Group #27217861 — subgroups

Subgroups — C₅₃×#10349 #27217861

Order 13568.

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

normal subgroups only

This group has 504 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
2	C₅₃×#1987	C₂	1	2	yes	maximal
3	C₅₃×#1956	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1296	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1279	C₂	1	2	yes	maximal
6	C₂×C₄⋊Q₈×C₅₃	—	2	2	no	—
7	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
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×#395	—	2	2	no	—
12	C₅₃×#389	—	2	2	no	—
13	C₅₃×#384	C₂²	1	1	yes	characteristic
14	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
15	C₅₃×#340	—	2	2	no	—
16	C₅₃×#340	—	2	2	no	—
17	C₂²×Q₈×C₅₃	—	2	2	no	—
18	C₂²×Q₈×C₅₃	—	2	1	no	—
19	C₂²×C₈×C₅₃	—	2	1	no	—
20	C₂²×C₈×C₅₃	—	2	2	no	—
21	C₄⋊Q₈×C₅₃	—	2	1	no	—
22	C₄⋊Q₈×C₅₃	—	2	2	no	—
23	C₄⋊Q₈×C₅₃	—	4	4	no	—
24	C₄⋊Q₈×C₅₃	—	4	4	no	—
25	C₄⋊Q₈×C₅₃	Q₈	1	2	yes	complemented
26	C₂×C₄⋊C₄×C₅₃	—	2	4	no	—
27	C₂×C₄⋊C₄×C₅₃	—	2	2	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	#103×C₅₃	—	4	4	no	—
37	C₄×C₈×C₅₃	—	2	2	no	—
38	#95×C₅₃	—	2	2	no	—
39	#95×C₅₃	—	2	2	no	—
40	C₂×Q₈×C₅₃	—	2	2	no	—
41	C₂×Q₈×C₅₃	—	4	8	no	—
42	C₂×Q₈×C₅₃	—	4	2	no	—
43	C₂×Q₈×C₅₃	—	4	2	no	—
44	C₂×Q₈×C₅₃	—	4	1	no	—
45	C₂×Q₈×C₅₃	—	8	2	no	—
46	C₂²×C₄×C₅₃	—	2	2	no	—
47	C₂²×C₄×C₅₃	—	2	2	no	—
48	C₂²×C₄×C₅₃	—	2	2	no	—
49	C₂²×C₄×C₅₃	—	4	1	no	—
50	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
52	C₂×C₈×C₅₃	—	2	2	no	—
53	C₂×C₈×C₅₃	—	4	1	no	—
54	C₂×C₈×C₅₃	—	4	1	no	—
55	C₂×C₈×C₅₃	—	4	2	no	—
56	C₂×C₈×C₅₃	—	4	2	no	—
57	C₂×C₈×C₅₃	—	4	2	no	—
58	C₄⋊C₄×C₅₃	—	2	2	no	—
59	C₄⋊C₄×C₅₃	—	2	2	no	—
60	C₄⋊C₄×C₅₃	—	2	4	no	—
61	C₄⋊C₄×C₅₃	—	4	8	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	SD₁₆	1	4	yes	—
67	C₄⋊C₄×C₅₃	Q₁₆	1	4	yes	—
68	C₄²×C₅₃	—	4	2	no	—
69	C₄²×C₅₃	—	4	2	no	—
70	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
71	C₄²×C₅₃	#41	1	2	yes	—
72	C₄²×C₅₃	#41	1	1	yes	characteristic
73	C₂³×C₅₃	#120	1	1	yes	characteristic
74	Q₈×C₅₃	—	4	4	no	—
75	Q₈×C₅₃	—	8	2	no	—
76	Q₈×C₅₃	—	8	8	no	—
77	C₂×C₄×C₅₃	—	2	1	no	—
78	C₂×C₄×C₅₃	—	2	1	no	—
79	C₂×C₄×C₅₃	—	2	4	no	—
80	C₂×C₄×C₅₃	—	2	4	no	—
81	C₂×C₄×C₅₃	—	4	2	no	—
82	C₂×C₄×C₅₃	—	4	2	no	—
83	C₂×C₄×C₅₃	—	4	2	no	—
84	C₂×C₄×C₅₃	—	4	2	no	—
