Group #30753211 — subgroups

Subgroups — #30753211

Order 17152.

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

normal subgroups only

This group has 470 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	#22565421	C₂	1	1	yes	maximal, characteristic
2	#22560205	C₂	1	1	yes	maximal, characteristic
3	#22558893	C₂	1	2	yes	maximal
4	#22558111	C₂	1	2	yes	maximal
5	C₆₇×#2982	C₂	1	1	yes	maximal, characteristic
6	C₂×#14524043	C₂²	1	1	yes	characteristic
7	#16863420	C₂²	1	2	yes	—
8	C₂×#14523962	—	2	1	no	—
9	#16863335	C₄	1	4	yes	—
10	#16863303	—	2	1	no	—
11	#16863302	—	2	2	no	—
12	#16863261	—	2	1	no	—
13	C₄×C₄⋊Dic₆₇	C₂²	1	1	yes	characteristic
14	#16863165	—	2	1	no	—
15	#16863143	—	2	4	no	—
16	C₂×C₄×Q₈×C₆₇	C₂²	1	1	yes	characteristic
17	C₈×Q₈×C₆₇	—	2	4	no	—
18	#431×C₆₇	—	2	2	no	—
19	#431×C₆₇	C₂²	1	2	yes	—
20	C₂²×Dic₁₃₄	—	2	1	no	—
21	C₂×#11761612	—	2	2	no	—
22	#14524043	—	2	2	no	—
23	#14524043	—	2	2	no	—
24	C₂×C₄⋊Dic₆₇	—	2	1	no	—
25	C₂×C₄⋊Dic₆₇	—	2	1	no	—
26	C₂×C₄⋊Dic₆₇	C₂×C₄	1	2	yes	—
27	C₂×C₄×Dic₆₇	—	2	1	no	—
28	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
29	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
30	#14523962	—	4	2	no	—
31	C₆₇⋊#95	—	2	4	no	—
32	C₆₇⋊#95	—	2	4	no	—
33	C₆₇⋊#95	C₂×C₄	1	2	yes	—
34	C₂²×Q₈×C₆₇	D₄	1	1	yes	characteristic
35	C₄×Q₈×C₆₇	—	2	2	no	—
36	C₄×Q₈×C₆₇	—	2	2	no	—
37	C₂×C₄⋊C₄×C₆₇	C₂×C₄	1	2	yes	—
38	C₂×C₄⋊C₄×C₆₇	D₄	1	1	yes	characteristic
39	C₂×C₄²×C₆₇	D₄	1	2	yes	—
40	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
41	C₄⋊C₈×C₆₇	—	2	4	no	—
42	C₄⋊C₈×C₆₇	D₄	1	4	yes	—
43	C₂²⋊C₈×C₆₇	—	2	2	no	—
44	C₄×C₈×C₆₇	—	2	4	no	—
45	C₄×C₈×C₆₇	—	2	2	no	—
46	C₂×Dic₁₃₄	—	4	4	no	—
47	C₂×Dic₁₃₄	—	4	1	no	—
48	C₂×Dic₁₃₄	—	4	1	no	—
49	C₂²×Dic₆₇	—	2	2	no	—
50	C₂²×Dic₆₇	—	2	2	no	—
51	#11761612	—	4	2	no	—
52	#11761612	—	4	2	no	—
53	C₂×C₆₇⋊C₈	—	4	2	no	—
54	C₄⋊Dic₆₇	—	2	4	no	—
55	C₄⋊Dic₆₇	—	4	2	no	—
56	C₄⋊Dic₆₇	—	4	2	no	—
57	C₄×Dic₆₇	—	4	2	no	—
58	C₂×Q₈×C₆₇	—	2	2	no	—
59	C₂×Q₈×C₆₇	—	2	2	no	—
60	C₂×Q₈×C₆₇	SD₁₆	1	2	yes	—
61	C₂×Q₈×C₆₇	Q₁₆	1	2	yes	—
62	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₆₇	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₆₇	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂×C₈×C₆₇	—	2	4	no	—
68	C₄⋊C₄×C₆₇	—	2	4	no	—
69	C₄⋊C₄×C₆₇	—	2	2	no	—
70	C₄²×C₆₇	—	2	2	no	—
71	C₄²×C₆₇	—	2	2	no	—
72	C₄²×C₆₇	C₂×D₄	1	2	yes	—
