Group #27217821 — subgroups

Subgroups — C₅₃×#10309 #27217821

Order 13568.

618 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 618 conjugacy classes of subgroups — too many to draw. Only its 186 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	C₂×C₅₃×#500	C₂	1	2	yes	maximal
2	C₂×C₅₃×#494	C₂	1	1	yes	maximal, characteristic
3	C₅₃×#1990	C₂	1	2	yes	maximal
4	C₅₃×#1956	C₂	1	1	yes	maximal, characteristic
5	C₅₃×#1273	C₂	1	1	yes	maximal, characteristic
6	C₂²×Q₁₆×C₅₃	—	2	2	no	—
7	C₂×C₄⋊Q₈×C₅₃	C₂²	1	1	yes	characteristic
8	C₂×C₄⋊Q₈×C₅₃	C₂²	1	2	yes	—
9	C₅₃×#500	—	2	4	no	—
10	C₅₃×#500	—	2	4	no	—
11	C₅₃×#494	—	2	4	no	—
12	C₂×#107×C₅₃	—	2	4	no	—
13	C₂×#103×C₅₃	—	2	2	no	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	2	yes	—
16	C₅₃×#384	C₂²	1	1	yes	characteristic
17	C₅₃×#340	—	2	4	no	—
18	C₂²×Q₈×C₅₃	—	2	2	no	—
19	C₂²×Q₈×C₅₃	—	2	1	no	—
20	C₂×Q₁₆×C₅₃	—	2	8	no	—
21	C₂×Q₁₆×C₅₃	—	4	4	no	—
22	C₂×Q₁₆×C₅₃	—	4	4	no	—
23	C₂²×C₈×C₅₃	D₄	1	2	yes	—
24	C₂²×C₈×C₅₃	D₄	1	4	yes	—
25	C₄⋊Q₈×C₅₃	—	2	2	no	—
26	C₄⋊Q₈×C₅₃	—	2	1	no	—
27	C₄⋊Q₈×C₅₃	—	2	2	no	—
28	C₄⋊Q₈×C₅₃	—	2	2	no	—
29	C₄⋊Q₈×C₅₃	—	2	2	no	—
30	C₄⋊Q₈×C₅₃	D₄	1	4	yes	—
31	C₄⋊Q₈×C₅₃	D₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
34	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
35	C₂×C₄⋊C₄×C₅₃	—	2	2	no	—
36	C₂×C₄⋊C₄×C₅₃	—	2	1	no	—
37	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
38	#107×C₅₃	—	2	16	no	—
39	#103×C₅₃	—	4	4	no	—
40	C₄×C₈×C₅₃	—	2	2	no	—
41	C₄×C₈×C₅₃	—	2	2	no	—
42	C₄×C₈×C₅₃	—	2	2	no	—
43	C₂×Q₈×C₅₃	—	2	4	no	—
44	C₂×Q₈×C₅₃	—	2	2	no	—
45	C₂×Q₈×C₅₃	—	4	1	no	—
46	C₂×Q₈×C₅₃	—	4	2	no	—
47	C₂×Q₈×C₅₃	—	4	2	no	—
48	C₂×Q₈×C₅₃	—	4	4	no	—
49	C₂×Q₈×C₅₃	—	4	4	no	—
50	C₂×Q₈×C₅₃	—	4	2	no	—
51	C₂²×C₄×C₅₃	—	4	2	no	—
52	C₂²×C₄×C₅₃	—	4	1	no	—
53	C₂²×C₄×C₅₃	—	4	1	no	—
54	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
55	C₂²×C₄×C₅₃	C₂×D₄	1	2	yes	—
56	Q₁₆×C₅₃	—	4	16	no	—
57	C₂×C₈×C₅₃	—	2	4	no	—
58	C₂×C₈×C₅₃	—	2	2	no	—
59	C₂×C₈×C₅₃	—	2	2	no	—
60	C₂×C₈×C₅₃	—	2	4	no	—
61	C₂×C₈×C₅₃	D₈	1	4	yes	—
62	C₂×C₈×C₅₃	Q₁₆	1	8	yes	complemented
63	C₄⋊C₄×C₅₃	—	2	4	no	—
64	C₄⋊C₄×C₅₃	—	2	4	no	—
65	C₄⋊C₄×C₅₃	—	2	4	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	—	4	4	no	—
71	C₄²×C₅₃	C₂×D₄	1	2	yes	—
72	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
