Subgroups — #28328576

Order 13568.

532 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291129	C₂	1	2	yes	maximal, complemented
2	#20291101	C₂	1	2	yes	maximal, complemented
3	#20291100	C₂	1	4	yes	maximal, complemented
4	#20291075	C₂	1	2	yes	maximal, complemented
5	#20291072	C₂	1	1	yes	maximal, characteristic, complemented
6	#20291048	C₂	1	2	yes	maximal, complemented
7	#20291047	C₂	1	1	yes	maximal, characteristic, complemented
8	#20283671	C₂	1	1	yes	maximal, characteristic
9	C₂×#13133292	C₂²	1	4	yes	—
10	C₂×#13133291	C₂²	1	2	yes	—
11	#16811544	C₂²	1	4	yes	—
12	C₄⋊#11719910	C₂²	1	1	yes	characteristic
13	C₄⋊#11719910	C₂²	1	1	yes	characteristic
14	C₄⋊#11719910	C₂²	1	2	yes	—
15	C₄⋊#11719910	C₂²	1	2	yes	—
16	#16811518	C₂²	1	4	yes	—
17	C₂²⋊#11719910	C₂²	1	2	yes	—
18	#16811509	C₂²	1	4	yes	—
19	C₄×#11719910	C₂²	1	2	yes	—
20	C₄×#11719945	C₄	1	1	yes	characteristic
21	#16810747	C₄	1	1	yes	characteristic
22	#16810747	C₂²	1	1	yes	characteristic
23	#16810688	C₄	1	1	yes	characteristic
24	#16810688	C₂²	1	1	yes	characteristic
25	#16810686	C₄	1	2	yes	—
26	#16810686	C₂²	1	2	yes	—
27	#16810674	C₄	1	2	yes	—
28	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
29	C₄×C₈×D₅₃	C₂²	1	2	yes	—
30	C₅₃×#431	C₄	1	1	yes	characteristic
31	#13133292	—	4	4	no	—
32	#13133291	—	4	2	no	—
33	C₂×#11719910	C₂³	1	4	yes	—
34	C₂×#11719910	C₂³	1	2	yes	—
35	C₂×#11719910	C₂³	1	2	yes	—
36	C₂×#11719945	C₈	1	1	yes	characteristic
37	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
38	C₂²×#88322	C₂³	1	1	yes	characteristic
39	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
40	C₂×C₈×D₅₃	C₂³	1	2	yes	—
41	C₄×Dic₁₀₆	C₈	1	1	yes	characteristic
42	C₄×Dic₁₀₆	C₂×C₄	1	1	yes	characteristic
43	C₄×D₂₁₂	C₈	1	1	yes	characteristic
44	C₄×D₂₁₂	C₂×C₄	1	1	yes	characteristic
45	C₄×C₂²⋊D₅₃	C₈	1	2	yes	—
46	C₄×C₂²⋊D₅₃	C₂×C₄	1	2	yes	—
47	#13133176	C₈	1	1	yes	characteristic
48	#13133174	C₈	1	2	yes	—
49	C₄²×D₅₃	C₂×C₄	1	2	yes	—
50	#13133146	C₂×C₄	1	2	yes	—
51	#13133146	C₂×C₄	1	2	yes	—
52	#13133129	C₂×C₄	1	2	yes	—
53	#13133129	C₂×C₄	1	2	yes	—
54	C₄×#88322	C₂×C₄	1	1	yes	characteristic
55	C₄×#88322	C₂×C₄	1	1	yes	characteristic
56	C₄×#88322	C₂³	1	1	yes	characteristic
57	C₄×#88322	C₂³	1	1	yes	characteristic
58	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
59	C₈×Dic₅₃	C₂³	1	2	yes	—
60	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
61	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
62	C₄⋊C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂²⋊C₈×C₅₃	C₂×C₄	1	2	yes	—
64	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
65	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
66	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂×C₂²⋊D₅₃	C₂×C₈	1	2	yes	—
