Subgroups — C₁₃×#33903 #15720242

Order 3328.

470 conjugacy classes of subgroups.

The subgroup lattice has 314 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3336	C₂	1	2	yes	maximal
2	C₁₃×#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2237	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2235	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2229	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2216	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2216	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic
10	C₁₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₁₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#552	—	2	2	no	—
13	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
16	C₁₃×#499	C₂²	1	2	yes	—
17	C₁₃×#485	—	2	2	no	—
18	C₁₃×#483	—	2	2	no	—
19	C₁₃×#479	—	2	2	no	—
20	C₁₃×#467	C₂²	1	2	yes	—
21	C₁₃×#467	C₂²	1	2	yes	—
22	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
23	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
24	C₁₃×#427	C₂²	1	2	yes	—
25	C₂×C₁₃×#107	C₂²	1	2	yes	—
26	C₁₃×#367	C₂²	1	2	yes	—
27	C₁₃×#367	C₂²	1	2	yes	—
28	C₁₃×#366	C₂²	1	1	yes	characteristic
29	C₁₃×#366	C₂²	1	1	yes	characteristic
30	C₁₃×#366	C₂²	1	1	yes	characteristic
31	C₁₃×#366	C₂²	1	2	yes	—
32	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₁₃×#358	C₂²	1	2	yes	—
34	C₁₃×#358	C₂²	1	2	yes	—
35	C₁₃×#358	C₂²	1	2	yes	—
36	C₁₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
37	C₁₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
39	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×#126	—	4	2	no	—
43	C₁₃×#125	—	2	2	no	—
44	C₁₃×#125	—	4	2	no	—
45	C₁₃×#123	—	4	2	no	—
46	C₁₃×C₂²⋊Q₈	—	2	2	no	—
47	C₁₃×C₂²⋊Q₈	—	2	2	no	—
48	C₄×Q₈×C₁₃	—	2	2	no	—
49	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
50	C₁₃×#117	—	2	2	no	—
51	C₁₃×#117	—	4	2	no	—
52	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
53	C₁₃×Q₃₂	—	2	2	no	—
54	C₂×C₁₃×C₁₆	—	2	2	no	—
55	C₂×C₁₃×C₁₆	—	2	2	no	—
56	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
60	C₁₃×#107	—	2	2	no	—
61	C₁₃×#107	C₂³	1	2	yes	—
62	C₁₃×#107	C₂³	1	2	yes	—
63	C₁₃×#107	C₂³	1	2	yes	—
64	C₁₃×#106	—	2	2	no	—
65	C₁₃×#106	—	2	2	no	—
66	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
67	C₁₃×#103	—	2	2	no	—
68	C₁₃×#103	—	2	2	no	—
69	C₁₃×#103	—	2	2	no	—
70	C₁₃×C₂²⋊C₈	D₄	1	1	yes	characteristic
71	C₁₃×C₂²⋊C₈	D₄	1	1	yes	characteristic
72	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	#33903	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂×Q₈×C₁₃	—	2	2	no	—
75	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	—	4	2	no	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₁₃×Q₁₆	—	2	2	no	—
80	C₁₃×Q₁₆	—	2	2	no	—
81	C₂×C₈×C₁₃	—	2	1	no	—
82	C₂×C₈×C₁₃	—	2	1	no	—
83	C₂×C₈×C₁₃	—	2	1	no	—
84	C₂×C₈×C₁₃	—	2	1	no	—
85	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₁₃×C₄⋊C₄	—	2	2	no	—
90	C₁₃×C₄⋊C₄	—	2	2	no	—
91	C₁₃×C₄⋊C₄	—	2	2	no	—
92	C₁₃×C₄⋊C₄	—	4	2	no	—
93	C₁₃×C₄⋊C₄	—	4	2	no	—
94	C₁₃×C₄⋊C₄	—	4	2	no	—
95	C₁₃×C₄⋊C₄	—	4	2	no	—
96	C₁₃×C₄⋊C₄	—	4	2	no	—
97	C₁₃×C₄⋊C₄	—	4	2	no	—
98	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
99	C₁₃×C₂²⋊C₄	—	4	2	no	—
100	C₁₃×C₂²⋊C₄	—	4	2	no	—
101	C₁₃×C₂²⋊C₄	—	4	2	no	—
102	C₁₃×C₂²⋊C₄	D₈	1	1	yes	characteristic
103	C₁₃×C₂²⋊C₄	D₈	1	1	yes	characteristic
104	C₄²×C₁₃	—	4	2	no	—
105	C₄²×C₁₃	—	4	2	no	—
106	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₁₃×C₁₆	—	2	1	no	—
108	C₁₃×C₁₆	—	2	1	no	—
109	C₁₃×C₁₆	—	2	1	no	—
110	C₁₃×C₁₆	#41	1	2	yes	—
111	#3336	C₂×C₁₃	1	2	yes	—
112	#2253	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#2237	C₂×C₁₃	1	2	yes	complemented
