Subgroups — C₂×C₁₃×#3261 #15747251

Order 3328.

1754 conjugacy classes of subgroups.

The subgroup lattice has 292 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#541	C₂	1	2	yes	maximal
2	C₁₃×#3261	C₂	1	16	yes	maximal, direct product factor
3	C₂×C₁₃×#479	C₂	1	4	yes	maximal, complemented
4	C₂×C₁₃×#470	C₂	1	2	yes	maximal
5	C₂×C₁₃×#467	C₂	1	2	yes	maximal
6	C₂×C₁₃×#462	C₂	1	4	yes	maximal, complemented
7	C₂×C₁₃×#424	C₂	1	1	yes	maximal, characteristic
8	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
9	C₂²×C₁₃×Q₁₆	C₂²	1	2	yes	—
10	C₂²×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
11	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₁₃×#541	—	2	8	no	—
13	C₁₃×#541	C₂²	1	16	yes	—
14	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
15	C₂×C₁₃×#126	—	2	4	no	—
16	C₂×C₁₃×#125	—	2	4	no	—
17	C₂×C₁₃×#125	C₂²	1	2	yes	—
18	C₂×C₁₃×C₂²⋊Q₈	C₂²	1	4	yes	—
19	C₂×C₄×Q₈×C₁₃	C₂²	1	4	yes	—
20	C₂×C₁₃×#117	C₂²	1	1	yes	characteristic
21	C₁₃×#479	C₂²	1	32	yes	complemented
22	C₁₃×#470	C₂²	1	16	yes	—
23	C₁₃×#467	C₂²	1	16	yes	—
24	C₁₃×#462	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	—
28	C₂×C₁₃×C₄⋊C₈	C₂²	1	4	yes	—
29	C₂×C₁₃×#103	C₂²	1	4	yes	—
30	C₂×C₁₃×#103	C₂²	1	4	yes	—
31	C₂²×Q₈×C₁₃	C₂³	1	4	yes	—
32	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
33	C₂×C₁₃×Q₁₆	—	4	4	no	—
34	C₂×C₁₃×Q₁₆	—	4	4	no	—
35	C₂×C₁₃×Q₁₆	C₂³	1	8	yes	—
36	C₂×C₁₃×Q₁₆	C₂³	1	8	yes	—
37	C₂×C₁₃×M₁₆	—	4	2	no	—
38	C₂×C₁₃×M₁₆	C₂³	1	4	yes	—
39	C₂²×C₈×C₁₃	—	4	2	no	—
40	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
41	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
43	C₁₃×C₄⋊Q₈	—	2	4	no	—
44	C₁₃×C₄⋊Q₈	C₂³	1	8	yes	—
45	C₁₃×#126	—	2	32	no	—
46	C₁₃×#125	—	2	32	no	—
47	C₁₃×#125	—	2	4	no	—
48	C₁₃×#125	C₂³	1	8	yes	—
49	C₁₃×C₂²⋊Q₈	—	2	8	no	—
50	C₁₃×C₂²⋊Q₈	C₂³	1	16	yes	—
51	C₄×Q₈×C₁₃	—	2	8	no	—
52	C₄×Q₈×C₁₃	C₂³	1	16	yes	—
53	C₁₃×#117	D₄	1	4	yes	—
54	C₁₃×#117	C₂³	1	4	yes	—
55	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
56	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
57	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
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	D₄	1	2	yes	—
63	C₂×C₄²×C₁₃	—	2	4	no	—
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×#103	C₂³	1	16	yes	—
69	C₁₃×#103	C₂³	1	16	yes	—
70	C₂×#3261	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
71	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
72	C₂×Q₈×C₁₃	—	2	4	no	—
73	C₂×Q₈×C₁₃	—	2	8	no	—
74	C₂×Q₈×C₁₃	—	2	8	no	—
75	C₂×Q₈×C₁₃	C₂⁴	1	8	yes	—
76	C₂²×C₄×C₁₃	—	2	2	no	—
