Subgroups — C₁₃×#3553 #13112611

Order 1664.

932 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	C₁₃×#564	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#563	C₂	1	2	yes	maximal, complemented
3	C₁₃×#562	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#554	C₂	1	4	yes	maximal, complemented
5	C₁₃×#553	C₂	1	4	yes	maximal, complemented
6	C₁₃×#552	C₂	1	4	yes	maximal, complemented
7	C₁₃×#541	C₂	1	4	yes	maximal, complemented
8	C₁₃×#539	C₂	1	4	yes	maximal, complemented
9	C₁₃×#539	C₂	1	4	yes	maximal, complemented
10	C₁₃×#529	C₂	1	2	yes	maximal, complemented
11	C₂×C₁₃×#125	C₂	1	1	yes	maximal, characteristic, complemented
12	C₁₃×C₄⋊Q₈	C₂²	1	4	yes	—
13	C₁₃×#126	C₂²	1	4	yes	—
14	C₁₃×#126	C₂²	1	4	yes	—
15	C₁₃×#126	C₂²	1	4	yes	—
16	C₁₃×#126	C₂²	1	4	yes	—
17	C₁₃×#126	C₂²	1	4	yes	—
18	C₁₃×#125	C₂²	1	4	yes	—
19	C₁₃×#125	C₂²	1	2	yes	—
20	C₁₃×#125	C₂²	1	1	yes	characteristic
21	C₁₃×#125	C₂²	1	1	yes	characteristic
22	C₁₃×#125	C₂²	1	2	yes	—
23	C₁₃×#125	C₂²	1	4	yes	—
24	C₁₃×#125	C₂²	1	4	yes	—
25	C₁₃×#125	C₂²	1	4	yes	—
26	C₁₃×#125	C₂²	1	4	yes	—
27	C₁₃×#125	C₂²	1	4	yes	—
28	C₁₃×#124	C₂²	1	2	yes	—
29	C₁₃×#123	C₂²	1	4	yes	—
30	C₁₃×#123	C₂²	1	4	yes	—
31	C₁₃×#123	C₂²	1	4	yes	—
32	C₁₃×#123	C₂²	1	4	yes	—
33	C₁₃×#123	C₂²	1	4	yes	—
34	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
35	C₁₃×C₂²⋊Q₈	C₂²	1	4	yes	—
36	C₁₃×C₂²⋊Q₈	C₂²	1	4	yes	—
37	C₁₃×C₂²⋊Q₈	C₂²	1	4	yes	—
38	C₁₃×C₂²⋊Q₈	C₂²	1	4	yes	—
39	C₁₃×C₂²⋊Q₈	C₂²	1	4	yes	—
40	C₁₃×C₂²⋊Q₈	C₂²	1	4	yes	—
41	C₁₃×#121	C₂²	1	2	yes	—
42	C₁₃×#121	C₂²	1	4	yes	—
43	C₁₃×#121	C₂²	1	4	yes	—
44	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
45	C₄×Q₈×C₁₃	C₂²	1	4	yes	—
46	C₄×D₄×C₁₃	C₂²	1	2	yes	—
47	C₄×D₄×C₁₃	C₂²	1	4	yes	—
48	C₄×D₄×C₁₃	C₂²	1	4	yes	—
49	C₄×D₄×C₁₃	C₂²	1	4	yes	—
50	C₄×D₄×C₁₃	C₂²	1	4	yes	—
51	C₁₃×#117	C₂²	1	4	yes	—
52	C₁₃×#117	C₂²	1	4	yes	—
53	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
54	C₂×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
55	C₂×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
56	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
57	C₂×Q₈×C₁₃	C₂³	1	4	yes	—
58	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
59	C₂×D₄×C₁₃	C₂³	1	4	yes	—
60	C₂×D₄×C₁₃	C₂³	1	4	yes	—
61	C₂×D₄×C₁₃	C₂³	1	2	yes	—
62	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
63	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
64	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
65	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
66	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
67	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
68	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
69	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
70	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
71	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
72	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
73	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
74	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
