Subgroups — C₁₃×#33638 #15719977

Order 3328.

514 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3163	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#2957	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2231	C₂	1	4	yes	maximal, complemented
4	C₁₃×#2230	C₂	1	4	yes	maximal, complemented
5	C₁₃×#2155	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₁₃×#365	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#2153	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2145	C₂	1	1	yes	maximal, characteristic
9	C₂×C₁₃×#360	C₂	1	1	yes	maximal, characteristic
10	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
11	C₁₃×#568	—	2	1	no	—
12	C₁₃×#544	—	2	1	no	—
13	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
15	C₁₃×#501	—	2	2	no	—
16	C₁₃×#499	C₂²	1	2	yes	complemented
17	C₁₃×#498	C₂²	1	2	yes	complemented
18	C₁₃×#444	—	2	2	no	—
19	C₁₃×#443	C₂²	1	4	yes	—
20	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
21	C₁₃×#428	—	2	1	no	—
22	C₁₃×#427	C₂²	1	1	yes	characteristic
23	C₂×C₁₃×#106	C₂²	1	1	yes	characteristic
24	C₁₃×#421	—	2	1	no	—
25	C₁₃×#405	C₂²	1	1	yes	characteristic
26	C₁₃×#365	C₂²	1	4	yes	complemented
27	C₁₃×#365	C₂²	1	2	yes	complemented
28	C₁₃×#365	C₂²	1	2	yes	complemented
29	C₁₃×#364	C₂²	1	4	yes	complemented
30	C₁₃×#360	C₂²	1	4	yes	—
31	C₁₃×#360	C₂²	1	2	yes	—
32	C₁₃×#360	C₂²	1	2	yes	—
33	C₂×C₁₃×#41	—	2	1	no	—
34	C₁₃×#135	—	2	1	no	—
35	C₁₃×#135	—	2	1	no	—
36	C₁₃×#135	Q₈	1	2	yes	complemented
37	C₁₃×#135	C₂³	1	2	yes	—
38	C₂×C₁₃×Q₁₆	Q₈	1	1	yes	characteristic, complemented
39	C₂×C₁₃×SD₁₆	—	2	1	no	—
40	C₂×C₁₃×D₈	Q₈	1	1	yes	characteristic, complemented
41	C₁₃×#131	—	2	4	no	—
42	C₁₃×#131	—	4	2	no	—
43	C₂×C₁₃×M₁₆	—	2	1	no	—
44	C₂×C₁₃×M₁₆	—	4	2	no	—
45	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×C₄⋊Q₈	—	2	2	no	—
48	C₁₃×C₄⋊Q₈	—	4	1	no	—
49	C₁₃×#125	—	2	2	no	—
50	C₁₃×#125	—	4	1	no	—
51	C₁₃×C₂²⋊Q₈	—	4	2	no	—
52	C₁₃×#117	—	2	1	no	—
53	C₁₃×#117	D₄	1	1	yes	characteristic
54	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
55	C₁₃×M₃₂	—	2	1	no	—
56	C₁₃×M₃₂	—	2	1	no	—
57	C₁₃×M₃₂	C₂³	1	2	yes	—
58	C₁₃×M₃₂	C₂³	1	2	yes	—
59	C₂×C₁₃×C₁₆	—	2	1	no	—
60	C₂×C₁₃×C₁₆	—	2	1	no	—
61	C₁₃×#108	—	2	1	no	—
62	C₁₃×#108	C₂³	1	2	yes	—
63	C₁₃×#107	—	2	1	no	—
64	C₁₃×#107	—	2	2	no	—
65	C₁₃×#106	—	2	1	no	—
66	C₁₃×#106	—	2	2	no	—
67	C₁₃×#106	C₂³	1	2	yes	—
68	C₁₃×#106	C₂³	1	1	yes	characteristic
69	C₁₃×#106	C₂³	1	1	yes	characteristic
70	C₁₃×#104	—	2	4	no	—
71	C₁₃×#97	D₄	1	2	yes	—
72	C₄×C₈×C₁₃	C₂³	1	2	yes	—
73	#33638	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₁₃×#41	—	2	2	no	—
75	C₁₃×#41	—	2	2	no	—
76	C₁₃×#41	—	4	1	no	—
77	C₁₃×#41	—	4	1	no	—
78	C₂×Q₈×C₁₃	—	2	1	no	—
79	C₂×Q₈×C₁₃	—	4	2	no	—
80	C₂×D₄×C₁₃	—	2	1	no	—
81	C₂×D₄×C₁₃	—	4	1	no	—
82	C₂²×C₄×C₁₃	—	4	1	no	—
83	C₂²×C₄×C₁₃	—	4	1	no	—
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₁₃×Q₁₆	—	2	1	no	—
86	C₁₃×Q₁₆	C₂×Q₈	1	2	yes	—
87	C₁₃×SD₁₆	—	2	2	no	—
88	C₁₃×SD₁₆	—	2	2	no	—
89	C₁₃×D₈	—	2	1	no	—
90	C₁₃×D₈	C₂×Q₈	1	2	yes	—
91	C₁₃×M₁₆	—	2	2	no	—
92	C₁₃×M₁₆	—	4	1	no	—
93	C₁₃×M₁₆	—	4	4	no	—
