Subgroups — #20291637

Order 6784.

388 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811675	C₂	1	2	yes	maximal, complemented
2	#16811656	C₂	1	2	yes	maximal, complemented
3	#16811655	C₂	1	2	yes	maximal, complemented
4	#16811653	C₂	1	2	yes	maximal, complemented
5	#16811645	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811644	C₂	1	1	yes	maximal, characteristic, complemented
7	#16811642	C₂	1	1	yes	maximal, characteristic, complemented
8	#16811627	C₂	1	1	yes	maximal, characteristic, complemented
9	#16811626	C₂	1	1	yes	maximal, characteristic, complemented
10	D₅₃⋊(C₄⋊C₈)	C₂	1	1	yes	maximal, characteristic, complemented
11	#16811091	C₂	1	1	yes	maximal, characteristic
12	C₂×#11719936	C₂²	1	2	yes	—
13	C₂×#11719940	C₂²	1	2	yes	—
14	C₂×D₅₃⋊C₈	C₂²	1	2	yes	—
15	#13133284	C₂²	1	2	yes	—
16	#13133284	C₂²	1	2	yes	—
17	C₄⋊#88324	C₂²	1	1	yes	characteristic
18	C₄⋊#88324	C₂²	1	1	yes	characteristic
19	C₄⋊#88324	C₂²	1	1	yes	characteristic
20	C₄⋊#88324	C₂²	1	2	yes	—
21	C₄⋊#88324	C₂²	1	1	yes	characteristic
22	#13133274	C₂²	1	2	yes	—
23	#13133274	C₂²	1	2	yes	—
24	C₂²⋊#88324	C₂²	1	2	yes	—
25	#13133272	C₂²	1	2	yes	—
26	#13133272	C₂²	1	2	yes	—
27	C₄×#88324	C₂²	1	2	yes	—
28	C₂×#11719945	C₄	1	1	yes	characteristic
29	#13133182	C₄	1	1	yes	characteristic
30	#13133182	C₂²	1	1	yes	characteristic
31	#13133179	C₄	1	1	yes	characteristic
32	#13133179	C₂²	1	1	yes	characteristic
33	#13133177	C₄	1	1	yes	characteristic
34	#13133177	C₄	1	1	yes	characteristic
35	#13133177	C₂²	1	1	yes	characteristic
36	#13133177	C₂²	1	1	yes	characteristic
37	#13133174	C₄	1	1	yes	characteristic
38	#13133174	C₄	1	1	yes	characteristic
39	C₂×C₄×Dic₅₃	C₂²	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	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₂²⋊D₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×Dic₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂²×Dic₅₃	C₂³	1	1	yes	characteristic
48	#11719945	—	2	4	no	—
49	C₂×D₂₁₂	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	C₂×C₄	1	1	yes	characteristic
54	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
55	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
56	C₂×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
57	#11719940	—	4	2	no	—
58	C₂×#88324	C₂³	1	2	yes	—
59	C₂×#88324	C₂³	1	2	yes	—
60	C₂×#88324	C₂³	1	2	yes	—
61	C₂×#88324	C₂³	1	2	yes	—
62	#11719936	—	4	2	no	—
63	D₅₃⋊C₈	—	4	2	no	—
64	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
66	C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
67	C₂²⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
68	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
69	C₄²×C₅₃	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	#11719921	C₂×C₄	1	1	yes	characteristic
74	#11719919	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	C₄×Dic₅₃	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	C₂×C₄	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	C₄×Dic₅₃	C₂³	1	1	yes	characteristic
