Subgroups — C₂×C₁₃×#1525 #15703975

Order 3328.

1402 conjugacy classes of subgroups.

The subgroup lattice has 298 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#532	C₂	1	1	yes	maximal, characteristic
2	C₂×C₄×C₁₃×M₁₆	C₂	1	2	yes	maximal, complemented
3	C₁₃×#1525	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₁₃×#329	C₂	1	4	yes	maximal, complemented
5	C₂²×C₁₃×#41	C₄	1	2	yes	—
6	C₂²×C₁₃×M₁₆	—	2	2	no	—
7	C₁₃×#532	—	2	4	no	—
8	C₁₃×#532	C₄	1	4	yes	—
9	C₁₃×#532	C₂²	1	4	yes	—
10	C₂×C₁₃×C₄⋊Q₈	C₄	1	1	yes	characteristic
11	C₂×C₁₃×#127	C₄	1	1	yes	characteristic
12	C₂×C₁₃×#124	C₂²	1	2	yes	—
13	C₂×C₁₃×#121	—	2	2	no	—
14	C₂×C₄×D₄×C₁₃	—	2	2	no	—
15	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
16	C₄×C₁₃×M₁₆	—	2	4	no	—
17	C₄×C₁₃×M₁₆	C₂²	1	8	yes	complemented
18	C₂×C₁₃×#97	C₂²	1	4	yes	—
19	C₂×C₄×C₈×C₁₃	—	2	2	no	—
20	C₁₃×#329	C₂²	1	16	yes	complemented
21	C₂×C₁₃×#41	—	2	4	no	—
22	C₂×C₁₃×#41	—	4	8	no	—
23	C₂×C₁₃×#41	C₂×C₄	1	8	yes	—
24	C₂²×Q₈×C₁₃	C₂×C₄	1	2	yes	—
25	C₂²×D₄×C₁₃	—	2	2	no	—
26	C₂²×D₄×C₁₃	—	2	2	no	—
27	C₂²×D₄×C₁₃	C₂×C₄	1	2	yes	—
28	C₂³×C₄×C₁₃	—	2	2	no	—
29	C₂³×C₄×C₁₃	C₂×C₄	1	1	yes	characteristic
30	C₂³×C₄×C₁₃	D₄	1	2	yes	—
31	C₂×C₁₃×M₁₆	—	2	16	no	—
32	C₂×C₁₃×M₁₆	—	2	8	no	—
33	C₂²×C₈×C₁₃	—	2	4	no	—
34	C₁₃×C₄⋊Q₈	—	2	2	no	—
35	C₁₃×C₄⋊Q₈	C₂×C₄	1	4	yes	—
36	C₁₃×#127	—	2	2	no	—
37	C₁₃×#127	C₂×C₄	1	4	yes	—
38	C₁₃×#124	—	2	4	no	—
39	C₁₃×#124	C₂×C₄	1	4	yes	—
40	C₁₃×#124	C₂³	1	4	yes	—
41	C₁₃×#121	—	2	8	no	—
42	C₁₃×#121	—	2	8	no	—
43	C₄×D₄×C₁₃	—	2	8	no	—
44	C₄×D₄×C₁₃	—	2	8	no	—
45	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
46	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
47	C₂×C₄²×C₁₃	C₂×C₄	1	1	yes	characteristic
48	C₂×C₄²×C₁₃	C₂×C₄	1	2	yes	—
49	C₂×C₄²×C₁₃	D₄	1	4	yes	—
50	C₂×C₄²×C₁₃	D₄	1	2	yes	—
51	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
53	C₁₃×#97	—	2	4	no	—
54	C₁₃×#97	C₂³	1	8	yes	—
55	C₄×C₈×C₁₃	—	2	8	no	—
56	C₂×#1525	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
57	C₂⁴×C₁₃	—	2	2	no	—
58	C₂⁴×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₁₃×#41	—	4	16	no	—
60	C₁₃×#41	—	4	16	no	—
61	C₂×Q₈×C₁₃	—	2	2	no	—
62	C₂×Q₈×C₁₃	—	4	4	no	—
63	C₂×Q₈×C₁₃	C₂²×C₄	1	4	yes	—
64	C₂×D₄×C₁₃	—	2	8	no	—
65	C₂×D₄×C₁₃	—	2	8	no	—
66	C₂×D₄×C₁₃	—	2	2	no	—
67	C₂×D₄×C₁₃	—	4	4	no	—
68	C₂×D₄×C₁₃	—	4	8	no	—
69	C₂×D₄×C₁₃	—	4	8	no	—
70	C₂×D₄×C₁₃	C₂²×C₄	1	4	yes	—
71	C₂²×C₄×C₁₃	—	2	4	no	—
72	C₂²×C₄×C₁₃	—	2	8	no	—
73	C₂²×C₄×C₁₃	—	2	2	no	—
74	C₂²×C₄×C₁₃	—	2	8	no	—
75	C₂²×C₄×C₁₃	—	4	8	no	—
76	C₂²×C₄×C₁₃	C₂²⋊C₄	1	4	yes	—
