Subgroups — C₂×C₁₃×#3150 #15746974

Order 3328.

3326 conjugacy classes of subgroups.

The subgroup lattice has 288 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×C₁₃×#135	C₂	1	4	yes	maximal, complemented
2	C₂×C₁₃×#532	C₂	1	2	yes	maximal, complemented
3	C₁₃×#3150	C₂	1	16	yes	maximal, direct product factor
4	C₂×C₁₃×#497	C₂	1	4	yes	maximal, complemented
5	C₂×C₁₃×#496	C₂	1	4	yes	maximal, complemented
6	C₂²×C₁₃×#97	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₁₃×#41	C₂²	1	4	yes	—
8	C₂×C₁₃×#135	—	2	32	no	—
9	C₂×C₁₃×#135	C₂²	1	32	yes	complemented
10	C₂²×C₁₃×Q₁₆	C₂²	1	4	yes	—
11	C₂²×C₁₃×SD₁₆	C₂²	1	8	yes	—
12	C₂²×C₁₃×D₈	C₂²	1	4	yes	—
13	C₂³×C₈×C₁₃	C₂²	1	2	yes	—
14	C₁₃×#532	—	2	8	no	—
15	C₁₃×#532	C₂²	1	16	yes	complemented
16	C₂×C₁₃×C₄⋊Q₈	C₂²	1	2	yes	—
17	C₂×C₁₃×#127	C₂²	1	2	yes	—
18	C₂×C₁₃×#124	C₂²	1	4	yes	—
19	C₂×C₁₃×#121	—	2	4	no	—
20	C₂×C₄×D₄×C₁₃	—	2	4	no	—
21	C₂²×C₄²×C₁₃	C₂²	1	1	yes	characteristic
22	C₁₃×#497	C₂²	1	32	yes	complemented
23	C₁₃×#496	C₂²	1	32	yes	complemented
24	C₂×C₁₃×#97	C₂²	1	4	yes	—
25	C₂×C₁₃×#97	C₂²	1	8	yes	complemented
26	C₂×C₁₃×#41	—	2	8	no	—
27	C₂×C₁₃×#41	—	2	32	no	—
28	C₂×C₁₃×#41	C₂³	1	16	yes	—
29	C₂²×Q₈×C₁₃	C₂³	1	4	yes	—
30	C₂²×D₄×C₁₃	—	2	4	no	—
31	C₂²×D₄×C₁₃	—	2	4	no	—
32	C₂²×D₄×C₁₃	C₂³	1	4	yes	—
33	C₂³×C₄×C₁₃	—	2	4	no	—
34	C₂³×C₄×C₁₃	D₄	1	2	yes	—
35	C₂³×C₄×C₁₃	C₂³	1	1	yes	characteristic
36	C₁₃×#135	—	2	128	no	—
37	C₂×C₁₃×Q₁₆	—	2	16	no	—
38	C₂×C₁₃×Q₁₆	C₂³	1	16	yes	—
39	C₂×C₁₃×SD₁₆	—	2	32	no	—
40	C₂×C₁₃×SD₁₆	C₂³	1	32	yes	—
41	C₂×C₁₃×D₈	—	2	16	no	—
42	C₂×C₁₃×D₈	C₂³	1	16	yes	—
43	C₂²×C₈×C₁₃	D₄	1	16	yes	complemented
44	C₂²×C₈×C₁₃	C₂³	1	4	yes	—
45	C₂²×C₈×C₁₃	C₂³	1	8	yes	—
46	C₁₃×C₄⋊Q₈	—	2	4	no	—
47	C₁₃×C₄⋊Q₈	C₂³	1	8	yes	—
48	C₁₃×#127	—	2	4	no	—
49	C₁₃×#127	C₂³	1	8	yes	—
50	C₁₃×#124	—	2	8	no	—
51	C₁₃×#124	C₂³	1	16	yes	—
52	C₁₃×#121	—	2	32	no	—
53	C₄×D₄×C₁₃	—	2	32	no	—
54	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
55	C₂×C₁₃×C₂²⋊C₄	—	2	8	no	—
56	C₂×C₄²×C₁₃	D₄	1	4	yes	—
57	C₂×C₄²×C₁₃	D₄	1	2	yes	—
58	C₂×C₄²×C₁₃	C₂³	1	2	yes	—