73	C₄²×C₆₇	#41	1	2	yes	—
74	C₂³×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
75	Q₈×C₆₇	—	2	4	no	—
76	Q₈×C₆₇	—	2	4	no	—
77	C₂×C₄×C₆₇	—	2	2	no	—
78	C₂×C₄×C₆₇	—	2	2	no	—
79	C₂×C₄×C₆₇	—	2	2	no	—
80	C₂×C₄×C₆₇	—	2	2	no	—
81	C₂×C₄×C₆₇	—	4	1	no	—
82	C₂×C₄×C₆₇	—	4	2	no	—
83	C₂×C₄×C₆₇	#103	1	4	yes	—
84	C₂×C₄×C₆₇	C₄×D₄	1	2	yes	—
85	C₂×C₄×C₆₇	#120	1	1	yes	characteristic
86	C₂×C₄×C₆₇	#121	1	2	yes	—
87	C₂×C₄×C₆₇	#123	1	1	yes	characteristic
88	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
89	C₂×C₄×C₆₇	C₂×Q₁₆	1	1	yes	characteristic
90	C₂×C₄×C₆₇	#135	1	2	yes	—
91	C₂×Dic₆₇	—	4	2	no	—
92	C₂×Dic₆₇	—	4	2	no	—
93	C₂×Dic₆₇	—	4	4	no	—
94	C₂×Dic₆₇	—	4	2	no	—
95	C₂×Dic₆₇	—	4	2	no	—
96	Dic₁₃₄	—	8	2	no	—
97	Dic₁₃₄	—	8	2	no	—
98	C₈×C₆₇	—	4	4	no	—
99	C₆₇⋊C₈	—	8	2	no	—
100	C₂²×C₆₇	—	2	1	no	—
101	C₂²×C₆₇	—	2	1	no	—
102	C₂²×C₆₇	#386	1	1	yes	characteristic
103	C₂²×C₆₇	C₂×#103	1	1	yes	characteristic
104	C₂²×C₆₇	#416	1	1	yes	characteristic
105	C₄×C₆₇	—	2	1	no	—
106	C₄×C₆₇	—	2	4	no	—
107	C₄×C₆₇	—	2	2	no	—
108	C₄×C₆₇	—	4	2	no	—
109	C₄×C₆₇	#454	1	2	yes	—
110	C₄×C₆₇	#465	1	2	yes	—
111	C₄×C₆₇	#467	1	2	yes	—
112	Dic₆₇	—	8	2	no	—
113	Dic₆₇	—	8	2	no	—
114	#9396	—	67	1	no	maximal, Sylow
115	C₂×C₆₇	—	2	1	no	—
116	C₂×C₆₇	—	2	1	no	—
117	C₂×C₆₇	#1888	1	1	yes	characteristic
118	C₂×C₆₇	#1892	1	1	yes	characteristic
119	C₂×C₆₇	#1915	1	1	yes	characteristic
120	#2982	D₆₇	1	1	yes	characteristic
121	#2568	—	67	1	no	—
122	#1801	—	67	1	no	—
123	#1549	—	67	2	no	—
124	#1303	—	67	2	no	—
125	C₆₇	#9396	1	1	yes	characteristic, complemented, Sylow
126	C₂×C₄⋊Q₈	—	67	1	no	—
127	C₂×C₄×Q₈	C₂×D₆₇	1	1	yes	characteristic
128	C₈×Q₈	—	2	4	no	—
129	#431	—	2	2	no	—
130	#431	C₂×D₆₇	1	2	yes	—
131	#423	—	67	2	no	—
132	C₂×#101	—	134	1	no	—
133	#400	—	67	4	no	—
134	#391	—	134	1	no	—
135	#391	—	134	2	no	—
136	#382	—	134	1	no	—
137	C₄×C₄⋊C₄	—	67	1	no	—
138	#343	—	134	1	no	—
139	#332	—	134	4	no	—
140	C₂²×Q₈	—	134	1	no	—
141	C₂²×Q₈	D₂₆₈	1	1	yes	characteristic
142	C₂×M₁₆	—	134	2	no	—
143	C₄⋊Q₈	—	134	2	no	—
144	C₄⋊Q₈	—	134	2	no	—
145	C₄×Q₈	—	2	2	no	—
146	C₄×Q₈	—	2	2	no	—
147	C₂×C₄⋊C₄	—	67	2	no	—
148	C₂×C₄⋊C₄	—	134	1	no	—
149	C₂×C₄⋊C₄	—	134	1	no	—
150	C₂×C₄⋊C₄	C₄×D₆₇	1	2	yes	—
151	C₂×C₄⋊C₄	D₂₆₈	1	1	yes	characteristic
152	C₂×C₄²	—	134	1	no	—
153	C₂×C₄²	C₂²⋊D₆₇	1	2	yes	—
154	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
155	C₄⋊C₈	—	2	4	no	—
