Subgroups — #28407528

Order 14464.

346 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#20299650	C₂	1	1	yes	maximal, characteristic, complemented
2	#20299648	C₂	1	1	yes	maximal, characteristic, complemented
3	#20299633	C₂	1	1	yes	maximal, characteristic, complemented
4	#20299631	C₂	1	1	yes	maximal, characteristic, complemented
5	#20299630	C₂	1	1	yes	maximal, characteristic, complemented
6	#20299630	C₂	1	1	yes	maximal, characteristic, complemented
7	#20299629	C₂	1	1	yes	maximal, characteristic, complemented
8	#20299627	C₂	1	1	yes	maximal, characteristic, complemented
9	#20299623	C₂	1	1	yes	maximal, characteristic, complemented
10	#20299622	C₂	1	1	yes	maximal, characteristic, complemented
11	#20299619	C₂	1	1	yes	maximal, characteristic, complemented
12	#20299605	C₂	1	1	yes	maximal, characteristic, complemented
13	#20299604	C₂	1	1	yes	maximal, characteristic, complemented
14	#20299601	C₂	1	1	yes	maximal, characteristic, complemented
15	#20299125	C₂	1	1	yes	maximal, characteristic
16	C₂×#14268657	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	1	yes	characteristic
20	C₂×#14268655	C₂²	1	1	yes	characteristic
21	C₂²×#11744753	C₂²	1	1	yes	characteristic
22	#16813689	C₂²	1	1	yes	characteristic
23	#16813689	C₂²	1	1	yes	characteristic
24	#16813689	C₂²	1	1	yes	characteristic
25	#16813689	C₂²	1	1	yes	characteristic
26	C₄⋊#11744753	C₂²	1	1	yes	characteristic
27	C₄⋊#11744753	C₂²	1	1	yes	characteristic
28	C₄⋊#11744753	C₂²	1	1	yes	characteristic
29	C₄⋊#11744753	C₂²	1	1	yes	characteristic
30	C₄⋊#11744753	C₂²	1	1	yes	characteristic
31	C₄⋊#11744753	C₂²	1	1	yes	characteristic
32	#16813679	C₂²	1	1	yes	characteristic
33	#16813679	C₂²	1	1	yes	characteristic
34	C₂²⋊#11744753	C₂²	1	1	yes	characteristic
35	C₂²⋊#11744753	C₂²	1	1	yes	characteristic
36	C₂²⋊#11744753	C₂²	1	1	yes	characteristic
37	C₂²⋊#11744753	C₂²	1	1	yes	characteristic
38	#16813677	C₂²	1	1	yes	characteristic
39	#16813677	C₂²	1	1	yes	characteristic
40	#16813677	C₂²	1	1	yes	characteristic
41	#16813677	C₂²	1	1	yes	characteristic
42	C₄×#11744753	C₂²	1	1	yes	characteristic
43	C₄×#11744753	C₂²	1	1	yes	characteristic
44	#16813617	C₄	1	1	yes	characteristic
45	#16813615	C₄	1	1	yes	characteristic
46	#16813612	C₄	1	1	yes	characteristic
47	#16813608	C₄	1	1	yes	characteristic
48	#16813605	C₄	1	1	yes	characteristic
49	#16813599	C₄	1	1	yes	characteristic
50	#16813594	C₄	1	1	yes	characteristic
51	#16813587	C₂²	1	1	yes	characteristic
52	D₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
53	#16813582	C₂²	1	1	yes	characteristic
54	#16813582	C₂²	1	1	yes	characteristic
55	#16813580	C₂²	1	1	yes	characteristic
56	#16813578	C₂²	1	1	yes	characteristic
57	C₂×C₄×Dic₁₁₃	C₂²	1	1	yes	characteristic
58	#123×C₁₁₃	C₄	1	1	yes	characteristic
59	#14268657	—	4	1	no	—
60	#14268656	—	4	1	no	—
61	#14268656	—	4	1	no	—
62	#14268656	—	4	1	no	—
63	#14268655	—	4	1	no	—
64	C₂×#11744753	—	4	1	no	—
