Subgroups — C₁₃×#33902 #15720241

Order 3328.

516 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3338	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#3334	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2254	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2235	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2233	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2226	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2225	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	C₂×C₁₃×#367	C₂	1	2	yes	maximal, complemented
12	C₁₃×#554	—	2	1	no	—
13	C₁₃×#547	—	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×#484	—	2	2	no	—
19	C₁₃×#480	—	2	2	no	—
20	C₁₃×#477	—	2	1	no	—
21	C₁₃×#474	—	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	C₂×C₁₃×#107	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	1	yes	characteristic
32	C₁₃×#367	C₂²	1	2	yes	—
33	C₁₃×#367	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×#41	—	4	1	no	—
42	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
43	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
44	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
45	C₁₃×C₄⋊Q₈	—	2	1	no	—
46	C₁₃×C₄⋊Q₈	—	2	1	no	—
47	C₁₃×#125	—	4	1	no	—
48	C₁₃×#123	—	4	2	no	—
49	C₁₃×C₂²⋊Q₈	—	2	2	no	—
50	C₁₃×C₂²⋊Q₈	—	4	1	no	—
51	C₁₃×C₂²⋊Q₈	—	4	2	no	—
52	C₁₃×#121	—	2	2	no	—
53	C₄×Q₈×C₁₃	—	2	1	no	—
54	C₄×D₄×C₁₃	—	2	1	no	—
55	C₄×D₄×C₁₃	—	4	1	no	—
56	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
57	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
58	C₂×C₁₃×C₄⋊C₄	—	2	2	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	2	no	—
65	C₁₃×#107	—	2	2	no	—
66	C₁₃×#107	—	2	1	no	—
67	C₁₃×#107	—	2	1	no	—
68	C₁₃×#107	C₂³	1	1	yes	characteristic
69	C₁₃×#107	C₂³	1	1	yes	characteristic
70	C₁₃×#107	C₂³	1	2	yes	—
71	C₁₃×#107	C₂³	1	2	yes	—
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	#33902	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₁₃×#41	—	8	4	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	—	2	1	no	—
85	C₂×D₄×C₁₃	—	4	2	no	—
86	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
87	C₂²×C₄×C₁₃	—	4	2	no	—
88	C₂²×C₄×C₁₃	—	4	2	no	—
89	C₂²×C₄×C₁₃	—	4	1	no	—
90	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
91	C₁₃×Q₁₆	—	2	1	no	—
92	C₁₃×Q₁₆	—	2	1	no	—
93	C₁₃×D₈	—	2	1	no	—
94	C₁₃×D₈	—	2	1	no	—
95	C₂×C₈×C₁₃	—	2	2	no	—
96	C₂×C₈×C₁₃	—	2	2	no	—
97	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
99	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
100	C₁₃×C₄⋊C₄	—	2	1	no	—
101	C₁₃×C₄⋊C₄	—	2	2	no	—
102	C₁₃×C₄⋊C₄	—	2	1	no	—
103	C₁₃×C₄⋊C₄	—	2	2	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	—	4	1	no	—
108	C₁₃×C₄⋊C₄	—	4	2	no	—
109	C₁₃×C₄⋊C₄	—	4	1	no	—
110	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
111	C₁₃×C₂²⋊C₄	—	4	2	no	—
112	C₁₃×C₂²⋊C₄	—	4	1	no	—
113	C₁₃×C₂²⋊C₄	—	4	2	no	—
114	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
115	C₄²×C₁₃	—	4	1	no	—
116	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
117	C₁₃×C₁₆	—	2	2	no	—
118	C₁₃×C₁₆	—	2	1	no	—
119	C₁₃×C₁₆	#41	1	2	yes	complemented
120	#3338	C₂×C₁₃	1	1	yes	characteristic, complemented
121	#3334	C₂×C₁₃	1	1	yes	characteristic
122	#2254	C₂×C₁₃	1	1	yes	characteristic, complemented
123	#2235	C₂×C₁₃	1	2	yes	complemented
124	#2233	C₂×C₁₃	1	2	yes	complemented
125	#2226	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#2225	C₂×C₁₃	1	1	yes	characteristic, complemented
127	#2217	C₂×C₁₃	1	2	yes	complemented
128	C₄×SD₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
129	D₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
130	C₂×#367	C₂×C₁₃	1	2	yes	complemented
131	C₂³×C₁₃	—	4	1	no	—
