Subgroups — C₁₃×#33961 #15720300

Order 3328.

520 conjugacy classes of subgroups.

The subgroup lattice has 356 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, complemented
2	C₁₃×#3393	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2249	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2248	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2211	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2210	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2209	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2206	C₂	1	1	yes	maximal, characteristic
9	Q₈×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₁₃×SD₃₂	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₁₃×SD₃₂	C₂	1	2	yes	maximal, complemented
12	C₁₃×#559	—	2	1	no	—
13	C₁₃×#556	—	2	1	no	—
14	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
15	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
16	C₂×C₁₃×SD₃₂	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
18	C₂×C₁₃×D₁₆	C₂²	1	2	yes	—
19	C₁₃×#495	C₂²	1	2	yes	—
20	C₁₃×#495	C₂²	1	2	yes	—
21	C₁₃×#494	C₂²	1	1	yes	characteristic
22	C₁₃×#493	C₂²	1	1	yes	characteristic
23	C₁₃×#464	—	2	2	no	—
24	C₁₃×#463	—	2	2	no	—
25	C₁₃×#461	—	2	1	no	—
26	C₁₃×#460	—	2	1	no	—
27	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
29	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
30	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
31	C₄×C₁₃×D₈	C₂²	1	2	yes	—
32	C₁₃×#367	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	2	yes	—
36	C₁₃×#358	C₂²	1	1	yes	characteristic
37	C₁₃×#357	C₂²	1	2	yes	—
38	C₁₃×#357	C₂²	1	1	yes	characteristic
39	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
40	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
41	C₂×C₁₃×Q₁₆	—	2	1	no	—
42	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
43	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
44	C₂×C₁₃×SD₁₆	—	2	2	no	—
45	C₂×C₁₃×SD₁₆	—	2	2	no	—
46	C₂×C₁₃×D₈	—	2	1	no	—
47	C₂×C₁₃×D₈	C₂³	1	2	yes	—
48	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
49	C₁₃×C₄⋊Q₈	—	2	1	no	—
50	C₁₃×#127	—	2	1	no	—
51	C₁₃×#126	—	4	2	no	—
52	C₁₃×#124	—	2	2	no	—
53	C₁₃×#124	—	2	2	no	—
54	C₁₃×#124	—	4	1	no	—
55	C₁₃×#123	—	4	2	no	—
56	C₁₃×C₂²⋊Q₈	—	4	1	no	—
57	C₄×Q₈×C₁₃	—	2	1	no	—
58	C₄×Q₈×C₁₃	—	2	2	no	—
59	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
60	C₄×D₄×C₁₃	—	2	1	no	—
61	C₄×D₄×C₁₃	—	2	2	no	—
62	C₄×D₄×C₁₃	—	4	1	no	—
63	C₁₃×#117	—	4	1	no	—
64	C₁₃×Q₃₂	—	4	2	no	—
65	C₁₃×SD₃₂	—	2	4	no	—
66	C₁₃×SD₃₂	—	2	2	no	—
67	C₁₃×SD₃₂	—	4	2	no	—
68	C₁₃×D₁₆	—	4	2	no	—
69	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
70	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
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×#102	—	2	2	no	—
79	C₁₃×#102	—	2	1	no	—
80	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
81	C₄×C₈×C₁₃	C₂³	1	2	yes	—
82	#33961	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	D₈	1	1	yes	characteristic
86	C₂×D₄×C₁₃	—	2	1	no	—
