Group #30774204 — subgroups

Subgroups — #30774204

Order 17152.

438 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 135 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	C₂²×Dic₅₃₆	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863199	C₂	1	2	yes	maximal
3	C₂×C₁₆×Dic₆₇	C₂	1	1	yes	maximal, characteristic
4	#22561068	C₂	1	2	yes	maximal
5	C₂×#366×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₆₈	C₂²	1	2	yes	—
7	C₂×Dic₅₃₆	—	2	2	no	—
8	C₂×Dic₅₃₆	—	2	2	no	—
9	C₂×Dic₅₃₆	C₄	1	4	yes	complemented
10	C₂²×C₆₇⋊C₁₆	C₂²	1	1	yes	characteristic
11	C₂×#14523966	—	2	2	no	—
12	C₂×C₈×Dic₆₇	C₂²	1	1	yes	characteristic
13	#16863301	—	2	2	no	—
14	#16863199	—	2	4	no	—
15	C₁₆×Dic₆₇	—	2	2	no	—
16	C₂²×C₁₆×C₆₇	C₂²	1	1	yes	characteristic
17	C₂×#107×C₆₇	C₂²	1	2	yes	—
18	#366×C₆₇	—	2	2	no	—
19	C₂²×Dic₁₃₄	—	2	2	no	—
20	C₂×Dic₂₆₈	—	2	2	no	—
21	C₂×Dic₂₆₈	—	2	2	no	—
22	C₂×Dic₂₆₈	—	2	2	no	—
23	C₂×Dic₂₆₈	—	2	2	no	—
24	C₂×Dic₂₆₈	—	2	2	no	—
25	C₂×Dic₂₆₈	C₂×C₄	1	4	yes	—
26	C₂²×C₆₇⋊C₈	D₄	1	1	yes	characteristic
27	C₂×C₄×Dic₆₇	D₄	1	1	yes	characteristic
28	Dic₅₃₆	—	2	8	no	—
29	C₂×C₆₇⋊C₁₆	—	2	1	no	—
30	C₂×C₆₇⋊C₁₆	—	2	1	no	—
31	C₂×C₆₇⋊C₁₆	D₄	1	2	yes	—
32	#14523966	—	4	4	no	—
33	C₈×Dic₆₇	—	2	1	no	—
34	C₈×Dic₆₇	—	2	1	no	—
35	C₆₇⋊#95	—	4	4	no	—
36	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
37	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
38	C₂×C₁₆×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
40	C₂×C₁₆×C₆₇	D₄	1	2	yes	—
41	#107×C₆₇	—	2	2	no	—
42	#107×C₆₇	—	2	2	no	—
43	C₂×Dic₁₃₄	—	2	4	no	—
44	C₂×Dic₁₃₄	—	4	2	no	—
45	C₂×Dic₁₃₄	—	4	2	no	—
46	C₂×Dic₁₃₄	—	4	2	no	—
47	C₂×Dic₁₃₄	—	4	2	no	—
48	C₂×Dic₁₃₄	—	4	2	no	—
49	C₂²×Dic₆₇	—	4	2	no	—
50	C₂²×Dic₆₇	D₈	1	2	yes	—
51	Dic₂₆₈	—	2	4	no	—
52	Dic₂₆₈	—	2	4	no	—
53	Dic₂₆₈	—	4	2	no	—
54	Dic₂₆₈	—	4	2	no	—
55	C₂×C₆₇⋊C₈	—	2	1	no	—
56	C₂×C₆₇⋊C₈	—	2	1	no	—
57	C₂×C₆₇⋊C₈	D₈	1	2	yes	—
58	C₄×Dic₆₇	—	2	1	no	—
59	C₄×Dic₆₇	—	2	1	no	—
60	C₆₇⋊C₁₆	—	2	2	no	—
61	C₂²×C₄×C₆₇	—	4	2	no	—
62	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
64	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
68	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
69	C₄⋊C₄×C₆₇	—	4	2	no	—
70	C₄⋊C₄×C₆₇	—	4	2	no	—
71	C₁₆×C₆₇	C₂²⋊C₄	1	4	yes	—
72	C₂³×C₆₇	C₂×D₈	1	1	yes	characteristic
73	C₂×C₄×C₆₇	—	8	2	no	—
74	C₂×C₄×C₆₇	—	8	4	no	—
75	C₂×C₄×C₆₇	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
