Subgroups — C₁₃×#22690 #15709029

Order 3328.

846 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#443	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2951	C₂	1	2	yes	maximal
3	C₁₃×#2912	C₂	1	2	yes	maximal, complemented
4	C₁₃×#1952	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1860	C₂	1	4	yes	maximal, complemented
6	C₁₃×#1846	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1815	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#131	C₂²	1	2	yes	—
11	C₁₃×#520	C₂²	1	4	yes	—
12	C₁₃×#443	C₄	1	8	yes	complemented
13	C₁₃×#440	—	2	4	no	—
14	C₄×C₁₃×Q₁₆	—	2	2	no	—
15	C₄×C₁₃×D₈	—	2	2	no	—
16	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
18	C₁₃×#428	—	2	4	no	—
19	C₁₃×#428	C₂²	1	4	yes	—
20	C₁₃×#427	—	2	2	no	—
21	C₁₃×#427	C₂²	1	1	yes	characteristic
22	C₂×C₁₃×#107	C₂²	1	2	yes	—
23	C₂×C₁₃×#106	—	2	2	no	—
24	C₂×C₁₃×#106	C₂²	1	2	yes	—
25	C₂×C₁₃×#104	C₂²	1	2	yes	—
26	C₁₃×#417	C₂²	1	4	yes	—
27	C₁₃×#405	C₂²	1	2	yes	—
28	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
29	C₁₃×#382	C₂²	1	2	yes	—
30	C₁₃×#343	C₂²	1	4	yes	—
31	C₁₃×#339	C₂²	1	2	yes	—
32	C₂×C₁₃×#41	C₂³	1	2	yes	—
33	C₁₃×#135	C₂×C₄	1	4	yes	—
34	C₁₃×#135	Q₈	1	4	yes	—
35	C₂×C₁₃×Q₁₆	D₄	1	1	yes	characteristic
36	C₂×C₁₃×SD₁₆	D₄	1	2	yes	—
37	C₂×C₁₃×D₈	D₄	1	1	yes	characteristic
38	C₁₃×#131	—	2	4	no	—
39	C₁₃×#131	C₂×C₄	1	8	yes	complemented
40	C₂×C₁₃×M₁₆	—	2	2	no	—
41	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
42	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
43	C₂²×C₈×C₁₃	—	2	2	no	—
44	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
46	C₄×Q₈×C₁₃	—	2	4	no	—
47	C₄×D₄×C₁₃	—	2	4	no	—
48	C₄×D₄×C₁₃	—	4	4	no	—
49	C₁₃×#117	—	2	4	no	—
50	C₁₃×#117	C₂³	1	2	yes	—
51	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
52	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
53	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
54	C₁₃×#108	—	2	2	no	—
55	C₁₃×#108	C₂×C₄	1	4	yes	complemented
56	C₁₃×#107	—	2	2	no	—
57	C₁₃×#107	—	2	2	no	—
58	C₁₃×#107	—	2	4	no	—
59	C₁₃×#107	D₄	1	2	yes	—
60	C₁₃×#106	—	2	2	no	—
61	C₁₃×#106	—	2	2	no	—
62	C₁₃×#106	—	2	4	no	—
63	C₁₃×#106	D₄	1	1	yes	characteristic
64	C₁₃×#106	D₄	1	1	yes	characteristic
65	C₁₃×#104	C₂×C₄	1	8	yes	complemented
66	C₁₃×#103	—	2	4	no	—
67	C₁₃×#102	—	2	4	no	—
68	C₁₃×#97	—	2	2	no	—
69	C₄×C₈×C₁₃	—	2	2	no	—
70	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	complemented
71	C₁₃×#95	—	4	2	no	—
72	#22690	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
73	C₁₃×#41	—	2	2	no	—
74	C₁₃×#41	—	2	2	no	—
75	C₁₃×#41	C₂²×C₄	1	4	yes	—
76	C₁₃×#41	C₂×Q₈	1	4	yes	—
77	C₂×Q₈×C₁₃	C₂×D₄	1	2	yes	—
78	C₂×D₄×C₁₃	—	2	2	no	—
79	C₂×D₄×C₁₃	C₂×D₄	1	2	yes	—
80	C₂²×C₄×C₁₃	—	2	2	no	—
81	C₂²×C₄×C₁₃	—	2	1	no	—
82	C₂²×C₄×C₁₃	—	2	4	no	—
83	C₂²×C₄×C₁₃	—	4	4	no	—
84	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
85	C₁₃×Q₁₆	C₄⋊C₄	1	4	yes	—
86	C₁₃×SD₁₆	C₄⋊C₄	1	8	yes	—
87	C₁₃×D₈	C₄⋊C₄	1	4	yes	—
