Subgroups — C₁₃×#25912 #15712251

Order 3328.

1078 conjugacy classes of subgroups.

The subgroup lattice has 296 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
2	C₂×C₁₃×#491	C₂	1	2	yes	maximal, complemented
3	C₁₃×#2857	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#2026	C₂	1	4	yes	maximal, complemented
5	C₁₃×#1992	C₂	1	2	yes	maximal, complemented
6	C₁₃×#1872	C₂	1	4	yes	maximal
7	C₁₃×#1749	C₂	1	1	yes	maximal, characteristic
8	C₂×C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
9	C₂×C₁₃×#125	—	2	8	no	—
10	C₂×C₁₃×#125	—	2	2	no	—
11	C₂×C₁₃×#125	C₂²	1	2	yes	—
12	C₂×C₁₃×#125	C₂²	1	2	yes	—
13	C₂×C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
14	C₂²×C₁₃×C₄⋊C₄	—	2	2	no	—
15	C₂²×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
17	C₁₃×#491	—	2	8	no	—
18	C₂×C₁₃×#103	C₂²	1	8	yes	—
19	C₂×C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
20	C₂×C₁₃×#97	C₂²	1	2	yes	—
21	C₁₃×#391	C₂²	1	2	yes	—
22	C₁₃×#389	—	2	2	no	—
23	C₁₃×#384	C₂²	1	1	yes	characteristic
24	C₁₃×#380	—	2	2	no	—
25	C₁₃×#340	C₂²	1	8	yes	—
26	C₁₃×#336	C₂²	1	2	yes	—
27	C₂²×Q₈×C₁₃	C₂³	1	2	yes	—
28	C₂³×C₄×C₁₃	—	2	2	no	—
29	C₂³×C₄×C₁₃	D₄	1	2	yes	—
30	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
31	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
32	C₁₃×C₄⋊Q₈	—	2	4	no	—
33	C₁₃×#125	—	2	16	no	—
34	C₁₃×#125	—	2	8	no	—
35	C₁₃×#125	—	2	4	no	—
36	C₁₃×#125	D₄	1	8	yes	—
37	C₁₃×C₂²⋊Q₈	—	2	8	no	—
38	C₂×C₁₃×C₄⋊C₄	—	2	16	no	—
39	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
40	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
41	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
42	C₂×C₁₃×C₄⋊C₄	—	2	8	no	—
43	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
44	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
45	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
46	C₂×C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
47	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
48	C₂×C₁₃×C₂²⋊C₄	—	2	2	no	—
49	C₂×C₄²×C₁₃	—	2	4	no	—
50	C₂×C₄²×C₁₃	D₄	1	2	yes	—
51	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×C₄²×C₁₃	C₂³	1	1	yes	characteristic
53	C₁₃×#103	—	2	16	no	—
54	C₁₃×C₂²⋊C₈	—	2	4	no	—
55	C₁₃×#97	—	2	4	no	—
56	C₁₃×#95	—	2	4	no	—
57	#25912	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
58	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
59	C₂×Q₈×C₁₃	—	2	4	no	—
60	C₂×Q₈×C₁₃	—	2	4	no	—
61	C₂×Q₈×C₁₃	—	2	4	no	—
62	C₂²×C₄×C₁₃	—	2	2	no	—
63	C₂²×C₄×C₁₃	—	2	2	no	—
