Subgroups — #13131406

Order 1664.

556 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#41×#261	C₂	1	1	yes	maximal, characteristic
2	D₁₃⋊#131	C₂	1	1	yes	maximal, characteristic
3	D₁₃×#131	C₂	1	1	yes	maximal, characteristic
4	#11719099	C₂	1	3	yes	maximal, complemented
5	M₁₆×#261	C₂	1	3	yes	maximal, complemented
6	#11719089	C₂	1	3	yes	maximal, complemented
7	C₂×#88100	C₂	1	3	yes	maximal, complemented
8	#41×D₁₃	C₂²	1	1	yes	characteristic
9	Q₈×#261	C₄	1	2	yes	—
10	#88240	C₄	1	2	yes	complemented
11	D₄×#261	C₄	1	6	yes	—
12	#88238	C₄	1	6	yes	complemented
13	#88237	C₂²	1	3	yes	—
14	C₂×C₄×#261	C₂²	1	3	yes	—
15	D₁₃⋊M₁₆	C₂²	1	3	yes	—
16	C₂×D₁₃⋊C₈	C₂²	1	3	yes	—
17	C₁₃×#131	C₄	1	1	yes	characteristic, complemented
18	#88202	C₄	1	1	yes	characteristic, complemented
19	#88161	C₄	1	3	yes	complemented
20	M₁₆×D₁₃	C₂²	1	3	yes	—
21	#88155	C₄	1	3	yes	complemented
22	C₂×C₈×D₁₃	C₂²	1	3	yes	—
23	#88100	C₂²	1	1	yes	characteristic
24	#88100	C₂²	1	3	yes	—
25	#88100	C₂²	1	3	yes	—
26	#88100	C₂²	1	3	yes	—
27	C₈×#261	C₂²	1	6	yes	—
28	C₂²×#261	C₂×C₄	1	6	yes	—
29	C₁₃×#41	C₂×C₄	1	1	yes	characteristic
30	#6131	C₂×C₄	1	1	yes	characteristic
31	Q₈×D₁₃	C₂×C₄	1	1	yes	characteristic
32	#6129	C₂×C₄	1	3	yes	—
33	D₄×D₁₃	C₂×C₄	1	3	yes	—
34	#6127	C₂×C₄	1	3	yes	—
35	C₂×C₄×D₁₃	C₂³	1	3	yes	—
36	C₂²⋊#261	C₂×C₄	1	6	yes	—
37	#6122	C₂×C₄	1	6	yes	complemented
38	C₂×#1057	C₂×C₄	1	6	yes	complemented
39	C₄⋊#261	C₂×C₄	1	6	yes	—
40	C₄×#261	C₂×C₄	1	6	yes	—
41	C₄×#261	C₂³	1	3	yes	—
42	C₄×#261	C₂³	1	1	yes	characteristic
43	#6118	C₂×C₄	1	6	yes	complemented
44	D₁₃⋊C₈	C₂×C₄	1	6	yes	complemented
45	D₁₃⋊C₈	C₂³	1	3	yes	—
46	D₁₃⋊C₈	C₂³	1	1	yes	characteristic
47	C₁₃×M₁₆	C₂×C₄	1	3	yes	complemented
48	C₂×C₈×C₁₃	C₂×C₄	1	3	yes	complemented
49	#6099	C₂×C₄	1	3	yes	complemented
50	C₂×#1055	C₂×C₄	1	3	yes	complemented
51	#6094	C₂×C₄	1	6	yes	complemented
52	C₈×D₁₃	C₂×C₄	1	3	yes	complemented
53	C₈×D₁₃	C₂×C₄	1	3	yes	complemented
54	C₈×D₁₃	C₂³	1	3	yes	—
55	C₈×D₁₃	C₂³	1	1	yes	characteristic
56	#2875	—	13	1	no	maximal, Sylow
57	C₂²×D₁₃	C₂²×C₄	1	3	yes	—
58	C₂×#261	—	4	6	no	—
59	C₂×#261	C₂²×C₄	1	2	yes	—
60	C₂×#261	C₂²×C₄	1	6	yes	—
61	Q₈×C₁₃	C₄²	1	1	yes	characteristic
62	D₄×C₁₃	C₄²	1	3	yes	—
63	C₂×C₄×C₁₃	C₂²×C₄	1	3	yes	—
64	C₂²⋊D₁₃	C₄²	1	6	yes	—
65	C₂×Dic₁₃	C₂²×C₄	1	3	yes	—
66	D₅₂	C₄²	1	3	yes	—
67	C₄×D₁₃	C₂²×C₄	1	3	yes	—
68	C₄×D₁₃	C₂²×C₄	1	3	yes	—
69	C₄×D₁₃	C₂²×C₄	1	3	yes	—
70	C₄×D₁₃	C₂⁴	1	1	yes	characteristic
71	Dic₂₆	C₄²	1	3	yes	—
72	#1057	C₂²×C₄	1	2	yes	—
73	#1057	C₂²×C₄	1	6	yes	—
74	C₈×C₁₃	C₂²×C₄	1	1	yes	characteristic
75	C₈×C₁₃	C₂²×C₄	1	3	yes	—
76	#1055	C₂²×C₄	1	1	yes	characteristic
