Subgroups — C₁₃×#33682 #15720021

Order 3328.

580 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3176	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#2948	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2234	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2228	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2206	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2184	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2177	C₂	1	2	yes	maximal, complemented
8	C₄×C₁₃×D₁₆	C₂	1	2	yes	maximal, complemented
9	C₁₃×#2164	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
11	C₁₃×#540	—	2	1	no	—
12	C₁₃×#518	—	2	1	no	—
13	C₂×C₁₃×D₁₆	C₂²	1	2	yes	—
14	C₂×C₁₃×D₁₆	C₂²	1	2	yes	—
15	C₁₃×#499	C₂²	1	1	yes	characteristic, complemented
16	C₁₃×#493	C₂²	1	1	yes	characteristic, complemented
17	C₁₃×#490	—	2	1	no	—
18	C₁₃×#469	—	2	1	no	—
19	C₁₃×#466	C₂²	1	2	yes	—
20	C₁₃×#442	—	2	1	no	—
21	C₁₃×#442	—	2	1	no	—
22	C₁₃×#442	—	2	1	no	—
23	C₄×C₁₃×D₈	C₂²	1	2	yes	—
24	C₄×C₁₃×D₈	C₂²	1	2	yes	—
25	C₄×C₁₃×D₈	C₂²	1	2	yes	complemented
26	C₁₃×#427	C₂²	1	1	yes	characteristic
27	C₁₃×#418	—	2	1	no	—
28	C₁₃×#405	C₂²	1	1	yes	characteristic
29	C₁₃×#367	C₂²	1	2	yes	complemented
30	C₁₃×#366	C₂²	1	2	yes	complemented
31	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	complemented
32	C₁₃×#357	C₂²	1	2	yes	—
33	C₁₃×#357	C₂²	1	2	yes	—
34	C₁₃×#357	C₂²	1	2	yes	complemented
35	C₁₃×#357	C₂²	1	2	yes	complemented
36	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
37	C₁₃×#346	C₂²	1	1	yes	characteristic, complemented
38	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic, complemented
39	C₂²×D₄×C₁₃	—	2	1	no	—
40	C₂×C₁₃×D₈	—	2	4	no	—
41	C₂×C₁₃×D₈	—	2	1	no	—
42	C₂×C₁₃×D₈	—	2	1	no	—
43	C₂×C₁₃×D₈	C₂³	1	2	yes	—
44	C₂×C₁₃×D₈	C₂³	1	2	yes	—
45	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×#127	—	2	1	no	—
48	C₁₃×#125	—	2	1	no	—
49	C₁₃×#123	—	4	2	no	—
50	C₁₃×#121	—	2	2	no	—
51	C₁₃×#121	—	4	2	no	—
52	C₄×D₄×C₁₃	—	2	2	no	—
53	C₄×D₄×C₁₃	—	2	2	no	—
54	C₄×D₄×C₁₃	—	2	2	no	—
55	C₄×D₄×C₁₃	—	4	2	no	—
56	C₁₃×#117	—	2	1	no	—
57	C₁₃×#117	D₄	1	1	yes	characteristic
58	C₂×C₁₃×C₂²⋊C₄	—	4	2	no	—
59	C₁₃×D₁₆	—	2	4	no	—
60	C₁₃×D₁₆	—	2	4	no	—
61	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
62	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
63	C₁₃×#107	C₂³	1	2	yes	—
64	C₁₃×#107	C₂³	1	2	yes	—
65	C₁₃×#106	—	2	1	no	—
66	C₁₃×#106	—	2	1	no	—
67	C₁₃×#102	—	2	2	no	—
68	C₁₃×#102	—	2	2	no	—
69	C₁₃×#102	—	2	2	no	—
70	C₁₃×#102	—	2	2	no	—
71	C₁₃×#97	D₄	1	1	yes	characteristic
72	C₁₃×#97	D₄	1	1	yes	characteristic
73	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
74	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
75	#33682	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂⁴×C₁₃	—	4	1	no	—
77	C₂×D₄×C₁₃	—	2	4	no	—
78	C₂×D₄×C₁₃	—	2	2	no	—
79	C₂×D₄×C₁₃	—	2	2	no	—
