Subgroups — C₁₃×#3212 #13112270

Order 1664.

400 conjugacy classes of subgroups.

The subgroup lattice has 290 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#529	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#495	C₂	1	2	yes	maximal, complemented
4	C₁₃×#488	C₂	1	2	yes	maximal, complemented
5	C₁₃×#462	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#461	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#460	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#459	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#452	C₂	1	2	yes	maximal, complemented
10	C₁₃×#449	C₂	1	2	yes	maximal, complemented
11	C₁₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	C₂²	1	2	yes	—
13	C₂×C₁₃×Q₁₆	C₂²	1	2	yes	—
14	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
15	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
16	C₂×C₁₃×D₈	C₂²	1	2	yes	—
17	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
18	C₁₃×#127	C₂²	1	1	yes	characteristic
19	C₁₃×#126	—	2	1	no	—
20	C₁₃×#125	C₂²	1	1	yes	characteristic
21	C₁₃×#124	C₂²	1	2	yes	—
22	C₁₃×#124	C₂²	1	1	yes	characteristic
23	C₁₃×#123	—	2	1	no	—
24	C₁₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
25	C₁₃×#121	—	2	1	no	—
26	C₁₃×#121	—	2	1	no	—
27	C₁₃×#121	C₂²	1	1	yes	characteristic
28	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
29	C₄×D₄×C₁₃	—	2	1	no	—
30	C₄×D₄×C₁₃	—	2	1	no	—
31	C₄×D₄×C₁₃	—	2	1	no	—
32	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
33	C₁₃×#117	—	2	1	no	—
34	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
35	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
36	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
37	C₁₃×#103	C₂²	1	2	yes	—
38	C₁₃×#103	C₂²	1	2	yes	—
39	C₁₃×#102	C₂²	1	2	yes	—
40	C₁₃×#102	C₂²	1	2	yes	—
41	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
42	C₄×C₈×C₁₃	C₂²	1	2	yes	—
43	C₁₃×#41	—	4	2	no	—
44	C₁₃×#41	—	4	2	no	—
45	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
46	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₂×D₄×C₁₃	—	2	1	no	—
48	C₂×D₄×C₁₃	—	2	2	no	—
49	C₂×D₄×C₁₃	—	2	1	no	—
50	C₂×D₄×C₁₃	—	2	1	no	—
51	C₂×D₄×C₁₃	C₂³	1	2	yes	—
52	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
53	C₂²×C₄×C₁₃	—	2	1	no	—
54	C₂²×C₄×C₁₃	—	2	2	no	—
55	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
58	C₁₃×Q₁₆	—	4	2	no	—
59	C₁₃×SD₁₆	—	4	2	no	—
60	C₁₃×SD₁₆	—	4	2	no	—
61	C₁₃×D₈	—	4	2	no	—
62	C₂×C₈×C₁₃	C₂³	1	2	yes	—
63	C₂×C₈×C₁₃	C₂³	1	2	yes	—
64	C₁₃×C₄⋊C₄	—	2	1	no	—
65	C₁₃×C₄⋊C₄	—	2	1	no	—
66	C₁₃×C₄⋊C₄	—	2	1	no	—
67	C₁₃×C₄⋊C₄	—	2	1	no	—
68	C₁₃×C₄⋊C₄	—	2	1	no	—
69	C₁₃×C₄⋊C₄	—	2	1	no	—
70	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
71	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
72	C₁₃×C₂²⋊C₄	—	2	1	no	—
73	C₁₃×C₂²⋊C₄	—	2	1	no	—
74	C₁₃×C₂²⋊C₄	—	2	1	no	—
75	C₁₃×C₂²⋊C₄	—	2	2	no	—
76	C₁₃×C₂²⋊C₄	—	2	2	no	—
77	C₁₃×C₂²⋊C₄	—	2	1	no	—
78	C₁₃×C₂²⋊C₄	—	2	1	no	—
79	C₄²×C₁₃	—	2	1	no	—
80	C₄²×C₁₃	D₄	1	1	yes	characteristic
81	C₄²×C₁₃	D₄	1	1	yes	characteristic
82	C₄²×C₁₃	C₂³	1	1	yes	characteristic
83	C₄²×C₁₃	C₂³	1	1	yes	characteristic
84	#3212	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
85	C₂³×C₁₃	—	2	1	no	—
86	C₂³×C₁₃	—	2	2	no	—
87	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
88	Q₈×C₁₃	—	2	1	no	—
89	Q₈×C₁₃	—	2	2	no	—
90	Q₈×C₁₃	—	2	1	no	—
91	Q₈×C₁₃	—	2	2	no	—
92	D₄×C₁₃	—	2	1	no	—
93	D₄×C₁₃	—	2	2	no	—
94	D₄×C₁₃	—	2	1	no	—
95	D₄×C₁₃	—	2	2	no	—
96	D₄×C₁₃	—	4	2	no	—
97	D₄×C₁₃	—	4	2	no	—
