Subgroups — #20283875

Order 6784.

364 conjugacy classes of subgroups.

The subgroup lattice has 244 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#16810754	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
3	#16810701	C₂	1	1	yes	maximal, characteristic, complemented
4	#16810695	C₂	1	2	yes	maximal, complemented
5	D₄×#88322	C₂	1	2	yes	maximal, complemented
6	D₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
7	#16810684	C₂	1	2	yes	maximal, complemented
8	#16810676	C₂	1	1	yes	maximal, characteristic, complemented
9	#16810650	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
11	C₅₃×#529	C₂	1	1	yes	maximal, characteristic
12	C₂×#11719917	C₂²	1	2	yes	—
13	C₂²×#88322	C₂²	1	2	yes	—
14	C₂²×#88322	C₂²	1	2	yes	—
15	C₄⋊#88322	C₂²	1	1	yes	characteristic
16	C₄⋊#88322	C₂²	1	2	yes	—
17	C₄⋊#88322	C₂²	1	2	yes	—
18	C₄⋊#88322	C₂²	1	1	yes	characteristic
19	C₄⋊#88322	C₂²	1	1	yes	characteristic
20	C₄⋊#88322	C₂²	1	2	yes	—
21	C₄⋊#88322	C₂²	1	1	yes	characteristic
22	C₂²⋊#88322	C₂²	1	1	yes	characteristic
23	C₂²⋊#88322	C₂²	1	1	yes	characteristic
24	C₂²⋊#88322	C₂²	1	2	yes	—
25	C₂²⋊#88322	C₂²	1	2	yes	—
26	#13133128	C₂²	1	2	yes	—
27	C₄×#88322	C₂²	1	1	yes	characteristic
28	C₄×#88322	C₂²	1	1	yes	characteristic
29	C₄×#88322	C₂²	1	2	yes	—
30	#126×C₅₃	C₄	1	2	yes	—
31	#125×C₅₃	C₄	1	1	yes	characteristic
32	#124×C₅₃	C₄	1	1	yes	characteristic
33	#123×C₅₃	C₄	1	2	yes	—
34	C₂²⋊Q₈×C₅₃	C₄	1	1	yes	characteristic
35	#121×C₅₃	C₄	1	1	yes	characteristic
36	C₄×Q₈×C₅₃	C₂²	1	1	yes	characteristic
37	C₄×D₄×C₅₃	C₂²	1	2	yes	—
38	C₄×D₄×C₅₃	C₂²	1	1	yes	characteristic
39	#117×C₅₃	C₂²	1	2	yes	—
40	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂×D₄×C₅₃	C₂×C₄	1	2	yes	—
43	C₂×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
47	C₂²×C₄×C₅₃	C₂³	1	2	yes	—
48	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
49	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
50	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
52	C₄⋊C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
54	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
55	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
56	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
57	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
58	C₂²⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
59	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
60	C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₄²×C₅₃	C₂³	1	1	yes	characteristic
62	C₄²×C₅₃	C₂³	1	1	yes	characteristic
63	C₄²×C₅₃	C₂³	1	2	yes	—
64	#11719917	—	4	2	no	—
65	C₂×#88322	—	4	2	no	—
66	C₂×#88322	—	4	2	no	—
67	C₂×#88322	C₂³	1	1	yes	characteristic
68	C₂×#88322	C₂³	1	1	yes	characteristic
69	C₂×#88322	C₂³	1	1	yes	characteristic
70	C₂×#88322	C₂³	1	2	yes	—
71	C₂×#88322	C₂³	1	1	yes	characteristic
72	C₂×#88322	C₂³	1	2	yes	—
73	C₂³×C₅₃	C₂²×C₄	1	2	yes	—
74	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