132	C₂³×C₁₃	C₂×D₈	1	2	yes	—
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	—	4	1	no	—
138	D₄×C₁₃	—	4	1	no	—
139	D₄×C₁₃	—	8	4	no	—
140	D₄×C₁₃	SD₃₂	1	4	yes	complemented
141	C₂×C₄×C₁₃	—	2	2	no	—
142	C₂×C₄×C₁₃	—	2	2	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	—	4	1	no	—
147	C₂×C₄×C₁₃	—	4	2	no	—
148	C₂×C₄×C₁₃	—	8	4	no	—
149	C₂×C₄×C₁₃	—	8	2	no	—
150	C₂×C₄×C₁₃	—	16	2	no	—
151	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
153	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
155	C₈×C₁₃	—	2	2	no	—
156	C₈×C₁₃	—	2	1	no	—
157	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
158	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
159	#554	—	2	1	no	—
160	#547	—	2	1	no	—
161	C₂×SD₃₂	C₂²×C₁₃	1	1	yes	characteristic
162	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
163	#500	C₂²×C₁₃	1	1	yes	characteristic
164	#500	C₂²×C₁₃	1	1	yes	characteristic
165	#484	—	2	2	no	—
166	#480	—	2	2	no	—
167	#477	—	2	1	no	—
168	#474	—	2	1	no	—
169	#467	C₂²×C₁₃	1	2	yes	—
170	#466	C₂²×C₁₃	1	2	yes	—
171	C₄×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
172	C₄×D₈	C₂²×C₁₃	1	1	yes	characteristic
173	C₈×D₄	C₂²×C₁₃	1	1	yes	characteristic
174	C₂×#107	C₂²×C₁₃	1	2	yes	—
175	C₂×#107	C₂²×C₁₃	1	2	yes	—
176	#367	C₂²×C₁₃	1	1	yes	characteristic
177	#367	C₂²×C₁₃	1	2	yes	—
178	#367	C₂²×C₁₃	1	2	yes	—
179	#367	C₂²×C₁₃	1	2	yes	—
180	#367	C₂²×C₁₃	1	2	yes	—
181	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
182	#358	C₂²×C₁₃	1	1	yes	characteristic
183	#358	C₂²×C₁₃	1	2	yes	—
184	#357	C₂²×C₁₃	1	1	yes	characteristic
185	#357	C₂²×C₁₃	1	2	yes	—
186	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
187	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
188	C₂²×C₁₃	—	2	2	no	—
189	C₂²×C₁₃	—	2	2	no	—
190	C₂²×C₁₃	—	8	1	no	—
191	C₂²×C₁₃	—	8	1	no	—
192	C₂²×C₁₃	—	8	1	no	—
193	C₂²×C₁₃	C₂×SD₃₂	1	4	yes	—
194	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
195	C₄×C₁₃	—	2	2	no	—
196	C₄×C₁₃	—	2	1	no	—
197	C₄×C₁₃	—	8	1	no	—
198	C₄×C₁₃	—	8	1	no	—
199	C₄×C₁₃	—	8	2	no	—
200	C₄×C₁₃	—	8	1	no	—
201	C₄×C₁₃	—	8	2	no	—
202	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
203	C₄×C₁₃	#547	1	1	yes	characteristic
204	C₄×C₁₃	#578	1	1	yes	characteristic
205	C₂×#41	—	4	1	no	—
206	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
207	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
208	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	2	1	no	—
211	#125	—	4	1	no	—
212	#123	—	4	2	no	—
213	C₂²⋊Q₈	—	2	2	no	—
214	C₂²⋊Q₈	—	4	1	no	—
215	C₂²⋊Q₈	—	4	2	no	—
216	#121	—	2	2	no	—
217	C₄×Q₈	—	2	1	no	—
218	C₄×D₄	—	2	1	no	—
219	C₄×D₄	—	4	1	no	—
220	C₄×D₄	D₄×C₁₃	1	1	yes	characteristic
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	SD₃₂	—	2	2	no	—
224	C₂×C₁₆	—	2	4	no	—
225	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
226	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
227	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
228	#107	—	2	2	no	—
229	#107	—	2	2	no	—
230	#107	—	2	1	no	—
231	#107	—	2	1	no	—
232	#107	C₂³×C₁₃	1	1	yes	characteristic
233	#107	C₂³×C₁₃	1	2	yes	—
234	#107	C₂³×C₁₃	1	1	yes	characteristic
235	#107	C₂³×C₁₃	1	2	yes	—
236	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
237	#103	—	2	1	no	—
238	#103	—	2	2	no	—
239	#102	—	2	1	no	—
240	#102	—	2	2	no	—
241	C₂²⋊C₈	D₄×C₁₃	1	2	yes	—
242	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
243	C₂×C₁₃	—	2	4	no	—
244	C₂×C₁₃	—	16	1	no	—
245	C₂×C₁₃	#3334	1	1	yes	characteristic
246	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
247	C₂×C₁₃	#3416	1	1	yes	characteristic
248	#41	—	8	4	no	—
249	C₂×Q₈	—	2	1	no	—
250	C₂×Q₈	—	4	1	no	—
251	C₂×Q₈	—	4	1	no	—
252	C₂×D₄	—	2	1	no	—
253	C₂×D₄	—	4	2	no	—
