Subgroups — C₁₃×#33906 #15720245

Order 3328.

494 conjugacy classes of subgroups.

The subgroup lattice has 326 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3339	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2253	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2234	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2233	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2228	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2215	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2215	C₂	1	1	yes	maximal, characteristic, complemented
8	C₄×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
9	D₄×C₁₃×C₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
11	C₂×C₁₃×#366	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#553	—	2	2	no	—
13	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
15	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
16	C₁₃×#499	C₂²	1	2	yes	—
17	C₁₃×#482	—	2	2	no	—
18	C₁₃×#481	—	2	2	no	—
19	C₁₃×#478	—	2	2	no	—
20	C₁₃×#466	C₂²	1	2	yes	—
21	C₁₃×#466	C₂²	1	2	yes	—
22	C₄×C₁₃×D₈	C₂²	1	2	yes	—
23	C₈×D₄×C₁₃	C₂²	1	1	yes	characteristic
24	C₁₃×#427	C₂²	1	2	yes	—
25	C₂×C₁₃×#107	C₂²	1	2	yes	—
26	C₁₃×#367	C₂²	1	2	yes	—
27	C₁₃×#367	C₂²	1	2	yes	—
28	C₁₃×#366	C₂²	1	1	yes	characteristic
29	C₁₃×#366	C₂²	1	1	yes	characteristic
30	C₁₃×#366	C₂²	1	1	yes	characteristic
31	C₁₃×#366	C₂²	1	2	yes	—
32	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
33	C₁₃×#357	C₂²	1	2	yes	—
34	C₁₃×#357	C₂²	1	2	yes	—
35	C₁₃×#357	C₂²	1	2	yes	—
36	C₁₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
37	C₁₃×C₂²⋊C₁₆	C₂²	1	1	yes	characteristic
38	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
39	C₂×C₁₃×D₈	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×#126	—	4	2	no	—
43	C₁₃×#125	—	2	2	no	—
44	C₁₃×#123	—	4	2	no	—
45	C₁₃×#121	—	2	2	no	—
46	C₁₃×#121	—	2	2	no	—
47	C₁₃×#121	—	4	2	no	—
48	C₄×D₄×C₁₃	—	2	2	no	—
49	C₄×D₄×C₁₃	—	4	2	no	—
50	C₄×D₄×C₁₃	D₄	1	1	yes	characteristic
51	C₁₃×#117	—	2	2	no	—
52	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
53	C₁₃×D₁₆	—	2	2	no	—
54	C₂×C₁₃×C₁₆	—	2	2	no	—
55	C₂×C₁₃×C₁₆	—	2	2	no	—
56	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
57	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
59	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
60	C₁₃×#107	—	2	2	no	—
61	C₁₃×#107	C₂³	1	2	yes	—
62	C₁₃×#107	C₂³	1	2	yes	—
63	C₁₃×#107	C₂³	1	2	yes	—
64	C₁₃×#106	—	2	2	no	—
65	C₁₃×#106	—	2	2	no	—
66	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
67	C₁₃×#102	—	2	2	no	—
68	C₁₃×#102	—	2	2	no	—
69	C₁₃×#102	—	2	2	no	—
70	C₁₃×C₂²⋊C₈	D₄	1	1	yes	characteristic
71	C₁₃×C₂²⋊C₈	D₄	1	1	yes	characteristic
72	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
73	#33906	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂×D₄×C₁₃	—	2	2	no	—
75	C₂×D₄×C₁₃	—	4	2	no	—
76	C₂×D₄×C₁₃	—	4	2	no	—
77	C₂×D₄×C₁₃	D₈	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	—	4	2	no	—
79	C₂²×C₄×C₁₃	—	4	2	no	—
80	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₁₃×D₈	—	2	2	no	—
83	C₁₃×D₈	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	1	no	—
85	C₂×C₈×C₁₃	—	2	1	no	—
86	C₂×C₈×C₁₃	—	2	1	no	—
87	C₂×C₈×C₁₃	—	2	1	no	—
88	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	2	2	no	—
94	C₁₃×C₄⋊C₄	—	2	2	no	—
95	C₁₃×C₄⋊C₄	—	4	2	no	—
96	C₁₃×C₄⋊C₄	—	4	2	no	—
97	C₁₃×C₄⋊C₄	—	4	2	no	—
98	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
99	C₁₃×C₂²⋊C₄	—	4	2	no	—
100	C₁₃×C₂²⋊C₄	—	4	2	no	—
101	C₁₃×C₂²⋊C₄	—	4	2	no	—
102	C₁₃×C₂²⋊C₄	—	4	2	no	—
103	C₁₃×C₂²⋊C₄	D₈	1	1	yes	characteristic
104	C₁₃×C₂²⋊C₄	D₈	1	1	yes	characteristic
105	C₄²×C₁₃	—	4	2	no	—
106	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₁₃×C₁₆	—	2	1	no	—
108	C₁₃×C₁₆	—	2	1	no	—
109	C₁₃×C₁₆	—	2	1	no	—
