Subgroups — #562×D₁₁₃ #28404696

Order 14464.

1152 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299346	C₂	1	1	yes	maximal, characteristic, complemented
2	#20299345	C₂	1	4	yes	maximal, complemented
3	#20299344	C₂	1	4	yes	maximal, complemented
4	#20299343	C₂	1	4	yes	maximal, complemented
5	#20299342	C₂	1	2	yes	maximal, complemented
6	#125×D₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
7	#124×D₁₁₃	C₂	1	2	yes	maximal, complemented
8	#123×D₁₁₃	C₂	1	4	yes	maximal, complemented
9	C₂²⋊Q₈×D₁₁₃	C₂	1	4	yes	maximal, complemented
10	#121×D₁₁₃	C₂	1	4	yes	maximal, complemented
11	#562×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×Q₈×D₁₁₃	C₂²	1	2	yes	—
13	C₂×D₄×D₁₁₃	C₂²	1	4	yes	—
14	C₂×D₄×D₁₁₃	C₂²	1	2	yes	—
15	C₂²×C₄×D₁₁₃	C₂²	1	4	yes	—
16	#16813614	C₂²	1	1	yes	characteristic, complemented
17	#16813613	C₂²	1	2	yes	complemented
18	#16813612	C₂²	1	4	yes	complemented
19	#16813611	C₂²	1	2	yes	complemented
20	#16813610	C₂²	1	2	yes	complemented
21	#16813609	C₂²	1	2	yes	complemented
22	#16813608	C₂²	1	4	yes	complemented
23	#16813608	C₂²	1	4	yes	complemented
24	#16813607	C₂²	1	4	yes	complemented
25	#16813607	C₂²	1	4	yes	complemented
26	#16813606	C₂²	1	4	yes	complemented
27	#16813605	C₂²	1	4	yes	complemented
28	#16813604	C₂²	1	4	yes	complemented
29	#16813604	C₂²	1	4	yes	complemented
30	#16813603	C₂²	1	4	yes	complemented
31	#16813602	C₂²	1	4	yes	complemented
32	#16813601	C₂²	1	4	yes	complemented
33	#16813600	C₂²	1	4	yes	complemented
34	#16813600	C₂²	1	4	yes	complemented
35	#16813599	C₂²	1	4	yes	complemented
36	#16813599	C₂²	1	4	yes	complemented
37	#16813598	C₂²	1	4	yes	complemented
38	#16813597	C₂²	1	4	yes	complemented
39	#16813596	C₂²	1	4	yes	complemented
40	#16813595	C₂²	1	4	yes	complemented
41	#16813595	C₂²	1	4	yes	complemented
42	#16813594	C₂²	1	4	yes	complemented
43	#16813594	C₂²	1	4	yes	complemented
44	#16813593	C₂²	1	4	yes	complemented
45	C₄⋊C₄×D₁₁₃	C₂²	1	4	yes	—
46	C₄⋊C₄×D₁₁₃	C₂²	1	4	yes	—
47	C₄⋊C₄×D₁₁₃	C₂²	1	2	yes	—
48	C₂²⋊C₄×D₁₁₃	C₂²	1	4	yes	—
49	C₂²⋊C₄×D₁₁₃	C₂²	1	4	yes	—
50	C₂²⋊C₄×D₁₁₃	C₂²	1	4	yes	—
51	C₄²×D₁₁₃	C₂²	1	1	yes	characteristic
52	#125×C₁₁₃	C₂²	1	1	yes	characteristic, complemented
53	#124×C₁₁₃	C₂²	1	2	yes	complemented
54	#123×C₁₁₃	C₂²	1	4	yes	complemented
55	C₂²⋊Q₈×C₁₁₃	C₂²	1	4	yes	complemented
56	#121×C₁₁₃	C₂²	1	4	yes	complemented
57	C₂³×D₁₁₃	C₂³	1	4	yes	—
58	C₂×Dic₂₂₆	C₂³	1	2	yes	—
59	C₂×Dic₂₂₆	C₂³	1	4	yes	—
60	Q₈×D₁₁₃	—	4	4	no	—
61	C₂×D₄₅₂	C₂³	1	2	yes	—
62	C₂×D₄₅₂	C₂³	1	4	yes	—
63	C₂×C₂²⋊D₁₁₃	C₂³	1	4	yes	—