73	C₄²×C₅₃	#41	1	1	yes	characteristic
74	C₂³×C₅₃	#127	1	1	yes	characteristic
75	Q₈×C₅₃	—	4	8	no	—
76	Q₈×C₅₃	—	4	4	no	—
77	Q₈×C₅₃	—	8	2	no	—
78	Q₈×C₅₃	—	8	4	no	—
79	C₂×C₄×C₅₃	—	4	4	no	—
80	C₂×C₄×C₅₃	—	4	2	no	—
81	C₂×C₄×C₅₃	—	8	1	no	—
82	C₂×C₄×C₅₃	—	8	2	no	—
83	C₂×C₄×C₅₃	—	8	2	no	—
84	C₂×C₄×C₅₃	—	8	1	no	—
85	C₂×C₄×C₅₃	—	8	1	no	—
86	C₂×C₄×C₅₃	—	8	2	no	—
87	C₂×C₄×C₅₃	#120	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#120	1	2	yes	—
89	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#121	1	2	yes	—
91	C₂×C₄×C₅₃	C₂×D₈	1	2	yes	—
92	C₂×C₄×C₅₃	C₂×Q₁₆	1	4	yes	—
93	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
94	C₂×C₄×C₅₃	#137	1	4	yes	—
95	C₈×C₅₃	—	2	4	no	—
96	C₈×C₅₃	—	2	8	no	—
97	#10309	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
98	C₂²×C₅₃	#392	1	1	yes	characteristic
99	C₂²×C₅₃	#493	1	1	yes	characteristic
100	C₂²×C₅₃	#494	1	2	yes	—
101	C₂²×C₅₃	#496	1	1	yes	characteristic
102	C₂²×C₅₃	#497	1	2	yes	—
103	C₄×C₅₃	—	8	4	no	—
104	C₄×C₅₃	—	8	2	no	—
105	C₄×C₅₃	—	16	1	no	—
106	C₄×C₅₃	#447	1	2	yes	—
107	C₄×C₅₃	#451	1	4	yes	—
108	C₄×C₅₃	#459	1	2	yes	—
109	C₄×C₅₃	#462	1	4	yes	—
110	C₂×#500	C₂×C₅₃	1	2	yes	—
111	C₂×#494	C₂×C₅₃	1	1	yes	characteristic
112	#1990	C₂×C₅₃	1	2	yes	—
113	#1956	C₂×C₅₃	1	1	yes	characteristic
114	#1273	C₂×C₅₃	1	1	yes	characteristic
115	C₂×C₅₃	#1713	1	1	yes	characteristic
116	C₂×C₅₃	#1722	1	1	yes	characteristic
117	C₂×C₅₃	#1723	1	2	yes	—
118	C₂×C₅₃	#2011	1	1	yes	characteristic
119	C₂×C₅₃	#2016	1	2	yes	—
120	C₂²×Q₁₆	—	2	2	no	—
121	C₂×C₄⋊Q₈	C₂²×C₅₃	1	2	yes	—
122	C₂×C₄⋊Q₈	C₂²×C₅₃	1	1	yes	characteristic
123	#500	—	2	4	no	—
124	#500	—	2	4	no	—
125	#494	—	2	4	no	—
126	C₂×#107	—	2	4	no	—
127	C₂×#103	—	2	2	no	—
128	C₂×C₄×C₈	C₂²×C₅₃	1	1	yes	characteristic
129	C₂×C₄×C₈	C₂²×C₅₃	1	2	yes	—
130	#384	C₂²×C₅₃	1	1	yes	characteristic
131	#340	—	2	4	no	—
132	C₅₃	#10309	1	1	yes	characteristic, direct product factor, Sylow
133	C₂²×Q₈	—	2	2	no	—
134	C₂²×Q₈	—	2	1	no	—
135	C₂×Q₁₆	—	2	8	no	—
136	C₂×Q₁₆	—	4	4	no	—
137	C₂×Q₁₆	—	4	4	no	—
138	C₂²×C₈	D₄×C₅₃	1	2	yes	—
139	C₂²×C₈	D₄×C₅₃	1	4	yes	—
140	C₄⋊Q₈	—	2	2	no	—
141	C₄⋊Q₈	—	2	2	no	—
142	C₄⋊Q₈	—	2	2	no	—
143	C₄⋊Q₈	—	2	2	no	—
144	C₄⋊Q₈	—	2	1	no	—
145	C₄⋊Q₈	D₄×C₅₃	1	4	yes	—
146	C₄⋊Q₈	D₄×C₅₃	1	2	yes	—
147	C₂×C₄⋊C₄	—	2	2	no	—
