Subgroups — C₁₃×#32859 #15719198

Order 3328.

708 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×#2860	C₂	1	2	yes	maximal, complemented
2	C₁₃×#2088	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2087	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×#1842	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×#126	—	2	2	no	—
12	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
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	2	yes	—
16	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
18	C₂×C₁₃×#102	C₂²	1	2	yes	—
19	C₂×C₁₃×#102	C₂²	1	2	yes	—
20	C₂×C₁₃×#102	C₂²	1	2	yes	—
21	C₁₃×#390	C₂²	1	2	yes	—
22	C₁₃×#390	C₂²	1	2	yes	—
23	C₁₃×#390	C₂²	1	2	yes	—
24	C₁₃×#389	C₂²	1	2	yes	—
25	C₁₃×#389	C₂²	1	2	yes	—
26	C₁₃×#389	C₂²	1	2	yes	—
27	C₁₃×#388	—	2	2	no	—
28	C₁₃×#388	—	2	2	no	—
29	C₁₃×#388	—	2	2	no	—
30	C₄³×C₁₃	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	C₁₃×#126	—	4	8	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
45	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
46	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
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×C₁₃×C₂²⋊C₄	—	2	2	no	—
54	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₁₃	D₄	1	2	yes	—
56	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
57	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
58	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
59	C₁₃×#107	—	4	1	no	—
60	C₁₃×#106	—	4	2	no	—
61	C₁₃×C₄⋊C₈	—	4	1	no	—
62	C₁₃×C₄⋊C₈	—	4	2	no	—
63	C₁₃×C₄⋊C₈	—	4	2	no	—
64	C₁₃×C₄⋊C₈	—	4	1	no	—
65	C₁₃×#102	—	4	2	no	—
66	C₁₃×#102	—	4	2	no	—
67	C₁₃×#102	—	4	2	no	—
68	C₁₃×#95	—	2	2	no	—
69	C₁₃×#95	—	2	2	no	—
70	C₁₃×#95	—	2	2	no	—
71	#32859	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₂⁴×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	—	2	2	no	—
79	C₂²×C₄×C₁₃	—	2	2	no	—
80	C₂²×C₄×C₁₃	—	2	2	no	—
81	C₂²×C₄×C₁₃	—	2	2	no	—
82	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
86	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
87	C₁₃×D₈	—	4	4	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	2	no	—
92	C₂×C₈×C₁₃	—	2	1	no	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	—	2	1	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	—	4	4	no	—
103	C₁₃×C₄⋊C₄	—	4	4	no	—
104	C₁₃×C₂²⋊C₄	—	4	4	no	—
105	C₁₃×C₂²⋊C₄	—	4	4	no	—
106	C₁₃×C₂²⋊C₄	—	4	4	no	—
107	C₁₃×C₂²⋊C₄	—	4	4	no	—
108	C₁₃×C₂²⋊C₄	—	4	4	no	—
109	C₁₃×C₂²⋊C₄	—	4	4	no	—
110	C₄²×C₁₃	—	2	2	no	—
111	C₄²×C₁₃	—	2	4	no	—
112	C₄²×C₁₃	—	2	2	no	—
113	C₄²×C₁₃	—	2	1	no	—
114	C₄²×C₁₃	—	2	2	no	—
115	C₄²×C₁₃	—	2	1	no	—
116	C₄²×C₁₃	—	4	1	no	—
117	#2860	C₂×C₁₃	1	2	yes	complemented
118	#2088	C₂×C₁₃	1	2	yes	complemented
119	#2087	C₂×C₁₃	1	1	yes	characteristic, complemented
120	#2067	C₂×C₁₃	1	2	yes	complemented
121	#2065	C₂×C₁₃	1	2	yes	complemented
122	#2065	C₂×C₁₃	1	2	yes	complemented
123	#2054	C₂×C₁₃	1	2	yes	—
124	#2054	C₂×C₁₃	1	1	yes	characteristic
125	#1842	C₂×C₁₃	1	1	yes	characteristic, complemented
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	—	4	2	no	—
132	C₂³×C₁₃	—	4	2	no	—
133	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
134	D₄×C₁₃	—	4	2	no	—
135	D₄×C₁₃	—	4	2	no	—
136	D₄×C₁₃	—	4	2	no	—
