Subgroups — C₂×#20288011 #28290804

Order 13568.

8518 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20288011	C₂	1	32	yes	maximal, direct product factor
2	C₂×#16811180	C₂	1	3	yes	maximal, complemented
3	C₂×#16811173	C₂	1	3	yes	maximal, complemented
4	C₂×#16811157	C₂	1	12	yes	maximal, complemented
5	C₂×#16811134	C₂	1	6	yes	maximal, complemented
6	C₂×C₄×#11719945	C₂	1	6	yes	maximal, complemented
7	C₂³×C₄²×C₅₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#11719945	C₂²	1	6	yes	—
9	#16811180	C₂²	1	48	yes	complemented
10	#16811173	C₂²	1	48	yes	complemented
11	C₂×#13133217	C₂²	1	3	yes	complemented
12	C₂×#13133214	C₂²	1	1	yes	characteristic, complemented
13	#16811157	C₂²	1	192	yes	complemented
14	C₂×#13133211	C₂²	1	8	yes	complemented
15	C₂×#13133209	C₂²	1	6	yes	complemented
16	C₂×#13133205	C₂²	1	6	yes	complemented
17	#16811134	C₂²	1	96	yes	complemented
18	C₂×#13133200	C₂²	1	12	yes	—
19	C₂×#13133196	C₂²	1	12	yes	—
20	C₂×#13133191	C₂²	1	12	yes	—
21	C₂×#13133186	C₂²	1	4	yes	—
22	C₄×#11719945	C₂²	1	96	yes	complemented
23	C₂×C₄×Dic₁₀₆	C₂²	1	12	yes	complemented
24	C₂×C₄×D₂₁₂	C₂²	1	12	yes	complemented
25	C₂×C₄×C₂²⋊D₅₃	C₂²	1	24	yes	—
26	C₂×#13133176	C₂²	1	6	yes	—
27	C₂×#13133174	C₂²	1	12	yes	complemented
28	C₂×C₄²×D₅₃	C₂²	1	4	yes	complemented
29	C₂⁴×C₄×C₅₃	C₂²	1	3	yes	—
30	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
31	C₂²×C₄²×C₅₃	C₂²	1	3	yes	—
32	C₂²×C₄²×C₅₃	C₂²	1	6	yes	—
33	C₂²×C₄²×C₅₃	C₂²	1	16	yes	complemented
34	C₂×#11719945	—	2	48	no	—
35	C₂×#11719945	C₂³	1	48	yes	—
36	C₂²×Dic₁₀₆	C₂³	1	6	yes	—
37	C₂²×D₂₁₂	C₂³	1	6	yes	—
38	C₂²×C₂²⋊D₅₃	C₂³	1	12	yes	—
39	C₂²×C₄×D₅₃	C₂³	1	12	yes	—
40	#13133217	C₂³	1	24	yes	complemented
41	#13133214	C₂³	1	8	yes	complemented
42	#13133211	C₂³	1	64	yes	complemented
43	#13133209	C₂³	1	48	yes	complemented
44	#13133205	C₂³	1	48	yes	complemented
45	#13133200	C₂³	1	96	yes	—
46	#13133196	C₂³	1	96	yes	—
47	#13133191	C₂³	1	96	yes	—
48	#13133186	C₂³	1	32	yes	—
49	C₄×Dic₁₀₆	C₂³	1	96	yes	complemented
50	C₄×D₂₁₂	C₂³	1	96	yes	complemented
51	C₄×C₂²⋊D₅₃	C₂³	1	192	yes	—
52	#13133176	C₂³	1	48	yes	—
53	#13133174	C₂³	1	96	yes	complemented
54	C₂×C₄⋊Dic₅₃	C₂³	1	12	yes	—
55	C₂×#11719919	C₂³	1	24	yes	—
56	C₂×C₂²⋊Dic₅₃	C₂³	1	12	yes	—
57	C₂×#11719921	C₂³	1	24	yes	—
58	C₂×C₄×Dic₅₃	C₂³	1	12	yes	—
59	C₄²×D₅₃	C₂³	1	32	yes	complemented
60	C₂⁵×C₅₃	C₂³	1	1	yes	characteristic
61	C₂³×C₄×C₅₃	—	2	24	no	—
