Subgroups — C₁₃×#34632 #15720971

Order 3328.

5346 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	C₂³×C₁₃×#41	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#525	C₂	1	3	yes	maximal, complemented
3	C₁₃×#2616	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2615	C₂	1	4	yes	maximal, complemented
5	C₁₃×#2614	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2613	C₂	1	6	yes	maximal, complemented
7	C₁₃×#2388	C₂	1	8	yes	maximal, complemented
8	C₁₃×#2280	C₂	1	3	yes	maximal, complemented
9	C₂²×C₁₃×#41	—	2	24	no	—
10	C₂²×C₁₃×#41	C₂²	1	4	yes	—
11	C₂²×C₁₃×#41	C₂²	1	4	yes	—
12	C₂²×C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₂³×Q₈×C₁₃	C₂²	1	1	yes	characteristic
14	C₂³×D₄×C₁₃	C₂²	1	3	yes	—
15	C₂⁴×C₄×C₁₃	C₂²	1	3	yes	—
16	C₁₃×#525	—	2	24	no	—
17	C₂×C₁₃×#123	C₂²	1	6	yes	—
18	C₂×C₁₃×C₂²⋊Q₈	C₂²	1	6	yes	complemented
19	C₂×C₁₃×#121	C₂²	1	6	yes	complemented
20	C₂×C₁₃×#120	C₂²	1	6	yes	—
21	C₂×C₄×D₄×C₁₃	C₂²	1	12	yes	complemented
22	C₂×C₁₃×#117	C₂²	1	3	yes	—
23	C₁₃×#395	C₂²	1	8	yes	complemented
24	C₁₃×#394	C₂²	1	24	yes	complemented
25	C₁₃×#393	C₂²	1	8	yes	complemented
26	C₁₃×#392	C₂²	1	24	yes	complemented
27	C₁₃×#381	C₂²	1	6	yes	—
28	C₁₃×#380	C₂²	1	12	yes	complemented
29	C₁₃×#379	C₂²	1	6	yes	—
30	C₄×C₁₃×C₂²⋊C₄	C₂²	1	12	yes	complemented
31	C₂⁵×C₁₃	C₂³	1	3	yes	—
32	C₂×C₁₃×#41	—	2	128	no	—
33	C₂×C₁₃×#41	—	2	48	no	—
34	C₂×C₁₃×#41	—	2	48	no	—
35	C₂×C₁₃×#41	—	2	12	no	—
36	C₂²×Q₈×C₁₃	—	2	12	no	—
37	C₂²×Q₈×C₁₃	C₂³	1	4	yes	—
38	C₂²×Q₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₂²×D₄×C₁₃	—	2	24	no	—
40	C₂²×D₄×C₁₃	—	2	12	no	—
41	C₂²×D₄×C₁₃	C₂³	1	12	yes	—
42	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
43	C₂²×D₄×C₁₃	C₂³	1	1	yes	characteristic
44	C₂²×D₄×C₁₃	C₂³	1	12	yes	—
45	C₂³×C₄×C₁₃	—	2	24	no	—
46	C₂³×C₄×C₁₃	D₄	1	24	yes	complemented
47	C₂³×C₄×C₁₃	C₂³	1	2	yes	—
48	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
49	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
50	C₂³×C₄×C₁₃	C₂³	1	6	yes	—
51	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
52	C₁₃×#123	—	2	24	no	—
53	C₁₃×C₂²⋊Q₈	—	2	24	no	—
54	C₁₃×#121	—	2	24	no	—
55	C₁₃×#120	—	2	24	no	—
56	C₄×D₄×C₁₃	—	2	48	no	—
57	C₁₃×#117	—	2	12	no	—
58	C₂×C₁₃×C₄⋊C₄	C₂³	1	6	yes	—
