Subgroups — C₁₃×#14694 #15701033

Order 3328.

7034 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂⁵×C₄×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₃×#120	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#386	C₂	1	6	yes	maximal, complemented
4	C₂×C₁₃×#379	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×C₂²⋊#95	C₂	1	6	yes	maximal, complemented
6	C₂⁶×C₁₃	C₂²	1	1	yes	characteristic
7	C₂³×D₄×C₁₃	C₂²	1	3	yes	—
8	C₂⁴×C₄×C₁₃	—	2	4	no	—
9	C₂⁴×C₄×C₁₃	—	2	24	no	—
10	C₂⁴×C₄×C₁₃	C₂²	1	6	yes	—
11	C₂×C₁₃×#120	—	2	4	no	—
12	C₂×C₁₃×#120	—	2	4	no	—
13	C₂×C₁₃×#120	C₄	1	8	yes	complemented
14	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	6	yes	—
15	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	3	yes	—
16	C₂²×C₁₃×C₂²⋊C₄	C₂²	1	4	yes	complemented
17	C₁₃×#386	—	2	24	no	—
18	C₁₃×#379	—	2	4	no	—
19	C₂×C₁₃×#95	C₂²	1	12	yes	complemented
20	C₂⁵×C₁₃	—	2	12	no	—
21	C₂⁵×C₁₃	—	2	12	no	—
22	C₂⁵×C₁₃	C₂×C₄	1	4	yes	—
23	C₂⁵×C₁₃	D₄	1	4	yes	—
24	C₂⁵×C₁₃	D₄	1	4	yes	—
25	C₂⁵×C₁₃	C₂³	1	1	yes	characteristic
26	C₂⁵×C₁₃	C₂³	1	3	yes	—
27	C₂²×D₄×C₁₃	—	2	12	no	—
28	C₂²×D₄×C₁₃	—	2	6	no	—
29	C₂²×D₄×C₁₃	—	2	6	no	—
30	C₂²×D₄×C₁₃	—	2	12	no	—
31	C₂²×D₄×C₁₃	C₂×C₄	1	12	yes	complemented
32	C₂³×C₄×C₁₃	—	2	96	no	—
33	C₂³×C₄×C₁₃	—	2	128	no	—
34	C₂³×C₄×C₁₃	—	2	12	no	—
35	C₂³×C₄×C₁₃	—	2	12	no	—
36	C₂³×C₄×C₁₃	—	2	48	no	—
37	C₂³×C₄×C₁₃	D₄	1	24	yes	complemented
38	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
39	C₂³×C₄×C₁₃	C₂³	1	3	yes	—
40	C₂³×C₄×C₁₃	C₂³	1	4	yes	—
41	C₁₃×#120	—	2	32	no	—
42	C₂×C₁₃×C₂²⋊C₄	—	2	6	no	—
43	C₂×C₁₃×C₂²⋊C₄	—	2	6	no	—
44	C₂×C₁₃×C₂²⋊C₄	—	2	12	no	—
45	C₂×C₁₃×C₂²⋊C₄	—	2	12	no	—
46	C₂×C₁₃×C₂²⋊C₄	—	2	12	no	—
47	C₂×C₁₃×C₂²⋊C₄	—	2	24	no	—
48	C₂×C₁₃×C₂²⋊C₄	C₂×C₄	1	12	yes	complemented
49	C₁₃×#95	—	2	24	no	—
50	#14694	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂⁴×C₁₃	—	2	32	no	—
52	C₂⁴×C₁₃	—	2	96	no	—
53	C₂⁴×C₁₃	—	2	48	no	—
54	C₂⁴×C₁₃	—	2	24	no	—
55	C₂⁴×C₁₃	—	2	48	no	—
56	C₂⁴×C₁₃	—	2	24	no	—
57	C₂⁴×C₁₃	—	2	12	no	—
58	C₂⁴×C₁₃	—	2	12	no	—
59	C₂⁴×C₁₃	—	2	6	no	—
60	C₂⁴×C₁₃	—	2	1	no	—
61	C₂⁴×C₁₃	—	2	1	no	—
62	C₂⁴×C₁₃	—	2	6	no	—
63	C₂⁴×C₁₃	C₂²⋊C₄	1	16	yes	complemented
64	C₂⁴×C₁₃	C₂²×C₄	1	6	yes	—
65	C₂⁴×C₁₃	C₂²×C₄	1	2	yes	complemented