65	C₂×#11744753	C₂³	1	1	yes	characteristic
66	C₂×#11744753	C₂³	1	1	yes	characteristic
67	C₂×#11744753	C₂³	1	1	yes	characteristic
68	C₂×#11744753	C₂³	1	1	yes	characteristic
69	C₂×#11744753	C₂³	1	1	yes	characteristic
70	C₂×#11744753	C₂³	1	1	yes	characteristic
71	C₂×#11744753	C₂³	1	1	yes	characteristic
72	C₂×#11744753	C₂³	1	1	yes	characteristic
73	C₂×Dic₂₂₆	C₂×C₄	1	1	yes	characteristic
74	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	1	yes	characteristic
75	C₂×C₂²⋊D₁₁₃	C₂×C₄	1	1	yes	characteristic
76	C₂²×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
77	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
78	C₂²×Dic₁₁₃	C₂³	1	1	yes	characteristic
79	C₂×C₄×D₁₁₃	C₂³	1	1	yes	characteristic
80	C₄⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
81	C₄⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
82	#14268626	C₂×C₄	1	1	yes	characteristic
83	#14268626	C₂×C₄	1	1	yes	characteristic
84	#14268626	C₂×C₄	1	1	yes	characteristic
85	#14268626	C₂×C₄	1	1	yes	characteristic
86	#14268626	C₂×C₄	1	1	yes	characteristic
87	#14268626	C₂×C₄	1	1	yes	characteristic
88	C₂²⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
89	C₂²⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
90	C₂²⋊Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
91	#14268624	C₂×C₄	1	1	yes	characteristic
92	#14268624	C₂×C₄	1	1	yes	characteristic
93	#14268624	C₂×C₄	1	1	yes	characteristic
94	#14268624	C₂×C₄	1	1	yes	characteristic
95	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
96	C₄×Dic₁₁₃	C₂×C₄	1	1	yes	characteristic
97	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
98	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
99	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
100	C₄×Dic₁₁₃	C₂³	1	1	yes	characteristic
101	C₂×D₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
102	C₂²×C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
103	C₄⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
104	C₄⋊C₄×C₁₁₃	C₂×C₄	1	1	yes	characteristic
105	C₂²⋊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	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
109	C₂³×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
110	C₂²×D₁₁₃	C₂²×C₄	1	1	yes	characteristic
111	D₄×C₁₁₃	—	4	1	no	—
112	C₂×C₄×C₁₁₃	—	4	1	no	—
113	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
114	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
115	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
116	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₁₃	C₂²×C₄	1	1	yes	characteristic
118	C₂²⋊D₁₁₃	—	4	1	no	—
119	C₂²⋊D₁₁₃	—	4	1	no	—
120	C₂×Dic₁₁₃	—	2	1	no	—
121	C₂×Dic₁₁₃	—	2	1	no	—
122	C₂×Dic₁₁₃	—	2	1	no	—
123	C₂×Dic₁₁₃	—	2	1	no	—
124	C₂×Dic₁₁₃	—	4	1	no	—
125	C₂×Dic₁₁₃	C₂²×C₄	1	1	yes	characteristic
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	C₂⁴	1	1	yes	characteristic
132	C₄×D₁₁₃	—	2	1	no	—
133	C₄×D₁₁₃	—	2	1	no	—
134	Dic₂₂₆	—	4	1	no	—
135	#11744753	—	2	1	no	—
