Subgroups — #27282635

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	#20275434	C₂	1	2	yes	maximal
2	C₂×#16810376	C₂	1	1	yes	maximal, characteristic
3	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic
4	#20272556	C₂	1	2	yes	maximal
5	C₄×#107×C₅₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊#88322	—	2	2	no	—
7	C₂×C₄⋊#88322	C₂²	1	2	yes	—
8	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
9	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
10	#16810378	—	2	2	no	—
11	#16810376	—	2	2	no	—
12	C₈×#88322	—	2	2	no	—
13	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
14	C₂×#107×C₅₃	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
16	C₄×C₄⋊C₄×C₅₃	C₂²	1	2	yes	—
17	C₅₃×#340	C₄	1	2	yes	—
18	C₂²×#88322	—	2	2	no	—
19	C₂²×#88322	—	2	1	no	—
20	C₂²×#88322	D₄	1	1	yes	characteristic
21	C₂²×#88322	D₄	1	1	yes	characteristic
22	C₄⋊#88322	—	2	2	no	—
23	C₄⋊#88322	—	2	2	no	—
24	C₄⋊#88322	—	4	4	no	—
25	C₄×#88322	—	2	1	no	—
26	C₄×#88322	—	2	1	no	—
27	C₄×#88322	—	2	1	no	—
28	C₄×#88322	Q₈	1	2	yes	—
29	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
30	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
32	C₂×C₄⋊C₄×C₅₃	C₂×C₄	1	2	yes	—
33	C₂×C₄²×C₅₃	—	2	2	no	—
34	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
35	#107×C₅₃	—	2	2	no	—
36	#107×C₅₃	C₈	1	4	yes	complemented
37	C₄×C₈×C₅₃	D₄	1	2	yes	—
38	C₄×C₈×C₅₃	Q₈	1	2	yes	—
39	#95×C₅₃	—	2	2	no	—
40	C₂²×C₄×C₅₃	—	2	2	no	—
41	C₂²×C₄×C₅₃	—	2	2	no	—
42	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
44	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
45	C₂×C₈×C₅₃	—	2	1	no	—
46	C₂×C₈×C₅₃	—	2	1	no	—
47	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
48	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
49	C₂×C₈×C₅₃	C₂×C₈	1	2	yes	—
50	C₂×C₈×C₅₃	M₁₆	1	2	yes	—
51	C₄⋊C₄×C₅₃	—	2	2	no	—
52	C₄⋊C₄×C₅₃	—	2	2	no	—
53	C₄⋊C₄×C₅₃	—	2	2	no	—
54	C₄⋊C₄×C₅₃	C₂×C₈	1	4	yes	—
55	C₄²×C₅₃	—	4	2	no	—
56	C₄²×C₅₃	—	4	2	no	—
57	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
58	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
59	C₄²×C₅₃	C₂×Q₈	1	1	yes	characteristic
60	C₄²×C₅₃	#41	1	1	yes	characteristic
61	C₂×#88322	—	2	2	no	—
62	C₂×#88322	—	2	2	no	—
63	C₂×#88322	—	2	1	no	—
64	C₂×#88322	—	2	1	no	—
65	C₂×#88322	—	2	2	no	—
66	C₂×#88322	—	2	1	no	—
67	C₂×#88322	—	2	1	no	—
68	C₂×#88322	—	4	2	no	—
69	C₂×#88322	—	4	2	no	—
70	C₂×#88322	—	4	2	no	—
71	C₂×#88322	D₈	1	2	yes	—
72	C₂×#88322	Q₁₆	1	2	yes	—
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	—	2	4	no	—
77	C₂×C₄×C₅₃	—	4	2	no	—
78	C₂×C₄×C₅₃	—	4	2	no	—
79	C₂×C₄×C₅₃	—	4	2	no	—
80	C₂×C₄×C₅₃	—	4	2	no	—
81	C₂×C₄×C₅₃	—	4	2	no	—
82	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
83	C₂×C₄×C₅₃	C₄×Q₈	1	1	yes	characteristic
84	C₂×C₄×C₅₃	#121	1	1	yes	characteristic
85	C₂×C₄×C₅₃	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₅₃	#125	1	1	yes	characteristic
87	C₂×C₄×C₅₃	C₄⋊Q₈	1	1	yes	characteristic
