Subgroups — #28407312

Order 14464.

544 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×D₁₁₃⋊M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×#14268656	C₂	1	1	yes	maximal, characteristic, complemented
3	#20299631	C₂	1	2	yes	maximal, complemented
4	#20299630	C₂	1	2	yes	maximal, complemented
5	#20299627	C₂	1	2	yes	maximal, complemented
6	#20299625	C₂	1	2	yes	maximal, complemented
7	#20299601	C₂	1	1	yes	maximal, characteristic, complemented
8	#20299570	C₂	1	1	yes	maximal, characteristic, complemented
9	#20299568	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×#14268656	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×#16813582	C₂	1	1	yes	maximal, characteristic
12	C₂×#14268657	C₂²	1	2	yes	complemented
13	D₁₁₃⋊M₁₆	—	2	4	no	—
14	C₂×#14268656	—	2	4	no	—
15	C₂×#14268656	C₂²	1	1	yes	characteristic
16	C₂×#14268656	C₂²	1	1	yes	characteristic
17	C₂×#14268656	C₂²	1	1	yes	characteristic
18	C₂×#14268656	C₂²	1	1	yes	characteristic
19	C₂×#14268656	C₂²	1	2	yes	complemented
20	C₂×#14268655	C₂²	1	2	yes	complemented
21	C₂²×#11744753	C₂²	1	2	yes	complemented
22	#16813689	C₂²	1	2	yes	complemented
23	C₄⋊#11744753	C₂²	1	2	yes	complemented
24	#16813679	C₂²	1	2	yes	complemented
25	#16813679	C₂²	1	2	yes	complemented
26	C₂²⋊#11744753	C₂²	1	2	yes	complemented
27	C₂²⋊#11744753	C₂²	1	2	yes	complemented
28	#16813677	C₂²	1	2	yes	complemented
29	C₄×#11744753	C₂²	1	2	yes	complemented
30	C₂²×C₂²⋊D₁₁₃	C₄	1	1	yes	characteristic
31	C₂³×Dic₁₁₃	C₂²	1	1	yes	characteristic
32	C₂²×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
33	#16813582	C₄	1	1	yes	characteristic
34	#16813582	C₄	1	1	yes	characteristic
35	#16813582	C₄	1	1	yes	characteristic
36	#16813582	C₄	1	1	yes	characteristic
37	#16813582	C₂²	1	1	yes	characteristic
38	#16813582	C₂²	1	1	yes	characteristic
39	#16813582	C₂²	1	1	yes	characteristic
40	#16813582	C₂²	1	1	yes	characteristic
41	C₂×#14268626	C₄	1	1	yes	characteristic
42	C₂×#14268624	C₄	1	1	yes	characteristic
43	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
44	C₂×C₂²⋊C₄×C₁₁₃	C₄	1	1	yes	characteristic
45	#14268657	—	2	4	no	—
46	#14268656	—	2	2	no	—
47	#14268656	—	2	2	no	—
48	#14268656	—	2	2	no	—
49	#14268656	—	2	4	no	—
50	#14268656	D₄	1	4	yes	complemented
51	#14268655	—	2	4	no	—
52	C₂×#11744753	—	2	4	no	—
53	C₂×#11744753	C₂³	1	2	yes	—
54	C₂×#11744753	C₂³	1	2	yes	—
55	C₂×#11744753	C₂³	1	2	yes	—
56	C₂×#11744753	C₂³	1	2	yes	—
57	C₂³×D₁₁₃	C₂×C₄	1	1	yes	characteristic
58	C₂×C₂²⋊D₁₁₃	—	2	4	no	—
59	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	1	yes	characteristic
60	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	1	yes	characteristic
61	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	1	yes	characteristic