77	C₂²×C₄×C₁₃	—	2	2	no	—
78	C₂²×C₄×C₁₃	—	2	4	no	—
79	C₂²×C₄×C₁₃	—	2	4	no	—
80	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
81	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
82	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
84	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
85	C₁₃×Q₁₆	—	4	8	no	—
86	C₁₃×Q₁₆	—	4	8	no	—
87	C₁₃×M₁₆	—	4	4	no	—
88	C₂×C₈×C₁₃	—	2	4	no	—
89	C₂×C₈×C₁₃	—	2	4	no	—
90	C₂×C₈×C₁₃	—	4	4	no	—
91	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
92	C₂×C₈×C₁₃	C₂⁴	1	4	yes	—
93	C₁₃×C₄⋊C₄	—	2	16	no	—
94	C₁₃×C₄⋊C₄	—	2	16	no	—
95	C₁₃×C₄⋊C₄	—	2	16	no	—
96	C₁₃×C₄⋊C₄	—	2	16	no	—
97	C₁₃×C₄⋊C₄	—	2	4	no	—
98	C₁₃×C₄⋊C₄	C₂×D₄	1	8	yes	—
99	C₁₃×C₄⋊C₄	C₂⁴	1	8	yes	—
100	C₁₃×C₂²⋊C₄	—	2	16	no	—
101	C₁₃×C₂²⋊C₄	C₂×D₄	1	8	yes	—
102	C₄²×C₁₃	—	2	16	no	—
103	C₄²×C₁₃	C₂×D₄	1	4	yes	—
104	C₄²×C₁₃	C₂⁴	1	4	yes	—
105	C₂×#541	C₂×C₁₃	1	2	yes	—
106	#3261	C₂×C₁₃	1	16	yes	direct product factor
107	C₂×#479	C₂×C₁₃	1	4	yes	complemented
108	C₂×#470	C₂×C₁₃	1	2	yes	—
109	C₂×#467	C₂×C₁₃	1	2	yes	—
110	C₂×#462	C₂×C₁₃	1	4	yes	complemented
111	C₂×#424	C₂×C₁₃	1	1	yes	characteristic
112	C₂³×C₁₃	—	2	2	no	—
113	C₂³×C₁₃	—	2	4	no	—
114	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
115	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
116	Q₈×C₁₃	—	2	8	no	—
117	Q₈×C₁₃	—	2	8	no	—
118	Q₈×C₁₃	—	4	8	no	—
119	C₂×C₄×C₁₃	—	2	2	no	—
120	C₂×C₄×C₁₃	—	2	8	no	—
121	C₂×C₄×C₁₃	—	2	4	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	2	no	—
124	C₂×C₄×C₁₃	—	2	8	no	—
125	C₂×C₄×C₁₃	—	2	8	no	—
126	C₂×C₄×C₁₃	—	4	8	no	—
127	C₂×C₄×C₁₃	—	4	8	no	—
128	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
129	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
130	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
131	C₂×C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
132	C₂×C₄×C₁₃	2₋¹⁺⁴	1	2	yes	—
133	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
134	C₈×C₁₃	—	2	4	no	—
135	C₈×C₁₃	—	2	4	no	—
136	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
137	C₂²×Q₁₆	C₂²×C₁₃	1	2	yes	—
138	C₂²×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
139	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
140	#541	—	2	8	no	—
141	#541	C₂²×C₁₃	1	16	yes	—
142	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
143	C₂×#126	—	2	4	no	—
144	C₂×#125	—	2	4	no	—
145	C₂×#125	C₂²×C₁₃	1	2	yes	—
146	C₂×C₂²⋊Q₈	C₂²×C₁₃	1	4	yes	—
147	C₂×C₄×Q₈	C₂²×C₁₃	1	4	yes	—
148	C₂×#117	C₂²×C₁₃	1	1	yes	characteristic
149	#479	C₂²×C₁₃	1	32	yes	complemented
150	#470	C₂²×C₁₃	1	16	yes	—
151	#467	C₂²×C₁₃	1	16	yes	—
