Subgroups — C₁₃×#33609 #15719948

Order 3328.

682 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#504	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#3059	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#3058	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#2220	C₂	1	4	yes	maximal, complemented
6	C₁₃×#2217	C₂	1	4	yes	maximal, complemented
7	C₁₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2141	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#2140	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#131	C₂²	1	1	yes	characteristic
13	C₁₃×#536	—	2	1	no	—
14	C₁₃×#535	—	2	1	no	—
15	C₂×C₁₃×SD₃₂	—	2	4	no	—
16	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
17	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
18	C₁₃×#504	—	2	4	no	—
19	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
20	C₂×C₁₃×M₃₂	C₂²	1	2	yes	complemented
21	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
22	C₂²×C₁₃×C₁₆	C₂²	1	2	yes	complemented
23	C₁₃×#470	—	2	2	no	—
24	C₁₃×#469	—	2	2	no	—
25	C₁₃×#467	C₂²	1	4	yes	complemented
26	C₁₃×#466	C₂²	1	4	yes	—
27	C₁₃×#427	C₂²	1	1	yes	characteristic
28	C₁₃×#427	C₂²	1	1	yes	characteristic
29	C₁₃×#419	—	2	1	no	—
30	C₁₃×#418	—	2	1	no	—
31	C₁₃×#367	C₂²	1	2	yes	complemented
32	C₁₃×#366	C₂²	1	2	yes	complemented
33	C₁₃×#358	C₂²	1	4	yes	complemented
34	C₁₃×#357	C₂²	1	4	yes	—
35	C₂×C₁₃×#41	D₄	1	1	yes	characteristic
36	C₂²×Q₈×C₁₃	—	2	1	no	—
37	C₂²×D₄×C₁₃	—	2	1	no	—
38	C₂×C₁₃×Q₁₆	—	2	2	no	—
39	C₂×C₁₃×Q₁₆	—	2	1	no	—
40	C₂×C₁₃×Q₁₆	—	2	2	no	—
41	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
42	C₂×C₁₃×D₈	—	2	2	no	—
43	C₂×C₁₃×D₈	—	2	1	no	—
44	C₂×C₁₃×D₈	—	2	2	no	—
45	C₂×C₁₃×D₈	C₂³	1	2	yes	—
46	C₁₃×#131	—	2	2	no	—
47	C₁₃×#131	—	2	2	no	—
48	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
49	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
50	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
51	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₁₃×C₄⋊Q₈	—	4	1	no	—
53	C₁₃×#127	—	4	1	no	—
54	C₁₃×#124	—	4	1	no	—
55	C₁₃×#124	—	4	1	no	—
56	C₁₃×#123	—	4	2	no	—
57	C₁₃×C₂²⋊Q₈	—	2	4	no	—
58	C₁₃×#121	—	2	4	no	—
59	C₁₃×#120	—	4	2	no	—
60	C₁₃×#117	—	2	1	no	—
61	C₁₃×#117	—	2	1	no	—
62	C₁₃×SD₃₂	—	2	4	no	—
63	C₁₃×SD₃₂	—	2	4	no	—
64	C₁₃×M₃₂	—	2	2	no	—
65	C₁₃×M₃₂	—	2	4	no	—
66	C₂×C₁₃×C₁₆	—	2	4	no	—
67	C₂×C₁₃×C₁₆	D₄	1	4	yes	complemented
68	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
69	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
70	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
71	C₁₃×#107	C₂³	1	2	yes	—
72	C₁₃×#107	C₂³	1	2	yes	—
73	C₁₃×#106	—	2	1	no	—
74	C₁₃×#106	—	2	1	no	—
75	C₁₃×#103	—	2	4	no	—
76	C₁₃×#102	—	2	4	no	—
77	#33609	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
78	C₂⁴×C₁₃	—	4	1	no	—
79	C₁₃×#41	—	2	2	no	—
80	C₁₃×#41	—	2	2	no	—
81	C₂×Q₈×C₁₃	—	2	2	no	—
82	C₂×Q₈×C₁₃	—	4	1	no	—
83	C₂×Q₈×C₁₃	—	4	2	no	—
84	C₂×Q₈×C₁₃	—	4	2	no	—
85	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
86	C₂×D₄×C₁₃	—	2	2	no	—
87	C₂×D₄×C₁₃	—	4	4	no	—
88	C₂×D₄×C₁₃	—	4	2	no	—
89	C₂×D₄×C₁₃	—	4	1	no	—
