Subgroups — C₁₃×#3060 #13112118

Order 1664.

484 conjugacy classes of subgroups.

The subgroup lattice has 212 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×#135	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×C₁₃×#131	C₂	1	1	yes	maximal, characteristic, complemented
3	C₁₃×#537	C₂	1	2	yes	maximal, complemented
4	C₁₃×#470	C₂	1	2	yes	maximal, complemented
5	C₁₃×#469	C₂	1	2	yes	maximal, complemented
6	C₁₃×#465	C₂	1	4	yes	maximal, complemented
7	C₂×C₁₃×#106	C₂	1	1	yes	maximal, characteristic, complemented
8	C₁₃×#417	C₂	1	2	yes	maximal, complemented
9	C₂×C₁₃×#41	C₂²	1	2	yes	—
10	C₂×C₁₃×#41	C₂²	1	1	yes	characteristic
11	C₁₃×#135	—	2	4	no	—
12	C₂×C₁₃×Q₁₆	C₂²	1	1	yes	characteristic
13	C₂×C₁₃×SD₁₆	C₂²	1	2	yes	—
14	C₂×C₁₃×D₈	C₂²	1	1	yes	characteristic
15	C₁₃×#131	—	2	4	no	—
16	C₂×C₁₃×M₁₆	C₂²	1	1	yes	characteristic
17	C₂×C₁₃×M₁₆	C₂²	1	2	yes	—
18	C₂²×C₈×C₁₃	C₂²	1	2	yes	—
19	C₂²×C₈×C₁₃	C₂²	1	1	yes	characteristic
20	C₁₃×C₂²⋊Q₈	—	2	2	no	—
21	C₁₃×C₂²⋊Q₈	C₂²	1	4	yes	—
22	C₁₃×#121	—	2	2	no	—
23	C₁₃×#121	—	2	4	no	—
24	C₁₃×#121	C₂²	1	4	yes	—
25	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
26	C₁₃×#106	C₂²	1	2	yes	—
27	C₁₃×#106	C₂²	1	2	yes	—
28	C₁₃×#103	C₂²	1	4	yes	—
29	C₁₃×#102	C₂²	1	4	yes	—
30	C₁₃×#41	—	2	2	no	—
31	C₁₃×#41	—	2	2	no	—
32	C₁₃×#41	—	2	4	no	—
33	C₁₃×#41	—	2	4	no	—
34	C₂×Q₈×C₁₃	D₄	1	1	yes	characteristic
35	C₂×Q₈×C₁₃	C₂³	1	2	yes	—
36	C₂×D₄×C₁₃	—	2	2	no	—
37	C₂×D₄×C₁₃	—	2	4	no	—
38	C₂×D₄×C₁₃	D₄	1	2	yes	—
39	C₂×D₄×C₁₃	D₄	1	1	yes	characteristic
40	C₂×D₄×C₁₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₃	—	2	2	no	—
42	C₂²×C₄×C₁₃	—	2	2	no	—
43	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
44	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
45	C₁₃×Q₁₆	—	2	2	no	—
46	C₁₃×SD₁₆	—	2	4	no	—
47	C₁₃×D₈	—	2	2	no	—
48	C₁₃×M₁₆	—	2	2	no	—
49	C₁₃×M₁₆	—	2	4	no	—
50	C₂×C₈×C₁₃	—	2	4	no	—
51	C₂×C₈×C₁₃	D₄	1	4	yes	complemented
52	C₂×C₈×C₁₃	C₂³	1	2	yes	—
53	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
54	C₂×C₈×C₁₃	C₂³	1	1	yes	characteristic
55	C₁₃×C₄⋊C₄	—	2	2	no	—
56	C₁₃×C₄⋊C₄	—	2	4	no	—
57	C₁₃×C₄⋊C₄	C₂³	1	4	yes	—
58	C₁₃×C₂²⋊C₄	—	2	4	no	—
59	C₁₃×C₂²⋊C₄	—	2	4	no	—
60	#3060	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
61	C₂³×C₁₃	—	2	2	no	—
62	C₂³×C₁₃	C₂×D₄	1	2	yes	—
63	C₂³×C₁₃	C₂×D₄	1	1	yes	characteristic
64	Q₈×C₁₃	—	2	2	no	—
65	Q₈×C₁₃	—	2	2	no	—
66	Q₈×C₁₃	—	2	2	no	—
67	D₄×C₁₃	—	2	4	no	—
68	D₄×C₁₃	—	2	2	no	—
69	D₄×C₁₃	—	2	2	no	—
70	D₄×C₁₃	—	2	2	no	—
71	D₄×C₁₃	—	4	4	no	—
72	D₄×C₁₃	—	8	4	no	—
73	C₂×C₄×C₁₃	—	2	2	no	—
74	C₂×C₄×C₁₃	—	2	2	no	—
75	C₂×C₄×C₁₃	—	2	4	no	—
76	C₂×C₄×C₁₃	—	2	2	no	—
77	C₂×C₄×C₁₃	—	4	4	no	—
78	C₂×C₄×C₁₃	—	8	2	no	—
79	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
80	C₂×C₄×C₁₃	C₂×D₄	1	1	yes	characteristic
81	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
82	C₂×C₄×C₁₃	#41	1	2	yes	—
83	C₂×C₄×C₁₃	C₂⁴	1	1	yes	characteristic
84	C₈×C₁₃	—	2	2	no	—
85	C₈×C₁₃	C₂×D₄	1	2	yes	—
86	C₈×C₁₃	C₂×D₄	1	2	yes	—
87	C₂×#135	C₂×C₁₃	1	1	yes	characteristic, complemented
88	C₂×#131	C₂×C₁₃	1	1	yes	characteristic, complemented
89	#537	C₂×C₁₃	1	2	yes	complemented
90	#470	C₂×C₁₃	1	2	yes	complemented
91	#469	C₂×C₁₃	1	2	yes	complemented
92	#465	C₂×C₁₃	1	4	yes	complemented
93	C₂×#106	C₂×C₁₃	1	1	yes	characteristic, complemented
94	#417	C₂×C₁₃	1	2	yes	complemented