59	C₂×C₁₃×C₄⋊C₄	C₂³	1	12	yes	—
60	C₂×C₁₃×C₂²⋊C₄	C₂³	1	6	yes	—
61	C₂×C₁₃×C₂²⋊C₄	C₂³	1	12	yes	—
62	C₂×C₁₃×C₂²⋊C₄	C₂³	1	12	yes	—
63	C₂×C₁₃×C₂²⋊C₄	C₂³	1	24	yes	—
64	C₂×C₄²×C₁₃	C₂³	1	6	yes	—
65	C₁₃×#95	C₂³	1	24	yes	—
66	#34632	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂⁴×C₁₃	—	2	12	no	—
68	C₂⁴×C₁₃	—	4	12	no	—
69	C₂⁴×C₁₃	C₂×D₄	1	12	yes	—
70	C₂⁴×C₁₃	C₂⁴	1	2	yes	—
71	C₂⁴×C₁₃	C₂⁴	1	6	yes	—
72	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
73	C₁₃×#41	—	2	128	no	—
74	C₁₃×#41	—	2	128	no	—
75	C₁₃×#41	—	4	16	no	—
76	C₂×Q₈×C₁₃	—	2	32	no	—
77	C₂×Q₈×C₁₃	—	2	24	no	—
78	C₂×Q₈×C₁₃	—	2	6	no	—
79	C₂×D₄×C₁₃	—	2	96	no	—
80	C₂×D₄×C₁₃	—	2	48	no	—
81	C₂×D₄×C₁₃	—	2	24	no	—
82	C₂×D₄×C₁₃	—	2	12	no	—
83	C₂×D₄×C₁₃	—	2	24	no	—
84	C₂×D₄×C₁₃	—	2	6	no	—
85	C₂×D₄×C₁₃	—	4	24	no	—
86	C₂²×C₄×C₁₃	—	2	12	no	—
87	C₂²×C₄×C₁₃	—	2	96	no	—
88	C₂²×C₄×C₁₃	—	2	24	no	—
89	C₂²×C₄×C₁₃	—	2	24	no	—
90	C₂²×C₄×C₁₃	—	2	12	no	—
91	C₂²×C₄×C₁₃	C₂×D₄	1	24	yes	—
92	C₂²×C₄×C₁₃	C₂×D₄	1	24	yes	—
93	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
94	C₂²×C₄×C₁₃	C₂×D₄	1	12	yes	—
95	C₂²×C₄×C₁₃	C₂⁴	1	12	yes	—
96	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
97	C₂²×C₄×C₁₃	C₂⁴	1	6	yes	—
98	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
99	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
100	C₁₃×C₄⋊C₄	—	2	12	no	—
101	C₁₃×C₄⋊C₄	—	2	24	no	—
102	C₁₃×C₂²⋊C₄	—	2	12	no	—
103	C₁₃×C₂²⋊C₄	—	2	24	no	—
104	C₁₃×C₂²⋊C₄	—	2	24	no	—
105	C₁₃×C₂²⋊C₄	—	4	24	no	—
106	C₄²×C₁₃	—	2	12	no	—
107	C₂³×#41	C₂×C₁₃	1	1	yes	characteristic, complemented
108	C₂×#525	C₂×C₁₃	1	3	yes	complemented
109	#2616	C₂×C₁₃	1	4	yes	complemented
110	#2615	C₂×C₁₃	1	4	yes	complemented
111	#2614	C₂×C₁₃	1	2	yes	complemented
112	#2613	C₂×C₁₃	1	6	yes	complemented
113	#2388	C₂×C₁₃	1	8	yes	complemented
114	#2280	C₂×C₁₃	1	3	yes	complemented
115	C₂³×C₁₃	—	2	4	no	—
116	C₂³×C₁₃	—	2	24	no	—
117	C₂³×C₁₃	—	2	12	no	—
118	C₂³×C₁₃	—	2	12	no	—
119	C₂³×C₁₃	—	2	12	no	—
120	C₂³×C₁₃	—	2	6	no	—
121	C₂³×C₁₃	—	2	8	no	—
122	C₂³×C₁₃	—	4	24	no	—
123	C₂³×C₁₃	—	4	48	no	—
124	C₂³×C₁₃	C₂²×D₄	1	12	yes	—
125	C₂³×C₁₃	C₂²×D₄	1	6	yes	—
126	C₂³×C₁₃	C₂⁵	1	1	yes	characteristic
127	Q₈×C₁₃	—	2	32	no	—
128	Q₈×C₁₃	—	4	4	no	—