90	C₂×D₄×C₁₃	—	4	2	no	—
91	C₂×D₄×C₁₃	D₈	1	2	yes	—
92	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
93	C₂²×C₄×C₁₃	—	4	1	no	—
94	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
95	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₁₃×Q₁₆	—	2	2	no	—
97	C₁₃×Q₁₆	—	2	2	no	—
98	C₁₃×Q₁₆	—	4	2	no	—
99	C₁₃×D₈	—	2	2	no	—
100	C₁₃×D₈	—	2	2	no	—
101	C₁₃×D₈	—	4	2	no	—
102	C₁₃×M₁₆	—	2	2	no	—
103	C₁₃×M₁₆	—	2	4	no	—
104	C₂×C₈×C₁₃	—	2	2	no	—
105	C₂×C₈×C₁₃	—	2	2	no	—
106	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
107	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
109	C₂×C₈×C₁₃	#41	1	2	yes	—
110	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
111	C₁₃×C₄⋊C₄	—	2	2	no	—
112	C₁₃×C₄⋊C₄	—	2	2	no	—
113	C₁₃×C₄⋊C₄	—	4	4	no	—
114	C₁₃×C₂²⋊C₄	—	4	1	no	—
115	C₁₃×C₂²⋊C₄	—	4	1	no	—
116	C₁₃×C₂²⋊C₄	—	4	4	no	—
117	C₁₃×C₂²⋊C₄	—	4	4	no	—
118	C₄²×C₁₃	—	4	1	no	—
119	C₄²×C₁₃	—	4	1	no	—
120	C₁₃×C₁₆	—	2	2	no	—
121	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
122	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
123	C₂²×SD₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
124	C₂×#504	C₂×C₁₃	1	1	yes	characteristic, complemented
125	#3059	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#3058	C₂×C₁₃	1	1	yes	characteristic
127	#2220	C₂×C₁₃	1	4	yes	complemented
128	#2217	C₂×C₁₃	1	4	yes	complemented
129	#2158	C₂×C₁₃	1	1	yes	characteristic, complemented
130	#2141	C₂×C₁₃	1	1	yes	characteristic, complemented
131	#2140	C₂×C₁₃	1	1	yes	characteristic, complemented
132	C₂³×C₁₃	—	4	2	no	—
133	C₂³×C₁₃	—	8	2	no	—
134	C₂³×C₁₃	—	16	2	no	—
135	C₂³×C₁₃	C₂×D₈	1	2	yes	—
136	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
137	Q₈×C₁₃	—	2	2	no	—
138	Q₈×C₁₃	—	4	2	no	—
139	Q₈×C₁₃	—	4	2	no	—
140	Q₈×C₁₃	—	8	2	no	—
141	D₄×C₁₃	—	2	4	no	—
142	D₄×C₁₃	—	2	2	no	—
143	D₄×C₁₃	—	4	2	no	—
144	D₄×C₁₃	—	4	2	no	—
145	D₄×C₁₃	—	8	2	no	—
146	D₄×C₁₃	—	16	4	no	—
147	C₂×C₄×C₁₃	—	2	2	no	—
148	C₂×C₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	—	4	2	no	—
150	C₂×C₄×C₁₃	—	4	2	no	—
151	C₂×C₄×C₁₃	—	4	2	no	—
152	C₂×C₄×C₁₃	—	8	2	no	—
153	C₂×C₄×C₁₃	#121	1	2	yes	—
154	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
155	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
156	C₂×C₄×C₁₃	#135	1	2	yes	—
157	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
158	C₈×C₁₃	—	2	2	no	—
159	C₈×C₁₃	#121	1	2	yes	—
160	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
161	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
162	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
163	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
164	C₂×#131	C₂²×C₁₃	1	1	yes	characteristic
165	#536	—	2	1	no	—
166	#535	—	2	1	no	—
167	C₂×SD₃₂	—	2	4	no	—
168	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
169	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
170	#504	—	2	4	no	—
171	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
172	C₂×M₃₂	C₂²×C₁₃	1	2	yes	complemented
173	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
174	C₂²×C₁₆	C₂²×C₁₃	1	2	yes	complemented
175	#470	—	2	2	no	—
