Subgroups — #13128637

Order 1664.

1100 conjugacy classes of subgroups.

The subgroup lattice has 326 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	#11719435	C₂	1	2	yes	maximal, complemented
3	#11719432	C₂	1	2	yes	maximal, complemented
4	#11719395	C₂	1	4	yes	maximal, complemented
5	#11719339	C₂	1	4	yes	maximal, complemented
6	D₁₃×C₂²⋊Q₈	C₂	1	4	yes	maximal, complemented
7	#11719284	C₂	1	2	yes	maximal, complemented
8	#11719284	C₂	1	4	yes	maximal, complemented
9	Q₈×D₅₂	C₂	1	2	yes	maximal, complemented
10	#11719226	C₂	1	2	yes	maximal
11	#11719224	C₂	1	2	yes	maximal, complemented
12	C₂×C₄×D₅₂	C₂	1	1	yes	maximal, characteristic
13	#11719175	C₂	1	1	yes	maximal, characteristic, complemented
14	C₂×Q₈×D₁₃	C₂²	1	4	yes	complemented
15	C₂²×D₅₂	C₂²	1	1	yes	characteristic
16	C₂²×C₄×D₁₃	C₂²	1	2	yes	—
17	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
18	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic, complemented
19	C₁₃×#125	C₂²	1	2	yes	complemented
20	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₄×Q₈×C₁₃	C₂²	1	2	yes	complemented
23	C₄×Q₈×C₁₃	C₂²	1	2	yes	complemented
24	C₁₃×#117	C₂²	1	2	yes	—
25	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
26	#88200	C₂²	1	4	yes	complemented
27	#88200	C₂²	1	8	yes	complemented
28	C₂×#6103	C₂²	1	2	yes	—
29	#88180	C₂²	1	2	yes	—
30	C₂×C₄⋊Dic₁₃	C₂²	1	1	yes	characteristic
31	#88178	C₂²	1	2	yes	—
32	#88178	C₂²	1	2	yes	—
33	#88151	C₂²	1	4	yes	complemented
34	#88151	C₂²	1	8	yes	complemented
35	#88150	C₂²	1	8	yes	complemented
36	#88150	C₂²	1	8	yes	complemented
37	#88150	C₂²	1	8	yes	complemented
38	#88147	C₂²	1	4	yes	—
39	#88147	C₂²	1	4	yes	—
40	C₄⋊C₄×D₁₃	C₂²	1	4	yes	complemented
41	C₄⋊C₄×D₁₃	C₂²	1	8	yes	complemented
42	#88144	C₂²	1	4	yes	complemented
43	#88142	C₂²	1	4	yes	complemented
44	C₂²⋊C₄×D₁₃	C₂²	1	8	yes	—
45	#88132	C₂²	1	8	yes	—
46	C₄×D₅₂	C₂²	1	2	yes	—
47	C₄×D₅₂	C₂²	1	2	yes	—
48	C₄×D₅₂	C₂²	1	2	yes	—
49	C₄×D₅₂	C₂²	1	2	yes	—
50	C₄×D₅₂	C₂²	1	4	yes	—
51	C₄×D₅₂	C₂²	1	4	yes	—
52	#88124	C₂²	1	2	yes	complemented
53	#88124	C₂²	1	4	yes	complemented
54	#88123	C₂²	1	1	yes	characteristic, complemented
55	#88123	C₂²	1	1	yes	characteristic, complemented
56	#88123	C₂²	1	4	yes	complemented
57	C₄×Dic₂₆	C₂²	1	2	yes	complemented
58	C₄×Dic₂₆	C₂²	1	2	yes	complemented
