Subgroups — C₂×#20272827 #27299023

Order 13568.

554 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#13133153	C₂	1	1	yes	maximal, characteristic
2	C₂²×#13133153	C₂	1	1	yes	maximal, characteristic
3	#20272827	C₂	1	4	yes	maximal, direct product factor
4	C₂²×M₃₂×C₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×#11719917	—	2	1	no	—
6	C₂²×#11719917	—	2	1	no	—
7	C₂×#13133153	—	2	1	no	—
8	C₂×#13133153	—	2	1	no	—
9	C₂×#13133153	—	2	1	no	—
10	C₂×#13133153	—	2	1	no	—
11	C₂×#13133153	—	2	1	no	—
12	C₂×#13133153	—	2	1	no	—
13	C₂×#13133153	C₄	1	2	yes	—
14	C₂×#13133153	C₄	1	2	yes	—
15	C₂×#13133153	C₄	1	2	yes	—
16	C₂×#13133153	C₄	1	2	yes	—
17	C₂×#13133153	C₂²	1	2	yes	—
18	C₂×#13133153	C₂²	1	2	yes	—
19	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×M₃₂×C₅₃	—	2	1	no	—
21	C₂×M₃₂×C₅₃	—	2	1	no	—
22	C₂×M₃₂×C₅₃	—	2	1	no	—
23	C₂×M₃₂×C₅₃	C₄	1	2	yes	—
24	C₂×M₃₂×C₅₃	C₄	1	2	yes	—
25	C₂×M₃₂×C₅₃	C₂²	1	2	yes	—
26	C₂²×C₁₆×C₅₃	—	2	1	no	—
27	C₂×#11719917	—	2	2	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	2	no	—
33	C₂×#11719917	—	4	1	no	—
34	C₂×#11719917	—	4	1	no	—
35	C₂×#11719917	—	4	1	no	—
36	C₂×#11719917	—	4	1	no	—
37	C₂×#11719917	—	4	1	no	—
38	C₂×#11719917	—	4	1	no	—
39	C₂²×#88322	—	4	1	no	—
40	C₂²×#88322	—	4	1	no	—
41	#13133153	—	2	2	no	—
42	#13133153	—	2	2	no	—
43	#13133153	—	2	2	no	—
44	#13133153	—	2	2	no	—
45	#13133153	—	2	2	no	—
46	#13133153	—	2	2	no	—
47	#13133153	C₂×C₄	1	4	yes	—
48	#13133153	C₂×C₄	1	4	yes	—
49	C₂³×C₄×C₅₃	D₄	1	1	yes	characteristic
50	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
57	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
58	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
59	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
60	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
61	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
62	C₂²×C₈×C₅₃	Q₈	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	C₂³	1	1	yes	characteristic
64	M₃₂×C₅₃	—	2	2	no	—
65	M₃₂×C₅₃	—	2	2	no	—
66	M₃₂×C₅₃	—	2	2	no	—
67	M₃₂×C₅₃	C₂×C₄	1	4	yes	—
68	C₂×C₁₆×C₅₃	—	2	2	no	—
69	C₂×C₁₆×C₅₃	—	2	2	no	—
70	C₂×C₁₆×C₅₃	—	2	2	no	—
71	C₂⁴×C₅₃	Q₁₆	1	1	yes	characteristic
72	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
74	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
75	C₂²×C₄×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
76	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₄⋊C₄	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	D₈	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
81	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	SD₁₆	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	Q₁₆	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
87	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
88	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
89	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
90	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
91	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
92	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
96	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₂×C₈×C₅₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
99	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