254	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
255	C₂²×C₄	—	4	2	no	—
256	C₂²×C₄	—	4	2	no	—
257	C₂²×C₄	—	4	1	no	—
258	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
259	Q₁₆	—	2	1	no	—
260	Q₁₆	—	2	1	no	—
261	D₈	—	2	1	no	—
262	D₈	—	2	1	no	—
263	C₂×C₈	—	2	2	no	—
264	C₂×C₈	—	2	2	no	—
265	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
266	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
267	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
268	C₄⋊C₄	—	2	1	no	—
269	C₄⋊C₄	—	2	2	no	—
270	C₄⋊C₄	—	2	1	no	—
271	C₄⋊C₄	—	2	2	no	—
272	C₄⋊C₄	—	4	2	no	—
273	C₄⋊C₄	—	4	2	no	—
274	C₄⋊C₄	—	4	1	no	—
275	C₄⋊C₄	—	4	1	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	1	no	—
278	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
279	C₂²⋊C₄	—	4	2	no	—
280	C₂²⋊C₄	—	4	1	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
283	C₄²	—	4	1	no	—
284	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
285	C₁₆	—	2	2	no	—
286	C₁₆	—	2	1	no	—
287	C₁₆	C₁₃×#41	1	2	yes	complemented
288	C₁₃	#33902	1	1	yes	characteristic, direct product factor, Sylow
289	C₂³	—	4	1	no	—
290	C₂³	C₂×C₁₃×D₈	1	2	yes	—
291	Q₈	—	4	1	no	—
292	Q₈	—	4	1	no	—
293	Q₈	—	8	2	no	—
294	Q₈	—	16	1	no	—
295	D₄	—	4	1	no	—
296	D₄	—	4	1	no	—
297	D₄	—	8	4	no	—
298	D₄	C₁₃×SD₃₂	1	4	yes	complemented
299	C₂×C₄	—	2	2	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	1	no	—
303	C₂×C₄	—	4	2	no	—
304	C₂×C₄	—	4	1	no	—
305	C₂×C₄	—	4	2	no	—
306	C₂×C₄	—	8	4	no	—
307	C₂×C₄	—	8	2	no	—
308	C₂×C₄	—	16	2	no	—
309	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
310	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
311	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
312	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
313	C₈	—	2	2	no	—
314	C₈	—	2	1	no	—
315	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
316	C₈	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
317	C₂²	—	2	2	no	—
318	C₂²	—	2	2	no	—
319	C₂²	—	8	1	no	—
320	C₂²	—	8	1	no	—
321	C₂²	—	8	1	no	—
322	C₂²	C₂×C₁₃×SD₃₂	1	4	yes	—
323	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
324	C₄	—	2	2	no	—
325	C₄	—	2	1	no	—
326	C₄	—	8	1	no	—
327	C₄	—	8	1	no	—
328	C₄	—	8	2	no	—
329	C₄	—	8	1	no	—
330	C₄	—	8	2	no	—
331	C₄	C₂×C₁₃×SD₃₂	1	2	yes	—
332	C₄	C₁₃×#547	1	1	yes	characteristic
333	C₄	C₁₃×#578	1	1	yes	characteristic
334	C₂	—	2	4	no	—
335	C₂	—	16	1	no	—
336	C₂	C₁₃×#3334	1	1	yes	characteristic
337	C₂	C₂²×C₁₃×SD₃₂	1	1	yes	characteristic
338	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×#3334	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×#3338	C₂	semidirect
3. C₁₃×#2254	C₂	semidirect
4. C₁₃×#2235	C₂	semidirect
5. C₁₃×#2233	C₂	semidirect
6. C₁₃×#2226	C₂	semidirect
7. C₁₃×#2225	C₂	semidirect
8. C₁₃×#2217	C₂	semidirect
9. C₄×C₁₃×SD₃₂	C₂	semidirect
10. D₄×C₁₃×C₁₆	C₂	semidirect
11. C₂×C₁₃×#367	C₂	semidirect
79. #33902	C₁₃	direct (N × H)
119. C₁₃×C₁₆	#41	semidirect
120. #3338	C₂×C₁₃	semidirect
122. #2254	C₂×C₁₃	semidirect
123. #2235	C₂×C₁₃	semidirect
124. #2233	C₂×C₁₃	semidirect
125. #2226	C₂×C₁₃	semidirect
126. #2225	C₂×C₁₃	semidirect
127. #2217	C₂×C₁₃	semidirect
128. C₄×SD₃₂	C₂×C₁₃	semidirect
129. D₄×C₁₆	C₂×C₁₃	semidirect
130. C₂×#367	C₂×C₁₃	semidirect
140. D₄×C₁₃	SD₃₂	semidirect
287. C₁₆	C₁₃×#41	semidirect
288. C₁₃	#33902	direct (N × H)
298. D₄	C₁₃×SD₃₂	semidirect

Maximal subgroups

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

subgroup	index
C₁₃×#3338	2
C₁₃×#3334	2
C₁₃×#2254	2
C₁₃×#2235	2
C₁₃×#2233	2
C₁₃×#2226	2
C₁₃×#2225	2
C₁₃×#2217	2
C₄×C₁₃×SD₃₂	2
D₄×C₁₃×C₁₆	2
C₂×C₁₃×#367	2
#33902	13

Subgroups — C13×#33902 #15720241

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33902 #15720241