Subgroups — C₄×#16811633 #28318365

Order 13568.

1182 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811633	C₂	1	2	yes	maximal
2	C₄×#13133296	C₂	1	2	yes	maximal
3	C₄×#13133286	C₂	1	4	yes	maximal, complemented
4	C₄×#13133285	C₂	1	4	yes	maximal, complemented
5	#20290401	C₂	1	2	yes	maximal
6	C₂×C₄×C₈×D₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂×#13133296	C₂²	1	4	yes	—
8	#16811633	C₄	1	16	yes	direct product factor
9	C₂×#13133286	C₂²	1	4	yes	—
10	C₂×#13133285	C₂²	1	4	yes	—
11	C₄×C₄⋊#6157	C₂²	1	8	yes	—
12	C₄×C₂²⋊#6157	—	2	4	no	—
13	#16811559	—	2	2	no	—
14	C₄²×#6157	—	2	2	no	—
15	#16811497	C₂²	1	8	yes	—
16	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
18	C₂×C₄²×D₅₃	C₂²	1	1	yes	characteristic
19	C₂×C₄×#88322	C₄	1	1	yes	characteristic, complemented
20	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic, complemented
21	C₂×C₈×Dic₅₃	C₄	1	1	yes	characteristic, complemented
22	C₄×C₈×D₅₃	C₄	1	2	yes	complemented
23	C₄×C₈×D₅₃	C₄	1	2	yes	complemented
24	C₄×C₈×D₅₃	C₂²	1	2	yes	—
25	C₄×C₈×D₅₃	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₅₃	C₄	1	1	yes	characteristic, complemented
27	#13133296	—	2	8	no	—
28	#13133296	C₂×C₄	1	16	yes	—
29	C₂×C₄⋊#6157	C₂³	1	8	yes	—
30	C₂×C₂²⋊#6157	—	2	4	no	—
31	C₂×C₄×#6157	—	2	4	no	—
32	C₂×C₄×#6157	—	2	8	no	—
33	#13133286	C₂×C₄	1	16	yes	complemented
34	#13133285	C₂×C₄	1	16	yes	complemented
35	(C₂×#6157)⋊C₄	—	2	8	no	—
36	C₂²×C₄×D₅₃	D₄	1	2	yes	—
37	C₂²×C₄×D₅₃	C₂³	1	1	yes	characteristic
38	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
39	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
40	C₂×C₈×D₅₃	—	2	4	no	—
41	C₂×C₈×D₅₃	C₂×C₄	1	2	yes	—
42	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×D₅₃	C₂×C₄	1	8	yes	—
45	C₂×C₈×D₅₃	C₂³	1	2	yes	—
46	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
47	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
48	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
49	C₂×C₄×Dic₅₃	Q₈	1	2	yes	—
50	C₄²×D₅₃	C₂×C₄	1	2	yes	—
51	C₄²×D₅₃	D₄	1	1	yes	characteristic
52	C₄²×D₅₃	D₄	1	1	yes	characteristic
53	C₄²×D₅₃	Q₈	1	1	yes	characteristic
54	C₄²×D₅₃	Q₈	1	1	yes	characteristic
55	C₄²×D₅₃	C₂³	1	2	yes	—
56	C₄×#88322	C₂×C₄	1	2	yes	—
57	C₄×#88322	C₂×C₄	1	2	yes	—
58	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
59	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
60	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
61	C₈×Dic₅₃	C₂×C₄	1	2	yes	—
62	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