129	D₄×C₁₃	—	2	96	no	—
130	D₄×C₁₃	—	4	48	no	—
131	D₄×C₁₃	—	4	8	no	—
132	D₄×C₁₃	—	4	4	no	—
133	C₂×C₄×C₁₃	—	2	24	no	—
134	C₂×C₄×C₁₃	—	2	12	no	—
135	C₂×C₄×C₁₃	—	2	48	no	—
136	C₂×C₄×C₁₃	—	2	48	no	—
137	C₂×C₄×C₁₃	—	2	12	no	—
138	C₂×C₄×C₁₃	—	2	6	no	—
139	C₂×C₄×C₁₃	—	2	24	no	—
140	C₂×C₄×C₁₃	—	4	8	no	—
141	C₂×C₄×C₁₃	#120	1	16	yes	—
142	C₂×C₄×C₁₃	C₂²×D₄	1	12	yes	—
143	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
144	C₂×C₄×C₁₃	C₂²×D₄	1	6	yes	—
145	C₂²×#41	—	2	24	no	—
146	C₂²×#41	C₂²×C₁₃	1	4	yes	—
147	C₂²×#41	C₂²×C₁₃	1	4	yes	—
148	C₂²×#41	C₂²×C₁₃	1	1	yes	characteristic
149	C₂³×Q₈	C₂²×C₁₃	1	1	yes	characteristic
150	C₂³×D₄	C₂²×C₁₃	1	3	yes	—
151	C₂⁴×C₄	C₂²×C₁₃	1	3	yes	—
152	#525	—	2	24	no	—
153	C₂×#123	C₂²×C₁₃	1	6	yes	—
154	C₂×C₂²⋊Q₈	C₂²×C₁₃	1	6	yes	complemented
155	C₂×#121	C₂²×C₁₃	1	6	yes	complemented
156	C₂×#120	C₂²×C₁₃	1	6	yes	—
157	C₂×C₄×D₄	C₂²×C₁₃	1	12	yes	complemented
158	C₂×#117	C₂²×C₁₃	1	3	yes	—
159	#395	C₂²×C₁₃	1	8	yes	complemented
160	#394	C₂²×C₁₃	1	24	yes	complemented
161	#393	C₂²×C₁₃	1	8	yes	complemented
162	#392	C₂²×C₁₃	1	24	yes	complemented
163	#381	C₂²×C₁₃	1	6	yes	—
164	#380	C₂²×C₁₃	1	12	yes	complemented
165	#379	C₂²×C₁₃	1	6	yes	—
166	C₄×C₂²⋊C₄	C₂²×C₁₃	1	12	yes	complemented
167	C₂²×C₁₃	—	2	12	no	—
168	C₂²×C₁₃	—	2	12	no	—
169	C₂²×C₁₃	—	2	12	no	—
170	C₂²×C₁₃	—	2	12	no	—
171	C₂²×C₁₃	—	4	48	no	—
172	C₂²×C₁₃	—	4	24	no	—
173	C₂²×C₁₃	—	4	2	no	—
174	C₂²×C₁₃	—	4	12	no	—
175	C₂²×C₁₃	C₂×#120	1	4	yes	—
176	C₂²×C₁₃	C₂³×D₄	1	3	yes	—
177	C₂²×C₁₃	#585	1	4	yes	—
178	C₄×C₁₃	—	2	12	no	—
179	C₄×C₁₃	—	4	2	no	—
180	C₄×C₁₃	—	4	12	no	—
181	C₄×C₁₃	C₂×#120	1	4	yes	—
182	C₄×C₁₃	C₂×#120	1	4	yes	—
183	C₂⁵	C₂³×C₁₃	1	3	yes	—
184	C₂×#41	—	2	128	no	—
185	C₂×#41	—	2	48	no	—
186	C₂×#41	—	2	48	no	—
187	C₂×#41	—	2	12	no	—
188	C₂²×Q₈	—	2	12	no	—
189	C₂²×Q₈	C₂³×C₁₃	1	1	yes	characteristic
190	C₂²×Q₈	C₂³×C₁₃	1	4	yes	—
191	C₂²×D₄	—	2	24	no	—
192	C₂²×D₄	—	2	12	no	—
193	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
194	C₂²×D₄	C₂³×C₁₃	1	1	yes	characteristic
195	C₂²×D₄	C₂³×C₁₃	1	12	yes	—
196	C₂²×D₄	C₂³×C₁₃	1	12	yes	—
197	C₂³×C₄	—	2	24	no	—
198	C₂³×C₄	D₄×C₁₃	1	24	yes	complemented
199	C₂³×C₄	C₂³×C₁₃	1	2	yes	—
200	C₂³×C₄	C₂³×C₁₃	1	6	yes	—
201	C₂³×C₄	C₂³×C₁₃	1	6	yes	—
