Subgroups — #13124468

Order 1664.

496 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#525	C₂	1	1	yes	maximal, characteristic
2	C₂×#88205	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#88178	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂²×#6099	C₂	1	1	yes	maximal, characteristic, complemented
5	#11718893	C₂	1	1	yes	maximal, characteristic, complemented
6	#11718713	C₂	1	2	yes	maximal, complemented
7	#11718712	C₂	1	2	yes	maximal, complemented
8	#11718703	C₂	1	2	yes	maximal, complemented
9	#11718702	C₂	1	2	yes	maximal, complemented
10	#11718665	C₂	1	1	yes	maximal, characteristic, complemented
11	#11718656	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₂²×Dic₂₆	C₂²	1	1	yes	characteristic
15	C₁₃×#123	—	2	1	no	—
16	C₁₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
18	C₁₃×#121	C₂²	1	1	yes	characteristic
19	C₁₃×#121	C₂²	1	1	yes	characteristic
20	C₁₃×#120	—	2	1	no	—
21	C₄×D₄×C₁₃	—	2	1	no	—
22	C₄×D₄×C₁₃	—	2	1	no	—
23	C₁₃×#117	C₂²	1	1	yes	characteristic
24	C₂×C₂²⋊Dic₁₃	—	2	1	no	—
25	#88205	—	2	4	no	—
26	C₂×#6107	C₂²	1	2	yes	complemented
27	C₂×#6105	C₂²	1	2	yes	complemented
28	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
29	#88178	—	2	1	no	—
30	#88178	—	2	1	no	—
31	#88178	C₂²	1	2	yes	—
32	#88178	C₂²	1	2	yes	—
33	C₂×#6101	—	2	1	no	—
34	C₂×#6099	—	2	4	no	—
35	C₂×#6099	C₂²	1	1	yes	characteristic
36	C₂×#6099	C₂²	1	2	yes	—
37	C₂×#6099	C₂²	1	1	yes	characteristic
38	C₂²×#1055	C₂²	1	2	yes	—
39	#88075	C₂²	1	2	yes	complemented
40	#88072	C₂²	1	2	yes	complemented
41	#88050	C₂²	1	2	yes	complemented
42	#88050	C₂²	1	2	yes	complemented
43	#88048	C₂²	1	2	yes	complemented
44	#88047	C₂²	1	2	yes	complemented
45	C₂⁴×C₁₃	D₄	1	1	yes	characteristic
46	C₁₃×#41	—	2	2	no	—
47	C₁₃×#41	—	2	2	no	—
48	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₂×D₄×C₁₃	—	2	1	no	—
50	C₂×D₄×C₁₃	—	2	1	no	—
51	C₂×D₄×C₁₃	—	2	1	no	—
52	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	—	2	2	no	—
54	C₂²×C₄×C₁₃	—	2	2	no	—
55	C₂²×C₄×C₁₃	—	2	1	no	—
56	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
59	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
62	C₂²×Dic₁₃	—	2	1	no	—
63	C₂²×Dic₁₃	—	2	1	no	—
64	C₂×Dic₂₆	—	2	2	no	—
65	C₂×Dic₂₆	—	2	2	no	—
66	C₂×Dic₂₆	—	2	1	no	—
67	C₂×Dic₂₆	C₂³	1	2	yes	—
68	C₁₃×C₄⋊C₄	—	2	1	no	—
69	C₁₃×C₄⋊C₄	—	2	1	no	—
70	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
71	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
72	C₁₃×C₂²⋊C₄	—	2	1	no	—
73	C₁₃×C₂²⋊C₄	—	2	1	no	—
74	C₁₃×C₂²⋊C₄	—	2	1	no	—
75	C₁₃×C₂²⋊C₄	—	2	1	no	—
