Subgroups — C₂×C₁₃×#3253 #15747242

Order 3328.

2426 conjugacy classes of subgroups.

The subgroup lattice has 324 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	2	yes	maximal, complemented
2	C₁₃×#3253	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₁₃×#476	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₃×#469	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₃×#459	C₂	1	4	yes	maximal, complemented
6	C₂×C₁₃×#423	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₁₃×#41	—	2	2	no	—
8	C₂²×C₁₃×D₈	C₂²	1	4	yes	—
9	C₂²×C₁₃×M₁₆	C₂²	1	2	yes	—
10	C₁₃×#532	—	2	8	no	—
11	C₁₃×#532	C₂²	1	16	yes	complemented
12	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
13	C₂×C₁₃×#127	C₂²	1	2	yes	—
14	C₂×C₁₃×#124	—	2	2	no	—
15	C₂×C₁₃×#121	—	2	2	no	—
16	C₂×C₁₃×#121	C₂²	1	4	yes	—
17	C₂×C₄×D₄×C₁₃	—	2	2	no	—
18	C₂×C₄×D₄×C₁₃	C₂²	1	4	yes	—
19	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×#476	C₂²	1	32	yes	complemented
21	C₁₃×#469	C₂²	1	32	yes	complemented
22	C₁₃×#459	C₂²	1	32	yes	complemented
23	C₂×C₁₃×#106	C₂²	1	4	yes	—
24	C₁₃×#423	C₂²	1	8	yes	complemented
25	C₂×C₁₃×C₄⋊C₈	C₂²	1	4	yes	—
26	C₂×C₁₃×#102	C₂²	1	8	yes	—
27	C₂×C₁₃×#41	—	2	16	no	—
28	C₂×C₁₃×#41	—	4	16	no	—
29	C₂²×Q₈×C₁₃	—	2	2	no	—
30	C₂²×D₄×C₁₃	—	2	4	no	—
31	C₂²×D₄×C₁₃	—	2	2	no	—
32	C₂²×D₄×C₁₃	—	2	2	no	—
33	C₂²×D₄×C₁₃	C₂³	1	4	yes	—
34	C₂³×C₄×C₁₃	—	2	4	no	—
35	C₂³×C₄×C₁₃	D₄	1	2	yes	—
36	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
37	C₂×C₁₃×D₈	—	4	8	no	—
38	C₂×C₁₃×D₈	C₂³	1	16	yes	—
39	C₂×C₁₃×M₁₆	—	4	4	no	—
40	C₂×C₁₃×M₁₆	C₂³	1	8	yes	—
41	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
42	C₁₃×C₄⋊Q₈	—	2	4	no	—
43	C₁₃×C₄⋊Q₈	C₂³	1	8	yes	—
44	C₁₃×#127	—	2	4	no	—
45	C₁₃×#127	C₂³	1	8	yes	—
46	C₁₃×#124	—	2	16	no	—
47	C₁₃×#121	—	2	16	no	—
48	C₁₃×#121	—	2	8	no	—
49	C₁₃×#121	C₂³	1	16	yes	—
50	C₄×D₄×C₁₃	—	2	16	no	—
51	C₄×D₄×C₁₃	—	2	8	no	—
52	C₄×D₄×C₁₃	C₂³	1	16	yes	—
53	C₂×C₁₃×C₄⋊C₄	—	2	2	no	—
54	C₂×C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
55	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
56	C₂×C₁₃×C₂²⋊C₄	—	2	4	no	—
57	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
58	C₂×C₄²×C₁₃	D₄	1	1	yes	characteristic
59	C₂×C₄²×C₁₃	D₄	1	4	yes	—
60	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
61	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
62	C₁₃×#106	C₂³	1	16	yes	—
