Subgroups — C₁₃×#3403 #13112461

Order 1664.

380 conjugacy classes of subgroups.

The subgroup lattice has 218 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	2	yes	maximal, complemented
2	C₁₃×#496	C₂	1	2	yes	maximal, complemented
3	C₁₃×#495	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#464	C₂	1	2	yes	maximal, complemented
5	C₁₃×#463	C₂	1	2	yes	maximal, complemented
6	C₁₃×#459	C₂	1	2	yes	maximal, complemented
7	C₁₃×#446	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#440	C₂	1	2	yes	maximal, complemented
9	C₄×C₁₃×D₈	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
11	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
12	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
13	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic
14	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic
15	C₂×C₁₃×D₈	C₂²	1	2	yes	—
16	C₁₃×#127	C₂²	1	2	yes	—
17	C₁₃×#124	—	2	2	no	—
18	C₁₃×#124	C₂²	1	2	yes	—
19	C₁₃×#124	C₂²	1	2	yes	—
20	C₁₃×#123	—	2	2	no	—
21	C₁₃×#123	—	2	2	no	—
22	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
23	C₄×Q₈×C₁₃	C₂²	1	1	yes	characteristic
24	C₄×D₄×C₁₃	—	2	2	no	—
25	C₄×D₄×C₁₃	C₂²	1	2	yes	—
26	C₄×D₄×C₁₃	C₂²	1	2	yes	—
27	C₁₃×#107	C₂²	1	1	yes	characteristic
28	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
29	C₁₃×C₄⋊C₈	C₂²	1	1	yes	characteristic
30	C₁₃×#103	C₂²	1	2	yes	—
31	C₁₃×#102	C₂²	1	2	yes	—
32	C₁₃×#102	C₂²	1	2	yes	—
33	C₁₃×#97	C₂²	1	2	yes	—
34	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
35	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
36	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
37	C₂×D₄×C₁₃	—	2	2	no	—
38	C₂×D₄×C₁₃	—	2	2	no	—
39	C₂×D₄×C₁₃	C₂³	1	2	yes	—
40	C₂×D₄×C₁₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₃	—	2	2	no	—
42	C₂²×C₄×C₁₃	—	2	2	no	—
43	C₁₃×Q₁₆	—	4	1	no	—
44	C₁₃×SD₁₆	—	2	4	no	—
45	C₁₃×SD₁₆	—	4	2	no	—
46	C₁₃×D₈	—	2	2	no	—
47	C₁₃×D₈	—	4	2	no	—
48	C₁₃×D₈	—	4	1	no	—
49	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
50	C₂×C₈×C₁₃	C₂³	1	2	yes	—
51	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
52	C₁₃×C₄⋊C₄	—	2	2	no	—
53	C₁₃×C₄⋊C₄	D₄	1	2	yes	—
54	C₁₃×C₄⋊C₄	D₄	1	1	yes	characteristic
55	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
56	C₁₃×C₂²⋊C₄	—	2	2	no	—
57	C₁₃×C₂²⋊C₄	—	2	2	no	—
58	C₁₃×C₂²⋊C₄	—	2	2	no	—
59	C₁₃×C₂²⋊C₄	—	2	2	no	—
60	C₁₃×C₂²⋊C₄	—	2	2	no	—
61	C₁₃×C₂²⋊C₄	—	2	2	no	—
62	C₄²×C₁₃	—	2	2	no	—
63	C₄²×C₁₃	C₂³	1	1	yes	characteristic
64	C₄²×C₁₃	C₂³	1	2	yes	—
65	#3403	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
66	C₂³×C₁₃	—	2	2	no	—
67	C₂³×C₁₃	—	2	2	no	—
68	Q₈×C₁₃	—	2	2	no	—
69	Q₈×C₁₃	—	2	2	no	—
70	Q₈×C₁₃	—	4	1	no	—
71	D₄×C₁₃	—	2	2	no	—
72	D₄×C₁₃	—	2	2	no	—
