Subgroups — #20291494

Order 6784.

452 conjugacy classes of subgroups.

The subgroup lattice has 250 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16811702	C₂	1	2	yes	maximal, complemented
2	#16811670	C₂	1	2	yes	maximal, complemented
3	#16811669	C₂	1	2	yes	maximal, complemented
4	#16811666	C₂	1	4	yes	maximal, complemented
5	C₂×#13133287	C₂	1	1	yes	maximal, characteristic, complemented
6	#16811620	C₂	1	2	yes	maximal, complemented
7	#16811586	C₂	1	1	yes	maximal, characteristic, complemented
8	#135×D₅₃	C₂	1	1	yes	maximal, characteristic
9	#13133305	C₂²	1	4	yes	complemented
10	#13133304	—	2	4	no	—
11	#13133304	C₂²	1	4	yes	complemented
12	C₂×#11719936	—	2	2	no	—
13	C₂×#11719936	C₂²	1	2	yes	—
14	D₅₃⋊M₁₆	—	2	2	no	—
15	D₅₃⋊M₁₆	C₂²	1	1	yes	characteristic
16	#13133296	C₂²	1	1	yes	characteristic
17	#13133287	C₂²	1	2	yes	complemented
18	#13133287	C₂²	1	2	yes	complemented
19	#13133282	C₂²	1	4	yes	complemented
20	#13133281	C₂²	1	4	yes	complemented
21	#13133271	C₂²	1	2	yes	complemented
22	C₈×#6157	C₂²	1	2	yes	complemented
23	#41×D₅₃	C₂²	1	2	yes	—
24	#13133240	C₂²	1	1	yes	characteristic
25	#13133239	C₄	1	1	yes	characteristic, complemented
26	#13133238	C₂²	1	2	yes	—
27	#13133237	C₄	1	2	yes	complemented
28	#13133236	C₂²	1	1	yes	characteristic
29	Q₁₆×D₅₃	C₄	1	1	yes	characteristic, complemented
30	SD₁₆×D₅₃	C₄	1	2	yes	complemented
31	D₈×D₅₃	C₄	1	1	yes	characteristic, complemented
32	C₂×C₈×D₅₃	C₂²	1	1	yes	characteristic
33	#135×C₅₃	C₄	1	1	yes	characteristic, complemented
34	#41×C₅₃	C₂×C₄	1	2	yes	—
35	#11719949	C₂³	1	2	yes	—
36	Q₈×D₅₃	C₂×C₄	1	2	yes	—
37	#11719947	—	2	2	no	—
38	#11719947	C₂³	1	2	yes	—
39	D₄×D₅₃	—	2	2	no	—
40	D₄×D₅₃	C₂×C₄	1	2	yes	—
41	#11719945	—	2	2	no	—
42	#11719945	C₂×C₄	1	2	yes	—
43	C₂×C₄×D₅₃	—	2	2	no	—
44	C₂×C₄×D₅₃	C₂³	1	1	yes	characteristic
45	C₂²⋊#6157	—	2	1	no	—
46	#11719940	—	2	4	no	—
47	#11719940	D₄	1	2	yes	—
48	C₂×#88324	—	2	2	no	—
49	C₂×#88324	—	2	4	no	—
50	C₄⋊#6157	—	2	1	no	—
51	C₄×#6157	C₂³	1	2	yes	—
52	#11719936	—	2	2	no	—
53	#11719936	—	2	4	no	—
54	#11719936	D₄	1	2	yes	—
55	#11719936	C₂³	1	4	yes	—
56	D₅₃⋊C₈	—	2	4	no	—
57	D₅₃⋊C₈	C₂³	1	2	yes	—
58	Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
59	SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
60	D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
62	#11719925	C₂×C₄	1	2	yes	—
63	#11719924	C₂×C₄	1	2	yes	—
64	#11719923	C₂×C₄	1	2	yes	—
65	#11719922	C₂×C₄	1	2	yes	—
66	C₂×#88322	C₂×C₄	1	1	yes	characteristic
67	Dic₂₁₂	C₂×C₄	1	1	yes	characteristic
68	D₄₂₄	C₂×C₄	1	1	yes	characteristic
69	#11719913	C₂×C₄	1	2	yes	—
70	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
71	C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
72	C₈×D₅₃	C₂³	1	1	yes	characteristic
73	C₈×D₅₃	C₂³	1	1	yes	characteristic