148	C₂×C₄⋊C₄	—	2	2	no	—
149	C₂×C₄⋊C₄	—	2	2	no	—
150	C₂×C₄⋊C₄	—	2	2	no	—
151	C₂×C₄⋊C₄	—	2	1	no	—
152	C₂×C₄²	C₂³×C₅₃	1	1	yes	characteristic
153	#107	—	2	16	no	—
154	#103	—	4	4	no	—
155	C₄×C₈	—	2	2	no	—
156	C₄×C₈	—	2	2	no	—
157	C₄×C₈	—	2	2	no	—
158	C₂×Q₈	—	2	4	no	—
159	C₂×Q₈	—	2	2	no	—
160	C₂×Q₈	—	4	1	no	—
161	C₂×Q₈	—	4	2	no	—
162	C₂×Q₈	—	4	2	no	—
163	C₂×Q₈	—	4	4	no	—
164	C₂×Q₈	—	4	4	no	—
165	C₂×Q₈	—	4	2	no	—
166	C₂²×C₄	—	4	2	no	—
167	C₂²×C₄	—	4	1	no	—
168	C₂²×C₄	—	4	1	no	—
169	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
170	C₂²×C₄	C₂×D₄×C₅₃	1	2	yes	—
171	Q₁₆	—	4	16	no	—
172	C₂×C₈	—	2	4	no	—
173	C₂×C₈	—	2	2	no	—
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	4	no	—
176	C₂×C₈	D₈×C₅₃	1	4	yes	—
177	C₂×C₈	Q₁₆×C₅₃	1	8	yes	complemented
178	C₄⋊C₄	—	2	4	no	—
179	C₄⋊C₄	—	2	4	no	—
180	C₄⋊C₄	—	2	4	no	—
181	C₄⋊C₄	—	4	2	no	—
182	C₄⋊C₄	—	4	2	no	—
183	C₄⋊C₄	—	4	2	no	—
184	C₄⋊C₄	—	4	2	no	—
185	C₄⋊C₄	—	4	4	no	—
186	C₄²	C₂×D₄×C₅₃	1	2	yes	—
187	C₄²	C₂×D₄×C₅₃	1	1	yes	characteristic
188	C₄²	#41×C₅₃	1	1	yes	characteristic
189	C₂³	#127×C₅₃	1	1	yes	characteristic
190	Q₈	—	4	8	no	—
191	Q₈	—	4	4	no	—
192	Q₈	—	8	2	no	—
193	Q₈	—	8	4	no	—
194	C₂×C₄	—	4	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	—	8	1	no	—
197	C₂×C₄	—	8	2	no	—
198	C₂×C₄	—	8	2	no	—
199	C₂×C₄	—	8	1	no	—
200	C₂×C₄	—	8	1	no	—
201	C₂×C₄	—	8	2	no	—
202	C₂×C₄	#120×C₅₃	1	1	yes	characteristic
203	C₂×C₄	#120×C₅₃	1	2	yes	—
204	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
205	C₂×C₄	#121×C₅₃	1	2	yes	—
206	C₂×C₄	C₂×D₈×C₅₃	1	2	yes	—
207	C₂×C₄	C₂×Q₁₆×C₅₃	1	4	yes	—
208	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
209	C₂×C₄	#137×C₅₃	1	4	yes	—
210	C₈	—	2	4	no	—
211	C₈	—	2	8	no	—
212	C₂²	C₅₃×#392	1	1	yes	characteristic
213	C₂²	C₅₃×#493	1	1	yes	characteristic
214	C₂²	C₅₃×#494	1	2	yes	—
215	C₂²	C₅₃×#496	1	1	yes	characteristic
216	C₂²	C₅₃×#497	1	2	yes	—
217	C₄	—	8	4	no	—
218	C₄	—	8	2	no	—
219	C₄	—	16	1	no	—
220	C₄	C₅₃×#447	1	2	yes	—
221	C₄	C₅₃×#451	1	4	yes	—
222	C₄	C₅₃×#459	1	2	yes	—
223	C₄	C₅₃×#462	1	4	yes	—
224	C₂	C₅₃×#1713	1	1	yes	characteristic
225	C₂	C₅₃×#1722	1	1	yes	characteristic
226	C₂	C₅₃×#1723	1	2	yes	—
227	C₂	C₅₃×#2011	1	1	yes	characteristic
228	C₂	C₅₃×#2016	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×C₅₃×#500