156	C₄⋊C₈	—	134	2	no	—
157	C₄⋊C₈	—	134	2	no	—
158	C₄⋊C₈	C₂²⋊D₆₇	1	4	yes	—
159	#101	—	268	2	no	—
160	C₂²⋊C₈	—	2	2	no	—
161	C₄×C₈	—	2	4	no	—
162	C₄×C₈	—	2	2	no	—
163	#95	—	67	2	no	—
164	#95	—	134	4	no	—
165	#95	—	134	4	no	—
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	268	4	no	—
169	C₂×Q₈	—	268	1	no	—
170	C₂×Q₈	—	268	1	no	—
171	C₂×Q₈	#11761617	1	2	yes	—
172	C₂×Q₈	Dic₂₆₈	1	2	yes	—
173	C₂²×C₄	—	134	2	no	—
174	C₂²×C₄	—	134	2	no	—
175	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
176	C₂²×C₄	(C₂×D₆₇)⋊C₄	1	2	yes	—
177	C₂²×C₄	C₂×C₄×D₆₇	1	1	yes	characteristic
178	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
179	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
180	M₁₆	—	268	2	no	—
181	M₁₆	—	268	2	no	—
182	C₂×C₈	—	2	4	no	—
183	C₂×C₈	—	268	2	no	—
184	C₄⋊C₄	—	2	4	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	268	2	no	—
187	C₄⋊C₄	—	268	2	no	—
188	C₄⋊C₄	—	134	4	no	—
189	C₄²	—	2	2	no	—
190	C₄²	—	2	2	no	—
191	C₄²	—	268	2	no	—
192	C₄²	C₂×C₂²⋊D₆₇	1	2	yes	—
193	C₄²	#11761628	1	2	yes	—
194	C₂³	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
195	Q₈	—	2	4	no	—
196	Q₈	—	2	4	no	—
197	Q₈	—	536	2	no	—
198	Q₈	—	536	2	no	—
199	C₂×C₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	4	1	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	268	2	no	—
206	C₂×C₄	—	268	4	no	—
207	C₂×C₄	—	268	2	no	—
208	C₂×C₄	—	268	2	no	—
209	C₂×C₄	—	268	2	no	—
210	C₂×C₄	#14523968	1	4	yes	—
211	C₂×C₄	C₄×C₂²⋊D₆₇	1	2	yes	—
212	C₂×C₄	#14524012	1	1	yes	characteristic
213	C₂×C₄	#14524017	1	2	yes	—
214	C₂×C₄	#14524026	1	1	yes	characteristic
215	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
216	C₂×C₄	C₂×Dic₂₆₈	1	1	yes	characteristic
217	C₂×C₄	#14524065	1	2	yes	—
218	C₈	—	4	4	no	—
219	C₈	—	536	2	no	—
220	C₂²	—	2	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	#16863275	1	1	yes	characteristic
223	C₂²	C₂×#14523968	1	1	yes	characteristic
224	C₂²	#16863393	1	1	yes	characteristic
225	C₄	—	2	1	no	—
226	C₄	—	2	4	no	—
227	C₄	—	2	2	no	—
228	C₄	—	4	2	no	—
229	C₄	—	536	2	no	—
230	C₄	—	536	2	no	—
231	C₄	#16863569	1	2	yes	—
232	C₄	#16863625	1	2	yes	—
233	C₄	#16863637	1	2	yes	—
234	C₂	—	2	1	no	—
235	C₂	—	2	1	no	—
236	C₂	#22560612	1	1	yes	characteristic
237	C₂	#22560632	1	1	yes	characteristic
238	C₂	#22560761	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22565421