62	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
63	C₂³×C₄×C₅₃	C₂³	1	6	yes	—
64	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
65	C₂³×C₄×C₅₃	C₂³	1	24	yes	—
66	C₂³×C₄×C₅₃	C₂³	1	3	yes	—
67	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
68	C₂×C₄²×C₅₃	C₂³	1	6	yes	—
69	C₂×C₄²×C₅₃	C₂³	1	3	yes	—
70	C₂×C₄²×C₅₃	C₂³	1	2	yes	—
71	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
72	C₂×C₄²×C₅₃	C₂³	1	24	yes	—
73	C₂×C₄²×C₅₃	C₂³	1	48	yes	—
74	C₂×C₄²×C₅₃	C₂³	1	24	yes	—
75	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
76	C₂⁴×C₅₃	—	2	12	no	—
77	C₂⁴×C₅₃	C₂⁴	1	4	yes	—
78	C₂⁴×C₅₃	C₂⁴	1	3	yes	—
79	C₂³×D₅₃	C₂⁴	1	4	yes	—
80	C₂²×C₄×C₅₃	—	2	12	no	—
81	C₂²×C₄×C₅₃	—	2	24	no	—
82	C₂²×C₄×C₅₃	—	2	24	no	—
83	C₂²×C₄×C₅₃	—	2	96	no	—
84	C₂²×C₄×C₅₃	#41	1	24	yes	complemented
85	C₂²×C₄×C₅₃	C₂⁴	1	6	yes	—
86	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
87	C₂²×C₄×C₅₃	C₂⁴	1	24	yes	—
88	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
89	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
90	C₂²×C₄×C₅₃	C₂⁴	1	3	yes	—
91	C₂²×C₄×C₅₃	C₂⁴	1	12	yes	—
92	C₂×C₂²⋊D₅₃	—	2	48	no	—
93	C₂×C₂²⋊D₅₃	C₂⁴	1	48	yes	—
94	C₂²×Dic₅₃	C₂⁴	1	12	yes	—
95	#11719945	—	2	192	no	—
96	C₂×D₂₁₂	—	2	24	no	—
97	C₂×D₂₁₂	C₂⁴	1	24	yes	—
98	C₂×C₄×D₅₃	—	2	48	no	—
99	C₂×C₄×D₅₃	C₂⁴	1	48	yes	—
100	C₂×Dic₁₀₆	—	2	24	no	—
101	C₂×Dic₁₀₆	C₂⁴	1	24	yes	—
102	C₄²×C₅₃	C₂⁴	1	4	yes	—
103	C₄²×C₅₃	C₂⁴	1	12	yes	—
104	C₄²×C₅₃	C₂⁴	1	24	yes	—
105	C₄²×C₅₃	C₂⁴	1	12	yes	—
106	C₄²×C₅₃	C₂⁴	1	8	yes	—
107	C₄²×C₅₃	C₂⁴	1	4	yes	—
108	C₂²⋊Dic₅₃	C₂⁴	1	48	yes	—
109	#11719921	C₂⁴	1	96	yes	—
110	C₄⋊Dic₅₃	C₂⁴	1	48	yes	—
111	#11719919	C₂⁴	1	96	yes	—
112	C₄×Dic₅₃	C₂⁴	1	48	yes	—
113	C₂³×C₅₃	—	2	32	no	—
114	C₂³×C₅₃	—	2	24	no	—
115	C₂³×C₅₃	—	2	12	no	—
116	C₂³×C₅₃	C₂×#41	1	12	yes	—
117	C₂³×C₅₃	C₂⁵	1	6	yes	—
118	C₂³×C₅₃	C₂⁵	1	1	yes	characteristic
119	C₂²×D₅₃	—	2	24	no	—
120	C₂²×D₅₃	C₂⁵	1	8	yes	—
121	C₂×C₄×C₅₃	—	2	24	no	—
122	C₂×C₄×C₅₃	—	2	12	no	—
123	C₂×C₄×C₅₃	—	2	48	no	—
124	C₂×C₄×C₅₃	—	2	48	no	—
125	C₂×C₄×C₅₃	C₂×#41	1	12	yes	—
126	C₂×C₄×C₅₃	C₂×#41	1	12	yes	—
127	C₂×C₄×C₅₃	C₂×#41	1	48	yes	complemented
128	C₂×C₄×C₅₃	C₂⁵	1	12	yes	—
129	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
130	C₂×C₄×C₅₃	C₂⁵	1	6	yes	—
131	C₂²⋊D₅₃	—	2	96	no	—
132	C₂×Dic₅₃	—	2	24	no	—
133	C₂×Dic₅₃	C₂⁵	1	24	yes	—
134	D₂₁₂	—	2	48	no	—
135	C₄×D₅₃	—	2	96	no	—
136	Dic₁₀₆	—	2	48	no	—
137	C₂×#3444	—	53	1	no	maximal, Sylow
