Subgroups — C₁₃×#29154 #15715493

Order 3328.

612 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#3160	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#443	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2031	C₂	1	4	yes	maximal, complemented
4	C₁₃×#1976	C₂	1	2	yes	maximal
5	C₁₃×#1833	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1814	C₂	1	2	yes	maximal, complemented
7	C₁₃×#1812	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1772	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×#135	C₂²	1	1	yes	characteristic
10	C₂×C₁₃×#131	C₂²	1	2	yes	—
11	C₁₃×#533	C₂²	1	2	yes	—
12	C₁₃×#500	—	2	1	no	—
13	C₁₃×#499	—	2	1	no	—
14	C₁₃×#499	—	2	1	no	—
15	C₁₃×#498	—	2	1	no	—
16	C₁₃×#443	—	2	1	no	—
17	C₁₃×#443	—	2	1	no	—
18	C₁₃×#443	—	2	1	no	—
19	C₁₃×#443	—	2	1	no	—
20	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
21	C₂×C₁₃×#108	C₂²	1	1	yes	characteristic
22	C₁₃×#427	C₂²	1	1	yes	characteristic
23	C₁₃×#427	C₂²	1	2	yes	—
24	C₁₃×#427	C₂²	1	2	yes	—
25	C₁₃×#421	C₂²	1	4	yes	—
26	C₂×C₁₃×#104	C₂²	1	2	yes	—
27	C₁₃×#410	C₂²	1	4	yes	—
28	C₁₃×#405	C₂²	1	2	yes	—
29	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
30	C₁₃×#344	C₂²	1	4	yes	—
31	C₁₃×#339	C₂²	1	2	yes	—
32	C₁₃×H(3,ℤ/4)	C₂²	1	4	yes	—
33	C₂×C₁₃×#41	C₂³	1	2	yes	—
34	C₁₃×#135	—	2	1	no	—
35	C₁₃×#135	—	2	1	no	—
36	C₁₃×#135	—	2	1	no	—
37	C₁₃×#135	—	2	1	no	—
38	C₂×C₁₃×Q₁₆	Q₈	1	1	yes	characteristic, complemented
39	C₂×C₁₃×SD₁₆	Q₈	1	2	yes	complemented
40	C₂×C₁₃×D₈	Q₈	1	1	yes	characteristic, complemented
41	C₁₃×#131	—	2	2	no	—
42	C₁₃×#131	—	2	2	no	—
43	C₁₃×#131	—	2	2	no	—
44	C₁₃×#131	—	2	2	no	—
45	C₂×C₁₃×M₁₆	—	2	2	no	—
46	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
47	C₂×C₁₃×M₁₆	C₂³	1	2	yes	—
48	C₂²×C₈×C₁₃	—	2	2	no	—
49	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
51	C₁₃×C₄⋊Q₈	—	2	2	no	—
52	C₁₃×#125	—	2	2	no	—
53	C₁₃×C₂²⋊Q₈	—	4	2	no	—
54	C₄×Q₈×C₁₃	—	4	2	no	—
55	C₁₃×#117	C₂³	1	2	yes	—
56	C₁₃×#117	C₂³	1	4	yes	—
57	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
58	C₁₃×#108	—	2	1	no	—
59	C₁₃×#108	—	2	1	no	—
60	C₁₃×#108	—	2	1	no	—
61	C₁₃×#108	—	2	1	no	—
62	C₁₃×#107	—	2	2	no	—
63	C₁₃×#107	—	2	2	no	—
64	C₁₃×#107	—	2	1	no	—
65	C₁₃×#106	—	2	2	no	—
66	C₁₃×#106	—	2	2	no	—
67	C₁₃×#106	—	2	1	no	—
68	C₁₃×#104	—	2	2	no	—
69	C₁₃×#104	—	2	2	no	—
70	C₁₃×#104	—	4	4	no	—
71	C₁₃×#99	—	4	4	no	—
72	C₁₃×#97	—	2	2	no	—
73	C₄×C₈×C₁₃	—	2	2	no	—
74	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
75	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
76	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
77	C₄×C₈×C₁₃	D₄	1	1	yes	characteristic
78	#29154	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
79	C₁₃×#41	—	2	2	no	—
80	C₁₃×#41	—	2	2	no	—
81	C₁₃×#41	—	2	2	no	—
82	C₁₃×#41	—	2	2	no	—
83	C₂×Q₈×C₁₃	—	4	2	no	—
84	C₂×Q₈×C₁₃	C₂×Q₈	1	2	yes	—
85	C₂×D₄×C₁₃	—	2	2	no	—
86	C₂×D₄×C₁₃	C₂×Q₈	1	2	yes	—
87	C₂²×C₄×C₁₃	—	2	1	no	—
88	C₂²×C₄×C₁₃	—	2	2	no	—
89	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₁₃×Q₁₆	—	2	1	no	—
91	C₁₃×Q₁₆	—	2	1	no	—
92	C₁₃×SD₁₆	—	2	2	no	—
93	C₁₃×SD₁₆	—	2	2	no	—
94	C₁₃×D₈	—	2	1	no	—
95	C₁₃×D₈	—	2	1	no	—
96	C₁₃×M₁₆	—	2	2	no	—
97	C₁₃×M₁₆	—	2	2	no	—
98	C₁₃×M₁₆	—	2	2	no	—
99	C₁₃×M₁₆	—	2	2	no	—
100	C₁₃×M₁₆	—	4	2	no	—
