Group #30774344 — subgroups

Subgroups — #30774344

Order 17152.

462 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 127 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	C₂×#16863776	C₂	1	1	yes	maximal, characteristic
2	C₂×#16863213	C₂	1	2	yes	maximal
3	#22561930	C₂	1	2	yes	maximal
4	#22560088	C₂	1	1	yes	maximal, characteristic
5	C₂×#364×C₆₇	C₂	1	1	yes	maximal, characteristic
6	C₂²×Dic₂₆₈	—	2	1	no	—
7	C₂²×Dic₂₆₈	C₂²	1	2	yes	—
8	C₂×#14524043	—	2	1	no	—
9	#16863776	—	2	2	no	—
10	#16863776	C₄	1	4	yes	complemented
11	C₂×#14523979	C₂²	1	1	yes	characteristic
12	#16863434	C₂²	1	1	yes	characteristic
13	#16863351	—	2	1	no	—
14	#16863213	—	2	4	no	—
15	#16863201	—	2	4	no	—
16	#16863156	—	2	1	no	—
17	C₂×M₃₂×C₆₇	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₆₇	C₂²	1	1	yes	characteristic
19	#364×C₆₇	—	2	2	no	—
20	C₂²×Dic₁₃₄	—	2	2	no	—
21	C₂×Dic₂₆₈	—	2	4	no	—
22	C₂×Dic₂₆₈	—	2	2	no	—
23	C₂×Dic₂₆₈	—	2	2	no	—
24	C₂×Dic₂₆₈	—	2	2	no	—
25	C₂×Dic₂₆₈	—	2	2	no	—
26	C₂×Dic₂₆₈	—	2	2	no	—
27	C₂×Dic₂₆₈	—	4	1	no	—
28	C₂×Dic₂₆₈	—	4	1	no	—
29	C₂×Dic₂₆₈	—	4	2	no	—
30	C₂×Dic₂₆₈	C₂×C₄	1	4	yes	—
31	C₂×#11761612	—	2	1	no	—
32	#14524043	—	2	4	no	—
33	#14524043	—	4	2	no	—
34	#14524002	—	2	1	no	—
35	C₂×C₄⋊Dic₆₇	—	4	2	no	—
36	#14523979	—	2	1	no	—
37	#14523979	D₄	1	2	yes	—
38	#14523977	D₄	1	2	yes	—
39	#14523975	—	2	1	no	—
40	#14523953	—	4	1	no	—
41	C₈×Dic₆₇	—	4	1	no	—
42	C₂²×C₈×C₆₇	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₆₇	C₂³	1	1	yes	characteristic
44	C₂×C₄²×C₆₇	D₄	1	1	yes	characteristic
45	M₃₂×C₆₇	—	2	2	no	—
46	M₃₂×C₆₇	—	2	2	no	—
47	C₂×C₁₆×C₆₇	—	2	2	no	—
48	C₄×C₈×C₆₇	C₂×C₄	1	2	yes	—
49	C₄×C₈×C₆₇	D₄	1	2	yes	—
50	C₂×Dic₁₃₄	—	2	4	no	—
51	C₂×Dic₁₃₄	—	4	2	no	—
52	C₂×Dic₁₃₄	—	4	2	no	—
53	C₂×Dic₁₃₄	—	4	2	no	—
54	C₂×Dic₁₃₄	—	4	2	no	—
55	C₂×Dic₁₃₄	—	4	2	no	—
56	C₂²×Dic₆₇	—	4	2	no	—
57	Dic₂₆₈	—	2	4	no	—
58	Dic₂₆₈	—	2	4	no	—
59	Dic₂₆₈	—	4	4	no	—
60	Dic₂₆₈	—	4	4	no	—
61	Dic₂₆₈	—	4	2	no	—
62	Dic₂₆₈	—	4	2	no	—
63	#11761612	—	2	2	no	—
64	#11761612	—	4	1	no	—
65	C₂×C₆₇⋊C₈	—	4	1	no	—
66	C₄⋊Dic₆₇	—	2	2	no	—
67	C₄⋊Dic₆₇	—	4	8	no	—
68	C₂²⋊Dic₆₇	—	4	1	no	—
69	C₄×Dic₆₇	—	4	1	no	—
70	C₂²×C₄×C₆₇	—	2	1	no	—
71	C₂²×C₄×C₆₇	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₆₇	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
74	C₂×C₈×C₆₇	D₈	1	1	yes	characteristic
75	C₂×C₈×C₆₇	SD₁₆	1	2	yes	—
76	C₂×C₈×C₆₇	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
78	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
79	C₂×C₈×C₆₇	C₂×D₄	1	1	yes	characteristic
80	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