C₂

yes

maximal, characteristic

#22560205

C₂

yes

maximal, characteristic

#22558893

C₂

yes

maximal

#22558111

C₂

yes

maximal

C₆₇×#2982

C₂

yes

maximal, characteristic

C₂×#14524043

C₂²

yes

characteristic

#16863420

C₂²

yes

—

C₂×#14523962

—

#16863335

C₄

yes

—

#16863303

—

#16863302

—

#16863261

—

C₄×C₄⋊Dic₆₇

C₂²

yes

characteristic

#16863165

—

#16863143

—

C₂×C₄×Q₈×C₆₇

C₂²

yes

characteristic

C₈×Q₈×C₆₇

—

#431×C₆₇

—

#431×C₆₇

C₂²

yes

—

C₂²×Dic₁₃₄

—

C₂×#11761612

—

#14524043

—

#14524043

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

—

C₂×C₄⋊Dic₆₇

C₂×C₄

yes

—

C₂×C₄×Dic₆₇

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

#14523962

—

C₆₇⋊#95

—

C₆₇⋊#95

—

C₆₇⋊#95

C₂×C₄

yes

—

C₂²×Q₈×C₆₇

D₄

yes

characteristic

C₄×Q₈×C₆₇

—

C₄×Q₈×C₆₇

—

C₂×C₄⋊C₄×C₆₇

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₆₇

D₄

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

C₄⋊C₈×C₆₇

—

C₄⋊C₈×C₆₇

D₄

yes

—

C₂²⋊C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₄×C₈×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

C₂²×Dic₆₇

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

—

C₂×Q₈×C₆₇

SD₁₆

yes

—

C₂×Q₈×C₆₇

Q₁₆

yes

—

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

C₂×C₈×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄⋊C₄×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

