Subgroups — C₁₃×#2684 #13111742

Order 1664.

688 conjugacy classes of subgroups.

The subgroup lattice has 306 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#124	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#120	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#397	C₂	1	2	yes	maximal, complemented
4	C₁₃×#394	C₂	1	2	yes	maximal, complemented
5	C₁₃×#393	C₂	1	2	yes	maximal, complemented
6	C₁₃×#388	C₂	1	2	yes	maximal, complemented
7	C₁₃×#386	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#386	C₂	1	2	yes	maximal, complemented
9	C₁₃×#379	C₂	1	2	yes	maximal, complemented
10	C₂⁵×C₁₃	C₂²	1	1	yes	characteristic
11	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic, complemented
12	C₂²×D₄×C₁₃	C₂²	1	2	yes	—
13	C₂²×D₄×C₁₃	C₂²	1	1	yes	characteristic
14	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic, complemented
15	C₁₃×#124	—	2	4	no	—
16	C₁₃×#120	—	2	4	no	—
17	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	complemented
18	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
19	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
20	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic
22	C₂×C₁₃×C₂²⋊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	2	yes	—
26	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	complemented
27	C₂×C₁₃×C₂²⋊C₄	C₂²	1	1	yes	characteristic, complemented
28	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic, complemented
29	C₁₃×#95	C₂²	1	2	yes	complemented
30	C₁₃×#95	C₂²	1	2	yes	complemented
31	C₁₃×#95	C₂²	1	2	yes	complemented
32	C₁₃×#95	C₂²	1	2	yes	complemented
33	C₂⁴×C₁₃	—	2	2	no	—
34	C₂⁴×C₁₃	—	2	4	no	—
35	C₂⁴×C₁₃	—	4	2	no	—
36	C₂⁴×C₁₃	—	4	1	no	—
37	C₂⁴×C₁₃	—	4	1	no	—
38	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
39	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
40	C₂⁴×C₁₃	C₂³	1	2	yes	—
41	C₂×Q₈×C₁₃	—	2	2	no	—
42	C₂×Q₈×C₁₃	—	4	1	no	—
43	C₂×Q₈×C₁₃	—	4	1	no	—
44	C₂×D₄×C₁₃	—	2	4	no	—
45	C₂×D₄×C₁₃	—	2	4	no	—
46	C₂×D₄×C₁₃	—	4	1	no	—
47	C₂×D₄×C₁₃	—	4	1	no	—
48	C₂×D₄×C₁₃	—	4	2	no	—
49	C₂×D₄×C₁₃	D₄	1	4	yes	—
50	C₂²×C₄×C₁₃	—	4	1	no	—
51	C₂²×C₄×C₁₃	—	4	2	no	—
52	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
53	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
54	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
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	C₁₃×C₄⋊C₄	—	4	2	no	—
59	C₁₃×C₂²⋊C₄	—	2	2	no	—
60	C₁₃×C₂²⋊C₄	—	2	4	no	—
61	C₁₃×C₂²⋊C₄	—	2	4	no	—
62	C₁₃×C₂²⋊C₄	—	2	4	no	—
63	C₁₃×C₂²⋊C₄	—	2	4	no	—
64	C₁₃×C₂²⋊C₄	—	4	2	no	—
65	C₁₃×C₂²⋊C₄	—	4	2	no	—
66	C₁₃×C₂²⋊C₄	—	4	2	no	—
67	C₁₃×C₂²⋊C₄	—	4	2	no	—
68	C₁₃×C₂²⋊C₄	—	4	1	no	—
69	C₄²×C₁₃	—	2	2	no	—
