Group #30773286 — subgroups

Subgroups — #30773286

Order 17152.

458 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 131 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	#22573312	C₂	1	1	yes	maximal, characteristic
2	#22561929	C₂	1	2	yes	maximal, complemented
3	#22560255	C₂	1	1	yes	maximal, characteristic
4	#22558310	C₂	1	2	yes	maximal, complemented
5	C₆₇×#3059	C₂	1	1	yes	maximal, characteristic
6	#16863813	—	2	4	no	—
7	C₂×#14523983	—	2	2	no	—
8	C₂×#14523983	C₂²	1	2	yes	—
9	#16863434	C₂²	1	2	yes	—
10	#16863370	—	2	1	no	—
11	#16863200	—	2	4	no	—
12	#16863153	—	2	1	no	—
13	C₂²×Q₁₆×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×#131×C₆₇	C₂²	1	1	yes	characteristic
15	#536×C₆₇	—	2	1	no	—
16	#470×C₆₇	C₄	1	4	yes	complemented
17	#467×C₆₇	—	2	2	no	—
18	#427×C₆₇	C₂²	1	1	yes	characteristic
19	#419×C₆₇	—	2	1	no	—
20	#14524002	—	2	2	no	—
21	#14523983	—	2	4	no	—
22	#14523983	—	2	2	no	—
23	#14523983	—	2	2	no	—
24	C₂×C₆₇⋊C₁₆	—	2	4	no	—
25	C₂×C₆₇⋊C₁₆	D₄	1	4	yes	—
26	#14523977	—	2	2	no	—
27	#14523975	—	2	2	no	—
28	#14523958	—	4	2	no	—
29	C₂×#41×C₆₇	D₄	1	1	yes	characteristic
30	C₂²×Q₈×C₆₇	—	2	1	no	—
31	C₂×Q₁₆×C₆₇	—	2	2	no	—
32	C₂×Q₁₆×C₆₇	—	2	1	no	—
33	C₂×Q₁₆×C₆₇	—	2	2	no	—
34	C₂×Q₁₆×C₆₇	C₂×C₄	1	2	yes	—
35	#131×C₆₇	—	2	2	no	—
36	#131×C₆₇	—	2	2	no	—
37	C₂×M₁₆×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×M₁₆×C₆₇	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₇	D₄	1	2	yes	—
40	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
41	C₄⋊Q₈×C₆₇	—	4	1	no	—
42	#124×C₆₇	—	4	1	no	—
43	#123×C₆₇	—	4	2	no	—
44	C₂²⋊Q₈×C₆₇	—	2	4	no	—
45	#117×C₆₇	—	2	1	no	—
46	#107×C₆₇	—	2	1	no	—
47	#106×C₆₇	C₂×C₄	1	2	yes	—
48	#103×C₆₇	—	2	4	no	—
49	C₄⋊Dic₆₇	—	4	2	no	—
50	C₂²⋊Dic₆₇	—	4	2	no	—
51	C₄×Dic₆₇	—	4	2	no	—
52	C₆₇⋊C₁₆	—	2	4	no	—
53	#41×C₆₇	—	2	2	no	—
54	#41×C₆₇	—	2	2	no	—
55	C₂×Q₈×C₆₇	—	2	2	no	—
56	C₂×Q₈×C₆₇	—	4	1	no	—
57	C₂×Q₈×C₆₇	—	4	2	no	—
58	C₂×Q₈×C₆₇	—	4	2	no	—
59	C₂×Q₈×C₆₇	SD₁₆	1	1	yes	characteristic
60	C₂×D₄×C₆₇	D₈	1	2	yes	—
61	C₂×D₄×C₆₇	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	—	4	1	no	—
63	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	Q₁₆×C₆₇	—	2	4	no	—
66	Q₁₆×C₆₇	—	4	1	no	—
67	Q₁₆×C₆₇	—	4	1	no	—
68	M₁₆×C₆₇	—	2	2	no	—
69	M₁₆×C₆₇	—	2	4	no	—
70	C₂×C₈×C₆₇	—	2	2	no	—
71	C₂×C₈×C₆₇	—	2	2	no	—
72	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₆₇	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	#41	1	2	yes	—