62	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	1	yes	characteristic
63	C₂²×Dic₁₁₃	—	2	2	no	—
64	C₂²×Dic₁₁₃	—	2	2	no	—
65	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
67	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
68	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
69	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
70	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
71	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
72	C₂×C₄×D₁₁₃	—	2	2	no	—
73	C₂×C₄×D₁₁₃	—	2	2	no	—
74	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
75	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
76	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
77	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
78	#14268626	C₂×C₄	1	1	yes	characteristic
79	#14268626	C₂×C₄	1	1	yes	characteristic
80	#14268626	C₂×C₄	1	1	yes	characteristic
81	#14268626	C₂×C₄	1	1	yes	characteristic
82	#14268624	C₂×C₄	1	1	yes	characteristic
83	#14268624	C₂×C₄	1	1	yes	characteristic
84	#14268624	C₂×C₄	1	1	yes	characteristic
85	#14268624	C₂×C₄	1	1	yes	characteristic
86	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
87	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
88	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
89	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
90	C₂⁴×C₁₁₃	C₂×C₄	1	1	yes	characteristic
91	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
92	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
93	C₂²⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
94	C₂²⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
95	C₂²⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
96	C₂²⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
97	C₂³×C₁₁₃	—	2	2	no	—
98	C₂³×C₁₁₃	—	2	2	no	—
99	C₂³×C₁₁₃	—	4	1	no	—
100	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
101	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
102	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
103	C₂²×D₁₁₃	—	2	2	no	—
104	C₂²×D₁₁₃	—	2	2	no	—
105	C₂²×D₁₁₃	—	4	1	no	—
106	C₂²×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
107	C₂²×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
108	C₂×C₄×C₁₁₃	—	2	2	no	—
109	C₂×C₄×C₁₁₃	—	4	1	no	—
110	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
111	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
112	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
113	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
114	C₂²⋊D₁₁₃	—	2	2	no	—
115	C₂²⋊D₁₁₃	—	2	2	no	—
116	C₂²⋊D₁₁₃	—	2	2	no	—
117	C₂²⋊D₁₁₃	M₁₆	1	4	yes	complemented
118	C₂×Dic₁₁₃	—	2	2	no	—
119	C₂×Dic₁₁₃	—	2	2	no	—
