Group #27216907 — subgroups

Subgroups — C₅₃×#9395 #27216907

Order 13568.

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

normal subgroups only

This group has 594 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×#2542	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1800	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1549	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×C₂²⋊Q₈×C₅₃	—	2	2	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×#399	C₄	1	4	yes	—
15	C₅₃×#390	—	2	1	no	—
16	C₅₃×#384	—	2	1	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	C₂²⋊Q₈×C₅₃	—	4	8	no	—
28	C₄×D₄×C₅₃	—	2	2	no	—
29	C₄×D₄×C₅₃	—	2	2	no	—
30	C₂×C₄⋊C₄×C₅₃	—	2	4	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	C₂×C₄	1	2	yes	—
47	C₂⁴×C₅₃	C₂²⋊C₄	1	2	yes	—
48	C₂×Q₈×C₅₃	—	4	4	no	—
49	C₂×Q₈×C₅₃	—	4	1	no	—
50	C₂×Q₈×C₅₃	—	4	1	no	—
51	C₂×D₄×C₅₃	—	2	2	no	—
52	C₂×D₄×C₅₃	—	2	2	no	—
53	C₂×D₄×C₅₃	SD₁₆	1	2	yes	—
54	C₂×D₄×C₅₃	Q₁₆	1	2	yes	—
55	C₂²×C₄×C₅₃	—	2	2	no	—
56	C₂²×C₄×C₅₃	—	2	2	no	—
57	C₂²×C₄×C₅₃	—	2	2	no	—
58	C₂²×C₄×C₅₃	—	2	2	no	—
59	C₂²×C₄×C₅₃	—	4	4	no	—
60	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
64	M₁₆×C₅₃	—	4	2	no	—
65	M₁₆×C₅₃	—	4	2	no	—
66	C₂×C₈×C₅₃	—	2	4	no	—
67	C₂×C₈×C₅₃	—	4	2	no	—
68	C₂×C₈×C₅₃	—	8	4	no	—
69	C₄⋊C₄×C₅₃	—	2	4	no	—
70	C₄⋊C₄×C₅₃	—	2	2	no	—
71	C₄⋊C₄×C₅₃	—	2	4	no	—
72	C₄⋊C₄×C₅₃	—	4	8	no	—
73	C₄⋊C₄×C₅₃	—	4	2	no	—
74	C₂²⋊C₄×C₅₃	—	2	4	no	—
75	C₂²⋊C₄×C₅₃	—	4	8	no	—
76	C₄²×C₅₃	—	4	2	no	—
77	C₄²×C₅₃	C₂×D₄	1	2	yes	—
78	C₄²×C₅₃	#41	1	2	yes	—
79	C₂³×C₅₃	—	2	2	no	—
80	C₂³×C₅₃	—	2	2	no	—
81	C₂³×C₅₃	—	4	2	no	—
82	C₂³×C₅₃	—	4	2	no	—
83	C₂³×C₅₃	#103	1	4	yes	—
84	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	Q₈×C₅₃	—	8	2	no	—
86	Q₈×C₅₃	—	8	2	no	—
87	D₄×C₅₃	—	2	4	no	—
88	D₄×C₅₃	—	2	4	no	—
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	—	2	4	no	—
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	—	4	1	no	—
93	C₂×C₄×C₅₃	—	4	2	no	—
94	C₂×C₄×C₅₃	—	4	2	no	—
95	C₂×C₄×C₅₃	—	4	4	no	—
96	C₂×C₄×C₅₃	—	4	2	no	—
97	C₂×C₄×C₅₃	—	4	4	no	—
98	C₂×C₄×C₅₃	—	4	2	no	—
99	C₂×C₄×C₅₃	—	8	2	no	—
100	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
101	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
102	C₂×C₄×C₅₃	#121	1	2	yes	—
103	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
104	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
