Subgroups — C₁₃×#33896 #15720235

Order 3328.

546 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3341	C₂	1	1	yes	maximal, characteristic
2	D₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2241	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2237	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2227	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#2220	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2208	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2185	C₂	1	2	yes	maximal, complemented
9	C₄×C₁₃×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#2169	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2141	C₂	1	2	yes	maximal, complemented
12	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
13	C₁₃×#555	—	2	1	no	—
14	D₄×Q₈×C₁₃	—	2	1	no	—
15	C₂×C₁₃×SD₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
17	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
18	C₁₃×#500	C₂²	1	1	yes	characteristic
19	C₁₃×#494	C₂²	1	1	yes	characteristic
20	C₁₃×#482	—	2	2	no	—
21	C₁₃×#476	—	2	1	no	—
22	C₁₃×#467	C₂²	1	2	yes	—
23	C₁₃×#466	C₂²	1	2	yes	—
24	C₁₃×#462	—	2	1	no	—
25	C₁₃×#451	—	2	2	no	—
26	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
27	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
28	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
29	C₁₃×#427	C₂²	1	2	yes	—
30	C₁₃×#367	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	1	yes	characteristic
36	C₁₃×#358	C₂²	1	2	yes	—
37	C₁₃×#358	C₂²	1	2	yes	—
38	C₁₃×#357	C₂²	1	1	yes	characteristic
39	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₁₃	—	2	2	no	—
42	C₂×C₁₃×Q₁₆	—	2	4	no	—
43	C₂×C₁₃×Q₁₆	—	2	2	no	—
44	C₂×C₁₃×Q₁₆	—	2	1	no	—
45	C₂×C₁₃×Q₁₆	—	4	2	no	—
46	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
47	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
48	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
50	C₁₃×C₄⋊Q₈	—	2	1	no	—
51	C₁₃×C₄⋊Q₈	—	2	1	no	—
52	C₁₃×C₄⋊Q₈	—	4	1	no	—
53	C₁₃×#124	—	4	2	no	—
54	C₁₃×C₂²⋊Q₈	—	2	2	no	—
55	C₁₃×C₂²⋊Q₈	—	4	2	no	—
56	C₁₃×#121	—	2	2	no	—
57	C₁₃×#121	—	4	1	no	—
58	C₄×Q₈×C₁₃	—	2	1	no	—
59	C₄×D₄×C₁₃	—	2	1	no	—
60	C₄×D₄×C₁₃	—	4	1	no	—
61	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
62	C₁₃×#117	—	2	2	no	—
63	C₁₃×#117	—	4	1	no	—
64	C₁₃×SD₃₂	—	2	2	no	—
65	C₁₃×SD₃₂	—	4	2	no	—
66	C₁₃×M₃₂	—	4	2	no	—
67	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
68	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
69	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
70	C₁₃×#107	—	2	1	no	—
71	C₁₃×#107	C₂³	1	1	yes	characteristic
72	C₁₃×#107	C₂³	1	2	yes	—
73	C₁₃×#107	C₂³	1	1	yes	characteristic
74	C₁₃×#106	—	2	2	no	—
75	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
76	C₁₃×#103	—	2	1	no	—
77	C₁₃×#103	—	2	2	no	—
78	C₁₃×#102	—	2	1	no	—
79	C₁₃×#102	—	2	2	no	—
80	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
81	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
82	#33896	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂×Q₈×C₁₃	—	2	1	no	—
84	C₂×Q₈×C₁₃	—	2	2	no	—
85	C₂×Q₈×C₁₃	—	4	4	no	—