138	C₂²×C₅₃	—	2	32	no	—
139	C₂²×C₅₃	—	2	12	no	—
140	C₂²×C₅₃	—	2	24	no	—
141	C₂²×C₅₃	C₂²×#41	1	12	yes	—
142	C₂²×C₅₃	C₂²×#41	1	6	yes	—
143	C₂²×C₅₃	C₂⁶	1	1	yes	characteristic
144	C₂×D₅₃	—	2	24	no	—
145	C₂×D₅₃	—	4	16	no	—
146	C₄×C₅₃	—	2	12	no	—
147	C₄×C₅₃	C₂²×#41	1	12	yes	—
148	C₄×C₅₃	C₂²×#41	1	12	yes	—
149	Dic₅₃	—	2	24	no	—
150	#3444	—	53	32	no	—
151	C₂×#533	—	53	3	no	—
152	C₂×#532	—	53	3	no	—
153	C₂×#529	—	53	12	no	—
154	C₂×#525	—	53	6	no	—
155	C₂×C₄×#41	—	53	6	no	—
156	C₂³×C₄²	D₅₃	1	1	yes	characteristic, complemented
157	C₂×C₅₃	—	2	12	no	—
158	C₂×C₅₃	#3444	1	4	yes	complemented
159	C₂×C₅₃	C₂³×#41	1	3	yes	—
160	D₅₃	—	4	8	no	—
161	C₂²×#41	—	53	6	no	—
162	C₂⁴×C₄	C₂×D₅₃	1	3	yes	—
163	#533	—	53	48	no	—
164	#532	—	53	48	no	—
165	C₂×C₄⋊Q₈	—	53	3	no	—
166	C₂×#127	—	53	1	no	—
167	#529	—	53	192	no	—
168	C₂×#126	—	53	8	no	—
169	C₂×#125	—	53	6	no	—
170	C₂×#124	—	53	6	no	—
171	#525	—	53	96	no	—
172	C₂×#123	—	53	12	no	—
173	C₂×C₂²⋊Q₈	—	53	12	no	—
174	C₂×#121	—	53	12	no	—
175	C₂×#120	—	53	4	no	—
176	C₄×#41	—	53	96	no	—
177	C₂×C₄×Q₈	—	53	12	no	—
178	C₂×C₄×D₄	—	53	24	no	—
179	C₂×C₄×D₄	—	53	12	no	—
180	C₂×#117	—	53	6	no	—
181	C₂×#117	—	53	12	no	—
182	C₂²×C₄²	—	53	4	no	—
183	C₂²×C₄²	C₂×D₅₃	1	6	yes	—
184	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
185	C₂²×C₄²	C₂×D₅₃	1	3	yes	—
186	C₂²×C₄²	C₂×D₅₃	1	16	yes	complemented
187	C₅₃	C₂×#3444	1	1	yes	characteristic, complemented, Sylow
188	C₂⁵	C₂²×D₅₃	1	1	yes	characteristic
189	C₂×#41	—	53	48	no	—
190	C₂×#41	—	106	48	no	—
191	C₂²×Q₈	—	53	6	no	—
192	C₂²×D₄	—	53	12	no	—
193	C₂²×D₄	—	53	6	no	—
194	C₂³×C₄	—	2	24	no	—
195	C₂³×C₄	—	53	12	no	—
196	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
197	C₂³×C₄	C₂²×D₅₃	1	6	yes	—
198	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
199	C₂³×C₄	C₂²×D₅₃	1	24	yes	—
200	C₂³×C₄	C₂²×D₅₃	1	3	yes	—
201	C₄⋊Q₈	—	53	24	no	—
202	#127	—	53	8	no	—
203	#126	—	53	64	no	—
204	#125	—	53	48	no	—
205	#124	—	53	48	no	—
206	#123	—	53	96	no	—
207	C₂²⋊Q₈	—	53	96	no	—
208	#121	—	53	96	no	—
209	#120	—	53	32	no	—
210	C₄×Q₈	—	53	96	no	—
211	C₄×D₄	—	53	192	no	—
212	C₄×D₄	—	53	96	no	—
213	#117	—	53	48	no	—
214	#117	—	53	96	no	—
215	C₂×C₄⋊C₄	—	53	12	no	—
216	C₂×C₄⋊C₄	—	53	24	no	—
217	C₂×C₂²⋊C₄	—	53	12	no	—
218	C₂×C₂²⋊C₄	—	53	24	no	—
219	C₂×C₄²	—	53	12	no	—
220	C₂×C₄²	—	53	32	no	—
221	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
222	C₂×C₄²	C₂²×D₅₃	1	6	yes	—
