Subgroups — C₁₃×#33739 #15720078

Order 3328.

508 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3175	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2249	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2243	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2220	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2216	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2215	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#2175	C₂	1	2	yes	maximal, complemented
9	C₄×C₁₃×Q₃₂	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×#542	—	2	2	no	—
13	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	—
15	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
17	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
18	C₁₃×#500	C₂²	1	2	yes	—
19	C₁₃×#495	C₂²	1	2	yes	—
20	C₁₃×#489	—	2	2	no	—
21	C₁₃×#468	—	2	2	no	—
22	C₁₃×#467	C₂²	1	2	yes	—
23	C₁₃×#466	C₂²	1	2	yes	—
24	C₁₃×#440	—	2	2	no	—
25	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
26	C₄×C₁₃×D₈	C₂²	1	2	yes	—
27	C₁₃×#405	C₂²	1	1	yes	characteristic
28	C₁₃×#366	C₂²	1	1	yes	characteristic
29	C₁₃×#366	C₂²	1	1	yes	characteristic
30	C₁₃×#366	C₂²	1	2	yes	—
31	C₁₃×#358	C₂²	1	2	yes	—
32	C₁₃×#358	C₂²	1	2	yes	—
33	C₁₃×#357	C₂²	1	2	yes	—
34	C₁₃×#357	C₂²	1	2	yes	—
35	C₁₃×#346	C₂²	1	1	yes	characteristic
36	C₁₃×#346	C₂²	1	1	yes	characteristic
37	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
38	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
39	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
40	C₂×C₁₃×SD₁₆	—	2	2	no	—
41	C₂×C₁₃×D₈	C₂³	1	2	yes	—
42	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×C₄⋊Q₈	—	2	2	no	—
45	C₁₃×#126	—	4	2	no	—
46	C₁₃×#124	—	2	2	no	—
47	C₁₃×#124	—	4	2	no	—
48	C₁₃×#123	—	4	2	no	—
49	C₁₃×C₂²⋊Q₈	—	2	2	no	—
50	C₁₃×C₂²⋊Q₈	—	4	2	no	—
51	C₁₃×#121	—	2	2	no	—
52	C₄×Q₈×C₁₃	—	2	2	no	—
53	C₄×D₄×C₁₃	—	2	2	no	—
54	C₁₃×#117	D₄	1	1	yes	characteristic
55	C₁₃×Q₃₂	—	2	2	no	—
56	C₁₃×SD₃₂	—	2	4	no	—
57	C₁₃×D₁₆	—	2	2	no	—
58	C₁₃×M₃₂	—	2	2	no	—
59	C₂×C₁₃×C₁₆	—	2	2	no	—
60	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
61	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
62	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
63	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
64	C₁₃×#107	—	2	2	no	—
65	C₁₃×#107	C₂³	1	2	yes	—
66	C₁₃×#107	C₂³	1	2	yes	—
67	C₁₃×#103	—	2	2	no	—
68	C₁₃×#103	—	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	#33739	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂×Q₈×C₁₃	—	2	2	no	—
77	C₂×Q₈×C₁₃	—	4	2	no	—
78	C₂×D₄×C₁₃	—	2	2	no	—
79	C₂×D₄×C₁₃	—	4	2	no	—
80	C₂²×C₄×C₁₃	—	4	2	no	—
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₁₃×Q₁₆	—	2	2	no	—
83	C₁₃×Q₁₆	—	2	2	no	—
84	C₁₃×SD₁₆	—	4	4	no	—
85	C₁₃×D₈	—	2	2	no	—
86	C₁₃×D₈	—	2	2	no	—
87	C₁₃×M₁₆	—	2	2	no	—
88	C₂×C₈×C₁₃	—	2	1	no	—
89	C₂×C₈×C₁₃	—	2	1	no	—
90	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₁₃×C₄⋊C₄	—	2	2	no	—
95	C₁₃×C₄⋊C₄	—	2	2	no	—
96	C₁₃×C₄⋊C₄	—	4	2	no	—
97	C₁₃×C₄⋊C₄	—	4	2	no	—
98	C₁₃×C₄⋊C₄	—	4	2	no	—
99	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
100	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
101	C₁₃×C₂²⋊C₄	—	4	2	no	—
102	C₁₃×C₂²⋊C₄	—	4	2	no	—
103	C₁₃×C₂²⋊C₄	—	4	2	no	—
104	C₁₃×C₂²⋊C₄	—	4	2	no	—
105	C₁₃×C₂²⋊C₄	—	4	2	no	—
106	C₁₃×C₂²⋊C₄	D₈	1	1	yes	characteristic
107	C₁₃×C₂²⋊C₄	D₈	1	1	yes	characteristic
108	C₄²×C₁₃	—	4	2	no	—
109	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₁₃×C₁₆	—	2	1	no	—
112	C₁₃×C₁₆	—	2	1	no	—
113	C₁₃×C₁₆	#41	1	2	yes	—
114	C₁₃×C₁₆	#41	1	2	yes	—
115	#3175	C₂×C₁₃	1	2	yes	complemented