80	C₂×D₄×C₁₃	—	4	2	no	—
81	C₂×D₄×C₁₃	—	4	2	no	—
82	C₂×D₄×C₁₃	—	4	1	no	—
83	C₂×D₄×C₁₃	—	4	2	no	—
84	C₂²×C₄×C₁₃	—	4	2	no	—
85	C₂²×C₄×C₁₃	—	4	2	no	—
86	C₂²×C₄×C₁₃	—	4	2	no	—
87	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₁₃×D₈	—	2	4	no	—
89	C₁₃×D₈	—	2	2	no	—
90	C₁₃×D₈	—	2	2	no	—
91	C₁₃×D₈	—	2	2	no	—
92	C₁₃×D₈	—	4	2	no	—
93	C₁₃×D₈	#41	1	4	yes	—
94	C₁₃×M₁₆	—	4	1	no	—
95	C₂×C₈×C₁₃	—	2	2	no	—
96	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	—	2	2	no	—
102	C₁₃×C₄⋊C₄	—	4	2	no	—
103	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
104	C₁₃×C₂²⋊C₄	—	4	1	no	—
105	C₁₃×C₂²⋊C₄	—	4	2	no	—
106	C₁₃×C₂²⋊C₄	—	4	2	no	—
107	C₁₃×C₂²⋊C₄	—	4	2	no	—
108	C₁₃×C₂²⋊C₄	—	4	2	no	—
109	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
110	C₄²×C₁₃	—	4	1	no	—
111	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₁₃×C₁₆	—	2	2	no	—
114	C₁₃×C₁₆	—	2	2	no	—
115	#3176	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#2948	C₂×C₁₃	1	1	yes	characteristic, complemented
117	#2234	C₂×C₁₃	1	2	yes	complemented
118	#2228	C₂×C₁₃	1	2	yes	complemented
119	#2206	C₂×C₁₃	1	2	yes	complemented
120	#2184	C₂×C₁₃	1	2	yes	complemented
121	#2177	C₂×C₁₃	1	2	yes	complemented
122	C₄×D₁₆	C₂×C₁₃	1	2	yes	complemented
123	#2164	C₂×C₁₃	1	1	yes	characteristic, complemented
124	C₂³×C₁₃	—	4	4	no	—
125	C₂³×C₁₃	—	4	2	no	—
126	C₂³×C₁₃	—	4	2	no	—
127	C₂³×C₁₃	—	16	2	no	—
128	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
129	D₄×C₁₃	—	2	4	no	—
130	D₄×C₁₃	—	4	4	no	—
131	D₄×C₁₃	—	4	2	no	—
132	D₄×C₁₃	—	4	2	no	—
133	D₄×C₁₃	—	4	2	no	—
134	D₄×C₁₃	—	16	2	no	—
135	D₄×C₁₃	—	16	2	no	—
136	C₂×C₄×C₁₃	—	2	1	no	—
137	C₂×C₄×C₁₃	—	2	1	no	—
138	C₂×C₄×C₁₃	—	4	2	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	—	16	2	no	—
141	C₂×C₄×C₁₃	—	16	2	no	—
142	C₂×C₄×C₁₃	—	16	2	no	—
143	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
147	C₈×C₁₃	—	2	1	no	—
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	C₂×#41	1	2	yes	—
151	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
152	#540	—	2	1	no	—
153	#518	—	2	1	no	—
154	C₂×D₁₆	C₂²×C₁₃	1	2	yes	—
155	C₂×D₁₆	C₂²×C₁₃	1	2	yes	—
156	#499	C₂²×C₁₃	1	1	yes	characteristic, complemented
157	#493	C₂²×C₁₃	1	1	yes	characteristic, complemented
158	#490	—	2	1	no	—
159	#469	—	2	1	no	—
160	#466	C₂²×C₁₃	1	2	yes	—
161	#442	—	2	1	no	—
162	#442	—	2	1	no	—
163	#442	—	2	1	no	—
164	C₄×D₈	C₂²×C₁₃	1	2	yes	—
165	C₄×D₈	C₂²×C₁₃	1	2	yes	—
166	C₄×D₈	C₂²×C₁₃	1	2	yes	complemented
167	#427	C₂²×C₁₃	1	1	yes	characteristic
168	#418	—	2	1	no	—
169	#405	C₂²×C₁₃	1	1	yes	characteristic
170	#367	C₂²×C₁₃	1	2	yes	complemented
171	#366	C₂²×C₁₃	1	2	yes	complemented
172	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	complemented
173	#357	C₂²×C₁₃	1	2	yes	—
174	#357	C₂²×C₁₃	1	2	yes	—
175	#357	C₂²×C₁₃	1	2	yes	complemented
176	#357	C₂²×C₁₃	1	2	yes	complemented
