Subgroups — C₂×M₁₆×#88324 #28313740

Order 13568.

1046 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16811639	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₄×#88324	C₂	1	1	yes	maximal, characteristic
3	M₁₆×#88324	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16811464	C₂	1	2	yes	maximal, complemented
5	C₂×C₈×#88324	C₂	1	2	yes	maximal, complemented
6	C₂×M₁₆×Dic₅₃	C₂	1	1	yes	maximal, characteristic
7	C₂³×#88324	C₄	1	2	yes	—
8	#16811639	C₄	1	4	yes	—
9	#16811639	C₂²	1	4	yes	—
10	C₂×C₄⋊#88324	C₄	1	2	yes	—
11	C₂×C₂²⋊#88324	C₄	1	2	yes	—
12	C₂×C₄×#88324	C₄	1	4	yes	—
13	C₂×C₄×#88324	C₄	1	2	yes	—
14	C₂×C₄×#88324	C₂²	1	2	yes	—
15	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
16	C₂×C₄×#88324	C₂²	1	1	yes	characteristic
17	C₂×C₄×#88324	C₂²	1	4	yes	—
18	#16811464	C₂²	1	8	yes	complemented
19	C₈×#88324	C₂²	1	8	yes	complemented
20	C₂²×#11719917	C₄	1	1	yes	characteristic
21	C₂²×C₄×Dic₅₃	C₂²	1	1	yes	characteristic
22	M₁₆×Dic₅₃	C₄	1	4	yes	—
23	M₁₆×Dic₅₃	C₂²	1	4	yes	—
24	C₂×#13133127	C₄	1	2	yes	—
25	C₂×C₈×Dic₅₃	C₂²	1	2	yes	—
26	C₂²×M₁₆×C₅₃	C₄	1	1	yes	characteristic
27	C₂²×#88324	—	2	8	no	—
28	C₂²×#88324	C₂×C₄	1	2	yes	—
29	C₂²×#88324	C₂×C₄	1	8	yes	—
30	C₂²×#88324	C₂×C₄	1	4	yes	—
31	C₂²×#88324	C₂×C₄	1	2	yes	—
32	C₄⋊#88324	C₂×C₄	1	8	yes	—
33	C₂²⋊#88324	C₂×C₄	1	8	yes	—
34	C₄×#88324	C₂×C₄	1	2	yes	—
35	C₄×#88324	C₂×C₄	1	8	yes	—
36	C₄×#88324	C₂×C₄	1	2	yes	—
37	C₄×#88324	C₂×C₄	1	4	yes	—
38	C₄×#88324	C₂³	1	2	yes	—
39	C₄×#88324	C₂³	1	2	yes	—
40	C₄×#88324	C₂³	1	4	yes	—
41	C₂³×Dic₅₃	C₂×C₄	1	1	yes	characteristic
42	C₂³×Dic₅₃	C₂×C₄	1	1	yes	characteristic
43	C₂×#11719917	C₈	1	8	yes	complemented
44	C₂×#11719917	C₂×C₄	1	4	yes	—
45	C₂²×#88322	C₂×C₄	1	2	yes	—
46	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
48	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
49	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
50	C₂×C₄×Dic₅₃	C₂×C₄	1	1	yes	characteristic
51	C₂×C₄×Dic₅₃	C₂×C₄	1	2	yes	—
52	C₂×C₄×Dic₅₃	C₂³	1	1	yes	characteristic
53	C₂×C₄×Dic₅₃	C₂³	1	2	yes	—
54	#13133127	C₂×C₄	1	8	yes	—
55	C₈×Dic₅₃	C₂×C₄	1	4	yes	—
56	C₈×Dic₅₃	C₂³	1	4	yes	—
57	C₂³×C₄×C₅₃	C₂×C₄	1	1	yes	characteristic
58	C₂×M₁₆×C₅₃	C₈	1	8	yes	complemented
59	C₂×M₁₆×C₅₃	C₂×C₄	1	4	yes	—
60	C₂²×C₈×C₅₃	C₂×C₄	1	2	yes	—
