Subgroups — C₂×(C₂×#6157)⋊C₁₆ #28310212

Order 13568.

726 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×D₅₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₈×#6157	C₂	1	1	yes	maximal, characteristic
3	(C₂×#6157)⋊C₁₆	C₂	1	4	yes	maximal, direct product factor
4	C₂²×C₁₆×D₅₃	C₂	1	1	yes	maximal, characteristic
5	C₂²×D₅₃⋊C₈	C₄	1	1	yes	characteristic
6	C₂²×C₄×#6157	C₄	1	1	yes	characteristic
7	C₂×D₅₃⋊C₁₆	—	2	2	no	—
8	C₂×D₅₃⋊C₁₆	—	2	2	no	—
9	C₂×D₅₃⋊C₁₆	C₄	1	2	yes	complemented
10	C₂×D₅₃⋊C₁₆	C₂²	1	2	yes	—
11	C₂²×#11719910	C₄	1	2	yes	complemented
12	C₂×C₈×#6157	—	2	2	no	—
13	C₂×C₈×#6157	—	2	2	no	—
14	C₂×C₈×#6157	C₄	1	2	yes	—
15	C₂×C₈×#6157	C₂²	1	2	yes	—
16	C₂²×C₈×D₅₃	C₂²	1	1	yes	characteristic
17	C₂²×#11719908	C₄	1	1	yes	characteristic, complemented
18	C₂×C₁₆×D₅₃	—	2	2	no	—
19	C₂×C₁₆×D₅₃	—	2	2	no	—
20	C₂×C₁₆×D₅₃	C₄	1	2	yes	complemented
21	C₂×C₁₆×D₅₃	C₂²	1	2	yes	—
22	C₂²×C₁₆×C₅₃	C₄	1	1	yes	characteristic, complemented
23	C₂³×#6157	C₈	1	2	yes	—
24	C₂×D₅₃⋊C₈	—	2	2	no	—
25	C₂×D₅₃⋊C₈	—	2	2	no	—
26	C₂×D₅₃⋊C₈	C₈	1	2	yes	—
27	C₂×D₅₃⋊C₈	C₂×C₄	1	2	yes	—
28	C₂²×#88324	C₈	1	2	yes	—
29	C₂×C₄×#6157	—	2	2	no	—
30	C₂×C₄×#6157	—	2	2	no	—
31	C₂×C₄×#6157	C₈	1	2	yes	—
32	C₂×C₄×#6157	C₂×C₄	1	2	yes	—
33	D₅₃⋊C₁₆	—	2	4	no	—
34	D₅₃⋊C₁₆	—	2	2	no	—
35	D₅₃⋊C₁₆	—	2	2	no	—
36	C₂×#11719910	—	2	4	no	—
37	C₂×#11719910	C₂×C₄	1	4	yes	complemented
38	C₈×#6157	—	2	4	no	—
39	C₈×#6157	—	2	2	no	—
40	C₈×#6157	—	2	2	no	—
41	C₂²×C₄×D₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂²×#88322	C₂×C₄	1	1	yes	characteristic
43	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
44	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
45	C₂×C₈×D₅₃	C₂×C₄	1	1	yes	characteristic
46	C₂×C₈×D₅₃	D₄	1	2	yes	—
47	C₂×C₈×D₅₃	D₄	1	2	yes	—
48	C₂×C₈×D₅₃	D₄	1	2	yes	—
49	C₂×C₈×D₅₃	Q₈	1	2	yes	—
50	C₂×C₈×D₅₃	C₂³	1	1	yes	characteristic
51	C₂×#11719908	—	2	2	no	—
52	C₂×#11719908	C₂×C₄	1	2	yes	complemented
53	C₁₆×D₅₃	—	2	4	no	—
54	C₁₆×D₅₃	—	2	2	no	—
55	C₁₆×D₅₃	—	2	2	no	—
56	C₂²×C₈×C₅₃	C₂×C₄	1	1	yes	characteristic
57	C₂×C₁₆×C₅₃	—	2	2	no	—
58	C₂×C₁₆×C₅₃	C₂×C₄	1	2	yes	complemented
59	C₂³×D₅₃	C₂×C₈	1	1	yes	characteristic
60	C₂²×#6157	—	2	4	no	—
61	C₂²×#6157	—	2	4	no	—
62	C₂²×#6157	C₁₆	1	8	yes	complemented
63	C₂²×#6157	C₂×C₈	1	4	yes	—
64	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
