Subgroups — #29567713

Order 15616.

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×#14519683	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839588	C₂	1	1	yes	maximal, characteristic
3	C₂×C₆₁×#366	C₂	1	1	yes	maximal, characteristic
4	C₂×#14519711	—	2	1	no	—
5	C₂×#14519683	—	2	1	no	—
6	C₂×#14519683	—	2	1	no	—
7	C₂×#14519683	C₄	1	2	yes	complemented
8	#16839588	—	2	1	no	—
9	#16839588	C₄	1	2	yes	complemented
10	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
11	C₂×#107×C₆₁	—	2	1	no	—
12	C₆₁×#366	—	2	1	no	—
13	C₆₁×#366	C₄	1	2	yes	complemented
14	C₂×C₄⋊Dic₆₁	—	4	1	no	—
15	#14519711	—	2	2	no	—
16	#14519711	—	4	1	no	—
17	#14519683	—	2	2	no	—
18	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
19	C₂×C₄⋊C₄×C₆₁	—	4	1	no	—
20	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
21	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
22	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
23	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
24	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₆₁	Q₈	1	1	yes	characteristic
26	#107×C₆₁	—	2	2	no	—
27	#107×C₆₁	—	4	1	no	—
28	C₂²×C₄×C₆₁	—	8	1	no	—
29	C₂²×C₄×C₆₁	D₈	1	1	yes	characteristic
30	C₂²×Dic₆₁	—	8	1	no	—
31	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
32	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
33	C₂×C₈×C₆₁	C₄⋊C₄	1	1	yes	characteristic
34	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
35	C₂×C₈×C₆₁	SD₁₆	1	1	yes	characteristic
36	C₂×C₈×C₆₁	Q₁₆	1	1	yes	characteristic
37	C₄⋊C₄×C₆₁	—	4	2	no	—
38	C₄⋊C₄×C₆₁	—	8	1	no	—
39	C₄⋊Dic₆₁	—	4	2	no	—
40	C₄⋊Dic₆₁	—	8	1	no	—
41	C₁₆×C₆₁	C₄²	1	1	yes	characteristic
42	C₁₆×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
43	C₁₆×C₆₁	C₄⋊C₄	1	1	yes	characteristic
44	C₁₆×C₆₁	C₄⋊C₄	1	1	yes	characteristic
45	C₂³×C₆₁	D₁₆	1	1	yes	characteristic
46	C₂×C₄×C₆₁	—	8	2	no	—
47	C₂×C₄×C₆₁	—	16	1	no	—
48	C₂×C₄×C₆₁	—	16	1	no	—
49	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
50	C₂×C₄×C₆₁	#102	1	1	yes	characteristic
51	C₂×C₄×C₆₁	#107	1	1	yes	characteristic
52	C₂×C₄×C₆₁	SD₃₂	1	1	yes	characteristic
53	C₂×C₄×C₆₁	SD₃₂	1	1	yes	characteristic
54	C₂×C₄×C₆₁	Q₃₂	1	1	yes	characteristic
55	C₂×Dic₆₁	—	8	2	no	—
56	C₂×Dic₆₁	—	16	1	no	—
57	C₂×Dic₆₁	—	16	1	no	—
58	C₈×C₆₁	#95	1	1	yes	characteristic
59	C₈×C₆₁	#103	1	1	yes	characteristic
60	C₈×C₆₁	#103	1	1	yes	characteristic
61	C₈×C₆₁	#106	1	1	yes	characteristic
62	#7471	—	61	1	no	maximal, Sylow
63	C₂²×C₆₁	#357	1	1	yes	characteristic
64	C₂²×C₆₁	#357	1	1	yes	characteristic
65	C₂²×C₆₁	#366	1	1	yes	characteristic
66	C₂²×C₆₁	D₃₂	1	1	yes	characteristic
67	C₂²×C₆₁	SD₆₄	1	1	yes	characteristic
68	C₂²×C₆₁	SD₆₄	1	1	yes	characteristic
69	C₂²×C₆₁	Q₆₄	1	1	yes	characteristic
70	C₄×C₆₁	—	16	2	no	—
71	C₄×C₆₁	#340	1	1	yes	characteristic
72	C₄×C₆₁	#358	1	1	yes	characteristic
73	C₄×C₆₁	#358	1	1	yes	characteristic
74	C₄×C₆₁	#367	1	1	yes	characteristic
75	Dic₆₁	—	16	2	no	—
76	C₂²×C₃₂	—	61	1	no	—
77	C₂×#366	—	61	1	no	—
78	C₂×#366	D₆₁	1	1	yes	characteristic
79	C₂×C₆₁	#1380	1	1	yes	characteristic
80	C₂×C₆₁	#1415	1	1	yes	characteristic
81	C₂×C₆₁	#1415	1	1	yes	characteristic
82	C₂×C₆₁	#1416	1	1	yes	characteristic
83	C₂×C₆₁	#1416	1	1	yes	characteristic
84	C₂×C₆₁	#1423	1	1	yes	characteristic
85	C₂×C₆₁	#1424	1	1	yes	characteristic
86	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
87	C₂×#107	—	2	1	no	—
