Group #30749067 — subgroups

Subgroups — #30749067

Order 17152.

350 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 115 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	#22558957	C₂	1	1	yes	maximal, characteristic
3	#22558772	C₂	1	2	yes	maximal
4	#22558067	C₂	1	2	yes	maximal
5	C₆₇×#1835	C₂	1	1	yes	maximal, characteristic
6	#16863953	C₂²	1	1	yes	characteristic
7	#16863526	—	2	1	no	—
8	#16863526	C₄	1	2	yes	complemented
9	C₈×Dic₁₃₄	—	2	1	no	—
10	C₈×Dic₁₃₄	C₄	1	2	yes	complemented
11	#16863449	C₂²	1	2	yes	—
12	#16863422	—	2	1	no	—
13	#16863420	C₂²	1	1	yes	characteristic
14	#16863351	—	2	1	no	—
15	#16863149	—	2	1	no	—
16	#16863147	—	2	1	no	—
17	#16863132	—	2	4	no	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	#382×C₆₇	C₂²	1	2	yes	—
20	#340×C₆₇	—	2	1	no	—
21	#340×C₆₇	—	2	1	no	—
22	C₂×#11761612	—	2	1	no	—
23	#14524043	Q₈	1	1	yes	characteristic
24	#14524043	Q₈	1	1	yes	characteristic
25	#14524035	—	2	1	no	—
26	#14524022	—	2	1	no	—
27	#14524022	D₄	1	2	yes	—
28	C₄×Dic₁₃₄	—	2	1	no	—
29	C₄×Dic₁₃₄	C₂×C₄	1	2	yes	—
30	#14524002	—	2	1	no	—
31	C₄⋊(C₆₇⋊C₈)	—	2	1	no	—
32	C₄⋊(C₆₇⋊C₈)	—	2	2	no	—
33	C₄⋊(C₆₇⋊C₈)	C₂×C₄	1	2	yes	—
34	Dic₆₇⋊C₈	—	2	2	no	—
35	Dic₆₇⋊C₈	—	2	2	no	—
36	C₂²⋊(C₆₇⋊C₈)	—	2	2	no	—
37	#14523953	—	2	2	no	—
38	C₄×C₆₇⋊C₈	—	2	2	no	—
39	C₈×Dic₆₇	—	2	2	no	—
40	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
42	C₂×C₄⋊C₄×C₆₇	—	2	2	no	—
43	C₂×C₄²×C₆₇	C₂³	1	1	yes	characteristic
44	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₆₇	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
47	C₄×C₈×C₆₇	D₄	1	1	yes	characteristic
48	#95×C₆₇	—	2	2	no	—
49	#95×C₆₇	—	2	2	no	—
50	C₂×Dic₁₃₄	—	2	1	no	—
51	C₂×Dic₁₃₄	C₄⋊C₄	1	2	yes	—
52	#11761612	—	4	2	no	—
53	C₂×C₆₇⋊C₈	—	2	2	no	—
54	C₂×C₆₇⋊C₈	—	4	2	no	—
55	C₄⋊Dic₆₇	—	2	1	no	—
56	C₄⋊Dic₆₇	—	2	1	no	—
57	C₄⋊Dic₆₇	—	2	1	no	—
58	C₄⋊Dic₆₇	C₄⋊C₄	1	2	yes	—
59	Dic₆₇⋊C₄	—	2	2	no	—
60	Dic₆₇⋊C₄	—	2	2	no	—
61	C₂²⋊Dic₆₇	—	2	2	no	—
62	C₄×Dic₆₇	—	2	2	no	—
63	C₂²×C₄×C₆₇	—	4	2	no	—
64	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₆₇	#41	1	1	yes	characteristic
67	C₂×C₈×C₆₇	—	2	2	no	—
68	C₂×C₈×C₆₇	—	2	2	no	—
69	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
70	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
72	C₂×C₈×C₆₇	C₂²⋊C₄	1	1	yes	characteristic
73	C₄⋊C₄×C₆₇	—	4	2	no	—
74	C₄⋊C₄×C₆₇	—	4	2	no	—