137	D₄×C₁₃	—	4	2	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	1	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	#135	1	2	yes	—
157	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
158	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
159	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
160	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
161	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
162	C₈×C₁₃	—	4	2	no	—
163	C₈×C₁₃	—	4	1	no	—
164	C₈×C₁₃	—	4	1	no	—
165	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
166	C₂×#126	—	2	2	no	—
167	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
168	C₂×#106	C₂²×C₁₃	1	2	yes	—
169	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
170	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
171	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
172	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
173	C₂×#102	C₂²×C₁₃	1	2	yes	—
174	C₂×#102	C₂²×C₁₃	1	2	yes	—
175	C₂×#102	C₂²×C₁₃	1	2	yes	—
176	#390	C₂²×C₁₃	1	2	yes	—
177	#390	C₂²×C₁₃	1	2	yes	—
178	#390	C₂²×C₁₃	1	2	yes	—
179	#389	C₂²×C₁₃	1	2	yes	—
180	#389	C₂²×C₁₃	1	2	yes	—
181	#389	C₂²×C₁₃	1	2	yes	—
182	#388	—	2	2	no	—
183	#388	—	2	2	no	—
184	#388	—	2	2	no	—
185	C₄³	C₂²×C₁₃	1	1	yes	characteristic
186	#340	C₂²×C₁₃	1	2	yes	—
187	#340	C₂²×C₁₃	1	1	yes	characteristic
188	#340	C₂²×C₁₃	1	1	yes	characteristic
189	#340	C₂²×C₁₃	1	2	yes	—
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	—	8	2	no	—
197	C₂²×C₁₃	#537	1	2	yes	—
198	C₂²×C₁₃	#537	1	1	yes	characteristic
199	C₂²×C₁₃	C₂×#135	1	1	yes	characteristic
200	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
201	C₂²×C₁₃	#575	1	1	yes	characteristic
202	C₄×C₁₃	—	2	4	no	—
203	C₄×C₁₃	—	2	2	no	—
204	C₄×C₁₃	—	4	2	no	—
205	C₄×C₁₃	—	4	1	no	—
206	C₄×C₁₃	—	8	2	no	—
207	C₄×C₁₃	—	8	2	no	—
208	C₄×C₁₃	—	8	2	no	—
209	C₄×C₁₃	#560	1	1	yes	characteristic
210	C₄×C₁₃	#563	1	2	yes	—
211	C₄×C₁₃	#563	1	1	yes	characteristic
212	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
213	C₂×D₈	—	4	2	no	—
214	C₂×D₈	—	4	1	no	—
215	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
216	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
217	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
218	#126	—	4	8	no	—
219	C₂×C₄⋊C₄	—	2	2	no	—
220	C₂×C₄⋊C₄	—	2	2	no	—
221	C₂×C₄⋊C₄	—	2	2	no	—
222	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
223	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
224	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
225	C₂×C₂²⋊C₄	—	2	2	no	—
226	C₂×C₂²⋊C₄	—	2	2	no	—
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	D₄×C₁₃	1	2	yes	—
232	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
233	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
234	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
235	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
236	#107	—	4	1	no	—
237	#106	—	4	2	no	—
238	C₄⋊C₈	—	4	1	no	—
239	C₄⋊C₈	—	4	2	no	—
240	C₄⋊C₈	—	4	2	no	—
241	C₄⋊C₈	—	4	1	no	—
242	#102	—	4	2	no	—
243	#102	—	4	2	no	—
244	#102	—	4	2	no	—
245	#95	—	2	2	no	—
246	#95	—	2	2	no	—
247	#95	—	2	2	no	—
248	C₂×C₁₃	—	16	2	no	—
249	C₂×C₁₃	#2823	1	1	yes	characteristic
250	C₂×C₁₃	#3247	1	2	yes	—
251	C₂×C₁₃	#3250	1	1	yes	characteristic
252	C₂×C₁₃	#3251	1	2	yes	—
253	C₂×C₁₃	#3253	1	1	yes	characteristic
