Subgroups — C₁₃×#3512 #13112570

Order 1664.

1032 conjugacy classes of subgroups.

The subgroup lattice has 272 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#553	C₂	1	4	yes	maximal, complemented
2	C₁₃×#553	C₂	1	8	yes	maximal, complemented
3	C₁₃×#552	C₂	1	4	yes	maximal, complemented
4	C₁₃×#541	C₂	1	2	yes	maximal, complemented
5	C₁₃×#540	C₂	1	2	yes	maximal, complemented
6	C₁₃×#537	C₂	1	2	yes	maximal, complemented
7	C₁₃×#532	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#529	C₂	1	4	yes	maximal, complemented
9	C₂×C₁₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#520	C₂	1	2	yes	maximal, complemented
11	C₄×C₁₃×#41	C₂	1	1	yes	maximal, characteristic, complemented
12	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×#41	C₂²	1	2	yes	—
14	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₁₃×#127	C₂²	1	1	yes	characteristic
16	C₁₃×#126	C₂²	1	8	yes	—
17	C₁₃×#126	C₂²	1	8	yes	—
18	C₁₃×#125	C₂²	1	2	yes	—
19	C₁₃×#125	C₂²	1	2	yes	—
20	C₁₃×#125	C₂²	1	4	yes	—
21	C₁₃×#125	C₂²	1	8	yes	—
22	C₁₃×#124	C₂²	1	2	yes	—
23	C₁₃×#123	C₂²	1	8	yes	—
24	C₁₃×#123	C₂²	1	8	yes	—
25	C₁₃×C₂²⋊Q₈	C₂²	1	4	yes	—
26	C₁₃×C₂²⋊Q₈	C₂²	1	4	yes	—
27	C₁₃×#121	C₂²	1	4	yes	—
28	C₁₃×#121	C₂²	1	4	yes	—
29	C₁₃×#121	C₂²	1	8	yes	—
30	C₁₃×#121	C₂²	1	4	yes	—
31	C₁₃×#121	C₂²	1	8	yes	—
32	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
33	C₄×Q₈×C₁₃	C₂²	1	4	yes	—
34	C₄×D₄×C₁₃	C₂²	1	4	yes	—
35	C₄×D₄×C₁₃	C₂²	1	2	yes	—
36	C₄×D₄×C₁₃	C₂²	1	4	yes	—
37	C₄×D₄×C₁₃	C₂²	1	8	yes	—
38	C₄×D₄×C₁₃	C₂²	1	8	yes	—
39	C₄×D₄×C₁₃	C₂²	1	4	yes	—
40	C₁₃×#117	C₂²	1	1	yes	characteristic
41	C₁₃×#117	C₂²	1	2	yes	—
42	C₁₃×#117	C₂²	1	4	yes	—
43	C₁₃×#117	C₂²	1	8	yes	—
44	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
45	C₂×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
46	C₂×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
47	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
48	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
49	C₁₃×#41	—	2	4	no	—
50	C₁₃×#41	—	2	8	no	—
51	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
53	C₂×D₄×C₁₃	C₂³	1	2	yes	—
54	C₂×D₄×C₁₃	C₂³	1	4	yes	—
55	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
56	C₂×D₄×C₁₃	C₂³	1	2	yes	—
57	C₂×D₄×C₁₃	C₂³	1	8	yes	—
58	C₂×D₄×C₁₃	C₂³	1	4	yes	—
59	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
60	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
61	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
63	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
64	C₂²×C₄×C₁₃	C₂³	1	8	yes	—
65	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
66	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
67	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
68	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
69	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
70	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
71	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
72	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
73	C₁₃×C₄⋊C₄	C₂³	1	8	yes	—
74	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
75	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
76	C₁₃×C₂²⋊C₄	C₂³	1	2	yes	—
77	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
