Group #20284496 — subgroups

Subgroups — #20284496

Order 6784.

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

normal subgroups only

This group has 388 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	#16811178	C₂	1	1	yes	maximal, characteristic, complemented
2	C₄×#11719947	C₂	1	1	yes	maximal, characteristic
3	#16810929	C₂	1	2	yes	maximal, complemented
4	#16810921	C₂	1	2	yes	maximal, complemented
5	#16810919	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810913	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810905	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810660	C₂	1	2	yes	maximal, complemented
9	C₄⋊C₈×D₅₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#16810616	C₂	1	2	yes	maximal
11	C₅₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×#11719947	C₂²	1	1	yes	characteristic
13	C₂×C₈×D₅₃	C₂²	1	2	yes	—
14	#13133217	C₂²	1	1	yes	characteristic
15	#13133208	C₂²	1	2	yes	—
16	#13133197	—	2	1	no	—
17	#13133193	—	2	1	no	—
18	Q₈×Dic₅₃	—	2	1	no	—
19	C₄×Dic₁₀₆	C₂²	1	1	yes	characteristic
20	#13133182	—	2	1	no	—
21	#13133182	C₂²	1	1	yes	characteristic
22	D₄×Dic₅₃	C₂²	1	1	yes	characteristic
23	C₄×C₂²⋊D₅₃	—	2	1	no	—
24	#13133177	—	2	1	no	—
25	#13133175	C₂²	1	1	yes	characteristic
26	#13133175	C₂²	1	2	yes	—
27	#13133173	—	2	1	no	—
28	C₂×C₄×Dic₅₃	—	2	1	no	—
29	C₄²×D₅₃	C₂²	1	1	yes	characteristic
30	#13133151	C₂²	1	2	yes	—
31	#13133148	C₂²	1	2	yes	—
32	#13133148	C₂²	1	2	yes	—
33	C₄⋊#88322	C₂²	1	1	yes	characteristic
34	#13133146	C₂²	1	2	yes	—
35	#13133142	C₂²	1	2	yes	—
36	#13133140	C₂²	1	2	yes	—
37	#13133138	C₂²	1	2	yes	—
38	C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
39	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
40	#107×C₅₃	C₂²	1	2	yes	—
41	C₄⋊C₈×C₅₃	C₂²	1	1	yes	characteristic
42	#102×C₅₃	C₂²	1	2	yes	—
43	C₂×Q₈×C₅₃	—	2	1	no	—
44	C₂×D₄×C₅₃	C₂³	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	—	2	1	no	—
46	C₂×C₂²⋊D₅₃	—	2	1	no	—
47	C₂²×Dic₅₃	—	2	1	no	—
48	C₂²×Dic₅₃	—	2	1	no	—
49	#11719947	—	2	2	no	—
50	#11719947	Q₈	1	4	yes	complemented
51	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
52	C₂×C₄×D₅₃	D₄	1	1	yes	characteristic
53	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
54	C₂×Dic₁₀₆	—	2	1	no	—
55	C₂×Dic₁₀₆	C₂³	1	1	yes	characteristic
56	C₂×C₈×C₅₃	C₂³	1	2	yes	—
57	C₄⋊C₄×C₅₃	—	2	1	no	—
58	C₄⋊C₄×C₅₃	C₂³	1	2	yes	—
59	C₄⋊C₄×C₅₃	C₂³	1	1	yes	characteristic
60	C₂²⋊C₄×C₅₃	—	2	1	no	—
61	C₄²×C₅₃	C₂³	1	1	yes	characteristic
62	C₂²⋊Dic₅₃	—	2	1	no	—
63	#11719921	—	2	1	no	—
64	#11719921	—	2	2	no	—
65	C₄⋊Dic₅₃	—	2	1	no	—
66	C₄⋊Dic₅₃	—	2	2	no	—
67	C₄⋊Dic₅₃	C₂³	1	2	yes	—
68	C₄⋊Dic₅₃	C₂³	1	1	yes	characteristic
