Group #30755734 — subgroups

Subgroups — #30755734

Order 17152.

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

normal subgroups only

This group has 530 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	#22570788	C₂	1	1	yes	maximal, characteristic
2	#22561036	C₂	1	2	yes	maximal, complemented
3	C₄×#14523953	C₂	1	1	yes	maximal, characteristic, complemented
4	#22558659	C₂	1	2	yes	maximal, complemented
5	C₆₇×#1566	C₂	1	1	yes	maximal, characteristic, complemented
6	#16863946	C₂²	1	2	yes	—
7	#16863801	C₄	1	1	yes	characteristic, complemented
8	#16863773	C₄	1	1	yes	characteristic, complemented
9	#16863743	C₄	1	2	yes	complemented
10	#16863632	—	2	2	no	—
11	C₄×D₅₃₆	—	2	2	no	—
12	C₂×#14523969	—	2	4	no	—
13	C₂×#14523953	—	2	2	no	—
14	C₂×C₄×C₆₇⋊C₈	C₂²	1	1	yes	characteristic
15	C₄²×Dic₆₇	C₂²	1	1	yes	characteristic
16	#16863123	—	2	4	no	—
17	#423×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	#334×C₆₇	—	2	2	no	—
20	C₂×#11761628	—	2	2	no	—
21	C₂×D₅₃₆	—	2	2	no	—
22	C₂²×C₆₇⋊C₈	—	2	1	no	—
23	#14524043	C₂×C₄	1	2	yes	—
24	#14524040	C₂×C₄	1	2	yes	—
25	#14524031	—	2	2	no	—
26	#14524017	—	2	2	no	—
27	#14524017	—	2	2	no	—
28	C₄×D₂₆₈	—	2	2	no	—
29	C₄×D₂₆₈	—	2	2	no	—
30	C₂×C₄×Dic₆₇	—	2	1	no	—
31	C₂×C₄×Dic₆₇	—	2	1	no	—
32	C₂×C₄×Dic₆₇	D₄	1	2	yes	—
33	#14523977	—	2	2	no	—
34	#14523969	—	4	8	no	—
35	#14523965	—	2	4	no	—
36	#14523953	—	2	8	no	—
37	C₄×C₆₇⋊C₈	—	2	1	no	—
38	C₄×C₆₇⋊C₈	D₄	1	2	yes	—
39	C₂×M₁₆×C₆₇	—	2	2	no	—
40	C₂²×C₈×C₆₇	D₄	1	2	yes	—
41	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
42	C₄⋊C₈×C₆₇	—	2	2	no	—
43	C₄⋊C₈×C₆₇	—	2	2	no	—
44	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₆₇	D₄	1	2	yes	—
47	#11761628	—	4	4	no	—
48	#11761628	—	4	4	no	—
49	C₂×Dic₁₃₄	—	2	2	no	—
50	C₂×D₂₆₈	—	2	2	no	—
51	C₂×D₂₆₈	—	2	2	no	—
52	C₂×D₂₆₈	—	2	2	no	—
53	C₂×C₂²⋊D₆₇	—	4	2	no	—
54	C₂²×Dic₆₇	—	2	2	no	—
55	C₂×C₄×D₆₇	—	4	2	no	—
56	D₅₃₆	—	4	4	no	—
57	C₂×C₆₇⋊C₈	—	2	4	no	—
58	C₂×C₆₇⋊C₈	—	2	2	no	—
59	C₄⋊Dic₆₇	—	2	2	no	—
60	C₄⋊Dic₆₇	—	2	2	no	—
61	(C₂×D₆₇)⋊C₄	—	4	2	no	—
62	(C₂×D₆₇)⋊C₄	—	4	2	no	—
63	C₄×Dic₆₇	—	2	4	no	—
64	C₄×Dic₆₇	—	2	4	no	—
65	C₄×Dic₆₇	—	4	2	no	—
66	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
69	M₁₆×C₆₇	—	4	4	no	—
70	C₂×C₈×C₆₇	—	2	4	no	—
71	C₂×C₈×C₆₇	—	4	2	no	—
72	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
74	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
75	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
76	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₆₇	#41	1	1	yes	characteristic