66	C₂⁴×C₁₃	C₂×D₄	1	6	yes	—
67	C₂⁴×C₁₃	C₂×D₄	1	6	yes	—
68	C₂⁴×C₁₃	C₂×D₄	1	12	yes	—
69	C₂⁴×C₁₃	#41	1	12	yes	—
70	C₂⁴×C₁₃	C₂⁴	1	1	yes	characteristic
71	C₂×D₄×C₁₃	—	2	24	no	—
72	C₂×D₄×C₁₃	—	2	48	no	—
73	C₂×D₄×C₁₃	—	4	12	no	—
74	C₂×D₄×C₁₃	—	4	24	no	—
75	C₂×D₄×C₁₃	—	4	12	no	—
76	C₂²×C₄×C₁₃	—	2	256	no	—
77	C₂²×C₄×C₁₃	—	2	96	no	—
78	C₂²×C₄×C₁₃	—	2	96	no	—
79	C₂²×C₄×C₁₃	—	2	48	no	—
80	C₂²×C₄×C₁₃	—	2	4	no	—
81	C₂²×C₄×C₁₃	—	2	12	no	—
82	C₂²×C₄×C₁₃	—	2	96	no	—
83	C₂²×C₄×C₁₃	—	2	6	no	—
84	C₂²×C₄×C₁₃	—	2	3	no	—
85	C₂²×C₄×C₁₃	—	2	3	no	—
86	C₂²×C₄×C₁₃	—	2	6	no	—
87	C₂²×C₄×C₁₃	—	2	12	no	—
88	C₂²×C₄×C₁₃	—	2	4	no	—
89	C₂²×C₄×C₁₃	—	2	12	no	—
90	C₂²×C₄×C₁₃	C₂²×C₄	1	6	yes	complemented
91	C₂²×C₄×C₁₃	C₂×D₄	1	24	yes	—
92	C₁₃×C₂²⋊C₄	—	2	24	no	—
93	C₁₃×C₂²⋊C₄	—	4	24	no	—
94	C₁₃×C₂²⋊C₄	—	4	16	no	—
95	C₂⁵×C₄	C₂×C₁₃	1	1	yes	characteristic, complemented
96	C₂²×#120	C₂×C₁₃	1	1	yes	characteristic
97	C₂×#386	C₂×C₁₃	1	6	yes	complemented
98	C₂×#379	C₂×C₁₃	1	1	yes	characteristic, complemented
99	C₂²⋊#95	C₂×C₁₃	1	6	yes	complemented
100	C₂³×C₁₃	—	2	32	no	—
101	C₂³×C₁₃	—	2	128	no	—
102	C₂³×C₁₃	—	2	96	no	—
103	C₂³×C₁₃	—	2	128	no	—
104	C₂³×C₁₃	—	2	24	no	—
105	C₂³×C₁₃	—	2	24	no	—
106	C₂³×C₁₃	—	2	12	no	—
107	C₂³×C₁₃	—	2	24	no	—
108	C₂³×C₁₃	—	2	12	no	—
109	C₂³×C₁₃	—	2	24	no	—
110	C₂³×C₁₃	—	2	48	no	—
111	C₂³×C₁₃	—	2	96	no	—
112	C₂³×C₁₃	—	2	2	no	—
113	C₂³×C₁₃	—	2	12	no	—
114	C₂³×C₁₃	—	4	4	no	—
115	C₂³×C₁₃	—	4	3	no	—
116	C₂³×C₁₃	—	4	6	no	—
117	C₂³×C₁₃	—	4	3	no	—
118	C₂³×C₁₃	—	4	12	no	—
119	C₂³×C₁₃	C₂×C₂²⋊C₄	1	12	yes	—
120	C₂³×C₁₃	C₄×D₄	1	24	yes	—
121	C₂³×C₁₃	#120	1	12	yes	—
122	C₂³×C₁₃	#121	1	24	yes	—
123	C₂³×C₁₃	#123	1	12	yes	—
124	C₂³×C₁₃	C₂³×C₄	1	1	yes	characteristic
125	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
126	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₂³×C₁₃	C₂²×D₄	1	6	yes	—
128	C₂³×C₁₃	C₂×#41	1	6	yes	—
129	D₄×C₁₃	—	4	48	no	—
130	C₂×C₄×C₁₃	—	2	192	no	—
131	C₂×C₄×C₁₃	—	2	24	no	—
132	C₂×C₄×C₁₃	—	2	24	no	—
133	C₂×C₄×C₁₃	—	2	6	no	—
134	C₂×C₄×C₁₃	—	2	12	no	—
135	C₂×C₄×C₁₃	—	4	3	no	—
136	C₂×C₄×C₁₃	—	4	6	no	—
137	C₂×C₄×C₁₃	—	4	3	no	—
138	C₂×C₄×C₁₃	—	4	4	no	—
139	C₂×C₄×C₁₃	—	4	12	no	—
140	C₂×C₄×C₁₃	—	4	12	no	—
141	C₂×C₄×C₁₃	#120	1	16	yes	complemented
