Subgroups — C₁₃×#2434 #13111492

Order 1664.

800 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂³×Q₈×C₁₃	C₂	1	1	yes	maximal, characteristic
2	C₂×C₁₃×C₄⋊Q₈	C₂	1	2	yes	maximal, complemented
3	C₂×C₁₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
4	C₁₃×#395	C₂	1	4	yes	maximal, complemented
5	C₁₃×#391	C₂	1	2	yes	maximal, complemented
6	C₁₃×#385	C₂	1	4	yes	maximal
7	C₄×C₁₃×C₂²⋊C₄	C₂	1	1	yes	maximal, characteristic
8	C₂²×Q₈×C₁₃	—	2	8	no	—
9	C₂²×Q₈×C₁₃	—	2	4	no	—
10	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
11	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
12	C₂²×Q₈×C₁₃	C₂²	1	2	yes	—
13	C₂³×C₄×C₁₃	C₂²	1	2	yes	—
14	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
15	C₁₃×C₄⋊Q₈	—	2	8	no	—
16	C₁₃×C₂²⋊Q₈	—	2	4	no	—
17	C₂×C₁₃×C₄⋊C₄	C₂²	1	8	yes	—
18	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
21	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₄²×C₁₃	C₂²	1	2	yes	—
23	C₁₃×#95	C₂²	1	2	yes	—
24	C₁₃×#95	C₂²	1	8	yes	—
25	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
26	C₂×Q₈×C₁₃	—	2	32	no	—
27	C₂×Q₈×C₁₃	—	2	8	no	—
28	C₂×Q₈×C₁₃	—	2	4	no	—
29	C₂×Q₈×C₁₃	—	2	4	no	—
30	C₂×Q₈×C₁₃	—	2	4	no	—
31	C₂×Q₈×C₁₃	—	2	2	no	—
32	C₂×Q₈×C₁₃	—	2	4	no	—
33	C₂×Q₈×C₁₃	D₄	1	8	yes	—
34	C₂²×C₄×C₁₃	—	2	4	no	—
35	C₂²×C₄×C₁₃	—	2	8	no	—
36	C₂²×C₄×C₁₃	—	2	4	no	—
37	C₂²×C₄×C₁₃	D₄	1	4	yes	—
38	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
39	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
41	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
43	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
44	C₁₃×C₄⋊C₄	—	2	2	no	—
45	C₁₃×C₄⋊C₄	—	2	4	no	—
46	C₁₃×C₄⋊C₄	—	2	16	no	—
47	C₁₃×C₂²⋊C₄	—	2	4	no	—
48	C₁₃×C₂²⋊C₄	Q₈	1	8	yes	complemented
49	C₄²×C₁₃	—	2	4	no	—
50	#2434	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
51	C₂³×C₁₃	—	2	2	no	—
52	C₂³×C₁₃	—	2	2	no	—
53	C₂³×C₁₃	C₂×D₄	1	2	yes	—
54	C₂³×C₁₃	C₂×Q₈	1	4	yes	—
55	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
56	Q₈×C₁₃	—	2	16	no	—
57	Q₈×C₁₃	—	2	16	no	—
58	Q₈×C₁₃	—	4	4	no	—
59	C₂×C₄×C₁₃	—	2	16	no	—
60	C₂×C₄×C₁₃	—	2	4	no	—
61	C₂×C₄×C₁₃	—	2	4	no	—
62	C₂×C₄×C₁₃	—	2	4	no	—
63	C₂×C₄×C₁₃	—	2	2	no	—
64	C₂×C₄×C₁₃	—	2	4	no	—
65	C₂×C₄×C₁₃	—	2	8	no	—
66	C₂×C₄×C₁₃	—	2	2	no	—
67	C₂×C₄×C₁₃	—	2	2	no	—
68	C₂×C₄×C₁₃	—	2	4	no	—
69	C₂×C₄×C₁₃	—	2	4	no	—
70	C₂×C₄×C₁₃	C₂×D₄	1	8	yes	—
71	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
72	C₂×C₄×C₁₃	C₂×D₄	1	4	yes	—
73	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
74	C₂×C₄×C₁₃	C₂×Q₈	1	8	yes	complemented
75	C₂³×Q₈	C₂×C₁₃	1	1	yes	characteristic
76	C₂×C₄⋊Q₈	C₂×C₁₃	1	2	yes	complemented
77	C₂×C₂²⋊Q₈	C₂×C₁₃	1	1	yes	characteristic
78	#395	C₂×C₁₃	1	4	yes	complemented
79	#391	C₂×C₁₃	1	2	yes	complemented
80	#385	C₂×C₁₃	1	4	yes	—
81	C₄×C₂²⋊C₄	C₂×C₁₃	1	1	yes	characteristic
82	C₂²×C₁₃	—	2	2	no	—
83	C₂²×C₁₃	—	2	2	no	—
84	C₂²×C₁₃	—	2	8	no	—
85	C₂²×C₁₃	C₄⋊Q₈	1	4	yes	complemented
86	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
87	C₂²×C₁₃	C₂²×D₄	1	2	yes	—
88	C₂²×C₁₃	C₂²×Q₈	1	2	yes	—
89	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
