Group #27301020 — subgroups

Subgroups — #27301020

Order 13568.

470 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 470 conjugacy classes of subgroups — too many to draw. Only its 135 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	#20283416	C₂	1	1	yes	maximal, characteristic
2	#20276336	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#16810404	C₂	1	2	yes	maximal, complemented
4	#20272802	C₂	1	2	yes	maximal, complemented
5	C₅₃×#3407	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂²×#11719912	C₂²	1	1	yes	characteristic
7	#16811130	C₄	1	1	yes	characteristic
8	C₂×#13133157	C₂²	1	2	yes	—
9	#16810699	—	2	2	no	—
10	#16810694	—	2	2	no	—
11	#16810651	C₄	1	1	yes	characteristic
12	#16810590	C₄	1	1	yes	characteristic
13	#16810586	C₄	1	1	yes	characteristic
14	#16810578	C₂²	1	1	yes	characteristic
15	#16810405	—	2	2	no	—
16	#16810404	—	2	4	no	—
17	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#504	—	2	4	no	—
19	C₂×M₃₂×C₅₃	—	2	2	no	—
20	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
21	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
22	C₂×#11719917	C₂×C₄	1	1	yes	characteristic
23	C₂×#11719912	—	2	2	no	—
24	C₂×#11719912	—	2	1	no	—
25	C₂×#11719912	—	2	1	no	—
26	C₂×#11719912	—	2	2	no	—
27	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
28	#13133198	C₈	1	2	yes	—
29	#13133193	—	2	1	no	—
30	#13133188	—	2	1	no	—
31	#13133185	C₈	1	2	yes	—
32	C₄×C₂²⋊D₅₃	—	2	2	no	—
33	#13133176	C₂×C₄	1	1	yes	characteristic
34	#13133157	—	2	2	no	—
35	#13133157	—	2	2	no	—
36	C₂×#11719908	—	2	2	no	—
37	#13133146	—	2	2	no	—
38	C₂²⋊#88322	—	2	2	no	—
39	#13133129	—	2	2	no	—
40	#13133127	—	2	1	no	—
41	C₈×Dic₅₃	—	2	1	no	—
42	C₂×#41×C₅₃	C₂×C₄	1	1	yes	characteristic
43	#131×C₅₃	—	2	2	no	—
44	#131×C₅₃	—	2	2	no	—
45	C₂×M₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
47	C₂²×C₈×C₅₃	D₄	1	2	yes	—
48	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
49	M₃₂×C₅₃	—	2	4	no	—
50	M₃₂×C₅₃	—	2	2	no	—
51	M₃₂×C₅₃	D₄	1	4	yes	complemented
52	C₂×C₁₆×C₅₃	—	2	4	no	—
53	C₂×C₁₆×C₅₃	—	2	2	no	—
54	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
55	#41×C₅₃	—	2	2	no	—
56	#41×C₅₃	—	2	2	no	—
57	C₂×Q₈×C₅₃	M₁₆	1	1	yes	characteristic
58	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
59	C₂×D₄×C₅₃	M₁₆	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₅₃	—	2	2	no	—
63	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
64	C₂×C₄×D₅₃	—	2	1	no	—
65	C₂×C₄×D₅₃	—	2	2	no	—
66	C₂×C₄×D₅₃	—	2	1	no	—
67	C₂×C₄×D₅₃	—	2	2	no	—
68	M₁₆×C₅₃	—	2	2	no	—
69	M₁₆×C₅₃	—	2	4	no	—
70	C₂×C₈×C₅₃	—	2	2	no	—
71	C₂×C₈×C₅₃	—	2	2	no	—
72	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
73	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	#41	1	2	yes	—
77	C₂²⋊Dic₅₃	—	2	2	no	—
