Subgroups — C₁₃×#3361 #13112419

Order 1664.

362 conjugacy classes of subgroups.

The subgroup lattice has 246 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#559	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#554	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#488	C₂	1	2	yes	maximal, complemented
4	C₁₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#478	C₂	1	2	yes	maximal, complemented
6	C₁₃×#477	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#463	C₂	1	2	yes	maximal, complemented
8	C₁₃×#459	C₂	1	1	yes	maximal, characteristic
9	C₈×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic, complemented
10	C₄×C₁₃×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
11	C₄×C₁₃×D₈	C₂	1	2	yes	maximal, complemented
12	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×D₈	C₂²	1	2	yes	—
14	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₁₃×#127	C₂²	1	1	yes	characteristic
16	C₁₃×#125	C₂²	1	2	yes	—
17	C₁₃×#124	—	2	1	no	—
18	C₁₃×#124	C₂²	1	2	yes	—
19	C₁₃×#123	—	2	2	no	—
20	C₁₃×C₂²⋊Q₈	—	2	1	no	—
21	C₁₃×C₂²⋊Q₈	—	2	2	no	—
22	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
23	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₁₃	—	2	1	no	—
25	C₄×D₄×C₁₃	C₂²	1	2	yes	—
26	C₄×D₄×C₁₃	C₂²	1	2	yes	—
27	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
28	C₂×C₁₃×C₄⋊C₄	—	2	1	no	—
29	C₁₃×#107	C₂²	1	2	yes	—
30	C₁₃×#106	C₂²	1	1	yes	characteristic
31	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
32	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
33	C₁₃×#103	C₂²	1	1	yes	characteristic
34	C₁₃×#103	C₂²	1	2	yes	—
35	C₁₃×#102	C₂²	1	1	yes	characteristic
36	C₁₃×#102	C₂²	1	2	yes	—
37	C₁₃×#102	C₂²	1	2	yes	—
38	C₁₃×#102	C₂²	1	2	yes	—
39	C₄×C₈×C₁₃	C₂²	1	2	yes	—
40	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
41	C₂×Q₈×C₁₃	—	2	1	no	—
42	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
43	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
44	C₂×D₄×C₁₃	—	2	1	no	—
45	C₂×D₄×C₁₃	—	2	1	no	—
46	C₂×D₄×C₁₃	C₂³	1	2	yes	—
47	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	—	2	1	no	—
49	C₂²×C₄×C₁₃	—	2	2	no	—
50	C₂²×C₄×C₁₃	—	2	2	no	—
51	C₁₃×SD₁₆	—	2	2	no	—
52	C₁₃×D₈	—	2	4	no	—
53	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
55	C₂×C₈×C₁₃	C₂³	1	2	yes	—
56	C₁₃×C₄⋊C₄	—	2	2	no	—
57	C₁₃×C₄⋊C₄	—	2	1	no	—
58	C₁₃×C₄⋊C₄	—	2	1	no	—
59	C₁₃×C₄⋊C₄	—	2	2	no	—
60	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
61	C₁₃×C₄⋊C₄	D₄	1	2	yes	—
62	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
63	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
64	C₁₃×C₄⋊C₄	C₂³	1	1	yes	characteristic
65	C₁₃×C₂²⋊C₄	—	2	2	no	—
66	C₁₃×C₂²⋊C₄	—	2	1	no	—
67	C₁₃×C₂²⋊C₄	—	2	2	no	—
68	C₁₃×C₂²⋊C₄	—	2	2	no	—
69	C₁₃×C₂²⋊C₄	—	2	2	no	—
70	C₄²×C₁₃	—	2	1	no	—
71	C₄²×C₁₃	C₂³	1	2	yes	—
72	C₄²×C₁₃	C₂³	1	1	yes	characteristic
73	#3361	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
74	C₂³×C₁₃	—	2	1	no	—
75	C₂³×C₁₃	—	2	2	no	—
76	Q₈×C₁₃	—	2	2	no	—
77	Q₈×C₁₃	—	2	1	no	—
78	Q₈×C₁₃	—	2	1	no	—
79	Q₈×C₁₃	—	8	1	no	—
80	D₄×C₁₃	—	2	2	no	—
81	D₄×C₁₃	—	2	1	no	—
82	D₄×C₁₃	—	2	2	no	—
83	D₄×C₁₃	—	4	2	no	—
84	D₄×C₁₃	—	8	1	no	—
