Subgroups — C₁₃×#18698 #15705037

Order 3328.

834 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×#3556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#3254	C₂	1	2	yes	maximal
3	C₁₃×#3131	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1636	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1626	C₂	1	4	yes	maximal, complemented
6	C₁₃×#1561	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1493	C₂	1	2	yes	maximal, complemented
8	C₁₃×#564	—	2	2	no	—
9	C₁₃×#557	—	2	2	no	—
10	C₁₃×#557	—	2	2	no	—
11	C₁₃×#557	C₂²	1	4	yes	—
12	C₁₃×#541	—	2	4	no	—
13	C₁₃×#541	—	2	2	no	—
14	C₁₃×#533	C₂²	1	2	yes	—
15	C₁₃×#533	C₂²	1	1	yes	characteristic
16	C₁₃×#532	C₂²	1	1	yes	characteristic
17	C₁₃×#488	—	2	2	no	—
18	C₁₃×#487	C₂²	1	2	yes	—
19	C₁₃×#486	C₂²	1	2	yes	—
20	C₁₃×#477	C₂²	1	4	yes	—
21	C₁₃×#474	—	2	2	no	—
22	C₁₃×#468	—	2	4	no	—
23	C₁₃×#461	—	2	2	no	—
24	C₁₃×#460	C₂²	1	4	yes	—
25	C₁₃×#423	C₂²	1	2	yes	—
26	C₁₃×#416	—	2	2	no	—
27	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₁₃×#332	C₂²	1	4	yes	—
29	C₁₃×Q₈⋊C₈	C₂²	1	8	yes	—
30	C₂×C₁₃×#41	—	2	1	no	—
31	C₂×C₁₃×SD₁₆	—	2	4	no	—
32	C₂×C₁₃×M₁₆	—	2	2	no	—
33	C₂²×C₈×C₁₃	—	2	1	no	—
34	C₁₃×C₄⋊Q₈	—	2	2	no	—
35	C₁₃×C₄⋊Q₈	—	2	4	no	—
36	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
37	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
38	C₁₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
39	C₁₃×C₄⋊Q₈	C₂³	1	2	yes	—
40	C₁₃×#127	C₂³	1	1	yes	characteristic
41	C₁₃×#126	—	2	8	no	—
42	C₁₃×#125	—	2	2	no	—
43	C₁₃×#125	—	2	8	no	—
44	C₁₃×#125	—	2	8	no	—
45	C₁₃×#125	—	2	4	no	—
46	C₁₃×#125	—	2	8	no	—
47	C₁₃×#125	—	2	2	no	—
48	C₁₃×#125	D₄	1	2	yes	—
49	C₁₃×#124	—	2	1	no	—
50	C₁₃×C₂²⋊Q₈	—	2	2	no	—
51	C₁₃×C₂²⋊Q₈	—	2	2	no	—
52	C₁₃×C₂²⋊Q₈	D₄	1	4	yes	—
53	C₁₃×#121	—	2	2	no	—
54	C₄×Q₈×C₁₃	—	2	4	no	—
55	C₄×Q₈×C₁₃	—	2	4	no	—
56	C₄×Q₈×C₁₃	—	2	4	no	—
57	C₄×Q₈×C₁₃	—	2	2	no	—
58	C₄×Q₈×C₁₃	—	2	2	no	—
59	C₄×Q₈×C₁₃	C₂³	1	4	yes	—
60	C₄×D₄×C₁₃	—	2	2	no	—
61	C₁₃×#117	—	2	2	no	—
62	C₁₃×#117	D₄	1	2	yes	—
63	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
64	C₁₃×#107	—	2	4	no	—
65	C₁₃×C₄⋊C₈	—	2	2	no	—
66	C₁₃×C₄⋊C₈	C₂³	1	4	yes	—
67	C₁₃×#103	—	2	4	no	—
68	C₁₃×#103	—	2	4	no	—
69	C₁₃×#102	—	2	4	no	—
70	C₁₃×#102	—	2	4	no	—
71	C₄×C₈×C₁₃	—	2	1	no	—
72	C₄×C₈×C₁₃	C₂³	1	2	yes	—
73	#18698	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₁₃×#41	—	4	4	no	—