86	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂²×D₅₃	C₂²×C₄	1	1	yes	characteristic
89	C₂×C₄×C₅₃	—	2	2	no	—
90	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
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	C₂²⋊D₅₃	—	2	2	no	—
97	C₂²⋊D₅₃	—	2	2	no	—
98	C₂×Dic₅₃	—	2	1	no	—
99	C₂×Dic₅₃	—	2	1	no	—
100	C₂×Dic₅₃	—	4	1	no	—
101	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
102	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
103	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
104	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
105	C₂×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
106	C₂×Dic₅₃	C₂⁴	1	1	yes	characteristic
107	D₂₁₂	—	2	2	no	—
108	C₄×D₅₃	—	2	2	no	—
109	C₄×D₅₃	—	2	1	no	—
110	C₄×D₅₃	—	2	1	no	—
111	C₄×D₅₃	—	2	1	no	—
112	C₄×D₅₃	—	2	1	no	—
113	C₄×D₅₃	—	4	1	no	—
114	C₄×D₅₃	—	4	1	no	—
115	C₄×D₅₃	M₁₆	1	4	yes	complemented
116	Dic₁₀₆	—	2	2	no	—
117	#88324	—	2	2	no	—
118	#88324	—	2	2	no	—
119	#88324	—	2	2	no	—
120	#88324	—	2	2	no	—
121	C₂²×C₅₃	—	2	1	no	—
122	C₂²×C₅₃	—	2	1	no	—
123	C₂²×C₅₃	—	2	1	no	—
124	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
125	C₂×D₅₃	—	2	1	no	—
126	C₂×D₅₃	—	2	1	no	—
127	C₂×D₅₃	—	2	1	no	—
128	C₂×D₅₃	—	2	1	no	—
129	C₂×D₅₃	—	2	1	no	—
130	C₂×D₅₃	C₂×M₁₆	1	2	yes	—
131	C₄×C₅₃	—	2	1	no	—
132	C₄×C₅₃	—	2	1	no	—
133	C₄×C₅₃	—	2	1	no	—
134	C₄×C₅₃	—	2	1	no	—
135	C₄×C₅₃	—	2	1	no	—
136	C₄×C₅₃	C₂×M₁₆	1	2	yes	—
137	Dic₅₃	—	2	1	no	—
138	Dic₅₃	—	2	1	no	—
139	Dic₅₃	—	2	1	no	—
140	Dic₅₃	—	2	1	no	—
141	Dic₅₃	C₂×M₁₆	1	2	yes	—
142	Dic₅₃	H(5,𝔽₂)	1	1	yes	characteristic
143	Dic₅₃	2₋¹⁺⁴	1	1	yes	characteristic
144	#2989	—	53	1	no	maximal, Sylow
145	C₂×C₅₃	—	4	1	no	—
146	C₂×C₅₃	#520	1	1	yes	characteristic
147	C₂×C₅₃	C₂²×M₁₆	1	1	yes	characteristic
148	C₂×C₅₃	#568	1	1	yes	characteristic
149	D₅₃	—	2	2	no	—
150	D₅₃	—	4	1	no	—
151	C₄×#41	—	53	1	no	—
152	#446	—	53	2	no	—
153	#436	—	53	2	no	—
154	#435	—	53	2	no	—
155	#435	—	53	2	no	—
156	#432	—	53	1	no	—
157	#432	—	53	1	no	—
158	#431	—	53	1	no	—
159	#423	—	53	1	no	—
160	#423	—	53	1	no	—
161	C₂×C₄⋊C₈	—	53	1	no	—
162	C₅₃	#2989	1	1	yes	characteristic, complemented, Sylow
163	C₂×#41	—	53	1	no	—
164	C₂×M₁₆	—	53	2	no	—
165	C₂×M₁₆	—	53	2	no	—
166	C₂²×C₈	—	53	2	no	—
167	C₄×Q₈	—	53	1	no	—
168	C₄×Q₈	—	53	1	no	—
169	C₄×D₄	—	53	1	no	—
170	C₄×D₄	—	53	1	no	—
171	C₄×D₄	—	53	1	no	—
172	C₄×D₄	—	53	1	no	—
173	C₄×D₄	—	53	1	no	—
174	C₄×D₄	—	53	1	no	—
175	#117	—	53	1	no	—
176	#117	—	53	1	no	—
177	#117	#6157	1	1	yes	characteristic
178	C₂×C₄²	—	53	1	no	—
179	C₂×C₄²	—	53	1	no	—
180	C₂×C₄²	—	53	1	no	—
181	C₄⋊C₈	—	53	1	no	—
182	C₄⋊C₈	—	53	1	no	—
183	C₄⋊C₈	—	53	2	no	—
184	C₄⋊C₈	—	53	1	no	—
185	C₄⋊C₈	—	53	2	no	—
186	C₄⋊C₈	—	53	2	no	—
187	C₄⋊C₈	—	53	1	no	—
188	C₂²⋊C₈	—	53	2	no	—
189	C₂²⋊C₈	—	53	2	no	—
190	C₂²⋊C₈	—	53	2	no	—
191	#97	—	53	2	no	—
192	#97	—	53	2	no	—
193	C₄×C₈	—	53	2	no	—
194	#41	—	106	4	no	—
195	C₂×Q₈	—	53	1	no	—
196	C₂×D₄	—	53	1	no	—
197	C₂×D₄	—	53	1	no	—
198	C₂×D₄	—	53	1	no	—
199	C₂²×C₄	—	53	1	no	—
200	C₂²×C₄	—	53	1	no	—
201	C₂²×C₄	—	53	1	no	—
202	C₂²×C₄	—	53	1	no	—
