Subgroups — C₁₃×#33953 #15720292

Order 3328.

484 conjugacy classes of subgroups.

The subgroup lattice has 322 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3383	C₂	1	1	yes	maximal, characteristic, complemented
2	Q₈×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2255	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2252	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2228	C₂	1	2	yes	maximal
6	C₁₃×#2227	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2225	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2224	C₂	1	1	yes	maximal, characteristic
9	C₁₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#2175	C₂	1	2	yes	maximal
11	C₄×C₁₃×SD₃₂	C₂	1	1	yes	maximal, characteristic
12	C₁₃×#557	—	2	1	no	—
13	D₄×Q₈×C₁₃	—	2	1	no	—
14	C₂×C₁₃×SD₃₂	C₂²	1	1	yes	characteristic
15	C₁₃×#500	C₂²	1	1	yes	characteristic
16	C₁₃×#500	C₂²	1	1	yes	characteristic
17	C₁₃×#500	C₂²	1	2	yes	—
18	C₁₃×#499	C₂²	1	2	yes	—
19	C₁₃×#479	—	2	2	no	—
20	C₁₃×#475	—	2	1	no	—
21	C₁₃×#474	—	2	1	no	—
22	C₁₃×#474	—	2	2	no	—
23	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
25	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
26	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
27	C₄×C₁₃×D₈	C₂²	1	2	yes	—
28	C₁₃×#367	C₂²	1	1	yes	characteristic
29	C₁₃×#367	C₂²	1	2	yes	—
30	C₁₃×#366	C₂²	1	2	yes	—
31	C₁₃×#366	C₂²	1	2	yes	—
32	C₁₃×#366	C₂²	1	2	yes	—
33	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
34	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
35	C₁₃×#358	C₂²	1	1	yes	characteristic
36	C₁₃×#358	C₂²	1	2	yes	—
37	C₁₃×#357	C₂²	1	1	yes	characteristic
38	C₁₃×#357	C₂²	1	2	yes	—
39	C₁₃×#346	C₂²	1	2	yes	—
40	C₄×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
41	C₂²×Q₈×C₁₃	—	4	1	no	—
42	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
43	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
44	C₁₃×C₄⋊Q₈	—	2	2	no	—
45	C₁₃×C₄⋊Q₈	—	2	1	no	—
46	C₁₃×C₄⋊Q₈	—	2	1	no	—
47	C₁₃×C₄⋊Q₈	—	4	1	no	—
48	C₁₃×#125	—	2	2	no	—
49	C₁₃×#125	—	4	2	no	—
50	C₁₃×C₂²⋊Q₈	—	4	1	no	—
51	C₁₃×C₂²⋊Q₈	—	4	2	no	—
52	C₄×Q₈×C₁₃	—	2	1	no	—
53	C₄×Q₈×C₁₃	—	2	2	no	—
54	C₄×Q₈×C₁₃	—	4	1	no	—
55	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
56	C₄×D₄×C₁₃	—	2	2	no	—
57	C₄×D₄×C₁₃	—	2	1	no	—
58	C₁₃×SD₃₂	Q₈	1	4	yes	complemented
59	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
60	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
61	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
62	C₁₃×#107	—	2	2	no	—
63	C₁₃×#107	—	2	1	no	—
64	C₁₃×#107	—	2	1	no	—
65	C₁₃×#107	C₂³	1	2	yes	—
66	C₁₃×#107	C₂³	1	1	yes	characteristic
67	C₁₃×#107	C₂³	1	1	yes	characteristic
68	C₁₃×#107	C₂³	1	2	yes	—
69	C₁₃×#106	—	2	2	no	—
70	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
71	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
72	C₁₃×#103	—	2	2	no	—
73	C₁₃×#103	—	2	1	no	—
74	C₁₃×#102	—	2	2	no	—