177	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
178	#346	C₂²×C₁₃	1	1	yes	characteristic, complemented
179	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic, complemented
180	C₂²×C₁₃	—	2	1	no	—
181	C₂²×C₁₃	—	2	2	no	—
182	C₂²×C₁₃	—	4	4	no	—
183	C₂²×C₁₃	—	8	2	no	—
184	C₂²×C₁₃	—	8	2	no	—
185	C₂²×C₁₃	—	8	2	no	—
186	C₂²×C₁₃	—	8	2	no	—
187	C₂²×C₁₃	—	8	2	no	—
188	C₂²×C₁₃	—	16	4	no	—
189	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
190	C₄×C₁₃	—	2	1	no	—
191	C₄×C₁₃	—	2	2	no	—
192	C₄×C₁₃	—	2	2	no	—
193	C₄×C₁₃	—	8	2	no	—
194	C₄×C₁₃	—	8	2	no	—
195	C₄×C₁₃	#525	1	1	yes	characteristic
196	C₄×C₁₃	#575	1	1	yes	characteristic
197	C₂²×D₄	—	2	1	no	—
198	C₂×D₈	—	2	4	no	—
199	C₂×D₈	—	2	1	no	—
200	C₂×D₈	—	2	1	no	—
201	C₂×D₈	C₂³×C₁₃	1	2	yes	—
202	C₂×D₈	C₂³×C₁₃	1	2	yes	—
203	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
205	#127	—	2	1	no	—
206	#125	—	2	1	no	—
207	#123	—	4	2	no	—
208	#121	—	2	2	no	—
209	#121	—	4	2	no	—
210	C₄×D₄	—	2	2	no	—
211	C₄×D₄	—	2	2	no	—
212	C₄×D₄	—	2	2	no	—
213	C₄×D₄	—	4	2	no	—
214	#117	—	2	1	no	—
215	#117	D₄×C₁₃	1	1	yes	characteristic
216	C₂×C₂²⋊C₄	—	4	2	no	—
217	D₁₆	—	2	4	no	—
218	D₁₆	—	2	4	no	—
219	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
220	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
221	#107	C₂³×C₁₃	1	2	yes	—
222	#107	C₂³×C₁₃	1	2	yes	—
223	#106	—	2	1	no	—
224	#106	—	2	1	no	—
225	#102	—	2	2	no	—
226	#102	—	2	2	no	—
227	#102	—	2	2	no	—
228	#102	—	2	2	no	—
229	#97	D₄×C₁₃	1	1	yes	characteristic
230	#97	D₄×C₁₃	1	1	yes	characteristic
231	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
232	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
233	C₂×C₁₃	—	4	1	no	—
234	C₂×C₁₃	—	8	4	no	—
235	C₂×C₁₃	—	16	2	no	—
236	C₂×C₁₃	#3100	1	1	yes	characteristic
237	C₂×C₁₃	#3415	1	2	yes	—
238	C₂⁴	—	4	1	no	—
239	C₂×D₄	—	2	4	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	—	4	2	no	—
243	C₂×D₄	—	4	2	no	—
244	C₂×D₄	—	4	1	no	—
245	C₂×D₄	—	4	2	no	—
246	C₂²×C₄	—	4	2	no	—
247	C₂²×C₄	—	4	2	no	—
248	C₂²×C₄	—	4	2	no	—
249	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
250	D₈	—	2	4	no	—
251	D₈	—	2	2	no	—
252	D₈	—	2	2	no	—
253	D₈	—	2	2	no	—
254	D₈	—	4	2	no	—
255	D₈	C₁₃×#41	1	4	yes	—
256	M₁₆	—	4	1	no	—
257	C₂×C₈	—	2	2	no	—
258	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
261	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	2	no	—
265	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
266	C₂²⋊C₄	—	4	1	no	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	4	2	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
272	C₄²	—	4	1	no	—
273	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
274	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
275	C₁₆	—	2	2	no	—
276	C₁₆	—	2	2	no	—
277	C₁₃	#33682	1	1	yes	characteristic, direct product factor, Sylow
278	C₂³	—	4	4	no	—
279	C₂³	—	4	2	no	—
