Subgroups — #28404436

Order 14464.

1012 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299334	C₂	1	2	yes	maximal, complemented
2	#20299323	C₂	1	2	yes	maximal, complemented
3	#20299318	C₂	1	2	yes	maximal, complemented
4	#20299317	C₂	1	4	yes	maximal, complemented
5	#20299305	C₂	1	2	yes	maximal, complemented
6	#20299288	C₂	1	4	yes	maximal, complemented
7	#20299193	C₂	1	4	yes	maximal, complemented
8	#20299169	C₂	1	2	yes	maximal, complemented
9	#20299155	C₂	1	1	yes	maximal, characteristic, complemented
10	#20299129	C₂	1	4	yes	maximal, complemented
11	#20299081	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×Dic₂₂₆	C₂	1	1	yes	maximal, characteristic, complemented
13	#532×C₁₁₃	C₂	1	1	yes	maximal, characteristic
14	C₂²×Dic₂₂₆	C₂²	1	1	yes	characteristic
15	#16813622	C₂²	1	1	yes	characteristic
16	#16813622	C₂²	1	1	yes	characteristic
17	#16813620	C₂²	1	4	yes	—
18	#16813615	C₂²	1	8	yes	—
19	#16813615	C₂²	1	8	yes	—
20	#16813614	C₂²	1	2	yes	—
21	#16813613	C₂²	1	4	yes	—
22	#16813612	C₂²	1	8	yes	—
23	#16813609	C₂²	1	4	yes	—
24	#16813609	C₂²	1	4	yes	—
25	#16813603	C₂²	1	8	yes	—
26	#16813603	C₂²	1	8	yes	—
27	#16813601	C₂²	1	4	yes	—
28	#16813601	C₂²	1	4	yes	—
29	#16813601	C₂²	1	2	yes	—
30	#16813601	C₂²	1	2	yes	—
31	#16813599	C₂²	1	8	yes	—
32	Q₈×Dic₁₁₃	C₂²	1	4	yes	—
33	C₄×Dic₂₂₆	C₂²	1	2	yes	—
34	C₄×Dic₂₂₆	C₂²	1	2	yes	—
35	C₄×Dic₂₂₆	C₂²	1	2	yes	—
36	C₄×Dic₂₂₆	C₂²	1	2	yes	—
37	C₄×Dic₂₂₆	C₂²	1	2	yes	—
38	C₄×Dic₂₂₆	C₂²	1	2	yes	—
39	#16813587	C₂²	1	4	yes	—
40	D₄×Dic₁₁₃	C₂²	1	4	yes	—
41	D₄×Dic₁₁₃	C₂²	1	4	yes	—
42	D₄×Dic₁₁₃	C₂²	1	8	yes	—
43	#16813581	C₂²	1	4	yes	—
44	#16813581	C₂²	1	2	yes	—
45	#16813578	C₂²	1	8	yes	—
46	C₂×C₄⋊Dic₁₁₃	C₂²	1	4	yes	—
47	C₂×C₄⋊Dic₁₁₃	C₂²	1	1	yes	characteristic
48	C₂×#14268626	C₂²	1	2	yes	—
49	C₂×C₄×Dic₁₁₃	C₂²	1	2	yes	—
50	C₂×#41×C₁₁₃	C₂²	1	2	yes	—
51	C₄⋊Q₈×C₁₁₃	C₂²	1	1	yes	characteristic
52	#127×C₁₁₃	C₂²	1	1	yes	characteristic
53	#124×C₁₁₃	C₂²	1	2	yes	—
54	#121×C₁₁₃	C₂²	1	2	yes	—
55	#121×C₁₁₃	C₂²	1	2	yes	—
56	C₄×D₄×C₁₁₃	C₂²	1	2	yes	—
57	C₄×D₄×C₁₁₃	C₂²	1	2	yes	—
58	C₂×C₄²×C₁₁₃	C₂²	1	1	yes	characteristic
59	C₂×Dic₂₂₆	—	2	4	no	—
60	C₂×Dic₂₂₆	C₂³	1	2	yes	—
61	C₂×Dic₂₂₆	C₂³	1	2	yes	—
62	C₂×Dic₂₂₆	C₂³	1	2	yes	—
63	C₂×Dic₂₂₆	C₂³	1	2	yes	—
64	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
65	C₂²×Dic₁₁₃	C₂³	1	2	yes	—
66	C₂²×Dic₁₁₃	C₂³	1	8	yes	—
67	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
68	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
69	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
70	C₄⋊Dic₁₁₃	C₂³	1	8	yes	—
71	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
72	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
73	C₄⋊Dic₁₁₃	C₂³	1	2	yes	—
