Subgroups — #28328596

Order 13568.

484 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291130	C₂	1	2	yes	maximal, complemented
2	#20291099	C₂	1	2	yes	maximal, complemented
3	#20291095	C₂	1	2	yes	maximal, complemented
4	#20291095	C₂	1	2	yes	maximal, complemented
5	#20291080	C₂	1	2	yes	maximal, complemented
6	#20291076	C₂	1	1	yes	maximal, characteristic, complemented
7	#20291050	C₂	1	2	yes	maximal, complemented
8	#20291047	C₂	1	1	yes	maximal, characteristic, complemented
9	#20283691	C₂	1	1	yes	maximal, characteristic
10	C₂×#13133291	C₂²	1	1	yes	characteristic
11	C₂×#13133291	C₂²	1	2	yes	—
12	C₂×#13133291	C₂²	1	1	yes	characteristic
13	C₂²×#11719910	C₂²	1	2	yes	—
14	C₄⋊#11719910	C₂²	1	1	yes	characteristic
15	C₄⋊#11719910	C₂²	1	1	yes	characteristic
16	C₄⋊#11719910	C₂²	1	2	yes	—
17	C₄⋊#11719910	C₂²	1	2	yes	—
18	C₄⋊#11719910	C₂²	1	2	yes	—
19	C₄⋊#11719910	C₂²	1	2	yes	—
20	C₂²⋊#11719910	C₂²	1	2	yes	—
21	C₂²⋊#11719910	C₂²	1	2	yes	—
22	C₂²⋊#11719910	C₂²	1	2	yes	—
23	#16811511	C₂²	1	2	yes	—
24	#16811511	C₂²	1	2	yes	—
25	C₄×#11719910	C₂²	1	2	yes	—
26	#16810754	C₄	1	1	yes	characteristic
27	Q₈×#88322	C₂²	1	1	yes	characteristic
28	#16810701	C₄	1	1	yes	characteristic
29	#16810695	C₄	1	2	yes	—
30	D₄×#88322	C₂²	1	1	yes	characteristic
31	D₄×#88322	C₂²	1	2	yes	—
32	#16810684	C₄	1	2	yes	—
33	#16810680	C₄	1	1	yes	characteristic
34	#16810579	C₂²	1	2	yes	—
35	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
36	C₅₃×#529	C₄	1	1	yes	characteristic
37	#13133291	—	4	1	no	—
38	#13133291	—	4	2	no	—
39	#13133291	—	4	1	no	—
40	C₂×#11719910	—	4	2	no	—
41	C₂×#11719910	C₂³	1	2	yes	—
42	C₂×#11719910	C₂³	1	2	yes	—
43	C₂×#11719910	C₂³	1	2	yes	—
44	C₂×#11719910	C₂³	1	2	yes	—
45	C₂×#11719917	C₂×C₄	1	2	yes	—
46	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
47	C₂²×#88322	C₂³	1	1	yes	characteristic
48	C₂²×#88322	C₂³	1	2	yes	—
49	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
50	C₄⋊#88322	C₂×C₄	1	2	yes	—
51	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
52	C₄⋊#88322	C₂×C₄	1	2	yes	—
53	C₂²⋊#88322	C₂×C₄	1	2	yes	—
54	C₂²⋊#88322	C₂×C₄	1	2	yes	—
55	C₂²⋊#88322	C₂×C₄	1	2	yes	—
56	#13133128	C₂×C₄	1	2	yes	—
57	#13133128	C₂×C₄	1	2	yes	—
58	C₄×#88322	C₂×C₄	1	1	yes	characteristic
59	C₄×#88322	C₂×C₄	1	1	yes	characteristic
60	C₄×#88322	C₂³	1	1	yes	characteristic
61	C₄×#88322	C₂³	1	1	yes	characteristic
62	C₄×#88322	C₂³	1	2	yes	—
63	#126×C₅₃	C₈	1	2	yes	—
64	#125×C₅₃	C₈	1	1	yes	characteristic
65	#124×C₅₃	C₈	1	1	yes	characteristic
66	#123×C₅₃	C₈	1	2	yes	—
67	C₂²⋊Q₈×C₅₃	C₈	1	1	yes	characteristic
68	#121×C₅₃	C₈	1	1	yes	characteristic
69	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
70	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
71	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
72	#117×C₅₃	C₂×C₄	1	2	yes	—
73	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