77	C₄⋊C₄×C₆₇	—	2	2	no	—
78	C₄⋊C₄×C₆₇	—	4	4	no	—
79	C₂²⋊C₄×C₆₇	—	4	4	no	—
80	C₂²⋊C₄×C₆₇	—	4	1	no	—
81	C₄²×C₆₇	—	4	1	no	—
82	C₂³×C₆₇	#102	1	2	yes	—
83	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
84	Q₈×C₆₇	—	2	2	no	—
85	Q₈×C₆₇	—	4	2	no	—
86	Q₈×C₆₇	—	4	2	no	—
87	Q₈×C₆₇	—	8	2	no	—
88	D₄×C₆₇	—	2	4	no	—
89	D₄×C₆₇	—	2	2	no	—
90	C₂×C₄×C₆₇	—	2	2	no	—
91	C₂×C₄×C₆₇	—	2	2	no	—
92	C₂×C₄×C₆₇	—	4	2	no	—
93	C₂×C₄×C₆₇	—	4	2	no	—
94	C₂×C₄×C₆₇	—	8	2	no	—
95	C₂×C₄×C₆₇	#102	1	2	yes	—
96	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₄×C₆₇	#121	1	2	yes	—
98	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
99	C₂×C₄×C₆₇	#135	1	2	yes	—
100	C₂×Dic₆₇	—	8	2	no	—
101	C₈×C₆₇	—	2	2	no	—
102	C₈×C₆₇	C₄×D₄	1	2	yes	—
103	C₈×C₆₇	#120	1	1	yes	characteristic
104	C₈×C₆₇	#123	1	1	yes	characteristic
105	C₂²×C₆₇	—	2	2	no	—
106	C₂²×C₆₇	—	4	1	no	—
107	C₂²×C₆₇	—	4	2	no	—
108	C₂²×C₆₇	C₂×#102	1	1	yes	characteristic
109	C₂²×C₆₇	#466	1	2	yes	—
110	C₄×C₆₇	—	2	2	no	—
111	C₄×C₆₇	—	8	2	no	—
112	C₄×C₆₇	—	8	2	no	—
113	C₄×C₆₇	#386	1	1	yes	characteristic
114	C₄×C₆₇	#416	1	1	yes	characteristic
115	C₄×C₆₇	#465	1	2	yes	—
116	Dic₆₇	—	16	2	no	—
117	#13075	—	67	1	no	maximal, Sylow
118	C₂×C₆₇	—	2	2	no	—
119	C₂×C₆₇	—	2	1	no	—
120	C₂×C₆₇	—	4	2	no	—
121	C₂×C₆₇	#1893	1	1	yes	characteristic
122	#3407	—	67	1	no	—
123	#3059	D₆₇	1	1	yes	characteristic
124	#2141	—	67	2	no	—
125	#1812	—	67	1	no	—
126	#1384	—	67	2	no	—
127	C₆₇	#13075	1	1	yes	characteristic, complemented, Sylow
128	C₂²×Q₁₆	C₂×D₆₇	1	1	yes	characteristic
129	C₂×#131	C₂×D₆₇	1	1	yes	characteristic
130	#536	—	2	1	no	—
131	#504	—	134	4	no	—
132	C₂×M₃₂	—	67	2	no	—
133	C₂×M₃₂	—	134	2	no	—
134	#470	Dic₆₇	1	4	yes	complemented
135	#467	—	2	2	no	—
136	#427	—	67	2	no	—
137	#427	C₂×D₆₇	1	1	yes	characteristic
138	#419	—	2	1	no	—
139	#410	—	134	1	no	—
140	#358	—	134	4	no	—
141	H(3,ℤ/4)	—	134	1	no	—
142	C₂×#41	C₂²⋊D₆₇	1	1	yes	characteristic
143	C₂²×Q₈	—	2	1	no	—
144	C₂×Q₁₆	—	2	2	no	—
145	C₂×Q₁₆	—	2	1	no	—
146	C₂×Q₁₆	—	2	2	no	—
147	C₂×Q₁₆	C₂×Dic₆₇	1	2	yes	—
148	#131	—	2	2	no	—
149	#131	—	2	2	no	—
150	C₂×M₁₆	C₂×Dic₆₇	1	2	yes	—
151	C₂×M₁₆	C₂²⋊D₆₇	1	1	yes	characteristic
152	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
153	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
154	C₄⋊Q₈	—	4	1	no	—
155	#124	—	4	1	no	—
