Group #27277966 — subgroups

Subgroups — #27277966

Order 13568.

454 conjugacy classes of subgroups; only the 131 normal subgroups are shown.

normal subgroups only

This group has 454 conjugacy classes of subgroups — too many to draw. Only its 131 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	#20283756	C₂	1	1	yes	maximal, characteristic
2	#20275510	C₂	1	1	yes	maximal, characteristic
3	#20273467	C₂	1	2	yes	maximal, complemented
4	#20272553	C₂	1	2	yes	maximal, complemented
5	C₄×#100×C₅₃	C₂	1	1	yes	maximal, characteristic
6	#16810701	—	2	2	no	—
7	#16810701	—	2	2	no	—
8	#16810676	C₂²	1	2	yes	—
9	C₂×C₄⋊#88322	C₂²	1	1	yes	characteristic
10	#16810587	—	2	2	no	—
11	C₂×C₂²⋊#88322	—	2	2	no	—
12	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
13	#16810376	—	2	2	no	—
14	#16810374	—	2	2	no	—
15	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×#100×C₅₃	C₄	1	2	yes	—
17	C₄×M₁₆×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#343	C₄	1	1	yes	characteristic
19	C₅₃×#343	C₄	1	1	yes	characteristic
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂²×#88322	—	2	1	no	—
23	C₂²×#88322	—	2	1	no	—
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	D₄	1	2	yes	—
26	C₄⋊#88322	D₄	1	2	yes	—
27	C₂²⋊#88322	—	2	4	no	—
28	C₂²⋊#88322	—	2	2	no	—
29	C₂²⋊#88322	—	4	4	no	—
30	C₄×#88322	—	2	2	no	—
31	C₄×#88322	—	2	2	no	—
32	C₂²×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂³×C₄×C₅₃	D₄	1	2	yes	—
34	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
36	C₄×D₄×C₅₃	—	2	2	no	—
37	C₄×D₄×C₅₃	—	2	2	no	—
38	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
39	C₂×C₂²⋊C₄×C₅₃	Q₈	1	2	yes	—
40	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
41	#100×C₅₃	—	2	4	no	—
42	#97×C₅₃	—	2	2	no	—
43	C₄×C₈×C₅₃	—	2	2	no	—
44	C₂⁴×C₅₃	C₄⋊C₄	1	2	yes	—
45	C₂×D₄×C₅₃	—	2	2	no	—
46	C₂×D₄×C₅₃	—	2	2	no	—
47	C₂×D₄×C₅₃	—	2	1	no	—
48	C₂×D₄×C₅₃	—	2	1	no	—
49	C₂²×C₄×C₅₃	—	2	2	no	—
50	C₂²×C₄×C₅₃	—	2	2	no	—
51	C₂²×C₄×C₅₃	—	4	2	no	—
52	C₂²×C₄×C₅₃	—	4	2	no	—
53	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
54	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
57	M₁₆×C₅₃	—	2	2	no	—
58	M₁₆×C₅₃	—	2	2	no	—
59	M₁₆×C₅₃	—	2	2	no	—
60	M₁₆×C₅₃	M₁₆	1	4	yes	complemented
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₄⋊C₄×C₅₃	—	2	2	no	—
64	C₂²⋊C₄×C₅₃	—	2	4	no	—
65	C₄²×C₅₃	C₂×D₄	1	2	yes	—
66	C₄²×C₅₃	#41	1	2	yes	—
67	#11719917	—	8	2	no	—
68	#11719917	—	8	2	no	—
69	C₂×#88322	—	2	2	no	—
70	C₂×#88322	—	2	2	no	—
71	C₂×#88322	—	2	2	no	—
72	C₂×#88322	—	4	1	no	—
73	C₂×#88322	—	4	1	no	—
74	C₂×#88322	—	4	2	no	—
75	C₂×#88322	—	4	1	no	—
76	C₂³×C₅₃	—	2	2	no	—
77	C₂³×C₅₃	—	2	2	no	—
78	C₂³×C₅₃	—	2	2	no	—
79	C₂³×C₅₃	—	4	2	no	—
80	C₂³×C₅₃	—	4	2	no	—
81	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
82	D₄×C₅₃	—	4	2	no	—
83	D₄×C₅₃	—	4	2	no	—
84	C₂×C₄×C₅₃	—	2	1	no	—
85	C₂×C₄×C₅₃	—	2	1	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	4	1	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	—	4	2	no	—
90	C₂×C₄×C₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	—	8	2	no	—
92	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
93	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
95	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂×M₁₆	1	2	yes	—
97	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
99	C₈×C₅₃	—	2	4	no	—
100	C₈×C₅₃	—	4	2	no	—
101	#88322	—	4	2	no	—
102	#88322	—	4	2	no	—
103	#88322	—	4	2	no	—
104	#88322	—	8	1	no	—
105	#8837	—	53	1	no	maximal, Sylow
106	C₂²×C₅₃	—	2	1	no	—
107	C₂²×C₅₃	—	2	1	no	—
108	C₂²×C₅₃	—	4	4	no	—
109	C₂²×C₅₃	—	4	2	no	—
110	C₂²×C₅₃	—	4	2	no	—
111	C₂²×C₅₃	—	4	2	no	—
112	C₂²×C₅₃	—	8	2	no	—
113	C₂²×C₅₃	#385	1	1	yes	characteristic
114	C₂²×C₅₃	#423	1	1	yes	characteristic
115	C₂²×C₅₃	#424	1	1	yes	characteristic
116	C₄×C₅₃	—	2	1	no	—
117	C₄×C₅₃	—	2	2	no	—
118	C₄×C₅₃	—	4	2	no	—
119	C₄×C₅₃	#435	1	2	yes	—
120	C₄×C₅₃	#436	1	2	yes	—
121	C₄×C₅₃	#453	1	1	yes	characteristic