88	C₁₃×M₁₆	—	2	2	no	—
89	C₁₃×M₁₆	—	2	8	no	—
90	C₁₃×M₁₆	—	2	2	no	—
91	C₁₃×M₁₆	—	2	2	no	—
92	C₁₃×M₁₆	C₂²×C₄	1	4	yes	—
93	C₂×C₈×C₁₃	—	2	2	no	—
94	C₂×C₈×C₁₃	—	2	8	no	—
95	C₂×C₈×C₁₃	—	2	2	no	—
96	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
98	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₈×C₁₃	C₂×Q₈	1	2	yes	—
103	C₂×C₈×C₁₃	#41	1	2	yes	—
104	C₁₃×C₄⋊C₄	—	2	4	no	—
105	C₁₃×C₄⋊C₄	—	2	4	no	—
106	C₁₃×C₄⋊C₄	—	4	4	no	—
107	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
108	C₁₃×C₄⋊C₄	C₂×D₄	1	2	yes	—
109	C₁₃×C₂²⋊C₄	—	2	2	no	—
110	C₁₃×C₂²⋊C₄	—	4	4	no	—
111	C₄²×C₁₃	—	2	2	no	—
112	C₄²×C₁₃	—	2	1	no	—
113	C₄²×C₁₃	—	4	4	no	—
114	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
115	C₂×#443	C₂×C₁₃	1	1	yes	characteristic
116	#2951	C₂×C₁₃	1	2	yes	—
117	#2912	C₂×C₁₃	1	2	yes	complemented
118	#1952	C₂×C₁₃	1	2	yes	complemented
119	#1860	C₂×C₁₃	1	4	yes	complemented
120	#1846	C₂×C₁₃	1	1	yes	characteristic, complemented
121	#1815	C₂×C₁₃	1	2	yes	complemented
122	#1772	C₂×C₁₃	1	1	yes	characteristic, complemented
123	C₂³×C₁₃	—	2	2	no	—
124	C₂³×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
125	Q₈×C₁₃	—	2	2	no	—
126	Q₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
127	D₄×C₁₃	—	2	4	no	—
128	D₄×C₁₃	—	2	4	no	—
129	D₄×C₁₃	—	2	2	no	—
130	D₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	4	no	—
133	C₂×C₄×C₁₃	—	2	4	no	—
134	C₂×C₄×C₁₃	—	2	2	no	—
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	4	4	no	—
139	C₂×C₄×C₁₃	—	8	4	no	—
140	C₂×C₄×C₁₃	—	16	4	no	—
141	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
146	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
147	C₈×C₁₃	—	2	4	no	—
148	C₈×C₁₃	—	4	2	no	—
149	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
150	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
151	C₈×C₁₃	C₄×D₄	1	2	yes	—
152	C₈×C₁₃	C₄×D₄	1	2	yes	—
153	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
154	C₂×#131	C₂²×C₁₃	1	2	yes	—
155	#520	C₂²×C₁₃	1	4	yes	—
156	#443	C₄×C₁₃	1	8	yes	complemented
157	#440	—	2	4	no	—
158	C₄×Q₁₆	—	2	2	no	—
159	C₄×D₈	—	2	2	no	—
160	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
161	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
162	#428	—	2	4	no	—
163	#428	C₂²×C₁₃	1	4	yes	—
164	#427	—	2	2	no	—
165	#427	C₂²×C₁₃	1	1	yes	characteristic
166	C₂×#107	C₂²×C₁₃	1	2	yes	—
167	C₂×#106	—	2	2	no	—
168	C₂×#106	C₂²×C₁₃	1	2	yes	—
169	C₂×#104	C₂²×C₁₃	1	2	yes	—
170	#417	C₂²×C₁₃	1	4	yes	—
171	#405	C₂²×C₁₃	1	2	yes	—
172	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
173	#382	C₂²×C₁₃	1	2	yes	—
174	#343	C₂²×C₁₃	1	4	yes	—
175	#339	C₂²×C₁₃	1	2	yes	—
176	C₂²×C₁₃	—	2	2	no	—
177	C₂²×C₁₃	—	2	4	no	—
178	C₂²×C₁₃	—	4	4	no	—
179	C₂²×C₁₃	#520	1	1	yes	characteristic
180	C₂²×C₁₃	D₄²	1	1	yes	characteristic
181	C₂²×C₁₃	#554	1	1	yes	characteristic
182	C₄×C₁₃	—	2	4	no	—
183	C₄×C₁₃	—	4	2	no	—
184	C₄×C₁₃	—	4	2	no	—
185	C₄×C₁₃	—	4	2	no	—