78	C₂³×C₆₇	#121	1	1	yes	characteristic
79	C₂²×D₆₇	—	4	2	no	—
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
84	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
85	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
86	C₂×C₄×C₆₇	#124	1	1	yes	characteristic
87	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
88	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	4	yes	—
89	C₂²⋊D₆₇	—	8	4	no	—
90	C₂×Dic₆₇	—	2	4	no	—
91	C₂×Dic₆₇	—	4	2	no	—
92	C₂×Dic₆₇	—	4	2	no	—
93	C₂×Dic₆₇	—	4	2	no	—
94	D₂₆₈	—	4	4	no	—
95	D₂₆₈	—	4	4	no	—
96	D₂₆₈	—	4	4	no	—
97	C₄×D₆₇	—	8	2	no	—
98	C₄×D₆₇	—	8	2	no	—
99	Dic₁₃₄	—	4	4	no	—
100	C₈×C₆₇	—	2	2	no	—
101	C₈×C₆₇	—	8	2	no	—
102	C₈×C₆₇	#104	1	4	yes	complemented
103	C₆₇⋊C₈	—	2	4	no	—
104	C₂²×C₆₇	—	2	1	no	—
105	C₂²×C₆₇	—	2	1	no	—
106	C₂²×C₆₇	#390	1	1	yes	characteristic
107	C₂²×C₆₇	#469	1	2	yes	—
108	C₂×D₆₇	—	8	2	no	—
109	C₂×D₆₇	—	8	2	no	—
110	C₂×D₆₇	—	8	2	no	—
111	C₄×C₆₇	—	2	1	no	—
112	C₄×C₆₇	—	2	1	no	—
113	C₄×C₆₇	C₂×#104	1	2	yes	—
114	C₄×C₆₇	#442	1	2	yes	—
115	C₄×C₆₇	#444	1	2	yes	—
116	C₄×C₆₇	#496	1	2	yes	—
117	Dic₆₇	—	4	4	no	—
118	Dic₆₇	—	8	2	no	—
119	#9849	—	67	1	no	maximal, Sylow
120	C₂×C₆₇	—	2	2	no	—
121	C₂×C₆₇	#1962	1	1	yes	characteristic
122	C₂×C₆₇	#1972	1	1	yes	characteristic
123	C₂×C₆₇	#1975	1	1	yes	characteristic
124	D₆₇	—	16	2	no	—
125	#3169	—	67	1	no	—
126	#1962	—	67	2	no	—
127	C₄×#97	—	67	1	no	—
128	#1566	D₆₇	1	1	yes	characteristic, complemented
129	#1492	—	67	2	no	—
130	C₆₇	#9849	1	1	yes	characteristic, complemented, Sylow
131	#532	—	67	2	no	—
132	#500	—	67	1	no	—
133	#493	—	67	1	no	—
134	#486	—	67	2	no	—
135	#466	—	134	2	no	—
136	C₄×D₈	—	134	2	no	—
137	#423	C₂×D₆₇	1	2	yes	—
138	C₂×#104	—	134	4	no	—
139	C₂×#97	—	134	2	no	—
140	C₂×C₄×C₈	—	67	1	no	—
141	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
142	C₄³	—	67	1	no	—
143	#334	—	2	2	no	—
144	D₄⋊C₈	—	134	4	no	—
145	C₂×#41	—	134	2	no	—
146	C₂×D₈	—	134	2	no	—
147	C₂×M₁₆	—	2	2	no	—
148	C₂²×C₈	—	134	1	no	—
149	C₂²×C₈	D₂₆₈	1	2	yes	—
150	C₄⋊Q₈	—	67	2	no	—
151	#127	—	67	2	no	—
152	#124	—	134	2	no	—
153	#121	—	134	2	no	—
154	#121	—	134	2	no	—
155	C₄×D₄	—	134	2	no	—
156	C₄×D₄	—	134	2	no	—
157	C₂×C₄²	—	67	2	no	—
158	C₂×C₄²	—	134	1	no	—
159	C₂×C₄²	—	134	1	no	—
160	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
161	#107	—	134	2	no	—
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	2	2	no	—
164	#104	—	268	8	no	—
165	#102	—	134	4	no	—
166	#97	—	134	8	no	—
167	C₄×C₈	—	67	2	no	—
168	C₄×C₈	—	134	1	no	—