75	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
76	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
77	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
78	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
79	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
80	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
81	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
82	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
83	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
84	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
85	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
86	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
87	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
88	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
89	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
90	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
91	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
92	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
93	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
94	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
97	#11719917	—	4	2	no	—
98	C₂×#88322	—	2	2	no	—
99	C₂×#88322	—	2	1	no	—
100	C₂×#88322	—	2	1	no	—
101	C₂×#88322	—	2	2	no	—
102	C₂×#88322	—	4	1	no	—
103	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
104	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
105	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
106	C₂×#88322	C₂²×C₄	1	2	yes	—
107	C₂×#88322	C₂²×C₄	1	2	yes	—
108	C₂×#88322	C₂⁴	1	1	yes	characteristic
109	#11719910	—	2	2	no	—
110	#11719910	—	2	2	no	—
111	#11719910	—	2	2	no	—
112	#11719910	—	2	2	no	—
113	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
114	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
115	Q₈×C₅₃	—	4	1	no	—
116	D₄×C₅₃	—	4	1	no	—
117	D₄×C₅₃	—	4	2	no	—
118	C₂×C₄×C₅₃	—	2	1	no	—
119	C₂×C₄×C₅₃	—	2	2	no	—
120	C₂×C₄×C₅₃	—	2	1	no	—
121	C₂×C₄×C₅₃	—	2	2	no	—
122	C₂×C₄×C₅₃	—	4	1	no	—
123	C₂×C₄×C₅₃	—	4	1	no	—
124	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
125	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
127	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
128	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
129	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
130	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
131	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
132	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
133	#88322	—	2	2	no	—
134	#88322	—	2	1	no	—
135	#88322	—	2	1	no	—
136	#88322	—	2	1	no	—
137	#88322	—	2	2	no	—
138	#88322	H(5,𝔽₂)	1	1	yes	characteristic
139	#88322	2₋¹⁺⁴	1	1	yes	characteristic
140	#33509	—	53	1	no	maximal, Sylow
141	C₂²×C₅₃	—	2	1	no	—
142	C₂²×C₅₃	—	2	1	no	—
143	C₂²×C₅₃	—	2	2	no	—
144	C₂²×C₅₃	—	2	2	no	—
145	C₂²×C₅₃	—	2	1	no	—
