Subgroups — C₁₃×#19063 #15705402

Order 3328.

806 conjugacy classes of subgroups.

The subgroup lattice has 298 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×#3235	C₂	1	2	yes	maximal
3	C₁₃×#3135	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1663	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1658	C₂	1	4	yes	maximal, complemented
6	C₁₃×#1537	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1531	C₂	1	2	yes	maximal, complemented
8	C₁₃×#564	C₂²	1	4	yes	—
9	C₁₃×#557	—	2	2	no	—
10	C₁₃×#557	—	2	2	no	—
11	C₁₃×#557	—	2	2	no	—
12	C₁₃×#541	—	2	2	no	—
13	C₁₃×#541	—	2	4	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×#501	C₂²	1	4	yes	—
18	C₁₃×#491	C₂²	1	2	yes	—
19	C₁₃×#490	C₂²	1	2	yes	—
20	C₁₃×#490	C₂²	1	4	yes	—
21	C₁₃×#489	—	2	2	no	—
22	C₁₃×#482	—	2	4	no	—
23	C₁₃×#476	—	2	2	no	—
24	C₁₃×#474	—	2	2	no	—
25	C₁₃×#432	C₂²	1	2	yes	—
26	C₁₃×#416	—	2	2	no	—
27	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
28	C₁₃×#330	C₂²	1	8	yes	—
29	C₁₃×#330	C₂²	1	4	yes	—
30	C₂×C₁₃×#41	—	2	1	no	—
31	C₂²×C₈×C₁₃	—	2	1	no	—
32	C₁₃×C₄⋊Q₈	—	2	4	no	—
33	C₁₃×C₄⋊Q₈	—	2	2	no	—
34	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
35	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
36	C₁₃×C₄⋊Q₈	D₄	1	2	yes	—
37	C₁₃×C₄⋊Q₈	C₂³	1	1	yes	characteristic
38	C₁₃×#127	C₂³	1	1	yes	characteristic
39	C₁₃×#126	—	2	8	no	—
40	C₁₃×#125	—	2	4	no	—
41	C₁₃×#125	—	2	2	no	—
42	C₁₃×#125	—	2	8	no	—
43	C₁₃×#125	—	2	8	no	—
44	C₁₃×#125	—	2	8	no	—
45	C₁₃×#125	C₂³	1	2	yes	—
46	C₁₃×#125	C₂³	1	4	yes	—
47	C₁₃×#124	—	2	1	no	—
48	C₁₃×C₂²⋊Q₈	—	2	2	no	—
49	C₁₃×C₂²⋊Q₈	—	2	2	no	—
50	C₁₃×C₂²⋊Q₈	—	2	2	no	—
51	C₁₃×#121	—	2	2	no	—
52	C₄×Q₈×C₁₃	—	2	4	no	—
53	C₄×Q₈×C₁₃	—	2	4	no	—
54	C₄×Q₈×C₁₃	—	2	4	no	—
55	C₄×Q₈×C₁₃	—	2	2	no	—
56	C₄×Q₈×C₁₃	—	2	2	no	—
57	C₄×Q₈×C₁₃	—	2	2	no	—
58	C₄×D₄×C₁₃	—	2	2	no	—
59	C₁₃×#117	D₄	1	2	yes	—
60	C₁₃×#117	D₄	1	4	yes	—
61	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
62	C₁₃×#107	—	2	4	no	—
63	C₁₃×#106	—	2	4	no	—
64	C₁₃×C₄⋊C₈	—	2	2	no	—
65	C₁₃×#103	—	2	4	no	—
66	C₁₃×#102	—	2	4	no	—
67	C₁₃×#102	—	2	4	no	—
68	C₁₃×#102	—	2	4	no	—
69	C₁₃×C₂²⋊C₈	—	2	2	no	—
70	C₁₃×#97	—	2	1	no	—
71	C₁₃×#97	C₂³	1	2	yes	—
72	C₁₃×#97	C₂³	1	4	yes	—
73	#19063	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	2	no	—
78	C₂×Q₈×C₁₃	—	2	1	no	—
79	C₂×D₄×C₁₃	—	2	2	no	—
80	C₂×D₄×C₁₃	—	2	1	no	—
81	C₂×D₄×C₁₃	—	4	1	no	—
82	C₂²×C₄×C₁₃	—	4	1	no	—
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	2	4	no	—
87	C₂×C₈×C₁₃	—	4	2	no	—
88	C₁₃×C₄⋊C₄	—	2	8	no	—
89	C₁₃×C₄⋊C₄	—	2	8	no	—
90	C₁₃×C₄⋊C₄	—	2	8	no	—
91	C₁₃×C₄⋊C₄	—	2	2	no	—
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	2	2	no	—
94	C₁₃×C₄⋊C₄	—	2	4	no	—
95	C₁₃×C₄⋊C₄	—	2	4	no	—
96	C₁₃×C₄⋊C₄	—	2	4	no	—
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
99	C₁₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
100	C₁₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
101	C₁₃×C₂²⋊C₄	—	2	2	no	—
