Group #27286627 — subgroups

Subgroups — #27286627

Order 13568.

574 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 574 conjugacy classes of subgroups — too many to draw. Only its 139 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×#16811028	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810833	C₂	1	2	yes	maximal
3	C₂×#16810368	C₂	1	2	yes	maximal
4	#20272550	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1984	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719925	—	2	2	no	—
7	C₂×#13133217	C₂²	1	2	yes	—
8	#16811028	—	2	4	no	—
9	#16810833	—	2	4	no	—
10	#16810833	—	2	4	no	—
11	C₂×#13133151	—	2	2	no	—
12	C₂×C₄⋊#88322	C₂²	1	2	yes	—
13	C₂×#13133140	—	2	2	no	—
14	#16810368	—	2	4	no	—
15	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#336	—	2	2	no	—
19	C₂²×Dic₁₀₆	—	2	2	no	—
20	C₂×#11719925	—	2	8	no	—
21	C₂×#11719925	—	4	4	no	—
22	C₂×#11719925	—	4	4	no	—
23	C₂²×#88322	—	2	2	no	—
24	#13133217	—	2	2	no	—
25	#13133217	—	2	2	no	—
26	#13133217	—	2	2	no	—
27	#13133217	D₄	1	4	yes	—
28	C₂×C₄⋊Dic₅₃	—	2	2	no	—
29	C₂×C₄⋊Dic₅₃	—	2	2	no	—
30	#13133151	—	2	8	no	—
31	C₄⋊#88322	—	2	2	no	—
32	C₄⋊#88322	D₄	1	4	yes	—
33	#13133140	—	2	8	no	—
34	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
35	C₂²×C₈×C₅₃	—	2	1	no	—
36	C₂²×C₈×C₅₃	D₄	1	2	yes	—
37	C₄×Q₈×C₅₃	—	2	2	no	—
38	C₄×Q₈×C₅₃	—	2	2	no	—
39	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
40	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
41	C₂×C₄²×C₅₃	—	2	1	no	—
42	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
43	C₄⋊C₈×C₅₃	—	2	2	no	—
44	C₄×C₈×C₅₃	—	2	2	no	—
45	C₂×Q₈×C₅₃	—	2	1	no	—
46	C₂×Q₈×C₅₃	—	2	2	no	—
47	C₂×Q₈×C₅₃	—	2	2	no	—
48	C₂×Q₈×C₅₃	D₈	1	2	yes	—
49	C₂²×C₄×C₅₃	—	2	1	no	—
50	C₂²×C₄×C₅₃	—	2	1	no	—
51	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
54	C₂²×Dic₅₃	—	4	2	no	—
55	C₂×Dic₁₀₆	—	2	4	no	—
56	C₂×Dic₁₀₆	—	4	2	no	—
57	C₂×Dic₁₀₆	—	4	2	no	—
58	C₂×Dic₁₀₆	—	4	2	no	—
59	C₂×Dic₁₀₆	—	4	2	no	—
60	C₂×Dic₁₀₆	—	4	2	no	—
61	C₂×C₈×C₅₃	—	2	2	no	—
62	C₂×C₈×C₅₃	—	2	2	no	—
63	C₂×C₈×C₅₃	—	4	1	no	—
64	C₂×C₈×C₅₃	—	4	2	no	—
65	C₂×C₈×C₅₃	Q₁₆	1	4	yes	complemented
66	C₄⋊C₄×C₅₃	—	2	2	no	—
67	C₄⋊C₄×C₅₃	—	4	2	no	—
68	C₄²×C₅₃	—	4	1	no	—
69	C₄²×C₅₃	—	4	1	no	—
70	C₄²×C₅₃	C₂×D₄	1	2	yes	—
71	C₄²×C₅₃	C₂×D₄	1	2	yes	—
72	#11719925	—	4	16	no	—
73	C₄⋊Dic₅₃	—	2	4	no	—
74	C₄⋊Dic₅₃	—	2	4	no	—
75	C₄⋊Dic₅₃	—	4	2	no	—
76	C₄⋊Dic₅₃	—	4	2	no	—
77	C₂×#88322	—	2	4	no	—
78	C₂×#88322	—	4	2	no	—
79	C₂×#88322	—	4	2	no	—
80	C₂³×C₅₃	#121	1	1	yes	characteristic
81	Q₈×C₅₃	—	2	4	no	—
