Subgroups — C₁₃×#33895 #15720234

Order 3328.

534 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×#3342	C₂	1	1	yes	maximal, characteristic
2	D₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2241	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2237	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2225	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#2219	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2209	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2188	C₂	1	2	yes	maximal, complemented
9	C₄×C₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#2169	C₂	1	1	yes	maximal, characteristic
11	C₁₃×#2141	C₂	1	2	yes	maximal, complemented
12	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
13	C₁₃×#556	—	2	1	no	—
14	D₄×Q₈×C₁₃	—	2	1	no	—
15	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×Q₃₂	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×#485	—	2	2	no	—
21	C₁₃×#475	—	2	1	no	—
22	C₁₃×#467	C₂²	1	2	yes	—
23	C₁₃×#467	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×Q₁₆	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	2	yes	—
31	C₁₃×#366	C₂²	1	1	yes	characteristic
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×#358	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×Q₁₆	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×#126	—	4	2	no	—
54	C₁₃×#124	—	4	1	no	—
55	C₁₃×C₂²⋊Q₈	—	2	2	no	—
56	C₁₃×C₂²⋊Q₈	—	2	2	no	—
57	C₁₃×C₂²⋊Q₈	—	4	2	no	—
58	C₄×Q₈×C₁₃	—	2	1	no	—
59	C₄×Q₈×C₁₃	—	2	1	no	—
60	C₄×Q₈×C₁₃	—	4	1	no	—
61	C₄×D₄×C₁₃	—	4	1	no	—
62	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
63	C₁₃×#117	—	2	2	no	—
64	C₁₃×Q₃₂	—	2	2	no	—
65	C₁₃×Q₃₂	—	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	1	no	—
78	C₁₃×#103	—	2	2	no	—
79	C₁₃×#103	—	2	2	no	—
80	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
81	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
82	#33895	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂×Q₈×C₁₃	—	2	1	no	—
84	C₂×Q₈×C₁₃	—	2	1	no	—
85	C₂×Q₈×C₁₃	—	2	2	no	—
86	C₂×Q₈×C₁₃	—	4	4	no	—
87	C₂×Q₈×C₁₃	—	4	1	no	—
88	C₂×Q₈×C₁₃	—	4	2	no	—
89	C₂×Q₈×C₁₃	—	8	2	no	—
90	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
91	C₂²×C₄×C₁₃	—	4	2	no	—
92	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
93	C₁₃×Q₁₆	—	2	1	no	—
94	C₁₃×Q₁₆	—	2	2	no	—
95	C₁₃×Q₁₆	—	2	1	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	C₂×C₈×C₁₃	—	2	2	no	—
102	C₂×C₈×C₁₃	—	2	2	no	—
103	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
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	—	2	2	no	—
109	C₁₃×C₄⋊C₄	—	4	1	no	—
110	C₁₃×C₄⋊C₄	—	4	1	no	—
111	C₁₃×C₄⋊C₄	—	4	1	no	—
112	C₁₃×C₄⋊C₄	—	4	2	no	—
113	C₁₃×C₄⋊C₄	—	4	2	no	—
114	C₁₃×C₄⋊C₄	—	4	1	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	D₈	1	2	yes	—
121	C₄²×C₁₃	—	4	2	no	—
122	C₄²×C₁₃	—	4	1	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	#3342	C₂×C₁₃	1	1	yes	characteristic
129	D₄×Q₁₆	C₂×C₁₃	1	1	yes	characteristic
130	#2241	C₂×C₁₃	1	1	yes	characteristic, complemented
131	#2237	C₂×C₁₃	1	2	yes	complemented
132	#2225	C₂×C₁₃	1	1	yes	characteristic, complemented
133	#2219	C₂×C₁₃	1	2	yes	complemented
134	#2209	C₂×C₁₃	1	1	yes	characteristic, complemented
135	#2188	C₂×C₁₃	1	2	yes	complemented
136	C₄×Q₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
137	#2169	C₂×C₁₃	1	1	yes	characteristic
138	#2141	C₂×C₁₃	1	2	yes	complemented