85	D₄×C₁₃	#41	1	2	yes	—
86	C₂×C₄×C₁₃	—	2	1	no	—
87	C₂×C₄×C₁₃	—	2	1	no	—
88	C₂×C₄×C₁₃	—	2	1	no	—
89	C₂×C₄×C₁₃	—	2	2	no	—
90	C₂×C₄×C₁₃	—	4	2	no	—
91	C₂×C₄×C₁₃	—	8	2	no	—
92	C₂×C₄×C₁₃	—	8	2	no	—
93	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
94	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
95	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
96	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
97	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
98	C₈×C₁₃	—	2	2	no	—
99	C₈×C₁₃	—	2	1	no	—
100	C₈×C₁₃	—	2	1	no	—
101	#559	C₂×C₁₃	1	1	yes	characteristic, complemented
102	#554	C₂×C₁₃	1	1	yes	characteristic, complemented
103	#488	C₂×C₁₃	1	2	yes	complemented
104	#486	C₂×C₁₃	1	1	yes	characteristic, complemented
105	#478	C₂×C₁₃	1	2	yes	complemented
106	#477	C₂×C₁₃	1	1	yes	characteristic, complemented
107	#463	C₂×C₁₃	1	2	yes	complemented
108	#459	C₂×C₁₃	1	1	yes	characteristic
109	C₈×Q₈	C₂×C₁₃	1	1	yes	characteristic, complemented
110	C₄×SD₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
111	C₄×D₈	C₂×C₁₃	1	2	yes	complemented
112	C₂²×C₁₃	—	2	2	no	—
113	C₂²×C₁₃	—	4	1	no	—
114	C₂²×C₁₃	—	4	1	no	—
115	C₂²×C₁₃	—	4	2	no	—
116	C₂²×C₁₃	—	4	2	no	—
117	C₂²×C₁₃	—	4	2	no	—
118	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
119	C₄×C₁₃	—	2	2	no	—
120	C₄×C₁₃	—	2	2	no	—
121	C₄×C₁₃	—	2	1	no	—
122	C₄×C₁₃	—	4	1	no	—
123	C₄×C₁₃	—	4	1	no	—
124	C₄×C₁₃	—	4	1	no	—
125	C₄×C₁₃	—	4	2	no	—
126	C₄×C₁₃	#135	1	2	yes	—
127	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
128	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
129	C₂×SD₁₆	C₂²×C₁₃	1	1	yes	characteristic
130	C₂×D₈	C₂²×C₁₃	1	2	yes	—
131	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
132	#127	C₂²×C₁₃	1	1	yes	characteristic
133	#125	C₂²×C₁₃	1	2	yes	—
134	#124	—	2	1	no	—
135	#124	C₂²×C₁₃	1	2	yes	—
136	#123	—	2	2	no	—
137	C₂²⋊Q₈	—	2	1	no	—
138	C₂²⋊Q₈	—	2	2	no	—
139	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
140	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
141	C₄×D₄	—	2	1	no	—
142	C₄×D₄	C₂²×C₁₃	1	2	yes	—
143	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
144	C₄×D₄	C₂²×C₁₃	1	2	yes	—
145	C₂×C₄⋊C₄	—	2	1	no	—
146	#107	C₂²×C₁₃	1	2	yes	—
147	#106	C₂²×C₁₃	1	1	yes	characteristic
148	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
149	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
150	#103	C₂²×C₁₃	1	1	yes	characteristic
151	#103	C₂²×C₁₃	1	2	yes	—
152	#102	C₂²×C₁₃	1	1	yes	characteristic
153	#102	C₂²×C₁₃	1	2	yes	—
154	#102	C₂²×C₁₃	1	2	yes	—
155	#102	C₂²×C₁₃	1	2	yes	—
156	C₄×C₈	C₂²×C₁₃	1	2	yes	—
157	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
158	C₂×C₁₃	—	4	2	no	—
159	C₂×C₁₃	—	8	2	no	—
160	C₂×C₁₃	#548	1	1	yes	characteristic
161	C₂×C₁₃	C₂×#135	1	1	yes	characteristic
162	C₂×C₁₃	#576	1	1	yes	characteristic
163	C₂×Q₈	—	2	1	no	—
164	C₂×Q₈	D₄×C₁₃	1	1	yes	characteristic
165	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
166	C₂×D₄	—	2	1	no	—
167	C₂×D₄	—	2	1	no	—
168	C₂×D₄	C₂³×C₁₃	1	2	yes	—
169	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
170	C₂²×C₄	—	2	1	no	—
171	C₂²×C₄	—	2	2	no	—
172	C₂²×C₄	—	2	2	no	—
173	SD₁₆	—	2	2	no	—
