Subgroups — C₁₃×#33942 #15720281

Order 3328.

486 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×#3396	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2243	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2240	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#2227	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#2224	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2211	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	Q₈²×C₁₃	—	2	1	no	—
14	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
16	C₁₃×#500	C₂²	1	1	yes	characteristic
17	C₁₃×#500	C₂²	1	2	yes	—
18	C₁₃×#495	C₂²	1	2	yes	—
19	C₁₃×#493	C₂²	1	1	yes	characteristic
20	C₁₃×#477	—	2	1	no	—
21	C₁₃×#477	—	2	2	no	—
22	C₁₃×#463	—	2	2	no	—
23	C₁₃×#460	—	2	1	no	—
24	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
25	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
26	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
27	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
28	C₄×C₁₃×D₈	C₂²	1	2	yes	—
29	C₁₃×#366	C₂²	1	1	yes	characteristic
30	C₁₃×#366	C₂²	1	2	yes	—
31	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
32	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
33	C₁₃×#358	C₂²	1	1	yes	characteristic
34	C₁₃×#358	C₂²	1	2	yes	—
35	C₁₃×#358	C₂²	1	1	yes	characteristic
36	C₁₃×#357	C₂²	1	2	yes	—
37	C₁₃×#357	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	C₂³	1	1	yes	characteristic
42	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
43	C₂×C₁₃×SD₁₆	—	2	2	no	—
44	C₂×C₁₃×D₈	—	2	1	no	—
45	C₂×C₁₃×D₈	C₂³	1	2	yes	—
46	C₁₃×C₄⋊Q₈	—	2	2	no	—
47	C₁₃×C₄⋊Q₈	—	2	1	no	—
48	C₁₃×C₄⋊Q₈	—	4	1	no	—
49	C₁₃×C₄⋊Q₈	—	4	2	no	—
50	C₁₃×#127	—	2	1	no	—
51	C₁₃×#124	—	2	2	no	—
52	C₁₃×#124	—	4	1	no	—
53	C₁₃×#123	—	4	2	no	—
54	C₄×Q₈×C₁₃	—	2	1	no	—
55	C₄×Q₈×C₁₃	—	2	2	no	—
56	C₄×Q₈×C₁₃	—	2	1	no	—
57	C₄×Q₈×C₁₃	—	4	1	no	—
58	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
59	C₄×D₄×C₁₃	—	2	2	no	—
60	C₄×D₄×C₁₃	—	4	1	no	—
61	C₁₃×Q₃₂	—	2	2	no	—
62	C₁₃×SD₃₂	—	2	4	no	—
63	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
65	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
66	C₁₃×#107	—	2	2	no	—
67	C₁₃×#107	—	2	1	no	—
68	C₁₃×#107	C₂³	1	2	yes	—
69	C₁₃×#107	C₂³	1	1	yes	characteristic
70	C₁₃×#107	C₂³	1	1	yes	characteristic
71	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
72	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
73	C₁₃×#103	—	2	1	no	—
74	C₁₃×#103	—	2	2	no	—
75	C₁₃×#103	—	2	1	no	—
76	C₁₃×#102	—	2	2	no	—
77	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₁₃	C₂³	1	2	yes	—
79	#33942	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂×Q₈×C₁₃	—	2	1	no	—
81	C₂×Q₈×C₁₃	—	2	1	no	—
82	C₂×Q₈×C₁₃	—	4	2	no	—
83	C₂×Q₈×C₁₃	—	4	1	no	—
84	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
85	C₂×D₄×C₁₃	—	2	2	no	—
86	C₂×D₄×C₁₃	—	4	1	no	—
87	C₂×D₄×C₁₃	—	4	1	no	—