69	#11719919	—	2	1	no	—
70	#11719919	—	2	1	no	—
71	#11719919	—	2	2	no	—
72	C₄×Dic₅₃	—	2	1	no	—
73	C₄×Dic₅₃	—	2	1	no	—
74	C₄×Dic₅₃	—	2	1	no	—
75	C₄×Dic₅₃	—	2	2	no	—
76	C₄×Dic₅₃	D₄	1	1	yes	characteristic
77	C₄×Dic₅₃	D₄	1	1	yes	characteristic
78	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
79	C₂×#88322	C₂³	1	2	yes	—
80	C₈×D₅₃	—	2	4	no	—
81	C₂³×C₅₃	—	2	1	no	—
82	C₂²×D₅₃	C₂×D₄	1	1	yes	characteristic
83	Q₈×C₅₃	—	4	2	no	—
84	D₄×C₅₃	—	2	1	no	—
85	D₄×C₅₃	C₂×Q₈	1	2	yes	—
86	C₂×C₄×C₅₃	—	2	1	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
92	C₂²⋊D₅₃	—	2	4	no	—
93	C₂×Dic₅₃	—	2	1	no	—
94	C₂×Dic₅₃	—	2	4	no	—
95	C₂×Dic₅₃	—	2	2	no	—
96	C₂×Dic₅₃	—	2	1	no	—
97	C₂×Dic₅₃	—	4	2	no	—
98	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
99	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
100	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
101	C₄×D₅₃	—	2	2	no	—
102	C₄×D₅₃	—	4	1	no	—
103	C₄×D₅₃	C₂×Q₈	1	2	yes	—
104	C₄×D₅₃	#41	1	2	yes	—
105	Dic₁₀₆	—	2	1	no	—
106	Dic₁₀₆	—	4	2	no	—
107	Dic₁₀₆	C₂×Q₈	1	2	yes	—
108	C₈×C₅₃	—	2	2	no	—
109	#88322	—	2	2	no	—
110	C₂²×C₅₃	—	2	2	no	—
111	C₂²×C₅₃	—	4	1	no	—
112	C₂²×C₅₃	—	4	1	no	—
113	C₂²×C₅₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×D₅₃	—	2	1	no	—
115	C₂×D₅₃	—	2	1	no	—
116	C₂×D₅₃	C₂²⋊Q₈	1	2	yes	—
117	C₄×C₅₃	—	2	1	no	—
118	C₄×C₅₃	—	4	1	no	—
119	C₄×C₅₃	—	4	2	no	—
120	C₄×C₅₃	#135	1	2	yes	—
121	C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
122	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
123	Dic₅₃	—	2	2	no	—
124	Dic₅₃	—	2	1	no	—
125	Dic₅₃	—	2	1	no	—
126	Dic₅₃	—	4	2	no	—
127	Dic₅₃	—	4	1	no	—
128	Dic₅₃	C₂²⋊Q₈	1	2	yes	—
129	#3076	—	53	1	no	maximal, Sylow
130	C₂×C₅₃	—	4	2	no	—
131	C₂×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
132	C₂×C₅₃	C₂×#135	1	1	yes	characteristic
133	C₂×C₅₃	#575	1	1	yes	characteristic
134	D₅₃	—	2	2	no	—
135	#533	—	53	1	no	—
136	C₄×#41	—	53	1	no	—
137	#479	—	53	2	no	—
138	#478	—	53	2	no	—
139	#477	—	53	1	no	—
140	#476	—	53	1	no	—
141	#475	—	53	1	no	—
142	#474	D₅₃	1	1	yes	characteristic, complemented
143	#427	—	53	2	no	—
144	C₂×C₄⋊C₈	—	53	1	no	—
145	#416	—	53	2	no	—
146	C₅₃	#3076	1	1	yes	characteristic, complemented, Sylow
147	C₂×#41	—	53	1	no	—
148	C₂²×C₈	—	53	2	no	—
149	C₄⋊Q₈	—	53	1	no	—
150	C₄⋊Q₈	C₂×D₅₃	1	1	yes	characteristic
151	#125	—	53	2	no	—
152	C₂²⋊Q₈	—	106	1	no	—
153	C₂²⋊Q₈	—	106	1	no	—
154	C₄×Q₈	—	53	1	no	—
155	C₄×Q₈	—	53	1	no	—
156	C₄×Q₈	—	106	1	no	—
157	C₄×Q₈	—	106	1	no	—
158	C₄×D₄	—	53	1	no	—
159	C₄×D₄	—	106	1	no	—
160	C₄×D₄	—	106	1	no	—
161	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
162	#117	—	53	1	no	—
163	#117	—	53	2	no	—
164	#117	—	106	1	no	—