110	C₁₃×C₁₆	#41	1	2	yes	—
111	#3339	C₂×C₁₃	1	2	yes	complemented
112	#2253	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#2234	C₂×C₁₃	1	2	yes	complemented
114	#2233	C₂×C₁₃	1	2	yes	complemented
115	#2228	C₂×C₁₃	1	2	yes	complemented
116	#2215	C₂×C₁₃	1	1	yes	characteristic, complemented
117	#2215	C₂×C₁₃	1	1	yes	characteristic, complemented
118	C₄×D₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
119	D₄×C₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
120	#2158	C₂×C₁₃	1	1	yes	characteristic, complemented
121	C₂×#366	C₂×C₁₃	1	1	yes	characteristic, complemented
122	C₂³×C₁₃	—	4	2	no	—
123	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
124	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
125	D₄×C₁₃	—	2	2	no	—
126	D₄×C₁₃	—	4	2	no	—
127	D₄×C₁₃	—	4	2	no	—
128	D₄×C₁₃	—	8	4	no	—
129	D₄×C₁₃	—	16	2	no	—
130	C₂×C₄×C₁₃	—	2	1	no	—
131	C₂×C₄×C₁₃	—	2	1	no	—
132	C₂×C₄×C₁₃	—	2	1	no	—
133	C₂×C₄×C₁₃	—	2	1	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	—	16	2	no	—
138	C₂×C₄×C₁₃	—	16	2	no	—
139	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
144	C₈×C₁₃	—	2	1	no	—
145	C₈×C₁₃	—	2	1	no	—
146	C₈×C₁₃	—	2	1	no	—
147	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
148	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
149	#553	—	2	2	no	—
150	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic
151	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
152	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
153	#499	C₂²×C₁₃	1	2	yes	—
154	#482	—	2	2	no	—
155	#481	—	2	2	no	—
156	#478	—	2	2	no	—
157	#466	C₂²×C₁₃	1	2	yes	—
158	#466	C₂²×C₁₃	1	2	yes	—
159	C₄×D₈	C₂²×C₁₃	1	2	yes	—
160	C₈×D₄	C₂²×C₁₃	1	1	yes	characteristic
161	#427	C₂²×C₁₃	1	2	yes	—
162	C₂×#107	C₂²×C₁₃	1	2	yes	—
163	#367	C₂²×C₁₃	1	2	yes	—
164	#367	C₂²×C₁₃	1	2	yes	—
165	#366	C₂²×C₁₃	1	1	yes	characteristic
166	#366	C₂²×C₁₃	1	1	yes	characteristic
167	#366	C₂²×C₁₃	1	2	yes	—
168	#366	C₂²×C₁₃	1	1	yes	characteristic
169	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
170	#357	C₂²×C₁₃	1	2	yes	—
171	#357	C₂²×C₁₃	1	2	yes	—
172	#357	C₂²×C₁₃	1	2	yes	—
173	C₂²⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
174	C₂²⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
175	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
176	C₂²×C₁₃	—	2	1	no	—
177	C₂²×C₁₃	—	2	1	no	—
178	C₂²×C₁₃	—	2	1	no	—
179	C₂²×C₁₃	—	2	1	no	—
180	C₂²×C₁₃	—	2	1	no	—
181	C₂²×C₁₃	—	8	2	no	—
182	C₂²×C₁₃	—	8	2	no	—
183	C₂²×C₁₃	—	8	2	no	—
184	C₂²×C₁₃	#508	1	2	yes	—
185	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
186	C₄×C₁₃	—	2	1	no	—
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	—	2	1	no	—
190	C₄×C₁₃	—	8	2	no	—
191	C₄×C₁₃	—	8	2	no	—
192	C₄×C₁₃	—	8	2	no	—
193	C₄×C₁₃	#547	1	1	yes	characteristic
194	C₄×C₁₃	#578	1	1	yes	characteristic
195	C₂×D₈	C₂³×C₁₃	1	2	yes	—
196	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
198	#126	—	4	2	no	—
199	#125	—	2	2	no	—
200	#123	—	4	2	no	—
201	#121	—	2	2	no	—
202	#121	—	2	2	no	—
203	#121	—	4	2	no	—
204	C₄×D₄	—	2	2	no	—
205	C₄×D₄	—	4	2	no	—
206	C₄×D₄	D₄×C₁₃	1	1	yes	characteristic
207	#117	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	D₁₆	—	2	2	no	—
210	C₂×C₁₆	—	2	2	no	—
211	C₂×C₁₆	—	2	2	no	—
212	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
213	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
214	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
215	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
216	#107	—	2	2	no	—
217	#107	C₂³×C₁₃	1	2	yes	—
218	#107	C₂³×C₁₃	1	2	yes	—
219	#107	C₂³×C₁₃	1	2	yes	—
220	#106	—	2	2	no	—
221	#106	—	2	2	no	—
222	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
223	#102	—	2	2	no	—
224	#102	—	2	2	no	—
225	#102	—	2	2	no	—
226	C₂²⋊C₈	D₄×C₁₃	1	1	yes	characteristic