69	#11719945	—	2	4	no	—
70	C₂×D₂₁₂	C₂×C₈	1	1	yes	characteristic
71	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
72	C₂×C₄×D₅₃	C₂×C₈	1	2	yes	—
73	C₂×C₄×D₅₃	C₂²×C₄	1	2	yes	—
74	C₂×Dic₁₀₆	C₂×C₈	1	1	yes	characteristic
75	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
77	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
78	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
79	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²⋊Dic₅₃	C₂×C₈	1	2	yes	—
82	#11719921	C₂×C₈	1	2	yes	—
83	#11719921	C₂×C₈	1	2	yes	—
84	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
85	C₄⋊Dic₅₃	C₂×C₈	1	1	yes	characteristic
86	#11719919	C₂×C₈	1	2	yes	—
87	#11719919	C₂×C₈	1	2	yes	—
88	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
89	C₄×Dic₅₃	C₂×C₈	1	2	yes	—
90	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
91	C₂×#88322	—	2	1	no	—
92	C₂×#88322	—	2	1	no	—
93	C₂×#88322	—	4	1	no	—
94	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
95	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
96	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
97	C₂×#88322	C₂⁴	1	1	yes	characteristic
98	C₈×D₅₃	—	2	2	no	—
99	C₈×D₅₃	—	2	2	no	—
100	C₈×D₅₃	—	4	2	no	—
101	#11719910	—	2	4	no	—
102	#11719910	—	2	2	no	—
103	#11719910	—	2	2	no	—
104	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
105	C₂²×D₅₃	C₂²×C₈	1	2	yes	—
106	C₂×C₄×C₅₃	—	2	1	no	—
107	C₂×C₄×C₅₃	—	2	1	no	—
108	C₂×C₄×C₅₃	—	2	2	no	—
109	C₂×C₄×C₅₃	—	2	2	no	—
110	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
111	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
112	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
113	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
114	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
115	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
116	C₂²⋊D₅₃	—	2	4	no	—
117	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
118	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
119	C₂×Dic₅₃	C₂²×C₈	1	2	yes	—
120	D₂₁₂	—	2	2	no	—
121	C₄×D₅₃	—	2	2	no	—
122	C₄×D₅₃	—	2	2	no	—
123	C₄×D₅₃	—	2	4	no	—
124	C₄×D₅₃	—	2	4	no	—
125	Dic₁₀₆	M₃₂	1	4	yes	complemented
126	C₈×C₅₃	—	2	2	no	—
127	C₈×C₅₃	—	2	2	no	—
128	#88322	—	2	1	no	—
129	#88322	—	2	1	no	—
130	#88322	—	2	1	no	—
131	#88322	H(5,𝔽₂)	1	1	yes	characteristic
132	#88322	2₋¹⁺⁴	1	1	yes	characteristic
133	#33505	—	53	1	no	maximal, Sylow
134	C₂²×C₅₃	—	2	1	no	—
135	C₂²×C₅₃	—	2	1	no	—
136	C₂²×C₅₃	—	2	1	no	—
137	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
138	C₂×D₅₃	—	2	2	no	—
139	C₂×D₅₃	—	2	2	no	—
140	C₂×D₅₃	—	2	2	no	—
141	C₄×C₅₃	—	2	1	no	—
142	C₄×C₅₃	—	2	1	no	—
143	C₄×C₅₃	—	2	1	no	—
