Subgroups — #28403869

Order 14464.

1212 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299155	C₂	1	1	yes	maximal, characteristic, complemented
2	#20299153	C₂	1	2	yes	maximal, complemented
3	#20299146	C₂	1	2	yes	maximal, complemented
4	#20299130	C₂	1	4	yes	maximal, complemented
5	#20299129	C₂	1	4	yes	maximal, complemented
6	#20299127	C₂	1	4	yes	maximal, complemented
7	#20299108	C₂	1	4	yes	maximal, complemented
8	#20299105	C₂	1	2	yes	maximal, complemented
9	#41×Dic₁₁₃	C₂	1	2	yes	maximal, complemented
10	C₄×#14268646	C₂	1	4	yes	maximal, complemented
11	C₂×C₄²×D₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
12	#532×C₁₁₃	C₂	1	1	yes	maximal, characteristic, complemented
13	C₂×#14268646	C₂²	1	4	yes	—
14	C₂²×C₄×D₁₁₃	C₂²	1	1	yes	characteristic
15	C₂²×C₄×D₁₁₃	C₂²	1	2	yes	—
16	#16813622	C₂²	1	2	yes	complemented
17	#16813615	C₂²	1	8	yes	complemented
18	#16813614	C₂²	1	2	yes	complemented
19	#16813613	C₂²	1	4	yes	complemented
20	#16813609	C₂²	1	4	yes	complemented
21	#16813604	C₂²	1	8	yes	—
22	#16813603	C₂²	1	4	yes	—
23	#16813602	C₂²	1	4	yes	—
24	#16813601	C₂²	1	4	yes	—
25	#16813598	C₂²	1	4	yes	—
26	#16813593	C₂²	1	4	yes	—
27	#16813593	C₂²	1	4	yes	—
28	#16813590	C₂²	1	4	yes	—
29	Q₈×Dic₁₁₃	C₂²	1	4	yes	complemented
30	C₄×Dic₂₂₆	C₂²	1	4	yes	complemented
31	#16813587	C₂²	1	4	yes	complemented
32	D₄×Dic₁₁₃	C₂²	1	8	yes	—
33	D₄×Dic₁₁₃	C₂²	1	4	yes	complemented
34	D₄×Dic₁₁₃	C₂²	1	4	yes	complemented
35	C₄×C₂²⋊D₁₁₃	C₂²	1	8	yes	—
36	#16813582	C₂²	1	8	yes	—
37	#16813581	C₂²	1	2	yes	—
38	#16813581	C₂²	1	4	yes	complemented
39	#16813580	C₂²	1	4	yes	—
40	#16813578	C₂²	1	8	yes	complemented
41	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
42	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
43	C₂×C₄×Dic₁₁₃	C₂²	1	4	yes	complemented
44	C₄²×D₁₁₃	C₂²	1	2	yes	—
45	C₄²×D₁₁₃	C₂²	1	2	yes	—
46	C₄²×D₁₁₃	C₂²	1	2	yes	—
47	C₄²×D₁₁₃	C₂²	1	2	yes	—
48	C₂×#41×C₁₁₃	C₂²	1	2	yes	—
49	C₄⋊Q₈×C₁₁₃	C₂²	1	1	yes	characteristic, complemented
50	#127×C₁₁₃	C₂²	1	1	yes	characteristic, complemented
51	#124×C₁₁₃	C₂²	1	2	yes	complemented
52	#121×C₁₁₃	C₂²	1	4	yes	—
53	C₄×D₄×C₁₁₃	C₂²	1	4	yes	complemented
54	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
55	C₂³×D₁₁₃	C₂³	1	1	yes	characteristic
56	#14268646	—	2	16	no	—
57	C₂×Dic₂₂₆	C₂³	1	4	yes	—
58	C₂×C₂²⋊D₁₁₃	C₂³	1	8	yes	—