227	C₂²⋊C₈	D₄×C₁₃	1	1	yes	characteristic
228	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
229	C₂×C₁₃	—	2	2	no	—
230	C₂×C₁₃	—	4	1	no	—
231	C₂×C₁₃	—	16	2	no	—
232	C₂×C₁₃	#3334	1	1	yes	characteristic
233	C₂×C₁₃	C₂×#508	1	1	yes	characteristic
234	C₂×C₁₃	#3415	1	1	yes	characteristic
235	C₂×D₄	—	2	2	no	—
236	C₂×D₄	—	4	2	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	C₁₃×D₈	1	1	yes	characteristic
239	C₂²×C₄	—	4	2	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
242	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
243	D₈	—	2	2	no	—
244	D₈	—	2	2	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
250	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
251	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
252	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	2	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	2	no	—
258	C₄⋊C₄	—	4	2	no	—
259	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
260	C₂²⋊C₄	—	4	2	no	—
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	C₁₃×D₈	1	1	yes	characteristic
265	C₂²⋊C₄	C₁₃×D₈	1	1	yes	characteristic
266	C₄²	—	4	2	no	—
267	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
268	C₁₆	—	2	1	no	—
269	C₁₆	—	2	1	no	—
270	C₁₆	—	2	1	no	—
271	C₁₆	C₁₃×#41	1	2	yes	—
272	C₁₃	#33906	1	1	yes	characteristic, direct product factor, Sylow
273	C₂³	—	4	2	no	—
274	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
275	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
276	D₄	—	2	2	no	—
277	D₄	—	4	2	no	—
278	D₄	—	4	2	no	—
279	D₄	—	8	4	no	—
280	D₄	—	16	2	no	—
281	C₂×C₄	—	2	1	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	16	2	no	—
289	C₂×C₄	—	16	2	no	—
290	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
291	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
292	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
293	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
294	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
295	C₈	—	2	1	no	—
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
299	C₈	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
300	C₂²	—	2	1	no	—
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	2	1	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	—	8	2	no	—
308	C₂²	C₁₃×#508	1	2	yes	—
309	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	2	1	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	—	8	2	no	—
317	C₄	C₁₃×#547	1	1	yes	characteristic
318	C₄	C₁₃×#578	1	1	yes	characteristic
319	C₂	—	2	2	no	—
320	C₂	—	4	1	no	—
321	C₂	—	16	2	no	—
322	C₂	C₁₃×#3334	1	1	yes	characteristic
323	C₂	C₂×C₁₃×#508	1	1	yes	characteristic
324	C₂	C₁₃×#3415	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
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×#3339	C₂	semidirect
2. C₁₃×#2253	C₂	semidirect
3. C₁₃×#2234	C₂	semidirect
4. C₁₃×#2233	C₂	semidirect
5. C₁₃×#2228	C₂	semidirect
6. C₁₃×#2215	C₂	semidirect
7. C₁₃×#2215	C₂	semidirect
8. C₄×C₁₃×D₁₆	C₂	semidirect
9. D₄×C₁₃×C₁₆	C₂	semidirect
10. C₁₃×#2158	C₂	semidirect
11. C₂×C₁₃×#366	C₂	semidirect
73. #33906	C₁₃	direct (N × H)
111. #3339	C₂×C₁₃	semidirect
112. #2253	C₂×C₁₃	semidirect
113. #2234	C₂×C₁₃	semidirect
114. #2233	C₂×C₁₃	semidirect
115. #2228	C₂×C₁₃	semidirect
116. #2215	C₂×C₁₃	semidirect
117. #2215	C₂×C₁₃	semidirect
118. C₄×D₁₆	C₂×C₁₃	semidirect
119. D₄×C₁₆	C₂×C₁₃	semidirect
120. #2158	C₂×C₁₃	semidirect
121. C₂×#366	C₂×C₁₃	semidirect
272. C₁₃	#33906	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×#3339	2
C₁₃×#2253	2
C₁₃×#2234	2
C₁₃×#2233	2
C₁₃×#2228	2
C₁₃×#2215	2
C₄×C₁₃×D₁₆	2
D₄×C₁₃×C₁₆	2
C₁₃×#2158	2
C₂×C₁₃×#366	2
#33906	13

Subgroups — C13×#33906 #15720245

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33906 #15720245