73	D₄×C₁₃	—	2	2	no	—
74	D₄×C₁₃	—	2	2	no	—
75	D₄×C₁₃	—	4	4	no	—
76	D₄×C₁₃	—	8	2	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	8	2	no	—
80	C₂×C₄×C₁₃	—	8	2	no	—
81	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
82	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
84	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
85	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
86	C₈×C₁₃	—	2	2	no	—
87	C₈×C₁₃	—	2	1	no	—
88	C₈×C₁₃	#41	1	2	yes	—
89	#559	C₂×C₁₃	1	2	yes	complemented
90	#496	C₂×C₁₃	1	2	yes	complemented
91	#495	C₂×C₁₃	1	1	yes	characteristic, complemented
92	#464	C₂×C₁₃	1	2	yes	complemented
93	#463	C₂×C₁₃	1	2	yes	complemented
94	#459	C₂×C₁₃	1	2	yes	complemented
95	#446	C₂×C₁₃	1	1	yes	characteristic, complemented
96	#440	C₂×C₁₃	1	2	yes	complemented
97	C₄×D₈	C₂×C₁₃	1	1	yes	characteristic, complemented
98	C₂²×C₁₃	—	4	2	no	—
99	C₂²×C₁₃	—	4	2	no	—
100	C₂²×C₁₃	—	4	2	no	—
101	C₂²×C₁₃	—	4	2	no	—
102	C₂²×C₁₃	—	4	2	no	—
103	C₂²×C₁₃	—	4	2	no	—
104	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
105	C₄×C₁₃	—	2	2	no	—
106	C₄×C₁₃	—	2	1	no	—
107	C₄×C₁₃	—	2	1	no	—
108	C₄×C₁₃	—	2	2	no	—
109	C₄×C₁₃	—	4	2	no	—
110	C₄×C₁₃	—	4	2	no	—
111	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
112	C₄×C₁₃	H(5,𝔽₂)	1	1	yes	characteristic
113	C₂×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
114	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
115	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
116	C₂×D₈	C₂²×C₁₃	1	1	yes	characteristic
117	C₂×D₈	C₂²×C₁₃	1	1	yes	characteristic
118	C₂×D₈	C₂²×C₁₃	1	2	yes	—
119	#127	C₂²×C₁₃	1	2	yes	—
120	#124	—	2	2	no	—
121	#124	C₂²×C₁₃	1	2	yes	—
122	#124	C₂²×C₁₃	1	2	yes	—
123	#123	—	2	2	no	—
124	#123	—	2	2	no	—
125	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
126	C₄×Q₈	C₂²×C₁₃	1	1	yes	characteristic
127	C₄×D₄	—	2	2	no	—
128	C₄×D₄	C₂²×C₁₃	1	2	yes	—
129	C₄×D₄	C₂²×C₁₃	1	2	yes	—
130	#107	C₂²×C₁₃	1	1	yes	characteristic
131	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
132	C₄⋊C₈	C₂²×C₁₃	1	1	yes	characteristic
133	#103	C₂²×C₁₃	1	2	yes	—
134	#102	C₂²×C₁₃	1	2	yes	—
135	#102	C₂²×C₁₃	1	2	yes	—
136	#97	C₂²×C₁₃	1	2	yes	—
137	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
138	C₂×C₁₃	—	8	2	no	—
139	C₂×C₁₃	—	8	2	no	—
140	C₂×C₁₃	#550	1	1	yes	characteristic
141	C₂×C₁₃	#575	1	1	yes	characteristic
142	C₂×C₁₃	#576	1	1	yes	characteristic
143	C₂×Q₈	D₄×C₁₃	1	1	yes	characteristic
144	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
145	C₂×D₄	—	2	2	no	—
146	C₂×D₄	—	2	2	no	—
147	C₂×D₄	C₂³×C₁₃	1	2	yes	—
148	C₂×D₄	C₂³×C₁₃	1	2	yes	—
149	C₂²×C₄	—	2	2	no	—
150	C₂²×C₄	—	2	2	no	—
151	Q₁₆	—	4	1	no	—
152	SD₁₆	—	2	4	no	—
