Subgroups — C₁₃×#32827 #15719166

Order 3328.

654 conjugacy classes of subgroups.

The subgroup lattice has 368 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2647	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#2641	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2092	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2071	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2067	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2065	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2059	C₂	1	2	yes	maximal, complemented
8	C₁₃×#2041	C₂	1	2	yes	maximal
9	C₁₃×#1832	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×#127	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#125	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#107	C₂²	1	2	yes	—
13	C₂×C₁₃×#106	C₂²	1	2	yes	—
14	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#102	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×#400	—	2	2	no	—
21	C₁₃×#398	—	2	2	no	—
22	C₁₃×#394	—	2	2	no	—
23	C₁₃×#394	—	2	2	no	—
24	C₁₃×#390	C₂²	1	2	yes	—
25	C₁₃×#390	C₂²	1	2	yes	—
26	C₁₃×#389	C₂²	1	2	yes	—
27	C₁₃×#389	C₂²	1	2	yes	—
28	C₁₃×#384	C₂²	1	1	yes	characteristic
29	C₁₃×#382	C₂²	1	1	yes	characteristic
30	C₁₃×#376	C₂²	1	1	yes	characteristic
31	C₁₃×#340	C₂²	1	2	yes	—
32	C₁₃×#340	C₂²	1	2	yes	—
33	C₁₃×#336	C₂²	1	2	yes	—
34	C₁₃×#336	C₂²	1	2	yes	—
35	C₂²×D₄×C₁₃	—	2	2	no	—
36	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
39	C₁₃×#127	—	4	1	no	—
40	C₁₃×#127	—	4	1	no	—
41	C₁₃×#125	—	4	1	no	—
42	C₁₃×#125	—	4	1	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
45	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
46	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
47	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
49	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
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×C₁₃×C₂²⋊C₄	—	2	2	no	—
54	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
55	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
57	C₁₃×#107	—	4	2	no	—
58	C₁₃×#106	—	4	2	no	—
59	C₁₃×C₄⋊C₈	—	4	1	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×#102	—	4	2	no	—
65	C₁₃×#95	—	2	2	no	—
66	C₁₃×#95	—	2	2	no	—
67	C₁₃×#95	—	2	2	no	—
68	C₁₃×#95	D₄	1	2	yes	—
69	C₁₃×#95	D₄	1	2	yes	—
70	#32827	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	C₂×D₄×C₁₃	—	4	4	no	—
79	C₂×D₄×C₁₃	—	8	2	no	—
80	C₂×D₄×C₁₃	—	8	2	no	—
81	C₂²×C₄×C₁₃	—	2	2	no	—
82	C₂²×C₄×C₁₃	—	2	2	no	—
83	C₂²×C₄×C₁₃	—	2	2	no	—
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₂×C₈×C₁₃	—	2	2	no	—
90	C₂×C₈×C₁₃	—	2	2	no	—
91	C₂×C₈×C₁₃	—	2	2	no	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	—	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	—	2	2	no	—
101	C₁₃×C₄⋊C₄	—	4	4	no	—
102	C₁₃×C₄⋊C₄	—	8	2	no	—
103	C₁₃×C₄⋊C₄	—	8	2	no	—
104	C₁₃×C₄⋊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	1	no	—
110	C₄²×C₁₃	—	2	1	no	—
111	C₄²×C₁₃	—	2	1	no	—
112	C₄²×C₁₃	—	2	1	no	—
113	C₄²×C₁₃	—	2	1	no	—
114	C₄²×C₁₃	—	2	1	no	—
115	#2647	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#2641	C₂×C₁₃	1	1	yes	characteristic
117	#2092	C₂×C₁₃	1	2	yes	complemented
118	#2071	C₂×C₁₃	1	2	yes	complemented
119	#2067	C₂×C₁₃	1	2	yes	complemented
120	#2065	C₂×C₁₃	1	2	yes	complemented
121	#2059	C₂×C₁₃	1	2	yes	complemented
122	#2041	C₂×C₁₃	1	2	yes	—
123	#1832	C₂×C₁₃	1	1	yes	characteristic, complemented
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	—	4	2	no	—
130	C₂³×C₁₃	—	4	2	no	—
131	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
132	D₄×C₁₃	—	4	2	no	—
133	D₄×C₁₃	—	4	2	no	—
134	D₄×C₁₃	—	4	2	no	—
135	D₄×C₁₃	—	4	2	no	—
