Subgroups — C₁₃×#33656 #15719995

Order 3328.

556 conjugacy classes of subgroups.

The subgroup lattice has 320 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3054	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#430	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2239	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2238	C₂	1	4	yes	maximal, complemented
5	C₁₃×#2155	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2148	C₂	1	2	yes	maximal, complemented
7	C₂×C₁₃×#349	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
9	C₂²×C₁₃×M₁₆	—	2	1	no	—
10	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₁₃×#525	—	2	1	no	—
12	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
13	C₁₃×#470	C₂²	1	2	yes	—
14	C₁₃×#469	C₂²	1	2	yes	—
15	C₁₃×#468	—	2	2	no	—
16	C₁₃×#430	—	2	2	no	—
17	C₁₃×#430	C₂²	1	4	yes	—
18	C₂×C₁₃×#108	—	2	1	no	—
19	C₂×C₁₃×#108	C₂²	1	2	yes	—
20	C₁₃×#427	C₂²	1	1	yes	characteristic
21	C₁₃×#416	—	2	1	no	—
22	C₁₃×#365	C₂²	1	8	yes	complemented
23	C₁₃×#362	C₂²	1	4	yes	complemented
24	C₁₃×#361	C₂²	1	4	yes	complemented
25	C₁₃×#349	C₂²	1	4	yes	—
26	C₂×C₁₃×#41	—	2	1	no	—
27	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
28	C₁₃×#135	—	2	2	no	—
29	C₁₃×#135	—	2	2	no	—
30	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
31	C₂×C₁₃×SD₁₆	—	2	1	no	—
32	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
33	C₂×C₁₃×M₁₆	—	2	2	no	—
34	C₂×C₁₃×M₁₆	—	2	2	no	—
35	C₂×C₁₃×M₁₆	—	2	4	no	—
36	C₂×C₁₃×M₁₆	—	2	2	no	—
37	C₂×C₁₃×M₁₆	—	4	2	no	—
38	C₂×C₁₃×M₁₆	—	4	1	no	—
39	C₂×C₁₃×M₁₆	D₄	1	2	yes	—
40	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₃	—	4	1	no	—
42	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
43	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×#123	—	4	1	no	—
45	C₁₃×C₂²⋊Q₈	—	2	2	no	—
46	C₁₃×#121	—	2	2	no	—
47	C₁₃×#120	—	4	1	no	—
48	C₄×D₄×C₁₃	—	4	2	no	—
49	C₁₃×#117	—	2	1	no	—
50	C₁₃×M₃₂	—	2	2	no	—
51	C₁₃×M₃₂	C₂³	1	4	yes	—
52	C₂×C₁₃×C₁₆	—	2	2	no	—
53	C₁₃×#108	—	2	4	no	—
54	C₁₃×#108	—	2	2	no	—
55	C₁₃×#108	C₂³	1	4	yes	—
56	C₁₃×#107	—	2	1	no	—
57	C₁₃×#106	C₂³	1	2	yes	—
58	C₁₃×#103	—	2	2	no	—
59	C₁₃×#102	—	2	2	no	—
60	#33656	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂⁴×C₁₃	SD₁₆	1	1	yes	characteristic
62	C₁₃×#41	—	4	1	no	—
63	C₁₃×#41	—	4	1	no	—
64	C₁₃×#41	—	4	1	no	—
65	C₁₃×#41	—	4	1	no	—
66	C₂×Q₈×C₁₃	—	2	1	no	—
67	C₂×D₄×C₁₃	—	2	1	no	—
68	C₂×D₄×C₁₃	—	4	2	no	—
69	C₂×D₄×C₁₃	—	4	1	no	—
70	C₂²×C₄×C₁₃	—	2	1	no	—
71	C₂²×C₄×C₁₃	—	2	1	no	—
72	C₂²×C₄×C₁₃	—	2	1	no	—
73	C₂²×C₄×C₁₃	—	2	1	no	—
74	C₂²×C₄×C₁₃	—	4	1	no	—
75	C₂²×C₄×C₁₃	SD₁₆	1	2	yes	—
76	C₂²×C₄×C₁₃	SD₁₆	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
78	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
