Subgroups — C₂×#20275506 #27387661

Order 13568.

858 conjugacy classes of subgroups.

The subgroup lattice has 396 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810672	C₂	1	2	yes	maximal
2	C₂²×#13133153	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810578	C₂	1	1	yes	maximal, characteristic
4	#20275506	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16810395	C₂	1	2	yes	maximal
6	C₂²×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₂²	1	2	yes	—
8	C₂²×#11719917	C₂²	1	1	yes	characteristic
9	C₂×#13133176	C₂²	1	1	yes	characteristic
10	#16810672	—	2	4	no	—
11	#16810672	C₂²	1	8	yes	—
12	C₂×#13133153	—	2	4	no	—
13	C₂×#13133153	—	2	1	no	—
14	C₂×#13133153	—	2	1	no	—
15	C₂×#13133153	—	2	2	no	—
16	C₂×#13133153	C₂²	1	4	yes	—
17	#16810578	—	2	2	no	—
18	#16810578	C₂²	1	4	yes	—
19	C₂×#13133127	—	2	1	no	—
20	C₂×C₈×Dic₅₃	—	2	1	no	—
21	#16810395	C₂²	1	8	yes	—
22	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
23	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
24	C₂×#108×C₅₃	C₄	1	4	yes	complemented
25	C₂×#108×C₅₃	C₄	1	4	yes	complemented
26	C₂×#108×C₅₃	C₂²	1	4	yes	—
27	C₂×#11719917	—	2	1	no	—
28	C₂×#11719917	—	2	2	no	—
29	C₂×#11719917	—	2	2	no	—
30	C₂×#11719917	—	2	2	no	—
31	C₂×#11719917	—	2	2	no	—
32	C₂×#11719917	—	2	1	no	—
33	C₂×#11719917	—	2	1	no	—
34	C₂×#11719917	—	2	2	no	—
35	C₂×#11719917	D₄	1	2	yes	—
36	C₂×#11719917	D₄	1	2	yes	—
37	C₂×#11719917	C₂³	1	4	yes	—
38	C₂×#11719917	C₂³	1	2	yes	—
39	C₂²×#88322	—	2	1	no	—
40	C₂²×#88322	—	2	2	no	—
41	#13133176	—	2	1	no	—
42	#13133176	—	2	1	no	—
43	#13133176	Q₈	1	2	yes	—
44	#13133176	C₂³	1	2	yes	—
45	C₂×C₄⋊Dic₅₃	—	2	1	no	—
46	C₂×C₂²⋊Dic₅₃	Q₈	1	2	yes	—
47	C₂×C₄×Dic₅₃	—	2	1	no	—
48	#13133153	—	2	4	no	—
49	#13133153	—	2	4	no	—
50	#13133153	—	2	16	no	—
51	#13133127	—	2	4	no	—
52	C₈×Dic₅₃	—	2	4	no	—
53	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
54	C₂×M₁₆×C₅₃	—	2	2	no	—
55	C₂×M₁₆×C₅₃	—	2	2	no	—
56	C₂×M₁₆×C₅₃	—	2	2	no	—
57	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
58	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
59	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
60	C₂²×C₈×C₅₃	—	2	2	no	—
61	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
62	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
63	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
64	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
65	C₂²×C₈×C₅₃	D₄	1	2	yes	—
66	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
67	C₂²×C₈×C₅₃	Q₈	1	2	yes	—
68	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
69	#108×C₅₃	C₂×C₄	1	16	yes	complemented
70	C₂⁴×C₅₃	C₂×Q₈	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂×Q₈	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
85	C₂²×Dic₅₃	—	2	2	no	—
86	M₁₆×C₅₃	—	2	4	no	—
87	M₁₆×C₅₃	—	2	4	no	—
88	M₁₆×C₅₃	—	2	4	no	—
89	M₁₆×C₅₃	C₂²×C₄	1	8	yes	—
90	C₂×C₈×C₅₃	—	2	4	no	—
91	C₂×C₈×C₅₃	—	2	4	no	—
92	C₂×C₈×C₅₃	—	2	4	no	—
93	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
94	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
95	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
96	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
97	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
98	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
99	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
100	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
101	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
102	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
103	C₂²⋊Dic₅₃	—	2	2	no	—
104	C₂²⋊Dic₅₃	C₂×Q₈	1	4	yes	—
105	C₄⋊Dic₅₃	—	2	2	no	—
106	C₄⋊Dic₅₃	—	2	2	no	—
107	C₄×Dic₅₃	—	2	2	no	—
108	C₄×Dic₅₃	—	2	2	no	—
109	#11719917	—	2	2	no	—
110	#11719917	—	2	2	no	—
111	#11719917	—	2	2	no	—