78	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
79	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
80	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
81	C₁₃×C₂²⋊C₄	C₂³	1	8	yes	—
82	C₄²×C₁₃	C₂³	1	1	yes	characteristic
83	C₄²×C₁₃	C₂³	1	2	yes	—
84	C₄²×C₁₃	C₂³	1	4	yes	—
85	C₄²×C₁₃	C₂³	1	1	yes	characteristic
86	C₄²×C₁₃	C₂³	1	2	yes	—
87	C₄²×C₁₃	C₂³	1	8	yes	—
88	#3512	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
89	C₂³×C₁₃	C₂⁴	1	4	yes	—
90	C₂³×C₁₃	C₂⁴	1	2	yes	—
91	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
92	Q₈×C₁₃	—	2	2	no	—
93	Q₈×C₁₃	—	2	4	no	—
94	D₄×C₁₃	—	2	4	no	—
95	D₄×C₁₃	—	2	8	no	—
96	D₄×C₁₃	—	2	2	no	—
97	D₄×C₁₃	—	2	4	no	—
98	D₄×C₁₃	—	4	8	no	—
99	D₄×C₁₃	—	4	4	no	—
100	C₂×C₄×C₁₃	—	2	8	no	—
101	C₂×C₄×C₁₃	—	2	4	no	—
102	C₂×C₄×C₁₃	—	4	4	no	—
103	C₂×C₄×C₁₃	—	4	8	no	—
104	C₂×C₄×C₁₃	—	4	2	no	—
105	C₂×C₄×C₁₃	—	4	4	no	—
106	C₂×C₄×C₁₃	#41	1	4	yes	—
107	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
108	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
109	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
110	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
111	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
112	C₂×C₄×C₁₃	C₂⁴	1	8	yes	—
113	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
114	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
115	#553	C₂×C₁₃	1	4	yes	complemented
116	#553	C₂×C₁₃	1	8	yes	complemented
117	#552	C₂×C₁₃	1	4	yes	complemented
118	#541	C₂×C₁₃	1	2	yes	complemented
119	#540	C₂×C₁₃	1	2	yes	complemented
120	#537	C₂×C₁₃	1	2	yes	complemented
121	#532	C₂×C₁₃	1	1	yes	characteristic, complemented
122	#529	C₂×C₁₃	1	4	yes	complemented
123	C₂×#125	C₂×C₁₃	1	1	yes	characteristic, complemented
124	#520	C₂×C₁₃	1	2	yes	complemented
125	C₄×#41	C₂×C₁₃	1	1	yes	characteristic, complemented
126	C₂²×C₁₃	—	2	2	no	—
127	C₂²×C₁₃	—	2	4	no	—
128	C₂²×C₁₃	—	2	2	no	—
129	C₂²×C₁₃	—	2	4	no	—
130	C₂²×C₁₃	—	2	4	no	—
131	C₂²×C₁₃	—	2	4	no	—
132	C₂²×C₁₃	C₂×#41	1	2	yes	—
133	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
134	C₄×C₁₃	—	2	4	no	—
135	C₄×C₁₃	—	2	2	no	—
136	C₄×C₁₃	—	2	2	no	—
137	C₄×C₁₃	—	2	4	no	—
138	C₄×C₁₃	—	2	8	no	—
139	C₄×C₁₃	—	2	2	no	—
140	C₄×C₁₃	C₂×#41	1	2	yes	—
141	C₄×C₁₃	C₂×#41	1	2	yes	—
142	C₂×#41	C₂²×C₁₃	1	1	yes	characteristic
143	C₂×#41	C₂²×C₁₃	1	2	yes	—
144	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
145	#127	C₂²×C₁₃	1	1	yes	characteristic
146	#126	C₂²×C₁₃	1	8	yes	—
147	#126	C₂²×C₁₃	1	8	yes	—
148	#125	C₂²×C₁₃	1	2	yes	—
149	#125	C₂²×C₁₃	1	2	yes	—
150	#125	C₂²×C₁₃	1	4	yes	—
151	#125	C₂²×C₁₃	1	8	yes	—
152	#124	C₂²×C₁₃	1	2	yes	—
153	#123	C₂²×C₁₃	1	8	yes	—
154	#123	C₂²×C₁₃	1	8	yes	—
155	C₂²⋊Q₈	C₂²×C₁₃	1	4	yes	—
156	C₂²⋊Q₈	C₂²×C₁₃	1	4	yes	—
157	#121	C₂²×C₁₃	1	4	yes	—
158	#121	C₂²×C₁₃	1	4	yes	—
159	#121	C₂²×C₁₃	1	8	yes	—
160	#121	C₂²×C₁₃	1	4	yes	—
161	#121	C₂²×C₁₃	1	8	yes	—
162	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
163	C₄×Q₈	C₂²×C₁₃	1	4	yes	—
164	C₄×D₄	C₂²×C₁₃	1	4	yes	—
165	C₄×D₄	C₂²×C₁₃	1	2	yes	—
166	C₄×D₄	C₂²×C₁₃	1	4	yes	—
167	C₄×D₄	C₂²×C₁₃	1	8	yes	—
168	C₄×D₄	C₂²×C₁₃	1	8	yes	—
169	C₄×D₄	C₂²×C₁₃	1	4	yes	—
170	#117	C₂²×C₁₃	1	1	yes	characteristic
171	#117	C₂²×C₁₃	1	2	yes	—
172	#117	C₂²×C₁₃	1	4	yes	—
173	#117	C₂²×C₁₃	1	8	yes	—
