Group #27216905 — subgroups

Subgroups — C₅₃×#9393 #27216905

Order 13568.

558 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 558 conjugacy classes of subgroups — too many to draw. Only its 154 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	C₅₃×#2972	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#2547	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1800	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1546	C₂	1	2	yes	maximal, complemented
5	C₅₃×#1303	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×#124×C₅₃	—	2	1	no	—
8	C₂×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
9	C₈×D₄×C₅₃	—	2	4	no	—
10	C₅₃×#431	C₂²	1	2	yes	—
11	C₅₃×#423	C₂²	1	2	yes	—
12	C₂×#100×C₅₃	—	2	1	no	—
13	C₂×C₂²⋊C₈×C₅₃	—	2	2	no	—
14	C₅₃×#397	C₄	1	4	yes	—
15	C₅₃×#391	—	2	1	no	—
16	C₅₃×#380	—	2	2	no	—
17	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#343	—	2	1	no	—
19	C₅₃×#332	—	2	4	no	—
20	C₂²×Q₈×C₅₃	—	2	1	no	—
21	C₂²×D₄×C₅₃	D₄	1	1	yes	characteristic
22	C₂³×C₄×C₅₃	D₄	1	2	yes	—
23	C₂×M₁₆×C₅₃	—	2	2	no	—
24	C₂²×C₈×C₅₃	—	2	4	no	—
25	C₄⋊Q₈×C₅₃	—	2	2	no	—
26	C₄⋊Q₈×C₅₃	—	2	2	no	—
27	#124×C₅₃	—	4	2	no	—
28	#124×C₅₃	—	4	2	no	—
29	C₄×D₄×C₅₃	—	2	2	no	—
30	C₄×D₄×C₅₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
33	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
34	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
35	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
36	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
37	C₂×C₄²×C₅₃	—	2	1	no	—
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₅₃	—	2	2	no	—
40	C₄⋊C₈×C₅₃	—	2	2	no	—
41	C₄⋊C₈×C₅₃	D₄	1	4	yes	—
42	#100×C₅₃	—	4	2	no	—
43	C₂²⋊C₈×C₅₃	—	2	4	no	—
44	C₂²⋊C₈×C₅₃	—	2	2	no	—
45	C₄×C₈×C₅₃	—	2	2	no	—
46	#95×C₅₃	—	2	4	no	—
47	#95×C₅₃	C₂×C₄	1	2	yes	—
48	C₂⁴×C₅₃	C₂²⋊C₄	1	2	yes	—
49	C₂×Q₈×C₅₃	—	4	2	no	—
50	C₂×Q₈×C₅₃	—	4	1	no	—
51	C₂×Q₈×C₅₃	—	4	1	no	—
52	C₂×Q₈×C₅₃	—	4	2	no	—
53	C₂×D₄×C₅₃	—	2	2	no	—
54	C₂×D₄×C₅₃	—	2	2	no	—
55	C₂×D₄×C₅₃	D₈	1	2	yes	—
56	C₂×D₄×C₅₃	SD₁₆	1	2	yes	—
57	C₂²×C₄×C₅₃	—	2	2	no	—
58	C₂²×C₄×C₅₃	—	2	2	no	—
59	C₂²×C₄×C₅₃	—	2	2	no	—
60	C₂²×C₄×C₅₃	—	2	2	no	—
61	C₂²×C₄×C₅₃	—	4	4	no	—
62	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	M₁₆×C₅₃	—	4	2	no	—
67	M₁₆×C₅₃	—	4	2	no	—
68	C₂×C₈×C₅₃	—	2	4	no	—
69	C₂×C₈×C₅₃	—	4	2	no	—
70	C₂×C₈×C₅₃	—	8	4	no	—
71	C₄⋊C₄×C₅₃	—	2	2	no	—
72	C₄⋊C₄×C₅₃	—	2	4	no	—
73	C₄⋊C₄×C₅₃	—	4	2	no	—
74	C₄⋊C₄×C₅₃	—	4	1	no	—
75	C₄⋊C₄×C₅₃	—	4	1	no	—
76	C₂²⋊C₄×C₅₃	—	2	4	no	—
77	C₂²⋊C₄×C₅₃	—	4	8	no	—
78	C₄²×C₅₃	—	4	1	no	—
79	C₄²×C₅₃	—	4	1	no	—
80	C₄²×C₅₃	C₂×D₄	1	2	yes	—
81	C₄²×C₅₃	#41	1	2	yes	—
82	C₂³×C₅₃	—	2	2	no	—
83	C₂³×C₅₃	—	2	2	no	—
84	C₂³×C₅₃	—	4	2	no	—
85	C₂³×C₅₃	—	4	2	no	—
86	C₂³×C₅₃	#102	1	4	yes	—
87	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
88	Q₈×C₅₃	—	8	2	no	—
89	Q₈×C₅₃	—	8	2	no	—
