Subgroups — C₁₃×#3181 #13112239

Order 1664.

366 conjugacy classes of subgroups.

The subgroup lattice has 260 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#529	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#501	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#497	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#491	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#489	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#467	C₂	1	2	yes	maximal, complemented
8	C₁₃×#465	C₂	1	2	yes	maximal
9	C₁₃×#441	C₂	1	2	yes	maximal, complemented
10	C₁₃×#440	C₂	1	2	yes	maximal, complemented
11	C₂×C₁₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×Q₁₆	C₂²	1	2	yes	—
13	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
14	C₂²×C₈×C₁₃	C₂²	1	2	yes	—
15	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₃×#126	—	2	1	no	—
18	C₁₃×#125	—	2	1	no	—
19	C₁₃×#125	C₂²	1	1	yes	characteristic
20	C₁₃×#124	C₂²	1	1	yes	characteristic
21	C₁₃×#123	—	2	1	no	—
22	C₁₃×C₂²⋊Q₈	—	2	1	no	—
23	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
24	C₁₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
25	C₁₃×#121	C₂²	1	1	yes	characteristic
26	C₄×Q₈×C₁₃	—	2	1	no	—
27	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
28	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
29	C₄×D₄×C₁₃	—	2	1	no	—
30	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
31	C₁₃×#117	—	2	1	no	—
32	C₁₃×#117	—	2	1	no	—
33	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
34	C₁₃×#107	C₂²	1	2	yes	—
35	C₁₃×#106	C₂²	1	2	yes	—
36	C₁₃×#103	C₂²	1	2	yes	—
37	C₁₃×#103	C₂²	1	2	yes	—
38	C₁₃×#103	C₂²	1	2	yes	—
39	C₁₃×#102	C₂²	1	2	yes	—
40	C₁₃×#97	C₂²	1	2	yes	—
41	C₁₃×#97	C₂²	1	1	yes	characteristic
42	C₁₃×#97	C₂²	1	1	yes	characteristic
43	C₂×Q₈×C₁₃	—	2	1	no	—
44	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
46	C₂×D₄×C₁₃	—	2	1	no	—
47	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
50	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
51	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
52	C₁₃×Q₁₆	—	2	4	no	—
53	C₁₃×SD₁₆	—	2	4	no	—
54	C₂×C₈×C₁₃	—	2	4	no	—
55	C₂×C₈×C₁₃	C₂³	1	2	yes	—
56	C₂×C₈×C₁₃	C₂³	1	2	yes	—
57	C₁₃×C₄⋊C₄	—	2	1	no	—
58	C₁₃×C₄⋊C₄	—	2	1	no	—
59	C₁₃×C₄⋊C₄	—	2	1	no	—
60	C₁₃×C₄⋊C₄	—	2	1	no	—
61	C₁₃×C₄⋊C₄	—	2	1	no	—
62	C₁₃×C₄⋊C₄	—	2	1	no	—
63	C₁₃×C₄⋊C₄	—	2	2	no	—
64	C₁₃×C₄⋊C₄	—	2	2	no	—
65	C₁₃×C₄⋊C₄	—	2	1	no	—
66	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
67	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
68	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
69	C₁₃×C₂²⋊C₄	—	2	2	no	—
70	C₁₃×C₂²⋊C₄	—	2	1	no	—
71	C₁₃×C₂²⋊C₄	—	2	1	no	—
72	C₁₃×C₂²⋊C₄	—	2	1	no	—
73	C₁₃×C₂²⋊C₄	—	2	1	no	—
74	C₁₃×C₂²⋊C₄	—	2	1	no	—
75	C₄²×C₁₃	—	2	1	no	—
76	C₄²×C₁₃	—	2	2	no	—
77	C₄²×C₁₃	D₄	1	1	yes	characteristic
78	C₄²×C₁₃	D₄	1	1	yes	characteristic
79	C₄²×C₁₃	C₂³	1	1	yes	characteristic
80	C₄²×C₁₃	C₂³	1	1	yes	characteristic
81	#3181	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
82	C₂³×C₁₃	—	2	1	no	—
83	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
84	Q₈×C₁₃	—	2	2	no	—
85	Q₈×C₁₃	—	2	2	no	—
86	Q₈×C₁₃	—	2	2	no	—
87	Q₈×C₁₃	—	4	2	no	—
88	D₄×C₁₃	—	2	2	no	—