78	C₂²⋊Dic₅₃	C₂×C₈	1	2	yes	—
79	#11719921	—	2	2	no	—
80	C₄⋊Dic₅₃	—	2	1	no	—
81	#11719919	—	2	2	no	—
82	C₄×Dic₅₃	—	2	1	no	—
83	#11719917	—	2	2	no	—
84	C₂×#88322	—	2	1	no	—
85	C₂×#88322	—	2	1	no	—
86	C₂×#88322	—	2	2	no	—
87	#11719912	—	4	1	no	—
88	#11719912	—	4	2	no	—
89	#11719912	—	4	1	no	—
90	C₁₆×C₅₃	—	2	4	no	—
91	#11719908	—	4	2	no	—
92	C₂³×C₅₃	C₂²⋊C₈	1	2	yes	—
93	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
94	C₂²×D₅₃	—	2	1	no	—
95	C₂²×D₅₃	—	2	2	no	—
96	C₂²×D₅₃	—	8	1	no	—
97	Q₈×C₅₃	—	2	2	no	—
98	D₄×C₅₃	—	2	4	no	—
99	D₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	2	2	no	—
101	C₂×C₄×C₅₃	—	2	2	no	—
102	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
103	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#131	1	2	yes	—
107	C₂²⋊D₅₃	—	8	2	no	—
108	C₂×Dic₅₃	—	2	1	no	—
109	C₂×Dic₅₃	—	2	2	no	—
110	C₂×Dic₅₃	—	2	2	no	—
111	C₄×D₅₃	—	4	1	no	—
112	C₄×D₅₃	—	4	1	no	—
113	C₄×D₅₃	—	4	2	no	—
114	C₈×C₅₃	—	2	2	no	—
115	C₈×C₅₃	#120	1	1	yes	characteristic
116	C₈×C₅₃	#121	1	2	yes	—
117	C₈×C₅₃	#123	1	1	yes	characteristic
118	#88322	—	4	1	no	—
119	#88322	—	4	1	no	—
120	#12989	—	53	1	no	maximal, Sylow
121	C₂²×C₅₃	—	2	2	no	—
122	C₂²×C₅₃	—	4	1	no	—
123	C₂²×C₅₃	—	4	2	no	—
124	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
125	C₂²×C₅₃	C₈×D₄	1	2	yes	—
126	C₂×D₅₃	—	4	1	no	—
127	C₂×D₅₃	—	4	1	no	—
128	C₂×D₅₃	—	4	1	no	—
129	C₂×D₅₃	—	8	2	no	—
130	C₄×C₅₃	—	2	2	no	—
131	C₄×C₅₃	#386	1	1	yes	characteristic
132	C₄×C₅₃	#408	1	1	yes	characteristic
133	C₄×C₅₃	#435	1	2	yes	—
134	Dic₅₃	—	4	2	no	—
135	Dic₅₃	—	4	1	no	—
136	#3407	D₅₃	1	1	yes	characteristic, complemented
137	#2930	—	53	1	no	—
138	#2115	—	53	1	no	—
139	C₂×#349	—	53	2	no	—
140	#1378	—	53	2	no	—
141	C₂×C₅₃	—	2	1	no	—
142	C₂×C₅₃	—	2	2	no	—
143	C₂×C₅₃	—	4	2	no	—
144	C₂×C₅₃	#1869	1	1	yes	characteristic
145	D₅₃	—	8	1	no	—
146	C₂×#131	C₂×D₅₃	1	1	yes	characteristic
147	C₂²×M₁₆	—	53	1	no	—
148	#525	—	53	1	no	—
149	#504	—	2	4	no	—
150	C₂×M₃₂	—	2	2	no	—
151	C₂×M₃₂	—	53	2	no	—
152	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
153	#436	—	106	2	no	—
154	#435	—	106	2	no	—
155	#424	—	53	1	no	—
156	#408	—	53	1	no	—
157	#407	—	53	1	no	—
158	#405	—	53	1	no	—
159	#349	—	106	2	no	—
160	#349	—	106	4	no	—
161	C₅₃	#12989	1	1	yes	characteristic, complemented, Sylow
162	C₂×#41	C₄×D₅₃	1	1	yes	characteristic
163	C₂³×C₄	—	53	1	no	—
164	#131	—	2	2	no	—
165	#131	—	2	2	no	—
166	C₂×M₁₆	—	53	1	no	—
167	C₂×M₁₆	—	106	1	no	—
168	C₂×M₁₆	—	106	2	no	—
169	C₂×M₁₆	—	106	2	no	—
170	C₂×M₁₆	—	106	1	no	—
171	C₂×M₁₆	C₄×D₅₃	1	1	yes	characteristic
172	C₂×M₁₆	D₂₁₂	1	2	yes	—
173	C₂²×C₈	—	53	1	no	—
174	C₂²×C₈	C₂²⋊D₅₃	1	2	yes	—
175	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
176	#123	—	53	2	no	—
177	C₂²⋊Q₈	—	106	1	no	—