152	#462	C₂²×C₁₃	1	32	yes	complemented
153	C₂×#107	C₂²×C₁₃	1	2	yes	—
154	C₂×#106	C₂²×C₁₃	1	2	yes	—
155	#424	C₂²×C₁₃	1	8	yes	—
156	C₂×C₄⋊C₈	C₂²×C₁₃	1	4	yes	—
157	C₂×#103	C₂²×C₁₃	1	4	yes	—
158	C₂×#103	C₂²×C₁₃	1	4	yes	—
159	C₂²×C₁₃	—	2	2	no	—
160	C₂²×C₁₃	—	2	4	no	—
161	C₂²×C₁₃	—	4	4	no	—
162	C₂²×C₁₃	#537	1	2	yes	—
163	C₂²×C₁₃	#578	1	4	yes	—
164	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	—	2	8	no	—
167	C₄×C₁₃	—	4	8	no	—
168	C₄×C₁₃	—	4	8	no	—
169	C₄×C₁₃	#537	1	2	yes	—
170	C₄×C₁₃	C₂×H(5,𝔽₂)	1	1	yes	characteristic
171	C₄×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
172	C₂²×Q₈	C₂³×C₁₃	1	4	yes	—
173	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
174	C₂×Q₁₆	—	4	4	no	—
175	C₂×Q₁₆	—	4	4	no	—
176	C₂×Q₁₆	C₂³×C₁₃	1	8	yes	—
177	C₂×Q₁₆	C₂³×C₁₃	1	8	yes	—
178	C₂×M₁₆	—	4	2	no	—
179	C₂×M₁₆	C₂³×C₁₃	1	4	yes	—
180	C₂²×C₈	—	4	2	no	—
181	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
182	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
183	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
184	C₄⋊Q₈	—	2	4	no	—
185	C₄⋊Q₈	C₂³×C₁₃	1	8	yes	—
186	#126	—	2	32	no	—
187	#125	—	2	32	no	—
188	#125	—	2	4	no	—
189	#125	C₂³×C₁₃	1	8	yes	—
190	C₂²⋊Q₈	—	2	8	no	—
191	C₂²⋊Q₈	C₂³×C₁₃	1	16	yes	—
192	C₄×Q₈	—	2	8	no	—
193	C₄×Q₈	C₂³×C₁₃	1	16	yes	—
194	#117	D₄×C₁₃	1	4	yes	—
195	#117	C₂³×C₁₃	1	4	yes	—
196	C₂×C₄⋊C₄	—	2	4	no	—
197	C₂×C₄⋊C₄	—	2	4	no	—
198	C₂×C₄⋊C₄	—	2	4	no	—
199	C₂×C₄⋊C₄	—	2	4	no	—
200	C₂×C₄⋊C₄	D₄×C₁₃	1	2	yes	—
201	C₂×C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
202	C₂×C₂²⋊C₄	—	2	4	no	—
203	C₂×C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
204	C₂×C₄²	—	2	4	no	—
205	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
206	#107	C₂³×C₁₃	1	8	yes	—
207	#106	C₂³×C₁₃	1	8	yes	—
208	C₄⋊C₈	C₂³×C₁₃	1	16	yes	—
209	#103	C₂³×C₁₃	1	16	yes	—
210	#103	C₂³×C₁₃	1	16	yes	—
211	C₂×C₁₃	—	4	2	no	—
212	C₂×C₁₃	#3261	1	4	yes	direct product factor
213	C₂×C₁₃	C₂×#537	1	1	yes	characteristic
214	C₂×C₁₃	C₂×#578	1	2	yes	—
215	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
216	C₂×Q₈	—	2	4	no	—
217	C₂×Q₈	—	2	8	no	—
218	C₂×Q₈	—	2	8	no	—
219	C₂×Q₈	C₂⁴×C₁₃	1	8	yes	—
220	C₂²×C₄	—	2	2	no	—
221	C₂²×C₄	—	2	2	no	—
222	C₂²×C₄	—	2	4	no	—
223	C₂²×C₄	—	2	4	no	—
224	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
225	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
226	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
227	C₂²×C₄	C₂⁴×C₁₃	1	2	yes	—
228	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
229	Q₁₆	—	4	8	no	—
230	Q₁₆	—	4	8	no	—
231	M₁₆	—	4	4	no	—