122	C₄×C₅₃	#455	1	1	yes	characteristic
123	#2975	—	53	1	no	—
124	#1954	—	53	1	no	—
125	C₄×#100	D₅₃	1	1	yes	characteristic
126	#1565	—	53	2	no	—
127	#1278	—	53	2	no	—
128	C₂×C₅₃	—	2	1	no	—
129	C₂×C₅₃	—	2	1	no	—
130	C₂×C₅₃	—	8	2	no	—
131	C₂×C₅₃	#1851	1	1	yes	characteristic
132	C₂×C₅₃	#1856	1	1	yes	characteristic
133	C₂×C₅₃	#1858	1	1	yes	characteristic
134	C₂×C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
135	#436	—	106	2	no	—
136	#436	—	106	2	no	—
137	#431	—	53	2	no	—
138	C₂×C₄⋊C₈	—	53	1	no	—
139	C₂×#100	Dic₅₃	1	2	yes	—
140	#407	—	106	2	no	—
141	C₂×C₂²⋊C₈	—	106	2	no	—
142	C₄×M₁₆	C₂×D₅₃	1	2	yes	—
143	C₂×C₄×C₈	—	53	1	no	—
144	#343	Dic₅₃	1	1	yes	characteristic
145	#343	Dic₅₃	1	1	yes	characteristic
146	#335	—	106	2	no	—
147	#334	—	106	2	no	—
148	C₅₃	#8837	1	1	yes	characteristic, complemented, Sylow
149	C₂²×D₄	C₂×Dic₅₃	1	1	yes	characteristic
150	C₂³×C₄	D₂₁₂	1	2	yes	—
151	C₂×M₁₆	—	106	2	no	—
152	C₂×M₁₆	—	106	2	no	—
153	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
154	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
155	C₂²×C₈	—	106	1	no	—
156	C₂²×C₈	—	106	1	no	—
157	C₄×D₄	—	2	2	no	—
158	C₄×D₄	—	2	2	no	—
159	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
160	C₂×C₂²⋊C₄	Dic₁₀₆	1	2	yes	—
161	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
162	C₄⋊C₈	—	53	2	no	—
163	C₄⋊C₈	—	53	2	no	—
164	C₄⋊C₈	—	106	2	no	—
165	#100	—	2	4	no	—
166	C₂²⋊C₈	—	106	4	no	—
167	C₂²⋊C₈	—	106	2	no	—
168	C₂²⋊C₈	—	212	4	no	—
169	#97	—	2	2	no	—
170	C₄×C₈	—	2	2	no	—
171	C₄×C₈	—	106	2	no	—
172	C₄×C₈	—	106	2	no	—
173	C₂⁴	C₄⋊Dic₅₃	1	2	yes	—
174	C₂×D₄	—	2	2	no	—
175	C₂×D₄	—	2	2	no	—
176	C₂×D₄	—	2	1	no	—
177	C₂×D₄	—	2	1	no	—
178	C₂²×C₄	—	2	2	no	—
179	C₂²×C₄	—	2	2	no	—
180	C₂²×C₄	—	4	2	no	—
181	C₂²×C₄	—	4	2	no	—
182	C₂²×C₄	C₄⋊Dic₅₃	1	2	yes	—
183	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
184	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
185	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
186	M₁₆	—	2	2	no	—
187	M₁₆	—	2	2	no	—
188	M₁₆	—	2	2	no	—
189	M₁₆	—	424	2	no	—
190	M₁₆	—	424	2	no	—
191	M₁₆	#11719917	1	4	yes	complemented
192	C₂×C₈	—	2	2	no	—
193	C₂×C₈	—	2	2	no	—
194	C₂×C₈	—	106	2	no	—
195	C₂×C₈	—	106	2	no	—
196	C₂×C₈	—	106	2	no	—
197	C₂×C₈	—	212	2	no	—
198	C₂×C₈	—	212	1	no	—
199	C₂×C₈	—	212	1	no	—
200	C₂×C₈	—	212	1	no	—
201	C₄⋊C₄	—	2	2	no	—
202	C₂²⋊C₄	—	2	4	no	—
203	C₄²	D₄×D₅₃	1	2	yes	—
204	C₄²	#11719947	1	2	yes	—
205	C₂³	—	2	2	no	—
206	C₂³	—	2	2	no	—
207	C₂³	—	2	2	no	—
208	C₂³	—	4	2	no	—
209	C₂³	—	4	2	no	—
210	C₂³	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
211	D₄	—	4	2	no	—
212	D₄	—	4	2	no	—
213	C₂×C₄	—	2	1	no	—
214	C₂×C₄	—	2	1	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	4	1	no	—
217	C₂×C₄	—	4	2	no	—
218	C₂×C₄	—	4	2	no	—
219	C₂×C₄	—	4	2	no	—
220	C₂×C₄	—	8	2	no	—
221	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
222	C₂×C₄	#13133187	1	1	yes	characteristic
223	C₂×C₄	#13133194	1	2	yes	—
224	C₂×C₄	#13133201	1	1	yes	characteristic
225	C₂×C₄	C₂×#11719917	1	2	yes	—
226	C₂×C₄	#13133224	1	1	yes	characteristic
227	C₂×C₄	#13133224	1	1	yes	characteristic
228	C₈	—	2	4	no	—
229	C₈	—	4	2	no	—
230	C₈	—	424	1	no	—
231	C₈	—	212	2	no	—
232	C₈	—	212	2	no	—
233	C₈	—	212	2	no	—
234	C₂²	—	2	1	no	—
235	C₂²	—	2	1	no	—
236	C₂²	—	4	4	no	—
237	C₂²	—	4	2	no	—
238	C₂²	—	4	2	no	—
239	C₂²	—	4	2	no	—
240	C₂²	—	8	2	no	—
241	C₂²	#16810498	1	1	yes	characteristic
242	C₂²	#16810647	1	1	yes	characteristic
243	C₂²	#16810650	1	1	yes	characteristic
244	C₄	—	2	1	no	—
245	C₄	—	2	2	no	—
246	C₄	—	4	2	no	—
247	C₄	#16810695	1	2	yes	—
248	C₄	#16810701	1	2	yes	—
249	C₄	#16810793	1	1	yes	characteristic
250	C₄	#16810803	1	1	yes	characteristic
251	C₂	—	2	1	no	—
252	C₂	—	2	1	no	—
253	C₂	—	8	2	no	—
254	C₂	#20274900	1	1	yes	characteristic
255	C₂	#20274927	1	1	yes	characteristic
256	C₂	#20274937	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20283756