88	C₂²×C₄×C₁₃	—	4	2	no	—
89	C₁₃×Q₁₆	—	2	1	no	—
90	C₁₃×Q₁₆	—	2	1	no	—
91	C₁₃×Q₁₆	—	2	1	no	—
92	C₁₃×Q₁₆	#41	1	2	yes	—
93	C₁₃×SD₁₆	—	8	2	no	—
94	C₁₃×D₈	—	2	2	no	—
95	C₁₃×D₈	—	2	2	no	—
96	C₁₃×D₈	—	8	1	no	—
97	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
98	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
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	—	4	1	no	—
104	C₁₃×C₄⋊C₄	—	4	2	no	—
105	C₁₃×C₄⋊C₄	—	4	1	no	—
106	C₁₃×C₄⋊C₄	—	4	2	no	—
107	C₁₃×C₄⋊C₄	—	4	1	no	—
108	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
109	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
110	C₁₃×C₂²⋊C₄	—	4	2	no	—
111	C₁₃×C₂²⋊C₄	—	4	2	no	—
112	C₁₃×C₂²⋊C₄	—	4	2	no	—
113	C₄²×C₁₃	—	4	1	no	—
114	C₄²×C₁₃	—	4	2	no	—
115	C₄²×C₁₃	—	4	1	no	—
116	C₄²×C₁₃	C₂×D₄	1	2	yes	—
117	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
118	C₁₃×C₁₆	—	2	2	no	—
119	C₁₃×C₁₆	—	2	1	no	—
120	C₁₃×C₁₆	—	2	1	no	—
121	#3396	C₂×C₁₃	1	1	yes	characteristic
122	Q₈×Q₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
123	#2243	C₂×C₁₃	1	2	yes	complemented
124	#2240	C₂×C₁₃	1	1	yes	characteristic
125	#2227	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#2224	C₂×C₁₃	1	2	yes	complemented
127	#2211	C₂×C₁₃	1	2	yes	complemented
128	#2207	C₂×C₁₃	1	1	yes	characteristic
129	#2183	C₂×C₁₃	1	1	yes	characteristic, complemented
130	#2175	C₂×C₁₃	1	2	yes	complemented
131	C₄×Q₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
132	C₂³×C₁₃	—	4	2	no	—
133	Q₈×C₁₃	—	2	2	no	—
134	Q₈×C₁₃	—	2	2	no	—
135	Q₈×C₁₃	—	4	1	no	—
136	Q₈×C₁₃	—	4	1	no	—
137	Q₈×C₁₃	—	4	1	no	—
138	Q₈×C₁₃	—	16	2	no	—
139	Q₈×C₁₃	—	16	1	no	—
140	D₄×C₁₃	—	4	2	no	—
141	D₄×C₁₃	—	4	2	no	—
142	D₄×C₁₃	—	8	2	no	—
143	D₄×C₁₃	—	16	1	no	—
144	C₂×C₄×C₁₃	—	4	1	no	—
145	C₂×C₄×C₁₃	—	4	1	no	—
146	C₂×C₄×C₁₃	—	4	1	no	—
147	C₂×C₄×C₁₃	—	4	2	no	—
148	C₂×C₄×C₁₃	—	4	1	no	—
149	C₂×C₄×C₁₃	—	16	2	no	—
150	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
151	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
152	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
153	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
154	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
155	C₈×C₁₃	—	2	1	no	—
156	C₈×C₁₃	—	2	2	no	—
157	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
158	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
159	#559	—	2	1	no	—
160	Q₈²	—	2	1	no	—
161	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
162	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
163	#500	C₂²×C₁₃	1	2	yes	—
164	#500	C₂²×C₁₃	1	1	yes	characteristic
165	#495	C₂²×C₁₃	1	2	yes	—
166	#493	C₂²×C₁₃	1	1	yes	characteristic
167	#477	—	2	1	no	—
168	#477	—	2	2	no	—
169	#463	—	2	2	no	—
170	#460	—	2	1	no	—
171	C₈×Q₈	C₂²×C₁₃	1	1	yes	characteristic
172	C₄×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
173	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
