Subgroups — #15750619

Order 3328.

190 conjugacy classes of subgroups.

The subgroup lattice has 166 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#88034	C₂	1	1	yes	maximal, characteristic
2	C₂×#11718096	C₂	1	1	yes	maximal, characteristic
3	C₂×C₁₃×#366	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₁₃×C₁₆	C₂²	1	1	yes	characteristic
5	C₂×C₁₃×#107	—	2	1	no	—
6	C₂×#88058	—	2	1	no	—
7	C₁₃×#366	—	2	1	no	—
8	C₁₃×#366	C₄	1	2	yes	complemented
9	#11718096	—	2	1	no	—
10	#11718096	C₄	1	2	yes	complemented
11	C₂×#88034	—	2	1	no	—
12	C₂×#88034	—	2	1	no	—
13	C₂×#88034	C₄	1	2	yes	complemented
14	C₂²×C₈×C₁₃	D₄	1	1	yes	characteristic
15	C₂×C₁₃×C₄⋊C₄	—	4	1	no	—
16	C₂×C₄⋊Dic₁₃	—	4	1	no	—
17	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
18	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
19	C₂×C₁₃×C₁₆	C₂×C₄	1	1	yes	characteristic
20	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
21	C₂×C₁₃×C₁₆	D₄	1	1	yes	characteristic
22	C₂×C₁₃×C₁₆	Q₈	1	1	yes	characteristic
23	C₁₃×#107	—	2	2	no	—
24	C₁₃×#107	—	4	1	no	—
25	#88058	—	2	2	no	—
26	#88058	—	4	1	no	—
27	#88034	—	2	2	no	—
28	#7471	—	13	1	no	maximal, Sylow
29	C₂²×C₄×C₁₃	—	8	1	no	—
30	C₂²×C₄×C₁₃	D₈	1	1	yes	characteristic
31	C₂²×Dic₁₃	—	8	1	no	—
32	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₁₃	C₂²⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₁₃	C₄⋊C₄	1	1	yes	characteristic
35	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
36	C₂×C₈×C₁₃	SD₁₆	1	1	yes	characteristic
37	C₂×C₈×C₁₃	Q₁₆	1	1	yes	characteristic
38	C₁₃×C₄⋊C₄	—	4	2	no	—
39	C₁₃×C₄⋊C₄	—	8	1	no	—
40	C₄⋊Dic₁₃	—	4	2	no	—
41	C₄⋊Dic₁₃	—	8	1	no	—
42	C₁₃×C₁₆	C₄²	1	1	yes	characteristic
43	C₁₃×C₁₆	C₂²⋊C₄	1	1	yes	characteristic
44	C₁₃×C₁₆	C₄⋊C₄	1	1	yes	characteristic
45	C₁₃×C₁₆	C₄⋊C₄	1	1	yes	characteristic
46	C₂²×C₃₂	—	13	1	no	—
47	C₂×#366	—	13	1	no	—
48	C₂×#366	D₁₃	1	1	yes	characteristic
49	C₂³×C₁₃	D₁₆	1	1	yes	characteristic
50	C₂×C₄×C₁₃	—	8	2	no	—
51	C₂×C₄×C₁₃	—	16	1	no	—
52	C₂×C₄×C₁₃	—	16	1	no	—
53	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
54	C₂×C₄×C₁₃	#102	1	1	yes	characteristic
55	C₂×C₄×C₁₃	#107	1	1	yes	characteristic
56	C₂×C₄×C₁₃	SD₃₂	1	1	yes	characteristic
57	C₂×C₄×C₁₃	SD₃₂	1	1	yes	characteristic
58	C₂×C₄×C₁₃	Q₃₂	1	1	yes	characteristic
59	C₂×Dic₁₃	—	8	2	no	—
60	C₂×Dic₁₃	—	16	1	no	—
61	C₂×Dic₁₃	—	16	1	no	—
62	C₈×C₁₃	#95	1	1	yes	characteristic
63	C₈×C₁₃	#103	1	1	yes	characteristic
64	C₈×C₁₃	#103	1	1	yes	characteristic
65	C₈×C₁₃	#106	1	1	yes	characteristic
66	C₂²×C₁₆	C₂×D₁₃	1	1	yes	characteristic
67	C₂×#107	—	2	1	no	—
68	C₂×#107	—	26	1	no	—
69	C₂×C₃₂	—	13	2	no	—
70	C₂×C₃₂	—	26	1	no	—
71	C₂×C₃₂	—	26	1	no	—
72	#366	—	2	1	no	—
73	#366	—	13	2	no	—
74	#366	—	26	1	no	—
75	#366	Dic₁₃	1	2	yes	complemented
76	C₂²×C₁₃	#357	1	1	yes	characteristic
77	C₂²×C₁₃	#357	1	1	yes	characteristic
78	C₂²×C₁₃	#366	1	1	yes	characteristic
79	C₂²×C₁₃	D₃₂	1	1	yes	characteristic
80	C₂²×C₁₃	SD₆₄	1	1	yes	characteristic
81	C₂²×C₁₃	SD₆₄	1	1	yes	characteristic
82	C₂²×C₁₃	Q₆₄	1	1	yes	characteristic
83	C₄×C₁₃	—	16	2	no	—
84	C₄×C₁₃	#340	1	1	yes	characteristic
85	C₄×C₁₃	#358	1	1	yes	characteristic
86	C₄×C₁₃	#358	1	1	yes	characteristic
87	C₄×C₁₃	#367	1	1	yes	characteristic
88	Dic₁₃	—	16	2	no	—
89	C₂²×C₈	C₂²⋊D₁₃	1	1	yes	characteristic
90	C₂×C₄⋊C₄	—	4	1	no	—