105	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
106	C₂×C₄×C₅₃	#135	1	2	yes	—
107	C₈×C₅₃	—	4	4	no	—
108	C₈×C₅₃	—	8	2	no	—
109	#9395	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
110	C₂²×C₅₃	—	2	1	no	—
111	C₂²×C₅₃	—	2	1	no	—
112	C₂²×C₅₃	—	2	4	no	—
113	C₂²×C₅₃	—	4	4	no	—
114	C₂²×C₅₃	—	4	2	no	—
115	C₂²×C₅₃	—	4	2	no	—
116	C₂²×C₅₃	—	8	2	no	—
117	C₂²×C₅₃	#386	1	1	yes	characteristic
118	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
119	C₂²×C₅₃	#416	1	1	yes	characteristic
120	C₄×C₅₃	—	2	1	no	—
121	C₄×C₅₃	—	2	2	no	—
122	C₄×C₅₃	—	4	4	no	—
123	C₄×C₅₃	—	4	2	no	—
124	C₄×C₅₃	—	8	2	no	—
125	C₄×C₅₃	#453	1	2	yes	—
126	C₄×C₅₃	#465	1	2	yes	—
127	C₄×C₅₃	#467	1	2	yes	—
128	#2972	C₂×C₅₃	1	1	yes	characteristic
129	#2542	C₂×C₅₃	1	1	yes	characteristic
130	#1800	C₂×C₅₃	1	1	yes	characteristic
131	#1549	C₂×C₅₃	1	2	yes	complemented
132	#1303	C₂×C₅₃	1	2	yes	complemented
133	C₂×C₅₃	—	2	1	no	—
134	C₂×C₅₃	—	2	1	no	—
135	C₂×C₅₃	—	4	4	no	—
136	C₂×C₅₃	#1887	1	1	yes	characteristic
137	C₂×C₅₃	#1892	1	1	yes	characteristic
138	C₂×C₅₃	#1915	1	1	yes	characteristic
139	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
140	C₂×C₂²⋊Q₈	—	2	2	no	—
141	C₂×C₄×D₄	C₂²×C₅₃	1	1	yes	characteristic
142	C₈×D₄	—	2	4	no	—
143	#431	C₂²×C₅₃	1	2	yes	—
144	#423	C₂²×C₅₃	1	2	yes	—
145	C₂×#100	—	2	1	no	—
146	C₂×C₂²⋊C₈	—	2	2	no	—
147	#399	C₄×C₅₃	1	4	yes	—
148	#390	—	2	1	no	—
149	#384	—	2	1	no	—
150	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
151	#343	—	2	1	no	—
152	#332	—	2	4	no	—
153	C₅₃	#9395	1	1	yes	characteristic, direct product factor, Sylow
154	C₂²×Q₈	—	2	1	no	—
155	C₂²×D₄	D₄×C₅₃	1	1	yes	characteristic
156	C₂³×C₄	D₄×C₅₃	1	2	yes	—
157	C₂×M₁₆	—	2	2	no	—
158	C₂²×C₈	—	2	4	no	—
159	C₄⋊Q₈	—	2	2	no	—
160	C₄⋊Q₈	—	2	2	no	—
161	C₂²⋊Q₈	—	4	8	no	—
162	C₄×D₄	—	2	2	no	—
163	C₄×D₄	—	2	2	no	—
164	C₂×C₄⋊C₄	—	2	1	no	—
165	C₂×C₄⋊C₄	—	2	1	no	—
166	C₂×C₄⋊C₄	—	2	4	no	—
167	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
168	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
169	C₂×C₂²⋊C₄	—	2	4	no	—
170	C₂×C₂²⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
171	C₂×C₄²	—	2	1	no	—
172	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
173	C₄⋊C₈	—	2	2	no	—
174	C₄⋊C₈	—	2	2	no	—
175	C₄⋊C₈	D₄×C₅₃	1	4	yes	—
176	#100	—	4	2	no	—
177	C₂²⋊C₈	—	2	4	no	—
178	C₂²⋊C₈	—	2	2	no	—
