Group #30753252 — subgroups

Subgroups — #30753252

Order 17152.

450 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 123 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	#22569294	C₂	1	1	yes	maximal, characteristic
2	#22560205	C₂	1	1	yes	maximal, characteristic
3	#22558894	C₂	1	2	yes	maximal
4	#22558112	C₂	1	2	yes	maximal
5	C₆₇×#2568	C₂	1	1	yes	maximal, characteristic
6	#16863527	—	2	4	no	—
7	#16863453	—	2	2	no	—
8	#16863449	C₂²	1	2	yes	—
9	#16863420	C₂²	1	2	yes	—
10	C₂×#14523962	—	2	1	no	—
11	#16863165	—	2	1	no	—
12	#16863142	—	2	4	no	—
13	C₂×C₄⋊Q₈×C₆₇	C₂²	1	1	yes	characteristic
14	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
15	#400×C₆₇	C₄	1	4	yes	—
16	#391×C₆₇	—	2	1	no	—
17	#391×C₆₇	—	2	2	no	—
18	#382×C₆₇	—	2	1	no	—
19	C₄×C₄⋊C₄×C₆₇	C₂²	1	1	yes	characteristic
20	C₂×#11761612	—	2	2	no	—
21	C₄⋊(C₆₇⋊C₈)	—	2	4	no	—
22	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
23	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
24	C₄⋊(C₆₇⋊C₈)	D₄	1	4	yes	—
25	#14523962	—	4	2	no	—
26	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
27	#14523954	—	2	4	no	—
28	C₄×C₆₇⋊C₈	—	2	2	no	—
29	C₂²×Q₈×C₆₇	—	2	1	no	—
30	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
31	C₄⋊Q₈×C₆₇	—	2	2	no	—
32	C₄⋊Q₈×C₆₇	—	2	2	no	—
33	C₄×Q₈×C₆₇	—	2	2	no	—
34	C₄×Q₈×C₆₇	—	2	2	no	—
35	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
36	C₂×C₄⋊C₄×C₆₇	—	2	1	no	—
37	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
39	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
40	C₂×C₄²×C₆₇	—	2	1	no	—
41	C₂×C₄²×C₆₇	D₄	1	2	yes	—
42	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
43	#95×C₆₇	—	2	4	no	—
44	#95×C₆₇	—	2	4	no	—
45	#95×C₆₇	C₂×C₄	1	2	yes	—
46	#11761612	—	4	2	no	—
47	#11761612	—	4	2	no	—
48	C₂×C₆₇⋊C₈	—	2	4	no	—
49	C₂×C₆₇⋊C₈	—	4	2	no	—
50	C₂×Q₈×C₆₇	—	2	2	no	—
51	C₂×Q₈×C₆₇	—	2	1	no	—
52	C₂×Q₈×C₆₇	—	2	1	no	—
53	C₂×Q₈×C₆₇	—	2	2	no	—
54	C₂×Q₈×C₆₇	—	4	4	no	—
55	C₂×Q₈×C₆₇	—	4	1	no	—
56	C₂×Q₈×C₆₇	—	4	1	no	—
57	C₂²×C₄×C₆₇	—	2	2	no	—
58	C₂²×C₄×C₆₇	—	2	2	no	—
59	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
64	C₄⋊C₄×C₆₇	—	2	4	no	—
65	C₄⋊C₄×C₆₇	—	2	2	no	—
66	C₄⋊C₄×C₆₇	—	2	4	no	—
67	C₄⋊C₄×C₆₇	—	4	2	no	—
68	C₄⋊C₄×C₆₇	—	4	2	no	—
69	C₄²×C₆₇	—	2	2	no	—
70	C₄²×C₆₇	—	2	2	no	—
71	C₄²×C₆₇	—	4	1	no	—
72	C₄²×C₆₇	—	4	1	no	—
73	C₄²×C₆₇	C₂×D₄	1	2	yes	—
74	C₄²×C₆₇	#41	1	2	yes	—
75	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
76	Q₈×C₆₇	—	4	2	no	—
77	Q₈×C₆₇	—	4	2	no	—
78	Q₈×C₆₇	—	8	2	no	—
79	Q₈×C₆₇	—	8	2	no	—
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	2	2	no	—
82	C₂×C₄×C₆₇	—	2	2	no	—
83	C₂×C₄×C₆₇	—	2	2	no	—
84	C₂×C₄×C₆₇	—	2	2	no	—
85	C₂×C₄×C₆₇	—	4	1	no	—
86	C₂×C₄×C₆₇	—	4	2	no	—
87	C₂×C₄×C₆₇	—	4	2	no	—
88	C₂×C₄×C₆₇	—	4	2	no	—
89	C₂×C₄×C₆₇	—	4	4	no	—
90	C₂×C₄×C₆₇	—	4	2	no	—
91	C₂×C₄×C₆₇	—	4	2	no	—
92	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
94	C₂×C₄×C₆₇	#121	1	2	yes	—
95	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
96	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×C₄×C₆₇	#137	1	1	yes	characteristic
98	C₂×C₄×C₆₇	#137	1	2	yes	—
99	C₆₇⋊C₈	—	4	4	no	—
100	C₆₇⋊C₈	—	8	2	no	—
101	C₂²×C₆₇	—	2	1	no	—
102	C₂²×C₆₇	—	2	1	no	—
103	C₂²×C₆₇	#386	1	1	yes	characteristic
104	C₂²×C₆₇	#417	1	1	yes	characteristic
105	C₂²×C₆₇	#419	1	1	yes	characteristic
106	C₄×C₆₇	—	2	1	no	—
107	C₄×C₆₇	—	2	2	no	—
108	C₄×C₆₇	—	4	2	no	—
109	C₄×C₆₇	—	4	2	no	—
110	C₄×C₆₇	—	8	2	no	—
111	C₄×C₆₇	—	8	2	no	—
112	C₄×C₆₇	#454	1	2	yes	—
113	C₄×C₆₇	#468	1	2	yes	—
114	C₄×C₆₇	#470	1	2	yes	—
115	#9404	—	67	1	no	maximal, Sylow