82	Q₈×C₅₃	—	4	2	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	—	2	2	no	—
85	C₂×C₄×C₅₃	—	4	1	no	—
86	C₂×C₄×C₅₃	—	4	1	no	—
87	C₂×C₄×C₅₃	—	4	1	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#120	1	2	yes	—
91	C₂×C₄×C₅₃	#121	1	2	yes	—
92	C₂×C₄×C₅₃	#127	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
95	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
96	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
97	C₂×C₄×C₅₃	#137	1	2	yes	—
98	C₂×Dic₅₃	—	4	4	no	—
99	C₂×Dic₅₃	—	8	2	no	—
100	C₂×Dic₅₃	—	8	2	no	—
101	Dic₁₀₆	—	4	4	no	—
102	Dic₁₀₆	—	4	4	no	—
103	Dic₁₀₆	—	8	2	no	—
104	Dic₁₀₆	—	8	2	no	—
105	C₈×C₅₃	—	2	4	no	—
106	C₈×C₅₃	—	8	1	no	—
107	#88322	—	4	4	no	—
108	#10265	—	53	1	no	maximal, Sylow
109	C₂²×C₅₃	#392	1	1	yes	characteristic
110	C₂²×C₅₃	#462	1	2	yes	—
111	C₂²×C₅₃	#466	1	1	yes	characteristic
112	C₂²×C₅₃	#469	1	1	yes	characteristic
113	C₂²×C₅₃	#473	1	2	yes	—
114	C₄×C₅₃	—	2	2	no	—
115	C₄×C₅₃	—	4	2	no	—
116	C₄×C₅₃	—	8	1	no	—
117	C₄×C₅₃	#447	1	2	yes	—
118	C₄×C₅₃	#449	1	2	yes	—
119	C₄×C₅₃	#451	1	2	yes	—
120	C₄×C₅₃	#494	1	1	yes	characteristic
121	C₄×C₅₃	#497	1	1	yes	characteristic
122	Dic₅₃	—	8	4	no	—
123	C₂×#500	—	53	1	no	—
124	C₂×#462	—	53	2	no	—
125	#1984	D₅₃	1	1	yes	characteristic
126	C₂×#332	—	53	2	no	—
127	#1277	—	53	1	no	—
128	C₂×C₅₃	#1674	1	2	yes	—
129	C₂×C₅₃	#1694	1	2	yes	—
130	C₂×C₅₃	#1999	1	1	yes	characteristic
131	C₂×C₅₃	#2016	1	1	yes	characteristic
132	C₂×C₅₃	#2020	1	1	yes	characteristic
133	C₂²×Q₁₆	—	106	2	no	—
134	C₂×C₄⋊Q₈	—	53	2	no	—
135	C₂×C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
136	#500	—	106	4	no	—
137	#462	—	106	4	no	—
138	#462	—	106	4	no	—
139	C₂×#107	—	106	2	no	—
140	C₂×C₄⋊C₈	—	53	2	no	—
141	C₂×C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
142	C₂×#103	—	106	2	no	—
143	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
144	#336	—	2	2	no	—
145	#332	—	106	4	no	—
146	C₅₃	#10265	1	1	yes	characteristic, complemented, Sylow
147	C₂²×Q₈	—	106	2	no	—
148	C₂²×Q₈	D₂₁₂	1	1	yes	characteristic
149	C₂×Q₁₆	—	106	8	no	—
150	C₂×Q₁₆	—	212	4	no	—
151	C₂×Q₁₆	—	212	4	no	—
152	C₂²×C₈	—	2	1	no	—
153	C₂²×C₈	—	106	2	no	—
154	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
155	C₄⋊Q₈	—	53	4	no	—
156	C₄⋊Q₈	—	106	2	no	—
157	C₄⋊Q₈	—	106	2	no	—
158	C₄⋊Q₈	—	106	2	no	—
159	C₄×Q₈	—	2	2	no	—
160	C₄×Q₈	—	2	2	no	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	—	106	2	no	—
163	C₂×C₄⋊C₄	—	106	2	no	—
164	C₂×C₄⋊C₄	D₂₁₂	1	1	yes	characteristic
165	C₂×C₄²	—	2	1	no	—
