Subgroups — C₁₃×#17342 #15703681

Order 3328.

688 conjugacy classes of subgroups.

The subgroup lattice has 300 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#2965	C₂	1	2	yes	maximal
2	C₁₃×#2922	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#1455	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#1454	C₂	1	2	yes	maximal, complemented
5	C₁₃×#1453	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1453	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#1452	C₂	1	2	yes	maximal, complemented
8	C₁₃×#1451	C₂	1	2	yes	maximal, complemented
9	C₁₃×#1451	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₃×#41	C₂²	1	1	yes	characteristic
11	C₁₃×#446	C₄	1	4	yes	—
12	C₁₃×#436	C₄	1	4	yes	—
13	C₁₃×#435	C₄	1	4	yes	—
14	C₁₃×#435	C₄	1	4	yes	—
15	C₁₃×#433	C₄	1	4	yes	—
16	C₁₃×#431	C₂²	1	2	yes	—
17	C₁₃×#431	C₂²	1	2	yes	—
18	C₁₃×#431	C₂²	1	2	yes	—
19	C₁₃×#431	C₂²	1	2	yes	—
20	C₁₃×#431	C₂²	1	2	yes	—
21	C₁₃×#424	C₄	1	2	yes	—
22	C₁₃×#408	C₄	1	2	yes	—
23	C₄×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
24	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
25	C₄×C₁₃×M₁₆	C₂²	1	2	yes	—
26	C₂×C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
27	C₂×C₄×C₈×C₁₃	C₂²	1	2	yes	—
28	C₁₃×#322	C₂²	1	2	yes	—
29	C₁₃×#322	C₂²	1	2	yes	—
30	C₁₃×#322	C₂²	1	1	yes	characteristic
31	C₁₃×#322	C₂²	1	2	yes	—
32	C₁₃×#322	C₂²	1	2	yes	—
33	C₁₃×#322	C₂²	1	2	yes	—
34	C₁₃×#322	C₂²	1	1	yes	characteristic
35	C₁₃×#322	C₂²	1	2	yes	—
36	C₈²×C₁₃	C₂²	1	1	yes	characteristic
37	C₈²×C₁₃	C₂²	1	1	yes	characteristic
38	C₂×C₁₃×#41	C₂×C₄	1	1	yes	characteristic
39	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
40	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
41	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
42	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
43	C₂²×C₈×C₁₃	C₂×C₄	1	4	yes	—
44	C₄×Q₈×C₁₃	C₂×C₄	1	2	yes	—
45	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
46	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
47	C₄×D₄×C₁₃	C₂×C₄	1	2	yes	—
48	C₁₃×#117	C₂×C₄	1	1	yes	characteristic
49	C₁₃×#117	C₂×C₄	1	2	yes	—
50	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
52	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
53	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
54	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
55	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
56	C₁₃×C₄⋊C₈	C₂×C₄	1	4	yes	—
57	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
58	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
59	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
60	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
61	C₁₃×C₂²⋊C₈	C₂×C₄	1	4	yes	—
62	C₁₃×#97	C₂×C₄	1	2	yes	—
63	C₁₃×#97	C₂×C₄	1	4	yes	—
64	C₁₃×#97	C₂×C₄	1	4	yes	—
65	C₄×C₈×C₁₃	C₂×C₄	1	2	yes	—
66	C₄×C₈×C₁₃	C₂×C₄	1	4	yes	—
67	C₄×C₈×C₁₃	C₂³	1	2	yes	—
68	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
69	C₄×C₈×C₁₃	C₂³	1	1	yes	characteristic
70	C₄×C₈×C₁₃	C₂³	1	2	yes	—
71	C₄×C₈×C₁₃	C₂³	1	2	yes	—
72	C₄×C₈×C₁₃	C₂³	1	2	yes	—