61	C₂⁴×C₅₃	C₄²	1	1	yes	characteristic
62	C₂²×C₄×C₅₃	C₄²	1	2	yes	—
63	C₂²×C₄×C₅₃	C₄²	1	1	yes	characteristic
64	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
65	C₂²×C₄×C₅₃	C₂×C₈	1	4	yes	—
66	C₂²×C₄×C₅₃	C₂²×C₄	1	2	yes	—
67	C₂²×C₄×C₅₃	C₂²×C₄	1	1	yes	characteristic
68	C₂²×Dic₅₃	—	2	4	no	—
69	C₂²×Dic₅₃	—	2	4	no	—
70	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
71	C₂²×Dic₅₃	C₄²	1	1	yes	characteristic
72	C₂²×Dic₅₃	C₄²	1	2	yes	—
73	C₂²×Dic₅₃	C₄²	1	2	yes	—
74	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
75	C₂²×Dic₅₃	C₂²×C₄	1	2	yes	—
76	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
77	C₂²×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
78	C₂×#88324	—	2	4	no	—
79	C₂×#88324	—	2	8	no	—
80	C₂×#88324	—	2	16	no	—
81	C₂×#88324	M₁₆	1	8	yes	direct product factor
82	C₂×#88324	C₂²×C₄	1	4	yes	—
83	C₂×#88324	C₂²×C₄	1	8	yes	—
84	C₂×#88324	C₂²×C₄	1	4	yes	—
85	M₁₆×C₅₃	C₂×C₈	1	16	yes	complemented
86	C₂×C₈×C₅₃	C₂×C₈	1	8	yes	complemented
87	C₂×C₈×C₅₃	C₂²×C₄	1	4	yes	—
88	C₄×Dic₅₃	C₄²	1	2	yes	—
89	C₄×Dic₅₃	C₄²	1	2	yes	—
90	C₄×Dic₅₃	C₄²	1	2	yes	—
91	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
92	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
93	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
94	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
95	C₄×Dic₅₃	C₂²×C₄	1	1	yes	characteristic
96	C₄×Dic₅₃	C₂²×C₄	1	2	yes	—
97	C₄×Dic₅₃	C₂⁴	1	1	yes	characteristic
98	#11719917	C₂×C₈	1	16	yes	complemented
99	C₂×#88322	C₂×C₈	1	8	yes	complemented
100	C₂×#88322	C₂²×C₄	1	4	yes	—
101	C₂³×C₅₃	—	2	2	no	—
102	C₂³×C₅₃	—	2	2	no	—
103	C₂³×C₅₃	C₄×C₈	1	4	yes	—
104	C₂³×C₅₃	C₂×C₄²	1	2	yes	—
105	C₂³×C₅₃	C₂×C₄²	1	1	yes	characteristic
106	C₂×C₄×C₅₃	C₄×C₈	1	8	yes	—
107	C₂×C₄×C₅₃	C₄×C₈	1	4	yes	—
108	C₂×C₄×C₅₃	C₂×C₄²	1	2	yes	—
109	C₂×C₄×C₅₃	C₂×C₄²	1	1	yes	characteristic
110	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
111	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
112	C₂×C₄×C₅₃	C₂²×C₈	1	2	yes	—
113	C₂×C₄×C₅₃	C₂²×C₈	1	4	yes	—
114	C₂×C₄×C₅₃	C₂³×C₄	1	1	yes	characteristic
115	C₂×Dic₅₃	—	2	2	no	—
116	C₂×Dic₅₃	—	2	2	no	—
117	C₂×Dic₅₃	—	2	4	no	—
118	C₂×Dic₅₃	—	2	4	no	—
119	C₂×Dic₅₃	—	2	4	no	—
120	C₂×Dic₅₃	—	2	4	no	—
121	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
