Group #30749111 — subgroups

Subgroups — #30749111

Order 17152.

346 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 119 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	#22569143	C₂	1	1	yes	maximal, characteristic
2	#22560347	C₂	1	1	yes	maximal, characteristic
3	#22558897	C₂	1	2	yes	maximal
4	#22558084	C₂	1	2	yes	maximal
5	C₂×#335×C₆₇	C₂	1	1	yes	maximal, characteristic
6	#16863953	C₂²	1	1	yes	characteristic
7	#16863526	—	2	1	no	—
8	#16863526	C₄	1	2	yes	—
9	C₈×Dic₁₃₄	—	2	1	no	—
10	C₈×Dic₁₃₄	C₄	1	2	yes	—
11	#16863422	—	2	1	no	—
12	#16863420	C₂²	1	2	yes	—
13	#16863420	C₂²	1	1	yes	characteristic
14	#16863351	—	2	1	no	—
15	#16863161	—	2	1	no	—
16	#16863161	—	2	1	no	—
17	#16863144	—	2	2	no	—
18	#16863143	—	2	2	no	—
19	C₂×C₄⋊C₈×C₆₇	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
21	#335×C₆₇	—	2	1	no	—
22	#335×C₆₇	—	2	1	no	—
23	C₂×#11761612	—	2	1	no	—
24	C₂×#11761612	—	2	2	no	—
25	#14524043	D₄	1	1	yes	characteristic
26	#14524043	D₄	1	1	yes	characteristic
27	#14524035	—	2	1	no	—
28	#14524022	—	2	1	no	—
29	#14524022	Q₈	1	2	yes	—
30	C₄×Dic₁₃₄	—	2	1	no	—
31	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
32	#14524002	—	2	1	no	—
33	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
34	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
35	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
36	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
37	Dic₆₇⋊C₈	—	2	2	no	—
38	Dic₆₇⋊C₈	—	2	2	no	—
39	#14523953	—	2	2	no	—
40	C₈×Dic₆₇	—	2	2	no	—
41	C₂²×C₈×C₆₇	—	2	2	no	—
42	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
45	C₄⋊C₈×C₆₇	—	2	2	no	—
46	C₄⋊C₈×C₆₇	—	2	2	no	—
47	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
48	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
49	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
50	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
51	C₂×Dic₁₃₄	—	2	1	no	—
52	C₂×Dic₁₃₄	C₄⋊C₄	1	2	yes	—
53	#11761612	—	4	2	no	—
54	#11761612	—	4	2	no	—
55	#11761612	—	4	2	no	—
56	C₂×C₆₇⋊C₈	—	2	2	no	—
57	C₂×C₆₇⋊C₈	—	4	2	no	—
58	C₄⋊Dic₆₇	—	2	1	no	—
59	C₄⋊Dic₆₇	—	2	1	no	—
60	C₄⋊Dic₆₇	—	2	1	no	—
61	C₄⋊Dic₆₇	C₄⋊C₄	1	2	yes	—
62	Dic₆₇⋊C₄	—	2	2	no	—
63	Dic₆₇⋊C₄	—	2	2	no	—
64	C₂²⋊Dic₆₇	—	2	2	no	—
65	C₄×Dic₆₇	—	2	2	no	—
66	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₆₇	C₂×Q₈	1	1	yes	characteristic
68	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
69	C₂×C₈×C₆₇	—	2	2	no	—
70	C₂×C₈×C₆₇	—	2	2	no	—
71	C₂×C₈×C₆₇	—	4	2	no	—
72	C₂×C₈×C₆₇	—	8	2	no	—
73	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
74	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
75	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
