Subgroups — C₁₃×#23843 #15710182

Order 3328.

646 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2330	C₂	1	2	yes	maximal
2	C₁₃×#2084	C₂	1	2	yes	maximal
3	C₁₃×#2083	C₂	1	2	yes	maximal
4	C₁₃×#2059	C₂	1	2	yes	maximal
5	C₁₃×#2059	C₂	1	2	yes	maximal
6	C₁₃×#1776	C₂	1	2	yes	maximal
7	C₄×C₁₃×#107	C₂	1	1	yes	maximal, characteristic
8	C₄×C₁₃×#106	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#1746	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×#125	C₂²	1	2	yes	—
11	C₂×C₁₃×#107	C₂²	1	2	yes	—
12	C₂×C₁₃×#107	C₂²	1	2	yes	—
13	C₂×C₁₃×#106	C₂²	1	2	yes	—
14	C₂×C₁₃×#106	C₂²	1	2	yes	—
15	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#389	—	2	2	no	—
18	C₁₃×#389	—	2	2	no	—
19	C₁₃×#389	C₂²	1	2	yes	—
20	C₁₃×#389	C₂²	1	2	yes	—
21	C₁₃×#387	—	2	2	no	—
22	C₁₃×#387	—	2	2	no	—
23	C₁₃×#384	C₂²	1	2	yes	—
24	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
25	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
26	C₁₃×#376	C₂²	1	1	yes	characteristic
27	C₁₃×#340	C₂²	1	1	yes	characteristic
28	C₁₃×#340	C₂²	1	1	yes	characteristic
29	C₁₃×#340	C₂²	1	1	yes	characteristic
30	C₁₃×#340	C₂²	1	1	yes	characteristic
31	C₁₃×#340	C₂²	1	2	yes	—
32	C₁₃×#340	C₂²	1	2	yes	—
33	C₁₃×#336	C₂²	1	2	yes	—
34	C₁₃×#336	C₂²	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
38	C₁₃×#125	—	2	4	no	—
39	C₁₃×#125	—	2	4	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
45	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
46	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
49	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃	—	2	2	no	—
51	C₂×C₄²×C₁₃	—	2	2	no	—
52	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
53	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₁₃×#107	—	2	4	no	—
55	C₁₃×#107	—	2	4	no	—
56	C₁₃×#106	—	2	4	no	—
57	C₁₃×#106	—	2	4	no	—
58	C₁₃×#97	Q₈	1	4	yes	—
59	C₄×C₈×C₁₃	Q₈	1	4	yes	—
60	C₁₃×#95	—	2	2	no	—
61	C₁₃×#95	—	2	2	no	—
62	C₁₃×#95	—	2	2	no	—
63	C₁₃×#95	—	2	2	no	—
64	C₁₃×#95	D₄	1	1	yes	characteristic
65	C₁₃×#95	D₄	1	2	yes	—
66	C₁₃×#95	D₄	1	1	yes	characteristic
67	#23843	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂²×C₄×C₁₃	—	2	2	no	—
69	C₂²×C₄×C₁₃	—	2	2	no	—
70	C₂²×C₄×C₁₃	—	2	2	no	—
71	C₂²×C₄×C₁₃	—	2	2	no	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
82	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
83	C₂×C₈×C₁₃	C₂×Q₈	1	4	yes	—
84	C₁₃×C₄⋊C₄	—	2	8	no	—
85	C₁₃×C₄⋊C₄	—	2	8	no	—
86	C₁₃×C₄⋊C₄	—	2	2	no	—
87	C₁₃×C₄⋊C₄	—	2	2	no	—
88	C₁₃×C₄⋊C₄	—	2	2	no	—
89	C₁₃×C₄⋊C₄	—	2	2	no	—
90	C₁₃×C₄⋊C₄	—	2	2	no	—
91	C₁₃×C₄⋊C₄	—	2	2	no	—
92	C₁₃×C₄⋊C₄	—	8	2	no	—
93	C₁₃×C₄⋊C₄	—	8	2	no	—
94	C₁₃×C₄⋊C₄	—	8	2	no	—
95	C₁₃×C₄⋊C₄	—	8	2	no	—
96	C₁₃×C₄⋊C₄	#41	1	4	yes	—
97	C₁₃×C₄⋊C₄	#41	1	4	yes	—
98	C₄²×C₁₃	—	2	2	no	—
99	C₄²×C₁₃	—	2	2	no	—
100	C₄²×C₁₃	—	8	2	no	—
101	C₄²×C₁₃	—	8	2	no	—
102	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
103	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
104	#2330	C₂×C₁₃	1	2	yes	—
105	#2084	C₂×C₁₃	1	2	yes	—
106	#2083	C₂×C₁₃	1	2	yes	—
107	#2059	C₂×C₁₃	1	2	yes	—
108	#2059	C₂×C₁₃	1	2	yes	—
