Subgroups — C₂×#20274864 #27384357

Order 13568.

730 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20274864	C₂	1	8	yes	maximal, direct product factor
2	C₂×#16810394	C₂	1	2	yes	maximal
3	C₂×#16810383	C₂	1	2	yes	maximal
4	C₂×#16810381	C₂	1	1	yes	maximal, characteristic
5	C₂×#16810380	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄×M₁₆×C₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂²×#11719917	C₂²	1	2	yes	—
8	C₂×#13133176	C₂²	1	2	yes	—
9	#16810394	C₂²	1	8	yes	—
10	#16810383	C₂²	1	8	yes	—
11	#16810381	C₂²	1	4	yes	—
12	#16810380	C₂²	1	4	yes	—
13	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
14	C₂²×M₁₆×C₅₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄²×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×M₁₆×C₅₃	C₄	1	4	yes	complemented
17	C₄×M₁₆×C₅₃	C₂²	1	4	yes	—
18	C₂×#97×C₅₃	C₄	1	2	yes	complemented
19	C₂×C₄×C₈×C₅₃	C₄	1	2	yes	complemented
20	C₂×#11719917	—	2	2	no	—
21	C₂×#11719917	—	2	2	no	—
22	C₂×#11719917	—	2	2	no	—
23	C₂×#11719917	—	2	2	no	—
24	C₂×#11719917	—	2	2	no	—
25	C₂×#11719917	C₂³	1	4	yes	—
26	C₂²×#88322	—	2	2	no	—
27	#13133176	—	2	2	no	—
28	#13133176	—	2	2	no	—
29	#13133176	—	2	2	no	—
30	#13133176	C₂³	1	4	yes	—
31	C₂×C₄⋊Dic₅₃	—	2	2	no	—
32	C₂×C₂²⋊Dic₅₃	—	2	2	no	—
33	C₂×C₄×Dic₅₃	—	2	2	no	—
34	C₂³×C₄×C₅₃	—	2	1	no	—
35	C₂³×C₄×C₅₃	C₂³	1	1	yes	characteristic
36	C₂×M₁₆×C₅₃	—	2	2	no	—
37	C₂×M₁₆×C₅₃	—	2	2	no	—
38	C₂×M₁₆×C₅₃	—	2	2	no	—
39	C₂×M₁₆×C₅₃	—	2	2	no	—
40	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
41	C₂×M₁₆×C₅₃	C₂×C₄	1	2	yes	—
42	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
43	C₂×M₁₆×C₅₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
45	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
46	C₂×C₄²×C₅₃	—	2	1	no	—
47	C₂×C₄²×C₅₃	—	2	1	no	—
48	C₂×C₄²×C₅₃	—	2	1	no	—
49	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄²×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
53	#97×C₅₃	C₂×C₄	1	8	yes	complemented
54	C₄×C₈×C₅₃	C₂×C₄	1	8	yes	complemented
55	C₂⁴×C₅₃	#41	1	1	yes	characteristic
56	C₂²×C₄×C₅₃	—	2	1	no	—
57	C₂²×C₄×C₅₃	—	2	2	no	—
58	C₂²×C₄×C₅₃	—	2	1	no	—
59	C₂²×C₄×C₅₃	—	2	2	no	—
60	C₂²×C₄×C₅₃	—	4	4	no	—
61	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
63	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	#41	1	2	yes	—
65	C₂²×C₄×C₅₃	#41	1	2	yes	—
66	C₂²×C₄×C₅₃	#41	1	2	yes	—
67	C₂²×C₄×C₅₃	#41	1	2	yes	—
68	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
69	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
70	C₂²×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
71	C₂²×Dic₅₃	—	4	2	no	—
72	M₁₆×C₅₃	—	2	4	no	—
73	M₁₆×C₅₃	—	2	4	no	—
74	M₁₆×C₅₃	—	2	4	no	—
75	M₁₆×C₅₃	—	2	4	no	—
76	C₂×C₈×C₅₃	—	2	2	no	—
77	C₂×C₈×C₅₃	—	2	2	no	—
78	C₂×C₈×C₅₃	—	2	2	no	—
79	C₂×C₈×C₅₃	—	2	2	no	—
80	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
81	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