232	C₂×C₈	—	2	4	no	—
233	C₂×C₈	—	2	4	no	—
234	C₂×C₈	—	4	4	no	—
235	C₂×C₈	C₂⁴×C₁₃	1	4	yes	—
236	C₂×C₈	C₂⁴×C₁₃	1	4	yes	—
237	C₄⋊C₄	—	2	16	no	—
238	C₄⋊C₄	—	2	16	no	—
239	C₄⋊C₄	—	2	16	no	—
240	C₄⋊C₄	—	2	16	no	—
241	C₄⋊C₄	—	2	4	no	—
242	C₄⋊C₄	C₂×D₄×C₁₃	1	8	yes	—
243	C₄⋊C₄	C₂⁴×C₁₃	1	8	yes	—
244	C₂²⋊C₄	—	2	16	no	—
245	C₂²⋊C₄	C₂×D₄×C₁₃	1	8	yes	—
246	C₄²	—	2	16	no	—
247	C₄²	C₂×D₄×C₁₃	1	4	yes	—
248	C₄²	C₂⁴×C₁₃	1	4	yes	—
249	C₁₃	C₂×#3261	1	1	yes	characteristic, direct product factor, Sylow
250	C₂³	—	2	2	no	—
251	C₂³	—	2	4	no	—
252	C₂³	C₂²×D₄×C₁₃	1	2	yes	—
253	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
254	Q₈	—	2	8	no	—
255	Q₈	—	2	8	no	—
256	Q₈	—	4	8	no	—
257	C₂×C₄	—	2	2	no	—
258	C₂×C₄	—	2	8	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	2	8	no	—
263	C₂×C₄	—	2	8	no	—
264	C₂×C₄	—	4	8	no	—
265	C₂×C₄	—	4	8	no	—
266	C₂×C₄	C₂²×D₄×C₁₃	1	8	yes	—
267	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
268	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
269	C₂×C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
270	C₂×C₄	C₁₃×2₋¹⁺⁴	1	2	yes	—
271	C₂×C₄	C₂⁵×C₁₃	1	1	yes	characteristic
272	C₈	—	2	4	no	—
273	C₈	—	2	4	no	—
274	C₂²	—	2	4	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	4	4	no	—
277	C₂²	C₁₃×#537	1	2	yes	—
278	C₂²	C₁₃×#578	1	4	yes	—
279	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
280	C₄	—	2	8	no	—
281	C₄	—	2	2	no	—
282	C₄	—	4	8	no	—
283	C₄	—	4	8	no	—
284	C₄	C₁₃×#537	1	2	yes	—
285	C₄	C₂×C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
286	C₄	C₂×C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
287	C₂	—	4	2	no	—
288	C₂	C₁₃×#3261	1	4	yes	direct product factor
289	C₂	C₂×C₁₃×#537	1	1	yes	characteristic
290	C₂	C₂×C₁₃×#578	1	2	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₃
prime radical	C₂⁴×C₁₃	C₂×D₄
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
2. C₁₃×#3261	C₂	direct (N × H)
3. C₂×C₁₃×#479	C₂	semidirect
6. C₂×C₁₃×#462	C₂	semidirect
21. C₁₃×#479	C₂²	semidirect
24. C₁₃×#462	C₂²	semidirect
70. C₂×#3261	C₁₃	direct (N × H)
106. #3261	C₂×C₁₃	direct (N × H)
107. C₂×#479	C₂×C₁₃	semidirect
110. C₂×#462	C₂×C₁₃	semidirect
149. #479	C₂²×C₁₃	semidirect
152. #462	C₂²×C₁₃	semidirect
212. C₂×C₁₃	#3261	direct (N × H)
249. C₁₃	C₂×#3261	direct (N × H)
288. C₂	C₁₃×#3261	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×#541	2
C₁₃×#3261	2
C₂×C₁₃×#479	2
C₂×C₁₃×#470	2
C₂×C₁₃×#467	2
C₂×C₁₃×#462	2
C₂×C₁₃×#424	2
C₂×#3261	13

Subgroups — C2×C13×#3261 #15747251

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#3261 #15747251