Subgroups — C₁₃×#33907 #15720246

Order 3328.

480 conjugacy classes of subgroups.

The subgroup lattice has 334 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, complemented
2	C₁₃×#3341	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×#2234	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2227	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2224	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2217	C₂	1	2	yes	maximal, complemented
9	C₄×C₁₃×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2158	C₂	1	2	yes	maximal, complemented
12	C₁₃×#556	—	2	1	no	—
13	C₁₃×#555	—	2	1	no	—
14	C₂×C₁₃×SD₃₂	C₂²	1	1	yes	characteristic
15	C₂²×C₁₃×C₁₆	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×#482	—	2	2	no	—
20	C₁₃×#476	—	2	1	no	—
21	C₁₃×#475	—	2	1	no	—
22	C₁₃×#467	C₂²	1	2	yes	—
23	C₁₃×#466	C₂²	1	2	yes	—
24	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
25	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
26	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
27	C₁₃×#427	C₂²	1	2	yes	—
28	C₁₃×#427	C₂²	1	2	yes	—
29	C₁₃×#367	C₂²	1	2	yes	—
30	C₁₃×#367	C₂²	1	2	yes	—
31	C₁₃×#367	C₂²	1	1	yes	characteristic
32	C₁₃×#366	C₂²	1	2	yes	—
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	2	yes	—
37	C₁₃×#357	C₂²	1	1	yes	characteristic
38	C₁₃×#357	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×D₈	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×#124	—	4	1	no	—
48	C₁₃×#124	—	4	2	no	—
49	C₁₃×C₂²⋊Q₈	—	2	2	no	—
50	C₁₃×#121	—	2	2	no	—
51	C₁₃×#121	—	4	1	no	—
52	C₄×Q₈×C₁₃	—	2	1	no	—
53	C₄×Q₈×C₁₃	—	4	1	no	—
54	C₄×D₄×C₁₃	—	2	1	no	—
55	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
56	C₁₃×#117	—	2	2	no	—
57	C₁₃×#117	—	2	2	no	—
58	C₁₃×#117	—	4	1	no	—
59	C₁₃×SD₃₂	—	2	2	no	—
60	C₂×C₁₃×C₁₆	—	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	1	no	—
65	C₁₃×#107	—	2	1	no	—
66	C₁₃×#107	C₂³	1	1	yes	characteristic
67	C₁₃×#107	C₂³	1	1	yes	characteristic
68	C₁₃×#107	C₂³	1	2	yes	—
69	C₁₃×#107	C₂³	1	2	yes	—
70	C₁₃×#106	—	2	2	no	—
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	2	no	—
75	C₁₃×#102	—	2	1	no	—
76	C₁₃×#102	—	2	2	no	—
77	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
78	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
79	#33907	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂×Q₈×C₁₃	—	2	1	no	—
81	C₂×Q₈×C₁₃	—	4	1	no	—
82	C₂×Q₈×C₁₃	—	4	1	no	—
83	C₂×D₄×C₁₃	—	2	1	no	—
84	C₂×D₄×C₁₃	—	4	2	no	—
85	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	—	4	1	no	—
87	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
88	C₁₃×Q₁₆	—	2	1	no	—
89	C₁₃×Q₁₆	—	2	1	no	—
90	C₁₃×D₈	—	2	1	no	—
91	C₁₃×D₈	—	2	1	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
96	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
97	C₁₃×C₄⋊C₄	—	2	1	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	2	1	no	—
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	—	4	1	no	—
102	C₁₃×C₄⋊C₄	—	4	1	no	—
103	C₁₃×C₄⋊C₄	—	4	2	no	—
104	C₁₃×C₄⋊C₄	—	4	1	no	—
105	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
106	C₁₃×C₂²⋊C₄	—	4	2	no	—
107	C₁₃×C₂²⋊C₄	—	4	2	no	—
108	C₁₃×C₂²⋊C₄	—	4	2	no	—
109	C₁₃×C₂²⋊C₄	—	4	2	no	—
110	C₁₃×C₂²⋊C₄	—	4	1	no	—
111	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
112	C₄²×C₁₃	—	4	2	no	—
113	C₄²×C₁₃	—	4	2	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, complemented
120	#3341	C₂×C₁₃	1	1	yes	characteristic
121	#2254	C₂×C₁₃	1	1	yes	characteristic, complemented
122	#2237	C₂×C₁₃	1	2	yes	complemented
123	#2234	C₂×C₁₃	1	2	yes	complemented
124	#2227	C₂×C₁₃	1	1	yes	characteristic, complemented