254	C₂⁴	—	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	—	2	2	no	—
261	C₂²×C₄	—	2	2	no	—
262	C₂²×C₄	—	2	2	no	—
263	C₂²×C₄	—	2	2	no	—
264	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
265	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
266	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
267	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
268	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
269	D₈	—	4	4	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	2	no	—
272	C₂×C₈	—	2	2	no	—
273	C₂×C₈	—	2	2	no	—
274	C₂×C₈	—	2	1	no	—
275	C₂×C₈	—	2	2	no	—
276	C₂×C₈	—	2	1	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	—	4	4	no	—
285	C₄⋊C₄	—	4	4	no	—
286	C₂²⋊C₄	—	4	4	no	—
287	C₂²⋊C₄	—	4	4	no	—
288	C₂²⋊C₄	—	4	4	no	—
289	C₂²⋊C₄	—	4	4	no	—
290	C₂²⋊C₄	—	4	4	no	—
291	C₂²⋊C₄	—	4	4	no	—
292	C₄²	—	2	2	no	—
293	C₄²	—	2	4	no	—
294	C₄²	—	2	2	no	—
295	C₄²	—	2	1	no	—
296	C₄²	—	2	2	no	—
297	C₄²	—	2	1	no	—
298	C₄²	—	4	1	no	—
299	C₁₃	#32859	1	1	yes	characteristic, direct product factor, Sylow
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	—	4	2	no	—
306	C₂³	—	4	2	no	—
307	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
308	D₄	—	4	2	no	—
309	D₄	—	4	2	no	—
310	D₄	—	4	2	no	—
311	D₄	—	4	2	no	—
312	C₂×C₄	—	2	2	no	—
313	C₂×C₄	—	2	1	no	—
314	C₂×C₄	—	2	2	no	—
315	C₂×C₄	—	2	1	no	—
316	C₂×C₄	—	2	2	no	—
317	C₂×C₄	—	2	1	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×#135	1	2	yes	—
331	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	4	yes	—
332	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
333	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
334	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
335	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
336	C₈	—	4	2	no	—
337	C₈	—	4	1	no	—
338	C₈	—	4	1	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	—	8	2	no	—
346	C₂²	C₁₃×#537	1	2	yes	—
347	C₂²	C₁₃×#537	1	1	yes	characteristic
348	C₂²	C₂×C₁₃×#135	1	1	yes	characteristic
349	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	2	yes	—
350	C₂²	C₁₃×#575	1	1	yes	characteristic
351	C₄	—	2	4	no	—
352	C₄	—	2	2	no	—
353	C₄	—	4	2	no	—
354	C₄	—	4	1	no	—
355	C₄	—	8	2	no	—
356	C₄	—	8	2	no	—
357	C₄	—	8	2	no	—
358	C₄	C₁₃×#560	1	1	yes	characteristic
359	C₄	C₁₃×#563	1	2	yes	—
360	C₄	C₁₃×#563	1	1	yes	characteristic
361	C₂	—	16	2	no	—
362	C₂	C₁₃×#2823	1	1	yes	characteristic
363	C₂	C₁₃×#3247	1	2	yes	—
364	C₂	C₁₃×#3250	1	1	yes	characteristic
365	C₂	C₁₃×#3251	1	2	yes	—
366	C₂	C₁₃×#3253	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 ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#2860	C₂	semidirect
2. C₁₃×#2088	C₂	semidirect
3. C₁₃×#2087	C₂	semidirect
4. C₁₃×#2067	C₂	semidirect
5. C₁₃×#2065	C₂	semidirect
6. C₁₃×#2065	C₂	semidirect
9. C₁₃×#1842	C₂	semidirect
71. #32859	C₁₃	direct (N × H)
117. #2860	C₂×C₁₃	semidirect
118. #2088	C₂×C₁₃	semidirect
119. #2087	C₂×C₁₃	semidirect
120. #2067	C₂×C₁₃	semidirect
121. #2065	C₂×C₁₃	semidirect
122. #2065	C₂×C₁₃	semidirect
125. #1842	C₂×C₁₃	semidirect
299. C₁₃	#32859	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×#2860	2
C₁₃×#2088	2
C₁₃×#2087	2
C₁₃×#2067	2
C₁₃×#2065	2
C₁₃×#2054	2
C₁₃×#1842	2
#32859	13

Subgroups — C13×#32859 #15719198

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32859 #15719198