166	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
167	#107	—	106	8	no	—
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	53	4	no	—
170	C₄⋊C₈	—	106	2	no	—
171	#103	—	106	8	no	—
172	C₄×C₈	—	2	2	no	—
173	C₂×Q₈	—	2	1	no	—
174	C₂×Q₈	—	2	2	no	—
175	C₂×Q₈	—	2	2	no	—
176	C₂×Q₈	—	106	4	no	—
177	C₂×Q₈	—	212	2	no	—
178	C₂×Q₈	—	212	2	no	—
179	C₂×Q₈	—	212	2	no	—
180	C₂×Q₈	—	212	2	no	—
181	C₂×Q₈	—	212	2	no	—
182	C₂×Q₈	D₄₂₄	1	2	yes	—
183	C₂²×C₄	—	2	1	no	—
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	212	2	no	—
186	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
187	C₂²×C₄	#11719945	1	1	yes	characteristic
188	C₂²×C₄	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
189	Q₁₆	—	212	16	no	—
190	C₂×C₈	—	2	2	no	—
191	C₂×C₈	—	2	2	no	—
192	C₂×C₈	—	4	1	no	—
193	C₂×C₈	—	4	2	no	—
194	C₂×C₈	—	106	4	no	—
195	C₂×C₈	—	212	2	no	—
196	C₂×C₈	—	212	2	no	—
197	C₂×C₈	#11719925	1	4	yes	complemented
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	106	4	no	—
201	C₄⋊C₄	—	106	4	no	—
202	C₄⋊C₄	—	212	2	no	—
203	C₄⋊C₄	—	212	2	no	—
204	C₄²	—	4	1	no	—
205	C₄²	—	4	1	no	—
206	C₄²	D₄×D₅₃	1	2	yes	—
207	C₄²	D₄×D₅₃	1	2	yes	—
208	C₂³	#13133191	1	1	yes	characteristic
209	Q₈	—	2	4	no	—
210	Q₈	—	4	2	no	—
211	Q₈	—	424	2	no	—
212	Q₈	—	424	2	no	—
213	Q₈	—	212	4	no	—
214	Q₈	—	212	4	no	—
215	C₂×C₄	—	2	2	no	—
216	C₂×C₄	—	2	2	no	—
217	C₂×C₄	—	4	1	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	2	no	—
221	C₂×C₄	—	424	2	no	—
222	C₂×C₄	—	424	2	no	—
223	C₂×C₄	—	212	4	no	—
224	C₂×C₄	#13133185	1	1	yes	characteristic
225	C₂×C₄	#13133187	1	2	yes	—
226	C₂×C₄	#13133190	1	2	yes	—
227	C₂×C₄	#13133213	1	1	yes	characteristic
228	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
229	C₂×C₄	C₂×#11719925	1	2	yes	—
230	C₂×C₄	#13133239	1	1	yes	characteristic
231	C₂×C₄	#13133245	1	2	yes	—
232	C₂×C₄	#13133247	1	2	yes	—
233	C₈	—	2	4	no	—
234	C₈	—	8	1	no	—
235	C₈	—	212	4	no	—
236	C₂²	#16810532	1	1	yes	characteristic
237	C₂²	#16810833	1	2	yes	—
238	C₂²	#16810859	1	1	yes	characteristic
239	C₂²	#16810876	1	1	yes	characteristic
240	C₂²	#16810898	1	2	yes	—
241	C₄	—	2	2	no	—
242	C₄	—	4	2	no	—
243	C₄	—	8	1	no	—
244	C₄	—	424	4	no	—
245	C₄	#16810759	1	2	yes	—
246	C₄	#16810770	1	2	yes	—
247	C₄	#16810778	1	2	yes	—
248	C₄	#16811002	1	1	yes	characteristic
249	C₄	#16811018	1	1	yes	characteristic
250	C₂	#20274109	1	2	yes	—
251	C₂	#20274233	1	2	yes	—
252	C₂	#20275713	1	1	yes	characteristic
253	C₂	#20275814	1	1	yes	characteristic
254	C₂	#20275836	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×#16811028

