Subgroups — C₁₃×#10828 #15697167

Order 3328.

642 conjugacy classes of subgroups.

The subgroup lattice has 312 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#405	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×#344	C₂	1	2	yes	maximal
3	C₂×C₁₃×#341	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#1735	C₂	1	1	yes	maximal, characteristic
5	C₁₃×#1733	C₂	1	2	yes	maximal
6	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
7	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#117	C₂²	1	2	yes	—
10	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
11	C₁₃×#405	—	2	2	no	—
12	C₁₃×#405	C₄	1	4	yes	complemented
13	C₂×C₁₃×#97	—	2	1	no	—
14	C₂×C₄×C₈×C₁₃	—	2	1	no	—
15	C₁₃×#344	C₄	1	8	yes	complemented
16	C₁₃×#341	—	2	1	no	—
17	C₁₃×#341	—	2	1	no	—
18	C₁₃×#340	—	2	2	no	—
19	C₁₃×#339	—	2	4	no	—
20	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
21	C₂×C₁₃×M₁₆	—	2	1	no	—
22	C₂×C₁₃×M₁₆	—	2	2	no	—
23	C₂×C₁₃×M₁₆	—	2	2	no	—
24	C₂×C₁₃×M₁₆	—	2	2	no	—
25	C₂×C₁₃×M₁₆	—	2	2	no	—
26	C₂×C₁₃×M₁₆	—	2	2	no	—
27	C₂×C₁₃×M₁₆	—	2	2	no	—
28	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
29	C₂×C₁₃×M₁₆	C₂×C₄	1	2	yes	—
30	C₂×C₁₃×M₁₆	C₂×C₄	1	4	yes	—
31	C₂²×C₈×C₁₃	—	2	1	no	—
32	C₂²×C₈×C₁₃	—	2	2	no	—
33	C₂²×C₈×C₁₃	—	2	2	no	—
34	C₂²×C₈×C₁₃	—	2	2	no	—
35	C₂²×C₈×C₁₃	C₂×C₄	1	2	yes	—
36	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
38	C₂²×C₈×C₁₃	D₄	1	2	yes	—
39	C₁₃×#117	—	2	1	no	—
40	C₁₃×#117	—	2	1	no	—
41	C₁₃×#117	—	2	2	no	—
42	C₁₃×#117	—	2	2	no	—
43	C₁₃×#117	—	2	2	no	—
44	C₁₃×#117	C₂×C₄	1	4	yes	—
45	C₁₃×#117	C₂×C₄	1	2	yes	—
46	C₁₃×#117	D₄	1	2	yes	—
47	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
48	C₂×C₁₃×C₄⋊C₄	Q₈	1	2	yes	—
49	C₂×C₁₃×C₂²⋊C₄	—	2	1	no	—
50	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
51	C₂×C₄²×C₁₃	—	2	1	no	—
52	C₂×C₄²×C₁₃	—	2	2	no	—
53	C₁₃×#97	—	2	4	no	—
54	C₄×C₈×C₁₃	—	2	4	no	—
55	#10828	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂⁴×C₁₃	#41	1	1	yes	characteristic
57	C₂²×C₄×C₁₃	—	2	2	no	—
58	C₂²×C₄×C₁₃	—	4	2	no	—
59	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
60	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
61	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₂²×C₄×C₁₃	C₂×Q₈	1	1	yes	characteristic
65	C₂²×C₄×C₁₃	#41	1	1	yes	characteristic
66	C₂²×C₄×C₁₃	#41	1	2	yes	—
67	C₂²×C₄×C₁₃	#41	1	2	yes	—
68	C₂²×C₄×C₁₃	#41	1	2	yes	—
69	C₁₃×M₁₆	—	2	2	no	—
70	C₁₃×M₁₆	—	2	4	no	—
71	C₁₃×M₁₆	—	2	4	no	—
72	C₁₃×M₁₆	—	2	4	no	—
73	C₁₃×M₁₆	—	4	2	no	—
74	C₁₃×M₁₆	—	4	2	no	—
75	C₁₃×M₁₆	C₄²	1	4	yes	—
76	C₂×C₈×C₁₃	—	2	2	no	—
77	C₂×C₈×C₁₃	—	2	2	no	—
78	C₂×C₈×C₁₃	—	2	2	no	—
79	C₂×C₈×C₁₃	—	2	4	no	—
80	C₂×C₈×C₁₃	—	2	4	no	—