223	C₂×C₄²	C₂²×D₅₃	1	24	yes	—
224	C₂×C₄²	C₂²×D₅₃	1	3	yes	—
225	C₂×C₄²	C₂²×D₅₃	1	2	yes	—
226	C₂×C₄²	C₂²×D₅₃	1	48	yes	—
227	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
228	C₂×C₄²	C₂²×D₅₃	1	24	yes	—
229	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
230	C₂⁴	—	2	12	no	—
231	C₂⁴	—	53	4	no	—
232	C₂⁴	C₂³×D₅₃	1	4	yes	—
233	C₂⁴	C₂³×D₅₃	1	3	yes	—
234	#41	—	106	192	no	—
235	C₂×Q₈	—	53	24	no	—
236	C₂×Q₈	—	106	24	no	—
237	C₂×D₄	—	53	48	no	—
238	C₂×D₄	—	53	24	no	—
239	C₂×D₄	—	106	48	no	—
240	C₂×D₄	—	106	24	no	—
241	C₂²×C₄	—	2	12	no	—
242	C₂²×C₄	—	2	24	no	—
243	C₂²×C₄	—	2	24	no	—
244	C₂²×C₄	—	2	96	no	—
245	C₂²×C₄	—	53	12	no	—
246	C₂²×C₄	—	53	48	no	—
247	C₂²×C₄	—	106	48	no	—
248	C₂²×C₄	#11719945	1	24	yes	complemented
249	C₂²×C₄	C₂³×D₅₃	1	6	yes	—
250	C₂²×C₄	C₂³×D₅₃	1	24	yes	—
251	C₂²×C₄	C₂³×D₅₃	1	24	yes	—
252	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
253	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
254	C₂²×C₄	C₂³×D₅₃	1	3	yes	—
255	C₂²×C₄	C₂³×D₅₃	1	12	yes	—
256	C₄⋊C₄	—	53	48	no	—
257	C₄⋊C₄	—	53	96	no	—
258	C₂²⋊C₄	—	53	48	no	—
259	C₂²⋊C₄	—	53	96	no	—
260	C₄²	—	53	48	no	—
261	C₄²	C₂³×D₅₃	1	4	yes	—
262	C₄²	C₂³×D₅₃	1	12	yes	—
263	C₄²	C₂³×D₅₃	1	24	yes	—
264	C₄²	C₂³×D₅₃	1	12	yes	—
265	C₄²	C₂³×D₅₃	1	8	yes	—
266	C₄²	C₂³×D₅₃	1	4	yes	—
267	C₂³	—	2	32	no	—
268	C₂³	—	2	24	no	—
269	C₂³	—	2	12	no	—
270	C₂³	—	53	8	no	—
271	C₂³	—	106	24	no	—
272	C₂³	C₂×#11719945	1	12	yes	—
273	C₂³	C₂⁴×D₅₃	1	6	yes	—
274	C₂³	C₂⁴×D₅₃	1	1	yes	characteristic
275	Q₈	—	106	48	no	—
276	D₄	—	106	96	no	—
277	D₄	—	106	48	no	—
278	C₂×C₄	—	2	24	no	—
279	C₂×C₄	—	2	12	no	—
280	C₂×C₄	—	2	48	no	—
281	C₂×C₄	—	2	48	no	—
282	C₂×C₄	—	53	24	no	—
283	C₂×C₄	—	106	24	no	—
284	C₂×C₄	—	106	96	no	—
285	C₂×C₄	C₂×#11719945	1	12	yes	—
286	C₂×C₄	C₂×#11719945	1	12	yes	—
287	C₂×C₄	C₂×#11719945	1	48	yes	complemented
288	C₂×C₄	C₂⁴×D₅₃	1	12	yes	—
289	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
290	C₂×C₄	C₂⁴×D₅₃	1	6	yes	—
291	C₂²	—	2	32	no	—
292	C₂²	—	2	24	no	—
293	C₂²	—	2	12	no	—
294	C₂²	—	106	24	no	—
295	C₂²	—	212	16	no	—
296	C₂²	C₂²×#11719945	1	12	yes	—
297	C₂²	C₂²×#11719945	1	6	yes	—
298	C₂²	C₂⁵×D₅₃	1	1	yes	characteristic
299	C₄	—	2	12	no	—
300	C₄	—	106	24	no	—
301	C₄	C₂²×#11719945	1	12	yes	—
302	C₄	C₂²×#11719945	1	12	yes	—
303	C₂	—	2	12	no	—
304	C₂	—	212	8	no	—
305	C₂	#20288011	1	4	yes	direct product factor
306	C₂	C₂³×#11719945	1	3	yes	—