102	C₁₃×C₂²⋊C₄	—	2	4	no	—
103	C₁₃×C₂²⋊C₄	—	4	2	no	—
104	C₄²×C₁₃	—	2	8	no	—
105	C₄²×C₁₃	—	2	4	no	—
106	C₄²×C₁₃	—	2	2	no	—
107	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₄²×C₁₃	C₂×D₄	1	2	yes	—
109	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
110	#3556	C₂×C₁₃	1	1	yes	characteristic, complemented
111	#3235	C₂×C₁₃	1	2	yes	—
112	#3135	C₂×C₁₃	1	1	yes	characteristic
113	#1663	C₂×C₁₃	1	4	yes	complemented
114	#1658	C₂×C₁₃	1	4	yes	complemented
115	#1537	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#1531	C₂×C₁₃	1	2	yes	complemented
117	C₂³×C₁₃	—	4	1	no	—
118	C₂³×C₁₃	#120	1	1	yes	characteristic
119	Q₈×C₁₃	—	4	4	no	—
120	Q₈×C₁₃	—	4	4	no	—
121	Q₈×C₁₃	—	4	2	no	—
122	Q₈×C₁₃	—	4	4	no	—
123	D₄×C₁₃	—	4	4	no	—
124	D₄×C₁₃	—	4	2	no	—
125	D₄×C₁₃	—	8	2	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	4	no	—
128	C₂×C₄×C₁₃	—	2	4	no	—
129	C₂×C₄×C₁₃	—	2	4	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	4	1	no	—
132	C₂×C₄×C₁₃	—	8	2	no	—
133	C₂×C₄×C₁₃	#120	1	2	yes	—
134	C₂×C₄×C₁₃	#120	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
136	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
137	C₈×C₁₃	—	4	4	no	—
138	C₈×C₁₃	—	4	2	no	—
139	#564	C₂²×C₁₃	1	4	yes	—
140	#557	—	2	2	no	—
141	#557	—	2	2	no	—
142	#557	—	2	2	no	—
143	#541	—	2	2	no	—
144	#541	—	2	4	no	—
145	#533	C₂²×C₁₃	1	1	yes	characteristic
146	#533	C₂²×C₁₃	1	2	yes	—
147	#532	C₂²×C₁₃	1	1	yes	characteristic
148	#501	C₂²×C₁₃	1	4	yes	—
149	#491	C₂²×C₁₃	1	2	yes	—
150	#490	C₂²×C₁₃	1	2	yes	—
151	#490	C₂²×C₁₃	1	4	yes	—
152	#489	—	2	2	no	—
153	#482	—	2	4	no	—
154	#476	—	2	2	no	—
155	#474	—	2	2	no	—
156	#432	C₂²×C₁₃	1	2	yes	—
157	#416	—	2	2	no	—
158	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
159	#330	C₂²×C₁₃	1	4	yes	—
160	#330	C₂²×C₁₃	1	8	yes	—
161	C₂²×C₁₃	—	2	1	no	—
162	C₂²×C₁₃	—	2	2	no	—
163	C₂²×C₁₃	—	8	1	no	—
164	C₂²×C₁₃	—	8	2	no	—
165	C₂²×C₁₃	C₂×#120	1	1	yes	characteristic
166	C₄×C₁₃	—	2	1	no	—
167	C₄×C₁₃	—	2	2	no	—
168	C₄×C₁₃	—	2	2	no	—
169	C₄×C₁₃	—	4	4	no	—
170	C₄×C₁₃	—	4	4	no	—
171	C₄×C₁₃	—	4	4	no	—
172	C₄×C₁₃	—	8	1	no	—
173	C₄×C₁₃	#454	1	2	yes	—
174	C₂×#41	—	2	1	no	—
175	C₂²×C₈	—	2	1	no	—
176	C₄⋊Q₈	—	2	4	no	—
177	C₄⋊Q₈	—	2	2	no	—
178	C₄⋊Q₈	D₄×C₁₃	1	2	yes	—
179	C₄⋊Q₈	D₄×C₁₃	1	2	yes	—
180	C₄⋊Q₈	D₄×C₁₃	1	2	yes	—
181	C₄⋊Q₈	C₂³×C₁₃	1	1	yes	characteristic
182	#127	C₂³×C₁₃	1	1	yes	characteristic
183	#126	—	2	8	no	—
184	#125	—	2	4	no	—
185	#125	—	2	2	no	—
186	#125	—	2	8	no	—
187	#125	—	2	8	no	—
188	#125	—	2	8	no	—
189	#125	C₂³×C₁₃	1	2	yes	—
190	#125	C₂³×C₁₃	1	4	yes	—
191	#124	—	2	1	no	—
192	C₂²⋊Q₈	—	2	2	no	—
193	C₂²⋊Q₈	—	2	2	no	—
194	C₂²⋊Q₈	—	2	2	no	—
195	#121	—	2	2	no	—
196	C₄×Q₈	—	2	4	no	—
197	C₄×Q₈	—	2	4	no	—
198	C₄×Q₈	—	2	4	no	—
199	C₄×Q₈	—	2	2	no	—
200	C₄×Q₈	—	2	2	no	—
201	C₄×Q₈	—	2	2	no	—
202	C₄×D₄	—	2	2	no	—
203	#117	D₄×C₁₃	1	2	yes	—
204	#117	D₄×C₁₃	1	4	yes	—
205	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