75	C₄²×C₆₇	C₂²×C₄	1	1	yes	characteristic
76	C₄²×C₆₇	C₂×D₄	1	1	yes	characteristic
77	C₄²×C₆₇	C₂×Q₈	1	1	yes	characteristic
78	C₄²×C₆₇	#41	1	1	yes	characteristic
79	C₂³×C₆₇	#121	1	1	yes	characteristic
80	C₂×C₄×C₆₇	—	4	1	no	—
81	C₂×C₄×C₆₇	—	4	1	no	—
82	C₂×C₄×C₆₇	—	8	2	no	—
83	C₂×C₄×C₆₇	—	8	2	no	—
84	C₂×C₄×C₆₇	—	8	2	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	C₂²⋊Q₈	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	2	no	—
96	Dic₁₃₄	—	4	2	no	—
97	C₈×C₆₇	—	2	1	no	—
98	C₈×C₆₇	—	2	1	no	—
99	C₈×C₆₇	—	2	1	no	—
100	C₈×C₆₇	—	2	1	no	—
101	C₆₇⋊C₈	—	4	2	no	—
102	C₆₇⋊C₈	—	8	2	no	—
103	C₂²×C₆₇	—	2	1	no	—
104	C₂²×C₆₇	—	2	1	no	—
105	C₂²×C₆₇	#380	1	1	yes	characteristic
106	C₂²×C₆₇	#464	1	2	yes	—
107	C₄×C₆₇	—	2	1	no	—
108	C₄×C₆₇	—	2	1	no	—
109	C₄×C₆₇	—	2	1	no	—
110	C₄×C₆₇	—	2	1	no	—
111	C₄×C₆₇	—	16	2	no	—
112	C₄×C₆₇	#416	1	1	yes	characteristic
113	C₄×C₆₇	#419	1	1	yes	characteristic
114	C₄×C₆₇	#479	1	2	yes	—
115	Dic₆₇	—	4	2	no	—
116	Dic₆₇	—	4	2	no	—
117	#8681	—	67	1	no	maximal, Sylow
118	C₂×C₆₇	—	2	2	no	—
119	C₂×C₆₇	#1805	1	1	yes	characteristic
120	C₂×C₆₇	#1812	1	1	yes	characteristic
121	C₂×C₆₇	#1814	1	1	yes	characteristic
122	#2960	—	67	1	no	—
123	#1835	D₆₇	1	1	yes	characteristic
124	#1564	—	67	1	no	—
125	#1518	—	67	2	no	—
126	#1286	—	67	2	no	—
127	C₆₇	#8681	1	1	yes	characteristic, complemented, Sylow
128	#533	—	67	1	no	—
129	#446	—	67	2	no	—
130	#446	—	134	1	no	—
131	C₈×Q₈	—	67	2	no	—
132	C₈×Q₈	—	134	1	no	—
133	#431	—	67	2	no	—
134	#424	—	134	1	no	—
135	#423	—	67	1	no	—
136	#405	—	134	1	no	—
137	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
138	#382	C₂×D₆₇	1	2	yes	—
139	#340	—	2	1	no	—
140	#340	—	2	1	no	—
141	#335	—	134	1	no	—
142	#334	—	134	1	no	—
143	#328	—	134	4	no	—
144	C₂×M₁₆	—	134	1	no	—
145	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
146	C₂²×C₈	C₂²⋊D₆₇	1	1	yes	characteristic
147	C₄⋊Q₈	—	67	1	no	—
148	C₄⋊Q₈	—	67	1	no	—
149	#125	—	134	1	no	—
150	C₂²⋊Q₈	—	67	2	no	—
151	C₂²⋊Q₈	—	134	1	no	—
152	C₄×Q₈	—	67	2	no	—
153	C₄×Q₈	—	134	1	no	—
154	#117	—	134	1	no	—
155	C₂×C₄⋊C₄	—	2	2	no	—
156	C₂×C₄²	C₂²×D₆₇	1	1	yes	characteristic
157	C₄⋊C₈	—	67	2	no	—
158	C₄⋊C₈	—	134	2	no	—
159	C₄⋊C₈	—	134	2	no	—
160	C₄⋊C₈	—	134	2	no	—
161	C₄⋊C₈	—	134	1	no	—
162	C₂²⋊C₈	—	134	2	no	—
163	#97	—	134	2	no	—
164	C₄×C₈	—	134	2	no	—
165	C₄×C₈	—	134	2	no	—
