Subgroups — C₁₃×#32692 #15719031

Order 3328.

610 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	C₁₃×#2584	C₂	1	2	yes	maximal
2	C₁₃×#2088	C₂	1	2	yes	maximal
3	C₁₃×#2059	C₂	1	2	yes	maximal
4	C₁₃×#2059	C₂	1	2	yes	maximal
5	C₁₃×#1985	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1944	C₂	1	1	yes	maximal, characteristic
7	C₁₃×#1944	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#1920	C₂	1	2	yes	maximal
9	C₁₃×#1919	C₂	1	2	yes	maximal
10	C₂×C₁₃×#125	C₂²	1	2	yes	—
11	C₂×C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#107	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×#106	C₂²	1	2	yes	—
15	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
16	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
18	C₂×C₁₃×#97	C₂²	1	1	yes	characteristic
19	C₁₃×#398	—	2	2	no	—
20	C₁₃×#395	—	2	2	no	—
21	C₁₃×#389	—	2	2	no	—
22	C₁₃×#389	—	2	2	no	—
23	C₁₃×#389	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	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
28	C₁₃×#340	C₂²	1	2	yes	—
29	C₁₃×#340	C₂²	1	2	yes	—
30	C₁₃×#340	C₂²	1	2	yes	—
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×Q₈×C₁₃	D₄	1	1	yes	characteristic
36	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
39	C₁₃×#125	—	4	2	no	—
40	C₁₃×#125	—	4	2	no	—
41	C₄×Q₈×C₁₃	—	4	1	no	—
42	C₄×Q₈×C₁₃	—	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	—	2	2	no	—
48	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
49	C₂×C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
50	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
51	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
52	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
53	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
54	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
55	C₂×C₄²×C₁₃	—	2	2	no	—
56	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
57	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
58	C₁₃×#107	—	4	1	no	—
59	C₁₃×#107	—	4	1	no	—
60	C₁₃×#106	—	4	2	no	—
61	C₁₃×#106	Q₈	1	4	yes	complemented
62	C₁₃×#106	Q₈	1	4	yes	complemented
63	C₁₃×C₄⋊C₈	—	4	1	no	—
64	C₁₃×#97	—	4	1	no	—
65	C₁₃×#95	—	2	2	no	—
66	C₁₃×#95	—	2	2	no	—
67	C₁₃×#95	—	2	2	no	—
68	C₁₃×#95	—	2	2	no	—
69	C₁₃×#95	—	2	2	no	—
70	#32692	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂×Q₈×C₁₃	—	4	1	no	—
72	C₂×Q₈×C₁₃	—	4	2	no	—
73	C₂²×C₄×C₁₃	—	2	2	no	—
74	C₂²×C₄×C₁₃	—	2	2	no	—
75	C₂²×C₄×C₁₃	—	2	2	no	—
76	C₂²×C₄×C₁₃	—	2	2	no	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
82	C₂×C₈×C₁₃	—	2	1	no	—
83	C₂×C₈×C₁₃	—	2	1	no	—
84	C₂×C₈×C₁₃	—	2	1	no	—
85	C₂×C₈×C₁₃	—	2	1	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	—	2	2	no	—
88	C₂×C₈×C₁₃	—	2	2	no	—
89	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
90	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
91	C₁₃×C₄⋊C₄	—	2	2	no	—
92	C₁₃×C₄⋊C₄	—	2	2	no	—
93	C₁₃×C₄⋊C₄	—	2	2	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	—	4	1	no	—
98	C₁₃×C₄⋊C₄	—	4	2	no	—
99	C₁₃×C₄⋊C₄	—	4	4	no	—
100	C₁₃×C₄⋊C₄	—	4	4	no	—
101	C₁₃×C₄⋊C₄	—	4	4	no	—
102	C₁₃×C₄⋊C₄	—	4	4	no	—
103	C₁₃×C₄⋊C₄	—	8	2	no	—
104	C₁₃×C₄⋊C₄	—	8	2	no	—
105	C₁₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
106	C₁₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
107	C₄²×C₁₃	—	2	1	no	—
108	C₄²×C₁₃	—	2	1	no	—
109	C₄²×C₁₃	—	2	2	no	—
110	C₄²×C₁₃	—	2	2	no	—
111	C₄²×C₁₃	—	8	2	no	—
112	#2584	C₂×C₁₃	1	2	yes	—
113	#2088	C₂×C₁₃	1	2	yes	—
114	#2059	C₂×C₁₃	1	2	yes	—
115	#2059	C₂×C₁₃	1	2	yes	—
