Subgroups — C₂×#20275316 #27386338

Order 13568.

854 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810650	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810579	C₂	1	1	yes	maximal, characteristic
3	#20275316	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16810388	C₂	1	1	yes	maximal, characteristic
5	C₂×#16810382	C₂	1	1	yes	maximal, characteristic
6	C₂×#16810382	C₂	1	1	yes	maximal, characteristic
7	C₂×#16810377	C₂	1	1	yes	maximal, characteristic
8	C₂×C₅₃×#427	C₂	1	1	yes	maximal, characteristic
9	C₂²×#11719917	C₂²	1	1	yes	characteristic
10	C₂²×#11719917	C₂²	1	1	yes	characteristic
11	C₂³×#88322	C₂²	1	1	yes	characteristic
12	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
13	#16810650	—	2	2	no	—
14	#16810650	C₂²	1	4	yes	—
15	C₂×C₄⋊#88322	—	2	1	no	—
16	C₂×C₄⋊#88322	—	2	1	no	—
17	#16810579	—	2	2	no	—
18	#16810579	C₂²	1	4	yes	—
19	C₂×#13133128	—	2	1	no	—
20	C₂×C₄×#88322	—	2	1	no	—
21	#16810388	C₂²	1	4	yes	—
22	#16810382	C₂²	1	4	yes	—
23	#16810382	C₂²	1	4	yes	—
24	#16810377	C₂²	1	4	yes	—
25	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
26	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
27	C₂×#117×C₅₃	C₂²	1	1	yes	characteristic
28	C₅₃×#427	C₄	1	4	yes	complemented
29	C₅₃×#427	C₂²	1	4	yes	—
30	C₂×#107×C₅₃	C₄	1	1	yes	characteristic, complemented
31	C₂×#107×C₅₃	C₄	1	1	yes	characteristic, complemented
32	C₂×#106×C₅₃	C₄	1	1	yes	characteristic, complemented
33	C₂×#106×C₅₃	C₄	1	1	yes	characteristic, complemented
34	C₂³×Dic₅₃	—	2	1	no	—
35	C₂×#11719917	—	2	1	no	—
36	C₂×#11719917	—	2	1	no	—
37	C₂×#11719917	—	2	2	no	—
38	C₂×#11719917	—	2	2	no	—
39	C₂×#11719917	—	2	2	no	—
40	C₂×#11719917	Q₈	1	4	yes	—
41	C₂×#11719917	C₂³	1	2	yes	—
42	C₂×#11719917	C₂³	1	2	yes	—
43	C₂²×#88322	—	2	1	no	—
44	C₂²×#88322	—	2	1	no	—
45	C₂²×#88322	—	2	2	no	—
46	C₂²×#88322	—	2	2	no	—
47	C₂²×#88322	D₄	1	1	yes	characteristic
48	C₂²×#88322	D₄	1	1	yes	characteristic
49	C₂²×#88322	D₄	1	2	yes	—
50	C₂²×#88322	C₂³	1	2	yes	—
51	C₂×C₄×Dic₅₃	—	2	1	no	—
52	C₂×C₄×Dic₅₃	—	2	2	no	—
53	C₂×C₄×Dic₅₃	—	2	2	no	—
54	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
55	C₄⋊#88322	—	2	4	no	—
56	C₄⋊#88322	—	2	4	no	—
57	#13133128	—	2	4	no	—
58	C₄×#88322	—	2	4	no	—
59	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
60	C₂²×C₈×C₅₃	—	2	2	no	—
61	C₂²×C₈×C₅₃	—	2	2	no	—
62	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
65	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
66	#117×C₅₃	—	2	1	no	—
67	#117×C₅₃	—	2	1	no	—
68	#117×C₅₃	—	2	1	no	—
69	#117×C₅₃	—	2	1	no	—
70	#117×C₅₃	C₂×C₄	1	2	yes	—
71	#117×C₅₃	C₂×C₄	1	2	yes	—
72	#117×C₅₃	C₂³	1	2	yes	—
73	#117×C₅₃	C₂³	1	2	yes	—
