Subgroups — #28328590

Order 13568.

502 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20291130	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×#11719910	C₂	1	1	yes	maximal, characteristic, complemented
3	#20291099	C₂	1	1	yes	maximal, characteristic, complemented
4	#20291095	C₂	1	2	yes	maximal, complemented
5	D₄×#11719910	C₂	1	1	yes	maximal, characteristic, complemented
6	D₄×#11719910	C₂	1	2	yes	maximal, complemented
7	#20291080	C₂	1	2	yes	maximal, complemented
8	#20291072	C₂	1	1	yes	maximal, characteristic, complemented
9	#20291050	C₂	1	2	yes	maximal, complemented
10	C₂×C₄⋊#11719910	C₂	1	1	yes	maximal, characteristic, complemented
11	#20283691	C₂	1	1	yes	maximal, characteristic
12	C₂×#13133291	C₂²	1	2	yes	—
13	C₂²×#11719910	C₂²	1	2	yes	—
14	C₂²×#11719910	C₂²	1	2	yes	—
15	C₄⋊#11719910	C₂²	1	1	yes	characteristic
16	C₄⋊#11719910	C₂²	1	1	yes	characteristic
17	C₄⋊#11719910	C₂²	1	2	yes	—
18	C₄⋊#11719910	C₂²	1	1	yes	characteristic
19	C₄⋊#11719910	C₂²	1	2	yes	—
20	C₄⋊#11719910	C₂²	1	2	yes	—
21	C₄⋊#11719910	C₂²	1	1	yes	characteristic
22	C₂²⋊#11719910	C₂²	1	1	yes	characteristic
23	C₂²⋊#11719910	C₂²	1	1	yes	characteristic
24	C₂²⋊#11719910	C₂²	1	2	yes	—
25	C₂²⋊#11719910	C₂²	1	2	yes	—
26	#16811511	C₂²	1	2	yes	—
27	C₄×#11719910	C₂²	1	1	yes	characteristic
28	C₄×#11719910	C₂²	1	1	yes	characteristic
29	C₄×#11719910	C₂²	1	2	yes	—
30	#16810754	C₄	1	1	yes	characteristic
31	Q₈×#88322	C₂²	1	1	yes	characteristic
32	#16810701	C₄	1	1	yes	characteristic
33	#16810695	C₄	1	2	yes	—
34	D₄×#88322	C₂²	1	2	yes	—
35	D₄×#88322	C₂²	1	1	yes	characteristic
36	#16810684	C₄	1	2	yes	—
37	#16810680	C₄	1	1	yes	characteristic
38	#16810579	C₂²	1	2	yes	—
39	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
40	C₅₃×#529	C₄	1	1	yes	characteristic
41	#13133291	—	4	2	no	—
42	C₂×#11719910	—	4	2	no	—
43	C₂×#11719910	—	4	2	no	—
44	C₂×#11719910	C₂³	1	1	yes	characteristic
45	C₂×#11719910	C₂³	1	2	yes	—
46	C₂×#11719910	C₂³	1	2	yes	—
47	C₂×#11719910	C₂³	1	1	yes	characteristic
48	C₂×#11719910	C₂³	1	1	yes	characteristic
49	C₂×#11719910	C₂³	1	1	yes	characteristic
50	C₂×#11719917	C₂×C₄	1	2	yes	—
51	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
52	C₂²×#88322	C₂³	1	1	yes	characteristic
53	C₂²×#88322	C₂³	1	2	yes	—
54	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
55	C₄⋊#88322	C₂×C₄	1	2	yes	—
56	C₄⋊#88322	C₂×C₄	1	1	yes	characteristic
57	C₄⋊#88322	C₂×C₄	1	2	yes	—
58	C₂²⋊#88322	C₂×C₄	1	2	yes	—
59	C₂²⋊#88322	C₂×C₄	1	2	yes	—
60	C₂²⋊#88322	C₂×C₄	1	2	yes	—
61	#13133128	C₂×C₄	1	2	yes	—
62	#13133128	C₂×C₄	1	2	yes	—
63	C₄×#88322	C₂×C₄	1	1	yes	characteristic
64	C₄×#88322	C₂×C₄	1	1	yes	characteristic
65	C₄×#88322	C₂³	1	1	yes	characteristic
66	C₄×#88322	C₂³	1	1	yes	characteristic
67	C₄×#88322	C₂³	1	2	yes	—
68	#126×C₅₃	C₈	1	2	yes	—
69	#125×C₅₃	C₈	1	1	yes	characteristic
70	#124×C₅₃	C₈	1	1	yes	characteristic
71	#123×C₅₃	C₈	1	2	yes	—
72	C₂²⋊Q₈×C₅₃	C₈	1	1	yes	characteristic
