Group #30755714 — subgroups

Subgroups — #30755714

Order 17152.

478 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 139 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	#22570795	C₂	1	1	yes	maximal, characteristic
2	#22561186	C₂	1	2	yes	maximal
3	#22558968	C₂	1	1	yes	maximal, characteristic
4	#22558665	C₂	1	2	yes	maximal
5	C₄²×C₈×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863953	C₂²	1	2	yes	—
7	#16863801	C₄	1	1	yes	characteristic, complemented
8	#16863776	C₄	1	1	yes	characteristic, complemented
9	#16863747	C₄	1	2	yes	complemented
10	#16863637	—	2	2	no	—
11	C₄×Dic₂₆₈	—	2	2	no	—
12	#16863420	C₂²	1	2	yes	—
13	#16863156	—	2	4	no	—
14	#16863149	—	2	2	no	—
15	#16863128	—	2	4	no	—
16	C₂×C₄×C₈×C₆₇	—	2	2	no	—
17	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	C₄³×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×Dic₂₆₈	—	2	2	no	—
21	C₂×#11761612	—	2	2	no	—
22	#14524043	C₂×C₄	1	2	yes	—
23	#14524043	C₂×C₄	1	2	yes	—
24	#14524035	—	2	2	no	—
25	#14524022	—	2	2	no	—
26	#14524022	—	2	2	no	—
27	C₄×Dic₁₃₄	—	2	2	no	—
28	C₄×Dic₁₃₄	—	2	2	no	—
29	#14524002	—	2	2	no	—
30	#14523977	—	2	2	no	—
31	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
32	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
33	#14523968	—	2	4	no	—
34	C₂²×C₈×C₆₇	—	2	1	no	—
35	C₂²×C₈×C₆₇	D₄	1	2	yes	—
36	C₂×C₄²×C₆₇	—	2	1	no	—
37	C₂×C₄²×C₆₇	—	2	1	no	—
38	C₂×C₄²×C₆₇	D₄	1	2	yes	—
39	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
40	C₄×C₈×C₆₇	—	2	8	no	—
41	C₄×C₈×C₆₇	—	2	1	no	—
42	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
43	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₆₇	D₄	1	2	yes	—
45	C₄×C₈×C₆₇	D₄	1	2	yes	—
46	C₂×Dic₁₃₄	—	2	2	no	—
47	C₂×Dic₁₃₄	—	2	2	no	—
48	Dic₂₆₈	—	4	4	no	—
49	#11761612	—	4	4	no	—
50	C₂×C₆₇⋊C₈	—	4	2	no	—
51	C₄⋊Dic₆₇	—	2	2	no	—
52	C₄⋊Dic₆₇	—	2	2	no	—
53	C₄⋊Dic₆₇	—	2	2	no	—
54	C₄⋊Dic₆₇	—	2	2	no	—
55	Dic₆₇⋊C₄	—	4	2	no	—
56	Dic₆₇⋊C₄	—	4	2	no	—
57	C₂²⋊Dic₆₇	—	4	2	no	—
58	C₄×Dic₆₇	—	4	2	no	—
59	C₂²×C₄×C₆₇	—	2	2	no	—
60	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
63	C₂×C₈×C₆₇	—	2	4	no	—
64	C₂×C₈×C₆₇	—	2	2	no	—
65	C₂×C₈×C₆₇	—	2	4	no	—
66	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
68	C₄²×C₆₇	—	2	4	no	—
69	C₄²×C₆₇	—	2	4	no	—
70	C₄²×C₆₇	—	2	2	no	—
71	C₄²×C₆₇	Q₁₆	1	4	yes	complemented
72	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
73	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
75	C₄²×C₆₇	#41	1	1	yes	characteristic
76	C₂³×C₆₇	#121	1	1	yes	characteristic
77	C₂×C₄×C₆₇	—	2	4	no	—
78	C₂×C₄×C₆₇	—	2	4	no	—
79	C₂×C₄×C₆₇	—	2	4	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	C₂×Q₁₆	1	2	yes	—
89	C₂×C₄×C₆₇	#135	1	2	yes	—
90	C₂×Dic₆₇	—	4	2	no	—
91	C₂×Dic₆₇	—	4	2	no	—
92	Dic₁₃₄	—	4	4	no	—
93	Dic₁₃₄	—	4	4	no	—
94	C₈×C₆₇	—	2	4	no	—
95	C₈×C₆₇	—	2	2	no	—
96	C₈×C₆₇	#104	1	4	yes	—
97	C₆₇⋊C₈	—	8	2	no	—
98	C₂²×C₆₇	—	2	1	no	—
99	C₂²×C₆₇	—	2	1	no	—
100	C₂²×C₆₇	#390	1	1	yes	characteristic
101	C₂²×C₆₇	#467	1	2	yes	—
102	C₄×C₆₇	—	2	8	no	—
103	C₄×C₆₇	—	2	1	no	—
104	C₄×C₆₇	—	2	1	no	—
105	C₄×C₆₇	C₂×#104	1	2	yes	—
106	C₄×C₆₇	C₄×Q₁₆	1	2	yes	—
107	C₄×C₆₇	#443	1	2	yes	—
108	C₄×C₆₇	#494	1	1	yes	characteristic
109	C₄×C₆₇	#495	1	1	yes	characteristic
110	Dic₆₇	—	8	2	no	—
111	Dic₆₇	—	8	2	no	—
112	#9845	—	67	1	no	maximal, Sylow
113	C₂×C₆₇	—	2	2	no	—
114	C₂×C₆₇	#1962	1	1	yes	characteristic
115	C₂×C₆₇	#1969	1	1	yes	characteristic
116	C₂×C₆₇	#1974	1	1	yes	characteristic
117	#3170	—	67	1	no	—
118	#1994	—	67	2	no	—