125	#2224	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#2217	C₂×C₁₃	1	2	yes	complemented
127	C₄×SD₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
128	D₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
129	#2158	C₂×C₁₃	1	2	yes	complemented
130	C₂³×C₁₃	—	4	1	no	—
131	C₂³×C₁₃	C₂×D₈	1	2	yes	—
132	Q₈×C₁₃	—	4	1	no	—
133	Q₈×C₁₃	—	4	1	no	—
134	Q₈×C₁₃	—	8	2	no	—
135	Q₈×C₁₃	—	16	1	no	—
136	D₄×C₁₃	—	2	2	no	—
137	D₄×C₁₃	—	4	1	no	—
138	D₄×C₁₃	—	4	1	no	—
139	D₄×C₁₃	—	16	2	no	—
140	C₂×C₄×C₁₃	—	2	2	no	—
141	C₂×C₄×C₁₃	—	2	2	no	—
142	C₂×C₄×C₁₃	—	4	1	no	—
143	C₂×C₄×C₁₃	—	4	1	no	—
144	C₂×C₄×C₁₃	—	4	2	no	—
145	C₂×C₄×C₁₃	—	4	2	no	—
146	C₂×C₄×C₁₃	—	4	1	no	—
147	C₂×C₄×C₁₃	—	8	2	no	—
148	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
149	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
152	C₈×C₁₃	—	2	2	no	—
153	C₈×C₁₃	—	2	1	no	—
154	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
155	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
156	#556	—	2	1	no	—
157	#555	—	2	1	no	—
158	C₂×SD₃₂	C₂²×C₁₃	1	1	yes	characteristic
159	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
160	#500	C₂²×C₁₃	1	1	yes	characteristic
161	#500	C₂²×C₁₃	1	1	yes	characteristic
162	#485	—	2	2	no	—
163	#482	—	2	2	no	—
164	#476	—	2	1	no	—
165	#475	—	2	1	no	—
166	#467	C₂²×C₁₃	1	2	yes	—
167	#466	C₂²×C₁₃	1	2	yes	—
168	C₄×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
169	C₄×D₈	C₂²×C₁₃	1	1	yes	characteristic
170	C₈×D₄	C₂²×C₁₃	1	1	yes	characteristic
171	#427	C₂²×C₁₃	1	2	yes	—
172	#427	C₂²×C₁₃	1	2	yes	—
173	#367	C₂²×C₁₃	1	1	yes	characteristic
174	#367	C₂²×C₁₃	1	2	yes	—
175	#367	C₂²×C₁₃	1	2	yes	—
176	#366	C₂²×C₁₃	1	2	yes	—
177	#366	C₂²×C₁₃	1	2	yes	—
178	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
179	#358	C₂²×C₁₃	1	1	yes	characteristic
180	#358	C₂²×C₁₃	1	2	yes	—
181	#357	C₂²×C₁₃	1	1	yes	characteristic
182	#357	C₂²×C₁₃	1	2	yes	—
183	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
184	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
185	C₂²×C₁₃	—	2	2	no	—
186	C₂²×C₁₃	—	2	2	no	—
187	C₂²×C₁₃	—	2	2	no	—
188	C₂²×C₁₃	—	8	1	no	—
189	C₂²×C₁₃	—	8	1	no	—
190	C₂²×C₁₃	—	8	1	no	—
191	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
192	C₄×C₁₃	—	2	2	no	—
193	C₄×C₁₃	—	2	1	no	—
194	C₄×C₁₃	—	8	1	no	—
195	C₄×C₁₃	—	8	1	no	—
196	C₄×C₁₃	—	8	2	no	—
197	C₄×C₁₃	—	8	1	no	—
198	C₄×C₁₃	—	8	2	no	—
199	C₄×C₁₃	#508	1	2	yes	—
200	C₄×C₁₃	#547	1	1	yes	characteristic
201	C₄×C₁₃	#578	1	1	yes	characteristic
202	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
203	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
205	C₄⋊Q₈	—	2	1	no	—
206	C₄⋊Q₈	—	2	1	no	—
207	#126	—	4	2	no	—
208	#124	—	4	1	no	—
209	#124	—	4	2	no	—
210	C₂²⋊Q₈	—	2	2	no	—
211	#121	—	2	2	no	—
212	#121	—	4	1	no	—
213	C₄×Q₈	—	2	1	no	—
214	C₄×Q₈	—	4	1	no	—
215	C₄×D₄	—	2	1	no	—
216	C₄×D₄	D₄×C₁₃	1	1	yes	characteristic
217	#117	—	2	2	no	—
218	#117	—	2	2	no	—
219	#117	—	4	1	no	—
220	SD₃₂	—	2	2	no	—
221	C₂×C₁₆	—	2	4	no	—
222	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
223	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
224	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
225	#107	—	2	1	no	—
226	#107	—	2	1	no	—
227	#107	C₂³×C₁₃	1	1	yes	characteristic
228	#107	C₂³×C₁₃	1	2	yes	—
229	#107	C₂³×C₁₃	1	1	yes	characteristic
230	#107	C₂³×C₁₃	1	2	yes	—
231	#106	—	2	2	no	—
232	#106	—	2	2	no	—
233	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
234	#103	—	2	1	no	—
235	#103	—	2	2	no	—
236	#102	—	2	1	no	—
237	#102	—	2	2	no	—