65	C₂²×Dic₅₃	C₂×C₈	1	1	yes	characteristic
66	C₂×C₄×D₅₃	C₄²	1	1	yes	characteristic
67	C₂×C₄×D₅₃	C₂²⋊C₄	1	2	yes	—
68	C₂×C₄×D₅₃	C₄⋊C₄	1	2	yes	—
69	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
70	C₂×C₄×D₅₃	C₂×C₈	1	1	yes	characteristic
71	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
72	C₂×C₄×D₅₃	M₁₆	1	2	yes	—
73	C₂×C₄×D₅₃	C₂²×C₄	1	1	yes	characteristic
74	C₂×#88324	—	2	4	no	—
75	C₂×#88324	C₂×C₈	1	4	yes	—
76	C₄×#6157	—	2	4	no	—
77	C₄×#6157	—	2	2	no	—
78	C₄×#6157	—	2	2	no	—
79	D₅₃⋊C₈	—	2	4	no	—
80	D₅₃⋊C₈	—	2	2	no	—
81	D₅₃⋊C₈	—	2	2	no	—
82	C₂×C₈×C₅₃	C₄²	1	1	yes	characteristic
83	C₂×C₈×C₅₃	C₂²⋊C₄	1	2	yes	—
84	C₂×C₈×C₅₃	C₄⋊C₄	1	2	yes	—
85	C₂×C₈×C₅₃	C₂²×C₄	1	1	yes	characteristic
86	C₂×#88322	C₄²	1	1	yes	characteristic
87	C₂×#88322	C₂²⋊C₄	1	2	yes	—
88	C₂×#88322	C₄⋊C₄	1	2	yes	—
89	C₂×#88322	C₂²×C₄	1	1	yes	characteristic
90	C₈×D₅₃	C₂²⋊C₄	1	2	yes	—
91	C₈×D₅₃	C₂²⋊C₄	1	2	yes	—
92	C₈×D₅₃	C₂²⋊C₄	1	2	yes	—
93	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
94	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
95	C₈×D₅₃	C₄⋊C₄	1	2	yes	—
96	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
97	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
98	C₈×D₅₃	C₂×D₄	1	1	yes	characteristic
99	C₈×D₅₃	C₂×Q₈	1	1	yes	characteristic
100	#11719910	—	2	4	no	—
101	C₁₆×C₅₃	—	2	2	no	—
102	#11719908	—	2	2	no	—
103	C₂³×C₅₃	C₄×C₈	1	1	yes	characteristic
104	C₂²×D₅₃	C₄×C₈	1	1	yes	characteristic
105	C₂²×D₅₃	C₂²⋊C₈	1	2	yes	—
106	C₂²×D₅₃	C₄⋊C₈	1	2	yes	—
107	C₂²×D₅₃	C₂×C₁₆	1	4	yes	—
108	C₂²×D₅₃	M₃₂	1	4	yes	—
109	C₂²×D₅₃	C₂²×C₈	1	1	yes	characteristic
110	C₂×#6157	—	2	16	no	—
111	C₂×#6157	—	2	4	no	—
112	C₂×#6157	—	2	4	no	—
113	C₂×#6157	C₂×C₁₆	1	8	yes	complemented
114	C₂×C₄×C₅₃	#95	1	2	yes	—
115	C₂×C₄×C₅₃	C₄×C₈	1	1	yes	characteristic
116	C₂×C₄×C₅₃	#97	1	2	yes	—
117	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
118	C₂×Dic₅₃	C₄×C₈	1	1	yes	characteristic
119	C₂×Dic₅₃	C₂²⋊C₈	1	2	yes	—
120	C₂×Dic₅₃	C₄⋊C₈	1	2	yes	—
121	C₂×Dic₅₃	C₂²×C₈	1	1	yes	characteristic
122	C₄×D₅₃	#95	1	2	yes	—
123	C₄×D₅₃	#97	1	2	yes	—
124	C₄×D₅₃	C₂²⋊C₈	1	2	yes	—
125	C₄×D₅₃	C₂²⋊C₈	1	2	yes	—
126	C₄×D₅₃	C₄⋊C₈	1	2	yes	—
127	C₄×D₅₃	C₄⋊C₈	1	2	yes	—
128	C₄×D₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
129	C₄×D₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
