Subgroups — C₂×C₁₃×#3259 #15747249

Order 3328.

2010 conjugacy classes of subgroups.

The subgroup lattice has 316 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#540	C₂	1	2	yes	maximal, complemented
2	C₁₃×#3259	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₁₃×#478	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₃×#469	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₃×#466	C₂	1	2	yes	maximal, complemented
6	C₂×C₁₃×#459	C₂	1	4	yes	maximal, complemented
7	C₂×C₁₃×#424	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
9	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
10	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₁₃×#540	—	2	8	no	—
13	C₁₃×#540	C₂²	1	16	yes	complemented
14	C₂×C₁₃×#127	C₂²	1	2	yes	—
15	C₂×C₁₃×#125	C₂²	1	2	yes	—
16	C₂×C₁₃×#123	—	2	4	no	—
17	C₂×C₁₃×#121	—	2	4	no	—
18	C₂×C₁₃×#121	C₂²	1	4	yes	—
19	C₂×C₄×D₄×C₁₃	C₂²	1	4	yes	—
20	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
21	C₁₃×#478	C₂²	1	32	yes	complemented
22	C₁₃×#469	C₂²	1	16	yes	complemented
23	C₁₃×#466	C₂²	1	16	yes	complemented
24	C₁₃×#459	C₂²	1	32	yes	complemented
25	C₂×C₁₃×#107	C₂²	1	2	yes	—
26	C₂×C₁₃×#106	C₂²	1	2	yes	—
27	C₁₃×#424	C₂²	1	8	yes	complemented
28	C₂×C₁₃×C₄⋊C₈	C₂²	1	4	yes	—
29	C₂×C₁₃×#102	C₂²	1	4	yes	—
30	C₂×C₁₃×#102	C₂²	1	4	yes	—
31	C₂²×D₄×C₁₃	—	2	4	no	—
32	C₂²×D₄×C₁₃	—	2	4	no	—
33	C₂²×D₄×C₁₃	C₂³	1	4	yes	—
34	C₂³×C₄×C₁₃	—	2	4	no	—
35	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
36	C₂×C₁₃×D₈	—	4	4	no	—
37	C₂×C₁₃×D₈	—	4	4	no	—
38	C₂×C₁₃×D₈	C₂³	1	8	yes	—
39	C₂×C₁₃×D₈	C₂³	1	8	yes	—
40	C₂×C₁₃×M₁₆	—	4	2	no	—
41	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
42	C₂²×C₈×C₁₃	—	4	2	no	—
43	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
44	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
45	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
46	C₁₃×#127	—	2	4	no	—
47	C₁₃×#127	C₂³	1	8	yes	—
48	C₁₃×#125	—	2	4	no	—
49	C₁₃×#125	C₂³	1	8	yes	—
50	C₁₃×#123	—	2	32	no	—
51	C₁₃×#121	—	2	8	no	—
52	C₁₃×#121	—	2	32	no	—
53	C₁₃×#121	C₂³	1	16	yes	—
54	C₄×D₄×C₁₃	—	2	8	no	—
55	C₄×D₄×C₁₃	C₂³	1	16	yes	—
56	C₁₃×#117	D₄	1	4	yes	—
57	C₁₃×#117	C₂³	1	4	yes	—
58	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
59	C₂×C₁₃×C₄⋊C₄	D₄	1	2	yes	—
60	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
61	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
62	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
63	C₂×C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
64	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
65	C₁₃×#107	C₂³	1	8	yes	—
66	C₁₃×#106	C₂³	1	8	yes	—
67	C₁₃×C₄⋊C₈	C₂³	1	16	yes	—
68	C₁₃×#102	C₂³	1	16	yes	—
69	C₁₃×#102	C₂³	1	16	yes	—
70	C₂×#3259	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂⁴×C₁₃	—	2	4	no	—
72	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
73	C₂×D₄×C₁₃	—	2	16	no	—
74	C₂×D₄×C₁₃	—	2	4	no	—
75	C₂×D₄×C₁₃	—	2	16	no	—
76	C₂×D₄×C₁₃	—	2	8	no	—
77	C₂×D₄×C₁₃	—	2	8	no	—
