Subgroups — C₁₃×#33871 #15720210

Order 3328.

556 conjugacy classes of subgroups.

The subgroup lattice has 360 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, complemented
2	C₁₃×#3287	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2240	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2234	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2225	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#2217	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2207	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2189	C₂	1	2	yes	maximal, complemented
9	C₄×C₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	D₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2138	C₂	1	2	yes	maximal, complemented
12	C₂×C₁₃×#135	C₂²	1	2	yes	—
13	C₁₃×#556	—	2	1	no	—
14	C₁₃×#550	—	2	1	no	—
15	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
17	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	—
18	C₁₃×#500	C₂²	1	1	yes	characteristic
19	C₁₃×#493	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×#466	C₂²	1	2	yes	—
24	C₁₃×#460	—	2	1	no	—
25	C₁₃×#449	—	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	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×C₂²⋊C₁₆	C₂²	1	2	yes	—
40	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
41	C₂×C₁₃×#41	—	2	2	no	—
42	C₁₃×#135	—	2	4	no	—
43	C₁₃×#135	—	4	2	no	—
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	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
50	C₁₃×C₄⋊Q₈	—	2	1	no	—
51	C₁₃×#127	—	2	1	no	—
52	C₁₃×#127	—	4	1	no	—
53	C₁₃×#126	—	4	2	no	—
54	C₁₃×#124	—	4	1	no	—
55	C₁₃×C₂²⋊Q₈	—	2	2	no	—
56	C₁₃×#121	—	2	2	no	—
57	C₁₃×#121	—	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×SD₃₂	—	4	2	no	—
66	C₂×C₁₃×C₁₆	—	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×#102	—	2	2	no	—
80	C₁₃×C₂²⋊C₈	D₄	1	2	yes	—
81	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
82	#33871	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₁₃×#41	—	4	4	no	—
84	C₁₃×#41	—	8	2	no	—
85	C₂×Q₈×C₁₃	—	2	1	no	—
86	C₂×Q₈×C₁₃	—	2	1	no	—
87	C₂×Q₈×C₁₃	—	4	1	no	—
88	C₂×D₄×C₁₃	—	2	2	no	—
89	C₂×D₄×C₁₃	—	4	2	no	—
90	C₂×D₄×C₁₃	—	4	2	no	—
91	C₂×D₄×C₁₃	—	4	1	no	—
92	C₂×D₄×C₁₃	—	4	1	no	—
93	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
94	C₂²×C₄×C₁₃	—	4	2	no	—
95	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
96	C₁₃×Q₁₆	—	2	1	no	—
97	C₁₃×Q₁₆	—	2	1	no	—
98	C₁₃×Q₁₆	—	2	1	no	—
99	C₁₃×Q₁₆	#41	1	2	yes	—
100	C₁₃×SD₁₆	—	4	4	no	—
101	C₁₃×D₈	—	2	2	no	—
102	C₁₃×D₈	—	2	2	no	—
103	C₁₃×D₈	—	8	1	no	—
104	C₂×C₈×C₁₃	—	2	2	no	—
105	C₂×C₈×C₁₃	—	2	2	no	—
106	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
108	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
109	C₁₃×C₄⋊C₄	—	2	1	no	—
110	C₁₃×C₄⋊C₄	—	2	1	no	—
111	C₁₃×C₄⋊C₄	—	2	2	no	—
112	C₁₃×C₄⋊C₄	—	4	1	no	—
113	C₁₃×C₄⋊C₄	—	4	1	no	—
114	C₁₃×C₄⋊C₄	—	4	2	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, complemented
129	#3287	C₂×C₁₃	1	1	yes	characteristic
130	#2240	C₂×C₁₃	1	1	yes	characteristic, complemented
131	#2234	C₂×C₁₃	1	2	yes	complemented
132	#2225	C₂×C₁₃	1	1	yes	characteristic, complemented
133	#2217	C₂×C₁₃	1	2	yes	complemented
134	#2207	C₂×C₁₃	1	1	yes	characteristic, complemented
135	#2189	C₂×C₁₃	1	2	yes	complemented
136	C₄×Q₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
137	D₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
138	#2138	C₂×C₁₃	1	2	yes	complemented
139	C₂³×C₁₃	—	4	2	no	—
140	C₂³×C₁₃	C₂×D₈	1	2	yes	—
141	Q₈×C₁₃	—	2	2	no	—
142	Q₈×C₁₃	—	4	1	no	—
143	Q₈×C₁₃	—	4	1	no	—
144	Q₈×C₁₃	—	4	1	no	—
145	Q₈×C₁₃	—	8	2	no	—
146	D₄×C₁₃	—	2	2	no	—
147	D₄×C₁₃	—	4	2	no	—
148	D₄×C₁₃	—	4	2	no	—