174	C₄×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
175	C₄×D₈	C₂²×C₁₃	1	2	yes	—
176	#366	C₂²×C₁₃	1	1	yes	characteristic
177	#366	C₂²×C₁₃	1	2	yes	—
178	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
179	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
180	#358	C₂²×C₁₃	1	1	yes	characteristic
181	#358	C₂²×C₁₃	1	1	yes	characteristic
182	#358	C₂²×C₁₃	1	2	yes	—
183	#357	C₂²×C₁₃	1	2	yes	—
184	#357	C₂²×C₁₃	1	2	yes	—
185	#357	C₂²×C₁₃	1	2	yes	—
186	#346	C₂²×C₁₃	1	2	yes	—
187	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
188	C₂²×C₁₃	—	8	2	no	—
189	C₂²×C₁₃	—	8	2	no	—
190	C₂²×C₁₃	—	8	2	no	—
191	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
192	C₄×C₁₃	—	2	2	no	—
193	C₄×C₁₃	—	2	2	no	—
194	C₄×C₁₃	—	2	1	no	—
195	C₄×C₁₃	—	4	2	no	—
196	C₄×C₁₃	—	8	1	no	—
197	C₄×C₁₃	—	8	1	no	—
198	C₄×C₁₃	—	8	1	no	—
199	C₄×C₁₃	—	8	2	no	—
200	C₄×C₁₃	#510	1	2	yes	—
201	C₄×C₁₃	#548	1	1	yes	characteristic
202	C₄×C₁₃	#577	1	1	yes	characteristic
203	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
204	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
205	C₂×SD₁₆	—	2	2	no	—
206	C₂×D₈	—	2	1	no	—
207	C₂×D₈	C₂³×C₁₃	1	2	yes	—
208	C₄⋊Q₈	—	2	2	no	—
209	C₄⋊Q₈	—	2	1	no	—
210	C₄⋊Q₈	—	4	1	no	—
211	C₄⋊Q₈	—	4	2	no	—
212	#127	—	2	1	no	—
213	#124	—	2	2	no	—
214	#124	—	4	1	no	—
215	#123	—	4	2	no	—
216	C₄×Q₈	—	2	2	no	—
217	C₄×Q₈	—	2	1	no	—
218	C₄×Q₈	—	2	1	no	—
219	C₄×Q₈	—	4	1	no	—
220	C₄×Q₈	D₄×C₁₃	1	1	yes	characteristic
221	C₄×D₄	—	2	2	no	—
222	C₄×D₄	—	4	1	no	—
223	Q₃₂	—	2	2	no	—
224	SD₃₂	—	2	4	no	—
225	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
226	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
227	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
228	#107	—	2	2	no	—
229	#107	—	2	1	no	—
230	#107	C₂³×C₁₃	1	2	yes	—
231	#107	C₂³×C₁₃	1	1	yes	characteristic
232	#107	C₂³×C₁₃	1	1	yes	characteristic
233	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
234	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
235	#103	—	2	1	no	—
236	#103	—	2	2	no	—
237	#103	—	2	1	no	—
238	#102	—	2	2	no	—
239	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
240	C₄×C₈	C₂³×C₁₃	1	2	yes	—
241	C₂×C₁₃	—	16	2	no	—
242	C₂×C₁₃	#3358	1	1	yes	characteristic
243	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
244	C₂×C₁₃	#3415	1	1	yes	characteristic
245	C₂×Q₈	—	2	1	no	—
246	C₂×Q₈	—	2	1	no	—
247	C₂×Q₈	—	4	2	no	—
248	C₂×Q₈	—	4	1	no	—
249	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	4	1	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂²×C₄	—	4	2	no	—
254	Q₁₆	—	2	1	no	—
255	Q₁₆	—	2	1	no	—
256	Q₁₆	—	2	1	no	—
257	Q₁₆	C₁₃×#41	1	2	yes	—
258	SD₁₆	—	8	2	no	—
259	D₈	—	2	2	no	—
260	D₈	—	2	2	no	—
261	D₈	—	8	1	no	—
262	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
263	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
264	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
