Group #27216903 — subgroups

Subgroups — C₅₃×#9391 #27216903

Order 13568.

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

normal subgroups only

This group has 538 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×#2563	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1800	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1548	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×C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
8	C₈×D₄×C₅₃	—	2	4	no	—
9	C₅₃×#431	C₂²	1	2	yes	—
10	C₅₃×#423	C₂²	1	2	yes	—
11	C₂×#100×C₅₃	—	2	1	no	—
12	C₂×C₂²⋊C₈×C₅₃	—	2	2	no	—
13	C₅₃×#393	C₄	1	4	yes	—
14	C₅₃×#390	—	2	1	no	—
15	C₅₃×#382	—	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×#333	—	2	4	no	—
20	C₂²×Q₈×C₅₃	C₂×C₄	1	2	yes	—
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	C₄×D₄×C₅₃	—	2	2	no	—
28	C₄×D₄×C₅₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
33	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
34	C₂×C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
35	C₂×C₄²×C₅₃	—	2	1	no	—
36	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
37	C₄⋊C₈×C₅₃	—	2	2	no	—
38	C₄⋊C₈×C₅₃	—	2	2	no	—
39	C₄⋊C₈×C₅₃	D₄	1	4	yes	—
40	#100×C₅₃	—	4	2	no	—
41	C₂²⋊C₈×C₅₃	—	2	4	no	—
42	C₂²⋊C₈×C₅₃	—	2	2	no	—
43	C₄×C₈×C₅₃	—	2	2	no	—
44	#95×C₅₃	—	2	4	no	—
45	#95×C₅₃	C₂×C₄	1	2	yes	—
46	C₂⁴×C₅₃	C₂²⋊C₄	1	2	yes	—
47	C₂×Q₈×C₅₃	—	2	4	no	—
48	C₂×Q₈×C₅₃	—	8	2	no	—
49	C₂×D₄×C₅₃	—	2	2	no	—
50	C₂×D₄×C₅₃	—	2	2	no	—
51	C₂×D₄×C₅₃	SD₁₆	1	2	yes	—
52	C₂×D₄×C₅₃	Q₁₆	1	2	yes	—
53	C₂²×C₄×C₅₃	—	2	2	no	—
54	C₂²×C₄×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	—	2	2	no	—
56	C₂²×C₄×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	—	4	4	no	—
58	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
59	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
62	M₁₆×C₅₃	—	4	2	no	—
63	M₁₆×C₅₃	—	4	2	no	—
64	C₂×C₈×C₅₃	—	2	4	no	—
65	C₂×C₈×C₅₃	—	4	2	no	—
66	C₂×C₈×C₅₃	—	8	4	no	—
67	C₄⋊C₄×C₅₃	—	2	2	no	—
68	C₄⋊C₄×C₅₃	—	4	2	no	—
69	C₄⋊C₄×C₅₃	—	4	1	no	—
70	C₄⋊C₄×C₅₃	—	4	2	no	—
71	C₄⋊C₄×C₅₃	—	4	1	no	—
72	C₂²⋊C₄×C₅₃	—	2	4	no	—
73	C₂²⋊C₄×C₅₃	—	8	4	no	—
74	C₄²×C₅₃	—	4	2	no	—
75	C₄²×C₅₃	C₂×D₄	1	2	yes	—
76	C₄²×C₅₃	#41	1	2	yes	—
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	#103	1	4	yes	—
82	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
83	Q₈×C₅₃	—	8	4	no	—
84	D₄×C₅₃	—	2	4	no	—
85	D₄×C₅₃	—	2	4	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	4	1	no	—
89	C₂×C₄×C₅₃	—	4	4	no	—
90	C₂×C₄×C₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	—	4	4	no	—