77	#1055	C₂²×C₄	1	3	yes	—
78	C₂×#131	—	13	1	no	—
79	C₂×#131	—	13	1	no	—
80	C₄×#41	—	13	1	no	—
81	#405	—	13	3	no	—
82	#405	—	13	3	no	—
83	C₄×M₁₆	—	13	3	no	—
84	C₂×#97	—	13	3	no	—
85	C₂²×C₁₃	C₂×C₄²	1	3	yes	—
86	C₂×D₁₃	—	2	6	no	—
87	C₂×D₁₃	C₂×C₄²	1	3	yes	—
88	C₂×D₁₃	C₂³×C₄	1	1	yes	characteristic
89	#261	—	2	2	no	—
90	#261	—	2	6	no	—
91	C₄×C₁₃	C₂×C₄²	1	3	yes	—
92	C₄×C₁₃	C₂³×C₄	1	1	yes	characteristic
93	Dic₁₃	C₂×C₄²	1	3	yes	—
94	Dic₁₃	C₂³×C₄	1	1	yes	characteristic
95	C₂×#41	—	13	1	no	—
96	#131	—	13	6	no	—
97	#131	—	13	2	no	—
98	#131	—	13	1	no	—
99	#131	—	13	3	no	—
100	#131	—	13	3	no	—
101	#131	#261	1	1	yes	characteristic, complemented
102	C₂×M₁₆	—	13	3	no	—
103	C₂×M₁₆	—	13	3	no	—
104	C₂²×C₈	—	13	3	no	—
105	C₂²×C₈	—	13	3	no	—
106	C₄×Q₈	—	13	2	no	—
107	C₄×D₄	—	13	6	no	—
108	#117	—	13	3	no	—
109	C₂×C₄²	—	13	3	no	—
110	#97	—	13	1	no	—
111	#97	—	13	3	no	—
112	#97	—	13	3	no	—
113	#97	—	13	3	no	—
114	C₄×C₈	—	13	6	no	—
115	C₂×C₁₃	—	2	3	no	—
116	C₂×C₁₃	C₂²×C₄²	1	1	yes	characteristic
117	D₁₃	—	2	3	no	—
118	D₁₃	#568	1	2	yes	—
119	#41	—	13	1	no	—
120	#41	—	13	3	no	—
121	#41	—	13	3	no	—
122	#41	C₂×#261	1	1	yes	characteristic
123	C₂×Q₈	—	13	1	no	—
124	C₂×D₄	—	13	3	no	—
125	C₂²×C₄	—	13	3	no	—
126	C₂²×C₄	—	13	6	no	—
127	M₁₆	—	13	6	no	—
128	M₁₆	—	13	6	no	—
129	M₁₆	—	13	3	no	—
130	M₁₆	—	13	6	no	—
131	M₁₆	C₂×#261	1	3	yes	complemented
132	C₂×C₈	—	13	3	no	—
133	C₂×C₈	—	13	1	no	—
134	C₂×C₈	—	13	6	no	—
135	C₂×C₈	—	13	6	no	—
136	C₂×C₈	—	13	3	no	—
137	C₂×C₈	—	13	3	no	—
138	C₂×C₈	—	13	1	no	—
139	C₂×C₈	—	13	3	no	—
140	C₂×C₈	—	13	3	no	—
141	C₂×C₈	C₂×#261	1	3	yes	complemented
142	C₄⋊C₄	—	13	6	no	—
143	C₂²⋊C₄	—	13	6	no	—
144	C₄²	—	13	3	no	—
145	C₄²	—	13	1	no	—
146	C₄²	—	13	6	no	—
147	C₁₃	#2875	1	1	yes	characteristic, complemented, Sylow
148	C₂³	—	13	3	no	—
149	Q₈	—	13	3	no	—
150	Q₈	C₄×#261	1	1	yes	characteristic
151	D₄	—	13	6	no	—
152	D₄	—	13	3	no	—
153	D₄	C₄×#261	1	3	yes	—
154	C₂×C₄	—	13	3	no	—
155	C₂×C₄	—	13	3	no	—
156	C₂×C₄	—	13	3	no	—
157	C₂×C₄	—	13	1	no	—
158	C₂×C₄	—	13	3	no	—
159	C₂×C₄	—	13	2	no	—
160	C₂×C₄	—	13	6	no	—
161	C₂×C₄	—	52	6	no	—
162	C₂×C₄	C₂²×#261	1	3	yes	—
163	C₈	—	13	2	no	—
164	C₈	—	13	6	no	—
165	C₈	—	13	1	no	—
166	C₈	—	13	3	no	—
167	C₈	C₂²×#261	1	1	yes	characteristic
168	C₈	C₂²×#261	1	3	yes	—
169	C₂²	—	13	1	no	—
170	C₂²	—	13	3	no	—
171	C₂²	—	26	6	no	—
172	C₂²	C₂×C₄×#261	1	3	yes	—
173	C₄	—	13	3	no	—
174	C₄	—	13	1	no	—
175	C₄	—	26	6	no	—
176	C₄	—	26	2	no	—
177	C₄	C₂×C₄×#261	1	3	yes	—
178	C₄	C₂³×#261	1	1	yes	characteristic
179	C₂	—	2	3	no	—
180	C₂	—	13	2	no	—
181	C₂	—	26	3	no	—
182	C₂	C₂²×C₄×#261	1	1	yes	characteristic

