Group #27217951 — subgroups

Subgroups — C₅₃×#10439 #27217951

Order 13568.

478 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 478 conjugacy classes of subgroups — too many to draw. Only its 146 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×#2582	C₂	1	1	yes	maximal, characteristic
2	C₅₃×#1845	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1803	C₂	1	2	yes	maximal
4	C₂×C₅₃×Q₈⋊C₈	C₂	1	2	yes	maximal
5	C₅₃×#1296	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×Q₈×C₅₃	C₂²	1	2	yes	—
7	C₂×#107×C₅₃	—	2	2	no	—
8	C₂×C₄⋊C₈×C₅₃	C₂²	1	2	yes	—
9	C₂×#103×C₅₃	—	2	2	no	—
10	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
11	C₅₃×#398	—	2	2	no	—
12	C₅₃×#391	—	2	2	no	—
13	C₅₃×#389	—	2	2	no	—
14	C₅₃×#384	C₂²	1	1	yes	characteristic
15	C₅₃×#384	C₂²	1	1	yes	characteristic
16	C₅₃×#340	—	2	2	no	—
17	C₅₃×Q₈⋊C₈	—	2	4	no	—
18	C₂²×Q₈×C₅₃	D₄	1	2	yes	—
19	C₂²×C₈×C₅₃	—	2	1	no	—
20	C₂²×C₈×C₅₃	—	2	2	no	—
21	C₄×Q₈×C₅₃	—	2	2	no	—
22	C₄×Q₈×C₅₃	—	2	2	no	—
23	C₄×Q₈×C₅₃	—	2	2	no	—
24	C₄×Q₈×C₅₃	—	2	2	no	—
25	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
26	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
27	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
28	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
29	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
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	#107×C₅₃	—	4	4	no	—
35	C₄⋊C₈×C₅₃	—	2	2	no	—
36	C₄⋊C₈×C₅₃	—	2	2	no	—
37	#103×C₅₃	—	4	4	no	—
38	C₄×C₈×C₅₃	—	2	1	no	—
39	C₄×C₈×C₅₃	D₄	1	2	yes	—
40	#95×C₅₃	—	2	4	no	—
41	#95×C₅₃	—	2	2	no	—
42	C₂×Q₈×C₅₃	—	2	2	no	—
43	C₂×Q₈×C₅₃	—	2	2	no	—
44	C₂×Q₈×C₅₃	—	2	2	no	—
45	C₂×Q₈×C₅₃	—	2	2	no	—
46	C₂×Q₈×C₅₃	—	2	2	no	—
47	C₂×Q₈×C₅₃	Q₁₆	1	4	yes	—
48	C₂²×C₄×C₅₃	—	2	2	no	—
49	C₂²×C₄×C₅₃	—	2	2	no	—
50	C₂²×C₄×C₅₃	—	2	2	no	—
51	C₂²×C₄×C₅₃	—	4	1	no	—
52	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₅₃	—	2	2	no	—
55	C₂×C₈×C₅₃	—	2	2	no	—
56	C₂×C₈×C₅₃	—	2	2	no	—
57	C₂×C₈×C₅₃	—	4	2	no	—
58	C₂×C₈×C₅₃	—	4	2	no	—
59	C₂×C₈×C₅₃	—	4	2	no	—
60	C₄⋊C₄×C₅₃	—	2	2	no	—
61	C₄⋊C₄×C₅₃	—	2	2	no	—
62	C₄⋊C₄×C₅₃	—	4	2	no	—
63	C₄⋊C₄×C₅₃	—	4	2	no	—
64	C₄⋊C₄×C₅₃	—	4	2	no	—
65	C₄⋊C₄×C₅₃	—	4	2	no	—
66	C₄⋊C₄×C₅₃	—	4	2	no	—
67	C₄⋊C₄×C₅₃	—	4	2	no	—
68	C₄⋊C₄×C₅₃	—	4	2	no	—
69	C₄⋊C₄×C₅₃	—	4	2	no	—
70	C₄⋊C₄×C₅₃	—	8	2	no	—
71	C₄²×C₅₃	—	4	2	no	—
72	C₄²×C₅₃	—	4	2	no	—
73	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₄²×C₅₃	#41	1	1	yes	characteristic
75	C₄²×C₅₃	#41	1	2	yes	—
