Subgroups — C₁₃×#2627 #13111685

Order 1664.

508 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₁₃×C₂²⋊Q₈	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₃×C₄⋊C₄	C₂	1	1	yes	maximal, characteristic
3	C₁₃×#400	C₂	1	2	yes	maximal, complemented
4	C₁₃×#398	C₂	1	4	yes	maximal, complemented
5	C₁₃×#395	C₂	1	2	yes	maximal, complemented
6	C₁₃×#385	C₂	1	2	yes	maximal
7	C₁₃×#381	C₂	1	2	yes	maximal
8	C₁₃×#380	C₂	1	1	yes	maximal, characteristic
9	C₂²×Q₈×C₁₃	C₂²	1	1	yes	characteristic
10	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
11	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
12	C₂³×C₄×C₁₃	C₂²	1	1	yes	characteristic
13	C₁₃×C₂²⋊Q₈	—	2	4	no	—
14	C₂×C₁₃×C₄⋊C₄	—	2	4	no	—
15	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
16	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
17	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
18	C₂×C₁₃×C₄⋊C₄	C₂²	1	1	yes	characteristic
19	C₂×C₁₃×C₄⋊C₄	C₂²	1	2	yes	—
20	C₂×C₁₃×C₄⋊C₄	C₂²	1	4	yes	—
21	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
22	C₂×C₁₃×C₂²⋊C₄	C₂²	1	2	yes	—
23	C₁₃×#95	C₂²	1	2	yes	—
24	C₁₃×#95	C₂²	1	4	yes	—
25	C₁₃×#95	C₂²	1	4	yes	—
26	C₁₃×#95	C₂²	1	4	yes	—
27	C₂⁴×C₁₃	C₂³	1	1	yes	characteristic
28	C₂×Q₈×C₁₃	—	2	2	no	—
29	C₂×Q₈×C₁₃	—	4	2	no	—
30	C₂²×C₄×C₁₃	—	2	2	no	—
31	C₂²×C₄×C₁₃	—	2	2	no	—
32	C₂²×C₄×C₁₃	—	2	2	no	—
33	C₂²×C₄×C₁₃	—	4	2	no	—
34	C₂²×C₄×C₁₃	—	4	2	no	—
35	C₂²×C₄×C₁₃	D₄	1	4	yes	—
36	C₂²×C₄×C₁₃	Q₈	1	4	yes	complemented
37	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
38	C₂²×C₄×C₁₃	C₂³	1	4	yes	—
39	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
40	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
41	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
42	C₂²×C₄×C₁₃	C₂³	1	1	yes	characteristic
43	C₂²×C₄×C₁₃	C₂³	1	2	yes	—
44	C₁₃×C₄⋊C₄	—	2	4	no	—
45	C₁₃×C₄⋊C₄	—	2	4	no	—
46	C₁₃×C₄⋊C₄	—	2	2	no	—
47	C₁₃×C₄⋊C₄	—	2	4	no	—
48	C₁₃×C₄⋊C₄	—	4	2	no	—
49	C₁₃×C₄⋊C₄	—	4	4	no	—
50	C₁₃×C₂²⋊C₄	—	2	4	no	—
51	C₁₃×C₂²⋊C₄	—	2	4	no	—
52	#2627	C₁₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
53	C₂³×C₁₃	—	2	2	no	—
54	C₂³×C₁₃	—	2	2	no	—
55	C₂³×C₁₃	C₂×D₄	1	2	yes	—
56	C₂³×C₁₃	C₂×Q₈	1	2	yes	—
57	C₂³×C₁₃	#41	1	2	yes	—
58	C₂³×C₁₃	C₂⁴	1	1	yes	characteristic
59	Q₈×C₁₃	—	4	4	no	—
60	C₂×C₄×C₁₃	—	2	8	no	—
61	C₂×C₄×C₁₃	—	2	2	no	—
62	C₂×C₄×C₁₃	—	2	2	no	—
63	C₂×C₄×C₁₃	—	2	2	no	—
64	C₂×C₄×C₁₃	—	2	2	no	—
