Subgroups — C₁₃×#3353 #13112411

Order 1664.

374 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×#556	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#547	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#500	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#484	C₂	1	2	yes	maximal, complemented
5	C₁₃×#482	C₂	1	2	yes	maximal, complemented
6	C₁₃×#475	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#474	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#468	C₂	1	2	yes	maximal, complemented
9	C₄×C₁₃×SD₁₆	C₂	1	1	yes	maximal, characteristic, complemented
10	C₁₃×#436	C₂	1	1	yes	maximal, characteristic, complemented
11	C₁₃×#428	C₂	1	2	yes	maximal, complemented
12	C₂×C₁₃×#41	—	2	1	no	—
13	C₂×C₁₃×SD₁₆	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
15	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
17	C₁₃×#126	—	2	2	no	—
18	C₁₃×#124	—	2	1	no	—
19	C₁₃×#123	—	2	2	no	—
20	C₁₃×C₂²⋊Q₈	—	2	1	no	—
21	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
22	C₁₃×#121	C₂²	1	2	yes	—
23	C₄×Q₈×C₁₃	—	2	1	no	—
24	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
25	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₁₃	C₂²	1	1	yes	characteristic
27	C₁₃×#117	C₂²	1	2	yes	—
28	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
29	C₁₃×#107	C₂²	1	2	yes	—
30	C₁₃×#107	C₂²	1	2	yes	—
31	C₁₃×#107	C₂²	1	2	yes	—
32	C₁₃×#106	C₂²	1	1	yes	characteristic
33	C₁₃×#106	C₂²	1	2	yes	—
34	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
35	C₁₃×#103	C₂²	1	1	yes	characteristic
36	C₁₃×#103	C₂²	1	2	yes	—
37	C₁₃×#102	C₂²	1	1	yes	characteristic
38	C₁₃×#102	C₂²	1	2	yes	—
39	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
40	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
41	C₁₃×#41	—	4	4	no	—
42	C₂×Q₈×C₁₃	—	2	1	no	—
43	C₂×Q₈×C₁₃	—	2	1	no	—
44	C₂×Q₈×C₁₃	C₂³	1	1	yes	characteristic
45	C₂×D₄×C₁₃	—	2	2	no	—
46	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
47	C₂×D₄×C₁₃	C₂³	1	1	yes	characteristic
48	C₂²×C₄×C₁₃	—	2	2	no	—
49	C₂²×C₄×C₁₃	—	2	1	no	—
50	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
51	C₁₃×SD₁₆	—	2	2	no	—
52	C₁₃×M₁₆	—	2	4	no	—
53	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₁₃	C₂³	1	2	yes	—
55	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
56	C₁₃×C₄⋊C₄	—	2	2	no	—
57	C₁₃×C₄⋊C₄	—	2	2	no	—
58	C₁₃×C₄⋊C₄	—	2	1	no	—
59	C₁₃×C₄⋊C₄	—	2	1	no	—
60	C₁₃×C₄⋊C₄	—	2	1	no	—
61	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
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×C₄⋊C₄	C₂³	1	2	yes	—
66	C₁₃×C₂²⋊C₄	—	2	2	no	—
67	C₁₃×C₂²⋊C₄	—	2	2	no	—
68	C₁₃×C₂²⋊C₄	—	2	1	no	—
69	C₁₃×C₂²⋊C₄	—	2	2	no	—
70	C₁₃×C₂²⋊C₄	D₄	1	2	yes	—
71	C₄²×C₁₃	—	2	2	no	—
72	C₄²×C₁₃	—	2	1	no	—
73	C₄²×C₁₃	C₂³	1	1	yes	characteristic
74	#3353	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
75	C₂³×C₁₃	—	2	1	no	—
76	C₂³×C₁₃	C₂×D₄	1	2	yes	—
77	Q₈×C₁₃	—	2	1	no	—