101	C₁₃×M₁₆	—	4	2	no	—
102	C₂×C₈×C₁₃	—	2	2	no	—
103	C₂×C₈×C₁₃	—	2	2	no	—
104	C₂×C₈×C₁₃	—	4	2	no	—
105	C₂×C₈×C₁₃	—	4	2	no	—
106	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
110	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
111	C₂×C₈×C₁₃	C₂×Q₈	1	1	yes	characteristic
112	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
113	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
114	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
115	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
116	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
117	C₂×C₈×C₁₃	#41	1	1	yes	characteristic
118	C₁₃×C₄⋊C₄	—	2	4	no	—
119	C₁₃×C₄⋊C₄	—	2	2	no	—
120	C₁₃×C₄⋊C₄	—	4	4	no	—
121	C₁₃×C₄⋊C₄	—	4	2	no	—
122	C₁₃×C₂²⋊C₄	—	2	2	no	—
123	C₁₃×C₂²⋊C₄	—	2	4	no	—
124	C₄²×C₁₃	—	2	1	no	—
125	C₄²×C₁₃	—	2	2	no	—
126	C₄²×C₁₃	—	2	4	no	—
127	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
128	C₄²×C₁₃	C₂×D₄	1	1	yes	characteristic
129	#3160	C₂×C₁₃	1	1	yes	characteristic, complemented
130	C₂×#443	C₂×C₁₃	1	1	yes	characteristic
131	#2031	C₂×C₁₃	1	4	yes	complemented
132	#1976	C₂×C₁₃	1	2	yes	—
133	#1833	C₂×C₁₃	1	2	yes	complemented
134	#1814	C₂×C₁₃	1	2	yes	complemented
135	#1812	C₂×C₁₃	1	2	yes	complemented
136	#1772	C₂×C₁₃	1	1	yes	characteristic, complemented
137	C₂³×C₁₃	—	2	2	no	—
138	C₂³×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
139	Q₈×C₁₃	—	2	2	no	—
140	Q₈×C₁₃	—	2	2	no	—
141	Q₈×C₁₃	—	8	4	no	—
142	D₄×C₁₃	—	2	2	no	—
143	D₄×C₁₃	—	2	2	no	—
144	D₄×C₁₃	—	4	2	no	—
145	D₄×C₁₃	—	4	2	no	—
146	C₂×C₄×C₁₃	—	2	1	no	—
147	C₂×C₄×C₁₃	—	2	1	no	—
148	C₂×C₄×C₁₃	—	2	2	no	—
149	C₂×C₄×C₁₃	—	4	2	no	—
150	C₂×C₄×C₁₃	—	4	2	no	—
151	C₂×C₄×C₁₃	—	4	2	no	—
152	C₂×C₄×C₁₃	—	4	4	no	—
153	C₂×C₄×C₁₃	—	4	2	no	—
154	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
155	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
156	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
157	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
158	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
159	C₂×C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
160	C₈×C₁₃	—	4	2	no	—
161	C₈×C₁₃	—	4	2	no	—
162	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
163	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
164	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
165	C₈×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
166	C₈×C₁₃	#124	1	1	yes	characteristic
167	C₈×C₁₃	#124	1	1	yes	characteristic
168	C₈×C₁₃	#124	1	1	yes	characteristic
169	C₈×C₁₃	#124	1	1	yes	characteristic
170	C₂×#135	C₂²×C₁₃	1	1	yes	characteristic
171	C₂×#131	C₂²×C₁₃	1	2	yes	—
172	#533	C₂²×C₁₃	1	2	yes	—
173	#500	—	2	1	no	—
174	#499	—	2	1	no	—
175	#499	—	2	1	no	—
176	#498	—	2	1	no	—
177	#443	—	2	1	no	—
178	#443	—	2	1	no	—
179	#443	—	2	1	no	—
180	#443	—	2	1	no	—
181	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
182	C₂×#108	C₂²×C₁₃	1	1	yes	characteristic
183	#427	C₂²×C₁₃	1	1	yes	characteristic
184	#427	C₂²×C₁₃	1	2	yes	—
185	#427	C₂²×C₁₃	1	2	yes	—
186	#421	C₂²×C₁₃	1	4	yes	—
187	C₂×#104	C₂²×C₁₃	1	2	yes	—
188	#410	C₂²×C₁₃	1	4	yes	—
189	#405	C₂²×C₁₃	1	2	yes	—
190	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
191	#344	C₂²×C₁₃	1	4	yes	—
192	#339	C₂²×C₁₃	1	2	yes	—