76	C₂³×C₅₃	#120	1	1	yes	characteristic
77	Q₈×C₅₃	—	2	4	no	—
78	Q₈×C₅₃	—	2	4	no	—
79	Q₈×C₅₃	—	4	2	no	—
80	Q₈×C₅₃	—	4	2	no	—
81	C₂×C₄×C₅₃	—	2	1	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	4	no	—
84	C₂×C₄×C₅₃	—	4	2	no	—
85	C₂×C₄×C₅₃	—	4	2	no	—
86	C₂×C₄×C₅₃	—	4	2	no	—
87	C₂×C₄×C₅₃	—	4	4	no	—
88	C₂×C₄×C₅₃	—	4	2	no	—
89	C₂×C₄×C₅₃	—	4	2	no	—
90	C₂×C₄×C₅₃	—	4	2	no	—
91	C₂×C₄×C₅₃	—	8	2	no	—
92	C₂×C₄×C₅₃	—	8	1	no	—
93	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#121	1	2	yes	—
95	C₂×C₄×C₅₃	#123	1	2	yes	—
96	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
98	C₂×C₄×C₅₃	#135	1	2	yes	—
99	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
100	C₂×C₄×C₅₃	#137	1	2	yes	—
101	C₈×C₅₃	—	2	4	no	—
102	C₈×C₅₃	—	8	2	no	—
103	#10439	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
104	C₂²×C₅₃	#394	1	1	yes	characteristic
105	C₂²×C₅₃	#449	1	2	yes	—
106	C₂²×C₅₃	#451	1	2	yes	—
107	C₂²×C₅₃	#453	1	1	yes	characteristic
108	C₂²×C₅₃	#456	1	1	yes	characteristic
109	C₄×C₅₃	—	2	1	no	—
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	—	4	4	no	—
112	C₄×C₅₃	—	8	2	no	—
113	C₄×C₅₃	—	8	2	no	—
114	C₄×C₅₃	—	16	1	no	—
115	C₄×C₅₃	#467	1	2	yes	—
116	C₄×C₅₃	#468	1	2	yes	—
117	C₄×C₅₃	#484	1	2	yes	—
118	C₄×C₅₃	#485	1	2	yes	—
119	#2582	C₂×C₅₃	1	1	yes	characteristic
120	#1845	C₂×C₅₃	1	1	yes	characteristic
121	#1803	C₂×C₅₃	1	2	yes	—
122	C₂×Q₈⋊C₈	C₂×C₅₃	1	2	yes	—
123	#1296	C₂×C₅₃	1	1	yes	characteristic
124	C₂×C₅₃	#1620	1	1	yes	characteristic
125	C₂×C₅₃	#1633	1	1	yes	characteristic
126	C₂×C₅₃	#1648	1	2	yes	—
127	C₂×C₅₃	#2036	1	2	yes	—
128	C₂×C₅₃	#2039	1	1	yes	characteristic
129	C₂×C₄×Q₈	C₂²×C₅₃	1	2	yes	—
130	C₂×#107	—	2	2	no	—
131	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
132	C₂×#103	—	2	2	no	—
133	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
134	#398	—	2	2	no	—
135	#391	—	2	2	no	—
136	#389	—	2	2	no	—
137	#384	C₂²×C₅₃	1	1	yes	characteristic
138	#384	C₂²×C₅₃	1	1	yes	characteristic
139	#340	—	2	2	no	—
140	Q₈⋊C₈	—	2	4	no	—
141	C₅₃	#10439	1	1	yes	characteristic, direct product factor, Sylow
142	C₂²×Q₈	D₄×C₅₃	1	2	yes	—
143	C₂²×C₈	—	2	1	no	—
144	C₂²×C₈	—	2	2	no	—
145	C₄×Q₈	—	2	2	no	—
146	C₄×Q₈	—	2	2	no	—
147	C₄×Q₈	—	2	2	no	—
148	C₄×Q₈	—	2	2	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄⋊C₄	—	2	2	no	—
152	C₂×C₄⋊C₄	—	2	1	no	—
153	C₂×C₄⋊C₄	—	2	2	no	—
154	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
155	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
