Subgroups — C₁₃×#33952 #15720291

Order 3328.

464 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	Q₈×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#3380	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×#2229	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2227	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2226	C₂	1	1	yes	maximal, characteristic
8	C₁₃×#2224	C₂	1	2	yes	maximal
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	Q₈²×C₁₃	—	2	1	no	—
13	C₁₃×#554	—	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×#478	—	2	2	no	—
20	C₁₃×#477	—	2	1	no	—
21	C₁₃×#477	—	2	2	no	—
22	C₁₃×#476	—	2	1	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	2	yes	—
34	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
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	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
42	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
43	C₁₃×C₄⋊Q₈	—	2	1	no	—
44	C₁₃×C₄⋊Q₈	—	2	2	no	—
45	C₁₃×C₄⋊Q₈	—	2	1	no	—
46	C₁₃×C₄⋊Q₈	—	4	1	no	—
47	C₁₃×C₄⋊Q₈	—	4	2	no	—
48	C₁₃×#125	—	2	2	no	—
49	C₁₃×C₂²⋊Q₈	—	4	1	no	—
50	C₁₃×C₂²⋊Q₈	—	4	2	no	—
51	C₄×Q₈×C₁₃	—	2	2	no	—
52	C₄×Q₈×C₁₃	—	2	1	no	—
53	C₄×Q₈×C₁₃	—	4	1	no	—
54	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
55	C₄×D₄×C₁₃	—	2	1	no	—
56	C₄×D₄×C₁₃	—	2	2	no	—
57	C₂×C₁₃×C₄⋊C₄	—	4	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	1	yes	characteristic
66	C₁₃×#107	C₂³	1	2	yes	—
67	C₁₃×#107	C₂³	1	2	yes	—
68	C₁₃×#107	C₂³	1	1	yes	characteristic
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	#33952	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	1	no	—
82	C₂×Q₈×C₁₃	—	4	2	no	—
83	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
84	C₂×D₄×C₁₃	—	2	1	no	—
85	C₂²×C₄×C₁₃	—	4	1	no	—
86	C₂²×C₄×C₁₃	—	4	2	no	—
87	C₁₃×Q₁₆	—	2	1	no	—
88	C₁₃×Q₁₆	C₂×Q₈	1	2	yes	—
89	C₁₃×D₈	—	2	1	no	—
90	C₁₃×D₈	C₂×Q₈	1	2	yes	—
91	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
92	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₁₃×C₄⋊C₄	—	2	1	no	—
95	C₁₃×C₄⋊C₄	—	2	2	no	—
96	C₁₃×C₄⋊C₄	—	2	2	no	—
97	C₁₃×C₄⋊C₄	—	2	1	no	—
98	C₁₃×C₄⋊C₄	—	4	2	no	—
99	C₁₃×C₄⋊C₄	—	4	1	no	—
100	C₁₃×C₄⋊C₄	—	4	1	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	D₈	1	1	yes	characteristic
106	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
107	C₁₃×C₂²⋊C₄	—	4	1	no	—
108	C₁₃×C₂²⋊C₄	—	4	2	no	—
109	C₄²×C₁₃	—	4	2	no	—
110	C₄²×C₁₃	—	4	1	no	—
111	C₄²×C₁₃	C₂×D₄	1	2	yes	—
112	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
113	C₁₃×C₁₆	—	2	1	no	—
114	C₁₃×C₁₆	—	2	2	no	—
115	C₁₃×C₁₆	C₂×Q₈	1	2	yes	—
116	Q₈×Q₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
117	#3380	C₂×C₁₃	1	1	yes	characteristic
118	#2255	C₂×C₁₃	1	2	yes	complemented
119	#2252	C₂×C₁₃	1	1	yes	characteristic, complemented
120	#2229	C₂×C₁₃	1	2	yes	complemented
121	#2227	C₂×C₁₃	1	1	yes	characteristic, complemented
122	#2226	C₂×C₁₃	1	1	yes	characteristic
123	#2224	C₂×C₁₃	1	2	yes	—
124	#2183	C₂×C₁₃	1	1	yes	characteristic, complemented
125	#2175	C₂×C₁₃	1	2	yes	—
126	C₄×SD₃₂	C₂×C₁₃	1	1	yes	characteristic
127	C₂³×C₁₃	—	4	1	no	—
128	Q₈×C₁₃	—	2	2	no	—
129	Q₈×C₁₃	—	2	2	no	—
130	Q₈×C₁₃	—	4	1	no	—
131	Q₈×C₁₃	—	16	2	no	—
132	Q₈×C₁₃	—	16	1	no	—
133	Q₈×C₁₃	—	16	1	no	—
134	D₄×C₁₃	—	2	2	no	—
135	D₄×C₁₃	—	4	1	no	—
136	C₂×C₄×C₁₃	—	4	1	no	—