179	C₄×C₈	—	2	2	no	—
180	#95	C₂×C₄×C₅₃	1	2	yes	—
181	C₂⁴	C₂²⋊C₄×C₅₃	1	2	yes	—
182	C₂×Q₈	—	4	4	no	—
183	C₂×Q₈	—	4	1	no	—
184	C₂×Q₈	—	4	1	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	—	2	2	no	—
187	C₂×D₄	SD₁₆×C₅₃	1	2	yes	—
188	C₂×D₄	Q₁₆×C₅₃	1	2	yes	—
189	C₂²×C₄	—	2	2	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	4	4	no	—
194	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
195	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
196	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
198	M₁₆	—	4	2	no	—
199	M₁₆	—	4	2	no	—
200	C₂×C₈	—	2	4	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	8	4	no	—
203	C₄⋊C₄	—	2	4	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	4	8	no	—
207	C₄⋊C₄	—	4	2	no	—
208	C₂²⋊C₄	—	2	4	no	—
209	C₂²⋊C₄	—	4	8	no	—
210	C₄²	—	4	2	no	—
211	C₄²	C₂×D₄×C₅₃	1	2	yes	—
212	C₄²	#41×C₅₃	1	2	yes	—
213	C₂³	—	2	2	no	—
214	C₂³	—	2	2	no	—
215	C₂³	—	4	2	no	—
216	C₂³	—	4	2	no	—
217	C₂³	#103×C₅₃	1	4	yes	—
218	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
219	Q₈	—	8	2	no	—
220	Q₈	—	8	2	no	—
221	D₄	—	2	4	no	—
222	D₄	—	2	4	no	—
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	4	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	2	no	—
231	C₂×C₄	—	4	4	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	—	8	2	no	—
234	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
235	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
236	C₂×C₄	#121×C₅₃	1	2	yes	—
237	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
238	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
239	C₂×C₄	C₂×Q₁₆×C₅₃	1	1	yes	characteristic
240	C₂×C₄	#135×C₅₃	1	2	yes	—
241	C₈	—	4	4	no	—
242	C₈	—	8	2	no	—
243	C₂²	—	2	1	no	—
244	C₂²	—	2	1	no	—
245	C₂²	—	2	4	no	—
246	C₂²	—	4	4	no	—
247	C₂²	—	4	2	no	—
248	C₂²	—	4	2	no	—
249	C₂²	—	8	2	no	—
250	C₂²	C₅₃×#386	1	1	yes	characteristic
251	C₂²	C₂×#103×C₅₃	1	1	yes	characteristic
252	C₂²	C₅₃×#416	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	2	1	no	—
255	C₄	—	4	4	no	—
256	C₄	—	4	2	no	—
257	C₄	—	8	2	no	—
258	C₄	C₅₃×#453	1	2	yes	—
259	C₄	C₅₃×#465	1	2	yes	—
260	C₄	C₅₃×#467	1	2	yes	—
261	C₂	—	2	1	no	—
262	C₂	—	2	1	no	—
263	C₂	—	4	4	no	—
264	C₂	C₅₃×#1887	1	1	yes	characteristic
265	C₂	C₅₃×#1892	1	1	yes	characteristic
266	C₂	C₅₃×#1915	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2972

