Subgroups — C₁₃×#60045 #15746384

Order 3328.

2490 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2804	C₂	1	3	yes	maximal, complemented
2	C₁₃×#2679	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2598	C₂	1	6	yes	maximal, complemented
4	C₁₃×#2572	C₂	1	6	yes	maximal, complemented
5	C₁₃×#2404	C₂	1	6	yes	maximal, complemented
6	C₁₃×#2358	C₂	1	6	yes	maximal, complemented
7	C₁₃×#2297	C₂	1	3	yes	maximal, complemented
8	C₂²×C₁₃×#41	C₂²	1	3	yes	—
9	C₂×C₁₃×#120	C₂²	1	6	yes	—
10	C₂×C₁₃×#120	C₂²	1	2	yes	complemented
11	C₂×C₄×D₄×C₁₃	C₂²	1	6	yes	—
12	C₂×C₁₃×#117	C₂²	1	3	yes	complemented
13	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	6	yes	complemented
14	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic, complemented
15	C₁₃×#400	C₂²	1	6	yes	complemented
16	C₁₃×#399	C₂²	1	2	yes	complemented
17	C₁₃×#394	C₂²	1	6	yes	—
18	C₁₃×#393	C₂²	1	12	yes	complemented
19	C₁₃×#392	C₂²	1	6	yes	—
20	C₁₃×#391	C₂²	1	3	yes	complemented
21	C₁₃×#391	C₂²	1	3	yes	complemented
22	C₁₃×#390	C₂²	1	3	yes	—
23	C₁₃×#390	C₂²	1	3	yes	—
24	C₁₃×#388	C₂²	1	12	yes	complemented
25	C₁₃×#386	C₂²	1	6	yes	—
26	C₁₃×#386	C₂²	1	12	yes	—
27	C₁₃×#386	C₂²	1	12	yes	complemented
28	C₁₃×#386	C₂²	1	6	yes	complemented
29	C₁₃×#385	C₂²	1	12	yes	complemented
30	C₁₃×#383	C₂²	1	6	yes	complemented
31	C₁₃×#381	C₂²	1	6	yes	complemented
32	C₄×C₁₃×C₂²⋊C₄	C₂²	1	12	yes	complemented
33	C₂⁵×C₁₃	C₂³	1	2	yes	—
34	C₂×C₁₃×#41	—	2	12	no	—
35	C₂×C₁₃×#41	—	4	6	no	—
36	C₂×C₁₃×#41	—	4	6	no	—
37	C₂²×Q₈×C₁₃	C₂³	1	3	yes	—
38	C₂²×D₄×C₁₃	C₂³	1	6	yes	—
39	C₂²×D₄×C₁₃	C₂³	1	6	yes	—
40	C₂²×D₄×C₁₃	C₂³	1	3	yes	—
41	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
42	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
43	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
44	C₁₃×#120	—	2	24	no	—
45	C₁₃×#120	—	4	4	no	—
46	C₄×D₄×C₁₃	—	4	12	no	—
47	C₁₃×#117	—	4	6	no	—
48	C₂×C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
49	C₂×C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
50	C₂×C₁₃×C₄⋊C₄	C₂³	1	3	yes	—
51	C₂×C₁₃×C₂²⋊C₄	—	4	12	no	—
52	C₂×C₁₃×C₂²⋊C₄	C₂³	1	12	yes	—
53	C₂×C₁₃×C₂²⋊C₄	C₂³	1	12	yes	—
54	C₂×C₁₃×C₂²⋊C₄	C₂³	1	6	yes	—
55	C₂×C₁₃×C₂²⋊C₄	C₂³	1	6	yes	—
56	C₂×C₁₃×C₂²⋊C₄	C₂³	1	12	yes	—
57	C₂×C₁₃×C₂²⋊C₄	C₂³	1	6	yes	—
58	C₂×C₄²×C₁₃	—	4	2	no	—
59	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
60	C₂×C₄²×C₁₃	C₂³	1	3	yes	—