64	C₂×C₂²⋊D₁₁₃	C₂³	1	4	yes	—
65	C₂×C₂²⋊D₁₁₃	C₂³	1	4	yes	—
66	D₄×D₁₁₃	—	4	8	no	—
67	D₄×D₁₁₃	—	4	4	no	—
68	C₂²×Dic₁₁₃	C₂³	1	4	yes	—
69	C₂×C₄×D₁₁₃	—	4	8	no	—
70	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
71	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
72	C₂×C₄×D₁₁₃	C₂³	1	4	yes	—
73	C₂×C₄×D₁₁₃	C₂³	1	4	yes	—
74	C₂×C₄×D₁₁₃	C₂³	1	4	yes	—
75	C₂×C₄×D₁₁₃	C₂³	1	4	yes	—
76	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
77	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
78	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
79	#14268626	C₂³	1	4	yes	—
80	#14268626	C₂³	1	4	yes	—
81	#14268626	C₂³	1	4	yes	—
82	#14268626	C₂³	1	4	yes	—
83	#14268626	C₂³	1	4	yes	—
84	C₂²⋊Dic₁₁₃	C₂³	1	4	yes	—
85	C₂²⋊Dic₁₁₃	C₂³	1	4	yes	—
86	C₂²⋊Dic₁₁₃	C₂³	1	4	yes	—
87	#14268624	C₂³	1	4	yes	—
88	#14268624	C₂³	1	4	yes	—
89	#14268624	C₂³	1	4	yes	—
90	#14268624	C₂³	1	4	yes	—
91	#14268624	C₂³	1	4	yes	—
92	#14268624	C₂³	1	4	yes	—
93	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
94	C₄×Dic₁₁₃	C₂³	1	2	yes	—
95	C₂×Q₈×C₁₁₃	C₂³	1	2	yes	—
96	C₂×D₄×C₁₁₃	C₂³	1	4	yes	—
97	C₂×D₄×C₁₁₃	C₂³	1	2	yes	—
98	C₂²×C₄×C₁₁₃	C₂³	1	4	yes	—
99	C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
100	C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
101	C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
102	C₂²⋊C₄×C₁₁₃	C₂³	1	4	yes	—
103	C₂²⋊C₄×C₁₁₃	C₂³	1	4	yes	—
104	C₂²⋊C₄×C₁₁₃	C₂³	1	4	yes	—
105	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
106	C₂³×C₁₁₃	C₂⁴	1	4	yes	—
107	C₂²×D₁₁₃	—	2	8	no	—
108	C₂²×D₁₁₃	—	2	8	no	—
109	C₂²×D₁₁₃	—	2	8	no	—
110	C₂²×D₁₁₃	C₂⁴	1	4	yes	—
111	C₂²×D₁₁₃	C₂⁴	1	1	yes	characteristic
112	Q₈×C₁₁₃	—	4	2	no	—
113	D₄×C₁₁₃	—	4	2	no	—
114	D₄×C₁₁₃	—	4	4	no	—
115	C₂×C₄×C₁₁₃	—	4	4	no	—
116	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
117	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
118	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
119	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
120	C₂²⋊D₁₁₃	—	4	4	no	—
121	C₂²⋊D₁₁₃	—	4	4	no	—
122	C₂²⋊D₁₁₃	—	4	4	no	—
123	C₂×Dic₁₁₃	—	4	4	no	—
124	C₂×Dic₁₁₃	C₂⁴	1	1	yes	characteristic
125	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
126	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
127	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
128	D₄₅₂	—	4	2	no	—
129	D₄₅₂	—	4	4	no	—
130	C₄×D₁₁₃	—	2	8	no	—
131	C₄×D₁₁₃	—	2	8	no	—
132	C₄×D₁₁₃	—	2	4	no	—
133	C₄×D₁₁₃	—	2	2	no	—
134	C₄×D₁₁₃	—	4	4	no	—
135	C₄×D₁₁₃	—	4	4	no	—
136	Dic₂₂₆	—	4	2	no	—
137	Dic₂₂₆	—	4	4	no	—
138	C₂²×C₁₁₃	—	2	4	no	—