122	C₂×Dic₅₃	C₂×C₄²	1	2	yes	—
123	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
124	C₂×Dic₅₃	C₂×C₄²	1	1	yes	characteristic
125	C₂×Dic₅₃	C₂×M₁₆	1	4	yes	—
126	C₂×Dic₅₃	#131	1	4	yes	—
127	C₂×Dic₅₃	C₂³×C₄	1	1	yes	characteristic
128	C₂×Dic₅₃	C₂³×C₄	1	1	yes	characteristic
129	#88324	—	2	4	no	—
130	#88324	—	2	8	no	—
131	#88324	C₂×M₁₆	1	8	yes	direct product factor
132	C₈×C₅₃	C₂²×C₈	1	8	yes	complemented
133	#88322	C₂²×C₈	1	8	yes	complemented
134	C₂×C₈×M₁₆	—	53	1	no	maximal, Sylow
135	C₂²×C₅₃	—	2	2	no	—
136	C₂²×C₅₃	—	2	2	no	—
137	C₂²×C₅₃	—	2	8	no	—
138	C₂²×C₅₃	C₂×C₄×C₈	1	4	yes	—
139	C₂²×C₅₃	C₂×C₄×C₈	1	2	yes	—
140	C₂²×C₅₃	C₄×M₁₆	1	2	yes	—
141	C₂²×C₅₃	#405	1	2	yes	—
142	C₂²×C₅₃	C₂²×C₄²	1	1	yes	characteristic
143	C₄×C₅₃	C₂×C₄×C₈	1	2	yes	—
144	C₄×C₅₃	C₂×C₄×C₈	1	4	yes	—
145	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
146	C₄×C₅₃	C₂³×C₈	1	1	yes	characteristic
147	Dic₅₃	—	2	2	no	—
148	Dic₅₃	—	2	2	no	—
149	Dic₅₃	C₄×M₁₆	1	2	yes	—
150	Dic₅₃	#405	1	2	yes	—
151	Dic₅₃	C₂²×M₁₆	1	2	yes	—
152	Dic₅₃	C₂×#131	1	2	yes	—
153	C₂×#431	—	53	1	no	—
154	C₂×C₄×M₁₆	—	53	1	no	—
155	C₂²×C₄×C₈	—	53	1	no	—
156	C₈×M₁₆	—	53	8	no	—
157	C₂×#322	—	53	2	no	—
158	C₂×C₈²	—	53	2	no	—
159	C₂×C₅₃	—	2	4	no	—
160	C₂×C₅₃	C₈×M₁₆	1	4	yes	complemented
161	C₂×C₅₃	C₂²×C₄×C₈	1	1	yes	characteristic
162	C₂×C₅₃	C₂×C₄×M₁₆	1	1	yes	characteristic
163	C₂×C₅₃	C₂×#405	1	1	yes	characteristic
164	C₂²×M₁₆	—	53	1	no	—
165	C₂²×M₁₆	#6157	1	1	yes	characteristic
166	C₂³×C₈	—	53	2	no	—
167	C₂²×C₄²	—	53	1	no	—
168	#431	—	53	4	no	—
169	#431	—	53	4	no	—
170	C₂×C₄⋊C₈	—	53	2	no	—
171	C₂×C₂²⋊C₈	—	53	2	no	—
172	C₄×M₁₆	—	53	4	no	—
173	C₄×M₁₆	—	53	4	no	—
174	C₂×#97	—	53	2	no	—
175	C₂×C₄×C₈	—	53	2	no	—
176	C₂×C₄×C₈	—	53	4	no	—
177	C₂×C₄×C₈	—	53	1	no	—
178	C₂×C₄×C₈	—	53	1	no	—
179	C₂×C₄×C₈	—	53	4	no	—
180	C₂×C₄×C₈	—	53	2	no	—
181	C₂×C₄×C₈	—	53	2	no	—
182	#322	—	53	8	no	—
183	C₈²	—	53	8	no	—
184	C₅₃	C₂×C₈×M₁₆	1	1	yes	characteristic, complemented, Sylow
185	C₂³×C₄	—	53	1	no	—
186	C₂³×C₄	—	53	1	no	—
187	C₂³×C₄	C₂×#6157	1	1	yes	characteristic
188	C₂×M₁₆	—	53	4	no	—
189	C₂×M₁₆	—	53	8	no	—
190	C₂×M₁₆	#88324	1	8	yes	direct product factor
191	C₂×M₁₆	C₂×#6157	1	4	yes	—
192	C₂²×C₈	—	53	2	no	—
193	C₂²×C₈	—	53	2	no	—
194	C₂²×C₈	—	53	8	no	—
195	C₂²×C₈	—	53	2	no	—
