Subgroups — #135×#88322 #27353471

Order 13568.

750 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#88322	C₂	1	2	yes	maximal, complemented
2	Q₁₆×#88322	C₂	1	2	yes	maximal, complemented
3	SD₁₆×#88322	C₂	1	4	yes	maximal, complemented
4	D₈×#88322	C₂	1	2	yes	maximal, complemented
5	#20273463	C₂	1	1	yes	maximal, characteristic, complemented
6	#20273097	C₂	1	2	yes	maximal, complemented
7	C₂×C₈×#88322	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×#135×C₅₃	C₂	1	1	yes	maximal, characteristic
9	Q₈×#88322	C₂²	1	4	yes	—
10	D₄×#88322	—	2	4	no	—
11	D₄×#88322	C₂²	1	4	yes	—
12	#16810676	—	2	2	no	—
13	#16810676	C₂²	1	2	yes	—
14	C₂×C₄×#88322	—	2	2	no	—
15	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
16	C₂×C₄×#88322	C₂²	1	1	yes	characteristic
17	#16810376	C₂²	1	2	yes	—
18	#16810374	C₂²	1	2	yes	—
19	Q₈⋊#88322	C₂²	1	4	yes	—
20	D₄⋊#88322	C₂²	1	4	yes	—
21	C₈×#88322	C₂²	1	2	yes	—
22	C₈×#88322	C₂²	1	2	yes	—
23	C₂×#135×C₅₃	C₄	1	1	yes	characteristic
24	C₄×#41×C₅₃	C₂²	1	2	yes	—
25	C₄×Q₁₆×C₅₃	C₄	1	1	yes	characteristic
26	C₄×Q₁₆×C₅₃	C₂²	1	1	yes	characteristic
27	C₄×SD₁₆×C₅₃	C₄	1	2	yes	—
28	C₄×SD₁₆×C₅₃	C₂²	1	2	yes	—
29	C₄×D₈×C₅₃	C₄	1	1	yes	characteristic
30	C₄×D₈×C₅₃	C₂²	1	1	yes	characteristic
31	C₅₃×#427	C₄	1	1	yes	characteristic
32	C₅₃×#416	C₄	1	2	yes	—
33	C₂×C₄×C₈×C₅₃	C₂²	1	1	yes	characteristic
34	C₂²×#88322	—	2	4	no	—
35	C₂²×#88322	—	2	1	no	—
36	C₂²×#88322	D₄	1	2	yes	—
37	C₄⋊#88322	—	2	4	no	—
38	C₄⋊#88322	C₂³	1	4	yes	—
39	C₂²⋊#88322	—	2	4	no	—
40	C₂²⋊#88322	—	2	2	no	—
41	C₄×#88322	—	2	2	no	—
42	C₄×#88322	—	2	1	no	—
43	C₄×#88322	—	2	4	no	—
44	C₄×#88322	D₄	1	1	yes	characteristic
45	C₄×#88322	D₄	1	1	yes	characteristic
46	C₄×#88322	C₂³	1	2	yes	—
47	C₄×#88322	C₂³	1	2	yes	—
48	C₂×#41×C₅₃	C₂×C₄	1	2	yes	—
49	#135×C₅₃	C₈	1	8	yes	complemented
50	C₂×Q₁₆×C₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×SD₁₆×C₅₃	C₂×C₄	1	2	yes	—
52	C₂×D₈×C₅₃	C₂×C₄	1	1	yes	characteristic
53	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
54	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
55	C₄×Q₈×C₅₃	C₂×C₄	1	2	yes	—
56	C₄×Q₈×C₅₃	C₂³	1	2	yes	—
57	C₄×D₄×C₅₃	—	2	2	no	—
58	C₄×D₄×C₅₃	—	2	2	no	—
59	C₄×D₄×C₅₃	C₂×C₄	1	2	yes	—
60	C₄×D₄×C₅₃	C₂³	1	2	yes	—
61	#117×C₅₃	—	2	2	no	—
62	#117×C₅₃	C₂×C₄	1	2	yes	—
63	C₂×C₄²×C₅₃	—	2	2	no	—
64	C₂×C₄²×C₅₃	C₂³	1	1	yes	characteristic
65	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
66	#107×C₅₃	C₂×C₄	1	1	yes	characteristic
67	#106×C₅₃	C₂×C₄	1	2	yes	—
68	#103×C₅₃	C₂×C₄	1	2	yes	—
69	#103×C₅₃	C₂×C₄	1	2	yes	—
70	#102×C₅₃	C₂×C₄	1	2	yes	—
71	#102×C₅₃	C₂×C₄	1	2	yes	—
72	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
73	C₄×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