70	#2684	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂³×C₁₃	—	2	2	no	—
72	C₂³×C₁₃	—	2	2	no	—
73	C₂³×C₁₃	—	2	2	no	—
74	C₂³×C₁₃	—	2	2	no	—
75	C₂³×C₁₃	—	2	1	no	—
76	C₂³×C₁₃	—	2	2	no	—
77	C₂³×C₁₃	—	2	2	no	—
78	C₂³×C₁₃	—	2	2	no	—
79	C₂³×C₁₃	—	2	2	no	—
80	C₂³×C₁₃	—	2	1	no	—
81	C₂³×C₁₃	—	2	2	no	—
82	C₂³×C₁₃	—	2	1	no	—
83	C₂³×C₁₃	—	2	1	no	—
84	C₂³×C₁₃	—	2	2	no	—
85	C₂³×C₁₃	—	4	8	no	—
86	C₂³×C₁₃	—	4	4	no	—
87	C₂³×C₁₃	—	4	4	no	—
88	C₂³×C₁₃	—	4	8	no	—
89	C₂³×C₁₃	—	4	4	no	—
90	C₂³×C₁₃	C₂×D₄	1	2	yes	—
91	C₂³×C₁₃	C₂×D₄	1	2	yes	—
92	C₂³×C₁₃	#41	1	4	yes	—
93	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
94	Q₈×C₁₃	—	4	4	no	—
95	D₄×C₁₃	—	4	4	no	—
96	D₄×C₁₃	—	4	8	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	2	2	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	2	2	no	—
102	C₂×C₄×C₁₃	—	2	2	no	—
103	C₂×C₄×C₁₃	—	2	2	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	—	4	4	no	—
115	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
116	C₂×#124	C₂×C₁₃	1	1	yes	characteristic, complemented
117	C₂×#120	C₂×C₁₃	1	1	yes	characteristic
118	#397	C₂×C₁₃	1	2	yes	complemented
119	#394	C₂×C₁₃	1	2	yes	complemented
120	#393	C₂×C₁₃	1	2	yes	complemented
121	#388	C₂×C₁₃	1	2	yes	complemented
122	#386	C₂×C₁₃	1	1	yes	characteristic, complemented
123	#386	C₂×C₁₃	1	2	yes	complemented
124	#379	C₂×C₁₃	1	2	yes	complemented
125	C₂²×C₁₃	—	2	2	no	—
126	C₂²×C₁₃	—	2	2	no	—
127	C₂²×C₁₃	—	2	4	no	—
128	C₂²×C₁₃	—	2	4	no	—
129	C₂²×C₁₃	—	2	4	no	—
130	C₂²×C₁₃	—	2	4	no	—
131	C₂²×C₁₃	—	2	4	no	—
132	C₂²×C₁₃	—	4	16	no	—
133	C₂²×C₁₃	—	4	2	no	—
134	C₂²×C₁₃	—	4	2	no	—
135	C₂²×C₁₃	—	4	2	no	—
136	C₂²×C₁₃	—	4	2	no	—
137	C₂²×C₁₃	—	4	2	no	—
138	C₂²×C₁₃	—	4	1	no	—
139	C₂²×C₁₃	—	4	1	no	—
140	C₂²×C₁₃	—	4	1	no	—
141	C₂²×C₁₃	—	4	1	no	—
142	C₂²×C₁₃	—	4	2	no	—
143	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
144	C₂²×C₁₃	C₂×#41	1	2	yes	—
145	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
146	C₂²×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
147	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
148	C₄×C₁₃	—	2	2	no	—
149	C₄×C₁₃	—	4	2	no	—
150	C₄×C₁₃	—	4	2	no	—
151	C₄×C₁₃	—	4	2	no	—
152	C₄×C₁₃	—	4	2	no	—
153	C₄×C₁₃	—	4	2	no	—
154	C₂⁵	C₂²×C₁₃	1	1	yes	characteristic
155	C₂²×Q₈	C₂²×C₁₃	1	1	yes	characteristic, complemented
156	C₂²×D₄	C₂²×C₁₃	1	2	yes	—
157	C₂²×D₄	C₂²×C₁₃	1	1	yes	characteristic
158	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic, complemented
159	#124	—	2	4	no	—
160	#120	—	2	4	no	—
161	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	complemented
162	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
163	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