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

#103

yes

—

C₂×C₄×C₆₇

C₄×D₄

yes

—

C₂×C₄×C₆₇

#120

yes

characteristic

C₂×C₄×C₆₇

#121

yes

—

C₂×C₄×C₆₇

#123

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

C₂×Q₁₆

yes

characteristic

C₂×C₄×C₆₇

#135

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₆₇⋊C₈

—

100

C₂²×C₆₇

—

101

C₂²×C₆₇

—

102

C₂²×C₆₇

#386

yes

characteristic

103

C₂²×C₆₇

C₂×#103

yes

characteristic

104

C₂²×C₆₇

#416

yes

characteristic

105

C₄×C₆₇

—

106

C₄×C₆₇

—

107

C₄×C₆₇

—

108

C₄×C₆₇

—

109

C₄×C₆₇

#454

yes

—

110

C₄×C₆₇

#465

yes

—

111

C₄×C₆₇

#467

yes

—

112

Dic₆₇

—

113

Dic₆₇

—

114

#9396

—

maximal, Sylow

115

C₂×C₆₇

—

116

C₂×C₆₇

—

117

C₂×C₆₇

#1888

yes

characteristic

118

C₂×C₆₇

#1892

yes

characteristic

119

C₂×C₆₇

#1915

yes

characteristic

120

#2982

D₆₇

yes

characteristic

121

#2568

—

122

#1801

—

123

#1549

—

124

#1303

—

125

C₆₇

#9396

yes

characteristic, complemented, Sylow

126

C₂×C₄⋊Q₈

—

127

C₂×C₄×Q₈

C₂×D₆₇

yes

characteristic

128

C₈×Q₈

—

129

#431

—

130

#431

C₂×D₆₇

yes

—

131

#423

—

132

C₂×#101

—

134

—

133

#400

—

134

#391

—

134

—

135

#391

—

134

—

136

#382

—

134

—

137

C₄×C₄⋊C₄

—

138

#343

—

134

—

139

#332

—

134

—

140

C₂²×Q₈

—

134

—

141

C₂²×Q₈

D₂₆₈

yes

characteristic

142

C₂×M₁₆

—

134

—

143

C₄⋊Q₈

—

134

—

144

C₄⋊Q₈

—

134

—

145

C₄×Q₈

—

146

C₄×Q₈

—

147

C₂×C₄⋊C₄

—

148

C₂×C₄⋊C₄

—

134

—

149

C₂×C₄⋊C₄

—

134

—

150

C₂×C₄⋊C₄

C₄×D₆₇

yes

—

151

C₂×C₄⋊C₄

D₂₆₈

yes

characteristic

152

C₂×C₄²

—

134

—

153

C₂×C₄²

C₂²⋊D₆₇

yes

—

154

C₂×C₄²

C₂²×D₆₇

yes

characteristic

155

C₄⋊C₈

—

156

C₄⋊C₈

—

134

—

157

C₄⋊C₈

—

134

—

158

C₄⋊C₈

C₂²⋊D₆₇

yes

—

159

#101

—

268

—

160

C₂²⋊C₈

—

161

C₄×C₈

—

162

C₄×C₈

—

163

#95

—

164

#95

—

134

—

165

#95

—

134

—

166

C₂×Q₈

—

167

C₂×Q₈

—

168

C₂×Q₈

—

268

—

169

C₂×Q₈

—

268

—

170

C₂×Q₈

—

268

—

171

C₂×Q₈

#11761617

yes

—

172

C₂×Q₈

Dic₂₆₈

yes

—

173

C₂²×C₄

—

134

—

174

C₂²×C₄

—

134

—

175

C₂²×C₄

(C₂×D₆₇)⋊C₄

yes

—

176

C₂²×C₄

(C₂×D₆₇)⋊C₄

yes

—

177

C₂²×C₄

C₂×C₄×D₆₇

yes

characteristic

178

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

179

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

180

M₁₆

—

268

—

181

M₁₆

—

268

—

182

C₂×C₈

—

183

C₂×C₈

—

268

—

184

C₄⋊C₄

—

185

C₄⋊C₄

—

186

C₄⋊C₄

—

268

—

187

C₄⋊C₄

—

268

—

188

C₄⋊C₄

—

134

—

189

C₄²

—

190

C₄²

—

191

C₄²

—

268

—

192

C₄²

C₂×C₂²⋊D₆₇

yes

—

193

C₄²

#11761628

yes

—

194

C₂³

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

195

Q₈

—

196

Q₈

—

197

Q₈

—

536

—

198

Q₈

—

536

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

268

—

206

C₂×C₄

—

268

—

207

C₂×C₄

—

268

—

208

C₂×C₄

—

268

—

209

C₂×C₄

—

268

—

210

C₂×C₄

#14523968

yes

—

211

C₂×C₄

C₄×C₂²⋊D₆₇

yes

—

212

C₂×C₄

#14524012

yes

characteristic

213

C₂×C₄

#14524017

yes

—

214

C₂×C₄

#14524026

yes

characteristic

215

C₂×C₄

C₂×#11761617

yes

characteristic

216

C₂×C₄

C₂×Dic₂₆₈

yes

characteristic

217

C₂×C₄

#14524065

yes

—

218

C₈

—

219

C₈

—

536

—

220

C₂²

—

221

C₂²

—

222

C₂²

#16863275

yes

characteristic

223

C₂²

C₂×#14523968

yes

characteristic

224

C₂²

#16863393

yes

characteristic

225

C₄

—

226

C₄

—

227

C₄

—

228

C₄

—

229

C₄

—

536

—

230

C₄

—

536

—

231

C₄

#16863569

yes

—

232

C₄

#16863625

yes

—

233

C₄

#16863637

yes

—

234

C₂

—

235

C₂

—

236

C₂

#22560612

yes

characteristic

237

C₂

#22560632

yes

characteristic

238

C₂

#22560761

yes

characteristic

	subgroup	G/H
center	C₂²	#16863275
commutator	C₂²×C₄×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#2982	C₂
socle	C₂²×C₆₇	#386
prime radical	C₂³×C₆₇	C₂×C₂²⋊C₄
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#2982	D₆₇
hypocenter	C₆₇	#9396
LCS	G ▷ C₂²×C₄×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2982

subgroup

G/H

center

C₂²

#16863275

commutator

C₂²×C₄×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#2982

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 ◁ #2982

N	H ≅ G/N	type
125. C₆₇	#9396	semidirect

H ≅ G/N

type

125. C₆₇

#9396

semidirect

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

subgroup	index
#22565421	2
#22560205	2
#22558893	2
#22558111	2
C₆₇×#2982	2
#9396	67

subgroup

index

C₆₇×#2982

Subgroups — #30753211

Conjugacy classes of subgroups

Complements

Maximal subgroups