Subgroups — C₁₃×#33936 #15720275

Order 3328.

474 conjugacy classes of subgroups.

The subgroup lattice has 330 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3393	C₂	1	1	yes	maximal, characteristic
2	Q₈×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#2243	C₂	1	2	yes	maximal
4	C₁₃×#2241	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#2226	C₂	1	1	yes	maximal, characteristic
6	C₁₃×#2225	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2211	C₂	1	2	yes	maximal
8	C₁₃×#2209	C₂	1	1	yes	maximal, characteristic, complemented
9	C₁₃×#2183	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#2176	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₃×SD₃₂	C₂	1	1	yes	maximal, characteristic
12	Q₈²×C₁₃	—	2	1	no	—
13	C₁₃×#556	—	2	1	no	—
14	C₂×C₁₃×Q₃₂	C₂²	1	2	yes	—
15	C₂×C₁₃×SD₃₂	C₂²	1	1	yes	characteristic
16	C₁₃×#500	C₂²	1	1	yes	characteristic
17	C₁₃×#500	C₂²	1	2	yes	—
18	C₁₃×#495	C₂²	1	2	yes	—
19	C₁₃×#494	C₂²	1	1	yes	characteristic
20	C₁₃×#477	—	2	1	no	—
21	C₁₃×#477	—	2	2	no	—
22	C₁₃×#464	—	2	2	no	—
23	C₁₃×#461	—	2	1	no	—
24	C₈×Q₈×C₁₃	C₂²	1	1	yes	characteristic
25	C₄×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
26	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
27	C₄×C₁₃×Q₁₆	C₂²	1	2	yes	—
28	C₄×C₁₃×D₈	C₂²	1	1	yes	characteristic
29	C₁₃×#367	C₂²	1	1	yes	characteristic
30	C₁₃×#367	C₂²	1	2	yes	—
31	C₁₃×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
32	C₁₃×C₄⋊C₁₆	C₂²	1	2	yes	—
33	C₁₃×#358	C₂²	1	2	yes	—
34	C₁₃×#358	C₂²	1	2	yes	—
35	C₁₃×#358	C₂²	1	2	yes	—
36	C₁₃×#358	C₂²	1	1	yes	characteristic
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	—	2	1	no	—
42	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
43	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
44	C₂×C₁₃×SD₁₆	—	2	2	no	—
45	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
46	C₁₃×C₄⋊Q₈	—	2	1	no	—
47	C₁₃×C₄⋊Q₈	—	2	2	no	—
48	C₁₃×C₄⋊Q₈	—	2	1	no	—
49	C₁₃×C₄⋊Q₈	—	4	1	no	—
50	C₁₃×C₄⋊Q₈	—	4	2	no	—
51	C₁₃×#126	—	4	2	no	—
52	C₁₃×#124	—	2	2	no	—
53	C₁₃×C₂²⋊Q₈	—	4	1	no	—
54	C₄×Q₈×C₁₃	—	2	2	no	—
55	C₄×Q₈×C₁₃	—	2	1	no	—
56	C₄×Q₈×C₁₃	—	2	2	no	—
57	C₄×Q₈×C₁₃	—	4	1	no	—
58	C₄×Q₈×C₁₃	D₄	1	1	yes	characteristic
59	C₄×D₄×C₁₃	—	2	1	no	—
60	C₁₃×#117	—	4	1	no	—
61	C₁₃×Q₃₂	—	2	4	no	—
62	C₁₃×SD₃₂	—	2	2	no	—
63	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
64	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
65	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
66	C₁₃×#107	—	2	1	no	—
67	C₁₃×#107	—	2	2	no	—
68	C₁₃×#107	C₂³	1	1	yes	characteristic
69	C₁₃×#107	C₂³	1	2	yes	—
70	C₁₃×#107	C₂³	1	1	yes	characteristic
71	C₁₃×C₄⋊C₈	D₄	1	1	yes	characteristic
72	C₁₃×C₄⋊C₈	D₄	1	2	yes	—
73	C₁₃×#103	—	2	2	no	—
74	C₁₃×#103	—	2	1	no	—
75	C₁₃×#103	—	2	2	no	—
76	C₁₃×#102	—	2	1	no	—
77	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