112	#11719917	—	2	4	no	—
113	#11719917	—	2	4	no	—
114	#11719917	—	4	2	no	—
115	#11719917	—	4	2	no	—
116	#11719917	C₂×D₄	1	4	yes	—
117	C₂×#88322	—	2	2	no	—
118	C₂×#88322	—	2	2	no	—
119	C₂×#88322	—	2	2	no	—
120	C₂×#88322	—	2	4	no	—
121	C₂×#88322	—	4	2	no	—
122	C₂×#88322	—	4	2	no	—
123	C₂³×C₅₃	—	4	2	no	—
124	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
125	C₂³×C₅₃	C₄×Q₈	1	1	yes	characteristic
126	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
127	C₂³×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
129	C₂³×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
130	C₂³×C₅₃	C₂²×Q₈	1	1	yes	characteristic
131	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
136	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
137	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
140	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
141	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
142	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
143	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
144	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
145	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
146	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
147	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
148	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
149	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
150	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
151	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
152	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
153	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
154	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
156	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
157	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
158	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
159	C₂×Dic₅₃	—	2	4	no	—
160	C₂×Dic₅₃	—	4	4	no	—
161	C₈×C₅₃	—	2	8	no	—
162	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
163	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
164	#88322	—	2	4	no	—
165	#88322	—	4	4	no	—
166	C₂×#1953	—	53	1	no	maximal, Sylow
167	C₂²×C₅₃	—	2	2	no	—
168	C₂²×C₅₃	—	2	2	no	—
169	C₂²×C₅₃	—	2	2	no	—
170	C₂²×C₅₃	—	4	1	no	—
171	C₂²×C₅₃	—	4	1	no	—
172	C₂²×C₅₃	—	4	1	no	—
173	C₂²×C₅₃	—	4	1	no	—
174	C₂²×C₅₃	#389	1	1	yes	characteristic
175	C₂²×C₅₃	#389	1	1	yes	characteristic
176	C₂²×C₅₃	#389	1	1	yes	characteristic
177	C₂²×C₅₃	#389	1	1	yes	characteristic
178	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
179	C₂²×C₅₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
180	C₂²×C₅₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
181	C₄×C₅₃	#389	1	1	yes	characteristic
182	C₄×C₅₃	#389	1	1	yes	characteristic
183	C₄×C₅₃	#389	1	1	yes	characteristic
184	C₄×C₅₃	#389	1	1	yes	characteristic
185	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
186	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
187	C₄×C₅₃	C₂×#121	1	1	yes	characteristic
188	C₄×C₅₃	C₂×#125	1	1	yes	characteristic
189	Dic₅₃	—	4	4	no	—
190	C₂×#430	—	53	2	no	—
191	C₂²×#108	—	53	1	no	—
192	C₂²×#108	D₅₃	1	1	yes	characteristic
193	C₂×#405	—	53	1	no	—
194	#1953	—	53	8	no	—
195	C₂×#344	—	53	2	no	—
196	C₂×C₅₃	—	2	1	no	—
197	C₂×C₅₃	—	2	1	no	—
198	C₂×C₅₃	—	2	1	no	—
199	C₂×C₅₃	—	2	1	no	—
200	C₂×C₅₃	—	2	1	no	—
201	C₂×C₅₃	—	2	1	no	—
202	C₂×C₅₃	#1953	1	2	yes	complemented
203	C₂×C₅₃	C₂×#389	1	1	yes	characteristic
204	C₂²×M₁₆	—	53	1	no	—
205	C₂²×M₁₆	—	53	2	no	—
206	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
207	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
208	C₂×#117	—	53	1	no	—
209	#430	—	53	8	no	—
210	#430	—	106	4	no	—
211	C₂×#108	—	53	4	no	—
212	C₂×#108	—	106	4	no	—
213	C₂×#108	—	106	1	no	—