79	C₁₃×Q₁₆	—	2	2	no	—
80	C₁₃×SD₁₆	—	4	2	no	—
81	C₁₃×D₈	—	2	2	no	—
82	C₁₃×M₁₆	—	2	4	no	—
83	C₁₃×M₁₆	—	2	4	no	—
84	C₁₃×M₁₆	—	2	4	no	—
85	C₁₃×M₁₆	—	4	4	no	—
86	C₁₃×M₁₆	—	4	2	no	—
87	C₂×C₈×C₁₃	—	2	1	no	—
88	C₂×C₈×C₁₃	—	2	1	no	—
89	C₂×C₈×C₁₃	—	4	4	no	—
90	C₂×C₈×C₁₃	—	4	2	no	—
91	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	#41	1	2	yes	—
94	C₂×C₈×C₁₃	#41	1	2	yes	—
95	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	4	2	no	—
98	C₁₃×C₂²⋊C₄	—	4	1	no	—
99	C₁₃×C₂²⋊C₄	—	4	2	no	—
100	C₁₃×C₂²⋊C₄	—	4	2	no	—
101	C₄²×C₁₃	—	4	1	no	—
102	C₁₃×C₁₆	—	2	4	no	—
103	#3054	C₂×C₁₃	1	1	yes	characteristic
104	C₂×#430	C₂×C₁₃	1	1	yes	characteristic, complemented
105	#2239	C₂×C₁₃	1	4	yes	complemented
106	#2238	C₂×C₁₃	1	4	yes	complemented
107	#2155	C₂×C₁₃	1	2	yes	complemented
108	#2148	C₂×C₁₃	1	2	yes	complemented
109	C₂×#349	C₂×C₁₃	1	1	yes	characteristic, complemented
110	C₂³×C₁₃	—	2	1	no	—
111	C₂³×C₁₃	—	2	1	no	—
112	C₂³×C₁₃	—	4	1	no	—
113	C₂³×C₁₃	—	8	1	no	—
114	C₂³×C₁₃	C₂×SD₁₆	1	2	yes	—
115	C₂³×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
116	Q₈×C₁₃	—	4	1	no	—
117	Q₈×C₁₃	—	4	1	no	—
118	D₄×C₁₃	—	4	1	no	—
119	D₄×C₁₃	—	4	1	no	—
120	D₄×C₁₃	—	8	4	no	—
121	D₄×C₁₃	—	8	1	no	—
122	D₄×C₁₃	—	8	1	no	—
123	C₂×C₄×C₁₃	—	2	1	no	—
124	C₂×C₄×C₁₃	—	2	1	no	—
125	C₂×C₄×C₁₃	—	2	2	no	—
126	C₂×C₄×C₁₃	—	2	2	no	—
127	C₂×C₄×C₁₃	—	2	2	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	4	2	no	—
130	C₂×C₄×C₁₃	—	4	1	no	—
131	C₂×C₄×C₁₃	—	8	1	no	—
132	C₂×C₄×C₁₃	—	8	1	no	—
133	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
134	C₂×C₄×C₁₃	#123	1	1	yes	characteristic
135	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
136	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
137	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
138	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
139	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
140	C₈×C₁₃	—	2	2	no	—
141	C₈×C₁₃	—	4	4	no	—
142	C₈×C₁₃	#123	1	2	yes	—
143	C₈×C₁₃	C₂×#41	1	2	yes	—
144	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
145	C₂²×M₁₆	—	2	1	no	—
146	C₂²×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
147	#525	—	2	1	no	—
148	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
149	#470	C₂²×C₁₃	1	2	yes	—
150	#469	C₂²×C₁₃	1	2	yes	—
151	#468	—	2	2	no	—
152	#430	—	2	2	no	—
153	#430	C₂²×C₁₃	1	4	yes	—
154	C₂×#108	—	2	1	no	—
155	C₂×#108	C₂²×C₁₃	1	2	yes	—
156	#427	C₂²×C₁₃	1	1	yes	characteristic
157	#416	—	2	1	no	—
158	#365	C₂²×C₁₃	1	8	yes	complemented
159	#362	C₂²×C₁₃	1	4	yes	complemented
160	#361	C₂²×C₁₃	1	4	yes	complemented
161	#349	C₂²×C₁₃	1	4	yes	—
162	C₂²×C₁₃	—	2	2	no	—
163	C₂²×C₁₃	—	4	1	no	—
164	C₂²×C₁₃	—	4	2	no	—
165	C₂²×C₁₃	—	4	2	no	—
166	C₂²×C₁₃	—	8	1	no	—
167	C₂²×C₁₃	—	8	2	no	—