130	C₄×D₅₃	C₂×M₁₆	1	1	yes	characteristic
131	C₄×D₅₃	C₂×M₁₆	1	1	yes	characteristic
132	#88324	—	2	4	no	—
133	C₈×C₅₃	#95	1	2	yes	—
134	C₈×C₅₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
135	C₈×C₅₃	C₂×C₄⋊C₄	1	1	yes	characteristic
136	#88322	#95	1	2	yes	—
137	#88322	C₂×C₂²⋊C₄	1	1	yes	characteristic
138	#88322	C₂×C₄⋊C₄	1	1	yes	characteristic
139	C₂×#1374	—	53	1	no	maximal, Sylow
140	C₂²×C₅₃	#336	1	2	yes	—
141	C₂²×C₅₃	C₄×C₁₆	1	2	yes	—
142	C₂²×C₅₃	#346	1	2	yes	—
143	C₂²×C₅₃	C₂×C₄×C₈	1	1	yes	characteristic
144	C₂×D₅₃	#336	1	2	yes	—
145	C₂×D₅₃	C₄×C₁₆	1	2	yes	—
146	C₂×D₅₃	#346	1	2	yes	—
147	C₂×D₅₃	C₂²⋊C₁₆	1	8	yes	—
148	C₂×D₅₃	C₄⋊C₁₆	1	8	yes	—
149	C₂×D₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
150	C₂×D₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
151	C₂×D₅₃	C₂²×C₁₆	1	2	yes	—
152	C₂×D₅₃	C₂×M₃₂	1	2	yes	—
153	#6157	—	2	8	no	—
154	C₄×C₅₃	#336	1	2	yes	—
155	C₄×C₅₃	C₂×#95	1	1	yes	characteristic
156	C₄×C₅₃	C₂×#97	1	1	yes	characteristic
157	Dic₅₃	#336	1	2	yes	—
158	Dic₅₃	C₂×C₂²⋊C₈	1	1	yes	characteristic
159	Dic₅₃	C₂×C₄⋊C₈	1	1	yes	characteristic
160	C₂³×C₁₆	—	53	1	no	—
161	C₂³×C₁₆	—	53	1	no	—
162	C₂²×C₄×C₈	—	53	1	no	—
163	#1374	—	53	4	no	—
164	C₂×C₅₃	#1374	1	4	yes	complemented
165	C₂×C₅₃	C₂×#336	1	1	yes	characteristic
166	C₂×C₅₃	C₂×C₄×C₁₆	1	1	yes	characteristic
167	C₂×C₅₃	C₂×#346	1	1	yes	characteristic
168	D₅₃	#1374	1	4	yes	complemented
169	D₅₃	C₂×C₂²⋊C₁₆	1	2	yes	—
170	D₅₃	C₂×C₄⋊C₁₆	1	2	yes	—
171	C₂³×C₈	—	53	1	no	—
172	C₂³×C₈	—	53	1	no	—
173	C₂²×C₄²	—	53	1	no	—
174	C₂²×C₁₆	—	53	2	no	—
175	C₂²×C₁₆	—	53	2	no	—
176	C₂²×C₁₆	—	53	2	no	—
177	C₂²×C₁₆	—	53	1	no	—
178	C₂²×C₁₆	—	53	2	no	—
179	C₂²×C₁₆	—	53	2	no	—
180	C₂²×C₁₆	—	106	2	no	—
181	C₂²×C₁₆	—	106	2	no	—
182	C₂²×C₁₆	—	106	2	no	—
183	C₂²×C₁₆	—	106	2	no	—
184	C₂²×C₁₆	#6157	1	1	yes	characteristic, complemented
185	C₂×C₄×C₈	—	53	2	no	—
186	C₂×C₄×C₈	—	53	2	no	—
187	C₂×C₄×C₈	—	106	2	no	—
188	C₂×C₄×C₈	—	106	2	no	—
189	C₅₃	C₂×#1374	1	1	yes	characteristic, complemented, Sylow
190	C₂³×C₄	—	53	1	no	—
191	C₂³×C₄	—	53	2	no	—
192	C₂²×C₈	—	53	1	no	—
193	C₂²×C₈	—	53	1	no	—
194	C₂²×C₈	—	53	1	no	—
195	C₂²×C₈	—	53	2	no	—
196	C₂²×C₈	—	53	2	no	—
197	C₂²×C₈	—	53	2	no	—
198	C₂²×C₈	—	53	2	no	—
199	C₂²×C₈	—	53	1	no	—
200	C₂²×C₈	—	53	1	no	—
