Subgroups — C₁₃×#2635 #13111693

Order 1664.

620 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#121	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₄×D₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#400	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#399	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#394	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#394	C₂	1	2	yes	maximal, complemented
7	C₁₃×#392	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#392	C₂	1	2	yes	maximal, complemented
9	C₁₃×#388	C₂	1	2	yes	maximal, complemented
10	C₁₃×#386	C₂	1	2	yes	maximal, complemented
11	C₁₃×#381	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
13	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
14	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
15	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
16	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
18	C₁₃×#121	—	2	4	no	—
19	C₄×D₄×C₁₃	—	2	4	no	—
20	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
23	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
24	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
25	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
26	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
27	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
28	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
29	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
30	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
31	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
32	C₁₃×#95	C₂²	1	2	yes	—
33	C₁₃×#95	C₂²	1	1	yes	characteristic
34	C₁₃×#95	C₂²	1	1	yes	characteristic
35	C₁₃×#95	C₂²	1	2	yes	—
36	C₁₃×#95	C₂²	1	2	yes	—
37	C₂⁴×C₁₃	C₂³	1	2	yes	—
38	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
39	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
40	C₂×D₄×C₁₃	—	2	2	no	—
41	C₂×D₄×C₁₃	—	2	2	no	—
42	C₂×D₄×C₁₃	—	2	4	no	—
43	C₂×D₄×C₁₃	—	2	4	no	—
44	C₂×D₄×C₁₃	—	2	2	no	—
45	C₂×D₄×C₁₃	—	2	2	no	—
46	C₂×D₄×C₁₃	—	4	2	no	—
47	C₂×D₄×C₁₃	—	4	2	no	—
48	C₂×D₄×C₁₃	—	4	2	no	—
49	C₂×D₄×C₁₃	—	4	2	no	—
50	C₂×D₄×C₁₃	—	4	2	no	—
51	C₂²×C₄×C₁₃	—	2	2	no	—
52	C₂²×C₄×C₁₃	—	4	2	no	—
53	C₂²×C₄×C₁₃	—	4	2	no	—
54	C₂²×C₄×C₁₃	D₄	1	4	yes	complemented
55	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
56	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
57	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
58	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
59	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
61	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
63	C₁₃×C₄⋊C₄	—	2	2	no	—
64	C₁₃×C₄⋊C₄	—	2	2	no	—
65	C₁₃×C₄⋊C₄	—	4	2	no	—
66	C₁₃×C₂²⋊C₄	—	2	2	no	—
67	C₁₃×C₂²⋊C₄	—	2	2	no	—
68	C₁₃×C₂²⋊C₄	—	2	4	no	—
69	C₁₃×C₂²⋊C₄	—	2	4	no	—
70	C₁₃×C₂²⋊C₄	—	4	2	no	—
71	C₁₃×C₂²⋊C₄	—	4	2	no	—
72	C₁₃×C₂²⋊C₄	—	4	2	no	—
73	C₁₃×C₂²⋊C₄	—	4	2	no	—
74	C₄²×C₁₃	—	2	2	no	—
75	#2635	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
76	C₂³×C₁₃	—	2	1	no	—
77	C₂³×C₁₃	—	2	1	no	—
78	C₂³×C₁₃	—	2	1	no	—
79	C₂³×C₁₃	—	2	1	no	—
80	C₂³×C₁₃	—	2	2	no	—
81	C₂³×C₁₃	—	2	2	no	—
82	C₂³×C₁₃	—	2	2	no	—
83	C₂³×C₁₃	—	2	2	no	—
84	C₂³×C₁₃	—	2	2	no	—
85	C₂³×C₁₃	—	2	2	no	—
86	C₂³×C₁₃	—	2	2	no	—
87	C₂³×C₁₃	—	2	2	no	—
88	C₂³×C₁₃	—	2	2	no	—
89	C₂³×C₁₃	—	2	2	no	—
90	C₂³×C₁₃	—	2	2	no	—
91	C₂³×C₁₃	C₂×D₄	1	2	yes	—
92	C₂³×C₁₃	#41	1	2	yes	—
93	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
94	D₄×C₁₃	—	2	8	no	—
95	D₄×C₁₃	—	4	4	no	—
96	D₄×C₁₃	—	4	8	no	—
97	D₄×C₁₃	—	4	8	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	1	no	—
103	C₂×C₄×C₁₃	—	2	1	no	—
104	C₂×C₄×C₁₃	—	2	2	no	—
105	C₂×C₄×C₁₃	—	2	2	no	—
106	C₂×C₄×C₁₃	—	2	2	no	—
107	C₂×C₄×C₁₃	—	2	2	no	—