95	C₂²×C₁₃	—	2	2	no	—
96	C₂²×C₁₃	—	2	2	no	—
97	C₂²×C₁₃	—	2	4	no	—
98	C₂²×C₁₃	—	4	2	no	—
99	C₂²×C₁₃	—	4	2	no	—
100	C₂²×C₁₃	—	4	2	no	—
101	C₂²×C₁₃	#121	1	2	yes	—
102	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
103	C₄×C₁₃	—	2	2	no	—
104	C₄×C₁₃	—	4	2	no	—
105	C₄×C₁₃	—	4	4	no	—
106	C₄×C₁₃	#121	1	2	yes	—
107	C₄×C₁₃	C₂²×D₄	1	1	yes	characteristic
108	C₄×C₁₃	C₂×#41	1	1	yes	characteristic
109	C₂×#41	C₂²×C₁₃	1	2	yes	—
110	C₂×#41	C₂²×C₁₃	1	1	yes	characteristic
111	#135	—	2	4	no	—
112	C₂×Q₁₆	C₂²×C₁₃	1	1	yes	characteristic
113	C₂×SD₁₆	C₂²×C₁₃	1	2	yes	—
114	C₂×D₈	C₂²×C₁₃	1	1	yes	characteristic
115	#131	—	2	4	no	—
116	C₂×M₁₆	C₂²×C₁₃	1	1	yes	characteristic
117	C₂×M₁₆	C₂²×C₁₃	1	2	yes	—
118	C₂²×C₈	C₂²×C₁₃	1	2	yes	—
119	C₂²×C₈	C₂²×C₁₃	1	1	yes	characteristic
120	C₂²⋊Q₈	—	2	2	no	—
121	C₂²⋊Q₈	C₂²×C₁₃	1	4	yes	—
122	#121	—	2	2	no	—
123	#121	—	2	4	no	—
124	#121	C₂²×C₁₃	1	4	yes	—
125	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
126	#106	C₂²×C₁₃	1	2	yes	—
127	#106	C₂²×C₁₃	1	2	yes	—
128	#103	C₂²×C₁₃	1	4	yes	—
129	#102	C₂²×C₁₃	1	4	yes	—
130	C₂×C₁₃	—	2	2	no	—
131	C₂×C₁₃	—	4	2	no	—
132	C₂×C₁₃	—	8	2	no	—
133	C₂×C₁₃	C₂×#121	1	1	yes	characteristic
134	C₂×C₁₃	#577	1	2	yes	—
135	#41	—	2	2	no	—
136	#41	—	2	2	no	—
137	#41	—	2	4	no	—
138	#41	—	2	4	no	—
139	C₂×Q₈	D₄×C₁₃	1	1	yes	characteristic
140	C₂×Q₈	C₂³×C₁₃	1	2	yes	—
141	C₂×D₄	—	2	2	no	—
142	C₂×D₄	—	2	4	no	—
143	C₂×D₄	D₄×C₁₃	1	2	yes	—
144	C₂×D₄	D₄×C₁₃	1	1	yes	characteristic
145	C₂×D₄	C₂³×C₁₃	1	2	yes	—
146	C₂²×C₄	—	2	2	no	—
147	C₂²×C₄	—	2	2	no	—
148	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
149	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
150	Q₁₆	—	2	2	no	—
151	SD₁₆	—	2	4	no	—
152	D₈	—	2	2	no	—
153	M₁₆	—	2	2	no	—
154	M₁₆	—	2	4	no	—
155	C₂×C₈	—	2	4	no	—
156	C₂×C₈	D₄×C₁₃	1	4	yes	complemented
157	C₂×C₈	C₂³×C₁₃	1	2	yes	—
158	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
159	C₂×C₈	C₂³×C₁₃	1	1	yes	characteristic
160	C₄⋊C₄	—	2	2	no	—
161	C₄⋊C₄	—	2	4	no	—
162	C₄⋊C₄	C₂³×C₁₃	1	4	yes	—
163	C₂²⋊C₄	—	2	4	no	—
164	C₂²⋊C₄	—	2	4	no	—
165	C₁₃	#3060	1	1	yes	characteristic, direct product factor, Sylow
166	C₂³	—	2	2	no	—
167	C₂³	C₂×D₄×C₁₃	1	2	yes	—
168	C₂³	C₂×D₄×C₁₃	1	1	yes	characteristic
169	Q₈	—	2	2	no	—
170	Q₈	—	2	2	no	—
171	Q₈	—	2	2	no	—
172	D₄	—	2	4	no	—
173	D₄	—	2	2	no	—
174	D₄	—	2	2	no	—
175	D₄	—	2	2	no	—
176	D₄	—	4	4	no	—
177	D₄	—	8	4	no	—
178	C₂×C₄	—	2	2	no	—
179	C₂×C₄	—	2	2	no	—
180	C₂×C₄	—	2	4	no	—
181	C₂×C₄	—	2	2	no	—
182	C₂×C₄	—	4	4	no	—
183	C₂×C₄	—	8	2	no	—
184	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
185	C₂×C₄	C₂×D₄×C₁₃	1	1	yes	characteristic
186	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
187	C₂×C₄	C₁₃×#41	1	2	yes	—
188	C₂×C₄	C₂⁴×C₁₃	1	1	yes	characteristic
189	C₈	—	2	2	no	—
190	C₈	C₂×D₄×C₁₃	1	2	yes	—
191	C₈	C₂×D₄×C₁₃	1	2	yes	—
192	C₂²	—	2	2	no	—
193	C₂²	—	2	2	no	—
194	C₂²	—	2	4	no	—
195	C₂²	—	4	2	no	—
196	C₂²	—	4	2	no	—
197	C₂²	—	4	2	no	—
198	C₂²	C₁₃×#121	1	2	yes	—
199	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
200	C₄	—	2	2	no	—
201	C₄	—	4	2	no	—
202	C₄	—	4	4	no	—
203	C₄	C₁₃×#121	1	2	yes	—
204	C₄	C₂²×D₄×C₁₃	1	1	yes	characteristic
205	C₄	C₂×C₁₃×#41	1	1	yes	characteristic
206	C₂	—	2	2	no	—
207	C₂	—	4	2	no	—
208	C₂	—	8	2	no	—
209	C₂	C₂×C₁₃×#121	1	1	yes	characteristic
210	C₂	C₁₃×#577	1	2	yes	—

