Subgroups — #20283410

Order 6784.

520 conjugacy classes of subgroups.

The subgroup lattice has 228 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#13133224	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂³×#88322	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810695	C₂	1	4	yes	maximal, complemented
4	D₄×#88322	C₂	1	4	yes	maximal, complemented
5	#16810650	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810590	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810587	C₂	1	1	yes	maximal, characteristic, complemented
8	#16810579	C₂	1	1	yes	maximal, characteristic, complemented
9	C₅₃×#525	C₂	1	1	yes	maximal, characteristic
10	#13133224	—	2	4	no	—
11	C₂×#11719917	C₂²	1	1	yes	characteristic
12	C₂×#11719917	C₂²	1	1	yes	characteristic
13	C₂×#11719917	C₂²	1	2	yes	complemented
14	C₂²×#88322	—	2	4	no	—
15	C₂²×#88322	C₂²	1	1	yes	characteristic
16	C₂²×#88322	C₂²	1	2	yes	—
17	C₂²×#88322	C₂²	1	2	yes	—
18	C₂²×#88322	C₂²	1	1	yes	characteristic
19	C₂²×#88322	C₂²	1	2	yes	complemented
20	C₄⋊#88322	C₂²	1	4	yes	complemented
21	C₂²⋊#88322	C₂²	1	4	yes	—
22	C₂²⋊#88322	C₂²	1	4	yes	complemented
23	#13133128	C₂²	1	2	yes	complemented
24	C₄×#88322	C₂²	1	2	yes	complemented
25	C₂×#41×C₅₃	C₂²	1	1	yes	characteristic
26	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
27	#123×C₅₃	C₄	1	2	yes	—
28	C₂²⋊Q₈×C₅₃	C₄	1	2	yes	—
29	#121×C₅₃	C₄	1	2	yes	—
30	#120×C₅₃	C₄	1	2	yes	—
31	C₄×D₄×C₅₃	C₂²	1	4	yes	—
32	#117×C₅₃	C₂²	1	1	yes	characteristic
33	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
34	#41×C₅₃	—	2	2	no	—
35	#41×C₅₃	—	2	2	no	—
36	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
37	C₂×D₄×C₅₃	C₂×C₄	1	4	yes	—
38	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
39	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
40	C₂²×C₄×C₅₃	—	2	2	no	—
41	C₂²×C₄×C₅₃	—	2	2	no	—
42	C₂²×C₄×C₅₃	C₂×C₄	1	2	yes	—
43	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
45	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
46	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
48	C₄⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
49	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
50	C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
51	C₂²⋊C₄×C₅₃	C₂×C₄	1	4	yes	—
52	C₄²×C₅₃	C₂³	1	2	yes	—
53	#11719917	—	2	2	no	—
54	#11719917	—	2	2	no	—
55	#11719917	—	2	4	no	—
56	C₂×#88322	—	2	2	no	—
57	C₂×#88322	—	2	4	no	—
58	C₂×#88322	—	2	4	no	—
59	C₂×#88322	—	2	4	no	—
60	C₂×#88322	D₄	1	4	yes	complemented
61	C₂×#88322	C₂³	1	2	yes	—
62	C₂×#88322	C₂³	1	2	yes	—
63	C₂×#88322	C₂³	1	2	yes	—
64	C₂×#88322	C₂³	1	1	yes	characteristic
65	C₂×#88322	C₂³	1	1	yes	characteristic
66	C₂³×C₅₃	—	2	2	no	—
67	C₂³×C₅₃	—	2	4	no	—
68	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
69	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
70	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
71	Q₈×C₅₃	—	2	2	no	—
72	D₄×C₅₃	—	2	4	no	—
73	D₄×C₅₃	—	2	8	no	—
74	D₄×C₅₃	—	2	2	no	—
75	C₂×C₄×C₅₃	—	2	2	no	—
76	C₂×C₄×C₅₃	—	2	4	no	—
77	C₂×C₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	2	no	—
81	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
82	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
83	C₂×C₄×C₅₃	C₂²×C₄	1	4	yes	—
84	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂×D₄	1	2	yes	—
86	C₂×C₄×C₅₃	#41	1	2	yes	—
87	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
88	#88322	—	2	2	no	—
89	#88322	—	2	2	no	—
90	#88322	—	2	2	no	—
91	#88322	C₂×D₄	1	2	yes	—
92	#88322	C₂×D₄	1	2	yes	—
93	C₂²×C₅₃	—	2	8	no	—
94	C₂²×C₅₃	—	2	2	no	—
95	C₂²×C₅₃	—	2	2	no	—
96	C₂²×C₅₃	—	2	2	no	—
97	C₂²×C₅₃	—	2	2	no	—
98	C₂²×C₅₃	—	2	4	no	—
99	C₂²×C₅₃	C₄×D₄	1	2	yes	—
100	C₂²×C₅₃	#131	1	4	yes	complemented
101	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
102	C₄×C₅₃	—	2	2	no	—
103	C₄×C₅₃	—	2	2	no	—
104	C₄×C₅₃	—	2	4	no	—
105	C₄×C₅₃	C₄×D₄	1	2	yes	—
106	C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
107	C₄×C₅₃	C₂×#41	1	1	yes	characteristic
108	#2929	—	53	1	no	maximal, Sylow
109	C₂×C₅₃	—	2	2	no	—
110	C₂×C₅₃	—	2	4	no	—
