Group #27220249 — subgroups

Subgroups — C₅₃×#12737 #27220249

Order 13568.

906 conjugacy classes of subgroups; only the 182 normal subgroups are shown.

normal subgroups only

This group has 906 conjugacy classes of subgroups — too many to draw. Only its 182 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	C₂³×Q₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₅₃×#362	C₂	1	4	yes	maximal, complemented
3	C₅₃×#2112	C₂	1	1	yes	maximal, characteristic
4	C₅₃×#1810	C₂	1	1	yes	maximal, characteristic
5	C₂³×Q₈×C₅₃	—	2	1	no	—
6	C₂²×Q₁₆×C₅₃	—	2	8	no	—
7	C₂²×Q₁₆×C₅₃	—	2	2	no	—
8	C₂²×Q₁₆×C₅₃	—	2	1	no	—
9	C₂²×Q₁₆×C₅₃	C₄	1	4	yes	—
10	C₂²×Q₁₆×C₅₃	C₂²	1	2	yes	—
11	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
12	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
13	C₂×#108×C₅₃	C₂²	1	2	yes	—
14	C₅₃×#407	—	2	1	no	—
15	C₅₃×#362	—	2	8	no	—
16	C₅₃×C₂²⋊C₁₆	—	2	2	no	—
17	C₅₃×#341	—	2	1	no	—
18	C₂²×Q₈×C₅₃	—	2	8	no	—
19	C₂²×Q₈×C₅₃	—	2	4	no	—
20	C₂²×Q₈×C₅₃	—	2	2	no	—
21	C₂²×Q₈×C₅₃	—	4	4	no	—
22	C₂²×Q₈×C₅₃	—	4	2	no	—
23	C₂²×Q₈×C₅₃	—	4	1	no	—
24	C₂³×C₄×C₅₃	—	4	1	no	—
25	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
26	C₂×Q₁₆×C₅₃	—	2	16	no	—
27	C₂×Q₁₆×C₅₃	—	2	8	no	—
28	C₂×Q₁₆×C₅₃	—	2	16	no	—
29	C₂×Q₁₆×C₅₃	—	2	4	no	—
30	C₂×Q₁₆×C₅₃	—	2	2	no	—
31	C₂×Q₁₆×C₅₃	—	2	4	no	—
32	C₂×Q₁₆×C₅₃	C₂×C₄	1	4	yes	—
33	C₂×Q₁₆×C₅₃	D₄	1	8	yes	—
34	C₂×M₁₆×C₅₃	—	2	2	no	—
35	C₂²×C₈×C₅₃	—	2	4	no	—
36	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
37	C₂²×C₈×C₅₃	D₄	1	2	yes	—
38	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
40	M₃₂×C₅₃	—	2	4	no	—
41	C₂×C₁₆×C₅₃	—	2	2	no	—
42	#108×C₅₃	—	2	4	no	—
43	C₂²⋊C₈×C₅₃	—	4	2	no	—
44	C₂⁴×C₅₃	SD₁₆	1	1	yes	characteristic
45	C₂×Q₈×C₅₃	—	2	16	no	—
46	C₂×Q₈×C₅₃	—	2	4	no	—
47	C₂×Q₈×C₅₃	—	2	8	no	—
48	C₂×Q₈×C₅₃	—	4	8	no	—
49	C₂×Q₈×C₅₃	—	4	8	no	—
50	C₂×Q₈×C₅₃	—	4	16	no	—
51	C₂×Q₈×C₅₃	—	4	4	no	—
52	C₂×Q₈×C₅₃	—	4	2	no	—
53	C₂×Q₈×C₅₃	—	4	4	no	—
54	C₂²×C₄×C₅₃	—	2	2	no	—
55	C₂²×C₄×C₅₃	—	2	2	no	—
56	C₂²×C₄×C₅₃	—	4	8	no	—
57	C₂²×C₄×C₅₃	—	4	4	no	—
58	C₂²×C₄×C₅₃	—	4	2	no	—
59	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
60	C₂²×C₄×C₅₃	D₈	1	2	yes	—
61	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
63	Q₁₆×C₅₃	—	2	16	no	—
64	Q₁₆×C₅₃	—	2	4	no	—
65	Q₁₆×C₅₃	—	2	8	no	—
66	Q₁₆×C₅₃	C₂²⋊C₄	1	8	yes	—
67	M₁₆×C₅₃	—	4	2	no	—
68	M₁₆×C₅₃	—	4	2	no	—
69	C₂×C₈×C₅₃	—	2	2	no	—
70	C₂×C₈×C₅₃	—	2	4	no	—