82	C₄²×C₅₃	—	2	1	no	—
83	C₄²×C₅₃	—	2	2	no	—
84	C₄²×C₅₃	—	2	1	no	—
85	C₄²×C₅₃	—	2	2	no	—
86	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
87	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
88	C₂²⋊Dic₅₃	—	2	4	no	—
89	C₂²⋊Dic₅₃	—	4	2	no	—
90	C₄⋊Dic₅₃	—	2	4	no	—
91	C₄⋊Dic₅₃	—	4	2	no	—
92	C₄×Dic₅₃	—	2	4	no	—
93	C₄×Dic₅₃	—	4	2	no	—
94	#11719917	—	2	4	no	—
95	#11719917	—	2	4	no	—
96	#11719917	—	4	2	no	—
97	#11719917	—	4	2	no	—
98	C₂×#88322	—	2	4	no	—
99	C₂×#88322	—	4	2	no	—
100	C₂×#88322	—	4	2	no	—
101	C₂³×C₅₃	—	4	2	no	—
102	C₂³×C₅₃	#117	1	1	yes	characteristic
103	C₂³×C₅₃	#117	1	1	yes	characteristic
104	C₂³×C₅₃	#126	1	2	yes	—
105	C₂³×C₅₃	#126	1	2	yes	—
106	C₂³×C₅₃	C₂×#41	1	1	yes	characteristic
107	C₂×C₄×C₅₃	—	2	4	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	—	4	4	no	—
111	C₂×C₄×C₅₃	—	4	4	no	—
112	C₂×C₄×C₅₃	—	4	2	no	—
113	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
114	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
115	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
116	C₂×C₄×C₅₃	#117	1	2	yes	—
117	C₂×C₄×C₅₃	#117	1	2	yes	—
118	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
119	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
120	C₂×C₄×C₅₃	#117	1	1	yes	characteristic
121	C₂×C₄×C₅₃	#126	1	2	yes	—
122	C₂×C₄×C₅₃	#126	1	2	yes	—
123	C₂×C₄×C₅₃	#126	1	2	yes	—
124	C₂×C₄×C₅₃	#126	1	2	yes	—
125	C₂×C₄×C₅₃	#126	1	2	yes	—
126	C₂×C₄×C₅₃	#126	1	2	yes	—
127	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
129	C₂×C₄×C₅₃	C₂×#41	1	2	yes	—
130	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
131	C₂×Dic₅₃	—	4	4	no	—
132	C₂×Dic₅₃	—	8	4	no	—
133	C₈×C₅₃	—	2	4	no	—
134	C₈×C₅₃	—	2	4	no	—
135	#88322	—	4	4	no	—
136	C₂×#1841	—	53	1	no	maximal, Sylow
137	C₂²×C₅₃	—	2	4	no	—
138	C₂²×C₅₃	—	2	2	no	—
139	C₂²×C₅₃	—	4	1	no	—
140	C₂²×C₅₃	—	4	1	no	—
141	C₂²×C₅₃	—	4	2	no	—
142	C₂²×C₅₃	#383	1	1	yes	characteristic
143	C₂²×C₅₃	#383	1	2	yes	—
144	C₂²×C₅₃	#383	1	1	yes	characteristic
145	C₂²×C₅₃	C₂×#117	1	1	yes	characteristic
146	C₂²×C₅₃	C₂×#126	1	2	yes	—
147	C₄×C₅₃	—	2	4	no	—
148	C₄×C₅₃	—	4	2	no	—
149	C₄×C₅₃	#383	1	1	yes	characteristic
150	C₄×C₅₃	#383	1	2	yes	—
151	C₄×C₅₃	#383	1	1	yes	characteristic
152	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
153	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
154	C₄×C₅₃	C₂×#126	1	2	yes	—
155	Dic₅₃	—	8	4	no	—
156	C₂×C₄×M₁₆	D₅₃	1	1	yes	characteristic
157	#1841	—	53	8	no	—
158	C₂×#344	—	53	2	no	—
159	C₂×#339	—	53	2	no	—
160	C₂×#338	—	53	1	no	—
161	C₂×H(3,ℤ/4)	—	53	1	no	—
162	C₂×C₅₃	—	2	2	no	—
163	C₂×C₅₃	—	2	1	no	—
164	C₂×C₅₃	—	2	2	no	—
165	C₂×C₅₃	—	2	1	no	—
166	C₂×C₅₃	#1841	1	2	yes	complemented
167	C₂×C₅₃	C₂×#383	1	1	yes	characteristic
168	C₂²×M₁₆	—	53	2	no	—
169	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
170	C₂²×M₁₆	C₂×D₅₃	1	1	yes	characteristic
171	C₂×#117	—	53	2	no	—
