Subgroups — C₁₃×#33717 #15720056

Order 3328.

918 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#509	C₂	1	4	yes	maximal, complemented
2	C₁₃×#3147	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×C₁₃×#493	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#2250	C₂	1	4	yes	maximal, complemented
5	C₁₃×#2248	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2246	C₂	1	2	yes	maximal, complemented
7	C₄×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#135	—	2	1	no	—
9	C₂×C₁₃×#135	C₂²	1	2	yes	—
10	C₂²×C₁₃×D₈	—	2	1	no	—
11	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
12	C₁₃×#532	—	2	1	no	—
13	C₂×C₁₃×#127	—	2	1	no	—
14	C₁₃×#509	—	2	16	no	—
15	C₂×C₁₃×SD₃₂	C₂²	1	8	yes	complemented
16	C₂×C₁₃×D₁₆	C₂²	1	8	yes	complemented
17	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
18	C₁₃×#495	—	2	1	no	—
19	C₁₃×#495	C₂²	1	2	yes	complemented
20	C₁₃×#494	C₂²	1	1	yes	characteristic, complemented
21	C₁₃×#493	—	2	1	no	—
22	C₁₃×#493	—	2	1	no	—
23	C₁₃×#493	C₂²	1	2	yes	—
24	C₁₃×#493	C₂²	1	2	yes	—
25	C₁₃×#493	C₂²	1	1	yes	characteristic, complemented
26	C₁₃×#492	—	2	1	no	—
27	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₁₃×#346	C₂²	1	2	yes	complemented
29	C₄×C₁₃×C₁₆	C₂²	1	2	yes	complemented
30	C₂×C₁₃×#41	—	2	2	no	—
31	C₂²×D₄×C₁₃	—	2	2	no	—
32	C₂²×D₄×C₁₃	—	4	2	no	—
33	C₁₃×#135	—	2	8	no	—
34	C₁₃×#135	—	4	4	no	—
35	C₂×C₁₃×Q₁₆	—	2	1	no	—
36	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
37	C₂×C₁₃×SD₁₆	—	2	2	no	—
38	C₂×C₁₃×SD₁₆	—	2	2	no	—
39	C₂×C₁₃×D₈	—	2	8	no	—
40	C₂×C₁₃×D₈	—	2	2	no	—
41	C₂×C₁₃×D₈	—	2	2	no	—
42	C₂×C₁₃×D₈	—	2	2	no	—
43	C₂×C₁₃×D₈	—	2	1	no	—
44	C₂×C₁₃×D₈	—	4	4	no	—
45	C₂×C₁₃×D₈	C₂³	1	4	yes	—
46	C₂×C₁₃×D₈	C₂³	1	2	yes	—
47	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
48	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
49	C₁₃×C₄⋊Q₈	—	2	1	no	—
50	C₁₃×#127	—	2	1	no	—
51	C₁₃×#127	—	2	2	no	—
52	C₁₃×#127	—	2	2	no	—
53	C₁₃×#127	—	4	2	no	—
54	C₁₃×#124	—	2	2	no	—
55	C₁₃×#121	—	4	2	no	—
56	C₄×D₄×C₁₃	—	4	2	no	—
57	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
58	C₁₃×SD₃₂	—	2	16	no	—
59	C₁₃×D₁₆	—	2	16	no	—
60	C₁₃×M₃₂	D₄	1	8	yes	complemented
61	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
62	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
63	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
64	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
65	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
66	#33717	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂⁴×C₁₃	—	4	2	no	—
68	C₁₃×#41	—	4	4	no	—
69	C₁₃×#41	—	4	4	no	—
70	C₂×Q₈×C₁₃	—	2	2	no	—
71	C₂×D₄×C₁₃	—	2	2	no	—
72	C₂×D₄×C₁₃	—	2	4	no	—
73	C₂×D₄×C₁₃	—	4	2	no	—
74	C₂×D₄×C₁₃	—	4	4	no	—
75	C₂×D₄×C₁₃	—	4	4	no	—
76	C₂×D₄×C₁₃	—	4	2	no	—
77	C₂×D₄×C₁₃	—	4	4	no	—