146	C₂²×C₅₃	—	2	2	no	—
147	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
148	C₄×C₅₃	—	2	1	no	—
149	C₄×C₅₃	—	2	1	no	—
150	C₄×C₅₃	—	2	1	no	—
151	C₄×C₅₃	—	2	1	no	—
152	C₄×C₅₃	—	2	2	no	—
153	C₄×C₅₃	—	2	2	no	—
154	C₄×C₅₃	—	2	1	no	—
155	C₄×C₅₃	—	2	2	no	—
156	C₄×C₅₃	#520	1	1	yes	characteristic
157	C₄×C₅₃	#568	1	1	yes	characteristic
158	#2966	—	53	1	no	—
159	#2183	—	53	2	no	—
160	#2169	—	53	2	no	—
161	#2168	—	53	2	no	—
162	#2168	—	53	2	no	—
163	#2164	—	53	2	no	—
164	#2163	—	53	1	no	—
165	#2151	—	53	2	no	—
166	#2150	—	53	1	no	—
167	C₂×C₅₃	—	4	1	no	—
168	C₂×C₅₃	—	4	2	no	—
169	C₂×C₅₃	#2988	1	1	yes	characteristic
170	C₂×C₅₃	#3407	1	1	yes	characteristic
171	C₂×C₅₃	#3407	1	1	yes	characteristic
172	#529	#6157	1	1	yes	characteristic
173	C₂×M₃₂	—	53	1	no	—
174	C₂×M₃₂	—	53	1	no	—
175	C₂×M₃₂	—	53	2	no	—
176	C₂²×C₁₆	—	53	2	no	—
177	#446	—	53	1	no	—
178	C₈×Q₈	—	53	1	no	—
179	#436	—	53	1	no	—
180	#435	—	53	2	no	—
181	C₈×D₄	—	53	1	no	—
182	C₈×D₄	—	53	2	no	—
183	#433	—	53	2	no	—
184	#432	—	53	1	no	—
185	#405	—	53	2	no	—
186	C₂×C₄×C₈	—	53	1	no	—
187	C₄⋊C₁₆	—	53	2	no	—
188	C₄⋊C₁₆	—	53	2	no	—
189	C₄⋊C₁₆	—	53	1	no	—
190	C₄⋊C₁₆	—	53	2	no	—
191	C₄⋊C₁₆	—	53	2	no	—
192	C₄⋊C₁₆	—	53	1	no	—
193	C₂²⋊C₁₆	—	53	2	no	—
194	C₂²⋊C₁₆	—	53	2	no	—
195	C₂²⋊C₁₆	—	53	2	no	—
196	#346	—	53	2	no	—
197	#346	—	53	2	no	—
198	C₄×C₁₆	—	53	2	no	—
199	C₅₃	#33509	1	1	yes	characteristic, complemented, Sylow
200	C₂×M₁₆	—	53	2	no	—
201	C₂²×C₈	—	53	1	no	—
202	C₂²×C₈	—	53	2	no	—
203	C₂²×C₈	—	53	1	no	—
204	#126	#88324	1	2	yes	—
205	#125	#88324	1	1	yes	characteristic
206	#124	#88324	1	1	yes	characteristic
207	#123	#88324	1	2	yes	—
208	C₂²⋊Q₈	#88324	1	1	yes	characteristic
209	#121	#88324	1	1	yes	characteristic
210	C₄×Q₈	C₂×#6157	1	1	yes	characteristic
211	C₄×D₄	C₂×#6157	1	2	yes	—
212	C₄×D₄	C₂×#6157	1	1	yes	characteristic
213	#117	C₂×#6157	1	2	yes	—
214	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
215	M₃₂	—	212	1	no	—
216	M₃₂	—	212	2	no	—
217	M₃₂	—	212	1	no	—
218	C₂×C₁₆	—	53	2	no	—
219	C₂×C₁₆	—	53	2	no	—
220	C₂×C₁₆	—	53	2	no	—
221	C₂×C₁₆	—	53	2	no	—
222	C₂×C₁₆	—	212	2	no	—
223	C₄⋊C₈	—	53	1	no	—
224	C₄⋊C₈	—	53	1	no	—
225	C₄⋊C₈	—	53	2	no	—
226	C₄⋊C₈	—	53	2	no	—
227	C₂²⋊C₈	—	53	2	no	—
228	C₂²⋊C₈	—	53	2	no	—
229	C₂²⋊C₈	—	53	2	no	—
230	#97	—	53	2	no	—
231	#97	—	53	2	no	—
232	C₄×C₈	—	53	1	no	—
233	C₄×C₈	—	53	1	no	—
234	C₄×C₈	—	53	2	no	—
235	C₄×C₈	—	53	1	no	—
236	C₄×C₈	—	53	1	no	—
237	C₂×Q₈	C₂×#88324	1	1	yes	characteristic
238	C₂×D₄	C₂×#88324	1	2	yes	—
239	C₂×D₄	C₂×#88324	1	1	yes	characteristic
240	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
241	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
242	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
243	C₂²×C₄	C₂²×#6157	1	2	yes	—
244	M₁₆	—	212	2	no	—
245	C₂×C₈	—	53	1	no	—
246	C₂×C₈	—	53	1	no	—
247	C₂×C₈	—	53	1	no	—
248	C₂×C₈	—	53	1	no	—
