Subgroups — C₁₃×#33706 #15720045

Order 3328.

598 conjugacy classes of subgroups.

The subgroup lattice has 282 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#495	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2243	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2242	C₂	1	4	yes	maximal, complemented
5	C₁₃×#2141	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2140	C₂	1	2	yes	maximal
7	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×SD₁₆	—	2	1	no	—
10	C₂²×C₁₃×D₈	C₂²	1	1	yes	characteristic
11	C₁₃×#533	—	2	1	no	—
12	C₂×C₁₃×#124	—	2	1	no	—
13	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
14	C₁₃×#500	C₂²	1	2	yes	complemented
15	C₁₃×#499	—	2	1	no	—
16	C₁₃×#499	C₂²	1	2	yes	complemented
17	C₁₃×#498	—	2	1	no	—
18	C₁₃×#495	—	2	1	no	—
19	C₁₃×#495	—	2	1	no	—
20	C₁₃×#495	C₂²	1	2	yes	—
21	C₁₃×#495	C₂²	1	2	yes	—
22	C₁₃×#427	—	2	1	no	—
23	C₁₃×#427	C₂²	1	2	yes	—
24	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
25	C₁₃×#358	C₂²	1	8	yes	complemented
26	C₁₃×#357	C₂²	1	8	yes	—
27	C₁₃×#346	C₂²	1	2	yes	complemented
28	C₄×C₁₃×C₁₆	C₂²	1	2	yes	complemented
29	C₂²×Q₈×C₁₃	—	2	1	no	—
30	C₂²×D₄×C₁₃	—	2	1	no	—
31	C₂×C₁₃×Q₁₆	—	2	1	no	—
32	C₂×C₁₃×Q₁₆	—	2	4	no	—
33	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
34	C₂×C₁₃×SD₁₆	—	2	2	no	—
35	C₂×C₁₃×SD₁₆	—	2	2	no	—
36	C₂×C₁₃×SD₁₆	—	4	4	no	—
37	C₂×C₁₃×D₈	—	2	1	no	—
38	C₂×C₁₃×D₈	—	2	4	no	—
39	C₂×C₁₃×D₈	C₂³	1	2	yes	—
40	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
41	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
42	C₁₃×C₄⋊Q₈	—	2	2	no	—
43	C₁₃×#125	—	2	2	no	—
44	C₁₃×#124	—	2	2	no	—
45	C₁₃×#124	—	2	2	no	—
46	C₁₃×#124	—	4	2	no	—
47	C₁₃×C₂²⋊Q₈	—	4	2	no	—
48	C₄×Q₈×C₁₃	—	4	2	no	—
49	C₁₃×#117	—	2	2	no	—
50	C₂×C₁₃×C₂²⋊C₄	—	4	2	no	—
51	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
52	C₁₃×M₃₂	—	2	4	no	—
53	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
54	C₁₃×#107	—	2	2	no	—
55	C₁₃×#107	C₂³	1	4	yes	—
56	C₁₃×#106	—	2	2	no	—
57	C₁₃×#106	—	2	2	no	—
58	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
59	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
60	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
61	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
62	#33706	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂⁴×C₁₃	—	4	1	no	—
64	C₂×Q₈×C₁₃	—	2	2	no	—
65	C₂×Q₈×C₁₃	—	4	2	no	—
66	C₂×Q₈×C₁₃	—	4	1	no	—
67	C₂×Q₈×C₁₃	—	4	2	no	—
68	C₂×Q₈×C₁₃	—	4	2	no	—
69	C₂×D₄×C₁₃	—	2	2	no	—
70	C₂×D₄×C₁₃	—	4	2	no	—
71	C₂×D₄×C₁₃	—	4	1	no	—
72	C₂×D₄×C₁₃	—	4	2	no	—
73	C₂²×C₄×C₁₃	—	4	1	no	—
74	C₂²×C₄×C₁₃	D₈	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₁₃×Q₁₆	—	2	4	no	—
77	C₁₃×Q₁₆	—	4	2	no	—
78	C₁₃×SD₁₆	—	4	4	no	—
79	C₁₃×SD₁₆	—	4	4	no	—
80	C₁₃×D₈	—	2	4	no	—
81	C₁₃×D₈	—	4	2	no	—
82	C₂×C₈×C₁₃	SD₁₆	1	4	yes	—
83	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₈×C₁₃	#41	1	4	yes	—
87	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
88	C₁₃×C₄⋊C₄	—	2	4	no	—
89	C₁₃×C₄⋊C₄	—	4	4	no	—
90	C₁₃×C₄⋊C₄	—	4	2	no	—
91	C₁₃×C₂²⋊C₄	—	4	2	no	—
92	C₁₃×C₂²⋊C₄	—	4	4	no	—
93	C₁₃×C₂²⋊C₄	—	4	4	no	—
94	C₄²×C₁₃	—	4	2	no	—
95	C₄²×C₁₃	D₈	1	2	yes	—
96	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
98	C₁₃×C₁₆	—	2	4	no	—
99	#3160	C₂×C₁₃	1	1	yes	characteristic, complemented
100	C₂×#495	C₂×C₁₃	1	1	yes	characteristic
101	#2243	C₂×C₁₃	1	4	yes	complemented