202	C₂³×C₄	C₂³×C₁₃	1	3	yes	—
203	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
204	#123	—	2	24	no	—
205	C₂²⋊Q₈	—	2	24	no	—
206	#121	—	2	24	no	—
207	#120	—	2	24	no	—
208	C₄×D₄	—	2	48	no	—
209	#117	—	2	12	no	—
210	C₂×C₄⋊C₄	C₂³×C₁₃	1	6	yes	—
211	C₂×C₄⋊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	12	yes	—
215	C₂×C₂²⋊C₄	C₂³×C₁₃	1	24	yes	—
216	C₂×C₄²	C₂³×C₁₃	1	6	yes	—
217	#95	C₂³×C₁₃	1	24	yes	—
218	C₂×C₁₃	—	2	4	no	—
219	C₂×C₁₃	—	4	12	no	—
220	C₂×C₁₃	—	8	2	no	—
221	C₂×C₁₃	C₂²×#120	1	1	yes	characteristic
222	C₂×C₁₃	#3449	1	6	yes	—
223	C₂⁴	—	2	12	no	—
224	C₂⁴	—	4	12	no	—
225	C₂⁴	C₂×D₄×C₁₃	1	12	yes	—
226	C₂⁴	C₂⁴×C₁₃	1	2	yes	—
227	C₂⁴	C₂⁴×C₁₃	1	6	yes	—
228	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
229	#41	—	2	128	no	—
230	#41	—	2	128	no	—
231	#41	—	4	16	no	—
232	C₂×Q₈	—	2	32	no	—
233	C₂×Q₈	—	2	24	no	—
234	C₂×Q₈	—	2	6	no	—
235	C₂×D₄	—	2	96	no	—
236	C₂×D₄	—	2	48	no	—
237	C₂×D₄	—	2	24	no	—
238	C₂×D₄	—	2	12	no	—
239	C₂×D₄	—	2	24	no	—
240	C₂×D₄	—	2	6	no	—
241	C₂×D₄	—	4	24	no	—
242	C₂²×C₄	—	2	12	no	—
243	C₂²×C₄	—	2	96	no	—
244	C₂²×C₄	—	2	24	no	—
245	C₂²×C₄	—	2	24	no	—
246	C₂²×C₄	—	2	12	no	—
247	C₂²×C₄	C₂×D₄×C₁₃	1	24	yes	—
248	C₂²×C₄	C₂×D₄×C₁₃	1	24	yes	—
249	C₂²×C₄	C₂×D₄×C₁₃	1	12	yes	—
250	C₂²×C₄	C₂×D₄×C₁₃	1	12	yes	—
251	C₂²×C₄	C₂⁴×C₁₃	1	12	yes	—
252	C₂²×C₄	C₂⁴×C₁₃	1	2	yes	—
253	C₂²×C₄	C₂⁴×C₁₃	1	6	yes	—
254	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
255	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
256	C₄⋊C₄	—	2	12	no	—
257	C₄⋊C₄	—	2	24	no	—
258	C₂²⋊C₄	—	2	12	no	—
259	C₂²⋊C₄	—	2	24	no	—
260	C₂²⋊C₄	—	2	24	no	—
261	C₂²⋊C₄	—	4	24	no	—
262	C₄²	—	2	12	no	—
263	C₁₃	#34632	1	1	yes	characteristic, direct product factor, Sylow
264	C₂³	—	2	4	no	—
265	C₂³	—	2	24	no	—
266	C₂³	—	2	12	no	—
267	C₂³	—	2	12	no	—
268	C₂³	—	2	12	no	—
269	C₂³	—	2	6	no	—
270	C₂³	—	2	8	no	—
271	C₂³	—	4	24	no	—
272	C₂³	—	4	48	no	—
273	C₂³	C₂²×D₄×C₁₃	1	12	yes	—
274	C₂³	C₂²×D₄×C₁₃	1	6	yes	—
275	C₂³	C₂⁵×C₁₃	1	1	yes	characteristic
276	Q₈	—	2	32	no	—
277	Q₈	—	4	4	no	—
278	D₄	—	2	96	no	—
279	D₄	—	4	48	no	—
280	D₄	—	4	8	no	—
281	D₄	—	4	4	no	—
282	C₂×C₄	—	2	24	no	—
283	C₂×C₄	—	2	12	no	—
284	C₂×C₄	—	2	48	no	—
285	C₂×C₄	—	2	48	no	—
286	C₂×C₄	—	2	12	no	—
