Subgroups — C₁₃×#22691 #15709030

Order 3328.

846 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×C₁₃×#443	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2952	C₂	1	2	yes	maximal
3	C₁₃×#2913	C₂	1	2	yes	maximal, complemented
4	C₁₃×#1951	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×#1816	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×#442	—	2	2	no	—
14	C₁₃×#441	—	2	2	no	—
15	C₄×C₁₃×SD₁₆	—	2	4	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	—	2	2	no	—
23	C₂×C₁₃×#107	C₂²	1	2	yes	—
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	2	yes	—
64	C₁₃×#104	C₂×C₄	1	8	yes	complemented
65	C₁₃×#103	—	2	4	no	—
66	C₁₃×#102	—	2	4	no	—
67	C₁₃×#97	—	2	2	no	—
68	C₄×C₈×C₁₃	—	2	2	no	—
69	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	complemented
70	C₁₃×#95	—	4	2	no	—
71	#22691	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₁₃×#41	—	2	2	no	—
73	C₁₃×#41	—	2	2	no	—
74	C₁₃×#41	C₂²×C₄	1	4	yes	—
75	C₁₃×#41	C₂×Q₈	1	4	yes	—
76	C₂×Q₈×C₁₃	C₂×D₄	1	2	yes	—
77	C₂×D₄×C₁₃	—	2	2	no	—
78	C₂×D₄×C₁₃	C₂×D₄	1	2	yes	—
79	C₂²×C₄×C₁₃	—	2	2	no	—
80	C₂²×C₄×C₁₃	—	2	1	no	—
81	C₂²×C₄×C₁₃	—	2	4	no	—
82	C₂²×C₄×C₁₃	—	4	4	no	—
83	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
84	C₁₃×Q₁₆	C₄⋊C₄	1	4	yes	—
85	C₁₃×SD₁₆	C₄⋊C₄	1	8	yes	—
86	C₁₃×D₈	C₄⋊C₄	1	4	yes	—
87	C₁₃×M₁₆	—	2	2	no	—
88	C₁₃×M₁₆	—	2	8	no	—
89	C₁₃×M₁₆	—	2	2	no	—
90	C₁₃×M₁₆	—	2	2	no	—
91	C₁₃×M₁₆	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₁₃	—	2	2	no	—
93	C₂×C₈×C₁₃	—	2	8	no	—
94	C₂×C₈×C₁₃	—	2	2	no	—
95	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
96	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
97	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
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×Q₈	1	2	yes	—
102	C₂×C₈×C₁₃	#41	1	2	yes	—
103	C₁₃×C₄⋊C₄	—	2	4	no	—
104	C₁₃×C₄⋊C₄	—	2	4	no	—
105	C₁₃×C₄⋊C₄	—	4	4	no	—
106	C₁₃×C₄⋊C₄	C₂×D₄	1	4	yes	—
107	C₁₃×C₂²⋊C₄	—	2	2	no	—
108	C₁₃×C₂²⋊C₄	—	4	4	no	—
109	C₄²×C₁₃	—	2	2	no	—
110	C₄²×C₁₃	—	2	1	no	—
111	C₄²×C₁₃	—	4	4	no	—
112	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
113	C₂×#443	C₂×C₁₃	1	1	yes	characteristic
114	#2952	C₂×C₁₃	1	2	yes	—
115	#2913	C₂×C₁₃	1	2	yes	complemented
116	#1951	C₂×C₁₃	1	2	yes	complemented
117	#1860	C₂×C₁₃	1	4	yes	complemented
118	#1846	C₂×C₁₃	1	1	yes	characteristic, complemented
119	#1816	C₂×C₁₃	1	2	yes	complemented
120	#1772	C₂×C₁₃	1	1	yes	characteristic, complemented
121	C₂³×C₁₃	—	2	2	no	—
122	C₂³×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
123	Q₈×C₁₃	—	2	2	no	—
124	Q₈×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
125	D₄×C₁₃	—	2	4	no	—
126	D₄×C₁₃	—	2	4	no	—
127	D₄×C₁₃	—	2	2	no	—
128	D₄×C₁₃	C₂×C₄⋊C₄	1	4	yes	—
129	C₂×C₄×C₁₃	—	2	2	no	—
130	C₂×C₄×C₁₃	—	2	4	no	—
131	C₂×C₄×C₁₃	—	2	4	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	4	no	—
134	C₂×C₄×C₁₃	—	4	2	no	—
135	C₂×C₄×C₁₃	—	4	4	no	—
136	C₂×C₄×C₁₃	—	8	4	no	—
137	C₂×C₄×C₁₃	—	16	4	no	—
138	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
140	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
144	C₈×C₁₃	—	2	4	no	—
145	C₈×C₁₃	—	4	2	no	—
146	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
147	C₈×C₁₃	C₂×C₄⋊C₄	1	2	yes	—
148	C₈×C₁₃	C₄×D₄	1	2	yes	—
149	C₈×C₁₃	C₄×D₄	1	2	yes	—