116	C₂×C₆₇	—	2	1	no	—
117	C₂×C₆₇	—	2	1	no	—
118	C₂×C₆₇	#1888	1	1	yes	characteristic
119	C₂×C₆₇	#1894	1	1	yes	characteristic
120	C₂×C₆₇	#1915	1	1	yes	characteristic
121	#2983	—	67	1	no	—
122	#2568	D₆₇	1	1	yes	characteristic
123	#1801	—	67	1	no	—
124	#1549	—	67	2	no	—
125	#1303	—	67	2	no	—
126	C₆₇	#9404	1	1	yes	characteristic, complemented, Sylow
127	C₂×C₄⋊Q₈	C₂×D₆₇	1	1	yes	characteristic
128	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
129	#446	—	134	4	no	—
130	#432	—	134	2	no	—
131	#431	—	67	2	no	—
132	#423	—	67	2	no	—
133	C₂×#101	—	134	1	no	—
134	#400	Dic₆₇	1	4	yes	—
135	#391	—	2	1	no	—
136	#391	—	2	2	no	—
137	#382	—	2	1	no	—
138	C₄×C₄⋊C₄	C₂×D₆₇	1	1	yes	characteristic
139	#343	—	134	1	no	—
140	#332	—	134	4	no	—
141	C₂²×Q₈	—	2	1	no	—
142	C₂²×Q₈	C₂²⋊D₆₇	1	1	yes	characteristic
143	C₂×M₁₆	—	134	2	no	—
144	C₄⋊Q₈	—	2	2	no	—
145	C₄⋊Q₈	—	2	2	no	—
146	C₄×Q₈	—	2	2	no	—
147	C₄×Q₈	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	1	no	—
149	C₂×C₄⋊C₄	—	2	1	no	—
150	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
151	C₂×C₄⋊C₄	C₂×Dic₆₇	1	2	yes	—
152	C₂×C₄⋊C₄	C₂²⋊D₆₇	1	1	yes	characteristic
153	C₂×C₄²	—	2	1	no	—
154	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
155	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
156	C₄⋊C₈	—	67	4	no	—
157	C₄⋊C₈	—	134	4	no	—
158	C₄⋊C₈	—	134	2	no	—
159	C₄⋊C₈	—	134	2	no	—
160	#101	—	268	2	no	—
161	C₂²⋊C₈	—	134	2	no	—
162	#97	—	134	4	no	—
163	C₄×C₈	—	134	2	no	—
164	#95	—	2	4	no	—
165	#95	—	2	4	no	—
166	#95	C₂×Dic₆₇	1	2	yes	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	2	1	no	—
169	C₂×Q₈	—	2	1	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	—	4	4	no	—
172	C₂×Q₈	—	4	1	no	—
173	C₂×Q₈	—	4	1	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	2	2	no	—
176	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
177	C₂²×C₄	C₂²⋊Dic₆₇	1	2	yes	—
178	C₂²×C₄	C₂²×Dic₆₇	1	1	yes	characteristic
179	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
180	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
181	M₁₆	—	268	2	no	—
182	M₁₆	—	268	2	no	—
183	C₂×C₈	—	268	2	no	—
184	C₂×C₈	—	134	4	no	—
185	C₄⋊C₄	—	2	4	no	—
186	C₄⋊C₄	—	2	2	no	—
187	C₄⋊C₄	—	2	4	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	2	2	no	—
192	C₄²	—	4	1	no	—
193	C₄²	—	4	1	no	—
194	C₄²	D₄×D₆₇	1	2	yes	—
195	C₄²	#11761627	1	2	yes	—
196	C₂³	C₂×C₂²⋊Dic₆₇	1	1	yes	characteristic
197	Q₈	—	4	2	no	—
198	Q₈	—	4	2	no	—
199	Q₈	—	8	2	no	—
200	Q₈	—	8	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	2	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	4	1	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	4	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	D₄×Dic₆₇	1	2	yes	—
214	C₂×C₄	#14524011	1	1	yes	characteristic
215	C₂×C₄	#14524015	1	2	yes	—
216	C₂×C₄	#14524024	1	1	yes	characteristic
217	C₂×C₄	#14524068	1	1	yes	characteristic
218	C₂×C₄	#14524073	1	1	yes	characteristic
219	C₂×C₄	#14524074	1	2	yes	—
220	C₈	—	268	4	no	—
221	C₈	—	536	2	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	1	no	—
224	C₂²	#16863276	1	1	yes	characteristic
225	C₂²	#16863397	1	1	yes	characteristic
226	C₂²	#16863405	1	1	yes	characteristic
227	C₄	—	2	1	no	—
228	C₄	—	2	2	no	—
229	C₄	—	4	2	no	—
230	C₄	—	4	2	no	—
231	C₄	—	8	2	no	—
232	C₄	—	8	2	no	—
233	C₄	#16863568	1	2	yes	—
234	C₄	#16863644	1	2	yes	—
235	C₄	#16863652	1	2	yes	—
236	C₂	—	2	1	no	—
237	C₂	—	2	1	no	—
238	C₂	#22560613	1	1	yes	characteristic
239	C₂	#22560643	1	1	yes	characteristic
240	C₂	#22560762	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22569294

