Group #31915325 — subgroups

Subgroups — #31915325

Order 18176.

320 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 99 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×#17985814	C₂	1	1	yes	maximal, characteristic
2	#22590061	C₂	1	1	yes	maximal, characteristic
3	#22589209	C₂	1	1	yes	maximal, characteristic
4	#22588227	C₂	1	1	yes	maximal, characteristic
5	#22586447	C₂	1	1	yes	maximal, characteristic
6	#22586419	C₂	1	1	yes	maximal, characteristic
7	C₂×#365×C₇₁	C₂	1	1	yes	maximal, characteristic
8	C₂²×Dic₂₈₄	C₂²	1	1	yes	characteristic
9	C₂×#14556068	C₂²	1	1	yes	characteristic
10	#17986037	C₂²	1	1	yes	characteristic
11	C₂×#14556051	—	2	1	no	—
12	#17985957	—	2	1	no	—
13	C₂×#14556038	C₂²	1	1	yes	characteristic
14	#17985904	—	2	1	no	—
15	#17985814	—	2	2	no	—
16	#17985804	C₄	1	4	yes	complemented
17	#17985760	—	2	1	no	—
18	C₂×M₃₂×C₇₁	C₂²	1	1	yes	characteristic
19	C₂×#108×C₇₁	C₂²	1	1	yes	characteristic
20	C₂×#106×C₇₁	C₂²	1	1	yes	characteristic
21	#365×C₇₁	—	2	2	no	—
22	C₂²×Dic₁₄₂	—	2	1	no	—
23	C₂×Dic₂₈₄	—	2	1	no	—
24	C₂×Dic₂₈₄	—	2	1	no	—
25	C₂×Dic₂₈₄	—	2	1	no	—
26	C₂×Dic₂₈₄	—	2	1	no	—
27	C₂×Dic₂₈₄	—	2	1	no	—
28	C₂×Dic₂₈₄	C₂×C₄	1	2	yes	—
29	C₂²×C₇₁⋊C₈	D₄	1	1	yes	characteristic
30	#14556087	—	2	1	no	—
31	C₂×C₄×Dic₇₁	D₄	1	1	yes	characteristic
32	#14556068	—	2	1	no	—
33	#14556068	—	2	1	no	—
34	C₂×C₇₁⋊C₁₆	—	2	1	no	—
35	#14556062	C₂×C₄	1	2	yes	—
36	#14556060	—	2	1	no	—
37	#14556051	—	4	2	no	—
38	#14556038	—	2	1	no	—
39	#14556038	—	2	1	no	—
40	#14556038	—	4	1	no	—
41	C₈×Dic₇₁	—	4	1	no	—
42	C₇₁⋊#95	—	4	2	no	—
43	C₂×M₁₆×C₇₁	—	2	1	no	—
44	C₂²×C₈×C₇₁	C₂³	1	1	yes	characteristic
45	C₂×C₄⋊C₄×C₇₁	—	2	1	no	—
46	M₃₂×C₇₁	—	2	1	no	—
47	M₃₂×C₇₁	—	2	1	no	—
48	C₂×C₁₆×C₇₁	C₂×C₄	1	2	yes	—
49	#108×C₇₁	—	2	1	no	—
50	#108×C₇₁	D₄	1	2	yes	—
51	#106×C₇₁	—	2	1	no	—
52	#106×C₇₁	—	2	1	no	—
53	C₂×Dic₁₄₂	—	2	2	no	—
54	C₂×Dic₁₄₂	—	4	1	no	—
55	C₂×Dic₁₄₂	—	4	1	no	—
56	C₂×Dic₁₄₂	—	4	1	no	—
57	C₂×Dic₁₄₂	—	4	1	no	—
58	C₂×Dic₁₄₂	—	4	1	no	—
59	C₂²×Dic₇₁	—	2	1	no	—
60	C₂²×Dic₇₁	—	4	1	no	—
61	Dic₂₈₄	—	2	2	no	—
62	Dic₂₈₄	—	2	2	no	—
63	Dic₂₈₄	—	4	1	no	—
64	Dic₂₈₄	—	4	1	no	—
65	C₂×C₇₁⋊C₈	—	2	1	no	—
66	C₂×C₇₁⋊C₈	—	2	1	no	—
67	C₂×C₇₁⋊C₈	SD₁₆	1	2	yes	—
68	C₄⋊Dic₇₁	—	2	2	no	—
69	C₂²⋊Dic₇₁	—	4	1	no	—
70	C₄×Dic₇₁	—	2	1	no	—
71	C₄×Dic₇₁	—	2	1	no	—
72	C₄×Dic₇₁	—	4	1	no	—
73	C₇₁⋊C₁₆	—	4	1	no	—
74	C₂²×C₄×C₇₁	—	4	1	no	—
75	C₂²×C₄×C₇₁	C₂×D₄	1	1	yes	characteristic
76	M₁₆×C₇₁	—	2	2	no	—
77	M₁₆×C₇₁	—	4	1	no	—
78	C₂×C₈×C₇₁	—	4	1	no	—
79	C₂×C₈×C₇₁	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₇₁	C₂×D₄	1	1	yes	characteristic
81	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