78	C₂×D₄×C₁₃	—	4	4	no	—
79	C₂×D₄×C₁₃	—	4	2	no	—
80	C₂×D₄×C₁₃	—	8	8	no	—
81	C₂²×C₄×C₁₃	—	4	2	no	—
82	C₂²×C₄×C₁₃	D₈	1	2	yes	—
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₁₃×Q₁₆	—	2	4	no	—
85	C₁₃×Q₁₆	—	4	2	no	—
86	C₁₃×SD₁₆	—	4	4	no	—
87	C₁₃×SD₁₆	—	4	4	no	—
88	C₁₃×D₈	—	2	4	no	—
89	C₁₃×D₈	—	2	8	no	—
90	C₁₃×D₈	—	4	4	no	—
91	C₁₃×D₈	—	4	4	no	—
92	C₁₃×D₈	—	4	4	no	—
93	C₁₃×D₈	—	4	2	no	—
94	C₂×C₈×C₁₃	D₈	1	4	yes	—
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂×D₄	1	4	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	C₂⁴	1	1	yes	characteristic
100	C₁₃×C₄⋊C₄	—	4	2	no	—
101	C₁₃×C₂²⋊C₄	—	4	4	no	—
102	C₄²×C₁₃	D₈	1	2	yes	—
103	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
104	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₁₃×C₁₆	C₂×D₄	1	8	yes	complemented
106	C₂×#509	C₂×C₁₃	1	4	yes	complemented
107	#3147	C₂×C₁₃	1	1	yes	characteristic, complemented
108	C₂×#493	C₂×C₁₃	1	1	yes	characteristic, complemented
109	#2250	C₂×C₁₃	1	4	yes	complemented
110	#2248	C₂×C₁₃	1	2	yes	complemented
111	#2246	C₂×C₁₃	1	2	yes	complemented
112	C₄×M₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
113	C₂³×C₁₃	—	4	2	no	—
114	C₂³×C₁₃	—	4	4	no	—
115	C₂³×C₁₃	—	8	8	no	—
116	C₂³×C₁₃	—	8	4	no	—
117	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
118	Q₈×C₁₃	—	4	2	no	—
119	Q₈×C₁₃	—	4	2	no	—
120	D₄×C₁₃	—	4	4	no	—
121	D₄×C₁₃	—	4	2	no	—
122	D₄×C₁₃	—	4	4	no	—
123	D₄×C₁₃	—	4	2	no	—
124	D₄×C₁₃	—	8	8	no	—
125	D₄×C₁₃	—	8	4	no	—
126	D₄×C₁₃	—	8	4	no	—
127	D₄×C₁₃	—	8	4	no	—
128	D₄×C₁₃	—	8	8	no	—
129	C₂×C₄×C₁₃	—	2	4	no	—
130	C₂×C₄×C₁₃	—	4	2	no	—
131	C₂×C₄×C₁₃	—	8	4	no	—
132	C₂×C₄×C₁₃	#127	1	2	yes	—
133	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
136	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
137	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
138	C₈×C₁₃	#127	1	2	yes	—
139	C₈×C₁₃	C₂×D₈	1	2	yes	—
140	C₈×C₁₃	C₂×D₈	1	2	yes	—
141	C₈×C₁₃	C₂²×D₄	1	2	yes	—
142	C₂×#135	—	2	1	no	—
143	C₂×#135	C₂²×C₁₃	1	2	yes	—
144	C₂²×D₈	—	2	1	no	—
145	C₂²×D₈	C₂²×C₁₃	1	2	yes	—
146	#532	—	2	1	no	—
147	C₂×#127	—	2	1	no	—
148	#509	—	2	16	no	—
149	C₂×SD₃₂	C₂²×C₁₃	1	8	yes	complemented
150	C₂×D₁₆	C₂²×C₁₃	1	8	yes	complemented
151	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
152	#495	—	2	1	no	—
153	#495	C₂²×C₁₃	1	2	yes	complemented
154	#494	C₂²×C₁₃	1	1	yes	characteristic, complemented
155	#493	—	2	1	no	—
156	#493	—	2	1	no	—
157	#493	C₂²×C₁₃	1	2	yes	—
158	#493	C₂²×C₁₃	1	2	yes	—
159	#493	C₂²×C₁₃	1	1	yes	characteristic, complemented
160	#492	—	2	1	no	—
161	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
162	#346	C₂²×C₁₃	1	2	yes	complemented
163	C₄×C₁₆	C₂²×C₁₃	1	2	yes	complemented
164	C₂²×C₁₃	—	2	2	no	—
165	C₂²×C₁₃	—	8	2	no	—
166	C₂²×C₁₃	—	8	4	no	—
167	C₂²×C₁₃	—	8	8	no	—
168	C₂²×C₁₃	—	8	4	no	—