75	C₁₃×#102	—	2	1	no	—
76	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
77	C₄×C₈×C₁₃	C₂³	1	2	yes	—
78	#33953	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₂×Q₈×C₁₃	—	2	1	no	—
80	C₂×Q₈×C₁₃	—	4	1	no	—
81	C₂×Q₈×C₁₃	—	4	2	no	—
82	C₂×Q₈×C₁₃	—	4	1	no	—
83	C₂×Q₈×C₁₃	—	8	4	no	—
84	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
85	C₂×D₄×C₁₃	—	2	1	no	—
86	C₂²×C₄×C₁₃	—	4	2	no	—
87	C₂²×C₄×C₁₃	—	4	1	no	—
88	C₁₃×Q₁₆	—	2	1	no	—
89	C₁₃×Q₁₆	C₂×Q₈	1	2	yes	complemented
90	C₁₃×D₈	—	2	1	no	—
91	C₁₃×D₈	C₂×Q₈	1	2	yes	—
92	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
95	C₁₃×C₄⋊C₄	—	2	2	no	—
96	C₁₃×C₄⋊C₄	—	2	1	no	—
97	C₁₃×C₄⋊C₄	—	2	1	no	—
98	C₁₃×C₄⋊C₄	—	2	2	no	—
99	C₁₃×C₄⋊C₄	—	4	1	no	—
100	C₁₃×C₄⋊C₄	—	4	2	no	—
101	C₁₃×C₄⋊C₄	—	4	2	no	—
102	C₁₃×C₄⋊C₄	—	4	2	no	—
103	C₁₃×C₄⋊C₄	—	4	1	no	—
104	C₁₃×C₄⋊C₄	—	4	2	no	—
105	C₁₃×C₄⋊C₄	—	4	1	no	—
106	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
107	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
108	C₁₃×C₂²⋊C₄	—	4	2	no	—
109	C₁₃×C₂²⋊C₄	—	4	1	no	—
110	C₄²×C₁₃	—	4	2	no	—
111	C₄²×C₁₃	—	4	1	no	—
112	C₄²×C₁₃	C₂×D₄	1	2	yes	—
113	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
114	C₁₃×C₁₆	—	2	1	no	—
115	C₁₃×C₁₆	—	2	2	no	—
116	C₁₃×C₁₆	C₂×Q₈	1	2	yes	complemented
117	#3383	C₂×C₁₃	1	1	yes	characteristic, complemented
118	Q₈×D₈	C₂×C₁₃	1	1	yes	characteristic
119	#2255	C₂×C₁₃	1	2	yes	complemented
120	#2252	C₂×C₁₃	1	1	yes	characteristic, complemented
121	#2228	C₂×C₁₃	1	2	yes	—
122	#2227	C₂×C₁₃	1	2	yes	complemented
123	#2225	C₂×C₁₃	1	1	yes	characteristic, complemented
124	#2224	C₂×C₁₃	1	1	yes	characteristic
125	#2183	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#2175	C₂×C₁₃	1	2	yes	—
127	C₄×SD₃₂	C₂×C₁₃	1	1	yes	characteristic
128	C₂³×C₁₃	—	4	1	no	—
129	Q₈×C₁₃	—	2	2	no	—
130	Q₈×C₁₃	—	2	2	no	—
131	Q₈×C₁₃	—	4	1	no	—
132	Q₈×C₁₃	—	8	4	no	—
133	Q₈×C₁₃	—	8	2	no	—
134	Q₈×C₁₃	—	8	2	no	—
135	D₄×C₁₃	—	2	2	no	—
136	D₄×C₁₃	—	4	1	no	—
137	C₂×C₄×C₁₃	—	4	1	no	—
138	C₂×C₄×C₁₃	—	4	1	no	—
139	C₂×C₄×C₁₃	—	4	2	no	—
140	C₂×C₄×C₁₃	—	4	2	no	—
141	C₂×C₄×C₁₃	—	4	1	no	—
142	C₂×C₄×C₁₃	—	8	4	no	—
143	C₂×C₄×C₁₃	—	8	2	no	—
144	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
145	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
147	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
149	C₈×C₁₃	—	2	2	no	—
150	C₈×C₁₃	—	2	1	no	—
151	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
152	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
153	#557	—	2	1	no	—
154	D₄×Q₈	—	2	1	no	—
155	C₂×SD₃₂	C₂²×C₁₃	1	1	yes	characteristic
156	#500	C₂²×C₁₃	1	1	yes	characteristic
157	#500	C₂²×C₁₃	1	2	yes	—
158	#500	C₂²×C₁₃	1	1	yes	characteristic
159	#499	C₂²×C₁₃	1	2	yes	—
160	#479	—	2	2	no	—
161	#475	—	2	1	no	—
162	#474	—	2	1	no	—
163	#474	—	2	2	no	—
164	C₈×Q₈	C₂²×C₁₃	1	1	yes	characteristic
165	C₄×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