75	Q₈×C₅₃	—	2	2	no	—
76	D₄×C₅₃	—	2	4	no	—
77	D₄×C₅₃	—	2	2	no	—
78	C₂×C₄×C₅₃	—	2	2	no	—
79	C₂×C₄×C₅₃	—	2	2	no	—
80	C₂×C₄×C₅₃	—	2	1	no	—
81	C₂×C₄×C₅₃	—	2	2	no	—
82	C₂×C₄×C₅₃	—	2	1	no	—
83	C₂×C₄×C₅₃	—	2	2	no	—
84	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
89	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
90	C₂×C₄×C₅₃	C₂²×C₄	1	2	yes	—
91	C₂×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂⁴	1	1	yes	characteristic
93	#88322	—	2	1	no	—
94	#88322	—	2	2	no	—
95	#88322	—	2	2	no	—
96	#88322	—	2	1	no	—
97	#88322	—	2	1	no	—
98	#88322	—	2	1	no	—
99	C₂²×C₅₃	—	2	1	no	—
100	C₂²×C₅₃	—	2	1	no	—
101	C₂²×C₅₃	—	2	2	no	—
102	C₂²×C₅₃	—	2	2	no	—
103	C₂²×C₅₃	—	2	2	no	—
104	C₂²×C₅₃	#131	1	2	yes	—
105	C₂²×C₅₃	C₂³×C₄	1	1	yes	characteristic
106	C₄×C₅₃	—	2	1	no	—
107	C₄×C₅₃	—	2	2	no	—
108	C₄×C₅₃	—	2	1	no	—
109	C₄×C₅₃	—	2	1	no	—
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	—	2	2	no	—
112	C₄×C₅₃	—	2	2	no	—
113	C₄×C₅₃	#131	1	2	yes	—
114	C₄×C₅₃	H(5,𝔽₂)	1	1	yes	characteristic
115	C₄×C₅₃	2₋¹⁺⁴	1	1	yes	characteristic
116	#2994	—	53	1	no	maximal, Sylow
117	C₂×C₅₃	—	2	2	no	—
118	C₂×C₅₃	—	4	2	no	—
119	C₂×C₅₃	#520	1	1	yes	characteristic
120	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
121	C₂×C₅₃	C₂×#131	1	1	yes	characteristic
122	#529	D₅₃	1	1	yes	characteristic
123	#446	—	53	1	no	—
124	C₈×Q₈	—	53	1	no	—
125	#436	—	53	1	no	—
126	#435	—	53	2	no	—
127	C₈×D₄	—	53	1	no	—
128	C₈×D₄	—	53	2	no	—
129	#433	—	53	2	no	—
130	#431	—	53	1	no	—
131	#424	—	53	2	no	—
132	C₂×C₄⋊C₈	—	53	1	no	—
133	C₅₃	#2994	1	1	yes	characteristic, complemented, Sylow
134	C₂×M₁₆	—	53	2	no	—
135	C₂²×C₈	—	53	2	no	—
136	C₂²×C₈	—	53	2	no	—
137	#126	Dic₅₃	1	2	yes	—
138	#125	Dic₅₃	1	1	yes	characteristic
139	#124	Dic₅₃	1	1	yes	characteristic
140	#123	Dic₅₃	1	2	yes	—
141	C₂²⋊Q₈	Dic₅₃	1	1	yes	characteristic
142	#121	Dic₅₃	1	1	yes	characteristic
143	C₄×Q₈	C₂×D₅₃	1	1	yes	characteristic
144	C₄×D₄	C₂×D₅₃	1	2	yes	—
145	C₄×D₄	C₂×D₅₃	1	1	yes	characteristic
146	#117	C₂×D₅₃	1	2	yes	—
147	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
148	C₄⋊C₈	—	53	2	no	—
149	C₄⋊C₈	—	53	1	no	—
150	C₄⋊C₈	—	53	1	no	—
151	C₄⋊C₈	—	53	1	no	—
152	C₄⋊C₈	—	53	2	no	—
153	C₄⋊C₈	—	53	2	no	—
154	C₄⋊C₈	—	53	1	no	—
155	C₂²⋊C₈	—	53	1	no	—
156	C₂²⋊C₈	—	53	1	no	—
157	C₂²⋊C₈	—	53	2	no	—
158	C₂²⋊C₈	—	53	2	no	—
159	#97	—	53	2	no	—
160	C₄×C₈	—	53	1	no	—
161	C₄×C₈	—	53	1	no	—
162	C₄×C₈	—	53	2	no	—
163	C₂×Q₈	C₂×Dic₅₃	1	1	yes	characteristic
164	C₂×D₄	C₂×Dic₅₃	1	2	yes	—
165	C₂×D₄	C₂×Dic₅₃	1	1	yes	characteristic
166	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
167	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
168	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
169	C₂²×C₄	C₂²×D₅₃	1	2	yes	—
170	M₁₆	—	212	2	no	—
171	C₂×C₈	—	53	1	no	—
172	C₂×C₈	—	53	1	no	—
173	C₂×C₈	—	53	1	no	—
174	C₂×C₈	—	53	2	no	—
175	C₂×C₈	—	53	1	no	—
176	C₂×C₈	—	53	2	no	—
177	C₂×C₈	—	212	2	no	—