92	C₂×C₄×C₅₃	—	4	2	no	—
93	C₂×C₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	—	4	2	no	—
95	C₂×C₄×C₅₃	—	4	2	no	—
96	C₂×C₄×C₅₃	—	8	2	no	—
97	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
98	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#121	1	2	yes	—
100	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
103	C₂×C₄×C₅₃	#135	1	2	yes	—
104	C₈×C₅₃	—	4	4	no	—
105	C₈×C₅₃	—	8	2	no	—
106	#9391	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
107	C₂²×C₅₃	—	2	1	no	—
108	C₂²×C₅₃	—	2	1	no	—
109	C₂²×C₅₃	—	2	4	no	—
110	C₂²×C₅₃	—	4	4	no	—
111	C₂²×C₅₃	—	4	2	no	—
112	C₂²×C₅₃	—	4	2	no	—
113	C₂²×C₅₃	—	8	2	no	—
114	C₂²×C₅₃	#386	1	1	yes	characteristic
115	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
116	C₂²×C₅₃	#416	1	1	yes	characteristic
117	C₄×C₅₃	—	2	1	no	—
118	C₄×C₅₃	—	2	2	no	—
119	C₄×C₅₃	—	4	2	no	—
120	C₄×C₅₃	—	8	2	no	—
121	C₄×C₅₃	—	8	2	no	—
122	C₄×C₅₃	#455	1	2	yes	—
123	C₄×C₅₃	#465	1	2	yes	—
124	C₄×C₅₃	#467	1	2	yes	—
125	#2972	C₂×C₅₃	1	1	yes	characteristic
126	#2563	C₂×C₅₃	1	1	yes	characteristic
127	#1800	C₂×C₅₃	1	1	yes	characteristic
128	#1548	C₂×C₅₃	1	2	yes	complemented
129	#1303	C₂×C₅₃	1	2	yes	complemented
130	C₂×C₅₃	—	2	1	no	—
131	C₂×C₅₃	—	2	1	no	—
132	C₂×C₅₃	—	4	4	no	—
133	C₂×C₅₃	#1885	1	1	yes	characteristic
134	C₂×C₅₃	#1892	1	1	yes	characteristic
135	C₂×C₅₃	#1914	1	1	yes	characteristic
136	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
137	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
138	C₈×D₄	—	2	4	no	—
139	#431	C₂²×C₅₃	1	2	yes	—
140	#423	C₂²×C₅₃	1	2	yes	—
141	C₂×#100	—	2	1	no	—
142	C₂×C₂²⋊C₈	—	2	2	no	—
143	#393	C₄×C₅₃	1	4	yes	—
144	#390	—	2	1	no	—
145	#382	—	2	1	no	—
146	#380	—	2	2	no	—
147	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
148	#343	—	2	1	no	—
149	#333	—	2	4	no	—
150	C₅₃	#9391	1	1	yes	characteristic, direct product factor, Sylow
151	C₂²×Q₈	C₂×C₄×C₅₃	1	2	yes	—
152	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
153	C₂³×C₄	D₄×C₅₃	1	2	yes	—
154	C₂×M₁₆	—	2	2	no	—
155	C₂²×C₈	—	2	4	no	—
156	C₄⋊Q₈	—	2	2	no	—
157	C₄⋊Q₈	—	2	2	no	—
158	C₄×D₄	—	2	2	no	—
159	C₄×D₄	—	2	2	no	—
160	C₂×C₄⋊C₄	—	2	1	no	—
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	—	2	1	no	—
163	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
164	C₂×C₂²⋊C₄	—	2	4	no	—
165	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
166	C₂×C₄²	—	2	1	no	—
167	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
168	C₄⋊C₈	—	2	2	no	—
169	C₄⋊C₈	—	2	2	no	—
170	C₄⋊C₈	D₄×C₅₃	1	4	yes	—
171	#100	—	4	2	no	—
172	C₂²⋊C₈	—	2	4	no	—