193	H(3,ℤ/4)	C₂²×C₁₃	1	4	yes	—
194	C₂²×C₁₃	—	4	1	no	—
195	C₂²×C₁₃	—	4	2	no	—
196	C₂²×C₁₃	—	8	2	no	—
197	C₂²×C₁₃	#535	1	1	yes	characteristic
198	C₂²×C₁₃	#554	1	1	yes	characteristic
199	C₂²×C₁₃	#555	1	1	yes	characteristic
200	C₄×C₁₃	—	2	2	no	—
201	C₄×C₁₃	—	4	1	no	—
202	C₄×C₁₃	—	4	2	no	—
203	C₄×C₁₃	—	8	2	no	—
204	C₄×C₁₃	—	8	4	no	—
205	C₄×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
206	C₄×C₁₃	C₂×#124	1	1	yes	characteristic
207	C₄×C₁₃	#554	1	1	yes	characteristic
208	C₄×C₁₃	#555	1	1	yes	characteristic
209	C₂×#41	C₂³×C₁₃	1	2	yes	—
210	#135	—	2	1	no	—
211	#135	—	2	1	no	—
212	#135	—	2	1	no	—
213	#135	—	2	1	no	—
214	C₂×Q₁₆	Q₈×C₁₃	1	1	yes	characteristic, complemented
215	C₂×SD₁₆	Q₈×C₁₃	1	2	yes	complemented
216	C₂×D₈	Q₈×C₁₃	1	1	yes	characteristic, complemented
217	#131	—	2	2	no	—
218	#131	—	2	2	no	—
219	#131	—	2	2	no	—
220	#131	—	2	2	no	—
221	C₂×M₁₆	—	2	2	no	—
222	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
223	C₂×M₁₆	C₂³×C₁₃	1	2	yes	—
224	C₂²×C₈	—	2	2	no	—
225	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
226	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
227	C₄⋊Q₈	—	2	2	no	—
228	#125	—	2	2	no	—
229	C₂²⋊Q₈	—	4	2	no	—
230	C₄×Q₈	—	4	2	no	—
231	#117	C₂³×C₁₃	1	4	yes	—
232	#117	C₂³×C₁₃	1	2	yes	—
233	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
234	#108	—	2	1	no	—
235	#108	—	2	1	no	—
236	#108	—	2	1	no	—
237	#108	—	2	1	no	—
238	#107	—	2	2	no	—
239	#107	—	2	2	no	—
240	#107	—	2	1	no	—
241	#106	—	2	2	no	—
242	#106	—	2	2	no	—
243	#106	—	2	1	no	—
244	#104	—	2	2	no	—
245	#104	—	2	2	no	—
246	#104	—	4	4	no	—
247	#99	—	4	4	no	—
248	#97	—	2	2	no	—
249	C₄×C₈	—	2	2	no	—
250	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
251	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
252	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
253	C₄×C₈	D₄×C₁₃	1	1	yes	characteristic
254	C₂×C₁₃	—	2	1	no	—
255	C₂×C₁₃	—	2	1	no	—
256	C₂×C₁₃	—	2	1	no	—
257	C₂×C₁₃	—	8	2	no	—
258	C₂×C₁₃	#2563	1	1	yes	characteristic
259	#41	—	2	2	no	—
260	#41	—	2	2	no	—
261	#41	—	2	2	no	—
262	#41	—	2	2	no	—
263	C₂×Q₈	—	4	2	no	—
264	C₂×Q₈	C₂×Q₈×C₁₃	1	2	yes	—
265	C₂×D₄	—	2	2	no	—
266	C₂×D₄	C₂×Q₈×C₁₃	1	2	yes	—
267	C₂²×C₄	—	2	1	no	—
268	C₂²×C₄	—	2	2	no	—
269	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
270	Q₁₆	—	2	1	no	—
271	Q₁₆	—	2	1	no	—
272	SD₁₆	—	2	2	no	—
273	SD₁₆	—	2	2	no	—
274	D₈	—	2	1	no	—
275	D₈	—	2	1	no	—
276	M₁₆	—	2	2	no	—
277	M₁₆	—	2	2	no	—
278	M₁₆	—	2	2	no	—
279	M₁₆	—	2	2	no	—
280	M₁₆	—	4	2	no	—
281	M₁₆	—	4	2	no	—
282	C₂×C₈	—	2	2	no	—
283	C₂×C₈	—	2	2	no	—
284	C₂×C₈	—	4	2	no	—
285	C₂×C₈	—	4	2	no	—
286	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
287	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
288	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
289	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
290	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
291	C₂×C₈	C₂×Q₈×C₁₃	1	1	yes	characteristic
292	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
293	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
294	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
295	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
296	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
