Group #30755223 — subgroups

Subgroups — #30755223

Order 17152.

370 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 115 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	#22570723	C₂	1	1	yes	maximal, characteristic
2	#22560340	C₂	1	1	yes	maximal, characteristic
3	#22558774	C₂	1	4	yes	maximal
4	C₆₇×#1452	C₂	1	1	yes	maximal, characteristic
5	#16863953	C₂²	1	2	yes	—
6	#16863801	C₄	1	2	yes	complemented
7	#16863798	—	2	2	no	—
8	#16863794	C₄	1	2	yes	complemented
9	#16863434	—	2	2	no	—
10	#16863262	C₂²	1	2	yes	—
11	#16863159	—	2	2	no	—
12	#16863134	—	2	8	no	—
13	#431×C₆₇	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
15	#322×C₆₇	—	2	2	no	—
16	#14524043	C₂×C₄	1	4	yes	—
17	#14524035	—	2	2	no	—
18	#14524022	—	2	4	no	—
19	C₄×Dic₁₃₄	—	2	4	no	—
20	#14524002	—	2	2	no	—
21	C₂×C₄⋊Dic₆₇	—	2	2	no	—
22	#14523977	—	2	4	no	—
23	#14523975	—	2	4	no	—
24	C₆₇⋊#95	—	2	4	no	—
25	C₂²×C₈×C₆₇	D₄	1	2	yes	—
26	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
27	C₄⋊C₈×C₆₇	—	2	2	no	—
28	C₂²⋊C₈×C₆₇	D₄	1	4	yes	—
29	C₄×C₈×C₆₇	—	2	2	no	—
30	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₆₇	D₄	1	2	yes	—
33	C₂×Dic₁₃₄	—	2	4	no	—
34	C₂²×Dic₆₇	—	4	2	no	—
35	C₄⋊Dic₆₇	—	2	4	no	—
36	C₄⋊Dic₆₇	—	2	4	no	—
37	C₄⋊Dic₆₇	—	4	4	no	—
38	Dic₆₇⋊C₄	—	4	4	no	—
39	C₂²⋊Dic₆₇	—	4	2	no	—
40	C₄×Dic₆₇	—	4	2	no	—
41	C₂²×C₄×C₆₇	C₂×D₄	1	2	yes	—
42	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
43	C₂×C₈×C₆₇	—	2	4	no	—
44	C₂×C₈×C₆₇	—	2	4	no	—
45	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
46	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
47	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
48	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
49	C₄²×C₆₇	#41	1	2	yes	—
50	C₂³×C₆₇	#127	1	1	yes	characteristic
51	C₂×C₄×C₆₇	—	4	2	no	—
52	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
53	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
54	C₂×C₄×C₆₇	#121	1	2	yes	—
55	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
56	C₂×C₄×C₆₇	#137	1	4	yes	—
57	C₂×Dic₆₇	—	4	4	no	—
58	C₂×Dic₆₇	—	8	4	no	—
59	C₂×Dic₆₇	—	8	2	no	—
60	Dic₁₃₄	—	8	4	no	—
61	C₈×C₆₇	—	2	2	no	—
62	C₈×C₆₇	—	2	2	no	—
63	C₈×C₆₇	—	4	4	no	—
64	C₂²×C₆₇	—	2	2	no	—
65	C₂²×C₆₇	#390	1	1	yes	characteristic
66	C₂²×C₆₇	#497	1	2	yes	—
67	C₄×C₆₇	—	2	2	no	—
68	C₄×C₆₇	—	2	2	no	—
69	C₄×C₆₇	#419	1	2	yes	—
70	C₄×C₆₇	#491	1	2	yes	—
71	Dic₆₇	—	8	4	no	—
72	Dic₆₇	—	16	2	no	—
73	#9759	—	67	1	no	maximal, Sylow