119	C₄²×C₈	D₆₇	1	1	yes	characteristic
120	#1567	—	67	1	no	—
121	#1493	—	67	2	no	—
122	C₆₇	#9845	1	1	yes	characteristic, complemented, Sylow
123	#533	—	67	2	no	—
124	#500	—	67	1	no	—
125	#494	—	67	1	no	—
126	#487	—	67	2	no	—
127	#467	—	134	2	no	—
128	C₄×Q₁₆	—	134	2	no	—
129	#423	—	67	2	no	—
130	C₂×C₄×C₈	—	2	2	no	—
131	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
132	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
133	C₄³	C₂×D₆₇	1	1	yes	characteristic
134	#339	—	134	4	no	—
135	#335	—	134	2	no	—
136	Q₈⋊C₈	—	134	4	no	—
137	C₂×Q₁₆	—	134	2	no	—
138	C₂×M₁₆	—	134	2	no	—
139	C₂²×C₈	—	2	1	no	—
140	C₂²×C₈	C₂²⋊D₆₇	1	2	yes	—
141	C₄⋊Q₈	—	67	2	no	—
142	C₄⋊Q₈	—	67	2	no	—
143	#125	—	134	2	no	—
144	C₂²⋊Q₈	—	134	2	no	—
145	C₂²⋊Q₈	—	134	2	no	—
146	C₄×Q₈	—	134	2	no	—
147	C₄×Q₈	—	134	2	no	—
148	#117	—	134	2	no	—
149	C₂×C₄²	—	2	1	no	—
150	C₂×C₄²	—	2	1	no	—
151	C₂×C₄²	D₂₆₈	1	2	yes	—
152	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
153	#107	—	134	2	no	—
154	C₄⋊C₈	—	134	2	no	—
155	C₄⋊C₈	—	134	2	no	—
156	#103	—	134	4	no	—
157	C₄×C₈	—	2	8	no	—
158	C₄×C₈	—	2	1	no	—
159	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
160	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
161	C₄×C₈	D₂₆₈	1	2	yes	—
162	C₄×C₈	D₂₆₈	1	2	yes	—
163	C₂×Q₈	—	134	2	no	—
164	C₂×Q₈	—	134	2	no	—
165	C₂²×C₄	—	2	2	no	—
166	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
167	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
168	C₂²×C₄	#11761628	1	1	yes	characteristic
169	Q₁₆	—	268	4	no	—
170	M₁₆	—	268	4	no	—
171	C₂×C₈	—	2	4	no	—
172	C₂×C₈	—	2	2	no	—
173	C₂×C₈	—	2	4	no	—
174	C₂×C₈	—	268	2	no	—
175	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
176	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
177	C₄⋊C₄	—	268	2	no	—
178	C₄⋊C₄	—	268	2	no	—
179	C₄⋊C₄	—	134	2	no	—
180	C₄⋊C₄	—	134	2	no	—
181	C₄⋊C₄	—	134	2	no	—
182	C₄⋊C₄	—	134	2	no	—
183	C₂²⋊C₄	—	268	2	no	—
184	C₄²	—	2	4	no	—
185	C₄²	—	2	4	no	—
186	C₄²	—	2	2	no	—
187	C₄²	—	268	2	no	—
188	C₄²	Dic₂₆₈	1	4	yes	complemented
189	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
190	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
191	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
192	C₄²	#11761628	1	1	yes	characteristic
193	C₂³	#14524017	1	1	yes	characteristic
194	Q₈	—	268	4	no	—
195	Q₈	—	268	4	no	—
196	C₂×C₄	—	2	4	no	—
197	C₂×C₄	—	2	4	no	—
198	C₂×C₄	—	2	4	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	268	2	no	—
202	C₂×C₄	—	268	2	no	—
203	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
204	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
205	C₂×C₄	C₄×D₂₆₈	1	1	yes	characteristic
206	C₂×C₄	#14524017	1	1	yes	characteristic
207	C₂×C₄	#14524031	1	1	yes	characteristic
208	C₂×C₄	#14524040	1	1	yes	characteristic
209	C₂×C₄	C₂×Dic₂₆₈	1	2	yes	—
210	C₂×C₄	#14524065	1	2	yes	—
211	C₈	—	2	4	no	—
212	C₈	—	2	2	no	—
213	C₈	—	536	2	no	—
214	C₈	#14523971	1	4	yes	—
215	C₂²	—	2	1	no	—
216	C₂²	—	2	1	no	—
217	C₂²	#16863298	1	1	yes	characteristic
218	C₂²	#16863637	1	2	yes	—
219	C₄	—	2	8	no	—
220	C₄	—	2	1	no	—
221	C₄	—	2	1	no	—
222	C₄	—	536	2	no	—
223	C₄	—	536	2	no	—
224	C₄	C₂×#14523971	1	2	yes	—
225	C₄	C₄×Dic₂₆₈	1	2	yes	—
226	C₄	#16863514	1	2	yes	—
227	C₄	#16863776	1	1	yes	characteristic
228	C₄	#16863780	1	1	yes	characteristic
229	C₂	—	2	2	no	—
230	C₂	#22561037	1	1	yes	characteristic
231	C₂	#22561070	1	1	yes	characteristic
232	C₂	#22561097	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22570795