88	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
89	C₂×C₄×C₅₃	C₂×M₁₆	1	1	yes	characteristic
90	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
91	C₂×C₄×C₅₃	#131	1	1	yes	characteristic
92	C₂×C₄×C₅₃	C₂×D₈	1	1	yes	characteristic
93	C₂×C₄×C₅₃	C₂×Q₁₆	1	1	yes	characteristic
94	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
95	C₂×C₄×C₅₃	#135	1	1	yes	characteristic
96	C₈×C₅₃	—	2	2	no	—
97	C₈×C₅₃	C₄⋊C₈	1	4	yes	complemented
98	#88322	—	2	4	no	—
99	#88322	—	2	2	no	—
100	#88322	—	2	2	no	—
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	—	4	4	no	—
113	C₄×C₅₃	—	8	2	no	—
114	C₄×C₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
115	C₄×C₅₃	#424	1	1	yes	characteristic
116	C₄×C₅₃	C₈×Q₈	1	1	yes	characteristic
117	C₄×C₅₃	#446	1	1	yes	characteristic
118	C₄×C₅₃	#466	1	1	yes	characteristic
119	C₄×C₅₃	#467	1	1	yes	characteristic
120	C₄×C₅₃	#499	1	1	yes	characteristic
121	C₄×C₅₃	#500	1	1	yes	characteristic
122	#1939	—	53	2	no	—
123	C₄×#107	D₅₃	1	1	yes	characteristic
124	C₂×#335	—	53	1	no	—
125	C₂×C₈²	—	53	1	no	—
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	Dic₅₃	1	1	yes	characteristic
135	C₂×#107	Dic₅₃	1	1	yes	characteristic
136	C₂×C₄⋊C₈	—	53	2	no	—
137	C₂×C₄⋊C₈	—	106	2	no	—
138	C₂×C₄×C₈	—	53	1	no	—
139	C₂×C₄×C₈	—	53	1	no	—
140	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
141	C₄×C₄⋊C₄	C₂×D₅₃	1	2	yes	—
142	#340	Dic₅₃	1	2	yes	—
143	#336	—	106	2	no	—
144	#335	—	106	2	no	—
145	C₈²	—	106	2	no	—
146	C₅₃	#9594	1	1	yes	characteristic, complemented, Sylow
147	C₂²×C₈	—	53	1	no	—
148	C₂²×C₈	—	53	1	no	—
149	C₂²×C₈	—	106	1	no	—
150	C₂²×C₈	—	106	2	no	—
151	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
152	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
153	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
154	C₂×C₄⋊C₄	C₂×Dic₅₃	1	2	yes	—
155	C₂×C₄²	—	2	2	no	—
156	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
157	#107	—	2	2	no	—
158	#107	#88322	1	4	yes	complemented
159	C₄⋊C₈	—	106	2	no	—
160	C₄⋊C₈	—	106	2	no	—
161	C₄⋊C₈	—	212	4	no	—
162	C₄×C₈	—	53	2	no	—
163	C₄×C₈	—	106	1	no	—
164	C₄×C₈	—	106	1	no	—
165	C₄×C₈	—	106	1	no	—
166	C₄×C₈	Dic₁₀₆	1	2	yes	—
167	C₄×C₈	D₂₁₂	1	2	yes	—
168	#95	—	2	2	no	—
169	C₂²×C₄	—	2	2	no	—
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
172	C₂²×C₄	#11719947	1	1	yes	characteristic
173	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
174	C₂×C₈	—	2	1	no	—
175	C₂×C₈	—	2	1	no	—
176	C₂×C₈	—	53	2	no	—
177	C₂×C₈	—	53	2	no	—
178	C₂×C₈	—	106	2	no	—
179	C₂×C₈	—	106	2	no	—
180	C₂×C₈	—	106	1	no	—
181	C₂×C₈	—	106	1	no	—
182	C₂×C₈	—	106	2	no	—
183	C₂×C₈	—	106	1	no	—
184	C₂×C₈	—	106	1	no	—
185	C₂×C₈	—	212	2	no	—
186	C₂×C₈	—	212	2	no	—
187	C₂×C₈	—	212	2	no	—
188	C₂×C₈	C₂×#88322	1	2	yes	—
189	C₂×C₈	#11719917	1	2	yes	—
190	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
191	C₂×C₈	C₄⋊Dic₅₃	1	2	yes	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	2	no	—