265	C₄⋊C₄	—	2	1	no	—
266	C₄⋊C₄	—	2	2	no	—
267	C₄⋊C₄	—	2	1	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₄⋊C₄	—	4	2	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	—	4	2	no	—
272	C₄⋊C₄	—	4	1	no	—
273	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
274	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₂²⋊C₄	—	4	2	no	—
277	C₂²⋊C₄	—	4	2	no	—
278	C₄²	—	4	1	no	—
279	C₄²	—	4	2	no	—
280	C₄²	—	4	1	no	—
281	C₄²	C₂×D₄×C₁₃	1	2	yes	—
282	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
283	C₁₆	—	2	2	no	—
284	C₁₆	—	2	1	no	—
285	C₁₆	—	2	1	no	—
286	C₁₃	#33942	1	1	yes	characteristic, direct product factor, Sylow
287	C₂³	—	4	2	no	—
288	Q₈	—	2	2	no	—
289	Q₈	—	2	2	no	—
290	Q₈	—	4	1	no	—
291	Q₈	—	4	1	no	—
292	Q₈	—	4	1	no	—
293	Q₈	—	16	2	no	—
294	Q₈	—	16	1	no	—
295	D₄	—	4	2	no	—
296	D₄	—	4	2	no	—
297	D₄	—	8	2	no	—
298	D₄	—	16	1	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	4	1	no	—
301	C₂×C₄	—	4	1	no	—
302	C₂×C₄	—	4	2	no	—
303	C₂×C₄	—	4	1	no	—
304	C₂×C₄	—	16	2	no	—
305	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
306	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
307	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
308	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
309	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
310	C₈	—	2	1	no	—
311	C₈	—	2	2	no	—
312	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
313	C₈	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
314	C₂²	—	8	2	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	8	2	no	—
317	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
318	C₄	—	2	2	no	—
319	C₄	—	2	2	no	—
320	C₄	—	2	1	no	—
321	C₄	—	4	2	no	—
322	C₄	—	8	1	no	—
323	C₄	—	8	1	no	—
324	C₄	—	8	1	no	—
325	C₄	—	8	2	no	—
326	C₄	C₁₃×#510	1	2	yes	—
327	C₄	C₁₃×#548	1	1	yes	characteristic
328	C₄	C₁₃×#577	1	1	yes	characteristic
329	C₂	—	16	2	no	—
330	C₂	C₁₃×#3358	1	1	yes	characteristic
331	C₂	C₂×C₁₃×#510	1	1	yes	characteristic
332	C₂	C₁₃×#3415	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. Q₈×C₁₃×Q₁₆	C₂	semidirect
3. C₁₃×#2243	C₂	semidirect
5. C₁₃×#2227	C₂	semidirect
6. C₁₃×#2224	C₂	semidirect
7. C₁₃×#2211	C₂	semidirect
9. C₁₃×#2183	C₂	semidirect
10. C₁₃×#2175	C₂	semidirect
11. C₄×C₁₃×Q₃₂	C₂	semidirect
79. #33942	C₁₃	direct (N × H)
122. Q₈×Q₁₆	C₂×C₁₃	semidirect
123. #2243	C₂×C₁₃	semidirect
125. #2227	C₂×C₁₃	semidirect
126. #2224	C₂×C₁₃	semidirect
127. #2211	C₂×C₁₃	semidirect
129. #2183	C₂×C₁₃	semidirect
130. #2175	C₂×C₁₃	semidirect
131. C₄×Q₃₂	C₂×C₁₃	semidirect
286. C₁₃	#33942	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
Q₈×C₁₃×Q₁₆	2
C₁₃×#2243	2
C₁₃×#2240	2
C₁₃×#2227	2
C₁₃×#2224	2
C₁₃×#2211	2
C₁₃×#2207	2
C₁₃×#2183	2
C₁₃×#2175	2
C₄×C₁₃×Q₃₂	2
#33942	13

Subgroups — C13×#33942 #15720281

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33942 #15720281