C₂

yes

maximal, characteristic

#22560205

C₂

yes

maximal, characteristic

#22558894

C₂

yes

maximal

#22558112

C₂

yes

maximal

C₆₇×#2568

C₂

yes

maximal, characteristic

#16863527

—

#16863453

—

#16863449

C₂²

yes

—

#16863420

C₂²

yes

—

C₂×#14523962

—

#16863165

—

#16863142

—

C₂×C₄⋊Q₈×C₆₇

C₂²

yes

characteristic

C₂×C₄×Q₈×C₆₇

C₂²

yes

characteristic

#400×C₆₇

C₄

yes

—

#391×C₆₇

—

#391×C₆₇

—

#382×C₆₇

—

C₄×C₄⋊C₄×C₆₇

C₂²

yes

characteristic

C₂×#11761612

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

D₄

yes

—

#14523962

—

C₂²⋊(C₆₇⋊C₈)

—

#14523954

—

C₄×C₆₇⋊C₈

—

C₂²×Q₈×C₆₇

—

C₂²×Q₈×C₆₇

D₄

yes

characteristic

C₄⋊Q₈×C₆₇

—

C₄⋊Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

—

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

#95×C₆₇

—

#95×C₆₇

—

#95×C₆₇

C₂×C₄

yes

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²⋊C₄

yes

—

C₂²×C₄×C₆₇

C₂²×C₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

—

C₄²×C₆₇

C₂×D₄

yes

—

C₄²×C₆₇

#41

yes

—

C₂³×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

Q₈×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

#120

yes

characteristic

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

—

C₆₇⋊C₈

—

100

C₆₇⋊C₈

—

101

C₂²×C₆₇

—

102

C₂²×C₆₇

—

103

C₂²×C₆₇

#386

yes

characteristic

104

C₂²×C₆₇

#417

yes

characteristic

105

C₂²×C₆₇

#419

yes

characteristic

106

C₄×C₆₇

—

107

C₄×C₆₇

—

108

C₄×C₆₇

—

109

C₄×C₆₇

—

110

C₄×C₆₇

—

111

C₄×C₆₇

—

112

C₄×C₆₇

#454

yes

—

113

C₄×C₆₇

#468

yes

—

114

C₄×C₆₇

#470

yes

—

115

#9404

—

maximal, Sylow

116

C₂×C₆₇

—

117

C₂×C₆₇

—

118

C₂×C₆₇

#1888

yes

characteristic

119

C₂×C₆₇

#1894

yes

characteristic

120

C₂×C₆₇

#1915

yes

characteristic

121

#2983

—

122

#2568

D₆₇

yes

characteristic

123

#1801

—

124

#1549

—

125

#1303

—

126

C₆₇

#9404

yes

characteristic, complemented, Sylow

127

C₂×C₄⋊Q₈

C₂×D₆₇

yes

characteristic

128

C₂×C₄×Q₈

C₂×D₆₇

yes

characteristic

129

#446

—

134

—

130

#432

—

134

—

131

#431

—

132

#423

—

133

C₂×#101

—

134

—

134

#400

Dic₆₇

yes

—

135

#391

—

136

#391

—

137

#382

—

138

C₄×C₄⋊C₄

C₂×D₆₇

yes

characteristic

139

#343

—

134

—

140

#332

—

134

—

141

C₂²×Q₈

—

142

C₂²×Q₈

C₂²⋊D₆₇

yes

characteristic

143

C₂×M₁₆

—

134

—

144

C₄⋊Q₈

—

145

C₄⋊Q₈

—

146

C₄×Q₈

—

147

C₄×Q₈

—

148

C₂×C₄⋊C₄

—

149

C₂×C₄⋊C₄

—

150

C₂×C₄⋊C₄

C₂×Dic₆₇

yes

—

151

C₂×C₄⋊C₄

C₂×Dic₆₇

yes

—

152

C₂×C₄⋊C₄

C₂²⋊D₆₇

yes

characteristic

153

C₂×C₄²

—

154

C₂×C₄²

C₂²⋊D₆₇

yes

—

155

