Subgroups — #29573783

Order 15616.

454 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445978	C₂	1	2	yes	maximal
2	#21445975	C₂	1	2	yes	maximal
3	#21445947	C₂	1	1	yes	maximal, characteristic
4	C₂×C₈×#92418	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1834	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519710	—	2	4	no	—
7	C₂×#14519708	—	2	4	no	—
8	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
9	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
10	#16839632	C₂²	1	2	yes	—
11	#16839525	—	2	2	no	—
12	C₈×#92418	C₄	1	4	yes	complemented
13	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
14	C₆₁×#384	C₂²	1	2	yes	—
15	C₆₁×#340	—	2	2	no	—
16	C₂²×#92418	D₄	1	2	yes	—
17	C₂²×#92418	D₄	1	2	yes	—
18	C₂×C₄⋊Dic₆₁	—	2	2	no	—
19	C₂×C₄⋊Dic₆₁	—	2	2	no	—
20	C₂×C₄⋊Dic₆₁	—	2	2	no	—
21	#14519710	—	4	8	no	—
22	#14519708	—	4	8	no	—
23	C₄×#92418	C₂×C₄	1	2	yes	—
24	C₄×#92418	C₂×C₄	1	2	yes	—
25	C₄×#92418	Q₈	1	2	yes	—
26	C₄×#92418	Q₈	1	2	yes	—
27	C₂²×C₈×C₆₁	D₄	1	2	yes	—
28	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
29	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
30	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
31	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
32	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₆₁	Q₈	1	2	yes	—
34	C₂²×C₄×C₆₁	—	4	2	no	—
35	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
37	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
38	C₂²×Dic₆₁	—	4	2	no	—
39	C₂×C₈×C₆₁	—	2	4	no	—
40	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
41	C₄⋊C₄×C₆₁	—	4	4	no	—
42	C₄⋊C₄×C₆₁	—	4	4	no	—
43	C₄⋊C₄×C₆₁	—	4	4	no	—
44	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
45	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
47	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
48	C₄⋊Dic₆₁	—	4	4	no	—
49	C₄⋊Dic₆₁	—	4	4	no	—
50	C₄⋊Dic₆₁	—	4	4	no	—
51	C₂×#92418	C₄⋊C₄	1	4	yes	—
52	C₂×#92418	C₄⋊C₄	1	4	yes	—
53	C₂×#92418	D₈	1	4	yes	—
54	C₂×#92418	SD₁₆	1	8	yes	—
55	C₂×#92418	Q₁₆	1	4	yes	—
56	C₂³×C₆₁	#127	1	1	yes	characteristic
57	C₂×C₄×C₆₁	—	8	2	no	—
58	C₂×C₄×C₆₁	—	8	2	no	—
59	C₂×C₄×C₆₁	—	8	2	no	—
60	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
61	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
63	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
65	C₂×C₄×C₆₁	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₆₁	C₂×D₈	1	2	yes	—
67	C₂×C₄×C₆₁	C₂×SD₁₆	1	4	yes	—
68	C₂×C₄×C₆₁	C₂×Q₁₆	1	2	yes	—
69	C₂×C₄×C₆₁	#135	1	4	yes	—
70	C₂×Dic₆₁	—	8	2	no	—
71	C₂×Dic₆₁	—	8	2	no	—
72	C₂×Dic₆₁	—	8	2	no	—
73	C₈×C₆₁	—	2	4	no	—
74	#92418	#106	1	8	yes	complemented
75	#92418	#107	1	8	yes	complemented
76	#8814	—	61	1	no	maximal, Sylow
77	C₂²×C₆₁	#384	1	1	yes	characteristic
78	C₂²×C₆₁	#492	1	2	yes	—
79	C₂²×C₆₁	#493	1	1	yes	characteristic
80	C₂²×C₆₁	#494	1	1	yes	characteristic
81	C₂²×C₆₁	#495	1	2	yes	—
82	C₄×C₆₁	—	16	2	no	—
83	C₄×C₆₁	C₂×#106	1	2	yes	—
84	C₄×C₆₁	C₂×#107	1	2	yes	—
85	C₄×C₆₁	#427	1	2	yes	—
86	C₄×C₆₁	#498	1	2	yes	—
87	C₄×C₆₁	#499	1	2	yes	—
88	C₄×C₆₁	#500	1	2	yes	—
89	Dic₆₁	—	16	2	no	—
90	#1845	—	61	2	no	—
91	#1844	—	61	2	no	—
92	#1834	—	61	1	no	—
93	#1834	D₆₁	1	1	yes	characteristic
94	C₂×C₈²	—	61	1	no	—
95	C₂×C₆₁	#1706	1	2	yes	—
96	C₂×C₆₁	#1707	1	2	yes	—
97	C₂×C₆₁	#1844	1	1	yes	characteristic
98	C₂×C₆₁	#1845	1	1	yes	characteristic
99	C₂×C₆₁	#1846	1	1	yes	characteristic
100	C₂×#107	—	122	4	no	—
101	C₂×#106	—	122	4	no	—