76	C₁₃×C₂²⋊C₄	—	2	1	no	—
77	C₄²×C₁₃	—	2	1	no	—
78	C₂²⋊Dic₁₃	—	2	2	no	—
79	C₂²⋊Dic₁₃	—	2	2	no	—
80	#6107	—	2	4	no	—
81	#6105	—	2	4	no	—
82	C₄⋊Dic₁₃	—	2	1	no	—
83	C₄⋊Dic₁₃	C₂³	1	2	yes	—
84	#6101	—	2	2	no	—
85	#6101	—	2	2	no	—
86	#6099	—	2	2	no	—
87	#6099	—	2	4	no	—
88	#6099	D₄	1	4	yes	complemented
89	C₂×#1055	—	2	4	no	—
90	C₂×#1055	C₂³	1	2	yes	—
91	C₂×#1055	C₂³	1	2	yes	—
92	#3056	—	13	1	no	maximal, Sylow
93	C₂³×C₁₃	—	2	2	no	—
94	C₂³×C₁₃	—	2	2	no	—
95	C₂³×C₁₃	—	2	1	no	—
96	C₂³×C₁₃	—	4	1	no	—
97	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
100	Q₈×C₁₃	—	2	1	no	—
101	Q₈×C₁₃	—	2	1	no	—
102	D₄×C₁₃	—	2	1	no	—
103	D₄×C₁₃	—	2	1	no	—
104	D₄×C₁₃	—	4	2	no	—
105	D₄×C₁₃	—	4	2	no	—
106	D₄×C₁₃	—	4	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	1	no	—
111	C₂×C₄×C₁₃	—	2	1	no	—
112	C₂×C₄×C₁₃	—	2	1	no	—
113	C₂×C₄×C₁₃	—	2	1	no	—
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	4	2	no	—
118	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
122	C₂×C₄×C₁₃	#41	1	2	yes	—
123	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
124	C₂×Dic₁₃	—	2	2	no	—
125	C₂×Dic₁₃	—	2	2	no	—
126	C₂×Dic₁₃	—	4	2	no	—
127	C₂×Dic₁₃	—	4	2	no	—
128	Dic₂₆	—	2	2	no	—
129	Dic₂₆	—	2	2	no	—
130	Dic₂₆	—	4	2	no	—
131	#1055	—	2	2	no	—
132	#1055	C₂×D₄	1	4	yes	—
133	C₂×#137	—	13	1	no	—
134	C₂²×M₁₆	—	13	1	no	—
135	#525	D₁₃	1	1	yes	characteristic
136	C₂×C₂²⋊Q₈	—	13	1	no	—
137	#470	—	13	2	no	—
138	#468	—	13	2	no	—
139	#467	—	13	2	no	—
140	#465	—	13	2	no	—
141	#428	—	13	1	no	—
142	#419	—	13	1	no	—
143	#417	—	13	1	no	—
144	C₂²×C₁₃	—	2	1	no	—
145	C₂²×C₁₃	—	2	1	no	—
146	C₂²×C₁₃	—	2	1	no	—
147	C₂²×C₁₃	—	2	1	no	—
148	C₂²×C₁₃	—	2	1	no	—
149	C₂²×C₁₃	—	2	1	no	—
150	C₂²×C₁₃	—	2	1	no	—
151	C₂²×C₁₃	—	4	4	no	—
152	C₂²×C₁₃	—	4	1	no	—
153	C₂²×C₁₃	—	4	1	no	—
154	C₂²×C₁₃	—	4	1	no	—
155	C₂²×C₁₃	#121	1	2	yes	—
156	C₂²×C₁₃	#137	1	2	yes	complemented
157	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
158	C₄×C₁₃	—	2	1	no	—
159	C₄×C₁₃	—	2	1	no	—
160	C₄×C₁₃	—	4	1	no	—
161	C₄×C₁₃	—	4	1	no	—
162	C₄×C₁₃	—	4	1	no	—
163	C₄×C₁₃	#121	1	2	yes	—
164	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
165	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
166	Dic₁₃	—	4	2	no	—
167	Dic₁₃	—	4	2	no	—
168	C₂×#41	C₂×D₁₃	1	1	yes	characteristic
169	C₂²×Q₈	—	13	1	no	—
170	C₂³×C₄	C₂×D₁₃	1	1	yes	characteristic
171	#137	—	26	4	no	—
172	C₂×Q₁₆	—	13	2	no	—
173	C₂×SD₁₆	—	13	2	no	—
174	C₂×M₁₆	—	13	1	no	—
175	C₂×M₁₆	—	13	1	no	—
176	C₂×M₁₆	—	13	2	no	—
177	C₂×M₁₆	—	26	4	no	—
178	C₂²×C₈	—	13	2	no	—
179	#123	—	2	1	no	—
180	C₂²⋊Q₈	—	13	2	no	—