59	C₂×C₄²×C₁₃	C₂³	1	4	yes	—
60	C₁₃×#97	C₂³	1	16	yes	—
61	C₂×#3150	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
62	C₂⁴×C₁₃	—	2	4	no	—
63	C₂⁴×C₁₃	C₂×D₄	1	1	yes	characteristic
64	C₁₃×#41	—	2	64	no	—
65	C₁₃×#41	—	4	32	no	—
66	C₂×Q₈×C₁₃	—	2	4	no	—
67	C₂×Q₈×C₁₃	—	2	16	no	—
68	C₂×Q₈×C₁₃	C₂⁴	1	8	yes	—
69	C₂×D₄×C₁₃	—	2	16	no	—
70	C₂×D₄×C₁₃	—	2	16	no	—
71	C₂×D₄×C₁₃	—	2	16	no	—
72	C₂×D₄×C₁₃	—	2	4	no	—
73	C₂×D₄×C₁₃	—	4	16	no	—
74	C₂×D₄×C₁₃	—	4	16	no	—
75	C₂×D₄×C₁₃	C₂⁴	1	8	yes	—
76	C₂²×C₄×C₁₃	—	2	4	no	—
77	C₂²×C₄×C₁₃	—	2	16	no	—
78	C₂²×C₄×C₁₃	—	4	4	no	—
79	C₂²×C₄×C₁₃	—	4	16	no	—
80	C₂²×C₄×C₁₃	C₂×D₄	1	4	yes	—
81	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
82	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
83	C₂²×C₄×C₁₃	C₂×D₄	1	8	yes	—
84	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂²×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
86	C₂²×C₄×C₁₃	C₂⁴	1	2	yes	—
87	C₁₃×Q₁₆	—	2	32	no	—
88	C₁₃×SD₁₆	—	2	64	no	—
89	C₁₃×D₈	—	2	32	no	—
90	C₂×C₈×C₁₃	C₂×D₄	1	16	yes	—
91	C₂×C₈×C₁₃	C₂×D₄	1	32	yes	complemented
92	C₂×C₈×C₁₃	C₂⁴	1	8	yes	—
93	C₁₃×C₄⋊C₄	—	2	16	no	—
94	C₁₃×C₂²⋊C₄	—	2	32	no	—
95	C₄²×C₁₃	C₂×D₄	1	8	yes	—
96	C₄²×C₁₃	C₂×D₄	1	4	yes	—
97	C₄²×C₁₃	C₂⁴	1	4	yes	—
98	C₂²×#135	C₂×C₁₃	1	4	yes	complemented
99	C₂×#532	C₂×C₁₃	1	2	yes	complemented
100	#3150	C₂×C₁₃	1	16	yes	direct product factor
101	C₂×#497	C₂×C₁₃	1	4	yes	complemented
102	C₂×#496	C₂×C₁₃	1	4	yes	complemented
103	C₂²×#97	C₂×C₁₃	1	1	yes	characteristic, complemented
104	C₂³×C₁₃	—	2	4	no	—
105	C₂³×C₁₃	—	2	8	no	—
106	C₂³×C₁₃	—	4	8	no	—
107	C₂³×C₁₃	—	4	16	no	—
108	C₂³×C₁₃	#127	1	4	yes	—
109	C₂³×C₁₃	C₂²×D₄	1	2	yes	—
110	C₂³×C₁₃	C₂²×D₄	1	1	yes	characteristic
111	Q₈×C₁₃	—	2	16	no	—
112	Q₈×C₁₃	—	4	8	no	—
113	D₄×C₁₃	—	2	16	no	—
114	D₄×C₁₃	—	4	32	no	—
115	D₄×C₁₃	—	4	32	no	—
116	D₄×C₁₃	—	4	8	no	—
117	C₂×C₄×C₁₃	—	2	8	no	—
118	C₂×C₄×C₁₃	—	2	8	no	—
119	C₂×C₄×C₁₃	—	4	8	no	—
120	C₂×C₄×C₁₃	—	4	8	no	—
121	C₂×C₄×C₁₃	—	4	32	no	—
122	C₂×C₄×C₁₃	#127	1	4	yes	—
123	C₂×C₄×C₁₃	#127	1	8	yes	—