137	C₂×C₄×C₁₃	—	4	2	no	—
138	C₂×C₄×C₁₃	—	4	1	no	—
139	C₂×C₄×C₁₃	—	4	1	no	—
140	C₂×C₄×C₁₃	—	4	2	no	—
141	C₂×C₄×C₁₃	—	8	2	no	—
142	C₂×C₄×C₁₃	—	16	2	no	—
143	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
144	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
146	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
148	C₈×C₁₃	—	2	2	no	—
149	C₈×C₁₃	—	2	1	no	—
150	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
151	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
152	Q₈²	—	2	1	no	—
153	#554	—	2	1	no	—
154	C₂×SD₃₂	C₂²×C₁₃	1	1	yes	characteristic
155	#500	C₂²×C₁₃	1	1	yes	characteristic
156	#500	C₂²×C₁₃	1	2	yes	—
157	#500	C₂²×C₁₃	1	1	yes	characteristic
158	#499	C₂²×C₁₃	1	2	yes	—
159	#478	—	2	2	no	—
160	#477	—	2	1	no	—
161	#477	—	2	2	no	—
162	#476	—	2	1	no	—
163	C₈×Q₈	C₂²×C₁₃	1	1	yes	characteristic
164	C₄×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
165	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
166	C₄×D₈	C₂²×C₁₃	1	1	yes	characteristic
167	C₄×D₈	C₂²×C₁₃	1	2	yes	—
168	#367	C₂²×C₁₃	1	1	yes	characteristic
169	#367	C₂²×C₁₃	1	2	yes	—
170	#366	C₂²×C₁₃	1	2	yes	—
171	#366	C₂²×C₁₃	1	2	yes	—
172	#366	C₂²×C₁₃	1	2	yes	—
173	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
174	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
175	#358	C₂²×C₁₃	1	1	yes	characteristic
176	#358	C₂²×C₁₃	1	2	yes	—
177	#357	C₂²×C₁₃	1	1	yes	characteristic
178	#357	C₂²×C₁₃	1	2	yes	—
179	#346	C₂²×C₁₃	1	2	yes	—
180	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
181	C₂²×C₁₃	—	4	2	no	—
182	C₂²×C₁₃	—	8	1	no	—
183	C₂²×C₁₃	—	8	1	no	—
184	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
185	C₄×C₁₃	—	2	2	no	—
186	C₄×C₁₃	—	2	2	no	—
187	C₄×C₁₃	—	2	1	no	—
188	C₄×C₁₃	—	4	2	no	—
189	C₄×C₁₃	—	8	1	no	—
190	C₄×C₁₃	—	8	2	no	—
191	C₄×C₁₃	—	8	1	no	—
192	C₄×C₁₃	—	8	2	no	—
193	C₄×C₁₃	#510	1	2	yes	—
194	C₄×C₁₃	D₄×Q₈	1	1	yes	characteristic
195	C₄×C₁₃	#578	1	1	yes	characteristic
196	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
197	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
198	C₄⋊Q₈	—	2	1	no	—
199	C₄⋊Q₈	—	2	2	no	—
200	C₄⋊Q₈	—	2	1	no	—
201	C₄⋊Q₈	—	4	1	no	—
202	C₄⋊Q₈	—	4	2	no	—
203	#125	—	2	2	no	—
204	C₂²⋊Q₈	—	4	1	no	—
205	C₂²⋊Q₈	—	4	2	no	—
206	C₄×Q₈	—	2	2	no	—
207	C₄×Q₈	—	2	1	no	—
208	C₄×Q₈	—	4	1	no	—
209	C₄×Q₈	D₄×C₁₃	1	1	yes	characteristic
210	C₄×D₄	—	2	1	no	—
211	C₄×D₄	—	2	2	no	—
212	C₂×C₄⋊C₄	—	4	1	no	—
213	SD₃₂	Q₈×C₁₃	1	4	yes	complemented
214	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
215	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
216	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
217	#107	—	2	2	no	—
218	#107	—	2	1	no	—
219	#107	—	2	1	no	—
220	#107	C₂³×C₁₃	1	1	yes	characteristic
221	#107	C₂³×C₁₃	1	2	yes	—
222	#107	C₂³×C₁₃	1	2	yes	—
223	#107	C₂³×C₁₃	1	1	yes	characteristic
224	#106	—	2	2	no	—
225	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
226	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
227	#103	—	2	2	no	—
228	#103	—	2	1	no	—
229	#102	—	2	2	no	—
230	#102	—	2	1	no	—
231	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
232	C₄×C₈	C₂³×C₁₃	1	2	yes	—
233	C₂×C₁₃	—	8	2	no	—
234	C₂×C₁₃	Q₈×D₈	1	1	yes	characteristic
235	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
