Subgroups — C₁₃×#33965 #15720304

Order 3328.

498 conjugacy classes of subgroups.

The subgroup lattice has 344 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3395	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#3393	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2251	C₂	1	2	yes	maximal
4	C₁₃×#2247	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2211	C₂	1	2	yes	maximal
6	C₁₃×#2209	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2209	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2208	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#2176	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₃×SD₃₂	C₂	1	1	yes	maximal, characteristic
12	C₁₃×#557	—	2	1	no	—
13	C₁₃×#556	—	2	1	no	—
14	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
15	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
16	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
17	C₂×C₁₃×SD₃₂	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
19	C₁₃×#495	C₂²	1	2	yes	—
20	C₁₃×#494	C₂²	1	1	yes	characteristic
21	C₁₃×#494	C₂²	1	1	yes	characteristic
22	C₁₃×#494	C₂²	1	2	yes	—
23	C₁₃×#464	—	2	2	no	—
24	C₁₃×#462	—	2	1	no	—
25	C₁₃×#462	—	2	2	no	—
26	C₁₃×#461	—	2	1	no	—
27	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
29	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
30	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
31	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
32	C₁₃×#367	C₂²	1	1	yes	characteristic
33	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₁₃×C₄⋊C₁₆	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×#346	C₂²	1	2	yes	—
40	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
41	C₂×C₁₃×Q₁₆	—	2	1	no	—
42	C₂×C₁₃×Q₁₆	—	2	2	no	—
43	C₂×C₁₃×Q₁₆	—	2	1	no	—
44	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
45	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
46	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
47	C₂×C₁₃×SD₁₆	—	2	2	no	—
48	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
49	C₁₃×C₄⋊Q₈	—	2	1	no	—
50	C₁₃×C₄⋊Q₈	—	2	1	no	—
51	C₁₃×C₄⋊Q₈	—	2	2	no	—
52	C₁₃×#126	—	4	2	no	—
53	C₁₃×#125	—	4	1	no	—
54	C₁₃×#125	—	4	2	no	—
55	C₁₃×#124	—	2	2	no	—
56	C₁₃×C₂²⋊Q₈	—	4	1	no	—
57	C₄×Q₈×C₁₃	—	2	1	no	—
58	C₄×Q₈×C₁₃	—	2	2	no	—
59	C₄×Q₈×C₁₃	—	2	2	no	—
60	C₄×Q₈×C₁₃	—	4	1	no	—
61	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
62	C₄×D₄×C₁₃	—	2	1	no	—
63	C₁₃×#117	—	4	1	no	—
64	C₁₃×Q₃₂	—	2	4	no	—
65	C₁₃×Q₃₂	—	4	2	no	—
66	C₁₃×Q₃₂	—	4	2	no	—
67	C₁₃×SD₃₂	—	2	2	no	—
68	C₁₃×SD₃₂	—	4	2	no	—
69	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
70	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
71	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
72	C₁₃×#107	C₂³	1	1	yes	characteristic
73	C₁₃×#107	C₂³	1	1	yes	characteristic
74	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
75	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
76	C₁₃×#103	—	2	2	no	—
77	C₁₃×#103	—	2	1	no	—
78	C₁₃×#103	—	2	2	no	—
79	C₁₃×#102	—	2	1	no	—