75	C₂×Q₈×C₁₃	—	2	2	no	—
76	C₂×Q₈×C₁₃	—	2	2	no	—
77	C₂×Q₈×C₁₃	—	2	1	no	—
78	C₂×Q₈×C₁₃	C₂×D₄	1	4	yes	—
79	C₂×D₄×C₁₃	—	2	1	no	—
80	C₂×D₄×C₁₃	—	2	2	no	—
81	C₂×D₄×C₁₃	—	4	1	no	—
82	C₂²×C₄×C₁₃	—	4	1	no	—
83	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₁₃×SD₁₆	—	8	4	no	—
86	C₁₃×M₁₆	—	8	2	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	—	2	4	no	—
89	C₂×C₈×C₁₃	—	4	2	no	—
90	C₁₃×C₄⋊C₄	—	2	8	no	—
91	C₁₃×C₄⋊C₄	—	2	8	no	—
92	C₁₃×C₄⋊C₄	—	2	8	no	—
93	C₁₃×C₄⋊C₄	—	2	2	no	—
94	C₁₃×C₄⋊C₄	—	2	2	no	—
95	C₁₃×C₄⋊C₄	—	2	2	no	—
96	C₁₃×C₄⋊C₄	—	2	4	no	—
97	C₁₃×C₄⋊C₄	—	2	4	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	2	4	no	—
100	C₁₃×C₄⋊C₄	—	2	2	no	—
101	C₁₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
102	C₁₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
103	C₁₃×C₂²⋊C₄	—	2	4	no	—
104	C₁₃×C₂²⋊C₄	—	2	2	no	—
105	C₁₃×C₂²⋊C₄	—	4	2	no	—
106	C₄²×C₁₃	—	2	8	no	—
107	C₄²×C₁₃	—	2	2	no	—
108	C₄²×C₁₃	—	2	4	no	—
109	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₄²×C₁₃	C₂×D₄	1	2	yes	—
111	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
112	#3556	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#3254	C₂×C₁₃	1	2	yes	—
114	#3131	C₂×C₁₃	1	1	yes	characteristic
115	#1636	C₂×C₁₃	1	4	yes	complemented
116	#1626	C₂×C₁₃	1	4	yes	complemented
117	#1561	C₂×C₁₃	1	1	yes	characteristic, complemented
118	#1493	C₂×C₁₃	1	2	yes	complemented
119	C₂³×C₁₃	—	4	1	no	—
120	C₂³×C₁₃	#120	1	1	yes	characteristic
121	Q₈×C₁₃	—	2	4	no	—
122	Q₈×C₁₃	—	2	4	no	—
123	Q₈×C₁₃	—	4	2	no	—
124	Q₈×C₁₃	—	4	2	no	—
125	Q₈×C₁₃	—	4	2	no	—
126	Q₈×C₁₃	—	4	4	no	—
127	D₄×C₁₃	—	4	2	no	—
128	D₄×C₁₃	—	4	2	no	—
129	D₄×C₁₃	—	4	2	no	—
130	D₄×C₁₃	—	8	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	2	4	no	—
135	C₂×C₄×C₁₃	—	2	4	no	—
136	C₂×C₄×C₁₃	—	2	4	no	—
137	C₂×C₄×C₁₃	—	4	1	no	—
138	C₂×C₄×C₁₃	—	8	2	no	—
139	C₂×C₄×C₁₃	#120	1	2	yes	—
140	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
143	C₈×C₁₃	—	4	2	no	—
144	C₈×C₁₃	—	4	4	no	—
145	#564	—	2	2	no	—
146	#557	—	2	2	no	—
147	#557	—	2	2	no	—
148	#557	C₂²×C₁₃	1	4	yes	—
149	#541	—	2	4	no	—
150	#541	—	2	2	no	—
151	#533	C₂²×C₁₃	1	2	yes	—
152	#533	C₂²×C₁₃	1	1	yes	characteristic
153	#532	C₂²×C₁₃	1	1	yes	characteristic
154	#488	—	2	2	no	—
155	#487	C₂²×C₁₃	1	2	yes	—
156	#486	C₂²×C₁₃	1	2	yes	—