236	C₂×C₁₃	#3415	1	1	yes	characteristic
237	C₂×Q₈	—	2	1	no	—
238	C₂×Q₈	—	4	1	no	—
239	C₂×Q₈	—	4	1	no	—
240	C₂×Q₈	—	4	2	no	—
241	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
242	C₂×D₄	—	2	1	no	—
243	C₂²×C₄	—	4	1	no	—
244	C₂²×C₄	—	4	2	no	—
245	Q₁₆	—	2	1	no	—
246	Q₁₆	C₂×Q₈×C₁₃	1	2	yes	—
247	D₈	—	2	1	no	—
248	D₈	C₂×Q₈×C₁₃	1	2	yes	—
249	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
250	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
251	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
252	C₄⋊C₄	—	2	1	no	—
253	C₄⋊C₄	—	2	2	no	—
254	C₄⋊C₄	—	2	2	no	—
255	C₄⋊C₄	—	2	1	no	—
256	C₄⋊C₄	—	4	2	no	—
257	C₄⋊C₄	—	4	1	no	—
258	C₄⋊C₄	—	4	1	no	—
259	C₄⋊C₄	—	4	2	no	—
260	C₄⋊C₄	—	4	2	no	—
261	C₄⋊C₄	—	4	1	no	—
262	C₄⋊C₄	—	4	2	no	—
263	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
264	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
265	C₂²⋊C₄	—	4	1	no	—
266	C₂²⋊C₄	—	4	2	no	—
267	C₄²	—	4	2	no	—
268	C₄²	—	4	1	no	—
269	C₄²	C₂×D₄×C₁₃	1	2	yes	—
270	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
271	C₁₆	—	2	1	no	—
272	C₁₆	—	2	2	no	—
273	C₁₆	C₂×Q₈×C₁₃	1	2	yes	—
274	C₁₃	#33952	1	1	yes	characteristic, direct product factor, Sylow
275	C₂³	—	4	1	no	—
276	Q₈	—	2	2	no	—
277	Q₈	—	2	2	no	—
278	Q₈	—	4	1	no	—
279	Q₈	—	16	2	no	—
280	Q₈	—	16	1	no	—
281	Q₈	—	16	1	no	—
282	D₄	—	2	2	no	—
283	D₄	—	4	1	no	—
284	C₂×C₄	—	4	1	no	—
285	C₂×C₄	—	4	2	no	—
286	C₂×C₄	—	4	1	no	—
287	C₂×C₄	—	4	1	no	—
288	C₂×C₄	—	4	2	no	—
289	C₂×C₄	—	8	2	no	—
290	C₂×C₄	—	16	2	no	—
291	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
292	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
293	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
294	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
295	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
296	C₈	—	2	2	no	—
297	C₈	—	2	1	no	—
298	C₈	C₂²×Q₈×C₁₃	1	1	yes	characteristic
299	C₈	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
300	C₂²	—	4	2	no	—
301	C₂²	—	8	1	no	—
302	C₂²	—	8	1	no	—
303	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
304	C₄	—	2	2	no	—
305	C₄	—	2	2	no	—
306	C₄	—	2	1	no	—
307	C₄	—	4	2	no	—
308	C₄	—	8	1	no	—
309	C₄	—	8	2	no	—
310	C₄	—	8	1	no	—
311	C₄	—	8	2	no	—
312	C₄	C₁₃×#510	1	2	yes	—
313	C₄	D₄×Q₈×C₁₃	1	1	yes	characteristic
314	C₄	C₁₃×#578	1	1	yes	characteristic
315	C₂	—	8	2	no	—
316	C₂	Q₈×C₁₃×D₈	1	1	yes	characteristic
317	C₂	C₂×C₁₃×#510	1	1	yes	characteristic
318	C₂	C₁₃×#3415	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. Q₈×C₁₃×Q₁₆	C₂	semidirect
3. C₁₃×#2255	C₂	semidirect
4. C₁₃×#2252	C₂	semidirect
5. C₁₃×#2229	C₂	semidirect
6. C₁₃×#2227	C₂	semidirect
9. C₁₃×#2183	C₂	semidirect
58. C₁₃×SD₃₂	Q₈	semidirect
78. #33952	C₁₃	direct (N × H)
116. Q₈×Q₁₆	C₂×C₁₃	semidirect
118. #2255	C₂×C₁₃	semidirect
119. #2252	C₂×C₁₃	semidirect
120. #2229	C₂×C₁₃	semidirect
121. #2227	C₂×C₁₃	semidirect
124. #2183	C₂×C₁₃	semidirect
213. SD₃₂	Q₈×C₁₃	semidirect
274. C₁₃	#33952	direct (N × H)

Maximal subgroups

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

subgroup	index
Q₈×C₁₃×Q₁₆	2
C₁₃×#3380	2
C₁₃×#2255	2
C₁₃×#2252	2
C₁₃×#2229	2
C₁₃×#2227	2
C₁₃×#2226	2
C₁₃×#2224	2
C₁₃×#2183	2
C₁₃×#2175	2
C₄×C₁₃×SD₃₂	2
#33952	13

Subgroups — C13×#33952 #15720291

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33952 #15720291