214	C₂×#108	—	106	1	no	—
215	C₂×#108	—	106	2	no	—
216	C₂×#108	Dic₅₃	1	4	yes	complemented
217	C₂×#108	Dic₅₃	1	4	yes	complemented
218	C₂×#108	C₂×D₅₃	1	4	yes	—
219	#405	—	53	4	no	—
220	#405	—	106	2	no	—
221	C₂×#97	—	106	1	no	—
222	C₂×C₄×C₈	—	106	1	no	—
223	#344	—	53	8	no	—
224	C₅₃	C₂×#1953	1	1	yes	characteristic, complemented, Sylow
225	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
226	C₂×M₁₆	—	2	2	no	—
227	C₂×M₁₆	—	2	2	no	—
228	C₂×M₁₆	—	2	2	no	—
229	C₂×M₁₆	—	53	2	no	—
230	C₂×M₁₆	—	53	2	no	—
231	C₂×M₁₆	—	53	2	no	—
232	C₂×M₁₆	—	53	4	no	—
233	C₂×M₁₆	—	106	1	no	—
234	C₂×M₁₆	—	106	1	no	—
235	C₂×M₁₆	—	106	1	no	—
236	C₂×M₁₆	—	106	2	no	—
237	C₂×M₁₆	—	106	2	no	—
238	C₂×M₁₆	—	106	2	no	—
239	C₂×M₁₆	—	106	2	no	—
240	C₂×M₁₆	—	106	2	no	—
241	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
242	C₂×M₁₆	C₂×Dic₅₃	1	4	yes	—
243	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
244	C₂²×C₈	—	2	2	no	—
245	C₂²×C₈	—	106	1	no	—
246	C₂²×C₈	—	106	2	no	—
247	C₂²×C₈	Dic₁₀₆	1	2	yes	—
248	C₂²×C₈	Dic₁₀₆	1	2	yes	—
249	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
250	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
251	C₂²×C₈	D₂₁₂	1	2	yes	—
252	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
253	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
254	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
255	#117	—	53	2	no	—
256	#117	—	53	2	no	—
257	#117	—	106	1	no	—
258	#117	—	106	1	no	—
259	C₂×C₄⋊C₄	—	106	1	no	—
260	C₂×C₂²⋊C₄	—	53	2	no	—
261	C₂×C₄²	—	106	1	no	—
262	#108	—	106	16	no	—
263	#108	—	106	4	no	—
264	#108	—	106	4	no	—
265	#108	C₂×Dic₅₃	1	16	yes	complemented
266	#97	—	106	4	no	—
267	C₄×C₈	—	106	4	no	—
268	C₂⁴	Q₈×D₅₃	1	1	yes	characteristic
269	C₂²×C₄	—	106	2	no	—
270	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
271	C₂²×C₄	C₂×Dic₁₀₆	1	1	yes	characteristic
272	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
273	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
274	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
275	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
276	C₂²×C₄	#11719947	1	1	yes	characteristic
277	C₂²×C₄	#11719947	1	1	yes	characteristic
278	C₂²×C₄	Q₈×D₅₃	1	1	yes	characteristic
279	C₂²×C₄	#11719949	1	1	yes	characteristic
280	C₂²×C₄	#11719949	1	1	yes	characteristic
281	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
282	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
283	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
284	M₁₆	—	2	4	no	—
285	M₁₆	—	2	4	no	—
286	M₁₆	—	2	4	no	—
287	M₁₆	—	53	4	no	—
288	M₁₆	—	106	2	no	—
289	M₁₆	—	106	2	no	—
290	M₁₆	—	106	2	no	—
291	M₁₆	—	106	4	no	—
292	M₁₆	—	106	4	no	—
293	M₁₆	—	212	2	no	—
294	M₁₆	—	212	2	no	—
295	M₁₆	C₂²×Dic₅₃	1	8	yes	—
296	C₂×C₈	—	2	4	no	—
297	C₂×C₈	—	2	4	no	—
298	C₂×C₈	—	2	4	no	—
299	C₂×C₈	—	106	2	no	—
300	C₂×C₈	—	106	2	no	—
301	C₂×C₈	—	106	2	no	—
302	C₂×C₈	—	106	4	no	—
303	C₂×C₈	—	212	2	no	—
304	C₂×C₈	—	212	2	no	—
305	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
306	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
307	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
308	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
309	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
310	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
311	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
312	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
313	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
314	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
315	C₄⋊C₄	—	106	2	no	—
316	C₄⋊C₄	—	106	2	no	—
317	C₂²⋊C₄	—	53	4	no	—
318	C₂²⋊C₄	—	106	2	no	—
319	C₄²	—	106	2	no	—
320	C₄²	—	106	2	no	—
321	C₂³	—	4	2	no	—
322	C₂³	Q₈×Dic₅₃	1	1	yes	characteristic
