Subgroups — C₁₃×#3171 #13112229

Order 1664.

408 conjugacy classes of subgroups.

The subgroup lattice has 276 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#529	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#501	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#496	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#490	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#489	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#469	C₂	1	2	yes	maximal, complemented
8	C₁₃×#468	C₂	1	2	yes	maximal, complemented
9	C₄×C₁₃×SD₁₆	C₂	1	2	yes	maximal, complemented
10	C₄×C₁₃×D₈	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₃×M₁₆	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	—	2	1	no	—
13	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
14	C₂×C₁₃×D₈	C₂²	1	2	yes	—
15	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
16	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₃×#127	C₂²	1	1	yes	characteristic
18	C₁₃×#126	—	2	1	no	—
19	C₁₃×#125	C₂²	1	1	yes	characteristic
20	C₁₃×#124	—	2	1	no	—
21	C₁₃×#124	C₂²	1	1	yes	characteristic
22	C₁₃×#123	—	2	1	no	—
23	C₁₃×C₂²⋊Q₈	C₂²	1	1	yes	characteristic
24	C₁₃×#121	—	2	1	no	—
25	C₁₃×#121	C₂²	1	2	yes	—
26	C₁₃×#121	C₂²	1	1	yes	characteristic
27	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
28	C₄×D₄×C₁₃	—	2	1	no	—
29	C₄×D₄×C₁₃	—	2	1	no	—
30	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
31	C₄×D₄×C₁₃	C₂²	1	2	yes	—
32	C₁₃×#117	—	2	1	no	—
33	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
34	C₁₃×#107	C₂²	1	2	yes	—
35	C₁₃×#106	C₂²	1	2	yes	—
36	C₁₃×#103	C₂²	1	2	yes	—
37	C₁₃×#102	C₂²	1	2	yes	—
38	C₁₃×#102	C₂²	1	2	yes	—
39	C₁₃×#102	C₂²	1	2	yes	—
40	C₁₃×#97	C₂²	1	1	yes	characteristic
41	C₁₃×#97	C₂²	1	1	yes	characteristic
42	C₄×C₈×C₁₃	C₂²	1	2	yes	—
43	C₁₃×#41	—	4	4	no	—
44	C₂×Q₈×C₁₃	—	2	1	no	—
45	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
46	C₂×D₄×C₁₃	—	2	1	no	—
47	C₂×D₄×C₁₃	—	2	2	no	—
48	C₂×D₄×C₁₃	—	2	1	no	—
49	C₂×D₄×C₁₃	—	2	1	no	—
50	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
51	C₂×D₄×C₁₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₁₃	—	2	1	no	—
53	C₂²×C₄×C₁₃	—	2	2	no	—
54	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
55	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
56	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
57	C₁₃×SD₁₆	—	2	4	no	—
58	C₁₃×D₈	—	2	4	no	—
59	C₁₃×M₁₆	—	2	4	no	—
60	C₂×C₈×C₁₃	C₂³	1	2	yes	—
61	C₂×C₈×C₁₃	C₂³	1	2	yes	—
62	C₁₃×C₄⋊C₄	—	2	1	no	—
63	C₁₃×C₄⋊C₄	—	2	1	no	—
64	C₁₃×C₄⋊C₄	—	2	1	no	—
65	C₁₃×C₄⋊C₄	—	2	1	no	—
66	C₁₃×C₄⋊C₄	—	2	1	no	—
67	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
68	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
69	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
70	C₁₃×C₂²⋊C₄	—	2	1	no	—
71	C₁₃×C₂²⋊C₄	—	2	1	no	—
72	C₁₃×C₂²⋊C₄	—	2	1	no	—
73	C₁₃×C₂²⋊C₄	—	2	2	no	—
74	C₁₃×C₂²⋊C₄	—	2	1	no	—
75	C₁₃×C₂²⋊C₄	—	2	2	no	—
76	C₁₃×C₂²⋊C₄	—	2	1	no	—
77	C₄²×C₁₃	—	2	1	no	—
78	C₄²×C₁₃	D₄	1	1	yes	characteristic
79	C₄²×C₁₃	D₄	1	1	yes	characteristic
80	C₄²×C₁₃	C₂³	1	1	yes	characteristic
81	C₄²×C₁₃	C₂³	1	1	yes	characteristic
82	#3171	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
83	C₂³×C₁₃	—	2	1	no	—
84	C₂³×C₁₃	—	2	2	no	—
85	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
86	Q₈×C₁₃	—	2	2	no	—
87	Q₈×C₁₃	—	4	2	no	—
88	D₄×C₁₃	—	2	2	no	—
89	D₄×C₁₃	—	2	2	no	—
90	D₄×C₁₃	—	2	2	no	—
91	D₄×C₁₃	—	4	2	no	—
92	D₄×C₁₃	—	4	2	no	—
93	D₄×C₁₃	—	4	2	no	—
94	D₄×C₁₃	—	4	4	no	—
