Group #30767722 — subgroups

Subgroups — #30767722

Order 17152.

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

normal subgroups only

This group has 204 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	#22562096	C₂	1	1	yes	maximal, characteristic
2	#22562052	C₂	1	2	yes	maximal, complemented
3	#22558277	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558186	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1313	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863828	C₄	1	1	yes	characteristic
7	#16863812	—	2	2	no	—
8	#16863514	C₂²	1	2	yes	—
9	#16863226	C₄	1	1	yes	characteristic
10	#16863217	C₂²	1	2	yes	—
11	#16863204	C₄	1	2	yes	—
12	#16863179	—	2	2	no	—
13	#346×C₆₇	C₂²	1	1	yes	characteristic
14	#346×C₆₇	C₂²	1	1	yes	characteristic
15	#322×C₆₇	C₂²	1	1	yes	characteristic
16	#14524065	C₂×C₄	1	2	yes	—
17	#14524049	—	2	2	no	—
18	Dic₅₃₆	C₈	1	1	yes	characteristic, complemented
19	#14523988	C₈	1	2	yes	complemented
20	D₁₀₇₂	C₈	1	1	yes	characteristic, complemented
21	#14523983	—	2	2	no	—
22	#14523982	—	4	2	no	—
23	C₁₆×D₆₇	—	4	2	no	—
24	#14523979	C₂×C₄	1	2	yes	—
25	#14523971	—	2	2	no	—
26	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
27	C₂×C₁₆×C₆₇	C₂×C₄	1	1	yes	characteristic
28	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
29	C₄×C₈×C₆₇	—	2	1	no	—
30	C₄×C₈×C₆₇	C₂³	1	1	yes	characteristic
31	#11761628	—	2	2	no	—
32	Dic₂₆₈	C₂×C₈	1	2	yes	—
33	#11761617	—	2	2	no	—
34	D₅₃₆	C₂×C₈	1	2	yes	—
35	#11761612	—	2	2	no	—
36	#11761611	—	4	2	no	—
37	C₈×D₆₇	—	4	2	no	—
38	C₆₇⋊C₁₆	—	4	2	no	—
39	C₂×C₈×C₆₇	—	4	1	no	—
40	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
41	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
42	C₄²×C₆₇	#41	1	1	yes	characteristic
43	C₁₆×C₆₇	—	2	1	no	—
44	C₁₆×C₆₇	—	2	2	no	—
45	C₁₆×C₆₇	C₂×C₈	1	1	yes	characteristic
46	C₁₆×C₆₇	C₂×C₈	1	1	yes	characteristic
47	C₂×C₄×C₆₇	—	2	1	no	—
48	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
49	C₂²⋊D₆₇	—	4	2	no	—
50	D₂₆₈	—	2	2	no	—
51	C₄×D₆₇	—	4	2	no	—
52	Dic₁₃₄	—	2	2	no	—
53	C₈×C₆₇	—	8	1	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	—	4	2	no	—
58	C₂²×C₆₇	#444	1	1	yes	characteristic
59	C₂×D₆₇	—	4	2	no	—