77	C₂×C₄×C₆₇	#127	1	1	yes	characteristic
78	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
79	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
80	C₂×C₄×C₆₇	#135	1	1	yes	characteristic
81	C₂×Dic₆₇	—	2	2	no	—
82	C₂×Dic₆₇	—	2	2	no	—
83	C₂×Dic₆₇	—	4	4	no	—
84	C₂×Dic₆₇	—	8	2	no	—
85	C₂×Dic₆₇	—	8	2	no	—
86	C₂×Dic₆₇	Q₃₂	1	4	yes	complemented
87	Dic₁₃₄	—	4	4	no	—
88	Dic₁₃₄	—	4	4	no	—
89	Dic₁₃₄	—	8	2	no	—
90	Dic₁₃₄	—	8	2	no	—
91	C₈×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
92	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
93	C₈×C₆₇	#121	1	1	yes	characteristic
94	C₈×C₆₇	#124	1	1	yes	characteristic
95	C₆₇⋊C₈	—	2	2	no	—
96	C₂²×C₆₇	C₄×D₈	1	1	yes	characteristic
97	C₂²×C₆₇	#466	1	1	yes	characteristic
98	C₂²×C₆₇	#493	1	1	yes	characteristic
99	C₂²×C₆₇	C₂×Q₃₂	1	2	yes	—
100	C₂²×C₆₇	#508	1	2	yes	—
101	C₄×C₆₇	—	16	2	no	—
102	C₄×C₆₇	#390	1	1	yes	characteristic
103	C₄×C₆₇	C₄×D₈	1	1	yes	characteristic
104	C₄×C₆₇	#466	1	1	yes	characteristic
105	C₄×C₆₇	#495	1	1	yes	characteristic
106	Dic₆₇	—	2	4	no	—
107	Dic₆₇	—	8	4	no	—
108	#13236	—	67	1	no	maximal, Sylow
109	C₂×C₆₇	#1970	1	1	yes	characteristic
110	C₂×C₆₇	C₄×Q₃₂	1	2	yes	—
111	C₂×C₆₇	#2216	1	2	yes	—
112	C₂×C₆₇	#2247	1	1	yes	characteristic
113	C₂×C₆₇	#2249	1	1	yes	characteristic
114	C₂²×Q₃₂	—	67	1	no	—
115	C₂×#366	D₆₇	1	1	yes	characteristic
116	C₂×#358	—	67	2	no	—
117	C₂×C₄×C₁₆	—	67	1	no	—
118	#1969	—	67	2	no	—
119	C₆₇	#13236	1	1	yes	characteristic, complemented, Sylow
120	C₂²×Q₁₆	—	67	2	no	—
121	C₂×Q₃₂	—	67	4	no	—
122	C₂×Q₃₂	—	134	2	no	—
123	C₂×Q₃₂	—	134	2	no	—
124	C₂²×C₁₆	—	67	1	no	—
125	C₂²×C₁₆	C₂×D₆₇	1	1	yes	characteristic
126	C₂×#107	C₂×D₆₇	1	2	yes	—
127	C₂×#103	—	134	2	no	—
128	C₂×C₄×C₈	—	67	1	no	—
129	#391	—	134	2	no	—
130	#366	—	2	2	no	—
131	#358	—	134	4	no	—
132	C₄×C₁₆	—	134	2	no	—
133	C₂²×Q₈	—	134	2	no	—
134	C₂×Q₁₆	—	67	4	no	—
135	C₂×Q₁₆	—	134	2	no	—
136	C₂×Q₁₆	—	134	2	no	—
137	C₂×Q₁₆	—	134	2	no	—
138	C₂×Q₁₆	—	134	2	no	—
139	C₂×Q₁₆	—	134	2	no	—
140	C₂²×C₈	—	67	1	no	—
141	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
142	C₂×C₄⋊C₄	—	2	2	no	—
143	C₂×C₄²	—	67	1	no	—
144	Q₃₂	—	134	8	no	—
145	C₂×C₁₆	—	67	2	no	—
146	C₂×C₁₆	—	134	1	no	—
147	C₂×C₁₆	—	134	1	no	—
148	C₂×C₁₆	C₄×D₆₇	1	2	yes	—
149	C₂×C₁₆	D₂₆₈	1	2	yes	—
150	C₂×C₁₆	C₂²⋊D₆₇	1	2	yes	—
151	#107	—	2	2	no	—
152	#107	—	2	2	no	—