124	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
125	C₂×C₄×C₁₃	C₂²×D₄	1	2	yes	—
126	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
127	C₂×C₄×C₁₃	C₂²×D₄	1	4	yes	—
128	C₂×C₄×C₁₃	C₂²×D₄	1	8	yes	—
129	C₂×C₄×C₁₃	C₂⁵	1	1	yes	characteristic
130	C₈×C₁₃	C₂²×D₄	1	16	yes	—
131	C₂²×#41	C₂²×C₁₃	1	4	yes	—
132	C₂×#135	—	2	32	no	—
133	C₂×#135	C₂²×C₁₃	1	32	yes	complemented
134	C₂²×Q₁₆	C₂²×C₁₃	1	4	yes	—
135	C₂²×SD₁₆	C₂²×C₁₃	1	8	yes	—
136	C₂²×D₈	C₂²×C₁₃	1	4	yes	—
137	C₂³×C₈	C₂²×C₁₃	1	2	yes	—
138	#532	—	2	8	no	—
139	#532	C₂²×C₁₃	1	16	yes	complemented
140	C₂×C₄⋊Q₈	C₂²×C₁₃	1	2	yes	—
141	C₂×#127	C₂²×C₁₃	1	2	yes	—
142	C₂×#124	C₂²×C₁₃	1	4	yes	—
143	C₂×#121	—	2	4	no	—
144	C₂×C₄×D₄	—	2	4	no	—
145	C₂²×C₄²	C₂²×C₁₃	1	1	yes	characteristic
146	#497	C₂²×C₁₃	1	32	yes	complemented
147	#496	C₂²×C₁₃	1	32	yes	complemented
148	C₂×#97	C₂²×C₁₃	1	4	yes	—
149	C₂×#97	C₂²×C₁₃	1	8	yes	complemented
150	C₂²×C₁₃	—	2	8	no	—
151	C₂²×C₁₃	—	2	4	no	—
152	C₂²×C₁₃	—	4	8	no	—
153	C₂²×C₁₃	—	4	16	no	—
154	C₂²×C₁₃	—	8	16	no	—
155	C₂²×C₁₃	C₂×#127	1	4	yes	—
156	C₂²×C₁₃	C₂×#127	1	2	yes	—
157	C₂²×C₁₃	#575	1	4	yes	—
158	C₂²×C₁₃	C₂³×D₄	1	1	yes	characteristic
159	C₄×C₁₃	—	2	8	no	—
160	C₄×C₁₃	—	4	8	no	—
161	C₄×C₁₃	C₂×#127	1	4	yes	—
162	C₄×C₁₃	C₂×#127	1	2	yes	—
163	C₄×C₁₃	C₂³×D₄	1	2	yes	—
164	C₂×#41	—	2	8	no	—
165	C₂×#41	—	2	32	no	—
166	C₂×#41	C₂³×C₁₃	1	16	yes	—
167	C₂²×Q₈	C₂³×C₁₃	1	4	yes	—
168	C₂²×D₄	—	2	4	no	—
169	C₂²×D₄	—	2	4	no	—
170	C₂²×D₄	C₂³×C₁₃	1	4	yes	—
171	C₂³×C₄	—	2	4	no	—
172	C₂³×C₄	D₄×C₁₃	1	2	yes	—
173	C₂³×C₄	C₂³×C₁₃	1	1	yes	characteristic
174	#135	—	2	128	no	—
175	C₂×Q₁₆	—	2	16	no	—
176	C₂×Q₁₆	C₂³×C₁₃	1	16	yes	—
177	C₂×SD₁₆	—	2	32	no	—
178	C₂×SD₁₆	C₂³×C₁₃	1	32	yes	—
179	C₂×D₈	—	2	16	no	—
180	C₂×D₈	C₂³×C₁₃	1	16	yes	—
181	C₂²×C₈	D₄×C₁₃	1	16	yes	complemented
182	C₂²×C₈	C₂³×C₁₃	1	4	yes	—
183	C₂²×C₈	C₂³×C₁₃	1	8	yes	—
184	C₄⋊Q₈	—	2	4	no	—
185	C₄⋊Q₈	C₂³×C₁₃	1	8	yes	—
186	#127	—	2	4	no	—
187	#127	C₂³×C₁₃	1	8	yes	—
188	#124	—	2	8	no	—
189	#124	C₂³×C₁₃	1	16	yes	—
190	#121	—	2	32	no	—
191	C₄×D₄	—	2	32	no	—
192	C₂×C₄⋊C₄	—	2	4	no	—