116	#2252	C₂×C₁₃	1	1	yes	characteristic, complemented
117	#2249	C₂×C₁₃	1	1	yes	characteristic, complemented
118	#2243	C₂×C₁₃	1	2	yes	complemented
119	#2220	C₂×C₁₃	1	2	yes	complemented
120	#2216	C₂×C₁₃	1	1	yes	characteristic, complemented
121	#2215	C₂×C₁₃	1	1	yes	characteristic
122	#2175	C₂×C₁₃	1	2	yes	complemented
123	C₄×Q₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
124	C₄×D₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
125	#2108	C₂×C₁₃	1	1	yes	characteristic, complemented
126	C₂³×C₁₃	—	4	2	no	—
127	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
128	Q₈×C₁₃	—	4	2	no	—
129	Q₈×C₁₃	—	4	2	no	—
130	Q₈×C₁₃	—	16	2	no	—
131	D₄×C₁₃	—	4	2	no	—
132	D₄×C₁₃	—	4	2	no	—
133	D₄×C₁₃	—	16	2	no	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	2	1	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	4	2	no	—
139	C₂×C₄×C₁₃	—	16	2	no	—
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	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
146	C₈×C₁₃	—	2	1	no	—
147	C₈×C₁₃	—	2	1	no	—
148	C₈×C₁₃	#135	1	2	yes	—
149	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
150	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
151	#542	—	2	2	no	—
152	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
153	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
154	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic
155	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
156	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
157	#500	C₂²×C₁₃	1	2	yes	—
158	#495	C₂²×C₁₃	1	2	yes	—
159	#489	—	2	2	no	—
160	#468	—	2	2	no	—
161	#467	C₂²×C₁₃	1	2	yes	—
162	#466	C₂²×C₁₃	1	2	yes	—
163	#440	—	2	2	no	—
164	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
165	C₄×D₈	C₂²×C₁₃	1	2	yes	—
166	#405	C₂²×C₁₃	1	1	yes	characteristic
167	#366	C₂²×C₁₃	1	1	yes	characteristic
168	#366	C₂²×C₁₃	1	1	yes	characteristic
169	#366	C₂²×C₁₃	1	2	yes	—
170	#358	C₂²×C₁₃	1	2	yes	—
171	#358	C₂²×C₁₃	1	2	yes	—
172	#357	C₂²×C₁₃	1	2	yes	—
173	#357	C₂²×C₁₃	1	2	yes	—
174	#346	C₂²×C₁₃	1	1	yes	characteristic
175	#346	C₂²×C₁₃	1	1	yes	characteristic
176	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
177	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
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	C₂²×D₈	1	1	yes	characteristic
185	C₄×C₁₃	—	2	1	no	—
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	#532	1	1	yes	characteristic
194	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
195	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
196	C₂×SD₁₆	—	2	2	no	—
197	C₂×D₈	C₂³×C₁₃	1	2	yes	—
198	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
199	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
200	C₄⋊Q₈	—	2	2	no	—
201	#126	—	4	2	no	—
202	#124	—	2	2	no	—
203	#124	—	4	2	no	—
204	#123	—	4	2	no	—
205	C₂²⋊Q₈	—	2	2	no	—
206	C₂²⋊Q₈	—	4	2	no	—
207	#121	—	2	2	no	—
208	C₄×Q₈	—	2	2	no	—
209	C₄×D₄	—	2	2	no	—
210	#117	D₄×C₁₃	1	1	yes	characteristic
211	Q₃₂	—	2	2	no	—
212	SD₃₂	—	2	4	no	—
213	D₁₆	—	2	2	no	—
214	M₃₂	—	2	2	no	—
215	C₂×C₁₆	—	2	2	no	—
216	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
217	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
218	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
219	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
220	#107	—	2	2	no	—
221	#107	C₂³×C₁₃	1	2	yes	—
222	#107	C₂³×C₁₃	1	2	yes	—
223	#103	—	2	2	no	—
224	#103	—	2	2	no	—
225	#102	—	2	2	no	—
226	#102	—	2	2	no	—
227	#97	D₄×C₁₃	1	1	yes	characteristic
228	#97	D₄×C₁₃	1	1	yes	characteristic
229	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