169	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
170	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
171	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
172	#41	—	268	4	no	—
173	#41	—	268	4	no	—
174	C₂×Q₈	—	134	2	no	—
175	C₂×D₄	—	268	2	no	—
176	C₂×D₄	—	134	2	no	—
177	C₂×D₄	—	134	2	no	—
178	C₂×D₄	—	134	2	no	—
179	C₂²×C₄	—	268	2	no	—
180	C₂²×C₄	—	134	2	no	—
181	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
182	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
183	C₂²×C₄	#11761629	1	1	yes	characteristic
184	D₈	—	268	4	no	—
185	M₁₆	—	4	4	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	4	2	no	—
188	C₂×C₈	—	134	4	no	—
189	C₂×C₈	—	134	2	no	—
190	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
191	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
192	C₄⋊C₄	—	134	2	no	—
193	C₄⋊C₄	—	134	2	no	—
194	C₂²⋊C₄	—	268	2	no	—
195	C₂²⋊C₄	—	268	2	no	—
196	C₄²	—	268	2	no	—
197	C₄²	—	134	4	no	—
198	C₄²	—	134	4	no	—
199	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
200	C₄²	D₄×D₆₇	1	1	yes	characteristic
201	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
202	C₄²	#11761629	1	1	yes	characteristic
203	C₂³	—	268	2	no	—
204	C₂³	#14524018	1	1	yes	characteristic
205	Q₈	—	268	4	no	—
206	D₄	—	268	4	no	—
207	D₄	—	268	4	no	—
208	D₄	—	268	4	no	—
209	D₄	—	536	4	no	—
210	C₂×C₄	—	2	2	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	268	2	no	—
213	C₂×C₄	—	268	2	no	—
214	C₂×C₄	—	268	2	no	—
215	C₂×C₄	—	536	2	no	—
216	C₂×C₄	—	536	2	no	—
217	C₂×C₄	—	134	4	no	—
218	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
219	C₂×C₄	#14524005	1	1	yes	characteristic
220	C₂×C₄	#14524005	1	1	yes	characteristic
221	C₂×C₄	#14524018	1	1	yes	characteristic
222	C₂×C₄	#14524032	1	1	yes	characteristic
223	C₂×C₄	#14524039	1	1	yes	characteristic
224	C₂×C₄	#14524070	1	4	yes	—
225	C₈	—	2	2	no	—
226	C₈	—	8	2	no	—
227	C₈	—	134	4	no	—
228	C₈	#14523969	1	4	yes	complemented
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	—	536	2	no	—
232	C₂²	—	536	2	no	—
233	C₂²	—	536	2	no	—
234	C₂²	#16863297	1	1	yes	characteristic
235	C₂²	#16863650	1	2	yes	—
236	C₄	—	2	1	no	—
237	C₄	—	2	1	no	—
238	C₄	—	268	4	no	—
239	C₄	—	536	2	no	—
240	C₄	C₂×#14523969	1	2	yes	—
241	C₄	#16863507	1	2	yes	—
242	C₄	#16863517	1	2	yes	—
243	C₄	#16863782	1	2	yes	—
244	C₂	—	2	2	no	—
245	C₂	—	1072	2	no	—
246	C₂	#22561036	1	1	yes	characteristic
247	C₂	#22561083	1	1	yes	characteristic
248	C₂	#22561100	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22570788