78	Q₈×C₁₃	—	2	1	no	—
79	Q₈×C₁₃	—	4	2	no	—
80	Q₈×C₁₃	—	4	2	no	—
81	D₄×C₁₃	—	2	2	no	—
82	D₄×C₁₃	—	2	1	no	—
83	D₄×C₁₃	—	2	1	no	—
84	D₄×C₁₃	—	4	4	no	—
85	C₂×C₄×C₁₃	—	2	2	no	—
86	C₂×C₄×C₁₃	—	2	2	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	—	2	1	no	—
91	C₂×C₄×C₁₃	—	2	2	no	—
92	C₂×C₄×C₁₃	—	4	4	no	—
93	C₂×C₄×C₁₃	—	4	2	no	—
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	#41	1	2	yes	complemented
101	#556	C₂×C₁₃	1	1	yes	characteristic, complemented
102	#547	C₂×C₁₃	1	1	yes	characteristic
103	#500	C₂×C₁₃	1	1	yes	characteristic, complemented
104	#484	C₂×C₁₃	1	2	yes	complemented
105	#482	C₂×C₁₃	1	2	yes	complemented
106	#475	C₂×C₁₃	1	1	yes	characteristic, complemented
107	#474	C₂×C₁₃	1	1	yes	characteristic, complemented
108	#468	C₂×C₁₃	1	2	yes	complemented
109	C₄×SD₁₆	C₂×C₁₃	1	1	yes	characteristic, complemented
110	#436	C₂×C₁₃	1	1	yes	characteristic, complemented
111	#428	C₂×C₁₃	1	2	yes	complemented
112	C₂²×C₁₃	—	2	2	no	—
113	C₂²×C₁₃	—	2	2	no	—
114	C₂²×C₁₃	—	2	2	no	—
115	C₂²×C₁₃	—	4	1	no	—
116	C₂²×C₁₃	—	4	1	no	—
117	C₂²×C₁₃	—	4	1	no	—
118	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
119	C₄×C₁₃	—	2	2	no	—
120	C₄×C₁₃	—	2	1	no	—
121	C₄×C₁₃	—	4	1	no	—
122	C₄×C₁₃	—	4	1	no	—
123	C₄×C₁₃	—	4	2	no	—
124	C₄×C₁₃	—	4	2	no	—
125	C₄×C₁₃	—	4	1	no	—
126	C₄×C₁₃	AGL(1,ℤ/8)	1	2	yes	—
127	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
128	C₄×C₁₃	2₋¹⁺⁴	1	1	yes	characteristic
129	C₂×#41	—	2	1	no	—
130	C₂×SD₁₆	C₂²×C₁₃	1	1	yes	characteristic
131	C₂×M₁₆	C₂²×C₁₃	1	2	yes	—
132	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
133	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
134	#126	—	2	2	no	—
135	#124	—	2	1	no	—
136	#123	—	2	2	no	—
137	C₂²⋊Q₈	—	2	1	no	—
138	C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	—
139	#121	C₂²×C₁₃	1	2	yes	—
140	C₄×Q₈	—	2	1	no	—
141	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
142	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
143	C₄×D₄	C₂²×C₁₃	1	1	yes	characteristic
144	#117	C₂²×C₁₃	1	2	yes	—
145	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
146	#107	C₂²×C₁₃	1	2	yes	—
147	#107	C₂²×C₁₃	1	2	yes	—
148	#107	C₂²×C₁₃	1	2	yes	—
149	#106	C₂²×C₁₃	1	1	yes	characteristic
150	#106	C₂²×C₁₃	1	2	yes	—
151	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
152	#103	C₂²×C₁₃	1	1	yes	characteristic
153	#103	C₂²×C₁₃	1	2	yes	—
154	#102	C₂²×C₁₃	1	1	yes	characteristic
155	#102	C₂²×C₁₃	1	2	yes	—
156	C₂²⋊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	1	no	—
160	C₂×C₁₃	#547	1	1	yes	characteristic
161	C₂×C₁₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
162	C₂×C₁₃	#578	1	1	yes	characteristic
163	#41	—	4	4	no	—
164	C₂×Q₈	—	2	1	no	—
165	C₂×Q₈	—	2	1	no	—
166	C₂×Q₈	C₂³×C₁₃	1	1	yes	characteristic
167	C₂×D₄	—	2	2	no	—
168	C₂×D₄	D₄×C₁₃	1	1	yes	characteristic
169	C₂×D₄	C₂³×C₁₃	1	1	yes	characteristic
170	C₂²×C₄	—	2	2	no	—