74	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
77	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
78	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
79	C₂×C₂²⋊C₄×C₅₃	—	2	1	no	—
80	C₂×C₄²×C₅₃	—	2	1	no	—
81	C₂×C₄²×C₅₃	—	2	1	no	—
82	#107×C₅₃	C₂×C₄	1	4	yes	complemented
83	#107×C₅₃	C₂×C₄	1	4	yes	complemented
84	#106×C₅₃	C₂×C₄	1	4	yes	complemented
85	#106×C₅₃	C₂×C₄	1	4	yes	complemented
86	C₂⁴×C₅₃	#41	1	1	yes	characteristic
87	C₂²×C₄×C₅₃	—	2	1	no	—
88	C₂²×C₄×C₅₃	—	2	1	no	—
89	C₂²×C₄×C₅₃	—	2	1	no	—
90	C₂²×C₄×C₅₃	—	2	1	no	—
91	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
94	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
95	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
96	C₂²×C₄×C₅₃	#41	1	2	yes	—
97	C₂²×C₄×C₅₃	#41	1	2	yes	—
98	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
99	C₂²×C₄×C₅₃	#41	1	2	yes	—
100	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
101	C₂²×Dic₅₃	—	2	4	no	—
102	C₂²×Dic₅₃	—	2	2	no	—
103	C₂²×Dic₅₃	—	4	2	no	—
104	C₂²×Dic₅₃	—	4	2	no	—
105	C₂×C₈×C₅₃	—	2	2	no	—
106	C₂×C₈×C₅₃	—	2	2	no	—
107	C₂×C₈×C₅₃	—	2	4	no	—
108	C₂×C₈×C₅₃	—	2	2	no	—
109	C₂×C₈×C₅₃	—	2	2	no	—
110	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
111	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
112	C₄⋊C₄×C₅₃	—	2	1	no	—
113	C₄⋊C₄×C₅₃	—	2	1	no	—
114	C₄⋊C₄×C₅₃	—	2	1	no	—
115	C₄⋊C₄×C₅₃	—	2	1	no	—
116	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
117	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
118	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
119	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
120	C₂²⋊C₄×C₅₃	—	2	2	no	—
121	C₂²⋊C₄×C₅₃	—	2	2	no	—
122	C₂²⋊C₄×C₅₃	—	2	2	no	—
123	C₂²⋊C₄×C₅₃	—	2	2	no	—
124	C₄²×C₅₃	—	2	2	no	—
125	C₄²×C₅₃	—	2	2	no	—
126	C₄²×C₅₃	—	2	2	no	—
127	C₄²×C₅₃	—	2	2	no	—
128	C₄×Dic₅₃	—	2	2	no	—
129	C₄×Dic₅₃	—	2	2	no	—
130	C₄×Dic₅₃	—	4	2	no	—
131	#11719917	—	2	4	no	—
132	#11719917	—	2	2	no	—
133	#11719917	—	2	2	no	—
134	#11719917	—	2	2	no	—
135	#11719917	—	4	2	no	—
136	#11719917	C₂×Q₈	1	4	yes	—
137	C₂×#88322	—	2	4	no	—
138	C₂×#88322	—	2	2	no	—
139	C₂×#88322	—	2	2	no	—
140	C₂×#88322	—	2	4	no	—
141	C₂×#88322	—	2	2	no	—
142	C₂×#88322	—	2	2	no	—
143	C₂×#88322	—	2	2	no	—
144	C₂×#88322	—	2	2	no	—
145	C₂×#88322	—	4	2	no	—
146	C₂×#88322	C₂×D₄	1	2	yes	—
147	C₂×#88322	C₂×D₄	1	2	yes	—
148	C₂³×C₅₃	—	2	2	no	—
149	C₂³×C₅₃	—	2	2	no	—
150	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
151	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
152	C₂³×C₅₃	#125	1	2	yes	—
153	C₂³×C₅₃	#126	1	2	yes	—
154	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
155	C₂×C₄×C₅₃	—	2	2	no	—
