Subgroups — C₁₃×#3210 #13112268

Order 1664.

394 conjugacy classes of subgroups.

The subgroup lattice has 226 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₁₃×#533	C₂	1	1	yes	maximal, characteristic, complemented
2	C₁₃×#532	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#494	C₂	1	1	yes	maximal, characteristic, complemented
4	C₁₃×#492	C₂	1	1	yes	maximal, characteristic, complemented
5	C₁₃×#487	C₂	1	1	yes	maximal, characteristic, complemented
6	C₁₃×#486	C₂	1	1	yes	maximal, characteristic, complemented
7	C₁₃×#464	C₂	1	4	yes	maximal, complemented
8	C₁₃×#452	C₂	1	4	yes	maximal, complemented
9	C₁₃×#431	C₂	1	1	yes	maximal, characteristic, complemented
10	C₂×C₁₃×#41	C₂²	1	2	yes	—
11	C₂×C₁₃×Q₁₆	C₂²	1	4	yes	—
12	C₂×C₁₃×SD₁₆	C₂²	1	4	yes	—
13	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
14	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
15	C₁₃×C₄⋊Q₈	C₂²	1	1	yes	characteristic
16	C₁₃×#127	C₂²	1	1	yes	characteristic
17	C₁₃×#125	—	2	1	no	—
18	C₁₃×#124	C₂²	1	2	yes	—
19	C₁₃×C₂²⋊Q₈	—	2	1	no	—
20	C₁₃×C₂²⋊Q₈	C₂²	1	2	yes	—
21	C₁₃×#121	—	2	1	no	—
22	C₁₃×#121	—	2	1	no	—
23	C₄×Q₈×C₁₃	—	2	1	no	—
24	C₄×Q₈×C₁₃	C₂²	1	2	yes	—
25	C₄×D₄×C₁₃	—	2	1	no	—
26	C₄×D₄×C₁₃	—	2	1	no	—
27	C₁₃×#117	—	2	1	no	—
28	C₂×C₄²×C₁₃	C₂²	1	1	yes	characteristic
29	C₁₃×C₄⋊C₈	C₂²	1	2	yes	—
30	C₁₃×#103	C₂²	1	4	yes	—
31	C₁₃×#102	C₂²	1	4	yes	—
32	C₁₃×C₂²⋊C₈	C₂²	1	2	yes	—
33	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
34	C₄×C₈×C₁₃	C₂²	1	1	yes	characteristic
35	C₁₃×#41	—	4	4	no	—
36	C₂×Q₈×C₁₃	—	2	1	no	—
37	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
38	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
39	C₂×D₄×C₁₃	—	2	2	no	—
40	C₂×D₄×C₁₃	—	2	1	no	—
41	C₂×D₄×C₁₃	—	2	1	no	—
42	C₂×D₄×C₁₃	C₂³	1	2	yes	—
43	C₂²×C₄×C₁₃	—	2	2	no	—
44	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
45	C₂²×C₄×C₁₃	D₄	1	1	yes	characteristic
46	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
47	C₁₃×Q₁₆	—	4	4	no	—
48	C₁₃×SD₁₆	—	4	4	no	—
49	C₂×C₈×C₁₃	C₂³	1	2	yes	—
50	C₂×C₈×C₁₃	C₂³	1	2	yes	—
51	C₁₃×C₄⋊C₄	—	2	1	no	—
52	C₁₃×C₄⋊C₄	—	2	1	no	—
53	C₁₃×C₄⋊C₄	—	2	1	no	—
54	C₁₃×C₄⋊C₄	—	2	1	no	—
55	C₁₃×C₄⋊C₄	—	2	1	no	—
56	C₁₃×C₄⋊C₄	—	2	2	no	—
57	C₁₃×C₄⋊C₄	—	2	2	no	—
58	C₁₃×C₄⋊C₄	C₂³	1	2	yes	—
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	D₄	1	1	yes	characteristic
64	C₄²×C₁₃	D₄	1	1	yes	characteristic
65	C₄²×C₁₃	C₂³	1	1	yes	characteristic
66	C₄²×C₁₃	C₂³	1	1	yes	characteristic
67	#3210	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
68	C₂³×C₁₃	—	2	2	no	—
69	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
70	Q₈×C₁₃	—	2	4	no	—
71	Q₈×C₁₃	—	2	4	no	—
72	Q₈×C₁₃	—	4	2	no	—
73	D₄×C₁₃	—	2	4	no	—
74	D₄×C₁₃	—	4	4	no	—
75	D₄×C₁₃	—	4	2	no	—
76	D₄×C₁₃	—	4	2	no	—
77	C₂×C₄×C₁₃	—	2	2	no	—
78	C₂×C₄×C₁₃	—	2	2	no	—
79	C₂×C₄×C₁₃	—	2	2	no	—