174	D₈	—	2	4	no	—
175	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
176	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
177	C₂×C₈	C₂³×C₁₃	1	2	yes	—
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	1	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	2	no	—
182	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
183	C₄⋊C₄	D₄×C₁₃	1	2	yes	—
184	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
185	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
186	C₄⋊C₄	C₂³×C₁₃	1	1	yes	characteristic
187	C₂²⋊C₄	—	2	2	no	—
188	C₂²⋊C₄	—	2	1	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	2	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₄²	—	2	1	no	—
193	C₄²	C₂³×C₁₃	1	2	yes	—
194	C₄²	C₂³×C₁₃	1	1	yes	characteristic
195	C₁₃	#3361	1	1	yes	characteristic, direct product factor, Sylow
196	C₂³	—	2	1	no	—
197	C₂³	—	2	2	no	—
198	Q₈	—	2	2	no	—
199	Q₈	—	2	1	no	—
200	Q₈	—	2	1	no	—
201	Q₈	—	8	1	no	—
202	D₄	—	2	2	no	—
203	D₄	—	2	1	no	—
204	D₄	—	2	2	no	—
205	D₄	—	4	2	no	—
206	D₄	—	8	1	no	—
207	D₄	C₁₃×#41	1	2	yes	—
208	C₂×C₄	—	2	1	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	4	2	no	—
213	C₂×C₄	—	8	2	no	—
214	C₂×C₄	—	8	2	no	—
215	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
216	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
217	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
218	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
219	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
220	C₈	—	2	2	no	—
221	C₈	—	2	1	no	—
222	C₈	—	2	1	no	—
223	C₂²	—	2	2	no	—
224	C₂²	—	4	1	no	—
225	C₂²	—	4	1	no	—
226	C₂²	—	4	2	no	—
227	C₂²	—	4	2	no	—
228	C₂²	—	4	2	no	—
229	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	2	no	—
232	C₄	—	2	1	no	—
233	C₄	—	4	1	no	—
234	C₄	—	4	1	no	—
235	C₄	—	4	1	no	—
236	C₄	—	4	2	no	—
237	C₄	C₁₃×#135	1	2	yes	—
238	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
239	C₄	C₁₃×2₋¹⁺⁴	1	1	yes	characteristic
240	C₂	—	4	2	no	—
241	C₂	—	8	2	no	—
242	C₂	C₁₃×#548	1	1	yes	characteristic
243	C₂	C₂×C₁₃×#135	1	1	yes	characteristic
244	C₂	C₁₃×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₁₃×#459	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×Q₈×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#559	C₂	semidirect
2. C₁₃×#554	C₂	semidirect
3. C₁₃×#488	C₂	semidirect
4. C₁₃×#486	C₂	semidirect
5. C₁₃×#478	C₂	semidirect
6. C₁₃×#477	C₂	semidirect
7. C₁₃×#463	C₂	semidirect
9. C₈×Q₈×C₁₃	C₂	semidirect
10. C₄×C₁₃×SD₁₆	C₂	semidirect
11. C₄×C₁₃×D₈	C₂	semidirect
73. #3361	C₁₃	direct (N × H)
101. #559	C₂×C₁₃	semidirect
102. #554	C₂×C₁₃	semidirect
103. #488	C₂×C₁₃	semidirect
104. #486	C₂×C₁₃	semidirect
105. #478	C₂×C₁₃	semidirect
106. #477	C₂×C₁₃	semidirect
107. #463	C₂×C₁₃	semidirect
109. C₈×Q₈	C₂×C₁₃	semidirect
110. C₄×SD₁₆	C₂×C₁₃	semidirect
111. C₄×D₈	C₂×C₁₃	semidirect
195. C₁₃	#3361	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×#559	2
C₁₃×#554	2
C₁₃×#488	2
C₁₃×#486	2
C₁₃×#478	2
C₁₃×#477	2
C₁₃×#463	2
C₁₃×#459	2
C₈×Q₈×C₁₃	2
C₄×C₁₃×SD₁₆	2
C₄×C₁₃×D₈	2
#3361	13

Subgroups — C13×#3361 #13112419

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3361 #13112419