153	#103	—	268	4	no	—
154	C₄×C₈	—	134	1	no	—
155	C₄×C₈	—	134	1	no	—
156	#95	—	268	4	no	—
157	C₂×Q₈	—	268	2	no	—
158	C₂×Q₈	—	268	2	no	—
159	C₂×Q₈	—	268	2	no	—
160	C₂×Q₈	—	268	2	no	—
161	C₂×Q₈	—	268	2	no	—
162	C₂×Q₈	—	134	4	no	—
163	C₂²×C₄	—	4	2	no	—
164	C₂²×C₄	—	268	2	no	—
165	C₂²×C₄	—	67	2	no	—
166	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
167	Q₁₆	—	268	2	no	—
168	Q₁₆	—	268	2	no	—
169	Q₁₆	—	134	4	no	—
170	Q₁₆	—	134	4	no	—
171	C₂×C₈	—	67	2	no	—
172	C₂×C₈	—	134	1	no	—
173	C₂×C₈	—	134	1	no	—
174	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
175	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
176	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
177	C₂×C₈	C₂×D₂₆₈	1	1	yes	characteristic
178	C₂×C₈	#11761629	1	1	yes	characteristic
179	C₂×C₈	#11761629	1	1	yes	characteristic
180	C₄⋊C₄	—	4	2	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄²	—	134	1	no	—
183	C₄²	—	134	1	no	—
184	C₁₆	—	134	2	no	—
185	C₁₆	(C₂×D₆₇)⋊C₄	1	4	yes	—
186	C₂³	D₈×D₆₇	1	1	yes	characteristic
187	Q₈	—	268	4	no	—
188	Q₈	—	268	4	no	—
189	Q₈	—	536	2	no	—
190	Q₈	—	536	2	no	—
191	C₂×C₄	—	8	2	no	—
192	C₂×C₄	—	8	4	no	—
193	C₂×C₄	—	268	4	no	—
194	C₂×C₄	—	536	2	no	—
195	C₂×C₄	—	536	2	no	—
196	C₂×C₄	—	67	4	no	—
197	C₂×C₄	—	134	2	no	—
198	C₂×C₄	—	134	2	no	—
199	C₂×C₄	#14524005	1	1	yes	characteristic
200	C₂×C₄	#14524018	1	1	yes	characteristic
201	C₂×C₄	#14524039	1	1	yes	characteristic
202	C₂×C₄	D₈×D₆₇	1	1	yes	characteristic
203	C₂×C₄	#14524066	1	1	yes	characteristic
204	C₂×C₄	#14524066	1	1	yes	characteristic
205	C₈	—	134	2	no	—
206	C₈	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
207	C₈	#14524005	1	1	yes	characteristic
208	C₈	#14524018	1	1	yes	characteristic
209	C₈	#14524032	1	1	yes	characteristic
210	C₂²	#16863478	1	1	yes	characteristic
211	C₂²	#16863633	1	1	yes	characteristic
212	C₂²	#16863771	1	1	yes	characteristic
213	C₂²	Q₃₂×D₆₇	1	2	yes	—
214	C₂²	#16863825	1	2	yes	—
215	C₄	—	16	2	no	—
216	C₄	—	536	4	no	—
217	C₄	—	134	4	no	—
218	C₄	#16863297	1	1	yes	characteristic
219	C₄	#16863478	1	1	yes	characteristic
220	C₄	#16863633	1	1	yes	characteristic
221	C₄	#16863781	1	1	yes	characteristic
222	C₂	#22561073	1	1	yes	characteristic
223	C₂	#22562067	1	2	yes	—
224	C₂	#22562301	1	2	yes	—
225	C₂	#22562460	1	1	yes	characteristic
226	C₂	#22562467	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×Dic₅₃₆