178	#121	—	106	1	no	—
179	#120	—	53	2	no	—
180	C₄×D₄	—	106	2	no	—
181	#117	—	53	1	no	—
182	M₃₂	—	2	4	no	—
183	M₃₂	—	2	2	no	—
184	M₃₂	—	106	2	no	—
185	M₃₂	—	106	2	no	—
186	M₃₂	C₂²⋊D₅₃	1	4	yes	complemented
187	C₂×C₁₆	—	2	4	no	—
188	C₂×C₁₆	—	2	2	no	—
189	C₂×C₁₆	—	106	2	no	—
190	C₄⋊C₈	—	106	2	no	—
191	C₂²⋊C₈	—	106	2	no	—
192	C₂²⋊C₈	—	106	2	no	—
193	#97	—	106	1	no	—
194	C₄×C₈	—	106	1	no	—
195	C₂⁴	—	53	1	no	—
196	#41	—	2	2	no	—
197	#41	—	2	2	no	—
198	C₂×Q₈	#11719912	1	1	yes	characteristic
199	C₂×D₄	—	106	2	no	—
200	C₂×D₄	C₈×D₅₃	1	2	yes	—
201	C₂×D₄	#11719912	1	1	yes	characteristic
202	C₂²×C₄	—	53	1	no	—
203	C₂²×C₄	—	106	1	no	—
204	C₂²×C₄	—	106	2	no	—
205	C₂²×C₄	—	106	1	no	—
206	C₂²×C₄	—	106	2	no	—
207	C₂²×C₄	#11719921	1	2	yes	—
208	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
209	M₁₆	—	2	2	no	—
210	M₁₆	—	2	4	no	—
211	M₁₆	—	106	2	no	—
212	M₁₆	—	212	1	no	—
213	M₁₆	—	212	2	no	—
214	M₁₆	—	212	1	no	—
215	C₂×C₈	—	2	2	no	—
216	C₂×C₈	—	2	2	no	—
217	C₂×C₈	—	106	1	no	—
218	C₂×C₈	—	106	1	no	—
219	C₂×C₈	—	106	2	no	—
220	C₂×C₈	#11719921	1	2	yes	—
221	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
222	C₂×C₈	#11719945	1	2	yes	—
223	C₂×C₈	C₂×C₂²⋊D₅₃	1	2	yes	—
224	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
225	C₄⋊C₄	—	106	2	no	—
226	C₄⋊C₄	—	106	1	no	—
227	C₂²⋊C₄	—	53	2	no	—
228	C₂²⋊C₄	—	106	2	no	—
229	C₂²⋊C₄	—	106	2	no	—
230	C₄²	—	106	1	no	—
231	C₁₆	—	2	4	no	—
232	C₁₆	—	212	2	no	—
233	C₂³	—	106	2	no	—
234	C₂³	—	106	1	no	—
235	C₂³	—	424	1	no	—
236	C₂³	#13133129	1	2	yes	—
237	C₂³	C₂×C₈×D₅₃	1	1	yes	characteristic
238	Q₈	—	2	2	no	—
239	D₄	—	2	4	no	—
240	D₄	—	2	2	no	—
241	D₄	—	424	2	no	—
242	C₂×C₄	—	2	2	no	—
243	C₂×C₄	—	2	2	no	—
244	C₂×C₄	—	106	2	no	—
245	C₂×C₄	—	106	1	no	—
246	C₂×C₄	—	106	2	no	—
247	C₂×C₄	—	212	1	no	—
248	C₂×C₄	—	212	1	no	—
249	C₂×C₄	—	212	2	no	—
250	C₂×C₄	#13133129	1	2	yes	—
251	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
252	C₂×C₄	C₄×C₂²⋊D₅₃	1	2	yes	—
253	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
254	C₂×C₄	#13133223	1	2	yes	—
255	C₈	—	2	2	no	—
256	C₈	—	212	1	no	—
257	C₈	—	212	1	no	—
258	C₈	#13133186	1	1	yes	characteristic
259	C₈	#13133191	1	2	yes	—
260	C₈	#13133200	1	1	yes	characteristic
261	C₂²	—	2	2	no	—
262	C₂²	—	4	1	no	—
263	C₂²	—	4	2	no	—
264	C₂²	—	424	2	no	—
265	C₂²	—	212	1	no	—
266	C₂²	—	212	1	no	—
267	C₂²	—	212	1	no	—
268	C₂²	C₂×#13133129	1	1	yes	characteristic
269	C₂²	C₈×C₂²⋊D₅₃	1	2	yes	—
270	C₄	—	2	2	no	—
271	C₄	—	212	2	no	—
272	C₄	—	212	1	no	—
273	C₄	#16810502	1	1	yes	characteristic
274	C₄	#16810589	1	1	yes	characteristic
275	C₄	#16810694	1	2	yes	—
276	C₂	—	2	1	no	—
277	C₂	—	2	2	no	—
278	C₂	—	4	2	no	—
279	C₂	—	424	1	no	—
280	C₂	#20275007	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

