Subgroups — C₁₃×#33793 #15720132

Order 3328.

538 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3169	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2254	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2240	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2233	C₂	1	4	yes	maximal, complemented
6	C₁₃×#2226	C₂	1	4	yes	maximal, complemented
7	C₁₃×#2162	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#532	—	2	1	no	—
9	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
10	C₁₃×#500	—	2	1	no	—
11	C₁₃×#500	—	2	1	no	—
12	C₁₃×#500	C₂²	1	2	yes	—
13	C₁₃×#500	C₂²	1	1	yes	characteristic
14	C₁₃×#500	C₂²	1	2	yes	—
15	C₁₃×#493	C₂²	1	1	yes	characteristic
16	C₁₃×#486	—	2	1	no	—
17	C₁₃×#466	—	2	1	no	—
18	C₁₃×#466	C₂²	1	2	yes	—
19	C₄×C₁₃×D₈	—	2	1	no	—
20	C₄×C₁₃×D₈	C₂²	1	2	yes	—
21	C₂×C₁₃×#107	—	2	1	no	—
22	C₂×C₁₃×#107	C₂²	1	2	yes	—
23	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₁₃×#367	C₂²	1	8	yes	—
25	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
26	C₁₃×#357	C₂²	1	8	yes	—
27	C₁₃×C₂²⋊C₁₆	C₂²	1	2	yes	—
28	C₄×C₁₃×C₁₆	C₂²	1	2	yes	—
29	C₂×C₁₃×#41	—	4	1	no	—
30	C₂²×Q₈×C₁₃	—	4	1	no	—
31	C₂×C₁₃×D₈	—	2	1	no	—
32	C₂×C₁₃×D₈	C₂³	1	2	yes	—
33	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
35	C₁₃×C₄⋊Q₈	—	2	1	no	—
36	C₁₃×C₄⋊Q₈	—	2	2	no	—
37	C₁₃×C₄⋊Q₈	—	2	2	no	—
38	C₁₃×C₄⋊Q₈	—	4	1	no	—
39	C₁₃×C₄⋊Q₈	—	4	1	no	—
40	C₁₃×#127	—	2	1	no	—
41	C₁₃×#124	—	4	1	no	—
42	C₁₃×#121	—	2	2	no	—
43	C₁₃×#121	—	4	1	no	—
44	C₄×D₄×C₁₃	—	2	2	no	—
45	C₄×D₄×C₁₃	—	4	1	no	—
46	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
47	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
48	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
49	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
50	C₁₃×#107	—	2	2	no	—
51	C₁₃×#107	—	2	2	no	—
52	C₁₃×#107	—	2	2	no	—
53	C₁₃×#107	—	2	1	no	—
54	C₁₃×#107	C₂³	1	4	yes	—
55	C₁₃×#107	C₂³	1	2	yes	—
56	C₁₃×#102	—	2	2	no	—
57	C₁₃×#102	—	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	#33793	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₁₃×#41	—	8	4	no	—
64	C₂×Q₈×C₁₃	—	4	1	no	—
65	C₂×Q₈×C₁₃	—	4	2	no	—
66	C₂×Q₈×C₁₃	—	4	2	no	—
67	C₂×Q₈×C₁₃	—	8	4	no	—
68	C₂×D₄×C₁₃	—	2	2	no	—
69	C₂×D₄×C₁₃	—	4	2	no	—
70	C₂×D₄×C₁₃	—	4	1	no	—
71	C₂×D₄×C₁₃	—	4	1	no	—
72	C₂²×C₄×C₁₃	—	4	2	no	—
73	C₂²×C₄×C₁₃	—	4	2	no	—
74	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₁₃×D₈	—	2	4	no	—
78	C₁₃×D₈	—	4	2	no	—
79	C₂×C₈×C₁₃	—	2	2	no	—
80	C₂×C₈×C₁₃	—	2	2	no	—
81	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
85	C₁₃×C₄⋊C₄	—	2	2	no	—
86	C₁₃×C₄⋊C₄	—	2	4	no	—
87	C₁₃×C₄⋊C₄	—	4	2	no	—
88	C₁₃×C₄⋊C₄	—	4	2	no	—
89	C₁₃×C₄⋊C₄	—	4	2	no	—
90	C₁₃×C₄⋊C₄	—	4	1	no	—
91	C₁₃×C₂²⋊C₄	—	4	2	no	—
92	C₁₃×C₂²⋊C₄	—	4	2	no	—
93	C₄²×C₁₃	D₈	1	1	yes	characteristic
94	C₄²×C₁₃	D₈	1	1	yes	characteristic
95	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₁₃×C₁₆	—	2	4	no	—
98	#3169	C₂×C₁₃	1	1	yes	characteristic
99	C₂×#500	C₂×C₁₃	1	1	yes	characteristic, complemented
100	#2254	C₂×C₁₃	1	2	yes	complemented
101	#2240	C₂×C₁₃	1	2	yes	complemented
102	#2233	C₂×C₁₃	1	4	yes	complemented
103	#2226	C₂×C₁₃	1	4	yes	complemented
104	#2162	C₂×C₁₃	1	1	yes	characteristic, complemented
105	C₂³×C₁₃	—	4	2	no	—