149	D₄×C₁₃	—	8	4	no	—
150	D₄×C₁₃	—	8	2	no	—
151	D₄×C₁₃	—	16	1	no	—
152	D₄×C₁₃	—	16	2	no	—
153	C₂×C₄×C₁₃	—	2	2	no	—
154	C₂×C₄×C₁₃	—	2	2	no	—
155	C₂×C₄×C₁₃	—	4	1	no	—
156	C₂×C₄×C₁₃	—	4	1	no	—
157	C₂×C₄×C₁₃	—	4	2	no	—
158	C₂×C₄×C₁₃	—	4	1	no	—
159	C₂×C₄×C₁₃	—	4	1	no	—
160	C₂×C₄×C₁₃	—	8	4	no	—
161	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
162	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
163	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
164	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
165	C₈×C₁₃	—	2	2	no	—
166	C₈×C₁₃	—	2	1	no	—
167	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
168	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
169	C₂×#135	C₂²×C₁₃	1	2	yes	—
170	#556	—	2	1	no	—
171	#550	—	2	1	no	—
172	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
173	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
174	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	—
175	#500	C₂²×C₁₃	1	1	yes	characteristic
176	#493	C₂²×C₁₃	1	1	yes	characteristic
177	#485	—	2	2	no	—
178	#475	—	2	1	no	—
179	#467	C₂²×C₁₃	1	2	yes	—
180	#466	C₂²×C₁₃	1	2	yes	—
181	#460	—	2	1	no	—
182	#449	—	2	2	no	—
183	C₄×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
184	C₄×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
185	C₈×D₄	C₂²×C₁₃	1	1	yes	characteristic
186	#427	C₂²×C₁₃	1	2	yes	—
187	#367	C₂²×C₁₃	1	2	yes	—
188	#366	C₂²×C₁₃	1	1	yes	characteristic
189	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
190	#358	C₂²×C₁₃	1	1	yes	characteristic
191	#358	C₂²×C₁₃	1	2	yes	—
192	#358	C₂²×C₁₃	1	1	yes	characteristic
193	#357	C₂²×C₁₃	1	2	yes	—
194	#357	C₂²×C₁₃	1	2	yes	—
195	#357	C₂²×C₁₃	1	2	yes	—
196	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
197	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
198	C₂²×C₁₃	—	2	2	no	—
199	C₂²×C₁₃	—	2	2	no	—
200	C₂²×C₁₃	—	2	2	no	—
201	C₂²×C₁₃	—	8	2	no	—
202	C₂²×C₁₃	—	8	2	no	—
203	C₂²×C₁₃	—	8	2	no	—
204	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
205	C₄×C₁₃	—	2	1	no	—
206	C₄×C₁₃	—	2	2	no	—
207	C₄×C₁₃	—	4	2	no	—
208	C₄×C₁₃	—	8	2	no	—
209	C₄×C₁₃	—	8	1	no	—
210	C₄×C₁₃	—	8	1	no	—
211	C₄×C₁₃	—	8	1	no	—
212	C₄×C₁₃	#508	1	2	yes	—
213	C₄×C₁₃	#546	1	1	yes	characteristic
214	C₄×C₁₃	#577	1	1	yes	characteristic
215	C₂×#41	—	2	2	no	—
216	#135	—	2	4	no	—
217	#135	—	4	2	no	—
218	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
219	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
220	C₂×SD₁₆	—	2	2	no	—
221	C₂×D₈	—	2	1	no	—
222	C₂×D₈	C₂³×C₁₃	1	2	yes	—
223	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
224	C₄⋊Q₈	—	2	1	no	—
225	#127	—	2	1	no	—
226	#127	—	4	1	no	—
227	#126	—	4	2	no	—
228	#124	—	4	1	no	—
229	C₂²⋊Q₈	—	2	2	no	—
230	#121	—	2	2	no	—
231	#121	—	4	2	no	—
232	C₄×Q₈	—	2	1	no	—
233	C₄×Q₈	—	2	1	no	—
234	C₄×Q₈	—	4	1	no	—
235	C₄×D₄	—	4	1	no	—
236	C₄×D₄	D₄×C₁₃	1	1	yes	characteristic
237	#117	—	2	2	no	—
238	Q₃₂	—	2	2	no	—
239	SD₃₂	—	4	2	no	—
240	C₂×C₁₆	—	4	2	no	—
241	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
242	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
243	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
244	#107	—	2	1	no	—
245	#107	C₂³×C₁₃	1	1	yes	characteristic
246	#107	C₂³×C₁₃	1	2	yes	—
247	#107	C₂³×C₁₃	1	1	yes	characteristic
248	#106	—	2	2	no	—
249	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
250	#103	—	2	1	no	—
251	#103	—	2	1	no	—
252	#103	—	2	2	no	—
253	#102	—	2	2	no	—
254	C₂²⋊C₈	D₄×C₁₃	1	2	yes	—
255	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
256	C₂×C₁₃	—	4	2	no	—
257	C₂×C₁₃	—	16	2	no	—
258	C₂×C₁₃	#3294	1	1	yes	characteristic
259	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