176	#469	—	2	2	no	—
177	#467	C₂²×C₁₃	1	4	yes	complemented
178	#466	C₂²×C₁₃	1	4	yes	—
179	#427	C₂²×C₁₃	1	1	yes	characteristic
180	#427	C₂²×C₁₃	1	1	yes	characteristic
181	#419	—	2	1	no	—
182	#418	—	2	1	no	—
183	#367	C₂²×C₁₃	1	2	yes	complemented
184	#366	C₂²×C₁₃	1	2	yes	complemented
185	#358	C₂²×C₁₃	1	4	yes	complemented
186	#357	C₂²×C₁₃	1	4	yes	—
187	C₂²×C₁₃	—	2	2	no	—
188	C₂²×C₁₃	—	2	4	no	—
189	C₂²×C₁₃	—	2	2	no	—
190	C₂²×C₁₃	—	8	4	no	—
191	C₂²×C₁₃	—	8	2	no	—
192	C₂²×C₁₃	—	16	4	no	—
193	C₂²×C₁₃	#466	1	2	yes	—
194	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
195	C₄×C₁₃	—	2	2	no	—
196	C₄×C₁₃	—	8	2	no	—
197	C₄×C₁₃	—	8	2	no	—
198	C₄×C₁₃	—	8	2	no	—
199	C₄×C₁₃	#466	1	2	yes	—
200	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
201	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
202	C₂×#41	D₄×C₁₃	1	1	yes	characteristic
203	C₂²×Q₈	—	2	1	no	—
204	C₂²×D₄	—	2	1	no	—
205	C₂×Q₁₆	—	2	2	no	—
206	C₂×Q₁₆	—	2	1	no	—
207	C₂×Q₁₆	—	2	2	no	—
208	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
209	C₂×D₈	—	2	2	no	—
210	C₂×D₈	—	2	1	no	—
211	C₂×D₈	—	2	2	no	—
212	C₂×D₈	C₂³×C₁₃	1	2	yes	—
213	#131	—	2	2	no	—
214	#131	—	2	2	no	—
215	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
216	C₂×M₁₆	D₄×C₁₃	1	2	yes	—
217	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
218	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
219	C₄⋊Q₈	—	4	1	no	—
220	#127	—	4	1	no	—
221	#124	—	4	1	no	—
222	#124	—	4	1	no	—
223	#123	—	4	2	no	—
224	C₂²⋊Q₈	—	2	4	no	—
225	#121	—	2	4	no	—
226	#120	—	4	2	no	—
227	#117	—	2	1	no	—
228	#117	—	2	1	no	—
229	SD₃₂	—	2	4	no	—
230	SD₃₂	—	2	4	no	—
231	M₃₂	—	2	2	no	—
232	M₃₂	—	2	4	no	—
233	C₂×C₁₆	—	2	4	no	—
234	C₂×C₁₆	D₄×C₁₃	1	4	yes	complemented
235	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
236	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
237	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
238	#107	C₂³×C₁₃	1	2	yes	—
239	#107	C₂³×C₁₃	1	2	yes	—
240	#106	—	2	1	no	—
241	#106	—	2	1	no	—
242	#103	—	2	4	no	—
243	#102	—	2	4	no	—
244	C₂×C₁₃	—	2	2	no	—
245	C₂×C₁₃	—	4	2	no	—
246	C₂×C₁₃	—	16	2	no	—
247	C₂×C₁₃	C₂×#466	1	1	yes	characteristic
248	C₂×C₁₃	#3416	1	2	yes	—
249	C₂⁴	—	4	1	no	—
250	#41	—	2	2	no	—
251	#41	—	2	2	no	—
252	C₂×Q₈	—	2	2	no	—
253	C₂×Q₈	—	4	1	no	—
254	C₂×Q₈	—	4	2	no	—
255	C₂×Q₈	—	4	2	no	—
256	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	4	4	no	—
259	C₂×D₄	—	4	2	no	—
260	C₂×D₄	—	4	1	no	—
261	C₂×D₄	—	4	2	no	—
262	C₂×D₄	C₁₃×D₈	1	2	yes	—
263	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
264	C₂²×C₄	—	4	1	no	—
265	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
266	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
267	Q₁₆	—	2	2	no	—
268	Q₁₆	—	2	2	no	—
269	Q₁₆	—	4	2	no	—
270	D₈	—	2	2	no	—
271	D₈	—	2	2	no	—
272	D₈	—	4	2	no	—
273	M₁₆	—	2	2	no	—
274	M₁₆	—	2	4	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
278	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
279	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
280	C₂×C₈	C₁₃×#41	1	2	yes	—
281	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	2	2	no	—
