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	C₄×C₁₆×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	1	yes	characteristic
20	C₂×C₆₇⋊C₁₆	D₄	1	1	yes	characteristic
21	C₄×C₆₇⋊C₈	—	2	1	no	—
22	#135×C₆₇	C₂×C₄	1	2	yes	—
23	#131×C₆₇	—	2	2	no	—
24	Q₃₂×C₆₇	C₈	1	1	yes	characteristic, complemented
25	SD₃₂×C₆₇	C₈	1	2	yes	complemented
26	D₁₆×C₆₇	C₈	1	1	yes	characteristic, complemented
27	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
29	#108×C₆₇	C₂×C₄	1	2	yes	—
30	#104×C₆₇	—	2	2	no	—
31	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
32	C₂×C₆₇⋊C₈	—	2	2	no	—
33	C₆₇⋊C₁₆	—	2	1	no	—
34	C₆₇⋊C₁₆	—	4	2	no	—
35	C₆₇⋊C₁₆	D₈	1	2	yes	—
36	#41×C₆₇	—	2	2	no	—
37	Q₁₆×C₆₇	C₂×C₈	1	2	yes	—
38	SD₁₆×C₆₇	—	2	2	no	—
39	D₈×C₆₇	C₂×C₈	1	2	yes	—
40	M₁₆×C₆₇	—	2	2	no	—
41	M₁₆×C₆₇	—	4	2	no	—
42	C₂×C₈×C₆₇	—	4	2	no	—
43	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
44	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
45	C₄²×C₆₇	#41	1	1	yes	characteristic
46	C₁₆×C₆₇	—	2	1	no	—
47	C₁₆×C₆₇	C₂×C₈	1	1	yes	characteristic
48	C₁₆×C₆₇	C₂×C₈	1	1	yes	characteristic
49	Q₈×C₆₇	—	2	2	no	—
50	D₄×C₆₇	—	2	2	no	—
51	D₄×C₆₇	—	4	2	no	—
52	C₂×C₄×C₆₇	—	2	1	no	—
53	C₂×C₄×C₆₇	—	4	2	no	—
54	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
55	C₈×C₆₇	—	4	2	no	—
56	C₈×C₆₇	C₂²×C₈	1	1	yes	characteristic
57	C₈×C₆₇	#131	1	1	yes	characteristic
58	C₈×C₆₇	C₂×D₈	1	1	yes	characteristic
59	C₈×C₆₇	#135	1	1	yes	characteristic
60	C₆₇⋊C₈	—	4	2	no	—
61	C₂²×C₆₇	—	4	2	no	—
62	C₂²×C₆₇	#443	1	1	yes	characteristic
63	C₄×C₆₇	—	2	2	no	—
64	C₄×C₆₇	—	4	2	no	—
65	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
66	C₄×C₆₇	C₄×D₈	1	1	yes	characteristic
67	#12047	—	67	1	no	maximal, Sylow
68	C₂×C₆₇	—	2	1	no	—
69	C₂×C₆₇	—	8	2	no	—
70	C₂×C₆₇	C₈×D₈	1	1	yes	characteristic
71	#2178	D₆₇	1	1	yes	characteristic
72	#2170	—	67	2	no	—
73	#1373	—	67	1	no	—
74	#1335	—	67	2	no	—
75	#1311	—	67	1	no	—
76	C₆₇	#12047	1	1	yes	characteristic, complemented, Sylow