156	C₂×C₄²	—	2	2	no	—
157	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
158	#107	—	4	4	no	—
159	C₄⋊C₈	—	2	2	no	—
160	C₄⋊C₈	—	2	2	no	—
161	#103	—	4	4	no	—
162	C₄×C₈	—	2	1	no	—
163	C₄×C₈	D₄×C₅₃	1	2	yes	—
164	#95	—	2	4	no	—
165	#95	—	2	2	no	—
166	C₂×Q₈	—	2	2	no	—
167	C₂×Q₈	—	2	2	no	—
168	C₂×Q₈	—	2	2	no	—
169	C₂×Q₈	—	2	2	no	—
170	C₂×Q₈	—	2	2	no	—
171	C₂×Q₈	Q₁₆×C₅₃	1	4	yes	—
172	C₂²×C₄	—	2	2	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	4	1	no	—
176	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
177	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	2	no	—
180	C₂×C₈	—	2	2	no	—
181	C₂×C₈	—	4	2	no	—
182	C₂×C₈	—	4	2	no	—
183	C₂×C₈	—	4	2	no	—
184	C₄⋊C₄	—	2	2	no	—
185	C₄⋊C₄	—	2	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	—	4	2	no	—
188	C₄⋊C₄	—	4	2	no	—
189	C₄⋊C₄	—	4	2	no	—
190	C₄⋊C₄	—	4	2	no	—
191	C₄⋊C₄	—	4	2	no	—
192	C₄⋊C₄	—	4	2	no	—
193	C₄⋊C₄	—	4	2	no	—
194	C₄⋊C₄	—	8	2	no	—
195	C₄²	—	4	2	no	—
196	C₄²	—	4	2	no	—
197	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
198	C₄²	#41×C₅₃	1	1	yes	characteristic
199	C₄²	#41×C₅₃	1	2	yes	—
200	C₂³	#120×C₅₃	1	1	yes	characteristic
201	Q₈	—	2	4	no	—
202	Q₈	—	2	4	no	—
203	Q₈	—	4	2	no	—
204	Q₈	—	4	2	no	—
205	C₂×C₄	—	2	1	no	—
206	C₂×C₄	—	2	1	no	—
207	C₂×C₄	—	2	4	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	—	4	4	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	4	2	no	—
214	C₂×C₄	—	4	2	no	—
215	C₂×C₄	—	8	2	no	—
216	C₂×C₄	—	8	1	no	—
217	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
218	C₂×C₄	#121×C₅₃	1	2	yes	—
219	C₂×C₄	#123×C₅₃	1	2	yes	—
220	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
221	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
222	C₂×C₄	#135×C₅₃	1	2	yes	—
223	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
224	C₂×C₄	#137×C₅₃	1	2	yes	—
225	C₈	—	2	4	no	—
226	C₈	—	8	2	no	—
227	C₂²	C₅₃×#394	1	1	yes	characteristic
228	C₂²	C₅₃×#449	1	2	yes	—
229	C₂²	C₅₃×#451	1	2	yes	—
230	C₂²	C₅₃×#453	1	1	yes	characteristic
231	C₂²	C₅₃×#456	1	1	yes	characteristic
232	C₄	—	2	1	no	—
233	C₄	—	2	1	no	—
234	C₄	—	4	4	no	—
235	C₄	—	8	2	no	—
236	C₄	—	8	2	no	—
237	C₄	—	16	1	no	—
238	C₄	C₅₃×#467	1	2	yes	—
239	C₄	C₅₃×#468	1	2	yes	—
240	C₄	C₅₃×#484	1	2	yes	—
241	C₄	C₅₃×#485	1	2	yes	—
242	C₂	C₅₃×#1620	1	1	yes	characteristic
243	C₂	C₅₃×#1633	1	1	yes	characteristic
244	C₂	C₅₃×#1648	1	2	yes	—
245	C₂	C₅₃×#2036	1	2	yes	—
246	C₂	C₅₃×#2039	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2582

