Group #30772392 — subgroups

Subgroups — #30772392

Order 17152.

414 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 414 conjugacy classes of subgroups — too many to draw. Only its 127 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	C₂×#16863434	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863220	C₂	1	2	yes	maximal, complemented
3	#22561820	C₂	1	1	yes	maximal, characteristic
4	#22558309	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1810	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524002	—	2	1	no	—
7	C₂×#14523983	C₂²	1	2	yes	—
8	#16863434	—	2	4	no	—
9	C₂×#14523977	—	2	1	no	—
10	C₂×#14523975	C₂²	1	2	yes	—
11	#16863220	—	2	4	no	—
12	C₂²⋊(C₆₇⋊C₁₆)	C₄	1	4	yes	complemented
13	C₂³×C₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
15	C₂×#108×C₆₇	C₂²	1	1	yes	characteristic
16	#407×C₆₇	—	2	1	no	—
17	#341×C₆₇	—	2	1	no	—
18	#14524002	—	2	4	no	—
19	#14524002	—	4	2	no	—
20	C₂×C₄⋊Dic₆₇	—	2	2	no	—
21	C₂×C₂²⋊Dic₆₇	—	4	1	no	—
22	C₂×C₄×Dic₆₇	—	4	1	no	—
23	#14523983	—	2	2	no	—
24	#14523983	—	2	2	no	—
25	C₂×C₆₇⋊C₁₆	C₂×C₄	1	4	yes	—
26	#14523977	—	2	4	no	—
27	#14523975	—	2	2	no	—
28	#14523975	D₄	1	4	yes	—
29	C₂³×C₄×C₆₇	D₄	1	1	yes	characteristic
30	C₂×M₁₆×C₆₇	—	2	1	no	—
31	C₂×M₁₆×C₆₇	—	2	1	no	—
32	C₂²×C₈×C₆₇	—	2	2	no	—
33	C₂²×C₈×C₆₇	—	2	2	no	—
34	C₂²×C₈×C₆₇	C₂×C₄	1	2	yes	—
35	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₆₇	Q₈	1	2	yes	—
37	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
38	#108×C₆₇	—	2	2	no	—
39	#108×C₆₇	—	2	2	no	—
40	C₂²⋊C₈×C₆₇	—	4	2	no	—
41	C₂²×Dic₆₇	—	4	2	no	—
42	C₄⋊Dic₆₇	—	2	4	no	—
43	C₄⋊Dic₆₇	—	4	2	no	—
44	C₂²⋊Dic₆₇	—	4	4	no	—
45	C₄×Dic₆₇	—	4	4	no	—
46	C₆₇⋊C₁₆	—	2	4	no	—
47	C₂⁴×C₆₇	SD₁₆	1	1	yes	characteristic
48	C₂²×C₄×C₆₇	—	2	1	no	—
49	C₂²×C₄×C₆₇	—	2	1	no	—
50	C₂²×C₄×C₆₇	—	2	1	no	—
51	C₂²×C₄×C₆₇	—	2	1	no	—
52	C₂²×C₄×C₆₇	C₄⋊C₄	1	2	yes	—
53	C₂²×C₄×C₆₇	SD₁₆	1	1	yes	characteristic
54	C₂²×C₄×C₆₇	SD₁₆	1	2	yes	—
55	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
56	M₁₆×C₆₇	—	4	1	no	—
57	M₁₆×C₆₇	—	4	1	no	—
58	M₁₆×C₆₇	—	4	1	no	—
59	M₁₆×C₆₇	—	4	1	no	—
60	C₂×C₈×C₆₇	—	2	1	no	—
61	C₂×C₈×C₆₇	—	2	1	no	—
62	C₂×C₈×C₆₇	—	2	4	no	—
63	C₂×C₈×C₆₇	—	2	4	no	—
64	C₂×C₈×C₆₇	—	4	1	no	—
65	C₂×C₈×C₆₇	—	4	1	no	—
66	C₂×C₈×C₆₇	C₄⋊C₄	1	2	yes	—
67	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
69	C₂×C₈×C₆₇	C₂×Q₈	1	1	yes	characteristic
70	C₂×C₈×C₆₇	#41	1	2	yes	—