86	C₂×Q₈×C₁₃	—	4	2	no	—
87	C₂×Q₈×C₁₃	—	4	1	no	—
88	C₂×Q₈×C₁₃	—	8	2	no	—
89	C₂×D₄×C₁₃	—	2	1	no	—
90	C₂×D₄×C₁₃	—	4	2	no	—
91	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
92	C₂²×C₄×C₁₃	—	4	2	no	—
93	C₂²×C₄×C₁₃	—	4	1	no	—
94	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
95	C₁₃×Q₁₆	—	2	2	no	—
96	C₁₃×Q₁₆	—	2	1	no	—
97	C₁₃×Q₁₆	—	2	2	no	—
98	C₁₃×Q₁₆	—	4	4	no	—
99	C₁₃×Q₁₆	—	8	1	no	—
100	C₁₃×Q₁₆	#41	1	2	yes	—
101	C₁₃×D₈	—	2	1	no	—
102	C₁₃×D₈	—	2	1	no	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	—	2	2	no	—
105	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
108	C₁₃×C₄⋊C₄	—	2	1	no	—
109	C₁₃×C₄⋊C₄	—	2	2	no	—
110	C₁₃×C₄⋊C₄	—	2	1	no	—
111	C₁₃×C₄⋊C₄	—	4	1	no	—
112	C₁₃×C₄⋊C₄	—	4	1	no	—
113	C₁₃×C₄⋊C₄	—	4	1	no	—
114	C₁₃×C₄⋊C₄	—	4	2	no	—
115	C₁₃×C₄⋊C₄	—	4	1	no	—
116	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
117	C₁₃×C₂²⋊C₄	—	4	2	no	—
118	C₁₃×C₂²⋊C₄	—	4	2	no	—
119	C₁₃×C₂²⋊C₄	—	4	2	no	—
120	C₁₃×C₂²⋊C₄	—	4	1	no	—
121	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
122	C₄²×C₁₃	—	4	2	no	—
123	C₄²×C₁₃	—	4	1	no	—
124	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
125	C₁₃×C₁₆	—	2	1	no	—
126	C₁₃×C₁₆	—	2	1	no	—
127	C₁₃×C₁₆	—	2	2	no	—
128	#3341	C₂×C₁₃	1	1	yes	characteristic
129	D₄×Q₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
130	#2241	C₂×C₁₃	1	1	yes	characteristic, complemented
131	#2237	C₂×C₁₃	1	2	yes	complemented
132	#2227	C₂×C₁₃	1	1	yes	characteristic, complemented
133	#2220	C₂×C₁₃	1	2	yes	complemented
134	#2208	C₂×C₁₃	1	1	yes	characteristic, complemented
135	#2185	C₂×C₁₃	1	2	yes	complemented
136	C₄×SD₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
137	#2169	C₂×C₁₃	1	1	yes	characteristic, complemented
138	#2141	C₂×C₁₃	1	2	yes	complemented
139	C₂³×C₁₃	—	4	1	no	—
140	C₂³×C₁₃	C₂×D₈	1	2	yes	—
141	Q₈×C₁₃	—	2	2	no	—
142	Q₈×C₁₃	—	4	1	no	—
143	Q₈×C₁₃	—	4	2	no	—
144	Q₈×C₁₃	—	4	2	no	—
145	Q₈×C₁₃	—	8	4	no	—
146	Q₈×C₁₃	—	16	1	no	—
147	D₄×C₁₃	—	2	2	no	—
148	D₄×C₁₃	—	4	1	no	—
149	D₄×C₁₃	—	4	1	no	—
150	D₄×C₁₃	—	16	2	no	—
151	C₂×C₄×C₁₃	—	2	2	no	—
152	C₂×C₄×C₁₃	—	2	2	no	—
153	C₂×C₄×C₁₃	—	4	1	no	—
154	C₂×C₄×C₁₃	—	4	1	no	—
155	C₂×C₄×C₁₃	—	4	1	no	—
156	C₂×C₄×C₁₃	—	4	2	no	—
157	C₂×C₄×C₁₃	—	4	2	no	—
158	C₂×C₄×C₁₃	—	8	4	no	—
159	C₂×C₄×C₁₃	—	8	2	no	—
160	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
161	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
163	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
164	C₈×C₁₃	—	2	2	no	—
165	C₈×C₁₃	—	2	1	no	—
166	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
167	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
168	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
169	#555	—	2	1	no	—
170	D₄×Q₈	—	2	1	no	—
171	C₂×SD₃₂	C₂²×C₁₃	1	1	yes	characteristic
172	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
173	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
174	#500	C₂²×C₁₃	1	1	yes	characteristic
175	#494	C₂²×C₁₃	1	1	yes	characteristic
176	#482	—	2	2	no	—
177	#476	—	2	1	no	—
178	#467	C₂²×C₁₃	1	2	yes	—
179	#466	C₂²×C₁₃	1	2	yes	—
180	#462	—	2	1	no	—