156	#123	—	4	2	no	—
157	C₂²⋊Q₈	—	2	4	no	—
158	#117	—	2	1	no	—
159	#117	—	134	2	no	—
160	M₃₂	—	134	4	no	—
161	M₃₂	—	134	2	no	—
162	M₃₂	—	134	2	no	—
163	C₂×C₁₆	—	67	4	no	—
164	C₂×C₁₆	—	134	4	no	—
165	#107	—	2	1	no	—
166	#107	—	134	2	no	—
167	#106	—	134	2	no	—
168	#106	C₂×Dic₆₇	1	2	yes	—
169	#103	—	2	4	no	—
170	#99	—	268	2	no	—
171	#41	—	2	2	no	—
172	#41	—	2	2	no	—
173	C₂×Q₈	—	2	2	no	—
174	C₂×Q₈	—	4	1	no	—
175	C₂×Q₈	—	4	2	no	—
176	C₂×Q₈	—	4	2	no	—
177	C₂×Q₈	#11761616	1	1	yes	characteristic
178	C₂×D₄	#11761613	1	2	yes	—
179	C₂×D₄	#11761616	1	1	yes	characteristic
180	C₂²×C₄	—	4	1	no	—
181	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
182	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
183	Q₁₆	—	2	4	no	—
184	Q₁₆	—	4	1	no	—
185	Q₁₆	—	4	1	no	—
186	M₁₆	—	2	2	no	—
187	M₁₆	—	2	4	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	2	no	—
190	C₂×C₈	C₂²⋊Dic₆₇	1	2	yes	—
191	C₂×C₈	C₂²×Dic₆₇	1	1	yes	characteristic
192	C₂×C₈	D₄×D₆₇	1	2	yes	—
193	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
194	C₂×C₈	#11761627	1	2	yes	—
195	C₄⋊C₄	—	2	2	no	—
196	C₄⋊C₄	—	4	4	no	—
197	C₄⋊C₄	—	268	2	no	—
198	C₂²⋊C₄	—	4	4	no	—
199	C₂²⋊C₄	—	4	1	no	—
200	C₂²⋊C₄	—	268	2	no	—
201	C₄²	—	4	1	no	—
202	C₄²	—	268	2	no	—
203	C₁₆	—	134	4	no	—
204	C₂³	#14523964	1	2	yes	—
205	C₂³	C₂×#11761613	1	1	yes	characteristic
206	Q₈	—	2	2	no	—
207	Q₈	—	4	2	no	—
208	Q₈	—	4	2	no	—
209	Q₈	—	8	2	no	—
210	D₄	—	2	4	no	—
211	D₄	—	2	2	no	—
212	C₂×C₄	—	2	2	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	4	2	no	—
216	C₂×C₄	—	8	2	no	—
217	C₂×C₄	—	536	2	no	—
218	C₂×C₄	#14523964	1	2	yes	—
219	C₂×C₄	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
220	C₂×C₄	#14524015	1	2	yes	—
221	C₂×C₄	C₂×#11761616	1	1	yes	characteristic
222	C₂×C₄	#14524063	1	2	yes	—
223	C₈	—	2	2	no	—
224	C₈	D₄×Dic₆₇	1	2	yes	—
225	C₈	#14524011	1	1	yes	characteristic
226	C₈	#14524024	1	1	yes	characteristic
227	C₂²	—	2	2	no	—
228	C₂²	—	4	1	no	—
229	C₂²	—	4	2	no	—
230	C₂²	C₂×#14523964	1	1	yes	characteristic
231	C₂²	#16863630	1	2	yes	—
232	C₄	—	2	2	no	—
233	C₄	—	8	2	no	—
234	C₄	—	8	2	no	—
235	C₄	—	1072	2	no	—
236	C₄	#16863276	1	1	yes	characteristic
237	C₄	#16863392	1	1	yes	characteristic
238	C₄	#16863623	1	2	yes	—
239	C₂	—	2	1	no	—
240	C₂	—	2	2	no	—
241	C₂	—	4	2	no	—
242	C₂	#22560638	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22573312