74	C₂²×D₅₃	—	2	2	no	—
75	C₂²×D₅₃	#41	1	1	yes	characteristic
76	C₂×#6157	—	2	2	no	—
77	Q₈×C₅₃	C₂²×C₄	1	2	yes	—
78	D₄×C₅₃	—	2	2	no	—
79	D₄×C₅₃	C₂²×C₄	1	2	yes	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
82	C₂²⋊D₅₃	—	2	2	no	—
83	C₂²⋊D₅₃	—	2	2	no	—
84	C₂²⋊D₅₃	—	2	2	no	—
85	C₂×Dic₅₃	—	2	2	no	—
86	C₂×Dic₅₃	C₂×D₄	1	1	yes	characteristic
87	D₂₁₂	—	2	2	no	—
88	D₂₁₂	C₂²×C₄	1	2	yes	—
89	C₄×D₅₃	—	2	2	no	—
90	C₄×D₅₃	—	2	2	no	—
91	C₄×D₅₃	—	2	2	no	—
92	C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
94	C₄×D₅₃	#41	1	1	yes	characteristic
95	C₄×D₅₃	C₂⁴	1	1	yes	characteristic
96	Dic₁₀₆	—	2	2	no	—
97	Dic₁₀₆	C₂²×C₄	1	2	yes	—
98	#88324	—	2	4	no	—
99	#88324	C₂×D₄	1	4	yes	—
100	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
102	#88322	C₂²×C₄	1	1	yes	characteristic
103	#88322	C₂²×C₄	1	1	yes	characteristic
104	C₂²×C₅₃	—	2	2	no	—
105	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
106	C₂×D₅₃	—	2	2	no	—
107	C₂×D₅₃	—	4	1	no	—
108	C₂×D₅₃	—	8	2	no	—
109	C₂×D₅₃	C₄×D₄	1	1	yes	characteristic
110	C₂×D₅₃	C₂×#41	1	1	yes	characteristic
111	#6157	—	4	2	no	—
112	C₄×C₅₃	—	2	2	no	—
113	C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
114	C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
115	Dic₅₃	—	2	2	no	—
116	Dic₅₃	C₄×D₄	1	1	yes	characteristic
117	Dic₅₃	C₂²×D₄	1	1	yes	characteristic
118	#2957	—	53	1	no	maximal, Sylow
119	C₂×C₅₃	—	2	1	no	—
120	C₂×C₅₃	—	4	2	no	—
121	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
122	D₅₃	—	2	1	no	—
123	D₅₃	—	2	1	no	—
124	D₅₃	—	4	2	no	—
125	C₂×#135	—	53	1	no	—
126	#568	—	53	2	no	—
127	#444	—	53	2	no	—
128	#444	—	53	2	no	—
129	#443	—	53	4	no	—
130	C₂×#108	—	53	1	no	—
131	#421	—	53	2	no	—
132	#405	—	53	1	no	—
133	C₅₃	#2957	1	1	yes	characteristic, complemented, Sylow
134	C₂×#41	—	53	2	no	—
135	#135	—	53	2	no	—
136	#135	—	53	1	no	—
137	#135	—	53	2	no	—
138	#135	—	53	1	no	—
139	#135	—	53	1	no	—
140	#135	#6157	1	1	yes	characteristic, complemented
141	C₂×Q₁₆	—	53	1	no	—
142	C₂×SD₁₆	—	53	2	no	—
143	C₂×D₈	—	53	1	no	—
144	#131	—	53	4	no	—
145	#131	—	53	4	no	—
146	#131	—	106	4	no	—
147	C₂×M₁₆	—	53	1	no	—
148	C₂×M₁₆	—	53	2	no	—
149	C₂×M₁₆	—	106	2	no	—
150	C₂×M₁₆	—	106	2	no	—
151	C₂²×C₈	—	53	1	no	—
152	#117	—	53	1	no	—
153	#108	—	53	2	no	—
154	#108	—	53	2	no	—
155	#104	—	53	4	no	—
156	#104	—	53	4	no	—
157	#97	—	53	2	no	—
158	C₄×C₈	—	53	2	no	—
159	#41	—	53	2	no	—
160	#41	—	53	2	no	—
161	#41	—	53	2	no	—
162	#41	—	106	2	no	—
163	#41	—	106	2	no	—
164	#41	C₂×#6157	1	2	yes	—
165	C₂×Q₈	—	53	2	no	—
166	C₂×D₄	—	53	2	no	—
167	C₂×D₄	—	106	2	no	—
168	C₂²×C₄	—	53	1	no	—
169	C₂²×C₄	—	106	2	no	—
170	Q₁₆	—	53	2	no	—
171	Q₁₆	—	53	1	no	—
172	Q₁₆	C₂×#6157	1	1	yes	characteristic
173	SD₁₆	—	53	2	no	—
174	SD₁₆	—	53	2	no	—
175	SD₁₆	—	53	2	no	—
