Subgroups — C₁₃×#29219 #15715558

Order 3328.

778 conjugacy classes of subgroups.

The subgroup lattice has 328 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2868	C₂	1	1	yes	maximal, characteristic
2	C₄×C₁₃×C₄⋊Q₈	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#2099	C₂	1	4	yes	maximal
4	C₁₃×#1990	C₂	1	2	yes	maximal
5	C₁₃×#1988	C₂	1	4	yes	maximal
6	C₁₃×#1834	C₂	1	2	yes	maximal
7	C₄×C₁₃×C₄⋊C₈	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
9	C₂×C₁₃×C₄⋊Q₈	—	2	1	no	—
10	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
11	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
12	C₂×C₄×Q₈×C₁₃	—	2	1	no	—
13	C₂×C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
15	C₁₃×#391	—	2	1	no	—
16	C₁₃×#389	—	2	1	no	—
17	C₁₃×#389	C₂²	1	2	yes	—
18	C₁₃×#384	—	2	1	no	—
19	C₁₃×#384	—	2	1	no	—
20	C₁₃×#384	C₂²	1	2	yes	—
21	C₁₃×#384	C₂²	1	1	yes	characteristic
22	C₁₃×#384	C₂²	1	2	yes	—
23	C₄×C₁₃×C₄⋊C₄	—	2	1	no	—
24	C₄×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
25	C₄³×C₁₃	C₂²	1	1	yes	characteristic
26	C₁₃×#340	C₂²	1	8	yes	—
27	C₁₃×#340	C₂²	1	8	yes	—
28	C₁₃×#336	C₂²	1	2	yes	—
29	C₂²×Q₈×C₁₃	—	2	1	no	—
30	C₂²×Q₈×C₁₃	—	2	2	no	—
31	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
32	C₁₃×C₄⋊Q₈	—	2	2	no	—
33	C₁₃×C₄⋊Q₈	—	2	8	no	—
34	C₁₃×C₄⋊Q₈	—	4	4	no	—
35	C₁₃×C₄⋊Q₈	—	4	4	no	—
36	C₁₃×C₄⋊Q₈	Q₈	1	4	yes	complemented
37	C₄×Q₈×C₁₃	—	4	4	no	—
38	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
45	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
46	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
47	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
49	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
50	C₂×C₄²×C₁₃	—	2	2	no	—
51	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
52	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
53	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
54	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
55	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
57	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
58	C₁₃×C₄⋊C₈	—	2	4	no	—
59	C₄×C₈×C₁₃	—	2	4	no	—
60	C₁₃×#95	—	2	2	no	—
61	C₁₃×#95	—	2	2	no	—
62	#29219	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
63	C₂×Q₈×C₁₃	—	2	4	no	—
64	C₂×Q₈×C₁₃	—	4	4	no	—
65	C₂×Q₈×C₁₃	—	4	2	no	—
66	C₂×Q₈×C₁₃	—	4	4	no	—
67	C₂×Q₈×C₁₃	—	4	4	no	—
68	C₂×Q₈×C₁₃	—	4	2	no	—
69	C₂²×C₄×C₁₃	—	2	2	no	—
70	C₂²×C₄×C₁₃	—	2	2	no	—
71	C₂²×C₄×C₁₃	—	2	4	no	—
72	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
74	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
79	C₂×C₈×C₁₃	—	2	4	no	—
80	C₂×C₈×C₁₃	—	2	4	no	—
81	C₂×C₈×C₁₃	—	2	4	no	—
82	C₁₃×C₄⋊C₄	—	2	4	no	—
83	C₁₃×C₄⋊C₄	—	2	2	no	—
84	C₁₃×C₄⋊C₄	—	2	8	no	—
85	C₁₃×C₄⋊C₄	—	2	4	no	—
86	C₁₃×C₄⋊C₄	—	4	4	no	—
87	C₁₃×C₄⋊C₄	—	4	4	no	—
88	C₁₃×C₄⋊C₄	—	4	2	no	—
89	C₁₃×C₄⋊C₄	—	4	4	no	—
90	C₁₃×C₄⋊C₄	—	4	4	no	—