61	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
62	C₁₃×#95	C₂³	1	12	yes	—
63	C₁₃×#95	C₂³	1	12	yes	—
64	C₁₃×#95	C₂³	1	6	yes	—
65	C₁₃×#95	C₂³	1	2	yes	—
66	C₁₃×#95	C₂³	1	12	yes	—
67	#60045	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂⁴×C₁₃	—	2	12	no	—
69	C₂⁴×C₁₃	—	4	6	no	—
70	C₂⁴×C₁₃	—	4	2	no	—
71	C₂⁴×C₁₃	C₂⁴	1	6	yes	—
72	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₁₃×#41	—	4	48	no	—
74	C₂×Q₈×C₁₃	—	2	6	no	—
75	C₂×Q₈×C₁₃	—	4	3	no	—
76	C₂×Q₈×C₁₃	—	4	3	no	—
77	C₂×D₄×C₁₃	—	2	24	no	—
78	C₂×D₄×C₁₃	—	2	12	no	—
79	C₂×D₄×C₁₃	—	2	12	no	—
80	C₂×D₄×C₁₃	—	4	6	no	—
81	C₂×D₄×C₁₃	—	4	3	no	—
82	C₂×D₄×C₁₃	—	4	3	no	—
83	C₂×D₄×C₁₃	—	4	6	no	—
84	C₂×D₄×C₁₃	#41	1	12	yes	complemented
85	C₂²×C₄×C₁₃	—	2	6	no	—
86	C₂²×C₄×C₁₃	—	2	6	no	—
87	C₂²×C₄×C₁₃	—	2	6	no	—
88	C₂²×C₄×C₁₃	—	2	12	no	—
89	C₂²×C₄×C₁₃	—	2	12	no	—
90	C₂²×C₄×C₁₃	—	4	6	no	—
91	C₂²×C₄×C₁₃	—	4	6	no	—
92	C₂²×C₄×C₁₃	—	4	12	no	—
93	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
94	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
95	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
96	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
97	C₂²×C₄×C₁₃	C₂⁴	1	3	yes	—
98	C₁₃×C₄⋊C₄	—	4	3	no	—
99	C₁₃×C₄⋊C₄	—	4	3	no	—
100	C₁₃×C₄⋊C₄	—	4	6	no	—
101	C₁₃×C₂²⋊C₄	—	2	12	no	—
102	C₁₃×C₂²⋊C₄	—	2	24	no	—
103	C₁₃×C₂²⋊C₄	—	2	24	no	—
104	C₁₃×C₂²⋊C₄	—	4	12	no	—
105	C₁₃×C₂²⋊C₄	—	4	6	no	—
106	C₁₃×C₂²⋊C₄	—	4	6	no	—
107	C₄²×C₁₃	—	2	12	no	—
108	C₄²×C₁₃	—	4	3	no	—
109	C₄²×C₁₃	—	4	1	no	—
110	#2804	C₂×C₁₃	1	3	yes	complemented
111	#2679	C₂×C₁₃	1	1	yes	characteristic, complemented
112	#2598	C₂×C₁₃	1	6	yes	complemented
113	#2572	C₂×C₁₃	1	6	yes	complemented
114	#2404	C₂×C₁₃	1	6	yes	complemented
115	#2358	C₂×C₁₃	1	6	yes	complemented
116	#2297	C₂×C₁₃	1	3	yes	complemented
117	C₂³×C₁₃	—	2	12	no	—
118	C₂³×C₁₃	—	2	3	no	—
119	C₂³×C₁₃	—	2	12	no	—
120	C₂³×C₁₃	—	2	3	no	—
121	C₂³×C₁₃	—	2	6	no	—
122	C₂³×C₁₃	—	2	6	no	—
123	C₂³×C₁₃	—	4	24	no	—
124	C₂³×C₁₃	—	8	12	no	—
125	C₂³×C₁₃	—	8	4	no	—
126	C₂³×C₁₃	C₂×#41	1	12	yes	—
127	C₂³×C₁₃	H(5,𝔽₂)	1	2	yes	—
128	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
129	Q₈×C₁₃	—	4	12	no	—
130	D₄×C₁₃	—	4	12	no	—
131	D₄×C₁₃	—	4	24	no	—
132	D₄×C₁₃	—	4	24	no	—
133	C₂×C₄×C₁₃	—	2	24	no	—
134	C₂×C₄×C₁₃	—	2	3	no	—
135	C₂×C₄×C₁₃	—	2	3	no	—
136	C₂×C₄×C₁₃	—	2	3	no	—
137	C₂×C₄×C₁₃	—	2	3	no	—
138	C₂×C₄×C₁₃	—	2	12	no	—