C₂

yes

maximal, characteristic

#22561036

C₂

yes

maximal, complemented

C₄×#14523953

C₂

yes

maximal, characteristic, complemented

#22558659

C₂

yes

maximal, complemented

C₆₇×#1566

C₂

yes

maximal, characteristic, complemented

#16863946

C₂²

yes

—

#16863801

C₄

yes

characteristic, complemented

#16863773

C₄

yes

characteristic, complemented

#16863743

C₄

yes

complemented

#16863632

—

C₄×D₅₃₆

—

C₂×#14523969

—

C₂×#14523953

—

C₂×C₄×C₆₇⋊C₈

C₂²

yes

characteristic

C₄²×Dic₆₇

C₂²

yes

characteristic

#16863123

—

#423×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#334×C₆₇

—

C₂×#11761628

—

C₂×D₅₃₆

—

C₂²×C₆₇⋊C₈

—

#14524043

C₂×C₄

yes

—

#14524040

C₂×C₄

yes

—

#14524031

—

#14524017

—

#14524017

—

C₄×D₂₆₈

—

C₄×D₂₆₈

—

C₂×C₄×Dic₆₇

—

C₂×C₄×Dic₆₇

—

C₂×C₄×Dic₆₇

D₄

yes

—

—

—

—

—

C₄×C₆₇⋊C₈

—

C₄×C₆₇⋊C₈

D₄

yes

—

C₂×M₁₆×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

C₂×C₄

yes

characteristic

C₄×C₈×C₆₇

D₄

yes

—

#11761628

—

#11761628

—

C₂×Dic₁₃₄

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×D₂₆₈

—

C₂×C₂²⋊D₆₇

—

C₂²×Dic₆₇

—

C₂×C₄×D₆₇

—

D₅₃₆

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

(C₂×D₆₇)⋊C₄

—

(C₂×D₆₇)⋊C₄

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

characteristic

M₁₆×C₆₇

—

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

C₂×D₄

yes

C₄²×C₆₇

yes

C₂³×C₆₇

yes

C₂²×D₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

characteristic

C₂×C₄×C₆₇

C₄×D₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₂²⋊D₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