142	C₂⁶	C₂²×C₁₃	1	1	yes	characteristic
143	C₂³×D₄	C₂²×C₁₃	1	3	yes	—
144	C₂⁴×C₄	—	2	4	no	—
145	C₂⁴×C₄	—	2	24	no	—
146	C₂⁴×C₄	C₂²×C₁₃	1	6	yes	—
147	C₂×#120	—	2	4	no	—
148	C₂×#120	—	2	4	no	—
149	C₂×#120	C₄×C₁₃	1	8	yes	complemented
150	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	6	yes	—
151	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	3	yes	—
152	C₂²×C₂²⋊C₄	C₂²×C₁₃	1	4	yes	complemented
153	#386	—	2	24	no	—
154	#379	—	2	4	no	—
155	C₂×#95	C₂²×C₁₃	1	12	yes	complemented
156	C₂²×C₁₃	—	2	128	no	—
157	C₂²×C₁₃	—	2	12	no	—
158	C₂²×C₁₃	—	2	48	no	—
159	C₂²×C₁₃	—	2	12	no	—
160	C₂²×C₁₃	—	2	24	no	—
161	C₂²×C₁₃	—	2	24	no	—
162	C₂²×C₁₃	—	2	48	no	—
163	C₂²×C₁₃	—	4	6	no	—
164	C₂²×C₁₃	—	4	8	no	—
165	C₂²×C₁₃	—	8	1	no	—
166	C₂²×C₁₃	—	8	4	no	—
167	C₂²×C₁₃	—	8	3	no	—
168	C₂²×C₁₃	#386	1	24	yes	complemented
169	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
170	C₂²×C₁₃	C₂×C₄×D₄	1	6	yes	—
171	C₂²×C₁₃	C₂×#120	1	4	yes	—
172	C₂²×C₁₃	C₂×#120	1	3	yes	—
173	C₂²×C₁₃	C₂×#121	1	6	yes	—
174	C₂²×C₁₃	C₂×#123	1	3	yes	—
175	C₂²×C₁₃	#525	1	12	yes	—
176	C₄×C₁₃	—	2	16	no	—
177	C₄×C₁₃	—	4	6	no	—
178	C₄×C₁₃	—	8	4	no	—
179	C₂⁵	—	2	12	no	—
180	C₂⁵	—	2	12	no	—
181	C₂⁵	C₂×C₄×C₁₃	1	4	yes	—
182	C₂⁵	D₄×C₁₃	1	4	yes	—
183	C₂⁵	D₄×C₁₃	1	4	yes	—
184	C₂⁵	C₂³×C₁₃	1	1	yes	characteristic
185	C₂⁵	C₂³×C₁₃	1	3	yes	—
186	C₂²×D₄	—	2	6	no	—
187	C₂²×D₄	—	2	6	no	—
188	C₂²×D₄	—	2	12	no	—
189	C₂²×D₄	—	2	12	no	—
190	C₂²×D₄	C₂×C₄×C₁₃	1	12	yes	complemented
191	C₂³×C₄	—	2	96	no	—
192	C₂³×C₄	—	2	128	no	—
193	C₂³×C₄	—	2	12	no	—
194	C₂³×C₄	—	2	12	no	—
195	C₂³×C₄	—	2	48	no	—
196	C₂³×C₄	D₄×C₁₃	1	24	yes	complemented
197	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
198	C₂³×C₄	C₂³×C₁₃	1	3	yes	—
199	C₂³×C₄	C₂³×C₁₃	1	4	yes	—
200	#120	—	2	32	no	—
201	C₂×C₂²⋊C₄	—	2	6	no	—
202	C₂×C₂²⋊C₄	—	2	6	no	—
203	C₂×C₂²⋊C₄	—	2	12	no	—
204	C₂×C₂²⋊C₄	—	2	12	no	—
205	C₂×C₂²⋊C₄	—	2	12	no	—
206	C₂×C₂²⋊C₄	—	2	24	no	—
207	C₂×C₂²⋊C₄	C₂×C₄×C₁₃	1	12	yes	complemented
208	#95	—	2	24	no	—
209	C₂×C₁₃	—	2	24	no	—
210	C₂×C₁₃	—	8	2	no	—
211	C₂×C₁₃	C₂×#386	1	3	yes	—
212	C₂×C₁₃	C₄×#120	1	4	yes	—
213	C₂×C₁₃	#2846	1	1	yes	characteristic
214	C₂×C₁₃	#2847	1	4	yes	—
215	C₂×C₁₃	#2849	1	3	yes	—
216	C₂⁴	—	2	32	no	—
217	C₂⁴	—	2	96	no	—