74	C₂×C₆₇	—	2	2	no	—
75	C₂×C₆₇	#1960	1	1	yes	characteristic
76	C₂×C₆₇	#1976	1	2	yes	—
77	#3160	—	67	1	no	—
78	#1835	—	67	1	no	—
79	#1518	—	67	4	no	—
80	#1452	D₆₇	1	1	yes	characteristic
81	C₆₇	#9759	1	1	yes	characteristic, complemented, Sylow
82	#533	—	67	2	no	—
83	#500	—	67	2	no	—
84	#499	—	134	2	no	—
85	#498	—	67	2	no	—
86	#431	C₂×D₆₇	1	2	yes	—
87	#427	—	134	2	no	—
88	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
89	#382	—	67	2	no	—
90	#340	—	134	2	no	—
91	#328	—	134	8	no	—
92	#322	—	2	2	no	—
93	C₂²×C₈	D₂₆₈	1	2	yes	—
94	C₄⋊Q₈	—	67	4	no	—
95	#125	—	134	2	no	—
96	C₂²⋊Q₈	—	134	4	no	—
97	C₄×Q₈	—	134	4	no	—
98	#117	—	134	2	no	—
99	C₂×C₄⋊C₄	—	134	2	no	—
100	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
101	#107	—	134	4	no	—
102	#106	—	134	4	no	—
103	C₄⋊C₈	—	2	2	no	—
104	C₂²⋊C₈	D₂₆₈	1	4	yes	—
105	C₄×C₈	—	2	2	no	—
106	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
107	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
108	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
109	#95	—	134	4	no	—
110	C₂×Q₈	—	134	4	no	—
111	C₂²×C₄	—	268	2	no	—
112	C₂²×C₄	C₂×D₂₆₈	1	2	yes	—
113	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
114	C₂×C₈	—	2	4	no	—
115	C₂×C₈	—	2	4	no	—
116	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
117	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
118	C₄⋊C₄	—	268	4	no	—
119	C₄⋊C₄	—	268	4	no	—
120	C₄⋊C₄	—	134	4	no	—
121	C₄⋊C₄	—	134	4	no	—
122	C₂²⋊C₄	—	268	2	no	—
123	C₄²	—	268	2	no	—
124	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
125	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
126	C₄²	#11761628	1	2	yes	—
127	C₂³	#14524040	1	1	yes	characteristic
128	Q₈	—	536	4	no	—
129	C₂×C₄	—	4	2	no	—
130	C₂×C₄	—	268	4	no	—
131	C₂×C₄	—	536	4	no	—
132	C₂×C₄	—	536	2	no	—
133	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
134	C₂×C₄	C₄×D₂₆₈	1	2	yes	—
135	C₂×C₄	#14524017	1	2	yes	—
136	C₂×C₄	#14524031	1	1	yes	characteristic
137	C₂×C₄	#14524076	1	4	yes	—
138	C₈	—	2	2	no	—
139	C₈	—	2	2	no	—
140	C₈	—	4	4	no	—
141	C₂²	—	2	2	no	—
142	C₂²	#16863298	1	1	yes	characteristic
143	C₂²	#16863790	1	2	yes	—
144	C₄	—	2	2	no	—
145	C₄	—	2	2	no	—
146	C₄	—	536	4	no	—
147	C₄	—	1072	2	no	—
148	C₄	#16863406	1	2	yes	—
149	C₄	#16863765	1	2	yes	—
150	C₂	—	2	2	no	—
151	C₂	#22561029	1	1	yes	characteristic
152	C₂	#22561107	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22570723