63	C₁₃×C₄⋊C₈	C₂³	1	16	yes	—
64	C₁₃×#102	C₂³	1	32	yes	—
65	C₂×#3253	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂⁴×C₁₃	—	2	4	no	—
67	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
68	C₁₃×#41	—	4	64	no	—
69	C₂×Q₈×C₁₃	—	2	8	no	—
70	C₂×Q₈×C₁₃	—	4	8	no	—
71	C₂×D₄×C₁₃	—	2	16	no	—
72	C₂×D₄×C₁₃	—	2	4	no	—
73	C₂×D₄×C₁₃	—	2	8	no	—
74	C₂×D₄×C₁₃	—	2	8	no	—
75	C₂×D₄×C₁₃	—	2	8	no	—
76	C₂×D₄×C₁₃	—	2	8	no	—
77	C₂×D₄×C₁₃	—	4	8	no	—
78	C₂×D₄×C₁₃	—	4	16	no	—
79	C₂×D₄×C₁₃	—	4	8	no	—
80	C₂×D₄×C₁₃	C₂⁴	1	8	yes	—
81	C₂²×C₄×C₁₃	—	2	8	no	—
82	C₂²×C₄×C₁₃	—	2	2	no	—
83	C₂²×C₄×C₁₃	—	2	2	no	—
84	C₂²×C₄×C₁₃	—	2	4	no	—
85	C₂²×C₄×C₁₃	—	2	16	no	—
86	C₂²×C₄×C₁₃	—	4	16	no	—
87	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
88	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
89	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
90	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
91	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
92	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
93	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
94	C₁₃×D₈	—	4	16	no	—
95	C₁₃×M₁₆	—	4	8	no	—
96	C₂×C₈×C₁₃	—	2	8	no	—
97	C₂×C₈×C₁₃	C₂⁴	1	8	yes	—
98	C₁₃×C₄⋊C₄	—	2	8	no	—
99	C₁₃×C₄⋊C₄	—	2	4	no	—
100	C₁₃×C₄⋊C₄	C₂⁴	1	8	yes	—
101	C₁₃×C₂²⋊C₄	—	2	16	no	—
102	C₁₃×C₂²⋊C₄	—	2	16	no	—
103	C₄²×C₁₃	C₂×D₄	1	4	yes	—
104	C₄²×C₁₃	C₂×D₄	1	4	yes	—
105	C₄²×C₁₃	C₂×D₄	1	4	yes	—
106	C₄²×C₁₃	C₂⁴	1	4	yes	—
107	C₂×#532	C₂×C₁₃	1	2	yes	complemented
108	#3253	C₂×C₁₃	1	16	yes	direct product factor
109	C₂×#476	C₂×C₁₃	1	4	yes	complemented
110	C₂×#469	C₂×C₁₃	1	4	yes	complemented
111	C₂×#459	C₂×C₁₃	1	4	yes	complemented
112	C₂×#423	C₂×C₁₃	1	1	yes	characteristic, complemented
113	C₂³×C₁₃	—	2	2	no	—
114	C₂³×C₁₃	—	2	4	no	—
115	C₂³×C₁₃	—	2	8	no	—
116	C₂³×C₁₃	—	4	8	no	—
117	C₂³×C₁₃	—	4	8	no	—
118	C₂³×C₁₃	—	4	8	no	—
119	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
120	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
121	Q₈×C₁₃	—	4	16	no	—
122	D₄×C₁₃	—	2	8	no	—
123	D₄×C₁₃	—	2	8	no	—
124	D₄×C₁₃	—	4	16	no	—
125	D₄×C₁₃	—	4	32	no	—
126	D₄×C₁₃	—	4	8	no	—
127	D₄×C₁₃	—	4	16	no	—
128	C₂×C₄×C₁₃	—	2	2	no	—
129	C₂×C₄×C₁₃	—	2	4	no	—
130	C₂×C₄×C₁₃	—	2	4	no	—
131	C₂×C₄×C₁₃	—	2	16	no	—
132	C₂×C₄×C₁₃	—	2	2	no	—
133	C₂×C₄×C₁₃	—	2	2	no	—
134	C₂×C₄×C₁₃	—	2	8	no	—
135	C₂×C₄×C₁₃	—	4	8	no	—