74	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
75	C₄×C₈×C₅₃	C₂³	1	1	yes	characteristic
76	#41×C₅₃	—	2	4	no	—
77	#41×C₅₃	C₂×C₈	1	8	yes	complemented
78	C₂×Q₈×C₅₃	C₂²×C₄	1	2	yes	—
79	C₂×D₄×C₅₃	—	2	2	no	—
80	C₂×D₄×C₅₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₅₃	—	2	2	no	—
82	C₂²×C₄×C₅₃	—	2	2	no	—
83	C₂²×C₄×C₅₃	—	2	2	no	—
84	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
85	C₂²×C₄×C₅₃	C₂×D₄	1	1	yes	characteristic
86	C₂²×C₄×C₅₃	#41	1	1	yes	characteristic
87	Q₁₆×C₅₃	C₂×C₈	1	4	yes	complemented
88	SD₁₆×C₅₃	C₂×C₈	1	8	yes	complemented
89	D₈×C₅₃	C₂×C₈	1	4	yes	complemented
90	C₂×C₈×C₅₃	—	2	2	no	—
91	C₂×C₈×C₅₃	C₂×C₈	1	4	yes	complemented
92	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₄⋊C₄×C₅₃	—	2	2	no	—
97	C₄⋊C₄×C₅₃	—	2	2	no	—
98	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
99	C₄⋊C₄×C₅₃	C₂²×C₄	1	2	yes	—
100	C₂²⋊C₄×C₅₃	—	2	2	no	—
101	C₂²⋊C₄×C₅₃	—	2	2	no	—
102	C₂²⋊C₄×C₅₃	—	2	2	no	—
103	C₄²×C₅₃	—	2	2	no	—
104	C₄²×C₅₃	—	2	2	no	—
105	C₄²×C₅₃	—	2	2	no	—
106	C₄²×C₅₃	C₂²×C₄	1	1	yes	characteristic
107	C₄²×C₅₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₅₃	#41	1	1	yes	characteristic
109	C₄²×C₅₃	C₂⁴	1	1	yes	characteristic
110	C₂×#88322	—	2	4	no	—
111	C₂×#88322	—	2	2	no	—
112	C₂×#88322	—	2	4	no	—
113	C₂×#88322	—	2	4	no	—
114	C₂×#88322	—	4	8	no	—
115	C₂×#88322	C₂×D₄	1	2	yes	—
116	C₂×#88322	C₂×D₄	1	2	yes	—
117	C₂³×C₅₃	—	2	2	no	—
118	C₂³×C₅₃	C₄×D₄	1	1	yes	characteristic
119	Q₈×C₅₃	—	2	2	no	—
120	Q₈×C₅₃	C₂²×C₈	1	4	yes	—
121	D₄×C₅₃	—	2	8	no	—
122	D₄×C₅₃	—	2	2	no	—
123	D₄×C₅₃	C₂²×C₈	1	4	yes	—
124	C₂×C₄×C₅₃	—	2	2	no	—
125	C₂×C₄×C₅₃	—	2	2	no	—
126	C₂×C₄×C₅₃	—	2	8	no	—
127	C₂×C₄×C₅₃	—	2	2	no	—
128	C₂×C₄×C₅₃	—	2	2	no	—
129	C₂×C₄×C₅₃	—	2	2	no	—
130	C₂×C₄×C₅₃	—	4	4	no	—
131	C₂×C₄×C₅₃	—	4	4	no	—
132	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
133	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
134	C₂×C₄×C₅₃	C₄×D₄	1	1	yes	characteristic
135	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
136	C₂×C₄×C₅₃	#131	1	2	yes	—
137	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
138	C₂×C₄×C₅₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₅₃	C₂×#41	1	1	yes	characteristic
140	C₈×C₅₃	—	2	1	no	—
141	C₈×C₅₃	—	2	1	no	—
142	C₈×C₅₃	C₂²×C₈	1	2	yes	—
143	C₈×C₅₃	C₂²×C₈	1	2	yes	—
144	#88322	—	2	4	no	—
145	#88322	—	2	2	no	—
146	#88322	—	4	4	no	—
147	#88322	#135	1	4	yes	direct product factor
148	C₈×#135	—	53	1	no	maximal, Sylow
149	C₂²×C₅₃	—	2	1	no	—
150	C₂²×C₅₃	—	2	1	no	—
151	C₂²×C₅₃	—	2	4	no	—
152	C₂²×C₅₃	—	4	2	no	—
153	C₂²×C₅₃	—	4	2	no	—
154	C₂²×C₅₃	C₈×D₄	1	2	yes	—
155	C₂²×C₅₃	C₂×C₄×D₄	1	1	yes	characteristic
156	C₄×C₅₃	—	2	4	no	—
157	C₄×C₅₃	—	2	1	no	—
158	C₄×C₅₃	—	2	1	no	—
159	C₄×C₅₃	—	4	2	no	—
160	C₄×C₅₃	—	4	2	no	—