116	#1985	C₂×C₁₃	1	1	yes	characteristic
117	#1944	C₂×C₁₃	1	1	yes	characteristic
118	#1944	C₂×C₁₃	1	1	yes	characteristic
119	#1920	C₂×C₁₃	1	2	yes	—
120	#1919	C₂×C₁₃	1	2	yes	—
121	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
122	Q₈×C₁₃	—	4	4	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	2	no	—
125	C₂×C₄×C₁₃	—	2	1	no	—
126	C₂×C₄×C₁₃	—	2	1	no	—
127	C₂×C₄×C₁₃	—	2	4	no	—
128	C₂×C₄×C₁₃	—	2	4	no	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	2	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	4	2	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	4	2	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	—	4	2	no	—
141	C₂×C₄×C₁₃	—	4	2	no	—
142	C₂×C₄×C₁₃	—	4	2	no	—
143	C₂×C₄×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	—	4	2	no	—
145	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
148	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
149	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
150	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
151	C₈×C₁₃	—	2	2	no	—
152	C₈×C₁₃	—	2	2	no	—
153	C₈×C₁₃	—	4	2	no	—
154	C₂×#125	C₂²×C₁₃	1	2	yes	—
155	C₂×C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
156	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
157	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
158	C₂×#106	C₂²×C₁₃	1	2	yes	—
159	C₂×#106	C₂²×C₁₃	1	1	yes	characteristic
160	C₂×#106	C₂²×C₁₃	1	1	yes	characteristic
161	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
162	C₂×#97	C₂²×C₁₃	1	1	yes	characteristic
163	#398	—	2	2	no	—
164	#395	—	2	2	no	—
165	#389	—	2	2	no	—
166	#389	—	2	2	no	—
167	#389	C₂²×C₁₃	1	2	yes	—
168	#389	C₂²×C₁₃	1	2	yes	—
169	#389	C₂²×C₁₃	1	2	yes	—
170	#389	C₂²×C₁₃	1	2	yes	—
171	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
172	#340	C₂²×C₁₃	1	2	yes	—
173	#340	C₂²×C₁₃	1	2	yes	—
174	#340	C₂²×C₁₃	1	2	yes	—
175	#340	C₂²×C₁₃	1	2	yes	—
176	#340	C₂²×C₁₃	1	2	yes	—
177	#336	C₂²×C₁₃	1	2	yes	—
178	#336	C₂²×C₁₃	1	2	yes	—
179	C₂²×C₁₃	#534	1	1	yes	characteristic
180	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
181	C₂²×C₁₃	D₄×Q₈	1	1	yes	characteristic
182	C₂²×C₁₃	#575	1	2	yes	—
183	C₂²×C₁₃	#577	1	2	yes	—
184	C₄×C₁₃	—	4	4	no	—
185	C₄×C₁₃	—	4	4	no	—
186	C₄×C₁₃	—	4	2	no	—
187	C₄×C₁₃	—	4	1	no	—
188	C₄×C₁₃	—	4	2	no	—
189	C₄×C₁₃	—	4	1	no	—
190	C₄×C₁₃	—	8	2	no	—
191	C₄×C₁₃	—	8	2	no	—
192	C₄×C₁₃	#544	1	1	yes	characteristic
193	C₄×C₁₃	#544	1	1	yes	characteristic
194	C₄×C₁₃	Q₈²	1	1	yes	characteristic
195	C₄×C₁₃	#563	1	1	yes	characteristic
196	C₂²×Q₈	D₄×C₁₃	1	1	yes	characteristic
197	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
198	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
199	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
200	#125	—	4	2	no	—
201	#125	—	4	2	no	—
202	C₄×Q₈	—	4	1	no	—
203	C₄×Q₈	—	4	1	no	—
204	C₂×C₄⋊C₄	—	2	2	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	2	no	—
207	C₂×C₄⋊C₄	—	2	2	no	—
208	C₂×C₄⋊C₄	—	2	2	no	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
211	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
212	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
213	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
214	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
215	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
216	C₂×C₄²	—	2	2	no	—
217	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
218	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
219	#107	—	4	1	no	—
220	#107	—	4	1	no	—
221	#106	—	4	2	no	—