218	C₂⁴	—	2	48	no	—
219	C₂⁴	—	2	24	no	—
220	C₂⁴	—	2	48	no	—
221	C₂⁴	—	2	24	no	—
222	C₂⁴	—	2	12	no	—
223	C₂⁴	—	2	12	no	—
224	C₂⁴	—	2	6	no	—
225	C₂⁴	—	2	1	no	—
226	C₂⁴	—	2	1	no	—
227	C₂⁴	—	2	6	no	—
228	C₂⁴	C₁₃×C₂²⋊C₄	1	16	yes	complemented
229	C₂⁴	C₂²×C₄×C₁₃	1	6	yes	—
230	C₂⁴	C₂²×C₄×C₁₃	1	2	yes	complemented
231	C₂⁴	C₂×D₄×C₁₃	1	6	yes	—
232	C₂⁴	C₂×D₄×C₁₃	1	6	yes	—
233	C₂⁴	C₂×D₄×C₁₃	1	12	yes	—
234	C₂⁴	C₁₃×#41	1	12	yes	—
235	C₂⁴	C₂⁴×C₁₃	1	1	yes	characteristic
236	C₂×D₄	—	2	24	no	—
237	C₂×D₄	—	2	48	no	—
238	C₂×D₄	—	4	12	no	—
239	C₂×D₄	—	4	24	no	—
240	C₂×D₄	—	4	12	no	—
241	C₂²×C₄	—	2	256	no	—
242	C₂²×C₄	—	2	96	no	—
243	C₂²×C₄	—	2	96	no	—
244	C₂²×C₄	—	2	48	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	2	12	no	—
247	C₂²×C₄	—	2	96	no	—
248	C₂²×C₄	—	2	6	no	—
249	C₂²×C₄	—	2	3	no	—
250	C₂²×C₄	—	2	3	no	—
251	C₂²×C₄	—	2	6	no	—
252	C₂²×C₄	—	2	12	no	—
253	C₂²×C₄	—	2	4	no	—
254	C₂²×C₄	—	2	12	no	—
255	C₂²×C₄	C₂²×C₄×C₁₃	1	6	yes	complemented
256	C₂²×C₄	C₂×D₄×C₁₃	1	24	yes	—
257	C₂²⋊C₄	—	2	24	no	—
258	C₂²⋊C₄	—	4	24	no	—
259	C₂²⋊C₄	—	4	16	no	—
260	C₁₃	#14694	1	1	yes	characteristic, direct product factor, Sylow
261	C₂³	—	2	32	no	—
262	C₂³	—	2	128	no	—
263	C₂³	—	2	128	no	—
264	C₂³	—	2	96	no	—
265	C₂³	—	2	96	no	—
266	C₂³	—	2	24	no	—
267	C₂³	—	2	12	no	—
268	C₂³	—	2	24	no	—
269	C₂³	—	2	24	no	—
270	C₂³	—	2	12	no	—
271	C₂³	—	2	24	no	—
272	C₂³	—	2	48	no	—
273	C₂³	—	2	2	no	—
274	C₂³	—	2	12	no	—
275	C₂³	—	4	4	no	—
276	C₂³	—	4	3	no	—
277	C₂³	—	4	6	no	—
278	C₂³	—	4	3	no	—
279	C₂³	—	4	12	no	—
280	C₂³	C₂×C₁₃×C₂²⋊C₄	1	12	yes	—
281	C₂³	C₄×D₄×C₁₃	1	24	yes	—
282	C₂³	C₁₃×#120	1	12	yes	—
283	C₂³	C₁₃×#121	1	24	yes	—
284	C₂³	C₁₃×#123	1	12	yes	—
285	C₂³	C₂³×C₄×C₁₃	1	1	yes	characteristic
286	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
287	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
288	C₂³	C₂²×D₄×C₁₃	1	6	yes	—
289	C₂³	C₂×C₁₃×#41	1	6	yes	—
290	D₄	—	4	48	no	—
291	C₂×C₄	—	2	192	no	—
292	C₂×C₄	—	2	24	no	—
293	C₂×C₄	—	2	24	no	—
294	C₂×C₄	—	2	6	no	—
295	C₂×C₄	—	2	12	no	—
296	C₂×C₄	—	4	3	no	—
297	C₂×C₄	—	4	6	no	—
298	C₂×C₄	—	4	3	no	—
299	C₂×C₄	—	4	4	no	—
300	C₂×C₄	—	4	12	no	—
301	C₂×C₄	—	4	12	no	—
302	C₂×C₄	C₁₃×#120	1	16	yes	complemented
303	C₂²	—	2	128	no	—