64	C₂×C₄²×C₅₃	C₂×C₄	1	1	yes	characteristic
65	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₅₃	C₂×C₄	1	2	yes	—
67	C₂³×D₅₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×#6157	—	2	8	no	—
69	C₂²×C₄×C₅₃	C₄⋊C₄	1	2	yes	—
70	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
71	C₂²×Dic₅₃	C₄⋊C₄	1	2	yes	—
72	C₂²×Dic₅₃	C₂×Q₈	1	1	yes	characteristic
73	C₂×C₄×D₅₃	—	2	8	no	—
74	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
75	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
76	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂×C₄×D₅₃	C₂²×C₄	1	4	yes	—
78	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
79	C₂×C₄×D₅₃	C₂×D₄	1	1	yes	characteristic
80	C₂×C₄×D₅₃	C₂×D₄	1	2	yes	—
81	C₂×C₄×D₅₃	C₂×Q₈	1	1	yes	characteristic
82	C₂×C₄×D₅₃	#41	1	4	yes	—
83	C₂×C₄×D₅₃	C₂⁴	1	1	yes	characteristic
84	C₂²⋊#6157	—	2	16	no	—
85	C₄⋊#6157	—	2	8	no	—
86	C₄⋊#6157	C₂²×C₄	1	16	yes	—
87	C₄×#6157	—	2	16	no	—
88	C₄×#6157	—	4	16	no	—
89	C₂×C₈×C₅₃	—	2	2	no	—
90	C₂×C₈×C₅₃	C₄²	1	4	yes	complemented
91	C₂×C₈×C₅₃	C₂²×C₄	1	2	yes	—
92	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
93	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
94	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
95	C₄²×C₅₃	C₄⋊C₄	1	1	yes	characteristic
96	C₄²×C₅₃	C₂²×C₄	1	2	yes	—
97	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
98	C₄×Dic₅₃	C₄⋊C₄	1	2	yes	—
99	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
100	C₄×Dic₅₃	C₄⋊C₄	1	1	yes	characteristic
101	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
102	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
103	C₄×Dic₅₃	C₂×Q₈	1	2	yes	—
104	C₂×#88322	—	2	2	no	—
105	C₂×#88322	C₄²	1	4	yes	complemented
106	C₂×#88322	C₂²×C₄	1	2	yes	—
107	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
108	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
109	C₈×D₅₃	—	2	4	no	—
110	C₈×D₅₃	—	2	4	no	—
111	C₈×D₅₃	C₄²	1	4	yes	complemented
112	C₈×D₅₃	C₄²	1	4	yes	complemented
113	C₈×D₅₃	C₂²×C₄	1	4	yes	—
114	C₈×D₅₃	C₂²×C₄	1	4	yes	—
115	C₂³×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
116	C₂²×D₅₃	—	2	4	no	—
117	C₂²×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂²×D₅₃	C₄×D₄	1	4	yes	—
119	C₂²×D₅₃	C₂²×D₄	1	1	yes	characteristic
120	C₂×#6157	—	2	16	no	—
121	C₂×#6157	—	4	16	no	—
122	C₂×C₄×C₅₃	—	2	4	no	—
123	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
124	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
125	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₂×C₄×C₅₃	C₂×C₄⋊C₄	1	2	yes	—
