Group #27220825 — subgroups

Subgroups — C₅₃×#13313 #27220825

Order 13568.

556 conjugacy classes of subgroups; only the 118 normal subgroups are shown.

normal subgroups only

This group has 556 conjugacy classes of subgroups — too many to draw. Only its 118 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	C₂²×D₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₅₃×#472	C₂	1	2	yes	maximal, complemented
3	C₂×C₅₃×#361	C₂	1	2	yes	maximal, complemented
4	C₅₃×#2113	C₂	1	1	yes	maximal, characteristic, complemented
5	C₅₃×#1381	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×AGL(1,ℤ/8)×C₅₃	—	2	2	no	—
7	C₂²×D₈×C₅₃	C₂²	1	2	yes	—
8	C₂×#131×C₅₃	C₂²	1	1	yes	characteristic
9	C₂×D₁₆×C₅₃	—	2	4	no	—
10	C₂×D₁₆×C₅₃	—	2	2	no	—
11	C₂×M₃₂×C₅₃	C₂²	1	1	yes	characteristic
12	C₂²×C₁₆×C₅₃	C₂²	1	1	yes	characteristic
13	C₅₃×#472	—	2	4	no	—
14	C₅₃×#472	—	2	4	no	—
15	C₂×#108×C₅₃	C₂²	1	2	yes	—
16	C₂×#100×C₅₃	—	2	2	no	—
17	C₅₃×#361	—	2	4	no	—
18	C₅₃×C₂²⋊C₁₆	—	2	2	no	—
19	C₂×#41×C₅₃	D₄	1	1	yes	characteristic
20	C₂²×D₄×C₅₃	—	2	2	no	—
21	AGL(1,ℤ/8)×C₅₃	—	2	8	no	—
22	AGL(1,ℤ/8)×C₅₃	—	4	4	no	—
23	C₂×SD₁₆×C₅₃	—	4	2	no	—
24	C₂×D₈×C₅₃	—	2	2	no	—
25	C₂×D₈×C₅₃	—	2	2	no	—
26	C₂×D₈×C₅₃	—	2	2	no	—
27	C₂×D₈×C₅₃	—	2	2	no	—
28	C₂×D₈×C₅₃	—	2	2	no	—
29	C₂×D₈×C₅₃	—	4	2	no	—
30	C₂×D₈×C₅₃	D₄	1	4	yes	—
31	#131×C₅₃	—	2	2	no	—
32	#131×C₅₃	—	2	2	no	—
33	C₂×M₁₆×C₅₃	—	2	1	no	—
34	C₂×M₁₆×C₅₃	—	2	2	no	—
35	C₂×M₁₆×C₅₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₅₃	—	2	1	no	—
37	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
38	D₁₆×C₅₃	—	2	8	no	—
39	M₃₂×C₅₃	—	2	2	no	—
40	C₂×C₁₆×C₅₃	—	2	2	no	—
41	C₂×C₁₆×C₅₃	—	2	1	no	—
42	C₂×C₁₆×C₅₃	D₄	1	2	yes	—
43	#108×C₅₃	—	2	2	no	—
44	#108×C₅₃	D₄	1	4	yes	—
45	#100×C₅₃	—	2	8	no	—
46	C₂⁴×C₅₃	—	4	2	no	—
47	#41×C₅₃	—	2	2	no	—
48	#41×C₅₃	—	2	2	no	—