95	C₂×C₄×C₁₃	—	2	2	no	—
96	C₂×C₄×C₁₃	—	2	2	no	—
97	C₂×C₄×C₁₃	—	2	1	no	—
98	C₂×C₄×C₁₃	—	2	1	no	—
99	C₂×C₄×C₁₃	—	2	1	no	—
100	C₂×C₄×C₁₃	—	2	2	no	—
101	C₂×C₄×C₁₃	—	4	1	no	—
102	C₂×C₄×C₁₃	—	4	2	no	—
103	C₂×C₄×C₁₃	—	4	4	no	—
104	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
106	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
107	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
108	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
109	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
110	C₈×C₁₃	—	2	2	no	—
111	C₈×C₁₃	#41	1	4	yes	complemented
112	#532	C₂×C₁₃	1	1	yes	characteristic, complemented
113	#529	C₂×C₁₃	1	1	yes	characteristic, complemented
114	#501	C₂×C₁₃	1	1	yes	characteristic, complemented
115	#496	C₂×C₁₃	1	1	yes	characteristic, complemented
116	#490	C₂×C₁₃	1	1	yes	characteristic, complemented
117	#489	C₂×C₁₃	1	1	yes	characteristic, complemented
118	#469	C₂×C₁₃	1	2	yes	complemented
119	#468	C₂×C₁₃	1	2	yes	complemented
120	C₄×SD₁₆	C₂×C₁₃	1	2	yes	complemented
121	C₄×D₈	C₂×C₁₃	1	2	yes	complemented
122	C₄×M₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
123	C₂²×C₁₃	—	2	1	no	—
124	C₂²×C₁₃	—	2	1	no	—
125	C₂²×C₁₃	—	2	1	no	—
126	C₂²×C₁₃	—	4	1	no	—
127	C₂²×C₁₃	—	4	2	no	—
128	C₂²×C₁₃	—	4	2	no	—
129	C₂²×C₁₃	—	4	1	no	—
130	C₂²×C₁₃	—	4	2	no	—
131	C₂²×C₁₃	—	4	1	no	—
132	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
133	C₄×C₁₃	—	2	1	no	—
134	C₄×C₁₃	—	2	1	no	—
135	C₄×C₁₃	—	2	1	no	—
136	C₄×C₁₃	—	2	1	no	—
137	C₄×C₁₃	—	4	1	no	—
138	C₄×C₁₃	—	4	1	no	—
139	C₄×C₁₃	—	4	2	no	—
140	C₄×C₁₃	—	4	1	no	—
141	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
142	C₄×C₁₃	C₂×#41	1	2	yes	—
143	C₂×#41	—	2	1	no	—
144	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
145	C₂×D₈	C₂²×C₁₃	1	2	yes	—
146	C₂×M₁₆	C₂²×C₁₃	1	2	yes	—
147	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
148	#127	C₂²×C₁₃	1	1	yes	characteristic
149	#126	—	2	1	no	—
150	#125	C₂²×C₁₃	1	1	yes	characteristic
151	#124	—	2	1	no	—
152	#124	C₂²×C₁₃	1	1	yes	characteristic
153	#123	—	2	1	no	—
154	C₂²⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
155	#121	—	2	1	no	—
156	#121	C₂²×C₁₃	1	1	yes	characteristic
157	#121	C₂²×C₁₃	1	2	yes	—
158	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
159	C₄×D₄	—	2	1	no	—
160	C₄×D₄	—	2	1	no	—
161	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
162	C₄×D₄	C₂²×C₁₃	1	2	yes	—
163	#117	—	2	1	no	—
164	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
165	#107	C₂²×C₁₃	1	2	yes	—
166	#106	C₂²×C₁₃	1	2	yes	—
167	#103	C₂²×C₁₃	1	2	yes	—
168	#102	C₂²×C₁₃	1	2	yes	—
169	#102	C₂²×C₁₃	1	2	yes	—
170	#102	C₂²×C₁₃	1	2	yes	—
171	#97	C₂²×C₁₃	1	1	yes	characteristic
172	#97	C₂²×C₁₃	1	1	yes	characteristic
173	C₄×C₈	C₂²×C₁₃	1	2	yes	—
174	C₂×C₁₃	—	4	1	no	—
175	C₂×C₁₃	—	8	2	no	—
176	C₂×C₁₃	—	8	1	no	—
177	C₂×C₁₃	#532	1	1	yes	characteristic
178	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
179	C₂×C₁₃	#575	1	1	yes	characteristic
180	#41	—	4	4	no	—
181	C₂×Q₈	—	2	1	no	—
182	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
183	C₂×D₄	—	2	1	no	—
184	C₂×D₄	—	2	2	no	—
185	C₂×D₄	—	2	1	no	—
186	C₂×D₄	—	2	1	no	—
187	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
188	C₂×D₄	C₂³×C₁₃	1	2	yes	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	2	no	—
191	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
192	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
193	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