59	C₂³×D₁₃	C₂³	1	2	yes	—
60	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
61	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
65	C₂²×Dic₁₃	C₂³	1	2	yes	—
66	Q₈×D₁₃	—	4	8	no	—
67	C₂×D₅₂	Q₈	1	8	yes	complemented
68	C₂×D₅₂	C₂³	1	2	yes	—
69	C₂×D₅₂	C₂³	1	2	yes	—
70	C₂×C₄×D₁₃	—	4	4	no	—
71	C₂×C₄×D₁₃	C₂³	1	4	yes	—
72	C₂×C₄×D₁₃	C₂³	1	4	yes	—
73	C₂×C₄×D₁₃	C₂³	1	8	yes	—
74	C₂×C₄×D₁₃	C₂³	1	8	yes	—
75	C₂×Dic₂₆	C₂³	1	2	yes	—
76	C₂×Dic₂₆	C₂³	1	2	yes	—
77	C₂×Dic₂₆	C₂³	1	8	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×C₄⋊C₄	C₂³	1	2	yes	—
83	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
84	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
85	C₄²×C₁₃	C₂³	1	1	yes	characteristic
86	C₄²×C₁₃	C₂³	1	1	yes	characteristic
87	C₄²×C₁₃	C₂³	1	1	yes	characteristic
88	C₄²×C₁₃	C₂³	1	4	yes	—
89	C₄²×C₁₃	C₂³	1	1	yes	characteristic
90	C₂²⋊Dic₁₃	C₂³	1	4	yes	—
91	#6103	C₂³	1	4	yes	—
92	#6103	C₂³	1	4	yes	—
93	#6103	C₂³	1	8	yes	—
94	#6103	C₂³	1	8	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	C₄⋊Dic₁₃	C₂³	1	2	yes	—
102	#6101	C₂³	1	4	yes	—
103	#6101	C₂³	1	4	yes	—
104	#6101	C₂³	1	8	yes	—
105	#6101	C₂³	1	8	yes	—
106	C₄×Dic₁₃	C₂³	1	4	yes	—
107	#3501	—	13	1	no	maximal, Sylow
108	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
109	C₂²×D₁₃	—	4	2	no	—
110	C₂²×D₁₃	—	4	2	no	—
111	C₂²×D₁₃	C₂×Q₈	1	8	yes	complemented
112	C₂²×D₁₃	C₂⁴	1	4	yes	—
113	Q₈×C₁₃	—	2	4	no	—
114	Q₈×C₁₃	—	4	2	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	4	1	no	—
117	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	complemented
118	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
121	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
122	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
123	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
126	C₂×Dic₁₃	—	4	2	no	—
127	C₂×Dic₁₃	C₂⁴	1	4	yes	—
128	C₂×Dic₁₃	C₂⁴	1	4	yes	—
129	C₂×Dic₁₃	C₂⁴	1	4	yes	—
130	D₅₂	C₂×Q₈	1	8	yes	complemented
131	D₅₂	C₂×Q₈	1	8	yes	complemented
132	C₄×D₁₃	—	2	8	no	—
133	C₄×D₁₃	—	4	4	no	—
134	C₄×D₁₃	—	4	8	no	—
135	C₄×D₁₃	—	4	8	no	—
136	Dic₂₆	—	2	4	no	—
137	Dic₂₆	—	4	2	no	—
138	Dic₂₆	—	4	8	no	—
139	#554	—	13	2	no	—
140	#554	—	13	4	no	—
141	#554	—	13	2	no	—
142	#554	—	13	4	no	—
143	D₄×Q₈	—	13	2	no	—
144	D₄×Q₈	—	13	2	no	—
145	#544	—	13	2	no	—
