Subgroups — C₂×#20274594 #27382098

Order 13568.

1102 conjugacy classes of subgroups.

The subgroup lattice has 326 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	C₂×#16810661	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810578	C₂	1	2	yes	maximal
4	#20274594	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16810383	C₂	1	2	yes	maximal
6	C₂²×C₄×C₈×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₂²	1	2	yes	—
8	C₂×#13133176	C₂²	1	2	yes	—
9	C₂×#13133153	C₄	1	8	yes	complemented
10	C₂×#13133153	C₂²	1	4	yes	—
11	#16810661	C₄	1	4	yes	complemented
12	#16810661	C₂²	1	4	yes	—
13	C₂×#13133151	C₄	1	2	yes	complemented
14	C₂×#13133149	C₄	1	2	yes	complemented
15	#16810578	—	2	4	no	—
16	#16810578	C₂²	1	8	yes	—
17	C₂×#13133127	—	2	2	no	—
18	C₂×C₈×Dic₅₃	—	2	2	no	—
19	#16810383	C₂²	1	8	yes	—
20	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
21	C₂³×C₈×C₅₃	C₂²	1	1	yes	characteristic
22	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₅₃	C₄	1	8	yes	complemented
24	C₂×C₄×C₈×C₅₃	C₂²	1	4	yes	—
25	C₂×#11719917	—	2	2	no	—
26	C₂×#11719917	—	2	4	no	—
27	C₂×#11719917	C₂×C₄	1	8	yes	—
28	C₂×#11719917	C₂³	1	4	yes	—
29	C₂²×#88322	—	2	2	no	—
30	#13133176	—	2	2	no	—
31	#13133176	—	2	2	no	—
32	#13133176	C₂×C₄	1	4	yes	—
33	#13133176	C₂³	1	4	yes	—
34	C₂×C₄⋊Dic₅₃	C₂×C₄	1	4	yes	—
35	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
36	C₂×C₄×Dic₅₃	—	2	2	no	—
37	#13133153	C₂×C₄	1	16	yes	complemented
38	#13133151	C₂×C₄	1	8	yes	complemented
39	#13133149	C₂×C₄	1	8	yes	complemented
40	#13133127	—	2	8	no	—
41	C₈×Dic₅₃	—	2	8	no	—
42	C₂³×C₄×C₅₃	—	2	1	no	—
43	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
48	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
49	C₂²×C₈×C₅₃	C₂×C₄	1	4	yes	—
50	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₅₃	D₄	1	2	yes	—
52	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	Q₈	1	4	yes	—
54	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
55	C₂²×C₈×C₅₃	C₂³	1	2	yes	—
56	C₂×C₄²×C₅₃	—	2	2	no	—
57	C₂×C₄²×C₅₃	—	2	1	no	—
58	C₂×C₄²×C₅₃	C₂×C₄	1	4	yes	—
59	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
60	C₄×C₈×C₅₃	C₂×C₄	1	16	yes	complemented
61	C₂⁴×C₅₃	#41	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	—	2	8	no	—
63	C₂²×C₄×C₅₃	—	2	4	no	—
64	C₂²×C₄×C₅₃	—	2	2	no	—
65	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
71	C₂²×C₄×C₅₃	C₂×Q₈	1	2	yes	—
72	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
73	C₂²×C₄×C₅₃	#41	1	2	yes	—
74	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
75	C₂²×Dic₅₃	—	2	4	no	—
76	C₂×C₈×C₅₃	C₄²	1	4	yes	—
77	C₂×C₈×C₅₃	C₄²	1	4	yes	—
78	C₂×C₈×C₅₃	C₄²	1	8	yes	—
79	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
80	C₂×C₈×C₅₃	C₄⋊C₄	1	8	yes	—
81	C₂×C₈×C₅₃	C₄⋊C₄	1	4	yes	—
82	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
85	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
86	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
87	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₅₃	C₂×D₄	1	2	yes	—
89	C₂×C₈×C₅₃	C₂×Q₈	1	4	yes	—
90	C₄²×C₅₃	—	2	4	no	—