136	D₄×C₁₃	—	8	8	no	—
137	C₂×C₄×C₁₃	—	2	2	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	2	1	no	—
140	C₂×C₄×C₁₃	—	2	1	no	—
141	C₂×C₄×C₁₃	—	2	2	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	—	2	2	no	—
146	C₂×C₄×C₁₃	—	2	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	—	4	2	no	—
155	C₂×C₄×C₁₃	—	4	2	no	—
156	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
157	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
158	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
159	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
160	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
161	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
162	C₈×C₁₃	—	4	2	no	—
163	C₈×C₁₃	—	4	2	no	—
164	C₂×#127	C₂²×C₁₃	1	1	yes	characteristic
165	C₂×#125	C₂²×C₁₃	1	1	yes	characteristic
166	C₂×#107	C₂²×C₁₃	1	2	yes	—
167	C₂×#106	C₂²×C₁₃	1	2	yes	—
168	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
169	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
170	C₂×#102	C₂²×C₁₃	1	2	yes	—
171	C₂×#102	C₂²×C₁₃	1	2	yes	—
172	C₂×#102	C₂²×C₁₃	1	2	yes	—
173	C₂×#102	C₂²×C₁₃	1	2	yes	—
174	#400	—	2	2	no	—
175	#398	—	2	2	no	—
176	#394	—	2	2	no	—
177	#394	—	2	2	no	—
178	#390	C₂²×C₁₃	1	2	yes	—
179	#390	C₂²×C₁₃	1	2	yes	—
180	#389	C₂²×C₁₃	1	2	yes	—
181	#389	C₂²×C₁₃	1	2	yes	—
182	#384	C₂²×C₁₃	1	1	yes	characteristic
183	#382	C₂²×C₁₃	1	1	yes	characteristic
184	#376	C₂²×C₁₃	1	1	yes	characteristic
185	#340	C₂²×C₁₃	1	2	yes	—
186	#340	C₂²×C₁₃	1	2	yes	—
187	#336	C₂²×C₁₃	1	2	yes	—
188	#336	C₂²×C₁₃	1	2	yes	—
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	—	8	2	no	—
194	C₂²×C₁₃	—	8	2	no	—
195	C₂²×C₁₃	—	8	2	no	—
196	C₂²×C₁₃	#535	1	1	yes	characteristic
197	C₂²×C₁₃	#536	1	1	yes	characteristic
198	C₂²×C₁₃	#537	1	1	yes	characteristic
199	C₂²×C₁₃	#576	1	2	yes	—
200	C₂²×C₁₃	#577	1	2	yes	—
201	C₄×C₁₃	—	4	2	no	—
202	C₄×C₁₃	—	4	2	no	—
203	C₄×C₁₃	—	4	1	no	—
204	C₄×C₁₃	—	4	1	no	—
205	C₄×C₁₃	—	8	2	no	—
206	C₄×C₁₃	—	8	2	no	—
207	C₄×C₁₃	—	8	2	no	—
208	C₄×C₁₃	#562	1	1	yes	characteristic
209	C₄×C₁₃	#562	1	1	yes	characteristic
210	C₄×C₁₃	#563	1	1	yes	characteristic
211	C₄×C₁₃	#563	1	1	yes	characteristic
212	C₂²×D₄	—	2	2	no	—
213	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
214	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
215	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
216	#127	—	4	1	no	—
217	#127	—	4	1	no	—
218	#125	—	4	1	no	—
219	#125	—	4	1	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	—	2	2	no	—
223	C₂×C₄⋊C₄	—	2	2	no	—
224	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
225	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
226	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
227	C₂×C₂²⋊C₄	—	2	2	no	—
228	C₂×C₂²⋊C₄	—	2	2	no	—
229	C₂×C₂²⋊C₄	—	2	2	no	—
230	C₂×C₂²⋊C₄	—	2	2	no	—
231	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
232	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
233	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
234	#107	—	4	2	no	—
235	#106	—	4	2	no	—
236	C₄⋊C₈	—	4	1	no	—
237	C₄⋊C₈	—	4	1	no	—
238	#102	—	4	2	no	—
239	#102	—	4	2	no	—
240	#102	—	4	2	no	—
241	#102	—	4	2	no	—
242	#95	—	2	2	no	—
243	#95	—	2	2	no	—
244	#95	—	2	2	no	—
245	#95	D₄×C₁₃	1	2	yes	—
246	#95	D₄×C₁₃	1	2	yes	—
247	C₂×C₁₃	—	16	2	no	—
248	C₂×C₁₃	#2820	1	1	yes	characteristic
249	C₂×C₁₃	#3224	1	2	yes	—
250	C₂×C₁₃	#3241	1	2	yes	—
251	C₂×C₁₃	#3259	1	1	yes	characteristic
252	C₂×C₁₃	#3260	1	1	yes	characteristic
253	C₂⁴	—	2	2	no	—
254	C₂×D₄	—	2	2	no	—
255	C₂×D₄	—	2	2	no	—
256	C₂×D₄	—	2	2	no	—
257	C₂×D₄	—	2	2	no	—
258	C₂×D₄	—	2	2	no	—
259	C₂×D₄	—	2	2	no	—
260	C₂×D₄	—	4	4	no	—
261	C₂×D₄	—	8	2	no	—
