Subgroups — C₁₃×#33742 #15720081

Order 3328.

508 conjugacy classes of subgroups.

The subgroup lattice has 308 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×#2254	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×#2219	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#2218	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#2217	C₂	1	2	yes	maximal, complemented
8	C₁₃×#2177	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#2176	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₁₃×SD₃₂	C₂	1	2	yes	maximal, 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×#367	C₂²	1	2	yes	—
29	C₁₃×#367	C₂²	1	1	yes	characteristic
30	C₁₃×#367	C₂²	1	1	yes	characteristic
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	#33742	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	2	yes	—
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	—	4	2	no	—
104	C₁₃×C₂²⋊C₄	—	4	2	no	—
105	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
106	C₄²×C₁₃	—	4	2	no	—
107	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₁₃×C₁₆	—	2	1	no	—
110	C₁₃×C₁₆	—	2	1	no	—
111	C₁₃×C₁₆	#41	1	2	yes	—
112	C₁₃×C₁₆	#41	1	2	yes	—
113	#3175	C₂×C₁₃	1	2	yes	complemented
114	#2254	C₂×C₁₃	1	1	yes	characteristic, complemented
115	#2249	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#2243	C₂×C₁₃	1	2	yes	complemented
117	#2219	C₂×C₁₃	1	1	yes	characteristic, complemented
118	#2218	C₂×C₁₃	1	1	yes	characteristic
119	#2217	C₂×C₁₃	1	2	yes	complemented
120	#2177	C₂×C₁₃	1	1	yes	characteristic, complemented
121	#2176	C₂×C₁₃	1	1	yes	characteristic, complemented
122	C₄×SD₃₂	C₂×C₁₃	1	2	yes	complemented
123	#2108	C₂×C₁₃	1	1	yes	characteristic, complemented
124	C₂³×C₁₃	—	4	2	no	—
125	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
126	Q₈×C₁₃	—	4	2	no	—
127	Q₈×C₁₃	—	4	2	no	—
128	Q₈×C₁₃	—	16	2	no	—
129	D₄×C₁₃	—	4	2	no	—
130	D₄×C₁₃	—	4	2	no	—
131	D₄×C₁₃	—	16	2	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	C₂×D₈	1	2	yes	—
139	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
140	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
142	C₈×C₁₃	—	2	1	no	—
143	C₈×C₁₃	—	2	1	no	—
144	C₈×C₁₃	#135	1	2	yes	—
145	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
146	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
147	#542	—	2	2	no	—
148	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic
149	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	—
150	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic
151	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
152	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
153	#500	C₂²×C₁₃	1	2	yes	—
154	#495	C₂²×C₁₃	1	2	yes	—
155	#489	—	2	2	no	—
156	#468	—	2	2	no	—
157	#467	C₂²×C₁₃	1	2	yes	—
158	#466	C₂²×C₁₃	1	2	yes	—
159	#440	—	2	2	no	—
160	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
161	C₄×D₈	C₂²×C₁₃	1	2	yes	—
162	#405	C₂²×C₁₃	1	1	yes	characteristic
163	#367	C₂²×C₁₃	1	2	yes	—
164	#367	C₂²×C₁₃	1	1	yes	characteristic
165	#367	C₂²×C₁₃	1	1	yes	characteristic
166	#358	C₂²×C₁₃	1	2	yes	—
167	#358	C₂²×C₁₃	1	2	yes	—
168	#357	C₂²×C₁₃	1	2	yes	—
169	#357	C₂²×C₁₃	1	2	yes	—
170	#346	C₂²×C₁₃	1	1	yes	characteristic
171	#346	C₂²×C₁₃	1	1	yes	characteristic
172	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
173	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
174	C₂²×C₁₃	—	2	2	no	—
175	C₂²×C₁₃	—	2	1	no	—
176	C₂²×C₁₃	—	8	2	no	—
177	C₂²×C₁₃	—	8	2	no	—
178	C₂²×C₁₃	—	8	2	no	—
179	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
180	C₄×C₁₃	—	2	2	no	—
181	C₄×C₁₃	—	2	2	no	—
182	C₄×C₁₃	—	2	1	no	—
183	C₄×C₁₃	—	8	2	no	—
184	C₄×C₁₃	—	8	2	no	—
185	C₄×C₁₃	—	8	2	no	—
186	C₄×C₁₃	#532	1	1	yes	characteristic
187	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
188	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
189	C₂×SD₁₆	—	2	2	no	—
190	C₂×D₈	C₂³×C₁₃	1	2	yes	—
191	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
192	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
193	C₄⋊Q₈	—	2	2	no	—
194	#126	—	4	2	no	—
195	#124	—	2	2	no	—
196	#124	—	4	2	no	—
197	#123	—	4	2	no	—
198	C₂²⋊Q₈	—	2	2	no	—
199	C₂²⋊Q₈	—	4	2	no	—
200	#121	—	2	2	no	—
201	C₄×Q₈	—	2	2	no	—
202	C₄×D₄	—	2	2	no	—