90	D₄×C₅₃	—	2	4	no	—
91	D₄×C₅₃	—	2	4	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	4	1	no	—
95	C₂×C₄×C₅₃	—	4	4	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	4	2	no	—
98	C₂×C₄×C₅₃	—	4	2	no	—
99	C₂×C₄×C₅₃	—	4	2	no	—
100	C₂×C₄×C₅₃	—	4	2	no	—
101	C₂×C₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	—	8	2	no	—
104	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
105	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#121	1	2	yes	—
107	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
108	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
109	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
110	C₂×C₄×C₅₃	#135	1	2	yes	—
111	C₈×C₅₃	—	4	4	no	—
112	C₈×C₅₃	—	8	2	no	—
113	#9393	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
114	C₂²×C₅₃	—	2	1	no	—
115	C₂²×C₅₃	—	2	1	no	—
116	C₂²×C₅₃	—	2	4	no	—
117	C₂²×C₅₃	—	4	4	no	—
118	C₂²×C₅₃	—	4	2	no	—
119	C₂²×C₅₃	—	4	2	no	—
120	C₂²×C₅₃	—	8	2	no	—
121	C₂²×C₅₃	#386	1	1	yes	characteristic
122	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
123	C₂²×C₅₃	#416	1	1	yes	characteristic
124	C₄×C₅₃	—	2	1	no	—
125	C₄×C₅₃	—	2	2	no	—
126	C₄×C₅₃	—	4	2	no	—
127	C₄×C₅₃	—	8	2	no	—
128	C₄×C₅₃	—	8	2	no	—
129	C₄×C₅₃	#455	1	2	yes	—
130	C₄×C₅₃	#465	1	2	yes	—
131	C₄×C₅₃	#466	1	2	yes	—
132	#2972	C₂×C₅₃	1	1	yes	characteristic
133	#2547	C₂×C₅₃	1	1	yes	characteristic
134	#1800	C₂×C₅₃	1	1	yes	characteristic
135	#1546	C₂×C₅₃	1	2	yes	complemented
136	#1303	C₂×C₅₃	1	2	yes	complemented
137	C₂×C₅₃	—	2	1	no	—
138	C₂×C₅₃	—	2	1	no	—
139	C₂×C₅₃	—	4	4	no	—
140	C₂×C₅₃	#1885	1	1	yes	characteristic
141	C₂×C₅₃	#1893	1	1	yes	characteristic
142	C₂×C₅₃	#1915	1	1	yes	characteristic
143	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
144	C₂×#124	—	2	1	no	—
145	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
146	C₈×D₄	—	2	4	no	—
147	#431	C₂²×C₅₃	1	2	yes	—
148	#423	C₂²×C₅₃	1	2	yes	—
149	C₂×#100	—	2	1	no	—
150	C₂×C₂²⋊C₈	—	2	2	no	—
151	#397	C₄×C₅₃	1	4	yes	—
152	#391	—	2	1	no	—
153	#380	—	2	2	no	—
154	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
155	#343	—	2	1	no	—
156	#332	—	2	4	no	—
157	C₅₃	#9393	1	1	yes	characteristic, direct product factor, Sylow
158	C₂²×Q₈	—	2	1	no	—
159	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
160	C₂³×C₄	D₄×C₅₃	1	2	yes	—
161	C₂×M₁₆	—	2	2	no	—
162	C₂²×C₈	—	2	4	no	—
163	C₄⋊Q₈	—	2	2	no	—
164	C₄⋊Q₈	—	2	2	no	—
165	#124	—	4	2	no	—
166	#124	—	4	2	no	—
167	C₄×D₄	—	2	2	no	—
168	C₄×D₄	—	2	2	no	—
169	C₂×C₄⋊C₄	—	2	1	no	—
170	C₂×C₄⋊C₄	—	2	1	no	—
171	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
172	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
173	C₂×C₂²⋊C₄	—	2	4	no	—
174	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
175	C₂×C₄²	—	2	1	no	—
176	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
177	C₄⋊C₈	—	2	2	no	—