85	C₂×C₄×C₅₃	—	4	2	no	—
86	C₂×C₄×C₅₃	—	4	4	no	—
87	C₂×C₄×C₅₃	—	4	2	no	—
88	C₂×C₄×C₅₃	—	8	1	no	—
89	C₂×C₄×C₅₃	—	8	1	no	—
90	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
92	C₂×C₄×C₅₃	#124	1	1	yes	characteristic
93	C₂×C₄×C₅₃	#124	1	2	yes	—
94	C₂×C₄×C₅₃	C₂×SD₁₆	1	2	yes	—
95	C₂×C₄×C₅₃	C₂×Q₁₆	1	2	yes	—
96	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
97	C₂×C₄×C₅₃	#137	1	2	yes	—
98	C₈×C₅₃	—	4	2	no	—
99	C₈×C₅₃	—	8	2	no	—
100	#10349	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
101	C₂²×C₅₃	#393	1	1	yes	characteristic
102	C₂²×C₅₃	#450	1	2	yes	—
103	C₂²×C₅₃	#451	1	2	yes	—
104	C₂²×C₅₃	#453	1	1	yes	characteristic
105	C₂²×C₅₃	#454	1	1	yes	characteristic
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	—	2	1	no	—
108	C₄×C₅₃	—	4	4	no	—
109	C₄×C₅₃	—	4	4	no	—
110	C₄×C₅₃	—	8	2	no	—
111	C₄×C₅₃	—	8	2	no	—
112	C₄×C₅₃	#475	1	2	yes	—
113	C₄×C₅₃	#477	1	2	yes	—
114	C₄×C₅₃	#487	1	2	yes	—
115	C₄×C₅₃	#489	1	2	yes	—
116	#2551	C₂×C₅₃	1	1	yes	characteristic
117	#1987	C₂×C₅₃	1	2	yes	—
118	#1956	C₂×C₅₃	1	1	yes	characteristic
119	#1296	C₂×C₅₃	1	1	yes	characteristic
120	#1279	C₂×C₅₃	1	2	yes	—
121	C₂×C₅₃	#1629	1	1	yes	characteristic
122	C₂×C₅₃	#1633	1	1	yes	characteristic
123	C₂×C₅₃	#1637	1	2	yes	—
124	C₂×C₅₃	#2024	1	2	yes	—
125	C₂×C₅₃	#2027	1	1	yes	characteristic
126	C₂×C₄⋊Q₈	—	2	2	no	—
127	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
128	C₂×C₄⋊C₈	C₂²×C₅₃	1	2	yes	—
129	C₂×#103	—	2	2	no	—
130	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
131	#395	—	2	2	no	—
132	#389	—	2	2	no	—
133	#384	C₂²×C₅₃	1	1	yes	characteristic
134	C₄×C₄⋊C₄	C₂²×C₅₃	1	2	yes	—
135	#340	—	2	2	no	—
136	#340	—	2	2	no	—
137	C₅₃	#10349	1	1	yes	characteristic, direct product factor, Sylow
138	C₂²×Q₈	—	2	2	no	—
139	C₂²×Q₈	—	2	1	no	—
140	C₂²×C₈	—	2	1	no	—
141	C₂²×C₈	—	2	2	no	—
142	C₄⋊Q₈	—	2	2	no	—
143	C₄⋊Q₈	—	2	1	no	—
144	C₄⋊Q₈	—	4	4	no	—
145	C₄⋊Q₈	—	4	4	no	—
146	C₄⋊Q₈	Q₈×C₅₃	1	2	yes	complemented
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	4	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
151	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
152	C₂×C₄⋊C₄	D₄×C₅₃	1	2	yes	—
153	C₂×C₄²	—	2	2	no	—
154	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
155	C₄⋊C₈	—	2	2	no	—
156	C₄⋊C₈	—	2	2	no	—