76	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
78	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₆₇	#41	1	1	yes	characteristic
81	C₂³×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₆₇	—	2	1	no	—
83	C₂×C₄×C₆₇	—	2	1	no	—
84	C₂×C₄×C₆₇	—	4	1	no	—
85	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
86	C₂×C₄×C₆₇	C₂×C₄⋊C₄	1	1	yes	characteristic
87	C₂×C₄×C₆₇	#117	1	1	yes	characteristic
88	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
89	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
90	C₂×C₄×C₆₇	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₆₇	#135	1	2	yes	—
92	C₂×C₄×C₆₇	#137	1	2	yes	—
93	C₂×Dic₆₇	—	2	2	no	—
94	C₂×Dic₆₇	—	2	2	no	—
95	Dic₁₃₄	—	4	1	no	—
96	Dic₁₃₄	—	4	1	no	—
97	Dic₁₃₄	—	4	2	no	—
98	C₈×C₆₇	—	2	1	no	—
99	C₈×C₆₇	—	2	1	no	—
100	C₈×C₆₇	—	2	1	no	—
101	C₈×C₆₇	—	2	1	no	—
102	C₈×C₆₇	—	8	2	no	—
103	C₆₇⋊C₈	—	4	2	no	—
104	C₆₇⋊C₈	—	8	2	no	—
105	C₂²×C₆₇	—	2	1	no	—
106	C₂²×C₆₇	—	2	1	no	—
107	C₂²×C₆₇	#380	1	1	yes	characteristic
108	C₂²×C₆₇	#479	1	2	yes	—
109	C₄×C₆₇	—	2	1	no	—
110	C₄×C₆₇	—	2	1	no	—
111	C₄×C₆₇	—	2	1	no	—
112	C₄×C₆₇	#416	1	1	yes	characteristic
113	C₄×C₆₇	#419	1	1	yes	characteristic
114	C₄×C₆₇	#464	1	2	yes	—
115	C₄×C₆₇	#472	1	1	yes	characteristic
116	C₄×C₆₇	#473	1	1	yes	characteristic
117	Dic₆₇	—	4	2	no	—
118	Dic₆₇	—	4	2	no	—
119	#8689	—	67	1	no	maximal, Sylow
120	C₂×C₆₇	—	2	2	no	—
121	C₂×C₆₇	#1805	1	1	yes	characteristic
122	C₂×C₆₇	#1813	1	1	yes	characteristic
123	C₂×C₆₇	#1816	1	1	yes	characteristic
124	#2960	—	67	1	no	—
125	#1837	—	67	1	no	—
126	C₂×#335	D₆₇	1	1	yes	characteristic
127	#1550	—	67	2	no	—
128	#1292	—	67	2	no	—
129	C₆₇	#8689	1	1	yes	characteristic, complemented, Sylow
130	#533	—	67	1	no	—
131	#446	—	67	2	no	—
132	#446	—	134	1	no	—
133	C₈×Q₈	—	67	2	no	—
134	C₈×Q₈	—	134	1	no	—
135	#424	—	134	1	no	—
136	#423	—	67	1	no	—
137	#423	—	67	2	no	—
138	C₂×C₄⋊C₈	C₂×D₆₇	1	2	yes	—
139	#405	—	134	1	no	—
140	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
141	#341	—	134	1	no	—
142	#341	—	134	1	no	—
143	#335	—	2	1	no	—
144	#335	—	2	1	no	—
145	#333	—	134	2	no	—
146	#332	—	134	2	no	—
147	C₂×M₁₆	—	134	2	no	—
148	C₂×M₁₆	—	134	1	no	—
149	C₂²×C₈	—	2	2	no	—
150	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
151	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
152	C₄⋊Q₈	—	67	1	no	—
153	C₄⋊Q₈	—	67	1	no	—
154	#125	—	134	1	no	—
155	C₂²⋊Q₈	—	67	2	no	—
156	C₂²⋊Q₈	—	134	1	no	—
157	C₄×Q₈	—	67	2	no	—
158	C₄×Q₈	—	134	1	no	—
159	#117	—	134	1	no	—
160	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
161	C₄⋊C₈	—	2	2	no	—
162	C₄⋊C₈	—	2	2	no	—
163	C₄⋊C₈	—	67	2	no	—
164	C₄⋊C₈	—	134	2	no	—