91	C₄²×C₅₃	—	2	2	no	—
92	C₄²×C₅₃	C₂²×C₄	1	4	yes	—
93	C₂²⋊Dic₅₃	—	2	4	no	—
94	C₂²⋊Dic₅₃	—	2	4	no	—
95	C₄⋊Dic₅₃	—	2	4	no	—
96	C₄⋊Dic₅₃	C₂²×C₄	1	8	yes	—
97	C₄×Dic₅₃	—	2	4	no	—
98	C₄×Dic₅₃	—	2	4	no	—
99	#11719917	—	2	8	no	—
100	#11719917	—	2	4	no	—
101	#11719917	C₂²×C₄	1	8	yes	—
102	C₂×#88322	—	2	8	no	—
103	C₂×#88322	—	2	4	no	—
104	C₂³×C₅₃	—	2	4	no	—
105	C₂³×C₅₃	#117	1	2	yes	—
106	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
107	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
108	C₂³×C₅₃	C₄×Q₈	1	2	yes	—
109	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
110	C₂×C₄×C₅₃	—	2	8	no	—
111	C₂×C₄×C₅₃	—	2	16	no	—
112	C₂×C₄×C₅₃	—	2	4	no	—
113	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
114	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
115	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
116	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
117	C₂×C₄×C₅₃	#117	1	2	yes	—
118	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
119	C₂×C₄×C₅₃	C₄×D₄	1	2	yes	—
120	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
121	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
122	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
123	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
124	C₂×C₄×C₅₃	C₄×Q₈	1	2	yes	—
125	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂²×Q₈	1	1	yes	characteristic
130	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
131	C₂×Dic₅₃	—	2	8	no	—
132	C₂×Dic₅₃	—	4	8	no	—
133	C₈×C₅₃	C₂×C₄²	1	4	yes	—
134	C₈×C₅₃	C₂×C₄²	1	4	yes	—
135	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
136	C₈×C₅₃	C₂×C₄⋊C₄	1	4	yes	—
137	#88322	—	2	8	no	—
138	C₂×#1772	—	53	1	no	maximal, Sylow
139	C₂²×C₅₃	—	2	2	no	—
140	C₂²×C₅₃	—	2	2	no	—
141	C₂²×C₅₃	—	2	4	no	—
142	C₂²×C₅₃	—	2	4	no	—
143	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
144	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
145	C₂²×C₅₃	#443	1	4	yes	—
146	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
147	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
148	C₂²×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
149	C₄×C₅₃	—	2	8	no	—
150	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
151	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
152	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
153	C₄×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
154	C₄×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
155	C₄×C₅₃	C₂×C₄×Q₈	1	1	yes	characteristic
156	Dic₅₃	—	4	8	no	—
157	C₂²×#108	—	53	1	no	—
158	C₂×#427	—	53	1	no	—
159	C₂×#405	—	53	2	no	—
160	C₂²×C₄×C₈	D₅₃	1	1	yes	characteristic
161	#1772	—	53	8	no	—
162	C₂×#339	—	53	2	no	—
163	C₂×C₅₃	—	2	1	no	—
164	C₂×C₅₃	—	2	1	no	—
165	C₂×C₅₃	—	2	2	no	—
166	C₂×C₅₃	#1772	1	4	yes	complemented
167	C₂×C₅₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
168	C₂×C₅₃	C₂×#443	1	2	yes	—
169	C₂²×M₁₆	—	53	2	no	—
170	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
171	C₂³×C₈	C₂×D₅₃	1	1	yes	characteristic
172	C₂×#117	—	53	2	no	—
173	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