136	#11744753	—	2	1	no	—
137	#11744753	—	2	1	no	—
138	#11744753	—	2	1	no	—
139	#11744753	—	2	1	no	—
140	#11744753	—	2	1	no	—
141	#11744753	—	2	1	no	—
142	#11744753	—	2	1	no	—
143	C₂²×C₁₁₃	—	2	1	no	—
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	C₂³×C₄	1	1	yes	characteristic
150	C₂×D₁₁₃	—	2	1	no	—
151	C₂×D₁₁₃	—	2	1	no	—
152	C₂×D₁₁₃	—	2	1	no	—
153	C₄×C₁₁₃	—	2	1	no	—
154	C₄×C₁₁₃	—	2	1	no	—
155	C₄×C₁₁₃	—	2	1	no	—
156	C₄×C₁₁₃	—	2	1	no	—
157	C₄×C₁₁₃	—	2	1	no	—
158	Dic₁₁₃	—	2	1	no	—
159	Dic₁₁₃	—	2	1	no	—
160	Dic₁₁₃	—	2	1	no	—
161	Dic₁₁₃	—	2	1	no	—
162	Dic₁₁₃	—	2	1	no	—
163	Dic₁₁₃	—	2	1	no	—
164	Dic₁₁₃	H(5,𝔽₂)	1	1	yes	characteristic
165	Dic₁₁₃	2₋¹⁺⁴	1	1	yes	characteristic
166	C₂×C₁₁₃	—	4	1	no	—
167	C₂×C₁₁₃	—	4	1	no	—
168	C₂×C₁₁₃	#520	1	1	yes	characteristic
169	C₂×C₁₁₃	#568	1	1	yes	characteristic
170	C₂×C₁₁₃	#568	1	1	yes	characteristic
171	D₁₁₃	—	4	1	no	—
172	#2995	—	113	1	no	maximal, Sylow
173	C₁₁₃	#2995	1	1	yes	characteristic, complemented, Sylow
174	#529	—	113	1	no	—
175	#446	—	113	1	no	—
176	#446	—	113	1	no	—
177	#436	—	113	1	no	—
178	#436	—	113	1	no	—
179	#435	—	113	1	no	—
180	#435	—	113	1	no	—
181	#435	—	113	1	no	—
182	#435	—	113	1	no	—
183	#433	—	113	1	no	—
184	#433	—	113	1	no	—
185	#432	—	113	1	no	—
186	#424	—	113	1	no	—
187	#424	—	113	1	no	—
188	#423	—	113	1	no	—
189	C₂×M₁₆	—	113	1	no	—
190	C₂×M₁₆	—	113	1	no	—
191	C₂×M₁₆	—	113	1	no	—
192	C₂×M₁₆	—	113	1	no	—
193	C₂²×C₈	—	113	1	no	—
194	C₂²×C₈	—	113	1	no	—
195	#126	—	113	1	no	—
196	#126	—	113	1	no	—
197	#125	—	113	1	no	—
198	#124	—	113	1	no	—
199	#123	—	113	1	no	—
200	#123	#90673	1	1	yes	characteristic
201	C₂²⋊Q₈	—	113	1	no	—
202	#121	—	113	1	no	—
203	C₄×Q₈	—	113	1	no	—
204	C₄×D₄	—	113	1	no	—
205	C₄×D₄	—	113	1	no	—
206	C₄×D₄	—	113	1	no	—
207	#117	—	113	1	no	—
208	#117	—	113	1	no	—
209	C₂×C₄²	—	113	1	no	—
210	C₄⋊C₈	—	113	1	no	—
211	C₄⋊C₈	—	113	1	no	—
212	C₄⋊C₈	—	113	1	no	—
213	C₄⋊C₈	—	113	1	no	—
214	C₄⋊C₈	—	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	C₂²⋊C₈	—	113	1	no	—
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	—	113	1	no	—
226	#97	—	113	1	no	—
227	#97	—	113	1	no	—
228	#97	—	113	1	no	—
229	#97	—	113	1	no	—
230	C₄×C₈	—	113	1	no	—
231	C₄×C₈	—	113	1	no	—
232	C₂×Q₈	—	113	1	no	—
233	C₂×D₄	—	113	1	no	—
234	C₂×D₄	—	113	1	no	—
235	C₂×D₄	C₂×#90673	1	1	yes	characteristic
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	C₂×#90673	1	1	yes	characteristic
241	M₁₆	—	452	1	no	—
242	M₁₆	—	452	1	no	—
243	M₁₆	—	452	1	no	—
244	M₁₆	—	452	1	no	—
245	C₂×C₈	—	113	1	no	—
246	C₂×C₈	—	113	1	no	—
247	C₂×C₈	—	113	1	no	—
248	C₂×C₈	—	113	1	no	—
249	C₂×C₈	—	113	1	no	—
250	C₂×C₈	—	113	1	no	—
251	C₂×C₈	—	113	1	no	—
252	C₂×C₈	—	113	1	no	—
253	C₂×C₈	—	452	1	no	—
254	C₂×C₈	—	452	1	no	—
255	C₄⋊C₄	—	113	1	no	—
256	C₄⋊C₄	—	113	1	no	—
257	C₄⋊C₄	—	113	1	no	—
258	C₄⋊C₄	—	113	1	no	—
259	C₄⋊C₄	—	113	1	no	—
260	C₄⋊C₄	—	113	1	no	—
261	C₄⋊C₄	—	113	1	no	—
262	C₄⋊C₄	—	113	1	no	—
