Subgroups — #20291407

Order 6784.

402 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811648	C₂	1	2	yes	maximal, complemented
2	#16811648	C₂	1	2	yes	maximal, complemented
3	#16811646	C₂	1	1	yes	maximal, characteristic, complemented
4	#16811645	C₂	1	1	yes	maximal, characteristic, complemented
5	#16811642	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811641	C₂	1	1	yes	maximal, characteristic, complemented
7	#16811629	C₂	1	2	yes	maximal, complemented
8	#16811595	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133274	C₂	1	1	yes	maximal, characteristic
10	#16811587	C₂	1	2	yes	maximal, complemented
11	#117×D₅₃	C₂	1	1	yes	maximal, characteristic
12	C₂×#11719940	C₂²	1	2	yes	—
13	C₂²×#88324	C₂²	1	2	yes	—
14	#13133284	C₂²	1	2	yes	complemented
15	#13133284	C₂²	1	2	yes	complemented
16	C₄⋊#88324	C₂²	1	2	yes	complemented
17	C₄⋊#88324	C₂²	1	2	yes	complemented
18	#13133274	C₂²	1	1	yes	characteristic
19	#13133274	C₂²	1	1	yes	characteristic
20	#13133274	C₂²	1	2	yes	—
21	#13133274	C₂²	1	1	yes	characteristic
22	#13133274	C₂²	1	1	yes	characteristic
23	#13133274	C₂²	1	2	yes	—
24	#13133272	C₂²	1	2	yes	complemented
25	#13133272	C₂²	1	2	yes	complemented
26	C₄×#88324	C₂²	1	2	yes	complemented
27	C₄×#88324	C₂²	1	2	yes	complemented
28	C₂²×C₄×D₅₃	C₂²	1	1	yes	characteristic
29	#13133176	C₄	1	1	yes	characteristic
30	#13133175	C₂²	1	1	yes	characteristic
31	#13133175	C₂²	1	1	yes	characteristic
32	#13133174	C₄	1	1	yes	characteristic
33	#13133174	C₄	1	1	yes	characteristic
34	#13133173	C₂²	1	1	yes	characteristic
35	#13133173	C₂²	1	1	yes	characteristic
36	C₄⋊C₄×D₅₃	C₄	1	1	yes	characteristic
37	C₄⋊C₄×D₅₃	C₄	1	1	yes	characteristic
38	C₂²⋊C₄×D₅₃	C₄	1	1	yes	characteristic
39	C₂²⋊C₄×D₅₃	C₄	1	1	yes	characteristic
40	C₄²×D₅₃	C₂²	1	1	yes	characteristic
41	C₄²×D₅₃	C₂²	1	1	yes	characteristic
42	#117×C₅₃	C₄	1	1	yes	characteristic
43	C₂³×D₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
46	C₂×C₄×D₅₃	—	2	4	no	—
47	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
52	C₂×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
54	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
55	#11719940	—	2	4	no	—
56	C₂×#88324	—	2	4	no	—
57	C₂×#88324	C₂³	1	2	yes	—
58	C₂×#88324	C₂³	1	2	yes	—
59	C₂×#88324	C₂³	1	2	yes	—
60	C₂×#88324	C₂³	1	2	yes	—
61	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
68	C₂²⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
69	#11719921	C₂×C₄	1	1	yes	characteristic
70	#11719921	C₂×C₄	1	1	yes	characteristic
71	#11719921	C₂×C₄	1	1	yes	characteristic
72	#11719921	C₂×C₄	1	1	yes	characteristic
73	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
74	C₄⋊Dic₅₃	C₂×C₄	1	1	yes	characteristic
75	#11719919	C₂×C₄	1	1	yes	characteristic
76	#11719919	C₂×C₄	1	1	yes	characteristic
77	#11719919	C₂×C₄	1	1	yes	characteristic
78	#11719919	C₂×C₄	1	1	yes	characteristic
79	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
80	C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
81	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
82	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