C₂

yes

maximal, characteristic

C₂×#16810833

C₂

yes

maximal

C₂×#16810368

C₂

yes

maximal

#20272550

C₂

yes

maximal, characteristic

C₅₃×#1984

C₂

yes

maximal, characteristic

C₂²×#11719925

—

C₂×#13133217

C₂²

yes

—

—

—

—

—

C₂×C₄⋊#88322

C₂²

yes

—

C₂×#13133140

—

#16810368

—

C₂×C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#336

—

C₂²×Dic₁₀₆

—

C₂×#11719925

—

C₂×#11719925

—

C₂×#11719925

—

C₂²×#88322

—

—

—

—

D₄

yes

—

C₂×C₄⋊Dic₅₃

—

C₂×C₄⋊Dic₅₃

—

#13133151

—

C₄⋊#88322

—

C₄⋊#88322

D₄

yes

—

#13133140

—

C₂²×Q₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

D₈

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

#41

yes

characteristic

C₂²×Dic₅₃

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

Q₁₆

yes

complemented

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

—

#11719925

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₂×#88322

—

C₂×#88322

—

C₂×#88322

—

C₂³×C₅₃

#121

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

#120

yes

characteristic

C₂×C₄×C₅₃

#120

yes

—

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

#127

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₂×C₄×C₅₃

#135

yes

characteristic

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

100

C₂×Dic₅₃

—

101

Dic₁₀₆

—

102

Dic₁₀₆

—

103

Dic₁₀₆

—

104

Dic₁₀₆

—

105

C₈×C₅₃

—

106

C₈×C₅₃

—

107

#88322

—

108

#10265

—

maximal, Sylow

109

C₂²×C₅₃

#392

yes

characteristic

110

C₂²×C₅₃

#462

yes

—

111

C₂²×C₅₃

#466

yes

characteristic

112

C₂²×C₅₃

#469

yes

characteristic

113

C₂²×C₅₃

#473

yes

—

114

C₄×C₅₃

—

115

C₄×C₅₃

—

116

C₄×C₅₃

—

117

C₄×C₅₃

#447

yes

—

118

C₄×C₅₃

#449

yes

—

119

C₄×C₅₃

#451

yes

—

120

C₄×C₅₃

#494

yes

characteristic

121

C₄×C₅₃

#497

yes

characteristic

122

Dic₅₃

—

123

C₂×#500

—

124

C₂×#462

—

125

#1984

D₅₃

yes

characteristic

126

C₂×#332

—

127

#1277

—

128

C₂×C₅₃

#1674

yes

—

129

C₂×C₅₃

#1694

yes

—

130

C₂×C₅₃

#1999

yes

characteristic

131

C₂×C₅₃

#2016

yes

characteristic

132

C₂×C₅₃

#2020

yes

characteristic

133

C₂²×Q₁₆

—

106

—

134

C₂×C₄⋊Q₈

—

135

C₂×C₄×Q₈

C₂×D₅₃

yes

characteristic

136

#500

—

106

—

137

#462

—

106

—

138

#462

—

106

—

139

C₂×#107

—

106

—

140

C₂×C₄⋊C₈

—

141

C₂×C₄⋊C₈

C₂×D₅₃

yes

characteristic

142

C₂×#103

—

106

—

143

C₂×C₄×C₈

C₂×D₅₃

yes

characteristic

144

#336

—

145

#332

—

106

—

146

C₅₃

#10265

yes

characteristic, complemented, Sylow

147

C₂²×Q₈

—

106

—

148

C₂²×Q₈

D₂₁₂

yes

characteristic

149

C₂×Q₁₆

—

106

—

150

C₂×Q₁₆

—

212

—

151

C₂×Q₁₆

—

212

—

152

C₂²×C₈

—

153

C₂²×C₈

—

106

—

154

C₂²×C₈

C₂²⋊D₅₃

yes

—

155

C₄⋊Q₈

—

156

C₄⋊Q₈

—

106

—

157

C₄⋊Q₈

—

106

—

158

C₄⋊Q₈

—

106

—

159

C₄×Q₈

—

160

C₄×Q₈

—

161

C₂×C₄⋊C₄

—

162

C₂×C₄⋊C₄

—

106

—

163

C₂×C₄⋊C₄

—

106

—

164

C₂×C₄⋊C₄

D₂₁₂

yes

characteristic

165

C₂×C₄²

—

166

C₂×C₄²

C₂²×D₅₃

yes

characteristic

167

#107

—

106

—

168

C₄⋊C₈

—

169

C₄⋊C₈

—

170