80	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₁₃	C₂³	1	2	yes	—
82	#33965	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂×Q₈×C₁₃	—	2	2	no	—
84	C₂×Q₈×C₁₃	—	2	1	no	—
85	C₂×Q₈×C₁₃	—	2	2	no	—
86	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
87	C₂×D₄×C₁₃	—	2	1	no	—
88	C₂²×C₄×C₁₃	—	4	1	no	—
89	C₁₃×Q₁₆	—	2	1	no	—
90	C₁₃×Q₁₆	—	2	2	no	—
91	C₁₃×Q₁₆	—	2	2	no	—
92	C₁₃×Q₁₆	—	2	2	no	—
93	C₁₃×Q₁₆	—	2	1	no	—
94	C₁₃×Q₁₆	—	2	2	no	—
95	C₁₃×Q₁₆	—	8	2	no	—
96	C₁₃×Q₁₆	—	8	1	no	—
97	C₁₃×Q₁₆	—	8	1	no	—
98	C₁₃×SD₁₆	—	8	2	no	—
99	C₁₃×D₈	—	2	1	no	—
100	C₁₃×D₈	—	2	1	no	—
101	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
103	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
104	C₁₃×C₄⋊C₄	—	2	1	no	—
105	C₁₃×C₄⋊C₄	—	2	1	no	—
106	C₁₃×C₄⋊C₄	—	4	1	no	—
107	C₁₃×C₄⋊C₄	—	4	2	no	—
108	C₁₃×C₄⋊C₄	—	4	2	no	—
109	C₁₃×C₄⋊C₄	—	4	1	no	—
110	C₁₃×C₄⋊C₄	—	4	2	no	—
111	C₁₃×C₄⋊C₄	—	4	1	no	—
112	C₁₃×C₄⋊C₄	—	4	2	no	—
113	C₁₃×C₄⋊C₄	—	4	1	no	—
114	C₁₃×C₄⋊C₄	—	4	1	no	—
115	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
116	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
117	C₁₃×C₂²⋊C₄	—	4	2	no	—
118	C₁₃×C₂²⋊C₄	—	4	1	no	—
119	C₁₃×C₂²⋊C₄	—	4	2	no	—
120	C₄²×C₁₃	—	4	2	no	—
121	C₄²×C₁₃	—	4	2	no	—
122	C₄²×C₁₃	—	4	1	no	—
123	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
124	C₄²×C₁₃	C₂×D₄	1	2	yes	—
125	C₁₃×C₁₆	—	2	2	no	—
126	C₁₃×C₁₆	—	2	1	no	—
127	C₁₃×C₁₆	#41	1	2	yes	—
128	#3395	C₂×C₁₃	1	1	yes	characteristic, complemented
129	#3393	C₂×C₁₃	1	1	yes	characteristic
130	#2251	C₂×C₁₃	1	2	yes	—
131	#2247	C₂×C₁₃	1	1	yes	characteristic, complemented
132	#2211	C₂×C₁₃	1	2	yes	—
133	#2209	C₂×C₁₃	1	2	yes	complemented
134	#2209	C₂×C₁₃	1	1	yes	characteristic, complemented
135	#2208	C₂×C₁₃	1	1	yes	characteristic
136	#2183	C₂×C₁₃	1	1	yes	characteristic, complemented
137	#2176	C₂×C₁₃	1	2	yes	complemented
138	C₄×SD₃₂	C₂×C₁₃	1	1	yes	characteristic
139	C₂³×C₁₃	—	4	1	no	—
140	Q₈×C₁₃	—	2	2	no	—
141	Q₈×C₁₃	—	4	1	no	—
142	Q₈×C₁₃	—	4	2	no	—
143	Q₈×C₁₃	—	4	2	no	—
144	Q₈×C₁₃	—	4	2	no	—
145	Q₈×C₁₃	—	4	1	no	—
146	Q₈×C₁₃	—	4	2	no	—
147	D₄×C₁₃	—	4	1	no	—
148	D₄×C₁₃	—	4	1	no	—
149	C₂×C₄×C₁₃	—	4	1	no	—
150	C₂×C₄×C₁₃	—	4	1	no	—
151	C₂×C₄×C₁₃	—	4	2	no	—
152	C₂×C₄×C₁₃	—	4	2	no	—
153	C₂×C₄×C₁₃	—	4	1	no	—
154	C₂×C₄×C₁₃	—	8	2	no	—
155	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
156	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
157	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
159	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
160	C₈×C₁₃	—	2	2	no	—
161	C₈×C₁₃	—	2	1	no	—
162	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
163	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
164	#557	—	2	1	no	—
165	#556	—	2	1	no	—
166	C₂×Q₃₂	C₂²×C₁₃	1	2	yes	—
167	C₂×Q₃₂	C₂²×C₁₃	1	2	yes	—
168	C₂×Q₃₂	C₂²×C₁₃	1	2	yes	—
169	C₂×SD₃₂	C₂²×C₁₃	1	1	yes	characteristic
170	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
171	#495	C₂²×C₁₃	1	2	yes	—
172	#494	C₂²×C₁₃	1	1	yes	characteristic