146	#543	—	13	4	no	—
147	#533	—	13	1	no	—
148	#533	D₁₃	1	1	yes	characteristic, complemented
149	C₂×C₂²⋊Q₈	—	13	4	no	—
150	#518	—	13	2	no	—
151	C₂×C₄×D₄	—	13	1	no	—
152	C₂²×C₁₃	—	2	1	no	—
153	C₂²×C₁₃	—	2	1	no	—
154	C₂²×C₁₃	C₂²×Q₈	1	2	yes	—
155	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
156	C₂×D₁₃	—	2	4	no	—
157	C₂×D₁₃	—	2	4	no	—
158	C₂×D₁₃	—	4	8	no	—
159	C₂×D₁₃	C₂²×Q₈	1	8	yes	—
160	C₄×C₁₃	—	2	4	no	—
161	C₄×C₁₃	—	2	1	no	—
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	—	2	4	no	—
164	C₄×C₁₃	—	2	1	no	—
165	C₄×C₁₃	C₂²×Q₈	1	2	yes	—
166	C₄×C₁₃	C₂²×Q₈	1	2	yes	—
167	C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
168	Dic₁₃	—	2	4	no	—
169	Dic₁₃	—	2	4	no	—
170	Dic₁₃	—	2	4	no	—
171	C₂²×Q₈	—	13	4	no	—
172	C₂²×D₄	—	13	1	no	—
173	C₂³×C₄	—	13	2	no	—
174	C₄⋊Q₈	—	13	4	no	—
175	C₄⋊Q₈	—	13	1	no	—
176	C₄⋊Q₈	—	13	1	no	—
177	C₄⋊Q₈	—	13	4	no	—
178	C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic, complemented
179	C₄⋊Q₈	C₂×D₁₃	1	1	yes	characteristic, complemented
180	#125	—	13	4	no	—
181	#125	—	13	2	no	—
182	#125	—	13	4	no	—
183	#125	C₂×D₁₃	1	2	yes	complemented
184	C₂²⋊Q₈	—	13	4	no	—
185	C₂²⋊Q₈	—	13	8	no	—
186	C₂²⋊Q₈	—	13	2	no	—
187	C₂²⋊Q₈	—	13	2	no	—
188	C₂²⋊Q₈	—	13	4	no	—
189	C₂²⋊Q₈	—	13	8	no	—
190	C₂²⋊Q₈	—	13	8	no	—
191	C₂²⋊Q₈	—	13	8	no	—
192	C₂²⋊Q₈	—	13	8	no	—
193	C₂²⋊Q₈	—	13	8	no	—
194	C₂²⋊Q₈	C₂×D₁₃	1	2	yes	—
195	C₂²⋊Q₈	C₂×D₁₃	1	2	yes	—
196	C₄×Q₈	—	13	2	no	—
197	C₄×Q₈	—	13	2	no	—
198	C₄×Q₈	C₂×D₁₃	1	2	yes	complemented
199	C₄×Q₈	C₂×D₁₃	1	2	yes	complemented
200	C₄×D₄	—	13	4	no	—
201	C₄×D₄	—	13	4	no	—
202	C₄×D₄	—	13	2	no	—
203	C₄×D₄	—	13	2	no	—
204	C₄×D₄	—	13	2	no	—
205	C₄×D₄	—	13	2	no	—
206	C₄×D₄	—	13	4	no	—
207	C₄×D₄	—	13	4	no	—
208	#117	—	13	2	no	—
209	#117	C₂×D₁₃	1	2	yes	—
210	C₂×C₄⋊C₄	—	13	1	no	—
211	C₂×C₄⋊C₄	—	13	8	no	—
212	C₂×C₄⋊C₄	—	13	4	no	—
213	C₂×C₂²⋊C₄	—	13	2	no	—
214	C₂×C₂²⋊C₄	—	13	8	no	—
215	C₂×C₄²	C₂×D₁₃	1	1	yes	characteristic
216	C₂×C₁₃	—	2	2	no	—
217	C₂×C₁₃	C₂³×Q₈	1	1	yes	characteristic
218	C₂×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
219	C₂×C₁₃	#585	1	1	yes	characteristic
220	D₁₃	—	2	8	no	—
221	C₂⁴	—	13	2	no	—
222	C₂×Q₈	—	13	2	no	—
223	C₂×Q₈	—	13	2	no	—
224	C₂×Q₈	—	13	8	no	—
225	C₂×Q₈	—	52	8	no	—
226	C₂×Q₈	C₂²×D₁₃	1	2	yes	—
227	C₂×Q₈	C₂²×D₁₃	1	2	yes	—