164	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
165	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
166	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
167	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
168	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
169	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
170	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	complemented
171	C₂×C₂²⋊C₄	C₂²×C₁₃	1	1	yes	characteristic, complemented
172	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic, complemented
173	#95	C₂²×C₁₃	1	2	yes	complemented
174	#95	C₂²×C₁₃	1	2	yes	complemented
175	#95	C₂²×C₁₃	1	2	yes	complemented
176	#95	C₂²×C₁₃	1	2	yes	complemented
177	C₂×C₁₃	—	4	2	no	—
178	C₂×C₁₃	—	4	2	no	—
179	C₂×C₁₃	—	4	4	no	—
180	C₂×C₁₃	#534	1	1	yes	characteristic
181	C₂×C₁₃	#538	1	2	yes	—
182	C₂×C₁₃	#546	1	2	yes	—
183	C₂×C₁₃	#551	1	1	yes	characteristic
184	C₂×C₁₃	H(3,𝔽₄)	1	1	yes	characteristic
185	C₂⁴	—	2	2	no	—
186	C₂⁴	—	2	4	no	—
187	C₂⁴	—	4	2	no	—
188	C₂⁴	—	4	1	no	—
189	C₂⁴	—	4	1	no	—
190	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
191	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
192	C₂⁴	C₂³×C₁₃	1	2	yes	—
193	C₂×Q₈	—	2	2	no	—
194	C₂×Q₈	—	4	1	no	—
195	C₂×Q₈	—	4	1	no	—
196	C₂×D₄	—	2	4	no	—
197	C₂×D₄	—	2	4	no	—
198	C₂×D₄	—	4	1	no	—
199	C₂×D₄	—	4	1	no	—
200	C₂×D₄	—	4	2	no	—
201	C₂×D₄	D₄×C₁₃	1	4	yes	—
202	C₂²×C₄	—	4	1	no	—
203	C₂²×C₄	—	4	2	no	—
204	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
205	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
206	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
207	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
208	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
209	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
210	C₄⋊C₄	—	4	2	no	—
211	C₂²⋊C₄	—	2	2	no	—
212	C₂²⋊C₄	—	2	4	no	—
213	C₂²⋊C₄	—	2	4	no	—
214	C₂²⋊C₄	—	2	4	no	—
215	C₂²⋊C₄	—	2	4	no	—
216	C₂²⋊C₄	—	4	2	no	—
217	C₂²⋊C₄	—	4	2	no	—
218	C₂²⋊C₄	—	4	2	no	—
219	C₂²⋊C₄	—	4	2	no	—
220	C₂²⋊C₄	—	4	1	no	—
221	C₄²	—	2	2	no	—
222	C₁₃	#2684	1	1	yes	characteristic, direct product factor, Sylow
223	C₂³	—	2	2	no	—
224	C₂³	—	2	2	no	—
225	C₂³	—	2	2	no	—
226	C₂³	—	2	2	no	—
227	C₂³	—	2	1	no	—
228	C₂³	—	2	2	no	—
229	C₂³	—	2	2	no	—
230	C₂³	—	2	2	no	—
231	C₂³	—	2	2	no	—
232	C₂³	—	2	1	no	—
233	C₂³	—	2	2	no	—
234	C₂³	—	2	1	no	—
235	C₂³	—	2	1	no	—
236	C₂³	—	2	2	no	—
237	C₂³	—	4	8	no	—
238	C₂³	—	4	4	no	—
239	C₂³	—	4	4	no	—
240	C₂³	—	4	8	no	—
241	C₂³	—	4	4	no	—
242	C₂³	C₂×D₄×C₁₃	1	2	yes	—
243	C₂³	C₂×D₄×C₁₃	1	2	yes	—
244	C₂³	C₁₃×#41	1	4	yes	—
245	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