C₄⋊C₈

—

106

—

171

#103

—

106

—

172

C₄×C₈

—

173

C₂×Q₈

—

174

C₂×Q₈

—

175

C₂×Q₈

—

176

C₂×Q₈

—

106

—

177

C₂×Q₈

—

212

—

178

C₂×Q₈

—

212

—

179

C₂×Q₈

—

212

—

180

C₂×Q₈

—

212

—

181

C₂×Q₈

—

212

—

182

C₂×Q₈

D₄₂₄

yes

—

183

C₂²×C₄

—

184

C₂²×C₄

—

185

C₂²×C₄

—

212

—

186

C₂²×C₄

C₂×D₂₁₂

yes

characteristic

187

C₂²×C₄

#11719945

yes

characteristic

188

C₂²×C₄

C₂×C₂²⋊D₅₃

yes

characteristic

189

Q₁₆

—

212

—

190

C₂×C₈

—

191

C₂×C₈

—

192

C₂×C₈

—

193

C₂×C₈

—

194

C₂×C₈

—

106

—

195

C₂×C₈

—

212

—

196

C₂×C₈

—

212

—

197

C₂×C₈

#11719925

yes

complemented

198

C₄⋊C₄

—

199

C₄⋊C₄

—

200

C₄⋊C₄

—

106

—

201

C₄⋊C₄

—

106

—

202

C₄⋊C₄

—

212

—

203

C₄⋊C₄

—

212

—

204

C₄²

—

205

C₄²

—

206

C₄²

D₄×D₅₃

yes

—

207

C₄²

D₄×D₅₃

yes

—

208

C₂³

#13133191

yes

characteristic

209

Q₈

—

210

Q₈

—

211

Q₈

—

424

—

212

Q₈

—

424

—

213

Q₈

—

212

—

214

Q₈

—

212

—

215

C₂×C₄

—

216

C₂×C₄

—

217

C₂×C₄

—

218

C₂×C₄

—

219

C₂×C₄

—

220

C₂×C₄

—

221

C₂×C₄

—

424

—

222

C₂×C₄

—

424

—

223

C₂×C₄

—

212

—

224

C₂×C₄

#13133185

yes

characteristic

225

C₂×C₄

#13133187

yes

—

226

C₂×C₄

#13133190

yes

—

227

C₂×C₄

#13133213

yes

characteristic

228

C₂×C₄

C₂×D₄₂₄

yes

characteristic

229

C₂×C₄

C₂×#11719925

yes

—

230

C₂×C₄

#13133239

yes

characteristic

231

C₂×C₄

#13133245

yes

—

232

C₂×C₄

#13133247

yes

—

233

C₈

—

234

C₈

—

235

C₈

—

212

—

236

C₂²

#16810532

yes

characteristic

237

C₂²

#16810833

yes

—

238

C₂²

#16810859

yes

characteristic

239

C₂²

#16810876

yes

characteristic

240

C₂²

#16810898

yes

—

241

C₄

—

242

C₄

—

243

C₄

—

244

C₄

—

424

—

245

C₄

#16810759

yes

—

246

C₄

#16810770

yes

—

247

C₄

#16810778

yes

—

248

C₄

#16811002

yes

characteristic

249

C₄

#16811018

yes

characteristic

250

C₂

#20274109

yes

—

251

C₂

#20274233

yes

—

252

C₂

#20275713

yes

characteristic

253

C₂

#20275814

yes

characteristic

254

C₂

#20275836

yes

characteristic

	subgroup	G/H
center	C₂³	#13133191
commutator, cosocle	C₂×C₄²×C₅₃	C₂³
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₅₃×#1984	C₂
socle, prime radical	C₂³×C₅₃	#121
hypercenter	#1984	D₅₃
hypocenter	C₅₃	#10265
LCS	G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1984

subgroup

G/H

center

C₂³

#13133191

commutator, cosocle

C₂×C₄²×C₅₃

C₂³

Frattini

C₂×C₄²

C₂²×D₅₃

Fitting

C₅₃×#1984

C₂

C₂³×C₅₃

D₅₃

C₅₃

G ▷ C₂×C₄²×C₅₃ ▷ C₂²×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1984

N	H ≅ G/N	type
65. C₂×C₈×C₅₃	Q₁₆	semidirect
146. C₅₃	#10265	semidirect
197. C₂×C₈	#11719925	semidirect

H ≅ G/N

type

65. C₂×C₈×C₅₃

Q₁₆

semidirect

146. C₅₃

#10265

semidirect

197. C₂×C₈

#11719925

semidirect

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

subgroup	index
C₂×#16811028	2
C₂×#16810833	2
C₂×#16810368	2
#20272550	2
C₅₃×#1984	2
#10265	53

subgroup

index

C₅₃×#1984

Subgroups — #27286627

Conjugacy classes of subgroups

Complements

Maximal subgroups