100	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
101	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
102	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
103	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
104	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
105	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
106	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
107	C₂×C₈×C₅₃	C₄⋊C₄	1	1	yes	characteristic
108	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
109	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
110	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₈×C₅₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₈×C₅₃	C₂×Q₈	1	1	yes	characteristic
114	#11719917	—	2	4	no	—
115	#11719917	—	2	4	no	—
116	#11719917	—	4	2	no	—
117	#11719917	—	4	2	no	—
118	#11719917	—	4	2	no	—
119	#11719917	—	4	2	no	—
120	#11719917	—	4	2	no	—
121	#11719917	—	4	2	no	—
122	C₂×#88322	—	4	2	no	—
123	C₂×#88322	—	4	2	no	—
124	C₂×#88322	—	4	2	no	—
125	C₂×#88322	—	4	2	no	—
126	C₂×#88322	—	4	2	no	—
127	C₂×#88322	—	4	2	no	—
128	C₁₆×C₅₃	—	2	4	no	—
129	C₂³×C₅₃	—	4	2	no	—
130	C₂³×C₅₃	#103	1	1	yes	characteristic
131	C₂³×C₅₃	#103	1	1	yes	characteristic
132	C₂³×C₅₃	#103	1	1	yes	characteristic
133	C₂³×C₅₃	#103	1	1	yes	characteristic
134	C₂³×C₅₃	#107	1	1	yes	characteristic
135	C₂³×C₅₃	#107	1	1	yes	characteristic
136	C₂³×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
137	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
138	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
139	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
140	C₂×C₄×C₅₃	#95	1	1	yes	characteristic
141	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
142	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
143	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
144	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
145	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
146	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
147	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
148	C₂×C₄×C₅₃	#102	1	1	yes	characteristic
149	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
150	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
151	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
152	C₂×C₄×C₅₃	#103	1	1	yes	characteristic
153	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
154	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
155	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
156	C₂×C₄×C₅₃	#106	1	1	yes	characteristic
157	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
158	C₂×C₄×C₅₃	#107	1	1	yes	characteristic
159	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
160	C₂×C₄×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
161	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
162	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
163	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
164	C₂×C₄×C₅₃	C₂×SD₁₆	1	1	yes	characteristic
165	C₈×C₅₃	#95	1	1	yes	characteristic
166	C₈×C₅₃	#95	1	1	yes	characteristic
167	C₈×C₅₃	#95	1	1	yes	characteristic
168	C₈×C₅₃	#95	1	1	yes	characteristic
169	C₈×C₅₃	C₂×C₄²	1	1	yes	characteristic