139	C₂²×C₁₁₃	—	2	4	no	—
140	C₂²×C₁₁₃	—	2	4	no	—
141	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
142	C₂×D₁₁₃	—	2	4	no	—
143	C₂×D₁₁₃	—	2	4	no	—
144	C₂×D₁₁₃	—	2	4	no	—
145	C₂×D₁₁₃	—	4	16	no	—
146	C₂×D₁₁₃	H(5,𝔽₂)	1	4	yes	—
147	C₂×D₁₁₃	2₋¹⁺⁴	1	2	yes	—
148	C₄×C₁₁₃	—	2	4	no	—
149	C₄×C₁₁₃	—	2	2	no	—
150	C₄×C₁₁₃	—	2	4	no	—
151	C₄×C₁₁₃	—	2	1	no	—
152	Dic₁₁₃	—	2	4	no	—
153	Dic₁₁₃	—	2	2	no	—
154	Dic₁₁₃	—	2	4	no	—
155	Dic₁₁₃	—	2	1	no	—
156	C₂×C₁₁₃	—	4	4	no	—
157	C₂×C₁₁₃	C₂×H(5,𝔽₂)	1	2	yes	—
158	C₂×C₁₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
159	D₁₁₃	—	4	4	no	—
160	D₁₁₃	#562	1	4	yes	direct product factor
161	C₂×#562	—	113	1	no	maximal, Sylow
162	C₁₁₃	C₂×#562	1	1	yes	characteristic, complemented, Sylow
163	#562	—	113	1	no	—
164	#562	—	113	2	no	—
165	#562	—	113	4	no	—
166	#562	—	113	4	no	—
167	#562	—	113	4	no	—
168	#562	D₁₁₃	1	1	yes	characteristic, direct product factor
169	C₂×#125	—	113	1	no	—
170	C₂×#124	—	113	2	no	—
171	C₂×#123	—	113	4	no	—
172	C₂×C₂²⋊Q₈	—	113	4	no	—
173	C₂×#121	—	113	4	no	—
174	C₂²×Q₈	—	113	2	no	—
175	C₂²×D₄	—	113	4	no	—
176	C₂²×D₄	—	113	2	no	—
177	C₂³×C₄	—	113	4	no	—
178	#125	—	113	4	no	—
179	#125	—	113	2	no	—
180	#125	—	113	1	no	—
181	#125	C₂×D₁₁₃	1	1	yes	characteristic, complemented
182	#124	—	113	2	no	—
183	#124	—	113	2	no	—
184	#124	—	113	4	no	—
185	#124	—	113	4	no	—
186	#124	—	113	2	no	—
187	#124	C₂×D₁₁₃	1	2	yes	complemented
188	#123	—	113	4	no	—
189	#123	—	113	4	no	—
190	#123	—	113	4	no	—
191	#123	—	113	4	no	—
192	#123	—	113	4	no	—
193	#123	—	113	4	no	—
194	#123	—	113	4	no	—
195	#123	C₂×D₁₁₃	1	4	yes	complemented
196	C₂²⋊Q₈	—	113	4	no	—
197	C₂²⋊Q₈	—	113	4	no	—
198	C₂²⋊Q₈	—	113	4	no	—
199	C₂²⋊Q₈	—	113	4	no	—
200	C₂²⋊Q₈	—	113	4	no	—
201	C₂²⋊Q₈	—	113	4	no	—
202	C₂²⋊Q₈	—	113	4	no	—
203	C₂²⋊Q₈	C₂×D₁₁₃	1	4	yes	complemented
204	#121	—	113	4	no	—
205	#121	—	113	4	no	—
206	#121	—	113	4	no	—
207	#121	—	113	4	no	—
208	#121	—	113	4	no	—
209	#121	—	113	4	no	—
210	#121	—	113	4	no	—
211	#121	C₂×D₁₁₃	1	4	yes	complemented
212	C₂×C₄⋊C₄	—	113	4	no	—
213	C₂×C₄⋊C₄	—	113	4	no	—
214	C₂×C₄⋊C₄	—	113	2	no	—
215	C₂×C₂²⋊C₄	—	113	4	no	—
216	C₂×C₂²⋊C₄	—	113	4	no	—
217	C₂×C₂²⋊C₄	—	113	4	no	—
218	C₂×C₄²	—	113	1	no	—
219	C₂⁴	—	113	4	no	—
220	C₂×Q₈	—	113	4	no	—
221	C₂×Q₈	—	113	2	no	—
222	C₂×Q₈	—	452	4	no	—
223	C₂×Q₈	C₂²×D₁₁₃	1	2	yes	—
224	C₂×D₄	—	113	4	no	—
225	C₂×D₄	—	113	4	no	—
226	C₂×D₄	—	113	4	no	—
227	C₂×D₄	—	113	4	no	—
228	C₂×D₄	—	113	2	no	—
229	C₂×D₄	—	452	8	no	—
230	C₂×D₄	—	452	4	no	—
231	C₂×D₄	C₂²×D₁₁₃	1	4	yes	—
232	C₂×D₄	C₂²×D₁₁₃	1	2	yes	—
