Subgroups — C₁₃×#33720 #15720059

Order 3328.

858 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×#509	C₂	1	2	yes	maximal, complemented
2	C₂×C₁₃×#508	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×C₁₃×D₁₆	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#3155	C₂	1	2	yes	maximal, complemented
5	C₁₃×#2250	C₂	1	4	yes	maximal, complemented
6	C₁₃×#2249	C₂	1	2	yes	maximal, complemented
7	C₁₃×#2246	C₂	1	2	yes	maximal, complemented
8	C₁₃×#2108	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×C₁₃×AGL(1,ℤ/8)	—	2	2	no	—
10	C₂×C₁₃×#135	C₂²	1	2	yes	—
11	C₂²×C₁₃×D₈	C₂²	1	2	yes	—
12	C₁₃×#535	—	2	2	no	—
13	C₁₃×#509	—	2	8	no	—
14	C₁₃×#508	—	2	4	no	—
15	C₂×C₁₃×Q₃₂	C₂²	1	1	yes	characteristic, complemented
16	C₂×C₁₃×SD₃₂	C₂²	1	4	yes	complemented
17	C₂×C₁₃×SD₃₂	C₂²	1	2	yes	complemented
18	C₂×C₁₃×D₁₆	—	2	4	no	—
19	C₂×C₁₃×D₁₆	C₂²	1	2	yes	—
20	C₂×C₁₃×D₁₆	C₂²	1	2	yes	—
21	C₂×C₁₃×D₁₆	C₂²	1	1	yes	characteristic, complemented
22	C₂×C₁₃×D₁₆	C₂²	1	4	yes	complemented
23	C₂×C₁₃×M₃₂	C₂²	1	1	yes	characteristic
24	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
25	C₁₃×#496	—	2	4	no	—
26	C₁₃×#495	C₂²	1	4	yes	complemented
27	C₁₃×#493	C₂²	1	4	yes	complemented
28	C₁₃×#405	C₂²	1	1	yes	characteristic
29	C₁₃×#346	C₂²	1	2	yes	complemented
30	C₄×C₁₃×C₁₆	C₂²	1	2	yes	complemented
31	C₂×C₁₃×#41	—	2	2	no	—
32	C₂²×D₄×C₁₃	—	2	2	no	—
33	C₁₃×AGL(1,ℤ/8)	—	4	8	no	—
34	C₁₃×#135	—	2	4	no	—
35	C₁₃×#135	—	2	4	no	—
36	C₂×C₁₃×Q₁₆	C₂³	1	2	yes	—
37	C₂×C₁₃×SD₁₆	—	2	4	no	—
38	C₂×C₁₃×SD₁₆	—	2	2	no	—
39	C₂×C₁₃×D₈	—	2	4	no	—
40	C₂×C₁₃×D₈	—	2	4	no	—
41	C₂×C₁₃×D₈	—	2	4	no	—
42	C₂×C₁₃×D₈	—	2	2	no	—
43	C₂×C₁₃×D₈	C₂³	1	2	yes	—
44	C₂×C₁₃×D₈	C₂³	1	4	yes	—
45	C₂×C₁₃×M₁₆	D₄	1	1	yes	characteristic
46	C₂²×C₈×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×#127	—	2	4	no	—
48	C₁₃×#124	—	2	4	no	—
49	C₁₃×#121	—	4	4	no	—
50	C₁₃×#120	—	4	4	no	—
51	C₁₃×#117	D₄	1	1	yes	characteristic
52	C₁₃×Q₃₂	—	2	2	no	—
53	C₁₃×SD₃₂	—	2	4	no	—
54	C₁₃×SD₃₂	—	2	8	no	—
55	C₁₃×D₁₆	—	2	4	no	—
56	C₁₃×D₁₆	—	2	4	no	—
57	C₁₃×D₁₆	—	2	2	no	—
58	C₁₃×D₁₆	—	2	8	no	—
59	C₁₃×M₃₂	D₄	1	4	yes	complemented
