Subgroups — C₁₃×#33604 #15719943

Order 3328.

690 conjugacy classes of subgroups.

The subgroup lattice has 294 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#508	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂²×C₁₃×M₃₂	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#3050	C₂	1	2	yes	maximal, complemented
4	C₁₃×#2220	C₂	1	4	yes	maximal, complemented
5	C₁₃×#2219	C₂	1	2	yes	maximal, complemented
6	C₁₃×#2218	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2158	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#2138	C₂	1	2	yes	maximal, complemented
9	C₂×C₁₃×#135	C₂²	1	2	yes	—
10	C₂³×C₈×C₁₃	C₂²	1	1	yes	characteristic
11	C₁₃×#525	—	2	2	no	—
12	C₁₃×#508	—	2	4	no	—
13	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic, complemented
14	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	complemented
15	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₁₃×M₃₂	—	2	4	no	—
17	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
18	C₂×C₁₃×M₃₂	C₂²	1	2	yes	—
19	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
20	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
21	C₁₃×#467	C₂²	1	4	yes	—
22	C₁₃×#466	C₂²	1	4	yes	—
23	C₁₃×#465	—	2	4	no	—
24	C₁₃×#427	C₂²	1	2	yes	—
25	C₁₃×#416	—	2	2	no	—
26	C₁₃×#367	C₂²	1	2	yes	complemented
27	C₁₃×#366	C₂²	1	2	yes	complemented
28	C₁₃×#358	C₂²	1	4	yes	complemented
29	C₁₃×#357	C₂²	1	4	yes	complemented
30	C₂×C₁₃×#41	—	2	2	no	—
31	C₂³×C₄×C₁₃	D₄	1	1	yes	characteristic
32	C₁₃×#135	—	2	4	no	—
33	C₁₃×#135	—	2	4	no	—
34	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
35	C₂×C₁₃×SD₁₆	—	2	2	no	—
36	C₂×C₁₃×D₈	C₂³	1	2	yes	—
37	C₂²×C₈×C₁₃	—	2	2	no	—
38	C₂²×C₈×C₁₃	—	2	2	no	—
39	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
40	C₂²×C₈×C₁₃	D₄	1	2	yes	—
41	C₂²×C₈×C₁₃	C₂³	1	2	yes	—
42	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
43	C₁₃×#123	—	4	2	no	—
44	C₁₃×C₂²⋊Q₈	—	2	4	no	—
45	C₁₃×#121	—	2	4	no	—
46	C₁₃×#120	—	4	2	no	—
47	C₄×D₄×C₁₃	—	4	4	no	—
48	C₁₃×#117	—	2	2	no	—
49	C₁₃×Q₃₂	—	2	2	no	—
50	C₁₃×SD₃₂	—	2	4	no	—
51	C₁₃×D₁₆	—	2	2	no	—
52	C₁₃×M₃₂	—	2	4	no	—
53	C₁₃×M₃₂	—	2	4	no	—
54	C₂×C₁₃×C₁₆	—	2	2	no	—
55	C₂×C₁₃×C₁₆	D₄	1	4	yes	complemented
56	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
57	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
58	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
59	C₁₃×#107	C₂³	1	4	yes	—
60	C₁₃×#106	—	2	2	no	—
61	C₁₃×#103	—	2	4	no	—
62	C₁₃×#102	—	2	4	no	—
63	#33604	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
64	C₂⁴×C₁₃	D₈	1	1	yes	characteristic
65	C₁₃×#41	—	4	4	no	—
66	C₁₃×#41	—	4	4	no	—
67	C₂×Q₈×C₁₃	—	2	2	no	—
68	C₂×D₄×C₁₃	—	2	2	no	—
69	C₂×D₄×C₁₃	—	4	4	no	—
70	C₂×D₄×C₁₃	—	4	2	no	—
71	C₂²×C₄×C₁₃	—	2	2	no	—
72	C₂²×C₄×C₁₃	—	2	2	no	—
73	C₂²×C₄×C₁₃	—	4	2	no	—
