Group #27300317 — subgroups

Subgroups — #27300317

Order 13568.

462 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 135 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×M₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810405	C₂	1	1	yes	maximal, characteristic
3	#20272796	C₂	1	2	yes	maximal, complemented
4	#20272794	C₂	1	2	yes	maximal, complemented
5	C₂×C₅₃×#349	C₂	1	1	yes	maximal, characteristic
6	C₂²×#11719917	C₄	1	1	yes	characteristic
7	C₂²×C₄×Dic₅₃	C₄	1	1	yes	characteristic
8	C₂×#13133157	C₂²	1	2	yes	—
9	M₁₆×Dic₅₃	—	2	2	no	—
10	M₁₆×Dic₅₃	—	2	2	no	—
11	C₂×#13133127	C₄	1	1	yes	characteristic
12	C₂×#13133127	C₂²	1	1	yes	characteristic
13	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic
14	C₂×C₈×Dic₅₃	C₂²	1	1	yes	characteristic
15	#16810405	—	2	2	no	—
16	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
18	C₅₃×#349	—	2	2	no	—
19	C₂³×Dic₅₃	—	2	1	no	—
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	—	2	2	no	—
23	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
24	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄×Dic₅₃	—	2	2	no	—
26	C₂×C₄×Dic₅₃	—	2	1	no	—
27	C₂×C₄×Dic₅₃	C₈	1	4	yes	—
28	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
30	#13133157	—	2	2	no	—
31	#13133157	—	2	2	no	—
32	C₂×#11719908	—	2	2	no	—
33	#13133127	—	2	2	no	—
34	#13133127	—	2	1	no	—
35	#13133127	—	2	1	no	—
36	C₈×Dic₅₃	—	2	2	no	—
37	C₈×Dic₅₃	—	2	1	no	—
38	C₈×Dic₅₃	—	2	1	no	—
39	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂×M₁₆×C₅₃	—	2	2	no	—
41	C₂×M₁₆×C₅₃	—	2	2	no	—
42	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
43	C₂×M₁₆×C₅₃	D₄	1	2	yes	—
44	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
46	M₃₂×C₅₃	—	2	2	no	—
47	M₃₂×C₅₃	—	2	2	no	—
48	C₂×C₁₆×C₅₃	—	2	2	no	—
49	C₂⁴×C₅₃	M₁₆	1	1	yes	characteristic
50	C₂²×C₄×C₅₃	—	2	1	no	—
51	C₂²×C₄×C₅₃	—	2	1	no	—
52	C₂²×C₄×C₅₃	—	2	1	no	—
53	C₂²×C₄×C₅₃	—	2	1	no	—
54	C₂²×C₄×C₅₃	C₂²⋊C₄	1	2	yes	—
55	C₂²×C₄×C₅₃	C₂×C₈	1	2	yes	—
56	C₂²×C₄×C₅₃	M₁₆	1	1	yes	characteristic
57	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
58	C₂²×Dic₅₃	—	2	1	no	—
59	C₂²×Dic₅₃	—	2	4	no	—
60	C₂²×Dic₅₃	—	2	1	no	—
61	C₂²×Dic₅₃	—	4	4	no	—
62	M₁₆×C₅₃	—	2	4	no	—
63	M₁₆×C₅₃	—	2	4	no	—
64	C₂×C₈×C₅₃	—	2	1	no	—
65	C₂×C₈×C₅₃	—	2	1	no	—
66	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
67	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
69	C₂×C₈×C₅₃	#41	1	2	yes	—
70	C₂×C₈×C₅₃	#41	1	2	yes	—
71	C₄×Dic₅₃	—	2	4	no	—
72	C₄×Dic₅₃	—	2	1	no	—
73	C₄×Dic₅₃	—	2	1	no	—
74	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
75	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
76	#11719917	—	2	8	no	—
77	C₂×#88322	—	2	1	no	—
78	C₂×#88322	—	2	1	no	—