157	#103	—	4	4	no	—
158	C₄×C₈	—	2	2	no	—
159	#95	—	2	2	no	—
160	#95	—	2	2	no	—
161	C₂×Q₈	—	2	2	no	—
162	C₂×Q₈	—	4	8	no	—
163	C₂×Q₈	—	4	2	no	—
164	C₂×Q₈	—	4	2	no	—
165	C₂×Q₈	—	4	1	no	—
166	C₂×Q₈	—	8	2	no	—
167	C₂²×C₄	—	2	2	no	—
168	C₂²×C₄	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	4	1	no	—
171	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
172	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	4	1	no	—
175	C₂×C₈	—	4	1	no	—
176	C₂×C₈	—	4	2	no	—
177	C₂×C₈	—	4	2	no	—
178	C₂×C₈	—	4	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	2	no	—
181	C₄⋊C₄	—	2	4	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	4	8	no	—
184	C₄⋊C₄	—	4	2	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	4	2	no	—
187	C₄⋊C₄	SD₁₆×C₅₃	1	4	yes	—
188	C₄⋊C₄	Q₁₆×C₅₃	1	4	yes	—
189	C₄²	—	4	2	no	—
190	C₄²	—	4	2	no	—
191	C₄²	C₂×Q₈×C₅₃	1	1	yes	characteristic
192	C₄²	#41×C₅₃	1	2	yes	—
193	C₄²	#41×C₅₃	1	1	yes	characteristic
194	C₂³	#120×C₅₃	1	1	yes	characteristic
195	Q₈	—	4	4	no	—
196	Q₈	—	8	2	no	—
197	Q₈	—	8	8	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	2	1	no	—
200	C₂×C₄	—	2	4	no	—
201	C₂×C₄	—	2	4	no	—
202	C₂×C₄	—	4	2	no	—
203	C₂×C₄	—	4	2	no	—
204	C₂×C₄	—	4	2	no	—
205	C₂×C₄	—	4	2	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	4	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	8	1	no	—
210	C₂×C₄	—	8	1	no	—
211	C₂×C₄	C₂²⋊Q₈×C₅₃	1	1	yes	characteristic
212	C₂×C₄	C₂²⋊Q₈×C₅₃	1	2	yes	—
213	C₂×C₄	#124×C₅₃	1	1	yes	characteristic
214	C₂×C₄	#124×C₅₃	1	2	yes	—
215	C₂×C₄	C₂×SD₁₆×C₅₃	1	2	yes	—
216	C₂×C₄	C₂×Q₁₆×C₅₃	1	2	yes	—
217	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
218	C₂×C₄	#137×C₅₃	1	2	yes	—
219	C₈	—	4	2	no	—
220	C₈	—	8	2	no	—
221	C₂²	C₅₃×#393	1	1	yes	characteristic
222	C₂²	C₅₃×#450	1	2	yes	—
223	C₂²	C₅₃×#451	1	2	yes	—
224	C₂²	C₅₃×#453	1	1	yes	characteristic
225	C₂²	C₅₃×#454	1	1	yes	characteristic
226	C₄	—	2	1	no	—
227	C₄	—	2	1	no	—
228	C₄	—	4	4	no	—
229	C₄	—	4	4	no	—
230	C₄	—	8	2	no	—
231	C₄	—	8	2	no	—
232	C₄	C₅₃×#475	1	2	yes	—
233	C₄	C₅₃×#477	1	2	yes	—
234	C₄	C₅₃×#487	1	2	yes	—
235	C₄	C₅₃×#489	1	2	yes	—
236	C₂	C₅₃×#1629	1	1	yes	characteristic
237	C₂	C₅₃×#1633	1	1	yes	characteristic
238	C₂	C₅₃×#1637	1	2	yes	—
239	C₂	C₅₃×#2024	1	2	yes	—
240	C₂	C₅₃×#2027	1	1	yes	characteristic