71	C₂×C₈×C₅₃	—	2	4	no	—
72	C₂×C₈×C₅₃	—	4	2	no	—
73	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
74	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂×D₄	1	4	yes	—
76	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₁₆×C₅₃	—	4	2	no	—
78	C₂³×C₅₃	—	2	4	no	—
79	C₂³×C₅₃	—	2	2	no	—
80	C₂³×C₅₃	#102	1	2	yes	—
81	C₂³×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
82	Q₈×C₅₃	—	2	8	no	—
83	Q₈×C₅₃	—	4	16	no	—
84	Q₈×C₅₃	—	4	4	no	—
85	Q₈×C₅₃	—	4	8	no	—
86	C₂×C₄×C₅₃	—	2	2	no	—
87	C₂×C₄×C₅₃	—	2	2	no	—
88	C₂×C₄×C₅₃	—	2	2	no	—
89	C₂×C₄×C₅₃	—	2	4	no	—
90	C₂×C₄×C₅₃	—	4	16	no	—
91	C₂×C₄×C₅₃	—	4	4	no	—
92	C₂×C₄×C₅₃	—	4	8	no	—
93	C₂×C₄×C₅₃	#102	1	2	yes	—
94	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#120	1	2	yes	—
96	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
97	C₂×C₄×C₅₃	AGL(1,ℤ/8)	1	2	yes	—
98	C₈×C₅₃	—	2	2	no	—
99	C₈×C₅₃	—	8	2	no	—
100	C₈×C₅₃	C₂×C₂²⋊C₄	1	2	yes	—
101	C₈×C₅₃	#120	1	2	yes	—
102	#12737	C₅₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂²×C₅₃	—	2	8	no	—
104	C₂²×C₅₃	—	2	2	no	—
105	C₂²×C₅₃	—	2	2	no	—
106	C₂²×C₅₃	—	2	2	no	—
107	C₂²×C₅₃	#362	1	4	yes	complemented
108	C₂²×C₅₃	C₂×#102	1	1	yes	characteristic
109	C₂²×C₅₃	#450	1	2	yes	—
110	C₄×C₅₃	—	2	2	no	—
111	C₄×C₅₃	—	4	8	no	—
112	C₄×C₅₃	#379	1	1	yes	characteristic
113	C₄×C₅₃	#418	1	1	yes	characteristic
114	C₄×C₅₃	#447	1	2	yes	—
115	C₂³×Q₁₆	C₂×C₅₃	1	1	yes	characteristic
116	C₂×#362	C₂×C₅₃	1	4	yes	complemented
117	#2112	C₂×C₅₃	1	1	yes	characteristic
118	#1810	C₂×C₅₃	1	1	yes	characteristic
119	C₂×C₅₃	—	2	4	no	—
120	C₂×C₅₃	—	2	2	no	—
121	C₂×C₅₃	#1786	1	1	yes	characteristic
122	C₂×C₅₃	C₂×#362	1	2	yes	—
123	C₂³×Q₈	—	2	1	no	—
124	C₂²×Q₁₆	—	2	8	no	—
125	C₂²×Q₁₆	—	2	2	no	—
126	C₂²×Q₁₆	—	2	1	no	—
127	C₂²×Q₁₆	C₄×C₅₃	1	4	yes	—
128	C₂²×Q₁₆	C₂²×C₅₃	1	2	yes	—
129	C₂³×C₈	C₂²×C₅₃	1	1	yes	characteristic
130	C₂×M₃₂	C₂²×C₅₃	1	2	yes	—
131	C₂×#108	C₂²×C₅₃	1	2	yes	—
132	#407	—	2	1	no	—
133	#362	—	2	8	no	—
134	C₂²⋊C₁₆	—	2	2	no	—
135	#341	—	2	1	no	—
136	C₅₃	#12737	1	1	yes	characteristic, direct product factor, Sylow
137	C₂²×Q₈	—	2	8	no	—
138	C₂²×Q₈	—	2	2	no	—
139	C₂²×Q₈	—	2	4	no	—
140	C₂²×Q₈	—	4	4	no	—
141	C₂²×Q₈	—	4	2	no	—
142	C₂²×Q₈	—	4	1	no	—
143	C₂³×C₄	—	4	1	no	—
144	C₂³×C₄	D₄×C₅₃	1	1	yes	characteristic
145	C₂×Q₁₆	—	2	8	no	—
146	C₂×Q₁₆	—	2	16	no	—
147	C₂×Q₁₆	—	2	16	no	—
148	C₂×Q₁₆	—	2	4	no	—
149	C₂×Q₁₆	—	2	2	no	—
150	C₂×Q₁₆	—	2	4	no	—