C₂

yes

maximal, characteristic

#20275510

C₂

yes

maximal, characteristic

#20273467

C₂

yes

maximal, complemented

#20272553

C₂

yes

maximal, complemented

C₄×#100×C₅₃

C₂

yes

maximal, characteristic

#16810701

—

#16810701

—

#16810676

C₂²

yes

—

C₂×C₄⋊#88322

C₂²

yes

characteristic

#16810587

—

C₂×C₂²⋊#88322

—

C₂×C₄×#88322

C₂²

yes

characteristic

#16810376

—

#16810374

—

C₂×C₄×D₄×C₅₃

C₂²

yes

characteristic

C₂×#100×C₅₃

C₄

yes

—

C₄×M₁₆×C₅₃

C₂²

yes

—

C₅₃×#343

C₄

yes

characteristic

C₅₃×#343

C₄

yes

characteristic

C₂×#11719917

—

C₂×#11719917

—

C₂²×#88322

—

C₂²×#88322

—

C₄⋊#88322

—

C₄⋊#88322

D₄

yes

—

C₄⋊#88322

D₄

yes

—

—

—

—

—

—

C₂²×D₄×C₅₃

C₂×C₄

yes

characteristic

C₂³×C₄×C₅₃

D₄

yes

—

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

C₂×M₁₆×C₅₃

C₂×C₄

yes

—

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂×C₄

yes

characteristic

C₂×C₂²⋊C₄×C₅₃

Q₈

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#100×C₅₃

—

#97×C₅₃

—

C₄×C₈×C₅₃

—

C₂⁴×C₅₃

C₄⋊C₄

yes

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₄⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×Q₈