79	C₂×#88322	—	2	1	no	—
80	C₂×#88322	—	2	1	no	—
81	C₂×#88322	M₁₆	1	2	yes	—
82	C₂×#88322	M₁₆	1	2	yes	—
83	C₁₆×C₅₃	—	4	2	no	—
84	#11719908	—	4	2	no	—
85	C₂³×C₅₃	—	2	1	no	—
86	C₂³×C₅₃	—	2	1	no	—
87	C₂³×C₅₃	—	4	2	no	—
88	C₂³×C₅₃	C₂²⋊C₈	1	2	yes	—
89	C₂³×C₅₃	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₅₃	—	2	1	no	—
91	C₂×C₄×C₅₃	—	2	1	no	—
92	C₂×C₄×C₅₃	—	2	2	no	—
93	C₂×C₄×C₅₃	—	2	2	no	—
94	C₂×C₄×C₅₃	—	2	2	no	—
95	C₂×C₄×C₅₃	—	2	2	no	—
96	C₂×C₄×C₅₃	C₂²⋊C₈	1	2	yes	—
97	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
99	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
100	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
101	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
102	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
103	C₂×Dic₅₃	—	2	4	no	—
104	C₂×Dic₅₃	—	2	2	no	—
105	C₂×Dic₅₃	—	2	2	no	—
106	C₂×Dic₅₃	—	4	8	no	—
107	C₂×Dic₅₃	—	4	2	no	—
108	C₈×C₅₃	—	2	2	no	—
109	C₈×C₅₃	#123	1	2	yes	—
110	C₈×C₅₃	#123	1	2	yes	—
111	#88322	—	2	2	no	—
112	#88322	—	2	2	no	—
113	#12849	—	53	1	no	maximal, Sylow
114	C₂²×C₅₃	—	2	2	no	—
115	C₂²×C₅₃	—	2	2	no	—
116	C₂²×C₅₃	—	4	2	no	—
117	C₂²×C₅₃	—	4	2	no	—
118	C₂²×C₅₃	—	4	2	no	—
119	C₂²×C₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
120	C₂²×C₅₃	#432	1	1	yes	characteristic
121	C₂²×C₅₃	#432	1	1	yes	characteristic
122	C₄×C₅₃	—	2	2	no	—
123	C₄×C₅₃	#381	1	1	yes	characteristic
124	C₄×C₅₃	#407	1	1	yes	characteristic
125	C₄×C₅₃	#431	1	1	yes	characteristic
126	C₄×C₅₃	#431	1	1	yes	characteristic
127	Dic₅₃	—	2	4	no	—
128	Dic₅₃	—	4	2	no	—
129	C₂×C₄×M₁₆	—	53	1	no	—
130	C₂×#349	—	53	1	no	—
131	C₂×#349	D₅₃	1	1	yes	characteristic
132	#1376	—	53	2	no	—
133	#1375	—	53	2	no	—
134	C₂×C₅₃	—	2	1	no	—
135	C₂×C₅₃	—	2	1	no	—
136	C₂×C₅₃	—	4	2	no	—
137	C₂×C₅₃	#1817	1	1	yes	characteristic
138	C₂×C₅₃	#2165	1	2	yes	—
139	C₂²×M₁₆	—	53	1	no	—
140	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
141	C₂²×C₄²	—	53	1	no	—
142	C₂×M₃₂	—	53	2	no	—
143	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
144	C₄×M₁₆	—	106	2	no	—
145	C₄×M₁₆	—	106	2	no	—
146	C₂×#97	—	53	1	no	—
147	C₂×#97	—	53	1	no	—
148	C₂×C₄×C₈	—	53	1	no	—
149	C₂×C₄×C₈	—	53	1	no	—
150	#349	—	2	2	no	—
151	#349	—	106	2	no	—
152	C₅₃	#12849	1	1	yes	characteristic, complemented, Sylow
153	C₂³×C₄	—	106	1	no	—
154	C₂³×C₄	C₄×D₅₃	1	1	yes	characteristic
155	C₂×M₁₆	—	2	2	no	—
156	C₂×M₁₆	—	2	2	no	—
157	C₂×M₁₆	—	106	2	no	—
158	C₂×M₁₆	—	106	2	no	—
159	C₂×M₁₆	—	106	2	no	—
160	C₂×M₁₆	D₂₁₂	1	2	yes	—
161	C₂×M₁₆	C₂²⋊D₅₃	1	2	yes	—
162	C₂²×C₈	—	53	1	no	—
163	C₂²×C₈	—	53	1	no	—
164	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
165	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
166	C₂×C₄²	—	53	1	no	—