71	C₂³×C₆₇	—	2	1	no	—
72	C₂³×C₆₇	—	2	1	no	—
73	C₂³×C₆₇	—	4	2	no	—
74	C₂³×C₆₇	#106	1	2	yes	—
75	C₂³×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₆₇	—	2	1	no	—
77	C₂×C₄×C₆₇	—	2	1	no	—
78	C₂×C₄×C₆₇	—	2	2	no	—
79	C₂×C₄×C₆₇	—	2	2	no	—
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	#106	1	2	yes	—
83	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
84	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
85	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
87	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
88	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
89	C₂×Dic₆₇	—	4	4	no	—
90	C₂×Dic₆₇	—	8	4	no	—
91	C₈×C₆₇	—	2	2	no	—
92	C₈×C₆₇	—	8	1	no	—
93	C₈×C₆₇	—	8	1	no	—
94	C₈×C₆₇	C₄×D₄	1	2	yes	—
95	C₈×C₆₇	C₂²⋊Q₈	1	2	yes	—
96	C₂²×C₆₇	—	2	2	no	—
97	C₂²×C₆₇	—	2	2	no	—
98	C₂²×C₆₇	—	2	4	no	—
99	C₂²×C₆₇	—	4	2	no	—
100	C₂²×C₆₇	—	4	2	no	—
101	C₂²×C₆₇	C₂×#106	1	1	yes	characteristic
102	C₂²×C₆₇	#450	1	1	yes	characteristic
103	C₂²×C₆₇	#483	1	1	yes	characteristic
104	C₄×C₆₇	—	2	2	no	—
105	C₄×C₆₇	#385	1	1	yes	characteristic
106	C₄×C₆₇	#428	1	1	yes	characteristic
107	C₄×C₆₇	#448	1	1	yes	characteristic
108	C₄×C₆₇	#481	1	1	yes	characteristic
109	Dic₆₇	—	8	4	no	—
110	#12930	—	67	1	no	maximal, Sylow
111	C₂×C₆₇	—	2	1	no	—
112	C₂×C₆₇	—	2	1	no	—
113	C₂×C₆₇	—	4	2	no	—
114	C₂×C₆₇	#1853	1	1	yes	characteristic
115	C₂×C₆₇	#2155	1	2	yes	—
116	C₂×#427	—	67	1	no	—
117	C₂×#365	—	67	2	no	—
118	#2112	—	67	1	no	—
119	#1810	D₆₇	1	1	yes	characteristic
120	#1383	—	67	2	no	—
121	C₆₇	#12930	1	1	yes	characteristic, complemented, Sylow
122	C₂³×C₈	C₂×D₆₇	1	1	yes	characteristic
123	C₂×#117	—	134	1	no	—
124	C₂×M₃₂	—	67	2	no	—
125	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
126	C₂×#108	C₂×D₆₇	1	1	yes	characteristic
127	#427	—	134	4	no	—
128	C₂×#107	—	134	1	no	—
129	C₂×#106	—	67	2	no	—
130	#407	—	2	1	no	—
131	#365	—	134	4	no	—
132	C₂²⋊C₁₆	—	67	4	no	—
133	#341	—	2	1	no	—
134	C₂³×C₄	C₂²⋊D₆₇	1	1	yes	characteristic
135	C₂×M₁₆	—	2	1	no	—
136	C₂×M₁₆	—	2	1	no	—
137	C₂²×C₈	—	2	2	no	—
138	C₂²×C₈	—	2	2	no	—
139	C₂²×C₈	Dic₁₃₄	1	2	yes	—
140	C₂²×C₈	C₄×D₆₇	1	2	yes	—
141	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
142	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
143	#117	—	268	2	no	—
144	#117	—	134	4	no	—
145	C₂×C₄⋊C₄	—	134	2	no	—
146	C₂×C₂²⋊C₄	—	268	1	no	—
147	C₂×C₄²	—	268	1	no	—
