Subgroups — C₁₃×#32864 #15719203

Order 3328.

630 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2644	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2088	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2086	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2067	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2065	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2065	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2054	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#2054	C₂	1	2	yes	maximal
9	C₁₃×#1843	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#106	C₂²	1	2	yes	—
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
15	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
17	C₂×C₁₃×#102	C₂²	1	2	yes	—
18	C₂×C₁₃×#102	C₂²	1	2	yes	—
19	C₂×C₁₃×#102	C₂²	1	2	yes	—
20	C₁₃×#397	—	2	2	no	—
21	C₁₃×#397	—	2	2	no	—
22	C₁₃×#397	—	2	2	no	—
23	C₁₃×#396	—	2	2	no	—
24	C₁₃×#390	C₂²	1	2	yes	—
25	C₁₃×#390	C₂²	1	2	yes	—
26	C₁₃×#390	C₂²	1	2	yes	—
27	C₁₃×#389	C₂²	1	2	yes	—
28	C₁₃×#389	C₂²	1	2	yes	—
29	C₁₃×#389	C₂²	1	2	yes	—
30	C₁₃×#376	C₂²	1	1	yes	characteristic
31	C₁₃×#340	C₂²	1	2	yes	—
32	C₁₃×#340	C₂²	1	1	yes	characteristic
33	C₁₃×#340	C₂²	1	1	yes	characteristic
34	C₁₃×#340	C₂²	1	2	yes	—
35	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
36	C₂×C₁₃×D₈	—	4	2	no	—
37	C₂×C₁₃×D₈	—	4	1	no	—
38	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
41	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
44	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
45	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
46	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
48	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
49	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
50	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
51	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
52	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
53	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
55	C₁₃×#107	—	4	1	no	—
56	C₁₃×#106	—	4	2	no	—
57	C₁₃×C₄⋊C₈	—	4	1	no	—
58	C₁₃×C₄⋊C₈	—	4	2	no	—
59	C₁₃×C₄⋊C₈	—	4	2	no	—
60	C₁₃×C₄⋊C₈	—	4	1	no	—
61	C₁₃×#102	—	4	2	no	—
62	C₁₃×#102	—	4	2	no	—
63	C₁₃×#102	—	4	2	no	—
64	C₁₃×#95	—	2	2	no	—
65	C₁₃×#95	—	2	2	no	—
66	C₁₃×#95	—	2	2	no	—
67	C₁₃×#95	D₄	1	2	yes	—
68	C₁₃×#95	D₄	1	1	yes	characteristic
69	C₁₃×#95	D₄	1	1	yes	characteristic
70	#32864	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂⁴×C₁₃	—	2	2	no	—
72	C₂×D₄×C₁₃	—	2	2	no	—
73	C₂×D₄×C₁₃	—	2	2	no	—
74	C₂×D₄×C₁₃	—	2	2	no	—
75	C₂×D₄×C₁₃	—	2	2	no	—
76	C₂×D₄×C₁₃	—	2	2	no	—
77	C₂×D₄×C₁₃	—	2	2	no	—
78	C₂²×C₄×C₁₃	—	2	2	no	—
79	C₂²×C₄×C₁₃	—	2	2	no	—
80	C₂²×C₄×C₁₃	—	2	2	no	—
81	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
86	C₁₃×D₈	—	4	4	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	—	2	1	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	—	2	1	no	—