59	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
60	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
61	C₂²×Dic₁₁₃	C₂³	1	8	yes	—
62	C₂×C₄×D₁₁₃	—	2	4	no	—
63	C₂×C₄×D₁₁₃	—	2	8	no	—
64	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
65	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
66	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
67	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
68	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
69	C₂×C₄×D₁₁₃	C₂³	1	2	yes	—
70	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
71	C₄⋊Dic₁₁₃	C₂³	1	4	yes	—
72	C₄⋊Dic₁₁₃	C₂³	1	8	yes	—
73	#14268626	C₂³	1	8	yes	—
74	#14268626	C₂³	1	8	yes	—
75	C₂²⋊Dic₁₁₃	C₂³	1	8	yes	—
76	C₂²⋊Dic₁₁₃	C₂³	1	4	yes	—
77	C₂²⋊Dic₁₁₃	C₂³	1	8	yes	—
78	#14268624	C₂³	1	8	yes	—
79	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
80	C₄×Dic₁₁₃	C₂³	1	2	yes	—
81	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
82	C₄×Dic₁₁₃	C₂³	1	2	yes	—
83	C₄×Dic₁₁₃	C₂³	1	2	yes	—
84	C₄×Dic₁₁₃	C₂³	1	2	yes	—
85	C₄×Dic₁₁₃	C₂³	1	2	yes	—
86	C₄×Dic₁₁₃	C₂³	1	8	yes	—
87	C₄×Dic₁₁₃	C₂³	1	4	yes	—
88	#41×C₁₁₃	—	2	8	no	—
89	C₂×Q₈×C₁₁₃	C₂³	1	2	yes	—
90	C₂×D₄×C₁₁₃	C₂³	1	4	yes	—
91	C₂×D₄×C₁₁₃	C₂³	1	2	yes	—
92	C₂×D₄×C₁₁₃	C₂³	1	4	yes	—
93	C₂²×C₄×C₁₁₃	C₂³	1	2	yes	—
94	C₂²×C₄×C₁₁₃	C₂³	1	4	yes	—
95	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
96	C₄⋊C₄×C₁₁₃	C₂³	1	4	yes	—
97	C₂²⋊C₄×C₁₁₃	C₂³	1	8	yes	—
98	C₄²×C₁₁₃	C₂³	1	2	yes	—
99	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
100	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
101	C₂³×C₁₁₃	C₂⁴	1	4	yes	—
102	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
103	C₂²×D₁₁₃	—	2	2	no	—
104	C₂²×D₁₁₃	—	2	4	no	—
105	C₂²×D₁₁₃	C₂⁴	1	2	yes	—
106	Q₈×C₁₁₃	—	2	4	no	—
107	D₄×C₁₁₃	—	2	8	no	—
108	D₄×C₁₁₃	—	2	4	no	—
109	D₄×C₁₁₃	—	2	8	no	—
110	C₂×C₄×C₁₁₃	—	2	8	no	—
111	C₂×C₄×C₁₁₃	—	2	4	no	—
112	C₂×C₄×C₁₁₃	#41	1	4	yes	complemented
113	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
114	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
115	C₂×C₄×C₁₁₃	C₂⁴	1	2	yes	—
116	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
117	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
118	C₂²⋊D₁₁₃	—	2	16	no	—
119	C₂×Dic₁₁₃	—	2	2	no	—
120	C₂×Dic₁₁₃	—	2	4	no	—
121	C₂×Dic₁₁₃	—	2	16	no	—