165	C₂×C₄²	—	53	1	no	—
166	C₂×C₄²	—	106	1	no	—
167	#107	—	53	2	no	—
168	#107	C₂×D₅₃	1	2	yes	—
169	#106	—	53	2	no	—
170	#106	—	53	2	no	—
171	C₄⋊C₈	—	53	2	no	—
172	C₄⋊C₈	—	53	1	no	—
173	C₄⋊C₈	C₂×D₅₃	1	1	yes	characteristic
174	#103	—	53	2	no	—
175	#103	—	53	2	no	—
176	#102	—	53	2	no	—
177	#102	C₂×D₅₃	1	2	yes	—
178	#41	—	53	4	no	—
179	#41	—	106	2	no	—
180	C₂×Q₈	—	2	1	no	—
181	C₂×Q₈	—	53	1	no	—
182	C₂×Q₈	—	106	1	no	—
183	C₂×D₄	—	106	1	no	—
184	C₂×D₄	C₂²×D₅₃	1	1	yes	characteristic
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	53	1	no	—
187	C₂²×C₄	—	53	1	no	—
188	C₂²×C₄	—	53	1	no	—
189	C₂²×C₄	—	106	1	no	—
190	C₂²×C₄	—	106	1	no	—
191	C₂×C₈	—	53	2	no	—
192	C₂×C₈	—	106	4	no	—
193	C₂×C₈	C₂²×D₅₃	1	2	yes	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	—	53	2	no	—
196	C₄⋊C₄	—	53	1	no	—
197	C₄⋊C₄	—	106	1	no	—
198	C₄⋊C₄	—	106	2	no	—
199	C₄⋊C₄	—	106	2	no	—
200	C₄⋊C₄	—	106	1	no	—
201	C₄⋊C₄	—	106	1	no	—
202	C₄⋊C₄	C₂²×D₅₃	1	2	yes	—
203	C₄⋊C₄	C₂²×D₅₃	1	1	yes	characteristic
204	C₂²⋊C₄	—	2	1	no	—
205	C₂²⋊C₄	—	106	1	no	—
206	C₂²⋊C₄	—	106	2	no	—
207	C₂²⋊C₄	—	106	1	no	—
208	C₄²	—	53	1	no	—
209	C₄²	—	53	1	no	—
210	C₄²	—	53	1	no	—
211	C₄²	—	106	1	no	—
212	C₄²	—	106	1	no	—
213	C₄²	—	106	2	no	—
214	C₄²	—	106	1	no	—
215	C₄²	C₂²×D₅₃	1	1	yes	characteristic
216	C₂³	—	2	1	no	—
217	C₂³	—	53	1	no	—
218	Q₈	—	4	2	no	—
219	Q₈	—	53	2	no	—
220	Q₈	—	106	1	no	—
221	Q₈	—	212	2	no	—
222	D₄	—	2	1	no	—
223	D₄	—	106	4	no	—
224	D₄	Q₈×D₅₃	1	2	yes	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	53	1	no	—
229	C₂×C₄	—	53	2	no	—
230	C₂×C₄	—	53	2	no	—
231	C₂×C₄	—	53	1	no	—
232	C₂×C₄	—	53	1	no	—
233	C₂×C₄	—	106	4	no	—
234	C₂×C₄	—	106	1	no	—
235	C₂×C₄	—	106	2	no	—
236	C₂×C₄	—	106	1	no	—
237	C₂×C₄	—	106	2	no	—
238	C₂×C₄	—	212	1	no	—
239	C₂×C₄	—	212	2	no	—
240	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
241	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
242	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
243	C₈	—	2	2	no	—
244	C₈	—	106	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	53	2	no	—
249	C₂²	—	106	1	no	—
250	C₂²	—	106	1	no	—
251	C₂²	C₂×D₄×D₅₃	1	1	yes	characteristic
252	C₄	—	2	1	no	—
253	C₄	—	4	1	no	—
254	C₄	—	4	2	no	—
255	C₄	—	53	2	no	—
256	C₄	—	106	2	no	—
257	C₄	—	106	1	no	—
258	C₄	—	106	1	no	—
259	C₄	—	212	1	no	—
260	C₄	—	212	2	no	—
261	C₄	#13133236	1	2	yes	—
262	C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
263	C₄	#41×D₅₃	1	1	yes	characteristic
264	C₂	—	4	2	no	—
265	C₂	—	106	2	no	—
266	C₂	C₂²⋊Q₈×D₅₃	1	1	yes	characteristic
267	C₂	C₂×#13133236	1	1	yes	characteristic
268	C₂	#16811422	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#16811178