195	C₄⋊C₄	C₂×#88322	1	4	yes	—
196	C₄²	—	4	2	no	—
197	C₄²	—	4	2	no	—
198	C₄²	C₂×Dic₁₀₆	1	1	yes	characteristic
199	C₄²	C₂×D₂₁₂	1	1	yes	characteristic
200	C₄²	Q₈×D₅₃	1	1	yes	characteristic
201	C₄²	#11719949	1	1	yes	characteristic
202	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
203	C₂×C₄	—	2	1	no	—
204	C₂×C₄	—	2	1	no	—
205	C₂×C₄	—	2	4	no	—
206	C₂×C₄	—	4	2	no	—
207	C₂×C₄	—	4	2	no	—
208	C₂×C₄	—	4	2	no	—
209	C₂×C₄	—	4	2	no	—
210	C₂×C₄	—	4	2	no	—
211	C₂×C₄	C₂×C₄⋊Dic₅₃	1	1	yes	characteristic
212	C₂×C₄	Q₈×Dic₅₃	1	1	yes	characteristic
213	C₂×C₄	#13133192	1	1	yes	characteristic
214	C₂×C₄	#13133197	1	1	yes	characteristic
215	C₂×C₄	#13133208	1	1	yes	characteristic
216	C₂×C₄	#13133216	1	1	yes	characteristic
217	C₂×C₄	C₂²×#88322	1	1	yes	characteristic
218	C₂×C₄	C₂×#11719917	1	1	yes	characteristic
219	C₂×C₄	#13133224	1	1	yes	characteristic
220	C₂×C₄	#13133224	1	1	yes	characteristic
221	C₂×C₄	D₈×D₅₃	1	1	yes	characteristic
222	C₂×C₄	Q₁₆×D₅₃	1	1	yes	characteristic
223	C₂×C₄	#13133236	1	1	yes	characteristic
224	C₂×C₄	#13133240	1	1	yes	characteristic
225	C₈	—	2	2	no	—
226	C₈	—	106	4	no	—
227	C₈	—	106	2	no	—
228	C₈	—	106	2	no	—
229	C₈	—	424	2	no	—
230	C₈	C₄⋊#88322	1	4	yes	complemented
231	C₂²	#16810522	1	1	yes	characteristic
232	C₂²	D₄×#88322	1	1	yes	characteristic
233	C₂²	#16810701	1	1	yes	characteristic
234	C₂²	D₈×Dic₅₃	1	1	yes	characteristic
235	C₂²	Q₁₆×Dic₅₃	1	1	yes	characteristic
236	C₂²	#16810740	1	1	yes	characteristic
237	C₂²	#16810740	1	1	yes	characteristic
238	C₄	—	2	1	no	—
239	C₄	—	2	1	no	—
240	C₄	—	4	4	no	—
241	C₄	—	8	2	no	—
242	C₄	C₂×C₄⋊#88322	1	1	yes	characteristic
243	C₄	#16810650	1	1	yes	characteristic
244	C₄	Q₈×#88322	1	1	yes	characteristic
245	C₄	#16810754	1	1	yes	characteristic
246	C₄	#16810860	1	1	yes	characteristic
247	C₄	#16810865	1	1	yes	characteristic
248	C₄	#16811024	1	1	yes	characteristic
249	C₄	#16811027	1	1	yes	characteristic
250	C₂	D₈×#88322	1	1	yes	characteristic
251	C₂	Q₁₆×#88322	1	1	yes	characteristic
252	C₂	#20273600	1	1	yes	characteristic
253	C₂	#20273612	1	1	yes	characteristic
254	C₂	#20275434	1	1	yes	characteristic
255	C₂	#20275449	1	1	yes	characteristic
256	C₂	#20275464	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₅₃
commutator	C₂×C₄×C₅₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₅₃
Fitting	C₄×#107×C₅₃	C₂
socle, prime radical	C₂³×C₅₃	C₄×D₄
cosocle	C₂×C₄²×C₅₃	C₂³
hypercenter	C₄×#107	D₅₃
hypocenter	C₅₃	#9594
LCS	G ▷ C₂×C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ C₄×#107

Complements

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

N	H ≅ G/N	type
36. #107×C₅₃	C₈	semidirect
97. C₈×C₅₃	C₄⋊C₈	semidirect
146. C₅₃	#9594	semidirect
158. #107	#88322	semidirect
230. C₈	C₄⋊#88322	semidirect

Maximal subgroups

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

subgroup	index
#20275434	2
C₂×#16810376	2
C₂×C₈×#88322	2
#20272556	2
C₄×#107×C₅₃	2
#9594	53