C₂×C₄²

C₂²×D₆₇

yes

characteristic

156

C₄⋊C₈

—

157

C₄⋊C₈

—

134

—

158

C₄⋊C₈

—

134

—

159

C₄⋊C₈

—

134

—

160

#101

—

268

—

161

C₂²⋊C₈

—

134

—

162

#97

—

134

—

163

C₄×C₈

—

134

—

164

#95

—

165

#95

—

166

#95

C₂×Dic₆₇

yes

—

167

C₂×Q₈

—

168

C₂×Q₈

—

169

C₂×Q₈

—

170

C₂×Q₈

—

171

C₂×Q₈

—

172

C₂×Q₈

—

173

C₂×Q₈

—

174

C₂²×C₄

—

175

C₂²×C₄

—

176

C₂²×C₄

C₂²⋊Dic₆₇

yes

—

177

C₂²×C₄

C₂²⋊Dic₆₇

yes

—

178

C₂²×C₄

C₂²×Dic₆₇

yes

characteristic

179

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

180

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

181

M₁₆

—

268

—

182

M₁₆

—

268

—

183

C₂×C₈

—

268

—

184

C₂×C₈

—

134

—

185

C₄⋊C₄

—

186

C₄⋊C₄

—

187

C₄⋊C₄

—

188

C₄⋊C₄

—

189

C₄⋊C₄

—

190

C₄²

—

191

C₄²

—

192

C₄²

—

193

C₄²

—

194

C₄²

D₄×D₆₇

yes

—

195

C₄²

#11761627

yes

—

196

C₂³

C₂×C₂²⋊Dic₆₇

yes

characteristic

197

Q₈

—

198

Q₈

—

199

Q₈

—

200

Q₈

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

213

C₂×C₄

D₄×Dic₆₇

yes

—

214

C₂×C₄

#14524011

yes

characteristic

215

C₂×C₄

#14524015

yes

—

216

C₂×C₄

#14524024

yes

characteristic

217

C₂×C₄

#14524068

yes

characteristic

218

C₂×C₄

#14524073

yes

characteristic

219

C₂×C₄

#14524074

yes

—

220

C₈

—

268

—

221

C₈

—

536

—

222

C₂²

—

223

C₂²

—

224

C₂²

#16863276

yes

characteristic

225

C₂²

#16863397

yes

characteristic

226

C₂²

#16863405

yes

characteristic

227

C₄

—

228

C₄

—

229

C₄

—

230

C₄

—

231

C₄

—

232

C₄

—

233

C₄

#16863568

yes

—

234

C₄

#16863644

yes

—

235

C₄

#16863652

yes

—

236

C₂

—

237

C₂

—

238

C₂

#22560613

yes

characteristic

239

C₂

#22560643

yes

characteristic

240

C₂

#22560762

yes

characteristic

	subgroup	G/H
center	C₂²	#16863276
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2568	C₂
socle	C₂²×C₆₇	#386
prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2568	D₆₇
hypocenter	C₆₇	#9404
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2568

subgroup

G/H

center

C₂²

#16863276

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#2568

C₂

socle

C₂²×C₆₇

#386

prime radical

C₂³×C₆₇

C₂×C₂²⋊C₄

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2568

N	H ≅ G/N	type
126. C₆₇	#9404	semidirect

H ≅ G/N

type

126. C₆₇

#9404

semidirect

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

subgroup	index
#22569294	2
#22560205	2
#22558894	2
#22558112	2
C₆₇×#2568	2
#9404	67

subgroup

index

C₆₇×#2568

Subgroups — #30753252

Conjugacy classes of subgroups

Complements

Maximal subgroups