151	C₂×Q₁₆	C₂×C₄×C₅₃	1	4	yes	—
152	C₂×Q₁₆	D₄×C₅₃	1	8	yes	—
153	C₂×M₁₆	—	2	2	no	—
154	C₂²×C₈	—	2	4	no	—
155	C₂²×C₈	C₂×C₄×C₅₃	1	2	yes	—
156	C₂²×C₈	D₄×C₅₃	1	2	yes	—
157	C₂²×C₈	D₄×C₅₃	1	1	yes	characteristic
158	C₂²×C₈	C₂³×C₅₃	1	1	yes	characteristic
159	M₃₂	—	2	4	no	—
160	C₂×C₁₆	—	2	2	no	—
161	#108	—	2	4	no	—
162	C₂²⋊C₈	—	4	2	no	—
163	C₂⁴	SD₁₆×C₅₃	1	1	yes	characteristic
164	C₂×Q₈	—	2	16	no	—
165	C₂×Q₈	—	2	4	no	—
166	C₂×Q₈	—	2	8	no	—
167	C₂×Q₈	—	4	8	no	—
168	C₂×Q₈	—	4	8	no	—
169	C₂×Q₈	—	4	16	no	—
170	C₂×Q₈	—	4	4	no	—
171	C₂×Q₈	—	4	2	no	—
172	C₂×Q₈	—	4	4	no	—
173	C₂²×C₄	—	2	2	no	—
174	C₂²×C₄	—	2	2	no	—
175	C₂²×C₄	—	4	8	no	—
176	C₂²×C₄	—	4	4	no	—
177	C₂²×C₄	—	4	2	no	—
178	C₂²×C₄	C₂²⋊C₄×C₅₃	1	2	yes	—
179	C₂²×C₄	D₈×C₅₃	1	2	yes	—
180	C₂²×C₄	SD₁₆×C₅₃	1	1	yes	characteristic
181	C₂²×C₄	C₂×D₄×C₅₃	1	1	yes	characteristic
182	Q₁₆	—	2	16	no	—
183	Q₁₆	—	2	4	no	—
184	Q₁₆	—	2	8	no	—
185	Q₁₆	C₂²⋊C₄×C₅₃	1	8	yes	—
186	M₁₆	—	4	2	no	—
187	M₁₆	—	4	2	no	—
188	C₂×C₈	—	2	2	no	—
189	C₂×C₈	—	2	4	no	—
190	C₂×C₈	—	2	4	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	C₂²⋊C₄×C₅₃	1	2	yes	—
193	C₂×C₈	C₂²×C₄×C₅₃	1	1	yes	characteristic
194	C₂×C₈	C₂×D₄×C₅₃	1	4	yes	—
195	C₂×C₈	C₂×D₄×C₅₃	1	1	yes	characteristic
196	C₁₆	—	4	2	no	—
197	C₂³	—	2	4	no	—
198	C₂³	—	2	2	no	—
199	C₂³	#102×C₅₃	1	2	yes	—
200	C₂³	C₂×SD₁₆×C₅₃	1	1	yes	characteristic
201	Q₈	—	2	8	no	—
202	Q₈	—	4	16	no	—
203	Q₈	—	4	4	no	—
204	Q₈	—	4	8	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	—	2	2	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	4	no	—
209	C₂×C₄	—	4	16	no	—
210	C₂×C₄	—	4	4	no	—
211	C₂×C₄	—	4	8	no	—
212	C₂×C₄	#102×C₅₃	1	2	yes	—
213	C₂×C₄	C₂×C₂²⋊C₄×C₅₃	1	1	yes	characteristic
214	C₂×C₄	#120×C₅₃	1	2	yes	—
215	C₂×C₄	C₂×D₈×C₅₃	1	1	yes	characteristic
216	C₂×C₄	AGL(1,ℤ/8)×C₅₃	1	2	yes	—
217	C₈	—	2	2	no	—
218	C₈	—	8	2	no	—
219	C₈	C₂×C₂²⋊C₄×C₅₃	1	2	yes	—
220	C₈	#120×C₅₃	1	2	yes	—
221	C₂²	—	2	8	no	—
222	C₂²	—	2	2	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	C₅₃×#362	1	4	yes	complemented
226	C₂²	C₂×#102×C₅₃	1	1	yes	characteristic
227	C₂²	C₅₃×#450	1	2	yes	—
228	C₄	—	2	2	no	—
229	C₄	—	4	8	no	—
230	C₄	C₅₃×#379	1	1	yes	characteristic
231	C₄	C₅₃×#418	1	1	yes	characteristic
232	C₄	C₅₃×#447	1	2	yes	—
233	C₂	—	2	4	no	—
234	C₂	—	2	2	no	—
235	C₂	C₅₃×#1786	1	1	yes	characteristic
236	C₂	C₂×C₅₃×#362	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂³×Q₁₆×C₅₃