233	C₂²×C₄	—	113	4	no	—
234	C₂²×C₄	—	113	4	no	—
235	C₂²×C₄	—	113	4	no	—
236	C₂²×C₄	—	113	1	no	—
237	C₂²×C₄	—	113	2	no	—
238	C₂²×C₄	—	113	4	no	—
239	C₂²×C₄	—	113	4	no	—
240	C₂²×C₄	—	452	8	no	—
241	C₂²×C₄	C₂²×D₁₁₃	1	4	yes	—
242	C₄⋊C₄	—	113	4	no	—
243	C₄⋊C₄	—	113	4	no	—
244	C₄⋊C₄	—	113	4	no	—
245	C₄⋊C₄	—	113	4	no	—
246	C₄⋊C₄	—	113	4	no	—
247	C₄⋊C₄	—	113	4	no	—
248	C₄⋊C₄	—	113	4	no	—
249	C₄⋊C₄	—	113	2	no	—
250	C₄⋊C₄	C₂²×D₁₁₃	1	4	yes	—
251	C₄⋊C₄	C₂²×D₁₁₃	1	4	yes	—
252	C₄⋊C₄	C₂²×D₁₁₃	1	2	yes	—
253	C₂²⋊C₄	—	113	4	no	—
254	C₂²⋊C₄	—	113	4	no	—
255	C₂²⋊C₄	—	113	4	no	—
256	C₂²⋊C₄	—	113	4	no	—
257	C₂²⋊C₄	—	113	4	no	—
258	C₂²⋊C₄	—	113	4	no	—
259	C₂²⋊C₄	—	113	4	no	—
260	C₂²⋊C₄	—	113	4	no	—
261	C₂²⋊C₄	—	113	4	no	—
262	C₂²⋊C₄	C₂²×D₁₁₃	1	4	yes	—
263	C₂²⋊C₄	C₂²×D₁₁₃	1	4	yes	—
264	C₂²⋊C₄	C₂²×D₁₁₃	1	4	yes	—
265	C₄²	—	113	2	no	—
266	C₄²	—	113	1	no	—
267	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
268	C₂³	—	113	1	no	—
269	C₂³	—	113	4	no	—
270	C₂³	—	226	8	no	—
271	C₂³	—	226	8	no	—
272	C₂³	—	226	8	no	—
273	C₂³	C₂³×D₁₁₃	1	4	yes	—
274	Q₈	—	4	2	no	—
275	Q₈	—	452	4	no	—
276	Q₈	—	452	2	no	—
277	D₄	—	4	4	no	—
278	D₄	—	4	2	no	—
279	D₄	—	452	4	no	—
280	D₄	—	452	2	no	—
281	D₄	—	452	4	no	—
282	D₄	—	452	4	no	—
283	D₄	—	452	4	no	—
284	C₂×C₄	—	4	4	no	—
285	C₂×C₄	—	113	1	no	—
286	C₂×C₄	—	113	2	no	—
287	C₂×C₄	—	113	4	no	—
288	C₂×C₄	—	113	4	no	—
289	C₂×C₄	—	452	4	no	—
290	C₂×C₄	—	452	4	no	—
291	C₂×C₄	—	452	4	no	—
292	C₂×C₄	—	226	8	no	—
293	C₂×C₄	—	226	8	no	—
294	C₂×C₄	—	226	4	no	—
295	C₂×C₄	—	226	2	no	—
296	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
297	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
298	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
299	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
300	C₂²	—	2	4	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	2	4	no	—
303	C₂²	—	113	4	no	—
304	C₂²	—	113	2	no	—
305	C₂²	—	452	16	no	—
306	C₂²	—	226	4	no	—
307	C₂²	—	226	4	no	—
308	C₂²	—	226	4	no	—
309	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
310	C₄	—	2	4	no	—
311	C₄	—	2	4	no	—
312	C₄	—	2	2	no	—
313	C₄	—	2	1	no	—
314	C₄	—	226	4	no	—
315	C₄	—	226	4	no	—
316	C₄	—	226	2	no	—
317	C₄	—	226	1	no	—
318	C₂	—	4	4	no	—
319	C₂	—	113	4	no	—
320	C₂	—	452	4	no	—
321	C₂	H(5,𝔽₂)×D₁₁₃	1	2	yes	—
322	C₂	2₋¹⁺⁴×D₁₁₃	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₁₃
commutator, socle, cosocle	C₂²×C₁₁₃	C₂⁵
Fitting	#562×C₁₁₃	C₂
hypercenter	#562	D₁₁₃
hypocenter	C₁₁₃	C₂×#562
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ #562