C₂

yes

maximal, characteristic

C₅₃×#1845

C₂

yes

maximal, characteristic

C₅₃×#1803

C₂

yes

maximal

C₂×C₅₃×Q₈⋊C₈

C₂

yes

maximal

C₅₃×#1296

C₂

yes

maximal, characteristic

C₂×C₄×Q₈×C₅₃

C₂²

yes

—

C₂×#107×C₅₃

—

C₂×C₄⋊C₈×C₅₃

C₂²

yes

—

C₂×#103×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#398

—

C₅₃×#391

—

C₅₃×#389

—

C₅₃×#384

C₂²

yes

characteristic

C₅₃×#384

C₂²

yes

characteristic

C₅₃×#340

—

C₅₃×Q₈⋊C₈

—

C₂²×Q₈×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₄×Q₈×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#107×C₅₃

—

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

#103×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

D₄

yes

—

#95×C₅₃

—

#95×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

Q₁₆

yes

—

C₂²×C₄×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

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄⋊C₄×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×D₄

yes

C₄²×C₅₃

yes

C₄²×C₅₃

yes

—

C₂³×C₅₃

#120

yes

characteristic

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×C₅₃

—

Q₈×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

—

C₂×C₄×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

#123

yes

—

C₂×C₄×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₂×C₄×C₅₃

#135

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

100

C₂×C₄×C₅₃

#137

yes

—

101

C₈×C₅₃

—

102

C₈×C₅₃

—

103

#10439

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

104

C₂²×C₅₃

#394

yes

characteristic

105

C₂²×C₅₃

#449

yes

—

106

C₂²×C₅₃

#451

yes

—

107

C₂²×C₅₃

#453

yes

characteristic

108

C₂²×C₅₃

#456

yes

characteristic

109

C₄×C₅₃

—

110

C₄×C₅₃

—

111

C₄×C₅₃

—

112

C₄×C₅₃

—

113

C₄×C₅₃

—

114

C₄×C₅₃

—

115

C₄×C₅₃

#467

yes

—

116

C₄×C₅₃

#468

yes

—

117

C₄×C₅₃

#484

yes

—

118

C₄×C₅₃

#485

yes

—

119

#2582

C₂×C₅₃

yes

characteristic

120

#1845

C₂×C₅₃

yes

characteristic

121

#1803

C₂×C₅₃

yes

—

122

C₂×Q₈⋊C₈

C₂×C₅₃

yes

—

123

#1296

C₂×C₅₃

yes

characteristic

124

C₂×C₅₃

#1620

yes

characteristic

125

C₂×C₅₃

#1633

yes

characteristic

126

C₂×C₅₃

#1648

yes

—

127

C₂×C₅₃

#2036

yes

—

128

C₂×C₅₃

#2039

yes

characteristic

129

C₂×C₄×Q₈

C₂²×C₅₃

yes

—

130

C₂×#107

—

131

C₂×C₄⋊C₈

C₂²×C₅₃

yes

—

132

C₂×#103

—

133

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

134

#398

—

135

#391

—

136

#389

—

137

#384

C₂²×C₅₃

yes

characteristic

138

#384

C₂²×C₅₃

yes

characteristic

139

#340

—

140

Q₈⋊C₈

—

141

C₅₃

#10439

yes

characteristic, direct product factor, Sylow

142

C₂²×Q₈

D₄×C₅₃

yes

—

143

C₂²×C₈

—

144

C₂²×C₈

—

145

C₄×Q₈

—

146

C₄×Q₈

—

147

C₄×Q₈

—

148

C₄×Q₈

—

149

C₂×C₄⋊C₄

—

150

C₂×C₄⋊C₄

—

151