60	C₄×C₆₇	—	4	1	no	—
61	C₄×C₆₇	C₈×D₄	1	1	yes	characteristic
62	C₄×C₆₇	#442	1	1	yes	characteristic
63	Dic₆₇	—	4	2	no	—
64	#12068	—	67	1	no	maximal, Sylow
65	C₂×C₆₇	—	2	1	no	—
66	C₂×C₆₇	#1580	1	1	yes	characteristic
67	D₆₇	—	8	2	no	—
68	#2179	—	67	1	no	—
69	#2171	—	67	2	no	—
70	#1369	—	67	1	no	—
71	#1336	—	67	2	no	—
72	#1313	D₆₇	1	1	yes	characteristic, complemented
73	C₆₇	#12068	1	1	yes	characteristic, complemented, Sylow
74	#508	—	67	1	no	—
75	#504	—	134	2	no	—
76	#443	—	67	2	no	—
77	#368	—	67	1	no	—
78	#364	—	67	2	no	—
79	#359	—	67	2	no	—
80	#350	—	134	2	no	—
81	#346	C₂×D₆₇	1	1	yes	characteristic
82	#346	C₂×D₆₇	1	1	yes	characteristic
83	#322	C₂×D₆₇	1	1	yes	characteristic
84	#135	—	67	2	no	—
85	#131	—	134	2	no	—
86	Q₃₂	—	67	1	no	—
87	SD₃₂	—	67	2	no	—
88	D₁₆	—	67	1	no	—
89	M₃₂	—	268	2	no	—
90	M₃₂	—	134	2	no	—
91	C₂×C₁₆	—	268	2	no	—
92	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
93	C₂×C₁₆	C₄×D₆₇	1	1	yes	characteristic
94	C₂×C₁₆	D₂₆₈	1	2	yes	—
95	#108	—	67	2	no	—
96	#104	—	134	2	no	—
97	C₄×C₈	—	2	1	no	—
98	C₄×C₈	C₂²×D₆₇	1	1	yes	characteristic
99	#41	—	134	2	no	—
100	Q₁₆	—	67	2	no	—
101	SD₁₆	—	134	2	no	—
102	D₈	—	67	2	no	—
103	M₁₆	—	268	2	no	—
104	M₁₆	—	134	2	no	—
105	C₂×C₈	—	4	1	no	—
106	C₂×C₈	—	268	2	no	—
107	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
108	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
109	C₄²	#11761628	1	1	yes	characteristic
110	C₁₆	—	2	1	no	—
111	C₁₆	—	2	2	no	—
112	C₁₆	—	268	2	no	—
113	C₁₆	C₈×D₆₇	1	1	yes	characteristic
114	C₁₆	C₈×D₆₇	1	1	yes	characteristic
115	Q₈	—	134	2	no	—
116	D₄	—	268	2	no	—
117	D₄	—	134	2	no	—
118	C₂×C₄	—	2	1	no	—
119	C₂×C₄	—	268	2	no	—
120	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
121	C₈	—	8	1	no	—
122	C₈	—	268	2	no	—
123	C₈	C₂×C₈×D₆₇	1	1	yes	characteristic
124	C₈	#14524049	1	1	yes	characteristic
125	C₈	#14524071	1	2	yes	—
126	C₂²	—	268	2	no	—
127	C₂²	#16863519	1	1	yes	characteristic
128	C₄	—	4	1	no	—
129	C₄	—	268	2	no	—
130	C₄	C₈×D₂₆₈	1	1	yes	characteristic
131	C₄	#16863509	1	1	yes	characteristic
132	C₂	—	2	1	no	—
133	C₂	—	536	2	no	—
134	C₂	#22559034	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22562096

