Group #27216910 — subgroups

Subgroups — C₅₃×#9398 #27216910

Order 13568.

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

normal subgroups only

This group has 574 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×#2982	C₂	1	1	yes	maximal, characteristic, complemented
2	C₅₃×#2548	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1801	C₂	1	1	yes	maximal, characteristic, complemented
4	C₅₃×#1545	C₂	1	2	yes	maximal
5	C₅₃×#1303	C₂	1	2	yes	maximal, complemented
6	C₂×#127×C₅₃	C₂²	1	1	yes	characteristic
7	C₂×#124×C₅₃	—	2	1	no	—
8	C₂×C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
9	C₈×Q₈×C₅₃	—	2	4	no	—
10	C₅₃×#431	—	2	2	no	—
11	C₅₃×#431	C₂²	1	2	yes	—
12	C₅₃×#423	C₂²	1	2	yes	—
13	C₂×#101×C₅₃	—	2	1	no	—
14	C₅₃×#394	C₄	1	4	yes	—
15	C₅₃×#390	—	2	1	no	—
16	C₅₃×#390	—	2	2	no	—
17	C₄×C₄⋊C₄×C₅₃	C₂²	1	1	yes	characteristic
18	C₅₃×#343	—	2	1	no	—
19	C₅₃×#331	—	2	4	no	—
20	C₂²×Q₈×C₅₃	D₄	1	1	yes	characteristic
21	C₂²×D₄×C₅₃	—	2	1	no	—
22	C₂²×D₄×C₅₃	—	2	1	no	—
23	C₂²×D₄×C₅₃	C₂×C₄	1	2	yes	—
24	C₂×M₁₆×C₅₃	—	2	2	no	—
25	#127×C₅₃	—	2	2	no	—
26	#127×C₅₃	—	2	2	no	—
27	#124×C₅₃	—	4	2	no	—
28	#124×C₅₃	—	4	2	no	—
29	C₄×Q₈×C₅₃	—	2	2	no	—
30	C₄×Q₈×C₅₃	—	2	2	no	—
31	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
32	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄⋊C₄×C₅₃	D₄	1	1	yes	characteristic
34	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
35	C₂×C₂²⋊C₄×C₅₃	—	2	4	no	—
36	C₂×C₄²×C₅₃	—	2	1	no	—
37	C₂×C₄²×C₅₃	D₄	1	2	yes	—
38	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₅₃	—	2	4	no	—
40	C₄⋊C₈×C₅₃	—	2	2	no	—
41	C₄⋊C₈×C₅₃	—	2	2	no	—
42	C₄⋊C₈×C₅₃	D₄	1	4	yes	—
43	#101×C₅₃	—	4	2	no	—
44	C₂²⋊C₈×C₅₃	—	2	2	no	—
45	C₄×C₈×C₅₃	—	2	4	no	—
46	C₄×C₈×C₅₃	—	2	2	no	—
47	#95×C₅₃	C₂×C₄	1	2	yes	—
48	C₂⁴×C₅₃	—	2	2	no	—
49	C₂×Q₈×C₅₃	—	2	2	no	—
50	C₂×Q₈×C₅₃	—	2	2	no	—
51	C₂×Q₈×C₅₃	D₈	1	2	yes	—
52	C₂×Q₈×C₅₃	SD₁₆	1	2	yes	—
53	C₂×D₄×C₅₃	—	2	4	no	—
54	C₂×D₄×C₅₃	—	4	4	no	—
55	C₂×D₄×C₅₃	—	4	1	no	—
56	C₂×D₄×C₅₃	—	4	2	no	—
57	C₂×D₄×C₅₃	—	4	1	no	—
58	C₂×D₄×C₅₃	—	4	1	no	—
59	C₂×D₄×C₅₃	—	4	1	no	—
60	C₂×D₄×C₅₃	—	4	2	no	—
61	C₂×D₄×C₅₃	—	8	2	no	—
62	C₂²×C₄×C₅₃	—	2	2	no	—
63	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
64	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
68	M₁₆×C₅₃	—	4	2	no	—
69	M₁₆×C₅₃	—	4	2	no	—
70	C₂×C₈×C₅₃	—	2	4	no	—
71	C₂×C₈×C₅₃	—	4	2	no	—
72	C₄⋊C₄×C₅₃	—	2	4	no	—
73	C₄⋊C₄×C₅₃	—	2	2	no	—
74	C₄⋊C₄×C₅₃	—	4	2	no	—
75	C₂²⋊C₄×C₅₃	—	4	8	no	—
76	C₂²⋊C₄×C₅₃	—	8	4	no	—
77	C₄²×C₅₃	—	2	2	no	—
78	C₄²×C₅₃	—	2	2	no	—
79	C₄²×C₅₃	—	4	1	no	—
80	C₄²×C₅₃	—	4	1	no	—
81	C₄²×C₅₃	C₂×D₄	1	2	yes	—
82	C₄²×C₅₃	#41	1	2	yes	—
83	C₂³×C₅₃	—	4	4	no	—
84	C₂³×C₅₃	—	4	4	no	—
85	C₂³×C₅₃	—	4	2	no	—