178	C₄⋊C₈	—	2	2	no	—
179	C₄⋊C₈	D₄×C₅₃	1	4	yes	—
180	#100	—	4	2	no	—
181	C₂²⋊C₈	—	2	4	no	—
182	C₂²⋊C₈	—	2	2	no	—
183	C₄×C₈	—	2	2	no	—
184	#95	—	2	4	no	—
185	#95	C₂×C₄×C₅₃	1	2	yes	—
186	C₂⁴	C₂²⋊C₄×C₅₃	1	2	yes	—
187	C₂×Q₈	—	4	2	no	—
188	C₂×Q₈	—	4	1	no	—
189	C₂×Q₈	—	4	1	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂×D₄	—	2	2	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	D₈×C₅₃	1	2	yes	—
194	C₂×D₄	SD₁₆×C₅₃	1	2	yes	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	—	2	2	no	—
197	C₂²×C₄	—	2	2	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	—	4	4	no	—
200	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
201	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
202	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
203	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
204	M₁₆	—	4	2	no	—
205	M₁₆	—	4	2	no	—
206	C₂×C₈	—	2	4	no	—
207	C₂×C₈	—	4	2	no	—
208	C₂×C₈	—	8	4	no	—
209	C₄⋊C₄	—	2	2	no	—
210	C₄⋊C₄	—	2	4	no	—
211	C₄⋊C₄	—	4	2	no	—
212	C₄⋊C₄	—	4	1	no	—
213	C₄⋊C₄	—	4	1	no	—
214	C₂²⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	—	4	8	no	—
216	C₄²	—	4	1	no	—
217	C₄²	—	4	1	no	—
218	C₄²	C₂×D₄×C₅₃	1	2	yes	—
219	C₄²	#41×C₅₃	1	2	yes	—
220	C₂³	—	2	2	no	—
221	C₂³	—	2	2	no	—
222	C₂³	—	4	2	no	—
223	C₂³	—	4	2	no	—
224	C₂³	#102×C₅₃	1	4	yes	—
225	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
226	Q₈	—	8	2	no	—
227	Q₈	—	8	2	no	—
228	D₄	—	2	4	no	—
229	D₄	—	2	4	no	—
230	C₂×C₄	—	2	2	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	4	1	no	—
233	C₂×C₄	—	4	4	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	2	no	—
236	C₂×C₄	—	4	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	2	no	—
241	C₂×C₄	—	8	2	no	—
242	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
243	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
244	C₂×C₄	#121×C₅₃	1	2	yes	—
245	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
246	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
247	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
248	C₂×C₄	#135×C₅₃	1	2	yes	—
249	C₈	—	4	4	no	—
250	C₈	—	8	2	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	2	1	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	4	4	no	—
255	C₂²	—	4	2	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	8	2	no	—
258	C₂²	C₅₃×#386	1	1	yes	characteristic
259	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
260	C₂²	C₅₃×#416	1	1	yes	characteristic
261	C₄	—	2	2	no	—
262	C₄	—	2	1	no	—
263	C₄	—	4	2	no	—
264	C₄	—	8	2	no	—
265	C₄	—	8	2	no	—
266	C₄	C₅₃×#455	1	2	yes	—
267	C₄	C₅₃×#465	1	2	yes	—
268	C₄	C₅₃×#466	1	2	yes	—
269	C₂	—	2	1	no	—
270	C₂	—	2	1	no	—
271	C₂	—	4	4	no	—
272	C₂	C₅₃×#1885	1	1	yes	characteristic
273	C₂	C₅₃×#1893	1	1	yes	characteristic
274	C₂	C₅₃×#1915	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2972