156	C₂×C₄×C₅₃	—	2	2	no	—
157	C₂×C₄×C₅₃	—	2	2	no	—
158	C₂×C₄×C₅₃	—	2	2	no	—
159	C₂×C₄×C₅₃	—	4	2	no	—
160	C₂×C₄×C₅₃	—	4	2	no	—
161	C₂×C₄×C₅₃	—	4	2	no	—
162	C₂×C₄×C₅₃	—	4	2	no	—
163	C₂×C₄×C₅₃	#117	1	2	yes	—
164	C₂×C₄×C₅₃	#117	1	2	yes	—
165	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
166	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
167	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
168	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
169	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
170	C₂×C₄×C₅₃	C₂²⋊Q₈	1	2	yes	—
171	C₂×C₄×C₅₃	#123	1	2	yes	—
172	C₂×C₄×C₅₃	#123	1	2	yes	—
173	C₂×C₄×C₅₃	#125	1	2	yes	—
174	C₂×C₄×C₅₃	#126	1	2	yes	—
175	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
176	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
177	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
178	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
179	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
180	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
181	C₂×Dic₅₃	—	2	4	no	—
182	C₂×Dic₅₃	—	4	8	no	—
183	C₂×Dic₅₃	—	4	2	no	—
184	C₂×Dic₅₃	—	4	2	no	—
185	C₈×C₅₃	—	2	2	no	—
186	C₈×C₅₃	—	2	2	no	—
187	#88322	—	2	4	no	—
188	#88322	—	2	2	no	—
189	#88322	—	2	2	no	—
190	#88322	—	4	2	no	—
191	C₂×#1922	—	53	1	no	maximal, Sylow
192	C₂²×C₅₃	—	2	4	no	—
193	C₂²×C₅₃	—	2	4	no	—
194	C₂²×C₅₃	—	2	1	no	—
195	C₂²×C₅₃	—	2	1	no	—
196	C₂²×C₅₃	—	2	1	no	—
197	C₂²×C₅₃	—	2	1	no	—
198	C₂²×C₅₃	#387	1	2	yes	—
199	C₂²×C₅₃	#387	1	2	yes	—
200	C₂²×C₅₃	#443	1	2	yes	—
201	C₂²×C₅₃	#444	1	2	yes	—
202	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
203	C₂²×C₅₃	C₂×#125	1	1	yes	characteristic
204	C₂²×C₅₃	C₂×#126	1	1	yes	characteristic
205	C₄×C₅₃	—	4	2	no	—
206	C₄×C₅₃	—	4	2	no	—
207	C₄×C₅₃	—	4	2	no	—
208	C₄×C₅₃	—	4	2	no	—
209	C₄×C₅₃	#387	1	2	yes	—
210	C₄×C₅₃	#387	1	2	yes	—
211	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
212	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
213	C₄×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
214	C₄×C₅₃	C₂×#123	1	1	yes	characteristic
215	Dic₅₃	—	4	4	no	—
216	C₂×#427	D₅₃	1	1	yes	characteristic
217	C₂×#424	—	53	1	no	—
218	C₂×#405	—	53	1	no	—
219	#1922	—	53	8	no	—
220	C₂×#341	—	53	1	no	—
221	C₂×#339	—	53	1	no	—
222	C₂×#339	—	53	1	no	—
223	C₂×#336	—	53	1	no	—
224	C₂×C₅₃	—	2	1	no	—
225	C₂×C₅₃	—	2	1	no	—
226	C₂×C₅₃	—	2	2	no	—
227	C₂×C₅₃	#1922	1	4	yes	complemented
228	C₂×C₅₃	C₂×#387	1	1	yes	characteristic
229	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
230	C₂×C₅₃	C₂×#444	1	1	yes	characteristic
231	C₂²×M₁₆	—	53	1	no	—
232	C₂²×M₁₆	—	53	1	no	—
233	C₂³×C₈	—	53	1	no	—