122	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
123	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
124	C₂×Dic₁₁₃	C₂⁴	1	8	yes	—
125	C₂×Dic₁₁₃	C₂⁴	1	2	yes	—
126	C₄×D₁₁₃	—	2	4	no	—
127	C₄×D₁₁₃	—	2	4	no	—
128	C₄×D₁₁₃	—	2	4	no	—
129	C₄×D₁₁₃	—	2	4	no	—
130	C₄×D₁₁₃	—	2	4	no	—
131	C₄×D₁₁₃	#41	1	8	yes	complemented
132	Dic₂₂₆	—	2	8	no	—
133	C₂²×C₁₁₃	—	2	2	no	—
134	C₂²×C₁₁₃	—	2	8	no	—
135	C₂²×C₁₁₃	—	2	4	no	—
136	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
137	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
138	C₂×D₁₁₃	—	2	8	no	—
139	C₂×D₁₁₃	—	2	2	no	—
140	C₂×D₁₁₃	—	2	2	no	—
141	C₂×D₁₁₃	C₂×#41	1	4	yes	—
142	C₄×C₁₁₃	—	2	2	no	—
143	C₄×C₁₁₃	—	2	4	no	—
144	C₄×C₁₁₃	C₂×#41	1	4	yes	—
145	C₄×C₁₁₃	C₂×#41	1	2	yes	—
146	C₄×C₁₁₃	C₂×#41	1	2	yes	—
147	Dic₁₁₃	—	2	8	no	—
148	Dic₁₁₃	—	2	2	no	—
149	Dic₁₁₃	—	2	2	no	—
150	Dic₁₁₃	C₂×#41	1	4	yes	—
151	C₂×C₁₁₃	—	2	2	no	—
152	C₂×C₁₁₃	—	4	4	no	—
153	C₂×C₁₁₃	C₂²×#41	1	2	yes	—
154	C₂×C₁₁₃	C₂²×#41	1	1	yes	characteristic
155	D₁₁₃	—	2	4	no	—
156	#3444	—	113	1	no	maximal, Sylow
157	C₁₁₃	#3444	1	1	yes	characteristic, complemented, Sylow
158	#533	—	113	2	no	—
159	#533	—	113	1	no	—
160	#532	—	113	2	no	—
161	#532	D₁₁₃	1	1	yes	characteristic, complemented
162	#529	—	113	4	no	—
163	#529	—	113	4	no	—
164	#529	—	113	4	no	—
165	#525	—	113	2	no	—
166	#525	—	113	4	no	—
167	C₄×#41	—	113	2	no	—
168	C₄×#41	—	113	4	no	—
169	C₂²×C₄²	—	113	1	no	—
170	C₂×#41	—	113	4	no	—
171	C₂×#41	C₂×D₁₁₃	1	2	yes	—
172	C₂³×C₄	—	113	2	no	—
173	C₂³×C₄	—	113	1	no	—
174	C₄⋊Q₈	—	113	2	no	—
175	C₄⋊Q₈	C₂×D₁₁₃	1	1	yes	characteristic, complemented
176	#127	C₂×D₁₁₃	1	1	yes	characteristic, complemented
177	#126	—	113	8	no	—
178	#125	—	113	4	no	—
179	#125	—	113	2	no	—
180	#124	—	113	4	no	—
181	#124	C₂×D₁₁₃	1	2	yes	complemented
182	#123	—	113	4	no	—
183	#123	—	113	8	no	—
184	C₂²⋊Q₈	—	113	4	no	—
185	C₂²⋊Q₈	—	113	4	no	—
186	C₂²⋊Q₈	—	113	4	no	—
187	#121	—	113	4	no	—
188	#121	—	113	4	no	—
189	#121	C₂×D₁₁₃	1	4	yes	—
190	#120	—	113	4	no	—
191	C₄×Q₈	—	113	4	no	—
192	C₄×Q₈	—	113	4	no	—
193	C₄×Q₈	—	113	4	no	—
194	C₄×D₄	—	113	8	no	—
195	C₄×D₄	—	113	4	no	—
196	C₄×D₄	—	113	4	no	—
197	C₄×D₄	—	113	8	no	—
198	C₄×D₄	—	113	8	no	—
199	C₄×D₄	C₂×D₁₁₃	1	4	yes	complemented
200	#117	—	113	2	no	—
201	#117	—	113	4	no	—
202	#117	—	113	4	no	—
203	#117	—	113	8	no	—
204	C₂×C₄²	—	113	1	no	—