C₂

yes

maximal, characteristic

C₅₃×#2547

C₂

yes

maximal, characteristic

C₅₃×#1800

C₂

yes

maximal, characteristic

C₅₃×#1546

C₂

yes

maximal, complemented

C₅₃×#1303

C₂

yes

maximal, complemented

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₂×#124×C₅₃

—

C₂×C₄×D₄×C₅₃

C₂²

yes

characteristic

C₈×D₄×C₅₃

—

C₅₃×#431

C₂²

yes

—

C₅₃×#423

C₂²

yes

—

C₂×#100×C₅₃

—

C₂×C₂²⋊C₈×C₅₃

—

C₅₃×#397

C₄

yes

—

C₅₃×#391

—

C₅₃×#380

—

C₄×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₅₃×#343

—

C₅₃×#332

—

C₂²×Q₈×C₅₃

—

C₂²×D₄×C₅₃

D₄

yes

characteristic

C₂³×C₄×C₅₃

D₄

yes

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

#124×C₅₃

—

#124×C₅₃

—

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

characteristic

C₂×C₂²⋊C₄×C₅₃

—

C₂×C₂²⋊C₄×C₅₃

C₂×C₄

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

D₄

yes

—

#100×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

#95×C₅₃

—

#95×C₅₃

C₂×C₄

yes

—

C₂⁴×C₅₃

C₂²⋊C₄

yes

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

D₈

yes

—

C₂×D₄×C₅₃

SD₁₆

yes

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#102

yes

—

C₂³×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

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

—

100

C₂×C₄×C₅₃

—

101

C₂×C₄×C₅₃

—

102

C₂×C₄×C₅₃

—

103

C₂×C₄×C₅₃

—

104

C₂×C₄×C₅₃

C₄×D₄

yes

—

105

C₂×C₄×C₅₃

#120

yes

characteristic

106

C₂×C₄×C₅₃

#121

yes

—

107

C₂×C₄×C₅₃

#123

yes

characteristic

108

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

109

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

110

C₂×C₄×C₅₃

#135

yes

—

111

C₈×C₅₃

—

112

C₈×C₅₃

—

113

#9393

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

114

C₂²×C₅₃

—

115

C₂²×C₅₃

—

116

C₂²×C₅₃

—

117

C₂²×C₅₃

—

118

C₂²×C₅₃

—

119

C₂²×C₅₃

—

120

C₂²×C₅₃

—

121

C₂²×C₅₃

#386

yes

characteristic

122

C₂²×C₅₃

C₂×#102

yes

characteristic

123

C₂²×C₅₃

#416

yes

characteristic

124

C₄×C₅₃

—

125

C₄×C₅₃

—

126

C₄×C₅₃

—

127

C₄×C₅₃

—

128

C₄×C₅₃

—

129

C₄×C₅₃

#455

yes

—

130

C₄×C₅₃

#465

yes

—

131

C₄×C₅₃

#466

yes

—

132

#2972

C₂×C₅₃

yes

characteristic

133

#2547

C₂×C₅₃

yes

characteristic

134

#1800

C₂×C₅₃

yes

characteristic

135

#1546

C₂×C₅₃

yes

complemented

136

#1303

C₂×C₅₃

yes

complemented

137

C₂×C₅₃

—

138

C₂×C₅₃

—

139

C₂×C₅₃

—

140

C₂×C₅₃

#1885

yes

characteristic

141

C₂×C₅₃

#1893

yes

characteristic

142

C₂×C₅₃

#1915

yes

characteristic

143

C₂×C₄⋊Q₈

C₂²×C₅₃

yes

characteristic

144

C₂×#124

—

145

C₂×C₄×D₄

C₂²×C₅₃

yes

characteristic

146

C₈×D₄

—

147

#431

C₂²×C₅₃

yes

—

148

#423

C₂²×C₅₃

yes

—

149

C₂×#100

—

150

C₂×C₂²⋊C₈

—

151

#397

C₄×C₅₃

yes

—

152

#391

—

153

#380

—

154

C₄×C₄⋊C₄

C₂²×C₅₃

yes

characteristic

155

#343

—

156

#332

—

157

C₅₃

#9393

yes

characteristic, direct product factor, Sylow

158

C₂²×Q₈

—

159

C₂²×D₄

D₄×C₅₃

yes

characteristic

160

C₂³×C₄

D₄×C₅₃

yes

—

161

C₂×M₁₆

—

162

C₂²×C₈

—

163

C₄⋊Q₈

—

164

C₄⋊Q₈

—

165

#124

—

166

#124

—

167

C₄×D₄

—

168

C₄×D₄

—

169

C₂×C₄⋊C₄

—

170

C₂×C₄⋊C₄

—

171

C₂×C₄⋊C₄

C₂×C₄×C₅₃

yes

—

172

C₂×C₄⋊C₄

D₄×C₅₃

yes

characteristic

173

C₂×C₂²⋊C₄

—

174

C₂×C₂²⋊C₄

C₂×C₄×C₅₃

