Group #27217860 — subgroups

Subgroups — C₅₃×#10348 #27217860

Order 13568.

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

normal subgroups only

This group has 588 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×#2551	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#1982	C₂	1	2	yes	maximal
3	C₅₃×#1957	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1296	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1279	C₂	1	2	yes	maximal, complemented
6	C₂×C₄⋊Q₈×C₅₃	—	2	2	no	—
7	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
9	C₂×#102×C₅₃	—	2	2	no	—
10	C₂×#102×C₅₃	—	2	2	no	—
11	C₂×#102×C₅₃	—	2	2	no	—
12	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#395	—	2	2	no	—
14	C₅₃×#389	—	2	2	no	—
15	C₅₃×#384	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
17	C₂²×Q₈×C₅₃	—	2	2	no	—
18	C₂²×D₄×C₅₃	—	2	1	no	—
19	C₂²×D₄×C₅₃	—	2	2	no	—
20	C₂²×C₈×C₅₃	—	2	1	no	—
21	C₂²×C₈×C₅₃	—	2	2	no	—
22	C₄⋊Q₈×C₅₃	—	4	4	no	—
23	C₄⋊Q₈×C₅₃	—	4	4	no	—
24	#127×C₅₃	—	2	2	no	—
25	#127×C₅₃	—	2	1	no	—
26	#127×C₅₃	Q₈	1	2	yes	complemented
27	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
30	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
31	C₂×C₄⋊C₄×C₅₃	D₄	1	2	yes	—
32	C₂×C₄²×C₅₃	—	2	2	no	—
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₄⋊C₈×C₅₃	—	2	2	no	—
35	C₄⋊C₈×C₅₃	—	2	2	no	—
36	#102×C₅₃	—	4	4	no	—
37	#102×C₅₃	—	4	4	no	—
38	#102×C₅₃	—	4	4	no	—
39	C₄×C₈×C₅₃	—	2	2	no	—
40	#95×C₅₃	—	2	2	no	—
41	#95×C₅₃	—	2	2	no	—
42	C₂⁴×C₅₃	—	4	1	no	—
43	C₂×Q₈×C₅₃	—	4	8	no	—
44	C₂×Q₈×C₅₃	—	8	2	no	—
45	C₂×D₄×C₅₃	—	2	2	no	—
46	C₂×D₄×C₅₃	—	2	4	no	—
47	C₂×D₄×C₅₃	—	4	2	no	—
48	C₂×D₄×C₅₃	—	4	2	no	—
49	C₂×D₄×C₅₃	—	4	1	no	—
50	C₂×D₄×C₅₃	—	4	2	no	—
51	C₂×D₄×C₅₃	—	4	2	no	—
52	C₂×D₄×C₅₃	—	4	2	no	—
53	C₂×D₄×C₅₃	—	4	2	no	—
54	C₂×D₄×C₅₃	—	4	2	no	—
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	C₂×D₄	1	1	yes	characteristic
59	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
60	C₂×C₈×C₅₃	—	2	2	no	—
61	C₂×C₈×C₅₃	—	4	1	no	—
62	C₂×C₈×C₅₃	—	4	1	no	—
63	C₂×C₈×C₅₃	—	4	2	no	—
64	C₂×C₈×C₅₃	—	4	2	no	—
65	C₂×C₈×C₅₃	—	4	2	no	—
66	C₄⋊C₄×C₅₃	—	2	2	no	—
67	C₄⋊C₄×C₅₃	—	2	2	no	—
68	C₄⋊C₄×C₅₃	—	4	8	no	—
69	C₄⋊C₄×C₅₃	—	4	2	no	—
70	C₄⋊C₄×C₅₃	—	4	2	no	—
71	C₄⋊C₄×C₅₃	—	4	2	no	—
72	C₄⋊C₄×C₅₃	D₈	1	4	yes	—
73	C₄⋊C₄×C₅₃	SD₁₆	1	4	yes	—
74	C₄²×C₅₃	—	4	2	no	—
75	C₄²×C₅₃	—	4	2	no	—
76	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
77	C₄²×C₅₃	#41	1	2	yes	—
78	C₄²×C₅₃	#41	1	1	yes	characteristic
79	C₂³×C₅₃	—	4	2	no	—
80	C₂³×C₅₃	—	8	2	no	—
81	C₂³×C₅₃	—	8	2	no	—
82	C₂³×C₅₃	—	8	1	no	—
83	C₂³×C₅₃	—	8	1	no	—
84	C₂³×C₅₃	#120	1	1	yes	characteristic
85	Q₈×C₅₃	—	8	8	no	—
86	D₄×C₅₃	—	4	4	no	—
87	D₄×C₅₃	—	4	4	no	—
88	D₄×C₅₃	—	4	4	no	—
89	D₄×C₅₃	—	8	2	no	—
90	D₄×C₅₃	—	8	2	no	—
91	D₄×C₅₃	—	8	2	no	—
92	C₂×C₄×C₅₃	—	2	1	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	2	4	no	—
95	C₂×C₄×C₅₃	—	2	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	4	no	—
101	C₂×C₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
103	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
104	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
105	C₂×C₄×C₅₃	#124	1	2	yes	—
106	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
107	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
108	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
109	C₂×C₄×C₅₃	#137	1	2	yes	—
110	C₈×C₅₃	—	4	2	no	—
111	C₈×C₅₃	—	8	2	no	—
112	#10348	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
113	C₂²×C₅₃	—	8	4	no	—
114	C₂²×C₅₃	—	8	2	no	—
115	C₂²×C₅₃	—	16	1	no	—
116	C₂²×C₅₃	—	16	1	no	—
117	C₂²×C₅₃	—	16	2	no	—
118	C₂²×C₅₃	#393	1	1	yes	characteristic
119	C₂²×C₅₃	#447	1	2	yes	—
120	C₂²×C₅₃	#448	1	2	yes	—
121	C₂²×C₅₃	#453	1	1	yes	characteristic
122	C₂²×C₅₃	#454	1	1	yes	characteristic
123	C₄×C₅₃	—	2	1	no	—
124	C₄×C₅₃	—	2	1	no	—
125	C₄×C₅₃	—	4	4	no	—
126	C₄×C₅₃	—	4	4	no	—
127	C₄×C₅₃	—	8	2	no	—
128	C₄×C₅₃	#474	1	2	yes	—
129	C₄×C₅₃	#476	1	2	yes	—
130	C₄×C₅₃	#486	1	2	yes	—
131	C₄×C₅₃	#489	1	2	yes	—
132	#2551	C₂×C₅₃	1	1	yes	characteristic, complemented
133	#1982	C₂×C₅₃	1	2	yes	—
