Subgroups — #28399584

Order 14464.

406 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₄×#14268646	C₂	1	1	yes	maximal, characteristic
2	#20298727	C₂	1	1	yes	maximal, characteristic, complemented
3	#20298726	C₂	1	1	yes	maximal, characteristic, complemented
4	#20298672	C₂	1	1	yes	maximal, characteristic, complemented
5	#20298670	C₂	1	1	yes	maximal, characteristic, complemented
6	#20298667	C₂	1	1	yes	maximal, characteristic, complemented
7	C₈×C₂²⋊D₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#20298661	C₂	1	1	yes	maximal, characteristic, complemented
9	#20298654	C₂	1	1	yes	maximal, characteristic, complemented
10	#20298652	C₂	1	1	yes	maximal, characteristic, complemented
11	#20298652	C₂	1	1	yes	maximal, characteristic, complemented
12	M₁₆×Dic₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#16813532	C₂	1	1	yes	maximal, characteristic, complemented
14	#97×D₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
15	#435×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
16	C₂×#14268646	C₄	1	1	yes	characteristic
17	C₂×#14268631	C₂²	1	1	yes	characteristic
18	C₂²×#11744751	C₂²	1	1	yes	characteristic
19	C₂×C₈×D₁₁₃	C₂²	1	1	yes	characteristic
20	C₂×C₈×D₁₁₃	C₂²	1	1	yes	characteristic
21	C₄×Dic₂₂₆	C₂²	1	1	yes	characteristic
22	#16813587	C₄	1	1	yes	characteristic
23	D₄×Dic₁₁₃	C₄	1	1	yes	characteristic
24	C₄×C₂²⋊D₁₁₃	C₂²	1	1	yes	characteristic
25	C₄×C₂²⋊D₁₁₃	C₂²	1	1	yes	characteristic
26	#16813582	C₄	1	1	yes	characteristic
27	#16813582	C₄	1	1	yes	characteristic
28	#16813580	C₄	1	1	yes	characteristic
29	#16813578	C₄	1	1	yes	characteristic
30	#16813578	C₄	1	1	yes	characteristic
31	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
32	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
33	C₄²×D₁₁₃	C₂²	1	1	yes	characteristic
34	C₄⋊#11744751	C₂²	1	1	yes	characteristic
35	#16813551	C₂²	1	1	yes	characteristic
36	#16813551	C₂²	1	1	yes	characteristic
37	#16813551	C₂²	1	1	yes	characteristic
38	#16813551	C₂²	1	1	yes	characteristic
39	C₂²⋊#11744751	C₂²	1	1	yes	characteristic
40	C₂²⋊#11744751	C₂²	1	1	yes	characteristic
41	#16813534	C₂²	1	1	yes	characteristic
42	#16813534	C₂²	1	1	yes	characteristic
43	#16813533	C₂²	1	1	yes	characteristic
44	#16813532	C₂²	1	1	yes	characteristic
45	#16813532	C₂²	1	1	yes	characteristic
46	#16813532	C₂²	1	1	yes	characteristic
47	#16813532	C₂²	1	1	yes	characteristic
48	#16813532	C₂²	1	1	yes	characteristic
49	#16813532	C₂²	1	1	yes	characteristic
50	C₈×Dic₁₁₃	C₂²	1	1	yes	characteristic
51	C₈×Dic₁₁₃	C₂²	1	1	yes	characteristic
52	C₂×M₁₆×C₁₁₃	C₂²	1	1	yes	characteristic
53	C₂²×C₈×C₁₁₃	C₂²	1	1	yes	characteristic
54	C₄×D₄×C₁₁₃	C₂²	1	1	yes	characteristic
55	C₄⋊C₈×C₁₁₃	C₂²	1	1	yes	characteristic
56	C₂²⋊C₈×C₁₁₃	C₂²	1	1	yes	characteristic
57	C₂²⋊C₈×C₁₁₃	C₂²	1	1	yes	characteristic
58	#97×C₁₁₃	C₂²	1	1	yes	characteristic
59	#14268646	—	2	4	no	—
60	C₂×Dic₂₂₆	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	C₂×C₄	1	1	yes	characteristic
64	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
65	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
66	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
67	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
68	C₂×C₄×D₁₁₃	C₂×C₄	1	1	yes	characteristic
69	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
70	#14268631	—	2	2	no	—
71	C₂×#11744751	—	2	2	no	—
72	C₂×#11744751	C₂³	1	1	yes	characteristic
73	C₂×#11744751	C₂³	1	1	yes	characteristic
74	C₂×#11744751	C₂³	1	1	yes	characteristic
75	C₂×#11744751	C₂³	1	1	yes	characteristic
76	C₈×D₁₁₃	—	2	2	no	—
77	C₈×D₁₁₃	—	2	2	no	—
78	C₄⋊Dic₁₁₃	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	#14268626	C₂×C₄	1	1	yes	characteristic
83	C₂²⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
84	C₂²⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
85	#14268624	C₂×C₄	1	1	yes	characteristic
86	#14268624	C₂×C₄	1	1	yes	characteristic
87	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
88	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
89	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