228	C₂×D₄	—	13	8	no	—
229	C₂×D₄	—	13	2	no	—
230	C₂×D₄	—	13	2	no	—
231	C₂²×C₄	—	13	2	no	—
232	C₂²×C₄	—	13	4	no	—
233	C₂²×C₄	—	13	8	no	—
234	C₂²×C₄	—	13	8	no	—
235	C₂²×C₄	—	13	4	no	—
236	C₂²×C₄	—	52	4	no	—
237	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
238	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
239	C₂²×C₄	C₂²×D₁₃	1	1	yes	characteristic
240	C₄⋊C₄	—	13	4	no	—
241	C₄⋊C₄	—	13	2	no	—
242	C₄⋊C₄	—	13	2	no	—
243	C₄⋊C₄	—	13	2	no	—
244	C₄⋊C₄	—	13	8	no	—
245	C₄⋊C₄	—	13	8	no	—
246	C₄⋊C₄	—	13	8	no	—
247	C₄⋊C₄	—	13	2	no	—
248	C₄⋊C₄	—	13	2	no	—
249	C₄⋊C₄	—	13	4	no	—
250	C₄⋊C₄	—	13	2	no	—
251	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
252	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
253	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
254	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
255	C₄⋊C₄	C₂²×D₁₃	1	4	yes	—
256	C₄⋊C₄	C₂²×D₁₃	1	2	yes	—
257	C₂²⋊C₄	—	13	4	no	—
258	C₂²⋊C₄	—	13	8	no	—
259	C₂²⋊C₄	—	13	8	no	—
260	C₂²⋊C₄	—	13	4	no	—
261	C₂²⋊C₄	—	13	4	no	—
262	C₂²⋊C₄	C₂²×D₁₃	1	4	yes	—
263	C₄²	—	13	4	no	—
264	C₄²	C₂²×D₁₃	1	1	yes	characteristic
265	C₄²	C₂²×D₁₃	1	1	yes	characteristic
266	C₄²	C₂²×D₁₃	1	1	yes	characteristic
267	C₄²	C₂²×D₁₃	1	1	yes	characteristic
268	C₄²	C₂²×D₁₃	1	4	yes	—
269	C₁₃	#3501	1	1	yes	characteristic, complemented, Sylow
270	C₂³	—	13	4	no	—
271	C₂³	—	13	8	no	—
272	C₂³	—	52	2	no	—
273	C₂³	—	52	2	no	—
274	C₂³	C₂³×D₁₃	1	1	yes	characteristic
275	Q₈	—	2	4	no	—
276	Q₈	—	4	2	no	—
277	Q₈	—	26	4	no	—
278	Q₈	—	52	2	no	—
279	Q₈	—	52	8	no	—
280	D₄	—	13	8	no	—
281	D₄	—	13	8	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	4	1	no	—
284	C₂×C₄	—	13	4	no	—
285	C₂×C₄	—	13	4	no	—
286	C₂×C₄	—	13	4	no	—
287	C₂×C₄	—	26	8	no	—
288	C₂×C₄	—	52	8	no	—
289	C₂×C₄	—	52	4	no	—
290	C₂×C₄	—	52	8	no	—
291	C₂×C₄	—	52	2	no	—
292	C₂×C₄	Q₈×D₁₃	1	4	yes	complemented
293	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
294	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
295	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
296	C₂×C₄	C₂³×D₁₃	1	4	yes	—
297	C₂×C₄	C₂³×D₁₃	1	4	yes	—
298	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
299	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
300	C₂×C₄	C₂³×D₁₃	1	1	yes	characteristic