167	C₂×C₄²	—	53	1	no	—
168	C₂×C₄²	—	53	4	no	—
169	C₂×C₄²	—	106	1	no	—
170	C₂×C₄²	—	106	2	no	—
171	M₃₂	—	2	2	no	—
172	M₃₂	—	2	2	no	—
173	M₃₂	—	106	2	no	—
174	M₃₂	—	106	2	no	—
175	C₂×C₁₆	—	2	2	no	—
176	C₂×C₁₆	—	106	2	no	—
177	#97	—	106	1	no	—
178	#97	—	106	1	no	—
179	#97	—	106	2	no	—
180	C₄×C₈	—	106	1	no	—
181	C₄×C₈	—	106	1	no	—
182	C₄×C₈	—	106	2	no	—
183	C₂⁴	#11719912	1	1	yes	characteristic
184	C₂²×C₄	—	2	1	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	1	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	—	106	1	no	—
189	C₂²×C₄	—	106	4	no	—
190	C₂²×C₄	—	106	1	no	—
191	C₂²×C₄	—	212	4	no	—
192	C₂²×C₄	C₈×D₅₃	1	2	yes	—
193	C₂²×C₄	#11719912	1	1	yes	characteristic
194	C₂²×C₄	#11719921	1	2	yes	—
195	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
196	M₁₆	—	2	4	no	—
197	M₁₆	—	2	4	no	—
198	M₁₆	—	106	8	no	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	53	2	no	—
202	C₂×C₈	—	53	2	no	—
203	C₂×C₈	—	106	1	no	—
204	C₂×C₈	—	106	1	no	—
205	C₂×C₈	—	106	1	no	—
206	C₂×C₈	—	106	1	no	—
207	C₂×C₈	#11719921	1	2	yes	—
208	C₂×C₈	C₂×D₂₁₂	1	1	yes	characteristic
209	C₂×C₈	#11719947	1	2	yes	—
210	C₂×C₈	#11719947	1	2	yes	—
211	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
212	C₄²	—	53	2	no	—
213	C₄²	—	53	2	no	—
214	C₄²	—	106	4	no	—
215	C₄²	—	106	1	no	—
216	C₄²	—	106	1	no	—
217	C₁₆	—	4	2	no	—
218	C₁₆	—	212	2	no	—
219	C₂³	—	2	1	no	—
220	C₂³	—	2	1	no	—
221	C₂³	—	4	2	no	—
222	C₂³	#13133129	1	2	yes	—
223	C₂³	C₂×#11719912	1	1	yes	characteristic
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	106	4	no	—
231	C₂×C₄	—	106	2	no	—
232	C₂×C₄	—	106	2	no	—
233	C₂×C₄	—	212	8	no	—
234	C₂×C₄	—	212	2	no	—
235	C₂×C₄	#13133129	1	2	yes	—
236	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
237	C₂×C₄	#13133173	1	1	yes	characteristic
238	C₂×C₄	#13133173	1	1	yes	characteristic
239	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
240	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
241	C₂×C₄	M₁₆×D₅₃	1	1	yes	characteristic
242	C₈	—	2	2	no	—
243	C₈	—	106	2	no	—
244	C₈	—	106	2	no	—
245	C₈	#13133198	1	2	yes	—
246	C₈	#13133201	1	2	yes	—
247	C₂²	—	2	2	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	4	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	C₂×#13133129	1	1	yes	characteristic
253	C₂²	#16810677	1	1	yes	characteristic
254	C₂²	#16810677	1	1	yes	characteristic
255	C₄	—	2	2	no	—
256	C₄	—	106	4	no	—
257	C₄	—	212	2	no	—
258	C₄	#16810484	1	1	yes	characteristic
259	C₄	#16810586	1	1	yes	characteristic
260	C₄	#16810673	1	1	yes	characteristic
261	C₄	#16810673	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	4	2	no	—
265	C₂	#20274781	1	1	yes	characteristic
266	C₂	#20276524	1	2	yes	—