127	C₂×C₄×C₅₃	#117	1	2	yes	—
128	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
129	C₂×Dic₅₃	—	2	4	no	—
130	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
131	C₂×Dic₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
132	C₂×Dic₅₃	C₂×C₄⋊C₄	1	2	yes	—
133	C₂×Dic₅₃	C₄×Q₈	1	4	yes	—
134	C₂×Dic₅₃	C₂²×Q₈	1	1	yes	characteristic
135	C₄×D₅₃	—	2	4	no	—
136	C₄×D₅₃	—	2	4	no	—
137	C₄×D₅₃	—	2	4	no	—
138	C₄×D₅₃	C₂×C₄²	1	2	yes	—
139	C₄×D₅₃	#117	1	2	yes	—
140	C₄×D₅₃	C₄×D₄	1	4	yes	—
141	C₄×D₅₃	C₄×Q₈	1	4	yes	—
142	C₄×D₅₃	#135	1	8	yes	—
143	C₄×D₅₃	C₂³×C₄	1	2	yes	—
144	C₄×D₅₃	C₂×#41	1	2	yes	—
145	C₈×C₅₃	—	2	2	no	—
146	C₈×C₅₃	C₂×C₄²	1	2	yes	—
147	C₈×C₅₃	C₂×C₄²	1	2	yes	—
148	#88322	—	2	2	no	—
149	#88322	C₂×C₄²	1	2	yes	—
150	#88322	C₂×C₄²	1	2	yes	—
151	C₄×#427	—	53	1	no	maximal, Sylow
152	C₂²×C₅₃	—	2	2	no	—
153	C₂²×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
154	C₂²×C₅₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
155	C₂×D₅₃	—	2	2	no	—
156	C₂×D₅₃	—	2	8	no	—
157	C₂×D₅₃	C₄×C₄⋊C₄	1	2	yes	—
158	C₂×D₅₃	C₂×C₄×D₄	1	2	yes	—
159	C₂×D₅₃	C₂×#135	1	4	yes	—
160	#6157	—	4	16	no	—
161	C₄×C₅₃	—	2	2	no	—
162	C₄×C₅₃	C₄×C₄⋊C₄	1	2	yes	—
163	C₄×C₅₃	#427	1	4	yes	complemented
164	C₄×C₅₃	C₂²×C₄²	1	1	yes	characteristic
165	C₄×C₅₃	C₂×#117	1	1	yes	characteristic
166	Dic₅₃	—	2	2	no	—
167	Dic₅₃	C₄×C₄⋊C₄	1	2	yes	—
168	Dic₅₃	#427	1	4	yes	complemented
169	Dic₅₃	C₂×C₄×Q₈	1	2	yes	—
170	C₂×#427	—	53	2	no	—
171	C₂²×C₄×C₈	—	53	1	no	—
172	C₄×#117	—	53	2	no	—
173	C₄×#107	—	53	4	no	—
174	C₄×#106	—	53	4	no	—
175	#1735	—	53	2	no	—
176	C₂×C₅₃	—	2	2	no	—
177	C₂×C₅₃	C₂×C₄×C₄⋊C₄	1	1	yes	characteristic
178	C₂×C₅₃	C₂×#427	1	2	yes	—
179	D₅₃	—	2	2	no	—
180	D₅₃	C₄×#135	1	4	yes	—
181	C₂³×C₈	—	53	1	no	—
182	C₂³×C₈	—	53	1	no	—
183	C₂×#117	—	53	4	no	—
184	C₂²×C₄²	—	53	1	no	—
185	#427	—	53	16	no	—
186	C₂×#107	—	53	4	no	—
187	C₂×#106	—	53	4	no	—
188	C₂×C₄×C₈	—	53	1	no	—
189	C₂×C₄×C₈	—	53	1	no	—
190	C₂×C₄×C₈	—	53	1	no	—
191	C₂×C₄×C₈	—	53	2	no	—
192	C₂×C₄×C₈	—	53	2	no	—
193	C₂×C₄×C₈	—	53	2	no	—
194	C₂×C₄×C₈	—	53	2	no	—
195	C₂×C₄×C₈	#6157	1	1	yes	characteristic, complemented
196	C₄×C₄⋊C₄	—	53	8	no	—
197	C₄×C₂²⋊C₄	—	106	4	no	—
198	#376	—	106	2	no	—
199	C₄³	—	106	2	no	—
200	#340	—	53	8	no	—
201	C₅₃	C₄×#427	1	1	yes	characteristic, complemented, Sylow
202	C₂³×C₄	—	53	1	no	—
203	C₂³×C₄	—	53	2	no	—
204	C₂²×C₈	—	53	1	no	—
205	C₂²×C₈	—	53	8	no	—
206	C₂²×C₈	—	53	1	no	—
207	C₂²×C₈	—	53	1	no	—
208	C₂²×C₈	—	53	1	no	—
209	C₂²×C₈	—	53	1	no	—