165	C₄⋊C₈	—	134	2	no	—
166	C₄⋊C₈	—	134	2	no	—
167	C₄⋊C₈	—	134	1	no	—
168	C₄⋊C₈	—	134	2	no	—
169	#97	—	134	2	no	—
170	C₄×C₈	—	134	2	no	—
171	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
172	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
173	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
174	C₄×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
175	C₂×Q₈	—	67	2	no	—
176	C₂×Q₈	—	134	1	no	—
177	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
178	C₂²×C₄	Q₈×D₆₇	1	1	yes	characteristic
179	C₂²×C₄	#11761627	1	1	yes	characteristic
180	M₁₆	—	268	2	no	—
181	M₁₆	—	268	2	no	—
182	M₁₆	—	268	2	no	—
183	C₂×C₈	—	2	2	no	—
184	C₂×C₈	—	2	2	no	—
185	C₂×C₈	—	4	2	no	—
186	C₂×C₈	—	8	2	no	—
187	C₂×C₈	—	268	2	no	—
188	C₂×C₈	—	134	2	no	—
189	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
190	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
191	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
192	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
193	C₄⋊C₄	—	67	2	no	—
194	C₄⋊C₄	—	134	2	no	—
195	C₄⋊C₄	—	134	1	no	—
196	C₄⋊C₄	—	134	1	no	—
197	C₄⋊C₄	—	134	1	no	—
198	C₄⋊C₄	—	134	2	no	—
199	C₂²⋊C₄	—	134	2	no	—
200	C₄²	—	134	2	no	—
201	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
202	C₄²	D₄×D₆₇	1	1	yes	characteristic
203	C₄²	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
204	C₄²	#11761629	1	1	yes	characteristic
205	C₂³	#14524023	1	1	yes	characteristic
206	Q₈	—	268	2	no	—
207	Q₈	—	268	1	no	—
208	Q₈	—	268	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	4	1	no	—
212	C₂×C₄	—	134	2	no	—
213	C₂×C₄	—	134	2	no	—
214	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
215	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
216	C₂×C₄	#14524001	1	1	yes	characteristic
217	C₂×C₄	#14524014	1	1	yes	characteristic
218	C₂×C₄	#14524018	1	1	yes	characteristic
219	C₂×C₄	#14524019	1	1	yes	characteristic
220	C₂×C₄	#14524065	1	2	yes	—
221	C₂×C₄	#14524076	1	2	yes	—
222	C₈	—	2	1	no	—
223	C₈	—	2	1	no	—
224	C₈	—	2	1	no	—
225	C₈	—	2	1	no	—
226	C₈	—	8	2	no	—
227	C₈	—	268	2	no	—
228	C₈	—	536	2	no	—
229	C₂²	—	2	1	no	—
230	C₂²	—	2	1	no	—
231	C₂²	#16863253	1	1	yes	characteristic
232	C₂²	#16863704	1	2	yes	—
233	C₄	—	2	1	no	—
234	C₄	—	2	1	no	—
235	C₄	—	2	1	no	—
236	C₄	—	268	2	no	—
237	C₄	—	268	2	no	—
238	C₄	#16863393	1	1	yes	characteristic
239	C₄	#16863406	1	1	yes	characteristic
240	C₄	#16863619	1	2	yes	—
241	C₄	#16863663	1	1	yes	characteristic
242	C₄	#16863668	1	1	yes	characteristic
243	C₂	—	2	2	no	—
244	C₂	#22560225	1	1	yes	characteristic
245	C₂	#22560256	1	1	yes	characteristic
246	C₂	#22560274	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22569143

