Subgroups — C₁₃×#33518 #15719857

Order 3328.

1124 conjugacy classes of subgroups.

The subgroup lattice has 340 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×C₁₃×Q₁₆	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#510	C₂	1	2	yes	maximal, complemented
3	C₁₃×#3050	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2188	C₂	1	4	yes	maximal, complemented
5	C₁₃×#2185	C₂	1	4	yes	maximal, complemented
6	C₁₃×#2141	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2112	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂³×Q₈×C₁₃	—	2	1	no	—
9	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
10	C₂²×C₁₃×Q₁₆	—	2	8	no	—
11	C₂²×C₁₃×Q₁₆	—	2	1	no	—
12	C₂²×C₁₃×Q₁₆	—	2	2	no	—
13	C₂²×C₁₃×Q₁₆	C₂²	1	4	yes	—
14	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
15	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
16	C₁₃×#525	—	2	1	no	—
17	C₁₃×#510	—	2	8	no	—
18	C₂×C₁₃×Q₃₂	C₂²	1	4	yes	complemented
19	C₂×C₁₃×SD₃₂	C₂²	1	4	yes	complemented
20	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
21	C₁₃×#467	C₂²	1	2	yes	—
22	C₁₃×#466	C₂²	1	2	yes	—
23	C₁₃×#465	—	2	2	no	—
24	C₁₃×#427	C₂²	1	1	yes	characteristic
25	C₁₃×#416	—	2	1	no	—
26	C₁₃×#358	C₂²	1	8	yes	complemented
27	C₁₃×C₂²⋊C₁₆	C₂²	1	4	yes	—
28	C₂×C₁₃×#41	—	2	1	no	—
29	C₂²×Q₈×C₁₃	—	2	8	no	—
30	C₂²×Q₈×C₁₃	—	2	4	no	—
31	C₂²×Q₈×C₁₃	—	2	2	no	—
32	C₂²×Q₈×C₁₃	—	4	4	no	—
33	C₂²×Q₈×C₁₃	—	4	2	no	—
34	C₂²×Q₈×C₁₃	—	4	1	no	—
35	C₂³×C₄×C₁₃	—	4	1	no	—
36	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
37	C₁₃×#135	—	2	4	no	—
38	C₂×C₁₃×Q₁₆	—	2	8	no	—
39	C₂×C₁₃×Q₁₆	—	2	16	no	—
40	C₂×C₁₃×Q₁₆	—	2	16	no	—
41	C₂×C₁₃×Q₁₆	—	2	4	no	—
42	C₂×C₁₃×Q₁₆	—	2	2	no	—
43	C₂×C₁₃×Q₁₆	—	2	4	no	—
44	C₂×C₁₃×Q₁₆	D₄	1	8	yes	complemented
45	C₂×C₁₃×Q₁₆	C₂³	1	1	yes	characteristic
46	C₂×C₁₃×Q₁₆	C₂³	1	4	yes	—
47	C₂×C₁₃×SD₁₆	—	2	1	no	—
48	C₂×C₁₃×D₈	C₂³	1	1	yes	characteristic
49	C₂²×C₈×C₁₃	—	2	4	no	—
50	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
51	C₂²×C₈×C₁₃	D₄	1	2	yes	—
52	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
53	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
54	C₁₃×#123	—	4	1	no	—
55	C₁₃×C₂²⋊Q₈	—	2	2	no	—
56	C₁₃×#121	—	2	2	no	—
57	C₁₃×#120	—	4	1	no	—
58	C₄×D₄×C₁₃	—	4	2	no	—
59	C₁₃×#117	—	2	1	no	—
60	C₁₃×Q₃₂	—	2	8	no	—
61	C₁₃×SD₃₂	—	2	8	no	—