81	C₂×C₈×C₁₃	—	2	2	no	—
82	C₂×C₈×C₁₃	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	4	no	—
84	C₂×C₈×C₁₃	—	4	2	no	—
85	C₂×C₈×C₁₃	—	4	2	no	—
86	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
87	C₂×C₈×C₁₃	C₂²⋊C₄	1	2	yes	—
88	C₁₃×C₄⋊C₄	—	2	2	no	—
89	C₁₃×C₄⋊C₄	—	2	4	no	—
90	C₁₃×C₄⋊C₄	—	4	2	no	—
91	C₁₃×C₄⋊C₄	C₄⋊C₄	1	4	yes	—
92	C₁₃×C₂²⋊C₄	—	2	2	no	—
93	C₁₃×C₂²⋊C₄	—	2	2	no	—
94	C₁₃×C₂²⋊C₄	—	2	4	no	—
95	C₁₃×C₂²⋊C₄	—	4	2	no	—
96	C₄²×C₁₃	—	2	2	no	—
97	C₄²×C₁₃	—	2	2	no	—
98	C₄²×C₁₃	—	2	4	no	—
99	C₄²×C₁₃	—	4	2	no	—
100	C₂×#405	C₂×C₁₃	1	1	yes	characteristic
101	C₂×#344	C₂×C₁₃	1	2	yes	—
102	C₂×#341	C₂×C₁₃	1	1	yes	characteristic
103	#1735	C₂×C₁₃	1	1	yes	characteristic
104	#1733	C₂×C₁₃	1	2	yes	—
105	C₂³×C₁₃	—	4	2	no	—
106	C₂³×C₁₃	#117	1	1	yes	characteristic
107	C₂³×C₁₃	#117	1	2	yes	—
108	C₂³×C₁₃	#121	1	1	yes	characteristic
109	C₂³×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
110	C₂³×C₁₃	#126	1	2	yes	—
111	C₂×C₄×C₁₃	—	2	4	no	—
112	C₂×C₄×C₁₃	—	4	4	no	—
113	C₂×C₄×C₁₃	—	4	4	no	—
114	C₂×C₄×C₁₃	—	8	4	no	—
115	C₂×C₄×C₁₃	C₂×C₄²	1	1	yes	characteristic
116	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
117	C₂×C₄×C₁₃	C₂×C₄⋊C₄	1	1	yes	characteristic
118	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
119	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
120	C₂×C₄×C₁₃	#117	1	1	yes	characteristic
121	C₂×C₄×C₁₃	#117	1	2	yes	—
122	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
123	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
124	C₂×C₄×C₁₃	C₄×D₄	1	2	yes	—
125	C₂×C₄×C₁₃	C₄×Q₈	1	2	yes	—
126	C₂×C₄×C₁₃	#121	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂²⋊Q₈	1	1	yes	characteristic
128	C₂×C₄×C₁₃	#123	1	2	yes	—
129	C₂×C₄×C₁₃	#123	1	2	yes	—
130	C₂×C₄×C₁₃	#124	1	2	yes	—
131	C₂×C₄×C₁₃	#125	1	2	yes	—
132	C₂×C₄×C₁₃	#126	1	2	yes	—
133	C₈×C₁₃	—	2	4	no	—
134	C₈×C₁₃	—	2	2	no	—
135	C₈×C₁₃	—	2	2	no	—
136	C₈×C₁₃	—	4	4	no	—
137	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
138	C₂²×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
139	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
140	C₂×#117	C₂²×C₁₃	1	2	yes	—
141	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
142	#405	—	2	2	no	—
143	#405	C₄×C₁₃	1	4	yes	complemented
144	C₂×#97	—	2	1	no	—
145	C₂×C₄×C₈	—	2	1	no	—
146	#344	C₄×C₁₃	1	8	yes	complemented
147	#341	—	2	1	no	—
148	#341	—	2	1	no	—
149	#340	—	2	2	no	—
150	#339	—	2	4	no	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	—	2	2	no	—
153	C₂²×C₁₃	—	2	2	no	—
154	C₂²×C₁₃	—	4	2	no	—
155	C₂²×C₁₃	—	4	2	no	—
156	C₂²×C₁₃	#376	1	1	yes	characteristic
157	C₂²×C₁₃	#380	1	1	yes	characteristic