150	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
151	C₂×#131	C₂²×C₁₃	1	2	yes	—
152	#520	C₂²×C₁₃	1	4	yes	—
153	#443	C₄×C₁₃	1	8	yes	complemented
154	#442	—	2	2	no	—
155	#441	—	2	2	no	—
156	C₄×SD₁₆	—	2	4	no	—
157	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
158	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
159	#428	—	2	4	no	—
160	#428	C₂²×C₁₃	1	4	yes	—
161	#427	—	2	2	no	—
162	#427	C₂²×C₁₃	1	1	yes	characteristic
163	C₂×#107	—	2	2	no	—
164	C₂×#107	C₂²×C₁₃	1	2	yes	—
165	C₂×#106	C₂²×C₁₃	1	2	yes	—
166	C₂×#104	C₂²×C₁₃	1	2	yes	—
167	#417	C₂²×C₁₃	1	4	yes	—
168	#405	C₂²×C₁₃	1	2	yes	—
169	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
170	#382	C₂²×C₁₃	1	2	yes	—
171	#343	C₂²×C₁₃	1	4	yes	—
172	#339	C₂²×C₁₃	1	2	yes	—
173	C₂²×C₁₃	—	2	2	no	—
174	C₂²×C₁₃	—	2	4	no	—
175	C₂²×C₁₃	—	4	2	no	—
176	C₂²×C₁₃	—	4	2	no	—
177	C₂²×C₁₃	#520	1	1	yes	characteristic
178	C₂²×C₁₃	D₄²	1	1	yes	characteristic
179	C₂²×C₁₃	#554	1	1	yes	characteristic
180	C₄×C₁₃	—	2	4	no	—
181	C₄×C₁₃	—	4	2	no	—
182	C₄×C₁₃	—	4	4	no	—
183	C₄×C₁₃	—	8	4	no	—
184	C₄×C₁₃	C₂²×C₄⋊C₄	1	1	yes	characteristic
185	C₄×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
186	C₄×C₁₃	#547	1	1	yes	characteristic
187	C₄×C₁₃	D₄×Q₈	1	1	yes	characteristic
188	C₂×#41	C₂³×C₁₃	1	2	yes	—
189	#135	C₂×C₄×C₁₃	1	4	yes	—
190	#135	Q₈×C₁₃	1	4	yes	—
191	C₂×Q₁₆	D₄×C₁₃	1	1	yes	characteristic
192	C₂×SD₁₆	D₄×C₁₃	1	2	yes	—
193	C₂×D₈	D₄×C₁₃	1	1	yes	characteristic
194	#131	—	2	4	no	—
195	#131	C₂×C₄×C₁₃	1	8	yes	complemented
196	C₂×M₁₆	—	2	2	no	—
197	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
198	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
199	C₂²×C₈	—	2	2	no	—
200	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
201	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
202	C₄×Q₈	—	2	4	no	—
203	C₄×D₄	—	2	4	no	—
204	C₄×D₄	—	4	4	no	—
205	#117	—	2	4	no	—
206	#117	C₂³×C₁₃	1	2	yes	—
207	C₂×C₄⋊C₄	—	2	4	no	—
208	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
209	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
210	#108	—	2	2	no	—
211	#108	C₂×C₄×C₁₃	1	4	yes	complemented
212	#107	—	2	2	no	—
213	#107	—	2	2	no	—
214	#107	—	2	4	no	—
215	#107	D₄×C₁₃	1	2	yes	—
216	#106	—	2	2	no	—
217	#106	—	2	2	no	—
218	#106	—	2	4	no	—
219	#106	D₄×C₁₃	1	2	yes	—
220	#104	C₂×C₄×C₁₃	1	8	yes	complemented
221	#103	—	2	4	no	—
222	#102	—	2	4	no	—
223	#97	—	2	2	no	—
224	C₄×C₈	—	2	2	no	—
225	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	complemented
226	#95	—	4	2	no	—
227	C₂×C₁₃	—	2	1	no	—
228	C₂×C₁₃	—	2	1	no	—
229	C₂×C₁₃	—	4	4	no	—
230	C₂×C₁₃	D₄×C₄⋊C₄	1	1	yes	characteristic
231	C₂×C₁₃	#3290	1	2	yes	—
232	#41	—	2	2	no	—
233	#41	—	2	2	no	—
234	#41	C₂²×C₄×C₁₃	1	4	yes	—
235	#41	C₂×Q₈×C₁₃	1	4	yes	—
236	C₂×Q₈	C₂×D₄×C₁₃	1	2	yes	—
237	C₂×D₄	—	2	2	no	—
238	C₂×D₄	C₂×D₄×C₁₃	1	2	yes	—
239	C₂²×C₄	—	2	2	no	—
240	C₂²×C₄	—	2	1	no	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	4	4	no	—
243	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
244	Q₁₆	C₁₃×C₄⋊C₄	1	4	yes	—
245	SD₁₆	C₁₃×C₄⋊C₄	1	8	yes	—
246	D₈	C₁₃×C₄⋊C₄	1	4	yes	—
247	M₁₆	—	2	2	no	—
248	M₁₆	—	2	8	no	—
249	M₁₆	—	2	2	no	—
250	M₁₆	—	2	2	no	—
251	M₁₆	C₂²×C₄×C₁₃	1	4	yes	—
252	C₂×C₈	—	2	2	no	—
253	C₂×C₈	—	2	8	no	—
254	C₂×C₈	—	2	2	no	—