157	#477	C₂²×C₁₃	1	4	yes	—
158	#474	—	2	2	no	—
159	#468	—	2	4	no	—
160	#461	—	2	2	no	—
161	#460	C₂²×C₁₃	1	4	yes	—
162	#423	C₂²×C₁₃	1	2	yes	—
163	#416	—	2	2	no	—
164	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
165	#332	C₂²×C₁₃	1	4	yes	—
166	Q₈⋊C₈	C₂²×C₁₃	1	8	yes	—
167	C₂²×C₁₃	—	2	2	no	—
168	C₂²×C₁₃	—	2	1	no	—
169	C₂²×C₁₃	—	8	2	no	—
170	C₂²×C₁₃	—	8	1	no	—
171	C₂²×C₁₃	C₂×#120	1	1	yes	characteristic
172	C₄×C₁₃	—	2	2	no	—
173	C₄×C₁₃	—	2	1	no	—
174	C₄×C₁₃	—	4	4	no	—
175	C₄×C₁₃	—	4	4	no	—
176	C₄×C₁₃	—	4	4	no	—
177	C₄×C₁₃	—	8	1	no	—
178	C₄×C₁₃	#454	1	2	yes	—
179	C₄×C₁₃	#577	1	2	yes	—
180	C₄×C₁₃	#578	1	2	yes	—
181	C₂×#41	—	2	1	no	—
182	C₂×SD₁₆	—	2	4	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂²×C₈	—	2	1	no	—
185	C₄⋊Q₈	—	2	2	no	—
186	C₄⋊Q₈	—	2	4	no	—
187	C₄⋊Q₈	D₄×C₁₃	1	2	yes	—
188	C₄⋊Q₈	D₄×C₁₃	1	2	yes	—
189	C₄⋊Q₈	C₂³×C₁₃	1	1	yes	characteristic
190	C₄⋊Q₈	C₂³×C₁₃	1	2	yes	—
191	#127	C₂³×C₁₃	1	1	yes	characteristic
192	#126	—	2	8	no	—
193	#125	—	2	2	no	—
194	#125	—	2	8	no	—
195	#125	—	2	8	no	—
196	#125	—	2	4	no	—
197	#125	—	2	8	no	—
198	#125	—	2	2	no	—
199	#125	D₄×C₁₃	1	2	yes	—
200	#124	—	2	1	no	—
201	C₂²⋊Q₈	—	2	2	no	—
202	C₂²⋊Q₈	—	2	2	no	—
203	C₂²⋊Q₈	D₄×C₁₃	1	4	yes	—
204	#121	—	2	2	no	—
205	C₄×Q₈	—	2	4	no	—
206	C₄×Q₈	—	2	4	no	—
207	C₄×Q₈	—	2	4	no	—
208	C₄×Q₈	—	2	2	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	C₂³×C₁₃	1	4	yes	—
211	C₄×D₄	—	2	2	no	—
212	#117	—	2	2	no	—
213	#117	D₄×C₁₃	1	2	yes	—
214	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
215	#107	—	2	4	no	—
216	C₄⋊C₈	—	2	2	no	—
217	C₄⋊C₈	C₂³×C₁₃	1	4	yes	—
218	#103	—	2	4	no	—
219	#103	—	2	4	no	—
220	#102	—	2	4	no	—
221	#102	—	2	4	no	—
222	C₄×C₈	—	2	1	no	—
223	C₄×C₈	C₂³×C₁₃	1	2	yes	—
224	C₂×C₁₃	—	4	1	no	—
225	C₂×C₁₃	—	16	1	no	—
226	C₂×C₁₃	#3013	1	2	yes	—
227	C₂×C₁₃	C₂×#454	1	1	yes	characteristic
228	#41	—	4	4	no	—
229	C₂×Q₈	—	2	2	no	—
230	C₂×Q₈	—	2	2	no	—
231	C₂×Q₈	—	2	1	no	—
232	C₂×Q₈	C₂×D₄×C₁₃	1	4	yes	—
233	C₂×D₄	—	2	1	no	—
234	C₂×D₄	—	2	2	no	—
235	C₂×D₄	—	4	1	no	—
236	C₂²×C₄	—	4	1	no	—
237	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
238	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
239	SD₁₆	—	8	4	no	—
240	M₁₆	—	8	2	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	4	2	no	—
244	C₄⋊C₄	—	2	8	no	—
245	C₄⋊C₄	—	2	8	no	—
246	C₄⋊C₄	—	2	8	no	—
247	C₄⋊C₄	—	2	2	no	—