153	SD₁₆	—	4	2	no	—
154	D₈	—	2	2	no	—
155	D₈	—	4	2	no	—
156	D₈	—	4	1	no	—
157	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
158	C₂×C₈	C₂³×C₁₃	1	2	yes	—
159	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	D₄×C₁₃	1	2	yes	—
162	C₄⋊C₄	D₄×C₁₃	1	1	yes	characteristic
163	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
164	C₂²⋊C₄	—	2	2	no	—
165	C₂²⋊C₄	—	2	2	no	—
166	C₂²⋊C₄	—	2	2	no	—
167	C₂²⋊C₄	—	2	2	no	—
168	C₂²⋊C₄	—	2	2	no	—
169	C₂²⋊C₄	—	2	2	no	—
170	C₄²	—	2	2	no	—
171	C₄²	C₂³×C₁₃	1	1	yes	characteristic
172	C₄²	C₂³×C₁₃	1	2	yes	—
173	C₁₃	#3403	1	1	yes	characteristic, direct product factor, Sylow
174	C₂³	—	2	2	no	—
175	C₂³	—	2	2	no	—
176	Q₈	—	2	2	no	—
177	Q₈	—	2	2	no	—
178	Q₈	—	4	1	no	—
179	D₄	—	2	2	no	—
180	D₄	—	2	2	no	—
181	D₄	—	2	2	no	—
182	D₄	—	2	2	no	—
183	D₄	—	4	4	no	—
184	D₄	—	8	2	no	—
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	2	2	no	—
187	C₂×C₄	—	8	2	no	—
188	C₂×C₄	—	8	2	no	—
189	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
190	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
191	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
192	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
193	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
194	C₈	—	2	2	no	—
195	C₈	—	2	1	no	—
196	C₈	C₁₃×#41	1	2	yes	—
197	C₂²	—	4	2	no	—
198	C₂²	—	4	2	no	—
199	C₂²	—	4	2	no	—
200	C₂²	—	4	2	no	—
201	C₂²	—	4	2	no	—
202	C₂²	—	4	2	no	—
203	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
204	C₄	—	2	2	no	—
205	C₄	—	2	1	no	—
206	C₄	—	2	1	no	—
207	C₄	—	2	2	no	—
208	C₄	—	4	2	no	—
209	C₄	—	4	2	no	—
210	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
211	C₄	C₁₃×H(5,𝔽₂)	1	1	yes	characteristic
212	C₂	—	8	2	no	—
213	C₂	—	8	2	no	—
214	C₂	C₁₃×#550	1	1	yes	characteristic
215	C₂	C₁₃×#575	1	1	yes	characteristic
216	C₂	C₁₃×#576	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
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×#496	C₂	semidirect
3. C₁₃×#495	C₂	semidirect
4. C₁₃×#464	C₂	semidirect
5. C₁₃×#463	C₂	semidirect
6. C₁₃×#459	C₂	semidirect
7. C₁₃×#446	C₂	semidirect
8. C₁₃×#440	C₂	semidirect
9. C₄×C₁₃×D₈	C₂	semidirect
65. #3403	C₁₃	direct (N × H)
89. #559	C₂×C₁₃	semidirect
90. #496	C₂×C₁₃	semidirect
91. #495	C₂×C₁₃	semidirect
92. #464	C₂×C₁₃	semidirect
93. #463	C₂×C₁₃	semidirect
94. #459	C₂×C₁₃	semidirect
95. #446	C₂×C₁₃	semidirect
96. #440	C₂×C₁₃	semidirect
97. C₄×D₈	C₂×C₁₃	semidirect
173. C₁₃	#3403	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×#496	2
C₁₃×#495	2
C₁₃×#464	2
C₁₃×#463	2
C₁₃×#459	2
C₁₃×#446	2
C₁₃×#440	2
C₄×C₁₃×D₈	2
#3403	13

Subgroups — C13×#3403 #13112461

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3403 #13112461