75	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
76	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
77	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
78	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
79	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
80	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
81	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
82	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
83	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
84	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
85	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
86	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
87	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
88	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
89	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
90	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
91	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
92	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
93	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
94	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
95	C₁₃×C₂²⋊C₄	C₂³	1	4	yes	—
96	C₄²×C₁₃	C₂³	1	1	yes	characteristic
97	C₄²×C₁₃	C₂³	1	2	yes	—
98	C₄²×C₁₃	C₂³	1	1	yes	characteristic
99	C₄²×C₁₃	C₂³	1	4	yes	—
100	C₄²×C₁₃	C₂³	1	4	yes	—
101	C₄²×C₁₃	C₂³	1	4	yes	—
102	#3553	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
103	C₂³×C₁₃	C₂⁴	1	4	yes	—
104	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
105	Q₈×C₁₃	—	4	4	no	—
106	Q₈×C₁₃	—	4	2	no	—
107	D₄×C₁₃	—	2	8	no	—
108	D₄×C₁₃	—	4	4	no	—
109	D₄×C₁₃	—	4	2	no	—
110	C₂×C₄×C₁₃	—	2	2	no	—
111	C₂×C₄×C₁₃	—	4	4	no	—
112	C₂×C₄×C₁₃	—	4	2	no	—
113	C₂×C₄×C₁₃	—	4	4	no	—
114	C₂×C₄×C₁₃	—	4	4	no	—
115	C₂×C₄×C₁₃	—	4	4	no	—
116	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
117	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
118	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
119	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
120	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
121	C₂×C₄×C₁₃	C₂⁴	1	2	yes	—
122	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
123	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
124	C₂×C₄×C₁₃	C₂⁴	1	4	yes	—
125	#564	C₂×C₁₃	1	1	yes	characteristic, complemented
126	#563	C₂×C₁₃	1	2	yes	complemented
127	#562	C₂×C₁₃	1	1	yes	characteristic, complemented
128	#554	C₂×C₁₃	1	4	yes	complemented
129	#553	C₂×C₁₃	1	4	yes	complemented
130	#552	C₂×C₁₃	1	4	yes	complemented
131	#541	C₂×C₁₃	1	4	yes	complemented
132	#539	C₂×C₁₃	1	4	yes	complemented
133	#539	C₂×C₁₃	1	4	yes	complemented
134	#529	C₂×C₁₃	1	2	yes	complemented
135	C₂×#125	C₂×C₁₃	1	1	yes	characteristic, complemented
136	C₂²×C₁₃	—	2	2	no	—
137	C₂²×C₁₃	—	2	4	no	—
138	C₂²×C₁₃	—	2	4	no	—
139	C₂²×C₁₃	—	2	4	no	—
140	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
141	C₂²×C₁₃	C₂⁵	1	1	yes	characteristic
142	C₄×C₁₃	—	2	1	no	—
143	C₄×C₁₃	—	2	1	no	—