91	C₂×C₄⋊C₄	—	52	1	no	—
92	C₂×C₁₆	Dic₂₆	1	1	yes	characteristic
93	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
94	C₂×C₁₆	C₄×D₁₃	1	1	yes	characteristic
95	C₂×C₁₆	D₅₂	1	1	yes	characteristic
96	C₂×C₁₆	C₂×Dic₁₃	1	1	yes	characteristic
97	C₂×C₁₆	C₂²⋊D₁₃	1	1	yes	characteristic
98	#107	—	2	2	no	—
99	#107	—	4	1	no	—
100	#107	—	26	2	no	—
101	#107	—	52	1	no	—
102	C₃₂	—	26	2	no	—
103	C₂×C₁₃	#1380	1	1	yes	characteristic
104	C₂×C₁₃	#1415	1	1	yes	characteristic
105	C₂×C₁₃	#1415	1	1	yes	characteristic
106	C₂×C₁₃	#1416	1	1	yes	characteristic
107	C₂×C₁₃	#1416	1	1	yes	characteristic
108	C₂×C₁₃	#1423	1	1	yes	characteristic
109	C₂×C₁₃	#1424	1	1	yes	characteristic
110	C₂²×C₄	—	8	1	no	—
111	C₂²×C₄	—	104	1	no	—
112	C₂²×C₄	#6104	1	1	yes	characteristic
113	C₂×C₈	#6101	1	1	yes	characteristic
114	C₂×C₈	#6103	1	1	yes	characteristic
115	C₂×C₈	#6105	1	1	yes	characteristic
116	C₂×C₈	#6106	1	1	yes	characteristic
117	C₂×C₈	#6107	1	1	yes	characteristic
118	C₂×C₈	C₂²⋊Dic₁₃	1	1	yes	characteristic
119	C₄⋊C₄	—	4	2	no	—
120	C₄⋊C₄	—	8	1	no	—
121	C₄⋊C₄	—	52	2	no	—
122	C₄⋊C₄	—	104	1	no	—
123	C₁₆	C₄×Dic₁₃	1	1	yes	characteristic
124	C₁₆	#6101	1	1	yes	characteristic
125	C₁₆	C₄⋊Dic₁₃	1	1	yes	characteristic
126	C₁₆	#6103	1	1	yes	characteristic
127	C₁₃	#7471	1	1	yes	characteristic, complemented, Sylow
128	C₂³	#88066	1	1	yes	characteristic
129	C₂×C₄	—	8	2	no	—
130	C₂×C₄	—	16	1	no	—
131	C₂×C₄	—	16	1	no	—
132	C₂×C₄	—	104	2	no	—
133	C₂×C₄	—	208	1	no	—
134	C₂×C₄	—	208	1	no	—
135	C₂×C₄	#88047	1	1	yes	characteristic
136	C₂×C₄	#88049	1	1	yes	characteristic
137	C₂×C₄	#88067	1	1	yes	characteristic
138	C₂×C₄	#88068	1	1	yes	characteristic
139	C₂×C₄	#88069	1	1	yes	characteristic
140	C₂×C₄	#88072	1	1	yes	characteristic
141	C₈	#88048	1	1	yes	characteristic
142	C₈	#88050	1	1	yes	characteristic
143	C₈	#88071	1	1	yes	characteristic
144	C₈	#88075	1	1	yes	characteristic
145	C₂²	#11718081	1	1	yes	characteristic
146	C₂²	#11718083	1	1	yes	characteristic
147	C₂²	#11718111	1	1	yes	characteristic
148	C₂²	#11718112	1	1	yes	characteristic
149	C₂²	#11718113	1	1	yes	characteristic
150	C₂²	#11718114	1	1	yes	characteristic
151	C₂²	#11718154	1	1	yes	characteristic
152	C₄	—	16	2	no	—
153	C₄	—	208	2	no	—
154	C₄	#11718082	1	1	yes	characteristic
155	C₄	#11718084	1	1	yes	characteristic
156	C₄	#11718121	1	1	yes	characteristic
157	C₄	#11718156	1	1	yes	characteristic
158	C₂	#13112927	1	1	yes	characteristic
159	C₂	#13113013	1	1	yes	characteristic
160	C₂	#13113014	1	1	yes	characteristic
161	C₂	#13113016	1	1	yes	characteristic
162	C₂	#13113017	1	1	yes	characteristic
163	C₂	#13113035	1	1	yes	characteristic
164	C₂	#13113037	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#88066
commutator	C₁₃×C₁₆	C₄²
Frattini	C₂²×C₁₆	C₂×D₁₃
Fitting	C₂×C₁₃×#366	C₂
socle, prime radical	C₂³×C₁₃	D₁₆
cosocle	C₂²×C₁₃×C₁₆	C₂²
hypercenter	C₂×#366	D₁₃
hypocenter	C₁₃	#7471
LCS	G ▷ C₁₃×C₁₆ ▷ C₈×C₁₃ ▷ C₄×C₁₃ ▷ C₂×C₁₃ ▷ C₁₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂²×C₁₆ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
8. C₁₃×#366	C₄	semidirect
10. #11718096	C₄	semidirect
13. C₂×#88034	C₄	semidirect
75. #366	Dic₁₃	semidirect
127. C₁₃	#7471	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#88034	2
C₂×#11718096	2
C₂×C₁₃×#366	2
#7471	13