166	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
167	C₄×C₈	C₄×D₆₇	1	1	yes	characteristic
168	C₄×C₈	D₂₆₈	1	1	yes	characteristic
169	C₄×C₈	D₂₆₈	1	1	yes	characteristic
170	#95	—	2	2	no	—
171	#95	—	2	2	no	—
172	C₂×Q₈	—	67	2	no	—
173	C₂×Q₈	—	134	1	no	—
174	C₂²×C₄	—	4	2	no	—
175	C₂²×C₄	D₄×D₆₇	1	1	yes	characteristic
176	C₂²×C₄	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
177	C₂²×C₄	#11761627	1	1	yes	characteristic
178	M₁₆	—	268	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	268	2	no	—
182	C₂×C₈	—	134	2	no	—
183	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
184	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
185	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
186	C₂×C₈	(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	67	2	no	—
190	C₄⋊C₄	—	134	2	no	—
191	C₄⋊C₄	—	134	2	no	—
192	C₄⋊C₄	—	134	1	no	—
193	C₄⋊C₄	—	134	1	no	—
194	C₄⋊C₄	—	134	1	no	—
195	C₂²⋊C₄	—	134	2	no	—
196	C₄²	—	134	2	no	—
197	C₄²	C₂×C₄×D₆₇	1	1	yes	characteristic
198	C₄²	C₂×D₂₆₈	1	1	yes	characteristic
199	C₄²	Q₈×D₆₇	1	1	yes	characteristic
200	C₄²	#11761629	1	1	yes	characteristic
201	C₂³	#14524014	1	1	yes	characteristic
202	Q₈	—	268	2	no	—
203	Q₈	—	268	2	no	—
204	C₂×C₄	—	4	1	no	—
205	C₂×C₄	—	4	1	no	—
206	C₂×C₄	—	8	2	no	—
207	C₂×C₄	—	8	2	no	—
208	C₂×C₄	—	8	2	no	—
209	C₂×C₄	—	134	2	no	—
210	C₂×C₄	—	134	2	no	—
211	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
212	C₂×C₄	C₄⋊C₄×D₆₇	1	1	yes	characteristic
213	C₂×C₄	#14524001	1	1	yes	characteristic
214	C₂×C₄	#14524018	1	1	yes	characteristic
215	C₂×C₄	#14524019	1	1	yes	characteristic
216	C₂×C₄	#14524023	1	1	yes	characteristic
217	C₂×C₄	#14524063	1	2	yes	—
218	C₂×C₄	#14524073	1	2	yes	—
219	C₈	—	2	1	no	—
220	C₈	—	2	1	no	—
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₈	—	268	2	no	—
224	C₈	—	536	2	no	—
225	C₂²	—	2	1	no	—
226	C₂²	—	2	1	no	—
227	C₂²	#16863253	1	1	yes	characteristic
228	C₂²	#16863615	1	2	yes	—
229	C₄	—	2	1	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	268	2	no	—
234	C₄	—	268	2	no	—
235	C₄	—	16	2	no	—
236	C₄	#16863390	1	1	yes	characteristic
237	C₄	#16863404	1	1	yes	characteristic
238	C₄	#16863700	1	2	yes	—
239	C₂	—	2	2	no	—
240	C₂	#22560221	1	1	yes	characteristic
241	C₂	#22560252	1	1	yes	characteristic
242	C₂	#22560260	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#22569143