108	C₂×C₄×C₁₃	—	2	2	no	—
109	C₂×C₄×C₁₃	—	2	2	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	4	4	no	—
116	C₂×C₄×C₁₃	—	4	4	no	—
117	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
118	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
119	C₂×C₄×C₁₃	#41	1	2	yes	—
120	C₂×#121	C₂×C₁₃	1	1	yes	characteristic, complemented
121	C₂×C₄×D₄	C₂×C₁₃	1	1	yes	characteristic, complemented
122	#400	C₂×C₁₃	1	1	yes	characteristic, complemented
123	#399	C₂×C₁₃	1	1	yes	characteristic, complemented
124	#394	C₂×C₁₃	1	1	yes	characteristic, complemented
125	#394	C₂×C₁₃	1	2	yes	complemented
126	#392	C₂×C₁₃	1	1	yes	characteristic, complemented
127	#392	C₂×C₁₃	1	2	yes	complemented
128	#388	C₂×C₁₃	1	2	yes	complemented
129	#386	C₂×C₁₃	1	2	yes	complemented
130	#381	C₂×C₁₃	1	1	yes	characteristic, complemented
131	C₂²×C₁₃	—	2	2	no	—
132	C₂²×C₁₃	—	2	2	no	—
133	C₂²×C₁₃	—	2	2	no	—
134	C₂²×C₁₃	—	2	2	no	—
135	C₂²×C₁₃	—	2	4	no	—
136	C₂²×C₁₃	—	4	1	no	—
137	C₂²×C₁₃	—	4	1	no	—
138	C₂²×C₁₃	—	4	2	no	—
139	C₂²×C₁₃	—	4	2	no	—
140	C₂²×C₁₃	—	4	2	no	—
141	C₂²×C₁₃	—	4	2	no	—
142	C₂²×C₁₃	—	4	2	no	—
143	C₂²×C₁₃	—	4	2	no	—
144	C₂²×C₁₃	—	4	2	no	—
145	C₂²×C₁₃	—	4	2	no	—
146	C₂²×C₁₃	—	4	2	no	—
147	C₂²×C₁₃	—	4	2	no	—
148	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
149	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
150	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
151	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
152	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
153	C₄×C₁₃	—	2	2	no	—
154	C₄×C₁₃	—	2	2	no	—
155	C₄×C₁₃	—	2	2	no	—
156	C₄×C₁₃	—	4	1	no	—
157	C₄×C₁₃	—	4	1	no	—
158	C₄×C₁₃	—	4	2	no	—
159	C₄×C₁₃	—	4	2	no	—
160	C₄×C₁₃	—	4	2	no	—
161	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
162	C₂²×D₄	C₂²×C₁₃	1	2	yes	—
163	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
164	C₂²×D₄	C₂²×C₁₃	1	2	yes	—
165	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
166	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
167	#121	—	2	4	no	—
168	C₄×D₄	—	2	4	no	—
169	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
170	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
171	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
172	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
173	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
174	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
175	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
176	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
177	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
178	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
179	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
180	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
181	#95	C₂²×C₁₃	1	2	yes	—
182	#95	C₂²×C₁₃	1	1	yes	characteristic
183	#95	C₂²×C₁₃	1	1	yes	characteristic
184	#95	C₂²×C₁₃	1	2	yes	—
185	#95	C₂²×C₁₃	1	2	yes	—
186	C₂×C₁₃	—	4	2	no	—
187	C₂×C₁₃	—	4	2	no	—
188	C₂×C₁₃	—	8	2	no	—
189	C₂×C₁₃	#525	1	1	yes	characteristic
190	C₂×C₁₃	#534	1	1	yes	characteristic
191	C₂×C₁₃	#535	1	1	yes	characteristic
192	C₂×C₁₃	#538	1	2	yes	—
193	C₂×C₁₃	#540	1	2	yes	—
194	C₂⁴	C₂³×C₁₃	1	2	yes	—
195	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
196	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
197	C₂×D₄	—	2	2	no	—
198	C₂×D₄	—	2	2	no	—
199	C₂×D₄	—	2	4	no	—
200	C₂×D₄	—	2	4	no	—
201	C₂×D₄	—	2	2	no	—
202	C₂×D₄	—	2	2	no	—
203	C₂×D₄	—	4	2	no	—
204	C₂×D₄	—	4	2	no	—
205	C₂×D₄	—	4	2	no	—
206	C₂×D₄	—	4	2	no	—
207	C₂×D₄	—	4	2	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	4	2	no	—
210	C₂²×C₄	—	4	2	no	—
211	C₂²×C₄	D₄×C₁₃	1	4	yes	complemented