81	C₂×C₈×C₆₇	#41	1	1	yes	characteristic
82	C₄²×C₆₇	—	2	1	no	—
83	C₄²×C₆₇	C₂²⋊C₄	1	2	yes	—
84	C₁₆×C₆₇	—	4	2	no	—
85	C₂³×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
86	C₂×C₄×C₆₇	—	2	4	no	—
87	C₂×C₄×C₆₇	—	2	2	no	—
88	C₂×C₄×C₆₇	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
90	C₂×C₄×C₆₇	#121	1	1	yes	characteristic
91	C₂×C₄×C₆₇	C₂×D₈	1	1	yes	characteristic
92	C₂×C₄×C₆₇	C₂×SD₁₆	1	1	yes	characteristic
93	C₂×C₄×C₆₇	AGL(1,ℤ/8)	1	1	yes	characteristic
94	C₂×Dic₆₇	—	4	4	no	—
95	C₂×Dic₆₇	—	4	2	no	—
96	C₂×Dic₆₇	—	8	2	no	—
97	C₂×Dic₆₇	—	8	2	no	—
98	Dic₁₃₄	—	4	4	no	—
99	Dic₁₃₄	—	4	4	no	—
100	Dic₁₃₄	—	8	2	no	—
101	Dic₁₃₄	—	8	2	no	—
102	C₈×C₆₇	#102	1	2	yes	—
103	C₈×C₆₇	#102	1	2	yes	—
104	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
105	C₈×C₆₇	C₄×D₄	1	1	yes	characteristic
106	C₈×C₆₇	#124	1	1	yes	characteristic
107	C₈×C₆₇	#127	1	1	yes	characteristic
108	C₆₇⋊C₈	—	4	2	no	—
109	C₂²×C₆₇	—	2	2	no	—
110	C₂²×C₆₇	#418	1	1	yes	characteristic
111	C₂²×C₆₇	#459	1	1	yes	characteristic
112	C₂²×C₆₇	#460	1	1	yes	characteristic
113	C₄×C₆₇	—	2	4	no	—
114	C₄×C₆₇	#390	1	1	yes	characteristic
115	C₄×C₆₇	C₂×#102	1	1	yes	characteristic
116	C₄×C₆₇	#459	1	1	yes	characteristic
117	C₄×C₆₇	#460	1	1	yes	characteristic
118	Dic₆₇	—	8	2	no	—
119	Dic₆₇	—	8	4	no	—
120	#13262	—	67	1	no	maximal, Sylow
121	C₂×C₆₇	—	2	1	no	—
122	C₂×C₆₇	—	2	1	no	—
123	C₂×C₆₇	#1982	1	1	yes	characteristic
124	C₂×C₆₇	#2214	1	2	yes	—
125	C₂×#494	—	67	1	no	—
126	C₂×#364	D₆₇	1	1	yes	characteristic
127	C₂×#362	—	67	2	no	—
128	#2141	—	67	2	no	—
129	#1772	—	67	1	no	—
130	C₆₇	#13262	1	1	yes	characteristic, complemented, Sylow
131	C₂²×Q₁₆	—	67	2	no	—
132	C₂²×Q₁₆	—	134	1	no	—
133	C₂×C₄⋊Q₈	—	134	1	no	—
134	C₂×M₃₂	C₂×D₆₇	1	2	yes	—
135	#494	—	67	4	no	—
136	#494	—	134	2	no	—
137	C₂×#108	—	67	1	no	—
138	#427	—	67	1	no	—
139	#405	—	134	1	no	—
140	C₂×C₄×C₈	C₂×D₆₇	1	1	yes	characteristic
141	#364	—	2	2	no	—
142	#362	—	134	4	no	—
143	#358	—	134	4	no	—
144	#339	—	134	1	no	—
145	C₂²×Q₈	—	134	2	no	—
146	C₂×Q₁₆	—	268	1	no	—
147	C₂×Q₁₆	—	268	1	no	—
148	C₂×Q₁₆	—	268	2	no	—
149	C₂×Q₁₆	—	67	4	no	—
150	C₂×Q₁₆	—	134	4	no	—
151	C₂×Q₁₆	—	134	2	no	—
152	C₂×Q₁₆	—	134	2	no	—
153	C₂×Q₁₆	—	134	2	no	—
154	C₂×Q₁₆	—	134	2	no	—
155	C₂×Q₁₆	—	134	2	no	—
156	C₂×M₁₆	—	134	1	no	—
157	C₂²×C₈	D₂₆₈	1	1	yes	characteristic
158	C₂²×C₈	C₂²×D₆₇	1	1	yes	characteristic
159	C₄⋊Q₈	—	268	2	no	—
160	C₄⋊Q₈	—	134	4	no	—
161	#117	—	134	1	no	—
162	C₂×C₄⋊C₄	—	268	2	no	—
163	C₂×C₄²	D₂₆₈	1	1	yes	characteristic
164	M₃₂	—	2	2	no	—
165	M₃₂	—	2	2	no	—