246	Q₈	—	4	4	no	—
247	D₄	—	4	4	no	—
248	D₄	—	4	8	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	1	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	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	—	4	4	no	—
267	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	2	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	4	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	4	16	no	—
276	C₂²	—	4	2	no	—
277	C₂²	—	4	2	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	4	2	no	—
280	C₂²	—	4	2	no	—
281	C₂²	—	4	1	no	—
282	C₂²	—	4	1	no	—
283	C₂²	—	4	1	no	—
284	C₂²	—	4	1	no	—
285	C₂²	—	4	2	no	—
286	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
287	C₂²	C₂×C₁₃×#41	1	2	yes	—
288	C₂²	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
289	C₂²	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
290	C₂²	C₁₃×H(5,𝔽₂)	1	2	yes	—
291	C₄	—	2	2	no	—
292	C₄	—	4	2	no	—
293	C₄	—	4	2	no	—
294	C₄	—	4	2	no	—
295	C₄	—	4	2	no	—
296	C₄	—	4	2	no	—
297	C₂	—	4	2	no	—
298	C₂	—	4	4	no	—
299	C₂	—	4	2	no	—
300	C₂	C₁₃×#534	1	1	yes	characteristic
301	C₂	C₁₃×#538	1	2	yes	—
302	C₂	C₁₃×#546	1	2	yes	—
303	C₂	C₁₃×#551	1	1	yes	characteristic
304	C₂	C₁₃×H(3,𝔽₄)	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#120	C₂
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×#124	C₂	semidirect
3. C₁₃×#397	C₂	semidirect
4. C₁₃×#394	C₂	semidirect
5. C₁₃×#393	C₂	semidirect
6. C₁₃×#388	C₂	semidirect
7. C₁₃×#386	C₂	semidirect
8. C₁₃×#386	C₂	semidirect
9. C₁₃×#379	C₂	semidirect
11. C₂²×Q₈×C₁₃	C₂²	semidirect
14. C₂³×C₄×C₁₃	C₂²	semidirect
17. C₂×C₁₃×C₄⋊C₄	C₂²	semidirect
26. C₂×C₁₃×C₂²⋊C₄	C₂²	semidirect
27. C₂×C₁₃×C₂²⋊C₄	C₂²	semidirect
28. C₂×C₄²×C₁₃	C₂²	semidirect
29. C₁₃×#95	C₂²	semidirect
30. C₁₃×#95	C₂²	semidirect
31. C₁₃×#95	C₂²	semidirect
32. C₁₃×#95	C₂²	semidirect
70. #2684	C₁₃	direct (N × H)
116. C₂×#124	C₂×C₁₃	semidirect
118. #397	C₂×C₁₃	semidirect
119. #394	C₂×C₁₃	semidirect
120. #393	C₂×C₁₃	semidirect
121. #388	C₂×C₁₃	semidirect
122. #386	C₂×C₁₃	semidirect
123. #386	C₂×C₁₃	semidirect
124. #379	C₂×C₁₃	semidirect
155. C₂²×Q₈	C₂²×C₁₃	semidirect
158. C₂³×C₄	C₂²×C₁₃	semidirect
161. C₂×C₄⋊C₄	C₂²×C₁₃	semidirect
170. C₂×C₂²⋊C₄	C₂²×C₁₃	semidirect
171. C₂×C₂²⋊C₄	C₂²×C₁₃	semidirect
172. C₂×C₄²	C₂²×C₁₃	semidirect
173. #95	C₂²×C₁₃	semidirect
174. #95	C₂²×C₁₃	semidirect
175. #95	C₂²×C₁₃	semidirect
176. #95	C₂²×C₁₃	semidirect
222. C₁₃	#2684	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×#124	2
C₂×C₁₃×#120	2
C₁₃×#397	2
C₁₃×#394	2
C₁₃×#393	2
C₁₃×#388	2
C₁₃×#386	2
C₁₃×#379	2
#2684	13

Subgroups — C13×#2684 #13111742

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2684 #13111742