Subgroups — #20275323

Order 6784.

292 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊#88322	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₂²⋊#88322	C₂	1	1	yes	maximal, characteristic
3	C₂×C₂²⋊#88322	C₂	1	1	yes	maximal, characteristic
4	C₂×C₄×#88322	C₂	1	1	yes	maximal, characteristic, complemented
5	#16810378	C₂	1	1	yes	maximal, characteristic, complemented
6	#16810378	C₂	1	1	yes	maximal, characteristic, complemented
7	C₄×C₂²⋊C₄×C₅₃	C₂	1	1	yes	maximal, characteristic
8	C₂²×#88322	C₂²	1	1	yes	characteristic
9	C₂²×#88322	C₂²	1	1	yes	characteristic
10	C₂²×#88322	C₂²	1	1	yes	characteristic
11	C₂²×#88322	C₂²	1	1	yes	characteristic
12	C₄⋊#88322	—	2	2	no	—
13	C₂²⋊#88322	—	2	2	no	—
14	C₂²⋊#88322	—	2	2	no	—
15	C₄×#88322	—	2	2	no	—
16	C₂³×C₄×C₅₃	C₂²	1	1	yes	characteristic
17	C₂×C₂²⋊C₄×C₅₃	C₄	1	1	yes	characteristic
18	C₂×C₂²⋊C₄×C₅₃	C₄	1	1	yes	characteristic
19	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
20	C₂×C₄²×C₅₃	C₂²	1	1	yes	characteristic
21	#95×C₅₃	C₄	1	1	yes	characteristic
22	#95×C₅₃	C₄	1	1	yes	characteristic
23	C₂⁴×C₅₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₄×C₅₃	—	2	1	no	—
25	C₂²×C₄×C₅₃	—	2	1	no	—
26	C₂²×C₄×C₅₃	—	2	1	no	—
27	C₂²×C₄×C₅₃	—	2	1	no	—
28	C₂²×C₄×C₅₃	—	2	1	no	—
29	C₂²×C₄×C₅₃	—	2	1	no	—
30	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
32	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
33	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
34	C₂²×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
35	C₂²×C₄×C₅₃	C₂³	1	1	yes	characteristic
36	C₂²⋊C₄×C₅₃	—	2	2	no	—
37	C₂²⋊C₄×C₅₃	C₈	1	4	yes	complemented
38	C₄²×C₅₃	—	2	1	no	—
39	C₄²×C₅₃	—	2	1	no	—
40	C₄²×C₅₃	—	2	1	no	—
41	C₄²×C₅₃	—	2	1	no	—
42	C₂×#88322	—	2	1	no	—
43	C₂×#88322	—	2	1	no	—
44	C₂×#88322	—	2	1	no	—
45	C₂×#88322	—	2	1	no	—
46	C₂×#88322	—	2	1	no	—
47	C₂×#88322	—	2	1	no	—
48	C₂×#88322	—	2	1	no	—
49	C₂×#88322	—	2	1	no	—
50	C₂×#88322	—	2	1	no	—
51	C₂×#88322	—	2	1	no	—
52	C₂×#88322	D₄	1	2	yes	—
53	C₂×#88322	D₄	1	2	yes	—
54	C₂³×C₅₃	—	2	1	no	—
55	C₂³×C₅₃	—	2	1	no	—
56	C₂³×C₅₃	—	2	1	no	—
57	C₂³×C₅₃	—	2	1	no	—
58	C₂³×C₅₃	—	2	1	no	—
59	C₂³×C₅₃	—	2	1	no	—
60	C₂³×C₅₃	C₂×C₈	1	2	yes	—
61	C₂³×C₅₃	C₂²×C₄	1	1	yes	characteristic
62	C₂×C₄×C₅₃	—	2	1	no	—
63	C₂×C₄×C₅₃	—	2	1	no	—
64	C₂×C₄×C₅₃	—	2	1	no	—
65	C₂×C₄×C₅₃	—	2	1	no	—
66	C₂×C₄×C₅₃	—	2	1	no	—
67	C₂×C₄×C₅₃	—	2	1	no	—
68	C₂×C₄×C₅₃	—	2	1	no	—
69	C₂×C₄×C₅₃	—	2	1	no	—
70	C₂×C₄×C₅₃	—	2	1	no	—
71	C₂×C₄×C₅₃	—	2	1	no	—
72	C₂×C₄×C₅₃	—	2	1	no	—
73	C₂×C₄×C₅₃	—	2	1	no	—
74	C₂×C₄×C₅₃	—	4	1	no	—
75	C₂×C₄×C₅₃	—	4	1	no	—
76	C₂×C₄×C₅₃	—	4	1	no	—
77	C₂×C₄×C₅₃	—	4	1	no	—
78	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
79	C₂×C₄×C₅₃	C₂×C₈	1	2	yes	—