80	C₂×C₄×C₁₃	—	2	2	no	—
81	C₂×C₄×C₁₃	—	2	2	no	—
82	C₂×C₄×C₁₃	—	2	2	no	—
83	C₂×C₄×C₁₃	—	4	4	no	—
84	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
85	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
86	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
87	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
88	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
89	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
90	C₈×C₁₃	—	2	2	no	—
91	C₈×C₁₃	—	2	2	no	—
92	#533	C₂×C₁₃	1	1	yes	characteristic, complemented
93	#532	C₂×C₁₃	1	1	yes	characteristic
94	#494	C₂×C₁₃	1	1	yes	characteristic, complemented
95	#492	C₂×C₁₃	1	1	yes	characteristic, complemented
96	#487	C₂×C₁₃	1	1	yes	characteristic, complemented
97	#486	C₂×C₁₃	1	1	yes	characteristic, complemented
98	#464	C₂×C₁₃	1	4	yes	complemented
99	#452	C₂×C₁₃	1	4	yes	complemented
100	#431	C₂×C₁₃	1	1	yes	characteristic, complemented
101	C₂²×C₁₃	—	2	1	no	—
102	C₂²×C₁₃	—	2	1	no	—
103	C₂²×C₁₃	—	2	1	no	—
104	C₂²×C₁₃	—	4	2	no	—
105	C₂²×C₁₃	—	4	2	no	—
106	C₂²×C₁₃	—	4	2	no	—
107	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₄×C₁₃	—	2	1	no	—
109	C₄×C₁₃	—	2	1	no	—
110	C₄×C₁₃	—	2	1	no	—
111	C₄×C₁₃	—	4	2	no	—
112	C₄×C₁₃	—	4	2	no	—
113	C₄×C₁₃	—	4	2	no	—
114	C₄×C₁₃	#135	1	2	yes	—
115	C₄×C₁₃	#137	1	2	yes	—
116	C₄×C₁₃	H(5,𝔽₂)	1	2	yes	—
117	C₂×#41	C₂²×C₁₃	1	2	yes	—
118	C₂×Q₁₆	C₂²×C₁₃	1	4	yes	—
119	C₂×SD₁₆	C₂²×C₁₃	1	4	yes	—
120	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
121	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
122	C₄⋊Q₈	C₂²×C₁₃	1	1	yes	characteristic
123	#127	C₂²×C₁₃	1	1	yes	characteristic
124	#125	—	2	1	no	—
125	#124	C₂²×C₁₃	1	2	yes	—
126	C₂²⋊Q₈	—	2	1	no	—
127	C₂²⋊Q₈	C₂²×C₁₃	1	2	yes	—
128	#121	—	2	1	no	—
129	#121	—	2	1	no	—
130	C₄×Q₈	—	2	1	no	—
131	C₄×Q₈	C₂²×C₁₃	1	2	yes	—
132	C₄×D₄	—	2	1	no	—
133	C₄×D₄	—	2	1	no	—
134	#117	—	2	1	no	—
135	C₂×C₄²	C₂²×C₁₃	1	1	yes	characteristic
136	C₄⋊C₈	C₂²×C₁₃	1	2	yes	—
137	#103	C₂²×C₁₃	1	4	yes	—
138	#102	C₂²×C₁₃	1	4	yes	—
139	C₂²⋊C₈	C₂²×C₁₃	1	2	yes	—
140	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
141	C₄×C₈	C₂²×C₁₃	1	1	yes	characteristic
142	C₂×C₁₃	—	4	1	no	—
143	C₂×C₁₃	—	8	2	no	—
144	C₂×C₁₃	#535	1	1	yes	characteristic
145	C₂×C₁₃	C₂×#135	1	1	yes	characteristic
146	C₂×C₁₃	C₂×#137	1	1	yes	characteristic
147	#41	—	4	4	no	—
148	C₂×Q₈	—	2	1	no	—
149	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
150	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
151	C₂×D₄	—	2	2	no	—
152	C₂×D₄	—	2	1	no	—
153	C₂×D₄	—	2	1	no	—
154	C₂×D₄	C₂³×C₁₃	1	2	yes	—
155	C₂²×C₄	—	2	2	no	—
156	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
157	C₂²×C₄	D₄×C₁₃	1	1	yes	characteristic
158	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
159	Q₁₆	—	4	4	no	—
160	SD₁₆	—	4	4	no	—
161	C₂×C₈	C₂³×C₁₃	1	2	yes	—