194	SD₁₆	—	2	4	no	—
195	D₈	—	2	4	no	—
196	M₁₆	—	2	4	no	—
197	C₂×C₈	C₂³×C₁₃	1	2	yes	—
198	C₂×C₈	C₂³×C₁₃	1	2	yes	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	—	2	1	no	—
201	C₄⋊C₄	—	2	1	no	—
202	C₄⋊C₄	—	2	1	no	—
203	C₄⋊C₄	—	2	1	no	—
204	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
205	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
206	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
207	C₂²⋊C₄	—	2	1	no	—
208	C₂²⋊C₄	—	2	1	no	—
209	C₂²⋊C₄	—	2	1	no	—
210	C₂²⋊C₄	—	2	2	no	—
211	C₂²⋊C₄	—	2	1	no	—
212	C₂²⋊C₄	—	2	2	no	—
213	C₂²⋊C₄	—	2	1	no	—
214	C₄²	—	2	1	no	—
215	C₄²	D₄×C₁₃	1	1	yes	characteristic
216	C₄²	D₄×C₁₃	1	1	yes	characteristic
217	C₄²	C₂³×C₁₃	1	1	yes	characteristic
218	C₄²	C₂³×C₁₃	1	1	yes	characteristic
219	C₁₃	#3171	1	1	yes	characteristic, direct product factor, Sylow
220	C₂³	—	2	1	no	—
221	C₂³	—	2	2	no	—
222	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
223	Q₈	—	2	2	no	—
224	Q₈	—	4	2	no	—
225	D₄	—	2	2	no	—
226	D₄	—	2	2	no	—
227	D₄	—	2	2	no	—
228	D₄	—	4	2	no	—
229	D₄	—	4	2	no	—
230	D₄	—	4	2	no	—
231	D₄	—	4	4	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	1	no	—
235	C₂×C₄	—	2	1	no	—
236	C₂×C₄	—	2	1	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	4	1	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
242	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
243	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
244	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
245	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
246	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
247	C₈	—	2	2	no	—
248	C₈	C₁₃×#41	1	4	yes	complemented
249	C₂²	—	2	1	no	—
250	C₂²	—	2	1	no	—
251	C₂²	—	2	1	no	—
252	C₂²	—	4	1	no	—
253	C₂²	—	4	2	no	—
254	C₂²	—	4	2	no	—
255	C₂²	—	4	1	no	—
256	C₂²	—	4	2	no	—
257	C₂²	—	4	1	no	—
258	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
259	C₄	—	2	1	no	—
260	C₄	—	2	1	no	—
261	C₄	—	2	1	no	—
262	C₄	—	2	1	no	—
263	C₄	—	4	1	no	—
264	C₄	—	4	1	no	—
265	C₄	—	4	2	no	—
266	C₄	—	4	1	no	—
267	C₄	C₁₃×AGL(1,ℤ/8)	1	2	yes	—
268	C₄	C₂×C₁₃×#41	1	2	yes	—
269	C₂	—	4	1	no	—
270	C₂	—	8	2	no	—
271	C₂	—	8	1	no	—
272	C₂	C₁₃×#532	1	1	yes	characteristic
273	C₂	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
274	C₂	C₁₃×#575	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄×C₁₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
socle	C₂²×C₁₃	C₂²×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#532	C₂	semidirect
2. C₁₃×#529	C₂	semidirect
3. C₁₃×#501	C₂	semidirect
4. C₁₃×#496	C₂	semidirect
5. C₁₃×#490	C₂	semidirect
6. C₁₃×#489	C₂	semidirect
7. C₁₃×#469	C₂	semidirect
8. C₁₃×#468	C₂	semidirect
9. C₄×C₁₃×SD₁₆	C₂	semidirect
10. C₄×C₁₃×D₈	C₂	semidirect
11. C₄×C₁₃×M₁₆	C₂	semidirect
82. #3171	C₁₃	direct (N × H)
111. C₈×C₁₃	#41	semidirect
112. #532	C₂×C₁₃	semidirect
113. #529	C₂×C₁₃	semidirect
114. #501	C₂×C₁₃	semidirect
115. #496	C₂×C₁₃	semidirect
116. #490	C₂×C₁₃	semidirect
117. #489	C₂×C₁₃	semidirect
118. #469	C₂×C₁₃	semidirect
119. #468	C₂×C₁₃	semidirect
120. C₄×SD₁₆	C₂×C₁₃	semidirect
121. C₄×D₈	C₂×C₁₃	semidirect
122. C₄×M₁₆	C₂×C₁₃	semidirect
219. C₁₃	#3171	direct (N × H)
248. C₈	C₁₃×#41	semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₁₃×#532	2
C₁₃×#529	2
C₁₃×#501	2
C₁₃×#496	2
C₁₃×#490	2
C₁₃×#489	2
C₁₃×#469	2
C₁₃×#468	2
C₄×C₁₃×SD₁₆	2
C₄×C₁₃×D₈	2
C₄×C₁₃×M₁₆	2
#3171	13

Subgroups — C13×#3171 #13112229

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3171 #13112229