#20283416

C₂

yes

maximal, characteristic

#20276336

C₂

yes

maximal, characteristic, complemented

C₂×#16810404

C₂

yes

maximal, complemented

#20272802

C₂

yes

maximal, complemented

C₅₃×#3407

C₂

yes

maximal, characteristic, complemented

C₂²×#11719912

C₂²

yes

characteristic

#16811130

C₄

yes

characteristic

C₂×#13133157

C₂²

yes

—

#16810699

—

#16810694

—

#16810651

C₄

yes

characteristic

#16810590

C₄

yes

characteristic

#16810586

C₄

yes

characteristic

#16810578

C₂²

yes

characteristic

#16810405

—

#16810404

—

C₂×#131×C₅₃

C₂²

yes

characteristic

C₅₃×#504

—

C₂×M₃₂×C₅₃

—

C₂×M₃₂×C₅₃

C₂²

yes

—

C₂²×C₄×D₅₃

C₂×C₄

yes

characteristic

C₂×#11719917

C₂×C₄

yes

—

—

—

—

C₂²×#88322

C₂×C₄

yes

characteristic

#13133198

C₈

yes

—

#13133193

—

#13133188

—

#13133185

C₈

yes

—

C₄×C₂²⋊D₅₃

—

#13133176

C₂×C₄

yes

—

—

—

—

—

—

—

C₈×Dic₅₃

—

C₂×#41×C₅₃

C₂×C₄

yes

characteristic

#131×C₅₃

—

#131×C₅₃

—

C₂×M₁₆×C₅₃

C₂×C₄

yes

characteristic

C₂×M₁₆×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

M₃₂×C₅₃

—

M₃₂×C₅₃

—

M₃₂×C₅₃

D₄

yes

complemented

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂³×D₅₃

C₂×C₈

yes

characteristic

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

M₁₆

yes

characteristic

C₂×D₄×C₅₃

C₂×C₈

yes

—

C₂×D₄×C₅₃

M₁₆

yes

characteristic

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₂²⋊D₅₃

—

C₂²×Dic₅₃

C₂×C₈

yes

characteristic

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

C₂×C₄×D₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₂²⋊Dic₅₃

—

C₂²⋊Dic₅₃

C₂×C₈

yes

—

#11719921

—

C₄⋊Dic₅₃

—

#11719919

—

C₄×Dic₅₃

—

—

—

—

—

—

—

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

C₂²⋊C₈

yes

—

C₂³×C₅₃

C₂²×C₈

yes

characteristic

C₂²×D₅₃

—

C₂²×D₅₃

—

C₂²×D₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

100

C₂×C₄×C₅₃

—

101

C₂×C₄×C₅₃

—

102

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

103

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

104

C₂×C₄×C₅₃

C₄×D₄

yes

—

105

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

106

C₂×C₄×C₅₃

#131

yes

—

107

C₂²⋊D₅₃

—

108

C₂×Dic₅₃

—

109

C₂×Dic₅₃

—

110

C₂×Dic₅₃

—

111

C₄×D₅₃

—

112

C₄×D₅₃

—

113

C₄×D₅₃

—

114

C₈×C₅₃

—

115

C₈×C₅₃

#120

yes

characteristic

116

C₈×C₅₃

#121

yes

—

117

C₈×C₅₃

#123

yes

characteristic

118

#88322

—

119

#88322

—

120

#12989

—

maximal, Sylow

121

C₂²×C₅₃

—

122

C₂²×C₅₃

—

123

C₂²×C₅₃