102	C₂×C₄×C₈	—	61	1	no	—
103	C₂×C₄×C₈	—	61	1	no	—
104	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
105	#384	—	61	2	no	—
106	#384	C₂×D₆₁	1	2	yes	—
107	#340	—	2	2	no	—
108	#340	—	122	2	no	—
109	C₈²	—	61	4	no	—
110	C₆₁	#8814	1	1	yes	characteristic, complemented, Sylow
111	C₂²×C₈	—	61	2	no	—
112	C₂²×C₈	—	61	2	no	—
113	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	2	2	no	—
117	C₂×C₄⋊C₄	—	122	2	no	—
118	C₂×C₄⋊C₄	—	122	2	no	—
119	C₂×C₄⋊C₄	—	122	2	no	—
120	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
121	#107	—	244	8	no	—
122	#106	—	244	8	no	—
123	C₄×C₈	—	61	2	no	—
124	C₄×C₈	—	61	2	no	—
125	C₄×C₈	—	61	2	no	—
126	C₄×C₈	—	61	2	no	—
127	C₄×C₈	Dic₁₂₂	1	2	yes	—
128	C₄×C₈	C₄×D₆₁	1	2	yes	—
129	C₂²×C₄	—	4	2	no	—
130	C₂²×C₄	—	244	2	no	—
131	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
132	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
133	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
134	C₂×C₈	—	2	4	no	—
135	C₂×C₈	—	61	4	no	—
136	C₂×C₈	—	61	4	no	—
137	C₂×C₈	—	61	4	no	—
138	C₂×C₈	—	61	8	no	—
139	C₂×C₈	—	61	4	no	—
140	C₂×C₈	#11758196	1	4	yes	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄⋊C₄	—	4	4	no	—
143	C₄⋊C₄	—	4	4	no	—
144	C₄⋊C₄	—	244	4	no	—
145	C₄⋊C₄	—	244	4	no	—
146	C₄⋊C₄	—	244	4	no	—
147	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
148	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
149	C₄²	Q₈×D₆₁	1	1	yes	characteristic
150	C₄²	Q₈×D₆₁	1	1	yes	characteristic
151	C₂³	#14519773	1	1	yes	characteristic
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	—	8	2	no	—
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	—	488	2	no	—
156	C₂×C₄	—	488	2	no	—
157	C₂×C₄	—	488	2	no	—
158	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
159	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
160	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
161	C₂×C₄	#14519775	1	1	yes	characteristic
162	C₂×C₄	#14519776	1	1	yes	characteristic
163	C₂×C₄	#14519776	1	1	yes	characteristic
164	C₂×C₄	D₈×D₆₁	1	2	yes	—
165	C₂×C₄	SD₁₆×D₆₁	1	4	yes	—
166	C₂×C₄	Q₁₆×D₆₁	1	2	yes	—
167	C₂×C₄	#14519797	1	4	yes	—
168	C₈	—	2	4	no	—
169	C₈	—	61	8	no	—
170	C₈	—	61	8	no	—
171	C₂²	#16839631	1	1	yes	characteristic
172	C₂²	#16840133	1	2	yes	—
173	C₂²	#16840137	1	1	yes	characteristic
174	C₂²	#16840140	1	1	yes	characteristic
175	C₂²	#16840144	1	2	yes	—
176	C₄	—	16	2	no	—
177	C₄	—	976	2	no	—
178	C₄	#106×D₆₁	1	2	yes	—
179	C₄	#107×D₆₁	1	2	yes	—
180	C₄	#16839798	1	2	yes	—
181	C₄	#16840159	1	2	yes	—
182	C₄	#16840161	1	2	yes	—
183	C₄	#16840166	1	2	yes	—
184	C₂	#21445410	1	2	yes	—
185	C₂	#21445415	1	2	yes	—
186	C₂	#21445974	1	1	yes	characteristic
187	C₂	#21445977	1	1	yes	characteristic
188	C₂	#21445981	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519773
commutator	C₄²×C₆₁	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₆₁
Fitting	C₆₁×#1834	C₂
socle, prime radical	C₂³×C₆₁	#127
cosocle	C₂×C₄²×C₆₁	C₂³
hypercenter	#1834	D₆₁
hypocenter	C₆₁	#8814
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1834

Complements

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

N	H ≅ G/N	type
12. C₈×#92418	C₄	semidirect
74. #92418	#106	semidirect
75. #92418	#107	semidirect
110. C₆₁	#8814	semidirect

Maximal subgroups

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

subgroup	index
#21445978	2
#21445975	2
#21445947	2
C₂×C₈×#92418	2
C₆₁×#1834	2
#8814	61