169	C₂²×C₁₃	—	16	8	no	—
170	C₂²×C₁₃	#493	1	2	yes	—
171	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
172	C₄×C₁₃	—	2	2	no	—
173	C₄×C₁₃	—	8	2	no	—
174	C₄×C₁₃	#493	1	2	yes	—
175	C₄×C₁₃	#509	1	4	yes	complemented
176	C₄×C₁₃	C₂×#127	1	1	yes	characteristic
177	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
178	C₂×#41	—	2	2	no	—
179	C₂²×D₄	—	2	2	no	—
180	C₂²×D₄	—	4	2	no	—
181	#135	—	2	8	no	—
182	#135	—	4	4	no	—
183	C₂×Q₁₆	—	2	1	no	—
184	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
185	C₂×SD₁₆	—	2	2	no	—
186	C₂×SD₁₆	—	2	2	no	—
187	C₂×D₈	—	2	8	no	—
188	C₂×D₈	—	2	2	no	—
189	C₂×D₈	—	2	2	no	—
190	C₂×D₈	—	2	2	no	—
191	C₂×D₈	—	2	1	no	—
192	C₂×D₈	—	4	4	no	—
193	C₂×D₈	C₂³×C₁₃	1	4	yes	—
194	C₂×D₈	C₂³×C₁₃	1	2	yes	—
195	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
197	C₄⋊Q₈	—	2	1	no	—
198	#127	—	2	2	no	—
199	#127	—	2	2	no	—
200	#127	—	2	1	no	—
201	#127	—	4	2	no	—
202	#124	—	2	2	no	—
203	#121	—	4	2	no	—
204	C₄×D₄	—	4	2	no	—
205	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
206	SD₃₂	—	2	16	no	—
207	D₁₆	—	2	16	no	—
208	M₃₂	D₄×C₁₃	1	8	yes	complemented
209	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
210	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
211	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
212	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
213	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
214	C₂×C₁₃	—	2	2	no	—
215	C₂×C₁₃	—	16	2	no	—
216	C₂×C₁₃	—	16	4	no	—
217	C₂×C₁₃	C₂×#493	1	1	yes	characteristic
218	C₂×C₁₃	C₂×#509	1	2	yes	—
219	C₂⁴	—	4	2	no	—
220	#41	—	4	4	no	—
221	#41	—	4	4	no	—
222	C₂×Q₈	—	2	2	no	—
223	C₂×D₄	—	2	2	no	—
224	C₂×D₄	—	2	4	no	—
225	C₂×D₄	—	4	2	no	—
226	C₂×D₄	—	4	4	no	—
227	C₂×D₄	—	4	4	no	—
228	C₂×D₄	—	4	2	no	—
229	C₂×D₄	—	4	4	no	—
230	C₂×D₄	—	4	4	no	—
231	C₂×D₄	—	4	2	no	—
232	C₂×D₄	—	8	8	no	—
233	C₂²×C₄	—	4	2	no	—
234	C₂²×C₄	C₁₃×D₈	1	2	yes	—
235	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
236	Q₁₆	—	2	4	no	—
237	Q₁₆	—	4	2	no	—
238	SD₁₆	—	4	4	no	—
239	SD₁₆	—	4	4	no	—
240	D₈	—	2	4	no	—
241	D₈	—	2	8	no	—
242	D₈	—	4	4	no	—
243	D₈	—	4	4	no	—
244	D₈	—	4	4	no	—
245	D₈	—	4	2	no	—
246	C₂×C₈	C₁₃×D₈	1	4	yes	—
247	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
248	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
249	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
250	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
251	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
252	C₄⋊C₄	—	4	2	no	—
253	C₂²⋊C₄	—	4	4	no	—
254	C₄²	C₁₃×D₈	1	2	yes	—
255	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
256	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
257	C₁₆	C₂×D₄×C₁₃	1	8	yes	complemented
258	C₁₃	#33717	1	1	yes	characteristic, direct product factor, Sylow
259	C₂³	—	4	2	no	—