94	C₁₃×M₁₆	SD₁₆	1	2	yes	—
95	C₁₃×M₁₆	SD₁₆	1	2	yes	—
96	C₂×C₈×C₁₃	—	4	1	no	—
97	C₂×C₈×C₁₃	—	4	4	no	—
98	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
99	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
100	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
101	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
102	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
103	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
104	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
105	C₁₃×C₄⋊C₄	—	2	1	no	—
106	C₁₃×C₄⋊C₄	—	2	2	no	—
107	C₁₃×C₄⋊C₄	—	2	1	no	—
108	C₁₃×C₄⋊C₄	—	2	1	no	—
109	C₁₃×C₄⋊C₄	—	4	2	no	—
110	C₁₃×C₄⋊C₄	—	4	2	no	—
111	C₁₃×C₄⋊C₄	—	4	2	no	—
112	C₁₃×C₄⋊C₄	—	4	1	no	—
113	C₁₃×C₂²⋊C₄	—	2	1	no	—
114	C₁₃×C₂²⋊C₄	—	4	1	no	—
115	C₄²×C₁₃	—	4	1	no	—
116	C₄²×C₁₃	C₂×D₄	1	2	yes	—
117	C₁₃×C₁₆	—	2	2	no	—
118	C₁₃×C₁₆	—	2	2	no	—
119	#3163	C₂×C₁₃	1	1	yes	characteristic, complemented
120	#2957	C₂×C₁₃	1	1	yes	characteristic
121	#2231	C₂×C₁₃	1	4	yes	complemented
122	#2230	C₂×C₁₃	1	4	yes	complemented
123	#2155	C₂×C₁₃	1	1	yes	characteristic, complemented
124	C₂×#365	C₂×C₁₃	1	1	yes	characteristic, complemented
125	#2153	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#2145	C₂×C₁₃	1	1	yes	characteristic
127	C₂×#360	C₂×C₁₃	1	1	yes	characteristic
128	C₂³×C₁₃	—	4	1	no	—
129	C₂³×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
130	Q₈×C₁₃	—	2	2	no	—
131	Q₈×C₁₃	—	4	1	no	—
132	Q₈×C₁₃	—	16	2	no	—
133	D₄×C₁₃	—	2	2	no	—
134	D₄×C₁₃	—	4	2	no	—
135	D₄×C₁₃	—	4	2	no	—
136	D₄×C₁₃	—	4	1	no	—
137	C₂×C₄×C₁₃	—	2	2	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	—	4	1	no	—
143	C₂×C₄×C₁₃	—	4	2	no	—
144	C₂×C₄×C₁₃	—	16	1	no	—
145	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
146	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
147	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
148	C₂×C₄×C₁₃	C₂×SD₁₆	1	1	yes	characteristic
149	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	1	yes	characteristic
150	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
151	C₈×C₁₃	—	4	2	no	—
152	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
153	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
154	C₈×C₁₃	C₂×SD₁₆	1	4	yes	—
155	C₈×C₁₃	C₂²×Q₈	1	1	yes	characteristic
156	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
157	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
158	#568	—	2	1	no	—
159	#544	—	2	1	no	—
160	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
161	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
162	#501	—	2	2	no	—
163	#499	C₂²×C₁₃	1	2	yes	complemented
164	#498	C₂²×C₁₃	1	2	yes	complemented
165	#444	—	2	2	no	—
166	#443	C₂²×C₁₃	1	4	yes	—
167	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
168	#428	—	2	1	no	—
169	#427	C₂²×C₁₃	1	1	yes	characteristic
170	C₂×#106	C₂²×C₁₃	1	1	yes	characteristic
171	#421	—	2	1	no	—
172	#405	C₂²×C₁₃	1	1	yes	characteristic
173	#365	C₂²×C₁₃	1	4	yes	complemented
174	#365	C₂²×C₁₃	1	2	yes	complemented
175	#365	C₂²×C₁₃	1	2	yes	complemented
176	#364	C₂²×C₁₃	1	4	yes	complemented
