Group #30775515 — subgroups

Subgroups — #30775515

Order 17152.

416 conjugacy classes of subgroups; only the 81 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 81 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	#22560423	C₂	1	3	yes	maximal
2	#22558336	C₂	1	3	yes	maximal
3	C₆₇×#2811	C₂	1	1	yes	maximal, characteristic
4	#16863874	C₂²	1	1	yes	characteristic
5	#16863436	—	2	3	no	—
6	C₂×#14523962	C₂²	1	3	yes	—
7	#16863156	—	2	3	no	—
8	#395×C₆₇	—	2	3	no	—
9	#391×C₆₇	—	2	3	no	—
10	#391×C₆₇	C₂²	1	3	yes	—
11	C₂×#11761612	—	2	3	no	—
12	Q₈×Dic₆₇	—	4	1	no	—
13	Q₈×Dic₆₇	—	4	1	no	—
14	#14524002	—	2	3	no	—
15	#14523976	—	4	3	no	—
16	#14523974	—	4	3	no	—
17	#14523962	—	2	3	no	—
18	#14523962	Q₈	1	6	yes	complemented
19	C₂²×Q₈×C₆₇	—	2	1	no	—
20	C₂²×Q₈×C₆₇	—	2	1	no	—
21	C₂²×Q₈×C₆₇	C₂³	1	1	yes	characteristic
22	C₂×C₄⋊C₄×C₆₇	—	2	3	no	—
23	C₂×C₄⋊C₄×C₆₇	D₄	1	3	yes	—
24	C₂×C₄²×C₆₇	D₄	1	3	yes	—
25	#95×C₆₇	—	2	3	no	—
26	#95×C₆₇	—	2	6	no	—
27	#11761612	—	2	6	no	—
28	#11761612	—	4	3	no	—
29	C₂×C₆₇⋊C₈	—	4	3	no	—
30	C₄⋊Dic₆₇	—	2	3	no	—
31	C₄⋊Dic₆₇	—	2	3	no	—
32	C₂²⋊Dic₆₇	—	4	1	no	—
33	C₄×Dic₆₇	—	4	3	no	—
34	C₂×Q₈×C₆₇	—	2	4	no	—
35	C₂×Q₈×C₆₇	—	4	1	no	—
36	C₂×Q₈×C₆₇	—	4	1	no	—
37	C₂×Q₈×C₆₇	—	8	1	no	—
38	C₂×Q₈×C₆₇	—	8	1	no	—
39	C₂×Q₈×C₆₇	—	8	2	no	—
40	C₂×Q₈×C₆₇	—	8	1	no	—
41	C₂×Q₈×C₆₇	C₂×Q₈	1	3	yes	—
42	C₂×Q₈×C₆₇	#41	1	1	yes	characteristic
43	C₂²×C₄×C₆₇	—	2	3	no	—
44	C₂²×C₄×C₆₇	—	2	3	no	—
45	C₂²×C₄×C₆₇	C₂×D₄	1	3	yes	—
46	C₄⋊C₄×C₆₇	—	4	3	no	—
47	C₄⋊C₄×C₆₇	—	8	3	no	—
48	C₄²×C₆₇	—	4	3	no	—
49	C₂³×C₆₇	#120	1	1	yes	characteristic
50	Q₈×C₆₇	—	4	3	no	—
51	Q₈×C₆₇	—	4	1	no	—
52	Q₈×C₆₇	—	8	4	no	—
53	Q₈×C₆₇	—	8	4	no	—
54	C₂×C₄×C₆₇	—	2	3	no	—
55	C₂×C₄×C₆₇	—	2	3	no	—
56	C₂×C₄×C₆₇	—	2	6	no	—
57	C₂×C₄×C₆₇	—	4	3	no	—
58	C₂×C₄×C₆₇	—	4	3	no	—
59	C₂×C₄×C₆₇	—	4	6	no	—
60	C₂×C₄×C₆₇	—	4	3	no	—
61	C₂×C₄×C₆₇	C₂²⋊Q₈	1	3	yes	—
62	C₂×C₄×C₆₇	C₄⋊Q₈	1	3	yes	—
63	C₂×Dic₆₇	—	4	1	no	—
64	C₂×Dic₆₇	—	4	1	no	—
65	C₆₇⋊C₈	—	4	6	no	—
66	C₂²×C₆₇	—	2	1	no	—
67	C₂²×C₆₇	—	2	1	no	—
68	C₂²×C₆₇	#395	1	1	yes	characteristic
69	C₂²×C₆₇	B(4,𝔽₂)	1	1	yes	characteristic
70	C₂²×C₆₇	#458	1	1	yes	characteristic
71	C₄×C₆₇	—	2	3	no	—
72	C₄×C₆₇	—	2	3	no	—
73	C₄×C₆₇	—	4	6	no	—
74	C₄×C₆₇	—	8	3	no	—
75	Dic₆₇	—	8	1	no	—
76	Dic₆₇	—	8	1	no	—
