Subgroups — C₁₃×#33820 #15720159

Order 3328.

528 conjugacy classes of subgroups.

The subgroup lattice has 332 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3170	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#3169	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2227	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2224	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2220	C₂	1	4	yes	maximal, complemented
6	C₁₃×#2208	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2207	C₂	1	2	yes	maximal, complemented
8	C₁₃×#2150	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#533	—	2	1	no	—
10	C₁₃×#532	—	2	1	no	—
11	C₂×C₁₃×SD₃₂	C₂²	1	4	yes	—
12	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
13	C₁₃×#500	C₂²	1	1	yes	characteristic
14	C₁₃×#500	C₂²	1	1	yes	characteristic
15	C₁₃×#494	C₂²	1	1	yes	characteristic
16	C₁₃×#493	C₂²	1	1	yes	characteristic
17	C₁₃×#487	—	2	1	no	—
18	C₁₃×#486	—	2	1	no	—
19	C₁₃×#467	—	2	1	no	—
20	C₁₃×#467	C₂²	1	2	yes	—
21	C₁₃×#466	—	2	1	no	—
22	C₁₃×#466	C₂²	1	2	yes	—
23	C₄×C₁₃×Q₁₆	—	2	1	no	—
24	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
25	C₄×C₁₃×D₈	—	2	1	no	—
26	C₄×C₁₃×D₈	C₂²	1	2	yes	—
27	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₁₃×#366	C₂²	1	4	yes	—
29	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
30	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
31	C₁₃×#358	C₂²	1	4	yes	—
32	C₁₃×#357	C₂²	1	4	yes	—
33	C₂×C₁₃×#41	—	4	1	no	—
34	C₂×C₁₃×Q₁₆	—	2	1	no	—
35	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
36	C₂×C₁₃×D₈	—	2	1	no	—
37	C₂×C₁₃×D₈	C₂³	1	2	yes	—
38	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
40	C₁₃×C₄⋊Q₈	—	2	1	no	—
41	C₁₃×C₄⋊Q₈	—	2	1	no	—
42	C₁₃×C₄⋊Q₈	—	2	1	no	—
43	C₁₃×#127	—	2	1	no	—
44	C₁₃×#125	—	4	1	no	—
45	C₁₃×#124	—	4	1	no	—
46	C₁₃×C₂²⋊Q₈	—	2	2	no	—
47	C₁₃×C₂²⋊Q₈	—	4	1	no	—
48	C₁₃×#121	—	2	2	no	—
49	C₁₃×#121	—	4	1	no	—
50	C₄×Q₈×C₁₃	—	2	2	no	—
51	C₄×Q₈×C₁₃	—	4	1	no	—
52	C₄×D₄×C₁₃	—	2	2	no	—
53	C₄×D₄×C₁₃	—	4	1	no	—
54	C₁₃×#117	—	4	1	no	—
55	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
56	C₁₃×SD₃₂	—	4	4	no	—
57	C₁₃×M₃₂	—	4	2	no	—
58	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
59	C₁₃×#107	—	2	1	no	—
60	C₁₃×#107	—	2	1	no	—
61	C₁₃×#107	C₂³	1	2	yes	—
62	C₁₃×#107	C₂³	1	2	yes	—
63	C₁₃×#103	—	2	2	no	—
64	C₁₃×#103	—	2	2	no	—
65	C₁₃×#102	—	2	2	no	—
66	C₁₃×#102	—	2	2	no	—
67	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
68	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
71	#33820	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
72	C₁₃×#41	—	8	4	no	—
73	C₂×Q₈×C₁₃	—	2	2	no	—
74	C₂×Q₈×C₁₃	—	4	1	no	—
75	C₂×Q₈×C₁₃	—	4	1	no	—
76	C₂×D₄×C₁₃	—	2	2	no	—
77	C₂×D₄×C₁₃	—	4	2	no	—
78	C₂×D₄×C₁₃	—	4	1	no	—
79	C₂×D₄×C₁₃	—	4	1	no	—
80	C₂²×C₄×C₁₃	—	4	2	no	—
