Subgroups — #13129247

Order 1664.

1002 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	#11719437	C₂	1	2	yes	maximal, complemented
3	#11719432	C₂	1	2	yes	maximal, complemented
4	#11719404	C₂	1	4	yes	maximal, complemented
5	#11719348	C₂	1	4	yes	maximal, complemented
6	#11719333	C₂	1	4	yes	maximal, complemented
7	#11719290	C₂	1	2	yes	maximal, complemented
8	#11719289	C₂	1	2	yes	maximal, complemented
9	Q₈×D₅₂	C₂	1	2	yes	maximal, complemented
10	Q₈×Dic₂₆	C₂	1	2	yes	maximal, complemented
11	#11719239	C₂	1	4	yes	maximal, complemented
12	#11719182	C₂	1	1	yes	maximal, characteristic
13	#11719175	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×Q₈×D₁₃	C₂²	1	4	yes	—
15	C₂×#6127	C₂²	1	2	yes	—
16	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₁₃×#125	C₂²	1	2	yes	—
19	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
20	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
22	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
23	C₁₃×#117	C₂²	1	2	yes	—
24	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
25	#88201	C₂²	1	4	yes	—
26	#88200	C₂²	1	4	yes	—
27	Q₈×Dic₁₃	C₂²	1	4	yes	—
28	#88184	C₂²	1	2	yes	—
29	#88184	C₂²	1	2	yes	—
30	#88180	C₂²	1	2	yes	—
31	#88178	C₂²	1	2	yes	—
32	#88178	C₂²	1	2	yes	—
33	#88152	C₂²	1	8	yes	—
34	#88152	C₂²	1	8	yes	—
35	#88151	C₂²	1	4	yes	—
36	#88151	C₂²	1	8	yes	—
37	#88150	C₂²	1	8	yes	—
38	#88148	C₂²	1	8	yes	—
39	#88147	C₂²	1	4	yes	—
40	#88146	C₂²	1	4	yes	—
41	#88142	C₂²	1	4	yes	—
42	#88142	C₂²	1	8	yes	—
43	#88139	C₂²	1	8	yes	—
44	#88137	C₂²	1	8	yes	—
45	#88129	C₂²	1	2	yes	—
46	#88129	C₂²	1	4	yes	—
47	#88128	C₂²	1	1	yes	characteristic
48	C₄×D₅₂	C₂²	1	2	yes	—
49	C₄×D₅₂	C₂²	1	2	yes	—
50	C₄×D₅₂	C₂²	1	4	yes	—
51	#88124	C₂²	1	2	yes	—
52	#88123	C₂²	1	1	yes	characteristic
53	#88123	C₂²	1	1	yes	characteristic
54	#88123	C₂²	1	1	yes	characteristic
55	#88123	C₂²	1	4	yes	—
56	C₄×Dic₂₆	C₂²	1	2	yes	—
57	C₄×Dic₂₆	C₂²	1	2	yes	—
58	C₄×Dic₂₆	C₂²	1	4	yes	—
59	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
60	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
64	C₂×C₂²⋊D₁₃	C₂³	1	4	yes	—
65	Q₈×D₁₃	—	4	8	no	—
66	#6127	—	4	4	no	—
67	C₂×D₅₂	C₂³	1	2	yes	—
68	C₂×D₅₂	C₂³	1	2	yes	—
69	C₂×D₅₂	C₂³	1	2	yes	—
70	C₂×C₄×D₁₃	C₂³	1	4	yes	—
71	C₂×C₄×D₁₃	C₂³	1	8	yes	—
72	C₂×Dic₂₆	C₂³	1	2	yes	—
73	C₂×Dic₂₆	C₂³	1	2	yes	—
74	C₂×Dic₂₆	C₂³	1	2	yes	—
75	C₂×Dic₂₆	C₂³	1	8	yes	—
76	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
77	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
78	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
79	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
80	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
81	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
82	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
83	C₄²×C₁₃	C₂³	1	1	yes	characteristic
84	C₄²×C₁₃	C₂³	1	1	yes	characteristic
85	C₄²×C₁₃	C₂³	1	4	yes	—
86	C₄²×C₁₃	C₂³	1	1	yes	characteristic
87	C₄²×C₁₃	C₂³	1	1	yes	characteristic
88	C₂²⋊Dic₁₃	C₂³	1	4	yes	—
89	#6103	C₂³	1	4	yes	—
90	#6103	C₂³	1	4	yes	—
91	#6103	C₂³	1	8	yes	—
92	#6103	C₂³	1	8	yes	—
93	#6103	C₂³	1	8	yes	—
94	C₄⋊Dic₁₃	C₂³	1	2	yes	—
95	C₄⋊Dic₁₃	C₂³	1	2	yes	—
96	C₄⋊Dic₁₃	C₂³	1	2	yes	—
97	C₄⋊Dic₁₃	C₂³	1	2	yes	—
98	C₄⋊Dic₁₃	C₂³	1	2	yes	—
99	C₄⋊Dic₁₃	C₂³	1	2	yes	—
100	C₄⋊Dic₁₃	C₂³	1	8	yes	—
101	#6101	C₂³	1	4	yes	—
102	#6101	C₂³	1	4	yes	—