172	C₂²×C₄²	C₂×D₅₃	1	1	yes	characteristic
173	C₄×M₁₆	Dic₅₃	1	4	yes	complemented
174	C₄×M₁₆	C₂×D₅₃	1	4	yes	—
175	C₂×#97	Dic₅₃	1	2	yes	complemented
176	C₂×C₄×C₈	Dic₅₃	1	2	yes	complemented
177	#344	—	53	8	no	—
178	#339	—	53	8	no	—
179	#338	—	53	4	no	—
180	H(3,ℤ/4)	—	53	4	no	—
181	C₅₃	C₂×#1841	1	1	yes	characteristic, complemented, Sylow
182	C₂³×C₄	—	2	1	no	—
183	C₂³×C₄	C₂²×D₅₃	1	1	yes	characteristic
184	C₂×M₁₆	—	2	2	no	—
185	C₂×M₁₆	—	2	2	no	—
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	53	4	no	—
189	C₂×M₁₆	—	106	2	no	—
190	C₂×M₁₆	—	106	2	no	—
191	C₂×M₁₆	—	106	2	no	—
192	C₂×M₁₆	—	106	2	no	—
193	C₂×M₁₆	—	106	2	no	—
194	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
195	C₂×M₁₆	C₂×Dic₅₃	1	2	yes	—
196	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
197	C₂×M₁₆	C₂²×D₅₃	1	2	yes	—
198	C₂²×C₈	—	106	2	no	—
199	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
200	C₂²×C₈	C₂×Dic₅₃	1	2	yes	—
201	#117	—	53	4	no	—
202	#117	—	106	2	no	—
203	#117	—	106	2	no	—
204	#117	—	106	2	no	—
205	C₂×C₄⋊C₄	—	106	2	no	—
206	C₂×C₂²⋊C₄	—	106	2	no	—
207	C₂×C₄²	—	2	1	no	—
208	C₂×C₄²	—	2	1	no	—
209	C₂×C₄²	—	2	1	no	—
210	C₂×C₄²	—	106	2	no	—
211	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
212	C₂×C₄²	C₂×Dic₅₃	1	1	yes	characteristic
213	C₂×C₄²	C₂×Dic₅₃	1	2	yes	—
214	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
215	#97	C₂×Dic₅₃	1	8	yes	complemented
216	C₄×C₈	C₂×Dic₅₃	1	8	yes	complemented
217	C₂⁴	#11719945	1	1	yes	characteristic
218	C₂²×C₄	—	2	1	no	—
219	C₂²×C₄	—	2	2	no	—
220	C₂²×C₄	—	2	1	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	4	4	no	—
223	C₂²×C₄	—	212	2	no	—
224	C₂²×C₄	#11719945	1	1	yes	characteristic
225	C₂²×C₄	#11719945	1	2	yes	—
226	C₂²×C₄	#11719945	1	2	yes	—
227	C₂²×C₄	#11719945	1	2	yes	—
228	C₂²×C₄	#11719945	1	2	yes	—
229	C₂²×C₄	#11719945	1	1	yes	characteristic
230	C₂²×C₄	#11719945	1	1	yes	characteristic
231	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
232	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
233	C₂²×C₄	C₂³×D₅₃	1	1	yes	characteristic
234	M₁₆	—	2	4	no	—
235	M₁₆	—	2	4	no	—
236	M₁₆	—	2	4	no	—
237	M₁₆	—	2	4	no	—
238	M₁₆	—	106	4	no	—
239	M₁₆	—	106	4	no	—
240	M₁₆	—	212	2	no	—
241	M₁₆	—	212	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	—	2	2	no	—
245	C₂×C₈	—	2	2	no	—
246	C₂×C₈	—	106	4	no	—
247	C₂×C₈	—	212	2	no	—
248	C₂×C₈	—	212	2	no	—
249	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
250	C₂×C₈	C₂²×Dic₅₃	1	4	yes	—
251	C₄⋊C₄	—	106	4	no	—
252	C₄⋊C₄	—	212	2	no	—
253	C₂²⋊C₄	—	106	4	no	—
254	C₂²⋊C₄	—	212	2	no	—
255	C₄²	—	2	1	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	1	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	106	4	no	—
260	C₄²	—	212	2	no	—
261	C₄²	C₂²×Dic₅₃	1	2	yes	—
262	C₄²	C₂²×Dic₅₃	1	2	yes	—
263	C₂³	—	4	2	no	—
264	C₂³	#13133176	1	1	yes	characteristic
265	C₂³	#13133176	1	1	yes	characteristic