134	#1957	C₂×C₅₃	1	1	yes	characteristic
135	#1296	C₂×C₅₃	1	1	yes	characteristic, complemented
136	#1279	C₂×C₅₃	1	2	yes	complemented
137	C₂×C₅₃	—	16	2	no	—
138	C₂×C₅₃	#1619	1	1	yes	characteristic
139	C₂×C₅₃	#1623	1	1	yes	characteristic
140	C₂×C₅₃	#1627	1	2	yes	—
141	C₂×C₅₃	#2023	1	2	yes	—
142	C₂×C₅₃	#2027	1	1	yes	characteristic
143	C₂×C₄⋊Q₈	—	2	2	no	—
144	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
145	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
146	C₂×#102	—	2	2	no	—
147	C₂×#102	—	2	2	no	—
148	C₂×#102	—	2	2	no	—
149	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
150	#395	—	2	2	no	—
151	#389	—	2	2	no	—
152	#384	C₂²×C₅₃	1	1	yes	characteristic
153	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
154	C₅₃	#10348	1	1	yes	characteristic, direct product factor, Sylow
155	C₂²×Q₈	—	2	2	no	—
156	C₂²×D₄	—	2	1	no	—
157	C₂²×D₄	—	2	2	no	—
158	C₂²×C₈	—	2	1	no	—
159	C₂²×C₈	—	2	2	no	—
160	C₄⋊Q₈	—	4	4	no	—
161	C₄⋊Q₈	—	4	4	no	—
162	#127	—	2	2	no	—
163	#127	—	2	1	no	—
164	#127	Q₈×C₅₃	1	2	yes	complemented
165	C₂×C₄⋊C₄	—	2	4	no	—
166	C₂×C₄⋊C₄	—	2	2	no	—
167	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
168	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
169	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
170	C₂×C₄²	—	2	2	no	—
171	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
172	C₄⋊C₈	—	2	2	no	—
173	C₄⋊C₈	—	2	2	no	—
174	#102	—	4	4	no	—
175	#102	—	4	4	no	—
176	#102	—	4	4	no	—
177	C₄×C₈	—	2	2	no	—
178	#95	—	2	2	no	—
179	#95	—	2	2	no	—
180	C₂⁴	—	4	1	no	—
181	C₂×Q₈	—	4	8	no	—
182	C₂×Q₈	—	8	2	no	—
183	C₂×D₄	—	2	2	no	—
184	C₂×D₄	—	2	4	no	—
185	C₂×D₄	—	4	2	no	—
186	C₂×D₄	—	4	2	no	—
187	C₂×D₄	—	4	1	no	—
188	C₂×D₄	—	4	2	no	—
189	C₂×D₄	—	4	2	no	—
190	C₂×D₄	—	4	2	no	—
191	C₂×D₄	—	4	2	no	—
192	C₂×D₄	—	4	2	no	—
193	C₂²×C₄	—	2	2	no	—
194	C₂²×C₄	—	2	2	no	—
195	C₂²×C₄	—	2	2	no	—
196	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
197	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
198	C₂×C₈	—	2	2	no	—
199	C₂×C₈	—	4	1	no	—
200	C₂×C₈	—	4	1	no	—
201	C₂×C₈	—	4	2	no	—
202	C₂×C₈	—	4	2	no	—
203	C₂×C₈	—	4	2	no	—
204	C₄⋊C₄	—	2	2	no	—
205	C₄⋊C₄	—	2	2	no	—
206	C₄⋊C₄	—	4	2	no	—
207	C₄⋊C₄	—	4	8	no	—
208	C₄⋊C₄	—	4	2	no	—
209	C₄⋊C₄	—	4	2	no	—
210	C₄⋊C₄	D₈×C₅₃	1	4	yes	—
211	C₄⋊C₄	SD₁₆×C₅₃	1	4	yes	—
212	C₄²	—	4	2	no	—
213	C₄²	—	4	2	no	—
214	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
215	C₄²	#41×C₅₃	1	2	yes	—
216	C₄²	#41×C₅₃	1	1	yes	characteristic
217	C₂³	—	4	2	no	—
218	C₂³	—	8	2	no	—
219	C₂³	—	8	2	no	—
220	C₂³	—	8	1	no	—
221	C₂³	—	8	1	no	—
222	C₂³	#120×C₅₃	1	1	yes	characteristic
223	Q₈	—	8	8	no	—
224	D₄	—	4	4	no	—
225	D₄	—	4	4	no	—
226	D₄	—	4	4	no	—
227	D₄	—	8	2	no	—
228	D₄	—	8	2	no	—
229	D₄	—	8	2	no	—
230	C₂×C₄	—	2	1	no	—
231	C₂×C₄	—	2	1	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	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	4	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
241	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
242	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
243	C₂×C₄	#124×C₅₃	1	2	yes	—
244	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
245	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
246	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
247	C₂×C₄	#137×C₅₃	1	2	yes	—
248	C₈	—	4	2	no	—
249	C₈	—	8	2	no	—
250	C₂²	—	8	4	no	—
251	C₂²	—	8	2	no	—
252	C₂²	—	16	1	no	—
253	C₂²	—	16	1	no	—
254	C₂²	—	16	2	no	—
255	C₂²	C₅₃×#393	1	1	yes	characteristic
256	C₂²	C₅₃×#447	1	2	yes	—
257	C₂²	C₅₃×#448	1	2	yes	—
258	C₂²	C₅₃×#453	1	1	yes	characteristic
259	C₂²	C₅₃×#454	1	1	yes	characteristic
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	4	4	no	—
263	C₄	—	4	4	no	—
264	C₄	—	8	2	no	—
265	C₄	C₅₃×#474	1	2	yes	—
266	C₄	C₅₃×#476	1	2	yes	—
267	C₄	C₅₃×#486	1	2	yes	—
268	C₄	C₅₃×#489	1	2	yes	—
269	C₂	—	16	2	no	—
270	C₂	C₅₃×#1619	1	1	yes	characteristic
271	C₂	C₅₃×#1623	1	1	yes	characteristic
272	C₂	C₅₃×#1627	1	2	yes	—
273	C₂	C₅₃×#2023	1	2	yes	—
274	C₂	C₅₃×#2027	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2551

