Subgroups — C₂×C₁₃×#2025 #15709787

Order 3328.

1518 conjugacy classes of subgroups.

The subgroup lattice has 342 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₃×#102	C₂	1	2	yes	maximal
3	C₂×C₁₃×#390	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2025	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₁₃×#340	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#336	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂³×D₄×C₁₃	C₂²	1	1	yes	characteristic
8	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
9	C₂×C₁₃×#125	—	2	8	no	—
10	C₂×C₁₃×#125	—	2	2	no	—
11	C₂×C₁₃×#125	C₂²	1	4	yes	complemented
12	C₂²×C₁₃×C₄⋊C₄	—	2	2	no	—
13	C₂²×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
14	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
15	C₂²×C₁₃×C₂²⋊C₄	—	2	2	no	—
16	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×#102	—	2	8	no	—
18	C₂×C₁₃×#102	C₂²	1	8	yes	—
19	C₁₃×#390	—	2	2	no	—
20	C₁₃×#390	C₂²	1	4	yes	—
21	C₁₃×#340	C₂²	1	8	yes	complemented
22	C₁₃×#336	C₂²	1	4	yes	complemented
23	C₂⁵×C₁₃	—	2	1	no	—
24	C₂²×D₄×C₁₃	—	2	4	no	—
25	C₂²×D₄×C₁₃	—	2	1	no	—
26	C₂²×D₄×C₁₃	—	2	4	no	—
27	C₂²×D₄×C₁₃	—	2	2	no	—
28	C₂²×D₄×C₁₃	Q₈	1	4	yes	—
29	C₂²×D₄×C₁₃	C₂³	1	2	yes	—
30	C₂³×C₄×C₁₃	—	2	2	no	—
31	C₂³×C₄×C₁₃	—	2	1	no	—
32	C₂³×C₄×C₁₃	D₄	1	2	yes	—
33	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
34	C₂²×C₈×C₁₃	—	2	4	no	—
35	C₂²×C₈×C₁₃	—	2	4	no	—
36	C₂²×C₈×C₁₃	—	2	4	no	—
37	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
38	C₁₃×#125	—	2	16	no	—
39	C₁₃×#125	—	2	16	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	16	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
45	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
46	C₂×C₁₃×C₄⋊C₄	D₄	1	8	yes	—
47	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
48	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
49	C₂×C₁₃×C₂²⋊C₄	—	2	8	no	—
50	C₂×C₁₃×C₂²⋊C₄	—	4	8	no	—
51	C₂×C₄²×C₁₃	—	2	4	no	—
52	C₂×C₄²×C₁₃	D₄	1	2	yes	—
53	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
54	C₁₃×#102	—	2	16	no	—
55	C₂×#2025	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
56	C₂⁴×C₁₃	—	2	4	no	—
57	C₂⁴×C₁₃	—	2	2	no	—
58	C₂⁴×C₁₃	—	4	2	no	—
59	C₂⁴×C₁₃	—	4	2	no	—
60	C₂⁴×C₁₃	—	4	4	no	—
61	C₂⁴×C₁₃	—	4	4	no	—
62	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