176	SD₁₆	C₂×#6157	1	2	yes	—
177	D₈	—	53	2	no	—
178	D₈	—	53	1	no	—
179	D₈	C₂×#6157	1	1	yes	characteristic
180	M₁₆	—	53	2	no	—
181	M₁₆	—	53	2	no	—
182	M₁₆	—	53	4	no	—
183	M₁₆	—	106	4	no	—
184	M₁₆	—	106	4	no	—
185	M₁₆	—	106	2	no	—
186	C₂×C₈	—	53	2	no	—
187	C₂×C₈	—	53	1	no	—
188	C₂×C₈	—	53	1	no	—
189	C₂×C₈	—	53	1	no	—
190	C₂×C₈	—	53	1	no	—
191	C₂×C₈	—	53	1	no	—
192	C₂×C₈	—	106	4	no	—
193	C₂×C₈	—	106	4	no	—
194	C₂×C₈	—	106	2	no	—
195	C₂×C₈	C₂×#6157	1	1	yes	characteristic
196	C₄⋊C₄	—	106	1	no	—
197	C₂²⋊C₄	—	106	1	no	—
198	C₄²	—	53	2	no	—
199	C₂³	—	53	1	no	—
200	C₂³	—	106	2	no	—
201	Q₈	—	53	2	no	—
202	Q₈	—	106	2	no	—
203	Q₈	C₂²×#6157	1	2	yes	—
204	D₄	—	2	2	no	—
205	D₄	—	53	2	no	—
206	D₄	—	106	2	no	—
207	D₄	—	106	2	no	—
208	D₄	—	106	2	no	—
209	D₄	—	106	2	no	—
210	D₄	C₂²×#6157	1	2	yes	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	53	1	no	—
213	C₂×C₄	—	53	1	no	—
214	C₂×C₄	—	53	1	no	—
215	C₂×C₄	—	53	1	no	—
216	C₂×C₄	—	53	1	no	—
217	C₂×C₄	—	106	2	no	—
218	C₂×C₄	—	106	2	no	—
219	C₂×C₄	—	106	2	no	—
220	C₂×C₄	—	106	2	no	—
221	C₂×C₄	—	106	2	no	—
222	C₂×C₄	C₂²×#6157	1	1	yes	characteristic
223	C₈	—	53	4	no	—
224	C₈	—	53	1	no	—
225	C₈	—	53	1	no	—
226	C₈	—	106	4	no	—
227	C₈	C₂²×#6157	1	1	yes	characteristic
228	C₈	C₂²×#6157	1	1	yes	characteristic
229	C₂²	—	2	2	no	—
230	C₂²	—	53	1	no	—
231	C₂²	—	53	1	no	—
232	C₂²	—	106	2	no	—
233	C₂²	—	424	2	no	—
234	C₂²	—	212	1	no	—
235	C₂²	D₄×#6157	1	1	yes	characteristic
236	C₄	—	2	2	no	—
237	C₄	—	53	1	no	—
238	C₄	—	53	1	no	—
239	C₄	—	106	2	no	—
240	C₄	—	212	2	no	—
241	C₄	D₄×#6157	1	1	yes	characteristic
242	C₄	C₂³×#6157	1	1	yes	characteristic
243	C₂	—	2	1	no	—
244	C₂	—	4	2	no	—
245	C₂	—	106	1	no	—
246	C₂	—	106	1	no	—
247	C₂	—	212	2	no	—
248	C₂	C₂×D₄×#6157	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂×D₄×#6157
commutator	C₄×C₅₃	C₂³×C₄
Frattini	C₄	C₂³×#6157
Fitting	#135×C₅₃	C₄
socle	C₂×C₅₃	C₂×C₄×D₄
hypercenter	#135	#6157
hypocenter	C₅₃	#2957
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂ ◁ C₂×C₄ ◁ #135

Complements

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

N	H ≅ G/N	type
1. #16811702	C₂	semidirect
2. #16811670	C₂	semidirect
3. #16811669	C₂	semidirect
4. #16811666	C₂	semidirect
5. C₂×#13133287	C₂	semidirect
6. #16811620	C₂	semidirect
7. #16811586	C₂	semidirect
9. #13133305	C₂²	semidirect
11. #13133304	C₂²	semidirect
17. #13133287	C₂²	semidirect
18. #13133287	C₂²	semidirect
19. #13133282	C₂²	semidirect
20. #13133281	C₂²	semidirect
21. #13133271	C₂²	semidirect
22. C₈×#6157	C₂²	semidirect
25. #13133239	C₄	semidirect
27. #13133237	C₄	semidirect
29. Q₁₆×D₅₃	C₄	semidirect
30. SD₁₆×D₅₃	C₄	semidirect
31. D₈×D₅₃	C₄	semidirect
33. #135×C₅₃	C₄	semidirect
133. C₅₃	#2957	semidirect
140. #135	#6157	semidirect