Group #30758900 — subgroups

Subgroups — #30758900

Order 17152.

484 conjugacy classes of subgroups; only the 93 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 93 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	#22561032	C₂	1	3	yes	maximal
2	#22558090	C₂	1	3	yes	maximal, complemented
3	C₆₇×#2844	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#14524040	C₂²	1	1	yes	characteristic
5	C₂×#14523963	—	2	3	no	—
6	C₂×#14523963	—	2	3	no	—
7	C₂×#14523954	C₂²	1	3	yes	—
8	#400×C₆₇	—	2	3	no	—
9	#395×C₆₇	—	2	3	no	—
10	#382×C₆₇	C₂²	1	3	yes	—
11	C₂²×D₂₆₈	—	2	3	no	—
12	C₂²×C₆₇⋊C₈	—	2	3	no	—
13	#14524040	—	2	3	no	—
14	#14524040	Q₈	1	2	yes	complemented
15	#14523963	—	4	6	no	—
16	#14523963	—	4	6	no	—
17	#14523954	—	2	6	no	—
18	C₂²×Q₈×C₆₇	—	2	1	no	—
19	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
20	C₂×C₄⋊C₄×C₆₇	D₄	1	3	yes	—
21	C₂×C₄⋊C₄×C₆₇	D₄	1	3	yes	—
22	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
23	#95×C₆₇	—	2	3	no	—
24	#95×C₆₇	—	2	1	no	—
25	#95×C₆₇	—	2	6	no	—
26	C₂³×D₆₇	—	4	1	no	—
27	C₂×D₂₆₈	—	2	6	no	—
28	C₂×D₂₆₈	—	4	3	no	—
29	C₂×D₂₆₈	—	4	3	no	—
30	C₂×D₂₆₈	—	4	3	no	—
31	C₂×D₂₆₈	—	4	3	no	—
32	C₂×D₂₆₈	—	4	3	no	—
33	C₂×C₆₇⋊C₈	—	2	6	no	—
34	C₂×C₆₇⋊C₈	—	4	3	no	—
35	C₂×C₆₇⋊C₈	—	4	3	no	—
36	C₂×Q₈×C₆₇	—	8	3	no	—
37	C₂²×C₄×C₆₇	—	2	3	no	—
38	C₂²×C₄×C₆₇	—	2	3	no	—
39	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
40	C₄⋊C₄×C₆₇	—	4	3	no	—
41	C₄⋊C₄×C₆₇	—	4	3	no	—
42	C₄⋊C₄×C₆₇	—	8	3	no	—
43	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
44	C₄²×C₆₇	#41	1	3	yes	—
45	C₂³×C₆₇	#120	1	1	yes	characteristic
46	C₂²×D₆₇	—	4	2	no	—
47	C₂²×D₆₇	—	8	3	no	—
48	C₂²×D₆₇	—	8	3	no	—
49	Q₈×C₆₇	—	8	4	no	—
50	C₂×C₄×C₆₇	—	2	3	no	—
51	C₂×C₄×C₆₇	—	2	3	no	—
52	C₂×C₄×C₆₇	—	4	3	no	—
53	C₂×C₄×C₆₇	—	4	3	no	—
54	C₂×C₄×C₆₇	—	4	6	no	—
55	C₂×C₄×C₆₇	—	4	6	no	—
56	C₂×C₄×C₆₇	C₂²⋊Q₈	1	3	yes	—
57	C₂×C₄×C₆₇	#124	1	3	yes	—
58	D₂₆₈	—	4	6	no	—
59	D₂₆₈	—	4	6	no	—
60	D₂₆₈	—	8	3	no	—
61	D₂₆₈	—	8	3	no	—
62	C₆₇⋊C₈	—	4	6	no	—
63	C₂²×C₆₇	#393	1	1	yes	characteristic
64	C₂²×C₆₇	#453	1	3	yes	—
65	C₂²×C₆₇	#454	1	3	yes	—
66	C₂×D₆₇	—	8	6	no	—
67	C₂×D₆₇	—	16	1	no	—
68	C₂×D₆₇	—	16	3	no	—
69	C₄×C₆₇	—	2	3	no	—
70	C₄×C₆₇	—	2	3	no	—