63	C₂×D₄×C₁₃	—	2	16	no	—
64	C₂×D₄×C₁₃	—	2	4	no	—
65	C₂×D₄×C₁₃	—	2	4	no	—
66	C₂×D₄×C₁₃	—	2	4	no	—
67	C₂×D₄×C₁₃	—	2	2	no	—
68	C₂×D₄×C₁₃	—	4	2	no	—
69	C₂×D₄×C₁₃	—	4	8	no	—
70	C₂×D₄×C₁₃	—	4	2	no	—
71	C₂×D₄×C₁₃	C₂×Q₈	1	4	yes	—
72	C₂²×C₄×C₁₃	—	2	4	no	—
73	C₂²×C₄×C₁₃	—	2	8	no	—
74	C₂²×C₄×C₁₃	—	2	4	no	—
75	C₂²×C₄×C₁₃	—	2	4	no	—
76	C₂²×C₄×C₁₃	—	2	4	no	—
77	C₂²×C₄×C₁₃	—	2	2	no	—
78	C₂²×C₄×C₁₃	—	4	4	no	—
79	C₂²×C₄×C₁₃	—	4	4	no	—
80	C₂²×C₄×C₁₃	—	4	2	no	—
81	C₂²×C₄×C₁₃	—	4	4	no	—
82	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
83	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
85	C₂²×C₄×C₁₃	C₂×Q₈	1	2	yes	—
86	C₂²×C₄×C₁₃	#41	1	4	yes	—
87	C₂²×C₄×C₁₃	#41	1	2	yes	—
88	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
89	C₂×C₈×C₁₃	—	2	4	no	—
90	C₂×C₈×C₁₃	—	2	4	no	—
91	C₂×C₈×C₁₃	—	2	4	no	—
92	C₂×C₈×C₁₃	—	4	8	no	—
93	C₁₃×C₄⋊C₄	—	2	32	no	—
94	C₁₃×C₄⋊C₄	—	2	8	no	—
95	C₁₃×C₄⋊C₄	—	2	4	no	—
96	C₁₃×C₄⋊C₄	—	2	4	no	—
97	C₁₃×C₄⋊C₄	—	4	2	no	—
98	C₁₃×C₄⋊C₄	C₂×D₄	1	8	yes	—
99	C₁₃×C₂²⋊C₄	—	4	16	no	—
100	C₄²×C₁₃	—	2	4	no	—
101	C₄²×C₁₃	—	2	4	no	—
102	C₂²×#125	C₂×C₁₃	1	1	yes	characteristic, complemented
103	C₂²×#102	C₂×C₁₃	1	2	yes	—
104	C₂×#390	C₂×C₁₃	1	1	yes	characteristic
105	#2025	C₂×C₁₃	1	8	yes	direct product factor
106	C₂×#340	C₂×C₁₃	1	2	yes	complemented
107	C₂×#336	C₂×C₁₃	1	1	yes	characteristic, complemented
108	C₂³×C₁₃	—	2	4	no	—
109	C₂³×C₁₃	—	4	2	no	—
110	C₂³×C₁₃	—	4	2	no	—
111	C₂³×C₁₃	—	4	4	no	—
112	C₂³×C₁₃	—	4	4	no	—
113	C₂³×C₁₃	—	4	8	no	—
114	C₂³×C₁₃	—	4	16	no	—
115	C₂³×C₁₃	—	8	8	no	—
116	C₂³×C₁₃	—	8	4	no	—
117	C₂³×C₁₃	—	8	4	no	—
118	C₂³×C₁₃	#120	1	2	yes	—
119	C₂³×C₁₃	C₂²⋊Q₈	1	2	yes	—
120	C₂³×C₁₃	#124	1	2	yes	—
121	C₂³×C₁₃	#135	1	4	yes	—
122	C₂³×C₁₃	AGL(1,ℤ/8)	1	4	yes	—
123	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
124	D₄×C₁₃	—	2	8	no	—
125	D₄×C₁₃	—	4	4	no	—
126	D₄×C₁₃	—	4	8	no	—
127	C₂×C₄×C₁₃	—	2	4	no	—
128	C₂×C₄×C₁₃	—	2	8	no	—
129	C₂×C₄×C₁₃	—	2	4	no	—
130	C₂×C₄×C₁₃	—	2	4	no	—
131	C₂×C₄×C₁₃	—	4	4	no	—
132	C₂×C₄×C₁₃	—	4	4	no	—
133	C₂×C₄×C₁₃	—	4	8	no	—
134	C₂×C₄×C₁₃	—	4	16	no	—
135	C₂×C₄×C₁₃	—	4	2	no	—
136	C₂×C₄×C₁₃	—	4	4	no	—
137	C₂×C₄×C₁₃	—	8	4	no	—
138	C₂×C₄×C₁₃	#120	1	2	yes	—
139	C₂×C₄×C₁₃	C₂²⋊Q₈	1	8	yes	—
140	C₂×C₄×C₁₃	C₂²⋊Q₈	1	2	yes	—
141	C₂×C₄×C₁₃	#124	1	8	yes	—
142	C₂×C₄×C₁₃	#124	1	2	yes	—
143	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