144	C₄×C₅₃	C₂×M₃₂	1	2	yes	—
145	C₄×C₅₃	#504	1	2	yes	—
146	C₄×C₅₃	#520	1	1	yes	characteristic
147	C₄×C₅₃	#568	1	1	yes	characteristic
148	Dic₅₃	—	2	2	no	—
149	Dic₅₃	—	2	2	no	—
150	Dic₅₃	C₂×M₃₂	1	4	yes	—
151	C₈×#41	—	53	1	no	—
152	Q₈×C₁₆	—	53	2	no	—
153	#2169	—	53	4	no	—
154	#2169	—	53	2	no	—
155	#2162	—	53	1	no	—
156	#2162	—	53	2	no	—
157	#2150	—	53	1	no	—
158	#2150	—	53	2	no	—
159	C₂×C₅₃	—	4	1	no	—
160	C₂×C₅₃	#2988	1	1	yes	characteristic
161	C₂×C₅₃	C₂²×M₃₂	1	1	yes	characteristic
162	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
163	D₅₃	—	4	2	no	—
164	C₄×#41	—	53	1	no	—
165	C₂×M₃₂	—	53	2	no	—
166	C₂×M₃₂	—	53	4	no	—
167	C₈×Q₈	—	53	1	no	—
168	C₈×Q₈	—	53	1	no	—
169	C₈×D₄	—	53	1	no	—
170	C₈×D₄	—	53	1	no	—
171	C₈×D₄	—	53	2	no	—
172	C₈×D₄	—	53	2	no	—
173	#431	—	53	2	no	—
174	#431	#6157	1	1	yes	characteristic
175	C₂×C₄×C₈	—	53	1	no	—
176	C₂×C₄×C₈	—	53	2	no	—
177	C₄⋊C₁₆	—	53	1	no	—
178	C₄⋊C₁₆	—	53	2	no	—
179	C₄⋊C₁₆	—	53	2	no	—
180	C₄⋊C₁₆	—	53	1	no	—
181	C₄⋊C₁₆	—	53	4	no	—
182	C₂²⋊C₁₆	—	53	2	no	—
183	C₂²⋊C₁₆	—	53	4	no	—
184	C₄×C₁₆	—	53	2	no	—
185	C₄×C₁₆	—	53	4	no	—
186	C₅₃	#33505	1	1	yes	characteristic, complemented, Sylow
187	C₂×#41	—	53	1	no	—
188	C₂²×C₈	—	53	1	no	—
189	C₂²×C₈	—	53	1	no	—
190	C₂²×C₈	—	53	2	no	—
191	C₂²×C₈	—	53	2	no	—
192	C₄×Q₈	—	53	1	no	—
193	C₄×Q₈	—	53	1	no	—
194	C₄×D₄	—	53	2	no	—
195	C₄×D₄	—	53	2	no	—
196	C₄×D₄	—	53	1	no	—
197	C₄×D₄	—	53	1	no	—
198	#117	—	53	1	no	—
199	#117	—	53	2	no	—
200	C₂×C₄²	—	53	2	no	—
201	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
202	M₃₂	—	212	2	no	—
203	M₃₂	—	212	4	no	—
204	C₂×C₁₆	—	53	2	no	—
205	C₂×C₁₆	—	53	2	no	—
206	C₂×C₁₆	—	53	4	no	—
207	C₄⋊C₈	—	53	2	no	—
208	C₄⋊C₈	—	53	2	no	—
209	C₄⋊C₈	C₂×#6157	1	1	yes	characteristic
210	C₄⋊C₈	C₂×#6157	1	1	yes	characteristic
211	C₂²⋊C₈	—	53	2	no	—
212	C₂²⋊C₈	—	53	2	no	—
213	C₂²⋊C₈	C₂×#6157	1	2	yes	—
214	C₄×C₈	—	53	2	no	—
215	C₄×C₈	—	53	2	no	—
216	C₄×C₈	—	53	1	no	—
217	C₄×C₈	—	53	1	no	—
218	C₄×C₈	—	53	1	no	—
219	C₄×C₈	—	53	1	no	—
220	C₄×C₈	C₂×#6157	1	2	yes	—
221	#41	—	106	4	no	—
222	C₂×Q₈	—	53	1	no	—
223	C₂×D₄	—	53	2	no	—
224	C₂×D₄	—	53	1	no	—
225	C₂²×C₄	—	53	2	no	—
226	C₂²×C₄	—	53	2	no	—
227	C₂²×C₄	—	53	2	no	—
228	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
229	C₂²×C₄	D₅₃⋊C₈	1	1	yes	characteristic
230	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
231	C₂×C₈	—	53	1	no	—
232	C₂×C₈	—	53	1	no	—
233	C₂×C₈	—	53	1	no	—
234	C₂×C₈	—	53	1	no	—
235	C₂×C₈	—	106	2	no	—
236	C₂×C₈	—	106	2	no	—
237	C₂×C₈	—	106	1	no	—
238	C₂×C₈	—	106	1	no	—
239	C₂×C₈	—	212	2	no	—
240	C₂×C₈	—	212	1	no	—
241	C₂×C₈	C₂²×#6157	1	2	yes	—
242	C₂×C₈	C₂²×#6157	1	2	yes	—
243	C₄⋊C₄	—	53	2	no	—
244	C₄⋊C₄	—	53	2	no	—