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

M₁₆

yes

complemented

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

—

—

—

—

—

—

—

—

—

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×C₄⋊C₄

yes

characteristic

D₄×C₅₃

—

D₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

#120

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

#123

yes

characteristic

C₂×C₄×C₅₃

C₂×M₁₆

yes

—

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

C₈×C₅₃

—

100

C₈×C₅₃

—

101

#88322

—

102

#88322

—

103

#88322

—

104

#88322

—

105

#8837

—

maximal, Sylow

106

C₂²×C₅₃

—

107

C₂²×C₅₃

—

108

C₂²×C₅₃

—

109

C₂²×C₅₃

—

110

C₂²×C₅₃

—

111

C₂²×C₅₃

—

112

C₂²×C₅₃

—

113

C₂²×C₅₃

#385

yes

characteristic

114

C₂²×C₅₃

#423

yes

characteristic

115

C₂²×C₅₃

#424

yes

characteristic

116

C₄×C₅₃

—

117

C₄×C₅₃

—

118

C₄×C₅₃

—

119

C₄×C₅₃

#435

yes

—

120

C₄×C₅₃

#436

yes

—

121

C₄×C₅₃

#453

yes

characteristic

122

C₄×C₅₃

#455

yes

characteristic

123

#2975

—

124

#1954

—

125

C₄×#100

D₅₃

yes

characteristic

126

#1565

—

127

#1278

—

128

C₂×C₅₃

—

129

C₂×C₅₃

—

130

C₂×C₅₃

—

131

C₂×C₅₃

#1851

yes

characteristic

132

C₂×C₅₃

#1856

yes

characteristic

133

C₂×C₅₃

#1858

yes

characteristic

134

C₂×C₄×D₄

C₂×D₅₃

yes

characteristic

135

#436

—

106

—

136

#436

—

106

—

137

#431

—

138

C₂×C₄⋊C₈

—

139

C₂×#100

Dic₅₃

yes

—

140

#407

—

106

—

141

C₂×C₂²⋊C₈

—

106

—

142

C₄×M₁₆

C₂×D₅₃

yes

—

143

C₂×C₄×C₈

—

144

#343

Dic₅₃

yes

characteristic

145

#343

Dic₅₃

yes

characteristic

146

#335

—

106

—

147

#334

—

106

—

148

C₅₃

#8837

yes

characteristic, complemented, Sylow

149

C₂²×D₄

C₂×Dic₅₃

yes

characteristic

150

C₂³×C₄

D₂₁₂

yes

—

151

C₂×M₁₆

—

106

—

152

C₂×M₁₆

—

106

—

153

C₂×M₁₆

C₂×Dic₅₃

yes

—

154

C₂×M₁₆

C₂×Dic₅₃

yes

—

155

C₂²×C₈

—

106

—

156

C₂²×C₈

—

106

—

157

C₄×D₄

—

158

C₄×D₄

—

159

C₂×C₄⋊C₄

C₂×Dic₅₃

yes

characteristic

160

C₂×C₂²⋊C₄

Dic₁₀₆

yes

—

161

C₂×C₄²

C₂²×D₅₃

yes

characteristic

162

C₄⋊C₈

—

163

C₄⋊C₈

—

164

C₄⋊C₈

—

106

—

165

#100

—

166

C₂²⋊C₈

—

106

—

167

C₂²⋊C₈

—

106

—

168

C₂²⋊C₈

—

212

—

169

#97

—

170

C₄×C₈

—

171

C₄×C₈

—

106

—

172

C₄×C₈

—

106

—

173

C₂⁴

C₄⋊Dic₅₃

yes

—

174

C₂×D₄

—

175

C₂×D₄

—

176

C₂×D₄

—

177

C₂×D₄

—

178

C₂²×C₄

—

179

C₂²×C₄

—

180

C₂²×C₄

—

181

C₂²×C₄

—

182

C₂²×C₄

C₄⋊Dic₅₃

yes

—

183

C₂²×C₄

C₂×Dic₁₀₆

yes

characteristic

184

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

185

C₂²×C₄

C₂²×Dic₅₃

yes

characteristic

186

M₁₆

—

187

M₁₆

—

188

M₁₆

—

189

M₁₆

—

424

—

190

M₁₆

—