173	#494	C₂²×C₁₃	1	1	yes	characteristic
174	#494	C₂²×C₁₃	1	2	yes	—
175	#464	—	2	2	no	—
176	#462	—	2	1	no	—
177	#462	—	2	2	no	—
178	#461	—	2	1	no	—
179	C₈×Q₈	C₂²×C₁₃	1	1	yes	characteristic
180	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
181	C₄×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
182	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
183	C₄×D₈	C₂²×C₁₃	1	1	yes	characteristic
184	#367	C₂²×C₁₃	1	1	yes	characteristic
185	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
186	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
187	#358	C₂²×C₁₃	1	1	yes	characteristic
188	#358	C₂²×C₁₃	1	2	yes	—
189	#358	C₂²×C₁₃	1	2	yes	—
190	#357	C₂²×C₁₃	1	1	yes	characteristic
191	#346	C₂²×C₁₃	1	2	yes	—
192	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
193	C₂²×C₁₃	—	8	1	no	—
194	C₂²×C₁₃	—	8	1	no	—
195	C₂²×C₁₃	—	8	1	no	—
196	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
197	C₄×C₁₃	—	2	2	no	—
198	C₄×C₁₃	—	2	2	no	—
199	C₄×C₁₃	—	2	1	no	—
200	C₄×C₁₃	—	8	2	no	—
201	C₄×C₁₃	—	8	1	no	—
202	C₄×C₁₃	—	8	2	no	—
203	C₄×C₁₃	—	8	1	no	—
204	C₄×C₁₃	—	8	1	no	—
205	C₄×C₁₃	#510	1	2	yes	—
206	C₄×C₁₃	#550	1	1	yes	characteristic
207	C₄×C₁₃	#576	1	1	yes	characteristic
208	C₂×Q₁₆	—	2	1	no	—
209	C₂×Q₁₆	—	2	2	no	—
210	C₂×Q₁₆	—	2	1	no	—
211	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
212	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
213	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
214	C₂×SD₁₆	—	2	2	no	—
215	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
216	C₄⋊Q₈	—	2	1	no	—
217	C₄⋊Q₈	—	2	1	no	—
218	C₄⋊Q₈	—	2	2	no	—
219	#126	—	4	2	no	—
220	#125	—	4	1	no	—
221	#125	—	4	2	no	—
222	#124	—	2	2	no	—
223	C₂²⋊Q₈	—	4	1	no	—
224	C₄×Q₈	—	2	2	no	—
225	C₄×Q₈	—	2	1	no	—
226	C₄×Q₈	—	2	2	no	—
227	C₄×Q₈	—	4	1	no	—
228	C₄×Q₈	D₄×C₁₃	1	1	yes	characteristic
229	C₄×D₄	—	2	1	no	—
230	#117	—	4	1	no	—
231	Q₃₂	—	2	4	no	—
232	Q₃₂	—	4	2	no	—
233	Q₃₂	—	4	2	no	—
234	SD₃₂	—	2	2	no	—
235	SD₃₂	—	4	2	no	—
236	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
237	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
238	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
239	#107	C₂³×C₁₃	1	1	yes	characteristic
240	#107	C₂³×C₁₃	1	1	yes	characteristic
241	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
242	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
243	#103	—	2	2	no	—
244	#103	—	2	1	no	—
245	#103	—	2	2	no	—
246	#102	—	2	1	no	—
247	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
248	C₄×C₈	C₂³×C₁₃	1	2	yes	—
249	C₂×C₁₃	—	16	1	no	—
250	C₂×C₁₃	#3391	1	1	yes	characteristic
251	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
252	C₂×C₁₃	#3417	1	1	yes	characteristic
253	C₂×Q₈	—	2	2	no	—
254	C₂×Q₈	—	2	1	no	—
255	C₂×Q₈	—	2	2	no	—
256	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
257	C₂×D₄	—	2	1	no	—
258	C₂²×C₄	—	4	1	no	—
259	Q₁₆	—	2	1	no	—
260	Q₁₆	—	2	2	no	—
261	Q₁₆	—	2	2	no	—
262	Q₁₆	—	2	2	no	—
263	Q₁₆	—	2	1	no	—
264	Q₁₆	—	2	2	no	—
265	Q₁₆	—	8	2	no	—
266	Q₁₆	—	8	1	no	—