222	#106	Q₈×C₁₃	1	4	yes	complemented
223	#106	Q₈×C₁₃	1	4	yes	complemented
224	C₄⋊C₈	—	4	1	no	—
225	#97	—	4	1	no	—
226	#95	—	2	2	no	—
227	#95	—	2	2	no	—
228	#95	—	2	2	no	—
229	#95	—	2	2	no	—
230	#95	—	2	2	no	—
231	C₂×C₁₃	#2734	1	1	yes	characteristic
232	C₂×C₁₃	#3197	1	1	yes	characteristic
233	C₂×C₁₃	#3204	1	1	yes	characteristic
234	C₂×C₁₃	#3388	1	2	yes	—
235	C₂×C₁₃	#3388	1	2	yes	—
236	C₂×Q₈	—	4	1	no	—
237	C₂×Q₈	—	4	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	2	no	—
241	C₂²×C₄	—	2	2	no	—
242	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
243	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
244	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
245	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
246	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
247	C₂×C₈	—	2	1	no	—
248	C₂×C₈	—	2	1	no	—
249	C₂×C₈	—	2	1	no	—
250	C₂×C₈	—	2	1	no	—
251	C₂×C₈	—	2	2	no	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	2	no	—
254	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
255	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
256	C₄⋊C₄	—	2	2	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	2	2	no	—
259	C₄⋊C₄	—	2	2	no	—
260	C₄⋊C₄	—	2	2	no	—
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	4	no	—
265	C₄⋊C₄	—	4	4	no	—
266	C₄⋊C₄	—	4	4	no	—
267	C₄⋊C₄	—	4	4	no	—
268	C₄⋊C₄	—	8	2	no	—
269	C₄⋊C₄	—	8	2	no	—
270	C₄⋊C₄	C₂×Q₈×C₁₃	1	4	yes	—
271	C₄⋊C₄	C₂×Q₈×C₁₃	1	4	yes	—
272	C₄²	—	2	1	no	—
273	C₄²	—	2	1	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	2	no	—
276	C₄²	—	8	2	no	—
277	C₁₃	#32692	1	1	yes	characteristic, direct product factor, Sylow
278	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
279	Q₈	—	4	4	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	1	no	—
283	C₂×C₄	—	2	1	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	2	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	4	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	2	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
303	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
304	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
305	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
306	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
307	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
308	C₈	—	2	2	no	—
309	C₈	—	2	2	no	—
310	C₈	—	4	2	no	—
311	C₂²	C₁₃×#534	1	1	yes	characteristic
312	C₂²	D₄×Q₈×C₁₃	1	1	yes	characteristic
313	C₂²	D₄×Q₈×C₁₃	1	1	yes	characteristic
314	C₂²	C₁₃×#575	1	2	yes	—
315	C₂²	C₁₃×#577	1	2	yes	—
316	C₄	—	4	4	no	—
317	C₄	—	4	4	no	—
318	C₄	—	4	2	no	—
319	C₄	—	4	1	no	—
320	C₄	—	4	2	no	—
321	C₄	—	4	1	no	—
322	C₄	—	8	2	no	—
323	C₄	—	8	2	no	—
324	C₄	C₁₃×#544	1	1	yes	characteristic
325	C₄	C₁₃×#544	1	1	yes	characteristic
326	C₄	Q₈²×C₁₃	1	1	yes	characteristic
327	C₄	C₁₃×#563	1	1	yes	characteristic
328	C₂	C₁₃×#2734	1	1	yes	characteristic
329	C₂	C₁₃×#3197	1	1	yes	characteristic
330	C₂	C₁₃×#3204	1	1	yes	characteristic
331	C₂	C₁₃×#3388	1	2	yes	—
332	C₂	C₁₃×#3388	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
61. C₁₃×#106	Q₈	semidirect
62. C₁₃×#106	Q₈	semidirect
70. #32692	C₁₃	direct (N × H)
222. #106	Q₈×C₁₃	semidirect
223. #106	Q₈×C₁₃	semidirect
277. C₁₃	#32692	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×#2584	2
C₁₃×#2088	2
C₁₃×#2059	2
C₁₃×#1985	2
C₁₃×#1944	2
C₁₃×#1920	2
C₁₃×#1919	2
#32692	13

Subgroups — C13×#32692 #15719031

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#32692 #15719031