Complements

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

N	H ≅ G/N	type
1. #20299346	C₂	semidirect
2. #20299345	C₂	semidirect
3. #20299344	C₂	semidirect
4. #20299343	C₂	semidirect
5. #20299342	C₂	semidirect
6. #125×D₁₁₃	C₂	semidirect
7. #124×D₁₁₃	C₂	semidirect
8. #123×D₁₁₃	C₂	semidirect
9. C₂²⋊Q₈×D₁₁₃	C₂	semidirect
10. #121×D₁₁₃	C₂	semidirect
11. #562×C₁₁₃	C₂	semidirect
16. #16813614	C₂²	semidirect
17. #16813613	C₂²	semidirect
18. #16813612	C₂²	semidirect
19. #16813611	C₂²	semidirect
20. #16813610	C₂²	semidirect
21. #16813609	C₂²	semidirect
22. #16813608	C₂²	semidirect
23. #16813608	C₂²	semidirect
24. #16813607	C₂²	semidirect
25. #16813607	C₂²	semidirect
26. #16813606	C₂²	semidirect
27. #16813605	C₂²	semidirect
28. #16813604	C₂²	semidirect
29. #16813604	C₂²	semidirect
30. #16813603	C₂²	semidirect
31. #16813602	C₂²	semidirect
32. #16813601	C₂²	semidirect
33. #16813600	C₂²	semidirect
34. #16813600	C₂²	semidirect
35. #16813599	C₂²	semidirect
36. #16813599	C₂²	semidirect
37. #16813598	C₂²	semidirect
38. #16813597	C₂²	semidirect
39. #16813596	C₂²	semidirect
40. #16813595	C₂²	semidirect
41. #16813595	C₂²	semidirect
42. #16813594	C₂²	semidirect
43. #16813594	C₂²	semidirect
44. #16813593	C₂²	semidirect
52. #125×C₁₁₃	C₂²	semidirect
53. #124×C₁₁₃	C₂²	semidirect
54. #123×C₁₁₃	C₂²	semidirect
55. C₂²⋊Q₈×C₁₁₃	C₂²	semidirect
56. #121×C₁₁₃	C₂²	semidirect
160. D₁₁₃	#562	direct (N × H)
162. C₁₁₃	C₂×#562	semidirect
168. #562	D₁₁₃	direct (N × H)
181. #125	C₂×D₁₁₃	semidirect
187. #124	C₂×D₁₁₃	semidirect
195. #123	C₂×D₁₁₃	semidirect
203. C₂²⋊Q₈	C₂×D₁₁₃	semidirect
211. #121	C₂×D₁₁₃	semidirect

Maximal subgroups

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

subgroup	index
#20299346	2
#20299345	2
#20299344	2
#20299343	2
#20299342	2
#125×D₁₁₃	2
#124×D₁₁₃	2
#123×D₁₁₃	2
C₂²⋊Q₈×D₁₁₃	2
#121×D₁₁₃	2
#562×C₁₁₃	2
C₂×#562	113

Subgroups — #562×D113 #28404696

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — #562×D₁₁₃ #28404696