136	C₂×C₄×C₁₃	—	4	32	no	—
137	C₂×C₄×C₁₃	AGL(1,ℤ/8)	1	8	yes	—
138	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
139	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
140	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
141	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
142	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
143	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
144	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
145	C₈×C₁₃	—	2	8	no	—
146	C₂²×#41	—	2	2	no	—
147	C₂²×D₈	C₂²×C₁₃	1	4	yes	—
148	C₂²×M₁₆	C₂²×C₁₃	1	2	yes	—
149	#532	—	2	8	no	—
150	#532	C₂²×C₁₃	1	16	yes	complemented
151	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
152	C₂×#127	C₂²×C₁₃	1	2	yes	—
153	C₂×#124	—	2	2	no	—
154	C₂×#121	—	2	2	no	—
155	C₂×#121	C₂²×C₁₃	1	4	yes	—
156	C₂×C₄×D₄	—	2	2	no	—
157	C₂×C₄×D₄	C₂²×C₁₃	1	4	yes	—
158	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
159	#476	C₂²×C₁₃	1	32	yes	complemented
160	#469	C₂²×C₁₃	1	32	yes	complemented
161	#459	C₂²×C₁₃	1	32	yes	complemented
162	C₂×#106	C₂²×C₁₃	1	4	yes	—
163	#423	C₂²×C₁₃	1	8	yes	complemented
164	C₂×C₄⋊C₈	C₂²×C₁₃	1	4	yes	—
165	C₂×#102	C₂²×C₁₃	1	8	yes	—
166	C₂²×C₁₃	—	2	2	no	—
167	C₂²×C₁₃	—	2	4	no	—
168	C₂²×C₁₃	—	4	4	no	—
169	C₂²×C₁₃	—	4	8	no	—
170	C₂²×C₁₃	—	4	8	no	—
171	C₂²×C₁₃	—	4	8	no	—
172	C₂²×C₁₃	—	8	16	no	—
173	C₂²×C₁₃	#537	1	2	yes	—
174	C₂²×C₁₃	C₂×AGL(1,ℤ/8)	1	4	yes	—
175	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
176	C₄×C₁₃	—	2	2	no	—
177	C₄×C₁₃	—	2	4	no	—
178	C₄×C₁₃	—	4	8	no	—
179	C₄×C₁₃	#537	1	2	yes	—
180	C₄×C₁₃	C₂×AGL(1,ℤ/8)	1	8	yes	—
181	C₄×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
182	C₄×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
183	C₂×#41	—	2	16	no	—
184	C₂×#41	—	4	16	no	—
185	C₂²×Q₈	—	2	2	no	—
186	C₂²×D₄	—	2	4	no	—
187	C₂²×D₄	—	2	2	no	—
188	C₂²×D₄	—	2	2	no	—
189	C₂²×D₄	C₂³×C₁₃	1	4	yes	—
190	C₂³×C₄	—	2	4	no	—
191	C₂³×C₄	D₄×C₁₃	1	2	yes	—
192	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
193	C₂×D₈	—	4	8	no	—
194	C₂×D₈	C₂³×C₁₃	1	16	yes	—
195	C₂×M₁₆	—	4	4	no	—
196	C₂×M₁₆	C₂³×C₁₃	1	8	yes	—
197	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
198	C₄⋊Q₈	—	2	4	no	—
199	C₄⋊Q₈	C₂³×C₁₃	1	8	yes	—
200	#127	—	2	4	no	—
201	#127	C₂³×C₁₃	1	8	yes	—
202	#124	—	2	16	no	—
203	#121	—	2	16	no	—
204	#121	—	2	8	no	—
205	#121	C₂³×C₁₃	1	16	yes	—