86	C₂³×C₅₃	—	4	2	no	—
87	C₂³×C₅₃	—	4	2	no	—
88	C₂³×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
89	Q₈×C₅₃	—	2	4	no	—
90	Q₈×C₅₃	—	2	4	no	—
91	D₄×C₅₃	—	8	4	no	—
92	D₄×C₅₃	—	8	2	no	—
93	D₄×C₅₃	—	8	2	no	—
94	D₄×C₅₃	—	8	2	no	—
95	D₄×C₅₃	—	8	2	no	—
96	C₂×C₄×C₅₃	—	2	2	no	—
97	C₂×C₄×C₅₃	—	2	2	no	—
98	C₂×C₄×C₅₃	—	2	2	no	—
99	C₂×C₄×C₅₃	—	2	2	no	—
100	C₂×C₄×C₅₃	—	4	1	no	—
101	C₂×C₄×C₅₃	—	4	2	no	—
102	C₂×C₄×C₅₃	—	4	2	no	—
103	C₂×C₄×C₅₃	—	4	2	no	—
104	C₂×C₄×C₅₃	—	4	2	no	—
105	C₂×C₄×C₅₃	#102	1	4	yes	—
106	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
107	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
108	C₂×C₄×C₅₃	#121	1	2	yes	—
109	C₂×C₄×C₅₃	#123	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
112	C₂×C₄×C₅₃	#135	1	2	yes	—
113	C₈×C₅₃	—	4	4	no	—
114	C₈×C₅₃	—	8	2	no	—
115	#9398	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
116	C₂²×C₅₃	—	2	1	no	—
117	C₂²×C₅₃	—	2	1	no	—
118	C₂²×C₅₃	—	8	4	no	—
119	C₂²×C₅₃	—	8	4	no	—
120	C₂²×C₅₃	—	8	2	no	—
121	C₂²×C₅₃	—	8	2	no	—
122	C₂²×C₅₃	—	8	2	no	—
123	C₂²×C₅₃	#386	1	1	yes	characteristic
124	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
125	C₂²×C₅₃	#416	1	1	yes	characteristic
126	C₄×C₅₃	—	2	1	no	—
127	C₄×C₅₃	—	2	4	no	—
128	C₄×C₅₃	—	2	2	no	—
129	C₄×C₅₃	—	4	2	no	—
130	C₄×C₅₃	—	8	2	no	—
131	C₄×C₅₃	#454	1	2	yes	—
132	C₄×C₅₃	#465	1	2	yes	—
133	C₄×C₅₃	#466	1	2	yes	—
134	#2982	C₂×C₅₃	1	1	yes	characteristic, complemented
135	#2548	C₂×C₅₃	1	1	yes	characteristic
136	#1801	C₂×C₅₃	1	1	yes	characteristic, complemented
137	#1545	C₂×C₅₃	1	2	yes	—
138	#1303	C₂×C₅₃	1	2	yes	complemented
139	C₂×C₅₃	—	2	1	no	—
140	C₂×C₅₃	—	2	1	no	—
141	C₂×C₅₃	—	16	2	no	—
142	C₂×C₅₃	#1888	1	1	yes	characteristic
143	C₂×C₅₃	#1893	1	1	yes	characteristic
144	C₂×C₅₃	#1914	1	1	yes	characteristic
145	C₂×#127	C₂²×C₅₃	1	1	yes	characteristic
146	C₂×#124	—	2	1	no	—
147	C₂×C₄×Q₈	C₂²×C₅₃	1	1	yes	characteristic
148	C₈×Q₈	—	2	4	no	—
149	#431	—	2	2	no	—
150	#431	C₂²×C₅₃	1	2	yes	—
151	#423	C₂²×C₅₃	1	2	yes	—
152	C₂×#101	—	2	1	no	—
153	#394	C₄×C₅₃	1	4	yes	—
154	#390	—	2	1	no	—
155	#390	—	2	2	no	—
156	C₄×C₄⋊C₄	C₂²×C₅₃	1	1	yes	characteristic
157	#343	—	2	1	no	—
158	#331	—	2	4	no	—
159	C₅₃	#9398	1	1	yes	characteristic, direct product factor, Sylow
160	C₂²×Q₈	D₄×C₅₃	1	1	yes	characteristic
161	C₂²×D₄	—	2	1	no	—
162	C₂²×D₄	—	2	1	no	—
163	C₂²×D₄	C₂×C₄×C₅₃	1	2	yes	—
164	C₂×M₁₆	—	2	2	no	—
165	#127	—	2	2	no	—
166	#127	—	2	2	no	—
167	#124	—	4	2	no	—
168	#124	—	4	2	no	—
169	C₄×Q₈	—	2	2	no	—
170	C₄×Q₈	—	2	2	no	—
171	C₂×C₄⋊C₄	—	2	1	no	—
172	C₂×C₄⋊C₄	C₂×C₄×C₅₃	1	2	yes	—
173	C₂×C₄⋊C₄	D₄×C₅₃	1	1	yes	characteristic
174	C₂×C₂²⋊C₄	—	2	4	no	—
175	C₂×C₂²⋊C₄	—	2	4	no	—
176	C₂×C₄²	—	2	1	no	—