109	#1776	C₂×C₁₃	1	2	yes	—
110	C₄×#107	C₂×C₁₃	1	1	yes	characteristic
111	C₄×#106	C₂×C₁₃	1	1	yes	characteristic
112	#1746	C₂×C₁₃	1	1	yes	characteristic
113	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	4	no	—
120	C₂×C₄×C₁₃	—	2	4	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	4	2	no	—
127	C₂×C₄×C₁₃	—	4	2	no	—
128	C₂×C₄×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	4	2	no	—
132	C₂×C₄×C₁₃	—	4	2	no	—
133	C₂×C₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
136	C₂×C₄×C₁₃	C₄⋊Q₈	1	2	yes	—
137	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
140	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
141	C₈×C₁₃	—	2	2	no	—
142	C₈×C₁₃	—	2	4	no	—
143	C₈×C₁₃	—	2	2	no	—
144	C₂×#125	C₂²×C₁₃	1	2	yes	—
145	C₂×#107	C₂²×C₁₃	1	2	yes	—
146	C₂×#107	C₂²×C₁₃	1	2	yes	—
147	C₂×#106	C₂²×C₁₃	1	2	yes	—
148	C₂×#106	C₂²×C₁₃	1	2	yes	—
149	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
150	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
151	#389	—	2	2	no	—
152	#389	—	2	2	no	—
153	#389	C₂²×C₁₃	1	2	yes	—
154	#389	C₂²×C₁₃	1	2	yes	—
155	#387	—	2	2	no	—
156	#387	—	2	2	no	—
157	#384	C₂²×C₁₃	1	2	yes	—
158	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
159	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
160	#376	C₂²×C₁₃	1	1	yes	characteristic
161	#340	C₂²×C₁₃	1	1	yes	characteristic
162	#340	C₂²×C₁₃	1	1	yes	characteristic
163	#340	C₂²×C₁₃	1	2	yes	—
164	#340	C₂²×C₁₃	1	1	yes	characteristic
165	#340	C₂²×C₁₃	1	2	yes	—
166	#340	C₂²×C₁₃	1	1	yes	characteristic
167	#336	C₂²×C₁₃	1	2	yes	—
168	#336	C₂²×C₁₃	1	2	yes	—
169	C₂²×C₁₃	#525	1	2	yes	—
170	C₂²×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
171	C₂²×C₁₃	#576	1	1	yes	characteristic
172	C₂²×C₁₃	#577	1	2	yes	—
173	C₂²×C₁₃	#578	1	1	yes	characteristic
174	C₄×C₁₃	—	2	2	no	—
175	C₄×C₁₃	—	2	2	no	—
176	C₄×C₁₃	—	4	2	no	—
177	C₄×C₁₃	—	4	4	no	—
178	C₄×C₁₃	—	4	2	no	—
179	C₄×C₁₃	—	4	4	no	—
180	C₄×C₁₃	—	8	2	no	—
181	C₄×C₁₃	—	8	2	no	—
182	C₄×C₁₃	#533	1	1	yes	characteristic
183	C₄×C₁₃	#533	1	2	yes	—
184	C₄×C₁₃	#533	1	1	yes	characteristic
185	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
188	#125	—	2	4	no	—
189	#125	—	2	4	no	—
190	C₂×C₄⋊C₄	—	2	2	no	—
191	C₂×C₄⋊C₄	—	2	2	no	—
192	C₂×C₄⋊C₄	—	2	2	no	—
193	C₂×C₄⋊C₄	—	2	2	no	—
194	C₂×C₄⋊C₄	—	2	2	no	—
195	C₂×C₄⋊C₄	—	2	2	no	—
196	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
197	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
198	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
199	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
200	C₂×C₄²	—	2	2	no	—
201	C₂×C₄²	—	2	2	no	—
202	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
203	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
204	#107	—	2	4	no	—
205	#107	—	2	4	no	—
206	#106	—	2	4	no	—
207	#106	—	2	4	no	—
208	#97	Q₈×C₁₃	1	4	yes	—
209	C₄×C₈	Q₈×C₁₃	1	4	yes	—
210	#95	—	2	2	no	—
211	#95	—	2	2	no	—
212	#95	—	2	2	no	—
213	#95	—	2	2	no	—
214	#95	D₄×C₁₃	1	1	yes	characteristic
215	#95	D₄×C₁₃	1	2	yes	—
216	#95	D₄×C₁₃	1	1	yes	characteristic
217	C₂×C₁₃	#2399	1	1	yes	characteristic