144	C₄×C₁₃	—	2	4	no	—
145	C₄×C₁₃	—	2	2	no	—
146	C₄×C₁₃	—	2	4	no	—
147	C₄×C₁₃	—	2	4	no	—
148	C₄×C₁₃	—	2	4	no	—
149	C₄×C₁₃	—	2	2	no	—
150	C₄×C₁₃	—	2	4	no	—
151	C₄⋊Q₈	C₂²×C₁₃	1	4	yes	—
152	#126	C₂²×C₁₃	1	4	yes	—
153	#126	C₂²×C₁₃	1	4	yes	—
154	#126	C₂²×C₁₃	1	4	yes	—
155	#126	C₂²×C₁₃	1	4	yes	—
156	#126	C₂²×C₁₃	1	4	yes	—
157	#125	C₂²×C₁₃	1	4	yes	—
158	#125	C₂²×C₁₃	1	2	yes	—
159	#125	C₂²×C₁₃	1	1	yes	characteristic
160	#125	C₂²×C₁₃	1	1	yes	characteristic
161	#125	C₂²×C₁₃	1	2	yes	—
162	#125	C₂²×C₁₃	1	4	yes	—
163	#125	C₂²×C₁₃	1	4	yes	—
164	#125	C₂²×C₁₃	1	4	yes	—
165	#125	C₂²×C₁₃	1	4	yes	—
166	#125	C₂²×C₁₃	1	4	yes	—
167	#124	C₂²×C₁₃	1	2	yes	—
168	#123	C₂²×C₁₃	1	4	yes	—
169	#123	C₂²×C₁₃	1	4	yes	—
170	#123	C₂²×C₁₃	1	4	yes	—
171	#123	C₂²×C₁₃	1	4	yes	—
172	#123	C₂²×C₁₃	1	4	yes	—
173	C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	—
174	C₂²⋊Q₈	C₂²×C₁₃	1	4	yes	—
175	C₂²⋊Q₈	C₂²×C₁₃	1	4	yes	—
176	C₂²⋊Q₈	C₂²×C₁₃	1	4	yes	—
177	C₂²⋊Q₈	C₂²×C₁₃	1	4	yes	—
178	C₂²⋊Q₈	C₂²×C₁₃	1	4	yes	—
179	C₂²⋊Q₈	C₂²×C₁₃	1	4	yes	—
180	#121	C₂²×C₁₃	1	2	yes	—
181	#121	C₂²×C₁₃	1	4	yes	—
182	#121	C₂²×C₁₃	1	4	yes	—
183	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
184	C₄×Q₈	C₂²×C₁₃	1	4	yes	—
185	C₄×D₄	C₂²×C₁₃	1	2	yes	—
186	C₄×D₄	C₂²×C₁₃	1	4	yes	—
187	C₄×D₄	C₂²×C₁₃	1	4	yes	—
188	C₄×D₄	C₂²×C₁₃	1	4	yes	—
189	C₄×D₄	C₂²×C₁₃	1	4	yes	—
190	#117	C₂²×C₁₃	1	4	yes	—
191	#117	C₂²×C₁₃	1	4	yes	—
192	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
193	C₂×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
194	C₂×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
195	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
196	C₂×C₁₃	—	2	2	no	—
197	C₂×C₁₃	—	4	4	no	—
198	C₂×C₁₃	C₂×2₋¹⁺⁴	1	1	yes	characteristic
199	C₂×C₁₃	#585	1	2	yes	—
200	C₂×Q₈	C₂³×C₁₃	1	4	yes	—
201	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
202	C₂×D₄	C₂³×C₁₃	1	4	yes	—
203	C₂×D₄	C₂³×C₁₃	1	4	yes	—
204	C₂×D₄	C₂³×C₁₃	1	2	yes	—
205	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
206	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
207	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
208	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
209	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
210	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
211	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
212	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
213	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
214	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
215	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
216	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
217	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
218	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
219	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
220	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
221	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
222	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
