Subgroups — C₁₃×#22810 #15709149

Order 3328.

862 conjugacy classes of subgroups.

The subgroup lattice has 324 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2357	C₂	1	2	yes	maximal, complemented
3	C₁₃×#1970	C₂	1	2	yes	maximal
4	C₁₃×#1926	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#1804	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₃×#107	C₂	1	2	yes	maximal, complemented
8	C₈×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
10	C₂×C₄×D₄×C₁₃	C₂²	1	2	yes	—
11	C₄×C₁₃×D₈	C₄	1	8	yes	complemented
12	C₂×C₁₃×#107	C₂²	1	2	yes	—
13	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#106	C₂²	1	2	yes	—
15	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#102	C₂²	1	4	yes	—
17	C₂×C₁₃×#102	C₂²	1	2	yes	—
18	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
19	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×#390	C₂²	1	4	yes	—
21	C₁₃×#388	—	2	4	no	—
22	C₁₃×#382	C₂²	1	2	yes	—
23	C₁₃×#381	—	2	2	no	—
24	C₄×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
25	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₄×C₁₃×C₂²⋊C₄	—	2	2	no	—
27	C₁₃×#340	C₂²	1	4	yes	—
28	C₁₃×#336	C₂²	1	2	yes	—
29	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
30	C₂³×C₄×C₁₃	—	2	2	no	—
31	C₂×C₁₃×D₈	—	2	2	no	—
32	C₂×C₁₃×D₈	—	2	2	no	—
33	C₂×C₁₃×D₈	C₂×C₄	1	4	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
37	C₄×D₄×C₁₃	—	2	4	no	—
38	C₄×D₄×C₁₃	C₂×C₄	1	8	yes	—
39	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
40	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
41	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
42	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
43	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
44	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
45	C₂×C₄²×C₁₃	—	2	4	no	—
46	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
47	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
48	C₁₃×#107	—	2	4	no	—
49	C₁₃×#107	C₂×C₄	1	4	yes	—
50	C₁₃×#106	—	4	2	no	—
51	C₁₃×C₄⋊C₈	—	2	2	no	—
52	C₁₃×#102	—	4	4	no	—
53	C₁₃×#102	C₂×C₄	1	8	yes	—
54	C₄×C₈×C₁₃	—	2	4	no	—
55	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
56	C₁₃×#95	—	2	2	no	—
57	C₁₃×#95	—	2	4	no	—
58	C₁₃×#95	—	2	2	no	—
59	C₁₃×#95	D₄	1	1	yes	characteristic
60	C₁₃×#95	D₄	1	1	yes	characteristic
61	#22810	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂⁴×C₁₃	—	2	2	no	—
63	C₂×D₄×C₁₃	—	2	2	no	—
64	C₂×D₄×C₁₃	—	2	2	no	—
65	C₂×D₄×C₁₃	—	2	2	no	—
66	C₂×D₄×C₁₃	—	2	2	no	—
67	C₂×D₄×C₁₃	—	2	2	no	—
68	C₂×D₄×C₁₃	C₂²×C₄	1	4	yes	—
69	C₂²×C₄×C₁₃	—	2	2	no	—
70	C₂²×C₄×C₁₃	—	2	8	no	—
71	C₂²×C₄×C₁₃	—	2	4	no	—
72	C₂²×C₄×C₁₃	—	8	4	no	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
78	C₁₃×D₈	—	2	8	no	—
79	C₂×C₈×C₁₃	—	2	1	no	—
80	C₂×C₈×C₁₃	—	2	1	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
85	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
86	C₂×C₈×C₁₃	#41	1	2	yes	—
87	C₂×C₈×C₁₃	#41	1	2	yes	—
88	C₁₃×C₄⋊C₄	—	2	4	no	—
89	C₁₃×C₄⋊C₄	—	2	2	no	—
90	C₁₃×C₄⋊C₄	—	2	4	no	—
91	C₁₃×C₄⋊C₄	—	2	4	no	—
92	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
93	C₁₃×C₂²⋊C₄	—	2	8	no	—
94	C₁₃×C₂²⋊C₄	—	4	8	no	—
95	C₁₃×C₂²⋊C₄	—	8	4	no	—
96	C₄²×C₁₃	—	2	2	no	—
97	C₄²×C₁₃	—	2	2	no	—
98	C₄²×C₁₃	—	8	4	no	—
99	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
100	C₄²×C₁₃	#41	1	2	yes	—
101	C₂×C₄×D₈	C₂×C₁₃	1	1	yes	characteristic
102	#2357	C₂×C₁₃	1	2	yes	complemented
103	#1970	C₂×C₁₃	1	2	yes	—
104	#1926	C₂×C₁₃	1	4	yes	complemented
105	#1846	C₂×C₁₃	1	1	yes	characteristic, complemented
106	#1804	C₂×C₁₃	1	2	yes	complemented
107	C₄×#107	C₂×C₁₃	1	2	yes	complemented
108	C₈×C₄⋊C₄	C₂×C₁₃	1	1	yes	characteristic, complemented
109	C₂³×C₁₃	—	2	4	no	—
110	C₂³×C₁₃	—	4	2	no	—
111	C₂³×C₁₃	—	4	2	no	—