C₂

yes

maximal, characteristic

#22562052

C₂

yes

maximal, complemented

#22558277

C₂

yes

maximal, characteristic, complemented

#22558186

C₂

yes

maximal, complemented

C₆₇×#1313

C₂

yes

maximal, characteristic, complemented

#16863828

C₄

yes

characteristic

#16863812

—

#16863514

C₂²

yes

—

#16863226

C₄

yes

characteristic

#16863217

C₂²

yes

—

#16863204

C₄

yes

—

#16863179

—

#346×C₆₇

C₂²

yes

characteristic

#346×C₆₇

C₂²

yes

characteristic

#322×C₆₇

C₂²

yes

characteristic

#14524065

C₂×C₄

yes

—

#14524049

—

Dic₅₃₆

C₈

yes

characteristic, complemented

#14523988

C₈

yes

complemented

D₁₀₇₂

C₈

yes

characteristic, complemented

#14523983

—

#14523982

—

C₁₆×D₆₇

—

#14523979

C₂×C₄

yes

—

#14523971

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₇

C₂×C₄

yes

characteristic

C₂×C₁₆×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

C₂³

yes

characteristic

#11761628

—

Dic₂₆₈

C₂×C₈

yes

—

#11761617

—

D₅₃₆

C₂×C₈

yes

—

#11761612

—

#11761611

—

C₈×D₆₇

—

C₆₇⋊C₁₆

—

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

—

C₁₆×C₆₇

C₂×C₈

yes

characteristic

C₁₆×C₆₇

C₂×C₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄×D₄

yes

characteristic

C₂²⋊D₆₇

—

D₂₆₈

—

C₄×D₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

C₂²×C₈

yes

C₈×C₆₇

yes

C₈×C₆₇

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

#444

yes

characteristic

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

C₈×D₄

yes

characteristic

C₄×C₆₇

#442

yes

characteristic

Dic₆₇

—

#12068

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1580

yes

characteristic

D₆₇

—

—

—

—

—

D₆₇

yes

characteristic, complemented

C₆₇

#12068

yes

characteristic, complemented, Sylow

#508

—

#504

—

134

—

—

—

—

—

—

134

—

#346

C₂×D₆₇

yes

characteristic

#346

C₂×D₆₇

yes

characteristic

#322

C₂×D₆₇

yes

characteristic

#135

—

#131

—

134

—

Q₃₂

—

SD₃₂

—

D₁₆

—

M₃₂

—

268

—

M₃₂

—

134

—

C₂×C₁₆

—

268

—

C₂×C₁₆

C₄×D₆₇

yes

characteristic

C₂×C₁₆

C₄×D₆₇

yes

characteristic

C₂×C₁₆

D₂₆₈

yes

—

#108

—

#104

—

134

—

C₄×C₈

—

C₄×C₈

C₂²×D₆₇

yes

characteristic

#41

—

134

—

100

Q₁₆

—

101

SD₁₆

—

134

—

102

D₈

—

103

M₁₆

—

268

—

104

M₁₆

—

134

—

105

C₂×C₈

—

106

C₂×C₈

—

268

—

107

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

108

C₂×C₈

C₂×D₂₆₈

yes

characteristic

109

C₄²

#11761628

yes

characteristic

110

C₁₆

—

111

C₁₆

—

112

C₁₆

—

268

—

113

C₁₆

C₈×D₆₇

yes

characteristic

114

C₁₆

C₈×D₆₇

yes

characteristic

115

Q₈

—

134

—

116

D₄

—

268

—

117

D₄

—

134

—

118

C₂×C₄

—

119

C₂×C₄

—

268

—

120

C₂×C₄

C₄×D₂₆₈

yes

characteristic

121

C₈

—

122

C₈

—

268

—

123

C₈

C₂×C₈×D₆₇

yes

characteristic

124

C₈

#14524049

yes

characteristic

125

C₈

#14524071

yes

—

126

C₂²

—

268

—

127

C₂²

#16863519

yes

characteristic

128

C₄

—

129

C₄

—

268

—

130

C₄

C₈×D₂₆₈

yes

characteristic

131

C₄

#16863509

yes

characteristic

132

C₂

—

133

C₂

—

536

—

134

C₂

#22559034

yes

characteristic

	subgroup	G/H
center	C₄	#16863509
commutator	C₈×C₆₇	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₇
Fitting	C₆₇×#1313	C₂
socle	C₂×C₆₇	#1580
prime radical	#16863828	C₄
cosocle	C₄×C₈×C₆₇	C₂³
hypercenter	#1313	D₆₇
hypocenter	C₆₇	#12068
LCS	G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₄ ◁ C₂×C₈ ◁ #346 ◁ #1313

subgroup

G/H

center

C₄

#16863509

commutator

C₈×C₆₇

C₂²×C₈

Frattini

C₄×C₈

C₂²×D₆₇

Fitting

C₆₇×#1313

C₂

C₂×C₆₇

C₄

C₄×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₄ ◁ C₂×C₈ ◁ #346 ◁ #1313

N	H ≅ G/N	type
2. #22562052	C₂	semidirect
3. #22558277	C₂	semidirect
4. #22558186	C₂	semidirect
5. C₆₇×#1313	C₂	semidirect
18. Dic₅₃₆	C₈	semidirect
19. #14523988	C₈	semidirect
20. D₁₀₇₂	C₈	semidirect
72. #1313	D₆₇	semidirect
73. C₆₇	#12068	semidirect

H ≅ G/N

type

2. #22562052

C₂

semidirect

3. #22558277

C₂

semidirect

4. #22558186

C₂

semidirect

5. C₆₇×#1313

C₂

semidirect

18. Dic₅₃₆

C₈

semidirect

19. #14523988

C₈

semidirect

20. D₁₀₇₂

C₈

semidirect

72. #1313

D₆₇

semidirect

73. C₆₇

#12068

semidirect

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

subgroup	index
#22562096	2
#22562052	2
#22558277	2
#22558186	2
C₆₇×#1313	2
#12068	67

subgroup

index

C₆₇×#1313

Subgroups — #30767722

Conjugacy classes of subgroups

Complements

Maximal subgroups