78	C₂×D₄×C₁₃	—	8	8	no	—
79	C₂×D₄×C₁₃	—	8	8	no	—
80	C₂×D₄×C₁₃	C₂⁴	1	8	yes	—
81	C₂²×C₄×C₁₃	—	2	2	no	—
82	C₂²×C₄×C₁₃	—	2	2	no	—
83	C₂²×C₄×C₁₃	—	2	4	no	—
84	C₂²×C₄×C₁₃	—	2	16	no	—
85	C₂²×C₄×C₁₃	—	8	8	no	—
86	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
87	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
88	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
90	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
91	C₁₃×D₈	—	4	8	no	—
92	C₁₃×D₈	—	4	8	no	—
93	C₁₃×M₁₆	—	4	4	no	—
94	C₂×C₈×C₁₃	—	2	4	no	—
95	C₂×C₈×C₁₃	—	2	4	no	—
96	C₂×C₈×C₁₃	—	4	4	no	—
97	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
98	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
99	C₁₃×C₄⋊C₄	—	2	16	no	—
100	C₁₃×C₄⋊C₄	—	2	4	no	—
101	C₁₃×C₄⋊C₄	C₂×D₄	1	8	yes	—
102	C₁₃×C₄⋊C₄	C₂⁴	1	8	yes	—
103	C₁₃×C₂²⋊C₄	—	2	16	no	—
104	C₁₃×C₂²⋊C₄	—	2	16	no	—
105	C₁₃×C₂²⋊C₄	C₂×D₄	1	8	yes	—
106	C₄²×C₁₃	C₂×D₄	1	4	yes	—
107	C₄²×C₁₃	C₂⁴	1	4	yes	—
108	C₂×#540	C₂×C₁₃	1	2	yes	complemented
109	#3259	C₂×C₁₃	1	16	yes	direct product factor
110	C₂×#478	C₂×C₁₃	1	4	yes	complemented
111	C₂×#469	C₂×C₁₃	1	2	yes	complemented
112	C₂×#466	C₂×C₁₃	1	2	yes	complemented
113	C₂×#459	C₂×C₁₃	1	4	yes	complemented
114	C₂×#424	C₂×C₁₃	1	1	yes	characteristic, complemented
115	C₂³×C₁₃	—	2	2	no	—
116	C₂³×C₁₃	—	2	4	no	—
117	C₂³×C₁₃	—	2	8	no	—
118	C₂³×C₁₃	—	4	8	no	—
119	C₂³×C₁₃	—	4	8	no	—
120	C₂³×C₁₃	—	4	8	no	—
121	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
122	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
123	D₄×C₁₃	—	2	8	no	—
124	D₄×C₁₃	—	2	8	no	—
125	D₄×C₁₃	—	4	8	no	—
126	D₄×C₁₃	—	8	16	no	—
127	D₄×C₁₃	—	8	16	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	8	no	—
130	C₂×C₄×C₁₃	—	2	4	no	—
131	C₂×C₄×C₁₃	—	2	2	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	8	no	—
134	C₂×C₄×C₁₃	—	4	8	no	—
135	C₂×C₄×C₁₃	—	8	16	no	—
136	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
137	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
138	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
139	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
140	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
141	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
142	C₈×C₁₃	—	2	4	no	—
143	C₈×C₁₃	—	2	4	no	—
144	C₂²×D₈	C₂²×C₁₃	1	2	yes	—
145	C₂²×D₈	C₂²×C₁₃	1	2	yes	—
146	C₂²×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
147	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
148	#540	—	2	8	no	—
149	#540	C₂²×C₁₃	1	16	yes	complemented
150	C₂×#127	C₂²×C₁₃	1	2	yes	—
151	C₂×#125	C₂²×C₁₃	1	2	yes	—
152	C₂×#123	—	2	4	no	—
153	C₂×#121	—	2	4	no	—
154	C₂×#121	C₂²×C₁₃	1	4	yes	—
155	C₂×C₄×D₄	C₂²×C₁₃	1	4	yes	—
156	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
157	#478	C₂²×C₁₃	1	32	yes	complemented
158	#469	C₂²×C₁₃	1	16	yes	complemented
159	#466	C₂²×C₁₃	1	16	yes	complemented
160	#459	C₂²×C₁₃	1	32	yes	complemented
161	C₂×#107	C₂²×C₁₃	1	2	yes	—
162	C₂×#106	C₂²×C₁₃	1	2	yes	—
163	#424	C₂²×C₁₃	1	8	yes	complemented