subgroup

G/H

class size

multiplicity

normal?

properties

C₂×M₁₆×Dic₅₃

C₂

yes

maximal, characteristic

C₂×#16810405

C₂

yes

maximal, characteristic

#20272796

C₂

yes

maximal, complemented

#20272794

C₂

yes

maximal, complemented

C₂×C₅₃×#349

C₂

yes

maximal, characteristic

C₂²×#11719917

C₄

yes

characteristic

C₂²×C₄×Dic₅₃

C₄

yes

characteristic

C₂×#13133157

C₂²

yes

—

M₁₆×Dic₅₃

—

M₁₆×Dic₅₃

—

C₂×#13133127

C₄

yes

characteristic

C₂×#13133127

C₂²

yes

characteristic

C₂×C₈×Dic₅₃

C₄

yes

characteristic

C₂×C₈×Dic₅₃

C₂²

yes

characteristic

#16810405

—

C₂²×M₁₆×C₅₃

C₂²

yes

characteristic

C₂×M₃₂×C₅₃

C₂²

yes

—

C₅₃×#349

—

C₂³×Dic₅₃

—

C₂×#11719917

—

C₂×#11719917

—

C₂×#11719917

—

C₂²×#88322

C₂×C₄

yes

characteristic

C₂²×#88322

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₃

—

C₂×C₄×Dic₅₃

—

C₂×C₄×Dic₅₃

C₈

yes

—

C₂×C₄×Dic₅₃

C₂×C₄

yes

characteristic

C₂×C₄×Dic₅₃

C₂×C₄

yes

—

—

—

—

—

—

C₈×Dic₅₃

—

C₈×Dic₅₃

—

C₈×Dic₅₃

—

C₂³×C₄×C₅₃

C₂×C₄

yes

characteristic

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

—

C₂×M₁₆×C₅₃

D₄

yes

—

C₂×M₁₆×C₅₃

D₄

yes

—

C₂²×C₈×C₅₃

C₂×C₄

yes

characteristic

C₂²×C₈×C₅₃

C₂³

yes

characteristic

M₃₂×C₅₃

—

M₃₂×C₅₃

—

C₂×C₁₆×C₅₃

—

C₂⁴×C₅₃

M₁₆

yes

characteristic

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

—

C₂²×C₄×C₅₃

C₂²⋊C₄

yes

—

C₂²×C₄×C₅₃

C₂×C₈

yes

—

C₂²×C₄×C₅₃

M₁₆

yes

characteristic

C₂²×C₄×C₅₃

C₂²×C₄

yes

characteristic

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

C₂²×Dic₅₃

—

M₁₆×C₅₃

—

M₁₆×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

—

C₂×C₈×C₅₃

C₂²⋊C₄

yes

—

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

C₂×D₄

yes

characteristic

C₂×C₈×C₅₃

#41

yes

—

C₂×C₈×C₅₃

#41

yes

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

—

C₄×Dic₅₃

C₂×C₈

yes

—

C₄×Dic₅₃

C₂×C₈

yes

—

—

—

—

—

—

M₁₆

yes

—

C₂×#88322

M₁₆

yes

—

C₁₆×C₅₃

—

#11719908

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

—

C₂³×C₅₃

C₂²⋊C₈

yes

—

C₂³×C₅₃

C₂×M₁₆

yes

characteristic

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

—

C₂×C₄×C₅₃

C₂²⋊C₈

yes

—

C₂×C₄×C₅₃

C₂×C₂²⋊C₄

yes

C₂×C₄×C₅₃

yes

C₂×C₄×C₅₃

yes

100

C₂×C₄×C₅₃

C₂²×C₈

yes

characteristic

101

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

102

C₂×C₄×C₅₃

C₂×M₁₆

yes

characteristic

103

C₂×Dic₅₃

—

104

C₂×Dic₅₃

—

105

C₂×Dic₅₃

—

106

C₂×Dic₅₃

—

107

C₂×Dic₅₃

—

108