C₂

yes

maximal, characteristic

#22561186

C₂

yes

maximal

#22558968

C₂

yes

maximal, characteristic

#22558665

C₂

yes

maximal

C₄²×C₈×C₆₇

C₂

yes

maximal, characteristic

#16863953

C₂²

yes

—

#16863801

C₄

yes

characteristic, complemented

#16863776

C₄

yes

characteristic, complemented

#16863747

C₄

yes

complemented

#16863637

—

C₄×Dic₂₆₈

—

#16863420

C₂²

yes

—

#16863156

—

#16863149

—

#16863128

—

C₂×C₄×C₈×C₆₇

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

C₄³×C₆₇

C₂²

yes

characteristic

C₂×Dic₂₆₈

—

C₂×#11761612

—

#14524043

C₂×C₄

yes

—

#14524043

C₂×C₄

yes

—

#14524035

—

#14524022

—

#14524022

—

C₄×Dic₁₃₄

—

C₄×Dic₁₃₄

—

#14524002

—

#14523977

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

#14523968

—

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

—

C₂×C₄²×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

—

C₄×C₈×C₆₇

D₄

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

Dic₂₆₈

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

Q₁₆

yes

complemented

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

C₄²×C₆₇

yes

C₂³×C₆₇

yes

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

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

C₂×Q₁₆

yes

—

C₂×C₄×C₆₇

#135

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

#104

yes

—

C₆₇⋊C₈

—

C₂²×C₆₇

—

C₂²×C₆₇

—

100

C₂²×C₆₇

#390

yes

characteristic

101

C₂²×C₆₇

#467

yes

—

102

C₄×C₆₇

—

103

C₄×C₆₇

—

104

C₄×C₆₇

—

105

C₄×C₆₇

C₂×#104

yes

—

106

C₄×C₆₇

C₄×Q₁₆

yes

—

107

C₄×C₆₇

#443

yes

—

108

C₄×C₆₇

#494

yes

characteristic

109

C₄×C₆₇

#495

yes

characteristic

110

Dic₆₇

—

111

Dic₆₇

—

112

#9845

—

maximal, Sylow

113

C₂×C₆₇

—

114

C₂×C₆₇

#1962

yes

characteristic

115

C₂×C₆₇

#1969

yes

characteristic

116

C₂×C₆₇

#1974

yes

characteristic

117

#3170

—

118

#1994

—

119

C₄²×C₈

D₆₇

yes

characteristic

120

#1567

—

121

#1493

—

122

C₆₇

#9845

yes

characteristic, complemented, Sylow

123

#533

—

124

#500

—

125

#494

—

126

#487

—

127

#467

—

134

—

128

C₄×Q₁₆

—

134

—

129

#423

—

130

C₂×C₄×C₈

—

131

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

132

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

133

C₄³

C₂×D₆₇

yes

characteristic

134

#339

—

134

—

135

#335

—

134

—

136

Q₈⋊C₈

—

134

—

137

C₂×Q₁₆

—

134

—

138

C₂×M₁₆

—

134

—

139

C₂²×C₈

—

140

C₂²×C₈

C₂²⋊D₆₇

yes

—

141

C₄⋊Q₈

—

142

C₄⋊Q₈

—

143

#125

—

134

—

144

C₂²⋊Q₈

—

134

—

145

C₂²⋊Q₈

—

134

—

146

C₄×Q₈

—

134

—

147

C₄×Q₈

—

134

—

148

#117

—

134

—

149

C₂×C₄²

—

150

C₂×C₄²

—

151

C₂×C₄²

D₂₆₈

yes

—

152

C₂×C₄²

C₂²×D₆₇

yes

characteristic

153

#107

—

134

—

154

C₄⋊C₈

—

134

—

155

C₄⋊C₈

—

134

—

156

#103

—

134

—

157

C₄×C₈

—

158

C₄×C₈

—

159

C₄×C₈

C₄×D₆₇

yes

characteristic

160

C₄×C₈

C₄×D₆₇

yes

characteristic

161

C₄×C₈

D₂₆₈

yes

—

162

C₄×C₈

D₂₆₈

yes

—

163

C₂×Q₈

—

134

—

164

C₂×Q₈

—

134

—

165

C₂²×C₄

—