112	C₂³×C₁₃	—	4	4	no	—
113	C₂³×C₁₃	—	4	4	no	—
114	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
115	D₄×C₁₃	—	2	4	no	—
116	D₄×C₁₃	—	2	4	no	—
117	D₄×C₁₃	—	4	2	no	—
118	D₄×C₁₃	—	4	2	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	4	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	4	4	no	—
128	C₂×C₄×C₁₃	—	4	4	no	—
129	C₂×C₄×C₁₃	—	4	4	no	—
130	C₂×C₄×C₁₃	—	4	4	no	—
131	C₂×C₄×C₁₃	—	8	8	no	—
132	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
133	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
134	C₂×C₄×C₁₃	#135	1	4	yes	—
135	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
137	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
138	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
139	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
140	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
141	C₈×C₁₃	—	2	2	no	—
142	C₈×C₁₃	—	4	2	no	—
143	C₈×C₁₃	#117	1	4	yes	—
144	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
145	C₂×C₄×D₄	C₂²×C₁₃	1	2	yes	—
146	C₄×D₈	C₄×C₁₃	1	8	yes	complemented
147	C₂×#107	C₂²×C₁₃	1	2	yes	—
148	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
149	C₂×#106	C₂²×C₁₃	1	2	yes	—
150	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
151	C₂×#102	C₂²×C₁₃	1	4	yes	—
152	C₂×#102	C₂²×C₁₃	1	2	yes	—
153	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
154	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
155	#390	C₂²×C₁₃	1	4	yes	—
156	#388	—	2	4	no	—
157	#382	C₂²×C₁₃	1	2	yes	—
158	#381	—	2	2	no	—
159	C₄×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
160	C₄×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
161	C₄×C₂²⋊C₄	—	2	2	no	—
162	#340	C₂²×C₁₃	1	4	yes	—
163	#336	C₂²×C₁₃	1	2	yes	—
164	C₂²×C₁₃	—	4	4	no	—
165	C₂²×C₁₃	—	4	4	no	—
166	C₂²×C₁₃	—	4	4	no	—
167	C₂²×C₁₃	—	8	4	no	—
168	C₂²×C₁₃	—	8	4	no	—
169	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
170	C₂²×C₁₃	#547	1	1	yes	characteristic
171	C₂²×C₁₃	#550	1	1	yes	characteristic
172	C₂²×C₁₃	C₂×#135	1	2	yes	—
173	C₂²×C₁₃	#576	1	2	yes	—
174	C₄×C₁₃	—	2	4	no	—
175	C₄×C₁₃	—	2	2	no	—
176	C₄×C₁₃	—	2	2	no	—
177	C₄×C₁₃	—	4	2	no	—
178	C₄×C₁₃	—	4	4	no	—
179	C₄×C₁₃	—	8	4	no	—
180	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
181	C₄×C₁₃	#520	1	1	yes	characteristic
182	C₄×C₁₃	#555	1	1	yes	characteristic
183	C₄×C₁₃	#556	1	1	yes	characteristic
184	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
185	C₂³×C₄	—	2	2	no	—
186	C₂×D₈	—	2	2	no	—
187	C₂×D₈	—	2	2	no	—
188	C₂×D₈	C₂×C₄×C₁₃	1	4	yes	—
189	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
190	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
191	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
192	C₄×D₄	—	2	4	no	—
193	C₄×D₄	C₂×C₄×C₁₃	1	8	yes	—
194	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
195	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
196	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
197	C₂×C₂²⋊C₄	—	2	2	no	—
198	C₂×C₂²⋊C₄	—	2	4	no	—
199	C₂×C₂²⋊C₄	—	2	4	no	—
200	C₂×C₄²	—	2	4	no	—
201	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
202	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
203	#107	—	2	4	no	—
204	#107	C₂×C₄×C₁₃	1	4	yes	—
205	#106	—	4	2	no	—
206	C₄⋊C₈	—	2	2	no	—
207	#102	—	4	4	no	—
208	#102	C₂×C₄×C₁₃	1	8	yes	—
209	C₄×C₈	—	2	4	no	—
210	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
211	#95	—	2	2	no	—
212	#95	—	2	4	no	—
213	#95	—	2	2	no	—
214	#95	D₄×C₁₃	1	1	yes	characteristic
215	#95	D₄×C₁₃	1	1	yes	characteristic
216	C₂×C₁₃	—	8	4	no	—
217	C₂×C₁₃	#2354	1	1	yes	characteristic
218	C₂×C₁₃	C₄×#135	1	1	yes	characteristic
219	C₂×C₁₃	#2948	1	1	yes	characteristic
220	C₂×C₁₃	#3339	1	2	yes	—
221	C₂×C₁₃	#3396	1	2	yes	—