248	C₄⋊C₄	—	2	2	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	—	2	2	no	—
253	C₄⋊C₄	—	2	4	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	C₂×D₄×C₁₃	1	4	yes	—
256	C₄⋊C₄	C₂×D₄×C₁₃	1	4	yes	—
257	C₂²⋊C₄	—	2	4	no	—
258	C₂²⋊C₄	—	2	2	no	—
259	C₂²⋊C₄	—	4	2	no	—
260	C₄²	—	2	8	no	—
261	C₄²	—	2	2	no	—
262	C₄²	—	2	4	no	—
263	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
264	C₄²	C₂×D₄×C₁₃	1	2	yes	—
265	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
266	C₁₃	#18698	1	1	yes	characteristic, direct product factor, Sylow
267	C₂³	—	4	1	no	—
268	C₂³	C₁₃×#120	1	1	yes	characteristic
269	Q₈	—	2	4	no	—
270	Q₈	—	2	4	no	—
271	Q₈	—	4	2	no	—
272	Q₈	—	4	4	no	—
273	Q₈	—	4	2	no	—
274	Q₈	—	4	2	no	—
275	D₄	—	4	2	no	—
276	D₄	—	4	2	no	—
277	D₄	—	4	2	no	—
278	D₄	—	8	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	4	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	4	1	no	—
286	C₂×C₄	—	8	2	no	—
287	C₂×C₄	C₁₃×#120	1	2	yes	—
288	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
289	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
290	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
291	C₈	—	4	2	no	—
292	C₈	—	4	4	no	—
293	C₂²	—	2	2	no	—
294	C₂²	—	2	1	no	—
295	C₂²	—	8	2	no	—
296	C₂²	—	8	1	no	—
297	C₂²	C₂×C₁₃×#120	1	1	yes	characteristic
298	C₄	—	2	2	no	—
299	C₄	—	2	1	no	—
300	C₄	—	4	4	no	—
301	C₄	—	4	4	no	—
302	C₄	—	4	4	no	—
303	C₄	—	8	1	no	—
304	C₄	C₁₃×#454	1	2	yes	—
305	C₄	C₁₃×#577	1	2	yes	—
306	C₄	C₁₃×#578	1	2	yes	—
307	C₂	—	4	1	no	—
308	C₂	—	16	1	no	—
309	C₂	C₁₃×#3013	1	2	yes	—
310	C₂	C₂×C₁₃×#454	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂×#120
commutator, Frattini, cosocle	C₄²	C₂⁴×C₁₃
prime radical	C₁₃×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3556	C₂	semidirect
4. C₁₃×#1636	C₂	semidirect
5. C₁₃×#1626	C₂	semidirect
6. C₁₃×#1561	C₂	semidirect
7. C₁₃×#1493	C₂	semidirect
73. #18698	C₁₃	direct (N × H)
112. #3556	C₂×C₁₃	semidirect
115. #1636	C₂×C₁₃	semidirect
116. #1626	C₂×C₁₃	semidirect
117. #1561	C₂×C₁₃	semidirect
118. #1493	C₂×C₁₃	semidirect
266. C₁₃	#18698	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×#3556	2
C₁₃×#3254	2
C₁₃×#3131	2
C₁₃×#1636	2
C₁₃×#1626	2
C₁₃×#1561	2
C₁₃×#1493	2
#18698	13

Subgroups — C13×#18698 #15705037

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#18698 #15705037