90	C₄×C₁₃	—	2	8	no	—
91	C₄×C₁₃	—	2	2	no	—
92	C₄×C₁₃	—	2	8	no	—
93	C₄×C₁₃	—	4	2	no	—
94	C₄×C₁₃	—	4	2	no	—
95	C₄×C₁₃	#120	1	4	yes	—
96	C₂²×Q₈	—	2	8	no	—
97	C₂²×Q₈	—	2	4	no	—
98	C₂²×Q₈	C₂²×C₁₃	1	1	yes	characteristic
99	C₂²×Q₈	C₂²×C₁₃	1	2	yes	—
100	C₂²×Q₈	C₂²×C₁₃	1	2	yes	—
101	C₂³×C₄	C₂²×C₁₃	1	2	yes	—
102	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
103	C₄⋊Q₈	—	2	8	no	—
104	C₂²⋊Q₈	—	2	4	no	—
105	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
106	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
107	C₂×C₄⋊C₄	C₂²×C₁₃	1	8	yes	—
108	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
109	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
110	C₂×C₄²	C₂²×C₁₃	1	2	yes	—
111	#95	C₂²×C₁₃	1	2	yes	—
112	#95	C₂²×C₁₃	1	8	yes	—
113	C₂×C₁₃	—	2	4	no	—
114	C₂×C₁₃	C₂×#120	1	1	yes	characteristic
115	C₂×C₁₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
116	C₂×C₁₃	#536	1	1	yes	characteristic
117	C₂×C₁₃	D₄×Q₈	1	4	yes	—
118	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
119	C₂×Q₈	—	2	32	no	—
120	C₂×Q₈	—	2	8	no	—
121	C₂×Q₈	—	2	4	no	—
122	C₂×Q₈	—	2	4	no	—
123	C₂×Q₈	—	2	4	no	—
124	C₂×Q₈	—	2	2	no	—
125	C₂×Q₈	—	2	4	no	—
126	C₂×Q₈	D₄×C₁₃	1	8	yes	—
127	C₂²×C₄	—	2	4	no	—
128	C₂²×C₄	—	2	8	no	—
129	C₂²×C₄	—	2	4	no	—
130	C₂²×C₄	D₄×C₁₃	1	4	yes	—
131	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
132	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
133	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
134	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
135	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
136	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
137	C₄⋊C₄	—	2	2	no	—
138	C₄⋊C₄	—	2	4	no	—
139	C₄⋊C₄	—	2	16	no	—
140	C₂²⋊C₄	—	2	4	no	—
141	C₂²⋊C₄	Q₈×C₁₃	1	8	yes	complemented
142	C₄²	—	2	4	no	—
143	C₁₃	#2434	1	1	yes	characteristic, direct product factor, Sylow
144	C₂³	—	2	2	no	—
145	C₂³	—	2	2	no	—
146	C₂³	C₂×D₄×C₁₃	1	2	yes	—
147	C₂³	C₂×Q₈×C₁₃	1	4	yes	—
148	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
149	Q₈	—	2	16	no	—
150	Q₈	—	2	16	no	—
151	Q₈	—	4	4	no	—
152	C₂×C₄	—	2	16	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	4	no	—
155	C₂×C₄	—	2	4	no	—
156	C₂×C₄	—	2	2	no	—
157	C₂×C₄	—	2	4	no	—
158	C₂×C₄	—	2	8	no	—
159	C₂×C₄	—	2	2	no	—
160	C₂×C₄	—	2	2	no	—
161	C₂×C₄	—	2	4	no	—
162	C₂×C₄	—	2	4	no	—
163	C₂×C₄	C₂×D₄×C₁₃	1	8	yes	—
164	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
165	C₂×C₄	C₂×D₄×C₁₃	1	4	yes	—
166	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
167	C₂×C₄	C₂×Q₈×C₁₃	1	8	yes	complemented
168	C₂²	—	2	2	no	—
169	C₂²	—	2	2	no	—
170	C₂²	—	2	8	no	—
171	C₂²	C₁₃×C₄⋊Q₈	1	4	yes	complemented
172	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
173	C₂²	C₂²×D₄×C₁₃	1	2	yes	—
174	C₂²	C₂²×Q₈×C₁₃	1	2	yes	—
175	C₂²	C₁₃×2₋¹⁺⁴	1	2	yes	—
176	C₄	—	2	8	no	—
177	C₄	—	2	2	no	—
178	C₄	—	2	8	no	—
179	C₄	—	4	2	no	—
180	C₄	—	4	2	no	—
181	C₄	C₁₃×#120	1	4	yes	—
182	C₂	—	2	4	no	—
183	C₂	C₂×C₁₃×#120	1	1	yes	characteristic
184	C₂	C₂×C₁₃×C₄⋊Q₈	1	1	yes	characteristic
185	C₂	C₁₃×#536	1	1	yes	characteristic
186	C₂	D₄×Q₈×C₁₃	1	4	yes	—