168	C₂²×C₁₃	—	8	1	no	—
169	C₂²×C₁₃	#481	1	1	yes	characteristic
170	C₂²×C₁₃	#481	1	1	yes	characteristic
171	C₂²×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
172	C₄×C₁₃	—	2	2	no	—
173	C₄×C₁₃	—	8	2	no	—
174	C₄×C₁₃	—	8	1	no	—
175	C₄×C₁₃	#481	1	1	yes	characteristic
176	C₄×C₁₃	#481	1	1	yes	characteristic
177	C₄×C₁₃	C₂×#123	1	1	yes	characteristic
178	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
179	C₂×#41	—	2	1	no	—
180	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
181	#135	—	2	2	no	—
182	#135	—	2	2	no	—
183	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
184	C₂×SD₁₆	—	2	1	no	—
185	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
186	C₂×M₁₆	—	2	2	no	—
187	C₂×M₁₆	—	2	2	no	—
188	C₂×M₁₆	—	2	2	no	—
189	C₂×M₁₆	—	2	4	no	—
190	C₂×M₁₆	—	4	2	no	—
191	C₂×M₁₆	—	4	1	no	—
192	C₂×M₁₆	D₄×C₁₃	1	2	yes	—
193	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
194	C₂²×C₈	—	4	1	no	—
195	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
196	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
197	#123	—	4	1	no	—
198	C₂²⋊Q₈	—	2	2	no	—
199	#121	—	2	2	no	—
200	#120	—	4	1	no	—
201	C₄×D₄	—	4	2	no	—
202	#117	—	2	1	no	—
203	M₃₂	—	2	2	no	—
204	M₃₂	C₂³×C₁₃	1	4	yes	—
205	C₂×C₁₆	—	2	2	no	—
206	#108	—	2	4	no	—
207	#108	—	2	2	no	—
208	#108	C₂³×C₁₃	1	4	yes	—
209	#107	—	2	1	no	—
210	#106	C₂³×C₁₃	1	2	yes	—
211	#103	—	2	2	no	—
212	#102	—	2	2	no	—
213	C₂×C₁₃	—	2	1	no	—
214	C₂×C₁₃	—	2	1	no	—
215	C₂×C₁₃	—	2	1	no	—
216	C₂×C₁₃	—	4	2	no	—
217	C₂×C₁₃	—	16	1	no	—
218	C₂×C₁₃	C₂×#481	1	1	yes	characteristic
219	C₂⁴	C₁₃×SD₁₆	1	1	yes	characteristic
220	#41	—	4	1	no	—
221	#41	—	4	1	no	—
222	#41	—	4	1	no	—
223	#41	—	4	1	no	—
224	C₂×Q₈	—	2	1	no	—
225	C₂×D₄	—	2	1	no	—
226	C₂×D₄	—	4	2	no	—
227	C₂×D₄	—	4	1	no	—
228	C₂²×C₄	—	2	1	no	—
229	C₂²×C₄	—	2	1	no	—
230	C₂²×C₄	—	2	1	no	—
231	C₂²×C₄	—	2	1	no	—
232	C₂²×C₄	—	4	1	no	—
233	C₂²×C₄	C₁₃×SD₁₆	1	2	yes	—
234	C₂²×C₄	C₁₃×SD₁₆	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
236	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
237	Q₁₆	—	2	2	no	—
238	SD₁₆	—	4	2	no	—
239	D₈	—	2	2	no	—
240	M₁₆	—	2	4	no	—
241	M₁₆	—	2	4	no	—
242	M₁₆	—	2	4	no	—
243	M₁₆	—	4	4	no	—
244	M₁₆	—	4	2	no	—
245	C₂×C₈	—	2	1	no	—
246	C₂×C₈	—	2	1	no	—
247	C₂×C₈	—	4	4	no	—
248	C₂×C₈	—	4	2	no	—
249	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
250	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
251	C₂×C₈	C₁₃×#41	1	2	yes	—
252	C₂×C₈	C₁₃×#41	1	2	yes	—
253	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₂²⋊C₄	—	4	1	no	—
257	C₂²⋊C₄	—	4	2	no	—
258	C₂²⋊C₄	—	4	2	no	—
259	C₄²	—	4	1	no	—
260	C₁₆	—	2	4	no	—
261	C₁₃	#33656	1	1	yes	characteristic, direct product factor, Sylow
262	C₂³	—	2	1	no	—