60	C₂×C₁₃×C₁₆	D₄	1	4	yes	complemented
61	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
62	C₂×C₁₃×C₁₆	C₂³	1	2	yes	—
63	C₂×C₁₃×C₁₆	C₂³	1	1	yes	characteristic
64	C₁₃×#97	D₄	1	2	yes	—
65	C₄×C₈×C₁₃	C₂³	1	2	yes	—
66	#33720	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
67	C₂⁴×C₁₃	—	4	2	no	—
68	C₁₃×#41	—	4	4	no	—
69	C₁₃×#41	—	4	4	no	—
70	C₂×Q₈×C₁₃	—	2	2	no	—
71	C₂×D₄×C₁₃	—	2	4	no	—
72	C₂×D₄×C₁₃	—	2	2	no	—
73	C₂×D₄×C₁₃	—	4	2	no	—
74	C₂×D₄×C₁₃	—	4	4	no	—
75	C₂×D₄×C₁₃	—	4	4	no	—
76	C₂×D₄×C₁₃	—	4	2	no	—
77	C₂×D₄×C₁₃	—	4	8	no	—
78	C₂²×C₄×C₁₃	—	4	2	no	—
79	C₂²×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
80	C₁₃×Q₁₆	—	2	2	no	—
81	C₁₃×Q₁₆	—	2	2	no	—
82	C₁₃×SD₁₆	—	4	4	no	—
83	C₁₃×SD₁₆	—	4	8	no	—
84	C₁₃×D₈	—	2	4	no	—
85	C₁₃×D₈	—	2	2	no	—
86	C₁₃×D₈	—	2	4	no	—
87	C₁₃×D₈	—	2	2	no	—
88	C₁₃×D₈	—	4	4	no	—
89	C₁₃×D₈	—	4	8	no	—
90	C₁₃×M₁₆	D₈	1	4	yes	—
91	C₂×C₈×C₁₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
93	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
94	C₂×C₈×C₁₃	C₂×D₄	1	2	yes	—
95	C₂×C₈×C₁₃	C₂⁴	1	1	yes	characteristic
96	C₁₃×C₄⋊C₄	D₈	1	2	yes	—
97	C₁₃×C₂²⋊C₄	—	4	8	no	—
98	C₁₃×C₂²⋊C₄	D₈	1	2	yes	—
99	C₄²×C₁₃	C₂×D₄	1	2	yes	—
100	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
101	C₁₃×C₁₆	C₂×D₄	1	2	yes	—
102	C₁₃×C₁₆	C₂×D₄	1	4	yes	—
103	C₂×#509	C₂×C₁₃	1	2	yes	complemented
104	C₂×#508	C₂×C₁₃	1	1	yes	characteristic, complemented
105	C₂²×D₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
106	#3155	C₂×C₁₃	1	2	yes	complemented
107	#2250	C₂×C₁₃	1	4	yes	complemented
108	#2249	C₂×C₁₃	1	2	yes	complemented
109	#2246	C₂×C₁₃	1	2	yes	complemented
110	#2108	C₂×C₁₃	1	1	yes	characteristic, complemented
111	C₂³×C₁₃	—	4	4	no	—
112	C₂³×C₁₃	—	4	2	no	—
113	C₂³×C₁₃	—	8	4	no	—
114	C₂³×C₁₃	—	16	4	no	—
115	C₂³×C₁₃	C₂×D₈	1	1	yes	characteristic
116	Q₈×C₁₃	—	4	2	no	—
117	Q₈×C₁₃	—	4	2	no	—
118	D₄×C₁₃	—	4	4	no	—
119	D₄×C₁₃	—	4	2	no	—
120	D₄×C₁₃	—	4	2	no	—
121	D₄×C₁₃	—	4	4	no	—
122	D₄×C₁₃	—	8	4	no	—
123	D₄×C₁₃	—	8	4	no	—
124	D₄×C₁₃	—	16	8	no	—
125	C₂×C₄×C₁₃	—	4	2	no	—
126	C₂×C₄×C₁₃	—	8	4	no	—
127	C₂×C₄×C₁₃	#127	1	2	yes	—
128	C₂×C₄×C₁₃	C₂×D₈	1	4	yes	—