73	C₄×C₈×C₁₃	C₂³	1	2	yes	—
74	#17342	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₁₃×#41	—	4	2	no	—
76	C₂×Q₈×C₁₃	C₄²	1	1	yes	characteristic
77	C₂×D₄×C₁₃	C₄²	1	1	yes	characteristic
78	C₂×D₄×C₁₃	C₄²	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
80	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
81	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
83	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
84	C₁₃×M₁₆	—	4	4	no	—
85	C₁₃×M₁₆	—	4	4	no	—
86	C₁₃×M₁₆	—	4	2	no	—
87	C₂×C₈×C₁₃	—	4	2	no	—
88	C₂×C₈×C₁₃	—	4	4	no	—
89	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
90	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
91	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
92	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
93	C₂×C₈×C₁₃	C₂²×C₄	1	2	yes	—
94	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
95	C₂×C₈×C₁₃	C₂²×C₄	1	4	yes	—
96	C₁₃×C₄⋊C₄	C₄²	1	2	yes	—
97	C₁₃×C₄⋊C₄	C₄²	1	2	yes	—
98	C₁₃×C₄⋊C₄	C₄²	1	2	yes	—
99	C₁₃×C₂²⋊C₄	C₄²	1	1	yes	characteristic
100	C₁₃×C₂²⋊C₄	C₄²	1	1	yes	characteristic
101	C₁₃×C₂²⋊C₄	C₄²	1	2	yes	—
102	C₁₃×C₂²⋊C₄	C₄²	1	2	yes	—
103	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
104	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
105	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
106	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
107	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
108	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
109	#2965	C₂×C₁₃	1	2	yes	—
110	#2922	C₂×C₁₃	1	1	yes	characteristic
111	#1455	C₂×C₁₃	1	1	yes	characteristic, complemented
112	#1454	C₂×C₁₃	1	2	yes	complemented
113	#1453	C₂×C₁₃	1	2	yes	complemented
114	#1453	C₂×C₁₃	1	1	yes	characteristic, complemented
115	#1452	C₂×C₁₃	1	2	yes	complemented
116	#1451	C₂×C₁₃	1	2	yes	complemented
117	#1451	C₂×C₁₃	1	2	yes	complemented
118	C₂³×C₁₃	C₂×C₄²	1	2	yes	—
119	C₂³×C₁₃	C₂×C₄²	1	1	yes	characteristic
120	Q₈×C₁₃	—	4	1	no	—
121	D₄×C₁₃	—	4	1	no	—
122	D₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	4	no	—
125	C₂×C₄×C₁₃	—	2	4	no	—
126	C₂×C₄×C₁₃	—	2	4	no	—
127	C₂×C₄×C₁₃	—	2	4	no	—
128	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
130	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
131	C₂×C₄×C₁₃	C₂×C₄²	1	2	yes	—
132	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
133	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
134	C₂×C₄×C₁₃	C₂³×C₄	1	2	yes	—
135	C₈×C₁₃	—	2	4	no	—
136	C₈×C₁₃	—	2	4	no	—
137	C₈×C₁₃	—	2	4	no	—
138	C₈×C₁₃	—	2	2	no	—
139	C₈×C₁₃	—	2	2	no	—
140	C₈×C₁₃	—	2	4	no	—
141	C₈×C₁₃	—	2	4	no	—
142	C₄×#41	C₂²×C₁₃	1	1	yes	characteristic
143	#446	C₄×C₁₃	1	4	yes	—
144	#436	C₄×C₁₃	1	4	yes	—
145	#435	C₄×C₁₃	1	4	yes	—
146	#435	C₄×C₁₃	1	4	yes	—
147	#433	C₄×C₁₃	1	4	yes	—
148	#431	C₂²×C₁₃	1	2	yes	—
149	#431	C₂²×C₁₃	1	2	yes	—
150	#431	C₂²×C₁₃	1	2	yes	—
151	#431	C₂²×C₁₃	1	2	yes	—
152	#431	C₂²×C₁₃	1	2	yes	—
153	#424	C₄×C₁₃	1	2	yes	—
154	#408	C₄×C₁₃	1	2	yes	—
155	C₄×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