90	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
91	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
92	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
93	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
94	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
95	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
96	C₂×D₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
97	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
98	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
99	M₁₆×C₁₁₃	—	2	2	no	—
100	C₂×C₈×C₁₁₃	—	2	2	no	—
101	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
102	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
103	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
104	C₂×C₈×C₁₁₃	C₂³	1	1	yes	characteristic
105	C₄⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
106	C₂²⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
107	C₂²⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
108	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
109	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
111	C₂²×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
112	D₄×C₁₁₃	—	2	2	no	—
113	C₂×C₄×C₁₁₃	—	2	1	no	—
114	C₂×C₄×C₁₁₃	—	2	1	no	—
115	C₂×C₄×C₁₁₃	—	2	1	no	—
116	C₂×C₄×C₁₁₃	—	2	1	no	—
117	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
119	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
120	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
121	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₁₁₃	—	2	2	no	—
123	C₂²⋊D₁₁₃	M₁₆	1	4	yes	complemented
124	C₂×Dic₁₁₃	—	2	2	no	—
125	C₂×Dic₁₁₃	—	2	2	no	—
126	C₂×Dic₁₁₃	—	4	1	no	—
127	C₂×Dic₁₁₃	—	4	1	no	—
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	C₂²×C₄	1	1	yes	characteristic
132	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
133	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
134	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
135	C₄×D₁₁₃	—	2	2	no	—
136	C₄×D₁₁₃	—	2	1	no	—
137	C₄×D₁₁₃	—	2	1	no	—
138	C₄×D₁₁₃	—	4	1	no	—
139	Dic₂₂₆	—	2	2	no	—
140	C₈×C₁₁₃	—	2	1	no	—
141	C₈×C₁₁₃	—	2	1	no	—
142	C₈×C₁₁₃	—	2	1	no	—
143	C₈×C₁₁₃	#41	1	2	yes	complemented
144	#11744751	—	2	1	no	—
145	#11744751	—	2	1	no	—
146	#11744751	—	2	1	no	—
147	#11744751	#41	1	2	yes	complemented
148	C₂²×C₁₁₃	—	2	1	no	—
149	C₂²×C₁₁₃	—	2	1	no	—
150	C₂²×C₁₁₃	—	2	1	no	—
151	C₂²×C₁₁₃	—	2	1	no	—
152	C₂²×C₁₁₃	—	2	1	no	—
153	C₂²×C₁₁₃	C₂×M₁₆	1	2	yes	—
154	C₂²×C₁₁₃	C₂³×C₄	1	1	yes	characteristic
155	C₂×D₁₁₃	—	2	1	no	—
156	C₂×D₁₁₃	—	2	1	no	—
157	C₂×D₁₁₃	C₂×M₁₆	1	2	yes	—
158	C₄×C₁₁₃	—	2	1	no	—
159	C₄×C₁₁₃	—	2	1	no	—
160	C₄×C₁₁₃	—	2	1	no	—
161	C₄×C₁₁₃	—	2	1	no	—
162	C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
163	C₄×C₁₁₃	C₂×#41	1	1	yes	characteristic
164	Dic₁₁₃	—	2	1	no	—
165	Dic₁₁₃	—	2	1	no	—
166	Dic₁₁₃	—	2	1	no	—
167	Dic₁₁₃	—	2	1	no	—
168	Dic₁₁₃	—	2	1	no	—
169	Dic₁₁₃	—	2	1	no	—
170	Dic₁₁₃	C₂×M₁₆	1	2	yes	—
171	C₂×C₁₁₃	—	2	2	no	—
172	C₂×C₁₁₃	—	4	1	no	—
173	C₂×C₁₁₃	C₄×#41	1	1	yes	characteristic
174	C₂×C₁₁₃	C₂²×M₁₆	1	1	yes	characteristic
175	C₂×C₁₁₃	#568	1	1	yes	characteristic
176	D₁₁₃	—	2	2	no	—
177	#2970	—	113	1	no	maximal, Sylow
178	C₁₁₃	#2970	1	1	yes	characteristic, complemented, Sylow
179	C₄×#41	—	113	1	no	—
180	#446	—	113	1	no	—
181	#446	—	113	1	no	—
182	#435	—	113	1	no	—
183	#435	—	113	1	no	—
184	#435	—	113	1	no	—
185	#435	D₁₁₃	1	1	yes	characteristic, complemented
186	C₈×D₄	—	113	1	no	—
187	C₈×D₄	—	113	1	no	—
188	#432	—	113	1	no	—
189	#432	—	113	1	no	—
190	#432	—	113	1	no	—
191	C₄×M₁₆	—	113	1	no	—
192	C₂×#97	—	113	1	no	—
193	C₂×#97	—	113	1	no	—
194	C₂×#41	—	113	1	no	—
195	C₂×M₁₆	—	113	1	no	—
196	C₂×M₁₆	C₂×D₁₁₃	1	1	yes	characteristic
197	C₂²×C₈	—	113	1	no	—
198	C₂²×C₈	—	113	1	no	—
199	C₂²×C₈	—	113	1	no	—
200	C₂²×C₈	C₂×D₁₁₃	1	1	yes	characteristic
201	C₄×Q₈	—	113	1	no	—
202	C₄×Q₈	—	113	1	no	—
203	C₄×D₄	—	113	1	no	—
204	C₄×D₄	—	113	1	no	—
205	C₄×D₄	—	113	1	no	—
206	C₄×D₄	—	113	1	no	—
207	C₄×D₄	—	113	1	no	—
208	C₄×D₄	C₂×D₁₁₃	1	1	yes	characteristic
209	#117	—	113	1	no	—