	subgroup	G/H
center	C₂×C₄²	C₂²×D₅₃
commutator, cosocle	C₂²×C₅₃	C₂⁶
Frattini	C₂²	C₂⁵×D₅₃
Fitting	C₂³×C₄²×C₅₃	C₂
socle	C₂³×C₅₃	C₂⁵
hypercenter	C₂³×C₄²	D₅₃
hypocenter	C₅₃	C₂×#3444
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄² ◁ C₂³×C₄²

Complements

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

N	H ≅ G/N	type
1. #20288011	C₂	direct (N × H)
2. C₂×#16811180	C₂	semidirect
3. C₂×#16811173	C₂	semidirect
4. C₂×#16811157	C₂	semidirect
5. C₂×#16811134	C₂	semidirect
6. C₂×C₄×#11719945	C₂	semidirect
7. C₂³×C₄²×C₅₃	C₂	semidirect
9. #16811180	C₂²	semidirect
10. #16811173	C₂²	semidirect
11. C₂×#13133217	C₂²	semidirect
12. C₂×#13133214	C₂²	semidirect
13. #16811157	C₂²	semidirect
14. C₂×#13133211	C₂²	semidirect
15. C₂×#13133209	C₂²	semidirect
16. C₂×#13133205	C₂²	semidirect
17. #16811134	C₂²	semidirect
22. C₄×#11719945	C₂²	semidirect
23. C₂×C₄×Dic₁₀₆	C₂²	semidirect
24. C₂×C₄×D₂₁₂	C₂²	semidirect
27. C₂×#13133174	C₂²	semidirect
28. C₂×C₄²×D₅₃	C₂²	semidirect
33. C₂²×C₄²×C₅₃	C₂²	semidirect
40. #13133217	C₂³	semidirect
41. #13133214	C₂³	semidirect
42. #13133211	C₂³	semidirect
43. #13133209	C₂³	semidirect
44. #13133205	C₂³	semidirect
49. C₄×Dic₁₀₆	C₂³	semidirect
50. C₄×D₂₁₂	C₂³	semidirect
53. #13133174	C₂³	semidirect
59. C₄²×D₅₃	C₂³	semidirect
84. C₂²×C₄×C₅₃	#41	semidirect
127. C₂×C₄×C₅₃	C₂×#41	semidirect
156. C₂³×C₄²	D₅₃	semidirect
158. C₂×C₅₃	#3444	semidirect
186. C₂²×C₄²	C₂×D₅₃	semidirect
187. C₅₃	C₂×#3444	semidirect
248. C₂²×C₄	#11719945	semidirect
287. C₂×C₄	C₂×#11719945	semidirect
305. C₂	#20288011	direct (N × H)

Maximal subgroups

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

subgroup	index
#20288011	2
C₂×#16811180	2
C₂×#16811173	2
C₂×#16811157	2
C₂×#16811134	2
C₂×C₄×#11719945	2
C₂³×C₄²×C₅₃	2
C₂×#3444	53

Subgroups — C2×#20288011 #28290804

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#20288011 #28290804