205	C₂×C₄²	—	113	2	no	—
206	C₂×C₄²	—	113	4	no	—
207	C₂×C₄²	—	113	2	no	—
208	C₂×C₄²	—	113	2	no	—
209	C₂×C₄²	—	113	2	no	—
210	C₂×C₄²	—	113	2	no	—
211	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
212	C₂⁴	—	113	1	no	—
213	#41	—	2	8	no	—
214	#41	—	226	16	no	—
215	C₂×Q₈	—	113	4	no	—
216	C₂×Q₈	C₂²×D₁₁₃	1	2	yes	—
217	C₂×D₄	—	113	8	no	—
218	C₂×D₄	C₂²×D₁₁₃	1	4	yes	—
219	C₂×D₄	C₂²×D₁₁₃	1	2	yes	—
220	C₂×D₄	C₂²×D₁₁₃	1	4	yes	—
221	C₂²×C₄	—	113	2	no	—
222	C₂²×C₄	—	113	2	no	—
223	C₂²×C₄	—	113	1	no	—
224	C₂²×C₄	—	113	8	no	—
225	C₂²×C₄	—	113	2	no	—
226	C₂²×C₄	—	113	2	no	—
227	C₂²×C₄	—	113	2	no	—
228	C₂²×C₄	—	113	2	no	—
229	C₂²×C₄	—	113	2	no	—
230	C₂²×C₄	—	226	8	no	—
231	C₂²×C₄	—	226	4	no	—
232	C₂²×C₄	C₂²×D₁₁₃	1	2	yes	—
233	C₂²×C₄	C₂²×D₁₁₃	1	4	yes	—
234	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
235	C₄⋊C₄	—	113	4	no	—
236	C₄⋊C₄	—	113	4	no	—
237	C₄⋊C₄	—	113	8	no	—
238	C₄⋊C₄	—	113	8	no	—
239	C₄⋊C₄	—	113	8	no	—
240	C₄⋊C₄	C₂²×D₁₁₃	1	4	yes	—
241	C₂²⋊C₄	—	113	4	no	—
242	C₂²⋊C₄	—	113	8	no	—
243	C₂²⋊C₄	—	113	8	no	—
244	C₂²⋊C₄	—	113	8	no	—
245	C₂²⋊C₄	C₂²×D₁₁₃	1	8	yes	—
246	C₄²	—	113	2	no	—
247	C₄²	—	113	2	no	—
248	C₄²	—	113	2	no	—
249	C₄²	—	113	2	no	—
250	C₄²	—	113	2	no	—
251	C₄²	—	113	8	no	—
252	C₄²	—	113	1	no	—
253	C₄²	—	113	1	no	—
254	C₄²	—	113	4	no	—
255	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
256	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
257	C₄²	C₂²×D₁₁₃	1	2	yes	—
258	C₂³	—	113	2	no	—
259	C₂³	—	226	4	no	—
260	C₂³	—	226	2	no	—
261	C₂³	C₂³×D₁₁₃	1	4	yes	—
262	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
263	Q₈	—	2	4	no	—
264	Q₈	—	226	8	no	—
265	D₄	—	2	8	no	—
266	D₄	—	2	8	no	—
267	D₄	—	2	4	no	—
268	D₄	—	226	16	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	8	no	—
271	C₂×C₄	—	113	2	no	—
272	C₂×C₄	—	113	8	no	—
273	C₂×C₄	—	113	2	no	—
274	C₂×C₄	—	113	2	no	—
275	C₂×C₄	—	113	8	no	—
276	C₂×C₄	—	226	4	no	—
277	C₂×C₄	—	226	4	no	—
278	C₂×C₄	—	226	4	no	—
279	C₂×C₄	—	226	16	no	—
280	C₂×C₄	—	226	2	no	—
281	C₂×C₄	—	226	4	no	—
282	C₂×C₄	—	226	4	no	—
283	C₂×C₄	—	226	4	no	—
284	C₂×C₄	#14268646	1	4	yes	complemented
285	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
286	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
287	C₂×C₄	C₂³×D₁₁₃	1	2	yes	—
288	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