170	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
171	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
172	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
173	#88322	—	4	4	no	—
174	#88322	—	4	4	no	—
175	C₂×#1388	—	53	1	no	maximal, Sylow
176	C₂²×C₅₃	—	2	2	no	—
177	C₂²×C₅₃	—	2	2	no	—
178	C₂²×C₅₃	—	2	2	no	—
179	C₂²×C₅₃	—	4	1	no	—
180	C₂²×C₅₃	—	4	1	no	—
181	C₂²×C₅₃	—	4	1	no	—
182	C₂²×C₅₃	—	4	1	no	—
183	C₂²×C₅₃	#340	1	1	yes	characteristic
184	C₂²×C₅₃	#340	1	1	yes	characteristic
185	C₂²×C₅₃	#340	1	1	yes	characteristic
186	C₂²×C₅₃	#340	1	1	yes	characteristic
187	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
188	C₂²×C₅₃	C₂×#103	1	1	yes	characteristic
189	C₂²×C₅₃	C₂×#107	1	1	yes	characteristic
190	C₄×C₅₃	#340	1	1	yes	characteristic
191	C₄×C₅₃	#340	1	1	yes	characteristic
192	C₄×C₅₃	#340	1	1	yes	characteristic
193	C₄×C₅₃	#340	1	1	yes	characteristic
194	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
195	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
196	C₄×C₅₃	C₂×#102	1	1	yes	characteristic
197	C₄×C₅₃	C₂×#106	1	1	yes	characteristic
198	C₂²×M₃₂	D₅₃	1	1	yes	characteristic
199	C₂²×#108	—	53	1	no	—
200	C₂²×#108	—	53	1	no	—
201	#1388	—	53	4	no	—
202	C₂×C₅₃	—	2	1	no	—
203	C₂×C₅₃	—	2	1	no	—
204	C₂×C₅₃	—	2	1	no	—
205	C₂×C₅₃	—	2	1	no	—
206	C₂×C₅₃	—	2	1	no	—
207	C₂×C₅₃	—	2	1	no	—
208	C₂×C₅₃	#1388	1	2	yes	complemented
209	C₂×C₅₃	C₂×#340	1	1	yes	characteristic
210	C₂²×M₁₆	—	106	1	no	—
211	C₂²×M₁₆	—	106	1	no	—
212	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
213	C₂×M₃₂	—	2	1	no	—
214	C₂×M₃₂	—	2	1	no	—
215	C₂×M₃₂	—	2	1	no	—
216	C₂×M₃₂	Dic₅₃	1	2	yes	—
217	C₂×M₃₂	Dic₅₃	1	2	yes	—
218	C₂×M₃₂	C₂×D₅₃	1	2	yes	—
219	C₂²×C₁₆	—	2	1	no	—
220	C₂×#108	—	53	2	no	—
221	C₂×#108	—	53	2	no	—
222	C₂×#108	—	53	2	no	—
223	C₂×#108	—	53	2	no	—
224	C₂×#108	—	53	2	no	—
225	C₂×#108	—	53	2	no	—
226	C₂×#108	—	106	1	no	—
227	C₂×#108	—	106	1	no	—
228	C₂×#108	—	106	1	no	—
229	C₂×#108	—	106	1	no	—
230	C₂×#108	—	106	1	no	—
231	C₂×#108	—	106	1	no	—
232	C₅₃	C₂×#1388	1	1	yes	characteristic, complemented, Sylow
233	C₂³×C₄	D₂₁₂	1	1	yes	characteristic
234	C₂×M₁₆	—	106	2	no	—
235	C₂×M₁₆	—	106	2	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	—	212	1	no	—
241	C₂×M₁₆	—	212	1	no	—
242	C₂×M₁₆	—	212	1	no	—
243	C₂×M₁₆	—	212	1	no	—
244	C₂×M₁₆	—	212	1	no	—
245	C₂×M₁₆	—	212	1	no	—
246	C₂²×C₈	—	212	1	no	—
247	C₂²×C₈	—	212	1	no	—
248	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
249	C₂²×C₈	Dic₁₀₆	1	1	yes	characteristic
250	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
251	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
252	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
253	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
254	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
255	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
256	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
257	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
258	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
259	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
260	C₂²×C₈	C₂²⋊D₅₃	1	1	yes	characteristic
261	C₂²×C₈	C₂²×D₅₃	1	1	yes	characteristic
262	M₃₂	—	2	2	no	—
263	M₃₂	—	2	2	no	—