83	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
84	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
85	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂²×D₅₃	—	2	2	no	—
87	C₂²×D₅₃	—	2	2	no	—
88	C₂²×D₅₃	—	4	1	no	—
89	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
90	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	—	2	2	no	—
92	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
97	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
98	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
99	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₅₃	#41	1	4	yes	—
104	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
105	C₄×D₅₃	—	2	2	no	—
106	C₄×D₅₃	—	2	2	no	—
107	C₄×D₅₃	—	2	2	no	—
108	C₄×D₅₃	—	2	2	no	—
109	C₄×D₅₃	—	2	2	no	—
110	C₄×D₅₃	—	2	2	no	—
111	C₄×D₅₃	—	4	1	no	—
112	C₄×D₅₃	—	4	1	no	—
113	#88324	—	2	2	no	—
114	#88324	—	2	2	no	—
115	#88324	—	2	2	no	—
116	#88324	—	2	2	no	—
117	C₂²×C₅₃	—	2	1	no	—
118	C₂²×C₅₃	—	2	1	no	—
119	C₂²×C₅₃	#117	1	2	yes	—
120	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
121	C₂×D₅₃	—	2	1	no	—
122	C₂×D₅₃	—	2	1	no	—
123	C₂×D₅₃	—	2	1	no	—
124	C₂×D₅₃	—	2	1	no	—
125	C₂×D₅₃	—	2	1	no	—
126	C₂×D₅₃	—	4	4	no	—
127	C₂×D₅₃	#131	1	2	yes	—
128	C₄×C₅₃	—	2	1	no	—
129	C₄×C₅₃	—	2	1	no	—
130	C₄×C₅₃	—	2	1	no	—
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	—	2	1	no	—
134	Dic₅₃	—	2	1	no	—
135	Dic₅₃	—	2	1	no	—
136	Dic₅₃	—	2	1	no	—
137	Dic₅₃	—	2	1	no	—
138	Dic₅₃	#117	1	2	yes	—
139	Dic₅₃	C₂×#41	1	2	yes	—
140	#2924	—	53	1	no	maximal, Sylow
141	C₂×C₅₃	—	2	2	no	—
142	C₂×C₅₃	C₂×#117	1	1	yes	characteristic
143	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
144	C₂×C₅₃	#568	1	1	yes	characteristic
145	D₅₃	—	2	2	no	—
146	D₅₃	—	4	1	no	—
147	C₂×#117	—	53	1	no	—
148	#433	—	53	2	no	—
149	#433	—	53	2	no	—
150	#432	—	53	1	no	—
151	#432	—	53	1	no	—
152	#431	—	53	1	no	—
153	#431	—	53	1	no	—
154	#424	—	53	2	no	—
155	#407	—	53	1	no	—
156	C₂×C₂²⋊C₈	—	53	1	no	—
157	#405	—	53	2	no	—
158	C₅₃	#2924	1	1	yes	characteristic, complemented, Sylow
159	C₂³×C₄	—	53	1	no	—
160	C₂×M₁₆	—	53	2	no	—
161	C₂²×C₈	—	53	2	no	—
162	#117	—	53	1	no	—
163	#117	—	53	1	no	—
164	#117	—	53	1	no	—
165	#117	—	53	1	no	—
166	#117	—	53	1	no	—
167	#117	—	53	1	no	—
168	#117	—	53	1	no	—
169	#117	#6157	1	1	yes	characteristic
170	C₂×C₄⋊C₄	—	53	1	no	—
171	C₂×C₄⋊C₄	—	53	1	no	—
172	C₂×C₂²⋊C₄	—	53	1	no	—
173	C₂×C₂²⋊C₄	—	53	1	no	—
174	C₂×C₄²	—	53	1	no	—
175	C₂×C₄²	—	53	1	no	—
176	C₄⋊C₈	—	53	2	no	—
177	C₄⋊C₈	—	53	2	no	—
178	C₄⋊C₈	—	53	2	no	—
179	C₄⋊C₈	—	53	2	no	—
180	C₂²⋊C₈	—	53	1	no	—
181	C₂²⋊C₈	—	53	1	no	—
182	C₂²⋊C₈	—	53	2	no	—
183	C₂²⋊C₈	—	53	1	no	—
184	C₂²⋊C₈	—	53	1	no	—
185	C₂²⋊C₈	—	53	2	no	—
186	#97	—	53	2	no	—
187	#97	—	53	2	no	—
188	C₄×C₈	—	53	2	no	—
189	C₄×C₈	—	53	2	no	—
190	C₂⁴	—	53	1	no	—
191	C₂²×C₄	—	53	1	no	—
192	C₂²×C₄	—	53	1	no	—
193	C₂²×C₄	—	53	1	no	—
194	C₂²×C₄	—	53	1	no	—
195	C₂²×C₄	—	53	1	no	—
196	C₂²×C₄	—	53	1	no	—
197	C₂²×C₄	—	53	1	no	—
198	C₂²×C₄	—	53	1	no	—
199	C₂²×C₄	—	53	1	no	—
200	C₂²×C₄	—	106	4	no	—
201	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
202	M₁₆	—	106	4	no	—
203	C₂×C₈	—	53	2	no	—