C₂

yes

maximal, characteristic

C₂×C₅₃×#362

C₂

yes

maximal, complemented

C₅₃×#2112

C₂

yes

maximal, characteristic

C₅₃×#1810

C₂

yes

maximal, characteristic

C₂³×Q₈×C₅₃

—

C₂²×Q₁₆×C₅₃

—

C₂²×Q₁₆×C₅₃

—

C₂²×Q₁₆×C₅₃

—

C₂²×Q₁₆×C₅₃

C₄

yes

—

C₂²×Q₁₆×C₅₃

C₂²

yes

—

C₂³×C₈×C₅₃

C₂²

yes

characteristic

C₂×M₃₂×C₅₃

C₂²

yes

—

C₂×#108×C₅₃

C₂²

yes

—

C₅₃×#407

—

C₅₃×#362

—

C₅₃×C₂²⋊C₁₆

—

C₅₃×#341

—

C₂²×Q₈×C₅₃

—

C₂²×Q₈×C₅₃

—

C₂²×Q₈×C₅₃

—

C₂²×Q₈×C₅₃

—

C₂²×Q₈×C₅₃

—

C₂²×Q₈×C₅₃

—

C₂³×C₄×C₅₃

—

C₂³×C₄×C₅₃

D₄

yes

characteristic

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

—

C₂×Q₁₆×C₅₃

C₂×C₄

yes

—

C₂×Q₁₆×C₅₃

D₄

yes

—

C₂×M₁₆×C₅₃

—

C₂²×C₈×C₅₃

—

C₂²×C₈×C₅₃

C₂×C₄

yes

—

C₂²×C₈×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

D₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

#108×C₅₃

—

C₂²⋊C₈×C₅₃

—

C₂⁴×C₅₃

SD₁₆

yes

characteristic

C₂×Q₈×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

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

D₈

yes

—

C₂²×C₄×C₅₃

SD₁₆

yes

characteristic

C₂²×C₄×C₅₃

C₂×D₄

yes

characteristic

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

—

Q₁₆×C₅₃

C₂²⋊C₄

yes

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂²×C₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₁₆×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