196	C₂²×C₈	—	53	4	no	—
197	C₂²×C₈	—	106	8	no	—
198	C₂²×C₈	C₂×#6157	1	2	yes	—
199	C₂×C₄²	—	53	1	no	—
200	C₂×C₄²	—	53	2	no	—
201	C₂×C₄²	—	53	1	no	—
202	C₂×C₄²	—	53	2	no	—
203	C₂×C₄²	—	53	2	no	—
204	C₂×C₄²	—	53	2	no	—
205	C₂×C₄²	—	53	1	no	—
206	C₂×C₄²	—	53	1	no	—
207	C₄⋊C₈	—	53	8	no	—
208	C₂²⋊C₈	—	53	8	no	—
209	#97	—	53	8	no	—
210	C₄×C₈	—	53	2	no	—
211	C₄×C₈	—	53	4	no	—
212	C₄×C₈	—	53	2	no	—
213	C₄×C₈	—	53	2	no	—
214	C₄×C₈	—	53	8	no	—
215	C₄×C₈	—	53	2	no	—
216	C₄×C₈	—	53	4	no	—
217	C₄×C₈	—	53	4	no	—
218	C₄×C₈	—	53	4	no	—
219	C₂⁴	C₄×#6157	1	1	yes	characteristic
220	C₂²×C₄	—	53	1	no	—
221	C₂²×C₄	—	53	1	no	—
222	C₂²×C₄	—	53	2	no	—
223	C₂²×C₄	—	53	2	no	—
224	C₂²×C₄	—	53	1	no	—
225	C₂²×C₄	—	53	1	no	—
226	C₂²×C₄	—	53	2	no	—
227	C₂²×C₄	—	53	2	no	—
228	C₂²×C₄	—	106	4	no	—
229	C₂²×C₄	—	106	4	no	—
230	C₂²×C₄	D₅₃⋊C₈	1	4	yes	—
231	C₂²×C₄	C₄×#6157	1	1	yes	characteristic
232	C₂²×C₄	C₄×#6157	1	2	yes	—
233	C₂²×C₄	C₂×#88324	1	4	yes	—
234	C₂²×C₄	C₂²×#6157	1	2	yes	—
235	C₂²×C₄	C₂²×#6157	1	1	yes	characteristic
236	M₁₆	—	53	16	no	—
237	M₁₆	C₂×#88324	1	16	yes	direct product factor
238	C₂×C₈	—	53	8	no	—
239	C₂×C₈	—	53	4	no	—
240	C₂×C₈	—	53	8	no	—
241	C₂×C₈	—	53	4	no	—
242	C₂×C₈	—	53	4	no	—
243	C₂×C₈	—	53	8	no	—
244	C₂×C₈	—	106	16	no	—
245	C₂×C₈	—	106	8	no	—
246	C₂×C₈	—	106	4	no	—
247	C₂×C₈	C₂×#88324	1	8	yes	complemented
248	C₂×C₈	C₂²×#6157	1	4	yes	—
249	C₄²	—	53	1	no	—
250	C₄²	—	53	1	no	—
251	C₄²	—	53	2	no	—
252	C₄²	—	53	2	no	—
253	C₄²	—	53	2	no	—
254	C₄²	—	53	2	no	—
255	C₄²	—	53	2	no	—
256	C₄²	—	53	1	no	—
257	C₄²	—	53	1	no	—
258	C₄²	—	53	2	no	—
259	C₂³	—	2	2	no	—
260	C₂³	—	2	2	no	—
261	C₂³	C₄×#88324	1	4	yes	—
262	C₂³	C₂×C₄×#6157	1	2	yes	—
263	C₂³	C₂×C₄×#6157	1	1	yes	characteristic
264	C₂×C₄	—	53	4	no	—
265	C₂×C₄	—	53	4	no	—
266	C₂×C₄	—	53	1	no	—
267	C₂×C₄	—	53	1	no	—
268	C₂×C₄	—	53	1	no	—
269	C₂×C₄	—	53	1	no	—
270	C₂×C₄	—	53	2	no	—
271	C₂×C₄	—	53	2	no	—
272	C₂×C₄	—	106	4	no	—
273	C₂×C₄	—	106	4	no	—
274	C₂×C₄	—	106	4	no	—
275	C₂×C₄	—	106	4	no	—
276	C₂×C₄	—	106	2	no	—
277	C₂×C₄	—	106	2	no	—
278	C₂×C₄	C₄×#88324	1	8	yes	—
279	C₂×C₄	C₄×#88324	1	4	yes	—
280	C₂×C₄	C₂×C₄×#6157	1	2	yes	—
281	C₂×C₄	C₂×C₄×#6157	1	1	yes	characteristic
282	C₂×C₄	C₂²×#88324	1	2	yes	—
283	C₂×C₄	C₂²×#88324	1	4	yes	—
284	C₂×C₄	C₂×D₅₃⋊C₈	1	2	yes	—