—

124

C₂²×C₅₃

C₂×C₂²⋊C₈

yes

characteristic

125

C₂²×C₅₃

C₈×D₄

yes

—

126

C₂×D₅₃

—

127

C₂×D₅₃

—

128

C₂×D₅₃

—

129

C₂×D₅₃

—

130

C₄×C₅₃

—

131

C₄×C₅₃

#386

yes

characteristic

132

C₄×C₅₃

#408

yes

characteristic

133

C₄×C₅₃

#435

yes

—

134

Dic₅₃

—

135

Dic₅₃

—

136

#3407

D₅₃

yes

characteristic, complemented

137

#2930

—

138

#2115

—

139

C₂×#349

—

140

#1378

—

141

C₂×C₅₃

—

142

C₂×C₅₃

—

143

C₂×C₅₃

—

144

C₂×C₅₃

#1869

yes

characteristic

145

D₅₃

—

146

C₂×#131

C₂×D₅₃

yes

characteristic

147

C₂²×M₁₆

—

148

#525

—

149

#504

—

150

C₂×M₃₂

—

151

C₂×M₃₂

—

152

C₂×M₃₂

C₂×D₅₃

yes

—

153

#436

—

106

—

154

#435

—

106

—

155

#424

—

156

#408

—

157

#407

—

158

#405

—

159

#349

—

106

—

160

#349

—

106

—

161

C₅₃

#12989

yes

characteristic, complemented, Sylow

162

C₂×#41

C₄×D₅₃

yes

characteristic

163

C₂³×C₄

—

164

#131

—

165

#131

—

166

C₂×M₁₆

—

167

C₂×M₁₆

—

106

—

168

C₂×M₁₆

—

106

—

169

C₂×M₁₆

—

106

—

170

C₂×M₁₆

—

106

—

171

C₂×M₁₆

C₄×D₅₃

yes

characteristic

172

C₂×M₁₆

D₂₁₂

yes

—

173

C₂²×C₈

—

174

C₂²×C₈

C₂²⋊D₅₃

yes

—

175

C₂²×C₈

C₂²×D₅₃

yes

characteristic

176

#123

—

177

C₂²⋊Q₈

—

106

—

178

#121

—

106

—

179

#120

—

180

C₄×D₄

—

106

—

181

#117

—

182

M₃₂

—

183

M₃₂

—

184

M₃₂

—

106

—

185

M₃₂

—

106

—

186

M₃₂

C₂²⋊D₅₃

yes

complemented

187

C₂×C₁₆

—

188

C₂×C₁₆

—

189

C₂×C₁₆

—

106

—

190

C₄⋊C₈

—

106

—

191

C₂²⋊C₈

—

106

—

192

C₂²⋊C₈

—

106

—

193

#97

—

106

—

194

C₄×C₈

—

106

—

195

C₂⁴

—

196

#41

—

197

#41

—

198

C₂×Q₈

#11719912

yes

characteristic

199

C₂×D₄

—

106

—

200

C₂×D₄

C₈×D₅₃

yes

—

201

C₂×D₄

#11719912

yes

characteristic

202

C₂²×C₄

—

203

C₂²×C₄

—

106

—

204

C₂²×C₄

—

106

—

205

C₂²×C₄

—

106

—

206

C₂²×C₄

—

106

—

207

C₂²×C₄

#11719921

yes

—

208

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

209

M₁₆

—

210

M₁₆

—

211

M₁₆

—

106

—

212

M₁₆

—

212

—

213

M₁₆

—

212

—

214

M₁₆

—

212

—

215

C₂×C₈

—

216

C₂×C₈

—

217

C₂×C₈

—

106

—

218

C₂×C₈

—

106

—

219

C₂×C₈

—

106

—

220

C₂×C₈

#11719921

yes

—

221

C₂×C₈

C₂×D₂₁₂

yes

characteristic

222

C₂×C₈

#11719945

yes

—

223

C₂×C₈

C₂×C₂²⋊D₅₃

yes

—

224

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

225

C₄⋊C₄

—

106

—

226

C₄⋊C₄

—

106

—

227

C₂²⋊C₄

—

228

C₂²⋊C₄

—

106

—

229

C₂²⋊C₄

—

106

—

230

C₄²

—

106