166	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
167	C₄×D₈	C₂²×C₁₃	1	1	yes	characteristic
168	C₄×D₈	C₂²×C₁₃	1	2	yes	—
169	#367	C₂²×C₁₃	1	1	yes	characteristic
170	#367	C₂²×C₁₃	1	2	yes	—
171	#366	C₂²×C₁₃	1	2	yes	—
172	#366	C₂²×C₁₃	1	2	yes	—
173	#366	C₂²×C₁₃	1	2	yes	—
174	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
175	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
176	#358	C₂²×C₁₃	1	1	yes	characteristic
177	#358	C₂²×C₁₃	1	2	yes	—
178	#357	C₂²×C₁₃	1	1	yes	characteristic
179	#357	C₂²×C₁₃	1	2	yes	—
180	#346	C₂²×C₁₃	1	2	yes	—
181	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
182	C₂²×C₁₃	—	4	2	no	—
183	C₂²×C₁₃	—	8	1	no	—
184	C₂²×C₁₃	—	8	1	no	—
185	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
186	C₄×C₁₃	—	2	2	no	—
187	C₄×C₁₃	—	2	2	no	—
188	C₄×C₁₃	—	2	1	no	—
189	C₄×C₁₃	—	4	2	no	—
190	C₄×C₁₃	—	8	2	no	—
191	C₄×C₁₃	—	8	1	no	—
192	C₄×C₁₃	—	8	2	no	—
193	C₄×C₁₃	—	8	1	no	—
194	C₄×C₁₃	#509	1	2	yes	—
195	C₄×C₁₃	D₄×Q₈	1	1	yes	characteristic
196	C₄×C₁₃	#578	1	1	yes	characteristic
197	C₂²×Q₈	—	4	1	no	—
198	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
199	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
200	C₄⋊Q₈	—	2	2	no	—
201	C₄⋊Q₈	—	2	1	no	—
202	C₄⋊Q₈	—	2	1	no	—
203	C₄⋊Q₈	—	4	1	no	—
204	#125	—	2	2	no	—
205	#125	—	4	2	no	—
206	C₂²⋊Q₈	—	4	1	no	—
207	C₂²⋊Q₈	—	4	2	no	—
208	C₄×Q₈	—	2	1	no	—
209	C₄×Q₈	—	2	2	no	—
210	C₄×Q₈	—	4	1	no	—
211	C₄×Q₈	D₄×C₁₃	1	1	yes	characteristic
212	C₄×D₄	—	2	2	no	—
213	C₄×D₄	—	2	1	no	—
214	SD₃₂	Q₈×C₁₃	1	4	yes	complemented
215	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
216	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
217	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
218	#107	—	2	2	no	—
219	#107	—	2	1	no	—
220	#107	—	2	1	no	—
221	#107	C₂³×C₁₃	1	2	yes	—
222	#107	C₂³×C₁₃	1	1	yes	characteristic
223	#107	C₂³×C₁₃	1	1	yes	characteristic
224	#107	C₂³×C₁₃	1	2	yes	—
225	#106	—	2	2	no	—
226	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
227	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
228	#103	—	2	2	no	—
229	#103	—	2	1	no	—
230	#102	—	2	2	no	—
231	#102	—	2	1	no	—
232	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
233	C₄×C₈	C₂³×C₁₃	1	2	yes	—
234	C₂×C₁₃	—	8	2	no	—
235	C₂×C₁₃	Q₈×D₈	1	1	yes	characteristic
236	C₂×C₁₃	C₂×#509	1	1	yes	characteristic
237	C₂×C₁₃	#3417	1	1	yes	characteristic
238	C₂×Q₈	—	2	1	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	4	2	no	—
241	C₂×Q₈	—	4	1	no	—
242	C₂×Q₈	—	8	4	no	—
243	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
244	C₂×D₄	—	2	1	no	—
245	C₂²×C₄	—	4	2	no	—
246	C₂²×C₄	—	4	1	no	—
247	Q₁₆	—	2	1	no	—
248	Q₁₆	C₂×Q₈×C₁₃	1	2	yes	complemented
249	D₈	—	2	1	no	—
250	D₈	C₂×Q₈×C₁₃	1	2	yes	—
251	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
252	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