120	C₂×Dic₁₁₃	—	2	2	no	—
121	C₂×Dic₁₁₃	—	2	2	no	—
122	C₂×Dic₁₁₃	—	2	1	no	—
123	C₂×Dic₁₁₃	—	2	1	no	—
124	C₂×Dic₁₁₃	—	2	1	no	—
125	C₂×Dic₁₁₃	—	2	1	no	—
126	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
127	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
128	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
129	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
130	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
131	C₂×Dic₁₁₃	C₂×D₄	1	2	yes	—
132	C₂×Dic₁₁₃	#41	1	2	yes	—
133	C₂×Dic₁₁₃	C₂⁴	1	1	yes	characteristic
134	C₄×D₁₁₃	—	2	2	no	—
135	C₄×D₁₁₃	—	2	2	no	—
136	C₄×D₁₁₃	—	2	1	no	—
137	C₄×D₁₁₃	—	2	1	no	—
138	C₄×D₁₁₃	—	2	1	no	—
139	C₄×D₁₁₃	—	2	1	no	—
140	#11744753	—	2	2	no	—
141	#11744753	—	2	2	no	—
142	#11744753	—	2	2	no	—
143	#11744753	C₂×D₄	1	4	yes	complemented
144	C₂²×C₁₁₃	—	2	1	no	—
145	C₂²×C₁₁₃	—	2	1	no	—
146	C₂²×C₁₁₃	—	2	1	no	—
147	C₂²×C₁₁₃	—	2	1	no	—
148	C₂²×C₁₁₃	—	2	1	no	—
149	C₂²×C₁₁₃	—	2	1	no	—
150	C₂²×C₁₁₃	—	2	1	no	—
151	C₂²×C₁₁₃	—	4	4	no	—
152	C₂²×C₁₁₃	C₄×D₄	1	2	yes	—
153	C₂²×C₁₁₃	C₂×M₁₆	1	2	yes	complemented
154	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
155	C₂×D₁₁₃	—	2	1	no	—
156	C₂×D₁₁₃	—	2	1	no	—
157	C₂×D₁₁₃	—	2	1	no	—
158	C₂×D₁₁₃	—	2	1	no	—
159	C₂×D₁₁₃	—	2	1	no	—
160	C₂×D₁₁₃	—	4	4	no	—
161	C₂×D₁₁₃	C₂×M₁₆	1	2	yes	complemented
162	C₄×C₁₁₃	—	2	1	no	—
163	C₄×C₁₁₃	—	2	1	no	—
164	C₄×C₁₁₃	—	2	1	no	—
165	C₄×C₁₁₃	—	2	1	no	—
166	Dic₁₁₃	—	2	1	no	—
167	Dic₁₁₃	—	2	1	no	—
168	Dic₁₁₃	—	2	1	no	—
169	Dic₁₁₃	C₄×D₄	1	2	yes	—
170	Dic₁₁₃	C₂×M₁₆	1	2	yes	complemented
171	Dic₁₁₃	C₂²×D₄	1	1	yes	characteristic
172	Dic₁₁₃	C₂×#41	1	1	yes	characteristic
173	C₂×C₁₁₃	—	2	2	no	—
174	C₂×C₁₁₃	—	2	2	no	—
175	C₂×C₁₁₃	—	4	1	no	—
176	C₂×C₁₁₃	C₂×C₄×D₄	1	1	yes	characteristic
177	C₂×C₁₁₃	C₂²×M₁₆	1	1	yes	characteristic
178	C₂×C₁₁₃	#568	1	1	yes	characteristic
179	D₁₁₃	—	2	2	no	—
180	D₁₁₃	—	4	1	no	—
181	D₄×M₁₆	—	113	1	no	maximal, Sylow
182	C₁₁₃	D₄×M₁₆	1	1	yes	characteristic, complemented, Sylow
183	C₂²×M₁₆	—	113	1	no	—
184	C₂²×M₁₆	—	113	1	no	—
185	C₂×C₄×D₄	—	113	1	no	—
186	#436	—	113	2	no	—
187	#435	—	113	2	no	—
188	#435	—	113	2	no	—
189	C₈×D₄	—	113	2	no	—
190	#423	—	113	1	no	—
191	#407	—	113	1	no	—
192	#407	—	113	1	no	—
193	C₄×M₁₆	—	113	1	no	—
194	C₂²×D₄	—	113	1	no	—
195	C₂³×C₄	—	113	1	no	—
196	C₂³×C₄	—	113	1	no	—
197	C₂×M₁₆	—	113	1	no	—
198	C₂×M₁₆	—	113	1	no	—
199	C₂×M₁₆	—	113	1	no	—
200	C₂×M₁₆	—	113	2	no	—
201	C₂×M₁₆	—	113	2	no	—
202	C₂×M₁₆	—	113	1	no	—
203	C₂×M₁₆	—	226	4	no	—
204	C₂×M₁₆	—	226	4	no	—
205	C₂²×C₈	—	113	2	no	—
206	C₂²×C₈	—	113	2	no	—
207	C₄×D₄	—	113	1	no	—
208	C₄×D₄	—	113	1	no	—
209	C₄×D₄	—	113	1	no	—
210	C₄×D₄	—	113	1	no	—
211	C₄×D₄	—	113	1	no	—
212	C₄×D₄	—	113	1	no	—