62	C₁₃×M₃₂	—	2	4	no	—
63	C₂×C₁₃×C₁₆	C₂³	1	4	yes	—
64	C₁₃×#107	C₂³	1	2	yes	—
65	C₁₃×#106	—	2	1	no	—
66	C₁₃×#103	—	2	2	no	—
67	C₁₃×#102	—	2	2	no	—
68	#33518	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
69	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
70	C₁₃×#41	—	4	2	no	—
71	C₁₃×#41	—	4	2	no	—
72	C₂×Q₈×C₁₃	—	2	16	no	—
73	C₂×Q₈×C₁₃	—	2	8	no	—
74	C₂×Q₈×C₁₃	—	2	4	no	—
75	C₂×Q₈×C₁₃	—	2	1	no	—
76	C₂×Q₈×C₁₃	—	4	8	no	—
77	C₂×Q₈×C₁₃	—	4	8	no	—
78	C₂×Q₈×C₁₃	—	4	16	no	—
79	C₂×Q₈×C₁₃	—	4	4	no	—
80	C₂×Q₈×C₁₃	—	4	2	no	—
81	C₂×Q₈×C₁₃	—	4	4	no	—
82	C₂×D₄×C₁₃	—	2	1	no	—
83	C₂×D₄×C₁₃	—	4	2	no	—
84	C₂×D₄×C₁₃	—	4	1	no	—
85	C₂²×C₄×C₁₃	—	2	2	no	—
86	C₂²×C₄×C₁₃	—	2	2	no	—
87	C₂²×C₄×C₁₃	—	4	8	no	—
88	C₂²×C₄×C₁₃	—	4	4	no	—
89	C₂²×C₄×C₁₃	—	4	2	no	—
90	C₂²×C₄×C₁₃	—	4	1	no	—
91	C₂²×C₄×C₁₃	D₈	1	2	yes	—
92	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
93	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
94	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
95	C₁₃×Q₁₆	—	2	16	no	—
96	C₁₃×Q₁₆	—	2	8	no	—
97	C₁₃×Q₁₆	—	2	4	no	—
98	C₁₃×Q₁₆	—	2	2	no	—
99	C₁₃×Q₁₆	C₂×D₄	1	8	yes	—
100	C₁₃×SD₁₆	—	4	2	no	—
101	C₁₃×D₈	—	2	2	no	—
102	C₂×C₈×C₁₃	—	2	2	no	—
103	C₂×C₈×C₁₃	—	2	4	no	—
104	C₂×C₈×C₁₃	—	2	4	no	—
105	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
106	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂×D₄	1	4	yes	—
108	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
109	C₁₃×C₄⋊C₄	—	2	2	no	—
110	C₁₃×C₄⋊C₄	—	4	2	no	—
111	C₁₃×C₂²⋊C₄	—	4	1	no	—
112	C₁₃×C₂²⋊C₄	—	4	2	no	—
113	C₁₃×C₂²⋊C₄	—	4	2	no	—
114	C₄²×C₁₃	—	4	1	no	—
115	C₁₃×C₁₆	—	2	4	no	—
116	C₂³×Q₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
117	C₂×#510	C₂×C₁₃	1	2	yes	complemented
118	#3050	C₂×C₁₃	1	1	yes	characteristic
119	#2188	C₂×C₁₃	1	4	yes	complemented
120	#2185	C₂×C₁₃	1	4	yes	complemented
121	#2141	C₂×C₁₃	1	2	yes	complemented
122	#2112	C₂×C₁₃	1	1	yes	characteristic, complemented
123	C₂³×C₁₃	—	2	2	no	—
124	C₂³×C₁₃	—	2	4	no	—
125	C₂³×C₁₃	—	4	1	no	—
126	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
127	C₂³×C₁₃	C₂×D₈	1	2	yes	—
128	Q₈×C₁₃	—	2	8	no	—
129	Q₈×C₁₃	—	4	16	no	—
130	Q₈×C₁₃	—	4	4	no	—
131	Q₈×C₁₃	—	4	8	no	—
132	Q₈×C₁₃	—	4	1	no	—
133	Q₈×C₁₃	—	4	1	no	—
134	D₄×C₁₃	—	4	1	no	—
135	D₄×C₁₃	—	4	1	no	—