#102

yes

—

C₂³×C₅₃

C₂×SD₁₆

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

#102

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

characteristic

C₂×C₄×C₅₃

#120

yes

—

C₂×C₄×C₅₃

C₂×D₈

yes

characteristic

C₂×C₄×C₅₃

AGL(1,ℤ/8)

yes

—

C₈×C₅₃

—

C₈×C₅₃

—

100

C₈×C₅₃

C₂×C₂²⋊C₄

yes

—

101

C₈×C₅₃

#120

yes

—

102

#12737

C₅₃

yes

maximal, characteristic, direct product factor, Sylow

103

C₂²×C₅₃

—

104

C₂²×C₅₃

—

105

C₂²×C₅₃

—

106

C₂²×C₅₃

—

107

C₂²×C₅₃

#362

yes

complemented

108

C₂²×C₅₃

C₂×#102

yes

characteristic

109

C₂²×C₅₃

#450

yes

—

110

C₄×C₅₃

—

111

C₄×C₅₃

—

112

C₄×C₅₃

#379

yes

characteristic

113

C₄×C₅₃

#418

yes

characteristic

114

C₄×C₅₃

#447

yes

—

115

C₂³×Q₁₆

C₂×C₅₃

yes

characteristic

116

C₂×#362

C₂×C₅₃

yes

complemented

117

#2112

C₂×C₅₃

yes

characteristic

118

#1810

C₂×C₅₃

yes

characteristic

119

C₂×C₅₃

—

120

C₂×C₅₃

—

121

C₂×C₅₃

#1786

yes

characteristic

122

C₂×C₅₃

C₂×#362

yes

—

123

C₂³×Q₈

—

124

C₂²×Q₁₆

—

125

C₂²×Q₁₆

—

126

C₂²×Q₁₆

—

127

C₂²×Q₁₆

C₄×C₅₃

yes

—

128

C₂²×Q₁₆

C₂²×C₅₃

yes

—

129

C₂³×C₈

C₂²×C₅₃

yes

characteristic

130

C₂×M₃₂

C₂²×C₅₃

yes

—

131

C₂×#108

C₂²×C₅₃

yes

—

132

#407

—

133

#362

—

134

C₂²⋊C₁₆

—

135

#341

—

136

C₅₃

#12737

yes

characteristic, direct product factor, Sylow

137

C₂²×Q₈

—

138

C₂²×Q₈

—

139

C₂²×Q₈

—

140

C₂²×Q₈

—

141

C₂²×Q₈

—

142

C₂²×Q₈

—

143

C₂³×C₄

—

144

C₂³×C₄

D₄×C₅₃

yes

characteristic

145

C₂×Q₁₆

—

146

C₂×Q₁₆

—

147

C₂×Q₁₆

—

148

C₂×Q₁₆

—

149

C₂×Q₁₆

—

150

C₂×Q₁₆

—

151

C₂×Q₁₆

C₂×C₄×C₅₃

yes

—

152

C₂×Q₁₆

D₄×C₅₃

yes

—

153

C₂×M₁₆

—

154

C₂²×C₈

—

155

C₂²×C₈

C₂×C₄×C₅₃

yes

—

156

C₂²×C₈

D₄×C₅₃

yes

—

157

C₂²×C₈

D₄×C₅₃

yes

characteristic

158

C₂²×C₈

C₂³×C₅₃

yes

characteristic

159

M₃₂

—

160

C₂×C₁₆

—

161

#108

—

162

C₂²⋊C₈

—

163

C₂⁴

SD₁₆×C₅₃

yes

characteristic

164

C₂×Q₈

—

165

C₂×Q₈

—

166

C₂×Q₈

—

167

C₂×Q₈

—

168

C₂×Q₈

—

169

C₂×Q₈

—

170

C₂×Q₈

—

171

C₂×Q₈

—

172

C₂×Q₈

—

173

C₂²×C₄

—

174

C₂²×C₄

—

175

C₂²×C₄

—

176

C₂²×C₄

—

177

C₂²×C₄

—

178

C₂²×C₄

C₂²⋊C₄×C₅₃

yes

—

179

C₂²×C₄

D₈×C₅₃

yes

—

180

C₂²×C₄

SD₁₆×C₅₃

yes

characteristic

181

C₂²×C₄

C₂×D₄×C₅₃

yes

characteristic

182

Q₁₆

—