102	#2242	C₂×C₁₃	1	4	yes	complemented
103	#2141	C₂×C₁₃	1	2	yes	complemented
104	#2140	C₂×C₁₃	1	2	yes	—
105	C₄×M₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
106	C₂³×C₁₃	—	4	2	no	—
107	C₂³×C₁₃	—	8	4	no	—
108	C₂³×C₁₃	—	8	2	no	—
109	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
110	Q₈×C₁₃	—	4	2	no	—
111	Q₈×C₁₃	—	4	2	no	—
112	Q₈×C₁₃	—	8	4	no	—
113	Q₈×C₁₃	—	8	2	no	—
114	D₄×C₁₃	—	4	2	no	—
115	D₄×C₁₃	—	4	2	no	—
116	D₄×C₁₃	—	8	2	no	—
117	C₂×C₄×C₁₃	—	4	2	no	—
118	C₂×C₄×C₁₃	—	4	4	no	—
119	C₂×C₄×C₁₃	—	4	2	no	—
120	C₂×C₄×C₁₃	—	8	2	no	—
121	C₂×C₄×C₁₃	#124	1	2	yes	—
122	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
123	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
124	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
126	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₈×C₁₃	#124	1	2	yes	—
128	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
129	C₈×C₁₃	C₂×SD₁₆	1	2	yes	—
130	C₈×C₁₃	C₂×#41	1	2	yes	—
131	C₂²×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
132	C₂²×SD₁₆	—	2	1	no	—
133	C₂²×D₈	C₂²×C₁₃	1	1	yes	characteristic
134	#533	—	2	1	no	—
135	C₂×#124	—	2	1	no	—
136	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
137	#500	C₂²×C₁₃	1	2	yes	complemented
138	#499	—	2	1	no	—
139	#499	C₂²×C₁₃	1	2	yes	complemented
140	#498	—	2	1	no	—
141	#495	—	2	1	no	—
142	#495	—	2	1	no	—
143	#495	C₂²×C₁₃	1	2	yes	—
144	#495	C₂²×C₁₃	1	2	yes	—
145	#427	—	2	1	no	—
146	#427	C₂²×C₁₃	1	2	yes	—
147	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
148	#358	C₂²×C₁₃	1	8	yes	complemented
149	#357	C₂²×C₁₃	1	8	yes	—
150	#346	C₂²×C₁₃	1	2	yes	complemented
151	C₄×C₁₆	C₂²×C₁₃	1	2	yes	complemented
152	C₂²×C₁₃	—	2	2	no	—
153	C₂²×C₁₃	—	8	2	no	—
154	C₂²×C₁₃	—	8	4	no	—
155	C₂²×C₁₃	—	16	4	no	—
156	C₂²×C₁₃	#486	1	2	yes	—
157	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
158	C₄×C₁₃	—	2	2	no	—
159	C₄×C₁₃	—	2	2	no	—
160	C₄×C₁₃	—	8	2	no	—
161	C₄×C₁₃	—	8	4	no	—
162	C₄×C₁₃	#486	1	2	yes	—
163	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
164	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
165	C₂²×Q₈	—	2	1	no	—
166	C₂²×D₄	—	2	1	no	—
167	C₂×Q₁₆	—	2	1	no	—
168	C₂×Q₁₆	—	2	4	no	—
169	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
170	C₂×SD₁₆	—	2	2	no	—
171	C₂×SD₁₆	—	2	2	no	—
172	C₂×SD₁₆	—	4	4	no	—
173	C₂×D₈	—	2	1	no	—
174	C₂×D₈	—	2	4	no	—
175	C₂×D₈	C₂³×C₁₃	1	2	yes	—
176	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
177	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
178	C₄⋊Q₈	—	2	2	no	—
179	#125	—	2	2	no	—
180	#124	—	2	2	no	—
181	#124	—	2	2	no	—
182	#124	—	4	2	no	—
183	C₂²⋊Q₈	—	4	2	no	—
184	C₄×Q₈	—	4	2	no	—
185	#117	—	2	2	no	—
186	C₂×C₂²⋊C₄	—	4	2	no	—
187	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
188	M₃₂	—	2	4	no	—
189	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
190	#107	—	2	2	no	—
191	#107	C₂³×C₁₃	1	4	yes	—
192	#106	—	2	2	no	—
193	#106	—	2	2	no	—
194	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
195	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
196	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
197	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
198	C₂×C₁₃	—	2	2	no	—
199	C₂×C₁₃	—	16	2	no	—
200	C₂×C₁₃	C₂×#486	1	1	yes	characteristic
201	C₂×C₁₃	#3414	1	2	yes	—
202	C₂⁴	—	4	1	no	—
203	C₂×Q₈	—	2	2	no	—
204	C₂×Q₈	—	4	2	no	—
205	C₂×Q₈	—	4	1	no	—