177	C₂×C₄²	D₄×C₅₃	1	2	yes	—
178	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
179	C₄⋊C₈	—	2	4	no	—
180	C₄⋊C₈	—	2	2	no	—
181	C₄⋊C₈	—	2	2	no	—
182	C₄⋊C₈	D₄×C₅₃	1	4	yes	—
183	#101	—	4	2	no	—
184	C₂²⋊C₈	—	2	2	no	—
185	C₄×C₈	—	2	4	no	—
186	C₄×C₈	—	2	2	no	—
187	#95	C₂×C₄×C₅₃	1	2	yes	—
188	C₂⁴	—	2	2	no	—
189	C₂×Q₈	—	2	2	no	—
190	C₂×Q₈	—	2	2	no	—
191	C₂×Q₈	D₈×C₅₃	1	2	yes	—
192	C₂×Q₈	SD₁₆×C₅₃	1	2	yes	—
193	C₂×D₄	—	2	4	no	—
194	C₂×D₄	—	4	4	no	—
195	C₂×D₄	—	4	1	no	—
196	C₂×D₄	—	4	2	no	—
197	C₂×D₄	—	4	1	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂×D₄	—	8	2	no	—
202	C₂²×C₄	—	2	2	no	—
203	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
204	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
205	C₂²×C₄	C₂²×C₄×C₅₃	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
208	M₁₆	—	4	2	no	—
209	M₁₆	—	4	2	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	—	4	2	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	2	2	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₂²⋊C₄	—	4	8	no	—
216	C₂²⋊C₄	—	8	4	no	—
217	C₄²	—	2	2	no	—
218	C₄²	—	2	2	no	—
219	C₄²	—	4	1	no	—
220	C₄²	—	4	1	no	—
221	C₄²	C₂×D₄×C₅₃	1	2	yes	—
222	C₄²	#41×C₅₃	1	2	yes	—
223	C₂³	—	4	4	no	—
224	C₂³	—	4	4	no	—
225	C₂³	—	4	2	no	—
226	C₂³	—	4	2	no	—
227	C₂³	—	4	2	no	—
228	C₂³	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
229	Q₈	—	2	4	no	—
230	Q₈	—	2	4	no	—
231	D₄	—	8	4	no	—
232	D₄	—	8	2	no	—
233	D₄	—	8	2	no	—
234	D₄	—	8	2	no	—
235	D₄	—	8	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	2	2	no	—
240	C₂×C₄	—	4	1	no	—
241	C₂×C₄	—	4	2	no	—
242	C₂×C₄	—	4	2	no	—
243	C₂×C₄	—	4	2	no	—
244	C₂×C₄	—	4	2	no	—
245	C₂×C₄	#102×C₅₃	1	4	yes	—
246	C₂×C₄	C₄×D₄×C₅₃	1	2	yes	—
247	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
248	C₂×C₄	#121×C₅₃	1	2	yes	—
249	C₂×C₄	#123×C₅₃	1	1	yes	characteristic
250	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
251	C₂×C₄	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
252	C₂×C₄	#135×C₅₃	1	2	yes	—
253	C₈	—	4	4	no	—
254	C₈	—	8	2	no	—
255	C₂²	—	2	1	no	—
256	C₂²	—	2	1	no	—
257	C₂²	—	8	4	no	—
258	C₂²	—	8	4	no	—
259	C₂²	—	8	2	no	—
260	C₂²	—	8	2	no	—
261	C₂²	—	8	2	no	—
262	C₂²	C₅₃×#386	1	1	yes	characteristic
263	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
264	C₂²	C₅₃×#416	1	1	yes	characteristic
265	C₄	—	2	2	no	—
266	C₄	—	2	1	no	—
267	C₄	—	2	4	no	—
268	C₄	—	4	2	no	—
269	C₄	—	8	2	no	—
270	C₄	C₅₃×#454	1	2	yes	—
271	C₄	C₅₃×#465	1	2	yes	—
272	C₄	C₅₃×#466	1	2	yes	—
273	C₂	—	2	1	no	—
274	C₂	—	2	1	no	—
275	C₂	—	16	2	no	—
276	C₂	C₅₃×#1888	1	1	yes	characteristic
277	C₂	C₅₃×#1893	1	1	yes	characteristic
278	C₂	C₅₃×#1914	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2982