161	C₄×C₅₃	C₈×D₄	1	2	yes	—
162	C₄×C₅₃	#443	1	2	yes	—
163	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
164	C₄×C₅₃	C₂×#131	1	1	yes	characteristic
165	C₄×C₅₃	C₂×#135	1	2	yes	—
166	C₈×#41	—	53	2	no	—
167	C₄×#135	D₅₃	1	1	yes	characteristic
168	C₈×Q₁₆	—	53	2	no	—
169	C₈×SD₁₆	—	53	4	no	—
170	C₈×D₈	—	53	2	no	—
171	#1564	—	53	1	no	—
172	#1476	—	53	2	no	—
173	C₂×C₈²	—	53	1	no	—
174	C₂×C₅₃	—	2	2	no	—
175	C₂×C₅₃	—	4	4	no	—
176	C₂×C₅₃	C₂×C₈×D₄	1	1	yes	characteristic
177	C₂×C₅₃	C₄×#135	1	1	yes	characteristic
178	C₂×C₅₃	C₂×#443	1	1	yes	characteristic
179	C₂×#135	Dic₅₃	1	1	yes	characteristic
180	C₄×#41	C₂×D₅₃	1	2	yes	—
181	C₈×Q₈	—	53	4	no	—
182	C₄×Q₁₆	Dic₅₃	1	1	yes	characteristic
183	C₄×Q₁₆	C₂×D₅₃	1	1	yes	characteristic
184	C₄×SD₁₆	Dic₅₃	1	2	yes	—
185	C₄×SD₁₆	C₂×D₅₃	1	2	yes	—
186	C₄×D₈	Dic₅₃	1	1	yes	characteristic
187	C₄×D₈	C₂×D₅₃	1	1	yes	characteristic
188	C₈×D₄	—	53	4	no	—
189	C₈×D₄	—	106	4	no	—
190	#431	—	53	2	no	—
191	#431	—	106	2	no	—
192	#427	Dic₅₃	1	1	yes	characteristic
193	#416	Dic₅₃	1	2	yes	—
194	C₂×C₄×C₈	—	53	1	no	—
195	C₂×C₄×C₈	—	53	1	no	—
196	C₂×C₄×C₈	—	106	2	no	—
197	C₂×C₄×C₈	C₂×D₅₃	1	1	yes	characteristic
198	#335	—	53	2	no	—
199	#334	—	53	2	no	—
200	Q₈⋊C₈	—	53	4	no	—
201	D₄⋊C₈	—	53	4	no	—
202	C₈²	—	53	2	no	—
203	C₈²	—	53	2	no	—
204	C₅₃	C₈×#135	1	1	yes	characteristic, complemented, Sylow
205	C₂×#41	C₂×Dic₅₃	1	2	yes	—
206	#135	#88322	1	8	yes	direct product factor
207	C₂×Q₁₆	C₂×Dic₅₃	1	1	yes	characteristic
208	C₂×SD₁₆	C₂×Dic₅₃	1	2	yes	—
209	C₂×D₈	C₂×Dic₅₃	1	1	yes	characteristic
210	C₂²×C₈	—	53	2	no	—
211	C₂²×C₈	—	106	4	no	—
212	C₂²×C₈	—	106	1	no	—
213	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
214	C₂²×C₈	C₂×Dic₅₃	1	1	yes	characteristic
215	C₄×Q₈	C₂×Dic₅₃	1	2	yes	—
216	C₄×Q₈	C₂²×D₅₃	1	2	yes	—
217	C₄×D₄	—	2	2	no	—
218	C₄×D₄	—	2	2	no	—
219	C₄×D₄	C₂×Dic₅₃	1	2	yes	—
220	C₄×D₄	C₂²×D₅₃	1	2	yes	—
221	#117	—	2	2	no	—
222	#117	C₂×Dic₅₃	1	2	yes	—
223	C₂×C₄²	—	2	2	no	—
224	C₂×C₄²	C₂²×D₅₃	1	1	yes	characteristic
225	#107	C₂×Dic₅₃	1	1	yes	characteristic
226	#107	C₂×Dic₅₃	1	1	yes	characteristic
227	#106	C₂×Dic₅₃	1	2	yes	—
228	C₄⋊C₈	—	53	4	no	—
229	C₄⋊C₈	—	106	4	no	—
230	#103	C₂×Dic₅₃	1	2	yes	—
231	#103	C₂×Dic₅₃	1	2	yes	—
232	#102	C₂×Dic₅₃	1	2	yes	—
233	#102	C₂×Dic₅₃	1	2	yes	—
234	C₂²⋊C₈	—	106	4	no	—
235	C₂²⋊C₈	—	106	2	no	—
236	C₄×C₈	—	53	1	no	—
237	C₄×C₈	—	53	1	no	—
238	C₄×C₈	—	53	2	no	—
239	C₄×C₈	—	53	2	no	—
240	C₄×C₈	—	106	4	no	—
241	C₄×C₈	—	106	2	no	—
242	C₄×C₈	—	106	1	no	—
243	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
244	C₄×C₈	C₂×Dic₅₃	1	1	yes	characteristic
245	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
246	C₄×C₈	C₂²×D₅₃	1	1	yes	characteristic
247	#41	—	2	4	no	—
