Subgroups — C₁₃×#22809 #15709148

Order 3328.

766 conjugacy classes of subgroups.

The subgroup lattice has 302 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic
2	C₁₃×#2356	C₂	1	2	yes	maximal
3	C₁₃×#1969	C₂	1	2	yes	maximal
4	C₁₃×#1927	C₂	1	4	yes	maximal
5	C₁₃×#1846	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#1805	C₂	1	2	yes	maximal
7	C₄×C₁₃×#107	C₂	1	2	yes	maximal
8	C₈×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
9	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
10	C₂×C₄×Q₈×C₁₃	C₂²	1	2	yes	—
11	C₄×C₁₃×Q₁₆	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×#103	C₂²	1	4	yes	—
17	C₂×C₁₃×#103	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×#391	C₂²	1	4	yes	—
21	C₁₃×#387	—	2	4	no	—
22	C₁₃×#383	—	2	2	no	—
23	C₁₃×#382	C₂²	1	2	yes	—
24	C₄×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
25	C₄×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
26	C₁₃×#376	—	2	2	no	—
27	C₁₃×#340	C₂²	1	4	yes	—
28	C₁₃×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
30	C₂×C₁₃×Q₁₆	—	2	2	no	—
31	C₂×C₁₃×Q₁₆	—	2	2	no	—
32	C₂×C₁₃×Q₁₆	C₂×C₄	1	4	yes	—
33	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
36	C₄×Q₈×C₁₃	—	2	4	no	—
37	C₄×Q₈×C₁₃	C₂×C₄	1	8	yes	—
38	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
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×C₄²×C₁₃	—	2	2	no	—
43	C₂×C₄²×C₁₃	—	2	4	no	—
44	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
45	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
46	C₁₃×#107	—	2	4	no	—
47	C₁₃×#107	C₂×C₄	1	4	yes	—
48	C₁₃×#106	—	4	2	no	—
49	C₁₃×C₄⋊C₈	—	2	2	no	—
50	C₁₃×#103	—	4	4	no	—
51	C₁₃×#103	C₂×C₄	1	8	yes	—
52	C₄×C₈×C₁₃	—	2	4	no	—
53	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
54	C₁₃×#95	—	2	4	no	—
55	C₁₃×#95	—	2	2	no	—
56	C₁₃×#95	—	2	2	no	—
57	C₁₃×#95	—	2	4	no	—
58	C₁₃×#95	—	2	4	no	—
59	C₁₃×#95	D₄	1	1	yes	characteristic
60	C₁₃×#95	D₄	1	1	yes	characteristic
61	#22809	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂×Q₈×C₁₃	—	2	2	no	—
63	C₂×Q₈×C₁₃	—	2	2	no	—
64	C₂×Q₈×C₁₃	—	2	2	no	—
65	C₂×Q₈×C₁₃	—	2	2	no	—
66	C₂×Q₈×C₁₃	—	2	2	no	—
67	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
68	C₂²×C₄×C₁₃	—	2	2	no	—
69	C₂²×C₄×C₁₃	—	2	2	no	—
70	C₂²×C₄×C₁₃	—	2	4	no	—
71	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
76	C₁₃×Q₁₆	—	2	8	no	—
77	C₂×C₈×C₁₃	—	2	1	no	—
78	C₂×C₈×C₁₃	—	2	1	no	—
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
83	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
84	C₂×C₈×C₁₃	#41	1	2	yes	—
85	C₂×C₈×C₁₃	#41	1	2	yes	—
86	C₁₃×C₄⋊C₄	—	2	8	no	—
87	C₁₃×C₄⋊C₄	—	2	4	no	—
88	C₁₃×C₄⋊C₄	—	2	2	no	—
89	C₁₃×C₄⋊C₄	—	2	4	no	—
90	C₁₃×C₄⋊C₄	—	2	4	no	—
91	C₁₃×C₄⋊C₄	C₂²×C₄	1	4	yes	—
92	C₄²×C₁₃	—	2	8	no	—
93	C₄²×C₁₃	—	2	2	no	—
94	C₄²×C₁₃	—	2	2	no	—
95	C₄²×C₁₃	—	4	8	no	—
96	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
97	C₄²×C₁₃	#41	1	2	yes	—
98	C₂×C₄×Q₁₆	C₂×C₁₃	1	1	yes	characteristic
99	#2356	C₂×C₁₃	1	2	yes	—
100	#1969	C₂×C₁₃	1	2	yes	—
101	#1927	C₂×C₁₃	1	4	yes	—
102	#1846	C₂×C₁₃	1	1	yes	characteristic
103	#1805	C₂×C₁₃	1	2	yes	—
104	C₄×#107	C₂×C₁₃	1	2	yes	—
105	C₈×C₄⋊C₄	C₂×C₁₃	1	1	yes	characteristic
106	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
107	Q₈×C₁₃	—	2	4	no	—
108	Q₈×C₁₃	—	2	4	no	—
109	Q₈×C₁₃	—	4	2	no	—