230	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
231	C₂×C₁₃	—	4	1	no	—
232	C₂×C₁₃	—	16	2	no	—
233	C₂×C₁₃	#3169	1	1	yes	characteristic
234	C₂×C₁₃	#3415	1	1	yes	characteristic
235	C₂×C₁₃	#3417	1	1	yes	characteristic
236	C₂×Q₈	—	2	2	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂×D₄	—	2	2	no	—
239	C₂×D₄	—	4	2	no	—
240	C₂²×C₄	—	4	2	no	—
241	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
242	Q₁₆	—	2	2	no	—
243	Q₁₆	—	2	2	no	—
244	SD₁₆	—	4	4	no	—
245	D₈	—	2	2	no	—
246	D₈	—	2	2	no	—
247	M₁₆	—	2	2	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
251	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
252	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
253	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
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	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
261	C₂²⋊C₄	—	4	2	no	—
262	C₂²⋊C₄	—	4	2	no	—
263	C₂²⋊C₄	—	4	2	no	—
264	C₂²⋊C₄	—	4	2	no	—
265	C₂²⋊C₄	—	4	2	no	—
266	C₂²⋊C₄	C₁₃×D₈	1	1	yes	characteristic
267	C₂²⋊C₄	C₁₃×D₈	1	1	yes	characteristic
268	C₄²	—	4	2	no	—
269	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
270	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
271	C₁₆	—	2	1	no	—
272	C₁₆	—	2	1	no	—
273	C₁₆	C₁₃×#41	1	2	yes	—
274	C₁₆	C₁₃×#41	1	2	yes	—
275	C₁₃	#33739	1	1	yes	characteristic, direct product factor, Sylow
276	C₂³	—	4	2	no	—
277	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
278	Q₈	—	4	2	no	—
279	Q₈	—	4	2	no	—
280	Q₈	—	16	2	no	—
281	D₄	—	4	2	no	—
282	D₄	—	4	2	no	—
283	D₄	—	16	2	no	—
284	C₂×C₄	—	2	1	no	—
285	C₂×C₄	—	2	1	no	—
286	C₂×C₄	—	4	2	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	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	C₂×C₁₃×D₈	1	1	yes	characteristic
295	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
296	C₈	—	2	1	no	—
297	C₈	—	2	1	no	—
298	C₈	C₁₃×#135	1	2	yes	—
299	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
300	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
301	C₂²	—	2	1	no	—
302	C₂²	—	2	1	no	—
303	C₂²	—	2	1	no	—
304	C₂²	—	8	2	no	—
305	C₂²	—	8	2	no	—
306	C₂²	—	8	2	no	—
307	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
308	C₄	—	2	1	no	—
309	C₄	—	2	1	no	—
310	C₄	—	2	1	no	—
311	C₄	—	2	1	no	—
312	C₄	—	2	1	no	—
313	C₄	—	8	2	no	—
314	C₄	—	8	2	no	—
315	C₄	—	8	2	no	—
316	C₄	C₁₃×#532	1	1	yes	characteristic
317	C₄	C₂×C₁₃×#135	1	1	yes	characteristic
318	C₂	—	4	1	no	—
319	C₂	—	16	2	no	—
320	C₂	C₁₃×#3169	1	1	yes	characteristic
321	C₂	C₁₃×#3415	1	1	yes	characteristic
322	C₂	C₁₃×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₁₃×#2215	C₂
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×#3175	C₂	semidirect
2. C₁₃×#2252	C₂	semidirect
3. C₁₃×#2249	C₂	semidirect
4. C₁₃×#2243	C₂	semidirect
5. C₁₃×#2220	C₂	semidirect
6. C₁₃×#2216	C₂	semidirect
8. C₁₃×#2175	C₂	semidirect
9. C₄×C₁₃×Q₃₂	C₂	semidirect
10. C₄×C₁₃×D₁₆	C₂	semidirect
11. C₁₃×#2108	C₂	semidirect
75. #33739	C₁₃	direct (N × H)
115. #3175	C₂×C₁₃	semidirect
116. #2252	C₂×C₁₃	semidirect
117. #2249	C₂×C₁₃	semidirect
118. #2243	C₂×C₁₃	semidirect
119. #2220	C₂×C₁₃	semidirect
120. #2216	C₂×C₁₃	semidirect
122. #2175	C₂×C₁₃	semidirect
123. C₄×Q₃₂	C₂×C₁₃	semidirect
124. C₄×D₁₆	C₂×C₁₃	semidirect
125. #2108	C₂×C₁₃	semidirect
275. C₁₃	#33739	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×#3175	2
C₁₃×#2252	2
C₁₃×#2249	2
C₁₃×#2243	2
C₁₃×#2220	2
C₁₃×#2216	2
C₁₃×#2215	2
C₁₃×#2175	2
C₄×C₁₃×Q₃₂	2
C₄×C₁₃×D₁₆	2
C₁₃×#2108	2
#33739	13

Subgroups — C13×#33739 #15720078

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33739 #15720078