C₂

yes

maximal, characteristic, complemented

C₅₃×#1982

C₂

yes

maximal

C₅₃×#1957

C₂

yes

maximal, characteristic

C₅₃×#1296

C₂

yes

maximal, characteristic, complemented

C₅₃×#1279

C₂

yes

maximal, complemented

C₂×C₄⋊Q₈×C₅₃

—

C₂×#127×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₅₃

C₂²

yes

—

C₂×#102×C₅₃

—

C₂×#102×C₅₃

—

C₂×#102×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#395

—

C₅₃×#389

—

C₅₃×#384

C₂²

yes

characteristic

C₄×C₄⋊C₄×C₅₃

C₂²

yes

—

C₂²×Q₈×C₅₃

—

C₂²×D₄×C₅₃

—

C₂²×D₄×C₅₃

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

#127×C₅₃

—

#127×C₅₃

—

#127×C₅₃

Q₈

yes

complemented

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

#102×C₅₃

—

#102×C₅₃

—

#102×C₅₃

—

C₄×C₈×C₅₃

—

#95×C₅₃

—

#95×C₅₃

—

C₂⁴×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×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

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

D₈

yes

—

C₄⋊C₄×C₅₃

SD₁₆

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

#41

yes

—

C₄²×C₅₃

#41

yes

characteristic

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#120

yes

characteristic

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×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