213	C₄×D₄	—	113	1	no	—
214	C₄×D₄	—	113	1	no	—
215	C₂×C₄⋊C₄	—	113	1	no	—
216	C₂×C₂²⋊C₄	—	113	1	no	—
217	C₂×C₂²⋊C₄	#90673	1	1	yes	characteristic
218	C₂×C₄²	—	113	1	no	—
219	C₄⋊C₈	—	113	2	no	—
220	C₄⋊C₈	—	113	2	no	—
221	C₂²⋊C₈	—	113	2	no	—
222	C₂²⋊C₈	—	113	2	no	—
223	C₂²⋊C₈	—	113	2	no	—
224	C₂²⋊C₈	—	113	2	no	—
225	#97	—	113	2	no	—
226	C₄×C₈	—	113	2	no	—
227	C₂⁴	—	113	1	no	—
228	C₂⁴	C₂×#90673	1	1	yes	characteristic
229	C₂×D₄	—	113	1	no	—
230	C₂×D₄	—	113	1	no	—
231	C₂×D₄	—	113	1	no	—
232	C₂×D₄	—	113	1	no	—
233	C₂×D₄	—	226	4	no	—
234	C₂²×C₄	—	113	1	no	—
235	C₂²×C₄	—	113	1	no	—
236	C₂²×C₄	—	113	1	no	—
237	C₂²×C₄	—	113	1	no	—
238	C₂²×C₄	—	113	1	no	—
239	C₂²×C₄	—	113	1	no	—
240	C₂²×C₄	—	113	1	no	—
241	C₂²×C₄	—	113	1	no	—
242	C₂²×C₄	—	113	1	no	—
243	C₂²×C₄	—	113	1	no	—
244	C₂²×C₄	—	113	1	no	—
245	C₂²×C₄	—	226	2	no	—
246	C₂²×C₄	—	226	2	no	—
247	C₂²×C₄	—	226	2	no	—
248	C₂²×C₄	—	226	2	no	—
249	C₂²×C₄	C₂×#90673	1	1	yes	characteristic
250	C₂²×C₄	C₂×#90673	1	1	yes	characteristic
251	M₁₆	—	113	4	no	—
252	M₁₆	—	226	4	no	—
253	M₁₆	—	226	4	no	—
254	M₁₆	—	226	2	no	—
255	M₁₆	—	226	2	no	—
256	M₁₆	—	226	2	no	—
257	C₂×C₈	—	113	2	no	—
258	C₂×C₈	—	113	2	no	—
259	C₂×C₈	—	113	2	no	—
260	C₂×C₈	—	113	2	no	—
261	C₂×C₈	—	226	4	no	—
262	C₂×C₈	—	226	4	no	—
263	C₄⋊C₄	—	113	1	no	—
264	C₄⋊C₄	—	113	1	no	—
265	C₄⋊C₄	—	113	1	no	—
266	C₄⋊C₄	—	113	1	no	—
267	C₂²⋊C₄	—	113	1	no	—
268	C₂²⋊C₄	—	113	1	no	—
269	C₂²⋊C₄	—	113	1	no	—
270	C₂²⋊C₄	—	113	1	no	—
271	C₂²⋊C₄	C₂×#90673	1	1	yes	characteristic
272	C₂²⋊C₄	C₂×#90673	1	1	yes	characteristic
273	C₂²⋊C₄	C₂×#90673	1	1	yes	characteristic
274	C₂²⋊C₄	C₂×#90673	1	1	yes	characteristic
275	C₄²	—	113	1	no	—
276	C₄²	—	113	1	no	—
277	C₄²	—	113	1	no	—
278	C₄²	—	113	1	no	—
279	C₂³	—	2	2	no	—
280	C₂³	—	2	2	no	—
281	C₂³	—	4	1	no	—
282	C₂³	—	113	1	no	—
283	C₂³	—	113	1	no	—
284	C₂³	—	452	1	no	—
285	C₂³	—	226	2	no	—
286	C₂³	—	226	2	no	—
287	C₂³	C₂²×#90673	1	1	yes	characteristic
288	C₂³	C₂²×#90673	1	1	yes	characteristic
289	C₂³	C₂²×#90673	1	1	yes	characteristic
290	D₄	—	113	4	no	—
291	D₄	—	226	2	no	—
292	D₄	—	226	2	no	—
293	D₄	—	226	2	no	—
294	C₂×C₄	—	2	2	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	113	2	no	—
297	C₂×C₄	—	113	2	no	—
298	C₂×C₄	—	113	1	no	—
299	C₂×C₄	—	113	1	no	—
300	C₂×C₄	—	113	1	no	—
301	C₂×C₄	—	113	1	no	—
302	C₂×C₄	—	113	1	no	—
303	C₂×C₄	—	113	1	no	—
304	C₂×C₄	—	226	1	no	—
305	C₂×C₄	—	226	1	no	—
306	C₂×C₄	—	226	1	no	—
307	C₂×C₄	—	226	1	no	—
308	C₂×C₄	—	226	2	no	—
309	C₂×C₄	—	226	2	no	—
310	C₂×C₄	—	226	1	no	—
311	C₂×C₄	—	226	1	no	—
312	C₂×C₄	—	226	1	no	—
313	C₂×C₄	—	226	1	no	—
314	C₂×C₄	—	226	2	no	—
315	C₂×C₄	—	226	2	no	—
316	C₂×C₄	—	226	2	no	—
317	C₂×C₄	—	226	2	no	—