181	C₂²⋊Q₈	—	13	2	no	—
182	C₂²⋊Q₈	—	26	1	no	—
183	C₂²⋊Q₈	—	26	1	no	—
184	C₂²⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
185	C₂²⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
186	#121	C₂×D₁₃	1	1	yes	characteristic
187	#121	C₂×D₁₃	1	1	yes	characteristic
188	#120	—	2	1	no	—
189	C₄×D₄	—	2	1	no	—
190	C₄×D₄	—	2	1	no	—
191	#117	C₂×D₁₃	1	1	yes	characteristic
192	C₂×C₄⋊C₄	—	13	1	no	—
193	C₂×C₄⋊C₄	—	26	1	no	—
194	C₂×C₂²⋊C₄	—	26	1	no	—
195	#107	—	13	2	no	—
196	#106	—	13	2	no	—
197	#103	—	13	2	no	—
198	#103	—	13	2	no	—
199	#103	—	13	2	no	—
200	#102	—	13	2	no	—
201	C₂×C₁₃	—	2	2	no	—
202	C₂×C₁₃	—	2	2	no	—
203	C₂×C₁₃	—	4	1	no	—
204	C₂×C₁₃	—	8	1	no	—
205	C₂×C₁₃	C₂×#121	1	1	yes	characteristic
206	C₂×C₁₃	C₂×#137	1	1	yes	characteristic
207	C₂×C₁₃	#575	1	1	yes	characteristic
208	C₂⁴	C₂²⋊D₁₃	1	1	yes	characteristic
209	#41	—	2	2	no	—
210	#41	—	2	2	no	—
211	C₂×Q₈	—	13	2	no	—
212	C₂×Q₈	—	26	1	no	—
213	C₂×Q₈	—	26	2	no	—
214	C₂×Q₈	—	26	2	no	—
215	C₂×Q₈	C₂²×D₁₃	1	1	yes	characteristic
216	C₂×D₄	—	2	1	no	—
217	C₂×D₄	—	2	1	no	—
218	C₂×D₄	—	2	1	no	—
219	C₂×D₄	C₂²×D₁₃	1	1	yes	characteristic
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	1	no	—
223	C₂²×C₄	—	26	1	no	—
224	C₂²×C₄	—	26	1	no	—
225	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
226	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
227	C₂²×C₄	C₂²⋊D₁₃	1	1	yes	characteristic
228	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
229	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
230	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
231	Q₁₆	—	26	4	no	—
232	SD₁₆	—	26	4	no	—
233	M₁₆	—	13	4	no	—
234	M₁₆	—	26	4	no	—
235	M₁₆	—	26	2	no	—
236	C₂×C₈	—	13	2	no	—
237	C₂×C₈	—	13	2	no	—
238	C₂×C₈	—	26	4	no	—
239	C₄⋊C₄	—	2	1	no	—
240	C₄⋊C₄	—	2	1	no	—
241	C₄⋊C₄	—	13	2	no	—
242	C₄⋊C₄	—	26	1	no	—
243	C₄⋊C₄	—	26	2	no	—
244	C₄⋊C₄	—	26	2	no	—
245	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
246	C₄⋊C₄	C₂²×D₁₃	1	1	yes	characteristic
247	C₂²⋊C₄	—	2	1	no	—
248	C₂²⋊C₄	—	2	1	no	—
249	C₂²⋊C₄	—	2	1	no	—
250	C₂²⋊C₄	—	2	1	no	—
251	C₂²⋊C₄	—	2	1	no	—
252	C₂²⋊C₄	—	26	2	no	—
253	C₂²⋊C₄	—	26	2	no	—
254	C₄²	—	2	1	no	—
255	C₁₃	#3056	1	1	yes	characteristic, complemented, Sylow
256	C₂³	—	2	2	no	—
257	C₂³	—	2	2	no	—
258	C₂³	—	2	1	no	—
259	C₂³	—	4	1	no	—
260	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
261	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
262	C₂³	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
263	Q₈	—	2	1	no	—
264	Q₈	—	2	1	no	—
265	Q₈	—	26	2	no	—
266	Q₈	—	26	2	no	—
267	Q₈	—	52	2	no	—