C₂

yes

maximal, characteristic

#22558957

C₂

yes

maximal, characteristic

#22558772

C₂

yes

maximal

#22558067

C₂

yes

maximal

C₆₇×#1835

C₂

yes

maximal, characteristic

#16863953

C₂²

yes

characteristic

#16863526

—

#16863526

C₄

yes

complemented

C₈×Dic₁₃₄

—

C₈×Dic₁₃₄

C₄

yes

complemented

#16863449

C₂²

yes

—

#16863422

—

#16863420

C₂²

yes

—

—

—

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#382×C₆₇

C₂²

yes

—

#340×C₆₇

—

#340×C₆₇

—

C₂×#11761612

—

#14524043

Q₈

yes

characteristic

#14524043

Q₈

yes

—

—

D₄

yes

—

C₄×Dic₁₃₄

—

C₄×Dic₁₃₄

C₂×C₄

yes

—

#14524002

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

—

C₄⋊(C₆₇⋊C₈)

C₂×C₄

yes

—

Dic₆₇⋊C₈

—

Dic₆₇⋊C₈

—

C₂²⋊(C₆₇⋊C₈)

—

#14523953

—

C₄×C₆₇⋊C₈

—

C₈×Dic₆₇

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂×C₄⋊C₄×C₆₇

—

C₂×C₄²×C₆₇

C₂³

yes

characteristic

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

#95×C₆₇

—

#95×C₆₇

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

C₄⋊C₄

yes

—

#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

—

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

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

C₄⋊C₄×C₆₇

—

C₄⋊C₄×C₆₇

—

C₄²×C₆₇

C₂²×C₄

yes

characteristic

C₄²×C₆₇

C₂×D₄

yes

characteristic

C₄²×C₆₇

C₂×Q₈

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

C₂×C₄⋊C₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₆₇

#135

yes

—

C₂×C₄×C₆₇

#137

yes

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

C₈×C₆₇

—

C₈×C₆₇

—

C₈×C₆₇

—

100

C₈×C₆₇

—

101

C₆₇⋊C₈

—

102

C₆₇⋊C₈

—

103

C₂²×C₆₇

—

104

C₂²×C₆₇

—

105

C₂²×C₆₇

#380

yes

characteristic

106

C₂²×C₆₇

#464

yes

—

107

C₄×C₆₇

—

108

C₄×C₆₇

—

109

C₄×C₆₇

—

110

C₄×C₆₇

—

111

C₄×C₆₇

—

112

C₄×C₆₇

#416

yes

characteristic

113

C₄×C₆₇

#419

yes

characteristic

114

C₄×C₆₇

#479

yes

—

115

Dic₆₇

—

116

Dic₆₇

—

117

#8681

—

maximal, Sylow

118

C₂×C₆₇

—

119

C₂×C₆₇

#1805

yes

characteristic

120

C₂×C₆₇

#1812

yes

characteristic

121

C₂×C₆₇

#1814

yes

characteristic

122

#2960

—

123

#1835

D₆₇

yes

characteristic

124

#1564

—

125

#1518

—

126

#1286

—

127

C₆₇

#8681

yes

characteristic, complemented, Sylow

128

#533

—

129

#446

—

130

#446

—

134

—

131

C₈×Q₈

—

132

C₈×Q₈

—

134

—

133

#431

—

134

#424

—

134

—

135

#423

—

136

#405

—

134

—

137

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

138

#382

C₂×D₆₇

yes

—

139

#340

—

140

#340

—

141

#335

—

134

—

142

#334

—

134

—

143

#328

—

134

—

144

C₂×M₁₆

—

134

—

145

C₂²×C₈

C₂²⋊D₆₇

yes

characteristic

146

C₂²×C₈

C₂²⋊D₆₇

yes

characteristic

147

C₄⋊Q₈

—

148

C₄⋊Q₈

—

149

#125

—

134

—

150

C₂²⋊Q₈

—

151

C₂²⋊Q₈

—

134

—

152

C₄×Q₈

—

153

C₄×Q₈

—

134

—

154

#117

—

134

—

155

C₂×C₄⋊C₄

—

156

C₂×C₄²

C₂²×D₆₇

yes

characteristic

157

C₄⋊C₈

—

158

C₄⋊C₈

—

134

—

159

C₄⋊C₈

—

134

—

160

C₄⋊C₈

—

134

—

161

C₄⋊C₈

—

134

—

162

C₂²⋊C₈

—

134

—

163

#97

—

134

—

164

C₄×C₈

—

134

—

165

C₄×C₈

—

134

—

166

C₄×C₈

C₄×D₆₇

yes

characteristic

167

C₄×C₈

C₄×D₆₇

yes