C₂

yes

maximal, characteristic

#22561929

C₂

yes

maximal, complemented

#22560255

C₂

yes

maximal, characteristic

#22558310

C₂

yes

maximal, complemented

C₆₇×#3059

C₂

yes

maximal, characteristic

#16863813

—

C₂×#14523983

—

C₂×#14523983

C₂²

yes

—

#16863434

C₂²

yes

—

#16863370

—

#16863200

—

#16863153

—

C₂²×Q₁₆×C₆₇

C₂²

yes

characteristic

C₂×#131×C₆₇

C₂²

yes

characteristic

#536×C₆₇

—

#470×C₆₇

C₄

yes

complemented

#467×C₆₇

—

#427×C₆₇

C₂²

yes

characteristic

#419×C₆₇

—

—

—

—

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

#14523977

—

#14523975

—

#14523958

—

C₂×#41×C₆₇

D₄

yes

characteristic

C₂²×Q₈×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

—

C₂×Q₁₆×C₆₇

C₂×C₄

yes

—

#131×C₆₇

—

#131×C₆₇

—

C₂×M₁₆×C₆₇

C₂×C₄

yes

—

C₂×M₁₆×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

D₄

yes

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₄⋊Q₈×C₆₇

—

#124×C₆₇

—

#123×C₆₇

—

C₂²⋊Q₈×C₆₇

—

#117×C₆₇

—

#107×C₆₇

—

#106×C₆₇

C₂×C₄

yes

—

#103×C₆₇

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

#41×C₆₇

—

#41×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

SD₁₆

yes

characteristic

C₂×D₄×C₆₇

D₈

yes

—

C₂×D₄×C₆₇

SD₁₆

yes

characteristic

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

Q₁₆×C₆₇

—

Q₁₆×C₆₇

—

Q₁₆×C₆₇

—

M₁₆×C₆₇

—

M₁₆×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

—

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

#41

yes

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₂²⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₂³×C₆₇

#102

yes

—

C₂³×C₆₇

C₂×D₈

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

D₄×C₆₇

—

D₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#102

yes

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

#135

yes

—

100

C₂×Dic₆₇

—

101

C₈×C₆₇

—

102

C₈×C₆₇

C₄×D₄

yes

—

103

C₈×C₆₇

#120

yes

characteristic

104

C₈×C₆₇

#123

yes

characteristic

105

C₂²×C₆₇

—

106

C₂²×C₆₇

—

107

C₂²×C₆₇

—

108

C₂²×C₆₇

C₂×#102

yes

characteristic

109

C₂²×C₆₇

#466

yes

—

110

C₄×C₆₇

—

111

C₄×C₆₇

—

112

C₄×C₆₇

—

113

C₄×C₆₇

#386

yes

characteristic

114

C₄×C₆₇

#416

yes

characteristic

115

C₄×C₆₇

#465

yes

—

116

Dic₆₇

—

117

#13075

—

maximal, Sylow

118

C₂×C₆₇

—

119

C₂×C₆₇

—

120

C₂×C₆₇

—

121

C₂×C₆₇

#1893

yes

characteristic

122

#3407

—

123

#3059

D₆₇

yes

characteristic

124

#2141

—

125

#1812

—

126

#1384

—

127

C₆₇

#13075

yes

characteristic, complemented, Sylow

128

C₂²×Q₁₆

C₂×D₆₇

yes

characteristic

129

C₂×#131

C₂×D₆₇

yes

characteristic

130

#536

—

131

#504

—

134

—

132

C₂×M₃₂

—

133

C₂×M₃₂

—

134

—

134

#470

Dic₆₇

yes

complemented

135

#467

—

136

#427

—

137

#427

C₂×D₆₇

yes

characteristic

138

#419

—

139

#410

—

134

—

140

#358

—

134

—

141

H(3,ℤ/4)