89	D₄×C₁₃	—	4	2	no	—
90	C₂×C₄×C₁₃	—	2	2	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	2	1	no	—
93	C₂×C₄×C₁₃	—	2	2	no	—
94	C₂×C₄×C₁₃	—	2	1	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	1	no	—
97	C₂×C₄×C₁₃	—	4	1	no	—
98	C₂×C₄×C₁₃	—	4	2	no	—
99	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
100	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
102	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
105	C₈×C₁₃	—	2	2	no	—
106	C₈×C₁₃	#41	1	4	yes	—
107	#533	C₂×C₁₃	1	1	yes	characteristic, complemented
108	#529	C₂×C₁₃	1	1	yes	characteristic
109	#501	C₂×C₁₃	1	1	yes	characteristic, complemented
110	#497	C₂×C₁₃	1	1	yes	characteristic, complemented
111	#491	C₂×C₁₃	1	1	yes	characteristic, complemented
112	#489	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#467	C₂×C₁₃	1	2	yes	complemented
114	#465	C₂×C₁₃	1	2	yes	—
115	#441	C₂×C₁₃	1	2	yes	complemented
116	#440	C₂×C₁₃	1	2	yes	complemented
117	C₂×#97	C₂×C₁₃	1	1	yes	characteristic, complemented
118	C₂²×C₁₃	—	2	1	no	—
119	C₂²×C₁₃	—	2	1	no	—
120	C₂²×C₁₃	—	4	1	no	—
121	C₂²×C₁₃	—	4	1	no	—
122	C₂²×C₁₃	—	4	1	no	—
123	C₂²×C₁₃	#137	1	2	yes	—
124	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
125	C₄×C₁₃	—	2	1	no	—
126	C₄×C₁₃	—	2	1	no	—
127	C₄×C₁₃	—	2	1	no	—
128	C₄×C₁₃	—	2	1	no	—
129	C₄×C₁₃	—	2	1	no	—
130	C₄×C₁₃	—	4	1	no	—
131	C₄×C₁₃	—	4	2	no	—
132	C₄×C₁₃	—	4	1	no	—
133	C₄×C₁₃	—	4	1	no	—
134	C₄×C₁₃	—	4	2	no	—
135	C₄×C₁₃	C₂×#41	1	2	yes	—
136	C₂×Q₁₆	C₂²×C₁₃	1	2	yes	—
137	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
138	C₂²×C₈	C₂²×C₁₃	1	2	yes	—
139	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
140	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
141	#126	—	2	1	no	—
142	#125	—	2	1	no	—
143	#125	C₂²×C₁₃	1	1	yes	characteristic
144	#124	C₂²×C₁₃	1	1	yes	characteristic
145	#123	—	2	1	no	—
146	C₂²⋊Q₈	—	2	1	no	—
147	C₂²⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
148	C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	—
149	#121	C₂²×C₁₃	1	1	yes	characteristic
150	C₄×Q₈	—	2	1	no	—
151	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
152	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
153	C₄×D₄	—	2	1	no	—
154	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
155	#117	—	2	1	no	—
156	#117	—	2	1	no	—
157	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
158	#107	C₂²×C₁₃	1	2	yes	—
159	#106	C₂²×C₁₃	1	2	yes	—
160	#103	C₂²×C₁₃	1	2	yes	—
161	#103	C₂²×C₁₃	1	2	yes	—
162	#103	C₂²×C₁₃	1	2	yes	—
163	#102	C₂²×C₁₃	1	2	yes	—
164	#97	C₂²×C₁₃	1	2	yes	—
165	#97	C₂²×C₁₃	1	1	yes	characteristic
166	#97	C₂²×C₁₃	1	1	yes	characteristic
167	C₂×C₁₃	—	2	2	no	—
168	C₂×C₁₃	—	8	1	no	—
169	C₂×C₁₃	#532	1	1	yes	characteristic
170	C₂×C₁₃	C₂×#137	1	1	yes	characteristic
171	C₂×C₁₃	#575	1	1	yes	characteristic
172	C₂×Q₈	—	2	1	no	—
173	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
174	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
175	C₂×D₄	—	2	1	no	—
176	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
177	C₂²×C₄	—	2	1	no	—
178	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
179	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
180	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