yes

—

175

C₂×C₄²

—

176

C₂×C₄²

C₂³×C₅₃

yes

characteristic

177

C₄⋊C₈

—

178

C₄⋊C₈

—

179

C₄⋊C₈

D₄×C₅₃

yes

—

180

#100

—

181

C₂²⋊C₈

—

182

C₂²⋊C₈

—

183

C₄×C₈

—

184

#95

—

185

#95

C₂×C₄×C₅₃

yes

—

186

C₂⁴

C₂²⋊C₄×C₅₃

yes

—

187

C₂×Q₈

—

188

C₂×Q₈

—

189

C₂×Q₈

—

190

C₂×Q₈

—

191

C₂×D₄

—

192

C₂×D₄

—

193

C₂×D₄

D₈×C₅₃

yes

—

194

C₂×D₄

SD₁₆×C₅₃

yes

—

195

C₂²×C₄

—

196

C₂²×C₄

—

197

C₂²×C₄

—

198

C₂²×C₄

—

199

C₂²×C₄

—

200

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

—

201

C₂²×C₄

C₂²×C₄×C₅₃

yes

characteristic

202

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

203

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

204

M₁₆

—

205

M₁₆

—

206

C₂×C₈

—

207

C₂×C₈

—

208

C₂×C₈

—

209

C₄⋊C₄

—

210

C₄⋊C₄

—

211

C₄⋊C₄

—

212

C₄⋊C₄

—

213

C₄⋊C₄

—

214

C₂²⋊C₄

—

215

C₂²⋊C₄

—

216

C₄²

—

217

C₄²

—

218

C₄²

C₂×D₄×C₅₃

yes

—

219

C₄²

#41×C₅₃

yes

—

220

C₂³

—

221

C₂³

—

222

C₂³

—

223

C₂³

—

224

C₂³

#102×C₅₃

yes

—

225

C₂³

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

226

Q₈

—

227

Q₈

—

228

D₄

—

229

D₄

—

230

C₂×C₄

—

231

C₂×C₄

—

232

C₂×C₄

—

233

C₂×C₄

—

234

C₂×C₄

—

235

C₂×C₄

—

236

C₂×C₄

—

237

C₂×C₄

—

238

C₂×C₄

—

239

C₂×C₄

—

240

C₂×C₄

—

241

C₂×C₄

—

242

C₂×C₄

C₄×D₄×C₅₃

yes

—

243

C₂×C₄

#120×C₅₃

yes

characteristic

244

C₂×C₄

#121×C₅₃

yes

—

245

C₂×C₄

#123×C₅₃

yes

characteristic

246

C₂×C₄

C₂×D₈×C₅₃

yes

characteristic

247

C₂×C₄

C₂×SD₁₆×C₅₃

yes

characteristic

248

C₂×C₄

#135×C₅₃

yes

—

249

C₈

—

250

C₈

—

251

C₂²

—

252

C₂²

—

253

C₂²

—

254

C₂²

—

255

C₂²

—

256

C₂²

—

257

C₂²

—

258

C₂²

C₅₃×#386

yes

characteristic

259

C₂²

C₂×#102×C₅₃

yes

characteristic

260

C₂²

C₅₃×#416

yes

characteristic

261

C₄

—

262

C₄

—

263

C₄

—

264

C₄

—

265

C₄

—

266

C₄

C₅₃×#455

yes

—

267

C₄

C₅₃×#465

yes

—

268

C₄

C₅₃×#466

yes

—

269

C₂

—

270

C₂

—

271

C₂

—

272

C₂

C₅₃×#1885

yes

characteristic

273

C₂

C₅₃×#1893

yes

characteristic

274

C₂

C₅₃×#1915

yes

characteristic

	subgroup	G/H
center, socle	C₂²×C₅₃	#386
commutator	C₂²×C₄	C₂²×C₄×C₅₃
Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂²×D₄×C₅₃	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#386

commutator

C₂²×C₄

C₂²×C₄×C₅₃

Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

prime radical

C₂²×D₄×C₅₃

D₄

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
4. C₅₃×#1546	C₂	semidirect
5. C₅₃×#1303	C₂	semidirect
113. #9393	C₅₃	direct (N × H)
135. #1546	C₂×C₅₃	semidirect
136. #1303	C₂×C₅₃	semidirect
157. C₅₃	#9393	direct (N × H)

H ≅ G/N

type

4. C₅₃×#1546

C₂

semidirect

5. C₅₃×#1303

C₂

semidirect

113. #9393

C₅₃

direct (N × H)

135. #1546

C₂×C₅₃

semidirect

136. #1303

C₂×C₅₃

semidirect

157. C₅₃

#9393

direct (N × H)

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

subgroup	index
C₅₃×#2972	2
C₅₃×#2547	2
C₅₃×#1800	2
C₅₃×#1546	2
C₅₃×#1303	2
#9393	53

subgroup

index

C₅₃×#2972

C₅₃×#2547

C₅₃×#1800

C₅₃×#1546

C₅₃×#1303

#9393

Subgroups — C₅₃×#9393 #27216905

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#9393 #27216905

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9393 #27216905