94	C₁₃×C₄⋊C₄	—	2	2	no	—
95	C₁₃×C₄⋊C₄	—	2	2	no	—
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	2	2	no	—
100	C₁₃×C₄⋊C₄	—	8	2	no	—
101	C₁₃×C₄⋊C₄	—	8	2	no	—
102	C₁₃×C₄⋊C₄	—	8	2	no	—
103	C₁₃×C₂²⋊C₄	—	8	2	no	—
104	C₁₃×C₂²⋊C₄	—	8	2	no	—
105	C₁₃×C₂²⋊C₄	—	8	2	no	—
106	C₁₃×C₂²⋊C₄	—	8	2	no	—
107	C₁₃×C₂²⋊C₄	—	8	2	no	—
108	C₁₃×C₂²⋊C₄	—	8	2	no	—
109	C₄²×C₁₃	—	2	2	no	—
110	C₄²×C₁₃	—	2	1	no	—
111	C₄²×C₁₃	—	2	2	no	—
112	C₄²×C₁₃	—	2	1	no	—
113	#2644	C₂×C₁₃	1	2	yes	complemented
114	#2088	C₂×C₁₃	1	2	yes	complemented
115	#2086	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#2067	C₂×C₁₃	1	2	yes	complemented
117	#2065	C₂×C₁₃	1	2	yes	complemented
118	#2065	C₂×C₁₃	1	2	yes	complemented
119	#2054	C₂×C₁₃	1	2	yes	—
120	#2054	C₂×C₁₃	1	1	yes	characteristic
121	#1843	C₂×C₁₃	1	1	yes	characteristic, complemented
122	C₂³×C₁₃	—	4	2	no	—
123	C₂³×C₁₃	—	4	2	no	—
124	C₂³×C₁₃	—	4	2	no	—
125	C₂³×C₁₃	—	4	2	no	—
126	C₂³×C₁₃	—	4	2	no	—
127	C₂³×C₁₃	—	4	2	no	—
128	C₂³×C₁₃	—	4	2	no	—
129	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
130	D₄×C₁₃	—	4	2	no	—
131	D₄×C₁₃	—	4	2	no	—
132	D₄×C₁₃	—	4	2	no	—
133	D₄×C₁₃	—	4	2	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	—	2	1	no	—
137	C₂×C₄×C₁₃	—	2	1	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	2	1	no	—
140	C₂×C₄×C₁₃	—	2	2	no	—
141	C₂×C₄×C₁₃	—	2	1	no	—
142	C₂×C₄×C₁₃	—	2	2	no	—
143	C₂×C₄×C₁₃	—	2	2	no	—
144	C₂×C₄×C₁₃	—	2	2	no	—
145	C₂×C₄×C₁₃	—	4	2	no	—
146	C₂×C₄×C₁₃	—	4	2	no	—
147	C₂×C₄×C₁₃	—	4	2	no	—
148	C₂×C₄×C₁₃	—	4	2	no	—
149	C₂×C₄×C₁₃	—	4	2	no	—
150	C₂×C₄×C₁₃	—	4	2	no	—
151	C₂×C₄×C₁₃	—	4	2	no	—
152	C₂×C₄×C₁₃	—	4	2	no	—
153	C₂×C₄×C₁₃	—	4	2	no	—
154	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
155	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
156	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
157	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
158	C₈×C₁₃	—	4	2	no	—
159	C₈×C₁₃	—	4	1	no	—
160	C₈×C₁₃	—	4	1	no	—
161	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
162	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
163	C₂×#106	C₂²×C₁₃	1	2	yes	—
164	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
165	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
166	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
167	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
168	C₂×#102	C₂²×C₁₃	1	2	yes	—
169	C₂×#102	C₂²×C₁₃	1	2	yes	—
170	C₂×#102	C₂²×C₁₃	1	2	yes	—
171	#397	—	2	2	no	—
172	#397	—	2	2	no	—
173	#397	—	2	2	no	—
174	#396	—	2	2	no	—
175	#390	C₂²×C₁₃	1	2	yes	—
176	#390	C₂²×C₁₃	1	2	yes	—
177	#390	C₂²×C₁₃	1	2	yes	—
178	#389	C₂²×C₁₃	1	2	yes	—
179	#389	C₂²×C₁₃	1	2	yes	—
180	#389	C₂²×C₁₃	1	2	yes	—
181	#376	C₂²×C₁₃	1	1	yes	characteristic
182	#340	C₂²×C₁₃	1	2	yes	—
183	#340	C₂²×C₁₃	1	1	yes	characteristic