77	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
78	C₂²×C₄×C₁₃	C₂²⋊C₄	1	2	yes	—
79	C₂²×C₄×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
80	C₂²×C₄×C₁₃	C₂²×C₄	1	1	yes	characteristic
81	C₂²×C₄×C₁₃	C₂²×C₄	1	2	yes	—
82	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
84	C₁₃×M₁₆	—	2	32	no	—
85	C₂×C₈×C₁₃	—	2	16	no	—
86	C₂×C₈×C₁₃	—	2	8	no	—
87	C₁₃×C₄⋊C₄	—	2	8	no	—
88	C₁₃×C₂²⋊C₄	—	2	8	no	—
89	C₁₃×C₂²⋊C₄	—	2	8	no	—
90	C₄²×C₁₃	C₂²⋊C₄	1	4	yes	—
91	C₄²×C₁₃	C₂²⋊C₄	1	2	yes	—
92	C₄²×C₁₃	C₂²×C₄	1	1	yes	characteristic
93	C₄²×C₁₃	C₂²×C₄	1	2	yes	—
94	C₄²×C₁₃	C₂×D₄	1	4	yes	—
95	C₄²×C₁₃	C₂×D₄	1	2	yes	—
96	C₄²×C₁₃	C₂⁴	1	1	yes	characteristic
97	C₂×#532	C₂×C₁₃	1	1	yes	characteristic
98	C₂×C₄×M₁₆	C₂×C₁₃	1	2	yes	complemented
99	#1525	C₂×C₁₃	1	8	yes	direct product factor
100	C₂×#329	C₂×C₁₃	1	4	yes	complemented
101	C₂³×C₁₃	—	2	2	no	—
102	C₂³×C₁₃	—	2	4	no	—
103	C₂³×C₁₃	—	2	4	no	—
104	C₂³×C₁₃	—	4	4	no	—
105	C₂³×C₁₃	—	4	8	no	—
106	C₂³×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
107	C₂³×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
108	Q₈×C₁₃	—	4	4	no	—
109	Q₈×C₁₃	—	4	4	no	—
110	D₄×C₁₃	—	4	4	no	—
111	D₄×C₁₃	—	4	4	no	—
112	D₄×C₁₃	—	4	16	no	—
113	D₄×C₁₃	—	4	16	no	—
114	C₂×C₄×C₁₃	—	2	2	no	—
115	C₂×C₄×C₁₃	—	2	4	no	—
116	C₂×C₄×C₁₃	—	2	4	no	—
117	C₂×C₄×C₁₃	—	2	4	no	—
118	C₂×C₄×C₁₃	—	2	8	no	—
119	C₂×C₄×C₁₃	—	2	8	no	—
120	C₂×C₄×C₁₃	—	2	4	no	—
121	C₂×C₄×C₁₃	—	4	4	no	—
122	C₂×C₄×C₁₃	—	4	16	no	—
123	C₂×C₄×C₁₃	#100	1	4	yes	—
124	C₂×C₄×C₁₃	#104	1	8	yes	—
125	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
126	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	2	yes	—
128	C₂×C₄×C₁₃	C₂×C₂²⋊C₄	1	4	yes	—
129	C₂×C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
130	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
131	C₈×C₁₃	—	2	16	no	—
132	C₂²×#41	C₄×C₁₃	1	2	yes	—
133	C₂²×M₁₆	—	2	2	no	—
134	#532	—	2	4	no	—
135	#532	C₄×C₁₃	1	4	yes	—
136	#532	C₂²×C₁₃	1	4	yes	—
137	C₂×C₄⋊Q₈	C₄×C₁₃	1	1	yes	characteristic
138	C₂×#127	C₄×C₁₃	1	1	yes	characteristic
139	C₂×#124	C₂²×C₁₃	1	2	yes	—
140	C₂×#121	—	2	2	no	—
141	C₂×C₄×D₄	—	2	2	no	—
142	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
143	C₄×M₁₆	—	2	4	no	—
144	C₄×M₁₆	C₂²×C₁₃	1	8	yes	complemented
145	C₂×#97	C₂²×C₁₃	1	4	yes	—
146	C₂×C₄×C₈	—	2	2	no	—
147	#329	C₂²×C₁₃	1	16	yes	complemented
148	C₂²×C₁₃	—	2	2	no	—
149	C₂²×C₁₃	—	2	4	no	—
150	C₂²×C₁₃	—	4	4	no	—
151	C₂²×C₁₃	—	4	4	no	—
152	C₂²×C₁₃	—	4	8	no	—
153	C₂²×C₁₃	—	8	8	no	—
154	C₂²×C₁₃	C₂×#100	1	2	yes	—
155	C₂²×C₁₃	C₂×#104	1	4	yes	—