136	D₄×C₁₃	—	8	2	no	—
137	D₄×C₁₃	—	8	4	no	—
138	C₂×C₄×C₁₃	—	2	2	no	—
139	C₂×C₄×C₁₃	—	2	2	no	—
140	C₂×C₄×C₁₃	—	2	2	no	—
141	C₂×C₄×C₁₃	—	2	4	no	—
142	C₂×C₄×C₁₃	—	4	16	no	—
143	C₂×C₄×C₁₃	—	4	4	no	—
144	C₂×C₄×C₁₃	—	4	8	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	#120	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	AGL(1,ℤ/8)	1	2	yes	—
152	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
153	C₈×C₁₃	—	2	2	no	—
154	C₈×C₁₃	#120	1	2	yes	—
155	C₈×C₁₃	C₂²×D₄	1	2	yes	—
156	C₂³×Q₈	—	2	1	no	—
157	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
158	C₂²×Q₁₆	—	2	8	no	—
159	C₂²×Q₁₆	—	2	1	no	—
160	C₂²×Q₁₆	—	2	2	no	—
161	C₂²×Q₁₆	C₂²×C₁₃	1	4	yes	—
162	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
163	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
164	#525	—	2	1	no	—
165	#510	—	2	8	no	—
166	C₂×Q₃₂	C₂²×C₁₃	1	4	yes	complemented
167	C₂×SD₃₂	C₂²×C₁₃	1	4	yes	complemented
168	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
169	#467	C₂²×C₁₃	1	2	yes	—
170	#466	C₂²×C₁₃	1	2	yes	—
171	#465	—	2	2	no	—
172	#427	C₂²×C₁₃	1	1	yes	characteristic
173	#416	—	2	1	no	—
174	#358	C₂²×C₁₃	1	8	yes	complemented
175	C₂²⋊C₁₆	C₂²×C₁₃	1	4	yes	—
176	C₂²×C₁₃	—	2	8	no	—
177	C₂²×C₁₃	—	2	2	no	—
178	C₂²×C₁₃	—	2	2	no	—
179	C₂²×C₁₃	—	2	2	no	—
180	C₂²×C₁₃	—	8	2	no	—
181	C₂²×C₁₃	—	8	1	no	—
182	C₂²×C₁₃	#447	1	2	yes	—
183	C₂²×C₁₃	#510	1	4	yes	complemented
184	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
185	C₄×C₁₃	—	2	2	no	—
186	C₄×C₁₃	—	4	8	no	—
187	C₄×C₁₃	—	8	2	no	—
188	C₄×C₁₃	—	8	1	no	—
189	C₄×C₁₃	#447	1	2	yes	—
190	C₄×C₁₃	C₂×#120	1	1	yes	characteristic
191	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
192	C₂×#41	—	2	1	no	—
193	C₂²×Q₈	—	2	8	no	—
194	C₂²×Q₈	—	2	4	no	—
195	C₂²×Q₈	—	2	2	no	—
196	C₂²×Q₈	—	4	4	no	—
197	C₂²×Q₈	—	4	2	no	—
198	C₂²×Q₈	—	4	1	no	—
199	C₂³×C₄	—	4	1	no	—
200	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
201	#135	—	2	4	no	—
202	C₂×Q₁₆	—	2	8	no	—
203	C₂×Q₁₆	—	2	16	no	—
204	C₂×Q₁₆	—	2	16	no	—
205	C₂×Q₁₆	—	2	4	no	—
206	C₂×Q₁₆	—	2	2	no	—
207	C₂×Q₁₆	—	2	4	no	—
208	C₂×Q₁₆	D₄×C₁₃	1	8	yes	complemented
209	C₂×Q₁₆	C₂³×C₁₃	1	1	yes	characteristic
210	C₂×Q₁₆	C₂³×C₁₃	1	4	yes	—
211	C₂×SD₁₆	—	2	1	no	—
212	C₂×D₈	C₂³×C₁₃	1	1	yes	characteristic
213	C₂²×C₈	—	2	4	no	—