162	C₂×C₈	C₂³×C₁₃	1	2	yes	—
163	C₄⋊C₄	—	2	1	no	—
164	C₄⋊C₄	—	2	1	no	—
165	C₄⋊C₄	—	2	1	no	—
166	C₄⋊C₄	—	2	1	no	—
167	C₄⋊C₄	—	2	1	no	—
168	C₄⋊C₄	—	2	2	no	—
169	C₄⋊C₄	—	2	2	no	—
170	C₄⋊C₄	C₂³×C₁₃	1	2	yes	—
171	C₂²⋊C₄	—	2	2	no	—
172	C₂²⋊C₄	—	2	2	no	—
173	C₂²⋊C₄	—	2	2	no	—
174	C₄²	—	2	2	no	—
175	C₄²	D₄×C₁₃	1	1	yes	characteristic
176	C₄²	D₄×C₁₃	1	1	yes	characteristic
177	C₄²	C₂³×C₁₃	1	1	yes	characteristic
178	C₄²	C₂³×C₁₃	1	1	yes	characteristic
179	C₁₃	#3210	1	1	yes	characteristic, direct product factor, Sylow
180	C₂³	—	2	2	no	—
181	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
182	Q₈	—	2	4	no	—
183	Q₈	—	2	4	no	—
184	Q₈	—	4	2	no	—
185	D₄	—	2	4	no	—
186	D₄	—	4	4	no	—
187	D₄	—	4	2	no	—
188	D₄	—	4	2	no	—
189	C₂×C₄	—	2	2	no	—
190	C₂×C₄	—	2	2	no	—
191	C₂×C₄	—	2	2	no	—
192	C₂×C₄	—	2	2	no	—
193	C₂×C₄	—	2	2	no	—
194	C₂×C₄	—	2	2	no	—
195	C₂×C₄	—	4	4	no	—
196	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
197	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
198	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
199	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
200	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
201	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
202	C₈	—	2	2	no	—
203	C₈	—	2	2	no	—
204	C₂²	—	2	1	no	—
205	C₂²	—	2	1	no	—
206	C₂²	—	2	1	no	—
207	C₂²	—	4	2	no	—
208	C₂²	—	4	2	no	—
209	C₂²	—	4	2	no	—
210	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
211	C₄	—	2	1	no	—
212	C₄	—	2	1	no	—
213	C₄	—	2	1	no	—
214	C₄	—	4	2	no	—
215	C₄	—	4	2	no	—
216	C₄	—	4	2	no	—
217	C₄	C₁₃×#135	1	2	yes	—
218	C₄	C₁₃×#137	1	2	yes	—
219	C₄	C₁₃×H(5,𝔽₂)	1	2	yes	—
220	C₂	—	4	1	no	—
221	C₂	—	8	2	no	—
222	C₂	C₁₃×#535	1	1	yes	characteristic
223	C₂	C₂×C₁₃×#135	1	1	yes	characteristic
224	C₂	C₂×C₁₃×#137	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂²×C₁₃	C₂²×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₁₃
prime radical	C₁₃×#532	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₁₃ ◁ C₂×C₄²×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
1. C₁₃×#533	C₂	semidirect
3. C₁₃×#494	C₂	semidirect
4. C₁₃×#492	C₂	semidirect
5. C₁₃×#487	C₂	semidirect
6. C₁₃×#486	C₂	semidirect
7. C₁₃×#464	C₂	semidirect
8. C₁₃×#452	C₂	semidirect
9. C₁₃×#431	C₂	semidirect
67. #3210	C₁₃	direct (N × H)
92. #533	C₂×C₁₃	semidirect
94. #494	C₂×C₁₃	semidirect
95. #492	C₂×C₁₃	semidirect
96. #487	C₂×C₁₃	semidirect
97. #486	C₂×C₁₃	semidirect
98. #464	C₂×C₁₃	semidirect
99. #452	C₂×C₁₃	semidirect
100. #431	C₂×C₁₃	semidirect
179. C₁₃	#3210	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×#533	2
C₁₃×#532	2
C₁₃×#494	2
C₁₃×#492	2
C₁₃×#487	2
C₁₃×#486	2
C₁₃×#464	2
C₁₃×#452	2
C₁₃×#431	2
#3210	13

Subgroups — C13×#3210 #13112268

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3210 #13112268