110	Q₈×C₁₃	—	4	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	2	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	—	2	4	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	2	2	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	4	2	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	4	4	no	—
123	C₂×C₄×C₁₃	—	4	4	no	—
124	C₂×C₄×C₁₃	—	4	4	no	—
125	C₂×C₄×C₁₃	—	4	4	no	—
126	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
127	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
128	C₂×C₄×C₁₃	#135	1	4	yes	—
129	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
132	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
133	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
134	C₂×C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
135	C₈×C₁₃	—	2	2	no	—
136	C₈×C₁₃	—	4	2	no	—
137	C₈×C₁₃	#117	1	4	yes	—
138	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
139	C₂×C₄×Q₈	C₂²×C₁₃	1	2	yes	—
140	C₄×Q₁₆	C₄×C₁₃	1	8	yes	complemented
141	C₂×#107	C₂²×C₁₃	1	2	yes	—
142	C₂×#107	C₂²×C₁₃	1	1	yes	characteristic
143	C₂×#106	C₂²×C₁₃	1	2	yes	—
144	C₂×C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
145	C₂×#103	C₂²×C₁₃	1	4	yes	—
146	C₂×#103	C₂²×C₁₃	1	2	yes	—
147	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
148	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
149	#391	C₂²×C₁₃	1	4	yes	—
150	#387	—	2	4	no	—
151	#383	—	2	2	no	—
152	#382	C₂²×C₁₃	1	2	yes	—
153	C₄×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
154	C₄×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
155	#376	—	2	2	no	—
156	#340	C₂²×C₁₃	1	4	yes	—
157	#336	C₂²×C₁₃	1	2	yes	—
158	C₂²×C₁₃	C₂×C₄×D₄	1	1	yes	characteristic
159	C₂²×C₁₃	#547	1	1	yes	characteristic
160	C₂²×C₁₃	#550	1	1	yes	characteristic
161	C₂²×C₁₃	C₂×#135	1	2	yes	—
162	C₂²×C₁₃	#578	1	2	yes	—
163	C₄×C₁₃	—	2	4	no	—
164	C₄×C₁₃	—	2	2	no	—
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	—	4	2	no	—
167	C₄×C₁₃	—	4	4	no	—
168	C₄×C₁₃	—	4	4	no	—
169	C₄×C₁₃	—	8	4	no	—
170	C₄×C₁₃	C₂×#117	1	1	yes	characteristic
171	C₄×C₁₃	#520	1	1	yes	characteristic
172	C₄×C₁₃	#555	1	1	yes	characteristic
173	C₄×C₁₃	#556	1	1	yes	characteristic
174	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
175	C₂×Q₁₆	—	2	2	no	—
176	C₂×Q₁₆	—	2	2	no	—
177	C₂×Q₁₆	C₂×C₄×C₁₃	1	4	yes	—
178	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
179	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
180	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
181	C₄×Q₈	—	2	4	no	—
182	C₄×Q₈	C₂×C₄×C₁₃	1	8	yes	—
183	C₂×C₄⋊C₄	—	2	2	no	—
184	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
185	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
186	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
187	C₂×C₄²	—	2	2	no	—
188	C₂×C₄²	—	2	4	no	—
189	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
190	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
191	#107	—	2	4	no	—
192	#107	C₂×C₄×C₁₃	1	4	yes	—
193	#106	—	4	2	no	—
194	C₄⋊C₈	—	2	2	no	—
195	#103	—	4	4	no	—
196	#103	C₂×C₄×C₁₃	1	8	yes	—
197	C₄×C₈	—	2	4	no	—
198	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
199	#95	—	2	2	no	—
200	#95	—	2	4	no	—
201	#95	—	2	4	no	—
202	#95	—	2	2	no	—
203	#95	—	2	4	no	—
204	#95	D₄×C₁₃	1	1	yes	characteristic
205	#95	D₄×C₁₃	1	1	yes	characteristic
206	C₂×C₁₃	#2354	1	1	yes	characteristic
207	C₂×C₁₃	C₄×#135	1	1	yes	characteristic
208	C₂×C₁₃	#2950	1	1	yes	characteristic
209	C₂×C₁₃	#3336	1	2	yes	—