297	C₂×C₈	C₁₃×#41	1	1	yes	characteristic
298	C₄⋊C₄	—	2	4	no	—
299	C₄⋊C₄	—	2	2	no	—
300	C₄⋊C₄	—	4	4	no	—
301	C₄⋊C₄	—	4	2	no	—
302	C₂²⋊C₄	—	2	2	no	—
303	C₂²⋊C₄	—	2	4	no	—
304	C₄²	—	2	1	no	—
305	C₄²	—	2	2	no	—
306	C₄²	—	2	4	no	—
307	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
308	C₄²	C₂×D₄×C₁₃	1	1	yes	characteristic
309	C₁₃	#29154	1	1	yes	characteristic, direct product factor, Sylow
310	C₂³	—	2	2	no	—
311	C₂³	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
312	Q₈	—	2	2	no	—
313	Q₈	—	2	2	no	—
314	Q₈	—	8	4	no	—
315	D₄	—	2	2	no	—
316	D₄	—	2	2	no	—
317	D₄	—	4	2	no	—
318	D₄	—	4	2	no	—
319	C₂×C₄	—	2	1	no	—
320	C₂×C₄	—	2	1	no	—
321	C₂×C₄	—	2	2	no	—
322	C₂×C₄	—	4	2	no	—
323	C₂×C₄	—	4	2	no	—
324	C₂×C₄	—	4	2	no	—
325	C₂×C₄	—	4	4	no	—
326	C₂×C₄	—	4	2	no	—
327	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
328	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
329	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
330	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
331	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
332	C₂×C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
333	C₈	—	4	2	no	—
334	C₈	—	4	2	no	—
335	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
336	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
337	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
338	C₈	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
339	C₈	C₁₃×#124	1	1	yes	characteristic
340	C₈	C₁₃×#124	1	1	yes	characteristic
341	C₈	C₁₃×#124	1	1	yes	characteristic
342	C₈	C₁₃×#124	1	1	yes	characteristic
343	C₂²	—	4	1	no	—
344	C₂²	—	4	2	no	—
345	C₂²	—	8	2	no	—
346	C₂²	C₁₃×#535	1	1	yes	characteristic
347	C₂²	C₁₃×#554	1	1	yes	characteristic
348	C₂²	C₁₃×#555	1	1	yes	characteristic
349	C₄	—	2	2	no	—
350	C₄	—	4	1	no	—
351	C₄	—	4	2	no	—
352	C₄	—	8	2	no	—
353	C₄	—	8	4	no	—
354	C₄	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
355	C₄	C₂×C₁₃×#124	1	1	yes	characteristic
356	C₄	C₁₃×#554	1	1	yes	characteristic
357	C₄	C₁₃×#555	1	1	yes	characteristic
358	C₂	—	2	1	no	—
359	C₂	—	2	1	no	—
360	C₂	—	2	1	no	—
361	C₂	—	8	2	no	—
362	C₂	C₁₃×#2563	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₁₃	#555
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
socle	C₂×C₁₃	#2563
prime radical	C₂×C₁₃×#135	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₁₃ ◁ C₂²×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#3160	C₂	semidirect
3. C₁₃×#2031	C₂	semidirect
5. C₁₃×#1833	C₂	semidirect
6. C₁₃×#1814	C₂	semidirect
7. C₁₃×#1812	C₂	semidirect
8. C₁₃×#1772	C₂	semidirect
38. C₂×C₁₃×Q₁₆	Q₈	semidirect
39. C₂×C₁₃×SD₁₆	Q₈	semidirect
40. C₂×C₁₃×D₈	Q₈	semidirect
78. #29154	C₁₃	direct (N × H)
129. #3160	C₂×C₁₃	semidirect
131. #2031	C₂×C₁₃	semidirect
133. #1833	C₂×C₁₃	semidirect
134. #1814	C₂×C₁₃	semidirect
135. #1812	C₂×C₁₃	semidirect
136. #1772	C₂×C₁₃	semidirect
214. C₂×Q₁₆	Q₈×C₁₃	semidirect
215. C₂×SD₁₆	Q₈×C₁₃	semidirect
216. C₂×D₈	Q₈×C₁₃	semidirect
309. C₁₃	#29154	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×#3160	2
C₂×C₁₃×#443	2
C₁₃×#2031	2
C₁₃×#1976	2
C₁₃×#1833	2
C₁₃×#1814	2
C₁₃×#1812	2
C₁₃×#1772	2
#29154	13

Subgroups — C13×#29154 #15715493

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29154 #15715493