234	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
235	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
236	C₂×#117	C₂×D₅₃	1	1	yes	characteristic
237	C₂²×C₄²	—	53	1	no	—
238	#427	Dic₅₃	1	4	yes	complemented
239	#427	C₂×D₅₃	1	4	yes	—
240	C₂×#107	Dic₅₃	1	1	yes	characteristic, complemented
241	C₂×#107	Dic₅₃	1	1	yes	characteristic, complemented
242	C₂×#106	Dic₅₃	1	1	yes	characteristic, complemented
243	C₂×#106	Dic₅₃	1	1	yes	characteristic, complemented
244	#424	—	53	4	no	—
245	#424	—	106	2	no	—
246	C₂×C₄⋊C₈	—	106	1	no	—
247	C₂×C₄⋊C₈	—	106	1	no	—
248	#405	—	53	4	no	—
249	#405	—	106	2	no	—
250	C₂×#97	—	106	1	no	—
251	C₂×C₄×C₈	—	106	1	no	—
252	#341	—	53	4	no	—
253	#339	—	53	4	no	—
254	#339	—	53	4	no	—
255	#336	—	53	4	no	—
256	C₅₃	C₂×#1922	1	1	yes	characteristic, complemented, Sylow
257	C₂³×C₄	—	106	1	no	—
258	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
259	C₂×M₁₆	—	53	4	no	—
260	C₂×M₁₆	—	53	2	no	—
261	C₂×M₁₆	—	53	2	no	—
262	C₂×M₁₆	—	106	2	no	—
263	C₂×M₁₆	—	106	1	no	—
264	C₂×M₁₆	—	106	1	no	—
265	C₂×M₁₆	—	106	2	no	—
266	C₂×M₁₆	—	106	2	no	—
267	C₂²×C₈	—	2	2	no	—
268	C₂²×C₈	—	2	2	no	—
269	C₂²×C₈	—	53	2	no	—
270	C₂²×C₈	—	53	1	no	—
271	C₂²×C₈	—	53	1	no	—
272	C₂²×C₈	—	53	2	no	—
273	C₂²×C₈	—	106	2	no	—
274	C₂²×C₈	—	106	1	no	—
275	C₂²×C₈	—	106	2	no	—
276	C₂²×C₈	—	106	1	no	—
277	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
278	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
279	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
280	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
281	#117	—	2	1	no	—
282	#117	—	2	1	no	—
283	#117	—	2	1	no	—
284	#117	—	2	1	no	—
285	#117	C₂×Dic₅₃	1	2	yes	—
286	#117	C₂×Dic₅₃	1	2	yes	—
287	#117	C₂²×D₅₃	1	2	yes	—
288	#117	C₂²×D₅₃	1	2	yes	—
289	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
290	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
291	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
292	C₂×C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
293	C₂×C₂²⋊C₄	—	2	1	no	—
294	C₂×C₂²⋊C₄	—	2	1	no	—
295	C₂×C₄²	—	2	1	no	—
296	C₂×C₄²	—	2	1	no	—
297	C₂×C₄²	—	53	2	no	—
298	C₂×C₄²	—	106	2	no	—
299	C₂×C₄²	—	106	2	no	—
300	C₂×C₄²	—	106	1	no	—
301	#107	C₂×Dic₅₃	1	4	yes	complemented
302	#107	C₂×Dic₅₃	1	4	yes	complemented
303	#106	C₂×Dic₅₃	1	4	yes	complemented
304	#106	C₂×Dic₅₃	1	4	yes	complemented
305	C₄⋊C₈	—	106	4	no	—
306	C₄⋊C₈	—	106	4	no	—
307	#97	—	106	4	no	—
308	C₄×C₈	—	106	4	no	—
309	C₂⁴	#11719947	1	1	yes	characteristic
310	C₂²×C₄	—	2	1	no	—
311	C₂²×C₄	—	2	1	no	—
312	C₂²×C₄	—	2	1	no	—
313	C₂²×C₄	—	2	1	no	—
314	C₂²×C₄	—	106	4	no	—
315	C₂²×C₄	—	106	2	no	—
316	C₂²×C₄	—	212	2	no	—