210	#117	—	113	1	no	—
211	#117	—	113	1	no	—
212	C₂×C₄²	—	113	1	no	—
213	C₂×C₄²	—	113	1	no	—
214	C₂×C₄²	—	113	1	no	—
215	C₄⋊C₈	—	113	1	no	—
216	C₄⋊C₈	—	113	1	no	—
217	C₄⋊C₈	—	113	1	no	—
218	C₄⋊C₈	—	113	1	no	—
219	C₄⋊C₈	—	113	1	no	—
220	C₄⋊C₈	C₂×D₁₁₃	1	1	yes	characteristic
221	C₂²⋊C₈	—	113	1	no	—
222	C₂²⋊C₈	—	113	1	no	—
223	C₂²⋊C₈	—	113	1	no	—
224	C₂²⋊C₈	—	113	1	no	—
225	C₂²⋊C₈	C₂×D₁₁₃	1	1	yes	characteristic
226	C₂²⋊C₈	C₂×D₁₁₃	1	1	yes	characteristic
227	#97	—	113	1	no	—
228	#97	—	113	1	no	—
229	#97	—	113	1	no	—
230	#97	—	113	1	no	—
231	#97	—	113	1	no	—
232	#97	—	113	1	no	—
233	#97	—	113	1	no	—
234	#97	C₂×D₁₁₃	1	1	yes	characteristic
235	C₄×C₈	—	113	1	no	—
236	C₄×C₈	—	113	1	no	—
237	#41	—	226	4	no	—
238	C₂×Q₈	—	113	1	no	—
239	C₂×D₄	—	113	1	no	—
240	C₂×D₄	—	113	1	no	—
241	C₂×D₄	C₄×D₁₁₃	1	1	yes	characteristic
242	C₂²×C₄	—	113	1	no	—
243	C₂²×C₄	—	113	1	no	—
244	C₂²×C₄	—	113	1	no	—
245	C₂²×C₄	—	113	1	no	—
246	C₂²×C₄	—	113	1	no	—
247	C₂²×C₄	—	113	1	no	—
248	C₂²×C₄	—	113	1	no	—
249	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
250	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
251	M₁₆	—	2	2	no	—
252	M₁₆	—	226	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	—	113	1	no	—
255	C₂×C₈	—	113	1	no	—
256	C₂×C₈	—	113	1	no	—
257	C₂×C₈	—	113	1	no	—
258	C₂×C₈	—	226	2	no	—
259	C₂×C₈	—	226	2	no	—
260	C₂×C₈	—	226	2	no	—
261	C₂×C₈	C₂²×D₁₁₃	1	1	yes	characteristic
262	C₂×C₈	C₂²×D₁₁₃	1	1	yes	characteristic
263	C₂×C₈	C₂²×D₁₁₃	1	1	yes	characteristic
264	C₂×C₈	C₂²×D₁₁₃	1	1	yes	characteristic
265	C₄⋊C₄	—	113	1	no	—
266	C₄⋊C₄	—	113	1	no	—
267	C₄⋊C₄	—	113	1	no	—
268	C₄⋊C₄	—	113	1	no	—
269	C₄⋊C₄	—	113	1	no	—
270	C₄⋊C₄	C₄×D₁₁₃	1	1	yes	characteristic
271	C₂²⋊C₄	—	113	1	no	—
272	C₂²⋊C₄	—	113	1	no	—
273	C₂²⋊C₄	—	113	1	no	—
274	C₂²⋊C₄	—	113	1	no	—
275	C₂²⋊C₄	C₄×D₁₁₃	1	1	yes	characteristic
276	C₂²⋊C₄	C₄×D₁₁₃	1	1	yes	characteristic
277	C₄²	—	113	1	no	—
278	C₄²	—	113	1	no	—
279	C₄²	—	113	1	no	—
280	C₄²	—	113	1	no	—
281	C₄²	—	113	1	no	—
282	C₄²	—	113	1	no	—
283	C₄²	—	113	1	no	—
284	C₄²	—	113	1	no	—
285	C₄²	—	113	1	no	—
286	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
287	C₂³	—	113	1	no	—
288	C₂³	C₂×C₄×D₁₁₃	1	1	yes	characteristic
289	C₂³	C₂×C₄×D₁₁₃	1	1	yes	characteristic
290	Q₈	—	226	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	113	4	no	—
293	D₄	—	226	2	no	—
294	C₂×C₄	—	2	1	no	—
295	C₂×C₄	—	2	1	no	—
296	C₂×C₄	—	2	1	no	—
297	C₂×C₄	—	2	1	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	—	113	1	no	—
305	C₂×C₄	—	452	1	no	—
306	C₂×C₄	—	452	1	no	—
307	C₂×C₄	—	452	1	no	—
308	C₂×C₄	—	226	2	no	—
309	C₂×C₄	—	226	2	no	—
310	C₂×C₄	—	226	2	no	—
311	C₂×C₄	—	226	1	no	—
312	C₂×C₄	—	226	1	no	—
313	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
314	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
315	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
316	C₂×C₄	C₂×C₄×D₁₁₃	1	1	yes	characteristic
317	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
318	C₈	—	2	1	no	—
319	C₈	—	2	1	no	—
320	C₈	—	2	1	no	—
321	C₈	—	113	2	no	—
322	C₈	—	226	1	no	—
323	C₈	—	226	1	no	—
324	C₈	—	226	1	no	—
325	C₈	#14268646	1	2	yes	complemented
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	—	113	2	no	—
332	C₂²	—	226	1	no	—
333	C₂²	—	226	1	no	—
334	C₂²	M₁₆×D₁₁₃	1	2	yes	—
335	C₂²	C₂²×C₄×D₁₁₃	1	1	yes	characteristic
336	C₄	—	2	1	no	—
337	C₄	—	2	1	no	—
338	C₄	—	2	1	no	—
339	C₄	—	2	1	no	—
340	C₄	—	113	2	no	—
341	C₄	—	226	1	no	—
342	C₄	—	226	1	no	—
343	C₄	—	226	1	no	—
344	C₄	—	226	1	no	—
345	C₄	—	226	1	no	—
346	C₄	—	226	1	no	—
347	C₄	#41×D₁₁₃	1	1	yes	characteristic
348	C₄	C₂×#14268646	1	1	yes	characteristic
349	C₂	—	2	2	no	—
350	C₂	—	4	1	no	—
351	C₂	—	226	2	no	—
352	C₂	C₄×#14268646	1	1	yes	characteristic
353	C₂	C₂×M₁₆×D₁₁₃	1	1	yes	characteristic
354	C₂	#20299363	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂×C₄	C₂³×D₁₁₃
commutator, socle	C₂²×C₁₁₃	C₂³×C₄
Fitting	#435×C₁₁₃	C₂
prime radical	C₂×#14268646	C₄
cosocle	C₂×C₄×C₁₁₃	C₂⁴
hypercenter	#435	D₁₁₃
hypocenter	C₁₁₃	#2970
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #435