214	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
215	C₂²×C₈	D₄×C₁₃	1	2	yes	—
216	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
217	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
218	#123	—	4	1	no	—
219	C₂²⋊Q₈	—	2	2	no	—
220	#121	—	2	2	no	—
221	#120	—	4	1	no	—
222	C₄×D₄	—	4	2	no	—
223	#117	—	2	1	no	—
224	Q₃₂	—	2	8	no	—
225	SD₃₂	—	2	8	no	—
226	M₃₂	—	2	4	no	—
227	C₂×C₁₆	C₂³×C₁₃	1	4	yes	—
228	#107	C₂³×C₁₃	1	2	yes	—
229	#106	—	2	1	no	—
230	#103	—	2	2	no	—
231	#102	—	2	2	no	—
232	C₂×C₁₃	—	2	2	no	—
233	C₂×C₁₃	—	2	4	no	—
234	C₂×C₁₃	—	16	1	no	—
235	C₂×C₁₃	C₂×#447	1	1	yes	characteristic
236	C₂×C₁₃	C₂×#510	1	2	yes	—
237	C₂⁴	C₁₃×D₈	1	1	yes	characteristic
238	#41	—	4	2	no	—
239	#41	—	4	2	no	—
240	C₂×Q₈	—	2	16	no	—
241	C₂×Q₈	—	2	8	no	—
242	C₂×Q₈	—	2	4	no	—
243	C₂×Q₈	—	2	1	no	—
244	C₂×Q₈	—	4	8	no	—
245	C₂×Q₈	—	4	16	no	—
246	C₂×Q₈	—	4	8	no	—
247	C₂×Q₈	—	4	4	no	—
248	C₂×Q₈	—	4	2	no	—
249	C₂×Q₈	—	4	4	no	—
250	C₂×D₄	—	2	1	no	—
251	C₂×D₄	—	4	2	no	—
252	C₂×D₄	—	4	1	no	—
253	C₂²×C₄	—	2	2	no	—
254	C₂²×C₄	—	2	2	no	—
255	C₂²×C₄	—	4	8	no	—
256	C₂²×C₄	—	4	4	no	—
257	C₂²×C₄	—	4	2	no	—
258	C₂²×C₄	—	4	1	no	—
259	C₂²×C₄	C₁₃×D₈	1	2	yes	—
260	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
261	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
262	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
263	Q₁₆	—	2	16	no	—
264	Q₁₆	—	2	8	no	—
265	Q₁₆	—	2	4	no	—
266	Q₁₆	—	2	2	no	—
267	Q₁₆	C₂×D₄×C₁₃	1	8	yes	—
268	SD₁₆	—	4	2	no	—
269	D₈	—	2	2	no	—
270	C₂×C₈	—	2	2	no	—
271	C₂×C₈	—	2	4	no	—
272	C₂×C₈	—	2	4	no	—
273	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
274	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
275	C₂×C₈	C₂×D₄×C₁₃	1	4	yes	—
276	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
277	C₄⋊C₄	—	2	2	no	—
278	C₄⋊C₄	—	4	2	no	—
279	C₂²⋊C₄	—	4	1	no	—
280	C₂²⋊C₄	—	4	2	no	—
281	C₂²⋊C₄	—	4	2	no	—
282	C₄²	—	4	1	no	—
283	C₁₆	—	2	4	no	—
284	C₁₃	#33518	1	1	yes	characteristic, direct product factor, Sylow
285	C₂³	—	2	2	no	—
286	C₂³	—	2	4	no	—
287	C₂³	—	4	1	no	—
288	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
289	C₂³	C₂×C₁₃×D₈	1	2	yes	—
290	Q₈	—	2	8	no	—
291	Q₈	—	4	16	no	—
292	Q₈	—	4	8	no	—
293	Q₈	—	4	4	no	—
294	Q₈	—	4	1	no	—
295	Q₈	—	4	1	no	—
296	D₄	—	4	1	no	—
297	D₄	—	4	1	no	—