210	C₂×C₁₃	#3393	1	2	yes	—
211	C₂×Q₈	—	2	2	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×Q₈	—	2	2	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×Q₈	—	2	2	no	—
216	C₂×Q₈	C₂²×C₄×C₁₃	1	4	yes	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	2	2	no	—
219	C₂²×C₄	—	2	4	no	—
220	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
221	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
222	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
223	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
224	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
225	Q₁₆	—	2	8	no	—
226	C₂×C₈	—	2	1	no	—
227	C₂×C₈	—	2	1	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	2	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
232	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
233	C₂×C₈	C₁₃×#41	1	2	yes	—
234	C₂×C₈	C₁₃×#41	1	2	yes	—
235	C₄⋊C₄	—	2	8	no	—
236	C₄⋊C₄	—	2	4	no	—
237	C₄⋊C₄	—	2	2	no	—
238	C₄⋊C₄	—	2	4	no	—
239	C₄⋊C₄	—	2	4	no	—
240	C₄⋊C₄	C₂²×C₄×C₁₃	1	4	yes	—
241	C₄²	—	2	8	no	—
242	C₄²	—	2	2	no	—
243	C₄²	—	2	2	no	—
244	C₄²	—	4	8	no	—
245	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
246	C₄²	C₁₃×#41	1	2	yes	—
247	C₁₃	#22809	1	1	yes	characteristic, direct product factor, Sylow
248	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
249	Q₈	—	2	4	no	—
250	Q₈	—	2	4	no	—
251	Q₈	—	4	2	no	—
252	Q₈	—	4	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	2	4	no	—
258	C₂×C₄	—	2	4	no	—
259	C₂×C₄	—	2	2	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	4	no	—
267	C₂×C₄	—	4	4	no	—
268	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
269	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
270	C₂×C₄	C₁₃×#135	1	4	yes	—
271	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
272	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
273	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
274	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
275	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
276	C₂×C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
277	C₈	—	2	2	no	—
278	C₈	—	4	2	no	—
279	C₈	C₁₃×#117	1	4	yes	—
280	C₂²	C₂×C₄×D₄×C₁₃	1	1	yes	characteristic
281	C₂²	C₁₃×#547	1	1	yes	characteristic
282	C₂²	C₁₃×#550	1	1	yes	characteristic
283	C₂²	C₂×C₁₃×#135	1	2	yes	—
284	C₂²	C₁₃×#578	1	2	yes	—
285	C₄	—	2	4	no	—
286	C₄	—	2	2	no	—
287	C₄	—	2	2	no	—
288	C₄	—	4	2	no	—
289	C₄	—	4	4	no	—
290	C₄	—	4	4	no	—
291	C₄	—	8	4	no	—
292	C₄	C₂×C₁₃×#117	1	1	yes	characteristic
293	C₄	C₁₃×#520	1	1	yes	characteristic
294	C₄	C₁₃×#555	1	1	yes	characteristic
295	C₄	C₁₃×#556	1	1	yes	characteristic
296	C₂	C₁₃×#2354	1	1	yes	characteristic
297	C₂	C₄×C₁₃×#135	1	1	yes	characteristic
298	C₂	C₁₃×#2950	1	1	yes	characteristic
299	C₂	C₁₃×#3336	1	2	yes	—
300	C₂	C₁₃×#3393	1	2	yes	—

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂²×D₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
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
11. C₄×C₁₃×Q₁₆	C₄	semidirect
61. #22809	C₁₃	direct (N × H)
140. C₄×Q₁₆	C₄×C₁₃	semidirect
247. C₁₃	#22809	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×Q₁₆	2
C₁₃×#2356	2
C₁₃×#1969	2
C₁₃×#1927	2
C₁₃×#1846	2
C₁₃×#1805	2
C₄×C₁₃×#107	2
C₈×C₁₃×C₄⋊C₄	2
#22809	13

Subgroups — C13×#22809 #15709148

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#22809 #15709148