206	C₂×Q₈	—	4	2	no	—
207	C₂×Q₈	—	4	2	no	—
208	C₂×D₄	—	2	2	no	—
209	C₂×D₄	—	4	2	no	—
210	C₂×D₄	—	4	1	no	—
211	C₂×D₄	—	4	2	no	—
212	C₂²×C₄	—	4	1	no	—
213	C₂²×C₄	C₁₃×D₈	1	2	yes	—
214	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
215	Q₁₆	—	2	4	no	—
216	Q₁₆	—	4	2	no	—
217	SD₁₆	—	4	4	no	—
218	SD₁₆	—	4	4	no	—
219	D₈	—	2	4	no	—
220	D₈	—	4	2	no	—
221	C₂×C₈	C₁₃×SD₁₆	1	4	yes	—
222	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
223	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
224	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
225	C₂×C₈	C₁₃×#41	1	4	yes	—
226	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
227	C₄⋊C₄	—	2	4	no	—
228	C₄⋊C₄	—	4	4	no	—
229	C₄⋊C₄	—	4	2	no	—
230	C₂²⋊C₄	—	4	2	no	—
231	C₂²⋊C₄	—	4	4	no	—
232	C₂²⋊C₄	—	4	4	no	—
233	C₄²	—	4	2	no	—
234	C₄²	C₁₃×D₈	1	2	yes	—
235	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
236	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
237	C₁₆	—	2	4	no	—
238	C₁₃	#33706	1	1	yes	characteristic, direct product factor, Sylow
239	C₂³	—	4	2	no	—
240	C₂³	—	8	4	no	—
241	C₂³	—	8	2	no	—
242	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
243	Q₈	—	4	2	no	—
244	Q₈	—	4	2	no	—
245	Q₈	—	8	4	no	—
246	Q₈	—	8	2	no	—
247	D₄	—	4	2	no	—
248	D₄	—	4	2	no	—
249	D₄	—	8	2	no	—
250	C₂×C₄	—	4	2	no	—
251	C₂×C₄	—	4	4	no	—
252	C₂×C₄	—	4	2	no	—
253	C₂×C₄	—	8	2	no	—
254	C₂×C₄	C₁₃×#124	1	2	yes	—
255	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
256	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
257	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
258	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
259	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
260	C₈	C₁₃×#124	1	2	yes	—
261	C₈	C₂×C₁₃×SD₁₆	1	2	yes	—
262	C₈	C₂×C₁₃×SD₁₆	1	2	yes	—
263	C₈	C₂×C₁₃×#41	1	2	yes	—
264	C₂²	—	2	2	no	—
265	C₂²	—	8	2	no	—
266	C₂²	—	8	4	no	—
267	C₂²	—	16	4	no	—
268	C₂²	C₁₃×#486	1	2	yes	—
269	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
270	C₄	—	2	2	no	—
271	C₄	—	2	2	no	—
272	C₄	—	8	2	no	—
273	C₄	—	8	4	no	—
274	C₄	C₁₃×#486	1	2	yes	—
275	C₄	C₂×C₁₃×#124	1	1	yes	characteristic
276	C₄	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
277	C₂	—	2	2	no	—
278	C₂	—	16	2	no	—
279	C₂	C₂×C₁₃×#486	1	1	yes	characteristic
280	C₂	C₁₃×#3414	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂²×C₁₃×D₈	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ C₂×C₄×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3160	C₂	semidirect
3. C₁₃×#2243	C₂	semidirect
4. C₁₃×#2242	C₂	semidirect
5. C₁₃×#2141	C₂	semidirect
7. C₄×C₁₃×M₃₂	C₂	semidirect
14. C₁₃×#500	C₂²	semidirect
16. C₁₃×#499	C₂²	semidirect
25. C₁₃×#358	C₂²	semidirect
27. C₁₃×#346	C₂²	semidirect
28. C₄×C₁₃×C₁₆	C₂²	semidirect
62. #33706	C₁₃	direct (N × H)
99. #3160	C₂×C₁₃	semidirect
101. #2243	C₂×C₁₃	semidirect
102. #2242	C₂×C₁₃	semidirect
103. #2141	C₂×C₁₃	semidirect
105. C₄×M₃₂	C₂×C₁₃	semidirect
137. #500	C₂²×C₁₃	semidirect
139. #499	C₂²×C₁₃	semidirect
148. #358	C₂²×C₁₃	semidirect
150. #346	C₂²×C₁₃	semidirect
151. C₄×C₁₆	C₂²×C₁₃	semidirect
238. C₁₃	#33706	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×#3160	2
C₂×C₁₃×#495	2
C₁₃×#2243	2
C₁₃×#2242	2
C₁₃×#2141	2
C₁₃×#2140	2
C₄×C₁₃×M₃₂	2
#33706	13

Subgroups — C13×#33706 #15720045

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33706 #15720045