184	#340	C₂²×C₁₃	1	1	yes	characteristic
185	#340	C₂²×C₁₃	1	2	yes	—
186	C₂²×C₁₃	—	8	2	no	—
187	C₂²×C₁₃	—	8	2	no	—
188	C₂²×C₁₃	—	8	2	no	—
189	C₂²×C₁₃	—	8	2	no	—
190	C₂²×C₁₃	—	8	2	no	—
191	C₂²×C₁₃	—	8	2	no	—
192	C₂²×C₁₃	—	8	2	no	—
193	C₂²×C₁₃	#537	1	2	yes	—
194	C₂²×C₁₃	#537	1	1	yes	characteristic
195	C₂²×C₁₃	#576	1	1	yes	characteristic
196	C₂²×C₁₃	#576	1	1	yes	characteristic
197	C₂²×C₁₃	#577	1	2	yes	—
198	C₄×C₁₃	—	4	2	no	—
199	C₄×C₁₃	—	4	2	no	—
200	C₄×C₁₃	—	4	1	no	—
201	C₄×C₁₃	—	4	1	no	—
202	C₄×C₁₃	—	8	2	no	—
203	C₄×C₁₃	—	8	2	no	—
204	C₄×C₁₃	—	8	2	no	—
205	C₄×C₁₃	#560	1	1	yes	characteristic
206	C₄×C₁₃	#563	1	2	yes	—
207	C₄×C₁₃	#563	1	1	yes	characteristic
208	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
209	C₂×D₈	—	4	1	no	—
210	C₂×D₈	—	4	2	no	—
211	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
212	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
213	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
214	C₂×C₄⋊C₄	—	2	2	no	—
215	C₂×C₄⋊C₄	—	2	2	no	—
216	C₂×C₄⋊C₄	—	2	2	no	—
217	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
218	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
219	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
220	C₂×C₂²⋊C₄	—	2	2	no	—
221	C₂×C₂²⋊C₄	—	2	2	no	—
222	C₂×C₂²⋊C₄	—	2	2	no	—
223	C₂×C₂²⋊C₄	—	2	2	no	—
224	C₂×C₂²⋊C₄	—	2	2	no	—
225	C₂×C₂²⋊C₄	—	2	2	no	—
226	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
227	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
228	#107	—	4	1	no	—
229	#106	—	4	2	no	—
230	C₄⋊C₈	—	4	1	no	—
231	C₄⋊C₈	—	4	2	no	—
232	C₄⋊C₈	—	4	2	no	—
233	C₄⋊C₈	—	4	1	no	—
234	#102	—	4	2	no	—
235	#102	—	4	2	no	—
236	#102	—	4	2	no	—
237	#95	—	2	2	no	—
238	#95	—	2	2	no	—
239	#95	—	2	2	no	—
240	#95	D₄×C₁₃	1	2	yes	—
241	#95	D₄×C₁₃	1	1	yes	characteristic
242	#95	D₄×C₁₃	1	1	yes	characteristic
243	C₂×C₁₃	—	16	2	no	—
244	C₂×C₁₃	#2823	1	1	yes	characteristic
245	C₂×C₁₃	#3257	1	2	yes	—
246	C₂×C₁₃	#3257	1	2	yes	—
247	C₂×C₁₃	#3259	1	1	yes	characteristic
248	C₂×C₁₃	#3263	1	1	yes	characteristic
249	C₂⁴	—	2	2	no	—
250	C₂×D₄	—	2	2	no	—
251	C₂×D₄	—	2	2	no	—
252	C₂×D₄	—	2	2	no	—
253	C₂×D₄	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂²×C₄	—	2	2	no	—
257	C₂²×C₄	—	2	2	no	—
258	C₂²×C₄	—	2	2	no	—
259	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
260	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
261	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
262	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
263	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
264	D₈	—	4	4	no	—
265	C₂×C₈	—	2	2	no	—
266	C₂×C₈	—	2	2	no	—
267	C₂×C₈	—	2	2	no	—
268	C₂×C₈	—	2	2	no	—
269	C₂×C₈	—	2	1	no	—
270	C₂×C₈	—	2	1	no	—
271	C₂×C₈	—	2	2	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	2	no	—
274	C₄⋊C₄	—	2	2	no	—
275	C₄⋊C₄	—	2	2	no	—
276	C₄⋊C₄	—	2	2	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	8	2	no	—