212	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
213	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
214	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
215	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
216	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
217	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
218	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
219	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
220	C₄⋊C₄	—	2	2	no	—
221	C₄⋊C₄	—	2	2	no	—
222	C₄⋊C₄	—	4	2	no	—
223	C₂²⋊C₄	—	2	2	no	—
224	C₂²⋊C₄	—	2	2	no	—
225	C₂²⋊C₄	—	2	4	no	—
226	C₂²⋊C₄	—	2	4	no	—
227	C₂²⋊C₄	—	4	2	no	—
228	C₂²⋊C₄	—	4	2	no	—
229	C₂²⋊C₄	—	4	2	no	—
230	C₂²⋊C₄	—	4	2	no	—
231	C₄²	—	2	2	no	—
232	C₁₃	#2635	1	1	yes	characteristic, direct product factor, Sylow
233	C₂³	—	2	1	no	—
234	C₂³	—	2	1	no	—
235	C₂³	—	2	1	no	—
236	C₂³	—	2	1	no	—
237	C₂³	—	2	2	no	—
238	C₂³	—	2	2	no	—
239	C₂³	—	2	2	no	—
240	C₂³	—	2	2	no	—
241	C₂³	—	2	2	no	—
242	C₂³	—	2	2	no	—
243	C₂³	—	2	2	no	—
244	C₂³	—	2	2	no	—
245	C₂³	—	2	2	no	—
246	C₂³	—	2	2	no	—
247	C₂³	—	2	2	no	—
248	C₂³	C₂×D₄×C₁₃	1	2	yes	—
249	C₂³	C₁₃×#41	1	2	yes	—
250	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
251	D₄	—	2	8	no	—
252	D₄	—	4	4	no	—
253	D₄	—	4	8	no	—
254	D₄	—	4	8	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	1	no	—
260	C₂×C₄	—	2	1	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	2	no	—
265	C₂×C₄	—	2	2	no	—
266	C₂×C₄	—	2	2	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	—	2	2	no	—
272	C₂×C₄	—	4	4	no	—
273	C₂×C₄	—	4	4	no	—
274	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
275	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
276	C₂×C₄	C₁₃×#41	1	2	yes	—
277	C₂²	—	2	2	no	—
278	C₂²	—	2	2	no	—
279	C₂²	—	2	2	no	—
280	C₂²	—	2	2	no	—
281	C₂²	—	2	4	no	—
282	C₂²	—	4	1	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	2	no	—
285	C₂²	—	4	2	no	—
286	C₂²	—	4	2	no	—
287	C₂²	—	4	2	no	—
288	C₂²	—	4	2	no	—
289	C₂²	—	4	2	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	4	2	no	—
292	C₂²	—	4	2	no	—
293	C₂²	—	4	2	no	—
294	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
295	C₂²	C₂×C₁₃×#41	1	1	yes	characteristic
296	C₂²	C₂×C₁₃×#41	1	1	yes	characteristic
297	C₂²	C₁₃×H(5,𝔽₂)	1	2	yes	—
298	C₂²	C₁₃×H(5,𝔽₂)	1	2	yes	—
299	C₄	—	2	2	no	—
300	C₄	—	2	2	no	—
301	C₄	—	2	2	no	—
302	C₄	—	4	1	no	—
303	C₄	—	4	1	no	—
304	C₄	—	4	2	no	—
305	C₄	—	4	2	no	—
306	C₄	—	4	2	no	—
307	C₂	—	4	2	no	—
308	C₂	—	4	2	no	—
309	C₂	—	8	2	no	—
310	C₂	C₁₃×#525	1	1	yes	characteristic
311	C₂	C₁₃×#534	1	1	yes	characteristic
312	C₂	C₁₃×#535	1	1	yes	characteristic
313	C₂	C₁₃×#538	1	2	yes	—
314	C₂	C₁₃×#540	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#121	C₂	semidirect
2. C₂×C₄×D₄×C₁₃	C₂	semidirect
3. C₁₃×#400	C₂	semidirect
4. C₁₃×#399	C₂	semidirect
5. C₁₃×#394	C₂	semidirect
6. C₁₃×#394	C₂	semidirect
7. C₁₃×#392	C₂	semidirect
8. C₁₃×#392	C₂	semidirect
9. C₁₃×#388	C₂	semidirect
10. C₁₃×#386	C₂	semidirect
11. C₁₃×#381	C₂	semidirect
54. C₂²×C₄×C₁₃	D₄	semidirect
75. #2635	C₁₃	direct (N × H)
120. C₂×#121	C₂×C₁₃	semidirect
121. C₂×C₄×D₄	C₂×C₁₃	semidirect
122. #400	C₂×C₁₃	semidirect
123. #399	C₂×C₁₃	semidirect
124. #394	C₂×C₁₃	semidirect
125. #394	C₂×C₁₃	semidirect
126. #392	C₂×C₁₃	semidirect
127. #392	C₂×C₁₃	semidirect
128. #388	C₂×C₁₃	semidirect
129. #386	C₂×C₁₃	semidirect
130. #381	C₂×C₁₃	semidirect
211. C₂²×C₄	D₄×C₁₃	semidirect
232. C₁₃	#2635	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₃×#121	2
C₂×C₄×D₄×C₁₃	2
C₁₃×#400	2
C₁₃×#399	2
C₁₃×#394	2
C₁₃×#392	2
C₁₃×#388	2
C₁₃×#386	2
C₁₃×#381	2
#2635	13

Subgroups — C13×#2635 #13111693

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2635 #13111693