73	#121×C₅₃	C₈	1	1	yes	characteristic
74	C₄×Q₈×C₅₃	C₂×C₄	1	1	yes	characteristic
75	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
76	C₄×D₄×C₅₃	C₂×C₄	1	1	yes	characteristic
77	#117×C₅₃	C₂×C₄	1	2	yes	—
78	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
79	C₂×Q₈×C₅₃	C₂×C₈	1	1	yes	characteristic
80	C₂×D₄×C₅₃	C₂×C₈	1	2	yes	—
81	C₂×D₄×C₅₃	C₂×C₈	1	1	yes	characteristic
82	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
83	C₂²×C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
84	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
86	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
87	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
88	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
89	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
90	C₄⋊C₄×C₅₃	C₂×C₈	1	1	yes	characteristic
91	C₄⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
92	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
93	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
94	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
95	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
96	C₂²⋊C₄×C₅₃	C₂×C₈	1	2	yes	—
97	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
98	C₄²×C₅₃	C₂×C₈	1	1	yes	characteristic
99	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
100	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
101	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
102	#11719917	—	4	2	no	—
103	C₂×#88322	—	2	1	no	—
104	C₂×#88322	—	2	2	no	—
105	C₂×#88322	—	2	1	no	—
106	C₂×#88322	—	2	2	no	—
107	C₂×#88322	—	4	1	no	—
108	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
109	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
110	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
111	C₂×#88322	C₂²×C₄	1	2	yes	—
112	C₂×#88322	C₂²×C₄	1	2	yes	—
113	C₂×#88322	C₂⁴	1	1	yes	characteristic
114	#11719910	—	2	1	no	—
115	#11719910	—	2	2	no	—
116	#11719910	—	2	2	no	—
117	#11719910	—	2	1	no	—
118	#11719910	—	2	1	no	—
119	#11719910	—	2	1	no	—
120	C₂³×C₅₃	C₂²×C₈	1	2	yes	—
121	C₂³×C₅₃	C₂²×C₈	1	1	yes	characteristic
122	Q₈×C₅₃	—	2	2	no	—
123	D₄×C₅₃	—	2	4	no	—
124	D₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	1	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	1	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	2	2	no	—
131	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
132	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
137	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
138	C₂×C₄×C₅₃	C₂²×C₈	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
140	#88322	—	2	1	no	—
141	#88322	—	2	2	no	—
142	#88322	—	2	1	no	—
143	#88322	—	2	1	no	—
144	#88322	—	2	2	no	—
145	#88322	H(5,𝔽₂)	1	1	yes	characteristic
146	#88322	2₋¹⁺⁴	1	1	yes	characteristic
147	#33508	—	53	1	no	maximal, Sylow
148	C₂²×C₅₃	—	2	1	no	—
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	—	2	2	no	—