78	C₄×C₈×C₁₃	C₂³	1	2	yes	—
79	#33936	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
80	C₂×Q₈×C₁₃	—	2	1	no	—
81	C₂×Q₈×C₁₃	—	2	2	no	—
82	C₂×Q₈×C₁₃	—	4	2	no	—
83	C₂×Q₈×C₁₃	—	4	1	no	—
84	C₂×Q₈×C₁₃	D₈	1	1	yes	characteristic
85	C₂×D₄×C₁₃	—	2	1	no	—
86	C₂²×C₄×C₁₃	—	4	1	no	—
87	C₁₃×Q₁₆	—	2	2	no	—
88	C₁₃×Q₁₆	—	2	1	no	—
89	C₁₃×Q₁₆	—	2	2	no	—
90	C₁₃×Q₁₆	—	8	1	no	—
91	C₁₃×Q₁₆	#41	1	2	yes	—
92	C₁₃×SD₁₆	—	8	2	no	—
93	C₁₃×D₈	—	2	1	no	—
94	C₁₃×D₈	—	2	1	no	—
95	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
97	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
98	C₁₃×C₄⋊C₄	—	2	1	no	—
99	C₁₃×C₄⋊C₄	—	2	1	no	—
100	C₁₃×C₄⋊C₄	—	2	2	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×C₄⋊C₄	—	4	2	no	—
106	C₁₃×C₄⋊C₄	—	4	1	no	—
107	C₁₃×C₄⋊C₄	—	4	1	no	—
108	C₁₃×C₄⋊C₄	D₈	1	1	yes	characteristic
109	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
110	C₁₃×C₂²⋊C₄	—	4	2	no	—
111	C₁₃×C₂²⋊C₄	—	4	1	no	—
112	C₁₃×C₂²⋊C₄	—	4	2	no	—
113	C₄²×C₁₃	—	4	2	no	—
114	C₄²×C₁₃	—	4	2	no	—
115	C₄²×C₁₃	—	4	1	no	—
116	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
117	C₄²×C₁₃	C₂×D₄	1	2	yes	—
118	C₁₃×C₁₆	—	2	1	no	—
119	C₁₃×C₁₆	—	2	2	no	—
120	C₁₃×C₁₆	—	2	1	no	—
121	#3393	C₂×C₁₃	1	1	yes	characteristic
122	Q₈×Q₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
123	#2243	C₂×C₁₃	1	2	yes	—
124	#2241	C₂×C₁₃	1	1	yes	characteristic, complemented
125	#2226	C₂×C₁₃	1	1	yes	characteristic
126	#2225	C₂×C₁₃	1	2	yes	complemented
127	#2211	C₂×C₁₃	1	2	yes	—
128	#2209	C₂×C₁₃	1	1	yes	characteristic, complemented
129	#2183	C₂×C₁₃	1	1	yes	characteristic, complemented
130	#2176	C₂×C₁₃	1	2	yes	complemented
131	C₄×SD₃₂	C₂×C₁₃	1	1	yes	characteristic
132	C₂³×C₁₃	—	4	1	no	—
133	Q₈×C₁₃	—	2	2	no	—
134	Q₈×C₁₃	—	2	2	no	—
135	Q₈×C₁₃	—	4	1	no	—
136	Q₈×C₁₃	—	4	2	no	—
137	Q₈×C₁₃	—	4	2	no	—
138	Q₈×C₁₃	—	16	2	no	—
139	Q₈×C₁₃	—	16	1	no	—
140	D₄×C₁₃	—	4	1	no	—
141	D₄×C₁₃	—	4	1	no	—
142	C₂×C₄×C₁₃	—	4	1	no	—
143	C₂×C₄×C₁₃	—	4	1	no	—
144	C₂×C₄×C₁₃	—	4	2	no	—
145	C₂×C₄×C₁₃	—	4	2	no	—
146	C₂×C₄×C₁₃	—	4	1	no	—
147	C₂×C₄×C₁₃	—	8	2	no	—
148	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
149	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
150	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
151	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
152	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
153	C₈×C₁₃	—	2	2	no	—
154	C₈×C₁₃	—	2	1	no	—
155	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
156	C₈×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
157	Q₈²	—	2	1	no	—
158	#556	—	2	1	no	—
159	C₂×Q₃₂	C₂²×C₁₃	1	2	yes	—
160	C₂×SD₃₂	C₂²×C₁₃	1	1	yes	characteristic
161	#500	C₂²×C₁₃	1	2	yes	—
162	#500	C₂²×C₁₃	1	1	yes	characteristic
163	#495	C₂²×C₁₃	1	2	yes	—
164	#494	C₂²×C₁₃	1	1	yes	characteristic
165	#477	—	2	1	no	—
166	#477	—	2	2	no	—
167	#464	—	2	2	no	—