114	#2235	C₂×C₁₃	1	2	yes	complemented
115	#2229	C₂×C₁₃	1	2	yes	complemented
116	#2216	C₂×C₁₃	1	1	yes	characteristic, complemented
117	#2216	C₂×C₁₃	1	1	yes	characteristic, complemented
118	C₄×Q₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
119	D₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic
120	#2158	C₂×C₁₃	1	1	yes	characteristic, complemented
121	C₂×#366	C₂×C₁₃	1	1	yes	characteristic, complemented
122	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
123	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
124	Q₈×C₁₃	—	4	2	no	—
125	Q₈×C₁₃	—	4	2	no	—
126	D₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	1	no	—
128	C₂×C₄×C₁₃	—	2	1	no	—
129	C₂×C₄×C₁₃	—	2	1	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	—	8	4	no	—
136	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
141	C₈×C₁₃	—	2	1	no	—
142	C₈×C₁₃	—	2	1	no	—
143	C₈×C₁₃	—	2	1	no	—
144	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
145	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
146	#552	—	2	2	no	—
147	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
148	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
149	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
150	#499	C₂²×C₁₃	1	2	yes	—
151	#485	—	2	2	no	—
152	#483	—	2	2	no	—
153	#479	—	2	2	no	—
154	#467	C₂²×C₁₃	1	2	yes	—
155	#467	C₂²×C₁₃	1	2	yes	—
156	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
157	C₈×D₄	C₂²×C₁₃	1	1	yes	characteristic
158	#427	C₂²×C₁₃	1	2	yes	—
159	C₂×#107	C₂²×C₁₃	1	2	yes	—
160	#367	C₂²×C₁₃	1	2	yes	—
161	#367	C₂²×C₁₃	1	2	yes	—
162	#366	C₂²×C₁₃	1	1	yes	characteristic
163	#366	C₂²×C₁₃	1	1	yes	characteristic
164	#366	C₂²×C₁₃	1	2	yes	—
165	#366	C₂²×C₁₃	1	1	yes	characteristic
166	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
167	#358	C₂²×C₁₃	1	2	yes	—
168	#358	C₂²×C₁₃	1	2	yes	—
169	#358	C₂²×C₁₃	1	2	yes	—
170	C₂²⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
171	C₂²⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
172	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
173	C₂²×C₁₃	—	2	1	no	—
174	C₂²×C₁₃	—	2	1	no	—
175	C₂²×C₁₃	—	2	1	no	—
176	C₂²×C₁₃	—	2	1	no	—
177	C₂²×C₁₃	—	2	1	no	—
178	C₂²×C₁₃	#508	1	2	yes	—
179	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
180	C₄×C₁₃	—	2	1	no	—
181	C₄×C₁₃	—	2	1	no	—
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	—	2	1	no	—
184	C₄×C₁₃	—	8	2	no	—
185	C₄×C₁₃	—	8	2	no	—
186	C₄×C₁₃	—	8	2	no	—
187	C₄×C₁₃	—	8	2	no	—
188	C₄×C₁₃	#547	1	1	yes	characteristic
189	C₄×C₁₃	#578	1	1	yes	characteristic
190	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
191	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
192	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
193	#126	—	4	2	no	—
194	#125	—	2	2	no	—
195	#125	—	4	2	no	—
196	#123	—	4	2	no	—
197	C₂²⋊Q₈	—	2	2	no	—
198	C₂²⋊Q₈	—	2	2	no	—
199	C₄×Q₈	—	2	2	no	—
200	C₄×D₄	D₄×C₁₃	1	1	yes	characteristic
201	#117	—	2	2	no	—
202	#117	—	4	2	no	—
203	C₂×C₄⋊C₄	—	2	2	no	—
204	Q₃₂	—	2	2	no	—
205	C₂×C₁₆	—	2	2	no	—
206	C₂×C₁₆	—	2	2	no	—
207	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
208	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
209	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
210	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
211	#107	—	2	2	no	—
212	#107	C₂³×C₁₃	1	2	yes	—
213	#107	C₂³×C₁₃	1	2	yes	—
214	#107	C₂³×C₁₃	1	2	yes	—
215	#106	—	2	2	no	—
216	#106	—	2	2	no	—
217	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
218	#103	—	2	2	no	—
219	#103	—	2	2	no	—
220	#103	—	2	2	no	—