74	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
75	C₂²×C₄×C₁₃	D₈	1	2	yes	—
76	C₂²×C₄×C₁₃	C₂×D₄	1	2	yes	—
77	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
78	C₁₃×Q₁₆	—	2	2	no	—
79	C₁₃×Q₁₆	—	2	2	no	—
80	C₁₃×SD₁₆	—	4	4	no	—
81	C₁₃×D₈	—	2	2	no	—
82	C₁₃×D₈	—	2	2	no	—
83	C₂×C₈×C₁₃	—	2	2	no	—
84	C₂×C₈×C₁₃	—	2	4	no	—
85	C₂×C₈×C₁₃	—	2	2	no	—
86	C₂×C₈×C₁₃	—	2	2	no	—
87	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
88	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
90	C₂×C₈×C₁₃	#41	1	2	yes	—
91	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
92	C₁₃×C₄⋊C₄	—	2	4	no	—
93	C₁₃×C₄⋊C₄	—	4	4	no	—
94	C₁₃×C₂²⋊C₄	—	4	2	no	—
95	C₁₃×C₂²⋊C₄	—	4	4	no	—
96	C₁₃×C₂²⋊C₄	—	4	4	no	—
97	C₄²×C₁₃	—	4	2	no	—
98	C₁₃×C₁₆	—	2	2	no	—
99	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
100	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
101	C₂×#508	C₂×C₁₃	1	1	yes	characteristic, complemented
102	C₂²×M₃₂	C₂×C₁₃	1	1	yes	characteristic, complemented
103	#3050	C₂×C₁₃	1	2	yes	complemented
104	#2220	C₂×C₁₃	1	4	yes	complemented
105	#2219	C₂×C₁₃	1	2	yes	complemented
106	#2218	C₂×C₁₃	1	2	yes	complemented
107	#2158	C₂×C₁₃	1	1	yes	characteristic, complemented
108	#2138	C₂×C₁₃	1	2	yes	complemented
109	C₂³×C₁₃	—	2	2	no	—
110	C₂³×C₁₃	—	4	2	no	—
111	C₂³×C₁₃	—	4	2	no	—
112	C₂³×C₁₃	C₂×D₈	1	2	yes	—
113	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
114	Q₈×C₁₃	—	4	2	no	—
115	Q₈×C₁₃	—	4	2	no	—
116	D₄×C₁₃	—	4	2	no	—
117	D₄×C₁₃	—	4	2	no	—
118	D₄×C₁₃	—	8	8	no	—
119	D₄×C₁₃	—	8	4	no	—
120	C₂×C₄×C₁₃	—	2	2	no	—
121	C₂×C₄×C₁₃	—	2	2	no	—
122	C₂×C₄×C₁₃	—	2	2	no	—
123	C₂×C₄×C₁₃	—	2	4	no	—
124	C₂×C₄×C₁₃	—	4	4	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	8	4	no	—
127	C₂×C₄×C₁₃	#121	1	2	yes	—
128	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
129	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
130	C₂×C₄×C₁₃	#135	1	2	yes	—
131	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
132	C₈×C₁₃	—	2	2	no	—
133	C₈×C₁₃	#121	1	2	yes	—
134	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
135	C₈×C₁₃	C₂×#41	1	1	yes	characteristic
136	C₂×#135	C₂²×C₁₃	1	2	yes	—
137	C₂³×C₈	C₂²×C₁₃	1	1	yes	characteristic
138	#525	—	2	2	no	—
139	#508	—	2	4	no	—
140	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic, complemented
141	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	complemented
142	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic, complemented
143	C₂×M₃₂	—	2	4	no	—
144	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
145	C₂×M₃₂	C₂²×C₁₃	1	2	yes	—
146	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
147	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
148	#467	C₂²×C₁₃	1	4	yes	—
149	#466	C₂²×C₁₃	1	4	yes	—
150	#465	—	2	4	no	—
151	#427	C₂²×C₁₃	1	2	yes	—