74	#14268626	C₂³	1	4	yes	—
75	#14268626	C₂³	1	4	yes	—
76	#14268626	C₂³	1	4	yes	—
77	#14268626	C₂³	1	8	yes	—
78	#14268626	C₂³	1	8	yes	—
79	#14268626	C₂³	1	4	yes	—
80	C₂²⋊Dic₁₁₃	C₂³	1	8	yes	—
81	C₂²⋊Dic₁₁₃	C₂³	1	4	yes	—
82	C₂²⋊Dic₁₁₃	C₂³	1	8	yes	—
83	C₂²⋊Dic₁₁₃	C₂³	1	8	yes	—
84	C₄×Dic₁₁₃	C₂³	1	4	yes	—
85	C₄×Dic₁₁₃	C₂³	1	4	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	2	yes	—
93	C₂×D₄×C₁₁₃	C₂³	1	2	yes	—
94	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
95	C₂²×C₄×C₁₁₃	C₂³	1	4	yes	—
96	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
97	C₂²×C₄×C₁₁₃	C₂³	1	1	yes	characteristic
98	C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
99	C₄⋊C₄×C₁₁₃	C₂³	1	2	yes	—
100	C₂²⋊C₄×C₁₁₃	C₂³	1	4	yes	—
101	C₂²⋊C₄×C₁₁₃	C₂³	1	4	yes	—
102	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
103	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
104	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
105	C₄²×C₁₁₃	C₂³	1	1	yes	characteristic
106	C₂³×C₁₁₃	C₂⁴	1	4	yes	—
107	C₂³×C₁₁₃	C₂⁴	1	1	yes	characteristic
108	Q₈×C₁₁₃	—	2	4	no	—
109	D₄×C₁₁₃	—	2	8	no	—
110	D₄×C₁₁₃	—	2	4	no	—
111	D₄×C₁₁₃	—	2	4	no	—
112	D₄×C₁₁₃	—	4	2	no	—
113	C₂×C₄×C₁₁₃	—	2	8	no	—
114	C₂×C₄×C₁₁₃	—	2	2	no	—
115	C₂×C₄×C₁₁₃	—	4	1	no	—
116	C₂×C₄×C₁₁₃	#41	1	4	yes	—
117	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
118	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
119	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
120	C₂×C₄×C₁₁₃	C₂⁴	1	4	yes	—
121	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
122	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
123	C₂×C₄×C₁₁₃	C₂⁴	1	1	yes	characteristic
124	C₂×Dic₁₁₃	—	2	4	no	—
125	C₂×Dic₁₁₃	—	4	8	no	—
126	C₂×Dic₁₁₃	—	4	2	no	—
127	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
128	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
129	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
130	C₂×Dic₁₁₃	C₂⁴	1	4	yes	—
131	Dic₂₂₆	—	2	4	no	—
132	Dic₂₂₆	—	2	4	no	—
133	Dic₂₂₆	—	2	4	no	—
134	Dic₂₂₆	—	4	2	no	—
135	C₂²×C₁₁₃	—	2	1	no	—
136	C₂²×C₁₁₃	—	2	1	no	—
137	C₂²×C₁₁₃	—	2	4	no	—
138	C₂²×C₁₁₃	—	2	4	no	—
139	C₂²×C₁₁₃	—	2	4	no	—
140	C₂²×C₁₁₃	C₂×#41	1	2	yes	—
141	C₂²×C₁₁₃	C₂⁵	1	1	yes	characteristic
142	C₄×C₁₁₃	—	2	4	no	—
143	C₄×C₁₁₃	—	2	1	no	—
144	C₄×C₁₁₃	—	2	1	no	—
145	C₄×C₁₁₃	—	2	1	no	—
146	C₄×C₁₁₃	C₂×#41	1	2	yes	—
147	C₄×C₁₁₃	C₂×#41	1	2	yes	—
148	C₄×C₁₁₃	2₋¹⁺⁴	1	2	yes	—
149	Dic₁₁₃	—	2	4	no	—
150	Dic₁₁₃	—	2	4	no	—
151	Dic₁₁₃	—	2	4	no	—
152	Dic₁₁₃	—	2	4	no	—
153	C₂×C₁₁₃	—	2	2	no	—
154	C₂×C₁₁₃	—	4	4	no	—
155	C₂×C₁₁₃	C₂²×#41	1	1	yes	characteristic
156	C₂×C₁₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
157	C₂×C₁₁₃	#585	1	1	yes	characteristic