174	C₂×#108	—	53	8	no	—
175	C₂×#108	—	53	4	no	—
176	#427	—	53	4	no	—
177	#427	—	53	4	no	—
178	C₂×#107	—	53	2	no	—
179	C₂×#106	—	53	2	no	—
180	#405	—	53	8	no	—
181	#405	—	106	4	no	—
182	C₂×#97	—	106	2	no	—
183	C₂×C₄×C₈	—	106	2	no	—
184	C₂×C₄×C₈	Dic₅₃	1	8	yes	complemented
185	C₂×C₄×C₈	C₂×D₅₃	1	4	yes	—
186	#339	—	53	8	no	—
187	C₅₃	C₂×#1772	1	1	yes	characteristic, complemented, Sylow
188	C₂³×C₄	—	2	1	no	—
189	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
190	C₂×M₁₆	—	53	8	no	—
191	C₂×M₁₆	—	53	4	no	—
192	C₂×M₁₆	—	106	2	no	—
193	C₂×M₁₆	—	106	4	no	—
194	C₂²×C₈	—	106	2	no	—
195	C₂²×C₈	Dic₁₀₆	1	4	yes	—
196	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
197	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
198	C₂²×C₈	C₄×D₅₃	1	2	yes	—
199	C₂²×C₈	C₄×D₅₃	1	4	yes	—
200	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
201	C₂²×C₈	D₂₁₂	1	2	yes	—
202	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
203	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
204	C₂²×C₈	C₂×Dic₅₃	1	4	yes	—
205	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
206	C₂²×C₈	C₂²×D₅₃	1	2	yes	—
207	#117	—	53	4	no	—
208	#117	—	53	4	no	—
209	#117	—	106	2	no	—
210	#117	—	106	2	no	—
211	C₂×C₄⋊C₄	—	53	4	no	—
212	C₂×C₂²⋊C₄	—	106	2	no	—
213	C₂×C₄²	—	2	1	no	—
214	C₂×C₄²	—	2	2	no	—
215	C₂×C₄²	—	106	2	no	—
216	C₂×C₄²	C₂×Dic₅₃	1	4	yes	—
217	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
218	#108	—	53	16	no	—
219	#107	—	53	8	no	—
220	#106	—	53	8	no	—
221	#97	—	106	8	no	—
222	C₄×C₈	—	106	8	no	—
223	C₄×C₈	C₂×Dic₅₃	1	16	yes	complemented
224	C₂⁴	#11719945	1	1	yes	characteristic
225	C₂²×C₄	—	2	8	no	—
226	C₂²×C₄	—	2	4	no	—
227	C₂²×C₄	—	2	2	no	—
228	C₂²×C₄	—	106	4	no	—
229	C₂²×C₄	C₂×Dic₁₀₆	1	2	yes	—
230	C₂²×C₄	C₂×C₄×D₅₃	1	2	yes	—
231	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
232	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
234	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
235	C₂²×C₄	#11719945	1	1	yes	characteristic
236	C₂²×C₄	#11719945	1	2	yes	—
237	C₂²×C₄	C₂²×Dic₅₃	1	2	yes	—
238	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
239	M₁₆	—	53	8	no	—
240	M₁₆	—	106	8	no	—
241	M₁₆	—	106	4	no	—
242	C₂×C₈	—	106	8	no	—
243	C₂×C₈	—	106	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	C₄⋊Dic₅₃	1	4	yes	—
248	C₂×C₈	C₄⋊Dic₅₃	1	8	yes	—
249	C₂×C₈	C₄⋊Dic₅₃	1	4	yes	—
250	C₂×C₈	C₂×Dic₁₀₆	1	4	yes	—
251	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
252	C₂×C₈	C₂×C₄×D₅₃	1	2	yes	—
253	C₂×C₈	C₂×C₄×D₅₃	1	4	yes	—
254	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
255	C₂×C₈	C₂×D₂₁₂	1	2	yes	—
256	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
257	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
258	C₄⋊C₄	—	53	8	no	—
259	C₄⋊C₄	—	106	4	no	—
260	C₂²⋊C₄	—	106	4	no	—
261	C₂²⋊C₄	—	106	4	no	—
262	C₄²	—	2	4	no	—
263	C₄²	—	2	2	no	—
264	C₄²	—	106	4	no	—
265	C₄²	—	106	4	no	—
266	C₄²	C₂²×Dic₅₃	1	4	yes	—
267	C₂³	—	2	4	no	—
268	C₂³	#13133176	1	2	yes	—
269	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