221	C₂²⋊C₈	D₄×C₁₃	1	1	yes	characteristic
222	C₂²⋊C₈	D₄×C₁₃	1	1	yes	characteristic
223	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
224	C₂×C₁₃	—	2	2	no	—
225	C₂×C₁₃	—	4	1	no	—
226	C₂×C₁₃	#3334	1	1	yes	characteristic
227	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
228	C₂×C₁₃	#3417	1	1	yes	characteristic
229	C₂×Q₈	—	2	2	no	—
230	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
231	C₂²×C₄	—	4	2	no	—
232	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
233	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
234	Q₁₆	—	2	2	no	—
235	Q₁₆	—	2	2	no	—
236	C₂×C₈	—	2	1	no	—
237	C₂×C₈	—	2	1	no	—
238	C₂×C₈	—	2	1	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
241	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
242	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
243	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	4	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	2	no	—
252	C₄⋊C₄	—	4	2	no	—
253	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
254	C₂²⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	C₁₃×D₈	1	1	yes	characteristic
258	C₂²⋊C₄	C₁₃×D₈	1	1	yes	characteristic
259	C₄²	—	4	2	no	—
260	C₄²	—	4	2	no	—
261	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
262	C₁₆	—	2	1	no	—
263	C₁₆	—	2	1	no	—
264	C₁₆	—	2	1	no	—
265	C₁₆	C₁₃×#41	1	2	yes	—
266	C₁₃	#33903	1	1	yes	characteristic, direct product factor, Sylow
267	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
268	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
269	Q₈	—	4	2	no	—
270	Q₈	—	4	2	no	—
271	D₄	—	2	2	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	8	4	no	—
281	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
282	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
283	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
284	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
285	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
286	C₈	—	2	1	no	—
287	C₈	—	2	1	no	—
288	C₈	—	2	1	no	—
289	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
290	C₈	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
291	C₂²	—	2	1	no	—
292	C₂²	—	2	1	no	—
293	C₂²	—	2	1	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	2	1	no	—
296	C₂²	C₁₃×#508	1	2	yes	—
297	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
298	C₄	—	2	1	no	—
299	C₄	—	2	1	no	—
300	C₄	—	2	1	no	—
301	C₄	—	2	1	no	—
302	C₄	—	8	2	no	—
303	C₄	—	8	2	no	—
304	C₄	—	8	2	no	—
305	C₄	—	8	2	no	—
306	C₄	C₁₃×#547	1	1	yes	characteristic
307	C₄	C₁₃×#578	1	1	yes	characteristic
308	C₂	—	2	2	no	—
309	C₂	—	4	1	no	—
310	C₂	C₁₃×#3334	1	1	yes	characteristic
311	C₂	C₂×C₁₃×#508	1	1	yes	characteristic
312	C₂	C₁₃×#3417	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
2. C₁₃×#2253	C₂	semidirect
3. C₁₃×#2237	C₂	semidirect
4. C₁₃×#2235	C₂	semidirect
5. C₁₃×#2229	C₂	semidirect
6. C₁₃×#2216	C₂	semidirect
7. C₁₃×#2216	C₂	semidirect
8. C₄×C₁₃×Q₃₂	C₂	semidirect
10. C₁₃×#2158	C₂	semidirect
11. C₂×C₁₃×#366	C₂	semidirect
73. #33903	C₁₃	direct (N × H)
112. #2253	C₂×C₁₃	semidirect
113. #2237	C₂×C₁₃	semidirect
114. #2235	C₂×C₁₃	semidirect
115. #2229	C₂×C₁₃	semidirect
116. #2216	C₂×C₁₃	semidirect
117. #2216	C₂×C₁₃	semidirect
118. C₄×Q₃₂	C₂×C₁₃	semidirect
120. #2158	C₂×C₁₃	semidirect
121. C₂×#366	C₂×C₁₃	semidirect
266. C₁₃	#33903	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×#3336	2
C₁₃×#2253	2
C₁₃×#2237	2
C₁₃×#2235	2
C₁₃×#2229	2
C₁₃×#2216	2
C₄×C₁₃×Q₃₂	2
D₄×C₁₃×C₁₆	2
C₁₃×#2158	2
C₂×C₁₃×#366	2
#33903	13

Subgroups — C13×#33903 #15720242

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33903 #15720242