144	C₂×C₄×C₁₃	C₂²×Q₈	1	1	yes	characteristic
145	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
146	C₂×C₄×C₁₃	C₂×#41	1	1	yes	characteristic
147	C₈×C₁₃	—	4	4	no	—
148	C₂³×D₄	C₂²×C₁₃	1	1	yes	characteristic
149	C₂³×C₈	C₂²×C₁₃	1	2	yes	—
150	C₂×#125	—	2	8	no	—
151	C₂×#125	—	2	2	no	—
152	C₂×#125	C₂²×C₁₃	1	4	yes	complemented
153	C₂²×C₄⋊C₄	—	2	2	no	—
154	C₂²×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
155	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
156	C₂²×C₂²⋊C₄	—	2	2	no	—
157	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
158	C₂×#102	—	2	8	no	—
159	C₂×#102	C₂²×C₁₃	1	8	yes	—
160	#390	—	2	2	no	—
161	#390	C₂²×C₁₃	1	4	yes	—
162	#340	C₂²×C₁₃	1	8	yes	complemented
163	#336	C₂²×C₁₃	1	4	yes	complemented
164	C₂²×C₁₃	—	4	8	no	—
165	C₂²×C₁₃	—	4	4	no	—
166	C₂²×C₁₃	—	8	2	no	—
167	C₂²×C₁₃	—	8	2	no	—
168	C₂²×C₁₃	—	8	4	no	—
169	C₂²×C₁₃	—	8	16	no	—
170	C₂²×C₁₃	#393	1	4	yes	—
171	C₂²×C₁₃	#449	1	8	yes	—
172	C₂²×C₁₃	#478	1	8	yes	—
173	C₂²×C₁₃	#488	1	4	yes	—
174	C₂²×C₁₃	#490	1	4	yes	—
175	C₂²×C₁₃	C₂×#120	1	1	yes	characteristic
176	C₂²×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
177	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
178	C₂²×C₁₃	C₂×#135	1	2	yes	—
179	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	2	yes	—
180	C₄×C₁₃	—	4	4	no	—
181	C₄×C₁₃	—	4	8	no	—
182	C₄×C₁₃	—	8	2	no	—
183	C₄×C₁₃	#393	1	4	yes	—
184	C₄×C₁₃	C₂×C₂²⋊Q₈	1	2	yes	—
185	C₄×C₁₃	C₂×#124	1	2	yes	—
186	C₂⁵	—	2	1	no	—
187	C₂²×D₄	—	2	4	no	—
188	C₂²×D₄	—	2	1	no	—
189	C₂²×D₄	—	2	4	no	—
190	C₂²×D₄	—	2	2	no	—
191	C₂²×D₄	Q₈×C₁₃	1	4	yes	—
192	C₂²×D₄	C₂³×C₁₃	1	2	yes	—
193	C₂³×C₄	—	2	2	no	—
194	C₂³×C₄	—	2	1	no	—
195	C₂³×C₄	D₄×C₁₃	1	2	yes	—
196	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
197	C₂²×C₈	—	2	4	no	—
198	C₂²×C₈	—	2	4	no	—
199	C₂²×C₈	—	2	4	no	—
200	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
201	#125	—	2	16	no	—
202	#125	—	2	16	no	—
203	C₂×C₄⋊C₄	—	2	16	no	—
204	C₂×C₄⋊C₄	—	2	8	no	—
205	C₂×C₄⋊C₄	—	2	2	no	—
206	C₂×C₄⋊C₄	—	2	4	no	—
207	C₂×C₄⋊C₄	—	2	4	no	—
208	C₂×C₄⋊C₄	—	2	1	no	—
209	C₂×C₄⋊C₄	D₄×C₁₃	1	8	yes	—
210	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
211	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
212	C₂×C₂²⋊C₄	—	2	8	no	—
213	C₂×C₂²⋊C₄	—	4	8	no	—
214	C₂×C₄²	—	2	4	no	—