245	C₄⋊C₄	—	53	1	no	—
246	C₄⋊C₄	—	53	1	no	—
247	C₂²⋊C₄	—	53	2	no	—
248	C₂²⋊C₄	—	53	2	no	—
249	C₂²⋊C₄	—	53	2	no	—
250	C₄²	—	53	2	no	—
251	C₄²	—	53	2	no	—
252	C₄²	—	53	2	no	—
253	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
254	C₄²	D₅₃⋊C₈	1	1	yes	characteristic
255	C₄²	C₂²×#6157	1	1	yes	characteristic
256	C₄²	C₂²×#6157	1	1	yes	characteristic
257	C₁₆	—	106	4	no	—
258	C₁₆	—	106	2	no	—
259	C₁₆	—	106	2	no	—
260	C₂³	—	53	2	no	—
261	C₂³	C₂×D₅₃⋊C₈	1	1	yes	characteristic
262	Q₈	—	53	4	no	—
263	D₄	—	106	4	no	—
264	D₄	—	106	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	no	—
267	C₂×C₄	—	2	1	no	—
268	C₂×C₄	—	2	1	no	—
269	C₂×C₄	—	53	2	no	—
270	C₂×C₄	—	53	2	no	—
271	C₂×C₄	—	53	2	no	—
272	C₂×C₄	—	106	4	no	—
273	C₂×C₄	—	106	4	no	—
274	C₂×C₄	—	106	2	no	—
275	C₂×C₄	—	106	2	no	—
276	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
277	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
278	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
279	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
280	C₂×C₄	C₂×D₅₃⋊C₈	1	1	yes	characteristic
281	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
282	C₈	—	2	2	no	—
283	C₈	—	2	2	no	—
284	C₈	—	53	1	no	—
285	C₈	—	53	1	no	—
286	C₈	—	106	1	no	—
287	C₈	—	106	1	no	—
288	C₈	—	106	1	no	—
289	C₂²	—	2	1	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	2	1	no	—
292	C₂²	—	106	2	no	—
293	C₂²	—	106	2	no	—
294	C₂²	—	106	2	no	—
295	C₂²	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
296	C₄	—	2	1	no	—
297	C₄	—	2	1	no	—
298	C₄	—	2	1	no	—
299	C₄	—	53	4	no	—
300	C₄	—	106	2	no	—
301	C₄	—	106	2	no	—
302	C₄	D₅₃⋊M₃₂	1	2	yes	—
303	C₄	#16811682	1	2	yes	—
304	C₄	#16811692	1	1	yes	characteristic
305	C₄	#16811702	1	1	yes	characteristic
306	C₂	—	4	1	no	—
307	C₂	—	212	2	no	—
308	C₂	#20291632	1	1	yes	characteristic
309	C₂	C₂×D₅₃⋊M₃₂	1	1	yes	characteristic
310	C₂	C₂×#16811682	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#431	C₄
prime radical	C₂×#11719945	C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	#431	#6157
hypocenter	C₅₃	#33505
LCS	G ▷ C₂²×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂×C₄ ◁ #431

Complements

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

N	H ≅ G/N	type
1. #20291129	C₂	semidirect
2. #20291101	C₂	semidirect
3. #20291100	C₂	semidirect
4. #20291075	C₂	semidirect
5. #20291072	C₂	semidirect
6. #20291048	C₂	semidirect
7. #20291047	C₂	semidirect
125. Dic₁₀₆	M₃₂	semidirect
186. C₅₃	#33505	semidirect

Maximal subgroups

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

subgroup	index
#20291129	2
#20291101	2
#20291100	2
#20291075	2
#20291072	2
#20291048	2
#20291047	2
#20283671	2
#33505	53