156	C₂²×C₁₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
157	C₄×C₁₃	—	2	1	no	—
158	C₄×C₁₃	—	2	1	no	—
159	C₄×C₁₃	—	2	2	no	—
160	C₄×C₁₃	—	2	2	no	—
161	C₄×C₁₃	—	4	4	no	—
162	C₄×C₁₃	C₂×#100	1	4	yes	—
163	C₄×C₁₃	C₂×#104	1	8	yes	—
164	C₂×#41	—	2	4	no	—
165	C₂×#41	—	4	8	no	—
166	C₂×#41	C₂×C₄×C₁₃	1	8	yes	—
167	C₂²×Q₈	C₂×C₄×C₁₃	1	2	yes	—
168	C₂²×D₄	—	2	2	no	—
169	C₂²×D₄	—	2	2	no	—
170	C₂²×D₄	C₂×C₄×C₁₃	1	2	yes	—
171	C₂³×C₄	—	2	2	no	—
172	C₂³×C₄	C₂×C₄×C₁₃	1	1	yes	characteristic
173	C₂³×C₄	D₄×C₁₃	1	2	yes	—
174	C₂×M₁₆	—	2	16	no	—
175	C₂×M₁₆	—	2	8	no	—
176	C₂²×C₈	—	2	4	no	—
177	C₄⋊Q₈	—	2	2	no	—
178	C₄⋊Q₈	C₂×C₄×C₁₃	1	4	yes	—
179	#127	—	2	2	no	—
180	#127	C₂×C₄×C₁₃	1	4	yes	—
181	#124	—	2	4	no	—
182	#124	C₂×C₄×C₁₃	1	4	yes	—
183	#124	C₂³×C₁₃	1	4	yes	—
184	#121	—	2	8	no	—
185	#121	—	2	8	no	—
186	C₄×D₄	—	2	8	no	—
187	C₄×D₄	—	2	8	no	—
188	C₂×C₄⋊C₄	—	2	2	no	—
189	C₂×C₂²⋊C₄	—	2	4	no	—
190	C₂×C₄²	C₂×C₄×C₁₃	1	1	yes	characteristic
191	C₂×C₄²	C₂×C₄×C₁₃	1	2	yes	—
192	C₂×C₄²	D₄×C₁₃	1	2	yes	—
193	C₂×C₄²	D₄×C₁₃	1	4	yes	—
194	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
195	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
196	#97	—	2	4	no	—
197	#97	C₂³×C₁₃	1	8	yes	—
198	C₄×C₈	—	2	8	no	—
199	C₂×C₁₃	—	4	2	no	—
200	C₂×C₁₃	—	8	4	no	—
201	C₂×C₁₃	#1525	1	4	yes	direct product factor
202	C₂×C₁₃	C₂²×#100	1	1	yes	characteristic
203	C₂×C₁₃	C₂²×#104	1	2	yes	—
204	C₂⁴	—	2	2	no	—
205	C₂⁴	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
206	#41	—	4	16	no	—
207	#41	—	4	16	no	—
208	C₂×Q₈	—	2	2	no	—
209	C₂×Q₈	—	4	4	no	—
210	C₂×Q₈	C₂²×C₄×C₁₃	1	4	yes	—
211	C₂×D₄	—	2	8	no	—
212	C₂×D₄	—	2	8	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	4	4	no	—
215	C₂×D₄	—	4	8	no	—
216	C₂×D₄	—	4	8	no	—
217	C₂×D₄	C₂²×C₄×C₁₃	1	4	yes	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	8	no	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	8	no	—
222	C₂²×C₄	—	4	8	no	—
223	C₂²×C₄	C₁₃×C₂²⋊C₄	1	4	yes	—
224	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
225	C₂²×C₄	C₁₃×C₂²⋊C₄	1	1	yes	characteristic
226	C₂²×C₄	C₁₃×C₂²⋊C₄	1	2	yes	—
227	C₂²×C₄	C₂²×C₄×C₁₃	1	1	yes	characteristic
228	C₂²×C₄	C₂²×C₄×C₁₃	1	2	yes	—
229	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
230	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
231	M₁₆	—	2	32	no	—
232	C₂×C₈	—	2	16	no	—
233	C₂×C₈	—	2	8	no	—
234	C₄⋊C₄	—	2	8	no	—
235	C₂²⋊C₄	—	2	8	no	—
236	C₂²⋊C₄	—	2	8	no	—
237	C₄²	C₁₃×C₂²⋊C₄	1	4	yes	—
238	C₄²	C₁₃×C₂²⋊C₄	1	2	yes	—
239	C₄²	C₂²×C₄×C₁₃	1	1	yes	characteristic