C₂

yes

maximal

C₂×C₅₃×#494

C₂

yes

maximal, characteristic

C₅₃×#1990

C₂

yes

maximal

C₅₃×#1956

C₂

yes

maximal, characteristic

C₅₃×#1273

C₂

yes

maximal, characteristic

C₂²×Q₁₆×C₅₃

—

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

characteristic

C₂×C₄⋊Q₈×C₅₃

C₂²

yes

—

C₅₃×#500

—

C₅₃×#500

—

C₅₃×#494

—

C₂×#107×C₅₃

—

C₂×#103×C₅₃

—

C₂×C₄×C₈×C₅₃

C₂²

yes

characteristic

C₂×C₄×C₈×C₅₃

C₂²

yes

—

C₅₃×#384

C₂²

yes

characteristic

C₅₃×#340

—

C₂²×Q₈×C₅₃

—

C₂²×Q₈×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

D₄

yes

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

—

C₄⋊Q₈×C₅₃

D₄

yes

—

C₄⋊Q₈×C₅₃

D₄

yes

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄⋊C₄×C₅₃

—

C₂×C₄²×C₅₃

C₂³

yes

characteristic

#107×C₅₃

—

#103×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₄×C₈×C₅₃

—

C₂×Q₈×C₅₃

—

C₂×Q₈×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

C₂×D₄

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

—

Q₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

D₈

yes

—

C₂×C₈×C₅₃

Q₁₆

yes

complemented

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

C₂×D₄

yes

—

C₄²×C₅₃

C₂×D₄

yes

C₄²×C₅₃

yes

C₂³×C₅₃

yes

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

#120

yes

characteristic

C₂×C₄×C₅₃

#120

yes

—

C₂×C₄×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

#121

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

—

C₂×C₄×C₅₃

C₂×Q₁₆

yes

—

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₂×C₄×C₅₃

#137

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

#10309

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

C₂²×C₅₃

yes

C₂²×C₅₃

yes

100

C₂²×C₅₃

#494

yes

—

101

C₂²×C₅₃

#496

yes

characteristic

102

C₂²×C₅₃

#497

yes

—

103

C₄×C₅₃

—

104

C₄×C₅₃

—

105

C₄×C₅₃

—

106

C₄×C₅₃

#447

yes

—

107

C₄×C₅₃

#451

yes

—

108

C₄×C₅₃

#459

yes

—

109

C₄×C₅₃

#462

yes

—

110

C₂×#500

C₂×C₅₃

yes

—

111

C₂×#494

C₂×C₅₃

yes

characteristic

112

#1990

C₂×C₅₃

yes

—

113

#1956

C₂×C₅₃

yes

characteristic

114

#1273

C₂×C₅₃

yes

characteristic

115

C₂×C₅₃

#1713

yes

characteristic

116

C₂×C₅₃

#1722

yes

characteristic

117

C₂×C₅₃

#1723

yes

—

118

C₂×C₅₃

#2011

yes

characteristic

119

C₂×C₅₃

#2016

yes

—

120

C₂²×Q₁₆

—

121

C₂×C₄⋊Q₈

C₂²×C₅₃

yes

—

122

C₂×C₄⋊Q₈

C₂²×C₅₃

yes

characteristic

123

#500

—

124

#500

—

125

#494

—

126

C₂×#107

—

127

C₂×#103

—

128

C₂×C₄×C₈

C₂²×C₅₃

yes

characteristic

129

C₂×C₄×C₈

C₂²×C₅₃

yes

—

130

#384

C₂²×C₅₃

yes

characteristic

131

#340

—

132

C₅₃

#10309

yes

characteristic, direct product factor, Sylow

133

C₂²×Q₈

—

134

C₂²×Q₈

—

135

C₂×Q₁₆

—

136

C₂×Q₁₆

—

137

C₂×Q₁₆

—

138

C₂²×C₈

D₄×C₅₃

yes

—

139

C₂²×C₈

D₄×C₅₃

yes

—