206	C₄×D₄	—	2	16	no	—
207	C₄×D₄	—	2	8	no	—
208	C₄×D₄	C₂³×C₁₃	1	16	yes	—
209	C₂×C₄⋊C₄	—	2	2	no	—
210	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
211	C₂×C₂²⋊C₄	—	2	4	no	—
212	C₂×C₂²⋊C₄	—	2	4	no	—
213	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
214	C₂×C₄²	D₄×C₁₃	1	1	yes	characteristic
215	C₂×C₄²	D₄×C₁₃	1	4	yes	—
216	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
217	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
218	#106	C₂³×C₁₃	1	16	yes	—
219	C₄⋊C₈	C₂³×C₁₃	1	16	yes	—
220	#102	C₂³×C₁₃	1	32	yes	—
221	C₂×C₁₃	—	4	2	no	—
222	C₂×C₁₃	—	8	8	no	—
223	C₂×C₁₃	#3253	1	4	yes	direct product factor
224	C₂×C₁₃	C₂×#537	1	1	yes	characteristic
225	C₂×C₁₃	C₂²×AGL(1,ℤ/8)	1	2	yes	—
226	C₂⁴	—	2	4	no	—
227	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
228	#41	—	4	64	no	—
229	C₂×Q₈	—	2	8	no	—
230	C₂×Q₈	—	4	8	no	—
231	C₂×D₄	—	2	16	no	—
232	C₂×D₄	—	2	4	no	—
233	C₂×D₄	—	2	8	no	—
234	C₂×D₄	—	2	8	no	—
235	C₂×D₄	—	2	8	no	—
236	C₂×D₄	—	2	8	no	—
237	C₂×D₄	—	4	8	no	—
238	C₂×D₄	—	4	16	no	—
239	C₂×D₄	—	4	8	no	—
240	C₂×D₄	C₂⁴×C₁₃	1	8	yes	—
241	C₂²×C₄	—	2	8	no	—
242	C₂²×C₄	—	2	2	no	—
243	C₂²×C₄	—	2	2	no	—
244	C₂²×C₄	—	2	4	no	—
245	C₂²×C₄	—	2	16	no	—
246	C₂²×C₄	—	4	16	no	—
247	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
248	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
249	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
250	C₂²×C₄	C₂×D₄×C₁₃	1	8	yes	—
251	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
252	C₂²×C₄	C₂⁴×C₁₃	1	2	yes	—
253	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
254	D₈	—	4	16	no	—
255	M₁₆	—	4	8	no	—
256	C₂×C₈	—	2	8	no	—
257	C₂×C₈	C₂⁴×C₁₃	1	8	yes	—
258	C₄⋊C₄	—	2	8	no	—
259	C₄⋊C₄	—	2	4	no	—
260	C₄⋊C₄	C₂⁴×C₁₃	1	8	yes	—
261	C₂²⋊C₄	—	2	16	no	—
262	C₂²⋊C₄	—	2	16	no	—
263	C₄²	C₂×D₄×C₁₃	1	4	yes	—
264	C₄²	C₂×D₄×C₁₃	1	4	yes	—
265	C₄²	C₂×D₄×C₁₃	1	4	yes	—
266	C₄²	C₂⁴×C₁₃	1	4	yes	—
267	C₁₃	C₂×#3253	1	1	yes	characteristic, direct product factor, Sylow
268	C₂³	—	2	2	no	—
269	C₂³	—	2	4	no	—
270	C₂³	—	2	8	no	—
271	C₂³	—	4	8	no	—
272	C₂³	—	4	8	no	—
273	C₂³	—	4	8	no	—
274	C₂³	C₂²×D₄×C₁₃	1	2	yes	—
275	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
276	Q₈	—	4	16	no	—
277	D₄	—	2	8	no	—
278	D₄	—	2	8	no	—
279	D₄	—	4	16	no	—
280	D₄	—	4	32	no	—
281	D₄	—	4	8	no	—
282	D₄	—	4	16	no	—
283	C₂×C₄	—	2	2	no	—
284	C₂×C₄	—	2	4	no	—