210	C₂²×C₈	—	53	1	no	—
211	C₂²×C₈	—	53	2	no	—
212	C₂²×C₈	—	53	2	no	—
213	C₂²×C₈	—	106	4	no	—
214	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
215	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
216	#117	—	53	16	no	—
217	#117	—	106	8	no	—
218	C₂×C₄⋊C₄	—	53	8	no	—
219	C₂×C₂²⋊C₄	—	106	4	no	—
220	C₂×C₄²	—	53	2	no	—
221	C₂×C₄²	—	53	1	no	—
222	C₂×C₄²	—	53	1	no	—
223	C₂×C₄²	—	53	1	no	—
224	C₂×C₄²	—	53	1	no	—
225	C₂×C₄²	—	53	1	no	—
226	C₂×C₄²	—	53	2	no	—
227	C₂×C₄²	—	53	2	no	—
228	C₂×C₄²	—	106	4	no	—
229	C₂×C₄²	—	106	8	no	—
230	C₂×C₄²	C₂×#6157	1	1	yes	characteristic
231	#107	—	53	16	no	—
232	#106	—	53	16	no	—
233	C₄×C₈	—	53	2	no	—
234	C₄×C₈	—	53	2	no	—
235	C₄×C₈	—	53	2	no	—
236	C₄×C₈	—	53	2	no	—
237	C₄×C₈	—	53	2	no	—
238	C₄×C₈	—	53	2	no	—
239	C₄×C₈	C₂×#6157	1	2	yes	—
240	C₄×C₈	C₂×#6157	1	2	yes	—
241	#95	—	106	8	no	—
242	C₂⁴	—	53	1	no	—
243	C₂²×C₄	—	53	1	no	—
244	C₂²×C₄	—	53	4	no	—
245	C₂²×C₄	—	53	1	no	—
246	C₂²×C₄	—	53	2	no	—
247	C₂²×C₄	—	53	1	no	—
248	C₂²×C₄	—	53	4	no	—
249	C₂²×C₄	—	53	1	no	—
250	C₂²×C₄	—	53	1	no	—
251	C₂²×C₄	—	53	2	no	—
252	C₂²×C₄	—	53	2	no	—
253	C₂²×C₄	—	53	2	no	—
254	C₂²×C₄	—	53	2	no	—
255	C₂²×C₄	—	106	8	no	—
256	C₂²×C₄	—	106	8	no	—
257	C₂²×C₄	C₄⋊#6157	1	2	yes	—
258	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
259	C₂×C₈	—	2	2	no	—
260	C₂×C₈	—	53	4	no	—
261	C₂×C₈	—	53	1	no	—
262	C₂×C₈	—	53	1	no	—
263	C₂×C₈	—	53	4	no	—
264	C₂×C₈	—	53	4	no	—
265	C₂×C₈	—	53	4	no	—
266	C₂×C₈	—	53	4	no	—
267	C₂×C₈	—	53	2	no	—
268	C₂×C₈	—	106	4	no	—
269	C₂×C₈	—	106	4	no	—
270	C₂×C₈	—	106	2	no	—
271	C₂×C₈	C₄×#6157	1	4	yes	complemented
272	C₂×C₈	C₂²×#6157	1	2	yes	—
273	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
274	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
275	C₄⋊C₄	—	53	16	no	—
276	C₄⋊C₄	—	106	8	no	—
277	C₂²⋊C₄	—	106	16	no	—
278	C₄²	—	53	1	no	—
279	C₄²	—	53	2	no	—
280	C₄²	—	53	2	no	—
281	C₄²	—	53	1	no	—
282	C₄²	—	53	2	no	—
283	C₄²	—	53	2	no	—
284	C₄²	—	53	2	no	—
285	C₄²	—	106	16	no	—
286	C₄²	—	212	16	no	—
287	C₄²	C₄⋊#6157	1	1	yes	characteristic
288	C₄²	C₄⋊#6157	1	1	yes	characteristic
289	C₄²	C₂²×#6157	1	2	yes	—
290	C₂³	—	53	1	no	—
291	C₂³	—	53	4	no	—
292	C₂³	—	53	1	no	—
293	C₂³	—	106	4	no	—
294	C₂³	C₂×C₄⋊#6157	1	1	yes	characteristic
295	C₂×C₄	—	2	4	no	—
296	C₂×C₄	—	53	4	no	—
297	C₂×C₄	—	53	2	no	—
298	C₂×C₄	—	53	1	no	—
299	C₂×C₄	—	53	1	no	—
300	C₂×C₄	—	53	4	no	—
301	C₂×C₄	—	53	4	no	—
302	C₂×C₄	—	53	8	no	—
303	C₂×C₄	—	53	2	no	—
304	C₂×C₄	—	53	2	no	—
305	C₂×C₄	—	53	2	no	—