204	C₂×C₈	—	53	2	no	—
205	C₂×C₈	—	53	2	no	—
206	C₂×C₈	—	53	2	no	—
207	C₂×C₈	—	106	4	no	—
208	C₄⋊C₄	—	53	1	no	—
209	C₄⋊C₄	—	53	1	no	—
210	C₄⋊C₄	—	53	1	no	—
211	C₄⋊C₄	—	53	1	no	—
212	C₄⋊C₄	—	53	1	no	—
213	C₄⋊C₄	—	53	1	no	—
214	C₄⋊C₄	C₂×#6157	1	1	yes	characteristic
215	C₄⋊C₄	C₂×#6157	1	1	yes	characteristic
216	C₂²⋊C₄	—	53	1	no	—
217	C₂²⋊C₄	—	53	1	no	—
218	C₂²⋊C₄	—	53	1	no	—
219	C₂²⋊C₄	—	53	1	no	—
220	C₂²⋊C₄	—	53	1	no	—
221	C₂²⋊C₄	—	53	1	no	—
222	C₂²⋊C₄	C₂×#6157	1	1	yes	characteristic
223	C₂²⋊C₄	C₂×#6157	1	1	yes	characteristic
224	C₄²	—	53	1	no	—
225	C₄²	—	53	1	no	—
226	C₄²	—	53	1	no	—
227	C₄²	—	53	1	no	—
228	C₄²	—	53	1	no	—
229	C₄²	—	53	1	no	—
230	C₄²	C₂×#6157	1	1	yes	characteristic
231	C₄²	C₂×#6157	1	1	yes	characteristic
232	C₂³	—	53	1	no	—
233	C₂³	—	53	1	no	—
234	C₂³	—	106	2	no	—
235	C₂³	—	106	2	no	—
236	C₂³	—	212	1	no	—
237	C₂³	C₂²×#6157	1	1	yes	characteristic
238	C₂×C₄	—	2	2	no	—
239	C₂×C₄	—	53	4	no	—
240	C₂×C₄	—	53	1	no	—
241	C₂×C₄	—	53	1	no	—
242	C₂×C₄	—	53	1	no	—
243	C₂×C₄	—	53	1	no	—
244	C₂×C₄	—	53	1	no	—
245	C₂×C₄	—	53	1	no	—
246	C₂×C₄	—	106	2	no	—
247	C₂×C₄	—	106	2	no	—
248	C₂×C₄	—	106	2	no	—
249	C₂×C₄	—	106	2	no	—
250	C₂×C₄	—	106	2	no	—
251	C₂×C₄	—	106	2	no	—
252	C₂×C₄	—	212	1	no	—
253	C₂×C₄	—	212	1	no	—
254	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
255	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
256	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
257	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
258	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
259	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
260	C₈	—	106	2	no	—
261	C₈	—	106	2	no	—
262	C₈	—	106	2	no	—
263	C₈	—	106	2	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	2	1	no	—
266	C₂²	—	53	2	no	—
267	C₂²	—	106	1	no	—
268	C₂²	—	106	1	no	—
269	C₂²	—	106	1	no	—
270	C₂²	—	106	1	no	—
271	C₂²	—	106	1	no	—
272	C₂²	—	212	4	no	—
273	C₂²	#13133296	1	2	yes	—
274	C₂²	C₂³×#6157	1	1	yes	characteristic
275	C₄	—	2	1	no	—
276	C₄	—	2	1	no	—
277	C₄	—	2	1	no	—
278	C₄	—	2	1	no	—
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	53	2	no	—
282	C₄	—	53	2	no	—
283	C₄	—	106	1	no	—
284	C₄	—	106	1	no	—
285	C₄	—	106	1	no	—
286	C₄	—	106	1	no	—
287	C₂	—	2	2	no	—
288	C₂	—	106	2	no	—
289	C₂	—	212	1	no	—
290	C₂	C₂×#13133296	1	1	yes	characteristic
291	C₂	D₅₃⋊#131	1	1	yes	characteristic
292	C₂	#16811702	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	#117×C₅₃	C₄
hypercenter	#117	#6157
hypocenter	C₅₃	#2924
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂² ◁ #117

Complements

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

N	H ≅ G/N	type
1. #16811648	C₂	semidirect
2. #16811648	C₂	semidirect
3. #16811646	C₂	semidirect
4. #16811645	C₂	semidirect
5. #16811642	C₂	semidirect
6. #16811641	C₂	semidirect
7. #16811629	C₂	semidirect
10. #16811587	C₂	semidirect
14. #13133284	C₂²	semidirect
15. #13133284	C₂²	semidirect
16. C₄⋊#88324	C₂²	semidirect
17. C₄⋊#88324	C₂²	semidirect
24. #13133272	C₂²	semidirect
25. #13133272	C₂²	semidirect
26. C₄×#88324	C₂²	semidirect
27. C₄×#88324	C₂²	semidirect
158. C₅₃	#2924	semidirect