91	C₁₃×C₄⋊C₄	—	4	4	no	—
92	C₁₃×C₄⋊C₄	—	4	4	no	—
93	C₁₃×C₄⋊C₄	—	4	2	no	—
94	C₁₃×C₄⋊C₄	C₂×Q₈	1	4	yes	—
95	C₄²×C₁₃	—	2	2	no	—
96	C₄²×C₁₃	—	2	2	no	—
97	C₄²×C₁₃	—	2	2	no	—
98	C₄²×C₁₃	—	4	4	no	—
99	C₄²×C₁₃	D₈	1	4	yes	—
100	C₄²×C₁₃	SD₁₆	1	4	yes	—
101	C₄²×C₁₃	C₂×Q₈	1	2	yes	—
102	C₄²×C₁₃	#41	1	2	yes	—
103	C₄²×C₁₃	#41	1	4	yes	—
104	#2868	C₂×C₁₃	1	1	yes	characteristic
105	C₄×C₄⋊Q₈	C₂×C₁₃	1	1	yes	characteristic
106	#2099	C₂×C₁₃	1	4	yes	—
107	#1990	C₂×C₁₃	1	2	yes	—
108	#1988	C₂×C₁₃	1	4	yes	—
109	#1834	C₂×C₁₃	1	2	yes	—
110	C₄×C₄⋊C₈	C₂×C₁₃	1	1	yes	characteristic
111	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
112	Q₈×C₁₃	—	4	8	no	—
113	Q₈×C₁₃	—	8	8	no	—
114	C₂×C₄×C₁₃	—	2	1	no	—
115	C₂×C₄×C₁₃	—	2	1	no	—
116	C₂×C₄×C₁₃	—	2	1	no	—
117	C₂×C₄×C₁₃	—	2	1	no	—
118	C₂×C₄×C₁₃	—	2	1	no	—
119	C₂×C₄×C₁₃	—	2	1	no	—
120	C₂×C₄×C₁₃	—	2	4	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	—	4	4	no	—
124	C₂×C₄×C₁₃	—	4	2	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	4	4	no	—
127	C₂×C₄×C₁₃	—	4	4	no	—
128	C₂×C₄×C₁₃	—	4	2	no	—
129	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
130	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
131	C₂×C₄×C₁₃	#124	1	2	yes	—
132	C₂×C₄×C₁₃	#124	1	2	yes	—
133	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
134	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
135	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
136	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
137	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
138	C₂×C₄×C₁₃	C₂×SD₁₆	1	2	yes	—
139	C₂×C₄×C₁₃	#137	1	2	yes	—
140	C₂×C₄×C₁₃	#137	1	2	yes	—
141	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
142	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
143	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
144	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
145	C₈×C₁₃	—	4	4	no	—
146	C₂×C₄⋊Q₈	—	2	1	no	—
147	C₂×C₄⋊Q₈	—	2	1	no	—
148	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
149	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
150	C₂×C₄×Q₈	—	2	1	no	—
151	C₂×C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
152	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
153	#391	—	2	1	no	—
154	#389	—	2	1	no	—
155	#389	C₂²×C₁₃	1	2	yes	—
156	#384	—	2	1	no	—
157	#384	—	2	1	no	—
158	#384	C₂²×C₁₃	1	1	yes	characteristic
159	#384	C₂²×C₁₃	1	2	yes	—
160	#384	C₂²×C₁₃	1	2	yes	—
161	C₄×C₄⋊C₄	—	2	1	no	—
162	C₄×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
163	C₄³	C₂²×C₁₃	1	1	yes	characteristic
164	#340	C₂²×C₁₃	1	8	yes	—
165	#340	C₂²×C₁₃	1	8	yes	—
166	#336	C₂²×C₁₃	1	2	yes	—
167	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
168	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
169	C₂²×C₁₃	#536	1	1	yes	characteristic
170	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
171	C₂²×C₁₃	C₂²×SD₁₆	1	1	yes	characteristic