characteristic

168

C₄×C₈

D₂₆₈

yes

characteristic

169

C₄×C₈

D₂₆₈

yes

characteristic

170

#95

—

171

#95

—

172

C₂×Q₈

—

173

C₂×Q₈

—

134

—

174

C₂²×C₄

—

175

C₂²×C₄

D₄×D₆₇

yes

characteristic

176

C₂²×C₄

C₂×C₂²⋊D₆₇

yes

characteristic

177

C₂²×C₄

#11761627

yes

characteristic

178

M₁₆

—

268

—

179

C₂×C₈

—

180

C₂×C₈

—

181

C₂×C₈

—

268

—

182

C₂×C₈

—

134

—

183

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

characteristic

184

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

characteristic

185

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

characteristic

186

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

characteristic

187

C₄⋊C₄

—

188

C₄⋊C₄

—

189

C₄⋊C₄

—

190

C₄⋊C₄

—

134

—

191

C₄⋊C₄

—

134

—

192

C₄⋊C₄

—

134

—

193

C₄⋊C₄

—

134

—

194

C₄⋊C₄

—

134

—

195

C₂²⋊C₄

—

134

—

196

C₄²

—

134

—

197

C₄²

C₂×C₄×D₆₇

yes

characteristic

198

C₄²

C₂×D₂₆₈

yes

characteristic

199

C₄²

Q₈×D₆₇

yes

characteristic

200

C₄²

#11761629

yes

characteristic

201

C₂³

#14524014

yes

characteristic

202

Q₈

—

268

—

203

Q₈

—

268

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

134

—

210

C₂×C₄

—

134

—

211

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

212

C₂×C₄

C₄⋊C₄×D₆₇

yes

characteristic

213

C₂×C₄

#14524001

yes

characteristic

214

C₂×C₄

#14524018

yes

characteristic

215

C₂×C₄

#14524019

yes

characteristic

216

C₂×C₄

#14524023

yes

characteristic

217

C₂×C₄

#14524063

yes

—

218

C₂×C₄

#14524073

yes

—

219

C₈

—

220

C₈

—

221

C₈

—

222

C₈

—

223

C₈

—

268

—

224

C₈

—

536

—

225

C₂²

—

226

C₂²

—

227

C₂²

#16863253

yes

characteristic

228

C₂²

#16863615

yes

—

229

C₄

—

230

C₄

—

231

C₄

—

232

C₄

—

233

C₄

—

268

—

234

C₄

—

268

—

235

C₄

—

236

C₄

#16863390

yes

characteristic

237

C₄

#16863404

yes

characteristic

238

C₄

#16863700

yes

—

239

C₂

—

240

C₂

#22560221

yes

characteristic

241

C₂

#22560252

yes

characteristic

242

C₂

#22560260

yes

characteristic

	subgroup	G/H
center	C₂²	#16863253
commutator	C₄²×C₆₇	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₇
Fitting	C₆₇×#1835	C₂
socle	C₂²×C₆₇	#380
prime radical	C₂³×C₆₇	#121
cosocle	C₂×C₄²×C₆₇	C₂³
hypercenter	#1835	D₆₇
hypocenter	C₆₇	#8681
LCS	G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

subgroup

G/H

center

C₂²

#16863253

commutator

C₄²×C₆₇

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₆₇

Fitting

C₆₇×#1835

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂×C₄²×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₄²×C₆₇ ▷ C₂²×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂×C₄² ◁ #1835

N	H ≅ G/N	type
8. #16863526	C₄	semidirect
10. C₈×Dic₁₃₄	C₄	semidirect
127. C₆₇	#8681	semidirect

H ≅ G/N

type

8. #16863526

C₄

semidirect

10. C₈×Dic₁₃₄

C₄

semidirect

127. C₆₇

#8681

semidirect

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

subgroup	index
#22569143	2
#22558957	2
#22558772	2
#22558067	2
C₆₇×#1835	2
#8681	67

subgroup

index

C₆₇×#1835

Subgroups — #30749067

Conjugacy classes of subgroups

Complements

Maximal subgroups