287	C₂×C₄	—	2	6	no	—
288	C₂×C₄	—	2	24	no	—
289	C₂×C₄	—	4	8	no	—
290	C₂×C₄	C₁₃×#120	1	16	yes	—
291	C₂×C₄	C₂²×D₄×C₁₃	1	12	yes	—
292	C₂×C₄	C₂²×D₄×C₁₃	1	6	yes	—
293	C₂×C₄	C₂²×D₄×C₁₃	1	6	yes	—
294	C₂²	—	2	12	no	—
295	C₂²	—	2	12	no	—
296	C₂²	—	2	12	no	—
297	C₂²	—	2	12	no	—
298	C₂²	—	4	48	no	—
299	C₂²	—	4	24	no	—
300	C₂²	—	4	2	no	—
301	C₂²	—	4	12	no	—
302	C₂²	C₂×C₁₃×#120	1	4	yes	—
303	C₂²	C₂³×D₄×C₁₃	1	3	yes	—
304	C₂²	C₁₃×#585	1	4	yes	—
305	C₄	—	2	12	no	—
306	C₄	—	4	2	no	—
307	C₄	—	4	12	no	—
308	C₄	C₂×C₁₃×#120	1	4	yes	—
309	C₄	C₂×C₁₃×#120	1	4	yes	—
310	C₂	—	2	4	no	—
311	C₂	—	4	12	no	—
312	C₂	—	8	2	no	—
313	C₂	C₂²×C₁₃×#120	1	1	yes	characteristic
314	C₂	C₁₃×#3449	1	6	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁵×C₁₃
socle	C₂³×C₁₃	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₁₃×#41	C₂	semidirect
2. C₂×C₁₃×#525	C₂	semidirect
3. C₁₃×#2616	C₂	semidirect
4. C₁₃×#2615	C₂	semidirect
5. C₁₃×#2614	C₂	semidirect
6. C₁₃×#2613	C₂	semidirect
7. C₁₃×#2388	C₂	semidirect
8. C₁₃×#2280	C₂	semidirect
18. C₂×C₁₃×C₂²⋊Q₈	C₂²	semidirect
19. C₂×C₁₃×#121	C₂²	semidirect
21. C₂×C₄×D₄×C₁₃	C₂²	semidirect
23. C₁₃×#395	C₂²	semidirect
24. C₁₃×#394	C₂²	semidirect
25. C₁₃×#393	C₂²	semidirect
26. C₁₃×#392	C₂²	semidirect
28. C₁₃×#380	C₂²	semidirect
30. C₄×C₁₃×C₂²⋊C₄	C₂²	semidirect
46. C₂³×C₄×C₁₃	D₄	semidirect
66. #34632	C₁₃	direct (N × H)
107. C₂³×#41	C₂×C₁₃	semidirect
108. C₂×#525	C₂×C₁₃	semidirect
109. #2616	C₂×C₁₃	semidirect
110. #2615	C₂×C₁₃	semidirect
111. #2614	C₂×C₁₃	semidirect
112. #2613	C₂×C₁₃	semidirect
113. #2388	C₂×C₁₃	semidirect
114. #2280	C₂×C₁₃	semidirect
154. C₂×C₂²⋊Q₈	C₂²×C₁₃	semidirect
155. C₂×#121	C₂²×C₁₃	semidirect
157. C₂×C₄×D₄	C₂²×C₁₃	semidirect
159. #395	C₂²×C₁₃	semidirect
160. #394	C₂²×C₁₃	semidirect
161. #393	C₂²×C₁₃	semidirect
162. #392	C₂²×C₁₃	semidirect
164. #380	C₂²×C₁₃	semidirect
166. C₄×C₂²⋊C₄	C₂²×C₁₃	semidirect
198. C₂³×C₄	D₄×C₁₃	semidirect
263. C₁₃	#34632	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂³×C₁₃×#41	2
C₂×C₁₃×#525	2
C₁₃×#2616	2
C₁₃×#2615	2
C₁₃×#2614	2
C₁₃×#2613	2
C₁₃×#2388	2
C₁₃×#2280	2
#34632	13

Subgroups — C13×#34632 #15720971

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#34632 #15720971