203	C₂²×C₄	—	53	1	no	—
204	C₂²×C₄	—	53	1	no	—
205	C₂²×C₄	—	53	1	no	—
206	C₂²×C₄	—	53	1	no	—
207	C₂²×C₄	C₂×#6157	1	1	yes	characteristic
208	M₁₆	—	212	2	no	—
209	M₁₆	—	212	2	no	—
210	C₂×C₈	—	53	2	no	—
211	C₂×C₈	—	53	2	no	—
212	C₂×C₈	—	53	2	no	—
213	C₂×C₈	—	53	2	no	—
214	C₂×C₈	—	212	2	no	—
215	C₄⋊C₄	—	53	1	no	—
216	C₄⋊C₄	—	53	1	no	—
217	C₄⋊C₄	—	53	1	no	—
218	C₄⋊C₄	—	53	1	no	—
219	C₄⋊C₄	C₂×#6157	1	1	yes	characteristic
220	C₄⋊C₄	C₂×#6157	1	1	yes	characteristic
221	C₂²⋊C₄	—	53	1	no	—
222	C₂²⋊C₄	—	53	1	no	—
223	C₂²⋊C₄	—	53	1	no	—
224	C₂²⋊C₄	—	53	1	no	—
225	C₂²⋊C₄	C₂×#6157	1	1	yes	characteristic
226	C₂²⋊C₄	C₂×#6157	1	1	yes	characteristic
227	C₄²	—	53	1	no	—
228	C₄²	—	53	1	no	—
229	C₄²	—	53	1	no	—
230	C₄²	—	53	1	no	—
231	C₄²	—	53	1	no	—
232	C₄²	—	53	1	no	—
233	C₄²	—	53	1	no	—
234	C₄²	—	53	1	no	—
235	C₄²	C₂×#6157	1	1	yes	characteristic
236	C₄²	C₂×#6157	1	1	yes	characteristic
237	C₂³	—	53	1	no	—
238	C₂³	—	53	1	no	—
239	C₂³	C₂²×#6157	1	1	yes	characteristic
240	Q₈	—	106	2	no	—
241	D₄	—	106	2	no	—
242	D₄	—	106	2	no	—
243	D₄	—	106	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	53	1	no	—
246	C₂×C₄	—	53	1	no	—
247	C₂×C₄	—	53	4	no	—
248	C₂×C₄	—	53	1	no	—
249	C₂×C₄	—	53	1	no	—
250	C₂×C₄	—	53	1	no	—
251	C₂×C₄	—	53	1	no	—
252	C₂×C₄	—	106	1	no	—
253	C₂×C₄	—	106	1	no	—
254	C₂×C₄	—	106	1	no	—
255	C₂×C₄	—	106	1	no	—
256	C₂×C₄	—	106	1	no	—
257	C₂×C₄	—	106	1	no	—
258	C₂×C₄	—	106	2	no	—
259	C₂×C₄	—	212	1	no	—
260	C₂×C₄	—	212	1	no	—
261	C₂×C₄	—	212	1	no	—
262	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
263	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
264	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
265	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
266	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
267	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
268	C₈	—	106	2	no	—
269	C₈	—	106	2	no	—
270	C₈	—	106	2	no	—
271	C₈	—	106	2	no	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	2	1	no	—
275	C₂²	—	53	2	no	—
276	C₂²	—	106	1	no	—
277	C₂²	—	106	1	no	—
278	C₂²	—	106	1	no	—
279	C₂²	—	106	1	no	—
280	C₂²	—	106	1	no	—
281	C₂²	C₂³×#6157	1	1	yes	characteristic
282	C₄	—	2	1	no	—
283	C₄	—	2	1	no	—
284	C₄	—	2	1	no	—
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	53	1	no	—
288	C₄	—	53	1	no	—
289	C₄	—	53	2	no	—
290	C₄	—	106	1	no	—
291	C₄	—	106	1	no	—
292	C₄	—	106	1	no	—
293	C₄	—	106	1	no	—
294	C₄	D₅₃⋊M₁₆	1	2	yes	—
295	C₂	—	4	1	no	—
296	C₂	—	106	2	no	—
297	C₂	—	212	1	no	—
298	C₂	#16811692	1	1	yes	characteristic
299	C₂	C₂×D₅₃⋊M₁₆	1	1	yes	characteristic
300	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	#2989
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. #16811675	C₂	semidirect
2. #16811656	C₂	semidirect
3. #16811655	C₂	semidirect
4. #16811653	C₂	semidirect
5. #16811645	C₂	semidirect
6. #16811644	C₂	semidirect
7. #16811642	C₂	semidirect
8. #16811627	C₂	semidirect
9. #16811626	C₂	semidirect
10. D₅₃⋊(C₄⋊C₈)	C₂	semidirect
115. C₄×D₅₃	M₁₆	semidirect
162. C₅₃	#2989	semidirect