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	1	no	—
91	C₂²×C₄×C₁₃	—	4	2	no	—
92	C₁₃×Q₁₆	—	2	2	no	—
93	C₁₃×Q₁₆	—	2	1	no	—
94	C₁₃×Q₁₆	—	2	2	no	—
95	C₁₃×Q₁₆	—	2	1	no	—
96	C₁₃×Q₁₆	—	8	1	no	—
97	C₁₃×SD₁₆	—	8	2	no	—
98	C₁₃×SD₁₆	—	8	2	no	—
99	C₁₃×D₈	—	2	2	no	—
100	C₁₃×D₈	—	2	1	no	—
101	C₁₃×D₈	—	2	1	no	—
102	C₁₃×D₈	—	2	2	no	—
103	C₁₃×D₈	—	8	1	no	—
104	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
107	C₁₃×C₄⋊C₄	—	2	1	no	—
108	C₁₃×C₄⋊C₄	—	2	1	no	—
109	C₁₃×C₄⋊C₄	—	4	1	no	—
110	C₁₃×C₄⋊C₄	—	4	2	no	—
111	C₁₃×C₄⋊C₄	—	4	1	no	—
112	C₁₃×C₄⋊C₄	—	4	1	no	—
113	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
114	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
115	C₁₃×C₂²⋊C₄	—	4	2	no	—
116	C₁₃×C₂²⋊C₄	—	4	2	no	—
117	C₁₃×C₂²⋊C₄	—	4	1	no	—
118	C₁₃×C₂²⋊C₄	—	4	2	no	—
119	C₁₃×C₂²⋊C₄	—	4	2	no	—
120	C₁₃×C₂²⋊C₄	—	4	2	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	2	no	—
126	C₁₃×C₁₆	—	2	1	no	—
127	C₁₃×C₁₆	#41	1	2	yes	—
128	#3396	C₂×C₁₃	1	1	yes	characteristic, complemented
129	#3393	C₂×C₁₃	1	1	yes	characteristic, complemented
130	#2249	C₂×C₁₃	1	2	yes	complemented
131	#2248	C₂×C₁₃	1	1	yes	characteristic, complemented
132	#2211	C₂×C₁₃	1	2	yes	complemented
133	#2210	C₂×C₁₃	1	2	yes	complemented
134	#2209	C₂×C₁₃	1	1	yes	characteristic, complemented
135	#2206	C₂×C₁₃	1	1	yes	characteristic
136	Q₈×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
137	C₄×SD₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
138	C₄×SD₃₂	C₂×C₁₃	1	2	yes	complemented
139	C₂³×C₁₃	—	4	1	no	—
140	C₂³×C₁₃	—	4	2	no	—
141	Q₈×C₁₃	—	2	2	no	—
142	Q₈×C₁₃	—	4	2	no	—
143	Q₈×C₁₃	—	4	1	no	—
144	Q₈×C₁₃	—	4	1	no	—
145	Q₈×C₁₃	—	4	2	no	—
146	D₄×C₁₃	—	4	1	no	—
147	D₄×C₁₃	—	4	2	no	—
148	D₄×C₁₃	—	4	2	no	—
149	D₄×C₁₃	—	4	1	no	—
150	D₄×C₁₃	—	8	2	no	—
151	D₄×C₁₃	—	16	1	no	—
152	C₂×C₄×C₁₃	—	4	1	no	—
153	C₂×C₄×C₁₃	—	4	1	no	—
154	C₂×C₄×C₁₃	—	4	1	no	—
155	C₂×C₄×C₁₃	—	4	2	no	—
156	C₂×C₄×C₁₃	—	8	2	no	—
157	C₂×C₄×C₁₃	—	16	2	no	—
158	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
159	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
160	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
161	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
162	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
163	C₈×C₁₃	—	2	1	no	—
164	C₈×C₁₃	—	2	2	no	—
165	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
166	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
167	#559	—	2	1	no	—
168	#556	—	2	1	no	—
169	C₂×Q₃₂	C₂²×C₁₃	1	2	yes	—
170	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
171	C₂×SD₃₂	C₂²×C₁₃	1	1	yes	characteristic
172	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
173	C₂×D₁₆	C₂²×C₁₃	1	2	yes	—
174	#495	C₂²×C₁₃	1	2	yes	—
175	#495	C₂²×C₁₃	1	2	yes	—
176	#494	C₂²×C₁₃	1	1	yes	characteristic
177	#493	C₂²×C₁₃	1	1	yes	characteristic
178	#464	—	2	2	no	—
179	#463	—	2	2	no	—
180	#461	—	2	1	no	—
181	#460	—	2	1	no	—
182	C₈×Q₈	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	1	yes	characteristic
186	C₄×D₈	C₂²×C₁₃	1	2	yes	—