subgroup

G/H

class size

multiplicity

normal?

properties

C₅₃×#2551

C₂

yes

maximal, characteristic

C₅₃×#1987

C₂

yes

maximal

C₅₃×#1956

C₂

yes

maximal, characteristic

C₅₃×#1296

C₂

yes

maximal, characteristic

C₅₃×#1279

C₂

yes

maximal

C₂×C₄⋊Q₈×C₅₃

—

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊C₈×C₅₃

C₂²

yes

—

C₂×#103×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₅₃×#395

—

C₅₃×#389

—

C₅₃×#384

C₂²

yes

characteristic

C₄×C₄⋊C₄×C₅₃

C₂²

yes

—

C₅₃×#340

—

C₅₃×#340

—

C₂²×Q₈×C₅₃

—

C₂²×Q₈×C₅₃

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

Q₈

yes

complemented

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×C₄⋊C₄×C₅₃

D₄

yes

—

C₂×C₄²×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

C₄⋊C₈×C₅₃

—

C₄⋊C₈×C₅₃

—

#103×C₅₃

—

C₄×C₈×C₅₃

—

#95×C₅₃

—

#95×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×C₅₃

—

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

SD₁₆

yes

—

C₄⋊C₄×C₅₃

Q₁₆

yes

—

C₄²×C₅₃

—

C₄²×C₅₃

—

C₄²×C₅₃

C₂×Q₈

yes

characteristic

C₄²×C₅₃

#41

yes

—

C₄²×C₅₃

yes

C₂³×C₅₃

yes

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

—

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

characteristic

C₂×C₄×C₅₃

C₂²⋊Q₈

yes

—

C₂×C₄×C₅₃

#124

yes

characteristic

C₂×C₄×C₅₃

#124

yes

—

C₂×C₄×C₅₃

C₂×SD₁₆

yes

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

100

#10349

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

101

C₂²×C₅₃

#393

yes

characteristic

102

C₂²×C₅₃

#450

yes

—

103

C₂²×C₅₃

#451

yes

—

104

C₂²×C₅₃

#453

yes

characteristic

105

C₂²×C₅₃

#454

yes

characteristic

106

C₄×C₅₃

—

107

C₄×C₅₃

—

108

C₄×C₅₃

—

109

C₄×C₅₃

—

110

C₄×C₅₃

—

111

C₄×C₅₃

—

112

C₄×C₅₃

#475

yes

—

113

C₄×C₅₃

#477

yes

—

114

C₄×C₅₃

#487

yes

—

115

C₄×C₅₃

#489

yes

—

116

#2551

C₂×C₅₃

yes

characteristic

117

#1987

C₂×C₅₃

yes

—

118

#1956

C₂×C₅₃

yes

characteristic

119

#1296

C₂×C₅₃

yes

characteristic

120

#1279

C₂×C₅₃

yes

—

121

C₂×C₅₃

#1629

yes

characteristic

122

C₂×C₅₃

#1633

yes

characteristic

123

C₂×C₅₃

#1637

yes

—

124

C₂×C₅₃

#2024

yes

—

125

C₂×C₅₃

#2027

yes

characteristic

126

C₂×C₄⋊Q₈

—

127

C₂×C₄⋊Q₈

C₂²×C₅₃

yes

characteristic

128

C₂×C₄⋊C₈

C₂²×C₅₃

yes

—

129

C₂×#103

—

130

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

131

#395

—

132

#389

—

133

#384

C₂²×C₅₃

yes

characteristic

134

C₄×C₄⋊C₄

C₂²×C₅₃

yes

—

135

#340

—

136

#340

—

137

C₅₃

#10349

yes

characteristic, direct product factor, Sylow

138

C₂²×Q₈

—

139

C₂²×Q₈

—

140

C₂²×C₈

—

141

C₂²×C₈

—

142

C₄⋊Q₈

—

143

C₄⋊Q₈

—

144

C₄⋊Q₈

—

145

C₄⋊Q₈