82	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
83	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
84	C₂×C₈×C₇₁	#41	1	1	yes	characteristic
85	C₄⋊C₄×C₇₁	—	4	1	no	—
86	C₄⋊C₄×C₇₁	—	4	1	no	—
87	C₁₆×C₇₁	—	2	2	no	—
88	C₂³×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
89	C₂×C₄×C₇₁	—	8	2	no	—
90	C₂×C₄×C₇₁	—	8	1	no	—
91	C₂×C₄×C₇₁	C₄×D₄	1	1	yes	characteristic
92	C₂×C₄×C₇₁	#121	1	1	yes	characteristic
93	C₂×C₄×C₇₁	#123	1	1	yes	characteristic
94	C₂×C₄×C₇₁	C₂×SD₁₆	1	1	yes	characteristic
95	C₂×C₄×C₇₁	#135	1	1	yes	characteristic
96	C₂×C₄×C₇₁	AGL(1,ℤ/8)	1	1	yes	characteristic
97	C₂×Dic₇₁	—	4	2	no	—
98	C₂×Dic₇₁	—	4	2	no	—
99	C₂×Dic₇₁	—	4	2	no	—
100	C₂×Dic₇₁	—	4	1	no	—
101	C₂×Dic₇₁	—	8	1	no	—
102	C₂×Dic₇₁	—	8	1	no	—
103	Dic₁₄₂	—	4	2	no	—
104	Dic₁₄₂	—	4	2	no	—
105	Dic₁₄₂	—	8	1	no	—
106	Dic₁₄₂	—	8	1	no	—
107	C₈×C₇₁	—	4	2	no	—
108	C₈×C₇₁	#117	1	1	yes	characteristic
109	C₈×C₇₁	C₄×D₄	1	1	yes	characteristic
110	C₈×C₇₁	#124	1	1	yes	characteristic
111	C₈×C₇₁	#126	1	1	yes	characteristic
112	C₇₁⋊C₈	—	2	2	no	—
113	C₂²×C₇₁	—	2	2	no	—
114	C₂²×C₇₁	#442	1	1	yes	characteristic
115	C₂²×C₇₁	#460	1	1	yes	characteristic
116	C₂²×C₇₁	#482	1	1	yes	characteristic
117	C₄×C₇₁	—	16	1	no	—
118	C₄×C₇₁	#388	1	1	yes	characteristic
119	C₄×C₇₁	C₄×SD₁₆	1	1	yes	characteristic
120	C₄×C₇₁	#463	1	1	yes	characteristic
121	C₄×C₇₁	#481	1	1	yes	characteristic
122	Dic₇₁	—	8	1	no	—
123	Dic₇₁	—	8	2	no	—
124	Dic₇₁	—	8	2	no	—
125	#13141	—	71	1	no	maximal, Sylow
126	C₂×C₇₁	—	2	1	no	—
127	C₂×C₇₁	—	2	1	no	—
128	C₂×C₇₁	#1926	1	1	yes	characteristic
129	C₂×C₇₁	#2180	1	2	yes	—
130	C₂×#365	D₇₁	1	1	yes	characteristic
131	C₂×#362	—	71	1	no	—
132	#2141	—	71	1	no	—
133	#1971	—	71	1	no	—
134	#1773	—	71	1	no	—
135	#1386	—	71	1	no	—
136	#1375	—	71	1	no	—
137	C₇₁	#13141	1	1	yes	characteristic, complemented, Sylow
138	C₂²×Q₁₆	—	71	1	no	—
139	C₂×M₃₂	—	71	1	no	—
140	C₂×M₃₂	C₂×D₇₁	1	1	yes	characteristic
141	C₂×#108	C₂×D₇₁	1	1	yes	characteristic
142	#427	—	71	1	no	—
143	C₂×#106	C₂×D₇₁	1	1	yes	characteristic
144	C₂×#103	—	142	1	no	—
145	#405	—	142	1	no	—
146	C₂×#97	—	71	1	no	—
147	#391	—	142	1	no	—
148	#365	—	2	2	no	—
149	#362	—	142	2	no	—
150	#358	—	71	4	no	—
151	#339	—	142	1	no	—
152	C₂²×Q₈	—	142	1	no	—
153	C₂×Q₁₆	—	71	2	no	—
154	C₂×Q₁₆	—	142	1	no	—
155	C₂×Q₁₆	—	142	1	no	—
156	C₂×Q₁₆	—	142	1	no	—
157	C₂×Q₁₆	—	142	1	no	—
158	C₂×Q₁₆	—	142	1	no	—
159	C₂×M₁₆	—	2	1	no	—
160	C₂²×C₈	—	71	1	no	—
161	C₂²×C₈	C₂²×D₇₁	1	1	yes	characteristic
162	#117	—	142	1	no	—
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄²	—	71	1	no	—
165	M₃₂	—	2	1	no	—
166	M₃₂	—	2	1	no	—
167	M₃₂	—	142	1	no	—