215	C₂×C₄²	D₄×C₁₃	1	2	yes	—
216	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
217	#102	—	2	16	no	—
218	C₂×C₁₃	—	8	4	no	—
219	C₂×C₁₃	#2025	1	8	yes	direct product factor
220	C₂×C₁₃	C₂×#393	1	1	yes	characteristic
221	C₂×C₁₃	C₂×#449	1	2	yes	—
222	C₂×C₁₃	C₂×#478	1	2	yes	—
223	C₂×C₁₃	C₂×#488	1	1	yes	characteristic
224	C₂×C₁₃	C₂×#490	1	1	yes	characteristic
225	C₂⁴	—	2	4	no	—
226	C₂⁴	—	2	2	no	—
227	C₂⁴	—	4	2	no	—
228	C₂⁴	—	4	2	no	—
229	C₂⁴	—	4	4	no	—
230	C₂⁴	—	4	4	no	—
231	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
232	C₂×D₄	—	2	16	no	—
233	C₂×D₄	—	2	4	no	—
234	C₂×D₄	—	2	4	no	—
235	C₂×D₄	—	2	4	no	—
236	C₂×D₄	—	2	2	no	—
237	C₂×D₄	—	4	2	no	—
238	C₂×D₄	—	4	2	no	—
239	C₂×D₄	—	4	8	no	—
240	C₂×D₄	C₂×Q₈×C₁₃	1	4	yes	—
241	C₂²×C₄	—	2	4	no	—
242	C₂²×C₄	—	2	8	no	—
243	C₂²×C₄	—	2	4	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	4	no	—
246	C₂²×C₄	—	2	2	no	—
247	C₂²×C₄	—	4	4	no	—
248	C₂²×C₄	—	4	4	no	—
249	C₂²×C₄	—	4	2	no	—
250	C₂²×C₄	—	4	4	no	—
251	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
252	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
253	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
254	C₂²×C₄	C₂×Q₈×C₁₃	1	2	yes	—
255	C₂²×C₄	C₁₃×#41	1	4	yes	—
256	C₂²×C₄	C₁₃×#41	1	2	yes	—
257	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
258	C₂×C₈	—	2	4	no	—
259	C₂×C₈	—	2	4	no	—
260	C₂×C₈	—	2	4	no	—
261	C₂×C₈	—	4	8	no	—
262	C₄⋊C₄	—	2	32	no	—
263	C₄⋊C₄	—	2	8	no	—
264	C₄⋊C₄	—	2	4	no	—
265	C₄⋊C₄	—	2	4	no	—
266	C₄⋊C₄	—	4	2	no	—
267	C₄⋊C₄	C₂×D₄×C₁₃	1	8	yes	—
268	C₂²⋊C₄	—	4	16	no	—
269	C₄²	—	2	4	no	—
270	C₄²	—	2	4	no	—
271	C₁₃	C₂×#2025	1	1	yes	characteristic, direct product factor, Sylow
272	C₂³	—	2	4	no	—
273	C₂³	—	4	8	no	—
274	C₂³	—	4	16	no	—
275	C₂³	—	4	2	no	—
276	C₂³	—	4	2	no	—
277	C₂³	—	4	4	no	—
278	C₂³	—	4	4	no	—
279	C₂³	—	8	8	no	—
280	C₂³	—	8	4	no	—
281	C₂³	—	8	4	no	—
282	C₂³	C₁₃×#120	1	2	yes	—
283	C₂³	C₁₃×C₂²⋊Q₈	1	2	yes	—
284	C₂³	C₁₃×#124	1	2	yes	—
285	C₂³	C₁₃×#135	1	4	yes	—
286	C₂³	C₁₃×AGL(1,ℤ/8)	1	4	yes	—
287	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
288	D₄	—	2	8	no	—
289	D₄	—	4	8	no	—
290	D₄	—	4	4	no	—
291	C₂×C₄	—	2	4	no	—
292	C₂×C₄	—	2	8	no	—
293	C₂×C₄	—	2	4	no	—
294	C₂×C₄	—	2	4	no	—
295	C₂×C₄	—	4	8	no	—
296	C₂×C₄	—	4	16	no	—
297	C₂×C₄	—	4	4	no	—
298	C₂×C₄	—	4	4	no	—
299	C₂×C₄	—	4	2	no	—