186	C₄×C₁₃	—	8	4	no	—
187	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
188	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
189	C₄×C₁₃	#547	1	1	yes	characteristic
190	C₄×C₁₃	D₄×Q₈	1	1	yes	characteristic
191	C₂×#41	C₂³×C₁₃	1	2	yes	—
192	#135	C₂×C₄×C₁₃	1	4	yes	—
193	#135	Q₈×C₁₃	1	4	yes	—
194	C₂×Q₁₆	D₄×C₁₃	1	1	yes	characteristic
195	C₂×SD₁₆	D₄×C₁₃	1	2	yes	—
196	C₂×D₈	D₄×C₁₃	1	1	yes	characteristic
197	#131	—	2	4	no	—
198	#131	C₂×C₄×C₁₃	1	8	yes	complemented
199	C₂×M₁₆	—	2	2	no	—
200	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
201	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
202	C₂²×C₈	—	2	2	no	—
203	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
204	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
205	C₄×Q₈	—	2	4	no	—
206	C₄×D₄	—	2	4	no	—
207	C₄×D₄	—	4	4	no	—
208	#117	—	2	4	no	—
209	#117	C₂³×C₁₃	1	2	yes	—
210	C₂×C₄⋊C₄	—	2	4	no	—
211	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
212	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
213	#108	—	2	2	no	—
214	#108	C₂×C₄×C₁₃	1	4	yes	complemented
215	#107	—	2	2	no	—
216	#107	—	2	2	no	—
217	#107	—	2	4	no	—
218	#107	D₄×C₁₃	1	2	yes	—
219	#106	—	2	2	no	—
220	#106	—	2	2	no	—
221	#106	—	2	4	no	—
222	#106	D₄×C₁₃	1	1	yes	characteristic
223	#106	D₄×C₁₃	1	1	yes	characteristic
224	#104	C₂×C₄×C₁₃	1	8	yes	complemented
225	#103	—	2	4	no	—
226	#102	—	2	4	no	—
227	#97	—	2	2	no	—
228	C₄×C₈	—	2	2	no	—
229	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	complemented
230	#95	—	4	2	no	—
231	C₂×C₁₃	—	2	1	no	—
232	C₂×C₁₃	—	2	1	no	—
233	C₂×C₁₃	—	4	4	no	—
234	C₂×C₁₃	D₄×C₄⋊C₄	1	1	yes	characteristic
235	C₂×C₁₃	#3290	1	2	yes	—
236	#41	—	2	2	no	—
237	#41	—	2	2	no	—
238	#41	C₂²×C₄×C₁₃	1	4	yes	—
239	#41	C₂×Q₈×C₁₃	1	4	yes	—
240	C₂×Q₈	C₂×D₄×C₁₃	1	2	yes	—
241	C₂×D₄	—	2	2	no	—
242	C₂×D₄	C₂×D₄×C₁₃	1	2	yes	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	1	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	4	4	no	—
247	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
248	Q₁₆	C₁₃×C₄⋊C₄	1	4	yes	—
249	SD₁₆	C₁₃×C₄⋊C₄	1	8	yes	—
250	D₈	C₁₃×C₄⋊C₄	1	4	yes	—
251	M₁₆	—	2	2	no	—
252	M₁₆	—	2	8	no	—
253	M₁₆	—	2	2	no	—
254	M₁₆	—	2	2	no	—
255	M₁₆	C₂²×C₄×C₁₃	1	4	yes	—
256	C₂×C₈	—	2	2	no	—
257	C₂×C₈	—	2	8	no	—
258	C₂×C₈	—	2	2	no	—
259	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
260	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
261	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
262	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
263	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
264	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
265	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
266	C₂×C₈	C₁₃×#41	1	2	yes	—
267	C₄⋊C₄	—	2	4	no	—
268	C₄⋊C₄	—	2	4	no	—
269	C₄⋊C₄	—	4	4	no	—
270	C₄⋊C₄	C₂×D₄×C₁₃	1	2	yes	—
271	C₄⋊C₄	C₂×D₄×C₁₃	1	2	yes	—
272	C₂²⋊C₄	—	2	2	no	—
273	C₂²⋊C₄	—	4	4	no	—
274	C₄²	—	2	2	no	—