285	C₂×C₄	—	2	4	no	—
286	C₂×C₄	—	2	16	no	—
287	C₂×C₄	—	2	2	no	—
288	C₂×C₄	—	2	2	no	—
289	C₂×C₄	—	2	8	no	—
290	C₂×C₄	—	4	8	no	—
291	C₂×C₄	—	4	32	no	—
292	C₂×C₄	C₁₃×AGL(1,ℤ/8)	1	8	yes	—
293	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
294	C₂×C₄	C₂²×D₄×C₁₃	1	4	yes	—
295	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
296	C₂×C₄	C₂²×D₄×C₁₃	1	4	yes	—
297	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
298	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
299	C₂×C₄	C₂⁵×C₁₃	1	1	yes	characteristic
300	C₈	—	2	8	no	—
301	C₂²	—	2	4	no	—
302	C₂²	—	2	2	no	—
303	C₂²	—	4	4	no	—
304	C₂²	—	4	8	no	—
305	C₂²	—	4	8	no	—
306	C₂²	—	4	8	no	—
307	C₂²	—	8	16	no	—
308	C₂²	C₁₃×#537	1	2	yes	—
309	C₂²	C₂×C₁₃×AGL(1,ℤ/8)	1	4	yes	—
310	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
311	C₄	—	2	4	no	—
312	C₄	—	2	2	no	—
313	C₄	—	4	8	no	—
314	C₄	C₁₃×#537	1	2	yes	—
315	C₄	C₂×C₁₃×AGL(1,ℤ/8)	1	8	yes	—
316	C₄	C₂×C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
317	C₄	C₂×C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
318	C₂	—	4	2	no	—
319	C₂	—	8	8	no	—
320	C₂	C₁₃×#3253	1	4	yes	direct product factor
321	C₂	C₂×C₁₃×#537	1	1	yes	characteristic
322	C₂	C₂²×C₁₃×AGL(1,ℤ/8)	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 ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#532	C₂	semidirect
2. C₁₃×#3253	C₂	direct (N × H)
3. C₂×C₁₃×#476	C₂	semidirect
4. C₂×C₁₃×#469	C₂	semidirect
5. C₂×C₁₃×#459	C₂	semidirect
6. C₂×C₁₃×#423	C₂	semidirect
11. C₁₃×#532	C₂²	semidirect
20. C₁₃×#476	C₂²	semidirect
21. C₁₃×#469	C₂²	semidirect
22. C₁₃×#459	C₂²	semidirect
24. C₁₃×#423	C₂²	semidirect
65. C₂×#3253	C₁₃	direct (N × H)
107. C₂×#532	C₂×C₁₃	semidirect
108. #3253	C₂×C₁₃	direct (N × H)
109. C₂×#476	C₂×C₁₃	semidirect
110. C₂×#469	C₂×C₁₃	semidirect
111. C₂×#459	C₂×C₁₃	semidirect
112. C₂×#423	C₂×C₁₃	semidirect
150. #532	C₂²×C₁₃	semidirect
159. #476	C₂²×C₁₃	semidirect
160. #469	C₂²×C₁₃	semidirect
161. #459	C₂²×C₁₃	semidirect
163. #423	C₂²×C₁₃	semidirect
223. C₂×C₁₃	#3253	direct (N × H)
267. C₁₃	C₂×#3253	direct (N × H)
320. C₂	C₁₃×#3253	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
C₁₃×#3253	2
C₂×C₁₃×#476	2
C₂×C₁₃×#469	2
C₂×C₁₃×#459	2
C₂×C₁₃×#423	2
C₂×#3253	13

Subgroups — C2×C13×#3253 #15747242

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#3253 #15747242