Subgroups — #27282634

Order 13568.

398 conjugacy classes of subgroups.

The subgroup lattice has 258 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20275433	C₂	1	2	yes	maximal
2	C₄×#13133151	C₂	1	1	yes	maximal, characteristic
3	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
4	#20272555	C₂	1	2	yes	maximal
5	C₂×C₈²×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#13133151	C₄	1	1	yes	characteristic
7	C₂×#13133151	C₄	1	1	yes	characteristic
8	C₂×C₄⋊#88322	C₂²	1	2	yes	—
9	C₂×#13133146	—	2	2	no	—
10	C₄×C₄⋊Dic₅₃	C₂²	1	2	yes	—
11	#16810386	C₄	1	2	yes	—
12	#16810377	—	2	2	no	—
13	#16810376	—	2	2	no	—
14	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
17	C₈²×C₅₃	—	2	2	no	—
18	C₂²×#88322	—	2	2	no	—
19	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
20	C₂×C₄⋊Dic₅₃	C₂×C₄	1	2	yes	—
21	C₂×C₄×Dic₅₃	—	2	2	no	—
22	#13133151	—	2	2	no	—
23	#13133151	C₈	1	4	yes	complemented
24	C₄⋊#88322	—	2	2	no	—
25	C₄⋊#88322	—	2	2	no	—
26	#13133146	—	4	4	no	—
27	#13133124	—	2	2	no	—
28	C₂²×C₈×C₅₃	—	2	1	no	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
32	C₂²×C₈×C₅₃	D₄	1	1	yes	characteristic
33	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
34	C₄×C₈×C₅₃	—	2	1	no	—
35	C₄×C₈×C₅₃	—	2	1	no	—
36	C₄×C₈×C₅₃	—	2	1	no	—
37	C₄×C₈×C₅₃	D₄	1	2	yes	—
38	C₄×C₈×C₅₃	Q₈	1	2	yes	—
39	C₄×C₈×C₅₃	Q₈	1	2	yes	—
40	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
41	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
42	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
43	C₂²×Dic₅₃	—	2	2	no	—
44	C₂²×Dic₅₃	—	2	2	no	—
45	C₂×C₈×C₅₃	—	2	2	no	—
46	C₂×C₈×C₅₃	—	2	2	no	—
47	C₂×C₈×C₅₃	—	2	1	no	—
48	C₂×C₈×C₅₃	—	2	1	no	—
49	C₂×C₈×C₅₃	—	2	2	no	—
50	C₂×C₈×C₅₃	—	2	1	no	—
51	C₂×C₈×C₅₃	—	2	1	no	—
52	C₂×C₈×C₅₃	—	2	1	no	—
53	C₂×C₈×C₅₃	—	2	1	no	—
54	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
55	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
56	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
57	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
58	C₂×C₈×C₅₃	D₈	1	2	yes	—
59	C₂×C₈×C₅₃	Q₁₆	1	2	yes	—
60	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
61	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
62	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
63	C₄²×C₅₃	#41	1	1	yes	characteristic
64	C₄⋊Dic₅₃	—	2	2	no	—
65	C₄⋊Dic₅₃	—	2	2	no	—
66	C₄⋊Dic₅₃	—	2	2	no	—
67	C₄⋊Dic₅₃	C₂×C₈	1	4	yes	—
68	C₄×Dic₅₃	—	4	2	no	—
69	C₄×Dic₅₃	—	4	2	no	—
70	C₂×#88322	—	4	2	no	—
71	C₂×#88322	—	4	2	no	—
72	C₂×#88322	—	4	2	no	—
73	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
74	C₂×C₄×C₅₃	—	2	1	no	—
75	C₂×C₄×C₅₃	—	2	1	no	—
76	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
77	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
78	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
79	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
81	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
82	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
85	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
86	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