268	D₄	—	2	1	no	—
269	D₄	—	2	1	no	—
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	D₄	—	4	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	1	no	—
278	C₂×C₄	—	2	1	no	—
279	C₂×C₄	—	2	1	no	—
280	C₂×C₄	—	2	1	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	4	2	no	—
284	C₂×C₄	—	26	2	no	—
285	C₂×C₄	—	26	2	no	—
286	C₂×C₄	—	52	2	no	—
287	C₂×C₄	—	52	2	no	—
288	C₂×C₄	D₄×D₁₃	1	2	yes	—
289	C₂×C₄	#6129	1	2	yes	—
290	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
291	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
292	C₂×C₄	C₂×C₂²⋊D₁₃	1	1	yes	characteristic
293	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
294	C₈	—	13	4	no	—
295	C₈	—	26	2	no	—
296	C₂²	—	2	1	no	—
297	C₂²	—	2	1	no	—
298	C₂²	—	2	1	no	—
299	C₂²	—	2	1	no	—
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	4	4	no	—
304	C₂²	—	4	1	no	—
305	C₂²	—	4	1	no	—
306	C₂²	—	4	1	no	—
307	C₂²	#88193	1	2	yes	—
308	C₂²	#88205	1	2	yes	complemented
309	C₂²	C₂²×C₂²⋊D₁₃	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	4	1	no	—
313	C₄	—	4	1	no	—
314	C₄	—	4	1	no	—
315	C₄	—	52	2	no	—
316	C₄	—	52	2	no	—
317	C₄	#88193	1	2	yes	—
318	C₄	C₂×D₄×D₁₃	1	1	yes	characteristic
319	C₄	C₂×#6129	1	1	yes	characteristic
320	C₂	—	2	2	no	—
321	C₂	—	2	2	no	—
322	C₂	—	4	1	no	—
323	C₂	—	8	1	no	—
324	C₂	C₂×#88193	1	1	yes	characteristic
325	C₂	#11719527	1	1	yes	characteristic
326	C₂	C₂×#88205	1	1	yes	characteristic

	subgroup	G/H
center	C₂²	C₂²×C₂²⋊D₁₃
commutator, cosocle	C₂×C₄×C₁₃	C₂⁴
Frattini	C₂×C₄	C₂³×D₁₃
Fitting, prime radical	C₁₃×#525	C₂
socle	C₂²×C₁₃	C₂²×D₄
hypercenter	#525	D₁₃
hypocenter	C₁₃	#3056
LCS	G ▷ C₂×C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ #525

Complements

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

N	H ≅ G/N	type
2. C₂×#88205	C₂	semidirect
3. C₂×#88178	C₂	semidirect
4. C₂²×#6099	C₂	semidirect
5. #11718893	C₂	semidirect
6. #11718713	C₂	semidirect
7. #11718712	C₂	semidirect
8. #11718703	C₂	semidirect
9. #11718702	C₂	semidirect
10. #11718665	C₂	semidirect
11. #11718656	C₂	semidirect
26. C₂×#6107	C₂²	semidirect
27. C₂×#6105	C₂²	semidirect
39. #88075	C₂²	semidirect
40. #88072	C₂²	semidirect
41. #88050	C₂²	semidirect
42. #88050	C₂²	semidirect
43. #88048	C₂²	semidirect
44. #88047	C₂²	semidirect
88. #6099	D₄	semidirect
156. C₂²×C₁₃	#137	semidirect
255. C₁₃	#3056	semidirect
308. C₂²	#88205	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#525	2
C₂×#88205	2
C₂×#88178	2
C₂²×#6099	2
#11718893	2
#11718713	2
#11718712	2
#11718703	2
#11718702	2
#11718665	2
#11718656	2
#3056	13