49	C₂×Q₈×C₅₃	D₈	1	1	yes	characteristic
50	C₂×D₄×C₅₃	—	2	1	no	—
51	C₂×D₄×C₅₃	—	2	4	no	—
52	C₂×D₄×C₅₃	—	4	2	no	—
53	C₂×D₄×C₅₃	—	4	2	no	—
54	C₂×D₄×C₅₃	—	4	2	no	—
55	C₂×D₄×C₅₃	—	4	2	no	—
56	C₂×D₄×C₅₃	—	4	2	no	—
57	C₂×D₄×C₅₃	D₈	1	1	yes	characteristic
58	C₂²×C₄×C₅₃	—	2	1	no	—
59	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
60	SD₁₆×C₅₃	—	4	8	no	—
61	D₈×C₅₃	—	2	4	no	—
62	D₈×C₅₃	—	2	4	no	—
63	D₈×C₅₃	—	4	8	no	—
64	D₈×C₅₃	—	4	2	no	—
65	D₈×C₅₃	—	4	2	no	—
66	M₁₆×C₅₃	—	2	2	no	—
67	M₁₆×C₅₃	—	2	4	no	—
68	M₁₆×C₅₃	—	4	2	no	—
69	M₁₆×C₅₃	—	4	2	no	—
70	C₂×C₈×C₅₃	—	2	1	no	—
71	C₂×C₈×C₅₃	—	4	2	no	—
72	C₂×C₈×C₅₃	—	4	1	no	—
73	C₂×C₈×C₅₃	—	4	1	no	—
74	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
75	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
77	C₂×C₈×C₅₃	#41	1	1	yes	characteristic
78	C₁₆×C₅₃	—	2	2	no	—
79	C₁₆×C₅₃	—	4	1	no	—
80	C₂³×C₅₃	—	2	1	no	—
81	C₂³×C₅₃	—	4	4	no	—
82	C₂³×C₅₃	—	8	2	no	—
83	C₂³×C₅₃	—	8	2	no	—
84	C₂³×C₅₃	—	16	4	no	—
85	C₂³×C₅₃	#135	1	1	yes	characteristic
86	Q₈×C₅₃	—	2	2	no	—
87	D₄×C₅₃	—	2	2	no	—
88	D₄×C₅₃	—	4	2	no	—
89	D₄×C₅₃	—	4	4	no	—
90	D₄×C₅₃	—	4	4	no	—
91	D₄×C₅₃	—	8	2	no	—
92	D₄×C₅₃	—	8	2	no	—
93	C₂×C₄×C₅₃	—	2	1	no	—
94	C₂×C₄×C₅₃	—	4	1	no	—
95	C₂×C₄×C₅₃	—	4	1	no	—
96	C₂×C₄×C₅₃	#120	1	2	yes	—
97	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
98	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
100	C₈×C₅₃	—	4	1	no	—
101	C₈×C₅₃	—	4	4	no	—
102	C₈×C₅₃	#120	1	1	yes	characteristic
103	C₈×C₅₃	#121	1	2	yes	—
104	C₈×C₅₃	#127	1	1	yes	characteristic
105	#13313	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
106	C₂²×C₅₃	—	2	2	no	—
107	C₂²×C₅₃	—	4	2	no	—
108	C₂²×C₅₃	—	4	1	no	—
109	C₂²×C₅₃	—	8	4	no	—
110	C₂²×C₅₃	—	16	4	no	—
111	C₂²×C₅₃	—	16	4	no	—
112	C₂²×C₅₃	—	16	4	no	—
113	C₂²×C₅₃	#449	1	2	yes	—
114	C₂²×C₅₃	#466	1	1	yes	characteristic
115	C₄×C₅₃	—	4	1	no	—