248	#41	C₂×#88322	1	8	yes	complemented
249	C₂×Q₈	C₂²×Dic₅₃	1	2	yes	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	C₂²×Dic₅₃	1	2	yes	—
252	C₂²×C₄	—	2	2	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	D₄×D₅₃	1	1	yes	characteristic
256	C₂²×C₄	#11719947	1	1	yes	characteristic
257	C₂²×C₄	C₂²×Dic₅₃	1	1	yes	characteristic
258	Q₁₆	C₂×#88322	1	4	yes	complemented
259	SD₁₆	C₂×#88322	1	8	yes	complemented
260	D₈	C₂×#88322	1	4	yes	complemented
261	C₂×C₈	—	2	2	no	—
262	C₂×C₈	—	53	2	no	—
263	C₂×C₈	—	53	2	no	—
264	C₂×C₈	—	106	4	no	—
265	C₂×C₈	—	106	4	no	—
266	C₂×C₈	—	106	2	no	—
267	C₂×C₈	—	106	4	no	—
268	C₂×C₈	—	212	8	no	—
269	C₂×C₈	C₂×#88322	1	4	yes	complemented
270	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
271	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
272	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
273	C₂×C₈	C₂²×Dic₅₃	1	1	yes	characteristic
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
277	C₄⋊C₄	C₂²×Dic₅₃	1	2	yes	—
278	C₂²⋊C₄	—	2	2	no	—
279	C₂²⋊C₄	—	2	2	no	—
280	C₂²⋊C₄	—	2	2	no	—
281	C₄²	—	2	2	no	—
282	C₄²	—	2	2	no	—
283	C₄²	—	2	2	no	—
284	C₄²	D₄×D₅₃	1	1	yes	characteristic
285	C₄²	#11719947	1	1	yes	characteristic
286	C₄²	C₂²×Dic₅₃	1	1	yes	characteristic
287	C₄²	C₂³×D₅₃	1	1	yes	characteristic
288	C₂³	—	2	2	no	—
289	C₂³	D₄×Dic₅₃	1	1	yes	characteristic
290	Q₈	—	2	2	no	—
291	Q₈	C₂²×#88322	1	4	yes	—
292	D₄	—	2	8	no	—
293	D₄	—	2	2	no	—
294	D₄	C₂²×#88322	1	4	yes	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	2	8	no	—
298	C₂×C₄	—	2	2	no	—
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	4	no	—
302	C₂×C₄	—	4	4	no	—
303	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
304	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
305	C₂×C₄	D₄×Dic₅₃	1	1	yes	characteristic
306	C₂×C₄	C₂²×#88322	1	2	yes	—
307	C₂×C₄	#13133224	1	2	yes	—
308	C₂×C₄	C₂³×Dic₅₃	1	1	yes	characteristic
309	C₂×C₄	C₂×D₄×D₅₃	1	1	yes	characteristic
310	C₂×C₄	C₂×#11719947	1	1	yes	characteristic
311	C₈	—	2	1	no	—
312	C₈	—	2	1	no	—
313	C₈	—	53	4	no	—
314	C₈	—	106	4	no	—
315	C₈	—	106	2	no	—
316	C₈	—	212	4	no	—
317	C₈	C₂²×#88322	1	2	yes	—
318	C₈	C₂²×#88322	1	2	yes	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	4	no	—
322	C₂²	—	4	2	no	—
323	C₂²	—	4	2	no	—
324	C₂²	D₄×#88322	1	2	yes	—
325	C₂²	C₂×D₄×Dic₅₃	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	4	no	—
329	C₄	—	4	2	no	—
330	C₄	—	4	2	no	—
331	C₄	D₄×#88322	1	2	yes	—
332	C₄	#16810740	1	2	yes	—
333	C₄	C₂³×#88322	1	1	yes	characteristic
334	C₄	C₂×#13133224	1	1	yes	characteristic
335	C₄	#135×D₅₃	1	2	yes	—
336	C₂	—	2	2	no	—
337	C₂	—	4	4	no	—
338	C₂	C₂×D₄×#88322	1	1	yes	characteristic
339	C₂	#135×Dic₅₃	1	1	yes	characteristic
340	C₂	C₂×#16810740	1	1	yes	characteristic