424

—

191

M₁₆

#11719917

yes

complemented

192

C₂×C₈

—

193

C₂×C₈

—

194

C₂×C₈

—

106

—

195

C₂×C₈

—

106

—

196

C₂×C₈

—

106

—

197

C₂×C₈

—

212

—

198

C₂×C₈

—

212

—

199

C₂×C₈

—

212

—

200

C₂×C₈

—

212

—

201

C₄⋊C₄

—

202

C₂²⋊C₄

—

203

C₄²

D₄×D₅₃

yes

—

204

C₄²

#11719947

yes

—

205

C₂³

—

206

C₂³

—

207

C₂³

—

208

C₂³

—

209

C₂³

—

210

C₂³

C₂×C₄⋊Dic₅₃

yes

characteristic

211

D₄

—

212

D₄

—

213

C₂×C₄

—

214

C₂×C₄

—

215

C₂×C₄

—

216

C₂×C₄

—

217

C₂×C₄

—

218

C₂×C₄

—

219

C₂×C₄

—

220

C₂×C₄

—

221

C₂×C₄

D₄×Dic₅₃

yes

—

222

C₂×C₄

#13133187

yes

characteristic

223

C₂×C₄

#13133194

yes

—

224

C₂×C₄

#13133201

yes

characteristic

225

C₂×C₄

C₂×#11719917

yes

—

226

C₂×C₄

#13133224

yes

characteristic

227

C₂×C₄

#13133224

yes

characteristic

228

C₈

—

229

C₈

—

230

C₈

—

424

—

231

C₈

—

212

—

232

C₈

—

212

—

233

C₈

—

212

—

234

C₂²

—

235

C₂²

—

236

C₂²

—

237

C₂²

—

238

C₂²

—

239

C₂²

—

240

C₂²

—

241

C₂²

#16810498

yes

characteristic

242

C₂²

#16810647

yes

characteristic

243

C₂²

#16810650

yes

characteristic

244

C₄

—

245

C₄

—

246

C₄

—

247

C₄

#16810695

yes

—

248

C₄

#16810701

yes

—

249

C₄

#16810793

yes

characteristic

250

C₄

#16810803

yes

characteristic

251

C₂

—

252

C₂

—

253

C₂

—

254

C₂

#20274900

yes

characteristic

255

C₂

#20274927

yes

characteristic

256

C₂

#20274937

yes

characteristic

	subgroup	G/H
center	C₂×C₄	#13133187
commutator	C₂²×C₄×C₅₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#100×C₅₃	C₂
socle	C₂²×C₅₃	#385
prime radical	C₂²×D₄×C₅₃	C₂×C₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#100	D₅₃
hypocenter	C₅₃	#8837
LCS	G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#100

subgroup

G/H

center

C₂×C₄

#13133187

commutator

C₂²×C₄×C₅₃

C₂²×C₄

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₄×#100×C₅₃

C₂

socle

C₂²×C₅₃

#385

prime radical

C₂²×D₄×C₅₃

C₂×C₄

cosocle

C₂×C₄²×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂²×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#100

N	H ≅ G/N	type
3. #20273467	C₂	semidirect
4. #20272553	C₂	semidirect
60. M₁₆×C₅₃	M₁₆	semidirect
148. C₅₃	#8837	semidirect
191. M₁₆	#11719917	semidirect

H ≅ G/N

type

3. #20273467

C₂

semidirect

4. #20272553

C₂

semidirect

60. M₁₆×C₅₃

M₁₆

semidirect

148. C₅₃

#8837

semidirect

191. M₁₆

#11719917

semidirect

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

subgroup	index
#20283756	2
#20275510	2
#20273467	2
#20272553	2
C₄×#100×C₅₃	2
#8837	53

subgroup

index

C₄×#100×C₅₃

Subgroups — #27277966

Conjugacy classes of subgroups

Complements

Maximal subgroups