116	C₄×C₅₃	#392	1	1	yes	characteristic
117	C₄×C₅₃	#447	1	2	yes	—
118	C₄×C₅₃	#469	1	1	yes	characteristic
119	C₂²×D₁₆	C₂×C₅₃	1	1	yes	characteristic, complemented
120	C₂×#472	C₂×C₅₃	1	2	yes	complemented
121	C₂×#361	C₂×C₅₃	1	2	yes	complemented
122	#2113	C₂×C₅₃	1	1	yes	characteristic, complemented
123	#1381	C₂×C₅₃	1	1	yes	characteristic, complemented
124	C₂×C₅₃	—	2	2	no	—
125	C₂×C₅₃	—	8	1	no	—
126	C₂×C₅₃	—	16	4	no	—
127	C₂×C₅₃	#1999	1	1	yes	characteristic
128	C₂×C₅₃	#2221	1	2	yes	—
129	C₂×AGL(1,ℤ/8)	—	2	2	no	—
130	C₂²×D₈	C₂²×C₅₃	1	2	yes	—
131	C₂×#131	C₂²×C₅₃	1	1	yes	characteristic
132	C₂×D₁₆	—	2	4	no	—
133	C₂×D₁₆	—	2	2	no	—
134	C₂×M₃₂	C₂²×C₅₃	1	1	yes	characteristic
135	C₂²×C₁₆	C₂²×C₅₃	1	1	yes	characteristic
136	#472	—	2	4	no	—
137	#472	—	2	4	no	—
138	C₂×#108	C₂²×C₅₃	1	2	yes	—
139	C₂×#100	—	2	2	no	—
140	#361	—	2	4	no	—
141	C₂²⋊C₁₆	—	2	2	no	—
142	C₅₃	#13313	1	1	yes	characteristic, direct product factor, Sylow
143	C₂×#41	D₄×C₅₃	1	1	yes	characteristic
144	C₂²×D₄	—	2	2	no	—
145	AGL(1,ℤ/8)	—	2	8	no	—
146	AGL(1,ℤ/8)	—	4	4	no	—
147	C₂×SD₁₆	—	4	2	no	—
148	C₂×D₈	—	2	2	no	—
149	C₂×D₈	—	2	2	no	—
150	C₂×D₈	—	2	2	no	—
151	C₂×D₈	—	2	2	no	—
152	C₂×D₈	—	2	2	no	—
153	C₂×D₈	—	4	2	no	—
154	C₂×D₈	D₄×C₅₃	1	4	yes	—
155	#131	—	2	2	no	—
156	#131	—	2	2	no	—
157	C₂×M₁₆	—	2	1	no	—
158	C₂×M₁₆	—	2	2	no	—
159	C₂×M₁₆	D₄×C₅₃	1	1	yes	characteristic
160	C₂²×C₈	—	2	1	no	—
161	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
162	D₁₆	—	2	8	no	—
163	M₃₂	—	2	2	no	—
164	C₂×C₁₆	—	2	2	no	—
165	C₂×C₁₆	—	2	1	no	—
166	C₂×C₁₆	D₄×C₅₃	1	2	yes	—
167	#108	—	2	2	no	—
168	#108	D₄×C₅₃	1	4	yes	—
169	#100	—	2	8	no	—
170	C₂⁴	—	4	2	no	—
171	#41	—	2	2	no	—
172	#41	—	2	2	no	—
173	C₂×Q₈	D₈×C₅₃	1	1	yes	characteristic
174	C₂×D₄	—	2	1	no	—
175	C₂×D₄	—	2	4	no	—
176	C₂×D₄	—	4	2	no	—
177	C₂×D₄	—	4	2	no	—
178	C₂×D₄	—	4	2	no	—