181	#451	—	2	2	no	—
182	C₄×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
183	C₄×D₈	C₂²×C₁₃	1	1	yes	characteristic
184	C₈×D₄	C₂²×C₁₃	1	1	yes	characteristic
185	#427	C₂²×C₁₃	1	2	yes	—
186	#367	C₂²×C₁₃	1	1	yes	characteristic
187	#366	C₂²×C₁₃	1	2	yes	—
188	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
189	#358	C₂²×C₁₃	1	1	yes	characteristic
190	#358	C₂²×C₁₃	1	2	yes	—
191	#358	C₂²×C₁₃	1	2	yes	—
192	#358	C₂²×C₁₃	1	2	yes	—
193	#358	C₂²×C₁₃	1	2	yes	—
194	#357	C₂²×C₁₃	1	1	yes	characteristic
195	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
196	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
197	C₂²×C₁₃	—	2	2	no	—
198	C₂²×C₁₃	—	2	2	no	—
199	C₂²×C₁₃	—	2	2	no	—
200	C₂²×C₁₃	—	8	1	no	—
201	C₂²×C₁₃	—	8	1	no	—
202	C₂²×C₁₃	—	8	1	no	—
203	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
204	C₄×C₁₃	—	2	1	no	—
205	C₄×C₁₃	—	2	2	no	—
206	C₄×C₁₃	—	4	2	no	—
207	C₄×C₁₃	—	8	1	no	—
208	C₄×C₁₃	—	8	1	no	—
209	C₄×C₁₃	—	8	2	no	—
210	C₄×C₁₃	—	8	2	no	—
211	C₄×C₁₃	#510	1	2	yes	—
212	C₄×C₁₃	#546	1	1	yes	characteristic
213	C₄×C₁₃	#577	1	1	yes	characteristic
214	C₂²×Q₈	—	2	2	no	—
215	C₂×Q₁₆	—	2	4	no	—
216	C₂×Q₁₆	—	2	2	no	—
217	C₂×Q₁₆	—	2	1	no	—
218	C₂×Q₁₆	—	4	2	no	—
219	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
220	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
221	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
222	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
223	C₄⋊Q₈	—	2	1	no	—
224	C₄⋊Q₈	—	2	1	no	—
225	C₄⋊Q₈	—	4	1	no	—
226	#124	—	4	2	no	—
227	C₂²⋊Q₈	—	2	2	no	—
228	C₂²⋊Q₈	—	4	2	no	—
229	#121	—	2	2	no	—
230	#121	—	4	1	no	—
231	C₄×Q₈	—	2	1	no	—
232	C₄×D₄	—	2	1	no	—
233	C₄×D₄	—	4	1	no	—
234	C₄×D₄	D₄×C₁₃	1	1	yes	characteristic
235	#117	—	2	2	no	—
236	#117	—	4	1	no	—
237	SD₃₂	—	2	2	no	—
238	SD₃₂	—	4	2	no	—
239	M₃₂	—	4	2	no	—
240	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
241	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
242	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
243	#107	—	2	1	no	—
244	#107	C₂³×C₁₃	1	1	yes	characteristic
245	#107	C₂³×C₁₃	1	2	yes	—
246	#107	C₂³×C₁₃	1	1	yes	characteristic
247	#106	—	2	2	no	—
248	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
249	#103	—	2	1	no	—
250	#103	—	2	2	no	—
251	#102	—	2	1	no	—
252	#102	—	2	2	no	—
253	C₂²⋊C₈	D₄×C₁₃	1	2	yes	—
254	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
255	C₂×C₁₃	—	4	2	no	—
256	C₂×C₁₃	—	16	1	no	—
257	C₂×C₁₃	#3294	1	1	yes	characteristic
258	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
259	C₂×C₁₃	#3416	1	1	yes	characteristic
260	C₂×Q₈	—	2	1	no	—
261	C₂×Q₈	—	2	2	no	—
262	C₂×Q₈	—	4	4	no	—
263	C₂×Q₈	—	4	2	no	—
264	C₂×Q₈	—	4	1	no	—
265	C₂×Q₈	—	8	2	no	—
266	C₂×D₄	—	2	1	no	—
267	C₂×D₄	—	4	2	no	—
268	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
269	C₂²×C₄	—	4	2	no	—
270	C₂²×C₄	—	4	1	no	—
271	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
272	Q₁₆	—	2	2	no	—
273	Q₁₆	—	2	1	no	—
274	Q₁₆	—	2	2	no	—
275	Q₁₆	—	4	4	no	—
276	Q₁₆	—	8	1	no	—
277	Q₁₆	C₁₃×#41	1	2	yes	—
278	D₈	—	2	1	no	—
279	D₈	—	2	1	no	—
