Subgroups — C₁₃×#33967 #15720306

Order 3328.

510 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3396	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#3395	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2251	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2248	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#2211	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2209	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2208	C₂	1	2	yes	maximal, complemented
8	C₁₃×#2207	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#2175	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#559	—	2	1	no	—
13	C₁₃×#557	—	2	1	no	—
14	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
16	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
17	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
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	2	yes	—
22	C₁₃×#493	C₂²	1	1	yes	characteristic
23	C₁₃×#463	—	2	2	no	—
24	C₁₃×#462	—	2	1	no	—
25	C₁₃×#462	—	2	2	no	—
26	C₁₃×#460	—	2	1	no	—
27	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
29	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
30	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
31	C₄×C₁₃×D₈	C₂²	1	2	yes	—
32	C₁₃×#366	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	1	yes	characteristic
37	C₁₃×#358	C₂²	1	2	yes	—
38	C₁₃×#357	C₂²	1	2	yes	—
39	C₁₃×#346	C₂²	1	2	yes	—
40	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
41	C₂×C₁₃×Q₁₆	—	2	2	no	—
42	C₂×C₁₃×Q₁₆	—	2	1	no	—
43	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
44	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
45	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
46	C₂×C₁₃×SD₁₆	—	2	2	no	—
47	C₂×C₁₃×D₈	—	2	1	no	—
48	C₂×C₁₃×D₈	C₂³	1	2	yes	—
49	C₁₃×C₄⋊Q₈	—	2	2	no	—
50	C₁₃×C₄⋊Q₈	—	2	1	no	—
51	C₁₃×#127	—	2	1	no	—
52	C₁₃×#125	—	4	1	no	—
53	C₁₃×#125	—	4	2	no	—
54	C₁₃×#124	—	2	2	no	—
55	C₁₃×#124	—	4	1	no	—
56	C₁₃×#123	—	4	2	no	—
57	C₄×Q₈×C₁₃	—	2	1	no	—
58	C₄×Q₈×C₁₃	—	2	2	no	—
59	C₄×Q₈×C₁₃	—	2	1	no	—
60	C₄×Q₈×C₁₃	—	4	1	no	—
61	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
62	C₄×D₄×C₁₃	—	2	2	no	—
63	C₄×D₄×C₁₃	—	4	1	no	—
64	C₁₃×Q₃₂	—	2	2	no	—
65	C₁₃×Q₃₂	—	4	2	no	—
66	C₁₃×SD₃₂	—	2	4	no	—
67	C₁₃×SD₃₂	—	4	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	1	no	—
79	C₁₃×#102	—	2	2	no	—
80	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₁₃	C₂³	1	2	yes	—
82	#33967	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	—	2	1	no	—
86	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
87	C₂×D₄×C₁₃	—	2	2	no	—
88	C₂×D₄×C₁₃	—	4	1	no	—
89	C₂×D₄×C₁₃	—	4	1	no	—
90	C₂²×C₄×C₁₃	—	4	2	no	—
91	C₁₃×Q₁₆	—	2	2	no	—
92	C₁₃×Q₁₆	—	2	1	no	—
93	C₁₃×Q₁₆	—	2	1	no	—
94	C₁₃×Q₁₆	—	2	1	no	—
95	C₁₃×Q₁₆	—	2	2	no	—
96	C₁₃×Q₁₆	—	2	1	no	—
97	C₁₃×Q₁₆	—	8	1	no	—
98	C₁₃×Q₁₆	—	8	2	no	—
99	C₁₃×SD₁₆	—	8	2	no	—
100	C₁₃×D₈	—	2	2	no	—
101	C₁₃×D₈	—	2	2	no	—
102	C₁₃×D₈	—	8	1	no	—
103	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
104	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
106	C₁₃×C₄⋊C₄	—	2	1	no	—
107	C₁₃×C₄⋊C₄	—	2	1	no	—
108	C₁₃×C₄⋊C₄	—	4	1	no	—
109	C₁₃×C₄⋊C₄	—	4	2	no	—
110	C₁₃×C₄⋊C₄	—	4	1	no	—
111	C₁₃×C₄⋊C₄	—	4	2	no	—
112	C₁₃×C₄⋊C₄	—	4	1	no	—
113	C₁₃×C₄⋊C₄	—	4	2	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	2	no	—
119	C₁₃×C₂²⋊C₄	—	4	2	no	—
120	C₄²×C₁₃	—	4	1	no	—
121	C₄²×C₁₃	—	4	2	no	—
122	C₄²×C₁₃	—	4	1	no	—
123	C₄²×C₁₃	C₂×D₄	1	2	yes	—
124	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
125	C₁₃×C₁₆	—	2	1	no	—
126	C₁₃×C₁₆	—	2	2	no	—
127	C₁₃×C₁₆	#41	1	2	yes	—
128	#3396	C₂×C₁₃	1	1	yes	characteristic
129	#3395	C₂×C₁₃	1	1	yes	characteristic, complemented
130	#2251	C₂×C₁₃	1	2	yes	complemented
131	#2248	C₂×C₁₃	1	1	yes	characteristic
132	#2211	C₂×C₁₃	1	2	yes	complemented