152	#416	—	2	2	no	—
153	#367	C₂²×C₁₃	1	2	yes	complemented
154	#366	C₂²×C₁₃	1	2	yes	complemented
155	#358	C₂²×C₁₃	1	4	yes	complemented
156	#357	C₂²×C₁₃	1	4	yes	complemented
157	C₂²×C₁₃	—	2	2	no	—
158	C₂²×C₁₃	—	2	4	no	—
159	C₂²×C₁₃	—	2	2	no	—
160	C₂²×C₁₃	—	4	4	no	—
161	C₂²×C₁₃	—	8	4	no	—
162	C₂²×C₁₃	—	8	2	no	—
163	C₂²×C₁₃	#466	1	2	yes	—
164	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
165	C₄×C₁₃	—	2	2	no	—
166	C₄×C₁₃	—	8	4	no	—
167	C₄×C₁₃	—	8	2	no	—
168	C₄×C₁₃	#466	1	2	yes	—
169	C₄×C₁₃	C₂×#121	1	1	yes	characteristic
170	C₄×C₁₃	C₂×#135	1	1	yes	characteristic
171	C₂×#41	—	2	2	no	—
172	C₂³×C₄	D₄×C₁₃	1	1	yes	characteristic
173	#135	—	2	4	no	—
174	#135	—	2	4	no	—
175	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
176	C₂×SD₁₆	—	2	2	no	—
177	C₂×D₈	C₂³×C₁₃	1	2	yes	—
178	C₂²×C₈	—	2	2	no	—
179	C₂²×C₈	—	2	2	no	—
180	C₂²×C₈	D₄×C₁₃	1	1	yes	characteristic
181	C₂²×C₈	D₄×C₁₃	1	2	yes	—
182	C₂²×C₈	C₂³×C₁₃	1	2	yes	—
183	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
184	#123	—	4	2	no	—
185	C₂²⋊Q₈	—	2	4	no	—
186	#121	—	2	4	no	—
187	#120	—	4	2	no	—
188	C₄×D₄	—	4	4	no	—
189	#117	—	2	2	no	—
190	Q₃₂	—	2	2	no	—
191	SD₃₂	—	2	4	no	—
192	D₁₆	—	2	2	no	—
193	M₃₂	—	2	4	no	—
194	M₃₂	—	2	4	no	—
195	C₂×C₁₆	—	2	2	no	—
196	C₂×C₁₆	D₄×C₁₃	1	4	yes	complemented
197	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
198	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
199	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
200	#107	C₂³×C₁₃	1	4	yes	—
201	#106	—	2	2	no	—
202	#103	—	2	4	no	—
203	#102	—	2	4	no	—
204	C₂×C₁₃	—	2	2	no	—
205	C₂×C₁₃	—	4	2	no	—
206	C₂×C₁₃	—	16	2	no	—
207	C₂×C₁₃	C₂×#466	1	1	yes	characteristic
208	C₂×C₁₃	#3414	1	2	yes	—
209	C₂⁴	C₁₃×D₈	1	1	yes	characteristic
210	#41	—	4	4	no	—
211	#41	—	4	4	no	—
212	C₂×Q₈	—	2	2	no	—
213	C₂×D₄	—	2	2	no	—
214	C₂×D₄	—	4	4	no	—
215	C₂×D₄	—	4	2	no	—
216	C₂²×C₄	—	2	2	no	—
217	C₂²×C₄	—	2	2	no	—
218	C₂²×C₄	—	4	2	no	—
219	C₂²×C₄	C₁₃×D₈	1	2	yes	—
220	C₂²×C₄	C₁₃×D₈	1	1	yes	characteristic
221	C₂²×C₄	C₂×D₄×C₁₃	1	2	yes	—
222	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
223	Q₁₆	—	2	2	no	—
224	Q₁₆	—	2	2	no	—
225	SD₁₆	—	4	4	no	—
226	D₈	—	2	2	no	—
227	D₈	—	2	2	no	—
228	C₂×C₈	—	2	2	no	—
229	C₂×C₈	—	2	4	no	—
230	C₂×C₈	—	2	2	no	—
231	C₂×C₈	—	2	2	no	—
232	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
233	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
234	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
235	C₂×C₈	C₁₃×#41	1	2	yes	—
236	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
237	C₄⋊C₄	—	2	4	no	—
238	C₄⋊C₄	—	4	4	no	—
239	C₂²⋊C₄	—	4	2	no	—
240	C₂²⋊C₄	—	4	4	no	—
241	C₂²⋊C₄	—	4	4	no	—
242	C₄²	—	4	2	no	—
243	C₁₆	—	2	2	no	—
244	C₁₆	C₂×D₄×C₁₃	1	2	yes	—
245	C₁₆	C₂×D₄×C₁₃	1	2	yes	—