164	C₂×C₄⋊C₈	C₂²×C₁₃	1	4	yes	—
165	C₂×#102	C₂²×C₁₃	1	4	yes	—
166	C₂×#102	C₂²×C₁₃	1	4	yes	—
167	C₂²×C₁₃	—	2	2	no	—
168	C₂²×C₁₃	—	2	4	no	—
169	C₂²×C₁₃	—	4	4	no	—
170	C₂²×C₁₃	—	4	8	no	—
171	C₂²×C₁₃	—	4	8	no	—
172	C₂²×C₁₃	—	4	8	no	—
173	C₂²×C₁₃	—	8	16	no	—
174	C₂²×C₁₃	#537	1	2	yes	—
175	C₂²×C₁₃	#576	1	4	yes	—
176	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
177	C₄×C₁₃	—	2	2	no	—
178	C₄×C₁₃	—	2	8	no	—
179	C₄×C₁₃	—	4	8	no	—
180	C₄×C₁₃	#537	1	2	yes	—
181	C₄×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
182	C₄×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
183	C₂²×D₄	—	2	4	no	—
184	C₂²×D₄	—	2	4	no	—
185	C₂²×D₄	C₂³×C₁₃	1	4	yes	—
186	C₂³×C₄	—	2	4	no	—
187	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
188	C₂×D₈	—	4	4	no	—
189	C₂×D₈	—	4	4	no	—
190	C₂×D₈	C₂³×C₁₃	1	8	yes	—
191	C₂×D₈	C₂³×C₁₃	1	8	yes	—
192	C₂×M₁₆	—	4	2	no	—
193	C₂×M₁₆	C₂³×C₁₃	1	4	yes	—
194	C₂²×C₈	—	4	2	no	—
195	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
196	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
197	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
198	#127	—	2	4	no	—
199	#127	C₂³×C₁₃	1	8	yes	—
200	#125	—	2	4	no	—
201	#125	C₂³×C₁₃	1	8	yes	—
202	#123	—	2	32	no	—
203	#121	—	2	8	no	—
204	#121	—	2	32	no	—
205	#121	C₂³×C₁₃	1	16	yes	—
206	C₄×D₄	—	2	8	no	—
207	C₄×D₄	C₂³×C₁₃	1	16	yes	—
208	#117	D₄×C₁₃	1	4	yes	—
209	#117	C₂³×C₁₃	1	4	yes	—
210	C₂×C₄⋊C₄	—	2	4	no	—
211	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
212	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
213	C₂×C₂²⋊C₄	—	2	4	no	—
214	C₂×C₂²⋊C₄	—	2	4	no	—
215	C₂×C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
216	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
217	#107	C₂³×C₁₃	1	8	yes	—
218	#106	C₂³×C₁₃	1	8	yes	—
219	C₄⋊C₈	C₂³×C₁₃	1	16	yes	—
220	#102	C₂³×C₁₃	1	16	yes	—
221	#102	C₂³×C₁₃	1	16	yes	—
222	C₂×C₁₃	—	4	2	no	—
223	C₂×C₁₃	—	8	8	no	—
224	C₂×C₁₃	#3259	1	4	yes	direct product factor
225	C₂×C₁₃	C₂×#537	1	1	yes	characteristic
226	C₂×C₁₃	C₂×#576	1	2	yes	—
227	C₂⁴	—	2	4	no	—
228	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
229	C₂×D₄	—	2	16	no	—
230	C₂×D₄	—	2	4	no	—
231	C₂×D₄	—	2	16	no	—
232	C₂×D₄	—	2	8	no	—
233	C₂×D₄	—	2	8	no	—
234	C₂×D₄	—	8	8	no	—
235	C₂×D₄	—	8	8	no	—
236	C₂×D₄	C₂⁴×C₁₃	1	8	yes	—
237	C₂²×C₄	—	2	2	no	—
238	C₂²×C₄	—	2	2	no	—
239	C₂²×C₄	—	2	4	no	—
240	C₂²×C₄	—	2	16	no	—
241	C₂²×C₄	—	8	8	no	—
242	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
243	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
244	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
245	C₂²×C₄	C₂⁴×C₁₃	1	2	yes	—
246	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
247	D₈	—	4	8	no	—
248	D₈	—	4	8	no	—
249	M₁₆	—	4	4	no	—
250	C₂×C₈	—	2	4	no	—
251	C₂×C₈	—	2	4	no	—
252	C₂×C₈	—	4	4	no	—
253	C₂×C₈	C₂⁴×C₁₃	1	4	yes	—
254	C₂×C₈	C₂⁴×C₁₃	1	4	yes	—
255	C₄⋊C₄	—	2	16	no	—
256	C₄⋊C₄	—	2	4	no	—
257	C₄⋊C₄	C₂×D₄×C₁₃	1	8	yes	—
258	C₄⋊C₄	C₂⁴×C₁₃	1	8	yes	—
