Subgroups — C₂×#20274602 #27382107

Order 13568.

1062 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16810661	C₂	1	1	yes	maximal, characteristic
2	C₂×#16810581	C₂	1	2	yes	maximal
3	C₂²×#13133127	C₂	1	1	yes	maximal, characteristic
4	#20274602	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16810384	C₂	1	2	yes	maximal
6	C₂²×#108×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×#88322	C₂²	1	1	yes	characteristic
8	C₂×#13133176	C₂²	1	2	yes	—
9	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
10	#16810661	C₄	1	4	yes	complemented
11	#16810661	C₂²	1	4	yes	—
12	C₂×#13133151	C₄	1	2	yes	complemented
13	C₂×#13133149	C₄	1	2	yes	complemented
14	#16810581	—	2	4	no	—
15	#16810581	C₂²	1	8	yes	—
16	C₂×#13133127	—	2	2	no	—
17	C₂×#13133127	C₄	1	8	yes	complemented
18	C₂×#13133127	C₂²	1	4	yes	—
19	C₂×C₈×Dic₅₃	—	2	2	no	—
20	#16810384	C₂²	1	8	yes	—
21	C₂²×M₁₆×C₅₃	C₂²	1	2	yes	—
22	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×#108×C₅₃	C₄	1	8	yes	complemented
24	C₂×#108×C₅₃	C₂²	1	4	yes	—
25	C₂³×Dic₅₃	—	2	1	no	—
26	C₂²×#88322	C₂×C₄	1	4	yes	—
27	C₂²×#88322	D₄	1	2	yes	—
28	C₂²×#88322	D₄	1	2	yes	—
29	C₂²×#88322	Q₈	1	4	yes	—
30	C₂²×#88322	C₂³	1	2	yes	—
31	#13133176	—	2	2	no	—
32	#13133176	—	2	2	no	—
33	#13133176	C₂×C₄	1	4	yes	—
34	#13133176	C₂³	1	4	yes	—
35	C₂×C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
36	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
37	C₂×C₄×Dic₅₃	—	2	2	no	—
38	C₂×C₄×Dic₅₃	—	2	2	no	—
39	C₂×C₄×Dic₅₃	—	2	1	no	—
40	C₂×C₄×Dic₅₃	C₂×C₄	1	4	yes	—
41	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
42	#13133151	C₂×C₄	1	8	yes	complemented
43	#13133149	C₂×C₄	1	8	yes	complemented
44	#13133127	—	2	8	no	—
45	#13133127	C₂×C₄	1	16	yes	complemented
46	C₈×Dic₅₃	—	2	8	no	—
47	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
48	C₂×M₁₆×C₅₃	—	2	2	no	—
49	C₂×M₁₆×C₅₃	—	2	4	no	—
50	C₂×M₁₆×C₅₃	C₂×C₄	1	8	yes	—
51	C₂×M₁₆×C₅₃	C₂³	1	4	yes	—
52	C₂²×C₈×C₅₃	—	2	2	no	—
53	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
54	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
56	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
57	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
58	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
59	#108×C₅₃	C₂×C₄	1	16	yes	complemented
60	C₂⁴×C₅₃	#41	1	1	yes	characteristic
61	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
62	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
63	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
68	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	#41	1	2	yes	—
70	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₅₃	—	2	4	no	—
72	C₂²×Dic₅₃	—	2	4	no	—
73	C₂²×Dic₅₃	—	2	2	no	—
74	C₂²×Dic₅₃	—	4	4	no	—
75	M₁₆×C₅₃	—	2	8	no	—
76	M₁₆×C₅₃	—	2	4	no	—
77	M₁₆×C₅₃	C₂²×C₄	1	8	yes	—
78	C₂×C₈×C₅₃	—	2	8	no	—
79	C₂×C₈×C₅₃	—	2	4	no	—
80	C₂×C₈×C₅₃	C₄²	1	4	yes	—
81	C₂×C₈×C₅₃	C₄²	1	4	yes	—
82	C₂×C₈×C₅₃	C₄²	1	8	yes	—
83	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