193	C₂×C₂²⋊C₄	—	2	8	no	—
194	C₂×C₄²	D₄×C₁₃	1	2	yes	—
195	C₂×C₄²	D₄×C₁₃	1	4	yes	—
196	C₂×C₄²	C₂³×C₁₃	1	2	yes	—
197	C₂×C₄²	C₂³×C₁₃	1	4	yes	—
198	#97	C₂³×C₁₃	1	16	yes	—
199	C₂×C₁₃	—	2	4	no	—
200	C₂×C₁₃	—	8	8	no	—
201	C₂×C₁₃	#3150	1	4	yes	direct product factor
202	C₂×C₁₃	C₂²×#127	1	1	yes	characteristic
203	C₂×C₁₃	C₂×#575	1	2	yes	—
204	C₂⁴	—	2	4	no	—
205	C₂⁴	C₂×D₄×C₁₃	1	1	yes	characteristic
206	#41	—	2	64	no	—
207	#41	—	4	32	no	—
208	C₂×Q₈	—	2	4	no	—
209	C₂×Q₈	—	2	16	no	—
210	C₂×Q₈	C₂⁴×C₁₃	1	8	yes	—
211	C₂×D₄	—	2	16	no	—
212	C₂×D₄	—	2	16	no	—
213	C₂×D₄	—	2	16	no	—
214	C₂×D₄	—	2	4	no	—
215	C₂×D₄	—	4	16	no	—
216	C₂×D₄	—	4	16	no	—
217	C₂×D₄	C₂⁴×C₁₃	1	8	yes	—
218	C₂²×C₄	—	2	4	no	—
219	C₂²×C₄	—	2	16	no	—
220	C₂²×C₄	—	4	4	no	—
221	C₂²×C₄	—	4	16	no	—
222	C₂²×C₄	C₂×D₄×C₁₃	1	4	yes	—
223	C₂²×C₄	C₂×D₄×C₁₃	1	8	yes	—
224	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
225	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
226	C₂²×C₄	C₂×D₄×C₁₃	1	8	yes	—
227	C₂²×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
228	C₂²×C₄	C₂⁴×C₁₃	1	2	yes	—
229	Q₁₆	—	2	32	no	—
230	SD₁₆	—	2	64	no	—
231	D₈	—	2	32	no	—
232	C₂×C₈	C₂×D₄×C₁₃	1	16	yes	—
233	C₂×C₈	C₂×D₄×C₁₃	1	32	yes	complemented
234	C₂×C₈	C₂⁴×C₁₃	1	8	yes	—
235	C₄⋊C₄	—	2	16	no	—
236	C₂²⋊C₄	—	2	32	no	—
237	C₄²	C₂×D₄×C₁₃	1	8	yes	—
238	C₄²	C₂×D₄×C₁₃	1	4	yes	—
239	C₄²	C₂⁴×C₁₃	1	4	yes	—
240	C₁₃	C₂×#3150	1	1	yes	characteristic, direct product factor, Sylow
241	C₂³	—	2	4	no	—
242	C₂³	—	2	8	no	—
243	C₂³	—	4	8	no	—
244	C₂³	—	4	16	no	—
245	C₂³	C₁₃×#127	1	4	yes	—
246	C₂³	C₂²×D₄×C₁₃	1	2	yes	—
247	C₂³	C₂²×D₄×C₁₃	1	1	yes	characteristic
248	Q₈	—	2	16	no	—
249	Q₈	—	4	8	no	—
250	D₄	—	2	16	no	—
251	D₄	—	4	32	no	—
252	D₄	—	4	32	no	—
253	D₄	—	4	8	no	—
254	C₂×C₄	—	2	8	no	—
255	C₂×C₄	—	2	8	no	—
256	C₂×C₄	—	4	8	no	—
257	C₂×C₄	—	4	8	no	—
258	C₂×C₄	—	4	32	no	—
259	C₂×C₄	C₁₃×#127	1	4	yes	—
260	C₂×C₄	C₁₃×#127	1	8	yes	—
261	C₂×C₄	C₂²×D₄×C₁₃	1	8	yes	—
262	C₂×C₄	C₂²×D₄×C₁₃	1	2	yes	—
263	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