C₂

yes

maximal, characteristic

#22560340

C₂

yes

maximal, characteristic

#22558774

C₂

yes

maximal

C₆₇×#1452

C₂

yes

maximal, characteristic

#16863953

C₂²

yes

—

#16863801

C₄

yes

complemented

#16863798

—

#16863794

C₄

yes

complemented

#16863434

—

#16863262

C₂²

yes

—

#16863159

—

#16863134

—

#431×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#322×C₆₇

—

#14524043

C₂×C₄

yes

—

#14524035

—

#14524022

—

C₄×Dic₁₃₄

—

#14524002

—

C₂×C₄⋊Dic₆₇

—

#14523977

—

#14523975

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₂²⋊C₈×C₆₇

D₄

yes

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

#127

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

#124

yes

characteristic

C₂×C₄×C₆₇

#137

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

#390

yes

characteristic

C₂²×C₆₇

#497

yes

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

#419

yes

—

C₄×C₆₇

#491

yes

—

Dic₆₇

—

Dic₆₇

—

#9759

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

#1960

yes

characteristic

C₂×C₆₇

#1976

yes

—

—

—

—

D₆₇

yes

characteristic

C₆₇

#9759

yes

characteristic, complemented, Sylow

#533

—

#500

—

#499

—

134

—

#498

—

#431

C₂×D₆₇

yes

—

#427

—

134

—

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

#382

—

#340

—

134

—

#328

—

134

—

#322

—

C₂²×C₈

D₂₆₈

yes

—

C₄⋊Q₈

—

#125

—

134

—

C₂²⋊Q₈

—

134

—

C₄×Q₈

—

134

—

#117

—

134

—

C₂×C₄⋊C₄

—

134

—

100

C₂×C₄²

C₂²×D₆₇

yes

characteristic

101

#107

—

134

—

102

#106

—

134

—

103

C₄⋊C₈

—

104

C₂²⋊C₈

D₂₆₈

yes

—

105

C₄×C₈

—

106

C₄×C₈

C₄×D₆₇

yes

characteristic

107

C₄×C₈

C₄×D₆₇

yes

characteristic

108

C₄×C₈

C₂²⋊D₆₇

yes

—

109

#95

—

134

—

110

C₂×Q₈

—

134

—

111

C₂²×C₄

—

268

—

112

C₂²×C₄

C₂×D₂₆₈

yes

—

113

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

114

C₂×C₈

—

115

C₂×C₈

—

116

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

117

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

118

C₄⋊C₄

—

268

—

119

C₄⋊C₄

—

268

—

120

C₄⋊C₄

—

134

—

121

C₄⋊C₄

—

134

—

122

C₂²⋊C₄

—

268

—

123

C₄²

—

268

—

124

C₄²

C₂×C₄×D₆₇

yes

characteristic

125

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

126

C₄²

#11761628

yes

—

127

C₂³

#14524040

yes

characteristic

128

Q₈

—

536

—

129

C₂×C₄

—

130

C₂×C₄

—

268

—

131

C₂×C₄

—

536

—

132

C₂×C₄

—

536

—

133

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

134

C₂×C₄

C₄×D₂₆₈

yes

—

135

C₂×C₄

#14524017

yes

—

136

C₂×C₄

#14524031

yes

characteristic

137

C₂×C₄

#14524076

yes

—

138

C₈

—

139

C₈

—

140

C₈

—

141

C₂²

—

142

C₂²

#16863298

yes

characteristic

143

C₂²

#16863790

yes

—

144

C₄

—

145

C₄

—

146

C₄

—

536

—

147

C₄

—

1072

—

148

C₄

#16863406

yes

—

149

C₄

#16863765

yes

—

150

C₂

—

151

C₂

#22561029

yes

characteristic

152

C₂

#22561107

yes

—

	subgroup	G/H
center	C₂²	#16863298
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1452	C₂
socle	C₂²×C₆₇	#390
prime radical	C₂³×C₆₇	#127
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1452	D₆₇
hypocenter	C₆₇	#9759
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

subgroup

G/H

center

C₂²

#16863298

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1452

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1452

N	H ≅ G/N	type
6. #16863801	C₄	semidirect
8. #16863794	C₄	semidirect
81. C₆₇	#9759	semidirect

H ≅ G/N

type

6. #16863801

C₄

semidirect

8. #16863794

C₄

semidirect

81. C₆₇

#9759

semidirect

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

subgroup	index
#22570723	2
#22560340	2
#22558774	2
C₆₇×#1452	2
#9759	67

subgroup

index

C₆₇×#1452

Subgroups — #30755223

Conjugacy classes of subgroups

Complements

Maximal subgroups