77	#13464	—	67	1	no	maximal, Sylow
78	C₂×C₆₇	—	2	1	no	—
79	C₂×C₆₇	—	2	1	no	—
80	C₂×C₆₇	#2062	1	1	yes	characteristic
81	C₂×C₆₇	#2202	1	1	yes	characteristic
82	C₂×C₆₇	#2205	1	1	yes	characteristic
83	#2811	D₆₇	1	1	yes	characteristic
84	#1857	—	67	3	no	—
85	#1394	—	67	3	no	—
86	C₆₇	#13464	1	1	yes	characteristic, complemented, Sylow
87	#519	—	67	1	no	—
88	#428	—	134	3	no	—
89	C₂×#101	—	67	3	no	—
90	#395	—	2	3	no	—
91	#391	—	2	3	no	—
92	#391	C₂×D₆₇	1	3	yes	—
93	#339	—	134	3	no	—
94	C₂²×Q₈	—	2	1	no	—
95	C₂²×Q₈	—	2	1	no	—
96	C₂²×Q₈	C₂²×D₆₇	1	1	yes	characteristic
97	C₂×M₁₆	—	134	3	no	—
98	C₄×Q₈	—	268	1	no	—
99	C₄×Q₈	—	268	1	no	—
100	#117	—	134	3	no	—
101	C₂×C₄⋊C₄	—	2	3	no	—
102	C₂×C₄⋊C₄	C₂²⋊D₆₇	1	3	yes	—
103	C₂×C₄²	C₂²⋊D₆₇	1	3	yes	—
104	#107	—	268	3	no	—
105	#106	—	268	3	no	—
106	#101	—	67	6	no	—
107	#101	—	134	3	no	—
108	#95	—	2	3	no	—
109	#95	—	2	6	no	—
110	C₂×Q₈	—	2	4	no	—
111	C₂×Q₈	—	4	1	no	—
112	C₂×Q₈	—	4	1	no	—
113	C₂×Q₈	—	8	1	no	—
114	C₂×Q₈	—	8	1	no	—
115	C₂×Q₈	—	8	2	no	—
116	C₂×Q₈	—	8	1	no	—
117	C₂×Q₈	Q₈×D₆₇	1	3	yes	—
118	C₂×Q₈	#11761629	1	1	yes	characteristic
119	C₂²×C₄	—	2	3	no	—
120	C₂²×C₄	—	2	3	no	—
121	C₂²×C₄	C₂×C₂²⋊D₆₇	1	3	yes	—
122	M₁₆	—	268	3	no	—
123	M₁₆	—	134	6	no	—
124	C₂×C₈	—	268	3	no	—
125	C₄⋊C₄	—	4	3	no	—
126	C₄⋊C₄	—	8	3	no	—
127	C₄⋊C₄	—	134	3	no	—
128	C₄⋊C₄	—	134	3	no	—
129	C₂²⋊C₄	—	268	1	no	—
130	C₄²	—	4	3	no	—
131	C₄²	—	268	3	no	—
132	C₂³	#14524012	1	1	yes	characteristic
133	Q₈	—	4	3	no	—
134	Q₈	—	4	1	no	—
135	Q₈	—	8	4	no	—
136	Q₈	—	8	4	no	—
137	C₂×C₄	—	2	3	no	—
138	C₂×C₄	—	2	3	no	—
139	C₂×C₄	—	2	6	no	—
140	C₂×C₄	—	4	3	no	—
141	C₂×C₄	—	4	3	no	—
142	C₂×C₄	—	4	6	no	—
143	C₂×C₄	—	4	3	no	—
144	C₂×C₄	—	268	1	no	—
145	C₂×C₄	—	268	1	no	—
146	C₂×C₄	#14524019	1	3	yes	—
147	C₂×C₄	#14524041	1	3	yes	—
148	C₈	—	268	6	no	—
149	C₂²	—	2	1	no	—
150	C₂²	—	2	1	no	—
151	C₂²	#16863315	1	1	yes	characteristic
152	C₂²	#16863583	1	1	yes	characteristic
153	C₂²	#16863586	1	1	yes	characteristic
154	C₄	—	2	3	no	—
155	C₄	—	2	3	no	—
156	C₄	—	4	6	no	—
157	C₄	—	8	3	no	—
158	C₄	—	536	1	no	—
159	C₄	—	536	1	no	—
160	C₂	—	2	1	no	—
161	C₂	—	2	1	no	—
162	C₂	#22561561	1	1	yes	characteristic
163	C₂	#22562226	1	1	yes	characteristic
164	C₂	#22562239	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22560423