203	#117	D₄×C₁₃	1	1	yes	characteristic
204	Q₃₂	—	2	2	no	—
205	SD₃₂	—	2	4	no	—
206	D₁₆	—	2	2	no	—
207	M₃₂	—	2	2	no	—
208	C₂×C₁₆	—	2	2	no	—
209	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
210	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
211	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
212	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
213	#107	—	2	2	no	—
214	#107	C₂³×C₁₃	1	2	yes	—
215	#107	C₂³×C₁₃	1	2	yes	—
216	#103	—	2	2	no	—
217	#103	—	2	2	no	—
218	#102	—	2	2	no	—
219	#102	—	2	2	no	—
220	#97	D₄×C₁₃	1	1	yes	characteristic
221	#97	D₄×C₁₃	1	1	yes	characteristic
222	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
223	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
224	C₂×C₁₃	—	4	1	no	—
225	C₂×C₁₃	—	16	2	no	—
226	C₂×C₁₃	#3169	1	1	yes	characteristic
227	C₂×C₁₃	#3416	1	2	yes	—
228	C₂×Q₈	—	2	2	no	—
229	C₂×Q₈	—	4	2	no	—
230	C₂×D₄	—	2	2	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂²×C₄	—	4	2	no	—
233	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
234	Q₁₆	—	2	2	no	—
235	Q₁₆	—	2	2	no	—
236	SD₁₆	—	4	4	no	—
237	D₈	—	2	2	no	—
238	D₈	—	2	2	no	—
239	M₁₆	—	2	2	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	1	no	—
242	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
243	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
244	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
245	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
246	C₄⋊C₄	—	2	2	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	4	2	no	—
249	C₄⋊C₄	—	4	2	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
252	C₂²⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	4	2	no	—
254	C₂²⋊C₄	—	4	2	no	—
255	C₂²⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	2	no	—
257	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
258	C₄²	—	4	2	no	—
259	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
260	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
261	C₁₆	—	2	1	no	—
262	C₁₆	—	2	1	no	—
263	C₁₆	C₁₃×#41	1	2	yes	—
264	C₁₆	C₁₃×#41	1	2	yes	—
265	C₁₃	#33742	1	1	yes	characteristic, direct product factor, Sylow
266	C₂³	—	4	2	no	—
267	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
268	Q₈	—	4	2	no	—
269	Q₈	—	4	2	no	—
270	Q₈	—	16	2	no	—
271	D₄	—	4	2	no	—
272	D₄	—	4	2	no	—
273	D₄	—	16	2	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	16	2	no	—
280	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
281	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
282	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
283	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
284	C₈	—	2	1	no	—
285	C₈	—	2	1	no	—
286	C₈	C₁₃×#135	1	2	yes	—
287	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
288	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
289	C₂²	—	2	2	no	—
290	C₂²	—	2	1	no	—
291	C₂²	—	8	2	no	—
292	C₂²	—	8	2	no	—
293	C₂²	—	8	2	no	—
294	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	2	1	no	—
298	C₄	—	8	2	no	—
299	C₄	—	8	2	no	—
300	C₄	—	8	2	no	—
301	C₄	C₁₃×#532	1	1	yes	characteristic
302	C₄	C₂×C₁₃×#135	1	1	yes	characteristic
303	C₂	—	4	1	no	—
304	C₂	—	16	2	no	—
305	C₂	C₁₃×#3169	1	1	yes	characteristic
306	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×#2218	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×#2254	C₂	semidirect
3. C₁₃×#2249	C₂	semidirect
4. C₁₃×#2243	C₂	semidirect
5. C₁₃×#2219	C₂	semidirect
7. C₁₃×#2217	C₂	semidirect
8. C₁₃×#2177	C₂	semidirect
9. C₁₃×#2176	C₂	semidirect
10. C₄×C₁₃×SD₃₂	C₂	semidirect
11. C₁₃×#2108	C₂	semidirect
75. #33742	C₁₃	direct (N × H)
113. #3175	C₂×C₁₃	semidirect
114. #2254	C₂×C₁₃	semidirect
115. #2249	C₂×C₁₃	semidirect
116. #2243	C₂×C₁₃	semidirect
117. #2219	C₂×C₁₃	semidirect
119. #2217	C₂×C₁₃	semidirect
120. #2177	C₂×C₁₃	semidirect
121. #2176	C₂×C₁₃	semidirect
122. C₄×SD₃₂	C₂×C₁₃	semidirect
123. #2108	C₂×C₁₃	semidirect
265. C₁₃	#33742	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×#2254	2
C₁₃×#2249	2
C₁₃×#2243	2
C₁₃×#2219	2
C₁₃×#2218	2
C₁₃×#2217	2
C₁₃×#2177	2
C₁₃×#2176	2
C₄×C₁₃×SD₃₂	2
C₁₃×#2108	2
#33742	13

Subgroups — C13×#33742 #15720081

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33742 #15720081