C₂

yes

maximal, characteristic, complemented

C₅₃×#2548

C₂

yes

maximal, characteristic

C₅₃×#1801

C₂

yes

maximal, characteristic, complemented

C₅₃×#1545

C₂

yes

maximal

C₅₃×#1303

C₂

yes

maximal, complemented

C₂×#127×C₅₃

C₂²

yes

characteristic

C₂×#124×C₅₃

—

C₂×C₄×Q₈×C₅₃

C₂²

yes

characteristic

C₈×Q₈×C₅₃

—

C₅₃×#431

—

C₅₃×#431

C₂²

yes

—

C₅₃×#423

C₂²

yes

—

C₂×#101×C₅₃

—

C₅₃×#394

C₄

yes

—

C₅₃×#390

—

C₅₃×#390

—

C₄×C₄⋊C₄×C₅₃

C₂²

yes

characteristic

C₅₃×#343

—

C₅₃×#331

—

C₂²×Q₈×C₅₃

D₄

yes

characteristic

C₂²×D₄×C₅₃

—

C₂²×D₄×C₅₃

—

C₂²×D₄×C₅₃

C₂×C₄

yes

—

C₂×M₁₆×C₅₃

—

#127×C₅₃

—

#127×C₅₃

—

#124×C₅₃

—

#124×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×Q₈×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×C₄²×C₅₃

—

C₂×C₄²×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

D₄

yes

—

#101×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

#95×C₅₃

C₂×C₄

yes

—

C₂⁴×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

D₈

yes

—

C₂×Q₈×C₅₃

SD₁₆

yes

—

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

yes

—

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

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₂²⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

