Subgroups — C₁₃×#33923 #15720262

Order 3328.

470 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×#3342	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#3338	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2254	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2237	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2235	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2227	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2225	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2219	C₂	1	2	yes	maximal, complemented
9	C₄×C₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#2169	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#2159	C₂	1	2	yes	maximal, complemented
12	C₁₃×#556	—	2	1	no	—
13	C₁₃×#554	—	2	1	no	—
14	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
16	C₁₃×#500	C₂²	1	1	yes	characteristic
17	C₁₃×#500	C₂²	1	1	yes	characteristic
18	C₁₃×#485	—	2	2	no	—
19	C₁₃×#484	—	2	2	no	—
20	C₁₃×#477	—	2	1	no	—
21	C₁₃×#475	—	2	1	no	—
22	C₁₃×#467	C₂²	1	2	yes	—
23	C₁₃×#467	C₂²	1	2	yes	—
24	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
25	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
26	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
27	C₁₃×#427	C₂²	1	2	yes	—
28	C₂×C₁₃×#107	C₂²	1	2	yes	—
29	C₁₃×#367	C₂²	1	2	yes	—
30	C₁₃×#367	C₂²	1	2	yes	—
31	C₁₃×#367	C₂²	1	2	yes	—
32	C₁₃×#366	C₂²	1	1	yes	characteristic
33	C₁₃×#366	C₂²	1	2	yes	—
34	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
35	C₁₃×#358	C₂²	1	1	yes	characteristic
36	C₁₃×#358	C₂²	1	1	yes	characteristic
37	C₁₃×#358	C₂²	1	2	yes	—
38	C₁₃×#358	C₂²	1	2	yes	—
39	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
41	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
42	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
44	C₁₃×C₄⋊Q₈	—	2	1	no	—
45	C₁₃×C₄⋊Q₈	—	2	1	no	—
46	C₁₃×#126	—	4	2	no	—
47	C₁₃×#125	—	4	1	no	—
48	C₁₃×#124	—	4	1	no	—
49	C₁₃×C₂²⋊Q₈	—	2	2	no	—
50	C₁₃×C₂²⋊Q₈	—	2	2	no	—
51	C₁₃×C₂²⋊Q₈	—	4	2	no	—
52	C₄×Q₈×C₁₃	—	2	1	no	—
53	C₄×Q₈×C₁₃	—	2	1	no	—
54	C₄×Q₈×C₁₃	—	4	1	no	—
55	C₄×D₄×C₁₃	—	4	1	no	—
56	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
57	C₁₃×#117	—	2	2	no	—
58	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
59	C₁₃×Q₃₂	—	2	2	no	—
60	C₁₃×M₃₂	—	2	4	no	—
61	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
62	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
63	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
64	C₁₃×#107	—	2	2	no	—
65	C₁₃×#107	—	2	1	no	—
66	C₁₃×#107	—	2	1	no	—
67	C₁₃×#107	C₂³	1	1	yes	characteristic
68	C₁₃×#107	C₂³	1	2	yes	—
69	C₁₃×#107	C₂³	1	1	yes	characteristic
70	C₁₃×#107	C₂³	1	2	yes	—
71	C₁₃×#106	—	2	2	no	—
72	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
73	C₁₃×#103	—	2	1	no	—
74	C₁₃×#103	—	2	1	no	—
75	C₁₃×#103	—	2	2	no	—
76	C₁₃×#103	—	2	2	no	—
77	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
78	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
79	#33923	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂×Q₈×C₁₃	—	2	1	no	—
81	C₂×Q₈×C₁₃	—	2	1	no	—
82	C₂×Q₈×C₁₃	—	4	1	no	—
83	C₂×Q₈×C₁₃	—	4	1	no	—
84	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	—	4	2	no	—
86	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
87	C₁₃×Q₁₆	—	2	1	no	—
88	C₁₃×Q₁₆	—	2	1	no	—
89	C₁₃×Q₁₆	—	2	1	no	—
90	C₁₃×Q₁₆	—	2	1	no	—
91	C₂×C₈×C₁₃	—	2	2	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
96	C₁₃×C₄⋊C₄	—	2	1	no	—
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	—	2	1	no	—
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₁₃×C₄⋊C₄	—	4	2	no	—
101	C₁₃×C₄⋊C₄	—	4	1	no	—
102	C₁₃×C₄⋊C₄	—	4	1	no	—
103	C₁₃×C₄⋊C₄	—	4	1	no	—
104	C₁₃×C₄⋊C₄	—	4	2	no	—
105	C₁₃×C₄⋊C₄	—	4	2	no	—
106	C₁₃×C₄⋊C₄	—	4	1	no	—
107	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