C₂

yes

maximal, characteristic

#22560347

C₂

yes

maximal, characteristic

#22558897

C₂

yes

maximal

#22558084

C₂

yes

maximal

C₂×#335×C₆₇

C₂

yes

maximal, characteristic

#16863953

C₂²

yes

characteristic

#16863526

—

#16863526

C₄

yes

—

C₈×Dic₁₃₄

—

C₈×Dic₁₃₄

C₄

yes

—

#16863422

—

#16863420

C₂²

yes

—

#16863420

C₂²

yes

—

—

—

—

—

C₂×C₄⋊C₈×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#335×C₆₇

—

#335×C₆₇

—

C₂×#11761612

—

C₂×#11761612

—

#14524043

D₄

yes

characteristic

#14524043

D₄

yes

—

—

Q₈

yes

—

C₄×Dic₁₃₄

—

C₄×Dic₁₃₄

C₂×C₄

yes

—

#14524002

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

C₂×C₄

yes

—

Dic₆₇⋊C₈

—

Dic₆₇⋊C₈

—

#14523953

—

C₈×Dic₆₇

—

C₂²×C₈×C₆₇

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

D₄

yes

characteristic

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

characteristic

C₄×C₈×C₆₇

D₄

yes

characteristic

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₄⋊C₄

yes

—

#11761612

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

C₄⋊C₄

yes

—

Dic₆₇⋊C₄

—

Dic₆₇⋊C₄

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂²×C₄×C₆₇

C₂×Q₈

yes

characteristic

C₂²×C₄×C₆₇

#41

yes

characteristic

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

—

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

characteristic

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

#41

yes

characteristic

C₂³×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₆₇

C₂×C₄⋊C₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

#135

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

100

C₈×C₆₇

—

101

C₈×C₆₇

—

102

C₈×C₆₇

—

103

C₆₇⋊C₈

—

104

C₆₇⋊C₈

—

105

C₂²×C₆₇

—

106

C₂²×C₆₇

—

107

C₂²×C₆₇

#380

yes

characteristic

108

C₂²×C₆₇

#479

yes

—

109

C₄×C₆₇

—

110

C₄×C₆₇

—

111

C₄×C₆₇

—

112

C₄×C₆₇

#416

yes

characteristic

113

C₄×C₆₇

#419

yes

characteristic

114

C₄×C₆₇

#464

yes

—

115

C₄×C₆₇

#472

yes

characteristic

116

C₄×C₆₇

#473

yes

characteristic

117

Dic₆₇

—

118

Dic₆₇

—

119

#8689

—

maximal, Sylow

120

C₂×C₆₇

—

121

C₂×C₆₇

#1805

yes

characteristic

122

C₂×C₆₇

#1813

yes

characteristic

123

C₂×C₆₇

#1816

yes

characteristic

124

#2960

—

125

#1837

—

126

C₂×#335

D₆₇

yes

characteristic

127

#1550

—

128

#1292

—

129

C₆₇

#8689

yes

characteristic, complemented, Sylow

130

#533

—

131

#446

—

132

#446

—

134

—

133

C₈×Q₈

—

134

C₈×Q₈

—

134

—

135

#424

—

134

—

136

#423

—

137

#423

—

138

C₂×C₄⋊C₈

C₂×D₆₇

yes

—

139

#405

—

134

—

140

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

141

#341

—

134

—

142

#341

—

134

—

143

#335

—

144

#335

—

145

#333

—

134

—

146

#332

—

134

—

147

C₂×M₁₆

—

134

—

148

C₂×M₁₆

—

134

—

149

C₂²×C₈

—

150

C₂²×C₈

D₂₆₈

yes

characteristic

151

C₂²×C₈

D₂₆₈

yes

characteristic

152

C₄⋊Q₈

—

153

C₄⋊Q₈

—

154

#125

—

134

—

155

C₂²⋊Q₈

—

156

C₂²⋊Q₈

—

134

—

157

C₄×Q₈

—

158

C₄×Q₈

—

134

—

159

#117

—

134

—

160

C₂×C₄²

C₂²×D₆₇

yes

characteristic

161

C₄⋊C₈

—

162

C₄⋊C₈

—

163

C₄⋊C₈

—

164

C₄⋊C₈

—

134

—

165

C₄⋊C₈

—

134

—

166

C₄⋊C₈

—

134

—

167

C₄⋊C₈

—

134