103	#6101	C₂³	1	8	yes	—
104	C₄×Dic₁₃	C₂³	1	4	yes	—
105	C₄×Dic₁₃	C₂³	1	8	yes	—
106	#3550	—	13	1	no	maximal, Sylow
107	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
108	C₂²×D₁₃	C₂⁴	1	4	yes	—
109	Q₈×C₁₃	—	2	4	no	—
110	Q₈×C₁₃	—	2	4	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
115	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
118	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
121	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
122	C₂²⋊D₁₃	—	4	4	no	—
123	C₂×Dic₁₃	C₂⁴	1	4	yes	—
124	C₂×Dic₁₃	C₂⁴	1	4	yes	—
125	C₂×Dic₁₃	C₂⁴	1	4	yes	—
126	D₅₂	—	2	4	no	—
127	D₅₂	—	2	4	no	—
128	D₅₂	—	4	2	no	—
129	C₄×D₁₃	—	2	8	no	—
130	C₄×D₁₃	—	4	8	no	—
131	Dic₂₆	—	2	4	no	—
132	Dic₂₆	—	2	4	no	—
133	Dic₂₆	—	4	2	no	—
134	Dic₂₆	—	4	8	no	—
135	#563	—	13	4	no	—
136	#559	—	13	2	no	—
137	Q₈²	—	13	2	no	—
138	#556	—	13	2	no	—
139	#556	—	13	2	no	—
140	D₄×Q₈	—	13	2	no	—
141	D₄×Q₈	—	13	2	no	—
142	#542	—	13	4	no	—
143	#542	—	13	4	no	—
144	#536	—	13	4	no	—
145	#533	—	13	1	no	—
146	#533	D₁₃	1	1	yes	characteristic, complemented
147	#532	—	13	1	no	—
148	C₂²×C₁₃	—	2	1	no	—
149	C₂²×C₁₃	—	2	1	no	—
150	C₂²×C₁₃	—	2	1	no	—
151	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
152	C₂×D₁₃	—	2	4	no	—
153	C₂×D₁₃	—	2	4	no	—
154	C₂×D₁₃	—	2	4	no	—
155	C₄×C₁₃	—	2	4	no	—
156	C₄×C₁₃	—	2	1	no	—
157	C₄×C₁₃	—	2	1	no	—
158	C₄×C₁₃	—	2	1	no	—
159	C₄×C₁₃	—	2	4	no	—
160	C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
161	C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
162	C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
163	Dic₁₃	—	2	4	no	—
164	Dic₁₃	—	2	4	no	—
165	Dic₁₃	—	2	4	no	—
166	C₂×#41	—	13	2	no	—
167	C₂²×Q₈	—	13	4	no	—
168	C₄⋊Q₈	—	13	8	no	—
169	C₄⋊Q₈	—	13	4	no	—
170	C₄⋊Q₈	—	13	1	no	—
171	C₄⋊Q₈	—	13	1	no	—
172	C₄⋊Q₈	—	13	1	no	—
173	C₄⋊Q₈	—	13	4	no	—
174	C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
175	C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic
176	#127	—	13	1	no	—
177	#126	—	13	8	no	—
178	#126	—	13	8	no	—
179	#125	—	13	2	no	—
180	#125	C₂×D₁₃	1	2	yes	—
181	#124	—	13	4	no	—
182	#124	—	13	8	no	—
183	#124	—	13	2	no	—
184	#124	—	13	4	no	—
185	#123	—	13	8	no	—
186	#123	—	13	8	no	—
187	C₂²⋊Q₈	—	13	4	no	—
188	C₂²⋊Q₈	—	13	2	no	—
189	C₂²⋊Q₈	—	13	2	no	—
190	C₂²⋊Q₈	—	13	4	no	—
191	C₂²⋊Q₈	—	13	8	no	—
192	C₂²⋊Q₈	—	13	8	no	—
193	C₂²⋊Q₈	C₂×D₁₃	1	2	yes	—
194	C₂²⋊Q₈	C₂×D₁₃	1	2	yes	—
195	#121	—	13	2	no	—
196	#121	—	13	2	no	—
197	C₄×Q₈	—	13	4	no	—
198	C₄×Q₈	—	13	2	no	—
199	C₄×Q₈	—	13	2	no	—
200	C₄×Q₈	—	13	4	no	—
201	C₄×Q₈	C₂×D₁₃	1	2	yes	—
202	C₄×Q₈	C₂×D₁₃	1	2	yes	—
203	C₄×D₄	—	13	4	no	—
204	C₄×D₄	—	13	2	no	—
205	C₄×D₄	—	13	2	no	—
206	C₄×D₄	—	13	4	no	—
207	#117	—	13	2	no	—
208	#117	—	13	4	no	—
209	#117	C₂×D₁₃	1	2	yes	—
210	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
211	C₂×C₁₃	—	4	1	no	—
212	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
213	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
214	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
215	D₁₃	—	4	4	no	—
216	#41	—	52	4	no	—
217	C₂×Q₈	—	13	2	no	—
218	C₂×Q₈	—	13	2	no	—
219	C₂×Q₈	—	13	8	no	—
220	C₂×Q₈	—	13	2	no	—
221	C₂×Q₈	—	52	8	no	—
222	C₂×Q₈	C₂²×D₁₃	1	2	yes	—
223	C₂×Q₈	C₂²×D₁₃	1	2	yes	—
224	C₂×D₄	—	13	4	no	—
225	C₂×D₄	—	13	2	no	—
226	C₂×D₄	—	13	2	no	—
227	C₂×D₄	—	13	2	no	—
228	C₂²×C₄	—	13	4	no	—
229	C₂²×C₄	—	13	8	no	—