264	M₃₂	—	2	2	no	—
265	M₃₂	C₂×Dic₅₃	1	4	yes	—
266	C₂×C₁₆	—	2	2	no	—
267	C₂×C₁₆	—	2	2	no	—
268	C₂×C₁₆	—	2	2	no	—
269	#108	—	53	4	no	—
270	#108	—	53	4	no	—
271	#108	—	106	2	no	—
272	#108	—	106	2	no	—
273	#108	—	106	2	no	—
274	#108	—	106	2	no	—
275	#108	—	106	2	no	—
276	#108	—	106	2	no	—
277	C₂⁴	Dic₂₁₂	1	1	yes	characteristic
278	C₂²×C₄	#11719913	1	1	yes	characteristic
279	C₂²×C₄	#11719913	1	1	yes	characteristic
280	C₂²×C₄	#11719913	1	1	yes	characteristic
281	C₂²×C₄	#11719913	1	1	yes	characteristic
282	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
283	C₂²×C₄	D₄₂₄	1	1	yes	characteristic
284	C₂²×C₄	Dic₂₁₂	1	1	yes	characteristic
285	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
286	C₂²×C₄	C₄⋊Dic₅₃	1	1	yes	characteristic
287	C₂²×C₄	#11719921	1	1	yes	characteristic
288	C₂²×C₄	#11719921	1	1	yes	characteristic
289	C₂²×C₄	#11719921	1	1	yes	characteristic
290	C₂²×C₄	#11719921	1	1	yes	characteristic
291	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
292	M₁₆	—	106	4	no	—
293	M₁₆	—	106	4	no	—
294	M₁₆	—	212	2	no	—
295	M₁₆	—	212	2	no	—
296	M₁₆	—	212	2	no	—
297	M₁₆	—	212	2	no	—
298	M₁₆	—	212	2	no	—
299	M₁₆	—	212	2	no	—
300	C₂×C₈	—	212	2	no	—
301	C₂×C₈	—	212	2	no	—
302	C₂×C₈	—	212	2	no	—
303	C₂×C₈	—	212	2	no	—
304	C₂×C₈	—	212	2	no	—
305	C₂×C₈	—	212	2	no	—
306	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
307	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
308	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
309	C₂×C₈	C₄×Dic₅₃	1	1	yes	characteristic
310	C₂×C₈	#11719919	1	1	yes	characteristic
311	C₂×C₈	#11719919	1	1	yes	characteristic
312	C₂×C₈	#11719919	1	1	yes	characteristic
313	C₂×C₈	#11719919	1	1	yes	characteristic
314	C₂×C₈	#11719919	1	1	yes	characteristic
315	C₂×C₈	#11719919	1	1	yes	characteristic
316	C₂×C₈	#11719919	1	1	yes	characteristic
317	C₂×C₈	#11719919	1	1	yes	characteristic
318	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
319	C₂×C₈	C₄⋊Dic₅₃	1	1	yes	characteristic
320	C₂×C₈	#11719921	1	1	yes	characteristic
321	C₂×C₈	#11719921	1	1	yes	characteristic
322	C₂×C₈	#11719921	1	1	yes	characteristic
323	C₂×C₈	#11719921	1	1	yes	characteristic
324	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
325	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
326	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
327	C₂×C₈	C₂²⋊Dic₅₃	1	1	yes	characteristic
328	C₂×C₈	C₂×Dic₁₀₆	1	1	yes	characteristic
329	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
330	C₂×C₈	C₂×C₄×D₅₃	1	1	yes	characteristic
331	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
332	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
333	C₂×C₈	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
334	C₁₆	—	2	4	no	—
335	C₂³	—	4	2	no	—
336	C₂³	#13133142	1	1	yes	characteristic
337	C₂³	#13133142	1	1	yes	characteristic
338	C₂³	#13133142	1	1	yes	characteristic
339	C₂³	#13133142	1	1	yes	characteristic
340	C₂³	#13133151	1	1	yes	characteristic
341	C₂³	#13133151	1	1	yes	characteristic
342	C₂³	C₂×Dic₂₁₂	1	1	yes	characteristic
343	C₂×C₄	#13133124	1	1	yes	characteristic
344	C₂×C₄	#13133124	1	1	yes	characteristic
345	C₂×C₄	#13133124	1	1	yes	characteristic
346	C₂×C₄	#13133124	1	1	yes	characteristic