84	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
87	C₂²⋊Dic₅₃	—	2	4	no	—
88	C₂²⋊Dic₅₃	—	2	4	no	—
89	C₄⋊Dic₅₃	—	2	4	no	—
90	C₄⋊Dic₅₃	C₂²×C₄	1	8	yes	—
91	C₄×Dic₅₃	—	2	4	no	—
92	C₄×Dic₅₃	—	2	4	no	—
93	C₄×Dic₅₃	—	2	4	no	—
94	C₄×Dic₅₃	—	2	2	no	—
95	C₄×Dic₅₃	C₂²×C₄	1	4	yes	—
96	C₂×#88322	C₄⋊C₄	1	8	yes	—
97	C₂×#88322	C₄⋊C₄	1	8	yes	—
98	C₂×#88322	C₂²×C₄	1	4	yes	—
99	C₂×#88322	C₂×D₄	1	4	yes	—
100	C₂×#88322	C₂×Q₈	1	4	yes	—
101	C₂³×C₅₃	—	2	4	no	—
102	C₂³×C₅₃	#117	1	2	yes	—
103	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
104	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
105	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
106	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
107	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
108	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
109	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
110	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
111	C₂×C₄×C₅₃	#117	1	2	yes	—
112	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
115	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
116	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
117	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
118	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
119	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
120	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
123	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
124	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
125	C₂×Dic₅₃	—	2	4	no	—
126	C₂×Dic₅₃	—	2	8	no	—
127	C₂×Dic₅₃	—	4	4	no	—
128	C₂×Dic₅₃	—	4	8	no	—
129	C₂×Dic₅₃	—	4	8	no	—
130	C₈×C₅₃	—	2	8	no	—
131	C₈×C₅₃	C₂×C₄²	1	4	yes	—
132	C₈×C₅₃	C₂×C₄²	1	4	yes	—
133	#88322	C₂×C₄⋊C₄	1	8	yes	—
134	C₂×#1773	—	53	1	no	maximal, Sylow
135	C₂²×C₅₃	—	2	2	no	—
136	C₂²×C₅₃	—	2	2	no	—
137	C₂²×C₅₃	—	2	4	no	—
138	C₂²×C₅₃	—	2	4	no	—
139	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
140	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
141	C₂²×C₅₃	#444	1	4	yes	—
142	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
143	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
144	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
145	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
146	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
147	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
148	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
149	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
150	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
151	Dic₅₃	—	4	4	no	—
152	Dic₅₃	—	4	8	no	—
153	C₂²×#108	D₅₃	1	1	yes	characteristic
154	C₂×#427	—	53	1	no	—
155	C₂×#405	—	53	2	no	—
156	C₂²×#97	—	53	1	no	—
157	#1773	—	53	8	no	—
158	C₂×#339	—	53	2	no	—
159	C₂×C₅₃	—	2	1	no	—
160	C₂×C₅₃	—	2	1	no	—
161	C₂×C₅₃	—	2	2	no	—
162	C₂×C₅₃	#1773	1	4	yes	complemented
163	C₂×C₅₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
164	C₂×C₅₃	C₂×#444	1	2	yes	—
165	C₂²×M₁₆	C₂×D₅₃	1	2	yes	—
166	C₂³×C₈	—	53	1	no	—