156	C₄×M₁₆	C₂²×C₁₃	1	2	yes	—
157	C₄×M₁₆	C₂²×C₁₃	1	2	yes	—
158	C₂×C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
159	C₂×C₄×C₈	C₂²×C₁₃	1	2	yes	—
160	#322	C₂²×C₁₃	1	2	yes	—
161	#322	C₂²×C₁₃	1	2	yes	—
162	#322	C₂²×C₁₃	1	1	yes	characteristic
163	#322	C₂²×C₁₃	1	2	yes	—
164	#322	C₂²×C₁₃	1	2	yes	—
165	#322	C₂²×C₁₃	1	2	yes	—
166	#322	C₂²×C₁₃	1	1	yes	characteristic
167	#322	C₂²×C₁₃	1	2	yes	—
168	C₈²	C₂²×C₁₃	1	1	yes	characteristic
169	C₈²	C₂²×C₁₃	1	1	yes	characteristic
170	C₂²×C₁₃	—	2	1	no	—
171	C₂²×C₁₃	—	2	2	no	—
172	C₂²×C₁₃	—	2	2	no	—
173	C₂²×C₁₃	—	2	2	no	—
174	C₂²×C₁₃	—	2	2	no	—
175	C₂²×C₁₃	C₂²×C₄²	1	1	yes	characteristic
176	C₄×C₁₃	—	2	1	no	—
177	C₄×C₁₃	—	2	2	no	—
178	C₄×C₁₃	—	2	2	no	—
179	C₄×C₁₃	—	2	2	no	—
180	C₄×C₁₃	—	2	2	no	—
181	C₄×C₁₃	#568	1	2	yes	—
182	C₄×C₁₃	#568	1	4	yes	—
183	C₂×#41	C₂×C₄×C₁₃	1	1	yes	characteristic
184	C₂×M₁₆	C₂×C₄×C₁₃	1	2	yes	—
185	C₂×M₁₆	C₂×C₄×C₁₃	1	4	yes	—
186	C₂×M₁₆	C₂×C₄×C₁₃	1	4	yes	—
187	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
188	C₂²×C₈	C₂×C₄×C₁₃	1	4	yes	—
189	C₄×Q₈	C₂×C₄×C₁₃	1	2	yes	—
190	C₄×D₄	C₂×C₄×C₁₃	1	2	yes	—
191	C₄×D₄	C₂×C₄×C₁₃	1	2	yes	—
192	C₄×D₄	C₂×C₄×C₁₃	1	2	yes	—
193	#117	C₂×C₄×C₁₃	1	1	yes	characteristic
194	#117	C₂×C₄×C₁₃	1	2	yes	—
195	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
196	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
197	C₄⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
198	C₄⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
199	C₄⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
200	C₄⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
201	C₄⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
202	C₂²⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
203	C₂²⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
204	C₂²⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
205	C₂²⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
206	C₂²⋊C₈	C₂×C₄×C₁₃	1	4	yes	—
207	#97	C₂×C₄×C₁₃	1	2	yes	—
208	#97	C₂×C₄×C₁₃	1	4	yes	—
209	#97	C₂×C₄×C₁₃	1	4	yes	—
210	C₄×C₈	C₂×C₄×C₁₃	1	2	yes	—
211	C₄×C₈	C₂×C₄×C₁₃	1	4	yes	—
212	C₄×C₈	C₂³×C₁₃	1	2	yes	—
213	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
214	C₄×C₈	C₂³×C₁₃	1	1	yes	characteristic
215	C₄×C₈	C₂³×C₁₃	1	2	yes	—
216	C₄×C₈	C₂³×C₁₃	1	2	yes	—
217	C₄×C₈	C₂³×C₁₃	1	2	yes	—
218	C₄×C₈	C₂³×C₁₃	1	2	yes	—
219	C₂×C₁₃	—	4	1	no	—
220	C₂×C₁₃	—	4	2	no	—
221	C₂×C₁₃	#2873	1	1	yes	characteristic
222	C₂×C₁₃	#2875	1	2	yes	—
223	#41	—	4	2	no	—
224	C₂×Q₈	C₄²×C₁₃	1	1	yes	characteristic
225	C₂×D₄	C₄²×C₁₃	1	1	yes	characteristic
226	C₂×D₄	C₄²×C₁₃	1	2	yes	—
227	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
228	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
229	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
230	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
231	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
232	M₁₆	—	4	4	no	—
233	M₁₆	—	4	4	no	—
234	M₁₆	—	4	2	no	—
235	C₂×C₈	—	4	2	no	—
236	C₂×C₈	—	4	4	no	—
237	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