	subgroup	G/H
center	C₄²	D₄×D₅₃
commutator	C₄×C₅₃	C₂³×C₈
Frattini	C₄²	C₂³×D₅₃
Fitting	C₄×#135×C₅₃	C₂
socle	C₂²×C₅₃	C₂×C₄×D₄
prime radical	C₂×#135×C₅₃	C₄
cosocle	C₄²×C₅₃	C₂⁴
hypercenter	C₄×#135	D₅₃
hypocenter	C₅₃	C₈×#135
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₄×#135

Complements

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

N	H ≅ G/N	type
1. #41×#88322	C₂	semidirect
2. Q₁₆×#88322	C₂	semidirect
3. SD₁₆×#88322	C₂	semidirect
4. D₈×#88322	C₂	semidirect
5. #20273463	C₂	semidirect
6. #20273097	C₂	semidirect
7. C₂×C₈×#88322	C₂	semidirect
49. #135×C₅₃	C₈	semidirect
77. #41×C₅₃	C₂×C₈	semidirect
87. Q₁₆×C₅₃	C₂×C₈	semidirect
88. SD₁₆×C₅₃	C₂×C₈	semidirect
89. D₈×C₅₃	C₂×C₈	semidirect
91. C₂×C₈×C₅₃	C₂×C₈	semidirect
147. #88322	#135	direct (N × H)
204. C₅₃	C₈×#135	semidirect
206. #135	#88322	direct (N × H)
248. #41	C₂×#88322	semidirect
258. Q₁₆	C₂×#88322	semidirect
259. SD₁₆	C₂×#88322	semidirect
260. D₈	C₂×#88322	semidirect
269. C₂×C₈	C₂×#88322	semidirect

Maximal subgroups

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

subgroup	index
#41×#88322	2
Q₁₆×#88322	2
SD₁₆×#88322	2
D₈×#88322	2
#20273463	2
#20273097	2
C₂×C₈×#88322	2
C₄×#135×C₅₃	2
C₈×#135	53