81	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
82	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₁₃×Q₁₆	—	2	2	no	—
85	C₁₃×Q₁₆	—	2	2	no	—
86	C₁₃×Q₁₆	—	4	2	no	—
87	C₁₃×D₈	—	2	2	no	—
88	C₁₃×D₈	—	2	2	no	—
89	C₁₃×D₈	—	4	2	no	—
90	C₂×C₈×C₁₃	—	2	1	no	—
91	C₂×C₈×C₁₃	—	2	1	no	—
92	C₂×C₈×C₁₃	—	4	1	no	—
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
97	C₁₃×C₄⋊C₄	—	2	2	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	4	1	no	—
100	C₁₃×C₄⋊C₄	—	4	1	no	—
101	C₁₃×C₄⋊C₄	—	4	1	no	—
102	C₁₃×C₄⋊C₄	—	4	2	no	—
103	C₁₃×C₄⋊C₄	—	4	2	no	—
104	C₁₃×C₄⋊C₄	—	4	1	no	—
105	C₁₃×C₂²⋊C₄	—	4	2	no	—
106	C₁₃×C₂²⋊C₄	—	4	2	no	—
107	C₁₃×C₂²⋊C₄	—	4	2	no	—
108	C₄²×C₁₃	—	4	2	no	—
109	C₄²×C₁₃	D₈	1	1	yes	characteristic
110	C₄²×C₁₃	D₈	1	1	yes	characteristic
111	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
112	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₁₃×C₁₆	—	2	4	no	—
114	#3170	C₂×C₁₃	1	1	yes	characteristic, complemented
115	#3169	C₂×C₁₃	1	1	yes	characteristic
116	#2227	C₂×C₁₃	1	2	yes	complemented
117	#2224	C₂×C₁₃	1	2	yes	complemented
118	#2220	C₂×C₁₃	1	4	yes	complemented
119	#2208	C₂×C₁₃	1	2	yes	complemented
120	#2207	C₂×C₁₃	1	2	yes	complemented
121	#2150	C₂×C₁₃	1	1	yes	characteristic, complemented
122	C₂³×C₁₃	—	4	2	no	—
123	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
124	Q₈×C₁₃	—	4	2	no	—
125	Q₈×C₁₃	—	4	2	no	—
126	Q₈×C₁₃	—	8	2	no	—
127	Q₈×C₁₃	—	8	2	no	—
128	D₄×C₁₃	—	4	2	no	—
129	D₄×C₁₃	—	4	2	no	—
130	D₄×C₁₃	—	8	2	no	—
131	D₄×C₁₃	—	8	4	no	—
132	D₄×C₁₃	—	8	2	no	—
133	C₂×C₄×C₁₃	—	2	1	no	—
134	C₂×C₄×C₁₃	—	2	1	no	—
135	C₂×C₄×C₁₃	—	2	2	no	—
136	C₂×C₄×C₁₃	—	2	2	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	4	2	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	—	8	4	no	—
141	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
144	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
147	C₈×C₁₃	—	2	1	no	—
148	C₈×C₁₃	—	2	1	no	—
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
151	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
152	#533	—	2	1	no	—
153	#532	—	2	1	no	—
154	C₂×SD₃₂	C₂²×C₁₃	1	4	yes	—
155	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
156	#500	C₂²×C₁₃	1	1	yes	characteristic
157	#500	C₂²×C₁₃	1	1	yes	characteristic
158	#494	C₂²×C₁₃	1	1	yes	characteristic
159	#493	C₂²×C₁₃	1	1	yes	characteristic
160	#487	—	2	1	no	—
161	#486	—	2	1	no	—
162	#467	—	2	1	no	—
163	#467	C₂²×C₁₃	1	2	yes	—
164	#466	—	2	1	no	—
165	#466	C₂²×C₁₃	1	2	yes	—
166	C₄×Q₁₆	—	2	1	no	—
167	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
168	C₄×D₈	—	2	1	no	—
169	C₄×D₈	C₂²×C₁₃	1	2	yes	—
170	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
171	#366	C₂²×C₁₃	1	4	yes	—
172	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