64	C₂²×C₄×C₁₃	—	2	8	no	—
65	C₂²×C₄×C₁₃	—	2	4	no	—
66	C₂²×C₄×C₁₃	—	2	4	no	—
67	C₂²×C₄×C₁₃	—	2	4	no	—
68	C₂²×C₄×C₁₃	—	2	2	no	—
69	C₂²×C₄×C₁₃	—	2	2	no	—
70	C₂²×C₄×C₁₃	—	4	2	no	—
71	C₂²×C₄×C₁₃	—	4	4	no	—
72	C₂²×C₄×C₁₃	—	4	4	no	—
73	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
74	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
75	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
76	C₂²×C₄×C₁₃	#41	1	4	yes	—
77	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
78	C₂×C₈×C₁₃	—	2	4	no	—
79	C₂×C₈×C₁₃	—	2	4	no	—
80	C₂×C₈×C₁₃	—	2	4	no	—
81	C₁₃×C₄⋊C₄	—	2	8	no	—
82	C₁₃×C₄⋊C₄	—	2	16	no	—
83	C₁₃×C₄⋊C₄	—	2	16	no	—
84	C₁₃×C₄⋊C₄	—	2	4	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	4	no	—
89	C₁₃×C₄⋊C₄	C₂×D₄	1	8	yes	—
90	C₁₃×C₂²⋊C₄	—	4	4	no	—
91	C₄²×C₁₃	—	2	4	no	—
92	C₄²×C₁₃	—	2	2	no	—
93	C₄²×C₁₃	C₂×D₄	1	4	yes	—
94	C₂²×#125	C₂×C₁₃	1	1	yes	characteristic
95	C₂×#491	C₂×C₁₃	1	2	yes	complemented
96	#2857	C₂×C₁₃	1	1	yes	characteristic
97	#2026	C₂×C₁₃	1	4	yes	complemented
98	#1992	C₂×C₁₃	1	2	yes	complemented
99	#1872	C₂×C₁₃	1	4	yes	—
100	#1749	C₂×C₁₃	1	1	yes	characteristic
101	C₂³×C₁₃	—	2	1	no	—
102	C₂³×C₁₃	—	2	1	no	—
103	C₂³×C₁₃	—	2	2	no	—
104	C₂³×C₁₃	#124	1	2	yes	—
105	C₂³×C₁₃	#137	1	4	yes	—
106	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
107	Q₈×C₁₃	—	4	8	no	—
108	C₂×C₄×C₁₃	—	2	1	no	—
109	C₂×C₄×C₁₃	—	2	1	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	2	2	no	—
112	C₂×C₄×C₁₃	—	2	8	no	—
113	C₂×C₄×C₁₃	—	2	2	no	—
114	C₂×C₄×C₁₃	—	2	8	no	—
115	C₂×C₄×C₁₃	—	2	2	no	—
116	C₂×C₄×C₁₃	—	2	2	no	—
117	C₂×C₄×C₁₃	—	4	8	no	—
118	C₂×C₄×C₁₃	—	4	16	no	—
119	C₂×C₄×C₁₃	—	4	4	no	—
120	C₂×C₄×C₁₃	—	4	4	no	—
121	C₂×C₄×C₁₃	—	4	2	no	—
122	C₂×C₄×C₁₃	—	4	2	no	—
123	C₂×C₄×C₁₃	—	4	2	no	—
124	C₂×C₄×C₁₃	—	4	2	no	—
125	C₂×C₄×C₁₃	—	4	4	no	—
126	C₂×C₄×C₁₃	#120	1	4	yes	—
127	C₂×C₄×C₁₃	#124	1	2	yes	—
128	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×#41	1	2	yes	—
130	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
131	C₈×C₁₃	—	4	4	no	—
132	C₂×C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
133	C₂×#125	—	2	8	no	—
134	C₂×#125	—	2	2	no	—
135	C₂×#125	C₂²×C₁₃	1	2	yes	—
136	C₂×#125	C₂²×C₁₃	1	2	yes	—
137	C₂×C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	—
138	C₂²×C₄⋊C₄	—	2	2	no	—
139	C₂²×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
140	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
141	#491	—	2	8	no	—
142	C₂×#103	C₂²×C₁₃	1	8	yes	—
143	C₂×C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