158	#3507	—	113	1	no	maximal, Sylow
159	C₁₁₃	#3507	1	1	yes	characteristic, complemented, Sylow
160	#559	—	113	2	no	—
161	#557	—	113	2	no	—
162	#556	—	113	4	no	—
163	#556	—	113	2	no	—
164	#554	—	113	2	no	—
165	#552	—	113	4	no	—
166	#539	—	113	4	no	—
167	#536	—	113	2	no	—
168	#533	—	113	1	no	—
169	#532	D₁₁₃	1	1	yes	characteristic
170	#529	—	113	4	no	—
171	#520	—	113	2	no	—
172	C₂×C₄×Q₈	—	113	1	no	—
173	C₂×#41	C₂×D₁₁₃	1	2	yes	—
174	C₂²×Q₈	—	113	1	no	—
175	C₄⋊Q₈	—	113	4	no	—
176	C₄⋊Q₈	—	113	1	no	—
177	C₄⋊Q₈	—	113	1	no	—
178	C₄⋊Q₈	C₂×D₁₁₃	1	1	yes	characteristic
179	#127	C₂×D₁₁₃	1	1	yes	characteristic
180	#126	—	113	8	no	—
181	#126	—	113	8	no	—
182	#125	—	113	4	no	—
183	#125	—	113	8	no	—
184	#125	—	113	2	no	—
185	#124	—	113	4	no	—
186	#124	—	113	4	no	—
187	#124	C₂×D₁₁₃	1	2	yes	—
188	#123	—	113	8	no	—
189	#123	—	113	8	no	—
190	C₂²⋊Q₈	—	113	2	no	—
191	C₂²⋊Q₈	—	113	2	no	—
192	C₂²⋊Q₈	—	113	4	no	—
193	C₂²⋊Q₈	—	113	4	no	—
194	C₂²⋊Q₈	—	113	8	no	—
195	#121	C₂×D₁₁₃	1	2	yes	—
196	#121	C₂×D₁₁₃	1	2	yes	—
197	C₄×Q₈	—	113	4	no	—
198	C₄×Q₈	—	113	4	no	—
199	C₄×Q₈	—	113	2	no	—
200	C₄×Q₈	—	113	2	no	—
201	C₄×Q₈	—	113	2	no	—
202	C₄×Q₈	—	113	2	no	—
203	C₄×Q₈	—	113	2	no	—
204	C₄×Q₈	—	113	2	no	—
205	C₄×D₄	—	113	8	no	—
206	C₄×D₄	—	113	4	no	—
207	C₄×D₄	—	113	4	no	—
208	C₄×D₄	C₂×D₁₁₃	1	2	yes	—
209	C₄×D₄	C₂×D₁₁₃	1	2	yes	—
210	#117	—	113	2	no	—
211	#117	—	113	4	no	—
212	#117	—	113	8	no	—
213	C₂×C₄⋊C₄	—	113	1	no	—
214	C₂×C₄⋊C₄	—	113	2	no	—
215	C₂×C₄⋊C₄	—	113	4	no	—
216	C₂×C₄²	—	113	2	no	—
217	C₂×C₄²	C₂×D₁₁₃	1	1	yes	characteristic
218	#41	—	2	8	no	—
219	C₂×Q₈	—	113	2	no	—
220	C₂×Q₈	—	113	2	no	—
221	C₂×Q₈	—	113	2	no	—
222	C₂×Q₈	—	113	2	no	—
223	C₂×Q₈	—	226	4	no	—
224	C₂×Q₈	C₂²×D₁₁₃	1	2	yes	—
225	C₂×D₄	C₂²×D₁₁₃	1	2	yes	—
226	C₂×D₄	C₂²×D₁₁₃	1	4	yes	—
227	C₂×D₄	C₂²×D₁₁₃	1	2	yes	—
228	C₂×D₄	C₂²×D₁₁₃	1	2	yes	—
229	C₂²×C₄	—	113	2	no	—
230	C₂²×C₄	—	113	8	no	—
231	C₂²×C₄	—	113	2	no	—
232	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
233	C₂²×C₄	C₂²×D₁₁₃	1	4	yes	—
234	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
235	C₂²×C₄	C₂²×D₁₁₃	1	1	yes	characteristic
236	C₄⋊C₄	—	113	2	no	—
237	C₄⋊C₄	—	113	2	no	—
238	C₄⋊C₄	—	113	2	no	—
239	C₄⋊C₄	—	113	8	no	—
240	C₄⋊C₄	—	113	4	no	—
241	C₄⋊C₄	—	113	4	no	—
242	C₄⋊C₄	—	113	2	no	—
243	C₄⋊C₄	—	113	4	no	—
244	C₄⋊C₄	—	113	8	no	—
245	C₄⋊C₄	—	113	2	no	—
246	C₄⋊C₄	—	113	4	no	—
247	C₄⋊C₄	—	113	8	no	—
248	C₄⋊C₄	—	113	2	no	—
249	C₄⋊C₄	C₂²×D₁₁₃	1	2	yes	—
250	C₄⋊C₄	C₂²×D₁₁₃	1	2	yes	—
251	C₂²⋊C₄	—	113	4	no	—
252	C₂²⋊C₄	—	113	8	no	—
253	C₂²⋊C₄	—	113	8	no	—
254	C₂²⋊C₄	—	113	8	no	—
255	C₂²⋊C₄	C₂²×D₁₁₃	1	4	yes	—
256	C₂²⋊C₄	C₂²×D₁₁₃	1	4	yes	—
257	C₄²	—	113	4	no	—
258	C₄²	—	113	4	no	—