129	C₂×C₄×C₁₃	C₂×D₈	1	2	yes	—
130	C₂×C₄×C₁₃	C₂×D₈	1	1	yes	characteristic
131	C₂×C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
132	C₈×C₁₃	#127	1	2	yes	—
133	C₈×C₁₃	C₂×D₈	1	4	yes	—
134	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
135	C₈×C₁₃	C₂²×D₄	1	1	yes	characteristic
136	C₂×AGL(1,ℤ/8)	—	2	2	no	—
137	C₂×#135	C₂²×C₁₃	1	2	yes	—
138	C₂²×D₈	C₂²×C₁₃	1	2	yes	—
139	#535	—	2	2	no	—
140	#509	—	2	8	no	—
141	#508	—	2	4	no	—
142	C₂×Q₃₂	C₂²×C₁₃	1	1	yes	characteristic, complemented
143	C₂×SD₃₂	C₂²×C₁₃	1	2	yes	complemented
144	C₂×SD₃₂	C₂²×C₁₃	1	4	yes	complemented
145	C₂×D₁₆	—	2	4	no	—
146	C₂×D₁₆	C₂²×C₁₃	1	2	yes	—
147	C₂×D₁₆	C₂²×C₁₃	1	2	yes	—
148	C₂×D₁₆	C₂²×C₁₃	1	1	yes	characteristic, complemented
149	C₂×D₁₆	C₂²×C₁₃	1	4	yes	complemented
150	C₂×M₃₂	C₂²×C₁₃	1	1	yes	characteristic
151	C₂²×C₁₆	C₂²×C₁₃	1	1	yes	characteristic
152	#496	—	2	4	no	—
153	#495	C₂²×C₁₃	1	4	yes	complemented
154	#493	C₂²×C₁₃	1	4	yes	complemented
155	#405	C₂²×C₁₃	1	1	yes	characteristic
156	#346	C₂²×C₁₃	1	2	yes	complemented
157	C₄×C₁₆	C₂²×C₁₃	1	2	yes	complemented
158	C₂²×C₁₃	—	2	2	no	—
159	C₂²×C₁₃	—	8	2	no	—
160	C₂²×C₁₃	—	8	4	no	—
161	C₂²×C₁₃	—	8	4	no	—
162	C₂²×C₁₃	—	8	8	no	—
163	C₂²×C₁₃	—	16	8	no	—
164	C₂²×C₁₃	#493	1	2	yes	—
165	C₂²×C₁₃	C₂²×D₈	1	1	yes	characteristic
166	C₄×C₁₃	—	2	4	no	—
167	C₄×C₁₃	—	8	2	no	—
168	C₄×C₁₃	#493	1	2	yes	—
169	C₄×C₁₃	C₂×#127	1	1	yes	characteristic
170	C₄×C₁₃	C₂²×D₈	1	1	yes	characteristic
171	C₂×#41	—	2	2	no	—
172	C₂²×D₄	—	2	2	no	—
173	AGL(1,ℤ/8)	—	4	8	no	—
174	#135	—	2	4	no	—
175	#135	—	2	4	no	—
176	C₂×Q₁₆	C₂³×C₁₃	1	2	yes	—
177	C₂×SD₁₆	—	2	2	no	—
178	C₂×SD₁₆	—	2	4	no	—
179	C₂×D₈	—	2	4	no	—
180	C₂×D₈	—	2	4	no	—
181	C₂×D₈	—	2	2	no	—
182	C₂×D₈	—	2	4	no	—
183	C₂×D₈	C₂³×C₁₃	1	4	yes	—
184	C₂×D₈	C₂³×C₁₃	1	2	yes	—
185	C₂×M₁₆	D₄×C₁₃	1	1	yes	characteristic
186	C₂²×C₈	C₂³×C₁₃	1	1	yes	characteristic
187	#127	—	2	4	no	—
188	#124	—	2	4	no	—
189	#121	—	4	4	no	—
190	#120	—	4	4	no	—
191	#117	D₄×C₁₃	1	1	yes	characteristic
192	Q₃₂	—	2	2	no	—
193	SD₃₂	—	2	4	no	—
194	SD₃₂	—	2	8	no	—
195	D₁₆	—	2	4	no	—
196	D₁₆	—	2	4	no	—
197	D₁₆	—	2	2	no	—
198	D₁₆	—	2	8	no	—