230	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
231	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
232	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
233	C₄⋊C₄	—	13	2	no	—
234	C₄⋊C₄	—	13	2	no	—
235	C₄⋊C₄	—	13	2	no	—
236	C₄⋊C₄	—	13	2	no	—
237	C₄⋊C₄	—	13	2	no	—
238	C₄⋊C₄	—	13	2	no	—
239	C₄⋊C₄	—	13	4	no	—
240	C₄⋊C₄	—	13	8	no	—
241	C₄⋊C₄	—	13	8	no	—
242	C₄⋊C₄	—	13	4	no	—
243	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
244	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
245	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
246	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
247	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
248	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
249	C₂²⋊C₄	—	13	4	no	—
250	C₂²⋊C₄	—	13	8	no	—
251	C₂²⋊C₄	—	13	4	no	—
252	C₂²⋊C₄	—	13	4	no	—
253	C₂²⋊C₄	—	13	8	no	—
254	C₂²⋊C₄	—	13	8	no	—
255	C₂²⋊C₄	C₂²×D₁₃	1	4	yes	—
256	C₄²	—	13	4	no	—
257	C₄²	—	13	8	no	—
258	C₄²	C₂²×D₁₃	1	1	yes	characteristic
259	C₄²	C₂²×D₁₃	1	1	yes	characteristic
260	C₄²	C₂²×D₁₃	1	1	yes	characteristic
261	C₄²	C₂²×D₁₃	1	1	yes	characteristic
262	C₄²	C₂²×D₁₃	1	4	yes	—
263	C₁₃	#3550	1	1	yes	characteristic, complemented, Sylow
264	C₂³	—	13	4	no	—
265	C₂³	C₂³×D₁₃	1	1	yes	characteristic
266	Q₈	—	2	4	no	—
267	Q₈	—	2	4	no	—
268	Q₈	—	26	4	no	—
269	Q₈	—	26	4	no	—
270	Q₈	—	52	8	no	—
271	Q₈	—	52	2	no	—
272	D₄	—	26	4	no	—
273	D₄	—	26	4	no	—
274	D₄	—	52	2	no	—
275	D₄	—	52	4	no	—
276	C₂×C₄	—	2	2	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	13	4	no	—
280	C₂×C₄	—	13	4	no	—
281	C₂×C₄	—	13	4	no	—
282	C₂×C₄	—	26	8	no	—
283	C₂×C₄	—	52	8	no	—
284	C₂×C₄	C₂³×D₁₃	1	4	yes	—
285	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
286	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
287	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
288	C₂×C₄	C₂³×D₁₃	1	4	yes	—
289	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
290	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
291	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	26	4	no	—
296	C₂²	—	26	4	no	—
297	C₂²	—	26	4	no	—
298	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	4	no	—
302	C₄	—	2	1	no	—
303	C₄	—	2	4	no	—
304	C₄	—	26	4	no	—
305	C₄	—	26	4	no	—
306	C₄	—	26	4	no	—
307	C₄	#88254	1	2	yes	—
308	C₄	#88256	1	2	yes	—
309	C₄	#88257	1	2	yes	—
310	C₂	—	4	1	no	—
311	C₂	—	52	4	no	—
312	C₂	C₂×#88254	1	1	yes	characteristic
313	C₂	C₂×#88256	1	1	yes	characteristic
314	C₂	C₂×#88257	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₁₃×#533	C₂
prime radical	#11719182	C₂
hypercenter	#533	D₁₃
hypocenter	C₁₃	#3550
LCS	G ▷ C₂²×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂² ◁ #533

Complements

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

N	H ≅ G/N	type
1. C₁₃×#533	C₂	semidirect
2. #11719437	C₂	semidirect
3. #11719432	C₂	semidirect
4. #11719404	C₂	semidirect
5. #11719348	C₂	semidirect
6. #11719333	C₂	semidirect
7. #11719290	C₂	semidirect
8. #11719289	C₂	semidirect
9. Q₈×D₅₂	C₂	semidirect
10. Q₈×Dic₂₆	C₂	semidirect
11. #11719239	C₂	semidirect
13. #11719175	C₂	semidirect
146. #533	D₁₃	semidirect
263. C₁₃	#3550	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#533	2
#11719437	2
#11719432	2
#11719404	2
#11719348	2
#11719333	2
#11719290	2
#11719289	2
Q₈×D₅₂	2
Q₈×Dic₂₆	2
#11719239	2
#11719182	2
#11719175	2
#3550	13