88	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
89	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
90	C₂×Dic₅₃	—	2	4	no	—
91	C₂×Dic₅₃	—	4	2	no	—
92	C₂×Dic₅₃	—	4	2	no	—
93	C₂×Dic₅₃	—	4	2	no	—
94	C₂×Dic₅₃	—	4	2	no	—
95	C₂×Dic₅₃	—	4	2	no	—
96	C₈×C₅₃	—	2	4	no	—
97	C₈×C₅₃	—	2	2	no	—
98	C₈×C₅₃	—	2	2	no	—
99	C₈×C₅₃	—	2	2	no	—
100	C₈×C₅₃	C₄⋊C₈	1	4	yes	complemented
101	#88322	—	8	2	no	—
102	#9594	—	53	1	no	maximal, Sylow
103	C₂²×C₅₃	#389	1	1	yes	characteristic
104	C₂²×C₅₃	C₈×D₄	1	1	yes	characteristic
105	C₂²×C₅₃	#436	1	1	yes	characteristic
106	C₂²×C₅₃	C₄×D₈	1	1	yes	characteristic
107	C₂²×C₅₃	C₄×Q₁₆	1	1	yes	characteristic
108	C₂²×C₅₃	#443	1	1	yes	characteristic
109	C₂²×C₅₃	#443	1	1	yes	characteristic
110	C₄×C₅₃	—	2	1	no	—
111	C₄×C₅₃	—	2	1	no	—
112	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
113	C₄×C₅₃	#424	1	1	yes	characteristic
114	C₄×C₅₃	C₈×Q₈	1	1	yes	characteristic
115	C₄×C₅₃	#446	1	1	yes	characteristic
116	C₄×C₅₃	#466	1	1	yes	characteristic
117	C₄×C₅₃	#467	1	1	yes	characteristic
118	C₄×C₅₃	#499	1	1	yes	characteristic
119	C₄×C₅₃	#500	1	1	yes	characteristic
120	Dic₅₃	—	4	4	no	—
121	Dic₅₃	—	8	2	no	—
122	#1939	—	53	2	no	—
123	C₄×#107	—	53	1	no	—
124	C₂×#335	—	53	1	no	—
125	C₂×C₈²	D₅₃	1	1	yes	characteristic
126	#1279	—	53	2	no	—
127	C₂×C₅₃	C₈×D₈	1	1	yes	characteristic
128	C₂×C₅₃	C₈×Q₁₆	1	1	yes	characteristic
129	C₂×C₅₃	#1589	1	1	yes	characteristic
130	C₂×C₅₃	#1591	1	1	yes	characteristic
131	C₂×C₅₃	#1939	1	1	yes	characteristic
132	C₂×C₅₃	#1942	1	1	yes	characteristic
133	C₂×C₅₃	#1945	1	1	yes	characteristic
134	C₂×#107	—	53	1	no	—
135	C₂×#107	—	53	1	no	—
136	C₂×C₄⋊C₈	—	53	2	no	—
137	C₂×C₄⋊C₈	—	106	2	no	—
138	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
139	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
140	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
141	C₄×C₄⋊C₄	—	53	2	no	—
142	#340	—	53	2	no	—
143	#336	—	106	2	no	—
144	#335	—	106	2	no	—
145	C₈²	—	2	2	no	—
146	C₅₃	#9594	1	1	yes	characteristic, complemented, Sylow
147	C₂²×C₈	—	2	1	no	—
148	C₂²×C₈	—	106	2	no	—
149	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
150	C₂²×C₈	C₄×D₅₃	1	1	yes	characteristic
151	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
152	C₂²×C₈	D₂₁₂	1	1	yes	characteristic
153	C₂×C₄⋊C₄	—	53	2	no	—
154	C₂×C₄⋊C₄	—	53	2	no	—
155	C₂×C₄²	—	106	2	no	—
156	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
157	#107	—	53	4	no	—
158	#107	—	106	2	no	—
159	C₄⋊C₈	—	106	2	no	—
160	C₄⋊C₈	—	106	2	no	—
161	C₄⋊C₈	—	212	4	no	—
162	C₄×C₈	—	2	1	no	—
163	C₄×C₈	—	2	1	no	—
164	C₄×C₈	—	2	1	no	—
165	C₄×C₈	Dic₁₀₆	1	2	yes	—
166	C₄×C₈	Dic₁₀₆	1	2	yes	—
167	C₄×C₈	C₂²⋊D₅₃	1	2	yes	—
168	#95	—	106	2	no	—
169	C₂²×C₄	—	106	2	no	—
170	C₂²×C₄	—	106	2	no	—
171	C₂²×C₄	C₂×C₄×D₅₃	1	1	yes	characteristic
172	C₂²×C₄	C₂×D₂₁₂	1	1	yes	characteristic
173	C₂²×C₄	#11719945	1	1	yes	characteristic
174	C₂×C₈	—	2	2	no	—
175	C₂×C₈	—	2	2	no	—
176	C₂×C₈	—	2	1	no	—
177	C₂×C₈	—	2	1	no	—
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	2	1	no	—
180	C₂×C₈	—	2	1	no	—