284	C₄⋊C₄	—	4	4	no	—
285	C₂²⋊C₄	—	4	1	no	—
286	C₂²⋊C₄	—	4	1	no	—
287	C₂²⋊C₄	—	4	4	no	—
288	C₂²⋊C₄	—	4	4	no	—
289	C₄²	—	4	1	no	—
290	C₄²	—	4	1	no	—
291	C₁₆	—	2	2	no	—
292	C₁₆	C₂×D₄×C₁₃	1	2	yes	—
293	C₁₆	C₂×D₄×C₁₃	1	2	yes	—
294	C₁₃	#33609	1	1	yes	characteristic, direct product factor, Sylow
295	C₂³	—	4	2	no	—
296	C₂³	—	8	2	no	—
297	C₂³	—	16	2	no	—
298	C₂³	C₂×C₁₃×D₈	1	2	yes	—
299	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
300	Q₈	—	2	2	no	—
301	Q₈	—	4	2	no	—
302	Q₈	—	4	2	no	—
303	Q₈	—	8	2	no	—
304	D₄	—	2	4	no	—
305	D₄	—	2	2	no	—
306	D₄	—	4	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	8	2	no	—
309	D₄	—	16	4	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	4	2	no	—
313	C₂×C₄	—	4	2	no	—
314	C₂×C₄	—	4	2	no	—
315	C₂×C₄	—	8	2	no	—
316	C₂×C₄	C₁₃×#121	1	2	yes	—
317	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
318	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
319	C₂×C₄	C₁₃×#135	1	2	yes	—
320	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
321	C₈	—	2	2	no	—
322	C₈	C₁₃×#121	1	2	yes	—
323	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
324	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
325	C₂²	—	2	2	no	—
326	C₂²	—	2	2	no	—
327	C₂²	—	2	4	no	—
328	C₂²	—	8	2	no	—
329	C₂²	—	8	4	no	—
330	C₂²	—	16	4	no	—
331	C₂²	C₁₃×#466	1	2	yes	—
332	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
333	C₄	—	2	2	no	—
334	C₄	—	8	2	no	—
335	C₄	—	8	2	no	—
336	C₄	—	8	2	no	—
337	C₄	C₁₃×#466	1	2	yes	—
338	C₄	C₂×C₁₃×#121	1	1	yes	characteristic
339	C₄	C₂×C₁₃×#135	1	1	yes	characteristic
340	C₂	—	2	2	no	—
341	C₂	—	4	2	no	—
342	C₂	—	16	2	no	—
343	C₂	C₂×C₁₃×#466	1	1	yes	characteristic
344	C₂	C₁₃×#3416	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₁₃×#3058	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₁₃×#41 ◁ C₂×C₁₃×#131 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₃×SD₃₂	C₂	semidirect
2. C₂×C₁₃×#504	C₂	semidirect
3. C₁₃×#3059	C₂	semidirect
5. C₁₃×#2220	C₂	semidirect
6. C₁₃×#2217	C₂	semidirect
7. C₁₃×#2158	C₂	semidirect
8. C₁₃×#2141	C₂	semidirect
9. C₁₃×#2140	C₂	semidirect
20. C₂×C₁₃×M₃₂	C₂²	semidirect
22. C₂²×C₁₃×C₁₆	C₂²	semidirect
25. C₁₃×#467	C₂²	semidirect
31. C₁₃×#367	C₂²	semidirect
32. C₁₃×#366	C₂²	semidirect
33. C₁₃×#358	C₂²	semidirect
67. C₂×C₁₃×C₁₆	D₄	semidirect
77. #33609	C₁₃	direct (N × H)
123. C₂²×SD₃₂	C₂×C₁₃	semidirect
124. C₂×#504	C₂×C₁₃	semidirect
125. #3059	C₂×C₁₃	semidirect
127. #2220	C₂×C₁₃	semidirect
128. #2217	C₂×C₁₃	semidirect
129. #2158	C₂×C₁₃	semidirect
130. #2141	C₂×C₁₃	semidirect
131. #2140	C₂×C₁₃	semidirect
172. C₂×M₃₂	C₂²×C₁₃	semidirect
174. C₂²×C₁₆	C₂²×C₁₃	semidirect
177. #467	C₂²×C₁₃	semidirect
183. #367	C₂²×C₁₃	semidirect
184. #366	C₂²×C₁₃	semidirect
185. #358	C₂²×C₁₃	semidirect
234. C₂×C₁₆	D₄×C₁₃	semidirect
294. C₁₃	#33609	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×C₁₃×SD₃₂	2
C₂×C₁₃×#504	2
C₁₃×#3059	2
C₁₃×#3058	2
C₁₃×#2220	2
C₁₃×#2217	2
C₁₃×#2158	2
C₁₃×#2141	2
C₁₃×#2140	2
#33609	13

Subgroups — C13×#33609 #15719948

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33609 #15719948