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	13	8	no	—
304	C₂²	—	26	4	no	—
305	C₂²	—	26	4	no	—
306	C₂²	—	52	8	no	—
307	C₂²	C₂×Q₈×D₁₃	1	2	yes	—
308	C₂²	C₂⁴×D₁₃	1	1	yes	characteristic
309	C₄	—	2	1	no	—
310	C₄	—	2	4	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	4	no	—
314	C₄	—	26	4	no	—
315	C₄	—	26	4	no	—
316	C₄	—	26	4	no	—
317	C₄	C₂×Q₈×D₁₃	1	2	yes	—
318	C₄	C₂×Q₈×D₁₃	1	2	yes	—
319	C₄	#88256	1	2	yes	—
320	C₂	—	2	2	no	—
321	C₂	—	26	8	no	—
322	C₂	C₂²×Q₈×D₁₃	1	1	yes	characteristic
323	C₂	C₂×#88256	1	1	yes	characteristic
324	C₂	#11719648	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₂²	C₂⁴×D₁₃
commutator, socle, cosocle	C₂²×C₁₃	C₂⁵
Fitting	C₁₃×#533	C₂
prime radical	C₂²×D₅₂	C₂²
hypercenter	#533	D₁₃
hypocenter	C₁₃	#3501
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. #11719435	C₂	semidirect
3. #11719432	C₂	semidirect
4. #11719395	C₂	semidirect
5. #11719339	C₂	semidirect
6. D₁₃×C₂²⋊Q₈	C₂	semidirect
7. #11719284	C₂	semidirect
8. #11719284	C₂	semidirect
9. Q₈×D₅₂	C₂	semidirect
11. #11719224	C₂	semidirect
13. #11719175	C₂	semidirect
14. C₂×Q₈×D₁₃	C₂²	semidirect
17. C₁₃×C₄⋊Q₈	C₂²	semidirect
18. C₁₃×C₄⋊Q₈	C₂²	semidirect
19. C₁₃×#125	C₂²	semidirect
22. C₄×Q₈×C₁₃	C₂²	semidirect
23. C₄×Q₈×C₁₃	C₂²	semidirect
26. #88200	C₂²	semidirect
27. #88200	C₂²	semidirect
33. #88151	C₂²	semidirect
34. #88151	C₂²	semidirect
35. #88150	C₂²	semidirect
36. #88150	C₂²	semidirect
37. #88150	C₂²	semidirect
40. C₄⋊C₄×D₁₃	C₂²	semidirect
41. C₄⋊C₄×D₁₃	C₂²	semidirect
42. #88144	C₂²	semidirect
43. #88142	C₂²	semidirect
52. #88124	C₂²	semidirect
53. #88124	C₂²	semidirect
54. #88123	C₂²	semidirect
55. #88123	C₂²	semidirect
56. #88123	C₂²	semidirect
57. C₄×Dic₂₆	C₂²	semidirect
58. C₄×Dic₂₆	C₂²	semidirect
67. C₂×D₅₂	Q₈	semidirect
111. C₂²×D₁₃	C₂×Q₈	semidirect
117. C₂×C₄×C₁₃	C₂×Q₈	semidirect
130. D₅₂	C₂×Q₈	semidirect
131. D₅₂	C₂×Q₈	semidirect
148. #533	D₁₃	semidirect
178. C₄⋊Q₈	C₂×D₁₃	semidirect
179. C₄⋊Q₈	C₂×D₁₃	semidirect
183. #125	C₂×D₁₃	semidirect
198. C₄×Q₈	C₂×D₁₃	semidirect
199. C₄×Q₈	C₂×D₁₃	semidirect
269. C₁₃	#3501	semidirect
292. C₂×C₄	Q₈×D₁₃	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
#11719435	2
#11719432	2
#11719395	2
#11719339	2
D₁₃×C₂²⋊Q₈	2
#11719284	2
Q₈×D₅₂	2
#11719226	2
#11719224	2
C₂×C₄×D₅₂	2
#11719175	2
#3501	13