C₂

yes

maximal, characteristic

C₂×#16863199

C₂

yes

maximal

C₂×C₁₆×Dic₆₇

C₂

yes

maximal, characteristic

#22561068

C₂

yes

maximal

C₂×#366×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₆₈

C₂²

yes

—

C₂×Dic₅₃₆

—

C₂×Dic₅₃₆

—

C₂×Dic₅₃₆

C₄

yes

complemented

C₂²×C₆₇⋊C₁₆

C₂²

yes

characteristic

C₂×#14523966

—

C₂×C₈×Dic₆₇

C₂²

yes

characteristic

#16863301

—

#16863199

—

C₁₆×Dic₆₇

—

C₂²×C₁₆×C₆₇

C₂²

yes

characteristic

C₂×#107×C₆₇

C₂²

yes

—

#366×C₆₇

—

C₂²×Dic₁₃₄

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

C₂×C₄

yes

—

C₂²×C₆₇⋊C₈

D₄

yes

characteristic

C₂×C₄×Dic₆₇

D₄

yes

characteristic

Dic₅₃₆

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

—

C₂×C₆₇⋊C₁₆

D₄

yes

—

#14523966

—

C₈×Dic₆₇

—

C₈×Dic₆₇

—

C₆₇⋊#95

—

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₆₇

—

C₂×C₁₆×C₆₇

C₂×C₄

yes

—

C₂×C₁₆×C₆₇

D₄

yes

—

C₂×C₁₆×C₆₇

D₄

yes

—

#107×C₆₇

—

#107×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

D₈

yes

—

Dic₂₆₈

—

Dic₂₆₈

—

Dic₂₆₈

—

Dic₂₆₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

D₈

yes

—

C₄×Dic₆₇

—

C₄×Dic₆₇

—

C₆₇⋊C₁₆

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂²×C₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂×D₄

yes

C₂×C₈×C₆₇

yes

C₂×C₈×C₆₇

yes

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₁₆×C₆₇

C₂²⋊C₄

yes

—

C₂³×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄×D₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