168	M₃₂	—	142	1	no	—
169	C₂×C₁₆	—	142	1	no	—
170	C₂×C₁₆	C₄×D₇₁	1	2	yes	—
171	#108	—	2	1	no	—
172	#108	C₂²⋊D₇₁	1	2	yes	—
173	#107	—	71	2	no	—
174	#106	—	2	1	no	—
175	#106	—	2	1	no	—
176	#106	—	142	1	no	—
177	#103	—	284	2	no	—
178	#97	—	284	1	no	—
179	#97	—	142	1	no	—
180	#97	—	142	1	no	—
181	C₄×C₈	—	284	1	no	—
182	#95	—	284	2	no	—
183	C₂×Q₈	—	284	1	no	—
184	C₂×Q₈	—	284	1	no	—
185	C₂×Q₈	—	284	1	no	—
186	C₂×Q₈	—	284	1	no	—
187	C₂×Q₈	—	284	1	no	—
188	C₂×Q₈	—	142	2	no	—
189	C₂²×C₄	—	4	1	no	—
190	C₂²×C₄	—	284	1	no	—
191	C₂²×C₄	—	142	1	no	—
192	C₂²×C₄	D₄×D₇₁	1	1	yes	characteristic
193	Q₁₆	—	284	1	no	—
194	Q₁₆	—	284	1	no	—
195	Q₁₆	—	142	2	no	—
196	Q₁₆	—	142	2	no	—
197	M₁₆	—	2	2	no	—
198	M₁₆	—	4	1	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	71	2	no	—
201	C₂×C₈	—	142	1	no	—
202	C₂×C₈	—	142	1	no	—
203	C₂×C₈	C₂×C₄×D₇₁	1	1	yes	characteristic
204	C₂×C₈	C₂×C₂²⋊D₇₁	1	1	yes	characteristic
205	C₂×C₈	#11765864	1	1	yes	characteristic
206	C₂×C₈	#11765865	1	1	yes	characteristic
207	C₂×C₈	#11765866	1	1	yes	characteristic
208	C₂×C₈	#11765866	1	1	yes	characteristic
209	C₄⋊C₄	—	4	1	no	—
210	C₄⋊C₄	—	4	1	no	—
211	C₄⋊C₄	—	142	2	no	—
212	C₂²⋊C₄	—	284	1	no	—
213	C₄²	—	284	1	no	—
214	C₄²	—	142	1	no	—
215	C₄²	—	142	1	no	—
216	C₁₆	—	2	2	no	—
217	C₁₆	—	284	1	no	—
218	C₂³	#14556155	1	1	yes	characteristic
219	Q₈	—	284	2	no	—
220	Q₈	—	284	2	no	—
221	Q₈	—	568	1	no	—
222	Q₈	—	568	1	no	—
223	C₂×C₄	—	8	2	no	—
224	C₂×C₄	—	8	1	no	—
225	C₂×C₄	—	284	2	no	—
226	C₂×C₄	—	284	1	no	—
227	C₂×C₄	—	284	2	no	—
228	C₂×C₄	—	284	2	no	—
229	C₂×C₄	—	568	1	no	—
230	C₂×C₄	—	568	1	no	—
231	C₂×C₄	#14556090	1	1	yes	characteristic
232	C₂×C₄	#14556100	1	1	yes	characteristic
233	C₂×C₄	#14556113	1	1	yes	characteristic
234	C₂×C₄	SD₁₆×D₇₁	1	1	yes	characteristic
235	C₂×C₄	#14556149	1	1	yes	characteristic
236	C₂×C₄	#14556155	1	1	yes	characteristic
237	C₈	—	4	2	no	—
238	C₈	—	142	2	no	—
239	C₈	#14556086	1	1	yes	characteristic
240	C₈	C₄×C₂²⋊D₇₁	1	1	yes	characteristic
241	C₈	#14556117	1	1	yes	characteristic
242	C₈	#14556123	1	1	yes	characteristic
243	C₂²	—	2	2	no	—
244	C₂²	#17986110	1	1	yes	characteristic
245	C₂²	#17986197	1	1	yes	characteristic
246	C₂²	#17986325	1	1	yes	characteristic
247	C₄	—	16	1	no	—
248	C₄	—	568	1	no	—
249	C₄	—	568	2	no	—
250	C₄	—	568	2	no	—
251	C₄	#17985889	1	1	yes	characteristic
252	C₄	#17986086	1	1	yes	characteristic
253	C₄	#17986216	1	1	yes	characteristic
254	C₄	#17986319	1	1	yes	characteristic
255	C₂	—	2	1	no	—
256	C₂	—	2	1	no	—
257	C₂	#22588966	1	1	yes	characteristic
258	C₂	#22590230	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#17985814