300	C₂×C₄	—	4	4	no	—
301	C₂×C₄	—	8	4	no	—
302	C₂×C₄	C₁₃×#120	1	2	yes	—
303	C₂×C₄	C₁₃×C₂²⋊Q₈	1	2	yes	—
304	C₂×C₄	C₁₃×C₂²⋊Q₈	1	8	yes	—
305	C₂×C₄	C₁₃×#124	1	2	yes	—
306	C₂×C₄	C₁₃×#124	1	8	yes	—
307	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
308	C₂×C₄	C₂²×Q₈×C₁₃	1	1	yes	characteristic
309	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
310	C₂×C₄	C₂×C₁₃×#41	1	1	yes	characteristic
311	C₈	—	4	4	no	—
312	C₂²	—	4	8	no	—
313	C₂²	—	4	4	no	—
314	C₂²	—	8	16	no	—
315	C₂²	—	8	2	no	—
316	C₂²	—	8	2	no	—
317	C₂²	—	8	4	no	—
318	C₂²	C₁₃×#393	1	4	yes	—
319	C₂²	C₁₃×#449	1	8	yes	—
320	C₂²	C₁₃×#478	1	8	yes	—
321	C₂²	C₁₃×#488	1	4	yes	—
322	C₂²	C₁₃×#490	1	4	yes	—
323	C₂²	C₂×C₁₃×#120	1	1	yes	characteristic
324	C₂²	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
325	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
326	C₂²	C₂×C₁₃×#135	1	2	yes	—
327	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	2	yes	—
328	C₄	—	4	4	no	—
329	C₄	—	4	8	no	—
330	C₄	—	8	2	no	—
331	C₄	C₁₃×#393	1	4	yes	—
332	C₄	C₂×C₁₃×C₂²⋊Q₈	1	2	yes	—
333	C₄	C₂×C₁₃×#124	1	2	yes	—
334	C₂	—	8	4	no	—
335	C₂	C₁₃×#2025	1	8	yes	direct product factor
336	C₂	C₂×C₁₃×#393	1	1	yes	characteristic
337	C₂	C₂×C₁₃×#449	1	2	yes	—
338	C₂	C₂×C₁₃×#478	1	2	yes	—
339	C₂	C₂×C₁₃×#488	1	1	yes	characteristic
340	C₂	C₂×C₁₃×#490	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂⁴×C₁₃	C₂×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴×C₁₃
prime radical	C₂³×D₄×C₁₃	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₃×#125	C₂	semidirect
4. C₁₃×#2025	C₂	direct (N × H)
5. C₂×C₁₃×#340	C₂	semidirect
6. C₂×C₁₃×#336	C₂	semidirect
11. C₂×C₁₃×#125	C₂²	semidirect
21. C₁₃×#340	C₂²	semidirect
22. C₁₃×#336	C₂²	semidirect
55. C₂×#2025	C₁₃	direct (N × H)
102. C₂²×#125	C₂×C₁₃	semidirect
105. #2025	C₂×C₁₃	direct (N × H)
106. C₂×#340	C₂×C₁₃	semidirect
107. C₂×#336	C₂×C₁₃	semidirect
152. C₂×#125	C₂²×C₁₃	semidirect
162. #340	C₂²×C₁₃	semidirect
163. #336	C₂²×C₁₃	semidirect
219. C₂×C₁₃	#2025	direct (N × H)
271. C₁₃	C₂×#2025	direct (N × H)
335. C₂	C₁₃×#2025	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×C₁₃×#125	2
C₂²×C₁₃×#102	2
C₂×C₁₃×#390	2
C₁₃×#2025	2
C₂×C₁₃×#340	2
C₂×C₁₃×#336	2
C₂×#2025	13

Subgroups — C2×C13×#2025 #15709787

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#2025 #15709787