158	C₂²×C₁₃	#383	1	1	yes	characteristic
159	C₂²×C₁₃	#387	1	2	yes	—
160	C₂²×C₁₃	#388	1	2	yes	—
161	C₄×C₁₃	—	4	4	no	—
162	C₄×C₁₃	—	8	4	no	—
163	C₄×C₁₃	C₄×C₂²⋊C₄	1	1	yes	characteristic
164	C₄×C₁₃	C₄×C₄⋊C₄	1	1	yes	characteristic
165	C₄×C₁₃	#381	1	1	yes	characteristic
166	C₄×C₁₃	#382	1	1	yes	characteristic
167	C₄×C₁₃	#387	1	2	yes	—
168	C₄×C₁₃	#388	1	2	yes	—
169	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
170	C₂×M₁₆	—	2	1	no	—
171	C₂×M₁₆	—	2	2	no	—
172	C₂×M₁₆	—	2	2	no	—
173	C₂×M₁₆	—	2	2	no	—
174	C₂×M₁₆	—	2	2	no	—
175	C₂×M₁₆	—	2	2	no	—
176	C₂×M₁₆	—	2	2	no	—
177	C₂×M₁₆	C₂×C₄×C₁₃	1	4	yes	—
178	C₂×M₁₆	C₂×C₄×C₁₃	1	4	yes	—
179	C₂×M₁₆	C₂×C₄×C₁₃	1	2	yes	—
180	C₂²×C₈	—	2	1	no	—
181	C₂²×C₈	—	2	2	no	—
182	C₂²×C₈	—	2	2	no	—
183	C₂²×C₈	—	2	2	no	—
184	C₂²×C₈	C₂×C₄×C₁₃	1	2	yes	—
185	C₂²×C₈	D₄×C₁₃	1	2	yes	—
186	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
187	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
188	#117	—	2	1	no	—
189	#117	—	2	1	no	—
190	#117	—	2	2	no	—
191	#117	—	2	2	no	—
192	#117	—	2	2	no	—
193	#117	C₂×C₄×C₁₃	1	4	yes	—
194	#117	C₂×C₄×C₁₃	1	2	yes	—
195	#117	D₄×C₁₃	1	2	yes	—
196	C₂×C₄⋊C₄	—	2	2	no	—
197	C₂×C₄⋊C₄	Q₈×C₁₃	1	2	yes	—
198	C₂×C₂²⋊C₄	—	2	1	no	—
199	C₂×C₂²⋊C₄	—	2	2	no	—
200	C₂×C₄²	—	2	1	no	—
201	C₂×C₄²	—	2	2	no	—
202	#97	—	2	4	no	—
203	C₄×C₈	—	2	4	no	—
204	C₂×C₁₃	—	2	2	no	—
205	C₂×C₁₃	—	2	1	no	—
206	C₂×C₁₃	—	2	1	no	—
207	C₂×C₁₃	—	2	2	no	—
208	C₂×C₁₃	#1434	1	1	yes	characteristic
209	C₂×C₁₃	#1814	1	2	yes	—
210	C₂⁴	C₁₃×#41	1	1	yes	characteristic
211	C₂²×C₄	—	2	2	no	—
212	C₂²×C₄	—	4	2	no	—
213	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
214	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
215	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
216	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
217	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
218	C₂²×C₄	C₂×Q₈×C₁₃	1	1	yes	characteristic
219	C₂²×C₄	C₁₃×#41	1	1	yes	characteristic
220	C₂²×C₄	C₁₃×#41	1	2	yes	—
221	C₂²×C₄	C₁₃×#41	1	2	yes	—
222	C₂²×C₄	C₁₃×#41	1	2	yes	—
223	M₁₆	—	2	2	no	—
224	M₁₆	—	2	4	no	—
225	M₁₆	—	2	4	no	—
226	M₁₆	—	2	4	no	—
227	M₁₆	—	4	2	no	—
228	M₁₆	—	4	2	no	—
229	M₁₆	C₄²×C₁₃	1	4	yes	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	—	2	2	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	2	4	no	—
235	C₂×C₈	—	2	2	no	—
236	C₂×C₈	—	2	2	no	—
237	C₂×C₈	—	2	4	no	—
238	C₂×C₈	—	4	2	no	—
239	C₂×C₈	—	4	2	no	—
240	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
241	C₂×C₈	C₁₃×C₂²⋊C₄	1	2	yes	—
242	C₄⋊C₄	—	2	2	no	—
243	C₄⋊C₄	—	2	4	no	—
244	C₄⋊C₄	—	4	2	no	—
245	C₄⋊C₄	C₁₃×C₄⋊C₄	1	4	yes	—
246	C₂²⋊C₄	—	2	2	no	—
247	C₂²⋊C₄	—	2	2	no	—