238	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
239	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
240	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
241	C₂×C₈	C₂²×C₄×C₁₃	1	2	yes	—
242	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
243	C₂×C₈	C₂²×C₄×C₁₃	1	4	yes	—
244	C₄⋊C₄	C₄²×C₁₃	1	2	yes	—
245	C₄⋊C₄	C₄²×C₁₃	1	2	yes	—
246	C₄⋊C₄	C₄²×C₁₃	1	2	yes	—
247	C₂²⋊C₄	C₄²×C₁₃	1	1	yes	characteristic
248	C₂²⋊C₄	C₄²×C₁₃	1	1	yes	characteristic
249	C₂²⋊C₄	C₄²×C₁₃	1	2	yes	—
250	C₂²⋊C₄	C₄²×C₁₃	1	2	yes	—
251	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
252	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
253	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
254	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
255	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
256	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
257	C₁₃	#17342	1	1	yes	characteristic, direct product factor, Sylow
258	C₂³	C₂×C₄²×C₁₃	1	2	yes	—
259	C₂³	C₂×C₄²×C₁₃	1	1	yes	characteristic
260	Q₈	—	4	1	no	—
261	D₄	—	4	1	no	—
262	D₄	—	4	2	no	—
263	C₂×C₄	—	2	2	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	2	4	no	—
266	C₂×C₄	—	2	4	no	—
267	C₂×C₄	—	2	4	no	—
268	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
269	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
270	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
271	C₂×C₄	C₂×C₄²×C₁₃	1	2	yes	—
272	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
273	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
274	C₂×C₄	C₂³×C₄×C₁₃	1	2	yes	—
275	C₈	—	2	4	no	—
276	C₈	—	2	4	no	—
277	C₈	—	2	4	no	—
278	C₈	—	2	2	no	—
279	C₈	—	2	2	no	—
280	C₈	—	2	4	no	—
281	C₈	—	2	4	no	—
282	C₂²	—	2	1	no	—
283	C₂²	—	2	2	no	—
284	C₂²	—	2	2	no	—
285	C₂²	—	2	2	no	—
286	C₂²	—	2	2	no	—
287	C₂²	C₂²×C₄²×C₁₃	1	1	yes	characteristic
288	C₄	—	2	1	no	—
289	C₄	—	2	2	no	—
290	C₄	—	2	2	no	—
291	C₄	—	2	2	no	—
292	C₄	—	2	2	no	—
293	C₄	C₁₃×#568	1	2	yes	—
294	C₄	C₁₃×#568	1	4	yes	—
295	C₂	—	4	1	no	—
296	C₂	—	4	2	no	—
297	C₂	C₁₃×#2873	1	1	yes	characteristic
298	C₂	C₁₃×#2875	1	2	yes	—

	subgroup	G/H
center	C₄²×C₁₃	C₂⁴
commutator	C₂²	C₂²×C₄²×C₁₃
Frattini, cosocle	C₄²	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂²×C₄²
prime radical	C₂×C₁₃×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
3. C₁₃×#1455	C₂	semidirect
4. C₁₃×#1454	C₂	semidirect
5. C₁₃×#1453	C₂	semidirect
6. C₁₃×#1453	C₂	semidirect
7. C₁₃×#1452	C₂	semidirect
8. C₁₃×#1451	C₂	semidirect
9. C₁₃×#1451	C₂	semidirect
74. #17342	C₁₃	direct (N × H)
111. #1455	C₂×C₁₃	semidirect
112. #1454	C₂×C₁₃	semidirect
113. #1453	C₂×C₁₃	semidirect
114. #1453	C₂×C₁₃	semidirect
115. #1452	C₂×C₁₃	semidirect
116. #1451	C₂×C₁₃	semidirect
117. #1451	C₂×C₁₃	semidirect
257. C₁₃	#17342	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×#2965	2
C₁₃×#2922	2
C₁₃×#1455	2
C₁₃×#1454	2
C₁₃×#1453	2
C₁₃×#1452	2
C₁₃×#1451	2
#17342	13

Subgroups — C13×#17342 #15703681

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#17342 #15703681