267	Q₁₆	—	8	1	no	—
268	SD₁₆	—	8	2	no	—
269	D₈	—	2	1	no	—
270	D₈	—	2	1	no	—
271	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
272	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
273	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	2	1	no	—
276	C₄⋊C₄	—	4	1	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₄⋊C₄	—	4	1	no	—
280	C₄⋊C₄	—	4	2	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	2	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
286	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
287	C₂²⋊C₄	—	4	2	no	—
288	C₂²⋊C₄	—	4	1	no	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₄²	—	4	2	no	—
291	C₄²	—	4	2	no	—
292	C₄²	—	4	1	no	—
293	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
294	C₄²	C₂×D₄×C₁₃	1	2	yes	—
295	C₁₆	—	2	2	no	—
296	C₁₆	—	2	1	no	—
297	C₁₆	C₁₃×#41	1	2	yes	—
298	C₁₃	#33965	1	1	yes	characteristic, direct product factor, Sylow
299	C₂³	—	4	1	no	—
300	Q₈	—	2	2	no	—
301	Q₈	—	4	1	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	4	1	no	—
306	Q₈	—	4	2	no	—
307	D₄	—	4	1	no	—
308	D₄	—	4	1	no	—
309	C₂×C₄	—	4	1	no	—
310	C₂×C₄	—	4	1	no	—
311	C₂×C₄	—	4	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	8	2	no	—
315	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
316	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
317	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
318	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
319	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
320	C₈	—	2	2	no	—
321	C₈	—	2	1	no	—
322	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
323	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
324	C₂²	—	8	1	no	—
325	C₂²	—	8	1	no	—
326	C₂²	—	8	1	no	—
327	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
328	C₄	—	2	2	no	—
329	C₄	—	2	2	no	—
330	C₄	—	2	1	no	—
331	C₄	—	8	2	no	—
332	C₄	—	8	1	no	—
333	C₄	—	8	2	no	—
334	C₄	—	8	1	no	—
335	C₄	—	8	1	no	—
336	C₄	C₁₃×#510	1	2	yes	—
337	C₄	C₁₃×#550	1	1	yes	characteristic
338	C₄	C₁₃×#576	1	1	yes	characteristic
339	C₂	—	16	1	no	—
340	C₂	C₁₃×#3391	1	1	yes	characteristic
341	C₂	C₂×C₁₃×#510	1	1	yes	characteristic
342	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×C₁₃×D₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×Q₈×C₁₃ ◁ C₈×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3395	C₂	semidirect
4. C₁₃×#2247	C₂	semidirect
6. C₁₃×#2209	C₂	semidirect
7. C₁₃×#2209	C₂	semidirect
9. C₁₃×#2183	C₂	semidirect
10. C₁₃×#2176	C₂	semidirect
82. #33965	C₁₃	direct (N × H)
128. #3395	C₂×C₁₃	semidirect
131. #2247	C₂×C₁₃	semidirect
133. #2209	C₂×C₁₃	semidirect
134. #2209	C₂×C₁₃	semidirect
136. #2183	C₂×C₁₃	semidirect
137. #2176	C₂×C₁₃	semidirect
298. C₁₃	#33965	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×#3395	2
C₁₃×#3393	2
C₁₃×#2251	2
C₁₃×#2247	2
C₁₃×#2211	2
C₁₃×#2209	2
C₁₃×#2208	2
C₁₃×#2183	2
C₁₃×#2176	2
C₄×C₁₃×SD₃₂	2
#33965	13

Subgroups — C13×#33965 #15720304

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33965 #15720304