	subgroup	G/H
center, socle	C₂³×C₁₃	C₂⁴
commutator, Frattini, cosocle	C₂³	C₂⁴×C₁₃
prime radical	C₂⁴×C₁₃	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₁₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×C₁₃×C₄⋊Q₈	C₂	semidirect
4. C₁₃×#395	C₂	semidirect
5. C₁₃×#391	C₂	semidirect
48. C₁₃×C₂²⋊C₄	Q₈	semidirect
50. #2434	C₁₃	direct (N × H)
74. C₂×C₄×C₁₃	C₂×Q₈	semidirect
76. C₂×C₄⋊Q₈	C₂×C₁₃	semidirect
78. #395	C₂×C₁₃	semidirect
79. #391	C₂×C₁₃	semidirect
85. C₂²×C₁₃	C₄⋊Q₈	semidirect
141. C₂²⋊C₄	Q₈×C₁₃	semidirect
143. C₁₃	#2434	direct (N × H)
167. C₂×C₄	C₂×Q₈×C₁₃	semidirect
171. C₂²	C₁₃×C₄⋊Q₈	semidirect

Maximal subgroups

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

subgroup	index
C₂³×Q₈×C₁₃	2
C₂×C₁₃×C₄⋊Q₈	2
C₂×C₁₃×C₂²⋊Q₈	2
C₁₃×#395	2
C₁₃×#391	2
C₁₃×#385	2
C₄×C₁₃×C₂²⋊C₄	2
#2434	13

Subgroups — C13×#2434 #13111492

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2434 #13111492