—

C₄²×C₅₃

#41

yes

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×C₅₃

—

D₄×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

—

105

C₂×C₄×C₅₃

#102

yes

—

106

C₂×C₄×C₅₃

C₄×D₄

yes

—

107

C₂×C₄×C₅₃

#120

yes

characteristic

108

C₂×C₄×C₅₃

#121

yes

—

109

C₂×C₄×C₅₃

#123

yes

characteristic

110

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

111

C₂×C₄×C₅₃

C₂×SD₁₆

yes

characteristic

112

C₂×C₄×C₅₃

#135

yes

—

113

C₈×C₅₃

—

114

C₈×C₅₃

—

115

#9398

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

116

C₂²×C₅₃

—

117

C₂²×C₅₃

—

118

C₂²×C₅₃

—

119

C₂²×C₅₃

—

120

C₂²×C₅₃

—

121

C₂²×C₅₃

—

122

C₂²×C₅₃

—

123

C₂²×C₅₃

#386

yes

characteristic

124

C₂²×C₅₃

C₂×#102

yes

characteristic

125

C₂²×C₅₃

#416

yes

characteristic

126

C₄×C₅₃

—

127

C₄×C₅₃

—

128

C₄×C₅₃

—

129

C₄×C₅₃

—

130

C₄×C₅₃

—

131

C₄×C₅₃

#454

yes

—

132

C₄×C₅₃

#465

yes

—

133

C₄×C₅₃

#466

yes

—

134

#2982

C₂×C₅₃

yes

characteristic, complemented

135

#2548

C₂×C₅₃

yes

characteristic

136

#1801

C₂×C₅₃

yes

characteristic, complemented

137

#1545

C₂×C₅₃

yes

—

138

#1303

C₂×C₅₃

yes

complemented

139

C₂×C₅₃

—

140

C₂×C₅₃

—

141

C₂×C₅₃

—

142

C₂×C₅₃

#1888

yes

characteristic

143

C₂×C₅₃

#1893

yes

characteristic

144

C₂×C₅₃

#1914

yes

characteristic

145

C₂×#127

C₂²×C₅₃

yes

characteristic

146

C₂×#124

—

147

C₂×C₄×Q₈

C₂²×C₅₃

yes

characteristic

148

C₈×Q₈

—

149

#431

—

150

#431

C₂²×C₅₃

yes

—

151

#423

C₂²×C₅₃

yes

—

152

C₂×#101

—

153

#394

C₄×C₅₃

yes

—

154

#390

—

155

#390

—

156

C₄×C₄⋊C₄

C₂²×C₅₃

yes

characteristic

157

#343

—

158

#331

—

159

C₅₃

#9398

yes

characteristic, direct product factor, Sylow

160

C₂²×Q₈

D₄×C₅₃

yes

characteristic

161

C₂²×D₄

—

162

C₂²×D₄

—

163

C₂²×D₄

C₂×C₄×C₅₃

yes

—

164

C₂×M₁₆

—

165

#127

—

166

#127

—

167

#124

—

168

#124

—

169

C₄×Q₈

—

170

C₄×Q₈

—

171

C₂×C₄⋊C₄

—

172

C₂×C₄⋊C₄

C₂×C₄×C₅₃

yes

—

173

C₂×C₄⋊C₄

D₄×C₅₃

yes

characteristic

174

C₂×C₂²⋊C₄

—

175

C₂×C₂²⋊C₄

—

176

C₂×C₄²

—

177

C₂×C₄²

D₄×C₅₃

yes

—

178

C₂×C₄²

C₂³×C₅₃

yes

characteristic

179

C₄⋊C₈

—

180

C₄⋊C₈

—

181

C₄⋊C₈

—

182

C₄⋊C₈

D₄×C₅₃

yes

—

183

#101

—

184

C₂²⋊C₈

—

185

C₄×C₈

—

186

C₄×C₈

—

187

#95

C₂×C₄×C₅₃

yes

—

188

C₂⁴

—

189

C₂×Q₈

—

190

C₂×Q₈

—

191

C₂×Q₈

D₈×C₅₃

yes

—

192

C₂×Q₈

SD₁₆×C₅₃

yes

—

193

C₂×D₄

—

194

C₂×D₄

—

195

C₂×D₄

—

196

C₂×D₄

—

197

C₂×D₄

—

198

C₂×D₄

—

199

C₂×D₄

—

200

C₂×D₄

—

201

C₂×D₄

—

202

C₂²×C₄

—

203

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

—

204