C₂

yes

maximal, characteristic

C₅₃×#2542

C₂

yes

maximal, characteristic

C₅₃×#1800

C₂

yes

maximal, characteristic

C₅₃×#1549

C₂

yes

maximal, complemented

C₅₃×#1303

C₂

yes

maximal, complemented

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₂×C₂²⋊Q₈×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×#399

C₄

yes

—

C₅₃×#390

—

C₅₃×#384

—

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

—

C₂²⋊Q₈×C₅₃

—

C₄×D₄×C₅₃

—

C₄×D₄×C₅₃

—

C₂×C₄⋊C₄×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

C₂×C₄

yes

—

C₂⁴×C₅₃

C₂²⋊C₄

yes

—

C₂×Q₈×C₅₃

—

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

—

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

—

100

C₂×C₄×C₅₃

C₄×D₄

yes

—

101

C₂×C₄×C₅₃

#120

yes

characteristic

102

C₂×C₄×C₅₃

#121

yes

—

103

C₂×C₄×C₅₃

#123

yes

characteristic

104

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

105

C₂×C₄×C₅₃

C₂×Q₁₆

yes

characteristic

106

C₂×C₄×C₅₃

#135

yes

—

107

C₈×C₅₃

—

108

C₈×C₅₃

—

109

#9395

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

110

C₂²×C₅₃

—

111

C₂²×C₅₃

—

112

C₂²×C₅₃

—

113

C₂²×C₅₃

—

114

C₂²×C₅₃

—

115

C₂²×C₅₃

—

116

C₂²×C₅₃

—

117

C₂²×C₅₃

#386

yes

characteristic

118

C₂²×C₅₃

C₂×#103

yes

characteristic

119

C₂²×C₅₃

#416

yes

characteristic

120

C₄×C₅₃

—

121

C₄×C₅₃

—

122

C₄×C₅₃

—

123

C₄×C₅₃

—

124

C₄×C₅₃

—

125

C₄×C₅₃

#453

yes

—

126

C₄×C₅₃

#465

yes

—

127

C₄×C₅₃

#467

yes

—

128

#2972

C₂×C₅₃

yes

characteristic

129

#2542

C₂×C₅₃

yes

characteristic

130

#1800

C₂×C₅₃

yes

characteristic

131

#1549

C₂×C₅₃

yes

complemented

132

#1303

C₂×C₅₃

yes

complemented

133

C₂×C₅₃

—

134

C₂×C₅₃

—

135

C₂×C₅₃

—

136

C₂×C₅₃

#1887

yes

characteristic

137

C₂×C₅₃

#1892

yes

characteristic

138

C₂×C₅₃

#1915

yes

characteristic

139

C₂×C₄⋊Q₈

C₂²×C₅₃

yes

characteristic

140

C₂×C₂²⋊Q₈

—

141

C₂×C₄×D₄

C₂²×C₅₃

yes

characteristic

142

C₈×D₄

—

143

#431

C₂²×C₅₃

yes

—

144

#423

C₂²×C₅₃

yes

—

145

C₂×#100

—

146

C₂×C₂²⋊C₈

—

147

#399

C₄×C₅₃

yes

—

148

#390

—

149

#384

—

150

C₄×C₄⋊C₄

C₂²×C₅₃

yes

characteristic

151

#343

—

152

#332

—

153

C₅₃

#9395

yes

characteristic, direct product factor, Sylow

154

C₂²×Q₈

—

155

C₂²×D₄

D₄×C₅₃

yes

characteristic

156

C₂³×C₄

D₄×C₅₃

yes

—

157

C₂×M₁₆

—

158

C₂²×C₈

—

159

C₄⋊Q₈

—

160

C₄⋊Q₈

—

161

C₂²⋊Q₈

—

162

C₄×D₄

—

163

C₄×D₄

—

164

C₂×C₄⋊C₄

—

165

C₂×C₄⋊C₄

—

166

C₂×C₄⋊C₄

—

167

C₂×C₄⋊C₄

C₂×C₄×C₅₃

yes

—

168

C₂×C₄⋊C₄

D₄×C₅₃

yes

characteristic

169

C₂×C₂²⋊C₄

—

170

C₂×C₂²⋊C₄

C₂×C₄×C₅₃

yes

—

171

C₂×C₄²

—

172

C₂×C₄²

C₂³×C₅₃

yes