144	C₂×#97	C₂²×C₁₃	1	2	yes	—
145	#391	C₂²×C₁₃	1	2	yes	—
146	#389	—	2	2	no	—
147	#384	C₂²×C₁₃	1	1	yes	characteristic
148	#380	—	2	2	no	—
149	#340	C₂²×C₁₃	1	8	yes	—
150	#336	C₂²×C₁₃	1	2	yes	—
151	C₂²×C₁₃	—	2	2	no	—
152	C₂²×C₁₃	—	2	2	no	—
153	C₂²×C₁₃	—	2	4	no	—
154	C₂²×C₁₃	—	2	4	no	—
155	C₂²×C₁₃	#491	1	4	yes	complemented
156	C₂²×C₁₃	C₂×#120	1	1	yes	characteristic
157	C₂²×C₁₃	C₂×#124	1	1	yes	characteristic
158	C₂²×C₁₃	#536	1	1	yes	characteristic
159	C₂²×C₁₃	C₂×#137	1	2	yes	—
160	C₂²×C₁₃	#575	1	2	yes	—
161	C₄×C₁₃	—	2	2	no	—
162	C₄×C₁₃	—	2	4	no	—
163	C₄×C₁₃	—	4	2	no	—
164	C₄×C₁₃	—	4	8	no	—
165	C₄×C₁₃	—	8	2	no	—
166	C₄×C₁₃	—	8	2	no	—
167	C₄×C₁₃	#548	1	4	yes	—
168	C₂²×Q₈	C₂³×C₁₃	1	2	yes	—
169	C₂³×C₄	—	2	2	no	—
170	C₂³×C₄	D₄×C₁₃	1	2	yes	—
171	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
172	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
173	C₄⋊Q₈	—	2	4	no	—
174	#125	—	2	16	no	—
175	#125	—	2	8	no	—
176	#125	—	2	4	no	—
177	#125	D₄×C₁₃	1	8	yes	—
178	C₂²⋊Q₈	—	2	8	no	—
179	C₂×C₄⋊C₄	—	2	16	no	—
180	C₂×C₄⋊C₄	—	2	4	no	—
181	C₂×C₄⋊C₄	—	2	4	no	—
182	C₂×C₄⋊C₄	—	2	2	no	—
183	C₂×C₄⋊C₄	—	2	8	no	—
184	C₂×C₄⋊C₄	—	2	2	no	—
185	C₂×C₄⋊C₄	—	2	2	no	—
186	C₂×C₄⋊C₄	—	2	2	no	—
187	C₂×C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
188	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
189	C₂×C₂²⋊C₄	—	2	2	no	—
190	C₂×C₄²	—	2	4	no	—
191	C₂×C₄²	D₄×C₁₃	1	2	yes	—
192	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
193	C₂×C₄²	C₂³×C₁₃	1	1	yes	characteristic
194	#103	—	2	16	no	—
195	C₂²⋊C₈	—	2	4	no	—
196	#97	—	2	4	no	—
197	#95	—	2	4	no	—
198	C₂×C₁₃	—	2	4	no	—
199	C₂×C₁₃	#2436	1	1	yes	characteristic
200	C₂×C₁₃	#3007	1	2	yes	—
201	C₂×C₁₃	C₂×#491	1	1	yes	characteristic
202	C₂×C₁₃	#3142	1	1	yes	characteristic
203	C₂×C₁₃	#3236	1	2	yes	—
204	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
205	C₂×Q₈	—	2	4	no	—
206	C₂×Q₈	—	2	4	no	—
207	C₂×Q₈	—	2	4	no	—
208	C₂²×C₄	—	2	2	no	—
209	C₂²×C₄	—	2	2	no	—
210	C₂²×C₄	—	2	8	no	—
211	C₂²×C₄	—	2	4	no	—
212	C₂²×C₄	—	2	4	no	—
213	C₂²×C₄	—	2	4	no	—
214	C₂²×C₄	—	2	2	no	—
215	C₂²×C₄	—	2	2	no	—
216	C₂²×C₄	—	4	2	no	—
217	C₂²×C₄	—	4	4	no	—
218	C₂²×C₄	—	4	4	no	—
219	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
220	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
221	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
222	C₂²×C₄	C₁₃×#41	1	4	yes	—
223	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
224	C₂×C₈	—	2	4	no	—
225	C₂×C₈	—	2	4	no	—