323	C₂³	Q₈×Dic₅₃	1	1	yes	characteristic
324	C₂³	#13133197	1	1	yes	characteristic
325	C₂³	#13133197	1	1	yes	characteristic
326	C₂³	#13133216	1	1	yes	characteristic
327	C₂³	#13133216	1	1	yes	characteristic
328	C₂³	C₂×Q₈×D₅₃	1	1	yes	characteristic
329	C₂×C₄	—	106	4	no	—
330	C₂×C₄	—	212	4	no	—
331	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
332	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
333	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
334	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
335	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
336	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
337	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
338	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
339	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
340	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
341	C₂×C₄	#13133192	1	1	yes	characteristic
342	C₂×C₄	#13133192	1	1	yes	characteristic
343	C₂×C₄	#13133192	1	1	yes	characteristic
344	C₂×C₄	#13133192	1	1	yes	characteristic
345	C₂×C₄	#13133197	1	1	yes	characteristic
346	C₂×C₄	#13133197	1	1	yes	characteristic
347	C₂×C₄	#13133208	1	1	yes	characteristic
348	C₂×C₄	#13133208	1	1	yes	characteristic
349	C₂×C₄	#13133208	1	1	yes	characteristic
350	C₂×C₄	#13133208	1	1	yes	characteristic
351	C₂×C₄	#13133216	1	1	yes	characteristic
352	C₂×C₄	#13133216	1	1	yes	characteristic
353	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
354	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
355	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
356	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
357	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
358	C₂×C₄	C₂×#11719949	1	1	yes	characteristic
359	C₈	—	2	8	no	—
360	C₈	—	106	4	no	—
361	C₈	—	212	4	no	—
362	C₈	C₂×C₄⋊Dic₅₃	1	4	yes	—
363	C₈	C₂×C₄⋊Dic₅₃	1	4	yes	—
364	C₂²	—	2	2	no	—
365	C₂²	—	2	2	no	—
366	C₂²	—	2	2	no	—
367	C₂²	—	4	1	no	—
368	C₂²	—	4	1	no	—
369	C₂²	—	4	1	no	—
370	C₂²	—	4	1	no	—
371	C₂²	#16810522	1	1	yes	characteristic
372	C₂²	#16810522	1	1	yes	characteristic
373	C₂²	#16810522	1	1	yes	characteristic
374	C₂²	#16810522	1	1	yes	characteristic
375	C₂²	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
376	C₂²	C₂×#13133197	1	1	yes	characteristic
377	C₂²	C₂×#13133216	1	1	yes	characteristic
378	C₄	—	212	4	no	—
379	C₄	#16810522	1	1	yes	characteristic
380	C₄	#16810522	1	1	yes	characteristic
381	C₄	#16810522	1	1	yes	characteristic
382	C₄	#16810522	1	1	yes	characteristic
383	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
384	C₄	C₂×D₄×Dic₅₃	1	1	yes	characteristic
385	C₄	C₂×#13133192	1	1	yes	characteristic
386	C₄	C₂×#13133208	1	1	yes	characteristic
387	C₂	—	2	1	no	—
388	C₂	—	2	1	no	—
389	C₂	—	2	1	no	—
390	C₂	—	2	1	no	—
391	C₂	—	2	1	no	—
392	C₂	—	2	1	no	—
393	C₂	#20275506	1	2	yes	direct product factor
394	C₂	C₂×#16810522	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133192
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂²×#108×C₅₃	C₂
socle	C₂²×C₅₃	#389
prime radical	C₂⁴×C₅₃	C₂×Q₈
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂²×#108	D₅₃
hypocenter	C₅₃	C₂×#1953
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
4. #20275506	C₂	direct (N × H)
24. C₂×#108×C₅₃	C₄	semidirect
25. C₂×#108×C₅₃	C₄	semidirect
69. #108×C₅₃	C₂×C₄	semidirect
202. C₂×C₅₃	#1953	semidirect
216. C₂×#108	Dic₅₃	semidirect
217. C₂×#108	Dic₅₃	semidirect
224. C₅₃	C₂×#1953	semidirect
265. #108	C₂×Dic₅₃	semidirect
393. C₂	#20275506	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×#16810672	2
C₂²×#13133153	2
C₂×#16810578	2
#20275506	2
C₂×#16810395	2
C₂²×#108×C₅₃	2
C₂×#1953	53

Subgroups — C2×#20275506 #27387661

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20275506 #27387661