173	C₂²⋊C₈	—	2	2	no	—
174	C₄×C₈	—	2	2	no	—
175	#95	—	2	4	no	—
176	#95	C₂×C₄×C₅₃	1	2	yes	—
177	C₂⁴	C₂²⋊C₄×C₅₃	1	2	yes	—
178	C₂×Q₈	—	2	4	no	—
179	C₂×Q₈	—	8	2	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	—	2	2	no	—
182	C₂×D₄	SD₁₆×C₅₃	1	2	yes	—
183	C₂×D₄	Q₁₆×C₅₃	1	2	yes	—
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	2	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	2	no	—
188	C₂²×C₄	—	4	4	no	—
189	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
190	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
191	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
192	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
193	M₁₆	—	4	2	no	—
194	M₁₆	—	4	2	no	—
195	C₂×C₈	—	2	4	no	—
196	C₂×C₈	—	4	2	no	—
197	C₂×C₈	—	8	4	no	—
198	C₄⋊C₄	—	2	2	no	—
199	C₄⋊C₄	—	4	2	no	—
200	C₄⋊C₄	—	4	1	no	—
201	C₄⋊C₄	—	4	2	no	—
202	C₄⋊C₄	—	4	1	no	—
203	C₂²⋊C₄	—	2	4	no	—
204	C₂²⋊C₄	—	8	4	no	—
205	C₄²	—	4	2	no	—
206	C₄²	C₂×D₄×C₅₃	1	2	yes	—
207	C₄²	#41×C₅₃	1	2	yes	—
208	C₂³	—	2	2	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	4	2	no	—
211	C₂³	—	4	2	no	—
212	C₂³	#103×C₅₃	1	4	yes	—
213	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
214	Q₈	—	8	4	no	—
215	D₄	—	2	4	no	—
216	D₄	—	2	4	no	—
217	C₂×C₄	—	2	2	no	—
218	C₂×C₄	—	2	2	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	—	4	4	no	—
221	C₂×C₄	—	4	2	no	—
222	C₂×C₄	—	4	2	no	—
223	C₂×C₄	—	4	4	no	—
224	C₂×C₄	—	4	2	no	—
225	C₂×C₄	—	4	2	no	—
226	C₂×C₄	—	4	2	no	—
227	C₂×C₄	—	8	2	no	—
228	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
229	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
230	C₂×C₄	#121×C₅₃	1	2	yes	—
231	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
232	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
233	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
234	C₂×C₄	#135×C₅₃	1	2	yes	—
235	C₈	—	4	4	no	—
236	C₈	—	8	2	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	2	1	no	—
239	C₂²	—	2	4	no	—
240	C₂²	—	4	4	no	—
241	C₂²	—	4	2	no	—
242	C₂²	—	4	2	no	—
243	C₂²	—	8	2	no	—
244	C₂²	C₅₃×#386	1	1	yes	characteristic
245	C₂²	C₂×#103×C₅₃	1	1	yes	characteristic
246	C₂²	C₅₃×#416	1	1	yes	characteristic
247	C₄	—	2	2	no	—
248	C₄	—	2	1	no	—
249	C₄	—	4	2	no	—
250	C₄	—	8	2	no	—
251	C₄	—	8	2	no	—
252	C₄	C₅₃×#455	1	2	yes	—
253	C₄	C₅₃×#465	1	2	yes	—
254	C₄	C₅₃×#467	1	2	yes	—
255	C₂	—	2	1	no	—
256	C₂	—	2	1	no	—
257	C₂	—	4	4	no	—
258	C₂	C₅₃×#1885	1	1	yes	characteristic
259	C₂	C₅₃×#1892	1	1	yes	characteristic
260	C₂	C₅₃×#1914	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2972