317	C₂²×C₄	—	212	2	no	—
318	C₂²×C₄	#11719945	1	2	yes	—
319	C₂²×C₄	#11719945	1	2	yes	—
320	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
321	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
322	C₂²×C₄	#11719947	1	1	yes	characteristic
323	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
324	C₂²×C₄	#11719949	1	2	yes	—
325	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
326	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
327	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
328	M₁₆	—	53	4	no	—
329	M₁₆	—	106	4	no	—
330	M₁₆	—	106	2	no	—
331	M₁₆	—	106	2	no	—
332	M₁₆	—	106	2	no	—
333	M₁₆	—	212	2	no	—
334	C₂×C₈	—	2	2	no	—
335	C₂×C₈	—	2	2	no	—
336	C₂×C₈	—	2	4	no	—
337	C₂×C₈	—	2	2	no	—
338	C₂×C₈	—	2	2	no	—
339	C₂×C₈	—	53	2	no	—
340	C₂×C₈	—	53	2	no	—
341	C₂×C₈	—	106	4	no	—
342	C₂×C₈	—	106	2	no	—
343	C₂×C₈	—	106	2	no	—
344	C₂×C₈	—	106	4	no	—
345	C₂×C₈	—	106	2	no	—
346	C₂×C₈	—	106	2	no	—
347	C₂×C₈	—	106	2	no	—
348	C₂×C₈	—	106	2	no	—
349	C₂×C₈	—	212	2	no	—
350	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
351	C₂×C₈	C₂²×Dic₅₃	1	2	yes	—
352	C₄⋊C₄	—	2	1	no	—
353	C₄⋊C₄	—	2	1	no	—
354	C₄⋊C₄	—	2	1	no	—
355	C₄⋊C₄	—	2	1	no	—
356	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
357	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
358	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
359	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
360	C₂²⋊C₄	—	2	2	no	—
361	C₂²⋊C₄	—	2	2	no	—
362	C₂²⋊C₄	—	2	2	no	—
363	C₂²⋊C₄	—	2	2	no	—
364	C₄²	—	2	2	no	—
365	C₄²	—	2	2	no	—
366	C₄²	—	2	2	no	—
367	C₄²	—	2	2	no	—
368	C₄²	—	106	2	no	—
369	C₄²	—	106	2	no	—
370	C₄²	—	212	2	no	—
371	C₂³	—	2	2	no	—
372	C₂³	—	2	2	no	—
373	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
374	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
375	C₂³	#13133207	1	2	yes	—
376	C₂³	#13133212	1	2	yes	—
377	C₂³	C₂×#11719947	1	1	yes	characteristic
378	C₂×C₄	—	2	2	no	—
379	C₂×C₄	—	2	2	no	—
380	C₂×C₄	—	2	2	no	—
381	C₂×C₄	—	2	2	no	—
382	C₂×C₄	—	4	2	no	—
383	C₂×C₄	—	4	2	no	—
384	C₂×C₄	—	4	2	no	—
385	C₂×C₄	—	4	2	no	—
386	C₂×C₄	—	106	4	no	—
387	C₂×C₄	—	212	8	no	—
388	C₂×C₄	—	212	2	no	—
389	C₂×C₄	—	212	2	no	—
390	C₂×C₄	#13133176	1	2	yes	—
391	C₂×C₄	#13133176	1	2	yes	—
392	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
393	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
394	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
395	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
396	C₂×C₄	#13133195	1	2	yes	—
397	C₂×C₄	#13133195	1	2	yes	—
398	C₂×C₄	#13133202	1	2	yes	—
399	C₂×C₄	#13133202	1	2	yes	—
400	C₂×C₄	#13133207	1	2	yes	—