D₂₆₈

—

D₂₆₈

—

D₂₆₈

—

C₄×D₆₇

—

C₄×D₆₇

—

Dic₁₃₄

—

100

C₈×C₆₇

—

101

C₈×C₆₇

—

102

C₈×C₆₇

#104

yes

complemented

103

C₆₇⋊C₈

—

104

C₂²×C₆₇

—

105

C₂²×C₆₇

—

106

C₂²×C₆₇

#390

yes

characteristic

107

C₂²×C₆₇

#469

yes

—

108

C₂×D₆₇

—

109

C₂×D₆₇

—

110

C₂×D₆₇

—

111

C₄×C₆₇

—

112

C₄×C₆₇

—

113

C₄×C₆₇

C₂×#104

yes

—

114

C₄×C₆₇

#442

yes

—

115

C₄×C₆₇

#444

yes

—

116

C₄×C₆₇

#496

yes

—

117

Dic₆₇

—

118

Dic₆₇

—

119

#9849

—

maximal, Sylow

120

C₂×C₆₇

—

121

C₂×C₆₇

#1962

yes

characteristic

122

C₂×C₆₇

#1972

yes

characteristic

123

C₂×C₆₇

#1975

yes

characteristic

124

D₆₇

—

125

#3169

—

126

#1962

—

127

C₄×#97

—

128

#1566

D₆₇

yes

characteristic, complemented

129

#1492

—

130

C₆₇

#9849

yes

characteristic, complemented, Sylow

131

#532

—

132

#500

—

133

#493

—

134

#486

—

135

#466

—

134

—

136

C₄×D₈

—

134

—

137

#423

C₂×D₆₇

yes

—

138

C₂×#104

—

134

—

139

C₂×#97

—

134

—

140

C₂×C₄×C₈

—

141

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

142

C₄³

—

143

#334

—

144

D₄⋊C₈

—

134

—

145

C₂×#41

—

134

—

146

C₂×D₈

—

134

—

147

C₂×M₁₆

—

148

C₂²×C₈

—

134

—

149

C₂²×C₈

D₂₆₈

yes

—

150

C₄⋊Q₈

—

151

#127

—

152

#124

—

134

—

153

#121

—

134

—

154

#121

—

134

—

155

C₄×D₄

—

134

—

156

C₄×D₄

—

134

—

157

C₂×C₄²

—

158

C₂×C₄²

—

134

—

159

C₂×C₄²

—

134

—

160

C₂×C₄²

C₂²×D₆₇

yes

characteristic

161

#107

—

134

—

162

C₄⋊C₈

—

163

C₄⋊C₈

—

164

#104

—

268

—

165

#102

—

134

—

166

#97

—

134

—

167

C₄×C₈

—

168

C₄×C₈

—

134

—

169

C₄×C₈

C₄×D₆₇

yes

characteristic

170

C₄×C₈

C₄×D₆₇

yes

characteristic

171

C₄×C₈

C₂²⋊D₆₇

yes

—

172

#41

—

268

—

173

#41

—

268

—

174

C₂×Q₈

—

134

—

175

C₂×D₄

—

268

—

176

C₂×D₄

—

134

—

177

C₂×D₄

—

134

—

178

C₂×D₄

—

134

—

179

C₂²×C₄

—

268

—

180

C₂²×C₄

—

134

—

181

C₂²×C₄

D₄×D₆₇

yes

characteristic

182

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

183

C₂²×C₄

#11761629

yes

characteristic

184

D₈

—

268

—

185

M₁₆

—

186

C₂×C₈

—

187

C₂×C₈

—

188

C₂×C₈

—

134

—

189

C₂×C₈

—

134

—

190

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

191

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

192

C₄⋊C₄

—

134

—

193

C₄⋊C₄

—

134

—

194

C₂²⋊C₄

—

268

—

195

C₂²⋊C₄

—

268

—

196

C₄²

—

268

—

197

C₄²

—

134

—

198

C₄²

—

134

—

199

C₄²

C₂×C₄×D₆₇

yes

characteristic

200

C₄²

D₄×D₆₇

yes

characteristic

201

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

202

C₄²

#11761629

yes

characteristic

203

C₂³

—

268

—

204

C₂³

#14524018

yes

characteristic

205

Q₈

—

268

—

206

D₄

—

268

—

207

D₄

—

268

—

208

D₄

—

268

—

209

D₄

—

536

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

268

—

213

C₂×C₄

—

268

—

214

C₂×C₄

—

268

—

215

C₂×C₄

—

536

—

216

C₂×C₄

—

536

—

217

C₂×C₄

—

134

—

218

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

219

C₂×C₄

#14524005

yes

characteristic

220

C₂×C₄

#14524005

yes

characteristic

221

C₂×C₄

#14524018

yes

characteristic

222

C₂×C₄

#14524032

yes

characteristic

223

C₂×C₄

#14524039

yes

characteristic

224

C₂×C₄

#14524070

yes

—

225

C₈

—

226

C₈

—

227

C₈

—

134

—

228

C₈

#14523969

yes

complemented

229

C₂²

—

230

C₂²

—

231

C₂²

—

536

—

232

C₂²

—

536

—

233

C₂²

—

536

—

234

C₂²

#16863297

yes

characteristic

235

C₂²

#16863650

yes

—

236

C₄

—

237

C₄

—

238

C₄

—

268

—

239

C₄

—

536

—

240

C₄

C₂×#14523969

yes

—

241

C₄

#16863507

yes

—

242

C₄

#16863517

yes

—

243

C₄

#16863782

yes

—

244

C₂

—

245

C₂

—

1072

—

246

C₂

#22561036

yes

characteristic

247

C₂

#22561083

yes

characteristic

248

C₂

#22561100

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14524039
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1566	C₂
socle	C₂²×C₆₇	#390
prime radical	#22570788	C₂
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1566	D₆₇
hypocenter	C₆₇	#9849
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

subgroup

G/H

center

C₂×C₄

#14524039

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1566

C₂

C₂²×C₆₇

C₂

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1566

N	H ≅ G/N	type
2. #22561036	C₂	semidirect
3. C₄×#14523953	C₂	semidirect
4. #22558659	C₂	semidirect
5. C₆₇×#1566	C₂	semidirect
7. #16863801	C₄	semidirect
8. #16863773	C₄	semidirect
9. #16863743	C₄	semidirect
102. C₈×C₆₇	#104	semidirect
128. #1566	D₆₇	semidirect
130. C₆₇	#9849	semidirect
228. C₈	#14523969	semidirect

H ≅ G/N

type

2. #22561036

C₂

semidirect

3. C₄×#14523953

C₂

semidirect

4. #22558659

C₂

semidirect

5. C₆₇×#1566

C₂

semidirect

7. #16863801

C₄

semidirect

8. #16863773

C₄

semidirect

9. #16863743

C₄

semidirect

102. C₈×C₆₇

#104

semidirect

128. #1566

D₆₇

semidirect

130. C₆₇

#9849

semidirect

228. C₈

#14523969

semidirect

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

subgroup	index
#22570788	2
#22561036	2
C₄×#14523953	2
#22558659	2
C₆₇×#1566	2
#9849	67

subgroup

index

C₆₇×#1566

Subgroups — #30755734

Conjugacy classes of subgroups

Complements

Maximal subgroups