C₂

yes

maximal, characteristic

C₅₃×#2563

C₂

yes

maximal, characteristic

C₅₃×#1800

C₂

yes

maximal, characteristic

C₅₃×#1548

C₂

yes

maximal, complemented

C₅₃×#1303

C₂

yes

maximal, complemented

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

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×#393

C₄

yes

—

C₅₃×#390

—

C₅₃×#382

—

C₅₃×#380

—

C₄×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₅₃×#343

—

C₅₃×#333

—

C₂²×Q₈×C₅₃

C₂×C₄

yes

—

C₂²×D₄×C₅₃

D₄

yes

characteristic

C₂³×C₄×C₅₃

D₄

yes

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

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×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

SD₁₆

yes

—

C₂×D₄×C₅₃

Q₁₆

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

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#103

yes

—

C₂³×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

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

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₄×D₄

yes

—

C₂×C₄×C₅₃

#120

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

100

C₂×C₄×C₅₃

#123

yes

characteristic

101

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

102

C₂×C₄×C₅₃

C₂×Q₁₆

yes

characteristic

103

C₂×C₄×C₅₃

#135

yes

—

104

C₈×C₅₃

—

105

C₈×C₅₃

—

106

#9391

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

107

C₂²×C₅₃

—

108

C₂²×C₅₃

—

109

C₂²×C₅₃

—

110

C₂²×C₅₃

—

111

C₂²×C₅₃

—

112

C₂²×C₅₃

—

113

C₂²×C₅₃

—

114

C₂²×C₅₃

#386

yes

characteristic

115

C₂²×C₅₃

C₂×#103

yes

characteristic

116

C₂²×C₅₃

#416

yes

characteristic

117

C₄×C₅₃

—

118

C₄×C₅₃

—

119

C₄×C₅₃

—

120

C₄×C₅₃

—

121

C₄×C₅₃

—

122

C₄×C₅₃

#455

yes

—

123

C₄×C₅₃

#465

yes

—

124

C₄×C₅₃

#467

yes

—

125

#2972

C₂×C₅₃

yes

characteristic

126

#2563

C₂×C₅₃

yes

characteristic

127

#1800

C₂×C₅₃

yes

characteristic

128

#1548

C₂×C₅₃

yes

complemented

129

#1303

C₂×C₅₃

yes

complemented

130

C₂×C₅₃

—

131

C₂×C₅₃

—

132

C₂×C₅₃

—

133

C₂×C₅₃

#1885

yes

characteristic

134

C₂×C₅₃

#1892

yes

characteristic

135

C₂×C₅₃

#1914

yes

characteristic

136

C₂×C₄⋊Q₈

C₂²×C₅₃

yes

characteristic

137

C₂×C₄×D₄

C₂²×C₅₃

yes

characteristic

138

C₈×D₄

—

139

#431

C₂²×C₅₃

yes

—

140

#423

C₂²×C₅₃

yes

—

141

C₂×#100

—

142

C₂×C₂²⋊C₈

—

143

#393

C₄×C₅₃

yes

—

144

#390

—

145

#382

—

146

#380

—

147

C₄×C₄⋊C₄

C₂²×C₅₃

yes

characteristic

148

#343

—

149

#333

—

150

C₅₃

#9391

yes

characteristic, direct product factor, Sylow

151

C₂²×Q₈

C₂×C₄×C₅₃

yes

—

152

C₂²×D₄

D₄×C₅₃

yes

characteristic

153

C₂³×C₄

D₄×C₅₃

yes

—

154

C₂×M₁₆

—

155

C₂²×C₈

—

156

C₄⋊Q₈

—

157

C₄⋊Q₈

—

158

C₄×D₄

—

159

C₄×D₄

—

160

C₂×C₄⋊C₄

—

161

C₂×C₄⋊C₄

—

162

C₂×C₄⋊C₄

—

163

C₂×C₄⋊C₄

D₄×C₅₃

yes

characteristic

164