187	#367	C₂²×C₁₃	1	1	yes	characteristic
188	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
189	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
190	#358	C₂²×C₁₃	1	2	yes	—
191	#358	C₂²×C₁₃	1	1	yes	characteristic
192	#357	C₂²×C₁₃	1	2	yes	—
193	#357	C₂²×C₁₃	1	1	yes	characteristic
194	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
195	C₄×C₁₆	C₂²×C₁₃	1	2	yes	—
196	C₂²×C₁₃	—	8	2	no	—
197	C₂²×C₁₃	—	8	1	no	—
198	C₂²×C₁₃	—	8	1	no	—
199	C₂²×C₁₃	—	8	2	no	—
200	C₂²×C₁₃	—	8	2	no	—
201	C₂²×C₁₃	—	8	1	no	—
202	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
203	C₄×C₁₃	—	2	2	no	—
204	C₄×C₁₃	—	2	2	no	—
205	C₄×C₁₃	—	2	1	no	—
206	C₄×C₁₃	—	8	1	no	—
207	C₄×C₁₃	—	8	2	no	—
208	C₄×C₁₃	—	8	1	no	—
209	C₄×C₁₃	—	8	1	no	—
210	C₄×C₁₃	#508	1	2	yes	—
211	C₄×C₁₃	#550	1	1	yes	characteristic
212	C₄×C₁₃	#576	1	1	yes	characteristic
213	C₂×Q₁₆	—	2	1	no	—
214	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
215	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
216	C₂×SD₁₆	—	2	2	no	—
217	C₂×SD₁₆	—	2	2	no	—
218	C₂×D₈	—	2	1	no	—
219	C₂×D₈	C₂³×C₁₃	1	2	yes	—
220	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
221	C₄⋊Q₈	—	2	1	no	—
222	#127	—	2	1	no	—
223	#126	—	4	2	no	—
224	#124	—	2	2	no	—
225	#124	—	2	2	no	—
226	#124	—	4	1	no	—
227	#123	—	4	2	no	—
228	C₂²⋊Q₈	—	4	1	no	—
229	C₄×Q₈	—	2	2	no	—
230	C₄×Q₈	—	2	1	no	—
231	C₄×Q₈	D₄×C₁₃	1	1	yes	characteristic
232	C₄×D₄	—	2	1	no	—
233	C₄×D₄	—	2	2	no	—
234	C₄×D₄	—	4	1	no	—
235	#117	—	4	1	no	—
236	Q₃₂	—	4	2	no	—
237	SD₃₂	—	2	4	no	—
238	SD₃₂	—	2	2	no	—
239	SD₃₂	—	4	2	no	—
240	D₁₆	—	4	2	no	—
241	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
242	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
243	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
244	#107	C₂³×C₁₃	1	1	yes	characteristic
245	#107	C₂³×C₁₃	1	1	yes	characteristic
246	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
247	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
248	#103	—	2	2	no	—
249	#103	—	2	1	no	—
250	#102	—	2	2	no	—
251	#102	—	2	1	no	—
252	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
253	C₄×C₈	C₂³×C₁₃	1	2	yes	—
254	C₂×C₁₃	—	16	1	no	—
255	C₂×C₁₃	—	16	2	no	—
256	C₂×C₁₃	#3391	1	1	yes	characteristic
257	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
258	C₂×C₁₃	#3416	1	1	yes	characteristic
259	C₂×Q₈	—	2	1	no	—
260	C₂×Q₈	—	2	2	no	—
261	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
262	C₂×D₄	—	2	1	no	—
263	C₂×D₄	—	2	2	no	—
264	C₂×D₄	—	4	1	no	—
265	C₂×D₄	—	4	1	no	—
266	C₂²×C₄	—	4	1	no	—
267	C₂²×C₄	—	4	2	no	—
268	Q₁₆	—	2	2	no	—
269	Q₁₆	—	2	1	no	—
270	Q₁₆	—	2	2	no	—
271	Q₁₆	—	2	1	no	—
272	Q₁₆	—	8	1	no	—
273	SD₁₆	—	8	2	no	—
274	SD₁₆	—	8	2	no	—
275	D₈	—	2	2	no	—
276	D₈	—	2	1	no	—
277	D₈	—	2	1	no	—
278	D₈	—	2	2	no	—
279	D₈	—	8	1	no	—
280	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
281	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
282	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
283	C₄⋊C₄	—	2	1	no	—
284	C₄⋊C₄	—	2	1	no	—
285	C₄⋊C₄	—	4	1	no	—