151	C₂²×C₅₃	—	2	2	no	—
152	C₂²×C₅₃	—	2	2	no	—
153	C₂²×C₅₃	#504	1	2	yes	—
154	C₂²×C₅₃	C₂³×C₈	1	1	yes	characteristic
155	C₄×C₅₃	—	2	1	no	—
156	C₄×C₅₃	—	2	1	no	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	—	2	2	no	—
159	C₄×C₅₃	—	2	2	no	—
160	C₄×C₅₃	—	2	1	no	—
161	C₄×C₅₃	—	2	2	no	—
162	C₄×C₅₃	#504	1	2	yes	—
163	C₄×C₅₃	#520	1	1	yes	characteristic
164	C₄×C₅₃	#568	1	1	yes	characteristic
165	#2966	—	53	1	no	—
166	#2183	—	53	1	no	—
167	Q₈×C₁₆	—	53	1	no	—
168	#2169	—	53	1	no	—
169	#2168	—	53	2	no	—
170	D₄×C₁₆	—	53	1	no	—
171	D₄×C₁₆	—	53	2	no	—
172	#2164	—	53	2	no	—
173	#2162	—	53	1	no	—
174	#2151	—	53	2	no	—
175	C₂×C₄⋊C₁₆	—	53	1	no	—
176	C₂×C₅₃	—	2	2	no	—
177	C₂×C₅₃	—	4	2	no	—
178	C₂×C₅₃	#2988	1	1	yes	characteristic
179	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
180	C₂×C₅₃	C₂×#504	1	1	yes	characteristic
181	#529	#6157	1	1	yes	characteristic
182	C₂×M₃₂	—	53	2	no	—
183	C₂²×C₁₆	—	53	2	no	—
184	C₂²×C₁₆	—	53	2	no	—
185	#446	—	53	1	no	—
186	C₈×Q₈	—	53	1	no	—
187	#436	—	53	1	no	—
188	#435	—	53	2	no	—
189	C₈×D₄	—	53	2	no	—
190	C₈×D₄	—	53	1	no	—
191	#433	—	53	2	no	—
192	#432	—	53	1	no	—
193	#405	—	53	2	no	—
194	C₂×C₄×C₈	—	53	1	no	—
195	C₄⋊C₁₆	—	53	2	no	—
196	C₄⋊C₁₆	—	53	1	no	—
197	C₄⋊C₁₆	—	53	1	no	—
198	C₄⋊C₁₆	—	53	1	no	—
199	C₄⋊C₁₆	—	53	2	no	—
200	C₄⋊C₁₆	—	53	2	no	—
201	C₄⋊C₁₆	—	53	1	no	—
202	C₂²⋊C₁₆	—	53	1	no	—
203	C₂²⋊C₁₆	—	53	1	no	—
204	C₂²⋊C₁₆	—	53	2	no	—
205	C₂²⋊C₁₆	—	53	2	no	—
206	#346	—	53	2	no	—
207	C₄×C₁₆	—	53	1	no	—
208	C₄×C₁₆	—	53	1	no	—
209	C₄×C₁₆	—	53	2	no	—
210	C₅₃	#33508	1	1	yes	characteristic, complemented, Sylow
211	C₂×M₁₆	—	53	2	no	—
212	C₂²×C₈	—	53	1	no	—
213	C₂²×C₈	—	53	2	no	—
214	C₂²×C₈	—	53	1	no	—
215	#126	#88324	1	2	yes	—
216	#125	#88324	1	1	yes	characteristic
217	#124	#88324	1	1	yes	characteristic
218	#123	#88324	1	2	yes	—
219	C₂²⋊Q₈	#88324	1	1	yes	characteristic
220	#121	#88324	1	1	yes	characteristic
221	C₄×Q₈	C₂×#6157	1	1	yes	characteristic
222	C₄×D₄	C₂×#6157	1	2	yes	—
223	C₄×D₄	C₂×#6157	1	1	yes	characteristic
224	#117	C₂×#6157	1	2	yes	—
225	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
226	M₃₂	—	212	2	no	—
227	C₂×C₁₆	—	53	1	no	—
228	C₂×C₁₆	—	53	1	no	—
229	C₂×C₁₆	—	53	1	no	—
230	C₂×C₁₆	—	53	2	no	—
231	C₂×C₁₆	—	53	1	no	—
232	C₂×C₁₆	—	53	2	no	—
233	C₂×C₁₆	—	212	2	no	—
234	C₂×C₁₆	—	212	2	no	—
235	C₄⋊C₈	—	53	1	no	—
236	C₄⋊C₈	—	53	1	no	—
237	C₄⋊C₈	—	53	2	no	—
238	C₄⋊C₈	—	53	2	no	—
239	C₂²⋊C₈	—	53	2	no	—
240	C₂²⋊C₈	—	53	2	no	—
241	C₂²⋊C₈	—	53	2	no	—
242	#97	—	53	2	no	—
243	#97	—	53	2	no	—
244	C₄×C₈	—	53	1	no	—
245	C₄×C₈	—	53	1	no	—
246	C₄×C₈	—	53	1	no	—
247	C₄×C₈	—	53	2	no	—
248	C₄×C₈	—	53	1	no	—
249	C₂×Q₈	C₂×#88324	1	1	yes	characteristic
250	C₂×D₄	C₂×#88324	1	2	yes	—
251	C₂×D₄	C₂×#88324	1	1	yes	characteristic
252	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
253	C₂²×C₄	C₂×#88324	1	1	yes	characteristic
254	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