280	C₂×C₈	—	2	2	no	—
281	C₂×C₈	—	2	2	no	—
282	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
283	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
284	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
285	C₄⋊C₄	—	2	1	no	—
286	C₄⋊C₄	—	2	2	no	—
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	—	4	1	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	2	no	—
292	C₄⋊C₄	—	4	1	no	—
293	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	4	1	no	—
298	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
299	C₄²	—	4	2	no	—
300	C₄²	—	4	1	no	—
301	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
302	C₁₆	—	2	1	no	—
303	C₁₆	—	2	1	no	—
304	C₁₆	—	2	2	no	—
305	C₁₃	#33896	1	1	yes	characteristic, direct product factor, Sylow
306	C₂³	—	4	1	no	—
307	C₂³	C₂×C₁₃×D₈	1	2	yes	—
308	Q₈	—	2	2	no	—
309	Q₈	—	4	1	no	—
310	Q₈	—	4	2	no	—
311	Q₈	—	4	2	no	—
312	Q₈	—	8	4	no	—
313	Q₈	—	16	1	no	—
314	D₄	—	2	2	no	—
315	D₄	—	4	1	no	—
316	D₄	—	4	1	no	—
317	D₄	—	16	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	1	no	—
322	C₂×C₄	—	4	1	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	8	4	no	—
326	C₂×C₄	—	8	2	no	—
327	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
328	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
329	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
330	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
331	C₈	—	2	2	no	—
332	C₈	—	2	1	no	—
333	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
334	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
335	C₂²	—	2	2	no	—
336	C₂²	—	2	2	no	—
337	C₂²	—	2	2	no	—
338	C₂²	—	8	1	no	—
339	C₂²	—	8	1	no	—
340	C₂²	—	8	1	no	—
341	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
342	C₄	—	2	2	no	—
343	C₄	—	2	1	no	—
344	C₄	—	4	2	no	—
345	C₄	—	8	2	no	—
346	C₄	—	8	1	no	—
347	C₄	—	8	1	no	—
348	C₄	—	8	2	no	—
349	C₄	C₁₃×#510	1	2	yes	—
350	C₄	C₁₃×#546	1	1	yes	characteristic
351	C₄	C₁₃×#577	1	1	yes	characteristic
352	C₂	—	4	2	no	—
353	C₂	—	16	1	no	—
354	C₂	C₁₃×#3294	1	1	yes	characteristic
355	C₂	C₂×C₁₃×#510	1	1	yes	characteristic
356	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
2. D₄×C₁₃×Q₁₆	C₂	semidirect
3. C₁₃×#2241	C₂	semidirect
4. C₁₃×#2237	C₂	semidirect
5. C₁₃×#2227	C₂	semidirect
6. C₁₃×#2220	C₂	semidirect
7. C₁₃×#2208	C₂	semidirect
8. C₁₃×#2185	C₂	semidirect
9. C₄×C₁₃×SD₃₂	C₂	semidirect
10. C₁₃×#2169	C₂	semidirect
11. C₁₃×#2141	C₂	semidirect
82. #33896	C₁₃	direct (N × H)
129. D₄×Q₁₆	C₂×C₁₃	semidirect
130. #2241	C₂×C₁₃	semidirect
131. #2237	C₂×C₁₃	semidirect
132. #2227	C₂×C₁₃	semidirect
133. #2220	C₂×C₁₃	semidirect
134. #2208	C₂×C₁₃	semidirect
135. #2185	C₂×C₁₃	semidirect
136. C₄×SD₃₂	C₂×C₁₃	semidirect
137. #2169	C₂×C₁₃	semidirect
138. #2141	C₂×C₁₃	semidirect
305. C₁₃	#33896	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×#3341	2
D₄×C₁₃×Q₁₆	2
C₁₃×#2241	2
C₁₃×#2237	2
C₁₃×#2227	2
C₁₃×#2220	2
C₁₃×#2208	2
C₁₃×#2185	2
C₄×C₁₃×SD₃₂	2
C₁₃×#2169	2
C₁₃×#2141	2
#33896	13

Subgroups — C13×#33896 #15720235

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33896 #15720235