65	C₂×C₄×C₁₃	—	2	2	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	—	2	4	no	—
71	C₂×C₄×C₁₃	—	2	4	no	—
72	C₂×C₄×C₁₃	—	2	4	no	—
73	C₂×C₄×C₁₃	—	2	4	no	—
74	C₂×C₄×C₁₃	—	4	4	no	—
75	C₂×C₄×C₁₃	—	4	4	no	—
76	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
77	C₂×C₄×C₁₃	C₂×D₄	1	2	yes	—
78	C₂×C₄×C₁₃	C₂×Q₈	1	4	yes	—
79	C₂×C₂²⋊Q₈	C₂×C₁₃	1	1	yes	characteristic
80	C₂²×C₄⋊C₄	C₂×C₁₃	1	1	yes	characteristic
81	#400	C₂×C₁₃	1	2	yes	complemented
82	#398	C₂×C₁₃	1	4	yes	complemented
83	#395	C₂×C₁₃	1	2	yes	complemented
84	#385	C₂×C₁₃	1	2	yes	—
85	#381	C₂×C₁₃	1	2	yes	—
86	#380	C₂×C₁₃	1	1	yes	characteristic
87	C₂²×C₁₃	—	2	2	no	—
88	C₂²×C₁₃	—	2	2	no	—
89	C₂²×C₁₃	—	2	4	no	—
90	C₂²×C₁₃	—	2	4	no	—
91	C₂²×C₁₃	C₂²⋊Q₈	1	4	yes	—
92	C₂²×C₁₃	C₂²×D₄	1	1	yes	characteristic
93	C₂²×C₁₃	C₂²×Q₈	1	1	yes	characteristic
94	C₂²×C₁₃	C₂×#41	1	1	yes	characteristic
95	C₂²×C₁₃	H(5,𝔽₂)	1	2	yes	—
96	C₂²×C₁₃	2₋¹⁺⁴	1	2	yes	—
97	C₄×C₁₃	—	2	4	no	—
98	C₄×C₁₃	—	2	2	no	—
99	C₄×C₁₃	—	2	2	no	—
100	C₄×C₁₃	—	4	2	no	—
101	C₄×C₁₃	—	4	2	no	—
102	C₄×C₁₃	—	4	2	no	—
103	C₄×C₁₃	—	4	4	no	—
104	C₂²×Q₈	C₂²×C₁₃	1	1	yes	characteristic
105	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
106	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
107	C₂³×C₄	C₂²×C₁₃	1	1	yes	characteristic
108	C₂²⋊Q₈	—	2	4	no	—
109	C₂×C₄⋊C₄	—	2	4	no	—
110	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
111	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
112	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
113	C₂×C₄⋊C₄	C₂²×C₁₃	1	1	yes	characteristic
114	C₂×C₄⋊C₄	C₂²×C₁₃	1	2	yes	—
115	C₂×C₄⋊C₄	C₂²×C₁₃	1	4	yes	—
116	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
117	C₂×C₂²⋊C₄	C₂²×C₁₃	1	2	yes	—
118	#95	C₂²×C₁₃	1	2	yes	—
119	#95	C₂²×C₁₃	1	4	yes	—
120	#95	C₂²×C₁₃	1	4	yes	—
121	#95	C₂²×C₁₃	1	4	yes	—
122	C₂×C₁₃	—	2	4	no	—
123	C₂×C₁₃	C₂×C₂²⋊Q₈	1	1	yes	characteristic
124	C₂×C₁₃	#534	1	1	yes	characteristic
125	C₂×C₁₃	#536	1	1	yes	characteristic
126	C₂×C₁₃	#539	1	2	yes	—
127	C₂×C₁₃	#544	1	2	yes	—
128	C₂⁴	C₂³×C₁₃	1	1	yes	characteristic
129	C₂×Q₈	—	2	2	no	—
130	C₂×Q₈	—	4	2	no	—
131	C₂²×C₄	—	2	2	no	—
132	C₂²×C₄	—	2	2	no	—
133	C₂²×C₄	—	2	2	no	—
134	C₂²×C₄	—	4	2	no	—
135	C₂²×C₄	—	4	2	no	—
136	C₂²×C₄	D₄×C₁₃	1	4	yes	—
137	C₂²×C₄	Q₈×C₁₃	1	4	yes	complemented
138	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
139	C₂²×C₄	C₂³×C₁₃	1	4	yes	—