177	#360	C₂²×C₁₃	1	4	yes	—
178	#360	C₂²×C₁₃	1	2	yes	—
179	#360	C₂²×C₁₃	1	2	yes	—
180	C₂²×C₁₃	—	4	1	no	—
181	C₂²×C₁₃	—	4	2	no	—
182	C₂²×C₁₃	—	16	1	no	—
183	C₂²×C₁₃	#476	1	1	yes	characteristic
184	C₂²×C₁₃	#476	1	1	yes	characteristic
185	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
186	C₄×C₁₃	—	4	2	no	—
187	C₄×C₁₃	—	4	2	no	—
188	C₄×C₁₃	—	8	1	no	—
189	C₄×C₁₃	—	8	2	no	—
190	C₄×C₁₃	—	8	1	no	—
191	C₄×C₁₃	#476	1	1	yes	characteristic
192	C₄×C₁₃	#476	1	1	yes	characteristic
193	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
194	C₄×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
195	C₂×#41	—	2	1	no	—
196	#135	—	2	1	no	—
197	#135	—	2	1	no	—
198	#135	Q₈×C₁₃	1	2	yes	complemented
199	#135	C₂³×C₁₃	1	2	yes	—
200	C₂×Q₁₆	Q₈×C₁₃	1	1	yes	characteristic, complemented
201	C₂×SD₁₆	—	2	1	no	—
202	C₂×D₈	Q₈×C₁₃	1	1	yes	characteristic, complemented
203	#131	—	2	4	no	—
204	#131	—	4	2	no	—
205	C₂×M₁₆	—	2	1	no	—
206	C₂×M₁₆	—	4	2	no	—
207	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
208	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
209	C₄⋊Q₈	—	2	2	no	—
210	C₄⋊Q₈	—	4	1	no	—
211	#125	—	2	2	no	—
212	#125	—	4	1	no	—
213	C₂²⋊Q₈	—	4	2	no	—
214	#117	—	2	1	no	—
215	#117	D₄×C₁₃	1	1	yes	characteristic
216	C₂×C₄⋊C₄	—	2	1	no	—
217	M₃₂	—	2	1	no	—
218	M₃₂	—	2	1	no	—
219	M₃₂	C₂³×C₁₃	1	2	yes	—
220	M₃₂	C₂³×C₁₃	1	2	yes	—
221	C₂×C₁₆	—	2	1	no	—
222	C₂×C₁₆	—	2	1	no	—
223	#108	—	2	1	no	—
224	#108	C₂³×C₁₃	1	2	yes	—
225	#107	—	2	1	no	—
226	#107	—	2	2	no	—
227	#106	—	2	1	no	—
228	#106	—	2	2	no	—
229	#106	C₂³×C₁₃	1	2	yes	—
230	#106	C₂³×C₁₃	1	1	yes	characteristic
231	#106	C₂³×C₁₃	1	1	yes	characteristic
232	#104	—	2	4	no	—
233	#97	D₄×C₁₃	1	2	yes	—
234	C₄×C₈	C₂³×C₁₃	1	2	yes	—
235	C₂×C₁₃	—	2	1	no	—
236	C₂×C₁₃	—	2	1	no	—
237	C₂×C₁₃	—	2	1	no	—
238	C₂×C₁₃	—	8	2	no	—
239	C₂×C₁₃	C₂×#476	1	1	yes	characteristic
240	#41	—	2	2	no	—
241	#41	—	2	2	no	—
242	#41	—	4	1	no	—
243	#41	—	4	1	no	—
244	C₂×Q₈	—	2	1	no	—
245	C₂×Q₈	—	4	2	no	—
246	C₂×D₄	—	2	1	no	—
247	C₂×D₄	—	4	1	no	—
248	C₂²×C₄	—	4	1	no	—
249	C₂²×C₄	—	4	1	no	—
250	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
251	Q₁₆	—	2	1	no	—
252	Q₁₆	C₂×Q₈×C₁₃	1	2	yes	—
253	SD₁₆	—	2	2	no	—
254	SD₁₆	—	2	2	no	—
255	D₈	—	2	1	no	—
256	D₈	C₂×Q₈×C₁₃	1	2	yes	—
257	M₁₆	—	2	2	no	—
258	M₁₆	—	4	1	no	—
259	M₁₆	—	4	4	no	—
260	M₁₆	C₁₃×SD₁₆	1	2	yes	—
261	M₁₆	C₁₃×SD₁₆	1	2	yes	—
262	C₂×C₈	—	4	1	no	—
263	C₂×C₈	—	4	4	no	—
264	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
265	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
266	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
267	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
268	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
269	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
270	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
271	C₄⋊C₄	—	2	1	no	—
272	C₄⋊C₄	—	2	2	no	—
273	C₄⋊C₄	—	2	1	no	—
274	C₄⋊C₄	—	2	1	no	—
275	C₄⋊C₄	—	4	2	no	—
276	C₄⋊C₄	—	4	2	no	—
277	C₄⋊C₄	—	4	2	no	—
278	C₄⋊C₄	—	4	1	no	—
279	C₂²⋊C₄	—	2	1	no	—
280	C₂²⋊C₄	—	4	1	no	—
281	C₄²	—	4	1	no	—