—

134

—

142

C₂×#41

C₂²⋊D₆₇

yes

characteristic

143

C₂²×Q₈

—

144

C₂×Q₁₆

—

145

C₂×Q₁₆

—

146

C₂×Q₁₆

—

147

C₂×Q₁₆

C₂×Dic₆₇

yes

—

148

#131

—

149

#131

—

150

C₂×M₁₆

C₂×Dic₆₇

yes

—

151

C₂×M₁₆

C₂²⋊D₆₇

yes

characteristic

152

C₂²×C₈

C₂²⋊D₆₇

yes

—

153

C₂²×C₈

C₂²×D₆₇

yes

characteristic

154

C₄⋊Q₈

—

155

#124

—

156

#123

—

157

C₂²⋊Q₈

—

158

#117

—

159

#117

—

134

—

160

M₃₂

—

134

—

161

M₃₂

—

134

—

162

M₃₂

—

134

—

163

C₂×C₁₆

—

164

C₂×C₁₆

—

134

—

165

#107

—

166

#107

—

134

—

167

#106

—

134

—

168

#106

C₂×Dic₆₇

yes

—

169

#103

—

170

#99

—

268

—

171

#41

—

172

#41

—

173

C₂×Q₈

—

174

C₂×Q₈

—

175

C₂×Q₈

—

176

C₂×Q₈

—

177

C₂×Q₈

#11761616

yes

characteristic

178

C₂×D₄

#11761613

yes

—

179

C₂×D₄

#11761616

yes

characteristic

180

C₂²×C₄

—

181

C₂²×C₄

C₂²⋊Dic₆₇

yes

—

182

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

183

Q₁₆

—

184

Q₁₆

—

185

Q₁₆

—

186

M₁₆

—

187

M₁₆

—

188

C₂×C₈

—

189

C₂×C₈

—

190

C₂×C₈

C₂²⋊Dic₆₇

yes

—

191

C₂×C₈

C₂²×Dic₆₇

yes

characteristic

192

C₂×C₈

D₄×D₆₇

yes

—

193

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

194

C₂×C₈

#11761627

yes

—

195

C₄⋊C₄

—

196

C₄⋊C₄

—

197

C₄⋊C₄

—

268

—

198

C₂²⋊C₄

—

199

C₂²⋊C₄

—

200

C₂²⋊C₄

—

268

—

201

C₄²

—

202

C₄²

—

268

—

203

C₁₆

—

134

—

204

C₂³

#14523964

yes

—

205

C₂³

C₂×#11761613

yes

characteristic

206

Q₈

—

207

Q₈

—

208

Q₈

—

209

Q₈

—

210

D₄

—

211

D₄

—

212

C₂×C₄

—

213

C₂×C₄

—

214

C₂×C₄

—

215

C₂×C₄

—

216

C₂×C₄

—

217

C₂×C₄

—

536

—

218

C₂×C₄

#14523964

yes

—

219

C₂×C₄

C₂×C₂²⋊Dic₆₇

yes

characteristic

220

C₂×C₄

#14524015

yes

—

221

C₂×C₄

C₂×#11761616

yes

characteristic

222

C₂×C₄

#14524063

yes

—

223

C₈

—

224

C₈

D₄×Dic₆₇

yes

—

225

C₈

#14524011

yes

characteristic

226

C₈

#14524024

yes

characteristic

227

C₂²

—

228

C₂²

—

229

C₂²

—

230

C₂²

C₂×#14523964

yes

characteristic

231

C₂²

#16863630

yes

—

232

C₄

—

233

C₄

—

234

C₄

—

235

C₄

—

1072

—

236

C₄

#16863276

yes

characteristic

237

C₄

#16863392

yes

characteristic

238

C₄

#16863623

yes

—

239

C₂

—

240

C₂

—

241

C₂

—

242

C₂

#22560638

yes

characteristic

	subgroup	G/H
center	C₂	#22560638
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₆₇×#3059	C₂
socle	C₂×C₆₇	#1893
prime radical	C₂×#41×C₆₇	D₄
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	#3059	D₆₇
hypocenter	C₆₇	#13075
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #3059

subgroup

G/H

center

C₂

#22560638

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₆₇×#3059

C₂

socle

C₂×C₆₇

#1893

prime radical

C₂×#41×C₆₇

D₄

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂ ◁ C₂²×C₄ ◁ C₂×#131 ◁ #3059

N	H ≅ G/N	type
2. #22561929	C₂	semidirect
4. #22558310	C₂	semidirect
16. #470×C₆₇	C₄	semidirect
127. C₆₇	#13075	semidirect
134. #470	Dic₆₇	semidirect

H ≅ G/N

type

2. #22561929

C₂

semidirect

4. #22558310

C₂

semidirect

16. #470×C₆₇

C₄

semidirect

127. C₆₇

#13075

semidirect

134. #470

Dic₆₇

semidirect

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

subgroup	index
#22573312	2
#22561929	2
#22560255	2
#22558310	2
C₆₇×#3059	2
#13075	67

subgroup

index

C₆₇×#3059

Subgroups — #30773286

Conjugacy classes of subgroups

Complements

Maximal subgroups