168	#461	—	2	1	no	—
169	C₈×Q₈	C₂²×C₁₃	1	1	yes	characteristic
170	C₄×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
171	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
172	C₄×Q₁₆	C₂²×C₁₃	1	2	yes	—
173	C₄×D₈	C₂²×C₁₃	1	1	yes	characteristic
174	#367	C₂²×C₁₃	1	1	yes	characteristic
175	#367	C₂²×C₁₃	1	2	yes	—
176	C₄⋊C₁₆	C₂²×C₁₃	1	1	yes	characteristic
177	C₄⋊C₁₆	C₂²×C₁₃	1	2	yes	—
178	#358	C₂²×C₁₃	1	1	yes	characteristic
179	#358	C₂²×C₁₃	1	2	yes	—
180	#358	C₂²×C₁₃	1	2	yes	—
181	#358	C₂²×C₁₃	1	2	yes	—
182	#357	C₂²×C₁₃	1	1	yes	characteristic
183	#357	C₂²×C₁₃	1	2	yes	—
184	#346	C₂²×C₁₃	1	2	yes	—
185	C₄×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
186	C₂²×C₁₃	—	8	1	no	—
187	C₂²×C₁₃	—	8	1	no	—
188	C₂²×C₁₃	—	8	1	no	—
189	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
190	C₄×C₁₃	—	2	2	no	—
191	C₄×C₁₃	—	2	2	no	—
192	C₄×C₁₃	—	2	1	no	—
193	C₄×C₁₃	—	4	2	no	—
194	C₄×C₁₃	—	8	2	no	—
195	C₄×C₁₃	—	8	1	no	—
196	C₄×C₁₃	—	8	2	no	—
197	C₄×C₁₃	—	8	1	no	—
198	C₄×C₁₃	#510	1	2	yes	—
199	C₄×C₁₃	#548	1	1	yes	characteristic
200	C₄×C₁₃	#577	1	1	yes	characteristic
201	C₂×Q₁₆	—	2	1	no	—
202	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
203	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
204	C₂×SD₁₆	—	2	2	no	—
205	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
206	C₄⋊Q₈	—	2	1	no	—
207	C₄⋊Q₈	—	2	2	no	—
208	C₄⋊Q₈	—	2	1	no	—
209	C₄⋊Q₈	—	4	1	no	—
210	C₄⋊Q₈	—	4	2	no	—
211	#126	—	4	2	no	—
212	#124	—	2	2	no	—
213	C₂²⋊Q₈	—	4	1	no	—
214	C₄×Q₈	—	2	2	no	—
215	C₄×Q₈	—	2	1	no	—
216	C₄×Q₈	—	2	2	no	—
217	C₄×Q₈	—	4	1	no	—
218	C₄×Q₈	D₄×C₁₃	1	1	yes	characteristic
219	C₄×D₄	—	2	1	no	—
220	#117	—	4	1	no	—
221	Q₃₂	—	2	4	no	—
222	SD₃₂	—	2	2	no	—
223	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
224	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
225	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
226	#107	—	2	1	no	—
227	#107	—	2	2	no	—
228	#107	C₂³×C₁₃	1	1	yes	characteristic
229	#107	C₂³×C₁₃	1	2	yes	—
230	#107	C₂³×C₁₃	1	1	yes	characteristic
231	C₄⋊C₈	D₄×C₁₃	1	1	yes	characteristic
232	C₄⋊C₈	D₄×C₁₃	1	2	yes	—
233	#103	—	2	2	no	—
234	#103	—	2	1	no	—
235	#103	—	2	2	no	—
236	#102	—	2	1	no	—
237	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
238	C₄×C₈	C₂³×C₁₃	1	2	yes	—
239	C₂×C₁₃	—	16	1	no	—
240	C₂×C₁₃	#3358	1	1	yes	characteristic
241	C₂×C₁₃	C₂×#510	1	1	yes	characteristic
242	C₂×C₁₃	#3416	1	1	yes	characteristic
243	C₂×Q₈	—	2	1	no	—
244	C₂×Q₈	—	2	2	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×Q₈	—	4	1	no	—
247	C₂×Q₈	C₁₃×D₈	1	1	yes	characteristic
248	C₂×D₄	—	2	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	Q₁₆	—	2	2	no	—
251	Q₁₆	—	2	1	no	—
252	Q₁₆	—	2	2	no	—
253	Q₁₆	—	8	1	no	—
254	Q₁₆	C₁₃×#41	1	2	yes	—
255	SD₁₆	—	8	2	no	—
256	D₈	—	2	1	no	—
257	D₈	—	2	1	no	—
258	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