264	C₂×C₄	C₂²×D₄×C₁₃	1	4	yes	—
265	C₂×C₄	C₂²×D₄×C₁₃	1	8	yes	—
266	C₂×C₄	C₂⁵×C₁₃	1	1	yes	characteristic
267	C₈	C₂²×D₄×C₁₃	1	16	yes	—
268	C₂²	—	2	4	no	—
269	C₂²	—	2	8	no	—
270	C₂²	—	4	16	no	—
271	C₂²	—	4	8	no	—
272	C₂²	—	8	16	no	—
273	C₂²	C₂×C₁₃×#127	1	4	yes	—
274	C₂²	C₂×C₁₃×#127	1	2	yes	—
275	C₂²	C₁₃×#575	1	4	yes	—
276	C₂²	C₂³×D₄×C₁₃	1	1	yes	characteristic
277	C₄	—	2	8	no	—
278	C₄	—	4	8	no	—
279	C₄	C₂×C₁₃×#127	1	4	yes	—
280	C₄	C₂×C₁₃×#127	1	2	yes	—
281	C₄	C₂³×D₄×C₁₃	1	2	yes	—
282	C₂	—	2	4	no	—
283	C₂	—	8	8	no	—
284	C₂	C₁₃×#3150	1	4	yes	direct product factor
285	C₂	C₂²×C₁₃×#127	1	1	yes	characteristic
286	C₂	C₂×C₁₃×#575	1	2	yes	—

	subgroup	G/H
center	C₂²×C₄×C₁₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁵×C₁₃
socle	C₂³×C₁₃	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄×C₁₃ ◁ C₂²×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂²×C₁₃×#135	C₂	semidirect
2. C₂×C₁₃×#532	C₂	semidirect
3. C₁₃×#3150	C₂	direct (N × H)
4. C₂×C₁₃×#497	C₂	semidirect
5. C₂×C₁₃×#496	C₂	semidirect
6. C₂²×C₁₃×#97	C₂	semidirect
9. C₂×C₁₃×#135	C₂²	semidirect
15. C₁₃×#532	C₂²	semidirect
22. C₁₃×#497	C₂²	semidirect
23. C₁₃×#496	C₂²	semidirect
25. C₂×C₁₃×#97	C₂²	semidirect
43. C₂²×C₈×C₁₃	D₄	semidirect
61. C₂×#3150	C₁₃	direct (N × H)
91. C₂×C₈×C₁₃	C₂×D₄	semidirect
98. C₂²×#135	C₂×C₁₃	semidirect
99. C₂×#532	C₂×C₁₃	semidirect
100. #3150	C₂×C₁₃	direct (N × H)
101. C₂×#497	C₂×C₁₃	semidirect
102. C₂×#496	C₂×C₁₃	semidirect
103. C₂²×#97	C₂×C₁₃	semidirect
133. C₂×#135	C₂²×C₁₃	semidirect
139. #532	C₂²×C₁₃	semidirect
146. #497	C₂²×C₁₃	semidirect
147. #496	C₂²×C₁₃	semidirect
149. C₂×#97	C₂²×C₁₃	semidirect
181. C₂²×C₈	D₄×C₁₃	semidirect
201. C₂×C₁₃	#3150	direct (N × H)
233. C₂×C₈	C₂×D₄×C₁₃	semidirect
240. C₁₃	C₂×#3150	direct (N × H)
284. C₂	C₁₃×#3150	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×#135	2
C₂×C₁₃×#532	2
C₁₃×#3150	2
C₂×C₁₃×#497	2
C₂×C₁₃×#496	2
C₂²×C₁₃×#97	2
C₂×#3150	13

Subgroups — C2×C13×#3150 #15746974

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×C₁₃×#3150 #15746974