166	C₂×C₁₆	—	2	2	no	—
167	#108	—	67	2	no	—
168	#108	—	134	1	no	—
169	#107	—	67	2	no	—
170	#106	—	134	1	no	—
171	#97	—	268	1	no	—
172	C₄×C₈	—	268	1	no	—
173	C₄×C₈	C₄×D₆₇	1	2	yes	—
174	C₄×C₈	C₂²⋊D₆₇	1	2	yes	—
175	C₂×Q₈	—	268	2	no	—
176	C₂×Q₈	—	268	2	no	—
177	C₂×Q₈	—	268	2	no	—
178	C₂×Q₈	—	268	2	no	—
179	C₂×Q₈	—	268	2	no	—
180	C₂×Q₈	—	134	4	no	—
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	268	2	no	—
183	C₂²×C₄	C₂×D₂₆₈	1	1	yes	characteristic
184	Q₁₆	—	268	4	no	—
185	Q₁₆	—	268	4	no	—
186	Q₁₆	—	268	2	no	—
187	Q₁₆	—	268	2	no	—
188	Q₁₆	—	134	4	no	—
189	Q₁₆	—	134	4	no	—
190	M₁₆	—	268	1	no	—
191	M₁₆	—	134	2	no	—
192	C₂×C₈	—	268	1	no	—
193	C₂×C₈	(C₂×D₆₇)⋊C₄	1	2	yes	—
194	C₂×C₈	D₅₃₆	1	1	yes	characteristic
195	C₂×C₈	D₅₃₆	1	1	yes	characteristic
196	C₂×C₈	#11761617	1	2	yes	—
197	C₂×C₈	C₂×C₄×D₆₇	1	1	yes	characteristic
198	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
199	C₂×C₈	D₄×D₆₇	1	1	yes	characteristic
200	C₂×C₈	C₂×C₂²⋊D₆₇	1	1	yes	characteristic
201	C₂×C₈	#11761629	1	1	yes	characteristic
202	C₂×C₈	#11761629	1	1	yes	characteristic
203	C₄⋊C₄	—	268	8	no	—
204	C₄⋊C₄	—	134	2	no	—
205	C₂²⋊C₄	—	268	1	no	—
206	C₄²	—	2	1	no	—
207	C₄²	—	268	1	no	—
208	C₄²	(C₂×D₆₇)⋊C₄	1	2	yes	—
209	C₁₆	—	4	2	no	—
210	C₂³	#14524071	1	1	yes	characteristic
211	Q₈	—	268	4	no	—
212	Q₈	—	268	4	no	—
213	Q₈	—	536	2	no	—
214	Q₈	—	536	2	no	—
215	C₂×C₄	—	2	4	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	268	2	no	—
218	C₂×C₄	—	268	4	no	—
219	C₂×C₄	—	536	2	no	—
220	C₂×C₄	—	536	2	no	—
221	C₂×C₄	C₂×(C₂×D₆₇)⋊C₄	1	1	yes	characteristic
222	C₂×C₄	#14524018	1	1	yes	characteristic
223	C₂×C₄	#14524018	1	1	yes	characteristic
224	C₂×C₄	C₂×D₅₃₆	1	1	yes	characteristic
225	C₂×C₄	C₂×#11761617	1	1	yes	characteristic
226	C₂×C₄	#14524071	1	1	yes	characteristic
227	C₈	—	268	2	no	—
228	C₈	#14523965	1	2	yes	—
229	C₈	#14523965	1	2	yes	—
230	C₈	#14524005	1	1	yes	characteristic
231	C₈	#14524005	1	1	yes	characteristic
232	C₈	#14524032	1	1	yes	characteristic
233	C₈	#14524039	1	1	yes	characteristic
234	C₂²	—	2	2	no	—
235	C₂²	#16863402	1	1	yes	characteristic
236	C₂²	#16863592	1	1	yes	characteristic
237	C₂²	#16863597	1	1	yes	characteristic
238	C₄	—	2	4	no	—
239	C₄	—	536	4	no	—
240	C₄	—	536	2	no	—
241	C₄	#16863297	1	1	yes	characteristic
242	C₄	C₂×#14523965	1	1	yes	characteristic
243	C₄	#16863592	1	1	yes	characteristic
244	C₄	#16863597	1	1	yes	characteristic
245	C₂	—	2	1	no	—
246	C₂	—	2	1	no	—
247	C₂	#22561135	1	1	yes	characteristic
248	C₂	#22562291	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16863776

