Group #30749649 — subgroups

Subgroups — #30749649

Order 17152.

306 conjugacy classes of subgroups; only the 155 normal subgroups are shown.

normal subgroups only

This group has 306 conjugacy classes of subgroups — too many to draw. Only its 155 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	#22560347	C₂	1	2	yes	maximal
2	#22558968	C₂	1	2	yes	maximal
3	#22558965	C₂	1	2	yes	maximal
4	C₂×#322×C₆₇	C₂	1	1	yes	maximal, characteristic
5	#16863420	C₂²	1	4	yes	—
6	#16863161	—	2	4	no	—
7	#16863149	—	2	4	no	—
8	#16863147	—	2	4	no	—
9	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
12	#322×C₆₇	C₄	1	4	yes	—
13	C₂×#11761612	—	2	4	no	—
14	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
15	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
16	C₂²×C₈×C₆₇	D₄	1	2	yes	—
17	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
18	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
19	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
20	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
21	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
22	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₆₇	D₄	1	2	yes	—
24	C₄×C₈×C₆₇	D₄	1	2	yes	—
25	C₄×C₈×C₆₇	Q₈	1	2	yes	—
26	#11761612	—	4	8	no	—
27	C₂×C₆₇⋊C₈	—	4	4	no	—
28	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
30	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
31	C₂×C₈×C₆₇	—	2	4	no	—
32	C₂×C₈×C₆₇	—	4	2	no	—
33	C₂×C₈×C₆₇	—	4	2	no	—
34	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
35	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
36	C₂×C₈×C₆₇	C₄⋊C₄	1	4	yes	—
37	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
38	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
39	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
40	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
41	C₂³×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
42	C₂×C₄×C₆₇	—	2	2	no	—
43	C₂×C₄×C₆₇	—	2	2	no	—
44	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
45	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
48	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
49	C₂×C₄×C₆₇	C₄⋊Q₈	1	1	yes	characteristic
50	C₂×C₄×C₆₇	#135	1	4	yes	—
51	C₈×C₆₇	—	2	4	no	—
52	C₈×C₆₇	—	2	4	no	—
53	C₈×C₆₇	—	2	4	no	—
54	C₆₇⋊C₈	—	8	4	no	—
55	C₂²×C₆₇	—	2	1	no	—
56	C₂²×C₆₇	—	2	1	no	—
57	C₂²×C₆₇	#384	1	1	yes	characteristic
58	C₂²×C₆₇	#499	1	2	yes	—
59	C₄×C₆₇	—	2	1	no	—
60	C₄×C₆₇	—	2	1	no	—
61	C₄×C₆₇	#427	1	2	yes	—
62	C₄×C₆₇	#430	1	2	yes	—
63	C₄×C₆₇	#430	1	2	yes	—
64	C₄×C₆₇	#495	1	2	yes	—
65	#8827	—	67	1	no	maximal, Sylow
66	C₂×C₆₇	—	2	2	no	—
67	C₂×C₆₇	#1846	1	1	yes	characteristic
68	C₂×C₆₇	#1849	1	1	yes	characteristic
69	C₂×C₆₇	#1849	1	1	yes	characteristic
70	#1837	—	67	2	no	—
71	#1567	—	67	2	no	—
72	#1566	—	67	2	no	—
73	C₂×#322	D₆₇	1	1	yes	characteristic
74	C₆₇	#8827	1	1	yes	characteristic, complemented, Sylow
75	#423	—	67	4	no	—
76	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
77	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
78	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
79	#341	—	134	4	no	—
80	#335	—	134	4	no	—
81	#334	—	134	4	no	—
82	#322	Dic₆₇	1	4	yes	—
83	C₂×M₁₆	—	134	4	no	—
84	C₂²×C₈	Dic₁₃₄	1	2	yes	—
85	C₂²×C₈	Dic₁₃₄	1	2	yes	—
86	C₂²×C₈	D₂₆₈	1	2	yes	—
87	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
88	C₄⋊C₈	—	134	4	no	—
89	C₄⋊C₈	—	134	4	no	—
90	C₄×C₈	Dic₁₃₄	1	2	yes	—
91	C₄×C₈	D₂₆₈	1	2	yes	—
92	C₄×C₈	D₂₆₈	1	2	yes	—
93	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
94	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
95	C₄×C₈	C₂×Dic₆₇	1	2	yes	—
96	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
97	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
98	C₂²×C₄	C₂×Dic₁₃₄	1	1	yes	characteristic
99	M₁₆	—	268	8	no	—
100	C₂×C₈	—	2	4	no	—
101	C₂×C₈	—	4	2	no	—
102	C₂×C₈	—	4	2	no	—
103	C₂×C₈	—	268	4	no	—
104	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
105	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
106	C₂×C₈	C₄⋊Dic₆₇	1	4	yes	—
107	C₄²	C₂²×Dic₆₇	1	1	yes	characteristic
108	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
109	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
110	C₄²	C₂×Dic₁₃₄	1	1	yes	characteristic
111	C₂³	#14524043	1	1	yes	characteristic
112	C₂×C₄	—	2	2	no	—
113	C₂×C₄	—	2	2	no	—
114	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
115	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
116	C₂×C₄	C₂×C₄⋊Dic₆₇	1	1	yes	characteristic
117	C₂×C₄	#14524040	1	1	yes	characteristic
118	C₂×C₄	#14524043	1	1	yes	characteristic
119	C₂×C₄	#14524043	1	1	yes	characteristic
120	C₂×C₄	#14524065	1	4	yes	—
121	C₈	—	2	4	no	—
122	C₈	—	2	4	no	—
123	C₈	—	2	4	no	—
124	C₈	—	536	4	no	—
125	C₂²	—	2	1	no	—
126	C₂²	—	2	1	no	—
127	C₂²	#16863266	1	1	yes	characteristic
128	C₂²	#16863798	1	2	yes	—
129	C₄	—	2	1	no	—
130	C₄	—	2	1	no	—
131	C₄	#16863434	1	2	yes	—
132	C₄	#16863445	1	2	yes	—
133	C₄	#16863445	1	2	yes	—
134	C₄	#16863780	1	2	yes	—
135	C₂	—	2	2	no	—
136	C₂	#22560373	1	1	yes	characteristic
137	C₂	#22560384	1	1	yes	characteristic
138	C₂	#22560384	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560347