108	C₁₃×C₂²⋊C₄	—	4	2	no	—
109	C₁₃×C₂²⋊C₄	—	4	2	no	—
110	C₁₃×C₂²⋊C₄	—	4	2	no	—
111	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
112	C₄²×C₁₃	—	4	2	no	—
113	C₄²×C₁₃	—	4	1	no	—
114	C₄²×C₁₃	—	4	1	no	—
115	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
116	C₁₃×C₁₆	—	2	2	no	—
117	C₁₃×C₁₆	—	2	1	no	—
118	C₁₃×C₁₆	#41	1	2	yes	—
119	#3342	C₂×C₁₃	1	1	yes	characteristic
120	#3338	C₂×C₁₃	1	1	yes	characteristic
121	#2254	C₂×C₁₃	1	1	yes	characteristic, complemented
122	#2237	C₂×C₁₃	1	2	yes	complemented
123	#2235	C₂×C₁₃	1	2	yes	complemented
124	#2227	C₂×C₁₃	1	1	yes	characteristic, complemented
125	#2225	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#2219	C₂×C₁₃	1	2	yes	complemented
127	C₄×Q₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
128	#2169	C₂×C₁₃	1	1	yes	characteristic
129	#2159	C₂×C₁₃	1	2	yes	complemented
130	C₂³×C₁₃	C₂×D₈	1	2	yes	—
131	Q₈×C₁₃	—	4	1	no	—
132	Q₈×C₁₃	—	4	1	no	—
133	Q₈×C₁₃	—	4	1	no	—
134	Q₈×C₁₃	—	4	1	no	—
135	Q₈×C₁₃	—	8	2	no	—
136	Q₈×C₁₃	—	16	1	no	—
137	D₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	2	2	no	—
140	C₂×C₄×C₁₃	—	4	1	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	—	4	1	no	—
143	C₂×C₄×C₁₃	—	4	1	no	—
144	C₂×C₄×C₁₃	—	4	1	no	—
145	C₂×C₄×C₁₃	—	4	2	no	—
146	C₂×C₄×C₁₃	—	16	2	no	—
147	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
148	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
149	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
151	C₈×C₁₃	—	2	2	no	—
152	C₈×C₁₃	—	2	1	no	—
153	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
154	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
155	#556	—	2	1	no	—
156	#554	—	2	1	no	—
157	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
158	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
159	#500	C₂²×C₁₃	1	1	yes	characteristic
160	#500	C₂²×C₁₃	1	1	yes	characteristic
161	#485	—	2	2	no	—
162	#484	—	2	2	no	—
163	#477	—	2	1	no	—
164	#475	—	2	1	no	—
165	#467	C₂²×C₁₃	1	2	yes	—
166	#467	C₂²×C₁₃	1	2	yes	—
167	C₄×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
168	C₄×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
169	C₈×D₄	C₂²×C₁₃	1	1	yes	characteristic
170	#427	C₂²×C₁₃	1	2	yes	—
171	C₂×#107	C₂²×C₁₃	1	2	yes	—
172	#367	C₂²×C₁₃	1	2	yes	—
173	#367	C₂²×C₁₃	1	2	yes	—
174	#367	C₂²×C₁₃	1	2	yes	—
175	#366	C₂²×C₁₃	1	1	yes	characteristic
176	#366	C₂²×C₁₃	1	2	yes	—
177	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
178	#358	C₂²×C₁₃	1	1	yes	characteristic
179	#358	C₂²×C₁₃	1	1	yes	characteristic
180	#358	C₂²×C₁₃	1	2	yes	—
181	#358	C₂²×C₁₃	1	2	yes	—
182	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
183	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
184	C₂²×C₁₃	—	2	2	no	—
185	C₂²×C₁₃	—	2	2	no	—
186	C₂²×C₁₃	—	2	2	no	—
187	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
188	C₄×C₁₃	—	2	2	no	—
189	C₄×C₁₃	—	2	1	no	—
190	C₄×C₁₃	—	8	1	no	—
191	C₄×C₁₃	—	8	1	no	—
192	C₄×C₁₃	—	8	2	no	—
193	C₄×C₁₃	—	8	1	no	—
194	C₄×C₁₃	—	8	1	no	—
195	C₄×C₁₃	—	8	2	no	—
196	C₄×C₁₃	#510	1	2	yes	—
197	C₄×C₁₃	#547	1	1	yes	characteristic
198	C₄×C₁₃	#578	1	1	yes	characteristic
199	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
200	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
201	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
202	C₄⋊Q₈	—	2	1	no	—
203	C₄⋊Q₈	—	2	1	no	—
204	#126	—	4	2	no	—
205	#125	—	4	1	no	—
206	#124	—	4	1	no	—
207	C₂²⋊Q₈	—	2	2	no	—
208	C₂²⋊Q₈	—	2	2	no	—
209	C₂²⋊Q₈	—	4	2	no	—
210	C₄×Q₈	—	2	1	no	—
211	C₄×Q₈	—	2	1	no	—
212	C₄×Q₈	—	4	1	no	—
213	C₄×D₄	—	4	1	no	—
214	C₄×D₄	D₄×C₁₃	1	1	yes	characteristic
215	#117	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	Q₃₂	—	2	2	no	—
218	M₃₂	—	2	4	no	—
219	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