C₂

yes

maximal, characteristic

C₂×#16863213

C₂

yes

maximal

#22561930

C₂

yes

maximal

#22560088

C₂

yes

maximal, characteristic

C₂×#364×C₆₇

C₂

yes

maximal, characteristic

C₂²×Dic₂₆₈

—

C₂²×Dic₂₆₈

C₂²

yes

—

C₂×#14524043

—

#16863776

—

#16863776

C₄

yes

complemented

C₂×#14523979

C₂²

yes

characteristic

#16863434

C₂²

yes

—

—

—

—

C₂×M₃₂×C₆₇

C₂²

yes

—

C₂×C₄×C₈×C₆₇

C₂²

yes

characteristic

#364×C₆₇

—

C₂²×Dic₁₃₄

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

—

C₂×Dic₂₆₈

C₂×C₄

yes

—

—

—

—

—

C₂×C₄⋊Dic₆₇

—

#14523979

—

#14523979

D₄

yes

—

#14523977

D₄

yes

—

#14523975

—

#14523953

—

C₈×Dic₆₇

—

C₂²×C₈×C₆₇

D₄

yes

characteristic

C₂²×C₈×C₆₇

C₂³

yes

characteristic

C₂×C₄²×C₆₇

D₄

yes

characteristic

M₃₂×C₆₇

—

M₃₂×C₆₇

—

C₂×C₁₆×C₆₇

—

C₄×C₈×C₆₇

C₂×C₄

yes

—

C₄×C₈×C₆₇

D₄

yes

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂×Dic₁₃₄

—

C₂²×Dic₆₇

—

Dic₂₆₈

—

Dic₂₆₈

—

Dic₂₆₈

—

Dic₂₆₈

—

Dic₂₆₈

—

Dic₂₆₈

—

#11761612

—

#11761612

—

C₂×C₆₇⋊C₈

—

C₄⋊Dic₆₇

—

C₄⋊Dic₆₇

—

C₂²⋊Dic₆₇

—

C₄×Dic₆₇

—

C₂²×C₄×C₆₇

—

C₂²×C₄×C₆₇

C₂×D₄

yes

characteristic

C₂×C₈×C₆₇

C₂²⋊C₄

yes

—

C₂×C₈×C₆₇

D₈

yes

characteristic

C₂×C₈×C₆₇

D₈

yes

characteristic

C₂×C₈×C₆₇

SD₁₆

yes

—

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

C₄²×C₆₇

—

C₄²×C₆₇

C₂²⋊C₄

yes

—

C₁₆×C₆₇

—

C₂³×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

—

C₂×C₄×C₆₇

C₂×C₂²⋊C₄

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

yes

C₂×C₄×C₆₇

C₂×D₈

yes

characteristic

C₂×C₄×C₆₇

C₂×SD₁₆

yes

characteristic

C₂×C₄×C₆₇

AGL(1,ℤ/8)