179	C₂×D₄	—	4	2	no	—
180	C₂×D₄	—	4	2	no	—
181	C₂×D₄	D₈×C₅₃	1	1	yes	characteristic
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
184	SD₁₆	—	4	8	no	—
185	D₈	—	2	4	no	—
186	D₈	—	2	4	no	—
187	D₈	—	4	8	no	—
188	D₈	—	4	2	no	—
189	D₈	—	4	2	no	—
190	M₁₆	—	2	2	no	—
191	M₁₆	—	2	4	no	—
192	M₁₆	—	4	2	no	—
193	M₁₆	—	4	2	no	—
194	C₂×C₈	—	2	1	no	—
195	C₂×C₈	—	4	2	no	—
196	C₂×C₈	—	4	1	no	—
197	C₂×C₈	—	4	1	no	—
198	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
199	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
200	C₂×C₈	C₂×D₄×C₅₃	1	2	yes	—
201	C₂×C₈	#41×C₅₃	1	1	yes	characteristic
202	C₁₆	—	2	2	no	—
203	C₁₆	—	4	1	no	—
204	C₂³	—	2	1	no	—
205	C₂³	—	4	4	no	—
206	C₂³	—	8	2	no	—
207	C₂³	—	8	2	no	—
208	C₂³	—	16	4	no	—
209	C₂³	#135×C₅₃	1	1	yes	characteristic
210	Q₈	—	2	2	no	—
211	D₄	—	2	2	no	—
212	D₄	—	4	2	no	—
213	D₄	—	4	4	no	—
214	D₄	—	4	4	no	—
215	D₄	—	8	2	no	—
216	D₄	—	8	2	no	—
217	C₂×C₄	—	2	1	no	—
218	C₂×C₄	—	4	1	no	—
219	C₂×C₄	—	4	1	no	—
220	C₂×C₄	#120×C₅₃	1	2	yes	—
221	C₂×C₄	#121×C₅₃	1	1	yes	characteristic
222	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
223	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
224	C₈	—	4	1	no	—
225	C₈	—	4	4	no	—
226	C₈	#120×C₅₃	1	1	yes	characteristic
227	C₈	#121×C₅₃	1	2	yes	—
228	C₈	#127×C₅₃	1	1	yes	characteristic
229	C₂²	—	2	2	no	—
230	C₂²	—	4	2	no	—
231	C₂²	—	4	1	no	—
232	C₂²	—	8	4	no	—
233	C₂²	—	16	4	no	—
234	C₂²	—	16	4	no	—
235	C₂²	—	16	4	no	—
236	C₂²	C₅₃×#449	1	2	yes	—
237	C₂²	C₅₃×#466	1	1	yes	characteristic
238	C₄	—	4	1	no	—
239	C₄	C₅₃×#392	1	1	yes	characteristic
240	C₄	C₅₃×#447	1	2	yes	—
241	C₄	C₅₃×#469	1	1	yes	characteristic
242	C₂	—	2	2	no	—
243	C₂	—	8	1	no	—
244	C₂	—	16	4	no	—
245	C₂	C₅₃×#1999	1	1	yes	characteristic
246	C₂	C₅₃×#2221	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂²×D₁₆×C₅₃