282	C₄²	C₂×D₄×C₁₃	1	2	yes	—
283	C₁₆	—	2	2	no	—
284	C₁₆	—	2	2	no	—
285	C₁₃	#33638	1	1	yes	characteristic, direct product factor, Sylow
286	C₂³	—	4	1	no	—
287	C₂³	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
288	Q₈	—	2	2	no	—
289	Q₈	—	4	1	no	—
290	Q₈	—	16	2	no	—
291	D₄	—	2	2	no	—
292	D₄	—	4	2	no	—
293	D₄	—	4	2	no	—
294	D₄	—	4	1	no	—
295	C₂×C₄	—	2	2	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	—	4	1	no	—
301	C₂×C₄	—	4	2	no	—
302	C₂×C₄	—	16	1	no	—
303	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
304	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
305	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
306	C₂×C₄	C₂×C₁₃×SD₁₆	1	1	yes	characteristic
307	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
308	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
309	C₈	—	4	2	no	—
310	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
311	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
312	C₈	C₂×C₁₃×SD₁₆	1	4	yes	—
313	C₈	C₂²×Q₈×C₁₃	1	1	yes	characteristic
314	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
315	C₂²	—	4	1	no	—
316	C₂²	—	4	2	no	—
317	C₂²	—	16	1	no	—
318	C₂²	C₁₃×#476	1	1	yes	characteristic
319	C₂²	C₁₃×#476	1	1	yes	characteristic
320	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
321	C₄	—	4	2	no	—
322	C₄	—	4	2	no	—
323	C₄	—	8	1	no	—
324	C₄	—	8	2	no	—
325	C₄	—	8	1	no	—
326	C₄	C₁₃×#476	1	1	yes	characteristic
327	C₄	C₁₃×#476	1	1	yes	characteristic
328	C₄	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
329	C₄	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
330	C₂	—	2	1	no	—
331	C₂	—	2	1	no	—
332	C₂	—	2	1	no	—
333	C₂	—	8	2	no	—
334	C₂	C₂×C₁₃×#476	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂×C₁₃	C₂×#476
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
prime radical	C₂×C₁₃×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3163	C₂	semidirect
3. C₁₃×#2231	C₂	semidirect
4. C₁₃×#2230	C₂	semidirect
5. C₁₃×#2155	C₂	semidirect
6. C₂×C₁₃×#365	C₂	semidirect
7. C₁₃×#2153	C₂	semidirect
16. C₁₃×#499	C₂²	semidirect
17. C₁₃×#498	C₂²	semidirect
26. C₁₃×#365	C₂²	semidirect
27. C₁₃×#365	C₂²	semidirect
28. C₁₃×#365	C₂²	semidirect
29. C₁₃×#364	C₂²	semidirect
36. C₁₃×#135	Q₈	semidirect
38. C₂×C₁₃×Q₁₆	Q₈	semidirect
40. C₂×C₁₃×D₈	Q₈	semidirect
73. #33638	C₁₃	direct (N × H)
119. #3163	C₂×C₁₃	semidirect
121. #2231	C₂×C₁₃	semidirect
122. #2230	C₂×C₁₃	semidirect
123. #2155	C₂×C₁₃	semidirect
124. C₂×#365	C₂×C₁₃	semidirect
125. #2153	C₂×C₁₃	semidirect
163. #499	C₂²×C₁₃	semidirect
164. #498	C₂²×C₁₃	semidirect
173. #365	C₂²×C₁₃	semidirect
174. #365	C₂²×C₁₃	semidirect
175. #365	C₂²×C₁₃	semidirect
176. #364	C₂²×C₁₃	semidirect
198. #135	Q₈×C₁₃	semidirect
200. C₂×Q₁₆	Q₈×C₁₃	semidirect
202. C₂×D₈	Q₈×C₁₃	semidirect
285. C₁₃	#33638	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×#3163	2
C₁₃×#2957	2
C₁₃×#2231	2
C₁₃×#2230	2
C₁₃×#2155	2
C₂×C₁₃×#365	2
C₁₃×#2153	2
C₁₃×#2145	2
C₂×C₁₃×#360	2
#33638	13

Subgroups — C13×#33638 #15719977

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33638 #15719977