C₂

yes

maximal, characteristic

#22590061

C₂

yes

maximal, characteristic

#22589209

C₂

yes

maximal, characteristic

#22588227

C₂

yes

maximal, characteristic

#22586447

C₂

yes

maximal, characteristic

#22586419

C₂

yes

maximal, characteristic

C₂×#365×C₇₁

C₂

yes

maximal, characteristic

C₂²×Dic₂₈₄

C₂²

yes

characteristic

C₂×#14556068

C₂²

yes

characteristic

#17986037

C₂²

yes

—

—

C₂²

yes

—

—

C₄

yes

complemented

#17985760

—

C₂×M₃₂×C₇₁

C₂²

yes

characteristic

C₂×#108×C₇₁

C₂²

yes

characteristic

C₂×#106×C₇₁

C₂²

yes

characteristic

#365×C₇₁

—

C₂²×Dic₁₄₂

—

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

—

C₂×Dic₂₈₄

C₂×C₄

yes

—

C₂²×C₇₁⋊C₈

D₄

yes

characteristic

#14556087

—

C₂×C₄×Dic₇₁

D₄

yes

characteristic

#14556068

—

#14556068

—

C₂×C₇₁⋊C₁₆

—

#14556062

C₂×C₄

yes

—

—

—

—

—

—

C₈×Dic₇₁

—

C₇₁⋊#95

—

C₂×M₁₆×C₇₁

—

C₂²×C₈×C₇₁

C₂³

yes

characteristic

C₂×C₄⋊C₄×C₇₁

—

M₃₂×C₇₁

—

M₃₂×C₇₁

—

C₂×C₁₆×C₇₁

C₂×C₄

yes

—

#108×C₇₁

—

#108×C₇₁

D₄

yes

—

#106×C₇₁

—

#106×C₇₁

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂×Dic₁₄₂

—

C₂²×Dic₇₁

—

C₂²×Dic₇₁

—

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

—

Dic₂₈₄

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

—

C₂×C₇₁⋊C₈

SD₁₆

yes

—

C₄⋊Dic₇₁

—

C₂²⋊Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₄×Dic₇₁

—

C₇₁⋊C₁₆

—

C₂²×C₄×C₇₁

—

C₂²×C₄×C₇₁

C₂×D₄

yes

characteristic

M₁₆×C₇₁

—

M₁₆×C₇₁

—

C₂×C₈×C₇₁

—

C₂×C₈×C₇₁

C₂²×C₄

yes

characteristic

C₂×C₈×C₇₁

C₂×D₄

yes

C₂×C₈×C₇₁

yes

C₂×C₈×C₇₁

yes

C₂×C₈×C₇₁

yes

C₂×C₈×C₇₁

yes

C₄⋊C₄×C₇₁

—

C₄⋊C₄×C₇₁

—

C₁₆×C₇₁

—

C₂³×C₇₁

AGL(1,ℤ/8)