223	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
224	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
225	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
226	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
227	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
228	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
229	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
230	C₂²⋊C₄	C₂³×C₁₃	1	4	yes	—
231	C₂²⋊C₄	C₂³×C₁₃	1	4	yes	—
232	C₂²⋊C₄	C₂³×C₁₃	1	4	yes	—
233	C₂²⋊C₄	C₂³×C₁₃	1	4	yes	—
234	C₂²⋊C₄	C₂³×C₁₃	1	4	yes	—
235	C₂²⋊C₄	C₂³×C₁₃	1	4	yes	—
236	C₂²⋊C₄	C₂³×C₁₃	1	4	yes	—
237	C₂²⋊C₄	C₂³×C₁₃	1	4	yes	—
238	C₂²⋊C₄	C₂³×C₁₃	1	4	yes	—
239	C₄²	C₂³×C₁₃	1	1	yes	characteristic
240	C₄²	C₂³×C₁₃	1	2	yes	—
241	C₄²	C₂³×C₁₃	1	1	yes	characteristic
242	C₄²	C₂³×C₁₃	1	4	yes	—
243	C₄²	C₂³×C₁₃	1	4	yes	—
244	C₄²	C₂³×C₁₃	1	4	yes	—
245	C₁₃	#3553	1	1	yes	characteristic, direct product factor, Sylow
246	C₂³	C₂⁴×C₁₃	1	4	yes	—
247	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
248	Q₈	—	4	4	no	—
249	Q₈	—	4	2	no	—
250	D₄	—	2	8	no	—
251	D₄	—	4	4	no	—
252	D₄	—	4	2	no	—
253	C₂×C₄	—	2	2	no	—
254	C₂×C₄	—	4	4	no	—
255	C₂×C₄	—	4	2	no	—
256	C₂×C₄	—	4	4	no	—
257	C₂×C₄	—	4	4	no	—
258	C₂×C₄	—	4	4	no	—
259	C₂×C₄	C₂⁴×C₁₃	1	2	yes	—
260	C₂×C₄	C₂⁴×C₁₃	1	4	yes	—
261	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
262	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
263	C₂×C₄	C₂⁴×C₁₃	1	4	yes	—
264	C₂×C₄	C₂⁴×C₁₃	1	2	yes	—
265	C₂×C₄	C₂⁴×C₁₃	1	4	yes	—
266	C₂×C₄	C₂⁴×C₁₃	1	4	yes	—
267	C₂×C₄	C₂⁴×C₁₃	1	4	yes	—
268	C₂²	—	2	2	no	—
269	C₂²	—	2	4	no	—
270	C₂²	—	2	4	no	—
271	C₂²	—	2	4	no	—
272	C₂²	C₁₃×2₋¹⁺⁴	1	2	yes	—
273	C₂²	C₂⁵×C₁₃	1	1	yes	characteristic
274	C₄	—	2	1	no	—
275	C₄	—	2	1	no	—
276	C₄	—	2	4	no	—
277	C₄	—	2	2	no	—
278	C₄	—	2	4	no	—
279	C₄	—	2	4	no	—
280	C₄	—	2	4	no	—
281	C₄	—	2	2	no	—
282	C₄	—	2	4	no	—
283	C₂	—	2	2	no	—
284	C₂	—	4	4	no	—
285	C₂	C₂×C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
286	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×#564	C₂	semidirect
2. C₁₃×#563	C₂	semidirect
3. C₁₃×#562	C₂	semidirect
4. C₁₃×#554	C₂	semidirect
5. C₁₃×#553	C₂	semidirect
6. C₁₃×#552	C₂	semidirect
7. C₁₃×#541	C₂	semidirect
8. C₁₃×#539	C₂	semidirect
9. C₁₃×#539	C₂	semidirect
10. C₁₃×#529	C₂	semidirect
11. C₂×C₁₃×#125	C₂	semidirect
102. #3553	C₁₃	direct (N × H)
125. #564	C₂×C₁₃	semidirect
126. #563	C₂×C₁₃	semidirect
127. #562	C₂×C₁₃	semidirect
128. #554	C₂×C₁₃	semidirect
129. #553	C₂×C₁₃	semidirect
130. #552	C₂×C₁₃	semidirect
131. #541	C₂×C₁₃	semidirect
132. #539	C₂×C₁₃	semidirect
133. #539	C₂×C₁₃	semidirect
134. #529	C₂×C₁₃	semidirect
135. C₂×#125	C₂×C₁₃	semidirect
245. C₁₃	#3553	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×#564	2
C₁₃×#563	2
C₁₃×#562	2
C₁₃×#554	2
C₁₃×#553	2
C₁₃×#552	2
C₁₃×#541	2
C₁₃×#539	2
C₁₃×#529	2
C₂×C₁₃×#125	2
#3553	13

Subgroups — C13×#3553 #13112611

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3553 #13112611