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

—

205

C₂²×C₄

C₂²×C₄×C₅₃

yes

characteristic

206

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

207

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

208

M₁₆

—

209

M₁₆

—

210

C₂×C₈

—

211

C₂×C₈

—

212

C₄⋊C₄

—

213

C₄⋊C₄

—

214

C₄⋊C₄

—

215

C₂²⋊C₄

—

216

C₂²⋊C₄

—

217

C₄²

—

218

C₄²

—

219

C₄²

—

220

C₄²

—

221

C₄²

C₂×D₄×C₅₃

yes

—

222

C₄²

#41×C₅₃

yes

—

223

C₂³

—

224

C₂³

—

225

C₂³

—

226

C₂³

—

227

C₂³

—

228

C₂³

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

229

Q₈

—

230

Q₈

—

231

D₄

—

232

D₄

—

233

D₄

—

234

D₄

—

235

D₄

—

236

C₂×C₄

—

237

C₂×C₄

—

238

C₂×C₄

—

239

C₂×C₄

—

240

C₂×C₄

—

241

C₂×C₄

—

242

C₂×C₄

—

243

C₂×C₄

—

244

C₂×C₄

—

245

C₂×C₄

#102×C₅₃

yes

—

246

C₂×C₄

C₄×D₄×C₅₃

yes

—

247

C₂×C₄

#120×C₅₃

yes

characteristic

248

C₂×C₄

#121×C₅₃

yes

—

249

C₂×C₄

#123×C₅₃

yes

characteristic

250

C₂×C₄

C₂×D₈×C₅₃

yes

characteristic

251

C₂×C₄

C₂×SD₁₆×C₅₃

yes

characteristic

252

C₂×C₄

#135×C₅₃

yes

—

253

C₈

—

254

C₈

—

255

C₂²

—

256

C₂²

—

257

C₂²

—

258

C₂²

—

259

C₂²

—

260

C₂²

—

261

C₂²

—

262

C₂²

C₅₃×#386

yes

characteristic

263

C₂²

C₂×#102×C₅₃

yes

characteristic

264

C₂²

C₅₃×#416

yes

characteristic

265

C₄

—

266

C₄

—

267

C₄

—

268

C₄

—

269

C₄

—

270

C₄

C₅₃×#454

yes

—

271

C₄

C₅₃×#465

yes

—

272

C₄

C₅₃×#466

yes

—

273

C₂

—

274

C₂

—

275

C₂

—

276

C₂

C₅₃×#1888

yes

characteristic

277

C₂

C₅₃×#1893

yes

characteristic

278

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	C₂×#127×C₅₃	C₂²
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

C₂×#127×C₅₃

C₂²

LCS

G ▷ C₂²×C₄ ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
1. C₅₃×#2982	C₂	semidirect
3. C₅₃×#1801	C₂	semidirect
5. C₅₃×#1303	C₂	semidirect
115. #9398	C₅₃	direct (N × H)
134. #2982	C₂×C₅₃	semidirect
136. #1801	C₂×C₅₃	semidirect
138. #1303	C₂×C₅₃	semidirect
159. C₅₃	#9398	direct (N × H)

H ≅ G/N

type

1. C₅₃×#2982

C₂

semidirect

3. C₅₃×#1801

C₂

semidirect

5. C₅₃×#1303

C₂

semidirect

115. #9398

C₅₃

direct (N × H)

134. #2982

C₂×C₅₃

semidirect

136. #1801

C₂×C₅₃

semidirect

138. #1303

C₂×C₅₃

semidirect

159. C₅₃

#9398

direct (N × H)

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

subgroup	index
C₅₃×#2982	2
C₅₃×#2548	2
C₅₃×#1801	2
C₅₃×#1545	2
C₅₃×#1303	2
#9398	53

subgroup

index

C₅₃×#2982

C₅₃×#2548

C₅₃×#1801

C₅₃×#1545

C₅₃×#1303

#9398

Subgroups — C₅₃×#9398 #27216910

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#9398 #27216910

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#9398 #27216910