71	C₄×C₆₇	—	8	3	no	—
72	C₄×C₆₇	—	8	3	no	—
73	#10358	—	67	1	no	maximal, Sylow
74	C₂×C₆₇	#1655	1	1	yes	characteristic
75	C₂×C₆₇	#1657	1	3	yes	—
76	C₂×C₆₇	#2027	1	3	yes	—
77	D₆₇	—	16	2	no	—
78	#2844	D₆₇	1	1	yes	characteristic, complemented
79	#1961	—	67	3	no	—
80	#1295	—	67	3	no	—
81	C₆₇	#10358	1	1	yes	characteristic, complemented, Sylow
82	C₂×#127	—	67	1	no	—
83	C₂×#102	—	134	3	no	—
84	C₂×#102	—	134	3	no	—
85	C₂×#97	—	67	3	no	—
86	#400	—	2	3	no	—
87	#395	—	2	3	no	—
88	#382	C₂×D₆₇	1	3	yes	—
89	C₂²×Q₈	—	2	1	no	—
90	C₂²×D₄	—	134	3	no	—
91	C₂²×C₈	—	134	3	no	—
92	#127	—	67	2	no	—
93	#127	—	134	3	no	—
94	C₂×C₄⋊C₄	—	2	3	no	—
95	C₂×C₄⋊C₄	C₂²⋊D₆₇	1	3	yes	—
96	C₂×C₄⋊C₄	C₂²⋊D₆₇	1	3	yes	—
97	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
98	#102	—	268	6	no	—
99	#102	—	268	6	no	—
100	#97	—	134	6	no	—
101	#95	—	2	3	no	—
102	#95	—	2	1	no	—
103	#95	—	2	6	no	—
104	C₂⁴	—	268	1	no	—
105	C₂×Q₈	—	8	3	no	—
106	C₂×D₄	—	268	3	no	—
107	C₂×D₄	—	268	3	no	—
108	C₂×D₄	—	268	3	no	—
109	C₂×D₄	—	268	3	no	—
110	C₂×D₄	—	268	3	no	—
111	C₂×D₄	—	134	6	no	—
112	C₂²×C₄	—	2	3	no	—
113	C₂²×C₄	—	2	3	no	—
114	C₂²×C₄	C₂×C₂²⋊D₆₇	1	3	yes	—
115	C₂×C₈	—	268	3	no	—
116	C₂×C₈	—	268	3	no	—
117	C₂×C₈	—	134	6	no	—
118	C₄⋊C₄	—	4	3	no	—
119	C₄⋊C₄	—	4	3	no	—
120	C₄⋊C₄	—	8	3	no	—
121	C₄²	Q₈×D₆₇	1	1	yes	characteristic
122	C₄²	#11761629	1	3	yes	—
123	C₂³	—	268	2	no	—
124	C₂³	—	536	3	no	—
125	C₂³	—	536	3	no	—
126	C₂³	#14524012	1	1	yes	characteristic
127	Q₈	—	8	4	no	—
128	D₄	—	268	6	no	—
129	D₄	—	268	6	no	—
130	D₄	—	536	3	no	—
131	D₄	—	536	3	no	—
132	C₂×C₄	—	2	3	no	—
133	C₂×C₄	—	2	3	no	—
134	C₂×C₄	—	4	3	no	—
135	C₂×C₄	—	4	3	no	—
136	C₂×C₄	—	4	6	no	—
137	C₂×C₄	—	4	6	no	—
138	C₂×C₄	#14524019	1	3	yes	—
139	C₂×C₄	#14524032	1	3	yes	—
140	C₈	—	268	6	no	—
141	C₂²	—	536	6	no	—
142	C₂²	—	1072	1	no	—
143	C₂²	—	1072	3	no	—
144	C₂²	#16863307	1	1	yes	characteristic
145	C₂²	#16863564	1	3	yes	—
146	C₂²	#16863569	1	3	yes	—
147	C₄	—	2	3	no	—
148	C₄	—	2	3	no	—
149	C₄	—	8	3	no	—
150	C₄	—	8	3	no	—
151	C₂	—	1072	2	no	—
152	C₂	#22559498	1	1	yes	characteristic
153	C₂	#22559507	1	3	yes	—
154	C₂	#22561362	1	3	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