201	C₂²×C₈	—	53	2	no	—
202	C₂²×C₈	—	53	2	no	—
203	C₂²×C₈	—	53	2	no	—
204	C₂²×C₈	—	106	2	no	—
205	C₂²×C₈	—	106	2	no	—
206	C₂²×C₈	C₂×#6157	1	1	yes	characteristic
207	C₂×C₄²	—	53	2	no	—
208	C₂×C₄²	—	53	2	no	—
209	C₂×C₄²	—	106	2	no	—
210	C₂×C₄²	—	106	2	no	—
211	C₂×C₁₆	—	2	2	no	—
212	C₂×C₁₆	—	53	4	no	—
213	C₂×C₁₆	—	53	2	no	—
214	C₂×C₁₆	—	106	2	no	—
215	C₂×C₁₆	—	106	2	no	—
216	C₂×C₁₆	—	106	4	no	—
217	C₂×C₁₆	—	106	4	no	—
218	C₂×C₁₆	—	106	4	no	—
219	C₂×C₁₆	—	106	2	no	—
220	C₂×C₁₆	—	106	2	no	—
221	C₂×C₁₆	—	106	2	no	—
222	C₂×C₁₆	C₂×#6157	1	2	yes	complemented
223	C₄×C₈	—	106	4	no	—
224	C₄×C₈	—	106	2	no	—
225	C₄×C₈	—	106	2	no	—
226	C₂⁴	—	53	1	no	—
227	C₂²×C₄	—	53	1	no	—
228	C₂²×C₄	—	53	2	no	—
229	C₂²×C₄	—	53	2	no	—
230	C₂²×C₄	—	53	1	no	—
231	C₂²×C₄	—	53	1	no	—
232	C₂²×C₄	—	53	2	no	—
233	C₂²×C₄	—	53	2	no	—
234	C₂²×C₄	—	53	1	no	—
235	C₂²×C₄	—	53	1	no	—
236	C₂²×C₄	—	53	8	no	—
237	C₂²×C₄	—	53	4	no	—
238	C₂²×C₄	—	106	4	no	—
239	C₂²×C₄	—	106	4	no	—
240	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
241	C₂×C₈	—	53	2	no	—
242	C₂×C₈	—	53	2	no	—
243	C₂×C₈	—	53	1	no	—
244	C₂×C₈	—	53	1	no	—
245	C₂×C₈	—	53	2	no	—
246	C₂×C₈	—	53	2	no	—
247	C₂×C₈	—	53	2	no	—
248	C₂×C₈	—	53	2	no	—
249	C₂×C₈	—	53	1	no	—
250	C₂×C₈	—	53	1	no	—
251	C₂×C₈	—	53	2	no	—
252	C₂×C₈	—	53	1	no	—
253	C₂×C₈	—	53	2	no	—
254	C₂×C₈	—	53	1	no	—
255	C₂×C₈	—	53	4	no	—
256	C₂×C₈	—	106	4	no	—
257	C₂×C₈	—	106	2	no	—
258	C₂×C₈	—	106	2	no	—
259	C₂×C₈	—	106	4	no	—
260	C₂×C₈	C₄×#6157	1	1	yes	characteristic
261	C₂×C₈	C₄⋊#6157	1	2	yes	—
262	C₂×C₈	C₂²⋊#6157	1	2	yes	—
263	C₂×C₈	C₂²×#6157	1	1	yes	characteristic
264	C₄²	—	106	2	no	—
265	C₄²	—	106	2	no	—
266	C₄²	—	106	4	no	—
267	C₁₆	—	2	2	no	—
268	C₁₆	—	106	4	no	—
269	C₁₆	—	106	2	no	—
270	C₂³	—	53	1	no	—
271	C₂³	—	53	1	no	—
272	C₂³	—	53	2	no	—
273	C₂³	—	53	2	no	—
274	C₂³	—	53	4	no	—
275	C₂³	—	53	4	no	—
276	C₂³	C₈×#6157	1	1	yes	characteristic
277	C₂×C₄	—	53	1	no	—
278	C₂×C₄	—	53	1	no	—
279	C₂×C₄	—	53	2	no	—
280	C₂×C₄	—	53	2	no	—
281	C₂×C₄	—	53	2	no	—
282	C₂×C₄	—	53	1	no	—
283	C₂×C₄	—	53	1	no	—
284	C₂×C₄	—	53	2	no	—
285	C₂×C₄	—	53	1	no	—
286	C₂×C₄	—	53	1	no	—
287	C₂×C₄	—	53	2	no	—
288	C₂×C₄	—	53	2	no	—
289	C₂×C₄	—	53	2	no	—
290	C₂×C₄	—	53	2	no	—
291	C₂×C₄	—	53	8	no	—
292	C₂×C₄	—	106	4	no	—
293	C₂×C₄	—	106	4	no	—
294	C₂×C₄	—	106	16	no	—