248	C₂²⋊C₄	—	2	4	no	—
249	C₂²⋊C₄	—	4	2	no	—
250	C₄²	—	2	2	no	—
251	C₄²	—	2	2	no	—
252	C₄²	—	2	4	no	—
253	C₄²	—	4	2	no	—
254	C₁₃	#10828	1	1	yes	characteristic, direct product factor, Sylow
255	C₂³	—	4	2	no	—
256	C₂³	C₁₃×#117	1	1	yes	characteristic
257	C₂³	C₁₃×#117	1	2	yes	—
258	C₂³	C₁₃×#121	1	1	yes	characteristic
259	C₂³	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
260	C₂³	C₁₃×#126	1	2	yes	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	4	no	—
264	C₂×C₄	—	8	4	no	—
265	C₂×C₄	C₂×C₄²×C₁₃	1	1	yes	characteristic
266	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
267	C₂×C₄	C₂×C₁₃×C₄⋊C₄	1	1	yes	characteristic
268	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
269	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
270	C₂×C₄	C₁₃×#117	1	1	yes	characteristic
271	C₂×C₄	C₁₃×#117	1	2	yes	—
272	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
273	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
274	C₂×C₄	C₄×D₄×C₁₃	1	2	yes	—
275	C₂×C₄	C₄×Q₈×C₁₃	1	2	yes	—
276	C₂×C₄	C₁₃×#121	1	1	yes	characteristic
277	C₂×C₄	C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
278	C₂×C₄	C₁₃×#123	1	2	yes	—
279	C₂×C₄	C₁₃×#123	1	2	yes	—
280	C₂×C₄	C₁₃×#124	1	2	yes	—
281	C₂×C₄	C₁₃×#125	1	2	yes	—
282	C₂×C₄	C₁₃×#126	1	2	yes	—
283	C₈	—	2	4	no	—
284	C₈	—	2	2	no	—
285	C₈	—	2	2	no	—
286	C₈	—	4	4	no	—
287	C₂²	—	2	2	no	—
288	C₂²	—	2	2	no	—
289	C₂²	—	2	2	no	—
290	C₂²	—	4	2	no	—
291	C₂²	—	4	2	no	—
292	C₂²	C₁₃×#376	1	1	yes	characteristic
293	C₂²	C₁₃×#380	1	1	yes	characteristic
294	C₂²	C₁₃×#383	1	1	yes	characteristic
295	C₂²	C₁₃×#387	1	2	yes	—
296	C₂²	C₁₃×#388	1	2	yes	—
297	C₄	—	4	4	no	—
298	C₄	—	8	4	no	—
299	C₄	C₄×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
300	C₄	C₄×C₁₃×C₄⋊C₄	1	1	yes	characteristic
301	C₄	C₁₃×#381	1	1	yes	characteristic
302	C₄	C₁₃×#382	1	1	yes	characteristic
303	C₄	C₁₃×#387	1	2	yes	—
304	C₄	C₁₃×#388	1	2	yes	—
305	C₂	—	2	2	no	—
306	C₂	—	2	1	no	—
307	C₂	—	2	2	no	—
308	C₂	—	2	1	no	—
309	C₂	C₁₃×#1434	1	1	yes	characteristic
310	C₂	C₁₃×#1814	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃	#121
commutator	C₂×C₄	C₂×C₄²×C₁₃
Frattini, cosocle	C₂³×C₄	C₂³×C₁₃
socle	C₂²×C₁₃	#380
prime radical	C₂⁴×C₁₃	#41
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
12. C₁₃×#405	C₄	semidirect
15. C₁₃×#344	C₄	semidirect
55. #10828	C₁₃	direct (N × H)
143. #405	C₄×C₁₃	semidirect
146. #344	C₄×C₁₃	semidirect
254. C₁₃	#10828	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂×C₁₃×#405	2
C₂×C₁₃×#344	2
C₂×C₁₃×#341	2
C₁₃×#1735	2
C₁₃×#1733	2
#10828	13

Subgroups — C13×#10828 #15697167

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#10828 #15697167