172	C₂²×C₁₃	C₂×#137	1	1	yes	characteristic
173	C₂²×C₁₃	C₂×#137	1	1	yes	characteristic
174	C₄×C₁₃	—	2	2	no	—
175	C₄×C₁₃	—	2	2	no	—
176	C₄×C₁₃	—	2	2	no	—
177	C₄×C₁₃	—	4	4	no	—
178	C₄×C₁₃	—	8	4	no	—
179	C₄×C₁₃	—	8	2	no	—
180	C₄×C₁₃	#474	1	4	yes	—
181	C₄×C₁₃	#475	1	4	yes	—
182	C₄×C₁₃	#486	1	4	yes	—
183	C₄×C₁₃	#556	1	2	yes	—
184	C₄×C₁₃	#557	1	2	yes	—
185	C₂²×Q₈	—	2	1	no	—
186	C₂²×Q₈	—	2	2	no	—
187	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
188	C₄⋊Q₈	—	2	2	no	—
189	C₄⋊Q₈	—	2	8	no	—
190	C₄⋊Q₈	—	4	4	no	—
191	C₄⋊Q₈	—	4	4	no	—
192	C₄⋊Q₈	Q₈×C₁₃	1	4	yes	complemented
193	C₄×Q₈	—	4	4	no	—
194	C₂×C₄⋊C₄	—	2	1	no	—
195	C₂×C₄⋊C₄	—	2	1	no	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	—	2	2	no	—
198	C₂×C₄⋊C₄	—	2	2	no	—
199	C₂×C₄⋊C₄	—	2	2	no	—
200	C₂×C₄⋊C₄	—	2	2	no	—
201	C₂×C₄⋊C₄	—	2	2	no	—
202	C₂×C₄⋊C₄	—	2	1	no	—
203	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
204	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
205	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
206	C₂×C₄²	—	2	2	no	—
207	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
208	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
209	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
210	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
211	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
212	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
213	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
214	C₄⋊C₈	—	2	4	no	—
215	C₄×C₈	—	2	4	no	—
216	#95	—	2	2	no	—
217	#95	—	2	2	no	—
218	C₂×C₁₃	#2568	1	1	yes	characteristic
219	C₂×C₁₃	C₂×#474	1	1	yes	characteristic
220	C₂×C₁₃	C₂×#475	1	1	yes	characteristic
221	C₂×C₁₃	C₂×#486	1	1	yes	characteristic
222	C₂×C₁₃	#3139	1	1	yes	characteristic
223	C₂×C₁₃	#3226	1	1	yes	characteristic
224	C₂×C₁₃	#3229	1	1	yes	characteristic
225	C₂×Q₈	—	2	4	no	—
226	C₂×Q₈	—	4	4	no	—
227	C₂×Q₈	—	4	2	no	—
228	C₂×Q₈	—	4	4	no	—
229	C₂×Q₈	—	4	4	no	—
230	C₂×Q₈	—	4	2	no	—
231	C₂²×C₄	—	2	2	no	—
232	C₂²×C₄	—	2	2	no	—
233	C₂²×C₄	—	2	4	no	—
234	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
235	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
236	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
237	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
238	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
239	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
240	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
241	C₂×C₈	—	2	4	no	—
242	C₂×C₈	—	2	4	no	—
243	C₂×C₈	—	2	4	no	—
244	C₄⋊C₄	—	2	4	no	—
245	C₄⋊C₄	—	2	2	no	—
246	C₄⋊C₄	—	2	8	no	—
247	C₄⋊C₄	—	2	4	no	—
248	C₄⋊C₄	—	4	4	no	—
249	C₄⋊C₄	—	4	4	no	—
250	C₄⋊C₄	—	4	2	no	—
251	C₄⋊C₄	—	4	4	no	—
252	C₄⋊C₄	—	4	4	no	—
253	C₄⋊C₄	—	4	4	no	—
254	C₄⋊C₄	—	4	4	no	—
255	C₄⋊C₄	—	4	2	no	—
256	C₄⋊C₄	C₂×Q₈×C₁₃	1	4	yes	—
257	C₄²	—	2	2	no	—
258	C₄²	—	2	2	no	—
259	C₄²	—	2	2	no	—
260	C₄²	—	4	4	no	—