295	C₂×C₄	(C₂×#6157)⋊C₄	1	2	yes	—
296	C₂×C₄	C₈×#6157	1	1	yes	characteristic
297	C₂×C₄	#13133271	1	2	yes	—
298	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
299	C₈	—	53	2	no	—
300	C₈	—	53	1	no	—
301	C₈	—	53	1	no	—
302	C₈	—	106	4	no	—
303	C₈	(C₂×#6157)⋊C₄	1	2	yes	—
304	C₈	C₂×C₂²⋊#6157	1	1	yes	characteristic
305	C₈	C₂×C₄⋊#6157	1	1	yes	characteristic
306	C₂²	—	53	1	no	—
307	C₂²	—	53	2	no	—
308	C₂²	—	53	1	no	—
309	C₂²	—	53	2	no	—
310	C₂²	—	53	2	no	—
311	C₂²	—	53	2	no	—
312	C₂²	—	53	2	no	—
313	C₂²	—	53	8	no	—
314	C₂²	—	53	8	no	—
315	C₂²	(C₂×#6157)⋊C₈	1	2	yes	—
316	C₂²	C₁₆×#6157	1	2	yes	—
317	C₂²	#16811510	1	2	yes	—
318	C₂²	C₂×C₈×#6157	1	1	yes	characteristic
319	C₄	—	53	1	no	—
320	C₄	—	53	2	no	—
321	C₄	—	53	1	no	—
322	C₄	—	106	8	no	—
323	C₄	(C₂×#6157)⋊C₈	1	2	yes	—
324	C₄	C₂×(C₂×#6157)⋊C₄	1	1	yes	characteristic
325	C₄	C₂×#13133271	1	1	yes	characteristic
326	C₂	—	53	2	no	—
327	C₂	—	53	4	no	—
328	C₂	—	53	2	no	—
329	C₂	(C₂×#6157)⋊C₁₆	1	4	yes	direct product factor
330	C₂	C₂×(C₂×#6157)⋊C₈	1	1	yes	characteristic
331	C₂	C₂×C₁₆×#6157	1	1	yes	characteristic
332	C₂	C₂×#16811510	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂×#6157
commutator	C₂×C₅₃	C₂×C₄×C₁₆
Frattini	C₂×C₈	C₂²×#6157
Fitting	C₂²×C₁₆×C₅₃	C₄
socle	C₂³×C₅₃	C₄×C₈
prime radical	C₂³×D₅₃	C₂×C₈
cosocle	C₂×C₈×D₅₃	C₂³
hypercenter	C₂²×C₁₆	#6157
hypocenter	C₅₃	C₂×#1374
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₈ ◁ C₂²×C₁₆

Complements

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

N	H ≅ G/N	type
3. (C₂×#6157)⋊C₁₆	C₂	direct (N × H)
9. C₂×D₅₃⋊C₁₆	C₄	semidirect
11. C₂²×#11719910	C₄	semidirect
17. C₂²×#11719908	C₄	semidirect
20. C₂×C₁₆×D₅₃	C₄	semidirect
22. C₂²×C₁₆×C₅₃	C₄	semidirect
37. C₂×#11719910	C₂×C₄	semidirect
52. C₂×#11719908	C₂×C₄	semidirect
58. C₂×C₁₆×C₅₃	C₂×C₄	semidirect
62. C₂²×#6157	C₁₆	semidirect
113. C₂×#6157	C₂×C₁₆	semidirect
164. C₂×C₅₃	#1374	semidirect
168. D₅₃	#1374	semidirect
184. C₂²×C₁₆	#6157	semidirect
189. C₅₃	C₂×#1374	semidirect
222. C₂×C₁₆	C₂×#6157	semidirect
329. C₂	(C₂×#6157)⋊C₁₆	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×D₅₃⋊C₁₆	2
C₂²×C₈×#6157	2
(C₂×#6157)⋊C₁₆	2
C₂²×C₁₆×D₅₃	2
C₂×#1374	53

Subgroups — C2×(C2×#6157)⋊C16 #28310212

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×(C₂×#6157)⋊C₁₆ #28310212