306	C₂×C₄	—	53	2	no	—
307	C₂×C₄	—	53	2	no	—
308	C₂×C₄	—	106	4	no	—
309	C₂×C₄	—	106	4	no	—
310	C₂×C₄	—	106	4	no	—
311	C₂×C₄	—	106	4	no	—
312	C₂×C₄	—	106	16	no	—
313	C₂×C₄	—	212	16	no	—
314	C₂×C₄	C₂×C₄×#6157	1	2	yes	—
315	C₂×C₄	C₂×C₄⋊#6157	1	2	yes	—
316	C₂×C₄	C₂×C₄⋊#6157	1	1	yes	characteristic
317	C₂×C₄	C₂×C₄⋊#6157	1	2	yes	—
318	C₂×C₄	#13133296	1	2	yes	—
319	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
320	C₈	—	2	2	no	—
321	C₈	—	53	2	no	—
322	C₈	—	53	2	no	—
323	C₈	—	106	2	no	—
324	C₈	C₂×C₄×#6157	1	2	yes	—
325	C₈	C₂×C₄×#6157	1	2	yes	—
326	C₂²	—	2	2	no	—
327	C₂²	—	53	4	no	—
328	C₂²	—	53	2	no	—
329	C₂²	—	53	2	no	—
330	C₂²	—	106	8	no	—
331	C₂²	—	106	2	no	—
332	C₂²	C₄×C₄⋊#6157	1	2	yes	—
333	C₂²	C₂²×C₄⋊#6157	1	1	yes	characteristic
334	C₄	—	2	2	no	—
335	C₄	—	53	4	no	—
336	C₄	—	53	2	no	—
337	C₄	—	53	2	no	—
338	C₄	—	106	2	no	—
339	C₄	—	212	16	no	—
340	C₄	C₄×C₄⋊#6157	1	2	yes	—
341	C₄	#16811633	1	4	yes	direct product factor
342	C₄	C₂²×C₄×#6157	1	1	yes	characteristic
343	C₄	C₂×#13133296	1	1	yes	characteristic
344	C₂	—	2	2	no	—
345	C₂	—	53	4	no	—
346	C₂	—	106	2	no	—
347	C₂	C₂×C₄×C₄⋊#6157	1	1	yes	characteristic
348	C₂	C₂×#16811633	1	2	yes	—

	subgroup	G/H
center	C₄²	C₄⋊#6157
commutator	C₄×C₅₃	C₂²×C₄²
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂×C₄×C₈×C₅₃	C₄
socle	C₂²×C₅₃	C₂²×C₄⋊C₄
prime radical	C₂³×D₅₃	C₂×D₄
cosocle	C₂×C₄×D₅₃	C₂⁴
hypercenter	C₂×C₄×C₈	#6157
hypocenter	C₅₃	C₄×#427
LCS	G ▷ C₄×C₅₃ ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₄² ◁ C₂×C₄² ◁ C₂×C₄×C₈

Complements

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

N	H ≅ G/N	type
3. C₄×#13133286	C₂	semidirect
4. C₄×#13133285	C₂	semidirect
8. #16811633	C₄	direct (N × H)
19. C₂×C₄×#88322	C₄	semidirect
20. C₂×C₈×Dic₅₃	C₄	semidirect
21. C₂×C₈×Dic₅₃	C₄	semidirect
22. C₄×C₈×D₅₃	C₄	semidirect
23. C₄×C₈×D₅₃	C₄	semidirect
26. C₂×C₄×C₈×C₅₃	C₄	semidirect
33. #13133286	C₂×C₄	semidirect
34. #13133285	C₂×C₄	semidirect
90. C₂×C₈×C₅₃	C₄²	semidirect
105. C₂×#88322	C₄²	semidirect
111. C₈×D₅₃	C₄²	semidirect
112. C₈×D₅₃	C₄²	semidirect
163. C₄×C₅₃	#427	semidirect
168. Dic₅₃	#427	semidirect
195. C₂×C₄×C₈	#6157	semidirect
201. C₅₃	C₄×#427	semidirect
271. C₂×C₈	C₄×#6157	semidirect
341. C₄	#16811633	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×#16811633	2
C₄×#13133296	2
C₄×#13133286	2
C₄×#13133285	2
#20290401	2
C₂×C₄×C₈×D₅₃	2
C₄×#427	53

Subgroups — C4×#16811633 #28318365

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₄×#16811633 #28318365