263	C₂³	—	2	1	no	—
264	C₂³	—	4	1	no	—
265	C₂³	—	8	1	no	—
266	C₂³	C₂×C₁₃×SD₁₆	1	2	yes	—
267	C₂³	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
268	Q₈	—	4	1	no	—
269	Q₈	—	4	1	no	—
270	D₄	—	4	1	no	—
271	D₄	—	4	1	no	—
272	D₄	—	8	4	no	—
273	D₄	—	8	1	no	—
274	D₄	—	8	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	1	no	—
277	C₂×C₄	—	2	2	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	2	no	—
280	C₂×C₄	—	2	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	1	no	—
283	C₂×C₄	—	8	1	no	—
284	C₂×C₄	—	8	1	no	—
285	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
286	C₂×C₄	C₁₃×#123	1	1	yes	characteristic
287	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
288	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
289	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
290	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
291	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
292	C₈	—	2	2	no	—
293	C₈	—	4	4	no	—
294	C₈	C₁₃×#123	1	2	yes	—
295	C₈	C₂×C₁₃×#41	1	2	yes	—
296	C₂²	—	2	2	no	—
297	C₂²	—	4	1	no	—
298	C₂²	—	4	2	no	—
299	C₂²	—	4	2	no	—
300	C₂²	—	8	1	no	—
301	C₂²	—	8	2	no	—
302	C₂²	—	8	1	no	—
303	C₂²	C₁₃×#481	1	1	yes	characteristic
304	C₂²	C₁₃×#481	1	1	yes	characteristic
305	C₂²	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	8	2	no	—
308	C₄	—	8	1	no	—
309	C₄	C₁₃×#481	1	1	yes	characteristic
310	C₄	C₁₃×#481	1	1	yes	characteristic
311	C₄	C₂×C₁₃×#123	1	1	yes	characteristic
312	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
313	C₂	—	2	1	no	—
314	C₂	—	2	1	no	—
315	C₂	—	2	1	no	—
316	C₂	—	4	2	no	—
317	C₂	—	16	1	no	—
318	C₂	C₂×C₁₃×#481	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃	#481
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
socle	C₂×C₁₃	C₂×#481
prime radical	C₁₃×#3054	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₂²×C₁₃×M₁₆ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#430	C₂	semidirect
3. C₁₃×#2239	C₂	semidirect
4. C₁₃×#2238	C₂	semidirect
5. C₁₃×#2155	C₂	semidirect
6. C₁₃×#2148	C₂	semidirect
7. C₂×C₁₃×#349	C₂	semidirect
22. C₁₃×#365	C₂²	semidirect
23. C₁₃×#362	C₂²	semidirect
24. C₁₃×#361	C₂²	semidirect
60. #33656	C₁₃	direct (N × H)
104. C₂×#430	C₂×C₁₃	semidirect
105. #2239	C₂×C₁₃	semidirect
106. #2238	C₂×C₁₃	semidirect
107. #2155	C₂×C₁₃	semidirect
108. #2148	C₂×C₁₃	semidirect
109. C₂×#349	C₂×C₁₃	semidirect
158. #365	C₂²×C₁₃	semidirect
159. #362	C₂²×C₁₃	semidirect
160. #361	C₂²×C₁₃	semidirect
261. C₁₃	#33656	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×#3054	2
C₂×C₁₃×#430	2
C₁₃×#2239	2
C₁₃×#2238	2
C₁₃×#2155	2
C₁₃×#2148	2
C₂×C₁₃×#349	2
#33656	13

Subgroups — C13×#33656 #15719995

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33656 #15719995