C₈×C₅₃

—

109

C₈×C₅₃

#123

yes

—

110

C₈×C₅₃

#123

yes

—

111

#88322

—

112

#88322

—

113

#12849

—

maximal, Sylow

114

C₂²×C₅₃

—

115

C₂²×C₅₃

—

116

C₂²×C₅₃

—

117

C₂²×C₅₃

—

118

C₂²×C₅₃

—

119

C₂²×C₅₃

C₂×C₂²⋊C₈

yes

characteristic

120

C₂²×C₅₃

#432

yes

characteristic

121

C₂²×C₅₃

#432

yes

characteristic

122

C₄×C₅₃

—

123

C₄×C₅₃

#381

yes

characteristic

124

C₄×C₅₃

#407

yes

characteristic

125

C₄×C₅₃

#431

yes

characteristic

126

C₄×C₅₃

#431

yes

characteristic

127

Dic₅₃

—

128

Dic₅₃

—

129

C₂×C₄×M₁₆

—

130

C₂×#349

—

131

C₂×#349

D₅₃

yes

characteristic

132

#1376

—

133

#1375

—

134

C₂×C₅₃

—

135

C₂×C₅₃

—

136

C₂×C₅₃

—

137

C₂×C₅₃

#1817

yes

characteristic

138

C₂×C₅₃

#2165

yes

—

139

C₂²×M₁₆

—

140

C₂²×M₁₆

C₂×D₅₃

yes

characteristic

141

C₂²×C₄²

—

142

C₂×M₃₂

—

143

C₂×M₃₂

C₂×D₅₃

yes

—

144

C₄×M₁₆

—

106

—

145

C₄×M₁₆

—

106

—

146

C₂×#97

—

147

C₂×#97

—

148

C₂×C₄×C₈

—

149

C₂×C₄×C₈

—

150

#349

—

151

#349

—

106

—

152

C₅₃

#12849

yes

characteristic, complemented, Sylow

153

C₂³×C₄

—

106

—

154

C₂³×C₄

C₄×D₅₃

yes

characteristic

155

C₂×M₁₆

—

156

C₂×M₁₆

—

157

C₂×M₁₆

—

106

—

158

C₂×M₁₆

—

106

—

159

C₂×M₁₆

—

106

—

160

C₂×M₁₆

D₂₁₂

yes

—

161

C₂×M₁₆

C₂²⋊D₅₃

yes

—

162

C₂²×C₈

—

163

C₂²×C₈

—

164

C₂²×C₈

C₄×D₅₃

yes

characteristic

165

C₂²×C₈

C₂²×D₅₃

yes

characteristic

166

C₂×C₄²

—

167

C₂×C₄²

—

168

C₂×C₄²

—

169

C₂×C₄²

—

106

—

170

C₂×C₄²

—

106

—

171

M₃₂

—

172

M₃₂

—

173

M₃₂

—

106

—

174

M₃₂

—

106

—

175

C₂×C₁₆

—

176

C₂×C₁₆

—

106

—

177

#97

—

106

—

178

#97

—

106

—

179

#97

—

106

—

180

C₄×C₈

—

106

—

181

C₄×C₈

—

106

—

182

C₄×C₈

—

106

—

183

C₂⁴

#11719912

yes

characteristic

184

C₂²×C₄

—

185

C₂²×C₄

—

186

C₂²×C₄

—

187

C₂²×C₄

—

188

C₂²×C₄

—

106

—

189

C₂²×C₄

—

106

—

190

C₂²×C₄

—

106

—

191

C₂²×C₄

—

212

—

192

C₂²×C₄

C₈×D₅₃

yes

—

193

C₂²×C₄

#11719912

yes

characteristic

194

C₂²×C₄

#11719921

yes

—

195

C₂²×C₄

C₂×C₄×D₅₃

yes

characteristic

196

M₁₆

—

197

M₁₆

—

198

M₁₆

—

106

—

199

C₂×C₈

—

200

C₂×C₈

—

201

C₂×C₈

—

202

C₂×C₈

—

203

C₂×C₈

—

106

—

204

C₂×C₈

—

106

—

205

C₂×C₈

—

106

—

206

C₂×C₈

—

106

—

207

C₂×C₈

#11719921

yes

—

208

C₂×C₈

C₂×D₂₁₂

yes

characteristic

209

C₂×C₈

#11719947

yes

—

210

C₂×C₈

#11719947

yes

—

211

C₂×C₈

C₂×C₂²⋊D₅₃