80	C₂×C₄×C₅₃	M₁₆	1	2	yes	—
81	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#41	1	1	yes	characteristic
87	#88322	—	2	2	no	—
88	#88322	—	2	2	no	—
89	#88322	—	4	1	no	—
90	#88322	—	4	1	no	—
91	C₂²×C₅₃	—	2	2	no	—
92	C₂²×C₅₃	—	2	2	no	—
93	C₂²×C₅₃	—	2	2	no	—
94	C₂²×C₅₃	—	4	1	no	—
95	C₂²×C₅₃	—	4	1	no	—
96	C₂²×C₅₃	—	4	1	no	—
97	C₂²×C₅₃	—	4	1	no	—
98	C₂²×C₅₃	#117	1	1	yes	characteristic
99	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
100	C₂²×C₅₃	C₄×D₄	1	1	yes	characteristic
101	C₂²×C₅₃	C₂²×C₈	1	1	yes	characteristic
102	C₂²×C₅₃	C₂×M₁₆	1	1	yes	characteristic
103	C₂²×C₅₃	#131	1	1	yes	characteristic
104	C₂²×C₅₃	#131	1	1	yes	characteristic
105	C₄×C₅₃	—	2	2	no	—
106	C₄×C₅₃	—	2	2	no	—
107	C₄×C₅₃	—	2	2	no	—
108	C₄×C₅₃	—	4	1	no	—
109	C₄×C₅₃	—	4	1	no	—
110	C₄×C₅₃	#121	1	1	yes	characteristic
111	C₄×C₅₃	#123	1	1	yes	characteristic
112	C₄×C₅₃	#124	1	1	yes	characteristic
113	C₄×C₅₃	#126	1	1	yes	characteristic
114	#1923	—	53	1	no	maximal, Sylow
115	C₂×C₅₃	—	4	2	no	—
116	C₂×C₅₃	#388	1	1	yes	characteristic
117	C₂×C₅₃	#431	1	1	yes	characteristic
118	C₂×C₅₃	#433	1	1	yes	characteristic
119	C₂×C₅₃	C₈×D₄	1	1	yes	characteristic
120	C₂×C₅₃	C₈×D₄	1	1	yes	characteristic
121	C₂×C₅₃	#435	1	1	yes	characteristic
122	C₂×C₅₃	#436	1	1	yes	characteristic
123	C₂×C₄⋊C₈	—	53	1	no	—
124	C₂×C₂²⋊C₈	—	53	1	no	—
125	C₂×C₂²⋊C₈	—	53	1	no	—
126	C₂×C₄×C₈	—	53	1	no	—
127	C₄×C₂²⋊C₄	D₅₃	1	1	yes	characteristic
128	#336	—	53	1	no	—
129	#336	—	53	1	no	—
130	C₅₃	#1923	1	1	yes	characteristic, complemented, Sylow
131	C₂³×C₄	C₂×D₅₃	1	1	yes	characteristic
132	C₂²×C₈	—	53	1	no	—
133	C₂²×C₈	—	53	1	no	—
134	C₂²×C₈	—	53	1	no	—
135	C₂²×C₈	—	53	1	no	—
136	C₂×C₂²⋊C₄	Dic₅₃	1	1	yes	characteristic
137	C₂×C₂²⋊C₄	Dic₅₃	1	1	yes	characteristic
138	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
139	C₂×C₄²	C₂×D₅₃	1	1	yes	characteristic
140	C₄⋊C₈	—	106	2	no	—
141	C₂²⋊C₈	—	106	2	no	—
142	C₂²⋊C₈	—	106	2	no	—
143	C₄×C₈	—	106	2	no	—
144	#95	Dic₅₃	1	1	yes	characteristic
145	#95	Dic₅₃	1	1	yes	characteristic
146	C₂⁴	C₂×Dic₅₃	1	1	yes	characteristic
147	C₂²×C₄	—	2	1	no	—
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	—	2	1	no	—
150	C₂²×C₄	—	2	1	no	—
151	C₂²×C₄	—	2	1	no	—
152	C₂²×C₄	—	2	1	no	—
153	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
154	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
155	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
156	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
157	C₂²×C₄	C₂×Dic₅₃	1	1	yes	characteristic
158	C₂²×C₄	C₂²×D₅₃	1	1	yes	characteristic
159	C₂×C₈	—	53	2	no	—
160	C₂×C₈	—	53	2	no	—
161	C₂×C₈	—	106	1	no	—
162	C₂×C₈	—	106	1	no	—
163	C₂×C₈	—	106	1	no	—
164	C₂×C₈	—	106	1	no	—
165	C₂×C₈	—	106	1	no	—