285	C₂×C₄	C₂×D₅₃⋊C₈	1	4	yes	—
286	C₂×C₄	C₂³×#6157	1	1	yes	characteristic
287	C₈	—	53	8	no	—
288	C₈	—	53	8	no	—
289	C₈	—	106	8	no	—
290	C₈	—	106	4	no	—
291	C₈	C₂²×#88324	1	8	yes	complemented
292	C₂²	—	2	8	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	2	no	—
295	C₂²	C₂×C₄×#88324	1	4	yes	—
296	C₂²	C₂×C₄×#88324	1	2	yes	—
297	C₂²	M₁₆×#6157	1	2	yes	—
298	C₂²	#16811589	1	2	yes	—
299	C₂²	C₂²×C₄×#6157	1	1	yes	characteristic
300	C₄	—	53	2	no	—
301	C₄	—	53	2	no	—
302	C₄	—	53	2	no	—
303	C₄	—	53	2	no	—
304	C₄	—	106	2	no	—
305	C₄	—	106	2	no	—
306	C₄	C₂×C₄×#88324	1	4	yes	—
307	C₄	C₂×C₄×#88324	1	2	yes	—
308	C₄	C₂³×#88324	1	1	yes	characteristic
309	C₄	C₂²×D₅₃⋊C₈	1	1	yes	characteristic
310	C₂	—	2	4	no	—
311	C₂	M₁₆×#88324	1	4	yes	direct product factor
312	C₂	C₂²×C₄×#88324	1	1	yes	characteristic
313	C₂	C₂×M₁₆×#6157	1	1	yes	characteristic
314	C₂	C₂×#16811589	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	C₂²×#6157
commutator	C₂×C₅₃	C₂²×C₄×C₈
Frattini	C₂×C₄	C₂³×#6157
Fitting	C₂²×M₁₆×C₅₃	C₄
socle	C₂³×C₅₃	C₂×C₄²
prime radical	C₂⁴×C₅₃	C₄²
cosocle	C₄×Dic₅₃	C₂⁴
hypercenter	C₂²×M₁₆	#6157
hypocenter	C₅₃	C₂×C₈×M₁₆
LCS	G ▷ C₂×C₅₃ ▷ C₅₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×M₁₆

Complements

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

N	H ≅ G/N	type
3. M₁₆×#88324	C₂	direct (N × H)
4. C₂×#16811464	C₂	semidirect
5. C₂×C₈×#88324	C₂	semidirect
18. #16811464	C₂²	semidirect
19. C₈×#88324	C₂²	semidirect
43. C₂×#11719917	C₈	semidirect
58. C₂×M₁₆×C₅₃	C₈	semidirect
81. C₂×#88324	M₁₆	direct (N × H)
85. M₁₆×C₅₃	C₂×C₈	semidirect
86. C₂×C₈×C₅₃	C₂×C₈	semidirect
98. #11719917	C₂×C₈	semidirect
99. C₂×#88322	C₂×C₈	semidirect
131. #88324	C₂×M₁₆	direct (N × H)
132. C₈×C₅₃	C₂²×C₈	semidirect
133. #88322	C₂²×C₈	semidirect
160. C₂×C₅₃	C₈×M₁₆	semidirect
184. C₅₃	C₂×C₈×M₁₆	semidirect
190. C₂×M₁₆	#88324	direct (N × H)
237. M₁₆	C₂×#88324	direct (N × H)
247. C₂×C₈	C₂×#88324	semidirect
291. C₈	C₂²×#88324	semidirect
311. C₂	M₁₆×#88324	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×#16811639	2
C₂²×C₄×#88324	2
M₁₆×#88324	2
C₂×#16811464	2
C₂×C₈×#88324	2
C₂×M₁₆×Dic₅₃	2
C₂×C₈×M₁₆	53

Subgroups — C2×M16×#88324 #28313740

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×M₁₆×#88324 #28313740