174	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
175	C₂×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
176	C₂×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
177	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
178	C₂×C₄²	C₂²×C₁₃	1	2	yes	—
179	C₂×C₁₃	—	2	2	no	—
180	C₂×C₁₃	—	4	4	no	—
181	C₂×C₁₃	—	4	2	no	—
182	C₂×C₁₃	C₂²×#41	1	1	yes	characteristic
183	C₂×C₁₃	#585	1	2	yes	—
184	#41	—	2	4	no	—
185	#41	—	2	8	no	—
186	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
187	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
188	C₂×D₄	C₂³×C₁₃	1	2	yes	—
189	C₂×D₄	C₂³×C₁₃	1	4	yes	—
190	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
191	C₂×D₄	C₂³×C₁₃	1	2	yes	—
192	C₂×D₄	C₂³×C₁₃	1	8	yes	—
193	C₂×D₄	C₂³×C₁₃	1	4	yes	—
194	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
195	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
196	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
197	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
198	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
199	C₂²×C₄	C₂³×C₁₃	1	8	yes	—
200	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
201	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
202	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
203	C₄⋊C₄	C₂³×C₁₃	1	8	yes	—
204	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
205	C₄⋊C₄	C₂³×C₁₃	1	8	yes	—
206	C₄⋊C₄	C₂³×C₁₃	1	8	yes	—
207	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
208	C₄⋊C₄	C₂³×C₁₃	1	8	yes	—
209	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
210	C₂²⋊C₄	C₂³×C₁₃	1	4	yes	—
211	C₂²⋊C₄	C₂³×C₁₃	1	2	yes	—
212	C₂²⋊C₄	C₂³×C₁₃	1	4	yes	—
213	C₂²⋊C₄	C₂³×C₁₃	1	8	yes	—
214	C₂²⋊C₄	C₂³×C₁₃	1	8	yes	—
215	C₂²⋊C₄	C₂³×C₁₃	1	4	yes	—
216	C₂²⋊C₄	C₂³×C₁₃	1	8	yes	—
217	C₄²	C₂³×C₁₃	1	1	yes	characteristic
218	C₄²	C₂³×C₁₃	1	2	yes	—
219	C₄²	C₂³×C₁₃	1	4	yes	—
220	C₄²	C₂³×C₁₃	1	1	yes	characteristic
221	C₄²	C₂³×C₁₃	1	2	yes	—
222	C₄²	C₂³×C₁₃	1	8	yes	—
223	C₁₃	#3512	1	1	yes	characteristic, direct product factor, Sylow
224	C₂³	C₂⁴×C₁₃	1	4	yes	—
225	C₂³	C₂⁴×C₁₃	1	2	yes	—
226	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
227	Q₈	—	2	2	no	—
228	Q₈	—	2	4	no	—
229	D₄	—	2	4	no	—
230	D₄	—	2	8	no	—
231	D₄	—	2	2	no	—
232	D₄	—	2	4	no	—
233	D₄	—	4	8	no	—
234	D₄	—	4	4	no	—
235	C₂×C₄	—	2	8	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	4	4	no	—
238	C₂×C₄	—	4	8	no	—
239	C₂×C₄	—	4	2	no	—
240	C₂×C₄	—	4	4	no	—
241	C₂×C₄	C₁₃×#41	1	4	yes	—
242	C₂×C₄	C₂⁴×C₁₃	1	2	yes	—
243	C₂×C₄	C₂⁴×C₁₃	1	4	yes	—
244	C₂×C₄	C₂⁴×C₁₃	1	2	yes	—
245	C₂×C₄	C₂⁴×C₁₃	1	4	yes	—
246	C₂×C₄	C₂⁴×C₁₃	1	2	yes	—
247	C₂×C₄	C₂⁴×C₁₃	1	8	yes	—
248	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
249	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
250	C₂²	—	2	2	no	—
251	C₂²	—	2	4	no	—
252	C₂²	—	2	2	no	—
253	C₂²	—	2	4	no	—
254	C₂²	—	2	4	no	—
255	C₂²	—	2	4	no	—
256	C₂²	C₂×C₁₃×#41	1	2	yes	—
257	C₂²	C₂⁵×C₁₃	1	1	yes	characteristic
258	C₄	—	2	4	no	—
259	C₄	—	2	2	no	—
260	C₄	—	2	2	no	—
261	C₄	—	2	4	no	—
262	C₄	—	2	8	no	—
263	C₄	—	2	2	no	—
264	C₄	C₂×C₁₃×#41	1	2	yes	—
265	C₄	C₂×C₁₃×#41	1	2	yes	—
266	C₂	—	2	2	no	—
267	C₂	—	4	4	no	—
268	C₂	—	4	2	no	—
269	C₂	C₂²×C₁₃×#41	1	1	yes	characteristic
270	C₂	C₁₃×#585	1	2	yes	—