255	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
256	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
257	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
258	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
260	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
261	C₂×C₈	C₂×Q₈×C₁₃	1	2	yes	—
262	C₂×C₈	C₁₃×#41	1	2	yes	—
263	C₄⋊C₄	—	2	4	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	4	4	no	—
266	C₄⋊C₄	C₂×D₄×C₁₃	1	4	yes	—
267	C₂²⋊C₄	—	2	2	no	—
268	C₂²⋊C₄	—	4	4	no	—
269	C₄²	—	2	2	no	—
270	C₄²	—	2	1	no	—
271	C₄²	—	4	4	no	—
272	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
273	C₁₃	#22691	1	1	yes	characteristic, direct product factor, Sylow
274	C₂³	—	2	2	no	—
275	C₂³	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
276	Q₈	—	2	2	no	—
277	Q₈	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
278	D₄	—	2	4	no	—
279	D₄	—	2	4	no	—
280	D₄	—	2	2	no	—
281	D₄	C₂×C₁₃×C₄⋊C₄	1	4	yes	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	4	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	2	no	—
286	C₂×C₄	—	2	4	no	—
287	C₂×C₄	—	4	2	no	—
288	C₂×C₄	—	4	4	no	—
289	C₂×C₄	—	8	4	no	—
290	C₂×C₄	—	16	4	no	—
291	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
292	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
293	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
294	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
295	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
296	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
297	C₈	—	2	4	no	—
298	C₈	—	4	2	no	—
299	C₈	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
300	C₈	C₂×C₁₃×C₄⋊C₄	1	2	yes	—
301	C₈	C₄×D₄×C₁₃	1	2	yes	—
302	C₈	C₄×D₄×C₁₃	1	2	yes	—
303	C₂²	—	2	2	no	—
304	C₂²	—	2	4	no	—
305	C₂²	—	4	2	no	—
306	C₂²	—	4	2	no	—
307	C₂²	C₁₃×#520	1	1	yes	characteristic
308	C₂²	D₄²×C₁₃	1	1	yes	characteristic
309	C₂²	C₁₃×#554	1	1	yes	characteristic
310	C₄	—	2	4	no	—
311	C₄	—	4	2	no	—
312	C₄	—	4	4	no	—
313	C₄	—	8	4	no	—
314	C₄	C₂²×C₁₃×C₄⋊C₄	1	1	yes	characteristic
315	C₄	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
316	C₄	C₁₃×#547	1	1	yes	characteristic
317	C₄	D₄×Q₈×C₁₃	1	1	yes	characteristic
318	C₂	—	2	1	no	—
319	C₂	—	2	1	no	—
320	C₂	—	4	4	no	—
321	C₂	D₄×C₁₃×C₄⋊C₄	1	1	yes	characteristic
322	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×#2913	C₂	semidirect
4. C₁₃×#1951	C₂	semidirect
5. C₁₃×#1860	C₂	semidirect
6. C₁₃×#1846	C₂	semidirect
7. C₁₃×#1816	C₂	semidirect
8. C₁₃×#1772	C₂	semidirect
12. C₁₃×#443	C₄	semidirect
39. C₁₃×#131	C₂×C₄	semidirect
55. C₁₃×#108	C₂×C₄	semidirect
64. C₁₃×#104	C₂×C₄	semidirect
69. C₄×C₈×C₁₃	C₂×C₄	semidirect
71. #22691	C₁₃	direct (N × H)
115. #2913	C₂×C₁₃	semidirect
116. #1951	C₂×C₁₃	semidirect
117. #1860	C₂×C₁₃	semidirect
118. #1846	C₂×C₁₃	semidirect
119. #1816	C₂×C₁₃	semidirect
120. #1772	C₂×C₁₃	semidirect
153. #443	C₄×C₁₃	semidirect
195. #131	C₂×C₄×C₁₃	semidirect
211. #108	C₂×C₄×C₁₃	semidirect
220. #104	C₂×C₄×C₁₃	semidirect
225. C₄×C₈	C₂×C₄×C₁₃	semidirect
273. C₁₃	#22691	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×#2952	2
C₁₃×#2913	2
C₁₃×#1951	2
C₁₃×#1860	2
C₁₃×#1846	2
C₁₃×#1816	2
C₁₃×#1772	2
#22691	13

Subgroups — C13×#22691 #15709030

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#22691 #15709030