259	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
260	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
261	C₄⋊C₄	—	2	1	no	—
262	C₄⋊C₄	—	2	1	no	—
263	C₄⋊C₄	—	2	2	no	—
264	C₄⋊C₄	—	4	1	no	—
265	C₄⋊C₄	—	4	2	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	—	4	1	no	—
268	C₄⋊C₄	—	4	2	no	—
269	C₄⋊C₄	—	4	1	no	—
270	C₄⋊C₄	—	4	1	no	—
271	C₄⋊C₄	C₁₃×D₈	1	1	yes	characteristic
272	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
273	C₂²⋊C₄	—	4	2	no	—
274	C₂²⋊C₄	—	4	1	no	—
275	C₂²⋊C₄	—	4	2	no	—
276	C₄²	—	4	2	no	—
277	C₄²	—	4	2	no	—
278	C₄²	—	4	1	no	—
279	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
280	C₄²	C₂×D₄×C₁₃	1	2	yes	—
281	C₁₆	—	2	1	no	—
282	C₁₆	—	2	2	no	—
283	C₁₆	—	2	1	no	—
284	C₁₃	#33936	1	1	yes	characteristic, direct product factor, Sylow
285	C₂³	—	4	1	no	—
286	Q₈	—	2	2	no	—
287	Q₈	—	2	2	no	—
288	Q₈	—	4	1	no	—
289	Q₈	—	4	2	no	—
290	Q₈	—	4	2	no	—
291	Q₈	—	16	2	no	—
292	Q₈	—	16	1	no	—
293	D₄	—	4	1	no	—
294	D₄	—	4	1	no	—
295	C₂×C₄	—	4	1	no	—
296	C₂×C₄	—	4	1	no	—
297	C₂×C₄	—	4	2	no	—
298	C₂×C₄	—	4	2	no	—
299	C₂×C₄	—	4	1	no	—
300	C₂×C₄	—	8	2	no	—
301	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
302	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
303	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
304	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
305	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
306	C₈	—	2	2	no	—
307	C₈	—	2	1	no	—
308	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
309	C₈	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
310	C₂²	—	8	1	no	—
311	C₂²	—	8	1	no	—
312	C₂²	—	8	1	no	—
313	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
314	C₄	—	2	2	no	—
315	C₄	—	2	2	no	—
316	C₄	—	2	1	no	—
317	C₄	—	4	2	no	—
318	C₄	—	8	2	no	—
319	C₄	—	8	1	no	—
320	C₄	—	8	2	no	—
321	C₄	—	8	1	no	—
322	C₄	C₁₃×#510	1	2	yes	—
323	C₄	C₁₃×#548	1	1	yes	characteristic
324	C₄	C₁₃×#577	1	1	yes	characteristic
325	C₂	—	16	1	no	—
326	C₂	C₁₃×#3358	1	1	yes	characteristic
327	C₂	C₂×C₁₃×#510	1	1	yes	characteristic
328	C₂	C₁₃×#3416	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
2. Q₈×C₁₃×Q₁₆	C₂	semidirect
4. C₁₃×#2241	C₂	semidirect
6. C₁₃×#2225	C₂	semidirect
8. C₁₃×#2209	C₂	semidirect
9. C₁₃×#2183	C₂	semidirect
10. C₁₃×#2176	C₂	semidirect
79. #33936	C₁₃	direct (N × H)
122. Q₈×Q₁₆	C₂×C₁₃	semidirect
124. #2241	C₂×C₁₃	semidirect
126. #2225	C₂×C₁₃	semidirect
128. #2209	C₂×C₁₃	semidirect
129. #2183	C₂×C₁₃	semidirect
130. #2176	C₂×C₁₃	semidirect
284. C₁₃	#33936	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×#3393	2
Q₈×C₁₃×Q₁₆	2
C₁₃×#2243	2
C₁₃×#2241	2
C₁₃×#2226	2
C₁₃×#2225	2
C₁₃×#2211	2
C₁₃×#2209	2
C₁₃×#2183	2
C₁₃×#2176	2
C₄×C₁₃×SD₃₂	2
#33936	13

Subgroups — C13×#33936 #15720275

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33936 #15720275