77	#508	Dic₆₇	1	1	yes	characteristic
78	#504	—	134	2	no	—
79	#443	C₂×D₆₇	1	2	yes	—
80	#368	Dic₆₇	1	1	yes	characteristic
81	#364	—	67	2	no	—
82	#359	Dic₆₇	1	2	yes	—
83	#350	—	134	2	no	—
84	C₄×C₁₆	—	67	1	no	—
85	C₄×C₁₆	C₂×D₆₇	1	1	yes	characteristic
86	C₈²	—	67	1	no	—
87	#135	C₂×Dic₆₇	1	2	yes	—
88	#131	—	2	2	no	—
89	Q₃₂	C₆₇⋊C₈	1	1	yes	characteristic, complemented
90	SD₃₂	C₆₇⋊C₈	1	2	yes	complemented
91	D₁₆	C₆₇⋊C₈	1	1	yes	characteristic, complemented
92	M₃₂	—	268	2	no	—
93	M₃₂	—	134	2	no	—
94	C₂×C₁₆	—	268	2	no	—
95	C₂×C₁₆	—	67	1	no	—
96	C₂×C₁₆	—	67	1	no	—
97	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
98	C₂×C₁₆	C₂×Dic₆₇	1	1	yes	characteristic
99	#108	C₂×Dic₆₇	1	2	yes	—
100	#104	—	2	2	no	—
101	C₄×C₈	—	134	1	no	—
102	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
103	#41	—	2	2	no	—
104	Q₁₆	C₂×C₆₇⋊C₈	1	2	yes	—
105	SD₁₆	—	2	2	no	—
106	D₈	C₂×C₆₇⋊C₈	1	2	yes	—
107	M₁₆	—	2	2	no	—
108	M₁₆	—	4	2	no	—
109	C₂×C₈	—	4	2	no	—
110	C₂×C₈	—	134	2	no	—
111	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
112	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
113	C₄²	#11761627	1	1	yes	characteristic
114	C₁₆	—	2	1	no	—
115	C₁₆	—	268	2	no	—
116	C₁₆	—	67	2	no	—
117	C₁₆	—	134	1	no	—
118	C₁₆	C₂×C₆₇⋊C₈	1	1	yes	characteristic
119	C₁₆	C₂×C₆₇⋊C₈	1	1	yes	characteristic
120	Q₈	—	2	2	no	—
121	D₄	—	2	2	no	—
122	D₄	—	4	2	no	—
123	C₂×C₄	—	2	1	no	—
124	C₂×C₄	—	4	2	no	—
125	C₂×C₄	D₄×Dic₆₇	1	1	yes	characteristic
126	C₈	—	4	2	no	—
127	C₈	—	268	2	no	—
128	C₈	C₂²×C₆₇⋊C₈	1	1	yes	characteristic
129	C₈	#14524050	1	1	yes	characteristic
130	C₈	D₈×D₆₇	1	1	yes	characteristic
131	C₈	#14524062	1	1	yes	characteristic
132	C₂²	—	4	2	no	—
133	C₂²	#16863513	1	1	yes	characteristic
134	C₄	—	2	2	no	—
135	C₄	—	4	2	no	—
136	C₄	D₄×C₆₇⋊C₈	1	1	yes	characteristic
137	C₄	D₈×Dic₆₇	1	1	yes	characteristic
138	C₂	—	2	1	no	—
139	C₂	—	8	2	no	—
140	C₂	D₈×C₆₇⋊C₈	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562046