98	D₄×C₁₃	—	4	4	no	—
99	D₄×C₁₃	—	4	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	1	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	1	no	—
105	C₂×C₄×C₁₃	—	2	1	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	4	1	no	—
108	C₂×C₄×C₁₃	—	4	4	no	—
109	C₂×C₄×C₁₃	—	4	2	no	—
110	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
111	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
115	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
116	C₈×C₁₃	—	2	2	no	—
117	C₈×C₁₃	—	2	2	no	—
118	#532	C₂×C₁₃	1	1	yes	characteristic, complemented
119	#529	C₂×C₁₃	1	1	yes	characteristic, complemented
120	#495	C₂×C₁₃	1	2	yes	complemented
121	#488	C₂×C₁₃	1	2	yes	complemented
122	#462	C₂×C₁₃	1	1	yes	characteristic, complemented
123	#461	C₂×C₁₃	1	1	yes	characteristic, complemented
124	#460	C₂×C₁₃	1	1	yes	characteristic, complemented
125	#459	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#452	C₂×C₁₃	1	2	yes	complemented
127	#449	C₂×C₁₃	1	2	yes	complemented
128	#431	C₂×C₁₃	1	1	yes	characteristic, complemented
129	C₂²×C₁₃	—	2	1	no	—
130	C₂²×C₁₃	—	2	1	no	—
131	C₂²×C₁₃	—	2	1	no	—
132	C₂²×C₁₃	—	4	2	no	—
133	C₂²×C₁₃	—	4	1	no	—
134	C₂²×C₁₃	—	4	1	no	—
135	C₂²×C₁₃	—	4	2	no	—
136	C₂²×C₁₃	—	4	2	no	—
137	C₂²×C₁₃	—	4	1	no	—
138	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
139	C₄×C₁₃	—	2	1	no	—
140	C₄×C₁₃	—	2	1	no	—
141	C₄×C₁₃	—	2	1	no	—
142	C₄×C₁₃	—	2	1	no	—
143	C₄×C₁₃	—	4	2	no	—
144	C₄×C₁₃	—	4	1	no	—
145	C₄×C₁₃	—	4	1	no	—
146	C₄×C₁₃	—	4	1	no	—
147	C₄×C₁₃	#135	1	2	yes	—
148	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
149	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
150	C₂×#41	C₂²×C₁₃	1	2	yes	—
151	C₂×Q₁₆	C₂²×C₁₃	1	2	yes	—
152	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
153	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
154	C₂×D₈	C₂²×C₁₃	1	2	yes	—
155	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
156	#127	C₂²×C₁₃	1	1	yes	characteristic
157	#126	—	2	1	no	—
158	#125	C₂²×C₁₃	1	1	yes	characteristic
159	#124	C₂²×C₁₃	1	1	yes	characteristic
160	#124	C₂²×C₁₃	1	2	yes	—
161	#123	—	2	1	no	—
162	C₂²⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
163	#121	—	2	1	no	—
164	#121	—	2	1	no	—
165	#121	C₂²×C₁₃	1	1	yes	characteristic
166	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
167	C₄×D₄	—	2	1	no	—
168	C₄×D₄	—	2	1	no	—
169	C₄×D₄	—	2	1	no	—
170	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
171	#117	—	2	1	no	—
172	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
173	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
174	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
175	#103	C₂²×C₁₃	1	2	yes	—
176	#103	C₂²×C₁₃	1	2	yes	—
177	#102	C₂²×C₁₃	1	2	yes	—
178	#102	C₂²×C₁₃	1	2	yes	—
179	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
180	C₄×C₈	C₂²×C₁₃	1	2	yes	—
181	C₂×C₁₃	—	4	1	no	—
182	C₂×C₁₃	—	8	2	no	—
183	C₂×C₁₃	—	8	1	no	—
184	C₂×C₁₃	#535	1	1	yes	characteristic
185	C₂×C₁₃	C₂×#135	1	1	yes	characteristic
186	C₂×C₁₃	#575	1	1	yes	characteristic
187	#41	—	4	2	no	—
188	#41	—	4	2	no	—
189	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
190	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
191	C₂×D₄	—	2	1	no	—
192	C₂×D₄	—	2	2	no	—
193	C₂×D₄	—	2	1	no	—
194	C₂×D₄	—	2	1	no	—
195	C₂×D₄	C₂³×C₁₃	1	2	yes	—
196	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	2	2	no	—
199	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
200	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