139	C₂³×C₁₃	C₂×D₈	1	2	yes	—
140	Q₈×C₁₃	—	2	2	no	—
141	Q₈×C₁₃	—	4	1	no	—
142	Q₈×C₁₃	—	4	2	no	—
143	Q₈×C₁₃	—	4	1	no	—
144	Q₈×C₁₃	—	4	2	no	—
145	Q₈×C₁₃	—	4	1	no	—
146	Q₈×C₁₃	—	8	4	no	—
147	Q₈×C₁₃	—	8	2	no	—
148	D₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	—	2	2	no	—
150	C₂×C₄×C₁₃	—	2	2	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	—	4	2	no	—
157	C₂×C₄×C₁₃	—	8	4	no	—
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	2	no	—
163	C₈×C₁₃	—	2	1	no	—
164	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
165	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
166	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
167	#556	—	2	1	no	—
168	D₄×Q₈	—	2	1	no	—
169	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
170	C₂×Q₃₂	C₂²×C₁₃	1	2	yes	—
171	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
172	#500	C₂²×C₁₃	1	1	yes	characteristic
173	#494	C₂²×C₁₃	1	1	yes	characteristic
174	#485	—	2	2	no	—
175	#475	—	2	1	no	—
176	#467	C₂²×C₁₃	1	2	yes	—
177	#467	C₂²×C₁₃	1	2	yes	—
178	#462	—	2	1	no	—
179	#451	—	2	2	no	—
180	C₄×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
181	C₄×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
182	C₈×D₄	C₂²×C₁₃	1	1	yes	characteristic
183	#427	C₂²×C₁₃	1	2	yes	—
184	#367	C₂²×C₁₃	1	2	yes	—
185	#366	C₂²×C₁₃	1	1	yes	characteristic
186	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
187	#358	C₂²×C₁₃	1	1	yes	characteristic
188	#358	C₂²×C₁₃	1	2	yes	—
189	#358	C₂²×C₁₃	1	2	yes	—
190	#358	C₂²×C₁₃	1	2	yes	—
191	#358	C₂²×C₁₃	1	1	yes	characteristic
192	#358	C₂²×C₁₃	1	2	yes	—
193	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
194	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
195	C₂²×C₁₃	—	2	2	no	—
196	C₂²×C₁₃	—	2	2	no	—
197	C₂²×C₁₃	—	2	2	no	—
198	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
199	C₄×C₁₃	—	2	1	no	—
200	C₄×C₁₃	—	2	2	no	—
201	C₄×C₁₃	—	4	2	no	—
202	C₄×C₁₃	—	8	2	no	—
203	C₄×C₁₃	—	8	1	no	—
204	C₄×C₁₃	—	8	1	no	—
205	C₄×C₁₃	—	8	2	no	—
206	C₄×C₁₃	—	8	1	no	—
207	C₄×C₁₃	#510	1	2	yes	—
208	C₄×C₁₃	#546	1	1	yes	characteristic
209	C₄×C₁₃	#577	1	1	yes	characteristic
210	C₂²×Q₈	—	2	2	no	—
211	C₂×Q₁₆	—	2	4	no	—
212	C₂×Q₁₆	—	2	2	no	—
213	C₂×Q₁₆	—	2	1	no	—
214	C₂×Q₁₆	—	4	2	no	—
215	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
216	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
217	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
218	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
219	C₄⋊Q₈	—	2	1	no	—
220	C₄⋊Q₈	—	2	1	no	—
221	C₄⋊Q₈	—	4	1	no	—
222	#126	—	4	2	no	—
223	#124	—	4	1	no	—
224	C₂²⋊Q₈	—	2	2	no	—
225	C₂²⋊Q₈	—	2	2	no	—
226	C₂²⋊Q₈	—	4	2	no	—
227	C₄×Q₈	—	2	1	no	—
228	C₄×Q₈	—	2	1	no	—
229	C₄×Q₈	—	4	1	no	—
230	C₄×D₄	—	4	1	no	—
231	C₄×D₄	D₄×C₁₃	1	1	yes	characteristic
232	#117	—	2	2	no	—
233	Q₃₂	—	2	2	no	—
234	Q₃₂	—	4	2	no	—
235	M₃₂	—	4	2	no	—
236	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
237	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
238	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
239	#107	—	2	1	no	—
240	#107	C₂³×C₁₃	1	1	yes	characteristic
241	#107	C₂³×C₁₃	1	2	yes	—
242	#107	C₂³×C₁₃	1	1	yes	characteristic
243	#106	—	2	2	no	—
244	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
245	#103	—	2	1	no	—
246	#103	—	2	1	no	—
247	#103	—	2	2	no	—
248	#103	—	2	2	no	—
249	C₂²⋊C₈	D₄×C₁₃	1	2	yes	—
250	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