C₂

yes

maximal, complemented

#22558285

C₂

yes

maximal, characteristic, complemented

#22558182

C₂

yes

maximal, complemented

#22558125

C₂

yes

maximal, characteristic, complemented

C₆₇×#2178

C₂

yes

maximal, characteristic

#16863813

—

#16863217

C₂²

yes

—

#16863180

—

C₄×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₈×C₆₇⋊C₈

C₂²

yes

characteristic

#508×C₆₇

C₄

yes

characteristic

#443×C₆₇

C₂²

yes

—

#368×C₆₇

C₄

yes

characteristic

#359×C₆₇

C₄

yes

—

C₄×C₁₆×C₆₇

C₂²

yes

characteristic

#14523983

—

#14523983

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

D₄

yes

characteristic

C₂×C₆₇⋊C₁₆

D₄

yes

characteristic

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

—

C₆₇⋊C₁₆

D₈

yes

—

#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

characteristic

C₈×C₆₇

#131

yes

characteristic

C₈×C₆₇

C₂×D₈

yes

characteristic

C₈×C₆₇

#135

yes

characteristic

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

#443

yes

characteristic

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₈×D₄

yes

characteristic

C₄×C₆₇

C₄×D₈

yes

characteristic

#12047

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

C₈×D₈

yes

characteristic

#2178

D₆₇

yes

—

—

—

—

C₆₇

#12047

yes

characteristic, complemented, Sylow

#508

Dic₆₇

yes

characteristic

#504

—

134

—

#443

C₂×D₆₇

yes

—

#368

Dic₆₇

yes

characteristic

#364

—

#359

Dic₆₇

yes

—

#350

—

134

—

C₄×C₁₆

—

C₄×C₁₆

C₂×D₆₇

yes

characteristic

C₈²

—

#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×C₁₆

C₂×Dic₆₇

yes

characteristic

C₂×C₁₆

C₂×Dic₆₇

yes

characteristic

#108

C₂×Dic₆₇

yes

—

100

#104

—

101

C₄×C₈

—

134

—

102

C₄×C₈

C₂²×D₆₇

yes

characteristic

103

#41

—

104

Q₁₆

C₂×C₆₇⋊C₈

yes

—

105

SD₁₆

—

106

D₈

C₂×C₆₇⋊C₈

yes

—

107

M₁₆

—

108

M₁₆

—

109

C₂×C₈

—

110

C₂×C₈

—

134

—

111

C₂×C₈

C₂²×Dic₆₇

yes

characteristic

112

C₂×C₈

D₄×D₆₇

yes

characteristic

113

C₄²

#11761627

yes

characteristic

114

C₁₆

—

115

C₁₆

—

268

—

116

C₁₆

—

117

C₁₆

—

134

—

118

C₁₆

C₂×C₆₇⋊C₈

yes

characteristic

119

C₁₆

C₂×C₆₇⋊C₈

yes

characteristic

120

Q₈

—

121

D₄

—

122

D₄

—

123

C₂×C₄

—

124

C₂×C₄

—

125

C₂×C₄

D₄×Dic₆₇

yes

characteristic

126

C₈

—

127

C₈

—

268

—

128

C₈

C₂²×C₆₇⋊C₈

yes

characteristic

129

C₈

#14524050

yes

characteristic

130

C₈

D₈×D₆₇

yes

characteristic

131

C₈

#14524062

yes

characteristic

132

C₂²

—

133

C₂²

#16863513

yes

characteristic

134

C₄

—

135

C₄

—

136

C₄

D₄×C₆₇⋊C₈

yes

characteristic

137

C₄

D₈×Dic₆₇

yes

characteristic

138

C₂

—

139

C₂

—

140

C₂

D₈×C₆₇⋊C₈

yes

characteristic

	subgroup	G/H
center	C₈	D₈×D₆₇
commutator	C₈×C₆₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#2178	C₂
socle	C₂×C₆₇	C₈×D₈
prime radical	#508×C₆₇	C₄
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#2178	D₆₇
hypocenter	C₆₇	#12047
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2178

subgroup

G/H

center

C₈

D₈×D₆₇

commutator

C₈×C₆₇

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#2178

C₂

socle

C₂×C₆₇

C₈×D₈

prime radical

#508×C₆₇

C₄

cosocle

C₄×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₈ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #2178

N	H ≅ G/N	type
1. #22562046	C₂	semidirect
2. #22558285	C₂	semidirect
3. #22558182	C₂	semidirect
4. #22558125	C₂	semidirect
24. Q₃₂×C₆₇	C₈	semidirect
25. SD₃₂×C₆₇	C₈	semidirect
26. D₁₆×C₆₇	C₈	semidirect
76. C₆₇	#12047	semidirect
89. Q₃₂	C₆₇⋊C₈	semidirect
90. SD₃₂	C₆₇⋊C₈	semidirect
91. D₁₆	C₆₇⋊C₈	semidirect

H ≅ G/N

type

1. #22562046

C₂

semidirect

2. #22558285

C₂

semidirect

3. #22558182

C₂

semidirect

4. #22558125

C₂

semidirect

24. Q₃₂×C₆₇

C₈

semidirect

25. SD₃₂×C₆₇

C₈

semidirect

26. D₁₆×C₆₇

C₈

semidirect

76. C₆₇

#12047

semidirect

89. Q₃₂

C₆₇⋊C₈

semidirect

90. SD₃₂

C₆₇⋊C₈

semidirect

91. D₁₆

C₆₇⋊C₈

semidirect

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

subgroup	index
#22562046	2
#22558285	2
#22558182	2
#22558125	2
C₆₇×#2178	2
#12047	67

subgroup

index

C₆₇×#2178

Subgroups — #30767592

Conjugacy classes of subgroups

Complements

Maximal subgroups