C₂

yes

maximal, characteristic, complemented

C₂×C₅₃×#472

C₂

yes

maximal, complemented

C₂×C₅₃×#361

C₂

yes

maximal, complemented

C₅₃×#2113

C₂

yes

maximal, characteristic, complemented

C₅₃×#1381

C₂

yes

maximal, characteristic, complemented

C₂×AGL(1,ℤ/8)×C₅₃

—

C₂²×D₈×C₅₃

C₂²

yes

—

C₂×#131×C₅₃

C₂²

yes

characteristic

C₂×D₁₆×C₅₃

—

C₂×D₁₆×C₅₃

—

C₂×M₃₂×C₅₃

C₂²

yes

characteristic

C₂²×C₁₆×C₅₃

C₂²

yes

characteristic

C₅₃×#472

—

C₅₃×#472

—

C₂×#108×C₅₃

C₂²

yes

—

C₂×#100×C₅₃

—

C₅₃×#361

—

C₅₃×C₂²⋊C₁₆

—

C₂×#41×C₅₃

D₄

yes

characteristic

C₂²×D₄×C₅₃

—

AGL(1,ℤ/8)×C₅₃

—

AGL(1,ℤ/8)×C₅₃

—

C₂×SD₁₆×C₅₃

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

—

C₂×D₈×C₅₃

D₄

yes

—

#131×C₅₃

—

#131×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

C₂³

yes

characteristic

D₁₆×C₅₃

—

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂×C₁₆×C₅₃

D₄

yes

—

#108×C₅₃

—

#108×C₅₃

D₄

yes

—

#100×C₅₃

—

C₂⁴×C₅₃

—

#41×C₅₃

—

#41×C₅₃

—

C₂×Q₈×C₅₃

D₈

yes

characteristic

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

—

C₂×D₄×C₅₃

D₈

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

SD₁₆×C₅₃

—

D₈×C₅₃

—

D₈×C₅₃

—

D₈×C₅₃

—

D₈×C₅₃

—

D₈×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

#41

yes

characteristic

C₁₆×C₅₃

—

C₁₆×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#135

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

#120

yes

—

C₂×C₄×C₅₃

#121

yes

characteristic

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

100

C₈×C₅₃

—

101

C₈×C₅₃

—

102

C₈×C₅₃

#120

yes

characteristic

103

C₈×C₅₃

#121

yes

—

104

C₈×C₅₃

#127

yes

characteristic

105

#13313

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

106

C₂²×C₅₃

—

107

C₂²×C₅₃

—

108

C₂²×C₅₃

—

109

C₂²×C₅₃

—

110

C₂²×C₅₃

—

111

C₂²×C₅₃

—

112

C₂²×C₅₃

—

113

C₂²×C₅₃

#449

yes

—

114

C₂²×C₅₃

#466

yes

characteristic

115

C₄×C₅₃

—

116

C₄×C₅₃

#392

yes

characteristic

117

C₄×C₅₃

#447

yes

—

118

C₄×C₅₃

#469

yes

characteristic

119

C₂²×D₁₆

C₂×C₅₃

yes

characteristic, complemented

120

C₂×#472

C₂×C₅₃

yes

complemented

121

C₂×#361

C₂×C₅₃

yes

complemented

122

#2113

C₂×C₅₃

yes

characteristic, complemented

123

#1381

C₂×C₅₃

yes

characteristic, complemented

124

C₂×C₅₃

—

125

C₂×C₅₃

—

126

C₂×C₅₃

—

127

C₂×C₅₃

#1999

yes

characteristic

128

C₂×C₅₃

#2221

yes

—

129

C₂×AGL(1,ℤ/8)

—

130

C₂²×D₈

C₂²×C₅₃

yes

—

131

C₂×#131

C₂²×C₅₃

yes

characteristic

132

C₂×D₁₆

—

133

C₂×D₁₆

—

134

C₂×M₃₂

C₂²×C₅₃

yes

characteristic

135

C₂²×C₁₆

C₂²×C₅₃

yes

characteristic

136

#472

—

137

#472

—

138

C₂×#108

C₂²×C₅₃

yes

—

139

C₂×#100

—

140

#361

—

141

C₂²⋊C₁₆

—

142

C₅₃

#13313

yes

characteristic, direct product factor, Sylow

143

C₂×#41

D₄×C₅₃

yes

characteristic

144

C₂²×D₄

—

145

AGL(1,ℤ/8)

—

146

AGL(1,ℤ/8)