304	C₂²	—	2	24	no	—
305	C₂²	—	2	48	no	—
306	C₂²	—	2	12	no	—
307	C₂²	—	2	48	no	—
308	C₂²	—	2	12	no	—
309	C₂²	—	2	24	no	—
310	C₂²	—	4	6	no	—
311	C₂²	—	4	8	no	—
312	C₂²	—	8	1	no	—
313	C₂²	—	8	4	no	—
314	C₂²	—	8	3	no	—
315	C₂²	C₁₃×#386	1	24	yes	complemented
316	C₂²	C₂²×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
317	C₂²	C₂×C₄×D₄×C₁₃	1	6	yes	—
318	C₂²	C₂×C₁₃×#120	1	3	yes	—
319	C₂²	C₂×C₁₃×#120	1	4	yes	—
320	C₂²	C₂×C₁₃×#121	1	6	yes	—
321	C₂²	C₂×C₁₃×#123	1	3	yes	—
322	C₂²	C₁₃×#525	1	12	yes	—
323	C₄	—	2	16	no	—
324	C₄	—	4	6	no	—
325	C₄	—	8	4	no	—
326	C₂	—	2	24	no	—
327	C₂	—	8	2	no	—
328	C₂	C₂×C₁₃×#386	1	3	yes	—
329	C₂	C₄×C₁₃×#120	1	4	yes	—
330	C₂	C₁₃×#2846	1	1	yes	characteristic
331	C₂	C₁₃×#2847	1	4	yes	—
332	C₂	C₁₃×#2849	1	3	yes	—

	subgroup	G/H
center, socle	C₂⁴×C₁₃	C₂⁴
commutator	C₂³	C₂³×C₄×C₁₃
Frattini, cosocle	C₂⁴	C₂⁴×C₁₃
prime radical	C₂²×C₁₃×#120	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂⁵×C₄×C₁₃	C₂	semidirect
3. C₂×C₁₃×#386	C₂	semidirect
4. C₂×C₁₃×#379	C₂	semidirect
5. C₁₃×C₂²⋊#95	C₂	semidirect
13. C₂×C₁₃×#120	C₄	semidirect
16. C₂²×C₁₃×C₂²⋊C₄	C₂²	semidirect
19. C₂×C₁₃×#95	C₂²	semidirect
31. C₂²×D₄×C₁₃	C₂×C₄	semidirect
37. C₂³×C₄×C₁₃	D₄	semidirect
48. C₂×C₁₃×C₂²⋊C₄	C₂×C₄	semidirect
50. #14694	C₁₃	direct (N × H)
63. C₂⁴×C₁₃	C₂²⋊C₄	semidirect
65. C₂⁴×C₁₃	C₂²×C₄	semidirect
90. C₂²×C₄×C₁₃	C₂²×C₄	semidirect
95. C₂⁵×C₄	C₂×C₁₃	semidirect
97. C₂×#386	C₂×C₁₃	semidirect
98. C₂×#379	C₂×C₁₃	semidirect
99. C₂²⋊#95	C₂×C₁₃	semidirect
141. C₂×C₄×C₁₃	#120	semidirect
149. C₂×#120	C₄×C₁₃	semidirect
152. C₂²×C₂²⋊C₄	C₂²×C₁₃	semidirect
155. C₂×#95	C₂²×C₁₃	semidirect
168. C₂²×C₁₃	#386	semidirect
190. C₂²×D₄	C₂×C₄×C₁₃	semidirect
196. C₂³×C₄	D₄×C₁₃	semidirect
207. C₂×C₂²⋊C₄	C₂×C₄×C₁₃	semidirect
228. C₂⁴	C₁₃×C₂²⋊C₄	semidirect
230. C₂⁴	C₂²×C₄×C₁₃	semidirect
255. C₂²×C₄	C₂²×C₄×C₁₃	semidirect
260. C₁₃	#14694	direct (N × H)
302. C₂×C₄	C₁₃×#120	semidirect
315. C₂²	C₁₃×#386	semidirect

Maximal subgroups

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

subgroup	index
C₂⁵×C₄×C₁₃	2
C₂²×C₁₃×#120	2
C₂×C₁₃×#386	2
C₂×C₁₃×#379	2
C₁₃×C₂²⋊#95	2
#14694	13

Subgroups — C13×#14694 #15701033

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#14694 #15701033