148	M₃₂	—	134	2	no	—
149	M₃₂	—	134	2	no	—
150	C₂×C₁₆	—	67	4	no	—
151	#108	—	2	2	no	—
152	#108	—	2	2	no	—
153	#107	—	134	4	no	—
154	#106	—	67	4	no	—
155	#106	—	134	2	no	—
156	C₂²⋊C₈	—	4	2	no	—
157	C₂⁴	#11761615	1	1	yes	characteristic
158	C₂²×C₄	—	2	1	no	—
159	C₂²×C₄	—	2	1	no	—
160	C₂²×C₄	—	2	1	no	—
161	C₂²×C₄	—	2	1	no	—
162	C₂²×C₄	—	268	2	no	—
163	C₂²×C₄	Dic₆₇⋊C₄	1	2	yes	—
164	C₂²×C₄	#11761615	1	1	yes	characteristic
165	C₂²×C₄	#11761616	1	2	yes	—
166	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
167	M₁₆	—	4	1	no	—
168	M₁₆	—	4	1	no	—
169	M₁₆	—	4	1	no	—
170	M₁₆	—	4	1	no	—
171	C₂×C₈	—	2	1	no	—
172	C₂×C₈	—	2	1	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	2	4	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	4	1	no	—
177	C₂×C₈	Dic₆₇⋊C₄	1	2	yes	—
178	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
179	C₂×C₈	D₄×D₆₇	1	2	yes	—
180	C₂×C₈	C₂×Dic₁₃₄	1	1	yes	characteristic
181	C₂×C₈	#11761627	1	2	yes	—
182	C₄⋊C₄	—	268	2	no	—
183	C₄⋊C₄	—	134	4	no	—
184	C₂²⋊C₄	—	268	4	no	—
185	C₄²	—	268	4	no	—
186	C₁₆	—	134	4	no	—
187	C₂³	—	2	1	no	—
188	C₂³	—	2	1	no	—
189	C₂³	—	4	2	no	—
190	C₂³	#14523974	1	2	yes	—
191	C₂³	C₂×#11761615	1	1	yes	characteristic
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	2	2	no	—
196	C₂×C₄	—	2	2	no	—
197	C₂×C₄	—	2	2	no	—
198	C₂×C₄	—	268	4	no	—
199	C₂×C₄	—	536	4	no	—
200	C₂×C₄	#14523974	1	2	yes	—
201	C₂×C₄	C₂×Dic₆₇⋊C₄	1	1	yes	characteristic
202	C₂×C₄	#14524011	1	1	yes	characteristic
203	C₂×C₄	#14524024	1	1	yes	characteristic
204	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
205	C₂×C₄	#14524068	1	1	yes	characteristic
206	C₂×C₄	#14524073	1	1	yes	characteristic
207	C₈	—	2	2	no	—
208	C₈	—	8	1	no	—
209	C₈	—	8	1	no	—
210	C₈	(C₂²⋊D₆₇)⋊C₄	1	2	yes	—
211	C₈	#14524020	1	2	yes	—
212	C₂²	—	2	2	no	—
213	C₂²	—	2	2	no	—
214	C₂²	—	2	4	no	—
215	C₂²	—	4	2	no	—
216	C₂²	—	4	2	no	—
217	C₂²	C₂×#14523974	1	1	yes	characteristic
218	C₂²	#16863546	1	1	yes	characteristic
219	C₂²	#16863724	1	1	yes	characteristic
220	C₄	—	2	2	no	—
221	C₄	—	536	4	no	—
222	C₄	#16863268	1	1	yes	characteristic
223	C₄	#16863439	1	1	yes	characteristic
224	C₄	#16863534	1	1	yes	characteristic
225	C₄	#16863712	1	1	yes	characteristic
226	C₂	—	2	1	no	—
227	C₂	—	2	1	no	—
228	C₂	—	4	2	no	—
229	C₂	#22560401	1	1	yes	characteristic
230	C₂	#22561983	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863434