253	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	2	1	no	—
257	C₄⋊C₄	—	2	2	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	2	no	—
262	C₄⋊C₄	—	4	1	no	—
263	C₄⋊C₄	—	4	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
266	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
267	C₂²⋊C₄	—	4	2	no	—
268	C₂²⋊C₄	—	4	1	no	—
269	C₄²	—	4	2	no	—
270	C₄²	—	4	1	no	—
271	C₄²	C₂×D₄×C₁₃	1	2	yes	—
272	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
273	C₁₆	—	2	1	no	—
274	C₁₆	—	2	2	no	—
275	C₁₆	C₂×Q₈×C₁₃	1	2	yes	complemented
276	C₁₃	#33953	1	1	yes	characteristic, direct product factor, Sylow
277	C₂³	—	4	1	no	—
278	Q₈	—	2	2	no	—
279	Q₈	—	2	2	no	—
280	Q₈	—	4	1	no	—
281	Q₈	—	8	4	no	—
282	Q₈	—	8	2	no	—
283	Q₈	—	8	2	no	—
284	D₄	—	2	2	no	—
285	D₄	—	4	1	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	4	2	no	—
290	C₂×C₄	—	4	1	no	—
291	C₂×C₄	—	8	4	no	—
292	C₂×C₄	—	8	2	no	—
293	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
294	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
295	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
296	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
297	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
298	C₈	—	2	2	no	—
299	C₈	—	2	1	no	—
300	C₈	C₂²×Q₈×C₁₃	1	1	yes	characteristic
301	C₈	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
302	C₂²	—	4	2	no	—
303	C₂²	—	8	1	no	—
304	C₂²	—	8	1	no	—
305	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
306	C₄	—	2	2	no	—
307	C₄	—	2	2	no	—
308	C₄	—	2	1	no	—
309	C₄	—	4	2	no	—
310	C₄	—	8	2	no	—
311	C₄	—	8	1	no	—
312	C₄	—	8	2	no	—
313	C₄	—	8	1	no	—
314	C₄	C₁₃×#509	1	2	yes	—
315	C₄	D₄×Q₈×C₁₃	1	1	yes	characteristic
316	C₄	C₁₃×#578	1	1	yes	characteristic
317	C₂	—	8	2	no	—
318	C₂	Q₈×C₁₃×D₈	1	1	yes	characteristic
319	C₂	C₂×C₁₃×#509	1	1	yes	characteristic
320	C₂	C₁₃×#3417	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×C₁₃×D₈	C₂³
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×Q₈×C₁₃ ◁ C₈×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3383	C₂	semidirect
3. C₁₃×#2255	C₂	semidirect
4. C₁₃×#2252	C₂	semidirect
6. C₁₃×#2227	C₂	semidirect
7. C₁₃×#2225	C₂	semidirect
9. C₁₃×#2183	C₂	semidirect
58. C₁₃×SD₃₂	Q₈	semidirect
78. #33953	C₁₃	direct (N × H)
89. C₁₃×Q₁₆	C₂×Q₈	semidirect
116. C₁₃×C₁₆	C₂×Q₈	semidirect
117. #3383	C₂×C₁₃	semidirect
119. #2255	C₂×C₁₃	semidirect
120. #2252	C₂×C₁₃	semidirect
122. #2227	C₂×C₁₃	semidirect
123. #2225	C₂×C₁₃	semidirect
125. #2183	C₂×C₁₃	semidirect
214. SD₃₂	Q₈×C₁₃	semidirect
248. Q₁₆	C₂×Q₈×C₁₃	semidirect
275. C₁₆	C₂×Q₈×C₁₃	semidirect
276. C₁₃	#33953	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×#3383	2
Q₈×C₁₃×D₈	2
C₁₃×#2255	2
C₁₃×#2252	2
C₁₃×#2228	2
C₁₃×#2227	2
C₁₃×#2225	2
C₁₃×#2224	2
C₁₃×#2183	2
C₁₃×#2175	2
C₄×C₁₃×SD₃₂	2
#33953	13

Subgroups — C13×#33953 #15720292

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33953 #15720292