	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 ◁ C₂×C₁₃×#41 ◁ G

Complements

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

N	H ≅ G/N	type
1. C₂×C₁₃×#135	C₂	semidirect
2. C₂×C₁₃×#131	C₂	semidirect
3. C₁₃×#537	C₂	semidirect
4. C₁₃×#470	C₂	semidirect
5. C₁₃×#469	C₂	semidirect
6. C₁₃×#465	C₂	semidirect
7. C₂×C₁₃×#106	C₂	semidirect
8. C₁₃×#417	C₂	semidirect
51. C₂×C₈×C₁₃	D₄	semidirect
60. #3060	C₁₃	direct (N × H)
87. C₂×#135	C₂×C₁₃	semidirect
88. C₂×#131	C₂×C₁₃	semidirect
89. #537	C₂×C₁₃	semidirect
90. #470	C₂×C₁₃	semidirect
91. #469	C₂×C₁₃	semidirect
92. #465	C₂×C₁₃	semidirect
93. C₂×#106	C₂×C₁₃	semidirect
94. #417	C₂×C₁₃	semidirect
156. C₂×C₈	D₄×C₁₃	semidirect
165. C₁₃	#3060	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂×C₁₃×#135	2
C₂×C₁₃×#131	2
C₁₃×#537	2
C₁₃×#470	2
C₁₃×#469	2
C₁₃×#465	2
C₂×C₁₃×#106	2
C₁₃×#417	2
#3060	13

Subgroups — C13×#3060 #13112118

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#3060 #13112118