298	D₄	—	8	2	no	—
299	D₄	—	8	4	no	—
300	C₂×C₄	—	2	2	no	—
301	C₂×C₄	—	2	4	no	—
302	C₂×C₄	—	2	2	no	—
303	C₂×C₄	—	2	2	no	—
304	C₂×C₄	—	4	16	no	—
305	C₂×C₄	—	4	8	no	—
306	C₂×C₄	—	4	4	no	—
307	C₂×C₄	—	4	2	no	—
308	C₂×C₄	—	4	1	no	—
309	C₂×C₄	—	8	2	no	—
310	C₂×C₄	C₁₃×#120	1	2	yes	—
311	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
312	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
313	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
314	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
315	C₈	—	2	2	no	—
316	C₈	C₁₃×#120	1	2	yes	—
317	C₈	C₂²×D₄×C₁₃	1	2	yes	—
318	C₂²	—	2	8	no	—
319	C₂²	—	2	2	no	—
320	C₂²	—	2	2	no	—
321	C₂²	—	2	2	no	—
322	C₂²	—	8	2	no	—
323	C₂²	—	8	1	no	—
324	C₂²	C₁₃×#447	1	2	yes	—
325	C₂²	C₁₃×#510	1	4	yes	complemented
326	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
327	C₄	—	2	2	no	—
328	C₄	—	4	8	no	—
329	C₄	—	8	2	no	—
330	C₄	—	8	1	no	—
331	C₄	C₁₃×#447	1	2	yes	—
332	C₄	C₂×C₁₃×#120	1	1	yes	characteristic
333	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
334	C₂	—	2	2	no	—
335	C₂	—	2	4	no	—
336	C₂	—	16	1	no	—
337	C₂	C₂×C₁₃×#447	1	1	yes	characteristic
338	C₂	C₂×C₁₃×#510	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₁₃×#3050	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂³×C₁₃×Q₁₆	C₂	semidirect
2. C₂×C₁₃×#510	C₂	semidirect
4. C₁₃×#2188	C₂	semidirect
5. C₁₃×#2185	C₂	semidirect
6. C₁₃×#2141	C₂	semidirect
7. C₁₃×#2112	C₂	semidirect
18. C₂×C₁₃×Q₃₂	C₂²	semidirect
19. C₂×C₁₃×SD₃₂	C₂²	semidirect
26. C₁₃×#358	C₂²	semidirect
44. C₂×C₁₃×Q₁₆	D₄	semidirect
68. #33518	C₁₃	direct (N × H)
116. C₂³×Q₁₆	C₂×C₁₃	semidirect
117. C₂×#510	C₂×C₁₃	semidirect
119. #2188	C₂×C₁₃	semidirect
120. #2185	C₂×C₁₃	semidirect
121. #2141	C₂×C₁₃	semidirect
122. #2112	C₂×C₁₃	semidirect
166. C₂×Q₃₂	C₂²×C₁₃	semidirect
167. C₂×SD₃₂	C₂²×C₁₃	semidirect
174. #358	C₂²×C₁₃	semidirect
183. C₂²×C₁₃	#510	semidirect
208. C₂×Q₁₆	D₄×C₁₃	semidirect
284. C₁₃	#33518	direct (N × H)
325. C₂²	C₁₃×#510	semidirect

Maximal subgroups

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

subgroup	index
C₂³×C₁₃×Q₁₆	2
C₂×C₁₃×#510	2
C₁₃×#3050	2
C₁₃×#2188	2
C₁₃×#2185	2
C₁₃×#2141	2
C₁₃×#2112	2
#33518	13

Subgroups — C13×#33518 #15719857

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33518 #15719857