259	C₄²	—	113	8	no	—
260	C₄²	—	113	4	no	—
261	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
262	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
263	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
264	C₄²	C₂²×D₁₁₃	1	1	yes	characteristic
265	C₂³	C₂³×D₁₁₃	1	4	yes	—
266	C₂³	C₂³×D₁₁₃	1	1	yes	characteristic
267	Q₈	—	2	4	no	—
268	Q₈	—	452	2	no	—
269	Q₈	—	226	4	no	—
270	Q₈	—	226	4	no	—
271	Q₈	—	226	4	no	—
272	D₄	—	2	8	no	—
273	D₄	—	2	4	no	—
274	D₄	—	2	4	no	—
275	D₄	—	4	2	no	—
276	C₂×C₄	—	2	8	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	4	1	no	—
279	C₂×C₄	—	113	4	no	—
280	C₂×C₄	—	113	4	no	—
281	C₂×C₄	—	113	4	no	—
282	C₂×C₄	—	113	4	no	—
283	C₂×C₄	—	452	2	no	—
284	C₂×C₄	—	452	8	no	—
285	C₂×C₄	—	226	4	no	—
286	C₂×C₄	#14268646	1	4	yes	—
287	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
288	C₂×C₄	C₂³×D₁₁₃	1	4	yes	—
289	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
290	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
291	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
292	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
293	C₂×C₄	C₂³×D₁₁₃	1	1	yes	characteristic
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	—	2	4	no	—
297	C₂²	—	2	4	no	—
298	C₂²	—	2	4	no	—
299	C₂²	C₂×#14268646	1	2	yes	—
300	C₂²	C₂⁴×D₁₁₃	1	1	yes	characteristic
301	C₄	—	2	4	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	1	no	—
304	C₄	—	2	1	no	—
305	C₄	—	226	4	no	—
306	C₄	—	226	4	no	—
307	C₄	—	226	4	no	—
308	C₄	—	226	4	no	—
309	C₄	C₂×#14268646	1	2	yes	—
310	C₄	C₂×#14268646	1	2	yes	—
311	C₄	#16813669	1	2	yes	—
312	C₂	—	2	2	no	—
313	C₂	—	4	4	no	—
314	C₂	C₂²×#14268646	1	1	yes	characteristic
315	C₂	C₂×#16813669	1	1	yes	characteristic
316	C₂	#20299434	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₁₃
commutator, socle, cosocle	C₂²×C₁₁₃	C₂⁵
Fitting, prime radical	#532×C₁₁₃	C₂
hypercenter	#532	D₁₁₃
hypocenter	C₁₁₃	#3507
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ #532

Complements

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

N	H ≅ G/N	type
1. #20299334	C₂	semidirect
2. #20299323	C₂	semidirect
3. #20299318	C₂	semidirect
4. #20299317	C₂	semidirect
5. #20299305	C₂	semidirect
6. #20299288	C₂	semidirect
7. #20299193	C₂	semidirect
8. #20299169	C₂	semidirect
9. #20299155	C₂	semidirect
10. #20299129	C₂	semidirect
11. #20299081	C₂	semidirect
12. C₂×C₄×Dic₂₂₆	C₂	semidirect
159. C₁₁₃	#3507	semidirect

Maximal subgroups

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

subgroup	index
#20299334	2
#20299323	2
#20299318	2
#20299317	2
#20299305	2
#20299288	2
#20299193	2
#20299169	2
#20299155	2
#20299129	2
#20299081	2
C₂×C₄×Dic₂₂₆	2
#532×C₁₁₃	2
#3507	113