Subgroups — #29573784

Order 15616.

454 conjugacy classes of subgroups.

The subgroup lattice has 162 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445979	C₂	1	2	yes	maximal
2	#21445976	C₂	1	2	yes	maximal
3	#21445947	C₂	1	2	yes	maximal
4	C₂×C₈²×C₆₁	C₂	1	1	yes	maximal, characteristic
5	C₂×#14519711	—	2	4	no	—
6	C₂×#14519709	—	2	4	no	—
7	#16839632	C₂²	1	4	yes	—
8	#16839525	—	2	4	no	—
9	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
10	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
11	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
12	C₈²×C₆₁	C₄	1	4	yes	complemented
13	C₂×C₄⋊Dic₆₁	—	2	4	no	—
14	C₂×C₄⋊Dic₆₁	—	2	4	no	—
15	C₂×C₄⋊Dic₆₁	—	2	4	no	—
16	#14519711	—	4	8	no	—
17	#14519709	—	4	8	no	—
18	C₂²×C₈×C₆₁	D₄	1	2	yes	—
19	C₂²×C₈×C₆₁	D₄	1	2	yes	—
20	C₂²×C₈×C₆₁	D₄	1	2	yes	—
21	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
22	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
23	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
24	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
25	C₄×C₈×C₆₁	Q₈	1	2	yes	—
26	C₄×C₈×C₆₁	Q₈	1	2	yes	—
27	C₄×C₈×C₆₁	Q₈	1	2	yes	—
28	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
29	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
30	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
31	C₂²×Dic₆₁	—	4	4	no	—
32	C₂×C₈×C₆₁	—	2	4	no	—
33	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
34	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
35	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
36	C₂×C₈×C₆₁	D₈	1	4	yes	—
37	C₂×C₈×C₆₁	SD₁₆	1	8	yes	—
38	C₂×C₈×C₆₁	Q₁₆	1	4	yes	—
39	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
40	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
41	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
42	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
43	C₄⋊Dic₆₁	—	4	8	no	—
44	C₄⋊Dic₆₁	—	4	8	no	—
45	C₄⋊Dic₆₁	—	4	8	no	—
46	C₂³×C₆₁	#127	1	1	yes	characteristic
47	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
48	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
49	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
50	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
51	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
52	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
53	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
54	C₂×C₄×C₆₁	C₂×SD₁₆	1	4	yes	—
55	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
56	C₂×C₄×C₆₁	#135	1	4	yes	—
57	C₂×Dic₆₁	—	8	4	no	—
58	C₂×Dic₆₁	—	8	4	no	—
59	C₂×Dic₆₁	—	8	4	no	—
60	C₈×C₆₁	—	2	4	no	—
61	C₈×C₆₁	#106	1	8	yes	complemented
62	C₈×C₆₁	#107	1	8	yes	complemented
63	#8814	—	61	1	no	maximal, Sylow
64	C₂²×C₆₁	#384	1	1	yes	characteristic
65	C₂²×C₆₁	#492	1	2	yes	—
66	C₂²×C₆₁	#493	1	1	yes	characteristic
67	C₂²×C₆₁	#494	1	1	yes	characteristic
68	C₂²×C₆₁	#495	1	2	yes	—
69	C₄×C₆₁	C₂×#106	1	2	yes	—
70	C₄×C₆₁	C₂×#107	1	2	yes	—
71	C₄×C₆₁	#427	1	2	yes	—
72	C₄×C₆₁	#498	1	2	yes	—
73	C₄×C₆₁	#499	1	2	yes	—
74	C₄×C₆₁	#500	1	2	yes	—
75	Dic₆₁	—	16	4	no	—
76	#1845	—	61	2	no	—
77	#1844	—	61	2	no	—
78	#1834	—	61	2	no	—
79	C₂×C₈²	D₆₁	1	1	yes	characteristic
80	C₂×C₆₁	#1706	1	2	yes	—
81	C₂×C₆₁	#1707	1	2	yes	—
82	C₂×C₆₁	#1844	1	1	yes	characteristic
83	C₂×C₆₁	#1845	1	1	yes	characteristic
84	C₂×C₆₁	#1846	1	1	yes	characteristic
85	C₂×#107	—	122	4	no	—
86	C₂×#106	—	122	4	no	—