C₂

yes

maximal, characteristic, complemented

C₄×#11719947

C₂

yes

maximal, characteristic

#16810929

C₂

yes

maximal, complemented

#16810921

C₂

yes

maximal, complemented

#16810919

C₂

yes

maximal, characteristic, complemented

#16810913

C₂

yes

maximal, characteristic, complemented

#16810905

C₂

yes

maximal, characteristic, complemented

#16810660

C₂

yes

maximal, complemented

C₄⋊C₈×D₅₃

C₂

yes

maximal, characteristic, complemented

#16810616

C₂

yes

maximal

C₅₃×#474

C₂

yes

maximal, characteristic, complemented

C₂×#11719947

C₂²

yes

characteristic

C₂×C₈×D₅₃

C₂²

yes

—

#13133217

C₂²

yes

characteristic

#13133208

C₂²

yes

—

#13133197

—

#13133193

—

Q₈×Dic₅₃

—

C₄×Dic₁₀₆

C₂²

yes

characteristic

#13133182

—

#13133182

C₂²

yes

characteristic

D₄×Dic₅₃

C₂²

yes

characteristic

C₄×C₂²⋊D₅₃

—

#13133177

—

#13133175

C₂²

yes

characteristic

#13133175

C₂²

yes

—

#13133173

—

C₂×C₄×Dic₅₃

—

C₄²×D₅₃

C₂²

yes

characteristic

#13133151

C₂²

yes

—

#13133148

C₂²

yes

—

#13133148

C₂²

yes

—

C₄⋊#88322

C₂²

yes

characteristic

#13133146

C₂²

yes

—

#13133142

C₂²

yes

—

#13133140

C₂²

yes

—

#13133138

C₂²

yes

—

C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₄×D₄×C₅₃

C₂²

yes

characteristic

#107×C₅₃

C₂²

yes

—

C₄⋊C₈×C₅₃

C₂²

yes

characteristic

#102×C₅₃

C₂²

yes

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

C₂³

yes

characteristic

C₂²×C₄×C₅₃

—

C₂×C₂²⋊D₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

#11719947

—

#11719947

Q₈

yes

complemented

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

D₄

yes

characteristic

C₂×C₄×D₅₃

C₂³

yes

characteristic

C₂×Dic₁₀₆

—

C₂×Dic₁₀₆

C₂³

yes

characteristic

C₂×C₈×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

C₂³

yes

—

C₄⋊C₄×C₅₃

C₂³

yes

characteristic

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

C₂³

yes

characteristic

C₂²⋊Dic₅₃

—

#11719921

—

#11719921

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

—

C₄⋊Dic₅₃

C₂³

yes

—

C₄⋊Dic₅₃

C₂³

yes

—

—

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

D₄

yes

characteristic

C₄×Dic₅₃

C₂³

yes

characteristic

C₂×#88322

C₂³

yes

—

C₈×D₅₃

—

C₂³×C₅₃

—

C₂²×D₅₃

C₂×D₄

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

C₂×Q₈

yes

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂×C₄×C₅₃

C₂⁴

yes

characteristic

C₂²⋊D₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