C₂×C₂²⋊C₄

—

165

C₂×C₂²⋊C₄

C₂×C₄×C₅₃

yes

—

166

C₂×C₄²

—

167

C₂×C₄²

C₂³×C₅₃

yes

characteristic

168

C₄⋊C₈

—

169

C₄⋊C₈

—

170

C₄⋊C₈

D₄×C₅₃

yes

—

171

#100

—

172

C₂²⋊C₈

—

173

C₂²⋊C₈

—

174

C₄×C₈

—

175

#95

—

176

#95

C₂×C₄×C₅₃

yes

—

177

C₂⁴

C₂²⋊C₄×C₅₃

yes

—

178

C₂×Q₈

—

179

C₂×Q₈

—

180

C₂×D₄

—

181

C₂×D₄

—

182

C₂×D₄

SD₁₆×C₅₃

yes

—

183

C₂×D₄

Q₁₆×C₅₃

yes

—

184

C₂²×C₄

—

185

C₂²×C₄

—

186

C₂²×C₄

—

187

C₂²×C₄

—

188

C₂²×C₄

—

189

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

—

190

C₂²×C₄

C₂²×C₄×C₅₃

yes

characteristic

191

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

192

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

193

M₁₆

—

194

M₁₆

—

195

C₂×C₈

—

196

C₂×C₈

—

197

C₂×C₈

—

198

C₄⋊C₄

—

199

C₄⋊C₄

—

200

C₄⋊C₄

—

201

C₄⋊C₄

—

202

C₄⋊C₄

—

203

C₂²⋊C₄

—

204

C₂²⋊C₄

—

205

C₄²

—

206

C₄²

C₂×D₄×C₅₃

yes

—

207

C₄²

#41×C₅₃

yes

—

208

C₂³

—

209

C₂³

—

210

C₂³

—

211

C₂³

—

212

C₂³

#103×C₅₃

yes

—

213

C₂³

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

214

Q₈

—

215

D₄

—

216

D₄

—

217

C₂×C₄

—

218

C₂×C₄

—

219

C₂×C₄

—

220

C₂×C₄

—

221

C₂×C₄

—

222

C₂×C₄

—

223

C₂×C₄

—

224

C₂×C₄

—

225

C₂×C₄

—

226

C₂×C₄

—

227

C₂×C₄

—

228

C₂×C₄

C₄×D₄×C₅₃

yes

—

229

C₂×C₄

#120×C₅₃

yes

characteristic

230

C₂×C₄

#121×C₅₃

yes

—

231

C₂×C₄

#123×C₅₃

yes

characteristic

232

C₂×C₄

C₂×SD₁₆×C₅₃

yes

characteristic

233

C₂×C₄

C₂×Q₁₆×C₅₃

yes

characteristic

234

C₂×C₄

#135×C₅₃

yes

—

235

C₈

—

236

C₈

—

237

C₂²

—

238

C₂²

—

239

C₂²

—

240

C₂²

—

241

C₂²

—

242

C₂²

—

243

C₂²

—

244

C₂²

C₅₃×#386

yes

characteristic

245

C₂²

C₂×#103×C₅₃

yes

characteristic

246

C₂²

C₅₃×#416

yes

characteristic

247

C₄

—

248

C₄

—

249

C₄

—

250

C₄

—

251

C₄

—

252

C₄

C₅₃×#455

yes

—

253

C₄

C₅₃×#465

yes

—

254

C₄

C₅₃×#467

yes

—

255

C₂

—

256

C₂

—

257

C₂

—

258

C₂

C₅₃×#1885

yes

characteristic

259

C₂

C₅₃×#1892

yes

characteristic

260

C₂

C₅₃×#1914

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×#1548	C₂	semidirect
5. C₅₃×#1303	C₂	semidirect
106. #9391	C₅₃	direct (N × H)
128. #1548	C₂×C₅₃	semidirect
129. #1303	C₂×C₅₃	semidirect
150. C₅₃	#9391	direct (N × H)

H ≅ G/N

type

4. C₅₃×#1548

C₂

semidirect

5. C₅₃×#1303

C₂

semidirect

106. #9391

C₅₃

direct (N × H)

128. #1548

C₂×C₅₃

semidirect

129. #1303

C₂×C₅₃

semidirect

150. C₅₃

#9391

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×#2563	2
C₅₃×#1800	2
C₅₃×#1548	2
C₅₃×#1303	2
#9391	53

subgroup

index

C₅₃×#2972

C₅₃×#2563

C₅₃×#1800

C₅₃×#1548

C₅₃×#1303

#9391

Subgroups — C₅₃×#9391 #27216903

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#9391 #27216903

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9391 #27216903