106	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
107	Q₈×C₁₃	—	8	4	no	—
108	Q₈×C₁₃	—	8	2	no	—
109	Q₈×C₁₃	—	8	4	no	—
110	D₄×C₁₃	—	4	2	no	—
111	D₄×C₁₃	—	4	2	no	—
112	D₄×C₁₃	—	8	4	no	—
113	D₄×C₁₃	—	8	2	no	—
114	D₄×C₁₃	—	8	2	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	2	no	—
118	C₂×C₄×C₁₃	—	4	2	no	—
119	C₂×C₄×C₁₃	—	4	4	no	—
120	C₂×C₄×C₁₃	—	8	4	no	—
121	C₂×C₄×C₁₃	—	8	4	no	—
122	C₂×C₄×C₁₃	SD₃₂	1	4	yes	—
123	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
124	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
125	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
126	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
129	C₈×C₁₃	—	2	1	no	—
130	C₈×C₁₃	—	2	1	no	—
131	C₈×C₁₃	#137	1	2	yes	—
132	C₈×C₁₃	2₋¹⁺⁴	1	2	yes	—
133	#532	—	2	1	no	—
134	C₂×C₄⋊Q₈	—	2	1	no	—
135	#500	—	2	1	no	—
136	#500	—	2	1	no	—
137	#500	C₂²×C₁₃	1	2	yes	—
138	#500	C₂²×C₁₃	1	2	yes	—
139	#500	C₂²×C₁₃	1	1	yes	characteristic
140	#493	C₂²×C₁₃	1	1	yes	characteristic
141	#486	—	2	1	no	—
142	#466	—	2	1	no	—
143	#466	C₂²×C₁₃	1	2	yes	—
144	C₄×D₈	—	2	1	no	—
145	C₄×D₈	C₂²×C₁₃	1	2	yes	—
146	C₂×#107	—	2	1	no	—
147	C₂×#107	C₂²×C₁₃	1	2	yes	—
148	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
149	#367	C₂²×C₁₃	1	8	yes	—
150	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
151	#357	C₂²×C₁₃	1	8	yes	—
152	C₂²⋊C₁₆	C₂²×C₁₃	1	2	yes	—
153	C₄×C₁₆	C₂²×C₁₃	1	2	yes	—
154	C₂²×C₁₃	—	2	1	no	—
155	C₂²×C₁₃	—	2	1	no	—
156	C₂²×C₁₃	—	8	2	no	—
157	C₂²×C₁₃	—	8	2	no	—
158	C₂²×C₁₃	—	8	2	no	—
159	C₂²×C₁₃	C₂×SD₃₂	1	2	yes	—
160	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
161	C₄×C₁₃	—	2	1	no	—
162	C₄×C₁₃	—	2	1	no	—
163	C₄×C₁₃	—	8	2	no	—
164	C₄×C₁₃	—	8	4	no	—
165	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
166	C₄×C₁₃	C₂×SD₃₂	1	2	yes	—
167	C₄×C₁₃	#509	1	2	yes	—
168	C₄×C₁₃	#536	1	1	yes	characteristic
169	C₄×C₁₃	C₂×#137	1	1	yes	characteristic
170	C₂×#41	—	4	1	no	—
171	C₂²×Q₈	—	4	1	no	—
172	C₂×D₈	—	2	1	no	—
173	C₂×D₈	C₂³×C₁₃	1	2	yes	—
174	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
175	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
176	C₄⋊Q₈	—	2	2	no	—
177	C₄⋊Q₈	—	2	2	no	—
178	C₄⋊Q₈	—	2	1	no	—
179	C₄⋊Q₈	—	4	1	no	—
180	C₄⋊Q₈	—	4	1	no	—
181	#127	—	2	1	no	—
182	#124	—	4	1	no	—
183	#121	—	2	2	no	—
184	#121	—	4	1	no	—
185	C₄×D₄	—	2	2	no	—
186	C₄×D₄	—	4	1	no	—
187	C₂×C₄⋊C₄	—	2	2	no	—
188	C₂×C₄⋊C₄	—	4	1	no	—
189	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
190	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
191	#107	—	2	2	no	—
192	#107	—	2	2	no	—
193	#107	—	2	2	no	—
194	#107	—	2	1	no	—
195	#107	C₂³×C₁₃	1	4	yes	—
196	#107	C₂³×C₁₃	1	2	yes	—
197	#102	—	2	2	no	—
198	#102	—	2	2	no	—
199	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
200	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
201	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
202	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
203	C₂×C₁₃	—	2	2	no	—
204	C₂×C₁₃	—	16	2	no	—
205	C₂×C₁₃	#3226	1	1	yes	characteristic
206	C₂×C₁₃	C₂²×SD₃₂	1	1	yes	characteristic
207	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
208	#41	—	8	4	no	—
209	C₂×Q₈	—	4	1	no	—
210	C₂×Q₈	—	4	2	no	—
211	C₂×Q₈	—	4	2	no	—