261	C₄²	C₁₃×D₈	1	4	yes	—
262	C₄²	C₁₃×SD₁₆	1	4	yes	—
263	C₄²	C₂×Q₈×C₁₃	1	2	yes	—
264	C₄²	C₁₃×#41	1	2	yes	—
265	C₄²	C₁₃×#41	1	4	yes	—
266	C₁₃	#29219	1	1	yes	characteristic, direct product factor, Sylow
267	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
268	Q₈	—	4	8	no	—
269	Q₈	—	8	8	no	—
270	C₂×C₄	—	2	1	no	—
271	C₂×C₄	—	2	1	no	—
272	C₂×C₄	—	2	1	no	—
273	C₂×C₄	—	2	1	no	—
274	C₂×C₄	—	2	1	no	—
275	C₂×C₄	—	2	1	no	—
276	C₂×C₄	—	2	4	no	—
277	C₂×C₄	—	4	2	no	—
278	C₂×C₄	—	4	2	no	—
279	C₂×C₄	—	4	4	no	—
280	C₂×C₄	—	4	2	no	—
281	C₂×C₄	—	4	2	no	—
282	C₂×C₄	—	4	4	no	—
283	C₂×C₄	—	4	4	no	—
284	C₂×C₄	—	4	2	no	—
285	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
286	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
287	C₂×C₄	C₁₃×#124	1	2	yes	—
288	C₂×C₄	C₁₃×#124	1	2	yes	—
289	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
290	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
291	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
292	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
293	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
294	C₂×C₄	C₂×C₁₃×SD₁₆	1	2	yes	—
295	C₂×C₄	C₁₃×#137	1	2	yes	—
296	C₂×C₄	C₁₃×#137	1	2	yes	—
297	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
298	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
299	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
300	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
301	C₈	—	4	4	no	—
302	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
303	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
304	C₂²	C₁₃×#536	1	1	yes	characteristic
305	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
306	C₂²	C₂²×C₁₃×SD₁₆	1	1	yes	characteristic
307	C₂²	C₂×C₁₃×#137	1	1	yes	characteristic
308	C₂²	C₂×C₁₃×#137	1	1	yes	characteristic
309	C₄	—	2	2	no	—
310	C₄	—	2	2	no	—
311	C₄	—	2	2	no	—
312	C₄	—	4	4	no	—
313	C₄	—	8	4	no	—
314	C₄	—	8	2	no	—
315	C₄	C₁₃×#474	1	4	yes	—
316	C₄	C₁₃×#475	1	4	yes	—
317	C₄	C₁₃×#486	1	4	yes	—
318	C₄	C₁₃×#556	1	2	yes	—
319	C₄	C₁₃×#557	1	2	yes	—
320	C₂	C₁₃×#2568	1	1	yes	characteristic
321	C₂	C₂×C₁₃×#474	1	1	yes	characteristic
322	C₂	C₂×C₁₃×#475	1	1	yes	characteristic
323	C₂	C₂×C₁₃×#486	1	1	yes	characteristic
324	C₂	C₁₃×#3139	1	1	yes	characteristic
325	C₂	C₁₃×#3226	1	1	yes	characteristic
326	C₂	C₁₃×#3229	1	1	yes	characteristic

	subgroup	G/H
center, socle, prime radical	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₄³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
36. C₁₃×C₄⋊Q₈	Q₈	semidirect
62. #29219	C₁₃	direct (N × H)
192. C₄⋊Q₈	Q₈×C₁₃	semidirect
266. C₁₃	#29219	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×#2868	2
C₄×C₁₃×C₄⋊Q₈	2
C₁₃×#2099	2
C₁₃×#1990	2
C₁₃×#1988	2
C₁₃×#1834	2
C₄×C₁₃×C₄⋊C₈	2
#29219	13

Subgroups — C13×#29219 #15715558

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#29219 #15715558