166	C₂×C₈	—	106	1	no	—
167	C₂×C₈	—	106	1	no	—
168	C₂×C₈	—	106	1	no	—
169	C₂×C₈	—	106	1	no	—
170	C₂×C₈	—	106	1	no	—
171	C₂²⋊C₄	—	2	2	no	—
172	C₂²⋊C₄	#88322	1	4	yes	complemented
173	C₄²	—	2	1	no	—
174	C₄²	—	2	1	no	—
175	C₄²	—	2	1	no	—
176	C₄²	—	2	1	no	—
177	C₂³	—	2	1	no	—
178	C₂³	—	2	1	no	—
179	C₂³	—	2	1	no	—
180	C₂³	—	2	1	no	—
181	C₂³	—	2	1	no	—
182	C₂³	—	2	1	no	—
183	C₂³	C₂×#88322	1	2	yes	—
184	C₂³	C₂²×Dic₅₃	1	1	yes	characteristic
185	C₂×C₄	—	2	1	no	—
186	C₂×C₄	—	2	1	no	—
187	C₂×C₄	—	2	1	no	—
188	C₂×C₄	—	2	1	no	—
189	C₂×C₄	—	2	1	no	—
190	C₂×C₄	—	2	1	no	—
191	C₂×C₄	—	2	1	no	—
192	C₂×C₄	—	2	1	no	—
193	C₂×C₄	—	2	1	no	—
194	C₂×C₄	—	2	1	no	—
195	C₂×C₄	—	2	1	no	—
196	C₂×C₄	—	2	1	no	—
197	C₂×C₄	—	4	1	no	—
198	C₂×C₄	—	4	1	no	—
199	C₂×C₄	—	4	1	no	—
200	C₂×C₄	—	4	1	no	—
201	C₂×C₄	C₂×#88322	1	2	yes	—
202	C₂×C₄	C₂×#88322	1	2	yes	—
203	C₂×C₄	#11719917	1	2	yes	—
204	C₂×C₄	#11719945	1	1	yes	characteristic
205	C₂×C₄	#11719945	1	1	yes	characteristic
206	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
207	C₂×C₄	D₄×D₅₃	1	1	yes	characteristic
208	C₂×C₄	#11719947	1	1	yes	characteristic
209	C₂×C₄	#11719947	1	1	yes	characteristic
210	C₈	—	106	2	no	—
211	C₈	—	106	2	no	—
212	C₈	—	212	1	no	—
213	C₈	—	212	1	no	—
214	C₂²	—	2	2	no	—
215	C₂²	—	2	2	no	—
216	C₂²	—	2	2	no	—
217	C₂²	—	4	1	no	—
218	C₂²	—	4	1	no	—
219	C₂²	—	4	1	no	—
220	C₂²	—	4	1	no	—
221	C₂²	#13133176	1	1	yes	characteristic
222	C₂²	D₄×Dic₅₃	1	1	yes	characteristic
223	C₂²	D₄×Dic₅₃	1	1	yes	characteristic
224	C₂²	C₂²×#88322	1	1	yes	characteristic
225	C₂²	C₂×#11719917	1	1	yes	characteristic
226	C₂²	#13133224	1	1	yes	characteristic
227	C₂²	#13133224	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	2	2	no	—
230	C₄	—	2	2	no	—
231	C₄	—	4	1	no	—
232	C₄	—	4	1	no	—
233	C₄	#13133190	1	1	yes	characteristic
234	C₄	#13133199	1	1	yes	characteristic
235	C₄	#13133203	1	1	yes	characteristic
236	C₄	#13133210	1	1	yes	characteristic
237	C₂	—	4	2	no	—
238	C₂	#16810516	1	1	yes	characteristic
239	C₂	#16810676	1	1	yes	characteristic
240	C₂	#16810684	1	1	yes	characteristic
241	C₂	D₄×#88322	1	1	yes	characteristic
242	C₂	D₄×#88322	1	1	yes	characteristic
243	C₂	#16810695	1	1	yes	characteristic
244	C₂	#16810701	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂²×D₅₃
commutator	C₂²×C₅₃	C₂²×C₈
Fitting	C₄×C₂²⋊C₄×C₅₃	C₂
socle	C₂³×C₅₃	C₂²×C₄
hypercenter	C₄×C₂²⋊C₄	D₅₃
hypocenter	C₅₃	#1923
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₄×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×C₄⋊#88322	C₂	semidirect
4. C₂×C₄×#88322	C₂	semidirect
5. #16810378	C₂	semidirect
6. #16810378	C₂	semidirect
37. C₂²⋊C₄×C₅₃	C₈	semidirect
130. C₅₃	#1923	semidirect
172. C₂²⋊C₄	#88322	semidirect