yes

characteristic

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

—

C₂×C₄×C₇₁

C₄×D₄

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

C₂×SD₁₆

yes

C₂×C₄×C₇₁

yes

C₂×C₄×C₇₁

yes

C₂×Dic₇₁

—

C₂×Dic₇₁

—

C₂×Dic₇₁

—

100

C₂×Dic₇₁

—

101

C₂×Dic₇₁

—

102

C₂×Dic₇₁

—

103

Dic₁₄₂

—

104

Dic₁₄₂

—

105

Dic₁₄₂

—

106

Dic₁₄₂

—

107

C₈×C₇₁

—

108

C₈×C₇₁

#117

yes

characteristic

109

C₈×C₇₁

C₄×D₄

yes

characteristic

110

C₈×C₇₁

#124

yes

characteristic

111

C₈×C₇₁

#126

yes

characteristic

112

C₇₁⋊C₈

—

113

C₂²×C₇₁

—

114

C₂²×C₇₁

#442

yes

characteristic

115

C₂²×C₇₁

#460

yes

characteristic

116

C₂²×C₇₁

#482

yes

characteristic

117

C₄×C₇₁

—

118

C₄×C₇₁

#388

yes

characteristic

119

C₄×C₇₁

C₄×SD₁₆

yes

characteristic

120

C₄×C₇₁

#463

yes

characteristic

121

C₄×C₇₁

#481

yes

characteristic

122

Dic₇₁

—

123

Dic₇₁

—

124

Dic₇₁

—

125

#13141

—

maximal, Sylow

126

C₂×C₇₁

—

127

C₂×C₇₁

—

128

C₂×C₇₁

#1926

yes

characteristic

129

C₂×C₇₁

#2180

yes

—

130

C₂×#365

D₇₁

yes

characteristic

131

C₂×#362

—

132

#2141

—

133

#1971

—

134

#1773

—

135

#1386

—

136

#1375

—

137

C₇₁

#13141

yes

characteristic, complemented, Sylow

138

C₂²×Q₁₆

—

139

C₂×M₃₂

—

140

C₂×M₃₂

C₂×D₇₁

yes

characteristic

141

C₂×#108

C₂×D₇₁

yes

characteristic

142

#427

—

143

C₂×#106

C₂×D₇₁

yes

characteristic

144

C₂×#103

—

142

—

145

#405

—

142

—

146

C₂×#97

—

147

#391

—

142

—

148

#365

—

149

#362

—

142

—

150

#358

—

151

#339

—

142

—

152

C₂²×Q₈

—

142

—

153

C₂×Q₁₆

—

154

C₂×Q₁₆

—

142

—

155

C₂×Q₁₆

—

142

—

156

C₂×Q₁₆

—

142

—

157

C₂×Q₁₆

—

142

—

158

C₂×Q₁₆

—

142

—

159

C₂×M₁₆

—

160

C₂²×C₈

—

161

C₂²×C₈

C₂²×D₇₁

yes

characteristic

162

#117

—

142

—

163

C₂×C₄⋊C₄

—

164

C₂×C₄²

—

165

M₃₂

—

166

M₃₂

—

167

M₃₂

—

142

—

168

M₃₂

—

142

—

169

C₂×C₁₆

—

142

—

170

C₂×C₁₆

C₄×D₇₁

yes

—

171

#108

—

172

#108

C₂²⋊D₇₁

yes

—

173

#107

—

174

#106

—

175

#106

—

176

#106

—

142

—

177

#103

—

284

—

178

#97

—

284

—

179

#97

—

142

—

180

#97

—

142

—

181

C₄×C₈

—

284

—

182

#95

—

284

—

183

C₂×Q₈

—

284

—

184

C₂×Q₈

—

284

—

185

C₂×Q₈

—

284

—

186

C₂×Q₈

—

284

—

187

C₂×Q₈

—

284

—

188

C₂×Q₈

—

142

—

189

C₂²×C₄

—

190

C₂²×C₄

—

284

—

191

C₂²×C₄

—

142

—

192

C₂²×C₄

D₄×D₇₁

yes

characteristic

193

Q₁₆

—

284

—

194

Q₁₆

—

284

—

195

Q₁₆

—

142

—

196

Q₁₆

—

142

—

197

M₁₆

—

198

M₁₆

—

199

C₂×C₈

—

200

C₂×C₈

—

201

C₂×C₈

—

142

—

202

C₂×C₈

—

142

—

203

C₂×C₈

C₂×C₄×D₇₁

yes