240	C₄²	C₂²×C₄×C₁₃	1	2	yes	—
241	C₄²	C₂×D₄×C₁₃	1	4	yes	—
242	C₄²	C₂×D₄×C₁₃	1	2	yes	—
243	C₄²	C₂⁴×C₁₃	1	1	yes	characteristic
244	C₁₃	C₂×#1525	1	1	yes	characteristic, direct product factor, Sylow
245	C₂³	—	2	2	no	—
246	C₂³	—	2	4	no	—
247	C₂³	—	2	4	no	—
248	C₂³	—	4	4	no	—
249	C₂³	—	4	8	no	—
250	C₂³	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
251	C₂³	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
252	Q₈	—	4	4	no	—
253	Q₈	—	4	4	no	—
254	D₄	—	4	4	no	—
255	D₄	—	4	4	no	—
256	D₄	—	4	16	no	—
257	D₄	—	4	16	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	4	no	—
261	C₂×C₄	—	2	4	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	2	4	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	—	4	16	no	—
267	C₂×C₄	C₁₃×#100	1	4	yes	—
268	C₂×C₄	C₁₃×#104	1	8	yes	—
269	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
270	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
271	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	2	yes	—
272	C₂×C₄	C₂×C₁₃×C₂²⋊C₄	1	4	yes	—
273	C₂×C₄	C₂³×C₄×C₁₃	1	1	yes	characteristic
274	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
275	C₈	—	2	16	no	—
276	C₂²	—	2	2	no	—
277	C₂²	—	2	4	no	—
278	C₂²	—	4	4	no	—
279	C₂²	—	4	4	no	—
280	C₂²	—	4	8	no	—
281	C₂²	—	8	8	no	—
282	C₂²	C₂×C₁₃×#100	1	2	yes	—
283	C₂²	C₂×C₁₃×#104	1	4	yes	—
284	C₂²	C₂²×C₁₃×C₂²⋊C₄	1	1	yes	characteristic
285	C₄	—	2	1	no	—
286	C₄	—	2	1	no	—
287	C₄	—	2	2	no	—
288	C₄	—	2	2	no	—
289	C₄	—	4	4	no	—
290	C₄	C₂×C₁₃×#100	1	4	yes	—
291	C₄	C₂×C₁₃×#104	1	8	yes	—
292	C₂	—	4	2	no	—
293	C₂	—	8	4	no	—
294	C₂	C₁₃×#1525	1	4	yes	direct product factor
295	C₂	C₂²×C₁₃×#100	1	1	yes	characteristic
296	C₂	C₂²×C₁₃×#104	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂²⋊C₄
commutator	C₂×C₄	C₂³×C₄×C₁₃
Frattini, cosocle	C₄²	C₂⁴×C₁₃
socle	C₂³×C₁₃	C₂×C₂²⋊C₄
prime radical	C₂×C₁₃×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₄×C₁₃×M₁₆	C₂	semidirect
3. C₁₃×#1525	C₂	direct (N × H)
4. C₂×C₁₃×#329	C₂	semidirect
17. C₄×C₁₃×M₁₆	C₂²	semidirect
20. C₁₃×#329	C₂²	semidirect
56. C₂×#1525	C₁₃	direct (N × H)
98. C₂×C₄×M₁₆	C₂×C₁₃	semidirect
99. #1525	C₂×C₁₃	direct (N × H)
100. C₂×#329	C₂×C₁₃	semidirect
144. C₄×M₁₆	C₂²×C₁₃	semidirect
147. #329	C₂²×C₁₃	semidirect
201. C₂×C₁₃	#1525	direct (N × H)
244. C₁₃	C₂×#1525	direct (N × H)
294. C₂	C₁₃×#1525	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×#532	2
C₂×C₄×C₁₃×M₁₆	2
C₁₃×#1525	2
C₂×C₁₃×#329	2
C₂×#1525	13

Subgroups — C2×C13×#1525 #15703975

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#1525 #15703975