212	C₂×Q₈	—	8	4	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	4	2	no	—
215	C₂×D₄	—	4	1	no	—
216	C₂×D₄	—	4	1	no	—
217	C₂²×C₄	—	4	2	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
220	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
222	D₈	—	2	4	no	—
223	D₈	—	4	2	no	—
224	C₂×C₈	—	2	2	no	—
225	C₂×C₈	—	2	2	no	—
226	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
227	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
228	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
229	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
230	C₄⋊C₄	—	2	2	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	4	2	no	—
233	C₄⋊C₄	—	4	2	no	—
234	C₄⋊C₄	—	4	2	no	—
235	C₄⋊C₄	—	4	1	no	—
236	C₂²⋊C₄	—	4	2	no	—
237	C₂²⋊C₄	—	4	2	no	—
238	C₄²	C₁₃×D₈	1	1	yes	characteristic
239	C₄²	C₁₃×D₈	1	1	yes	characteristic
240	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
241	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
242	C₁₆	—	2	4	no	—
243	C₁₃	#33793	1	1	yes	characteristic, direct product factor, Sylow
244	C₂³	—	4	2	no	—
245	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
246	Q₈	—	8	4	no	—
247	Q₈	—	8	2	no	—
248	Q₈	—	8	4	no	—
249	D₄	—	4	2	no	—
250	D₄	—	4	2	no	—
251	D₄	—	8	4	no	—
252	D₄	—	8	2	no	—
253	D₄	—	8	2	no	—
254	C₂×C₄	—	2	2	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	—	8	4	no	—
260	C₂×C₄	—	8	4	no	—
261	C₂×C₄	C₁₃×SD₃₂	1	4	yes	—
262	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
263	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
264	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
265	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
266	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
267	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
268	C₈	—	2	1	no	—
269	C₈	—	2	1	no	—
270	C₈	C₁₃×#137	1	2	yes	—
271	C₈	C₁₃×2₋¹⁺⁴	1	2	yes	—
272	C₂²	—	2	1	no	—
273	C₂²	—	2	1	no	—
274	C₂²	—	8	2	no	—
275	C₂²	—	8	2	no	—
276	C₂²	—	8	2	no	—
277	C₂²	C₂×C₁₃×SD₃₂	1	2	yes	—
278	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
279	C₄	—	2	1	no	—
280	C₄	—	2	1	no	—
281	C₄	—	8	2	no	—
282	C₄	—	8	4	no	—
283	C₄	C₂×C₁₃×SD₃₂	1	2	yes	—
284	C₄	C₂×C₁₃×SD₃₂	1	2	yes	—
285	C₄	C₁₃×#509	1	2	yes	—
286	C₄	C₁₃×#536	1	1	yes	characteristic
287	C₄	C₂×C₁₃×#137	1	1	yes	characteristic
288	C₂	—	2	2	no	—
289	C₂	—	16	2	no	—
290	C₂	C₁₃×#3226	1	1	yes	characteristic
291	C₂	C₂²×C₁₃×SD₃₂	1	1	yes	characteristic
292	C₂	C₂×C₁₃×#509	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₁₃×#3169	C₂
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
2. C₂×C₁₃×#500	C₂	semidirect
3. C₁₃×#2254	C₂	semidirect
4. C₁₃×#2240	C₂	semidirect
5. C₁₃×#2233	C₂	semidirect
6. C₁₃×#2226	C₂	semidirect
7. C₁₃×#2162	C₂	semidirect
62. #33793	C₁₃	direct (N × H)
99. C₂×#500	C₂×C₁₃	semidirect
100. #2254	C₂×C₁₃	semidirect
101. #2240	C₂×C₁₃	semidirect
102. #2233	C₂×C₁₃	semidirect
103. #2226	C₂×C₁₃	semidirect
104. #2162	C₂×C₁₃	semidirect
243. C₁₃	#33793	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×#3169	2
C₂×C₁₃×#500	2
C₁₃×#2254	2
C₁₃×#2240	2
C₁₃×#2233	2
C₁₃×#2226	2
C₁₃×#2162	2
#33793	13

Subgroups — C13×#33793 #15720132

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33793 #15720132