270	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
271	C₂³	C₄×Dic₁₀₆	1	2	yes	—
272	C₂³	C₂×#11719945	1	1	yes	characteristic
273	C₂×C₄	—	2	16	no	—
274	C₂×C₄	—	2	8	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	106	8	no	—
277	C₂×C₄	—	212	8	no	—
278	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
279	C₂×C₄	C₂×C₄×Dic₅₃	1	2	yes	—
280	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
281	C₂×C₄	C₂×C₄⋊Dic₅₃	1	2	yes	—
282	C₂×C₄	#13133176	1	2	yes	—
283	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
284	C₂×C₄	C₄×D₂₁₂	1	2	yes	—
285	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
286	C₂×C₄	C₄×D₂₁₂	1	1	yes	characteristic
287	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
288	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
289	C₂×C₄	C₄×Dic₁₀₆	1	2	yes	—
290	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
291	C₂×C₄	C₂²×C₄×D₅₃	1	1	yes	characteristic
292	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₂₁₂	1	1	yes	characteristic
294	C₂×C₄	C₂²×Dic₁₀₆	1	1	yes	characteristic
295	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
296	C₈	—	106	8	no	—
297	C₈	C₂×C₄×Dic₅₃	1	4	yes	—
298	C₈	C₂×C₄×Dic₅₃	1	4	yes	—
299	C₈	C₂×C₄⋊Dic₅₃	1	4	yes	—
300	C₈	C₂×C₄⋊Dic₅₃	1	4	yes	—
301	C₂²	—	2	4	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	2	no	—
305	C₂²	C₄×C₄⋊Dic₅₃	1	2	yes	—
306	C₂²	C₄×C₄⋊Dic₅₃	1	2	yes	—
307	C₂²	#16810741	1	4	yes	—
308	C₂²	C₂×#13133176	1	1	yes	characteristic
309	C₂²	C₂×C₄×D₂₁₂	1	1	yes	characteristic
310	C₂²	C₂×C₄×Dic₁₀₆	1	1	yes	characteristic
311	C₄	—	2	8	no	—
312	C₄	—	212	8	no	—
313	C₄	C₄×C₄⋊Dic₅₃	1	2	yes	—
314	C₄	C₄×C₄⋊Dic₅₃	1	2	yes	—
315	C₄	C₂²×C₄×Dic₅₃	1	1	yes	characteristic
316	C₄	C₂²×C₄⋊Dic₅₃	1	1	yes	characteristic
317	C₄	C₂×C₄×D₂₁₂	1	1	yes	characteristic
318	C₄	C₂×C₄×Dic₁₀₆	1	1	yes	characteristic
319	C₂	—	2	1	no	—
320	C₂	—	2	1	no	—
321	C₂	—	2	2	no	—
322	C₂	#20274594	1	4	yes	direct product factor
323	C₂	C₂×C₄×C₄⋊Dic₅₃	1	1	yes	characteristic
324	C₂	C₂×#16810741	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	D₂₁₂
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂²×C₄×C₈×C₅₃	C₂
socle	C₂³×C₅₃	C₄×D₄
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂²×C₄×C₈	D₅₃
hypocenter	C₅₃	C₂×#1772
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈

Complements

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

N	H ≅ G/N	type
4. #20274594	C₂	direct (N × H)
9. C₂×#13133153	C₄	semidirect
11. #16810661	C₄	semidirect
13. C₂×#13133151	C₄	semidirect
14. C₂×#13133149	C₄	semidirect
23. C₂×C₄×C₈×C₅₃	C₄	semidirect
37. #13133153	C₂×C₄	semidirect
38. #13133151	C₂×C₄	semidirect
39. #13133149	C₂×C₄	semidirect
60. C₄×C₈×C₅₃	C₂×C₄	semidirect
166. C₂×C₅₃	#1772	semidirect
184. C₂×C₄×C₈	Dic₅₃	semidirect
187. C₅₃	C₂×#1772	semidirect
223. C₄×C₈	C₂×Dic₅₃	semidirect
322. C₂	#20274594	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
C₂×#16810661	2
C₂×#16810578	2
#20274594	2
C₂×#16810383	2
C₂²×C₄×C₈×C₅₃	2
C₂×#1772	53

Subgroups — C2×#20274594 #27382098

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274594 #27382098