260	C₂×C₁₃	#3416	1	1	yes	characteristic
261	#41	—	4	4	no	—
262	#41	—	8	2	no	—
263	C₂×Q₈	—	2	1	no	—
264	C₂×Q₈	—	2	1	no	—
265	C₂×Q₈	—	4	1	no	—
266	C₂×D₄	—	2	2	no	—
267	C₂×D₄	—	4	2	no	—
268	C₂×D₄	—	4	2	no	—
269	C₂×D₄	—	4	1	no	—
270	C₂×D₄	—	4	1	no	—
271	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
272	C₂²×C₄	—	4	2	no	—
273	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
274	Q₁₆	—	2	1	no	—
275	Q₁₆	—	2	1	no	—
276	Q₁₆	—	2	1	no	—
277	Q₁₆	C₁₃×#41	1	2	yes	—
278	SD₁₆	—	4	4	no	—
279	D₈	—	2	2	no	—
280	D₈	—	2	2	no	—
281	D₈	—	8	1	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
285	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
286	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
287	C₄⋊C₄	—	2	1	no	—
288	C₄⋊C₄	—	2	1	no	—
289	C₄⋊C₄	—	2	2	no	—
290	C₄⋊C₄	—	4	1	no	—
291	C₄⋊C₄	—	4	1	no	—
292	C₄⋊C₄	—	4	2	no	—
293	C₄⋊C₄	—	4	1	no	—
294	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₂²⋊C₄	—	4	2	no	—
298	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
299	C₄²	—	4	2	no	—
300	C₄²	—	4	1	no	—
301	C₄²	—	4	1	no	—
302	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
303	C₁₆	—	2	1	no	—
304	C₁₆	—	2	1	no	—
305	C₁₆	—	2	2	no	—
306	C₁₃	#33871	1	1	yes	characteristic, direct product factor, Sylow
307	C₂³	—	4	2	no	—
308	C₂³	C₂×C₁₃×D₈	1	2	yes	—
309	Q₈	—	2	2	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	4	1	no	—
312	Q₈	—	4	1	no	—
313	Q₈	—	8	2	no	—
314	D₄	—	2	2	no	—
315	D₄	—	4	2	no	—
316	D₄	—	4	2	no	—
317	D₄	—	8	4	no	—
318	D₄	—	8	2	no	—
319	D₄	—	16	1	no	—
320	D₄	—	16	2	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	2	2	no	—
323	C₂×C₄	—	4	1	no	—
324	C₂×C₄	—	4	1	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	1	no	—
327	C₂×C₄	—	4	1	no	—
328	C₂×C₄	—	8	4	no	—
329	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
330	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
331	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
332	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
333	C₈	—	2	2	no	—
334	C₈	—	2	1	no	—
335	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
336	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
337	C₂²	—	2	2	no	—
338	C₂²	—	2	2	no	—
339	C₂²	—	2	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
344	C₄	—	2	2	no	—
345	C₄	—	2	1	no	—
346	C₄	—	4	2	no	—
347	C₄	—	8	1	no	—
348	C₄	—	8	2	no	—
349	C₄	—	8	1	no	—
350	C₄	—	8	1	no	—
351	C₄	C₁₃×#508	1	2	yes	—
352	C₄	C₁₃×#546	1	1	yes	characteristic
353	C₄	C₁₃×#577	1	1	yes	characteristic
354	C₂	—	4	2	no	—
355	C₂	—	16	2	no	—
356	C₂	C₁₃×#3294	1	1	yes	characteristic
357	C₂	C₂×C₁₃×#508	1	1	yes	characteristic
358	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×#3287	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×#3342	C₂	semidirect
3. C₁₃×#2240	C₂	semidirect
4. C₁₃×#2234	C₂	semidirect
5. C₁₃×#2225	C₂	semidirect
6. C₁₃×#2217	C₂	semidirect
7. C₁₃×#2207	C₂	semidirect
8. C₁₃×#2189	C₂	semidirect
9. C₄×C₁₃×Q₃₂	C₂	semidirect
10. D₄×C₁₃×C₁₆	C₂	semidirect
11. C₁₃×#2138	C₂	semidirect
82. #33871	C₁₃	direct (N × H)
128. #3342	C₂×C₁₃	semidirect
130. #2240	C₂×C₁₃	semidirect
131. #2234	C₂×C₁₃	semidirect
132. #2225	C₂×C₁₃	semidirect
133. #2217	C₂×C₁₃	semidirect
134. #2207	C₂×C₁₃	semidirect
135. #2189	C₂×C₁₃	semidirect
136. C₄×Q₃₂	C₂×C₁₃	semidirect
137. D₄×C₁₆	C₂×C₁₃	semidirect
138. #2138	C₂×C₁₃	semidirect
306. C₁₃	#33871	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
C₁₃×#3287	2
C₁₃×#2240	2
C₁₃×#2234	2
C₁₃×#2225	2
C₁₃×#2217	2
C₁₃×#2207	2
C₁₃×#2189	2
C₄×C₁₃×Q₃₂	2
D₄×C₁₃×C₁₆	2
C₁₃×#2138	2
#33871	13

Subgroups — C13×#33871 #15720210

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33871 #15720210