222	C₂⁴	—	2	2	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	2	2	no	—
225	C₂×D₄	—	2	2	no	—
226	C₂×D₄	—	2	2	no	—
227	C₂×D₄	—	2	2	no	—
228	C₂×D₄	C₂²×C₄×C₁₃	1	4	yes	—
229	C₂²×C₄	—	2	2	no	—
230	C₂²×C₄	—	2	8	no	—
231	C₂²×C₄	—	2	4	no	—
232	C₂²×C₄	—	8	4	no	—
233	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
234	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
237	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
238	D₈	—	2	8	no	—
239	C₂×C₈	—	2	1	no	—
240	C₂×C₈	—	2	1	no	—
241	C₂×C₈	—	2	2	no	—
242	C₂×C₈	—	2	2	no	—
243	C₂×C₈	—	2	2	no	—
244	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
245	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
246	C₂×C₈	C₁₃×#41	1	2	yes	—
247	C₂×C₈	C₁₃×#41	1	2	yes	—
248	C₄⋊C₄	—	2	4	no	—
249	C₄⋊C₄	—	2	2	no	—
250	C₄⋊C₄	—	2	4	no	—
251	C₄⋊C₄	—	2	4	no	—
252	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
253	C₂²⋊C₄	—	2	8	no	—
254	C₂²⋊C₄	—	4	8	no	—
255	C₂²⋊C₄	—	8	4	no	—
256	C₄²	—	2	2	no	—
257	C₄²	—	2	2	no	—
258	C₄²	—	8	4	no	—
259	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
260	C₄²	C₁₃×#41	1	2	yes	—
261	C₁₃	#22810	1	1	yes	characteristic, direct product factor, Sylow
262	C₂³	—	2	4	no	—
263	C₂³	—	4	2	no	—
264	C₂³	—	4	2	no	—
265	C₂³	—	4	4	no	—
266	C₂³	—	4	4	no	—
267	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
268	D₄	—	2	4	no	—
269	D₄	—	2	4	no	—
270	D₄	—	4	2	no	—
271	D₄	—	4	2	no	—
272	C₂×C₄	—	2	2	no	—
273	C₂×C₄	—	2	2	no	—
274	C₂×C₄	—	2	2	no	—
275	C₂×C₄	—	2	2	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	4	4	no	—
281	C₂×C₄	—	4	4	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	8	8	no	—
285	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
286	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
287	C₂×C₄	C₁₃×#135	1	4	yes	—
288	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
289	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
290	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
291	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
292	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
293	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
294	C₈	—	2	2	no	—
295	C₈	—	4	2	no	—
296	C₈	C₁₃×#117	1	4	yes	—
297	C₂²	—	4	4	no	—
298	C₂²	—	4	4	no	—
299	C₂²	—	4	4	no	—
300	C₂²	—	8	4	no	—
301	C₂²	—	8	4	no	—
302	C₂²	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
303	C₂²	C₁₃×#547	1	1	yes	characteristic
304	C₂²	C₁₃×#550	1	1	yes	characteristic
305	C₂²	C₂×C₁₃×#135	1	2	yes	—
306	C₂²	C₁₃×#576	1	2	yes	—
307	C₄	—	2	4	no	—
308	C₄	—	2	2	no	—
309	C₄	—	2	2	no	—
310	C₄	—	4	2	no	—
311	C₄	—	4	4	no	—
312	C₄	—	8	4	no	—
313	C₄	C₂×C₁₃×#117	1	1	yes	characteristic
314	C₄	C₁₃×#520	1	1	yes	characteristic
315	C₄	C₁₃×#555	1	1	yes	characteristic
316	C₄	C₁₃×#556	1	1	yes	characteristic
317	C₂	—	8	4	no	—
318	C₂	C₁₃×#2354	1	1	yes	characteristic
319	C₂	C₄×C₁₃×#135	1	1	yes	characteristic
320	C₂	C₁₃×#2948	1	1	yes	characteristic
321	C₂	C₁₃×#3339	1	2	yes	—
322	C₂	C₁₃×#3396	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₁₃×#2357	C₂	semidirect
4. C₁₃×#1926	C₂	semidirect
5. C₁₃×#1846	C₂	semidirect
6. C₁₃×#1804	C₂	semidirect
7. C₄×C₁₃×#107	C₂	semidirect
8. C₈×C₁₃×C₄⋊C₄	C₂	semidirect
11. C₄×C₁₃×D₈	C₄	semidirect
61. #22810	C₁₃	direct (N × H)
102. #2357	C₂×C₁₃	semidirect
104. #1926	C₂×C₁₃	semidirect
105. #1846	C₂×C₁₃	semidirect
106. #1804	C₂×C₁₃	semidirect
107. C₄×#107	C₂×C₁₃	semidirect
108. C₈×C₄⋊C₄	C₂×C₁₃	semidirect
146. C₄×D₈	C₄×C₁₃	semidirect
261. C₁₃	#22810	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₄×C₁₃×D₈	2
C₁₃×#2357	2
C₁₃×#1970	2
C₁₃×#1926	2
C₁₃×#1846	2
C₁₃×#1804	2
C₄×C₁₃×#107	2
C₈×C₁₃×C₄⋊C₄	2
#22810	13

Subgroups — C13×#22810 #15709149

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#22810 #15709149