206	#107	—	2	4	no	—
207	#106	—	2	4	no	—
208	C₄⋊C₈	—	2	2	no	—
209	#103	—	2	4	no	—
210	#102	—	2	4	no	—
211	#102	—	2	4	no	—
212	#102	—	2	4	no	—
213	C₂²⋊C₈	—	2	2	no	—
214	#97	—	2	1	no	—
215	#97	C₂³×C₁₃	1	2	yes	—
216	#97	C₂³×C₁₃	1	4	yes	—
217	C₂×C₁₃	—	4	1	no	—
218	C₂×C₁₃	—	16	1	no	—
219	C₂×C₁₃	C₂×#454	1	1	yes	characteristic
220	C₂×C₁₃	#3020	1	2	yes	—
221	#41	—	4	4	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×Q₈	—	2	2	no	—
224	C₂×Q₈	—	2	2	no	—
225	C₂×Q₈	—	2	1	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	1	no	—
228	C₂×D₄	—	4	1	no	—
229	C₂²×C₄	—	4	1	no	—
230	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
231	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	4	2	no	—
235	C₄⋊C₄	—	2	8	no	—
236	C₄⋊C₄	—	2	8	no	—
237	C₄⋊C₄	—	2	8	no	—
238	C₄⋊C₄	—	2	2	no	—
239	C₄⋊C₄	—	2	2	no	—
240	C₄⋊C₄	—	2	2	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	—	2	4	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	2	2	no	—
245	C₄⋊C₄	C₂×D₄×C₁₃	1	4	yes	—
246	C₄⋊C₄	C₂×D₄×C₁₃	1	4	yes	—
247	C₄⋊C₄	C₂×D₄×C₁₃	1	4	yes	—
248	C₂²⋊C₄	—	2	2	no	—
249	C₂²⋊C₄	—	2	4	no	—
250	C₂²⋊C₄	—	4	2	no	—
251	C₄²	—	2	8	no	—
252	C₄²	—	2	4	no	—
253	C₄²	—	2	2	no	—
254	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
255	C₄²	C₂×D₄×C₁₃	1	2	yes	—
256	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
257	C₁₃	#19063	1	1	yes	characteristic, direct product factor, Sylow
258	C₂³	—	4	1	no	—
259	C₂³	C₁₃×#120	1	1	yes	characteristic
260	Q₈	—	4	4	no	—
261	Q₈	—	4	4	no	—
262	Q₈	—	4	2	no	—
263	Q₈	—	4	4	no	—
264	D₄	—	4	4	no	—
265	D₄	—	4	2	no	—
266	D₄	—	8	2	no	—
267	C₂×C₄	—	2	2	no	—
268	C₂×C₄	—	2	4	no	—
269	C₂×C₄	—	2	4	no	—
270	C₂×C₄	—	2	4	no	—
271	C₂×C₄	—	4	2	no	—
272	C₂×C₄	—	4	1	no	—
273	C₂×C₄	—	8	2	no	—
274	C₂×C₄	C₁₃×#120	1	2	yes	—
275	C₂×C₄	C₁₃×#120	1	1	yes	characteristic
276	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
277	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
278	C₈	—	4	4	no	—
279	C₈	—	4	2	no	—
280	C₂²	—	2	1	no	—
281	C₂²	—	2	2	no	—
282	C₂²	—	8	1	no	—
283	C₂²	—	8	2	no	—
284	C₂²	C₂×C₁₃×#120	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	4	4	no	—
289	C₄	—	4	4	no	—
290	C₄	—	4	4	no	—
291	C₄	—	8	1	no	—
292	C₄	C₁₃×#454	1	2	yes	—
293	C₂	—	4	1	no	—
294	C₂	—	16	1	no	—
295	C₂	C₂×C₁₃×#454	1	1	yes	characteristic
296	C₂	C₁₃×#3020	1	2	yes	—

	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×#1663	C₂	semidirect
5. C₁₃×#1658	C₂	semidirect
6. C₁₃×#1537	C₂	semidirect
7. C₁₃×#1531	C₂	semidirect
73. #19063	C₁₃	direct (N × H)
110. #3556	C₂×C₁₃	semidirect
113. #1663	C₂×C₁₃	semidirect
114. #1658	C₂×C₁₃	semidirect
115. #1537	C₂×C₁₃	semidirect
116. #1531	C₂×C₁₃	semidirect
257. C₁₃	#19063	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×#3235	2
C₁₃×#3135	2
C₁₃×#1663	2
C₁₃×#1658	2
C₁₃×#1537	2
C₁₃×#1531	2
#19063	13

Subgroups — C13×#19063 #15705402

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#19063 #15705402