characteristic

173

C₄⋊C₈

—

174

C₄⋊C₈

—

175

C₄⋊C₈

D₄×C₅₃

yes

—

176

#100

—

177

C₂²⋊C₈

—

178

C₂²⋊C₈

—

179

C₄×C₈

—

180

#95

C₂×C₄×C₅₃

yes

—

181

C₂⁴

C₂²⋊C₄×C₅₃

yes

—

182

C₂×Q₈

—

183

C₂×Q₈

—

184

C₂×Q₈

—

185

C₂×D₄

—

186

C₂×D₄

—

187

C₂×D₄

SD₁₆×C₅₃

yes

—

188

C₂×D₄

Q₁₆×C₅₃

yes

—

189

C₂²×C₄

—

190

C₂²×C₄

—

191

C₂²×C₄

—

192

C₂²×C₄

—

193

C₂²×C₄

—

194

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

—

195

C₂²×C₄

C₂²×C₄×C₅₃

yes

characteristic

196

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

197

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

198

M₁₆

—

199

M₁₆

—

200

C₂×C₈

—

201

C₂×C₈

—

202

C₂×C₈

—

203

C₄⋊C₄

—

204

C₄⋊C₄

—

205

C₄⋊C₄

—

206

C₄⋊C₄

—

207

C₄⋊C₄

—

208

C₂²⋊C₄

—

209

C₂²⋊C₄

—

210

C₄²

—

211

C₄²

C₂×D₄×C₅₃

yes

—

212

C₄²

#41×C₅₃

yes

—

213

C₂³

—

214

C₂³

—

215

C₂³

—

216

C₂³

—

217

C₂³

#103×C₅₃

yes

—

218

C₂³

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

219

Q₈

—

220

Q₈

—

221

D₄

—

222

D₄

—

223

C₂×C₄

—

224

C₂×C₄

—

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

—

234

C₂×C₄

C₄×D₄×C₅₃

yes

—

235

C₂×C₄

#120×C₅₃

yes

characteristic

236

C₂×C₄

#121×C₅₃

yes

—

237

C₂×C₄

#123×C₅₃

yes

characteristic

238

C₂×C₄

C₂×SD₁₆×C₅₃

yes

characteristic

239

C₂×C₄

C₂×Q₁₆×C₅₃

yes

characteristic

240

C₂×C₄

#135×C₅₃

yes

—

241

C₈

—

242

C₈

—

243

C₂²

—

244

C₂²

—

245

C₂²

—

246

C₂²

—

247

C₂²

—

248

C₂²

—

249

C₂²

—

250

C₂²

C₅₃×#386

yes

characteristic

251

C₂²

C₂×#103×C₅₃

yes

characteristic

252

C₂²

C₅₃×#416

yes

characteristic

253

C₄

—

254

C₄

—

255

C₄

—

256

C₄

—

257

C₄

—

258

C₄

C₅₃×#453

yes

—

259

C₄

C₅₃×#465

yes

—

260

C₄

C₅₃×#467

yes

—

261

C₂

—

262

C₂

—

263

C₂

—

264

C₂

C₅₃×#1887

yes

characteristic

265

C₂

C₅₃×#1892

yes

characteristic

266

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×#1549	C₂	semidirect
5. C₅₃×#1303	C₂	semidirect
109. #9395	C₅₃	direct (N × H)
131. #1549	C₂×C₅₃	semidirect
132. #1303	C₂×C₅₃	semidirect
153. C₅₃	#9395	direct (N × H)

H ≅ G/N

type

4. C₅₃×#1549

C₂

semidirect

5. C₅₃×#1303

C₂

semidirect

109. #9395

C₅₃

direct (N × H)

131. #1549

C₂×C₅₃

semidirect

132. #1303

C₂×C₅₃

semidirect

153. C₅₃

#9395

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×#2542	2
C₅₃×#1800	2
C₅₃×#1549	2
C₅₃×#1303	2
#9395	53

subgroup

index

C₅₃×#2972

C₅₃×#2542

C₅₃×#1800

C₅₃×#1549

C₅₃×#1303

#9395

Subgroups — C₅₃×#9395 #27216907

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#9395 #27216907

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9395 #27216907