140	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
141	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
142	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
143	C₂²×C₄	C₂³×C₁₃	1	1	yes	characteristic
144	C₂²×C₄	C₂³×C₁₃	1	2	yes	—
145	C₄⋊C₄	—	2	4	no	—
146	C₄⋊C₄	—	2	4	no	—
147	C₄⋊C₄	—	2	2	no	—
148	C₄⋊C₄	—	2	4	no	—
149	C₄⋊C₄	—	4	2	no	—
150	C₄⋊C₄	—	4	4	no	—
151	C₂²⋊C₄	—	2	4	no	—
152	C₂²⋊C₄	—	2	4	no	—
153	C₁₃	#2627	1	1	yes	characteristic, direct product factor, Sylow
154	C₂³	—	2	2	no	—
155	C₂³	—	2	2	no	—
156	C₂³	C₂×D₄×C₁₃	1	2	yes	—
157	C₂³	C₂×Q₈×C₁₃	1	2	yes	—
158	C₂³	C₁₃×#41	1	2	yes	—
159	C₂³	C₂⁴×C₁₃	1	1	yes	characteristic
160	Q₈	—	4	4	no	—
161	C₂×C₄	—	2	8	no	—
162	C₂×C₄	—	2	2	no	—
163	C₂×C₄	—	2	2	no	—
164	C₂×C₄	—	2	2	no	—
165	C₂×C₄	—	2	2	no	—
166	C₂×C₄	—	2	2	no	—
167	C₂×C₄	—	2	2	no	—
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	2	4	no	—
170	C₂×C₄	—	2	4	no	—
171	C₂×C₄	—	2	4	no	—
172	C₂×C₄	—	2	4	no	—
173	C₂×C₄	—	2	4	no	—
174	C₂×C₄	—	2	4	no	—
175	C₂×C₄	—	4	4	no	—
176	C₂×C₄	—	4	4	no	—
177	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
178	C₂×C₄	C₂×D₄×C₁₃	1	2	yes	—
179	C₂×C₄	C₂×Q₈×C₁₃	1	4	yes	—
180	C₂²	—	2	2	no	—
181	C₂²	—	2	2	no	—
182	C₂²	—	2	4	no	—
183	C₂²	—	2	4	no	—
184	C₂²	C₁₃×C₂²⋊Q₈	1	4	yes	—
185	C₂²	C₂²×D₄×C₁₃	1	1	yes	characteristic
186	C₂²	C₂²×Q₈×C₁₃	1	1	yes	characteristic
187	C₂²	C₂×C₁₃×#41	1	1	yes	characteristic
188	C₂²	C₁₃×H(5,𝔽₂)	1	2	yes	—
189	C₂²	C₁₃×2₋¹⁺⁴	1	2	yes	—
190	C₄	—	2	4	no	—
191	C₄	—	2	2	no	—
192	C₄	—	2	2	no	—
193	C₄	—	4	2	no	—
194	C₄	—	4	2	no	—
195	C₄	—	4	2	no	—
196	C₄	—	4	4	no	—
197	C₂	—	2	4	no	—
198	C₂	C₂×C₁₃×C₂²⋊Q₈	1	1	yes	characteristic
199	C₂	C₁₃×#534	1	1	yes	characteristic
200	C₂	C₁₃×#536	1	1	yes	characteristic
201	C₂	C₁₃×#539	1	2	yes	—
202	C₂	C₁₃×#544	1	2	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
3. C₁₃×#400	C₂	semidirect
4. C₁₃×#398	C₂	semidirect
5. C₁₃×#395	C₂	semidirect
36. C₂²×C₄×C₁₃	Q₈	semidirect
52. #2627	C₁₃	direct (N × H)
81. #400	C₂×C₁₃	semidirect
82. #398	C₂×C₁₃	semidirect
83. #395	C₂×C₁₃	semidirect
137. C₂²×C₄	Q₈×C₁₃	semidirect
153. C₁₃	#2627	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×C₂²⋊Q₈	2
C₂²×C₁₃×C₄⋊C₄	2
C₁₃×#400	2
C₁₃×#398	2
C₁₃×#395	2
C₁₃×#385	2
C₁₃×#381	2
C₁₃×#380	2
#2627	13

Subgroups — C13×#2627 #13111685

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₁₃×#2627 #13111685