199	M₃₂	D₄×C₁₃	1	4	yes	complemented
200	C₂×C₁₆	D₄×C₁₃	1	4	yes	complemented
201	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
202	C₂×C₁₆	C₂³×C₁₃	1	2	yes	—
203	C₂×C₁₆	C₂³×C₁₃	1	1	yes	characteristic
204	#97	D₄×C₁₃	1	2	yes	—
205	C₄×C₈	C₂³×C₁₃	1	2	yes	—
206	C₂×C₁₃	—	2	2	no	—
207	C₂×C₁₃	—	16	2	no	—
208	C₂×C₁₃	—	16	4	no	—
209	C₂×C₁₃	C₂×#493	1	1	yes	characteristic
210	C₂×C₁₃	#3415	1	2	yes	—
211	C₂⁴	—	4	2	no	—
212	#41	—	4	4	no	—
213	#41	—	4	4	no	—
214	C₂×Q₈	—	2	2	no	—
215	C₂×D₄	—	2	4	no	—
216	C₂×D₄	—	2	2	no	—
217	C₂×D₄	—	4	2	no	—
218	C₂×D₄	—	4	4	no	—
219	C₂×D₄	—	4	4	no	—
220	C₂×D₄	—	4	2	no	—
221	C₂×D₄	—	4	8	no	—
222	C₂²×C₄	—	4	2	no	—
223	C₂²×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
224	Q₁₆	—	2	2	no	—
225	Q₁₆	—	2	2	no	—
226	SD₁₆	—	4	4	no	—
227	SD₁₆	—	4	8	no	—
228	D₈	—	2	4	no	—
229	D₈	—	2	2	no	—
230	D₈	—	2	4	no	—
231	D₈	—	2	2	no	—
232	D₈	—	4	4	no	—
233	D₈	—	4	8	no	—
234	M₁₆	C₁₃×D₈	1	4	yes	—
235	C₂×C₈	C₂×D₄×C₁₃	1	1	yes	characteristic
236	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
237	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
238	C₂×C₈	C₂×D₄×C₁₃	1	2	yes	—
239	C₂×C₈	C₂⁴×C₁₃	1	1	yes	characteristic
240	C₄⋊C₄	C₁₃×D₈	1	2	yes	—
241	C₂²⋊C₄	—	4	8	no	—
242	C₂²⋊C₄	C₁₃×D₈	1	2	yes	—
243	C₄²	C₂×D₄×C₁₃	1	2	yes	—
244	C₁₆	C₂×D₄×C₁₃	1	2	yes	—
245	C₁₆	C₂×D₄×C₁₃	1	2	yes	—
246	C₁₆	C₂×D₄×C₁₃	1	4	yes	—
247	C₁₃	#33720	1	1	yes	characteristic, direct product factor, Sylow
248	C₂³	—	4	4	no	—
249	C₂³	—	4	2	no	—
250	C₂³	—	8	4	no	—
251	C₂³	—	16	4	no	—
252	C₂³	C₂×C₁₃×D₈	1	1	yes	characteristic
253	Q₈	—	4	2	no	—
254	Q₈	—	4	2	no	—
255	D₄	—	4	4	no	—
256	D₄	—	4	2	no	—
257	D₄	—	4	2	no	—
258	D₄	—	4	4	no	—
259	D₄	—	8	4	no	—
260	D₄	—	8	4	no	—
261	D₄	—	16	8	no	—
262	C₂×C₄	—	4	2	no	—
263	C₂×C₄	—	8	4	no	—
264	C₂×C₄	C₁₃×#127	1	2	yes	—
265	C₂×C₄	C₂×C₁₃×D₈	1	4	yes	—
266	C₂×C₄	C₂×C₁₃×D₈	1	1	yes	characteristic
267	C₂×C₄	C₂×C₁₃×D₈	1	2	yes	—
268	C₂×C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
269	C₈	C₁₃×#127	1	2	yes	—
270	C₈	C₂×C₁₃×D₈	1	4	yes	—
271	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
272	C₈	C₂²×D₄×C₁₃	1	1	yes	characteristic