140

C₄⋊Q₈

—

141

C₄⋊Q₈

—

142

C₄⋊Q₈

—

143

C₄⋊Q₈

—

144

C₄⋊Q₈

—

145

C₄⋊Q₈

D₄×C₅₃

yes

—

146

C₄⋊Q₈

D₄×C₅₃

yes

—

147

C₂×C₄⋊C₄

—

148

C₂×C₄⋊C₄

—

149

C₂×C₄⋊C₄

—

150

C₂×C₄⋊C₄

—

151

C₂×C₄⋊C₄

—

152

C₂×C₄²

C₂³×C₅₃

yes

characteristic

153

#107

—

154

#103

—

155

C₄×C₈

—

156

C₄×C₈

—

157

C₄×C₈

—

158

C₂×Q₈

—

159

C₂×Q₈

—

160

C₂×Q₈

—

161

C₂×Q₈

—

162

C₂×Q₈

—

163

C₂×Q₈

—

164

C₂×Q₈

—

165

C₂×Q₈

—

166

C₂²×C₄

—

167

C₂²×C₄

—

168

C₂²×C₄

—

169

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

170

C₂²×C₄

C₂×D₄×C₅₃

yes

—

171

Q₁₆

—

172

C₂×C₈

—

173

C₂×C₈

—

174

C₂×C₈

—

175

C₂×C₈

—

176

C₂×C₈

D₈×C₅₃

yes

—

177

C₂×C₈

Q₁₆×C₅₃

yes

complemented

178

C₄⋊C₄

—

179

C₄⋊C₄

—

180

C₄⋊C₄

—

181

C₄⋊C₄

—

182

C₄⋊C₄

—

183

C₄⋊C₄

—

184

C₄⋊C₄

—

185

C₄⋊C₄

—

186

C₄²

C₂×D₄×C₅₃

yes

—

187

C₄²

C₂×D₄×C₅₃

yes

characteristic

188

C₄²

#41×C₅₃

yes

characteristic

189

C₂³

#127×C₅₃

yes

characteristic

190

Q₈

—

191

Q₈

—

192

Q₈

—

193

Q₈

—

194

C₂×C₄

—

195

C₂×C₄

—

196

C₂×C₄

—

197

C₂×C₄

—

198

C₂×C₄

—

199

C₂×C₄

—

200

C₂×C₄

—

201

C₂×C₄

—

202

C₂×C₄

#120×C₅₃

yes

characteristic

203

C₂×C₄

#120×C₅₃

yes

—

204

C₂×C₄

#121×C₅₃

yes

characteristic

205

C₂×C₄

#121×C₅₃

yes

—

206

C₂×C₄

C₂×D₈×C₅₃

yes

—

207

C₂×C₄

C₂×Q₁₆×C₅₃

yes

—

208

C₂×C₄

AGL(1,ℤ/8)×C₅₃

yes

—

209

C₂×C₄

#137×C₅₃

yes

—

210

C₈

—

211

C₈

—

212

C₂²

C₅₃×#392

yes

characteristic

213

C₂²

C₅₃×#493

yes

characteristic

214

C₂²

C₅₃×#494

yes

—

215

C₂²

C₅₃×#496

yes

characteristic

216

C₂²

C₅₃×#497

yes

—

217

C₄

—

218

C₄

—

219

C₄

—

220

C₄

C₅₃×#447

yes

—

221

C₄

C₅₃×#451

yes

—

222

C₄

C₅₃×#459

yes

—

223

C₄

C₅₃×#462

yes

—

224

C₂

C₅₃×#1713

yes

characteristic

225

C₂

C₅₃×#1722

yes

characteristic

226

C₂

C₅₃×#1723

yes

—

227

C₂

C₅₃×#2011

yes

characteristic

228

C₂

C₅₃×#2016

yes

—

	subgroup	G/H
center, socle, prime radical	C₂³×C₅₃	#127
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

#127

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
62. C₂×C₈×C₅₃	Q₁₆	semidirect
97. #10309	C₅₃	direct (N × H)
132. C₅₃	#10309	direct (N × H)
177. C₂×C₈	Q₁₆×C₅₃	semidirect

H ≅ G/N

type

62. C₂×C₈×C₅₃

Q₁₆

semidirect

97. #10309

C₅₃

direct (N × H)

132. C₅₃

#10309

direct (N × H)

177. C₂×C₈

Q₁₆×C₅₃

semidirect

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

subgroup	index
C₂×C₅₃×#500	2
C₂×C₅₃×#494	2
C₅₃×#1990	2
C₅₃×#1956	2
C₅₃×#1273	2
#10309	53

subgroup

index

C₂×C₅₃×#500

C₂×C₅₃×#494

C₅₃×#1990

C₅₃×#1956

C₅₃×#1273

#10309

Subgroups — C₅₃×#10309 #27217821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#10309 #27217821

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#10309 #27217821