C₂²⋊Q₈

yes

characteristic

103

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

104

C₂×C₄×C₅₃

#124

yes

characteristic

105

C₂×C₄×C₅₃

#124

yes

—

106

C₂×C₄×C₅₃

C₂×D₈

yes

—

107

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

108

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

109

C₂×C₄×C₅₃

#137

yes

—

110

C₈×C₅₃

—

111

C₈×C₅₃

—

112

#10348

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

113

C₂²×C₅₃

—

114

C₂²×C₅₃

—

115

C₂²×C₅₃

—

116

C₂²×C₅₃

—

117

C₂²×C₅₃

—

118

C₂²×C₅₃

#393

yes

characteristic

119

C₂²×C₅₃

#447

yes

—

120

C₂²×C₅₃

#448

yes

—

121

C₂²×C₅₃

#453

yes

characteristic

122

C₂²×C₅₃

#454

yes

characteristic

123

C₄×C₅₃

—

124

C₄×C₅₃

—

125

C₄×C₅₃

—

126

C₄×C₅₃

—

127

C₄×C₅₃

—

128

C₄×C₅₃

#474

yes

—

129

C₄×C₅₃

#476

yes

—

130

C₄×C₅₃

#486

yes

—

131

C₄×C₅₃

#489

yes

—

132

#2551

C₂×C₅₃

yes

characteristic, complemented

133

#1982

C₂×C₅₃

yes

—

134

#1957

C₂×C₅₃

yes

characteristic

135

#1296

C₂×C₅₃

yes

characteristic, complemented

136

#1279

C₂×C₅₃

yes

complemented

137

C₂×C₅₃

—

138

C₂×C₅₃

#1619

yes

characteristic

139

C₂×C₅₃

#1623

yes

characteristic

140

C₂×C₅₃

#1627

yes

—

141

C₂×C₅₃

#2023

yes

—

142

C₂×C₅₃

#2027

yes

characteristic

143

C₂×C₄⋊Q₈

—

144

C₂×#127

C₂²×C₅₃

yes

characteristic

145

C₂×C₄⋊C₈

C₂²×C₅₃

yes

—

146

C₂×#102

—

147

C₂×#102

—

148

C₂×#102

—

149

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

150

#395

—

151

#389

—

152

#384

C₂²×C₅₃

yes

characteristic

153

C₄×C₄⋊C₄

C₂²×C₅₃

yes

—

154

C₅₃

#10348

yes

characteristic, direct product factor, Sylow

155

C₂²×Q₈

—

156

C₂²×D₄

—

157

C₂²×D₄

—

158

C₂²×C₈

—

159

C₂²×C₈

—

160

C₄⋊Q₈

—

161

C₄⋊Q₈

—

162

#127

—

163

#127

—

164

#127

Q₈×C₅₃

yes

complemented

165

C₂×C₄⋊C₄

—

166

C₂×C₄⋊C₄

—

167

C₂×C₄⋊C₄

D₄×C₅₃

yes

—

168

C₂×C₄⋊C₄

D₄×C₅₃

yes

—

169

C₂×C₄⋊C₄

D₄×C₅₃

yes

—

170

C₂×C₄²

—

171

C₂×C₄²

C₂³×C₅₃

yes

characteristic

172

C₄⋊C₈

—

173

C₄⋊C₈

—

174

#102

—

175

#102

—

176

#102

—

177

C₄×C₈

—

178

#95

—

179

#95

—

180

C₂⁴

—

181

C₂×Q₈

—

182

C₂×Q₈

—

183

C₂×D₄

—

184

C₂×D₄

—

185

C₂×D₄

—

186

C₂×D₄

—

187

C₂×D₄

—

188

C₂×D₄

—

189

C₂×D₄

—

190

C₂×D₄

—

191

C₂×D₄

—

192

C₂×D₄

—

193

C₂²×C₄

—

194

C₂²×C₄

—

195

C₂²×C₄

—

196

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

197

C₂²×C₄

C₂×D₄×C₅₃

yes

—

198

C₂×C₈

—

199

C₂×C₈

—

200

C₂×C₈

—

201

C₂×C₈

—

202

C₂×C₈

—

203

C₂×C₈

—

204

C₄⋊C₄

—

205

C₄⋊C₄

—

206

C₄⋊C₄

—

207

C₄⋊C₄

—

208

C₄⋊C₄