173	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
174	#358	C₂²×C₁₃	1	4	yes	—
175	#357	C₂²×C₁₃	1	4	yes	—
176	C₂²×C₁₃	—	2	1	no	—
177	C₂²×C₁₃	—	2	1	no	—
178	C₂²×C₁₃	—	2	1	no	—
179	C₂²×C₁₃	—	8	2	no	—
180	C₂²×C₁₃	—	8	2	no	—
181	C₂²×C₁₃	—	8	2	no	—
182	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
183	C₄×C₁₃	—	2	1	no	—
184	C₄×C₁₃	—	2	1	no	—
185	C₄×C₁₃	—	2	1	no	—
186	C₄×C₁₃	—	8	2	no	—
187	C₄×C₁₃	—	8	2	no	—
188	C₄×C₁₃	—	8	2	no	—
189	C₄×C₁₃	#509	1	2	yes	—
190	C₄×C₁₃	#510	1	2	yes	—
191	C₄×C₁₃	#537	1	1	yes	characteristic
192	C₄×C₁₃	#575	1	1	yes	characteristic
193	C₂×#41	—	4	1	no	—
194	C₂×Q₁₆	—	2	1	no	—
195	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
196	C₂×D₈	—	2	1	no	—
197	C₂×D₈	C₂³×C₁₃	1	2	yes	—
198	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
199	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
200	C₄⋊Q₈	—	2	1	no	—
201	C₄⋊Q₈	—	2	1	no	—
202	C₄⋊Q₈	—	2	1	no	—
203	#127	—	2	1	no	—
204	#125	—	4	1	no	—
205	#124	—	4	1	no	—
206	C₂²⋊Q₈	—	2	2	no	—
207	C₂²⋊Q₈	—	4	1	no	—
208	#121	—	2	2	no	—
209	#121	—	4	1	no	—
210	C₄×Q₈	—	2	2	no	—
211	C₄×Q₈	—	4	1	no	—
212	C₄×D₄	—	2	2	no	—
213	C₄×D₄	—	4	1	no	—
214	#117	—	4	1	no	—
215	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
216	SD₃₂	—	4	4	no	—
217	M₃₂	—	4	2	no	—
218	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
219	#107	—	2	1	no	—
220	#107	—	2	1	no	—
221	#107	C₂³×C₁₃	1	2	yes	—
222	#107	C₂³×C₁₃	1	2	yes	—
223	#103	—	2	2	no	—
224	#103	—	2	2	no	—
225	#102	—	2	2	no	—
226	#102	—	2	2	no	—
227	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
228	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
229	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
230	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
231	C₂×C₁₃	—	4	1	no	—
232	C₂×C₁₃	—	16	2	no	—
233	C₂×C₁₃	#3245	1	1	yes	characteristic
234	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
235	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
236	#41	—	8	4	no	—
237	C₂×Q₈	—	2	2	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×D₄	—	2	2	no	—
241	C₂×D₄	—	4	2	no	—
242	C₂×D₄	—	4	1	no	—
243	C₂×D₄	—	4	1	no	—
244	C₂²×C₄	—	4	2	no	—
245	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
246	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
247	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
248	Q₁₆	—	2	2	no	—
249	Q₁₆	—	2	2	no	—
250	Q₁₆	—	4	2	no	—
251	D₈	—	2	2	no	—
252	D₈	—	2	2	no	—
253	D₈	—	4	2	no	—
254	C₂×C₈	—	2	1	no	—
255	C₂×C₈	—	2	1	no	—
256	C₂×C₈	—	4	1	no	—
257	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
258	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
260	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
261	C₄⋊C₄	—	2	2	no	—
262	C₄⋊C₄	—	2	2	no	—