—

231

C₁₆

—

232

C₁₆

—

212

—

233

C₂³

—

106

—

234

C₂³

—

106

—

235

C₂³

—

424

—

236

C₂³

#13133129

yes

—

237

C₂³

C₂×C₈×D₅₃

yes

characteristic

238

Q₈

—

239

D₄

—

240

D₄

—

241

D₄

—

424

—

242

C₂×C₄

—

243

C₂×C₄

—

244

C₂×C₄

—

106

—

245

C₂×C₄

—

106

—

246

C₂×C₄

—

106

—

247

C₂×C₄

—

212

—

248

C₂×C₄

—

212

—

249

C₂×C₄

—

212

—

250

C₂×C₄

#13133129

yes

—

251

C₂×C₄

C₂×#11719921

yes

characteristic

252

C₂×C₄

C₄×C₂²⋊D₅₃

yes

—

253

C₂×C₄

C₂×#11719912

yes

characteristic

254

C₂×C₄

#13133223

yes

—

255

C₈

—

256

C₈

—

212

—

257

C₈

—

212

—

258

C₈

#13133186

yes

characteristic

259

C₈

#13133191

yes

—

260

C₈

#13133200

yes

characteristic

261

C₂²

—

262

C₂²

—

263

C₂²

—

264

C₂²

—

424

—

265

C₂²

—

212

—

266

C₂²

—

212

—

267

C₂²

—

212

—

268

C₂²

C₂×#13133129

yes

characteristic

269

C₂²

C₈×C₂²⋊D₅₃

yes

—

270

C₄

—

271

C₄

—

212

—

272

C₄

—

212

—

273

C₄

#16810502

yes

characteristic

274

C₄

#16810589

yes

characteristic

275

C₄

#16810694

yes

—

276

C₂

—

277

C₂

—

278

C₂

—

279

C₂

—

424

—

280

C₂

#20275007

yes

characteristic

	subgroup	G/H
center	C₄	#16810502
commutator	C₂³×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₅₃×#3407	C₂
socle	C₂×C₅₃	#1869
prime radical	#16811130	C₄
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	#3407	D₅₃
hypocenter	C₅₃	#12989
LCS	G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #3407

subgroup

G/H

center

C₄

#16810502

commutator

C₂³×C₅₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₅₃×#3407

C₂

C₂×C₅₃

C₄

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂³×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₄ ◁ C₂²×C₈ ◁ #3407

N	H ≅ G/N	type
2. #20276336	C₂	semidirect
3. C₂×#16810404	C₂	semidirect
4. #20272802	C₂	semidirect
5. C₅₃×#3407	C₂	semidirect
51. M₃₂×C₅₃	D₄	semidirect
136. #3407	D₅₃	semidirect
161. C₅₃	#12989	semidirect
186. M₃₂	C₂²⋊D₅₃	semidirect

H ≅ G/N

type

2. #20276336

C₂

semidirect

3. C₂×#16810404

C₂

semidirect

4. #20272802

C₂

semidirect

5. C₅₃×#3407

C₂

semidirect

51. M₃₂×C₅₃

D₄

semidirect

136. #3407

D₅₃

semidirect

161. C₅₃

#12989

semidirect

186. M₃₂

C₂²⋊D₅₃

semidirect

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

subgroup	index
#20283416	2
#20276336	2
C₂×#16810404	2
#20272802	2
C₅₃×#3407	2
#12989	53

subgroup

index

C₅₃×#3407

Subgroups — #27301020

Conjugacy classes of subgroups

Complements

Maximal subgroups