251	C₂×C₁₃	—	4	2	no	—
252	C₂×C₁₃	#3294	1	1	yes	characteristic
253	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
254	C₂×C₁₃	#3417	1	1	yes	characteristic
255	C₂×Q₈	—	2	1	no	—
256	C₂×Q₈	—	2	1	no	—
257	C₂×Q₈	—	2	2	no	—
258	C₂×Q₈	—	4	4	no	—
259	C₂×Q₈	—	4	1	no	—
260	C₂×Q₈	—	4	2	no	—
261	C₂×Q₈	—	8	2	no	—
262	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
263	C₂²×C₄	—	4	2	no	—
264	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
265	Q₁₆	—	2	1	no	—
266	Q₁₆	—	2	2	no	—
267	Q₁₆	—	2	1	no	—
268	Q₁₆	—	2	1	no	—
269	Q₁₆	—	2	2	no	—
270	Q₁₆	—	4	4	no	—
271	Q₁₆	—	8	1	no	—
272	Q₁₆	C₁₃×#41	1	2	yes	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
276	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
277	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
278	C₄⋊C₄	—	2	1	no	—
279	C₄⋊C₄	—	2	1	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	4	1	no	—
282	C₄⋊C₄	—	4	1	no	—
283	C₄⋊C₄	—	4	1	no	—
284	C₄⋊C₄	—	4	2	no	—
285	C₄⋊C₄	—	4	2	no	—
286	C₄⋊C₄	—	4	1	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
289	C₂²⋊C₄	—	4	2	no	—
290	C₂²⋊C₄	—	4	2	no	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
293	C₄²	—	4	2	no	—
294	C₄²	—	4	1	no	—
295	C₄²	—	4	1	no	—
296	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
297	C₁₆	—	2	1	no	—
298	C₁₆	—	2	1	no	—
299	C₁₆	—	2	2	no	—
300	C₁₃	#33895	1	1	yes	characteristic, direct product factor, Sylow
301	C₂³	C₂×C₁₃×D₈	1	2	yes	—
302	Q₈	—	2	2	no	—
303	Q₈	—	4	1	no	—
304	Q₈	—	4	2	no	—
305	Q₈	—	4	1	no	—
306	Q₈	—	4	2	no	—
307	Q₈	—	4	1	no	—
308	Q₈	—	8	4	no	—
309	Q₈	—	8	2	no	—
310	D₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	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	—	4	2	no	—
319	C₂×C₄	—	8	4	no	—
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	2	no	—
325	C₈	—	2	1	no	—
326	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
327	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
328	C₂²	—	2	2	no	—
329	C₂²	—	2	2	no	—
330	C₂²	—	2	2	no	—
331	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
332	C₄	—	2	2	no	—
333	C₄	—	2	1	no	—
334	C₄	—	4	2	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	1	no	—
337	C₄	—	8	2	no	—
338	C₄	—	8	1	no	—
339	C₄	—	8	1	no	—
340	C₄	C₁₃×#510	1	2	yes	—
341	C₄	C₁₃×#546	1	1	yes	characteristic
342	C₄	C₁₃×#577	1	1	yes	characteristic
343	C₂	—	4	2	no	—
344	C₂	C₁₃×#3294	1	1	yes	characteristic
345	C₂	C₂×C₁₃×#510	1	1	yes	characteristic
346	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×D₄×C₁₃	D₈
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
3. C₁₃×#2241	C₂	semidirect
4. C₁₃×#2237	C₂	semidirect
5. C₁₃×#2225	C₂	semidirect
6. C₁₃×#2219	C₂	semidirect
7. C₁₃×#2209	C₂	semidirect
8. C₁₃×#2188	C₂	semidirect
9. C₄×C₁₃×Q₃₂	C₂	semidirect
11. C₁₃×#2141	C₂	semidirect
82. #33895	C₁₃	direct (N × H)
130. #2241	C₂×C₁₃	semidirect
131. #2237	C₂×C₁₃	semidirect
132. #2225	C₂×C₁₃	semidirect
133. #2219	C₂×C₁₃	semidirect
134. #2209	C₂×C₁₃	semidirect
135. #2188	C₂×C₁₃	semidirect
136. C₄×Q₃₂	C₂×C₁₃	semidirect
138. #2141	C₂×C₁₃	semidirect
300. C₁₃	#33895	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×#3342	2
D₄×C₁₃×Q₁₆	2
C₁₃×#2241	2
C₁₃×#2237	2
C₁₃×#2225	2
C₁₃×#2219	2
C₁₃×#2209	2
C₁₃×#2188	2
C₄×C₁₃×Q₃₂	2
C₁₃×#2169	2
C₁₃×#2141	2
#33895	13

Subgroups — C13×#33895 #15720234

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33895 #15720234