133	#2209	C₂×C₁₃	1	1	yes	characteristic, complemented
134	#2208	C₂×C₁₃	1	2	yes	complemented
135	#2207	C₂×C₁₃	1	1	yes	characteristic
136	#2183	C₂×C₁₃	1	1	yes	characteristic, complemented
137	#2175	C₂×C₁₃	1	2	yes	complemented
138	C₄×Q₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
139	C₂³×C₁₃	—	4	2	no	—
140	Q₈×C₁₃	—	2	2	no	—
141	Q₈×C₁₃	—	4	2	no	—
142	Q₈×C₁₃	—	4	1	no	—
143	Q₈×C₁₃	—	4	1	no	—
144	Q₈×C₁₃	—	4	1	no	—
145	Q₈×C₁₃	—	4	2	no	—
146	Q₈×C₁₃	—	4	1	no	—
147	D₄×C₁₃	—	4	2	no	—
148	D₄×C₁₃	—	4	2	no	—
149	D₄×C₁₃	—	8	2	no	—
150	D₄×C₁₃	—	16	1	no	—
151	C₂×C₄×C₁₃	—	4	1	no	—
152	C₂×C₄×C₁₃	—	4	1	no	—
153	C₂×C₄×C₁₃	—	4	1	no	—
154	C₂×C₄×C₁₃	—	4	2	no	—
155	C₂×C₄×C₁₃	—	4	1	no	—
156	C₂×C₄×C₁₃	—	16	2	no	—
157	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
158	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
159	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
160	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
162	C₈×C₁₃	—	2	1	no	—
163	C₈×C₁₃	—	2	2	no	—
164	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
165	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
166	#559	—	2	1	no	—
167	#557	—	2	1	no	—
168	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
169	C₂×Q₃₂	C₂²×C₁₃	1	2	yes	—
170	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
171	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
172	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
173	#495	C₂²×C₁₃	1	2	yes	—
174	#494	C₂²×C₁₃	1	2	yes	—
175	#494	C₂²×C₁₃	1	1	yes	characteristic
176	#493	C₂²×C₁₃	1	1	yes	characteristic
177	#463	—	2	2	no	—
178	#462	—	2	1	no	—
179	#462	—	2	2	no	—
180	#460	—	2	1	no	—
181	C₈×Q₈	C₂²×C₁₃	1	1	yes	characteristic
182	C₄×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
183	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
184	C₄×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
185	C₄×D₈	C₂²×C₁₃	1	2	yes	—
186	#366	C₂²×C₁₃	1	1	yes	characteristic
187	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
188	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
189	#358	C₂²×C₁₃	1	1	yes	characteristic
190	#358	C₂²×C₁₃	1	1	yes	characteristic
191	#358	C₂²×C₁₃	1	2	yes	—
192	#357	C₂²×C₁₃	1	2	yes	—
193	#346	C₂²×C₁₃	1	2	yes	—
194	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
195	C₂²×C₁₃	—	8	2	no	—
196	C₂²×C₁₃	—	8	2	no	—
197	C₂²×C₁₃	—	8	2	no	—
198	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
199	C₄×C₁₃	—	2	2	no	—
200	C₄×C₁₃	—	2	2	no	—
201	C₄×C₁₃	—	2	1	no	—
202	C₄×C₁₃	—	8	1	no	—
203	C₄×C₁₃	—	8	2	no	—
204	C₄×C₁₃	—	8	1	no	—
205	C₄×C₁₃	—	8	1	no	—
206	C₄×C₁₃	—	8	1	no	—
207	C₄×C₁₃	#510	1	2	yes	—
208	C₄×C₁₃	#550	1	1	yes	characteristic
209	C₄×C₁₃	#576	1	1	yes	characteristic
210	C₂×Q₁₆	—	2	2	no	—
211	C₂×Q₁₆	—	2	1	no	—
212	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
213	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
214	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
215	C₂×SD₁₆	—	2	2	no	—
216	C₂×D₈	—	2	1	no	—
217	C₂×D₈	C₂³×C₁₃	1	2	yes	—
218	C₄⋊Q₈	—	2	2	no	—
219	C₄⋊Q₈	—	2	1	no	—
220	#127	—	2	1	no	—
221	#125	—	4	1	no	—
222	#125	—	4	2	no	—
223	#124	—	2	2	no	—
224	#124	—	4	1	no	—
225	#123	—	4	2	no	—
226	C₄×Q₈	—	2	2	no	—
227	C₄×Q₈	—	2	1	no	—
228	C₄×Q₈	—	2	1	no	—
229	C₄×Q₈	—	4	1	no	—
230	C₄×Q₈	D₄×C₁₃	1	1	yes	characteristic
231	C₄×D₄	—	2	2	no	—
232	C₄×D₄	—	4	1	no	—
233	Q₃₂	—	2	2	no	—
234	Q₃₂	—	4	2	no	—
235	SD₃₂	—	2	4	no	—
236	SD₃₂	—	4	2	no	—
237	SD₃₂	—	4	2	no	—
238	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
239	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
240	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
241	#107	C₂³×C₁₃	1	1	yes	characteristic
242	#107	C₂³×C₁₃	1	1	yes	characteristic
243	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