139	C₂×C₄×C₁₃	—	2	6	no	—
140	C₂×C₄×C₁₃	—	2	6	no	—
141	C₂×C₄×C₁₃	—	2	6	no	—
142	C₂×C₄×C₁₃	—	2	12	no	—
143	C₂×C₄×C₁₃	—	2	6	no	—
144	C₂×C₄×C₁₃	—	2	6	no	—
145	C₂×C₄×C₁₃	—	4	24	no	—
146	C₂×C₄×C₁₃	—	8	12	no	—
147	C₂×C₄×C₁₃	C₂×#41	1	6	yes	—
148	C₂×C₄×C₁₃	H(5,𝔽₂)	1	6	yes	—
149	C₂²×#41	C₂²×C₁₃	1	3	yes	—
150	C₂×#120	C₂²×C₁₃	1	6	yes	—
151	C₂×#120	C₂²×C₁₃	1	2	yes	complemented
152	C₂×C₄×D₄	C₂²×C₁₃	1	6	yes	—
153	C₂×#117	C₂²×C₁₃	1	3	yes	complemented
154	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	6	yes	complemented
155	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic, complemented
156	#400	C₂²×C₁₃	1	6	yes	complemented
157	#399	C₂²×C₁₃	1	2	yes	complemented
158	#394	C₂²×C₁₃	1	6	yes	—
159	#393	C₂²×C₁₃	1	12	yes	complemented
160	#392	C₂²×C₁₃	1	6	yes	—
161	#391	C₂²×C₁₃	1	3	yes	complemented
162	#391	C₂²×C₁₃	1	3	yes	complemented
163	#390	C₂²×C₁₃	1	3	yes	—
164	#390	C₂²×C₁₃	1	3	yes	—
165	#388	C₂²×C₁₃	1	12	yes	complemented
166	#386	C₂²×C₁₃	1	12	yes	—
167	#386	C₂²×C₁₃	1	6	yes	—
168	#386	C₂²×C₁₃	1	6	yes	complemented
169	#386	C₂²×C₁₃	1	12	yes	complemented
170	#385	C₂²×C₁₃	1	12	yes	complemented
171	#383	C₂²×C₁₃	1	6	yes	complemented
172	#381	C₂²×C₁₃	1	6	yes	complemented
173	C₄×C₂²⋊C₄	C₂²×C₁₃	1	12	yes	complemented
174	C₂²×C₁₃	—	2	24	no	—
175	C₂²×C₁₃	—	2	6	no	—
176	C₂²×C₁₃	—	2	12	no	—
177	C₂²×C₁₃	—	4	3	no	—
178	C₂²×C₁₃	—	4	6	no	—
179	C₂²×C₁₃	—	4	12	no	—
180	C₂²×C₁₃	—	4	6	no	—
181	C₂²×C₁₃	—	4	1	no	—
182	C₂²×C₁₃	—	8	16	no	—
183	C₂²×C₁₃	C₂²×#41	1	3	yes	—
184	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	3	yes	—
185	C₂²×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
186	C₄×C₁₃	—	2	6	no	—
187	C₄×C₁₃	—	2	6	no	—
188	C₄×C₁₃	—	4	6	no	—
189	C₄×C₁₃	—	4	6	no	—
190	C₄×C₁₃	—	4	6	no	—
191	C₂⁵	C₂³×C₁₃	1	2	yes	—
192	C₂×#41	—	2	12	no	—
193	C₂×#41	—	4	6	no	—
194	C₂×#41	—	4	6	no	—
195	C₂²×Q₈	C₂³×C₁₃	1	3	yes	—
196	C₂²×D₄	C₂³×C₁₃	1	6	yes	—
197	C₂²×D₄	C₂³×C₁₃	1	6	yes	—
198	C₂²×D₄	C₂³×C₁₃	1	3	yes	—
199	C₂³×C₄	C₂³×C₁₃	1	3	yes	—
200	C₂³×C₄	C₂³×C₁₃	1	6	yes	—
201	C₂³×C₄	C₂³×C₁₃	1	6	yes	—
202	#120	—	2	24	no	—
203	#120	—	4	4	no	—
204	C₄×D₄	—	4	12	no	—
205	#117	—	4	6	no	—
206	C₂×C₄⋊C₄	C₂³×C₁₃	1	3	yes	—
207	C₂×C₄⋊C₄	C₂³×C₁₃	1	3	yes	—
208	C₂×C₄⋊C₄	C₂³×C₁₃	1	6	yes	—
209	C₂×C₂²⋊C₄	—	4	12	no	—
210	C₂×C₂²⋊C₄	C₂³×C₁₃	1	12	yes	—
211	C₂×C₂²⋊C₄	C₂³×C₁₃	1	12	yes	—
212	C₂×C₂²⋊C₄	C₂³×C₁₃	1	6	yes	—