201	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
202	Q₁₆	—	4	2	no	—
203	SD₁₆	—	4	2	no	—
204	SD₁₆	—	4	2	no	—
205	D₈	—	4	2	no	—
206	C₂×C₈	C₂³×C₁₃	1	2	yes	—
207	C₂×C₈	C₂³×C₁₃	1	2	yes	—
208	C₄⋊C₄	—	2	1	no	—
209	C₄⋊C₄	—	2	1	no	—
210	C₄⋊C₄	—	2	1	no	—
211	C₄⋊C₄	—	2	1	no	—
212	C₄⋊C₄	—	2	1	no	—
213	C₄⋊C₄	—	2	1	no	—
214	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
215	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
216	C₂²⋊C₄	—	2	1	no	—
217	C₂²⋊C₄	—	2	1	no	—
218	C₂²⋊C₄	—	2	1	no	—
219	C₂²⋊C₄	—	2	2	no	—
220	C₂²⋊C₄	—	2	2	no	—
221	C₂²⋊C₄	—	2	1	no	—
222	C₂²⋊C₄	—	2	1	no	—
223	C₄²	—	2	1	no	—
224	C₄²	D₄×C₁₃	1	1	yes	characteristic
225	C₄²	D₄×C₁₃	1	1	yes	characteristic
226	C₄²	C₂³×C₁₃	1	1	yes	characteristic
227	C₄²	C₂³×C₁₃	1	1	yes	characteristic
228	C₁₃	#3212	1	1	yes	characteristic, direct product factor, Sylow
229	C₂³	—	2	1	no	—
230	C₂³	—	2	2	no	—
231	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
232	Q₈	—	2	1	no	—
233	Q₈	—	2	2	no	—
234	Q₈	—	2	1	no	—
235	Q₈	—	2	2	no	—
236	D₄	—	2	1	no	—
237	D₄	—	2	2	no	—
238	D₄	—	2	1	no	—
239	D₄	—	2	2	no	—
240	D₄	—	4	2	no	—
241	D₄	—	4	2	no	—
242	D₄	—	4	4	no	—
243	D₄	—	4	2	no	—
244	C₂×C₄	—	2	2	no	—
245	C₂×C₄	—	2	1	no	—
246	C₂×C₄	—	2	1	no	—
247	C₂×C₄	—	2	1	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	4	1	no	—
252	C₂×C₄	—	4	4	no	—
253	C₂×C₄	—	4	2	no	—
254	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
255	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
256	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
257	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
258	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
259	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
260	C₈	—	2	2	no	—
261	C₈	—	2	2	no	—
262	C₂²	—	2	1	no	—
263	C₂²	—	2	1	no	—
264	C₂²	—	2	1	no	—
265	C₂²	—	4	2	no	—
266	C₂²	—	4	1	no	—
267	C₂²	—	4	1	no	—
268	C₂²	—	4	2	no	—
269	C₂²	—	4	2	no	—
270	C₂²	—	4	1	no	—
271	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
272	C₄	—	2	1	no	—
273	C₄	—	2	1	no	—
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	4	2	no	—
277	C₄	—	4	1	no	—
278	C₄	—	4	1	no	—
279	C₄	—	4	1	no	—
280	C₄	C₁₃×#135	1	2	yes	—
281	C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
282	C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
283	C₂	—	4	1	no	—
284	C₂	—	8	2	no	—
285	C₂	—	8	1	no	—
286	C₂	C₁₃×#535	1	1	yes	characteristic
287	C₂	C₂×C₁₃×#135	1	1	yes	characteristic
288	C₂	C₁₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ 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×#532	C₂	semidirect
2. C₁₃×#529	C₂	semidirect
3. C₁₃×#495	C₂	semidirect
4. C₁₃×#488	C₂	semidirect
5. C₁₃×#462	C₂	semidirect
6. C₁₃×#461	C₂	semidirect
7. C₁₃×#460	C₂	semidirect
8. C₁₃×#459	C₂	semidirect
9. C₁₃×#452	C₂	semidirect
10. C₁₃×#449	C₂	semidirect
11. C₁₃×#431	C₂	semidirect
84. #3212	C₁₃	direct (N × H)
118. #532	C₂×C₁₃	semidirect
119. #529	C₂×C₁₃	semidirect
120. #495	C₂×C₁₃	semidirect
121. #488	C₂×C₁₃	semidirect
122. #462	C₂×C₁₃	semidirect
123. #461	C₂×C₁₃	semidirect
124. #460	C₂×C₁₃	semidirect
125. #459	C₂×C₁₃	semidirect
126. #452	C₂×C₁₃	semidirect
127. #449	C₂×C₁₃	semidirect
128. #431	C₂×C₁₃	semidirect
228. C₁₃	#3212	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×#532	2
C₁₃×#529	2
C₁₃×#495	2
C₁₃×#488	2
C₁₃×#462	2
C₁₃×#461	2
C₁₃×#460	2
C₁₃×#459	2
C₁₃×#452	2
C₁₃×#449	2
C₁₃×#431	2
#3212	13

Subgroups — C13×#3212 #13112270

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3212 #13112270