181	Q₁₆	—	2	4	no	—
182	SD₁₆	—	2	4	no	—
183	C₂×C₈	—	2	4	no	—
184	C₂×C₈	C₂³×C₁₃	1	2	yes	—
185	C₂×C₈	C₂³×C₁₃	1	2	yes	—
186	C₄⋊C₄	—	2	1	no	—
187	C₄⋊C₄	—	2	1	no	—
188	C₄⋊C₄	—	2	1	no	—
189	C₄⋊C₄	—	2	1	no	—
190	C₄⋊C₄	—	2	1	no	—
191	C₄⋊C₄	—	2	1	no	—
192	C₄⋊C₄	—	2	2	no	—
193	C₄⋊C₄	—	2	2	no	—
194	C₄⋊C₄	—	2	1	no	—
195	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
196	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
197	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
198	C₂²⋊C₄	—	2	2	no	—
199	C₂²⋊C₄	—	2	1	no	—
200	C₂²⋊C₄	—	2	1	no	—
201	C₂²⋊C₄	—	2	1	no	—
202	C₂²⋊C₄	—	2	1	no	—
203	C₂²⋊C₄	—	2	1	no	—
204	C₄²	—	2	1	no	—
205	C₄²	—	2	2	no	—
206	C₄²	D₄×C₁₃	1	1	yes	characteristic
207	C₄²	D₄×C₁₃	1	1	yes	characteristic
208	C₄²	C₂³×C₁₃	1	1	yes	characteristic
209	C₄²	C₂³×C₁₃	1	1	yes	characteristic
210	C₁₃	#3181	1	1	yes	characteristic, direct product factor, Sylow
211	C₂³	—	2	1	no	—
212	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
213	Q₈	—	2	2	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	2	2	no	—
216	Q₈	—	4	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	4	2	no	—
219	C₂×C₄	—	2	2	no	—
220	C₂×C₄	—	2	2	no	—
221	C₂×C₄	—	2	1	no	—
222	C₂×C₄	—	2	2	no	—
223	C₂×C₄	—	2	1	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	4	1	no	—
227	C₂×C₄	—	4	2	no	—
228	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
229	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
230	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
231	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
232	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
233	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
234	C₈	—	2	2	no	—
235	C₈	C₁₃×#41	1	4	yes	—
236	C₂²	—	2	1	no	—
237	C₂²	—	2	1	no	—
238	C₂²	—	4	1	no	—
239	C₂²	—	4	1	no	—
240	C₂²	—	4	1	no	—
241	C₂²	C₁₃×#137	1	2	yes	—
242	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
243	C₄	—	2	1	no	—
244	C₄	—	2	1	no	—
245	C₄	—	2	1	no	—
246	C₄	—	2	1	no	—
247	C₄	—	2	1	no	—
248	C₄	—	4	1	no	—
249	C₄	—	4	2	no	—
250	C₄	—	4	1	no	—
251	C₄	—	4	1	no	—
252	C₄	—	4	2	no	—
253	C₄	C₂×C₁₃×#41	1	2	yes	—
254	C₂	—	2	2	no	—
255	C₂	—	8	1	no	—
256	C₂	C₁₃×#532	1	1	yes	characteristic
257	C₂	C₂×C₁₃×#137	1	1	yes	characteristic
258	C₂	C₁₃×#575	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₁₃×#121	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#533	C₂	semidirect
3. C₁₃×#501	C₂	semidirect
4. C₁₃×#497	C₂	semidirect
5. C₁₃×#491	C₂	semidirect
6. C₁₃×#489	C₂	semidirect
7. C₁₃×#467	C₂	semidirect
9. C₁₃×#441	C₂	semidirect
10. C₁₃×#440	C₂	semidirect
11. C₂×C₁₃×#97	C₂	semidirect
81. #3181	C₁₃	direct (N × H)
107. #533	C₂×C₁₃	semidirect
109. #501	C₂×C₁₃	semidirect
110. #497	C₂×C₁₃	semidirect
111. #491	C₂×C₁₃	semidirect
112. #489	C₂×C₁₃	semidirect
113. #467	C₂×C₁₃	semidirect
115. #441	C₂×C₁₃	semidirect
116. #440	C₂×C₁₃	semidirect
117. C₂×#97	C₂×C₁₃	semidirect
210. C₁₃	#3181	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×#533	2
C₁₃×#529	2
C₁₃×#501	2
C₁₃×#497	2
C₁₃×#491	2
C₁₃×#489	2
C₁₃×#467	2
C₁₃×#465	2
C₁₃×#441	2
C₁₃×#440	2
C₂×C₁₃×#97	2
#3181	13

Subgroups — C13×#3181 #13112239

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3181 #13112239