226	C₂×C₈	—	2	4	no	—
227	C₄⋊C₄	—	2	8	no	—
228	C₄⋊C₄	—	2	16	no	—
229	C₄⋊C₄	—	2	16	no	—
230	C₄⋊C₄	—	2	4	no	—
231	C₄⋊C₄	—	2	4	no	—
232	C₄⋊C₄	—	4	4	no	—
233	C₄⋊C₄	—	4	4	no	—
234	C₄⋊C₄	—	4	4	no	—
235	C₄⋊C₄	C₂×D₄×C₁₃	1	8	yes	—
236	C₂²⋊C₄	—	4	4	no	—
237	C₄²	—	2	4	no	—
238	C₄²	—	2	2	no	—
239	C₄²	C₂×D₄×C₁₃	1	4	yes	—
240	C₁₃	#25912	1	1	yes	characteristic, direct product factor, Sylow
241	C₂³	—	2	1	no	—
242	C₂³	—	2	1	no	—
243	C₂³	—	2	2	no	—
244	C₂³	C₁₃×#124	1	2	yes	—
245	C₂³	C₁₃×#137	1	4	yes	—
246	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
247	Q₈	—	4	8	no	—
248	C₂×C₄	—	2	1	no	—
249	C₂×C₄	—	2	1	no	—
250	C₂×C₄	—	2	2	no	—
251	C₂×C₄	—	2	2	no	—
252	C₂×C₄	—	2	8	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	2	no	—
256	C₂×C₄	—	2	2	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	—	4	16	no	—
259	C₂×C₄	—	4	4	no	—
260	C₂×C₄	—	4	4	no	—
261	C₂×C₄	—	4	2	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	4	2	no	—
265	C₂×C₄	—	4	4	no	—
266	C₂×C₄	C₁₃×#120	1	4	yes	—
267	C₂×C₄	C₁₃×#124	1	2	yes	—
268	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
269	C₂×C₄	C₂×C₁₃×#41	1	2	yes	—
270	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
271	C₈	—	4	4	no	—
272	C₂²	—	2	4	no	—
273	C₂²	—	2	2	no	—
274	C₂²	—	2	2	no	—
275	C₂²	—	2	4	no	—
276	C₂²	C₁₃×#491	1	4	yes	complemented
277	C₂²	C₂×C₁₃×#120	1	1	yes	characteristic
278	C₂²	C₂×C₁₃×#124	1	1	yes	characteristic
279	C₂²	C₁₃×#536	1	1	yes	characteristic
280	C₂²	C₂×C₁₃×#137	1	2	yes	—
281	C₂²	C₁₃×#575	1	2	yes	—
282	C₄	—	2	4	no	—
283	C₄	—	2	2	no	—
284	C₄	—	4	2	no	—
285	C₄	—	4	8	no	—
286	C₄	—	8	2	no	—
287	C₄	—	8	2	no	—
288	C₄	C₁₃×#548	1	4	yes	—
289	C₂	—	2	4	no	—
290	C₂	C₁₃×#2436	1	1	yes	characteristic
291	C₂	C₁₃×#3007	1	2	yes	—
292	C₂	C₂×C₁₃×#491	1	1	yes	characteristic
293	C₂	C₁₃×#3142	1	1	yes	characteristic
294	C₂	C₁₃×#3236	1	2	yes	—

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

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×#491	C₂	semidirect
4. C₁₃×#2026	C₂	semidirect
5. C₁₃×#1992	C₂	semidirect
57. #25912	C₁₃	direct (N × H)
95. C₂×#491	C₂×C₁₃	semidirect
97. #2026	C₂×C₁₃	semidirect
98. #1992	C₂×C₁₃	semidirect
155. C₂²×C₁₃	#491	semidirect
240. C₁₃	#25912	direct (N × H)
276. C₂²	C₁₃×#491	semidirect

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
C₂×C₁₃×#491	2
C₁₃×#2857	2
C₁₃×#2026	2
C₁₃×#1992	2
C₁₃×#1872	2
C₁₃×#1749	2
#25912	13

Subgroups — C13×#25912 #15712251

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#25912 #15712251