167	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
168	C₂×#117	—	53	2	no	—
169	C₂²×C₄²	—	53	1	no	—
170	C₂×#108	Dic₅₃	1	8	yes	complemented
171	C₂×#108	C₂×D₅₃	1	4	yes	—
172	#427	—	53	4	no	—
173	#427	—	53	4	no	—
174	C₂×#107	—	53	2	no	—
175	C₂×#106	—	53	2	no	—
176	#405	—	53	8	no	—
177	#405	—	106	4	no	—
178	C₂×#97	—	53	8	no	—
179	C₂×#97	—	53	4	no	—
180	C₂×#97	—	106	2	no	—
181	C₂×C₄×C₈	—	106	2	no	—
182	#339	—	53	8	no	—
183	C₅₃	C₂×#1773	1	1	yes	characteristic, complemented, Sylow
184	C₂³×C₄	—	106	1	no	—
185	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	4	no	—
188	C₂×M₁₆	C₂×Dic₅₃	1	8	yes	—
189	C₂×M₁₆	C₂²×D₅₃	1	4	yes	—
190	C₂²×C₈	—	2	2	no	—
191	C₂²×C₈	—	53	4	no	—
192	C₂²×C₈	—	53	2	no	—
193	C₂²×C₈	—	53	2	no	—
194	C₂²×C₈	—	53	4	no	—
195	C₂²×C₈	—	53	2	no	—
196	C₂²×C₈	C₄×D₅₃	1	4	yes	—
197	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
198	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
199	C₂²×C₈	C₄×D₅₃	1	2	yes	—
200	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
201	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
202	#117	—	53	4	no	—
203	#117	—	53	4	no	—
204	#117	—	106	2	no	—
205	#117	—	106	2	no	—
206	C₂×C₄⋊C₄	—	53	4	no	—
207	C₂×C₂²⋊C₄	—	106	2	no	—
208	C₂×C₄²	—	53	4	no	—
209	C₂×C₄²	—	53	2	no	—
210	C₂×C₄²	—	106	2	no	—
211	C₂×C₄²	—	106	2	no	—
212	C₂×C₄²	—	106	1	no	—
213	#108	C₂×Dic₅₃	1	16	yes	complemented
214	#107	—	53	8	no	—
215	#106	—	53	8	no	—
216	#97	—	53	16	no	—
217	#97	—	106	8	no	—
218	C₄×C₈	—	106	8	no	—
219	C₂⁴	#11719949	1	1	yes	characteristic
220	C₂²×C₄	—	106	4	no	—
221	C₂²×C₄	—	106	4	no	—
222	C₂²×C₄	—	106	2	no	—
223	C₂²×C₄	—	212	4	no	—
224	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
225	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
226	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
227	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
228	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
229	C₂²×C₄	#11719947	1	2	yes	—
230	C₂²×C₄	Q₈×D₅₃	1	2	yes	—
231	C₂²×C₄	#11719949	1	1	yes	characteristic
232	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
233	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
234	M₁₆	—	2	8	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	C₂²×Dic₅₃	1	8	yes	—
237	C₂×C₈	—	2	8	no	—
238	C₂×C₈	—	2	4	no	—
239	C₂×C₈	—	53	8	no	—
240	C₂×C₈	—	53	8	no	—
241	C₂×C₈	—	53	4	no	—
242	C₂×C₈	—	53	4	no	—
243	C₂×C₈	—	53	4	no	—
244	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
245	C₂×C₈	C₄×Dic₅₃	1	4	yes	—
246	C₂×C₈	C₄×Dic₅₃	1	8	yes	—
247	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
248	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
249	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
250	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
251	C₄⋊C₄	—	53	8	no	—
252	C₄⋊C₄	—	106	4	no	—
253	C₂²⋊C₄	—	106	4	no	—
254	C₂²⋊C₄	—	106	4	no	—
255	C₄²	—	53	4	no	—
256	C₄²	—	106	4	no	—
257	C₄²	—	106	4	no	—
258	C₄²	—	106	4	no	—
259	C₄²	—	106	2	no	—
260	C₂³	—	2	4	no	—
261	C₂³	#13133175	1	2	yes	—