C₂

yes

maximal, characteristic

C₂×#16863220

C₂

yes

maximal, complemented

#22561820

C₂

yes

maximal, characteristic

#22558309

C₂

yes

maximal, complemented

C₆₇×#1810

C₂

yes

maximal, characteristic

C₂×#14524002

—

C₂×#14523983

C₂²

yes

—

#16863434

—

C₂×#14523977

—

C₂×#14523975

C₂²

yes

—

#16863220

—

C₂²⋊(C₆₇⋊C₁₆)

C₄

yes

complemented

C₂³×C₈×C₆₇

C₂²

yes

characteristic

C₂×#108×C₆₇

C₂²

yes

characteristic

C₂×#108×C₆₇

C₂²

yes

characteristic

#407×C₆₇

—

#341×C₆₇

—

#14524002

—

#14524002

—

C₂×C₄⋊Dic₆₇

—

C₂×C₂²⋊Dic₆₇

—

C₂×C₄×Dic₆₇

—

#14523983

—

#14523983

—

C₂×C₆₇⋊C₁₆

C₂×C₄

yes

—

#14523977

—

#14523975

—

#14523975

D₄

yes

—

C₂³×C₄×C₆₇

D₄

yes

characteristic

C₂×M₁₆×C₆₇

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

C₂×C₄

yes

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

Q₈

yes

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

#108×C₆₇

—

#108×C₆₇

—

C₂²⋊C₈×C₆₇

—

C₂²×Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₂⁴×C₆₇

SD₁₆

yes

characteristic

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₄⋊C₄

yes

—

C₂²×C₄×C₆₇

SD₁₆

yes

characteristic

C₂²×C₄×C₆₇

SD₁₆

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₄⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×Q₈