—

146

C₄⋊Q₈

Q₈×C₅₃

yes

complemented

147

C₂×C₄⋊C₄

—

148

C₂×C₄⋊C₄

—

149

C₂×C₄⋊C₄

—

150

C₂×C₄⋊C₄

D₄×C₅₃

yes

—

151

C₂×C₄⋊C₄

D₄×C₅₃

yes

—

152

C₂×C₄⋊C₄

D₄×C₅₃

yes

—

153

C₂×C₄²

—

154

C₂×C₄²

C₂³×C₅₃

yes

characteristic

155

C₄⋊C₈

—

156

C₄⋊C₈

—

157

#103

—

158

C₄×C₈

—

159

#95

—

160

#95

—

161

C₂×Q₈

—

162

C₂×Q₈

—

163

C₂×Q₈

—

164

C₂×Q₈

—

165

C₂×Q₈

—

166

C₂×Q₈

—

167

C₂²×C₄

—

168

C₂²×C₄

—

169

C₂²×C₄

—

170

C₂²×C₄

—

171

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

172

C₂²×C₄

C₂×D₄×C₅₃

yes

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

—

177

C₂×C₈

—

178

C₂×C₈

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₄⋊C₄

—

184

C₄⋊C₄

—

185

C₄⋊C₄

—

186

C₄⋊C₄

—

187

C₄⋊C₄

SD₁₆×C₅₃

yes

—

188

C₄⋊C₄

Q₁₆×C₅₃

yes

—

189

C₄²

—

190

C₄²

—

191

C₄²

C₂×Q₈×C₅₃

yes

characteristic

192

C₄²

#41×C₅₃

yes

—

193

C₄²

#41×C₅₃

yes

characteristic

194

C₂³

#120×C₅₃

yes

characteristic

195

Q₈

—

196

Q₈

—

197

Q₈

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

—

203

C₂×C₄

—

204

C₂×C₄

—

205

C₂×C₄

—

206

C₂×C₄

—

207

C₂×C₄

—

208

C₂×C₄

—

209

C₂×C₄

—

210

C₂×C₄

—

211

C₂×C₄

C₂²⋊Q₈×C₅₃

yes

characteristic

212

C₂×C₄

C₂²⋊Q₈×C₅₃

yes

—

213

C₂×C₄

#124×C₅₃

yes

characteristic

214

C₂×C₄

#124×C₅₃

yes

—

215

C₂×C₄

C₂×SD₁₆×C₅₃

yes

—

216

C₂×C₄

C₂×Q₁₆×C₅₃

yes

—

217

C₂×C₄

AGL(1,ℤ/8)×C₅₃

yes

—

218

C₂×C₄

#137×C₅₃

yes

—

219

C₈

—

220

C₈

—

221

C₂²

C₅₃×#393

yes

characteristic

222

C₂²

C₅₃×#450

yes

—

223

C₂²

C₅₃×#451

yes

—

224

C₂²

C₅₃×#453

yes

characteristic

225

C₂²

C₅₃×#454

yes

characteristic

226

C₄

—

227

C₄

—

228

C₄

—

229

C₄

—

230

C₄

—

231

C₄

—

232

C₄

C₅₃×#475

yes

—

233

C₄

C₅₃×#477

yes

—

234

C₄

C₅₃×#487

yes

—

235

C₄

C₅₃×#489

yes

—

236

C₂

C₅₃×#1629

yes

characteristic

237

C₂

C₅₃×#1633

yes

characteristic

238

C₂

C₅₃×#1637

yes

—

239

C₂

C₅₃×#2024

yes

—

240

C₂

C₅₃×#2027

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
25. C₄⋊Q₈×C₅₃	Q₈	semidirect
100. #10349	C₅₃	direct (N × H)
137. C₅₃	#10349	direct (N × H)
146. C₄⋊Q₈	Q₈×C₅₃	semidirect

H ≅ G/N

type

25. C₄⋊Q₈×C₅₃

Q₈

semidirect

100. #10349

C₅₃

direct (N × H)

137. C₅₃

#10349

direct (N × H)

146. C₄⋊Q₈

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×#1987	2
C₅₃×#1956	2
C₅₃×#1296	2
C₅₃×#1279	2
#10349	53

subgroup

index

C₅₃×#2551

C₅₃×#1987

C₅₃×#1956

C₅₃×#1296

C₅₃×#1279

#10349

Subgroups — C₅₃×#10349 #27217861

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#10349 #27217861

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10349 #27217861