C₂×D₈

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

Q₃₂

yes

complemented

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₈×C₆₇

C₄×D₄

yes

C₈×C₆₇

yes

C₈×C₆₇

yes

C₆₇⋊C₈

—

C₂²×C₆₇

C₄×D₈

yes

C₂²×C₆₇

yes

C₂²×C₆₇

yes

C₂²×C₆₇

C₂×Q₃₂

yes

—

100

C₂²×C₆₇

#508

yes

—

101

C₄×C₆₇

—

102

C₄×C₆₇

#390

yes

characteristic

103

C₄×C₆₇

C₄×D₈

yes

characteristic

104

C₄×C₆₇

#466

yes

characteristic

105

C₄×C₆₇

#495

yes

characteristic

106

Dic₆₇

—

107

Dic₆₇

—

108

#13236

—

maximal, Sylow

109

C₂×C₆₇

#1970

yes

characteristic

110

C₂×C₆₇

C₄×Q₃₂

yes

—

111

C₂×C₆₇

#2216

yes

—

112

C₂×C₆₇

#2247

yes

characteristic

113

C₂×C₆₇

#2249

yes

characteristic

114

C₂²×Q₃₂

—

115

C₂×#366

D₆₇

yes

characteristic

116

C₂×#358

—

117

C₂×C₄×C₁₆

—

118

#1969

—

119

C₆₇

#13236

yes

characteristic, complemented, Sylow

120

C₂²×Q₁₆

—

121

C₂×Q₃₂

—

122

C₂×Q₃₂

—

134

—

123

C₂×Q₃₂

—

134

—

124

C₂²×C₁₆

—

125

C₂²×C₁₆

C₂×D₆₇

yes

characteristic

126

C₂×#107

C₂×D₆₇

yes

—

127

C₂×#103

—

134

—

128

C₂×C₄×C₈

—

129

#391

—

134

—

130

#366

—

131

#358

—

134

—

132

C₄×C₁₆

—

134

—

133

C₂²×Q₈

—

134

—

134

C₂×Q₁₆

—

135

C₂×Q₁₆

—

134

—

136

C₂×Q₁₆

—

134

—

137

C₂×Q₁₆

—

134

—

138

C₂×Q₁₆

—

134

—

139

C₂×Q₁₆

—

134

—

140

C₂²×C₈

—

141

C₂²×C₈

C₂²×D₆₇

yes

characteristic

142

C₂×C₄⋊C₄

—

143

C₂×C₄²

—

144

Q₃₂

—

134

—

145

C₂×C₁₆

—

146

C₂×C₁₆

—

134

—

147

C₂×C₁₆

—

134

—

148

C₂×C₁₆

C₄×D₆₇

yes

—

149

C₂×C₁₆

D₂₆₈

yes

—

150

C₂×C₁₆

C₂²⋊D₆₇

yes

—

151

#107

—

152

#107

—

153

#103

—

268

—

154

C₄×C₈

—

134

—

155

C₄×C₈

—

134

—

156

#95

—

268

—

157

C₂×Q₈

—

268

—

158

C₂×Q₈

—

268

—

159

C₂×Q₈

—

268

—

160

C₂×Q₈

—

268

—

161

C₂×Q₈

—

268

—

162

C₂×Q₈

—

134

—

163

C₂²×C₄

—

164

C₂²×C₄

—

268

—

165

C₂²×C₄

—

166

C₂²×C₄

D₄×D₆₇

yes

characteristic

167

Q₁₆

—

268

—

168

Q₁₆

—

268

—

169

Q₁₆

—

134

—

170

Q₁₆

—

134

—

171

C₂×C₈

—

172

C₂×C₈

—

134

—

173

C₂×C₈

—

134

—

174

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

175

C₂×C₈

D₄×D₆₇

yes

characteristic

176

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

177

C₂×C₈

C₂×D₂₆₈

yes

characteristic

178

C₂×C₈

#11761629

yes

characteristic

179

C₂×C₈

#11761629

yes

characteristic

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄²

—

134

—

183

C₄²

—

134

—

184

C₁₆

—

134

—

185

C₁₆

(C₂×D₆₇)⋊C₄

yes

—

186

C₂³

D₈×D₆₇

yes

characteristic

187

Q₈

—

268

—

188

Q₈

—

268

—

189

Q₈

—

536

—

190

Q₈

—

536

—

191

C₂×C₄

—

192

C₂×C₄

—

193

C₂×C₄

—

268

—

194

C₂×C₄

—

536

—

195

C₂×C₄

—

536

—

196

C₂×C₄

—

197

C₂×C₄

—

134

—

198

C₂×C₄

—

134

—

199

C₂×C₄

#14524005

yes

characteristic

200

C₂×C₄

#14524018

yes

characteristic

201

C₂×C₄

#14524039

yes

characteristic

202

C₂×C₄

D₈×D₆₇

yes

characteristic

203

C₂×C₄

#14524066

yes

characteristic

204

C₂×C₄

#14524066

yes

characteristic

205

C₈

—

134

—

206

C₈

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

207

C₈

#14524005

yes

characteristic

208

C₈

#14524018

yes

characteristic

209

C₈

#14524032

yes

characteristic

210

C₂²

#16863478

yes

characteristic

211

C₂²

#16863633

yes

characteristic

212

C₂²

#16863771

yes

characteristic

213

C₂²

Q₃₂×D₆₇

yes

—

214

C₂²

#16863825

yes

—

215

C₄

—

216

C₄

—

536

—

217

C₄

—

134

—

218

C₄

#16863297

yes

characteristic

219

C₄

#16863478

yes

characteristic

220

C₄

#16863633

yes

characteristic

221

C₄

#16863781

yes

characteristic

222

C₂

#22561073

yes

characteristic

223

C₂

#22562067

yes

—

224

C₂

#22562301

yes

—

225

C₂

#22562460

yes

characteristic

226

C₂

#22562467

yes

characteristic

	subgroup	G/H
center	C₂³	D₈×D₆₇
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#366×C₆₇	C₂
socle, prime radical	C₂³×C₆₇	C₂×D₈
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#366	D₆₇
hypocenter	C₆₇	#13236
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

subgroup

G/H

center

C₂³

D₈×D₆₇

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₂×#366×C₆₇

C₂

socle, prime radical

C₂³×C₆₇

C₂×D₈

cosocle

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

N	H ≅ G/N	type
9. C₂×Dic₅₃₆	C₄	semidirect
86. C₂×Dic₆₇	Q₃₂	semidirect
119. C₆₇	#13236	semidirect

H ≅ G/N

type

9. C₂×Dic₅₃₆

C₄

semidirect

86. C₂×Dic₆₇

Q₃₂

semidirect

119. C₆₇

#13236

semidirect

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

subgroup	index
C₂²×Dic₅₃₆	2
C₂×#16863199	2
C₂×C₁₆×Dic₆₇	2
#22561068	2
C₂×#366×C₆₇	2
#13236	67

subgroup

index

C₂²×Dic₅₃₆

C₂×#16863199

C₂×C₁₆×Dic₆₇

#22561068

C₂×#366×C₆₇

#13236

Subgroups — #30774204

Conjugacy classes of subgroups

Complements

Maximal subgroups