262	C₂³	#13133179	1	1	yes	characteristic
263	C₂³	#13133179	1	1	yes	characteristic
264	C₂³	Q₈×Dic₅₃	1	2	yes	—
265	C₂³	C₂×#11719949	1	1	yes	characteristic
266	C₂×C₄	—	106	4	no	—
267	C₂×C₄	—	106	8	no	—
268	C₂×C₄	—	212	8	no	—
269	C₂×C₄	—	212	4	no	—
270	C₂×C₄	—	212	8	no	—
271	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
272	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
273	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
274	C₂×C₄	C₄⋊C₄×D₅₃	1	2	yes	—
275	C₂×C₄	#13133175	1	2	yes	—
276	C₂×C₄	#13133179	1	1	yes	characteristic
277	C₂×C₄	#13133179	1	1	yes	characteristic
278	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
279	C₂×C₄	D₄×Dic₅₃	1	2	yes	—
280	C₂×C₄	#13133182	1	2	yes	—
281	C₂×C₄	#13133182	1	2	yes	—
282	C₂×C₄	Q₈×Dic₅₃	1	2	yes	—
283	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
284	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
285	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
286	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
287	C₂×C₄	C₂×Q₈×D₅₃	1	1	yes	characteristic
288	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
289	C₈	—	2	8	no	—
290	C₈	—	53	8	no	—
291	C₈	C₂×C₄×Dic₅₃	1	4	yes	—
292	C₈	C₂×C₄×Dic₅₃	1	4	yes	—
293	C₂²	—	2	4	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	2	2	no	—
297	C₂²	C₄⋊C₄×Dic₅₃	1	2	yes	—
298	C₂²	C₄⋊C₄×Dic₅₃	1	2	yes	—
299	C₂²	#16810744	1	4	yes	—
300	C₂²	C₂×#13133175	1	1	yes	characteristic
301	C₂²	C₂×#13133179	1	1	yes	characteristic
302	C₂²	C₂×Q₈×Dic₅₃	1	1	yes	characteristic
303	C₄	—	212	4	no	—
304	C₄	—	212	8	no	—
305	C₄	C₄⋊C₄×Dic₅₃	1	2	yes	—
306	C₄	C₄⋊C₄×Dic₅₃	1	2	yes	—
307	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
308	C₄	C₂×C₄⋊C₄×D₅₃	1	1	yes	characteristic
309	C₄	C₂×D₄×Dic₅₃	1	1	yes	characteristic
310	C₄	C₂×#13133182	1	1	yes	characteristic
311	C₂	—	2	1	no	—
312	C₂	—	2	1	no	—
313	C₂	—	2	2	no	—
314	C₂	#20274602	1	4	yes	direct product factor
315	C₂	C₂×C₄⋊C₄×Dic₅₃	1	1	yes	characteristic
316	C₂	C₂×#16810744	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂²×#108×C₅₃	C₂
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂²×#108	D₅₃
hypocenter	C₅₃	C₂×#1773
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂³×C₄ ◁ C₂²×#108

Complements

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

N	H ≅ G/N	type
4. #20274602	C₂	direct (N × H)
10. #16810661	C₄	semidirect
12. C₂×#13133151	C₄	semidirect
13. C₂×#13133149	C₄	semidirect
17. C₂×#13133127	C₄	semidirect
23. C₂×#108×C₅₃	C₄	semidirect
42. #13133151	C₂×C₄	semidirect
43. #13133149	C₂×C₄	semidirect
45. #13133127	C₂×C₄	semidirect
59. #108×C₅₃	C₂×C₄	semidirect
162. C₂×C₅₃	#1773	semidirect
170. C₂×#108	Dic₅₃	semidirect
183. C₅₃	C₂×#1773	semidirect
213. #108	C₂×Dic₅₃	semidirect
314. C₂	#20274602	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×#16810661	2
C₂×#16810581	2
C₂²×#13133127	2
#20274602	2
C₂×#16810384	2
C₂²×#108×C₅₃	2
C₂×#1773	53

Subgroups — C2×#20274602 #27382107

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274602 #27382107