238	C₂²⋊C₈	D₄×C₁₃	1	2	yes	—
239	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
240	C₂×C₁₃	—	4	2	no	—
241	C₂×C₁₃	—	16	1	no	—
242	C₂×C₁₃	#3334	1	1	yes	characteristic
243	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
244	C₂×C₁₃	#3416	1	1	yes	characteristic
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	—	4	1	no	—
248	C₂×D₄	—	2	1	no	—
249	C₂×D₄	—	4	2	no	—
250	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
251	C₂²×C₄	—	4	1	no	—
252	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
253	Q₁₆	—	2	1	no	—
254	Q₁₆	—	2	1	no	—
255	D₈	—	2	1	no	—
256	D₈	—	2	1	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
261	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	2	1	no	—
265	C₄⋊C₄	—	2	2	no	—
266	C₄⋊C₄	—	4	1	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
271	C₂²⋊C₄	—	4	2	no	—
272	C₂²⋊C₄	—	4	2	no	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	2	no	—
275	C₂²⋊C₄	—	4	1	no	—
276	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
277	C₄²	—	4	2	no	—
278	C₄²	—	4	2	no	—
279	C₄²	—	4	1	no	—
280	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
281	C₁₆	—	2	2	no	—
282	C₁₆	—	2	1	no	—
283	C₁₆	C₁₃×#41	1	2	yes	—
284	C₁₃	#33907	1	1	yes	characteristic, direct product factor, Sylow
285	C₂³	—	4	1	no	—
286	C₂³	C₂×C₁₃×D₈	1	2	yes	—
287	Q₈	—	4	1	no	—
288	Q₈	—	4	1	no	—
289	Q₈	—	8	2	no	—
290	Q₈	—	16	1	no	—
291	D₄	—	2	2	no	—
292	D₄	—	4	1	no	—
293	D₄	—	4	1	no	—
294	D₄	—	16	2	no	—
295	C₂×C₄	—	2	2	no	—
296	C₂×C₄	—	2	2	no	—
297	C₂×C₄	—	4	1	no	—
298	C₂×C₄	—	4	1	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	8	2	no	—
303	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
304	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
305	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
306	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
307	C₈	—	2	2	no	—
308	C₈	—	2	1	no	—
309	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
310	C₈	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
311	C₂²	—	2	2	no	—
312	C₂²	—	2	2	no	—
313	C₂²	—	2	2	no	—
314	C₂²	—	8	1	no	—
315	C₂²	—	8	1	no	—
316	C₂²	—	8	1	no	—
317	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	1	no	—
320	C₄	—	8	1	no	—
321	C₄	—	8	1	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	1	no	—
324	C₄	—	8	2	no	—
325	C₄	C₁₃×#508	1	2	yes	—
326	C₄	C₁₃×#547	1	1	yes	characteristic
327	C₄	C₁₃×#578	1	1	yes	characteristic
328	C₂	—	4	2	no	—
329	C₂	—	16	1	no	—
330	C₂	C₁₃×#3334	1	1	yes	characteristic
331	C₂	C₂×C₁₃×#508	1	1	yes	characteristic
332	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	C₁₃×#3341	C₂
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
1. C₁₃×#3342	C₂	semidirect
3. C₁₃×#2254	C₂	semidirect
4. C₁₃×#2237	C₂	semidirect
5. C₁₃×#2234	C₂	semidirect
6. C₁₃×#2227	C₂	semidirect
7. C₁₃×#2224	C₂	semidirect
8. C₁₃×#2217	C₂	semidirect
9. C₄×C₁₃×SD₃₂	C₂	semidirect
10. D₄×C₁₃×C₁₆	C₂	semidirect
11. C₁₃×#2158	C₂	semidirect
79. #33907	C₁₃	direct (N × H)
119. #3342	C₂×C₁₃	semidirect
121. #2254	C₂×C₁₃	semidirect
122. #2237	C₂×C₁₃	semidirect
123. #2234	C₂×C₁₃	semidirect
124. #2227	C₂×C₁₃	semidirect
125. #2224	C₂×C₁₃	semidirect
126. #2217	C₂×C₁₃	semidirect
127. C₄×SD₃₂	C₂×C₁₃	semidirect
128. D₄×C₁₆	C₂×C₁₃	semidirect
129. #2158	C₂×C₁₃	semidirect
284. C₁₃	#33907	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×#3341	2
C₁₃×#2254	2
C₁₃×#2237	2
C₁₃×#2234	2
C₁₃×#2227	2
C₁₃×#2224	2
C₁₃×#2217	2
C₄×C₁₃×SD₃₂	2
D₄×C₁₃×C₁₆	2
C₁₃×#2158	2
#33907	13

Subgroups — C13×#33907 #15720246

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33907 #15720246