266	C₂³	#13133211	1	2	yes	—
267	C₂³	#13133211	1	2	yes	—
268	C₂³	C₂×#11719945	1	1	yes	characteristic
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	2	2	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	424	4	no	—
276	C₂×C₄	—	212	4	no	—
277	C₂×C₄	#13133176	1	1	yes	characteristic
278	C₂×C₄	#13133176	1	1	yes	characteristic
279	C₂×C₄	#13133176	1	1	yes	characteristic
280	C₂×C₄	#13133176	1	2	yes	—
281	C₂×C₄	#13133176	1	2	yes	—
282	C₂×C₄	#13133176	1	1	yes	characteristic
283	C₂×C₄	#13133176	1	1	yes	characteristic
284	C₂×C₄	#13133176	1	1	yes	characteristic
285	C₂×C₄	#13133211	1	2	yes	—
286	C₂×C₄	#13133211	1	2	yes	—
287	C₂×C₄	#13133211	1	2	yes	—
288	C₂×C₄	#13133211	1	2	yes	—
289	C₂×C₄	#13133211	1	2	yes	—
290	C₂×C₄	#13133211	1	2	yes	—
291	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
292	C₂×C₄	C₂×#11719945	1	2	yes	—
293	C₂×C₄	C₂×#11719945	1	2	yes	—
294	C₂×C₄	C₂×#11719945	1	1	yes	characteristic
295	C₈	—	2	4	no	—
296	C₈	—	2	4	no	—
297	C₈	—	212	4	no	—
298	C₂²	—	2	2	no	—
299	C₂²	—	2	4	no	—
300	C₂²	—	4	1	no	—
301	C₂²	—	4	1	no	—
302	C₂²	—	4	2	no	—
303	C₂²	#16810491	1	1	yes	characteristic
304	C₂²	#16810491	1	2	yes	—
305	C₂²	#16810491	1	1	yes	characteristic
306	C₂²	C₂×#13133176	1	1	yes	characteristic
307	C₂²	C₂×#13133211	1	2	yes	—
308	C₄	—	2	4	no	—
309	C₄	—	4	2	no	—
310	C₄	—	424	4	no	—
311	C₄	#16810491	1	1	yes	characteristic
312	C₄	#16810491	1	2	yes	—
313	C₄	#16810491	1	1	yes	characteristic
314	C₄	C₂×#13133176	1	1	yes	characteristic
315	C₄	C₂×#13133176	1	1	yes	characteristic
316	C₄	C₂×#13133211	1	2	yes	—
317	C₂	—	2	1	no	—
318	C₂	—	2	2	no	—
319	C₂	—	2	1	no	—
320	C₂	—	2	2	no	—
321	C₂	#20274864	1	2	yes	direct product factor
322	C₂	C₂×#16810491	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#13133176
commutator	C₂×C₄×C₅₃	C₂³×C₄
Frattini	C₂²×C₄	C₂³×D₅₃
Fitting	C₂×C₄×M₁₆×C₅₃	C₂
socle	C₂²×C₅₃	#383
prime radical	C₂⁴×C₅₃	#41
cosocle	C₂²×C₄×C₅₃	C₂⁴
hypercenter	C₂×C₄×M₁₆	D₅₃
hypocenter	C₅₃	C₂×#1841
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×M₁₆ ◁ C₂×C₄×M₁₆

Complements

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

N	H ≅ G/N	type
1. #20274864	C₂	direct (N × H)
16. C₄×M₁₆×C₅₃	C₄	semidirect
18. C₂×#97×C₅₃	C₄	semidirect
19. C₂×C₄×C₈×C₅₃	C₄	semidirect
53. #97×C₅₃	C₂×C₄	semidirect
54. C₄×C₈×C₅₃	C₂×C₄	semidirect
166. C₂×C₅₃	#1841	semidirect
173. C₄×M₁₆	Dic₅₃	semidirect
175. C₂×#97	Dic₅₃	semidirect
176. C₂×C₄×C₈	Dic₅₃	semidirect
181. C₅₃	C₂×#1841	semidirect
215. #97	C₂×Dic₅₃	semidirect
216. C₄×C₈	C₂×Dic₅₃	semidirect
321. C₂	#20274864	direct (N × H)

Maximal subgroups

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

subgroup	index
#20274864	2
C₂×#16810394	2
C₂×#16810383	2
C₂×#16810381	2
C₂×#16810380	2
C₂×C₄×M₁₆×C₅₃	2
C₂×#1841	53

Subgroups — C2×#20274864 #27384357

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20274864 #27384357