280	C₂³	—	4	2	no	—
281	C₂³	—	16	2	no	—
282	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
283	D₄	—	2	4	no	—
284	D₄	—	4	4	no	—
285	D₄	—	4	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	4	2	no	—
288	D₄	—	16	2	no	—
289	D₄	—	16	2	no	—
290	C₂×C₄	—	2	1	no	—
291	C₂×C₄	—	2	1	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	16	2	no	—
295	C₂×C₄	—	16	2	no	—
296	C₂×C₄	—	16	2	no	—
297	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
298	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
299	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
300	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
301	C₈	—	2	1	no	—
302	C₈	—	2	1	no	—
303	C₈	—	2	1	no	—
304	C₈	C₂×C₁₃×#41	1	2	yes	—
305	C₂²	—	2	1	no	—
306	C₂²	—	2	2	no	—
307	C₂²	—	4	4	no	—
308	C₂²	—	8	2	no	—
309	C₂²	—	8	2	no	—
310	C₂²	—	8	2	no	—
311	C₂²	—	8	2	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	16	4	no	—
314	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
315	C₄	—	2	1	no	—
316	C₄	—	2	2	no	—
317	C₄	—	2	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	2	no	—
320	C₄	C₁₃×#525	1	1	yes	characteristic
321	C₄	C₁₃×#575	1	1	yes	characteristic
322	C₂	—	4	1	no	—
323	C₂	—	8	4	no	—
324	C₂	—	16	2	no	—
325	C₂	C₁₃×#3100	1	1	yes	characteristic
326	C₂	C₁₃×#3415	1	2	yes	—

	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 ◁ C₁₃×#117 ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3176	C₂	semidirect
2. C₁₃×#2948	C₂	semidirect
3. C₁₃×#2234	C₂	semidirect
4. C₁₃×#2228	C₂	semidirect
5. C₁₃×#2206	C₂	semidirect
6. C₁₃×#2184	C₂	semidirect
7. C₁₃×#2177	C₂	semidirect
8. C₄×C₁₃×D₁₆	C₂	semidirect
9. C₁₃×#2164	C₂	semidirect
15. C₁₃×#499	C₂²	semidirect
16. C₁₃×#493	C₂²	semidirect
25. C₄×C₁₃×D₈	C₂²	semidirect
29. C₁₃×#367	C₂²	semidirect
30. C₁₃×#366	C₂²	semidirect
31. C₁₃×C₄⋊C₁₆	C₂²	semidirect
34. C₁₃×#357	C₂²	semidirect
35. C₁₃×#357	C₂²	semidirect
37. C₁₃×#346	C₂²	semidirect
38. C₄×C₁₃×C₁₆	C₂²	semidirect
75. #33682	C₁₃	direct (N × H)
115. #3176	C₂×C₁₃	semidirect
116. #2948	C₂×C₁₃	semidirect
117. #2234	C₂×C₁₃	semidirect
118. #2228	C₂×C₁₃	semidirect
119. #2206	C₂×C₁₃	semidirect
120. #2184	C₂×C₁₃	semidirect
121. #2177	C₂×C₁₃	semidirect
122. C₄×D₁₆	C₂×C₁₃	semidirect
123. #2164	C₂×C₁₃	semidirect
156. #499	C₂²×C₁₃	semidirect
157. #493	C₂²×C₁₃	semidirect
166. C₄×D₈	C₂²×C₁₃	semidirect
170. #367	C₂²×C₁₃	semidirect
171. #366	C₂²×C₁₃	semidirect
172. C₄⋊C₁₆	C₂²×C₁₃	semidirect
175. #357	C₂²×C₁₃	semidirect
176. #357	C₂²×C₁₃	semidirect
178. #346	C₂²×C₁₃	semidirect
179. C₄×C₁₆	C₂²×C₁₃	semidirect
277. C₁₃	#33682	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×#3176	2
C₁₃×#2948	2
C₁₃×#2234	2
C₁₃×#2228	2
C₁₃×#2206	2
C₁₃×#2184	2
C₁₃×#2177	2
C₄×C₁₃×D₁₆	2
C₁₃×#2164	2
#33682	13

Subgroups — C13×#33682 #15720021

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33682 #15720021