263	C₄⋊C₄	—	4	1	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	1	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	2	no	—
268	C₄⋊C₄	—	4	1	no	—
269	C₂²⋊C₄	—	4	2	no	—
270	C₂²⋊C₄	—	4	2	no	—
271	C₂²⋊C₄	—	4	2	no	—
272	C₄²	—	4	2	no	—
273	C₄²	C₁₃×D₈	1	1	yes	characteristic
274	C₄²	C₁₃×D₈	1	1	yes	characteristic
275	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
276	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
277	C₁₆	—	2	4	no	—
278	C₁₃	#33820	1	1	yes	characteristic, direct product factor, Sylow
279	C₂³	—	4	2	no	—
280	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
281	Q₈	—	4	2	no	—
282	Q₈	—	4	2	no	—
283	Q₈	—	8	2	no	—
284	Q₈	—	8	2	no	—
285	D₄	—	4	2	no	—
286	D₄	—	4	2	no	—
287	D₄	—	8	2	no	—
288	D₄	—	8	4	no	—
289	D₄	—	8	2	no	—
290	C₂×C₄	—	2	2	no	—
291	C₂×C₄	—	2	2	no	—
292	C₂×C₄	—	2	1	no	—
293	C₂×C₄	—	2	1	no	—
294	C₂×C₄	—	4	2	no	—
295	C₂×C₄	—	4	2	no	—
296	C₂×C₄	—	4	2	no	—
297	C₂×C₄	—	8	4	no	—
298	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
299	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
300	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
301	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
302	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
303	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
304	C₈	—	2	1	no	—
305	C₈	—	2	1	no	—
306	C₈	—	2	1	no	—
307	C₈	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
308	C₈	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
309	C₂²	—	2	1	no	—
310	C₂²	—	2	1	no	—
311	C₂²	—	2	1	no	—
312	C₂²	—	8	2	no	—
313	C₂²	—	8	2	no	—
314	C₂²	—	8	2	no	—
315	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
316	C₄	—	2	1	no	—
317	C₄	—	2	1	no	—
318	C₄	—	2	1	no	—
319	C₄	—	8	2	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	2	no	—
322	C₄	C₁₃×#509	1	2	yes	—
323	C₄	C₁₃×#510	1	2	yes	—
324	C₄	C₁₃×#537	1	1	yes	characteristic
325	C₄	C₁₃×#575	1	1	yes	characteristic
326	C₂	—	4	1	no	—
327	C₂	—	16	2	no	—
328	C₂	C₁₃×#3245	1	1	yes	characteristic
329	C₂	C₂×C₁₃×#509	1	1	yes	characteristic
330	C₂	C₂×C₁₃×#510	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
1. C₁₃×#3170	C₂	semidirect
3. C₁₃×#2227	C₂	semidirect
4. C₁₃×#2224	C₂	semidirect
5. C₁₃×#2220	C₂	semidirect
6. C₁₃×#2208	C₂	semidirect
7. C₁₃×#2207	C₂	semidirect
8. C₁₃×#2150	C₂	semidirect
71. #33820	C₁₃	direct (N × H)
114. #3170	C₂×C₁₃	semidirect
116. #2227	C₂×C₁₃	semidirect
117. #2224	C₂×C₁₃	semidirect
118. #2220	C₂×C₁₃	semidirect
119. #2208	C₂×C₁₃	semidirect
120. #2207	C₂×C₁₃	semidirect
121. #2150	C₂×C₁₃	semidirect
278. C₁₃	#33820	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×#3170	2
C₁₃×#3169	2
C₁₃×#2227	2
C₁₃×#2224	2
C₁₃×#2220	2
C₁₃×#2208	2
C₁₃×#2207	2
C₁₃×#2150	2
#33820	13

Subgroups — C13×#33820 #15720159

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33820 #15720159