286	C₄⋊C₄	—	4	2	no	—
287	C₄⋊C₄	—	4	1	no	—
288	C₄⋊C₄	—	4	1	no	—
289	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
290	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
291	C₂²⋊C₄	—	4	2	no	—
292	C₂²⋊C₄	—	4	2	no	—
293	C₂²⋊C₄	—	4	1	no	—
294	C₂²⋊C₄	—	4	2	no	—
295	C₂²⋊C₄	—	4	2	no	—
296	C₂²⋊C₄	—	4	2	no	—
297	C₄²	—	4	2	no	—
298	C₄²	—	4	1	no	—
299	C₄²	C₂×D₄×C₁₃	1	2	yes	—
300	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
301	C₁₆	—	2	2	no	—
302	C₁₆	—	2	1	no	—
303	C₁₆	C₁₃×#41	1	2	yes	—
304	C₁₃	#33961	1	1	yes	characteristic, direct product factor, Sylow
305	C₂³	—	4	1	no	—
306	C₂³	—	4	2	no	—
307	Q₈	—	2	2	no	—
308	Q₈	—	4	2	no	—
309	Q₈	—	4	1	no	—
310	Q₈	—	4	1	no	—
311	Q₈	—	4	2	no	—
312	D₄	—	4	1	no	—
313	D₄	—	4	2	no	—
314	D₄	—	4	2	no	—
315	D₄	—	4	1	no	—
316	D₄	—	8	2	no	—
317	D₄	—	16	1	no	—
318	C₂×C₄	—	4	1	no	—
319	C₂×C₄	—	4	1	no	—
320	C₂×C₄	—	4	1	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	8	2	no	—
323	C₂×C₄	—	16	2	no	—
324	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
325	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
326	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
327	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
328	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
329	C₈	—	2	1	no	—
330	C₈	—	2	2	no	—
331	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
332	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
333	C₂²	—	8	2	no	—
334	C₂²	—	8	1	no	—
335	C₂²	—	8	1	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	—	8	1	no	—
339	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
340	C₄	—	2	2	no	—
341	C₄	—	2	2	no	—
342	C₄	—	2	1	no	—
343	C₄	—	8	1	no	—
344	C₄	—	8	2	no	—
345	C₄	—	8	1	no	—
346	C₄	—	8	1	no	—
347	C₄	C₁₃×#508	1	2	yes	—
348	C₄	C₁₃×#550	1	1	yes	characteristic
349	C₄	C₁₃×#576	1	1	yes	characteristic
350	C₂	—	16	1	no	—
351	C₂	—	16	2	no	—
352	C₂	C₁₃×#3391	1	1	yes	characteristic
353	C₂	C₂×C₁₃×#508	1	1	yes	characteristic
354	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×#2206	C₂
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
1. C₁₃×#3396	C₂	semidirect
2. C₁₃×#3393	C₂	semidirect
3. C₁₃×#2249	C₂	semidirect
4. C₁₃×#2248	C₂	semidirect
5. C₁₃×#2211	C₂	semidirect
6. C₁₃×#2210	C₂	semidirect
7. C₁₃×#2209	C₂	semidirect
9. Q₈×C₁₃×C₁₆	C₂	semidirect
10. C₄×C₁₃×SD₃₂	C₂	semidirect
11. C₄×C₁₃×SD₃₂	C₂	semidirect
82. #33961	C₁₃	direct (N × H)
128. #3396	C₂×C₁₃	semidirect
129. #3393	C₂×C₁₃	semidirect
130. #2249	C₂×C₁₃	semidirect
131. #2248	C₂×C₁₃	semidirect
132. #2211	C₂×C₁₃	semidirect
133. #2210	C₂×C₁₃	semidirect
134. #2209	C₂×C₁₃	semidirect
136. Q₈×C₁₆	C₂×C₁₃	semidirect
137. C₄×SD₃₂	C₂×C₁₃	semidirect
138. C₄×SD₃₂	C₂×C₁₃	semidirect
304. C₁₃	#33961	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×#3393	2
C₁₃×#2249	2
C₁₃×#2248	2
C₁₃×#2211	2
C₁₃×#2210	2
C₁₃×#2209	2
C₁₃×#2206	2
Q₈×C₁₃×C₁₆	2
C₄×C₁₃×SD₃₂	2
#33961	13

Subgroups — C13×#33961 #15720300

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33961 #15720300