—

147

C₂×SD₁₆

—

148

C₂×D₈

—

149

C₂×D₈

—

150

C₂×D₈

—

151

C₂×D₈

—

152

C₂×D₈

—

153

C₂×D₈

—

154

C₂×D₈

D₄×C₅₃

yes

—

155

#131

—

156

#131

—

157

C₂×M₁₆

—

158

C₂×M₁₆

—

159

C₂×M₁₆

D₄×C₅₃

yes

characteristic

160

C₂²×C₈

—

161

C₂²×C₈

C₂³×C₅₃

yes

characteristic

162

D₁₆

—

163

M₃₂

—

164

C₂×C₁₆

—

165

C₂×C₁₆

—

166

C₂×C₁₆

D₄×C₅₃

yes

—

167

#108

—

168

#108

D₄×C₅₃

yes

—

169

#100

—

170

C₂⁴

—

171

#41

—

172

#41

—

173

C₂×Q₈

D₈×C₅₃

yes

characteristic

174

C₂×D₄

—

175

C₂×D₄

—

176

C₂×D₄

—

177

C₂×D₄

—

178

C₂×D₄

—

179

C₂×D₄

—

180

C₂×D₄

—

181

C₂×D₄

D₈×C₅₃

yes

characteristic

182

C₂²×C₄

—

183

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

184

SD₁₆

—

185

D₈

—

186

D₈

—

187

D₈

—

188

D₈

—

189

D₈

—

190

M₁₆

—

191

M₁₆

—

192

M₁₆

—

193

M₁₆

—

194

C₂×C₈

—

195

C₂×C₈

—

196

C₂×C₈

—

197

C₂×C₈

—

198

C₂×C₈

C₂×D₄×C₅₃

yes

—

199

C₂×C₈

C₂×D₄×C₅₃

yes

characteristic

200

C₂×C₈

C₂×D₄×C₅₃

yes

—

201

C₂×C₈

#41×C₅₃

yes

characteristic

202

C₁₆

—

203

C₁₆

—

204

C₂³

—

205

C₂³

—

206

C₂³

—

207

C₂³

—

208

C₂³

—

209

C₂³

#135×C₅₃

yes

characteristic

210

Q₈

—

211

D₄

—

212

D₄

—

213

D₄

—

214

D₄

—

215

D₄

—

216

D₄

—

217

C₂×C₄

—

218

C₂×C₄

—

219

C₂×C₄

—

220

C₂×C₄

#120×C₅₃

yes

—

221

C₂×C₄

#121×C₅₃

yes

characteristic

222

C₂×C₄

C₂×D₈×C₅₃

yes

characteristic

223

C₂×C₄

AGL(1,ℤ/8)×C₅₃

yes

—

224

C₈

—

225

C₈

—

226

C₈

#120×C₅₃

yes

characteristic

227

C₈

#121×C₅₃

yes

—

228

C₈

#127×C₅₃

yes

characteristic

229

C₂²

—

230

C₂²

—

231

C₂²

—

232

C₂²

—

233

C₂²

—

234

C₂²

—

235

C₂²

—

236

C₂²

C₅₃×#449

yes

—

237

C₂²

C₅₃×#466

yes

characteristic

238

C₄

—

239

C₄

C₅₃×#392

yes

characteristic

240

C₄

C₅₃×#447

yes

—

241

C₄

C₅₃×#469

yes

characteristic

242

C₂

—

243

C₂

—

244

C₂

—

245

C₂

C₅₃×#1999

yes

characteristic

246

C₂

C₅₃×#2221

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₃	#466
commutator, Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×#131×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

#466

commutator, Frattini, cosocle

C₂²×C₈

C₂³×C₅₃

LCS

G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₂²×C₄×C₅₃ ◁ C₂×#131×C₅₃ ◁ G

N	H ≅ G/N	type
1. C₂²×D₁₆×C₅₃	C₂	semidirect
2. C₂×C₅₃×#472	C₂	semidirect
3. C₂×C₅₃×#361	C₂	semidirect
4. C₅₃×#2113	C₂	semidirect
5. C₅₃×#1381	C₂	semidirect
105. #13313	C₅₃	direct (N × H)
119. C₂²×D₁₆	C₂×C₅₃	semidirect
120. C₂×#472	C₂×C₅₃	semidirect
121. C₂×#361	C₂×C₅₃	semidirect
122. #2113	C₂×C₅₃	semidirect
123. #1381	C₂×C₅₃	semidirect
142. C₅₃	#13313	direct (N × H)

H ≅ G/N

type

1. C₂²×D₁₆×C₅₃

C₂

semidirect

2. C₂×C₅₃×#472

C₂

semidirect

3. C₂×C₅₃×#361

C₂

semidirect

4. C₅₃×#2113

C₂

semidirect

5. C₅₃×#1381

C₂

semidirect

105. #13313

C₅₃

direct (N × H)

119. C₂²×D₁₆

C₂×C₅₃

semidirect

120. C₂×#472

C₂×C₅₃

semidirect

121. C₂×#361

C₂×C₅₃

semidirect

122. #2113

C₂×C₅₃

semidirect

123. #1381

C₂×C₅₃

semidirect

142. C₅₃

#13313

direct (N × H)

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

subgroup	index
C₂²×D₁₆×C₅₃	2
C₂×C₅₃×#472	2
C₂×C₅₃×#361	2
C₅₃×#2113	2
C₅₃×#1381	2
#13313	53

subgroup

index

C₂²×D₁₆×C₅₃

C₂×C₅₃×#472

C₂×C₅₃×#361

C₅₃×#2113

C₅₃×#1381

#13313

Subgroups — C₅₃×#13313 #27220825

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#13313 #27220825

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#13313 #27220825