Complements

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

N	H ≅ G/N	type
2. #20298727	C₂	semidirect
3. #20298726	C₂	semidirect
4. #20298672	C₂	semidirect
5. #20298670	C₂	semidirect
6. #20298667	C₂	semidirect
7. C₈×C₂²⋊D₁₁₃	C₂	semidirect
8. #20298661	C₂	semidirect
9. #20298654	C₂	semidirect
10. #20298652	C₂	semidirect
11. #20298652	C₂	semidirect
12. M₁₆×Dic₁₁₃	C₂	semidirect
13. C₂×#16813532	C₂	semidirect
14. #97×D₁₁₃	C₂	semidirect
15. #435×C₁₁₃	C₂	semidirect
123. C₂²⋊D₁₁₃	M₁₆	semidirect
143. C₈×C₁₁₃	#41	semidirect
147. #11744751	#41	semidirect
178. C₁₁₃	#2970	semidirect
185. #435	D₁₁₃	semidirect
325. C₈	#14268646	semidirect

Maximal subgroups

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

subgroup	index
C₄×#14268646	2
#20298727	2
#20298726	2
#20298672	2
#20298670	2
#20298667	2
C₈×C₂²⋊D₁₁₃	2
#20298661	2
#20298654	2
#20298652	2
M₁₆×Dic₁₁₃	2
C₂×#16813532	2
#97×D₁₁₃	2
#435×C₁₁₃	2
#2970	113