111	C₂×C₅₃	—	4	2	no	—
112	C₂×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
113	C₂×C₅₃	C₂×#131	1	2	yes	—
114	C₂×#131	—	53	1	no	—
115	C₂³×C₈	—	53	1	no	—
116	#525	D₅₃	1	1	yes	characteristic
117	#435	—	53	4	no	—
118	C₈×D₄	—	53	4	no	—
119	#424	—	53	1	no	—
120	#408	—	53	1	no	—
121	#407	—	53	1	no	—
122	#405	—	53	1	no	—
123	C₅₃	#2929	1	1	yes	characteristic, complemented, Sylow
124	C₂×#41	C₂×D₅₃	1	1	yes	characteristic
125	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
126	#131	—	106	4	no	—
127	C₂×M₁₆	—	53	1	no	—
128	C₂×M₁₆	—	53	2	no	—
129	C₂×M₁₆	—	53	1	no	—
130	C₂²×C₈	—	53	1	no	—
131	C₂²×C₈	—	53	1	no	—
132	C₂²×C₈	—	53	2	no	—
133	C₂²×C₈	—	53	2	no	—
134	C₂²×C₈	—	53	2	no	—
135	C₂²×C₈	—	106	4	no	—
136	#123	Dic₅₃	1	2	yes	—
137	C₂²⋊Q₈	Dic₅₃	1	2	yes	—
138	#121	Dic₅₃	1	2	yes	—
139	#120	Dic₅₃	1	2	yes	—
140	C₄×D₄	C₂×D₅₃	1	4	yes	—
141	#117	C₂×D₅₃	1	1	yes	characteristic
142	C₄⋊C₈	—	53	4	no	—
143	C₂²⋊C₈	—	53	4	no	—
144	C₂²⋊C₈	—	53	4	no	—
145	#97	—	53	2	no	—
146	C₄×C₈	—	53	2	no	—
147	C₂⁴	C₂×Dic₅₃	1	1	yes	characteristic
148	#41	—	2	2	no	—
149	#41	—	2	2	no	—
150	C₂×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
151	C₂×D₄	C₂×Dic₅₃	1	4	yes	—
152	C₂×D₄	C₂×Dic₅₃	1	2	yes	—
153	C₂×D₄	C₂×Dic₅₃	1	1	yes	characteristic
154	C₂²×C₄	—	2	2	no	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	C₂×Dic₅₃	1	2	yes	—
157	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
159	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
160	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
161	M₁₆	—	106	4	no	—
162	M₁₆	—	106	2	no	—
163	M₁₆	—	106	2	no	—
164	C₂×C₈	—	53	4	no	—
165	C₂×C₈	—	53	1	no	—
166	C₂×C₈	—	53	1	no	—
167	C₂×C₈	—	53	2	no	—
168	C₂×C₈	—	53	2	no	—
169	C₂×C₈	—	53	2	no	—
170	C₂×C₈	—	106	4	no	—
171	C₂×C₈	—	106	4	no	—
172	C₂×C₈	—	106	4	no	—
173	C₂×C₈	—	106	2	no	—
174	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
175	C₄⋊C₄	C₂×Dic₅₃	1	4	yes	—
176	C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
177	C₂²⋊C₄	C₂×Dic₅₃	1	4	yes	—
178	C₂²⋊C₄	C₂×Dic₅₃	1	4	yes	—
179	C₄²	C₂²×D₅₃	1	2	yes	—
180	C₂³	—	2	2	no	—
181	C₂³	—	2	4	no	—
182	C₂³	C₂²×Dic₅₃	1	2	yes	—
183	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
184	C₂³	C₂²×Dic₅₃	1	2	yes	—
185	Q₈	—	2	2	no	—
186	D₄	—	2	4	no	—
187	D₄	—	2	8	no	—
188	D₄	—	2	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	4	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	D₄×D₅₃	1	2	yes	—
196	C₂×C₄	#11719947	1	2	yes	—
197	C₂×C₄	C₂²×Dic₅₃	1	4	yes	—
198	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
199	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
200	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
201	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
202	C₈	—	53	2	no	—
203	C₈	—	53	2	no	—
204	C₈	—	106	2	no	—
205	C₈	—	106	2	no	—
206	C₈	—	106	2	no	—
207	C₂²	—	2	8	no	—
208	C₂²	—	2	2	no	—
209	C₂²	—	2	2	no	—
210	C₂²	—	2	2	no	—
211	C₂²	—	2	2	no	—
212	C₂²	—	2	4	no	—
213	C₂²	D₄×Dic₅₃	1	2	yes	—
214	C₂²	#13133224	1	4	yes	complemented
215	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
216	C₄	—	2	2	no	—
217	C₄	—	2	2	no	—
218	C₄	—	2	4	no	—
219	C₄	D₄×Dic₅₃	1	2	yes	—
220	C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
221	C₄	C₂×#11719947	1	1	yes	characteristic
222	C₂	—	2	2	no	—
223	C₂	—	2	4	no	—
224	C₂	—	4	2	no	—
225	C₂	C₂×D₄×Dic₅₃	1	1	yes	characteristic
226	C₂	C₂×#13133224	1	2	yes	—

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#525	C₂
hypercenter	#525	D₅₃
hypocenter	C₅₃	#2929
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
1. C₂×#13133224	C₂	semidirect
2. C₂³×#88322	C₂	semidirect
3. #16810695	C₂	semidirect
4. D₄×#88322	C₂	semidirect
5. #16810650	C₂	semidirect
6. #16810590	C₂	semidirect
7. #16810587	C₂	semidirect
8. #16810579	C₂	semidirect
13. C₂×#11719917	C₂²	semidirect
19. C₂²×#88322	C₂²	semidirect
20. C₄⋊#88322	C₂²	semidirect
22. C₂²⋊#88322	C₂²	semidirect
23. #13133128	C₂²	semidirect
24. C₄×#88322	C₂²	semidirect
60. C₂×#88322	D₄	semidirect
100. C₂²×C₅₃	#131	semidirect
123. C₅₃	#2929	semidirect
214. C₂²	#13133224	semidirect