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

—

C₂³×C₆₇

#106

yes

—

C₂³×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#106

yes

—

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

C₂×SD₁₆

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

C₄×D₄

yes

—

C₈×C₆₇

C₂²⋊Q₈

yes

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

C₂²×C₆₇

—

100

C₂²×C₆₇

—

101

C₂²×C₆₇

C₂×#106

yes

characteristic

102

C₂²×C₆₇

#450

yes

characteristic

103

C₂²×C₆₇

#483

yes

characteristic

104

C₄×C₆₇

—

105

C₄×C₆₇

#385

yes

characteristic

106

C₄×C₆₇

#428

yes

characteristic

107

C₄×C₆₇

#448

yes

characteristic

108

C₄×C₆₇

#481

yes

characteristic

109

Dic₆₇

—

110

#12930

—

maximal, Sylow

111

C₂×C₆₇

—

112

C₂×C₆₇

—

113

C₂×C₆₇

—

114

C₂×C₆₇

#1853

yes

characteristic

115

C₂×C₆₇

#2155

yes

—

116

C₂×#427

—

117

C₂×#365

—

118

#2112

—

119

#1810

D₆₇

yes

characteristic

120

#1383

—

121

C₆₇

#12930

yes

characteristic, complemented, Sylow

122

C₂³×C₈

C₂×D₆₇

yes

characteristic

123

C₂×#117

—

134

—

124

C₂×M₃₂

—

125

C₂×#108

C₂×D₆₇

yes

characteristic

126

C₂×#108

C₂×D₆₇

yes

characteristic

127

#427

—

134

—

128

C₂×#107

—

134

—

129

C₂×#106

—

130

#407

—

131

#365

—

134

—

132

C₂²⋊C₁₆

—

133

#341

—

134

C₂³×C₄

C₂²⋊D₆₇

yes

characteristic

135

C₂×M₁₆

—

136

C₂×M₁₆

—

137

C₂²×C₈

—

138

C₂²×C₈

—

139

C₂²×C₈

Dic₁₃₄

yes

—

140

C₂²×C₈

C₄×D₆₇

yes

—

141

C₂²×C₈

C₂²⋊D₆₇

yes

characteristic

142

C₂²×C₈

C₂²×D₆₇

yes

characteristic

143

#117

—

268

—

144

#117

—

134

—

145

C₂×C₄⋊C₄

—

134

—

146

C₂×C₂²⋊C₄

—

268

—

147

C₂×C₄²

—

268

—

148

M₃₂

—

134

—

149

M₃₂

—

134

—

150

C₂×C₁₆

—

151

#108

—

152

#108

—

153

#107

—

134

—

154

#106

—

155

#106

—

134

—

156

C₂²⋊C₈

—

157

C₂⁴

#11761615

yes

characteristic

158

C₂²×C₄

—

159

C₂²×C₄

—

160

C₂²×C₄

—

161

C₂²×C₄

—

162

C₂²×C₄

—

268

—

163

C₂²×C₄

Dic₆₇⋊C₄

yes

—

164

C₂²×C₄

#11761615

yes

characteristic

165

C₂²×C₄

#11761616

yes

—

166

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

167

M₁₆

—

168

M₁₆

—

169

M₁₆

—

170

M₁₆

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

—

177

C₂×C₈

Dic₆₇⋊C₄

yes

—

178

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

179

C₂×C₈

D₄×D₆₇

yes

—

180

C₂×C₈

C₂×Dic₁₃₄

yes

characteristic

181

C₂×C₈

#11761627

yes

—

182

C₄⋊C₄

—

268

—

183

C₄⋊C₄

—

134

—

184

C₂²⋊C₄

—

268

—

185

C₄²

—

268

—

186

C₁₆

—

134

—

187

C₂³

—

188

C₂³

—

189

C₂³

—

190

C₂³

#14523974

yes

—

191

C₂³

C₂×#11761615

yes

characteristic

192

C₂×C₄

—

193

C₂×C₄

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

268

—

199

C₂×C₄

—

536

—

200

C₂×C₄

#14523974

yes

—

201

C₂×C₄

C₂×Dic₆₇⋊C₄

yes

characteristic

202

C₂×C₄

#14524011

yes

characteristic

203

C₂×C₄

#14524024

yes

characteristic

204

C₂×C₄

C₂×#11761616

yes

characteristic

205

C₂×C₄

#14524068

yes

characteristic

206

C₂×C₄

#14524073

yes

characteristic

207

C₈

—

208

C₈

—

209

C₈

—

210

C₈

(C₂²⋊D₆₇)⋊C₄

yes

—

211

C₈

#14524020

yes

—

212

C₂²

—

213

C₂²

—

214

C₂²

—

215

C₂²

—

216

C₂²

—

217

C₂²

C₂×#14523974

yes

characteristic

218

C₂²

#16863546

yes

characteristic

219

C₂²

#16863724

yes

characteristic

220

C₄

—

221

C₄

—

536

—

222

C₄

#16863268

yes

characteristic

223

C₄

#16863439

yes

characteristic

224

C₄

#16863534

yes

characteristic

225

C₄

#16863712

yes

characteristic

226

C₂

—

227

C₂

—

228

C₂

—

229

C₂

#22560401

yes

characteristic

230

C₂

#22561983

yes

—

	subgroup	G/H
center	C₂²	C₂×#14523974
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#1810	C₂
socle	C₂²×C₆₇	C₂×#106
prime radical	C₂⁴×C₆₇	SD₁₆
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#1810	D₆₇
hypocenter	C₆₇	#12930
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1810

subgroup

G/H

center

C₂²

C₂×#14523974

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#1810

C₂

socle

C₂²×C₆₇

C₂×#106

prime radical

C₂⁴×C₆₇

SD₁₆

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ #1810

N	H ≅ G/N	type
2. C₂×#16863220	C₂	semidirect
4. #22558309	C₂	semidirect
12. C₂²⋊(C₆₇⋊C₁₆)	C₄	semidirect
121. C₆₇	#12930	semidirect

H ≅ G/N

type

2. C₂×#16863220

C₂

semidirect

4. #22558309

C₂

semidirect

12. C₂²⋊(C₆₇⋊C₁₆)

C₄

semidirect

121. C₆₇

#12930

semidirect

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

subgroup	index
C₂×#16863434	2
C₂×#16863220	2
#22561820	2
#22558309	2
C₆₇×#1810	2
#12930	67

subgroup

index

C₆₇×#1810

Subgroups — #30772392

Conjugacy classes of subgroups

Complements

Maximal subgroups