347	C₂×C₄	#13133139	1	1	yes	characteristic
348	C₂×C₄	#13133139	1	1	yes	characteristic
349	C₂×C₄	#13133139	1	1	yes	characteristic
350	C₂×C₄	#13133139	1	1	yes	characteristic
351	C₂×C₄	#13133139	1	1	yes	characteristic
352	C₂×C₄	#13133139	1	1	yes	characteristic
353	C₂×C₄	#13133139	1	1	yes	characteristic
354	C₂×C₄	#13133139	1	1	yes	characteristic
355	C₂×C₄	#13133142	1	1	yes	characteristic
356	C₂×C₄	#13133142	1	1	yes	characteristic
357	C₂×C₄	#13133142	1	1	yes	characteristic
358	C₂×C₄	#13133142	1	1	yes	characteristic
359	C₂×C₄	#13133149	1	1	yes	characteristic
360	C₂×C₄	#13133149	1	1	yes	characteristic
361	C₂×C₄	#13133149	1	1	yes	characteristic
362	C₂×C₄	#13133149	1	1	yes	characteristic
363	C₂×C₄	#13133151	1	1	yes	characteristic
364	C₂×C₄	#13133151	1	1	yes	characteristic
365	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
366	C₂×C₄	C₂×#11719921	1	1	yes	characteristic
367	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
368	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
369	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
370	C₂×C₄	C₂×#11719913	1	1	yes	characteristic
371	C₈	—	212	4	no	—
372	C₈	—	212	4	no	—
373	C₈	#13133124	1	1	yes	characteristic
374	C₈	#13133124	1	1	yes	characteristic
375	C₈	#13133124	1	1	yes	characteristic
376	C₈	#13133124	1	1	yes	characteristic
377	C₈	C₂×C₄×Dic₅₃	1	1	yes	characteristic
378	C₈	C₂×C₂²⋊Dic₅₃	1	1	yes	characteristic
379	C₈	C₂×#11719919	1	1	yes	characteristic
380	C₈	C₂×#11719919	1	1	yes	characteristic
381	C₂²	—	2	2	no	—
382	C₂²	—	2	2	no	—
383	C₂²	—	2	2	no	—
384	C₂²	—	4	1	no	—
385	C₂²	—	4	1	no	—
386	C₂²	—	4	1	no	—
387	C₂²	—	4	1	no	—
388	C₂²	#16810386	1	1	yes	characteristic
389	C₂²	#16810386	1	1	yes	characteristic
390	C₂²	#16810386	1	1	yes	characteristic
391	C₂²	#16810386	1	1	yes	characteristic
392	C₂²	C₂×#13133142	1	1	yes	characteristic
393	C₂²	C₂×#13133142	1	1	yes	characteristic
394	C₂²	C₂×#13133151	1	1	yes	characteristic
395	C₄	#16810386	1	1	yes	characteristic
396	C₄	#16810386	1	1	yes	characteristic
397	C₄	#16810386	1	1	yes	characteristic
398	C₄	#16810386	1	1	yes	characteristic
399	C₄	C₂×#13133124	1	1	yes	characteristic
400	C₄	C₂×#13133139	1	1	yes	characteristic
401	C₄	C₂×#13133139	1	1	yes	characteristic
402	C₄	C₂×#13133149	1	1	yes	characteristic
403	C₂	—	2	1	no	—
404	C₂	—	2	1	no	—
405	C₂	—	2	1	no	—
406	C₂	—	2	1	no	—
407	C₂	—	2	1	no	—
408	C₂	—	2	1	no	—
409	C₂	#20272827	1	2	yes	direct product factor
410	C₂	C₂×#16810386	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133139
commutator	C₈×C₅₃	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₅₃
Fitting	C₂²×M₃₂×C₅₃	C₂
socle	C₂²×C₅₃	#340
prime radical	C₂⁴×C₅₃	Q₁₆
cosocle	C₂²×C₈×C₅₃	C₂³
hypercenter	C₂²×M₃₂	D₅₃
hypocenter	C₅₃	C₂×#1388
LCS	G ▷ C₈×C₅₃ ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂²×M₃₂

Complements

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

N	H ≅ G/N	type
3. #20272827	C₂	direct (N × H)
208. C₂×C₅₃	#1388	semidirect
232. C₅₃	C₂×#1388	semidirect
409. C₂	#20272827	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#13133153	2
#20272827	2
C₂²×M₃₂×C₅₃	2
C₂×#1388	53

Subgroups — C2×#20272827 #27299023

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20272827 #27299023