220	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
221	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
222	#107	—	2	2	no	—
223	#107	—	2	1	no	—
224	#107	—	2	1	no	—
225	#107	C₂³×C₁₃	1	1	yes	characteristic
226	#107	C₂³×C₁₃	1	1	yes	characteristic
227	#107	C₂³×C₁₃	1	2	yes	—
228	#107	C₂³×C₁₃	1	2	yes	—
229	#106	—	2	2	no	—
230	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
231	#103	—	2	1	no	—
232	#103	—	2	1	no	—
233	#103	—	2	2	no	—
234	#103	—	2	2	no	—
235	C₂²⋊C₈	D₄×C₁₃	1	2	yes	—
236	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
237	C₂×C₁₃	—	4	2	no	—
238	C₂×C₁₃	#3334	1	1	yes	characteristic
239	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
240	C₂×C₁₃	#3416	1	1	yes	characteristic
241	C₂×Q₈	—	2	1	no	—
242	C₂×Q₈	—	2	1	no	—
243	C₂×Q₈	—	4	1	no	—
244	C₂×Q₈	—	4	1	no	—
245	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
248	Q₁₆	—	2	1	no	—
249	Q₁₆	—	2	1	no	—
250	Q₁₆	—	2	1	no	—
251	Q₁₆	—	2	1	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
255	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
256	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
257	C₄⋊C₄	—	2	1	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	1	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
269	C₂²⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
273	C₄²	—	4	2	no	—
274	C₄²	—	4	1	no	—
275	C₄²	—	4	1	no	—
276	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
277	C₁₆	—	2	2	no	—
278	C₁₆	—	2	1	no	—
279	C₁₆	C₁₃×#41	1	2	yes	—
280	C₁₃	#33923	1	1	yes	characteristic, direct product factor, Sylow
281	C₂³	C₂×C₁₃×D₈	1	2	yes	—
282	Q₈	—	4	1	no	—
283	Q₈	—	4	1	no	—
284	Q₈	—	4	1	no	—
285	Q₈	—	4	1	no	—
286	Q₈	—	8	2	no	—
287	Q₈	—	16	1	no	—
288	D₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	4	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	1	no	—
294	C₂×C₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	16	2	no	—
298	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
299	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
300	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
301	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
302	C₈	—	2	2	no	—
303	C₈	—	2	1	no	—
304	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
305	C₈	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
306	C₂²	—	2	2	no	—
307	C₂²	—	2	2	no	—
308	C₂²	—	2	2	no	—
309	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
310	C₄	—	2	2	no	—
311	C₄	—	2	1	no	—
312	C₄	—	8	1	no	—
313	C₄	—	8	1	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	1	no	—
316	C₄	—	8	1	no	—
317	C₄	—	8	2	no	—
318	C₄	C₁₃×#510	1	2	yes	—
319	C₄	C₁₃×#547	1	1	yes	characteristic
320	C₄	C₁₃×#578	1	1	yes	characteristic
321	C₂	—	4	2	no	—
322	C₂	C₁₃×#3334	1	1	yes	characteristic
323	C₂	C₂×C₁₃×#510	1	1	yes	characteristic
324	C₂	C₁₃×#3416	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×D₄×C₁₃	D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×D₄×C₁₃ ◁ C₈×D₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#2254	C₂	semidirect
4. C₁₃×#2237	C₂	semidirect
5. C₁₃×#2235	C₂	semidirect
6. C₁₃×#2227	C₂	semidirect
7. C₁₃×#2225	C₂	semidirect
8. C₁₃×#2219	C₂	semidirect
9. C₄×C₁₃×Q₃₂	C₂	semidirect
11. C₁₃×#2159	C₂	semidirect
79. #33923	C₁₃	direct (N × H)
121. #2254	C₂×C₁₃	semidirect
122. #2237	C₂×C₁₃	semidirect
123. #2235	C₂×C₁₃	semidirect
124. #2227	C₂×C₁₃	semidirect
125. #2225	C₂×C₁₃	semidirect
126. #2219	C₂×C₁₃	semidirect
127. C₄×Q₃₂	C₂×C₁₃	semidirect
129. #2159	C₂×C₁₃	semidirect
280. C₁₃	#33923	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#3342	2
C₁₃×#3338	2
C₁₃×#2254	2
C₁₃×#2237	2
C₁₃×#2235	2
C₁₃×#2227	2
C₁₃×#2225	2
C₁₃×#2219	2
C₄×C₁₃×Q₃₂	2
C₁₃×#2169	2
C₁₃×#2159	2
#33923	13

Subgroups — C13×#33923 #15720262

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33923 #15720262