88	C₂×#107	—	122	1	no	—
89	C₂×C₃₂	—	61	2	no	—
90	C₂×C₃₂	—	122	1	no	—
91	C₂×C₃₂	—	122	1	no	—
92	#366	—	2	1	no	—
93	#366	—	61	2	no	—
94	#366	—	122	1	no	—
95	#366	Dic₆₁	1	2	yes	complemented
96	C₆₁	#7471	1	1	yes	characteristic, complemented, Sylow
97	C₂²×C₈	C₂²⋊D₆₁	1	1	yes	characteristic
98	C₂×C₄⋊C₄	—	4	1	no	—
99	C₂×C₄⋊C₄	—	244	1	no	—
100	C₂×C₁₆	Dic₁₂₂	1	1	yes	characteristic
101	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
102	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
103	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
104	C₂×C₁₆	C₂×Dic₆₁	1	1	yes	characteristic
105	C₂×C₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
106	#107	—	2	2	no	—
107	#107	—	4	1	no	—
108	#107	—	122	2	no	—
109	#107	—	244	1	no	—
110	C₃₂	—	122	2	no	—
111	C₂²×C₄	—	8	1	no	—
112	C₂²×C₄	—	488	1	no	—
113	C₂²×C₄	#11758199	1	1	yes	characteristic
114	C₂×C₈	#11758196	1	1	yes	characteristic
115	C₂×C₈	#11758198	1	1	yes	characteristic
116	C₂×C₈	#11758200	1	1	yes	characteristic
117	C₂×C₈	#11758201	1	1	yes	characteristic
118	C₂×C₈	#11758202	1	1	yes	characteristic
119	C₂×C₈	C₂²⋊Dic₆₁	1	1	yes	characteristic
120	C₄⋊C₄	—	4	2	no	—
121	C₄⋊C₄	—	8	1	no	—
122	C₄⋊C₄	—	488	1	no	—
123	C₄⋊C₄	—	244	2	no	—
124	C₁₆	C₄×Dic₆₁	1	1	yes	characteristic
125	C₁₆	#11758196	1	1	yes	characteristic
126	C₁₆	C₄⋊Dic₆₁	1	1	yes	characteristic
127	C₁₆	#11758198	1	1	yes	characteristic
128	C₂³	#14519718	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	—	976	1	no	—
133	C₂×C₄	—	976	1	no	—
134	C₂×C₄	—	488	2	no	—
135	C₂×C₄	#14519697	1	1	yes	characteristic
136	C₂×C₄	#14519698	1	1	yes	characteristic
137	C₂×C₄	#14519710	1	1	yes	characteristic
138	C₂×C₄	#14519720	1	1	yes	characteristic
139	C₂×C₄	#14519721	1	1	yes	characteristic
140	C₂×C₄	#14519723	1	1	yes	characteristic
141	C₈	#14519684	1	1	yes	characteristic
142	C₈	#14519700	1	1	yes	characteristic
143	C₈	#14519701	1	1	yes	characteristic
144	C₈	#14519708	1	1	yes	characteristic
145	C₂²	#16839562	1	1	yes	characteristic
146	C₂²	#16839563	1	1	yes	characteristic
147	C₂²	#16839587	1	1	yes	characteristic
148	C₂²	#16839597	1	1	yes	characteristic
149	C₂²	#16839599	1	1	yes	characteristic
150	C₂²	#16839600	1	1	yes	characteristic
151	C₂²	#16839602	1	1	yes	characteristic
152	C₄	—	16	2	no	—
153	C₄	—	976	2	no	—
154	C₄	#16839524	1	1	yes	characteristic
155	C₄	#16839565	1	1	yes	characteristic
156	C₄	#16839566	1	1	yes	characteristic
157	C₄	#16839589	1	1	yes	characteristic
158	C₂	#21443910	1	1	yes	characteristic
159	C₂	#21443996	1	1	yes	characteristic
160	C₂	#21443997	1	1	yes	characteristic
161	C₂	#21443999	1	1	yes	characteristic
162	C₂	#21444000	1	1	yes	characteristic
163	C₂	#21444018	1	1	yes	characteristic
164	C₂	#21444020	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519718
commutator	C₁₆×C₆₁	C₄²
Frattini	C₂²×C₁₆	C₂×D₆₁
Fitting	C₂×C₆₁×#366	C₂
socle, prime radical	C₂³×C₆₁	D₁₆
cosocle	C₂²×C₁₆×C₆₁	C₂²
hypercenter	C₂×#366	D₆₁
hypocenter	C₆₁	#7471
LCS	G ▷ C₁₆×C₆₁ ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
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
7. C₂×#14519683	C₄	semidirect
9. #16839588	C₄	semidirect
13. C₆₁×#366	C₄	semidirect
95. #366	Dic₆₁	semidirect
96. C₆₁	#7471	semidirect

Maximal subgroups

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

subgroup	index
C₂²×#14519683	2
C₂×#16839588	2
C₂×C₆₁×#366	2
#7471	61