275	C₄²	—	2	1	no	—
276	C₄²	—	4	4	no	—
277	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
278	C₁₃	#22690	1	1	yes	characteristic, direct product factor, Sylow
279	C₂³	—	2	2	no	—
280	C₂³	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
281	Q₈	—	2	2	no	—
282	Q₈	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
283	D₄	—	2	4	no	—
284	D₄	—	2	4	no	—
285	D₄	—	2	2	no	—
286	D₄	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	4	no	—
289	C₂×C₄	—	2	4	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	2	no	—
293	C₂×C₄	—	4	2	no	—
294	C₂×C₄	—	4	4	no	—
295	C₂×C₄	—	8	4	no	—
296	C₂×C₄	—	16	4	no	—
297	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
298	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
299	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
300	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
301	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
302	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
303	C₈	—	2	4	no	—
304	C₈	—	4	2	no	—
305	C₈	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
306	C₈	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
307	C₈	C₄×D₄×C₁₃	1	2	yes	—
308	C₈	C₄×D₄×C₁₃	1	2	yes	—
309	C₂²	—	2	2	no	—
310	C₂²	—	2	4	no	—
311	C₂²	—	4	4	no	—
312	C₂²	C₁₃×#520	1	1	yes	characteristic
313	C₂²	D₄²×C₁₃	1	1	yes	characteristic
314	C₂²	C₁₃×#554	1	1	yes	characteristic
315	C₄	—	2	4	no	—
316	C₄	—	4	2	no	—
317	C₄	—	4	2	no	—
318	C₄	—	4	2	no	—
319	C₄	—	8	4	no	—
320	C₄	C₂²×C₁₃×C₄⋊C₄	1	1	yes	characteristic
321	C₄	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
322	C₄	C₁₃×#547	1	1	yes	characteristic
323	C₄	D₄×Q₈×C₁₃	1	1	yes	characteristic
324	C₂	—	2	1	no	—
325	C₂	—	2	1	no	—
326	C₂	—	4	4	no	—
327	C₂	D₄×C₁₃×C₄⋊C₄	1	1	yes	characteristic
328	C₂	C₁₃×#3290	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	D₄²
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#135	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#2912	C₂	semidirect
4. C₁₃×#1952	C₂	semidirect
5. C₁₃×#1860	C₂	semidirect
6. C₁₃×#1846	C₂	semidirect
7. C₁₃×#1815	C₂	semidirect
8. C₁₃×#1772	C₂	semidirect
12. C₁₃×#443	C₄	semidirect
39. C₁₃×#131	C₂×C₄	semidirect
55. C₁₃×#108	C₂×C₄	semidirect
65. C₁₃×#104	C₂×C₄	semidirect
70. C₄×C₈×C₁₃	C₂×C₄	semidirect
72. #22690	C₁₃	direct (N × H)
117. #2912	C₂×C₁₃	semidirect
118. #1952	C₂×C₁₃	semidirect
119. #1860	C₂×C₁₃	semidirect
120. #1846	C₂×C₁₃	semidirect
121. #1815	C₂×C₁₃	semidirect
122. #1772	C₂×C₁₃	semidirect
156. #443	C₄×C₁₃	semidirect
198. #131	C₂×C₄×C₁₃	semidirect
214. #108	C₂×C₄×C₁₃	semidirect
224. #104	C₂×C₄×C₁₃	semidirect
229. C₄×C₈	C₂×C₄×C₁₃	semidirect
278. C₁₃	#22690	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×C₁₃×#443	2
C₁₃×#2951	2
C₁₃×#2912	2
C₁₃×#1952	2
C₁₃×#1860	2
C₁₃×#1846	2
C₁₃×#1815	2
C₁₃×#1772	2
#22690	13

Subgroups — C13×#22690 #15709029

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#22690 #15709029