yes

characteristic

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

C₂×Dic₆₇

—

Dic₁₃₄

—

Dic₁₃₄

—

100

Dic₁₃₄

—

101

Dic₁₃₄

—

102

C₈×C₆₇

#102

yes

—

103

C₈×C₆₇

#102

yes

—

104

C₈×C₆₇

C₄×D₄

yes

characteristic

105

C₈×C₆₇

C₄×D₄

yes

characteristic

106

C₈×C₆₇

#124

yes

characteristic

107

C₈×C₆₇

#127

yes

characteristic

108

C₆₇⋊C₈

—

109

C₂²×C₆₇

—

110

C₂²×C₆₇

#418

yes

characteristic

111

C₂²×C₆₇

#459

yes

characteristic

112

C₂²×C₆₇

#460

yes

characteristic

113

C₄×C₆₇

—

114

C₄×C₆₇

#390

yes

characteristic

115

C₄×C₆₇

C₂×#102

yes

characteristic

116

C₄×C₆₇

#459

yes

characteristic

117

C₄×C₆₇

#460

yes

characteristic

118

Dic₆₇

—

119

Dic₆₇

—

120

#13262

—

maximal, Sylow

121

C₂×C₆₇

—

122

C₂×C₆₇

—

123

C₂×C₆₇

#1982

yes

characteristic

124

C₂×C₆₇

#2214

yes

—

125

C₂×#494

—

126

C₂×#364

D₆₇

yes

characteristic

127

C₂×#362

—

128

#2141

—

129

#1772

—

130

C₆₇

#13262

yes

characteristic, complemented, Sylow

131

C₂²×Q₁₆

—

132

C₂²×Q₁₆

—

134

—

133

C₂×C₄⋊Q₈

—

134

—

134

C₂×M₃₂

C₂×D₆₇

yes

—

135

#494

—

136

#494

—

134

—

137

C₂×#108

—

138

#427

—

139

#405

—

134

—

140

C₂×C₄×C₈

C₂×D₆₇

yes

characteristic

141

#364

—

142

#362

—

134

—

143

#358

—

134

—

144

#339

—

134

—

145

C₂²×Q₈

—

134

—

146

C₂×Q₁₆

—

268

—

147

C₂×Q₁₆

—

268

—

148

C₂×Q₁₆

—

268

—

149

C₂×Q₁₆

—

150

C₂×Q₁₆

—

134

—

151

C₂×Q₁₆

—

134

—

152

C₂×Q₁₆

—

134

—

153

C₂×Q₁₆

—

134

—

154

C₂×Q₁₆

—

134

—

155

C₂×Q₁₆

—

134

—

156

C₂×M₁₆

—

134

—

157

C₂²×C₈

D₂₆₈

yes

characteristic

158

C₂²×C₈

C₂²×D₆₇

yes

characteristic

159

C₄⋊Q₈

—

268

—

160

C₄⋊Q₈

—

134

—

161

#117

—

134

—

162

C₂×C₄⋊C₄

—

268

—

163

C₂×C₄²

D₂₆₈

yes

characteristic

164

M₃₂

—

165

M₃₂

—

166

C₂×C₁₆

—

167

#108

—

168

#108

—

134

—

169

#107

—

170

#106

—

134

—

171

#97

—

268

—

172

C₄×C₈

—

268

—

173

C₄×C₈

C₄×D₆₇

yes

—

174

C₄×C₈

C₂²⋊D₆₇

yes

—

175

C₂×Q₈

—

268

—

176

C₂×Q₈

—

268

—

177

C₂×Q₈

—

268

—

178

C₂×Q₈

—

268

—

179

C₂×Q₈

—

268

—

180

C₂×Q₈

—

134

—

181

C₂²×C₄

—

182

C₂²×C₄

—

268

—

183

C₂²×C₄

C₂×D₂₆₈

yes

characteristic

184

Q₁₆

—

268

—

185

Q₁₆

—

268

—

186

Q₁₆

—

268

—

187

Q₁₆

—

268

—

188

Q₁₆

—

134

—

189

Q₁₆

—

134

—

190

M₁₆

—

268

—

191

M₁₆

—

134

—

192

C₂×C₈

—

268

—

193

C₂×C₈

(C₂×D₆₇)⋊C₄

yes

—

194

C₂×C₈

D₅₃₆

yes

characteristic

195

C₂×C₈

D₅₃₆

yes

characteristic

196

C₂×C₈

#11761617

yes

—

197

C₂×C₈

C₂×C₄×D₆₇

yes