213	C₂×C₂²⋊C₄	C₂³×C₁₃	1	12	yes	—
214	C₂×C₂²⋊C₄	C₂³×C₁₃	1	6	yes	—
215	C₂×C₂²⋊C₄	C₂³×C₁₃	1	6	yes	—
216	C₂×C₄²	—	4	2	no	—
217	C₂×C₄²	C₂³×C₁₃	1	6	yes	—
218	C₂×C₄²	C₂³×C₁₃	1	3	yes	—
219	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
220	#95	C₂³×C₁₃	1	12	yes	—
221	#95	C₂³×C₁₃	1	12	yes	—
222	#95	C₂³×C₁₃	1	6	yes	—
223	#95	C₂³×C₁₃	1	2	yes	—
224	#95	C₂³×C₁₃	1	12	yes	—
225	C₂×C₁₃	—	4	2	no	—
226	C₂×C₁₃	—	4	12	no	—
227	C₂×C₁₃	#3481	1	3	yes	—
228	C₂×C₁₃	#3490	1	3	yes	—
229	C₂×C₁₃	#3549	1	1	yes	characteristic
230	C₂⁴	—	2	12	no	—
231	C₂⁴	—	4	6	no	—
232	C₂⁴	—	4	2	no	—
233	C₂⁴	C₂⁴×C₁₃	1	6	yes	—
234	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
235	#41	—	4	48	no	—
236	C₂×Q₈	—	2	6	no	—
237	C₂×Q₈	—	4	3	no	—
238	C₂×Q₈	—	4	3	no	—
239	C₂×D₄	—	2	24	no	—
240	C₂×D₄	—	2	12	no	—
241	C₂×D₄	—	2	12	no	—
242	C₂×D₄	—	4	6	no	—
243	C₂×D₄	—	4	3	no	—
244	C₂×D₄	—	4	3	no	—
245	C₂×D₄	—	4	6	no	—
246	C₂×D₄	C₁₃×#41	1	12	yes	complemented
247	C₂²×C₄	—	2	6	no	—
248	C₂²×C₄	—	2	6	no	—
249	C₂²×C₄	—	2	6	no	—
250	C₂²×C₄	—	2	12	no	—
251	C₂²×C₄	—	2	12	no	—
252	C₂²×C₄	—	4	6	no	—
253	C₂²×C₄	—	4	6	no	—
254	C₂²×C₄	—	4	12	no	—
255	C₂²×C₄	C₂⁴×C₁₃	1	6	yes	—
256	C₂²×C₄	C₂⁴×C₁₃	1	6	yes	—
257	C₂²×C₄	C₂⁴×C₁₃	1	6	yes	—
258	C₂²×C₄	C₂⁴×C₁₃	1	3	yes	—
259	C₂²×C₄	C₂⁴×C₁₃	1	3	yes	—
260	C₄⋊C₄	—	4	3	no	—
261	C₄⋊C₄	—	4	3	no	—
262	C₄⋊C₄	—	4	6	no	—
263	C₂²⋊C₄	—	2	12	no	—
264	C₂²⋊C₄	—	2	24	no	—
265	C₂²⋊C₄	—	2	24	no	—
266	C₂²⋊C₄	—	4	12	no	—
267	C₂²⋊C₄	—	4	6	no	—
268	C₂²⋊C₄	—	4	6	no	—
269	C₄²	—	2	12	no	—
270	C₄²	—	4	3	no	—
271	C₄²	—	4	1	no	—
272	C₁₃	#60045	1	1	yes	characteristic, direct product factor, Sylow
273	C₂³	—	2	12	no	—
274	C₂³	—	2	3	no	—
275	C₂³	—	2	12	no	—
276	C₂³	—	2	3	no	—
277	C₂³	—	2	6	no	—
278	C₂³	—	2	6	no	—
279	C₂³	—	4	24	no	—
280	C₂³	—	8	12	no	—
281	C₂³	—	8	4	no	—
282	C₂³	C₂×C₁₃×#41	1	12	yes	—
283	C₂³	C₁₃×H(5,𝔽₂)	1	2	yes	—
284	C₂³	C₂⁵×C₁₃	1	1	yes	characteristic
285	Q₈	—	4	12	no	—
286	D₄	—	4	12	no	—
287	D₄	—	4	24	no	—
288	D₄	—	4	24	no	—
289	C₂×C₄	—	2	24	no	—
290	C₂×C₄	—	2	3	no	—
291	C₂×C₄	—	2	3	no	—
292	C₂×C₄	—	2	3	no	—
293	C₂×C₄	—	2	3	no	—
294	C₂×C₄	—	2	12	no	—
295	C₂×C₄	—	2	6	no	—
296	C₂×C₄	—	2	6	no	—
297	C₂×C₄	—	2	6	no	—
298	C₂×C₄	—	2	12	no	—
299	C₂×C₄	—	2	6	no	—
300	C₂×C₄	—	2	6	no	—
301	C₂×C₄	—	4	24	no	—
302	C₂×C₄	—	8	12	no	—
303	C₂×C₄	C₂×C₁₃×#41	1	6	yes	—
304	C₂×C₄	C₁₃×H(5,𝔽₂)	1	6	yes	—
305	C₂²	—	2	24	no	—
306	C₂²	—	2	6	no	—