262	C₂×D₄	—	8	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	—	2	2	no	—
265	C₂²×C₄	—	2	2	no	—
266	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
268	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
269	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
270	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	2	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	2	no	—
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	2	2	no	—
279	C₄⋊C₄	—	2	2	no	—
280	C₄⋊C₄	—	2	2	no	—
281	C₄⋊C₄	—	2	2	no	—
282	C₄⋊C₄	—	2	2	no	—
283	C₄⋊C₄	—	4	4	no	—
284	C₄⋊C₄	—	8	2	no	—
285	C₄⋊C₄	—	8	2	no	—
286	C₄⋊C₄	—	8	2	no	—
287	C₂²⋊C₄	—	8	2	no	—
288	C₂²⋊C₄	—	8	2	no	—
289	C₂²⋊C₄	—	8	2	no	—
290	C₂²⋊C₄	—	8	2	no	—
291	C₄²	—	2	1	no	—
292	C₄²	—	2	1	no	—
293	C₄²	—	2	1	no	—
294	C₄²	—	2	1	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	2	1	no	—
297	C₁₃	#32827	1	1	yes	characteristic, direct product factor, Sylow
298	C₂³	—	4	2	no	—
299	C₂³	—	4	2	no	—
300	C₂³	—	4	2	no	—
301	C₂³	—	4	2	no	—
302	C₂³	—	4	2	no	—
303	C₂³	—	4	2	no	—
304	C₂³	—	4	2	no	—
305	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
306	D₄	—	4	2	no	—
307	D₄	—	4	2	no	—
308	D₄	—	4	2	no	—
309	D₄	—	4	2	no	—
310	D₄	—	8	8	no	—
311	C₂×C₄	—	2	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	1	no	—
315	C₂×C₄	—	2	2	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	2	no	—
318	C₂×C₄	—	2	2	no	—
319	C₂×C₄	—	2	2	no	—
320	C₂×C₄	—	2	2	no	—
321	C₂×C₄	—	4	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	2	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	—	4	2	no	—
328	C₂×C₄	—	4	2	no	—
329	C₂×C₄	—	4	2	no	—
330	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
331	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
332	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
333	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
334	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
335	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
336	C₈	—	4	2	no	—
337	C₈	—	4	2	no	—
338	C₂²	—	8	2	no	—
339	C₂²	—	8	2	no	—
340	C₂²	—	8	2	no	—
341	C₂²	—	8	2	no	—
342	C₂²	—	8	2	no	—
343	C₂²	—	8	2	no	—
344	C₂²	—	8	2	no	—
345	C₂²	C₁₃×#535	1	1	yes	characteristic
346	C₂²	C₁₃×#536	1	1	yes	characteristic
347	C₂²	C₁₃×#537	1	1	yes	characteristic
348	C₂²	C₁₃×#576	1	2	yes	—
349	C₂²	C₁₃×#577	1	2	yes	—
350	C₄	—	4	2	no	—
351	C₄	—	4	2	no	—
352	C₄	—	4	1	no	—
353	C₄	—	4	1	no	—
354	C₄	—	8	2	no	—
355	C₄	—	8	2	no	—
356	C₄	—	8	2	no	—
357	C₄	C₁₃×#562	1	1	yes	characteristic
358	C₄	C₁₃×#562	1	1	yes	characteristic
359	C₄	C₁₃×#563	1	1	yes	characteristic
360	C₄	C₁₃×#563	1	1	yes	characteristic
361	C₂	—	16	2	no	—
362	C₂	C₁₃×#2820	1	1	yes	characteristic
363	C₂	C₁₃×#3224	1	2	yes	—
364	C₂	C₁₃×#3241	1	2	yes	—
365	C₂	C₁₃×#3259	1	1	yes	characteristic
366	C₂	C₁₃×#3260	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#127	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×#2647	C₂	semidirect
3. C₁₃×#2092	C₂	semidirect
4. C₁₃×#2071	C₂	semidirect
5. C₁₃×#2067	C₂	semidirect
6. C₁₃×#2065	C₂	semidirect
7. C₁₃×#2059	C₂	semidirect
9. C₁₃×#1832	C₂	semidirect
70. #32827	C₁₃	direct (N × H)
115. #2647	C₂×C₁₃	semidirect
117. #2092	C₂×C₁₃	semidirect
118. #2071	C₂×C₁₃	semidirect
119. #2067	C₂×C₁₃	semidirect
120. #2065	C₂×C₁₃	semidirect
121. #2059	C₂×C₁₃	semidirect
123. #1832	C₂×C₁₃	semidirect
297. C₁₃	#32827	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×#2647	2
C₁₃×#2641	2
C₁₃×#2092	2
C₁₃×#2071	2
C₁₃×#2067	2
C₁₃×#2065	2
C₁₃×#2059	2
C₁₃×#2041	2
C₁₃×#1832	2
#32827	13

Subgroups — C13×#32827 #15719166

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32827 #15719166