181	C₂×C₈	—	2	1	no	—
182	C₂×C₈	—	2	1	no	—
183	C₂×C₈	—	212	2	no	—
184	C₂×C₈	—	212	2	no	—
185	C₂×C₈	—	212	2	no	—
186	C₂×C₈	C₈×D₅₃	1	2	yes	—
187	C₂×C₈	#11719912	1	2	yes	—
188	C₂×C₈	D₄₂₄	1	2	yes	—
189	C₂×C₈	Dic₂₁₂	1	2	yes	—
190	C₂×C₈	#11719919	1	2	yes	—
191	C₂×C₈	#11719919	1	2	yes	—
192	C₄⋊C₄	—	53	4	no	—
193	C₄⋊C₄	—	106	2	no	—
194	C₄⋊C₄	—	106	2	no	—
195	C₄⋊C₄	—	106	2	no	—
196	C₄²	—	212	2	no	—
197	C₄²	—	212	2	no	—
198	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
199	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
200	C₄²	#11719945	1	1	yes	characteristic
201	C₄²	C₂×C₂²⋊D₅₃	1	1	yes	characteristic
202	C₂³	C₄×D₂₁₂	1	1	yes	characteristic
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	106	4	no	—
206	C₂×C₄	—	212	2	no	—
207	C₂×C₄	—	212	2	no	—
208	C₂×C₄	—	212	2	no	—
209	C₂×C₄	—	212	2	no	—
210	C₂×C₄	—	212	2	no	—
211	C₂×C₄	C₂×#11719919	1	1	yes	characteristic
212	C₂×C₄	C₄×Dic₁₀₆	1	1	yes	characteristic
213	C₂×C₄	#13133191	1	1	yes	characteristic
214	C₂×C₄	#13133196	1	1	yes	characteristic
215	C₂×C₄	#13133209	1	1	yes	characteristic
216	C₂×C₄	#13133217	1	1	yes	characteristic
217	C₂×C₄	C₂×C₈×D₅₃	1	1	yes	characteristic
218	C₂×C₄	C₂×#11719912	1	1	yes	characteristic
219	C₂×C₄	#13133223	1	1	yes	characteristic
220	C₂×C₄	#13133223	1	1	yes	characteristic
221	C₂×C₄	C₂×D₄₂₄	1	1	yes	characteristic
222	C₂×C₄	C₂×Dic₂₁₂	1	1	yes	characteristic
223	C₂×C₄	#13133239	1	1	yes	characteristic
224	C₂×C₄	#13133239	1	1	yes	characteristic
225	C₈	—	2	4	no	—
226	C₈	—	2	2	no	—
227	C₈	—	2	2	no	—
228	C₈	—	2	2	no	—
229	C₈	—	424	2	no	—
230	C₈	#13133146	1	4	yes	complemented
231	C₂²	#16810521	1	1	yes	characteristic
232	C₂²	C₈×D₂₁₂	1	1	yes	characteristic
233	C₂²	#16810702	1	1	yes	characteristic
234	C₂²	C₄×D₄₂₄	1	1	yes	characteristic
235	C₂²	C₄×Dic₂₁₂	1	1	yes	characteristic
236	C₂²	#16810741	1	1	yes	characteristic
237	C₂²	#16810741	1	1	yes	characteristic
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	424	2	no	—
241	C₄	—	212	4	no	—
242	C₄	C₂×#13133146	1	1	yes	characteristic
243	C₄	#16810649	1	1	yes	characteristic
244	C₄	C₈×Dic₁₀₆	1	1	yes	characteristic
245	C₄	#16810753	1	1	yes	characteristic
246	C₄	#16810859	1	1	yes	characteristic
247	C₄	#16810864	1	1	yes	characteristic
248	C₄	#16811025	1	1	yes	characteristic
249	C₄	#16811028	1	1	yes	characteristic
250	C₂	C₈×D₄₂₄	1	1	yes	characteristic
251	C₂	C₈×Dic₂₁₂	1	1	yes	characteristic
252	C₂	#20273601	1	1	yes	characteristic
253	C₂	#20273613	1	1	yes	characteristic
254	C₂	#20275433	1	1	yes	characteristic
255	C₂	#20275448	1	1	yes	characteristic
256	C₂	#20275463	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂×D₂₁₂
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₂×C₈²×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₂×C₈²	D₅₃
hypocenter	C₅₃	#9594
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
23. #13133151	C₈	semidirect
100. C₈×C₅₃	C₄⋊C₈	semidirect
146. C₅₃	#9594	semidirect
230. C₈	#13133146	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
#20275433	2
C₄×#13133151	2
C₂×#16810376	2
#20272555	2
C₂×C₈²×C₅₃	2
#9594	53