—

168

C₄⋊C₈

—

134

—

169

#97

—

134

—

170

C₄×C₈

—

134

—

171

C₄×C₈

C₄×D₆₇

yes

characteristic

172

C₄×C₈

C₄×D₆₇

yes

characteristic

173

C₄×C₈

C₂²⋊D₆₇

yes

characteristic

174

C₄×C₈

C₂²⋊D₆₇

yes

characteristic

175

C₂×Q₈

—

176

C₂×Q₈

—

134

—

177

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

178

C₂²×C₄

Q₈×D₆₇

yes

characteristic

179

C₂²×C₄

#11761627

yes

characteristic

180

M₁₆

—

268

—

181

M₁₆

—

268

—

182

M₁₆

—

268

—

183

C₂×C₈

—

184

C₂×C₈

—

185

C₂×C₈

—

186

C₂×C₈

—

187

C₂×C₈

—

268

—

188

C₂×C₈

—

134

—

189

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

characteristic

190

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

characteristic

191

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

characteristic

192

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

characteristic

193

C₄⋊C₄

—

194

C₄⋊C₄

—

134

—

195

C₄⋊C₄

—

134

—

196

C₄⋊C₄

—

134

—

197

C₄⋊C₄

—

134

—

198

C₄⋊C₄

—

134

—

199

C₂²⋊C₄

—

134

—

200

C₄²

—

134

—

201

C₄²

C₂×C₄×D₆₇

yes

characteristic

202

C₄²

D₄×D₆₇

yes

characteristic

203

C₄²

C₂×C₂²⋊D₆₇

yes

characteristic

204

C₄²

#11761629

yes

characteristic

205

C₂³

#14524023

yes

characteristic

206

Q₈

—

268

—

207

Q₈

—

268

—

208

Q₈

—

268

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

134

—

213

C₂×C₄

—

134

—

214

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

215

C₂×C₄

C₄⋊C₄×D₆₇

yes

characteristic

216

C₂×C₄

#14524001

yes

characteristic

217

C₂×C₄

#14524014

yes

characteristic

218

C₂×C₄

#14524018

yes

characteristic

219

C₂×C₄

#14524019

yes

characteristic

220

C₂×C₄

#14524065

yes

—

221

C₂×C₄

#14524076

yes

—

222

C₈

—

223

C₈

—

224

C₈

—

225

C₈

—

226

C₈

—

227

C₈

—

268

—

228

C₈

—

536

—

229

C₂²

—

230

C₂²

—

231

C₂²

#16863253

yes

characteristic

232

C₂²

#16863704

yes

—

233

C₄

—

234

C₄

—

235

C₄

—

236

C₄

—

268

—

237

C₄

—

268

—

238

C₄

#16863393

yes

characteristic

239

C₄

#16863406

yes

characteristic

240

C₄

#16863619

yes

—

241

C₄

#16863663

yes

characteristic

242

C₄

#16863668

yes

characteristic

243

C₂

—

244

C₂

#22560225

yes

characteristic

245

C₂

#22560256

yes

characteristic

246

C₂

#22560274

yes

characteristic

	subgroup	G/H
center	C₂²	#16863253
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₂×#335×C₆₇	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂³×C₆₇	C₂²⋊Q₈
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	C₂×#335	D₆₇
hypocenter	C₆₇	#8689
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#335

subgroup

G/H

center

C₂²

#16863253

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₂×#335×C₆₇

C₂

socle

C₂²×C₆₇

#380

prime radical

C₂³×C₆₇

C₂²⋊Q₈

cosocle

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×#335

N	H ≅ G/N	type
129. C₆₇	#8689	semidirect

H ≅ G/N

type

129. C₆₇

#8689

semidirect

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

subgroup	index
#22569143	2
#22560347	2
#22558897	2
#22558084	2
C₂×#335×C₆₇	2
#8689	67

subgroup

index

C₂×#335×C₆₇

Subgroups — #30749111

Conjugacy classes of subgroups

Complements

Maximal subgroups