307	C₂²	—	2	12	no	—
308	C₂²	—	4	6	no	—
309	C₂²	—	4	1	no	—
310	C₂²	—	4	3	no	—
311	C₂²	—	4	6	no	—
312	C₂²	—	4	12	no	—
313	C₂²	—	8	16	no	—
314	C₂²	C₂²×C₁₃×#41	1	3	yes	—
315	C₂²	C₂×C₁₃×H(5,𝔽₂)	1	3	yes	—
316	C₂²	C₂×C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
317	C₄	—	2	6	no	—
318	C₄	—	2	6	no	—
319	C₄	—	4	6	no	—
320	C₄	—	4	6	no	—
321	C₄	—	4	6	no	—
322	C₂	—	4	12	no	—
323	C₂	—	4	2	no	—
324	C₂	C₁₃×#3481	1	3	yes	—
325	C₂	C₁₃×#3490	1	3	yes	—
326	C₂	C₁₃×#3549	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂⁵
commutator, Frattini, cosocle	C₂³	C₂⁵×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. C₁₃×#2804	C₂	semidirect
2. C₁₃×#2679	C₂	semidirect
3. C₁₃×#2598	C₂	semidirect
4. C₁₃×#2572	C₂	semidirect
5. C₁₃×#2404	C₂	semidirect
6. C₁₃×#2358	C₂	semidirect
7. C₁₃×#2297	C₂	semidirect
10. C₂×C₁₃×#120	C₂²	semidirect
12. C₂×C₁₃×#117	C₂²	semidirect
13. C₂²×C₁₃×C₂²⋊C₄	C₂²	semidirect
14. C₂²×C₄²×C₁₃	C₂²	semidirect
15. C₁₃×#400	C₂²	semidirect
16. C₁₃×#399	C₂²	semidirect
18. C₁₃×#393	C₂²	semidirect
20. C₁₃×#391	C₂²	semidirect
21. C₁₃×#391	C₂²	semidirect
24. C₁₃×#388	C₂²	semidirect
27. C₁₃×#386	C₂²	semidirect
28. C₁₃×#386	C₂²	semidirect
29. C₁₃×#385	C₂²	semidirect
30. C₁₃×#383	C₂²	semidirect
31. C₁₃×#381	C₂²	semidirect
32. C₄×C₁₃×C₂²⋊C₄	C₂²	semidirect
67. #60045	C₁₃	direct (N × H)
84. C₂×D₄×C₁₃	#41	semidirect
110. #2804	C₂×C₁₃	semidirect
111. #2679	C₂×C₁₃	semidirect
112. #2598	C₂×C₁₃	semidirect
113. #2572	C₂×C₁₃	semidirect
114. #2404	C₂×C₁₃	semidirect
115. #2358	C₂×C₁₃	semidirect
116. #2297	C₂×C₁₃	semidirect
151. C₂×#120	C₂²×C₁₃	semidirect
153. C₂×#117	C₂²×C₁₃	semidirect
154. C₂²×C₂²⋊C₄	C₂²×C₁₃	semidirect
155. C₂²×C₄²	C₂²×C₁₃	semidirect
156. #400	C₂²×C₁₃	semidirect
157. #399	C₂²×C₁₃	semidirect
159. #393	C₂²×C₁₃	semidirect
161. #391	C₂²×C₁₃	semidirect
162. #391	C₂²×C₁₃	semidirect
165. #388	C₂²×C₁₃	semidirect
168. #386	C₂²×C₁₃	semidirect
169. #386	C₂²×C₁₃	semidirect
170. #385	C₂²×C₁₃	semidirect
171. #383	C₂²×C₁₃	semidirect
172. #381	C₂²×C₁₃	semidirect
173. C₄×C₂²⋊C₄	C₂²×C₁₃	semidirect
246. C₂×D₄	C₁₃×#41	semidirect
272. C₁₃	#60045	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×#2804	2
C₁₃×#2679	2
C₁₃×#2598	2
C₁₃×#2572	2
C₁₃×#2404	2
C₁₃×#2358	2
C₁₃×#2297	2
#60045	13

Subgroups — C13×#60045 #15746384

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#60045 #15746384