166

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

167

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

168

C₂²×C₄

#11761628

yes

characteristic

169

Q₁₆

—

268

—

170

M₁₆

—

268

—

171

C₂×C₈

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

268

—

175

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

176

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

177

C₄⋊C₄

—

268

—

178

C₄⋊C₄

—

268

—

179

C₄⋊C₄

—

134

—

180

C₄⋊C₄

—

134

—

181

C₄⋊C₄

—

134

—

182

C₄⋊C₄

—

134

—

183

C₂²⋊C₄

—

268

—

184

C₄²

—

185

C₄²

—

186

C₄²

—

187

C₄²

—

268

—

188

C₄²

Dic₂₆₈

yes

complemented

189

C₄²

C₂×C₄×D₆₇

yes

characteristic

190

C₄²

C₂×D₂₆₈

yes

characteristic

191

C₄²

C₂×D₂₆₈

yes

characteristic

192

C₄²

#11761628

yes

characteristic

193

C₂³

#14524017

yes

characteristic

194

Q₈

—

268

—

195

Q₈

—

268

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

268

—

202

C₂×C₄

—

268

—

203

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

204

C₂×C₄

C₄×D₂₆₈

yes

characteristic

205

C₂×C₄

C₄×D₂₆₈

yes

characteristic

206

C₂×C₄

#14524017

yes

characteristic

207

C₂×C₄

#14524031

yes

characteristic

208

C₂×C₄

#14524040

yes

characteristic

209

C₂×C₄

C₂×Dic₂₆₈

yes

—

210

C₂×C₄

#14524065

yes

—

211

C₈

—

212

C₈

—

213

C₈

—

536

—

214

C₈

#14523971

yes

—

215

C₂²

—

216

C₂²

—

217

C₂²

#16863298

yes

characteristic

218

C₂²

#16863637

yes

—

219

C₄

—

220

C₄

—

221

C₄

—

222

C₄

—

536

—

223

C₄

—

536

—

224

C₄

C₂×#14523971

yes

—

225

C₄

C₄×Dic₂₆₈

yes

—

226

C₄

#16863514

yes

—

227

C₄

#16863776

yes

characteristic

228

C₄

#16863780

yes

characteristic

229

C₂

—

230

C₂

#22561037

yes

characteristic

231

C₂

#22561070

yes

characteristic

232

C₂

#22561097

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#14524040
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₄²×C₈×C₆₇	C₂
socle	C₂²×C₆₇	#390
prime radical	C₂³×C₆₇	#121
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₄²×C₈	D₆₇
hypocenter	C₆₇	#9845
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄²×C₈

subgroup

G/H

center

C₂×C₄

#14524040

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₄²×C₈×C₆₇

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

hypercenter

C₄²×C₈

D₆₇

hypocenter

C₆₇

#9845

LCS

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄²×C₈

N	H ≅ G/N	type
7. #16863801	C₄	semidirect
8. #16863776	C₄	semidirect
9. #16863747	C₄	semidirect
71. C₄²×C₆₇	Q₁₆	semidirect
122. C₆₇	#9845	semidirect
188. C₄²	Dic₂₆₈	semidirect

H ≅ G/N

type

7. #16863801

C₄

semidirect

8. #16863776

C₄

semidirect

9. #16863747

C₄

semidirect

71. C₄²×C₆₇

Q₁₆

semidirect

122. C₆₇

#9845

semidirect

188. C₄²

Dic₂₆₈

semidirect

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

subgroup	index
#22570795	2
#22561186	2
#22558968	2
#22558665	2
C₄²×C₈×C₆₇	2
#9845	67

subgroup

index

C₄²×C₈×C₆₇

Subgroups — #30755714

Conjugacy classes of subgroups

Complements

Maximal subgroups