—

C₂×Dic₅₃

C₂×D₄

yes

characteristic

C₂×Dic₅₃

C₂×D₄

yes

characteristic

100

C₂×Dic₅₃

C₂×D₄

yes

characteristic

101

C₄×D₅₃

—

102

C₄×D₅₃

—

103

C₄×D₅₃

C₂×Q₈

yes

—

104

C₄×D₅₃

#41

yes

—

105

Dic₁₀₆

—

106

Dic₁₀₆

—

107

Dic₁₀₆

C₂×Q₈

yes

—

108

C₈×C₅₃

—

109

#88322

—

110

C₂²×C₅₃

—

111

C₂²×C₅₃

—

112

C₂²×C₅₃

—

113

C₂²×C₅₃

C₂²×D₄

yes

characteristic

114

C₂×D₅₃

—

115

C₂×D₅₃

—

116

C₂×D₅₃

C₂²⋊Q₈

yes

—

117

C₄×C₅₃

—

118

C₄×C₅₃

—

119

C₄×C₅₃

—

120

C₄×C₅₃

#135

yes

—

121

C₄×C₅₃

C₂²×Q₈

yes

characteristic

122

C₄×C₅₃

C₂×#41

yes

characteristic

123

Dic₅₃

—

124

Dic₅₃

—

125

Dic₅₃

—

126

Dic₅₃

—

127

Dic₅₃

—

128

Dic₅₃

C₂²⋊Q₈

yes

—

129

#3076

—

maximal, Sylow

130

C₂×C₅₃

—

131

C₂×C₅₃

C₂×C₂²⋊Q₈

yes

characteristic

132

C₂×C₅₃

C₂×#135

yes

characteristic

133

C₂×C₅₃

#575

yes

characteristic

134

D₅₃

—

135

#533

—

136

C₄×#41

—

137

#479

—

138

#478

—

139

#477

—

140

#476

—

141

#475

—

142

#474

D₅₃

yes

characteristic, complemented

143

#427

—

144

C₂×C₄⋊C₈

—

145

#416

—

146

C₅₃

#3076

yes

characteristic, complemented, Sylow

147

C₂×#41

—

148

C₂²×C₈

—

149

C₄⋊Q₈

—

150

C₄⋊Q₈

C₂×D₅₃

yes

characteristic

151

#125

—

152

C₂²⋊Q₈

—

106

—

153

C₂²⋊Q₈

—

106

—

154

C₄×Q₈

—

155

C₄×Q₈

—

156

C₄×Q₈

—

106

—

157

C₄×Q₈

—

106

—

158

C₄×D₄

—

159

C₄×D₄

—

106

—

160

C₄×D₄

—

106

—

161

C₄×D₄

C₂×D₅₃

yes

characteristic

162

#117

—

163

#117

—

164

#117

—

106

—

165

C₂×C₄²

—

166

C₂×C₄²

—

106

—

167

#107

—

168

#107

C₂×D₅₃

yes

—

169

#106

—

170

#106

—

171

C₄⋊C₈

—

172

C₄⋊C₈

—

173

C₄⋊C₈

C₂×D₅₃

yes

characteristic

174

#103

—

175

#103

—

176

#102

—

177

#102

C₂×D₅₃

yes

—

178

#41

—

179

#41

—

106

—

180

C₂×Q₈

—

181

C₂×Q₈

—

182

C₂×Q₈

—

106

—

183

C₂×D₄

—

106

—

184

C₂×D₄

C₂²×D₅₃

yes

characteristic

185

C₂²×C₄

—

186

C₂²×C₄

—

187

C₂²×C₄

—

188

C₂²×C₄

—

189

C₂²×C₄

—

106

—

190

C₂²×C₄

—

106

—

191

C₂×C₈

—

192

C₂×C₈

—

106

—

193

C₂×C₈

C₂²×D₅₃

yes

—

194

C₄⋊C₄

—

195

C₄⋊C₄

—

196

C₄⋊C₄

—

197

C₄⋊C₄

—

106

—

198

C₄⋊C₄

—

106

—

199

C₄⋊C₄

—

106

—

200

C₄⋊C₄

—

106

—

201

C₄⋊C₄

—

106

—

202

C₄⋊C₄

C₂²×D₅₃

yes

—

203

C₄⋊C₄

C₂²×D₅₃

yes

characteristic

204

C₂²⋊C₄

—

205

C₂²⋊C₄

—

106

—

206

C₂²⋊C₄

—

106

—

207

C₂²⋊C₄

—

106

—

208

C₄²

—

209

C₄²

—

210

C₄²