273	C₂²	—	2	2	no	—
274	C₂²	—	8	2	no	—
275	C₂²	—	8	4	no	—
276	C₂²	—	8	4	no	—
277	C₂²	—	8	8	no	—
278	C₂²	—	16	8	no	—
279	C₂²	C₁₃×#493	1	2	yes	—
280	C₂²	C₂²×C₁₃×D₈	1	1	yes	characteristic
281	C₄	—	2	4	no	—
282	C₄	—	8	2	no	—
283	C₄	C₁₃×#493	1	2	yes	—
284	C₄	C₂×C₁₃×#127	1	1	yes	characteristic
285	C₄	C₂²×C₁₃×D₈	1	1	yes	characteristic
286	C₂	—	2	2	no	—
287	C₂	—	16	4	no	—
288	C₂	—	16	2	no	—
289	C₂	C₂×C₁₃×#493	1	1	yes	characteristic
290	C₂	C₁₃×#3415	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₈
commutator, Frattini, cosocle	C₂×C₈	C₂⁴×C₁₃
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₁₃×#117 ◁ C₁₃×#405 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#509	C₂	semidirect
2. C₂×C₁₃×#508	C₂	semidirect
3. C₂²×C₁₃×D₁₆	C₂	semidirect
4. C₁₃×#3155	C₂	semidirect
5. C₁₃×#2250	C₂	semidirect
6. C₁₃×#2249	C₂	semidirect
7. C₁₃×#2246	C₂	semidirect
8. C₁₃×#2108	C₂	semidirect
15. C₂×C₁₃×Q₃₂	C₂²	semidirect
16. C₂×C₁₃×SD₃₂	C₂²	semidirect
17. C₂×C₁₃×SD₃₂	C₂²	semidirect
21. C₂×C₁₃×D₁₆	C₂²	semidirect
22. C₂×C₁₃×D₁₆	C₂²	semidirect
26. C₁₃×#495	C₂²	semidirect
27. C₁₃×#493	C₂²	semidirect
29. C₁₃×#346	C₂²	semidirect
30. C₄×C₁₃×C₁₆	C₂²	semidirect
59. C₁₃×M₃₂	D₄	semidirect
60. C₂×C₁₃×C₁₆	D₄	semidirect
66. #33720	C₁₃	direct (N × H)
103. C₂×#509	C₂×C₁₃	semidirect
104. C₂×#508	C₂×C₁₃	semidirect
105. C₂²×D₁₆	C₂×C₁₃	semidirect
106. #3155	C₂×C₁₃	semidirect
107. #2250	C₂×C₁₃	semidirect
108. #2249	C₂×C₁₃	semidirect
109. #2246	C₂×C₁₃	semidirect
110. #2108	C₂×C₁₃	semidirect
142. C₂×Q₃₂	C₂²×C₁₃	semidirect
143. C₂×SD₃₂	C₂²×C₁₃	semidirect
144. C₂×SD₃₂	C₂²×C₁₃	semidirect
148. C₂×D₁₆	C₂²×C₁₃	semidirect
149. C₂×D₁₆	C₂²×C₁₃	semidirect
153. #495	C₂²×C₁₃	semidirect
154. #493	C₂²×C₁₃	semidirect
156. #346	C₂²×C₁₃	semidirect
157. C₄×C₁₆	C₂²×C₁₃	semidirect
199. M₃₂	D₄×C₁₃	semidirect
200. C₂×C₁₆	D₄×C₁₃	semidirect
247. C₁₃	#33720	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×#509	2
C₂×C₁₃×#508	2
C₂²×C₁₃×D₁₆	2
C₁₃×#3155	2
C₁₃×#2250	2
C₁₃×#2249	2
C₁₃×#2246	2
C₁₃×#2108	2
#33720	13

Subgroups — C13×#33720 #15720059

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#33720 #15720059