318	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
319	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
320	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
321	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
322	C₈	—	113	4	no	—
323	C₈	—	226	2	no	—
324	C₈	—	226	2	no	—
325	C₈	—	226	2	no	—
326	C₂²	—	2	1	no	—
327	C₂²	—	2	1	no	—
328	C₂²	—	2	1	no	—
329	C₂²	—	2	1	no	—
330	C₂²	—	2	1	no	—
331	C₂²	—	2	1	no	—
332	C₂²	—	2	1	no	—
333	C₂²	—	4	4	no	—
334	C₂²	—	113	2	no	—
335	C₂²	—	452	4	no	—
336	C₂²	—	226	1	no	—
337	C₂²	—	226	1	no	—
338	C₂²	—	226	1	no	—
339	C₂²	—	226	1	no	—
340	C₂²	—	226	1	no	—
341	C₂²	D₄×#90673	1	2	yes	—
342	C₂²	D₁₁₃⋊M₁₆	1	2	yes	complemented
343	C₂²	C₂³×#90673	1	1	yes	characteristic
344	C₄	—	2	1	no	—
345	C₄	—	2	1	no	—
346	C₄	—	2	1	no	—
347	C₄	—	2	1	no	—
348	C₄	—	113	1	no	—
349	C₄	—	113	1	no	—
350	C₄	—	113	2	no	—
351	C₄	—	113	2	no	—
352	C₄	—	226	1	no	—
353	C₄	—	226	1	no	—
354	C₄	—	226	1	no	—
355	C₂	—	2	2	no	—
356	C₂	—	2	2	no	—
357	C₂	—	4	1	no	—
358	C₂	—	452	1	no	—
359	C₂	—	226	2	no	—
360	C₂	C₂×D₄×#90673	1	1	yes	characteristic
361	C₂	C₂×D₁₁₃⋊M₁₆	1	1	yes	characteristic
362	C₂	#20299676	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#90673
commutator, socle	C₂²×C₁₁₃	C₂³×C₄
Fitting	C₂×C₂²⋊C₄×C₁₁₃	C₄
prime radical	C₂²×C₂²⋊D₁₁₃	C₄
cosocle	C₂×Dic₁₁₃	C₂⁴
hypercenter	C₂×C₂²⋊C₄	#90673
hypocenter	C₁₁₃	D₄×M₁₆
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ C₂×C₂²⋊C₄

Complements

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

N	H ≅ G/N	type
1. C₂×D₁₁₃⋊M₁₆	C₂	semidirect
2. C₂²×#14268656	C₂	semidirect
3. #20299631	C₂	semidirect
4. #20299630	C₂	semidirect
5. #20299627	C₂	semidirect
6. #20299625	C₂	semidirect
7. #20299601	C₂	semidirect
8. #20299570	C₂	semidirect
9. #20299568	C₂	semidirect
10. C₄×#14268656	C₂	semidirect
12. C₂×#14268657	C₂²	semidirect
19. C₂×#14268656	C₂²	semidirect
20. C₂×#14268655	C₂²	semidirect
21. C₂²×#11744753	C₂²	semidirect
22. #16813689	C₂²	semidirect
23. C₄⋊#11744753	C₂²	semidirect
24. #16813679	C₂²	semidirect
25. #16813679	C₂²	semidirect
26. C₂²⋊#11744753	C₂²	semidirect
27. C₂²⋊#11744753	C₂²	semidirect
28. #16813677	C₂²	semidirect
29. C₄×#11744753	C₂²	semidirect
50. #14268656	D₄	semidirect
117. C₂²⋊D₁₁₃	M₁₆	semidirect
143. #11744753	C₂×D₄	semidirect
153. C₂²×C₁₁₃	C₂×M₁₆	semidirect
161. C₂×D₁₁₃	C₂×M₁₆	semidirect
170. Dic₁₁₃	C₂×M₁₆	semidirect
182. C₁₁₃	D₄×M₁₆	semidirect
342. C₂²	D₁₁₃⋊M₁₆	semidirect

Maximal subgroups

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

subgroup	index
C₂×D₁₁₃⋊M₁₆	2
C₂²×#14268656	2
#20299631	2
#20299630	2
#20299627	2
#20299625	2
#20299601	2
#20299570	2
#20299568	2
C₄×#14268656	2
C₂×#16813582	2
D₄×M₁₆	113