yes

characteristic

212

C₄²

—

213

C₄²

—

214

C₄²

—

106

—

215

C₄²

—

106

—

216

C₄²

—

106

—

217

C₁₆

—

218

C₁₆

—

212

—

219

C₂³

—

220

C₂³

—

221

C₂³

—

222

C₂³

#13133129

yes

—

223

C₂³

C₂×#11719912

yes

characteristic

224

C₂×C₄

—

225

C₂×C₄

—

226

C₂×C₄

—

227

C₂×C₄

—

228

C₂×C₄

—

229

C₂×C₄

—

230

C₂×C₄

—

106

—

231

C₂×C₄

—

106

—

232

C₂×C₄

—

106

—

233

C₂×C₄

—

212

—

234

C₂×C₄

—

212

—

235

C₂×C₄

#13133129

yes

—

236

C₂×C₄

C₂×#11719921

yes

characteristic

237

C₂×C₄

#13133173

yes

characteristic

238

C₂×C₄

#13133173

yes

characteristic

239

C₂×C₄

C₂×C₈×D₅₃

yes

characteristic

240

C₂×C₄

M₁₆×D₅₃

yes

characteristic

241

C₂×C₄

M₁₆×D₅₃

yes

characteristic

242

C₈

—

243

C₈

—

106

—

244

C₈

—

106

—

245

C₈

#13133198

yes

—

246

C₈

#13133201

yes

—

247

C₂²

—

248

C₂²

—

249

C₂²

—

250

C₂²

—

251

C₂²

—

252

C₂²

C₂×#13133129

yes

characteristic

253

C₂²

#16810677

yes

characteristic

254

C₂²

#16810677

yes

characteristic

255

C₄

—

256

C₄

—

106

—

257

C₄

—

212

—

258

C₄

#16810484

yes

characteristic

259

C₄

#16810586

yes

characteristic

260

C₄

#16810673

yes

characteristic

261

C₄

#16810673

yes

characteristic

262

C₂

—

263

C₂

—

264

C₂

—

265

C₂

#20274781

yes

characteristic

266

C₂

#20276524

yes

—

	subgroup	G/H
center	C₂×C₄	#13133173
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂×C₅₃×#349	C₂
socle	C₂²×C₅₃	#432
prime radical	C₂⁴×C₅₃	M₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂×#349	D₅₃
hypocenter	C₅₃	#12849
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#349

subgroup

G/H

center

C₂×C₄

#13133173

commutator

C₂×C₄×C₅₃

C₂²×C₈

Frattini

C₂²×C₈

C₂²×D₅₃

Fitting

C₂×C₅₃×#349

C₂

socle

C₂²×C₅₃

#432

prime radical

C₂⁴×C₅₃

M₁₆

cosocle

C₂²×C₈×C₅₃

C₂³

D₅₃

C₅₃

G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃

UCS

C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×#349

N	H ≅ G/N	type
3. #20272796	C₂	semidirect
4. #20272794	C₂	semidirect
152. C₅₃	#12849	semidirect

H ≅ G/N

type

3. #20272796

C₂

semidirect

4. #20272794

C₂

semidirect

152. C₅₃

#12849

semidirect

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

subgroup	index
C₂×M₁₆×Dic₅₃	2
C₂×#16810405	2
#20272796	2
#20272794	2
C₂×C₅₃×#349	2
#12849	53

subgroup

index

C₂×M₁₆×Dic₅₃

C₂×C₅₃×#349

Subgroups — #27300317

Conjugacy classes of subgroups

Complements

Maximal subgroups