249	C₂×C₈	—	53	2	no	—
250	C₂×C₈	—	53	2	no	—
251	C₂×C₈	—	106	2	no	—
252	C₂×C₈	—	106	1	no	—
253	C₂×C₈	—	106	1	no	—
254	C₂×C₈	—	106	2	no	—
255	C₂×C₈	—	212	1	no	—
256	C₄⋊C₄	C₂×#88324	1	2	yes	—
257	C₄⋊C₄	C₂×#88324	1	2	yes	—
258	C₄⋊C₄	C₂×#88324	1	1	yes	characteristic
259	C₄⋊C₄	C₂×#88324	1	2	yes	—
260	C₄⋊C₄	C₂×#88324	1	1	yes	characteristic
261	C₄⋊C₄	C₂×#88324	1	2	yes	—
262	C₂²⋊C₄	C₂×#88324	1	2	yes	—
263	C₂²⋊C₄	C₂×#88324	1	2	yes	—
264	C₂²⋊C₄	C₂×#88324	1	2	yes	—
265	C₂²⋊C₄	C₂×#88324	1	2	yes	—
266	C₂²⋊C₄	C₂×#88324	1	2	yes	—
267	C₄²	C₂×#88324	1	1	yes	characteristic
268	C₄²	C₂×#88324	1	1	yes	characteristic
269	C₄²	C₂²×#6157	1	1	yes	characteristic
270	C₄²	C₂²×#6157	1	1	yes	characteristic
271	C₄²	C₂²×#6157	1	2	yes	—
272	C₁₆	—	106	2	no	—
273	C₁₆	—	106	2	no	—
274	C₁₆	—	106	2	no	—
275	C₁₆	—	106	2	no	—
276	C₂³	C₂²×#88324	1	2	yes	—
277	C₂³	C₂²×#88324	1	1	yes	characteristic
278	Q₈	—	4	1	no	—
279	D₄	—	4	1	no	—
280	D₄	—	4	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	2	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
288	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
289	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
290	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
291	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
292	C₂×C₄	C₂²×#88324	1	2	yes	—
293	C₂×C₄	C₂²×#88324	1	2	yes	—
294	C₂×C₄	C₂²×#88324	1	2	yes	—
295	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
296	C₈	—	53	1	no	—
297	C₈	—	53	1	no	—
298	C₈	—	106	1	no	—
299	C₈	—	106	2	no	—
300	C₈	—	106	1	no	—
301	C₈	—	106	1	no	—
302	C₈	—	106	2	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	2	no	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	1	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	C₂³×#88324	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	2	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	2	1	no	—
315	C₄	—	2	1	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	#16811692	1	1	yes	characteristic
319	C₄	#16811702	1	1	yes	characteristic
320	C₂	—	4	1	no	—
321	C₂	—	4	2	no	—
322	C₂	#20291630	1	1	yes	characteristic
323	C₂	#20291682	1	1	yes	characteristic
324	C₂	#20291682	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×#6157
commutator, socle	C₂²×C₅₃	C₂³×C₈
Fitting	C₅₃×#529	C₄
prime radical	#121×C₅₃	C₈
cosocle	C₂×#88322	C₂⁴
hypercenter	#529	#6157
hypocenter	C₅₃	#33509
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. #20291130	C₂	semidirect
2. #20291099	C₂	semidirect
3. #20291095	C₂	semidirect
4. #20291095	C₂	semidirect
5. #20291080	C₂	semidirect
6. #20291076	C₂	semidirect
7. #20291050	C₂	semidirect
8. #20291047	C₂	semidirect
199. C₅₃	#33509	semidirect

Maximal subgroups

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

subgroup	index
#20291130	2
#20291099	2
#20291095	2
#20291080	2
#20291076	2
#20291050	2
#20291047	2
#20283691	2
#33509	53