—

209

C₄⋊C₄

—

210

C₄⋊C₄

D₈×C₅₃

yes

—

211

C₄⋊C₄

SD₁₆×C₅₃

yes

—

212

C₄²

—

213

C₄²

—

214

C₄²

C₂×Q₈×C₅₃

yes

characteristic

215

C₄²

#41×C₅₃

yes

—

216

C₄²

#41×C₅₃

yes

characteristic

217

C₂³

—

218

C₂³

—

219

C₂³

—

220

C₂³

—

221

C₂³

—

222

C₂³

#120×C₅₃

yes

characteristic

223

Q₈

—

224

D₄

—

225

D₄

—

226

D₄

—

227

D₄

—

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

C₂²⋊Q₈×C₅₃

yes

characteristic

241

C₂×C₄

C₂²⋊Q₈×C₅₃

yes

—

242

C₂×C₄

#124×C₅₃

yes

characteristic

243

C₂×C₄

#124×C₅₃

yes

—

244

C₂×C₄

C₂×D₈×C₅₃

yes

—

245

C₂×C₄

C₂×SD₁₆×C₅₃

yes

—

246

C₂×C₄

AGL(1,ℤ/8)×C₅₃

yes

—

247

C₂×C₄

#137×C₅₃

yes

—

248

C₈

—

249

C₈

—

250

C₂²

—

251

C₂²

—

252

C₂²

—

253

C₂²

—

254

C₂²

—

255

C₂²

C₅₃×#393

yes

characteristic

256

C₂²

C₅₃×#447

yes

—

257

C₂²

C₅₃×#448

yes

—

258

C₂²

C₅₃×#453

yes

characteristic

259

C₂²

C₅₃×#454

yes

characteristic

260

C₄

—

261

C₄

—

262

C₄

—

263

C₄

—

264

C₄

—

265

C₄

C₅₃×#474

yes

—

266

C₄

C₅₃×#476

yes

—

267

C₄

C₅₃×#486

yes

—

268

C₄

C₅₃×#489

yes

—

269

C₂

—

270

C₂

C₅₃×#1619

yes

characteristic

271

C₂

C₅₃×#1623

yes

characteristic

272

C₂

C₅₃×#1627

yes

—

273

C₂

C₅₃×#2023

yes

—

274

C₂

C₅₃×#2027

yes

characteristic

	subgroup	G/H
center, socle	C₂³×C₅₃	#120
commutator, Frattini, cosocle	C₂×C₄²	C₂³×C₅₃
prime radical	C₂×#127×C₅₃	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂³×C₅₃

#120

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

prime radical

C₂×#127×C₅₃

C₂²

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
1. C₅₃×#2551	C₂	semidirect
4. C₅₃×#1296	C₂	semidirect
5. C₅₃×#1279	C₂	semidirect
26. #127×C₅₃	Q₈	semidirect
112. #10348	C₅₃	direct (N × H)
132. #2551	C₂×C₅₃	semidirect
135. #1296	C₂×C₅₃	semidirect
136. #1279	C₂×C₅₃	semidirect
154. C₅₃	#10348	direct (N × H)
164. #127	Q₈×C₅₃	semidirect

H ≅ G/N

type

1. C₅₃×#2551

C₂

semidirect

4. C₅₃×#1296

C₂

semidirect

5. C₅₃×#1279

C₂

semidirect

26. #127×C₅₃

Q₈

semidirect

112. #10348

C₅₃

direct (N × H)

132. #2551

C₂×C₅₃

semidirect

135. #1296

C₂×C₅₃

semidirect

136. #1279

C₂×C₅₃

semidirect

154. C₅₃

#10348

direct (N × H)

164. #127

Q₈×C₅₃

semidirect

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

subgroup	index
C₅₃×#2551	2
C₅₃×#1982	2
C₅₃×#1957	2
C₅₃×#1296	2
C₅₃×#1279	2
#10348	53

subgroup

index

C₅₃×#2551

C₅₃×#1982

C₅₃×#1957

C₅₃×#1296

C₅₃×#1279

#10348

Subgroups — C₅₃×#10348 #27217860

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#10348 #27217860

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10348 #27217860