—

211

C₄²

—

106

—

212

C₄²

—

106

—

213

C₄²

—

106

—

214

C₄²

—

106

—

215

C₄²

C₂²×D₅₃

yes

characteristic

216

C₂³

—

217

C₂³

—

218

Q₈

—

219

Q₈

—

220

Q₈

—

106

—

221

Q₈

—

212

—

222

D₄

—

223

D₄

—

106

—

224

D₄

Q₈×D₅₃

yes

—

225

C₂×C₄

—

226

C₂×C₄

—

227

C₂×C₄

—

228

C₂×C₄

—

229

C₂×C₄

—

230

C₂×C₄

—

231

C₂×C₄

—

232

C₂×C₄

—

233

C₂×C₄

—

106

—

234

C₂×C₄

—

106

—

235

C₂×C₄

—

106

—

236

C₂×C₄

—

106

—

237

C₂×C₄

—

106

—

238

C₂×C₄

—

212

—

239

C₂×C₄

—

212

—

240

C₂×C₄

D₄×D₅₃

yes

characteristic

241

C₂×C₄

D₄×D₅₃

yes

characteristic

242

C₂×C₄

C₂³×D₅₃

yes

characteristic

243

C₈

—

244

C₈

—

106

—

245

C₂²

—

246

C₂²

—

247

C₂²

—

248

C₂²

—

249

C₂²

—

106

—

250

C₂²

—

106

—

251

C₂²

C₂×D₄×D₅₃

yes

characteristic

252

C₄

—

253

C₄

—

254

C₄

—

255

C₄

—

256

C₄

—

106

—

257

C₄

—

106

—

258

C₄

—

106

—

259

C₄

—

212

—

260

C₄

—

212

—

261

C₄

#13133236

yes

—

262

C₄

C₂×Q₈×D₅₃

yes

characteristic

263

C₄

#41×D₅₃

yes

characteristic

264

C₂

—

265

C₂

—

106

—

266

C₂

C₂²⋊Q₈×D₅₃

yes

characteristic

267

C₂

C₂×#13133236

yes

characteristic

268

C₂

#16811422

yes

characteristic

	subgroup	G/H
center	C₂²	C₂×D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₅₃
Fitting	C₅₃×#474	C₂
socle	C₂²×C₅₃	C₂²×D₄
hypercenter	#474	D₅₃
hypocenter	C₅₃	#3076
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ #474

subgroup

G/H

center

C₂²

C₂×D₄×D₅₃

commutator

C₂×C₄×C₅₃

C₂⁴

Frattini

C₂×C₄

C₂³×D₅₃

Fitting

C₅₃×#474

C₂

socle

C₂²×C₅₃

C₂²×D₄

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂² ◁ C₄² ◁ #474

N	H ≅ G/N	type
1. #16811178	C₂	semidirect
3. #16810929	C₂	semidirect
4. #16810921	C₂	semidirect
5. #16810919	C₂	semidirect
6. #16810913	C₂	semidirect
7. #16810905	C₂	semidirect
8. #16810660	C₂	semidirect
9. C₄⋊C₈×D₅₃	C₂	semidirect
11. C₅₃×#474	C₂	semidirect
50. #11719947	Q₈	semidirect
142. #474	D₅₃	semidirect
146. C₅₃	#3076	semidirect

H ≅ G/N

type

1. #16811178

C₂

semidirect

3. #16810929

C₂

semidirect

4. #16810921

C₂

semidirect

5. #16810919

C₂

semidirect

6. #16810913

C₂

semidirect

7. #16810905

C₂

semidirect

8. #16810660

C₂

semidirect

9. C₄⋊C₈×D₅₃

C₂

semidirect

11. C₅₃×#474

C₂

semidirect

50. #11719947

Q₈

semidirect

142. #474

D₅₃

semidirect

146. C₅₃

#3076

semidirect

Subgroups — #20284496

Conjugacy classes of subgroups

Complements