259	C₂²⋊C₄	—	2	16	no	—
260	C₂²⋊C₄	—	2	16	no	—
261	C₂²⋊C₄	C₂×D₄×C₁₃	1	8	yes	—
262	C₄²	C₂×D₄×C₁₃	1	4	yes	—
263	C₄²	C₂⁴×C₁₃	1	4	yes	—
264	C₁₃	C₂×#3259	1	1	yes	characteristic, direct product factor, Sylow
265	C₂³	—	2	2	no	—
266	C₂³	—	2	4	no	—
267	C₂³	—	2	8	no	—
268	C₂³	—	4	8	no	—
269	C₂³	—	4	8	no	—
270	C₂³	—	4	8	no	—
271	C₂³	C₂²×D₄×C₁₃	1	2	yes	—
272	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
273	D₄	—	2	8	no	—
274	D₄	—	2	8	no	—
275	D₄	—	4	8	no	—
276	D₄	—	8	16	no	—
277	D₄	—	8	16	no	—
278	C₂×C₄	—	2	2	no	—
279	C₂×C₄	—	2	8	no	—
280	C₂×C₄	—	2	4	no	—
281	C₂×C₄	—	2	2	no	—
282	C₂×C₄	—	2	2	no	—
283	C₂×C₄	—	2	8	no	—
284	C₂×C₄	—	4	8	no	—
285	C₂×C₄	—	8	16	no	—
286	C₂×C₄	C₂²×D₄×C₁₃	1	8	yes	—
287	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
288	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
289	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
290	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
291	C₂×C₄	C₂⁵×C₁₃	1	1	yes	characteristic
292	C₈	—	2	4	no	—
293	C₈	—	2	4	no	—
294	C₂²	—	2	4	no	—
295	C₂²	—	2	2	no	—
296	C₂²	—	4	4	no	—
297	C₂²	—	4	8	no	—
298	C₂²	—	4	8	no	—
299	C₂²	—	4	8	no	—
300	C₂²	—	8	16	no	—
301	C₂²	C₁₃×#537	1	2	yes	—
302	C₂²	C₁₃×#576	1	4	yes	—
303	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
304	C₄	—	2	8	no	—
305	C₄	—	2	2	no	—
306	C₄	—	4	8	no	—
307	C₄	C₁₃×#537	1	2	yes	—
308	C₄	C₂×C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
309	C₄	C₂×C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
310	C₂	—	4	2	no	—
311	C₂	—	8	8	no	—
312	C₂	C₁₃×#3259	1	4	yes	direct product factor
313	C₂	C₂×C₁₃×#537	1	1	yes	characteristic
314	C₂	C₂×C₁₃×#576	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₃
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ C₂×C₁₃×#117 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#540	C₂	semidirect
2. C₁₃×#3259	C₂	direct (N × H)
3. C₂×C₁₃×#478	C₂	semidirect
4. C₂×C₁₃×#469	C₂	semidirect
5. C₂×C₁₃×#466	C₂	semidirect
6. C₂×C₁₃×#459	C₂	semidirect
7. C₂×C₁₃×#424	C₂	semidirect
13. C₁₃×#540	C₂²	semidirect
21. C₁₃×#478	C₂²	semidirect
22. C₁₃×#469	C₂²	semidirect
23. C₁₃×#466	C₂²	semidirect
24. C₁₃×#459	C₂²	semidirect
27. C₁₃×#424	C₂²	semidirect
70. C₂×#3259	C₁₃	direct (N × H)
108. C₂×#540	C₂×C₁₃	semidirect
109. #3259	C₂×C₁₃	direct (N × H)
110. C₂×#478	C₂×C₁₃	semidirect
111. C₂×#469	C₂×C₁₃	semidirect
112. C₂×#466	C₂×C₁₃	semidirect
113. C₂×#459	C₂×C₁₃	semidirect
114. C₂×#424	C₂×C₁₃	semidirect
149. #540	C₂²×C₁₃	semidirect
157. #478	C₂²×C₁₃	semidirect
158. #469	C₂²×C₁₃	semidirect
159. #466	C₂²×C₁₃	semidirect
160. #459	C₂²×C₁₃	semidirect
163. #424	C₂²×C₁₃	semidirect
224. C₂×C₁₃	#3259	direct (N × H)
264. C₁₃	C₂×#3259	direct (N × H)
312. C₂	C₁₃×#3259	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×#540	2
C₁₃×#3259	2
C₂×C₁₃×#478	2
C₂×C₁₃×#469	2
C₂×C₁₃×#466	2
C₂×C₁₃×#459	2
C₂×C₁₃×#424	2
C₂×#3259	13

Subgroups — C2×C13×#3259 #15747249

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#3259 #15747249