289	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
290	C₂²	—	2	2	no	—
291	C₂²	—	2	8	no	—
292	C₂²	—	2	4	no	—
293	C₂²	—	113	4	no	—
294	C₂²	—	226	8	no	—
295	C₂²	—	226	2	no	—
296	C₂²	—	226	2	no	—
297	C₂²	C₂×#14268646	1	2	yes	—
298	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
299	C₄	—	2	2	no	—
300	C₄	—	2	4	no	—
301	C₄	—	113	4	no	—
302	C₄	—	226	2	no	—
303	C₄	—	226	8	no	—
304	C₄	—	226	2	no	—
305	C₄	#41×D₁₁₃	1	4	yes	—
306	C₄	C₂×#14268646	1	2	yes	—
307	C₄	C₂×#14268646	1	2	yes	—
308	C₂	—	2	2	no	—
309	C₂	—	4	4	no	—
310	C₂	—	226	4	no	—
311	C₂	C₂×#41×D₁₁₃	1	2	yes	—
312	C₂	C₂²×#14268646	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂³×D₁₁₃
commutator, socle, cosocle	C₂²×C₁₁₃	C₂⁵
Frattini	C₂²	C₂⁴×D₁₁₃
Fitting	#532×C₁₁₃	C₂
hypercenter	#532	D₁₁₃
hypocenter	C₁₁₃	#3444
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂×C₄ ◁ #532

Complements

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

N	H ≅ G/N	type
1. #20299155	C₂	semidirect
2. #20299153	C₂	semidirect
3. #20299146	C₂	semidirect
4. #20299130	C₂	semidirect
5. #20299129	C₂	semidirect
6. #20299127	C₂	semidirect
7. #20299108	C₂	semidirect
8. #20299105	C₂	semidirect
9. #41×Dic₁₁₃	C₂	semidirect
10. C₄×#14268646	C₂	semidirect
11. C₂×C₄²×D₁₁₃	C₂	semidirect
12. #532×C₁₁₃	C₂	semidirect
16. #16813622	C₂²	semidirect
17. #16813615	C₂²	semidirect
18. #16813614	C₂²	semidirect
19. #16813613	C₂²	semidirect
20. #16813609	C₂²	semidirect
29. Q₈×Dic₁₁₃	C₂²	semidirect
30. C₄×Dic₂₂₆	C₂²	semidirect
31. #16813587	C₂²	semidirect
33. D₄×Dic₁₁₃	C₂²	semidirect
34. D₄×Dic₁₁₃	C₂²	semidirect
38. #16813581	C₂²	semidirect
40. #16813578	C₂²	semidirect
43. C₂×C₄×Dic₁₁₃	C₂²	semidirect
49. C₄⋊Q₈×C₁₁₃	C₂²	semidirect
50. #127×C₁₁₃	C₂²	semidirect
51. #124×C₁₁₃	C₂²	semidirect
53. C₄×D₄×C₁₁₃	C₂²	semidirect
112. C₂×C₄×C₁₁₃	#41	semidirect
131. C₄×D₁₁₃	#41	semidirect
157. C₁₁₃	#3444	semidirect
161. #532	D₁₁₃	semidirect
175. C₄⋊Q₈	C₂×D₁₁₃	semidirect
176. #127	C₂×D₁₁₃	semidirect
181. #124	C₂×D₁₁₃	semidirect
199. C₄×D₄	C₂×D₁₁₃	semidirect
284. C₂×C₄	#14268646	semidirect

Maximal subgroups

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

subgroup	index
#20299155	2
#20299153	2
#20299146	2
#20299130	2
#20299129	2
#20299127	2
#20299108	2
#20299105	2
#41×Dic₁₁₃	2
C₄×#14268646	2
C₂×C₄²×D₁₁₃	2
#532×C₁₁₃	2
#3444	113