218	C₂×C₁₃	#3119	1	2	yes	—
219	C₂×C₁₃	#3120	1	2	yes	—
220	C₂×C₁₃	#3163	1	1	yes	characteristic
221	C₂×C₁₃	#3164	1	1	yes	characteristic
222	C₂²×C₄	—	2	2	no	—
223	C₂²×C₄	—	2	2	no	—
224	C₂²×C₄	—	2	2	no	—
225	C₂²×C₄	—	2	2	no	—
226	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
227	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
228	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
229	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
230	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	2	no	—
234	C₂×C₈	—	2	2	no	—
235	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
236	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
237	C₂×C₈	C₂×Q₈×C₁₃	1	4	yes	—
238	C₄⋊C₄	—	2	8	no	—
239	C₄⋊C₄	—	2	8	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	2	no	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	2	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	8	2	no	—
247	C₄⋊C₄	—	8	2	no	—
248	C₄⋊C₄	—	8	2	no	—
249	C₄⋊C₄	—	8	2	no	—
250	C₄⋊C₄	C₁₃×#41	1	4	yes	—
251	C₄⋊C₄	C₁₃×#41	1	4	yes	—
252	C₄²	—	2	2	no	—
253	C₄²	—	2	2	no	—
254	C₄²	—	8	2	no	—
255	C₄²	—	8	2	no	—
256	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
257	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
258	C₁₃	#23843	1	1	yes	characteristic, direct product factor, Sylow
259	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	1	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	2	no	—
269	C₂×C₄	—	2	2	no	—
270	C₂×C₄	—	2	2	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	2	no	—
273	C₂×C₄	—	4	2	no	—
274	C₂×C₄	—	4	2	no	—
275	C₂×C₄	—	4	2	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	2	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	C₁₃×C₄⋊Q₈	1	2	yes	—
282	C₂×C₄	C₁₃×C₄⋊Q₈	1	2	yes	—
283	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
284	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
285	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
286	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
287	C₈	—	2	2	no	—
288	C₈	—	2	4	no	—
289	C₈	—	2	2	no	—
290	C₂²	C₁₃×#525	1	2	yes	—
291	C₂²	C₂×C₁₃×C₄⋊Q₈	1	1	yes	characteristic
292	C₂²	C₁₃×#576	1	1	yes	characteristic
293	C₂²	C₁₃×#577	1	2	yes	—
294	C₂²	C₁₃×#578	1	1	yes	characteristic
295	C₄	—	2	2	no	—
296	C₄	—	2	2	no	—
297	C₄	—	4	2	no	—
298	C₄	—	4	4	no	—
299	C₄	—	4	2	no	—
300	C₄	—	4	4	no	—
301	C₄	—	8	2	no	—
302	C₄	—	8	2	no	—
303	C₄	C₁₃×#533	1	1	yes	characteristic
304	C₄	C₁₃×#533	1	2	yes	—
305	C₄	C₁₃×#533	1	1	yes	characteristic
306	C₂	C₁₃×#2399	1	1	yes	characteristic
307	C₂	C₁₃×#3119	1	2	yes	—
308	C₂	C₁₃×#3120	1	2	yes	—
309	C₂	C₁₃×#3163	1	1	yes	characteristic
310	C₂	C₁₃×#3164	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₁₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
67. #23843	C₁₃	direct (N × H)
258. C₁₃	#23843	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×#2330	2
C₁₃×#2084	2
C₁₃×#2083	2
C₁₃×#2059	2
C₁₃×#1776	2
C₄×C₁₃×#107	2
C₄×C₁₃×#106	2
C₁₃×#1746	2
#23843	13

Subgroups — C13×#23843 #15710182

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#23843 #15710182