#22561032

C₂

yes

maximal

#22558090

C₂

yes

maximal, complemented

C₆₇×#2844

C₂

yes

maximal, characteristic, complemented

C₂×#14524040

C₂²

yes

—

—

C₂²

yes

—

#400×C₆₇

—

#395×C₆₇

—

#382×C₆₇

C₂²

yes

—

C₂²×D₂₆₈

—

C₂²×C₆₇⋊C₈

—

#14524040

—

#14524040

Q₈

yes

—

—

—

C₂²×Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

D₄

yes

—

C₂×C₄⋊C₄×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#95×C₆₇

—

#95×C₆₇

—

#95×C₆₇

—

C₂³×D₆₇

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂×Q₈

yes

characteristic

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

#120

yes

characteristic

C₂²×D₆₇

—

C₂²×D₆₇

—

C₂²×D₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

—

C₂×C₄×C₆₇

#124

yes

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₆₇⋊C₈

—

C₂²×C₆₇

#393

yes

characteristic

C₂²×C₆₇

#453

yes

—

C₂²×C₆₇

#454

yes

—

C₂×D₆₇

—

C₂×D₆₇

—

C₂×D₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

—

C₂×C₆₇

yes

C₂×C₆₇

yes

—

C₂×C₆₇

#2027

yes

—

D₆₇

—

#2844

D₆₇

yes

characteristic, complemented

#1961

—

#1295

—

C₆₇

#10358

yes

characteristic, complemented, Sylow

C₂×#127

—

C₂×#102

—

134

—

C₂×#102

—

134

—

C₂×#97

—

#400

—

#395

—

#382

C₂×D₆₇

yes

—

C₂²×Q₈

—

C₂²×D₄

—

134

—

C₂²×C₈

—

134

—

#127

—

#127

—

134

—

C₂×C₄⋊C₄

—

C₂×C₄⋊C₄

C₂²⋊D₆₇

yes

—

C₂×C₄⋊C₄

C₂²⋊D₆₇

yes

—

C₂×C₄²

C₂²×D₆₇

yes

characteristic

#102

—

268

—

#102

—

268

—

100

#97

—

134

—

101

#95

—

102

#95

—

103

#95

—

104

C₂⁴

—

268

—

105

C₂×Q₈

—

106

C₂×D₄

—

268

—

107

C₂×D₄

—

268

—

108

C₂×D₄

—

268

—

109

C₂×D₄

—

268

—

110

C₂×D₄

—

268

—

111

C₂×D₄

—

134

—

112

C₂²×C₄

—

113

C₂²×C₄

—

114

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

—

115

C₂×C₈

—

268

—

116

C₂×C₈

—

268

—

117

C₂×C₈

—

134

—

118

C₄⋊C₄

—

119

C₄⋊C₄

—

120

C₄⋊C₄

—

121

C₄²

Q₈×D₆₇

yes

characteristic

122

C₄²

#11761629

yes

—

123

C₂³

—

268

—

124

C₂³

—

536

—

125

C₂³

—

536

—

126

C₂³

#14524012

yes

characteristic

127

Q₈

—

128

D₄

—

268

—

129

D₄

—

268

—

130

D₄

—

536

—

131

D₄

—

536

—

132

C₂×C₄

—

133

C₂×C₄

—

134

C₂×C₄

—

135

C₂×C₄

—

136

C₂×C₄

—

137

C₂×C₄

—

138

C₂×C₄

#14524019

yes

—

139

C₂×C₄

#14524032

yes

—

140

C₈

—

268

—

141

C₂²

—

536

—

142

C₂²

—

1072

—

143

C₂²

—

1072

—

144

C₂²

#16863307

yes

characteristic

145

C₂²

#16863564

yes

—

146

C₂²

#16863569

yes

—

147

C₄

—

148

C₄

—

149

C₄

—

150

C₄

—

151

C₂

—

1072

—

152

C₂

#22559498

yes

characteristic

153

C₂

#22559507

yes

—

154

C₂

#22561362

yes

—

	subgroup	G/H
center	C₂³	#14524012
commutator, cosocle	C₂×C₄²×C₆₇	C₂³
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2844	C₂
socle	C₂³×C₆₇	#120
prime radical	C₂×#14524040	C₂²
hypercenter	#2844	D₆₇
hypocenter	C₆₇	#10358
LCS	G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2844

subgroup

G/H

center

C₂³

#14524012

commutator, cosocle

C₂×C₄²×C₆₇

C₂³

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#2844

C₂

C₂³×C₆₇

C₂²

D₆₇

C₆₇

G ▷ C₂×C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #2844

N	H ≅ G/N	type
2. #22558090	C₂	semidirect
3. C₆₇×#2844	C₂	semidirect
14. #14524040	Q₈	semidirect
78. #2844	D₆₇	semidirect
81. C₆₇	#10358	semidirect

H ≅ G/N

type

2. #22558090

C₂

semidirect

3. C₆₇×#2844

C₂

semidirect

14. #14524040

Q₈

semidirect

78. #2844

D₆₇

semidirect

81. C₆₇

#10358

semidirect

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

subgroup	index
#22561032	2
#22558090	2
C₆₇×#2844	2
#10358	67

subgroup

index

#22561032

#22558090

C₆₇×#2844

#10358

Subgroups — #30758900

Conjugacy classes of subgroups

Complements

Maximal subgroups