87	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
88	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
89	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
90	#384	—	61	4	no	—
91	#340	—	122	4	no	—
92	C₈²	Dic₆₁	1	4	yes	complemented
93	C₆₁	#8814	1	1	yes	characteristic, complemented, Sylow
94	C₂²×C₈	D₂₄₄	1	2	yes	—
95	C₂²×C₈	D₂₄₄	1	2	yes	—
96	C₂²×C₈	D₂₄₄	1	2	yes	—
97	C₂×C₄⋊C₄	—	122	4	no	—
98	C₂×C₄⋊C₄	—	122	4	no	—
99	C₂×C₄⋊C₄	—	122	4	no	—
100	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
101	#107	—	244	8	no	—
102	#106	—	244	8	no	—
103	C₄×C₈	Dic₁₂₂	1	2	yes	—
104	C₄×C₈	Dic₁₂₂	1	2	yes	—
105	C₄×C₈	Dic₁₂₂	1	2	yes	—
106	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
107	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
108	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
109	C₂²×C₄	—	244	4	no	—
110	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
111	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
112	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
113	C₂×C₈	—	2	4	no	—
114	C₂×C₈	#11758190	1	8	yes	—
115	C₂×C₈	D₄₈₈	1	4	yes	—
116	C₂×C₈	Dic₂₄₄	1	4	yes	—
117	C₂×C₈	C₄⋊Dic₆₁	1	4	yes	—
118	C₂×C₈	C₄⋊Dic₆₁	1	4	yes	—
119	C₂×C₈	C₄⋊Dic₆₁	1	4	yes	—
120	C₄⋊C₄	—	244	8	no	—
121	C₄⋊C₄	—	244	8	no	—
122	C₄⋊C₄	—	244	8	no	—
123	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
124	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
125	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
126	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
127	C₂³	#14519774	1	1	yes	characteristic
128	C₂×C₄	—	488	4	no	—
129	C₂×C₄	—	488	4	no	—
130	C₂×C₄	—	488	4	no	—
131	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
132	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
133	C₂×C₄	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
134	C₂×C₄	#14519777	1	1	yes	characteristic
135	C₂×C₄	#14519777	1	1	yes	characteristic
136	C₂×C₄	#14519777	1	1	yes	characteristic
137	C₂×C₄	C₂×D₄₈₈	1	2	yes	—
138	C₂×C₄	C₂×#11758190	1	4	yes	—
139	C₂×C₄	C₂×Dic₂₄₄	1	2	yes	—
140	C₂×C₄	#14519799	1	4	yes	—
141	C₈	—	2	4	no	—
142	C₈	#14519709	1	8	yes	complemented
143	C₈	#14519711	1	8	yes	complemented
144	C₂²	#16839632	1	1	yes	characteristic
145	C₂²	#16840135	1	2	yes	—
146	C₂²	#16840139	1	1	yes	characteristic
147	C₂²	#16840142	1	1	yes	characteristic
148	C₂²	#16840146	1	2	yes	—
149	C₄	—	976	4	no	—
150	C₄	C₂×#14519709	1	2	yes	—
151	C₄	C₂×#14519711	1	2	yes	—
152	C₄	#16839800	1	2	yes	—
153	C₄	#16840160	1	2	yes	—
154	C₄	#16840164	1	2	yes	—
155	C₄	#16840167	1	2	yes	—
156	C₂	#21445411	1	2	yes	—
157	C₂	#21445416	1	2	yes	—
158	C₂	#21445976	1	1	yes	characteristic
159	C₂	#21445979	1	1	yes	characteristic
160	C₂	#21445983	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519774
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₂×C₈²×C₆₁	C₂
socle, prime radical	C₂³×C₆₁	#127
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	C₂×C₈²	D₆₁
hypocenter	C₆₁	#8814
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
12. C₈²×C₆₁	C₄	semidirect
61. C₈×C₆₁	#106	semidirect
62. C₈×C₆₁	#107	semidirect
92. C₈²	Dic₆₁	semidirect
93. C₆₁	#8814	semidirect
142. C₈	#14519709	semidirect
143. C₈	#14519711	semidirect

Maximal subgroups

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

subgroup	index
#21445979	2
#21445976	2
#21445947	2
C₂×C₈²×C₆₁	2
#8814	61