characteristic

198

C₂×C₈

D₄×D₆₇

yes

characteristic

199

C₂×C₈

D₄×D₆₇

yes

characteristic

200

C₂×C₈

C₂×C₂²⋊D₆₇

yes

characteristic

201

C₂×C₈

#11761629

yes

characteristic

202

C₂×C₈

#11761629

yes

characteristic

203

C₄⋊C₄

—

268

—

204

C₄⋊C₄

—

134

—

205

C₂²⋊C₄

—

268

—

206

C₄²

—

207

C₄²

—

268

—

208

C₄²

(C₂×D₆₇)⋊C₄

yes

—

209

C₁₆

—

210

C₂³

#14524071

yes

characteristic

211

Q₈

—

268

—

212

Q₈

—

268

—

213

Q₈

—

536

—

214

Q₈

—

536

—

215

C₂×C₄

—

216

C₂×C₄

—

217

C₂×C₄

—

268

—

218

C₂×C₄

—

268

—

219

C₂×C₄

—

536

—

220

C₂×C₄

—

536

—

221

C₂×C₄

C₂×(C₂×D₆₇)⋊C₄

yes

characteristic

222

C₂×C₄

#14524018

yes

characteristic

223

C₂×C₄

#14524018

yes

characteristic

224

C₂×C₄

C₂×D₅₃₆

yes

characteristic

225

C₂×C₄

C₂×#11761617

yes

characteristic

226

C₂×C₄

#14524071

yes

characteristic

227

C₈

—

268

—

228

C₈

#14523965

yes

—

229

C₈

#14523965

yes

—

230

C₈

#14524005

yes

characteristic

231

C₈

#14524005

yes

characteristic

232

C₈

#14524032

yes

characteristic

233

C₈

#14524039

yes

characteristic

234

C₂²

—

235

C₂²

#16863402

yes

characteristic

236

C₂²

#16863592

yes

characteristic

237

C₂²

#16863597

yes

characteristic

238

C₄

—

239

C₄

—

536

—

240

C₄

—

536

—

241

C₄

#16863297

yes

characteristic

242

C₄

C₂×#14523965

yes

characteristic

243

C₄

#16863592

yes

characteristic

244

C₄

#16863597

yes

characteristic

245

C₂

—

246

C₂

—

247

C₂

#22561135

yes

characteristic

248

C₂

#22562291

yes

—

	subgroup	G/H
center	C₂²	#16863592
commutator	C₂×C₈×C₆₇	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₆₇
Fitting	C₂×#364×C₆₇	C₂
socle	C₂²×C₆₇	#459
prime radical	C₂³×C₆₇	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₆₇	C₂³
hypercenter	C₂×#364	D₆₇
hypocenter	C₆₇	#13262
LCS	G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

subgroup

G/H

center

C₂²

#16863592

commutator

C₂×C₈×C₆₇

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₆₇

Fitting

C₂×#364×C₆₇

C₂

C₂²×C₆₇

C₂³×C₆₇

C₂²×C₈×C₆₇

C₂³

D₆₇

C₆₇

G ▷ C₂×C₈×C₆₇ ▷ C₄×C₆₇ ▷ C₂×C₆₇ ▷ C₆₇

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×#364

N	H ≅ G/N	type
10. #16863776	C₄	semidirect
130. C₆₇	#13262	semidirect

H ≅ G/N

type

10. #16863776

C₄

semidirect

130. C₆₇

#13262

semidirect

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

subgroup	index
C₂×#16863776	2
C₂×#16863213	2
#22561930	2
#22560088	2
C₂×#364×C₆₇	2
#13262	67

subgroup

index

C₂×#364×C₆₇

Subgroups — #30774344

Conjugacy classes of subgroups

Complements

Maximal subgroups