401	C₂×C₄	#13133212	1	2	yes	—
402	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
403	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
404	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
405	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
406	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
407	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
408	C₈	—	2	2	no	—
409	C₈	—	2	2	no	—
410	C₈	—	106	4	no	—
411	C₈	—	106	2	no	—
412	C₈	—	106	2	no	—
413	C₈	—	212	2	no	—
414	C₂²	—	2	4	no	—
415	C₂²	—	2	4	no	—
416	C₂²	—	2	1	no	—
417	C₂²	—	2	1	no	—
418	C₂²	—	2	1	no	—
419	C₂²	—	2	1	no	—
420	C₂²	#16810509	1	2	yes	—
421	C₂²	#16810509	1	2	yes	—
422	C₂²	#16810740	1	2	yes	—
423	C₂²	#16810745	1	2	yes	—
424	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
425	C₂²	C₂×#13133207	1	1	yes	characteristic
426	C₂²	C₂×#13133212	1	1	yes	characteristic
427	C₄	—	4	2	no	—
428	C₄	—	4	2	no	—
429	C₄	—	4	2	no	—
430	C₄	—	4	2	no	—
431	C₄	—	212	4	no	—
432	C₄	#16810509	1	2	yes	—
433	C₄	#16810509	1	2	yes	—
434	C₄	C₂×#13133176	1	1	yes	characteristic
435	C₄	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
436	C₄	C₂×#13133195	1	1	yes	characteristic
437	C₄	C₂×#13133202	1	1	yes	characteristic
438	C₂	—	2	1	no	—
439	C₂	—	2	2	no	—
440	C₂	—	2	1	no	—
441	C₂	#20275316	1	4	yes	direct product factor
442	C₂	C₂×#16810509	1	1	yes	characteristic
443	C₂	C₂×#16810740	1	1	yes	characteristic
444	C₂	C₂×#16810745	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₅₃×#427	C₂
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×#427	D₅₃
hypocenter	C₅₃	C₂×#1922
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂×#427

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

N	H ≅ G/N	type
3. #20275316	C₂	direct (N × H)
28. C₅₃×#427	C₄	semidirect
30. C₂×#107×C₅₃	C₄	semidirect
31. C₂×#107×C₅₃	C₄	semidirect
32. C₂×#106×C₅₃	C₄	semidirect
33. C₂×#106×C₅₃	C₄	semidirect
82. #107×C₅₃	C₂×C₄	semidirect
83. #107×C₅₃	C₂×C₄	semidirect
84. #106×C₅₃	C₂×C₄	semidirect
85. #106×C₅₃	C₂×C₄	semidirect
227. C₂×C₅₃	#1922	semidirect
238. #427	Dic₅₃	semidirect
240. C₂×#107	Dic₅₃	semidirect
241. C₂×#107	Dic₅₃	semidirect
242. C₂×#106	Dic₅₃	semidirect
243. C₂×#106	Dic₅₃	semidirect
256. C₅₃	C₂×#1922	semidirect
301. #107	C₂×Dic₅₃	semidirect
302. #107	C₂×Dic₅₃	semidirect
303. #106	C₂×Dic₅₃	semidirect
304. #106	C₂×Dic₅₃	semidirect
441. C₂	#20275316	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16810650	2
C₂×#16810579	2
#20275316	2
C₂×#16810388	2
C₂×#16810382	2
C₂×#16810377	2
C₂×C₅₃×#427	2
C₂×#1922	53

Subgroups — C2×#20275316 #27386338

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20275316 #27386338