279	C₄⋊C₄	—	8	2	no	—
280	C₄⋊C₄	—	8	2	no	—
281	C₂²⋊C₄	—	8	2	no	—
282	C₂²⋊C₄	—	8	2	no	—
283	C₂²⋊C₄	—	8	2	no	—
284	C₂²⋊C₄	—	8	2	no	—
285	C₂²⋊C₄	—	8	2	no	—
286	C₂²⋊C₄	—	8	2	no	—
287	C₄²	—	2	1	no	—
288	C₄²	—	2	2	no	—
289	C₄²	—	2	1	no	—
290	C₄²	—	2	2	no	—
291	C₁₃	#32864	1	1	yes	characteristic, direct product factor, Sylow
292	C₂³	—	4	2	no	—
293	C₂³	—	4	2	no	—
294	C₂³	—	4	2	no	—
295	C₂³	—	4	2	no	—
296	C₂³	—	4	2	no	—
297	C₂³	—	4	2	no	—
298	C₂³	—	4	2	no	—
299	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
300	D₄	—	4	2	no	—
301	D₄	—	4	2	no	—
302	D₄	—	4	2	no	—
303	D₄	—	4	2	no	—
304	C₂×C₄	—	2	1	no	—
305	C₂×C₄	—	2	2	no	—
306	C₂×C₄	—	2	2	no	—
307	C₂×C₄	—	2	1	no	—
308	C₂×C₄	—	2	2	no	—
309	C₂×C₄	—	2	1	no	—
310	C₂×C₄	—	2	2	no	—
311	C₂×C₄	—	2	1	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	2	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	4	2	no	—
316	C₂×C₄	—	4	2	no	—
317	C₂×C₄	—	4	2	no	—
318	C₂×C₄	—	4	2	no	—
319	C₂×C₄	—	4	2	no	—
320	C₂×C₄	—	4	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
325	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
326	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
327	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
328	C₈	—	4	2	no	—
329	C₈	—	4	1	no	—
330	C₈	—	4	1	no	—
331	C₂²	—	8	2	no	—
332	C₂²	—	8	2	no	—
333	C₂²	—	8	2	no	—
334	C₂²	—	8	2	no	—
335	C₂²	—	8	2	no	—
336	C₂²	—	8	2	no	—
337	C₂²	—	8	2	no	—
338	C₂²	C₁₃×#537	1	1	yes	characteristic
339	C₂²	C₁₃×#537	1	2	yes	—
340	C₂²	C₁₃×#576	1	1	yes	characteristic
341	C₂²	C₁₃×#576	1	1	yes	characteristic
342	C₂²	C₁₃×#577	1	2	yes	—
343	C₄	—	4	1	no	—
344	C₄	—	4	2	no	—
345	C₄	—	4	2	no	—
346	C₄	—	4	1	no	—
347	C₄	—	8	2	no	—
348	C₄	—	8	2	no	—
349	C₄	—	8	2	no	—
350	C₄	C₁₃×#560	1	1	yes	characteristic
351	C₄	C₁₃×#563	1	1	yes	characteristic
352	C₄	C₁₃×#563	1	2	yes	—
353	C₂	—	16	2	no	—
354	C₂	C₁₃×#2823	1	1	yes	characteristic
355	C₂	C₁₃×#3257	1	2	yes	—
356	C₂	C₁₃×#3257	1	2	yes	—
357	C₂	C₁₃×#3259	1	1	yes	characteristic
358	C₂	C₁₃×#3263	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₁₃×#376 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#2644	C₂	semidirect
2. C₁₃×#2088	C₂	semidirect
3. C₁₃×#2086	C₂	semidirect
4. C₁₃×#2067	C₂	semidirect
5. C₁₃×#2065	C₂	semidirect
6. C₁₃×#2065	C₂	semidirect
9. C₁₃×#1843	C₂	semidirect
70. #32864	C₁₃	direct (N × H)
113. #2644	C₂×C₁₃	semidirect
114. #2088	C₂×C₁₃	semidirect
115. #2086	C₂×C₁₃	semidirect
116. #2067	C₂×C₁₃	semidirect
117. #2065	C₂×C₁₃	semidirect
118. #2065	C₂×C₁₃	semidirect
121. #1843	C₂×C₁₃	semidirect
291. C₁₃	#32864	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×#2644	2
C₁₃×#2088	2
C₁₃×#2086	2
C₁₃×#2067	2
C₁₃×#2065	2
C₁₃×#2054	2
C₁₃×#1843	2
#32864	13

Subgroups — C13×#32864 #15719203

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32864 #15719203