C₂×C₄⋊C₄

—

152

C₂×C₄⋊C₄

—

153

C₂×C₄⋊C₄

—

154

C₂×C₄⋊C₄

D₄×C₅₃

yes

—

155

C₂×C₄⋊C₄

D₄×C₅₃

yes

—

156

C₂×C₄²

—

157

C₂×C₄²

C₂³×C₅₃

yes

characteristic

158

#107

—

159

C₄⋊C₈

—

160

C₄⋊C₈

—

161

#103

—

162

C₄×C₈

—

163

C₄×C₈

D₄×C₅₃

yes

—

164

#95

—

165

#95

—

166

C₂×Q₈

—

167

C₂×Q₈

—

168

C₂×Q₈

—

169

C₂×Q₈

—

170

C₂×Q₈

—

171

C₂×Q₈

Q₁₆×C₅₃

yes

—

172

C₂²×C₄

—

173

C₂²×C₄

—

174

C₂²×C₄

—

175

C₂²×C₄

—

176

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

177

C₂²×C₄

C₂×D₄×C₅₃

yes

—

178

C₂×C₈

—

179

C₂×C₈

—

180

C₂×C₈

—

181

C₂×C₈

—

182

C₂×C₈

—

183

C₂×C₈

—

184

C₄⋊C₄

—

185

C₄⋊C₄

—

186

C₄⋊C₄

—

187

C₄⋊C₄

—

188

C₄⋊C₄

—

189

C₄⋊C₄

—

190

C₄⋊C₄

—

191

C₄⋊C₄

—

192

C₄⋊C₄

—

193

C₄⋊C₄

—

194

C₄⋊C₄

—

195

C₄²

—

196

C₄²

—

197

C₄²

C₂×D₄×C₅₃

yes

characteristic

198

C₄²

#41×C₅₃

yes

characteristic

199

C₄²

#41×C₅₃

yes

—

200

C₂³

#120×C₅₃

yes

characteristic

201

Q₈

—

202

Q₈

—

203

Q₈

—

204

Q₈

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

—

212

C₂×C₄

—

213

C₂×C₄

—

214

C₂×C₄

—

215

C₂×C₄

—

216

C₂×C₄

—

217

C₂×C₄

#121×C₅₃

yes

characteristic

218

C₂×C₄

#121×C₅₃

yes

—

219

C₂×C₄

#123×C₅₃

yes

—

220

C₂×C₄

#124×C₅₃

yes

characteristic

221

C₂×C₄

C₂×Q₁₆×C₅₃

yes

—

222

C₂×C₄

#135×C₅₃

yes

—

223

C₂×C₄

AGL(1,ℤ/8)×C₅₃

yes

—

224

C₂×C₄

#137×C₅₃

yes

—

225

C₈

—

226

C₈

—

227

C₂²

C₅₃×#394

yes

characteristic

228

C₂²

C₅₃×#449

yes

—

229

C₂²

C₅₃×#451

yes

—

230

C₂²

C₅₃×#453

yes

characteristic

231

C₂²

C₅₃×#456

yes

characteristic

232

C₄

—

233

C₄

—

234

C₄

—

235

C₄

—

236

C₄

—

237

C₄

—

238

C₄

C₅₃×#467

yes

—

239

C₄

C₅₃×#468

yes

—

240

C₄

C₅₃×#484

yes

—

241

C₄

C₅₃×#485

yes

—

242

C₂

C₅₃×#1620

yes

characteristic

243

C₂

C₅₃×#1633

yes

characteristic

244

C₂

C₅₃×#1648

yes

—

245

C₂

C₅₃×#2036

yes

—

246

C₂

C₅₃×#2039

yes

characteristic

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

subgroup

G/H

center, socle, prime radical

C₂³×C₅₃

#120

commutator, Frattini, cosocle

C₂×C₄²

C₂³×C₅₃

LCS

G ▷ C₂×C₄² ▷ C₂² ▷ C₁

UCS

C₁ ◁ C₂³×C₅₃ ◁ C₂×C₄²×C₅₃ ◁ G

N	H ≅ G/N	type
103. #10439	C₅₃	direct (N × H)
141. C₅₃	#10439	direct (N × H)

H ≅ G/N

type

103. #10439

C₅₃

direct (N × H)

141. C₅₃

#10439

direct (N × H)

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

subgroup	index
C₅₃×#2582	2
C₅₃×#1845	2
C₅₃×#1803	2
C₂×C₅₃×Q₈⋊C₈	2
C₅₃×#1296	2
#10439	53

subgroup

index

C₅₃×#2582

C₅₃×#1845

C₅₃×#1803

C₂×C₅₃×Q₈⋊C₈

C₅₃×#1296

#10439

Subgroups — C₅₃×#10439 #27217951

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#10439 #27217951

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10439 #27217951