	subgroup	G/H
center, Frattini	C₄	C₂³×#261
commutator, socle	C₂×C₁₃	C₂²×C₄²
Fitting	C₁₃×#131	C₄
prime radical	#41×D₁₃	C₂²
cosocle	C₄×D₁₃	C₂⁴
hypercenter	#131	#261
hypocenter	C₁₃	#2875
LCS	G ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₄ ◁ #131

Complements

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

N	H ≅ G/N	type
4. #11719099	C₂	semidirect
5. M₁₆×#261	C₂	semidirect
6. #11719089	C₂	semidirect
7. C₂×#88100	C₂	semidirect
10. #88240	C₄	semidirect
12. #88238	C₄	semidirect
17. C₁₃×#131	C₄	semidirect
18. #88202	C₄	semidirect
19. #88161	C₄	semidirect
21. #88155	C₄	semidirect
37. #6122	C₂×C₄	semidirect
38. C₂×#1057	C₂×C₄	semidirect
43. #6118	C₂×C₄	semidirect
44. D₁₃⋊C₈	C₂×C₄	semidirect
47. C₁₃×M₁₆	C₂×C₄	semidirect
48. C₂×C₈×C₁₃	C₂×C₄	semidirect
49. #6099	C₂×C₄	semidirect
50. C₂×#1055	C₂×C₄	semidirect
51. #6094	C₂×C₄	semidirect
52. C₈×D₁₃	C₂×C₄	semidirect
53. C₈×D₁₃	C₂×C₄	semidirect
101. #131	#261	semidirect
131. M₁₆	C₂×#261	semidirect
141. C₂×C₈	C₂×#261	semidirect
147. C₁₃	#2875	semidirect

Maximal subgroups

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

subgroup	index
#41×#261	2
D₁₃⋊#131	2
D₁₃×#131	2
#11719099	2
M₁₆×#261	2
#11719089	2
C₂×#88100	2
#2875	13