characteristic

204

C₂×C₈

C₂×C₂²⋊D₇₁

yes

characteristic

205

C₂×C₈

#11765864

yes

characteristic

206

C₂×C₈

#11765865

yes

characteristic

207

C₂×C₈

#11765866

yes

characteristic

208

C₂×C₈

#11765866

yes

characteristic

209

C₄⋊C₄

—

210

C₄⋊C₄

—

211

C₄⋊C₄

—

142

—

212

C₂²⋊C₄

—

284

—

213

C₄²

—

284

—

214

C₄²

—

142

—

215

C₄²

—

142

—

216

C₁₆

—

217

C₁₆

—

284

—

218

C₂³

#14556155

yes

characteristic

219

Q₈

—

284

—

220

Q₈

—

284

—

221

Q₈

—

568

—

222

Q₈

—

568

—

223

C₂×C₄

—

224

C₂×C₄

—

225

C₂×C₄

—

284

—

226

C₂×C₄

—

284

—

227

C₂×C₄

—

284

—

228

C₂×C₄

—

284

—

229

C₂×C₄

—

568

—

230

C₂×C₄

—

568

—

231

C₂×C₄

#14556090

yes

characteristic

232

C₂×C₄

#14556100

yes

characteristic

233

C₂×C₄

#14556113

yes

characteristic

234

C₂×C₄

SD₁₆×D₇₁

yes

characteristic

235

C₂×C₄

#14556149

yes

characteristic

236

C₂×C₄

#14556155

yes

characteristic

237

C₈

—

238

C₈

—

142

—

239

C₈

#14556086

yes

characteristic

240

C₈

C₄×C₂²⋊D₇₁

yes

characteristic

241

C₈

#14556117

yes

characteristic

242

C₈

#14556123

yes

characteristic

243

C₂²

—

244

C₂²

#17986110

yes

characteristic

245

C₂²

#17986197

yes

characteristic

246

C₂²

#17986325

yes

characteristic

247

C₄

—

248

C₄

—

568

—

249

C₄

—

568

—

250

C₄

—

568

—

251

C₄

#17985889

yes

characteristic

252

C₄

#17986086

yes

characteristic

253

C₄

#17986216

yes

characteristic

254

C₄

#17986319

yes

characteristic

255

C₂

—

256

C₂

—

257

C₂

#22588966

yes

characteristic

258

C₂

#22590230

yes

—

	subgroup	G/H
center	C₂²	#17986110
commutator	C₂×C₈×C₇₁	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₇₁
Fitting	C₂×#365×C₇₁	C₂
socle	C₂²×C₇₁	#442
prime radical	C₂³×C₇₁	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₇₁	C₂³
hypercenter	C₂×#365	D₇₁
hypocenter	C₇₁	#13141
LCS	G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#365

subgroup

G/H

center

C₂²

#17986110

commutator

C₂×C₈×C₇₁

C₂²×C₄

Frattini

C₂²×C₈

C₂²×D₇₁

Fitting

C₂×#365×C₇₁

C₂

C₂²×C₇₁

C₂³×C₇₁

C₂²×C₈×C₇₁

C₂³

D₇₁

C₇₁

G ▷ C₂×C₈×C₇₁ ▷ C₄×C₇₁ ▷ C₂×C₇₁ ▷ C₇₁

UCS

C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#365

N	H ≅ G/N	type
16. #17985804	C₄	semidirect
137. C₇₁	#13141	semidirect

H ≅ G/N

type

16. #17985804

C₄

semidirect

137. C₇₁

#13141

semidirect

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

subgroup	index
C₂×#17985814	2
#22590061	2
#22589209	2
#22588227	2
#22586447	2
#22586419	2
C₂×#365×C₇₁	2
#13141	71

subgroup

index

C₂×#365×C₇₁

Subgroups — #31915325

Conjugacy classes of subgroups

Complements

Maximal subgroups