246	C₁₃	#33604	1	1	yes	characteristic, direct product factor, Sylow
247	C₂³	—	2	2	no	—
248	C₂³	—	4	2	no	—
249	C₂³	—	4	2	no	—
250	C₂³	C₂×C₁₃×D₈	1	2	yes	—
251	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
252	Q₈	—	4	2	no	—
253	Q₈	—	4	2	no	—
254	D₄	—	4	2	no	—
255	D₄	—	4	2	no	—
256	D₄	—	8	8	no	—
257	D₄	—	8	4	no	—
258	C₂×C₄	—	2	2	no	—
259	C₂×C₄	—	2	4	no	—
260	C₂×C₄	—	2	2	no	—
261	C₂×C₄	—	2	2	no	—
262	C₂×C₄	—	4	4	no	—
263	C₂×C₄	—	4	2	no	—
264	C₂×C₄	—	8	4	no	—
265	C₂×C₄	C₁₃×#121	1	2	yes	—
266	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
267	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
268	C₂×C₄	C₁₃×#135	1	2	yes	—
269	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
270	C₈	—	2	2	no	—
271	C₈	C₁₃×#121	1	2	yes	—
272	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
273	C₈	C₂×C₁₃×#41	1	1	yes	characteristic
274	C₂²	—	2	2	no	—
275	C₂²	—	2	2	no	—
276	C₂²	—	2	4	no	—
277	C₂²	—	4	4	no	—
278	C₂²	—	8	2	no	—
279	C₂²	—	8	4	no	—
280	C₂²	C₁₃×#466	1	2	yes	—
281	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
282	C₄	—	2	2	no	—
283	C₄	—	8	2	no	—
284	C₄	—	8	4	no	—
285	C₄	C₁₃×#466	1	2	yes	—
286	C₄	C₂×C₁₃×#121	1	1	yes	characteristic
287	C₄	C₂×C₁₃×#135	1	1	yes	characteristic
288	C₂	—	2	2	no	—
289	C₂	—	4	2	no	—
290	C₂	—	16	2	no	—
291	C₂	C₂×C₁₃×#466	1	1	yes	characteristic
292	C₂	C₁₃×#3414	1	2	yes	—

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂²×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂³×C₄×C₁₃ ◁ C₂³×C₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#508	C₂	semidirect
2. C₂²×C₁₃×M₃₂	C₂	semidirect
3. C₁₃×#3050	C₂	semidirect
4. C₁₃×#2220	C₂	semidirect
5. C₁₃×#2219	C₂	semidirect
6. C₁₃×#2218	C₂	semidirect
7. C₁₃×#2158	C₂	semidirect
8. C₁₃×#2138	C₂	semidirect
13. C₂×C₁₃×Q₃₂	C₂²	semidirect
14. C₂×C₁₃×SD₃₂	C₂²	semidirect
15. C₂×C₁₃×D₁₆	C₂²	semidirect
26. C₁₃×#367	C₂²	semidirect
27. C₁₃×#366	C₂²	semidirect
28. C₁₃×#358	C₂²	semidirect
29. C₁₃×#357	C₂²	semidirect
55. C₂×C₁₃×C₁₆	D₄	semidirect
63. #33604	C₁₃	direct (N × H)
101. C₂×#508	C₂×C₁₃	semidirect
102. C₂²×M₃₂	C₂×C₁₃	semidirect
103. #3050	C₂×C₁₃	semidirect
104. #2220	C₂×C₁₃	semidirect
105. #2219	C₂×C₁₃	semidirect
106. #2218	C₂×C₁₃	semidirect
107. #2158	C₂×C₁₃	semidirect
108. #2138	C₂×C₁₃	semidirect
140. C₂×Q₃₂	C₂²×C₁₃	semidirect
141. C₂×SD₃₂	C₂²×C₁₃	semidirect
142. C₂×D₁₆	C₂²×C₁₃	semidirect
153. #367	C₂²×C₁₃	semidirect
154. #366	C₂²×C₁₃	semidirect
155. #358	C₂²×C₁₃	semidirect
156. #357	C₂²×C₁₃	semidirect
196. C₂×C₁₆	D₄×C₁₃	semidirect
246. C₁₃	#33604	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×#508	2
C₂²×C₁₃×M₃₂	2
C₁₃×#3050	2
C₁₃×#2220	2
C₁₃×#2219	2
C₁₃×#2218	2
C₁₃×#2158	2
C₁₃×#2138	2
#33604	13

Subgroups — C13×#33604 #15719943

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33604 #15719943