178	C₂×C₈	—	212	2	no	—
179	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
180	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
181	C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
182	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
183	C₄⋊C₄	C₂×Dic₅₃	1	1	yes	characteristic
184	C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
185	C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
186	C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
187	C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
188	C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
189	C₂²⋊C₄	C₂×Dic₅₃	1	2	yes	—
190	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
191	C₄²	C₂×Dic₅₃	1	1	yes	characteristic
192	C₄²	C₂²×D₅₃	1	1	yes	characteristic
193	C₄²	C₂²×D₅₃	1	2	yes	—
194	C₄²	C₂²×D₅₃	1	1	yes	characteristic
195	C₂³	C₂²×Dic₅₃	1	2	yes	—
196	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
197	Q₈	—	2	2	no	—
198	D₄	—	2	4	no	—
199	D₄	—	2	2	no	—
200	C₂×C₄	—	2	2	no	—
201	C₂×C₄	—	2	2	no	—
202	C₂×C₄	—	2	1	no	—
203	C₂×C₄	—	2	2	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	2	no	—
206	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
207	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
208	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
209	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
210	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
211	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
212	C₂×C₄	C₂²×Dic₅₃	1	2	yes	—
213	C₂×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
214	C₂×C₄	C₂³×D₅₃	1	1	yes	characteristic
215	C₈	—	106	1	no	—
216	C₈	—	106	2	no	—
217	C₈	—	106	2	no	—
218	C₈	—	106	1	no	—
219	C₈	—	106	1	no	—
220	C₈	—	106	1	no	—
221	C₂²	—	2	1	no	—
222	C₂²	—	2	1	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	#13133224	1	2	yes	—
227	C₂²	C₂³×Dic₅₃	1	1	yes	characteristic
228	C₄	—	2	1	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	1	no	—
231	C₄	—	2	1	no	—
232	C₄	—	2	1	no	—
233	C₄	—	2	2	no	—
234	C₄	—	2	2	no	—
235	C₄	#13133224	1	2	yes	—
236	C₄	#13133263	1	1	yes	characteristic
237	C₄	#13133264	1	1	yes	characteristic
238	C₂	—	2	2	no	—
239	C₂	—	4	2	no	—
240	C₂	#16811106	1	1	yes	characteristic
241	C₂	C₂×#13133224	1	1	yes	characteristic
242	C₂	C₂×#13133224	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₅₃
commutator, socle	C₂²×C₅₃	C₂³×C₄
Fitting	C₅₃×#529	C₂
hypercenter	#529	D₅₃
hypocenter	C₅₃	#2994
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #529

Complements

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

N	H ≅ G/N	type
1. #16810754	C₂	semidirect
2. Q₈×#88322	C₂	semidirect
3. #16810701	C₂	semidirect
4. #16810695	C₂	semidirect
5. D₄×#88322	C₂	semidirect
6. D₄×#88322	C₂	semidirect
7. #16810684	C₂	semidirect
8. #16810676	C₂	semidirect
9. #16810650	C₂	semidirect
10. C₂×C₄⋊#88322	C₂	semidirect
133. C₅₃	#2994	semidirect