255	C₂²×C₄	C₂²×#6157	1	2	yes	—
256	M₁₆	—	212	2	no	—
257	C₂×C₈	—	53	1	no	—
258	C₂×C₈	—	53	1	no	—
259	C₂×C₈	—	53	1	no	—
260	C₂×C₈	—	53	1	no	—
261	C₂×C₈	—	53	2	no	—
262	C₂×C₈	—	53	2	no	—
263	C₂×C₈	—	106	1	no	—
264	C₂×C₈	—	106	2	no	—
265	C₂×C₈	—	106	1	no	—
266	C₂×C₈	—	106	2	no	—
267	C₂×C₈	—	212	1	no	—
268	C₄⋊C₄	C₂×#88324	1	2	yes	—
269	C₄⋊C₄	C₂×#88324	1	2	yes	—
270	C₄⋊C₄	C₂×#88324	1	1	yes	characteristic
271	C₄⋊C₄	C₂×#88324	1	2	yes	—
272	C₄⋊C₄	C₂×#88324	1	1	yes	characteristic
273	C₄⋊C₄	C₂×#88324	1	2	yes	—
274	C₂²⋊C₄	C₂×#88324	1	2	yes	—
275	C₂²⋊C₄	C₂×#88324	1	2	yes	—
276	C₂²⋊C₄	C₂×#88324	1	2	yes	—
277	C₂²⋊C₄	C₂×#88324	1	2	yes	—
278	C₂²⋊C₄	C₂×#88324	1	2	yes	—
279	C₄²	C₂×#88324	1	1	yes	characteristic
280	C₄²	C₂×#88324	1	1	yes	characteristic
281	C₄²	C₂²×#6157	1	1	yes	characteristic
282	C₄²	C₂²×#6157	1	1	yes	characteristic
283	C₄²	C₂²×#6157	1	2	yes	—
284	C₁₆	—	106	1	no	—
285	C₁₆	—	106	2	no	—
286	C₁₆	—	106	2	no	—
287	C₁₆	—	106	1	no	—
288	C₁₆	—	106	1	no	—
289	C₁₆	—	106	1	no	—
290	C₂³	C₂²×#88324	1	2	yes	—
291	C₂³	C₂²×#88324	1	1	yes	characteristic
292	Q₈	—	2	2	no	—
293	D₄	—	2	4	no	—
294	D₄	—	2	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	1	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	1	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
302	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
303	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
304	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
305	C₂×C₄	C₂²×#88324	1	2	yes	—
306	C₂×C₄	C₂²×#88324	1	2	yes	—
307	C₂×C₄	C₂²×#88324	1	2	yes	—
308	C₂×C₄	C₂²×#88324	1	1	yes	characteristic
309	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
310	C₈	—	53	1	no	—
311	C₈	—	53	1	no	—
312	C₈	—	106	1	no	—
313	C₈	—	106	2	no	—
314	C₈	—	106	1	no	—
315	C₈	—	106	1	no	—
316	C₈	—	106	2	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	#16811681	1	2	yes	—
323	C₂²	C₂³×#88324	1	1	yes	characteristic
324	C₄	—	2	1	no	—
325	C₄	—	2	2	no	—
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	2	no	—
331	C₄	#16811681	1	2	yes	—
332	C₄	#16811692	1	1	yes	characteristic
333	C₄	#16811702	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	4	2	no	—
336	C₂	#20291630	1	1	yes	characteristic
337	C₂	C₂×#16811681	1	1	yes	characteristic
338	C₂	C₂×#16811681	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	#33508
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. Q₈×#11719910	C₂	semidirect
3. #20291099	C₂	semidirect
4. #20291095	C₂	semidirect
5. D₄×#11719910	C₂	semidirect
6. D₄×#11719910	C₂	semidirect
7. #20291080	C₂	semidirect
8. #20291072	C₂	semidirect
9. #20291050	C₂	semidirect
10. C₂×C₄⋊#11719910	C₂	semidirect
210. C₅₃	#33508	semidirect

Maximal subgroups

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

subgroup	index
#20291130	2
Q₈×#11719910	2
#20291099	2
#20291095	2
D₄×#11719910	2
#20291080	2
#20291072	2
#20291050	2
C₂×C₄⋊#11719910	2
#20283691	2
#33508	53