171	C₂²×C₄	—	2	1	no	—
172	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
173	SD₁₆	—	2	2	no	—
174	M₁₆	—	2	4	no	—
175	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
176	C₂×C₈	C₂³×C₁₃	1	2	yes	—
177	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
178	C₄⋊C₄	—	2	2	no	—
179	C₄⋊C₄	—	2	2	no	—
180	C₄⋊C₄	—	2	1	no	—
181	C₄⋊C₄	—	2	1	no	—
182	C₄⋊C₄	—	2	1	no	—
183	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
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	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
188	C₂²⋊C₄	—	2	2	no	—
189	C₂²⋊C₄	—	2	2	no	—
190	C₂²⋊C₄	—	2	1	no	—
191	C₂²⋊C₄	—	2	2	no	—
192	C₂²⋊C₄	D₄×C₁₃	1	2	yes	—
193	C₄²	—	2	2	no	—
194	C₄²	—	2	1	no	—
195	C₄²	C₂³×C₁₃	1	1	yes	characteristic
196	C₁₃	#3353	1	1	yes	characteristic, direct product factor, Sylow
197	C₂³	—	2	1	no	—
198	C₂³	C₂×D₄×C₁₃	1	2	yes	—
199	Q₈	—	2	1	no	—
200	Q₈	—	2	1	no	—
201	Q₈	—	4	2	no	—
202	Q₈	—	4	2	no	—
203	D₄	—	2	2	no	—
204	D₄	—	2	1	no	—
205	D₄	—	2	1	no	—
206	D₄	—	4	4	no	—
207	C₂×C₄	—	2	2	no	—
208	C₂×C₄	—	2	2	no	—
209	C₂×C₄	—	2	1	no	—
210	C₂×C₄	—	2	1	no	—
211	C₂×C₄	—	2	2	no	—
212	C₂×C₄	—	2	1	no	—
213	C₂×C₄	—	2	2	no	—
214	C₂×C₄	—	4	4	no	—
215	C₂×C₄	—	4	2	no	—
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	C₁₃×#41	1	2	yes	complemented
223	C₂²	—	2	2	no	—
224	C₂²	—	2	2	no	—
225	C₂²	—	2	2	no	—
226	C₂²	—	4	1	no	—
227	C₂²	—	4	1	no	—
228	C₂²	—	4	1	no	—
229	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
230	C₄	—	2	2	no	—
231	C₄	—	2	1	no	—
232	C₄	—	4	1	no	—
233	C₄	—	4	1	no	—
234	C₄	—	4	2	no	—
235	C₄	—	4	2	no	—
236	C₄	—	4	1	no	—
237	C₄	C₁₃×AGL(1,ℤ/8)	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	1	no	—
242	C₂	C₁₃×#547	1	1	yes	characteristic
243	C₂	C₂×C₁₃×AGL(1,ℤ/8)	1	1	yes	characteristic
244	C₂	C₁₃×#578	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₁₃×#547	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₄×D₄×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#556	C₂	semidirect
3. C₁₃×#500	C₂	semidirect
4. C₁₃×#484	C₂	semidirect
5. C₁₃×#482	C₂	semidirect
6. C₁₃×#475	C₂	semidirect
7. C₁₃×#474	C₂	semidirect
8. C₁₃×#468	C₂	semidirect
9. C₄×C₁₃×SD₁₆	C₂	semidirect
10. C₁₃×#436	C₂	semidirect
11. C₁₃×#428	C₂	semidirect
74. #3353	C₁₃	direct (N × H)
100. C₈×C₁₃	#41	semidirect
101. #556	C₂×C₁₃	semidirect
103. #500	C₂×C₁₃	semidirect
104. #484	C₂×C₁₃	semidirect
105. #482	C₂×C₁₃	semidirect
106. #475	C₂×C₁₃	semidirect
107. #474	C₂×C₁₃	semidirect
108. #468	C₂×C₁₃	semidirect
109. C₄×SD₁₆	C₂×C₁₃	semidirect
110. #436	C₂×C₁₃	semidirect
111. #428	C₂×C₁₃	semidirect
196. C₁₃	#3353	direct (N × H)
222. 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×#556	2
C₁₃×#547	2
C₁₃×#500	2
C₁₃×#484	2
C₁₃×#482	2
C₁₃×#475	2
C₁₃×#474	2
C₁₃×#468	2
C₄×C₁₃×SD₁₆	2
C₁₃×#436	2
C₁₃×#428	2
#3353	13

Subgroups — C13×#3353 #13112411

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3353 #13112411