244	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
245	#103	—	2	2	no	—
246	#103	—	2	1	no	—
247	#103	—	2	1	no	—
248	#102	—	2	2	no	—
249	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
250	C₄×C₈	C₂³×C₁₃	1	2	yes	—
251	C₂×C₁₃	—	16	2	no	—
252	C₂×C₁₃	#3391	1	1	yes	characteristic
253	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
254	C₂×C₁₃	#3416	1	1	yes	characteristic
255	C₂×Q₈	—	2	1	no	—
256	C₂×Q₈	—	2	2	no	—
257	C₂×Q₈	—	2	1	no	—
258	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂×D₄	—	4	1	no	—
262	C₂²×C₄	—	4	2	no	—
263	Q₁₆	—	2	2	no	—
264	Q₁₆	—	2	1	no	—
265	Q₁₆	—	2	1	no	—
266	Q₁₆	—	2	1	no	—
267	Q₁₆	—	2	2	no	—
268	Q₁₆	—	2	1	no	—
269	Q₁₆	—	8	1	no	—
270	Q₁₆	—	8	2	no	—
271	SD₁₆	—	8	2	no	—
272	D₈	—	2	2	no	—
273	D₈	—	2	2	no	—
274	D₈	—	8	1	no	—
275	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
276	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
277	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	4	1	no	—
281	C₄⋊C₄	—	4	2	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	2	no	—
284	C₄⋊C₄	—	4	1	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
288	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
289	C₂²⋊C₄	—	4	2	no	—
290	C₂²⋊C₄	—	4	2	no	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₄²	—	4	1	no	—
293	C₄²	—	4	2	no	—
294	C₄²	—	4	1	no	—
295	C₄²	C₂×D₄×C₁₃	1	2	yes	—
296	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
297	C₁₆	—	2	1	no	—
298	C₁₆	—	2	2	no	—
299	C₁₆	C₁₃×#41	1	2	yes	—
300	C₁₃	#33967	1	1	yes	characteristic, direct product factor, Sylow
301	C₂³	—	4	2	no	—
302	Q₈	—	2	2	no	—
303	Q₈	—	4	2	no	—
304	Q₈	—	4	1	no	—
305	Q₈	—	4	1	no	—
306	Q₈	—	4	1	no	—
307	Q₈	—	4	2	no	—
308	Q₈	—	4	1	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	8	2	no	—
312	D₄	—	16	1	no	—
313	C₂×C₄	—	4	1	no	—
314	C₂×C₄	—	4	1	no	—
315	C₂×C₄	—	4	1	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	1	no	—
318	C₂×C₄	—	16	2	no	—
319	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
320	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
321	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
322	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
323	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
324	C₈	—	2	1	no	—
325	C₈	—	2	2	no	—
326	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
327	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
328	C₂²	—	8	2	no	—
329	C₂²	—	8	2	no	—
330	C₂²	—	8	2	no	—
331	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	2	no	—
334	C₄	—	2	1	no	—
335	C₄	—	8	1	no	—
336	C₄	—	8	2	no	—
337	C₄	—	8	1	no	—
338	C₄	—	8	1	no	—
339	C₄	—	8	1	no	—
340	C₄	C₁₃×#510	1	2	yes	—
341	C₄	C₁₃×#550	1	1	yes	characteristic
342	C₄	C₁₃×#576	1	1	yes	characteristic
343	C₂	—	16	2	no	—
344	C₂	C₁₃×#3391	1	1	yes	characteristic
345	C₂	C₂×C₁₃×#510	1	1	yes	characteristic
346	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×#493	C₂²
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
2. C₁₃×#3395	C₂	semidirect
3. C₁₃×#2251	C₂	semidirect
5. C₁₃×#2211	C₂	semidirect
6. C₁₃×#2209	C₂	semidirect
7. C₁₃×#2208	C₂	semidirect
9. C₁₃×#2183	C₂	semidirect
10. C₁₃×#2175	C₂	semidirect
11. C₄×C₁₃×Q₃₂	C₂	semidirect
82. #33967	C₁₃	direct (N × H)
129. #3395	C₂×C₁₃	semidirect
130. #2251	C₂×C₁₃	semidirect
132. #2211	C₂×C₁₃	semidirect
133. #2209	C₂×C₁₃	semidirect
134. #2208	C₂×C₁₃	semidirect
136. #2183	C₂×C₁₃	semidirect
137. #2175	C₂×C₁₃	semidirect
138. C₄×Q₃₂	C₂×C₁₃	semidirect
300. C₁₃	#33967	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×#3396	2
C₁₃×#3395	2
C₁₃×#2251	2
C₁₃×#2248	2
C₁₃×#2211	2
C₁₃×#2209	2
C₁₃×#2208	2
C₁₃×#2207	2
C₁₃×#2183	2
C₁₃×#2175	2
C₄×C₁₃×Q₃₂	2
#33967	13

Subgroups — C13×#33967 #15720306

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33967 #15720306