C₂

yes

maximal

#22558336

C₂

yes

maximal

C₆₇×#2811

C₂

yes

maximal, characteristic

#16863874

C₂²

yes

characteristic

#16863436

—

C₂×#14523962

C₂²

yes

—

#16863156

—

#395×C₆₇

—

#391×C₆₇

—

#391×C₆₇

C₂²

yes

—

C₂×#11761612

—

Q₈×Dic₆₇

—

Q₈×Dic₆₇

—

—

—

—

—

Q₈

yes

complemented

C₂²×Q₈×C₆₇

—

C₂²×Q₈×C₆₇

—

C₂²×Q₈×C₆₇

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

D₄

yes

—

#95×C₆₇

—

#95×C₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

C₂×Q₈

yes

—

C₂×Q₈×C₆₇

#41

yes

characteristic

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

#120

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×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

C₄⋊Q₈

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

C₄×C₆₇

—

Dic₆₇

—

Dic₆₇

—

#13464

—

maximal, Sylow

C₂×C₆₇

—

C₂×C₆₇

—

C₂×C₆₇

yes

C₂×C₆₇

yes

C₂×C₆₇

yes

D₆₇

yes

characteristic

#1857

—

#1394

—

C₆₇

#13464

yes

characteristic, complemented, Sylow

#519

—

#428

—

134

—

—

—

—

C₂×D₆₇

yes

—

#339

—

134

—

C₂²×Q₈

—

C₂²×Q₈

—

C₂²×Q₈

C₂²×D₆₇

yes

characteristic

C₂×M₁₆

—

134

—

C₄×Q₈

—

268

—

C₄×Q₈

—

268

—

100

#117

—

134

—

101

C₂×C₄⋊C₄

—

102

C₂×C₄⋊C₄

C₂²⋊D₆₇

yes

—

103

C₂×C₄²

C₂²⋊D₆₇

yes

—

104

#107

—

268

—

105

#106

—

268

—

106

#101

—

107

#101

—

134

—

108

#95

—

109

#95

—

110

C₂×Q₈

—

111

C₂×Q₈

—

112

C₂×Q₈

—

113

C₂×Q₈

—

114

C₂×Q₈

—

115

C₂×Q₈

—

116

C₂×Q₈

—

117

C₂×Q₈

Q₈×D₆₇

yes

—

118

C₂×Q₈

#11761629

yes

characteristic

119

C₂²×C₄

—

120

C₂²×C₄

—

121

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

—

122

M₁₆

—

268

—

123

M₁₆

—

134

—

124

C₂×C₈

—

268

—

125

C₄⋊C₄

—

126

C₄⋊C₄

—

127

C₄⋊C₄

—

134

—

128

C₄⋊C₄

—

134

—

129

C₂²⋊C₄

—

268

—

130

C₄²

—

131

C₄²

—

268

—

132

C₂³

#14524012

yes

characteristic

133

Q₈

—

134

Q₈

—

135

Q₈

—

136

Q₈

—

137

C₂×C₄

—

138

C₂×C₄

—

139

C₂×C₄

—

140

C₂×C₄

—

141

C₂×C₄

—

142

C₂×C₄

—

143

C₂×C₄

—

144

C₂×C₄

—

268

—

145

C₂×C₄

—

268

—

146

C₂×C₄

#14524019

yes

—

147

C₂×C₄

#14524041

yes

—

148

C₈

—

268

—

149

C₂²

—

150

C₂²

—

151

C₂²

#16863315

yes

characteristic

152

C₂²

#16863583

yes

characteristic

153

C₂²

#16863586

yes

characteristic

154

C₄

—

155

C₄

—

156

C₄

—

157

C₄

—

158

C₄

—

536

—

159

C₄

—

536

—

160

C₂

—

161

C₂

—

162

C₂

#22561561

yes

characteristic

163

C₂

#22562226

yes

characteristic

164

C₂

#22562239

yes

characteristic

	subgroup	G/H
center	C₂²	#16863583
commutator, cosocle	C₂²×Q₈×C₆₇	C₂³
Frattini	C₂²×Q₈	C₂²×D₆₇
Fitting	C₆₇×#2811	C₂
socle	C₂²×C₆₇	B(4,𝔽₂)
prime radical	C₂³×C₆₇	#120
hypercenter	#2811	D₆₇
hypocenter	C₆₇	#13464
LCS	G ▷ C₂²×Q₈×C₆₇ ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ #2811

subgroup

G/H

center

C₂²

#16863583

commutator, cosocle

C₂²×Q₈×C₆₇

C₂³

Frattini

C₂²×Q₈

C₂²×D₆₇

Fitting

C₆₇×#2811

C₂

C₂²×C₆₇

C₂³×C₆₇

D₆₇

C₆₇

G ▷ C₂²×Q₈×C₆₇ ▷ C₂²×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂³ ◁ C₂²×Q₈ ◁ #2811

N	H ≅ G/N	type
18. #14523962	Q₈	semidirect
86. C₆₇	#13464	semidirect

H ≅ G/N

type

18. #14523962

Q₈

semidirect

86. C₆₇

#13464

semidirect

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

subgroup	index
#22560423	2
#22558336	2
C₆₇×#2811	2
#13464	67

subgroup

index

#22560423

#22558336

C₆₇×#2811

#13464

Subgroups — #30775515

Conjugacy classes of subgroups

Complements

Maximal subgroups