183

Q₁₆

—

184

Q₁₆

—

185

Q₁₆

C₂²⋊C₄×C₅₃

yes

—

186

M₁₆

—

187

M₁₆

—

188

C₂×C₈

—

189

C₂×C₈

—

190

C₂×C₈

—

191

C₂×C₈

—

192

C₂×C₈

C₂²⋊C₄×C₅₃

yes

—

193

C₂×C₈

C₂²×C₄×C₅₃

yes

characteristic

194

C₂×C₈

C₂×D₄×C₅₃

yes

—

195

C₂×C₈

C₂×D₄×C₅₃

yes

characteristic

196

C₁₆

—

197

C₂³

—

198

C₂³

—

199

C₂³

#102×C₅₃

yes

—

200

C₂³

C₂×SD₁₆×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

#102×C₅₃

yes

—

213

C₂×C₄

C₂×C₂²⋊C₄×C₅₃

yes

characteristic

214

C₂×C₄

#120×C₅₃

yes

—

215

C₂×C₄

C₂×D₈×C₅₃

yes

characteristic

216

C₂×C₄

AGL(1,ℤ/8)×C₅₃

yes

—

217

C₈

—

218

C₈

—

219

C₈

C₂×C₂²⋊C₄×C₅₃

yes

—

220

C₈

#120×C₅₃

yes

—

221

C₂²

—

222

C₂²

—

223

C₂²

—

224

C₂²

—

225

C₂²

C₅₃×#362

yes

complemented

226

C₂²

C₂×#102×C₅₃

yes

characteristic

227

C₂²

C₅₃×#450

yes

—

228

C₄

—

229

C₄

—

230

C₄

C₅₃×#379

yes

characteristic

231

C₄

C₅₃×#418

yes

characteristic

232

C₄

C₅₃×#447

yes

—

233

C₂

—

234

C₂

—

235

C₂

C₅₃×#1786

yes

characteristic

236

C₂

C₂×C₅₃×#362

yes

—

	subgroup	G/H
center, socle	C₂²×C₅₃	C₂×#102
commutator	C₂×C₈	C₂²×C₄×C₅₃
Frattini, cosocle	C₂²×C₈	C₂³×C₅₃
prime radical	C₂⁴×C₅₃	SD₁₆
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ C₂³×C₈×C₅₃ ◁ G

subgroup

G/H

center, socle

C₂²×C₅₃

C₂×#102

commutator

C₂×C₈

C₂²×C₄×C₅₃

Frattini, cosocle

C₂²×C₈

C₂³×C₅₃

prime radical

C₂⁴×C₅₃

SD₁₆

LCS

G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

UCS

C₁ ◁ C₂²×C₅₃ ◁ C₂³×C₄×C₅₃ ◁ C₂³×C₈×C₅₃ ◁ G

N	H ≅ G/N	type
2. C₂×C₅₃×#362	C₂	semidirect
102. #12737	C₅₃	direct (N × H)
107. C₂²×C₅₃	#362	semidirect
116. C₂×#362	C₂×C₅₃	semidirect
136. C₅₃	#12737	direct (N × H)
225. C₂²	C₅₃×#362	semidirect

H ≅ G/N

type

2. C₂×C₅₃×#362

C₂

semidirect

102. #12737

C₅₃

direct (N × H)

107. C₂²×C₅₃

#362

semidirect

116. C₂×#362

C₂×C₅₃

semidirect

136. C₅₃

#12737

direct (N × H)

225. C₂²

C₅₃×#362

semidirect

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

subgroup	index
C₂³×Q₁₆×C₅₃	2
C₂×C₅₃×#362	2
C₅₃×#2112	2
C₅₃×#1810	2
#12737	53

subgroup

index

C₂³×Q₁₆×C₅₃

C₂×C₅₃×#362

C₅₃×#2112

C₅₃×#1810

#12737

Subgroups — C₅₃×#12737 #27220249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C53×#12737 #27220249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₅₃×#12737 #27220249