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂⁵
commutator, Frattini, cosocle	C₂²	C₂⁵×C₁₃
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#553	C₂	semidirect
2. C₁₃×#553	C₂	semidirect
3. C₁₃×#552	C₂	semidirect
4. C₁₃×#541	C₂	semidirect
5. C₁₃×#540	C₂	semidirect
6. C₁₃×#537	C₂	semidirect
7. C₁₃×#532	C₂	semidirect
8. C₁₃×#529	C₂	semidirect
9. C₂×C₁₃×#125	C₂	semidirect
10. C₁₃×#520	C₂	semidirect
11. C₄×C₁₃×#41	C₂	semidirect
88. #3512	C₁₃	direct (N × H)
115. #553	C₂×C₁₃	semidirect
116. #553	C₂×C₁₃	semidirect
117. #552	C₂×C₁₃	semidirect
118. #541	C₂×C₁₃	semidirect
119. #540	C₂×C₁₃	semidirect
120. #537	C₂×C₁₃	semidirect
121. #532	C₂×C₁₃	semidirect
122. #529	C₂×C₁₃	semidirect
123. C₂×#125	C₂×C₁₃	semidirect
124. #520	C₂×C₁₃	semidirect
125. C₄×#41	C₂×C₁₃	semidirect
223. C₁₃	#3512	direct (N × H)

Maximal subgroups

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

subgroup	index
C₁₃×#553	2
C₁₃×#552	2
C₁₃×#541	2
C₁₃×#540	2
C₁₃×#537	2
C₁₃×#532	2
C₁₃×#529	2
C₂×C₁₃×#125	2
C₁₃×#520	2
C₄×C₁₃×#41	2
#3512	13

Subgroups — C13×#3512 #13112570

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3512 #13112570