260	C₂³	—	4	4	no	—
261	C₂³	—	8	8	no	—
262	C₂³	—	8	4	no	—
263	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
264	Q₈	—	4	2	no	—
265	Q₈	—	4	2	no	—
266	D₄	—	4	4	no	—
267	D₄	—	4	2	no	—
268	D₄	—	4	4	no	—
269	D₄	—	4	2	no	—
270	D₄	—	8	8	no	—
271	D₄	—	8	4	no	—
272	D₄	—	8	4	no	—
273	D₄	—	8	8	no	—
274	D₄	—	8	4	no	—
275	C₂×C₄	—	2	4	no	—
276	C₂×C₄	—	4	2	no	—
277	C₂×C₄	—	8	4	no	—
278	C₂×C₄	C₁₃×#127	1	2	yes	—
279	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
280	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
281	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
282	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
283	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
284	C₈	C₁₃×#127	1	2	yes	—
285	C₈	C₂×C₁₃×D₈	1	2	yes	—
286	C₈	C₂×C₁₃×D₈	1	2	yes	—
287	C₈	C₂²×D₄×C₁₃	1	2	yes	—
288	C₂²	—	2	2	no	—
289	C₂²	—	8	2	no	—
290	C₂²	—	8	4	no	—
291	C₂²	—	8	8	no	—
292	C₂²	—	8	4	no	—
293	C₂²	—	16	8	no	—
294	C₂²	C₁₃×#493	1	2	yes	—
295	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
296	C₄	—	2	2	no	—
297	C₄	—	8	2	no	—
298	C₄	C₁₃×#493	1	2	yes	—
299	C₄	C₁₃×#509	1	4	yes	complemented
300	C₄	C₂×C₁₃×#127	1	1	yes	characteristic
301	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
302	C₂	—	2	2	no	—
303	C₂	—	16	4	no	—
304	C₂	—	16	2	no	—
305	C₂	C₂×C₁₃×#493	1	1	yes	characteristic
306	C₂	C₂×C₁₃×#509	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
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. C₂×C₁₃×#509	C₂	semidirect
2. C₁₃×#3147	C₂	semidirect
3. C₂×C₁₃×#493	C₂	semidirect
4. C₁₃×#2250	C₂	semidirect
5. C₁₃×#2248	C₂	semidirect
6. C₁₃×#2246	C₂	semidirect
7. C₄×C₁₃×M₃₂	C₂	semidirect
15. C₂×C₁₃×SD₃₂	C₂²	semidirect
16. C₂×C₁₃×D₁₆	C₂²	semidirect
19. C₁₃×#495	C₂²	semidirect
20. C₁₃×#494	C₂²	semidirect
25. C₁₃×#493	C₂²	semidirect
28. C₁₃×#346	C₂²	semidirect
29. C₄×C₁₃×C₁₆	C₂²	semidirect
60. C₁₃×M₃₂	D₄	semidirect
66. #33717	C₁₃	direct (N × H)
105. C₁₃×C₁₆	C₂×D₄	semidirect
106. C₂×#509	C₂×C₁₃	semidirect
107. #3147	C₂×C₁₃	semidirect
108. C₂×#493	C₂×C₁₃	semidirect
109. #2250	C₂×C₁₃	semidirect
110. #2248	C₂×C₁₃	semidirect
111. #2246	C₂×C₁₃	semidirect
112. C₄×M₃₂	C₂×C₁₃	semidirect
149. C₂×SD₃₂	C₂²×C₁₃	semidirect
150. C₂×D₁₆	C₂²×C₁₃	semidirect
153. #495	C₂²×C₁₃	semidirect
154. #494	C₂²×C₁₃	semidirect
159. #493	C₂²×C₁₃	semidirect
162. #346	C₂²×C₁₃	semidirect
163. C₄×C₁₆	C₂²×C₁₃	semidirect
175. C₄×C₁₃	#509	semidirect
208. M₃₂	D₄×C₁₃	semidirect
257. C₁₆	C₂×D₄×C₁₃	semidirect
258. C₁₃	#33717	direct (N × H)
299. C₄	C₁₃×#509	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₃×#509	2
C₁₃×#3147	2
C₂×C₁₃×#493	2
C₁₃×#2250	2
C₁₃×#2248	2
C₁₃×#2246	2
C₄×C₁₃×M₃₂	2
#33717	13

Subgroups — C13×#33717 #15720056

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33717 #15720056