C₂

yes

maximal

#22558968

C₂

yes

maximal

#22558965

C₂

yes

maximal

C₂×#322×C₆₇

C₂

yes

maximal, characteristic

#16863420

C₂²

yes

—

#16863161

—

#16863149

—

#16863147

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#322×C₆₇

C₄

yes

—

C₂×#11761612

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂²×C₈×C₆₇

Q₈

yes

—

C₂²×C₈×C₆₇

Q₈

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

Q₈

yes

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₂³×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₄⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

#135

yes

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#384

yes

characteristic

C₂²×C₆₇

#499

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#427

yes

—

C₄×C₆₇

#430

yes

—

C₄×C₆₇

#430

yes

—

C₄×C₆₇

#495

yes

—

#8827

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

—

—

—

D₆₇

yes

characteristic

C₆₇

#8827

yes

characteristic, complemented, Sylow

#423

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#341

—

134

—

#335

—

134

—

#334

—

134

—

#322

Dic₆₇

yes

—

C₂×M₁₆

—

134

—

C₂²×C₈

Dic₁₃₄

yes

—

C₂²×C₈

Dic₁₃₄

yes

—

C₂²×C₈

D₂₆₈

yes

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

C₄⋊C₈

—

134

—

C₄⋊C₈

—

134

—

C₄×C₈

Dic₁₃₄

yes

—

C₄×C₈

D₂₆₈

yes

—

C₄×C₈

D₂₆₈

yes

—

C₄×C₈

C₂×Dic₆₇

yes

—

C₄×C₈

C₂×Dic₆₇

yes

—

C₄×C₈

C₂×Dic₆₇

yes

—

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

C₂²×C₄

C₂×Dic₁₃₄

yes

characteristic

C₂²×C₄

C₂×Dic₁₃₄

yes

characteristic

M₁₆

—

268

—

100

C₂×C₈

—

101

C₂×C₈

—

102

C₂×C₈

—

103

C₂×C₈

—

268

—

104

C₂×C₈

C₄⋊Dic₆₇

yes

—

105

C₂×C₈

C₄⋊Dic₆₇

yes

—

106

C₂×C₈

C₄⋊Dic₆₇

yes

—

107

C₄²

C₂²×Dic₆₇

yes

characteristic

108

C₄²

C₂×D₂₆₈

yes

characteristic

109

C₄²

C₂×D₂₆₈

yes

characteristic

110

C₄²

C₂×Dic₁₃₄

yes

characteristic

111

C₂³

#14524043

yes

characteristic

112

C₂×C₄

—

113

C₂×C₄

—

114

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

115

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

116

C₂×C₄

C₂×C₄⋊Dic₆₇

yes

characteristic

117

C₂×C₄

#14524040

yes

characteristic

118

C₂×C₄

#14524043

yes

characteristic

119

C₂×C₄

#14524043

yes

characteristic

120

C₂×C₄

#14524065

yes

—

121

C₈

—

122

C₈

—

123

C₈

—

124

C₈

—

536

—

125

C₂²

—

126

C₂²

—

127

C₂²

#16863266

yes

characteristic

128

C₂²

#16863798

yes

—

129

C₄

—

130

C₄

—

131

C₄

#16863434

yes

—

132

C₄

#16863445

yes

—

133

C₄

#16863445

yes

—

134

C₄

#16863780

yes

—

135

C₂

—

136

C₂

#22560373

yes

characteristic

137

C₂

#22560384

yes

characteristic

138

C₂

#22560384

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14524040
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#322×C₆₇	C₂
socle	C₂²×C₆₇	#384
prime radical	C₂³×C₆₇	C₄⋊Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#322	D₆₇
hypocenter	C₆₇	#8827
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

subgroup

G/H

center

C₂×C₄

#14524040

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₂×#322×C₆₇

C₂

socle

C₂²×C₆₇

#384

prime radical

C₂³×C₆₇

C₄⋊Q₈

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×#322

N	H ≅ G/N	type
74. C₆₇	#8827	semidirect

H ≅ G/N

type

74. C₆₇

#8827

semidirect

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

subgroup	index
#22560347	2
#22558968	2
#22558965	2
C₂×#322×C₆₇	2
#8827	67

subgroup

index

C₂×#322×C₆₇

Subgroups — #30749649

Conjugacy classes of subgroups

Complements

Maximal subgroups