263	C₄⋊C₄	C₂×#90673	1	1	yes	characteristic
264	C₄⋊C₄	C₂×#90673	1	1	yes	characteristic
265	C₂²⋊C₄	—	113	1	no	—
266	C₂²⋊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	—	113	1	no	—
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	C₄²	—	113	1	no	—
280	C₄²	—	113	1	no	—
281	C₂³	—	113	1	no	—
282	C₂³	C₂²×#90673	1	1	yes	characteristic
283	C₂³	C₂²×#90673	1	1	yes	characteristic
284	Q₈	—	452	1	no	—
285	D₄	—	4	1	no	—
286	D₄	—	452	1	no	—
287	D₄	—	452	1	no	—
288	C₂×C₄	—	4	1	no	—
289	C₂×C₄	—	113	1	no	—
290	C₂×C₄	—	113	1	no	—
291	C₂×C₄	—	113	1	no	—
292	C₂×C₄	—	113	1	no	—
293	C₂×C₄	—	113	1	no	—
294	C₂×C₄	—	113	1	no	—
295	C₂×C₄	—	113	1	no	—
296	C₂×C₄	—	452	1	no	—
297	C₂×C₄	—	226	1	no	—
298	C₂×C₄	—	226	1	no	—
299	C₂×C₄	—	226	1	no	—
300	C₂×C₄	—	226	1	no	—
301	C₂×C₄	—	226	1	no	—
302	C₂×C₄	—	226	1	no	—
303	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
304	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
305	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
306	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
307	C₂×C₄	C₂²×#90673	1	1	yes	characteristic
308	C₈	—	226	1	no	—
309	C₈	—	226	1	no	—
310	C₈	—	226	1	no	—
311	C₈	—	226	1	no	—
312	C₈	—	226	1	no	—
313	C₈	—	226	1	no	—
314	C₈	—	226	1	no	—
315	C₈	—	226	1	no	—
316	C₂²	—	2	1	no	—
317	C₂²	—	2	1	no	—
318	C₂²	—	2	1	no	—
319	C₂²	—	2	1	no	—
320	C₂²	—	2	1	no	—
321	C₂²	—	2	1	no	—
322	C₂²	—	226	1	no	—
323	C₂²	—	226	1	no	—
324	C₂²	—	226	1	no	—
325	C₂²	C₂³×#90673	1	1	yes	characteristic
326	C₄	—	2	1	no	—
327	C₄	—	2	1	no	—
328	C₄	—	2	1	no	—
329	C₄	—	2	1	no	—
330	C₄	—	2	1	no	—
331	C₄	—	113	1	no	—
332	C₄	—	113	1	no	—
333	C₄	—	226	1	no	—
334	C₄	—	226	1	no	—
335	C₄	—	226	1	no	—
336	C₄	—	226	1	no	—
337	C₄	—	226	1	no	—
338	C₄	—	226	1	no	—
339	C₂	—	4	1	no	—
340	C₂	—	4	1	no	—
341	C₂	—	452	1	no	—
342	C₂	#20299667	1	1	yes	characteristic
343	C₂	#20299676	1	1	yes	characteristic
344	C₂	#20299677	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂³×#90673
commutator, socle	C₂²×C₁₁₃	C₂³×C₄
Fitting	#123×C₁₁₃	C₄
prime radical	#16813594	C₄
cosocle	C₂×Dic₁₁₃	C₂⁴
hypercenter	#123	#90673
hypocenter	C₁₁₃	#2995
LCS	G ▷ C₂²×C₁₁₃ ▷ C₁₁₃
UCS	C₁ ◁ C₂² ◁ #123

Complements

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

N	H ≅ G/N	type
1. #20299650	C₂	semidirect
2. #20299648	C₂	semidirect
3. #20299633	C₂	semidirect
4. #20299631	C₂	semidirect
5. #20299630	C₂	semidirect
6. #20299630	C₂	semidirect
7. #20299629	C₂	semidirect
8. #20299627	C₂	semidirect
9. #20299623	C₂	semidirect
10. #20299622	C₂	semidirect
11. #20299619	C₂	semidirect
12. #20299605	C₂	semidirect
13. #20299604	C₂	semidirect
14. #20299601	C₂	semidirect
173. C₁₁₃	#2995	semidirect

Maximal subgroups

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

subgroup	index
#20299650	2
#20299648	2
#20299633	2
#20299631	2
#20299630	2
#20299629	2
#20299627	2
#20299623	2
#20299622	2
#20299619	2
#20299605	2
#20299604	2
#20299601	2
#20299125	2
#2995	113