Subgroups — #29573786

Order 15616.

422 conjugacy classes of subgroups.

The subgroup lattice has 188 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21445975	C₂	1	2	yes	maximal
2	#21445947	C₂	1	1	yes	maximal, characteristic
3	#21445946	C₂	1	2	yes	maximal
4	C₂×C₈×#92418	C₂	1	1	yes	maximal, characteristic
5	C₆₁×#1834	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519708	—	2	4	no	—
7	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
8	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
9	#16839632	C₂²	1	2	yes	—
10	#16839525	—	2	2	no	—
11	#16839524	—	2	4	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	#14519708	—	4	8	no	—
22	C₄×#92418	C₂×C₄	1	2	yes	—
23	C₄×#92418	C₂×C₄	1	2	yes	—
24	C₄×#92418	Q₈	1	2	yes	—
25	C₄×#92418	Q₈	1	2	yes	—
26	C₂²×C₈×C₆₁	D₄	1	2	yes	—
27	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
28	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
29	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
30	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
31	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₆₁	Q₈	1	2	yes	—
33	C₂²×C₄×C₆₁	—	4	2	no	—
34	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
36	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
37	C₂²×Dic₆₁	—	4	2	no	—
38	C₂×C₈×C₆₁	—	2	4	no	—
39	C₂×C₈×C₆₁	C₄⋊C₄	1	4	yes	—
40	C₄⋊C₄×C₆₁	—	4	4	no	—
41	C₄⋊C₄×C₆₁	—	4	4	no	—
42	C₄⋊C₄×C₆₁	—	4	4	no	—
43	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
44	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
45	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
46	C₄²×C₆₁	C₂×Q₈	1	1	yes	characteristic
47	C₄⋊Dic₆₁	—	4	4	no	—
48	C₄⋊Dic₆₁	—	4	4	no	—
49	C₄⋊Dic₆₁	—	4	4	no	—
50	C₂×#92418	—	2	2	no	—
51	C₂×#92418	—	2	2	no	—
52	C₂×#92418	C₄⋊C₄	1	4	yes	—
53	C₂×#92418	C₄⋊C₄	1	4	yes	—
54	C₂×#92418	SD₁₆	1	8	yes	—
55	C₂³×C₆₁	#127	1	1	yes	characteristic
56	C₂×C₄×C₆₁	—	8	2	no	—
57	C₂×C₄×C₆₁	—	8	2	no	—
58	C₂×C₄×C₆₁	—	8	2	no	—
59	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
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	C₄⋊Q₈	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	C₂×SD₁₆	1	4	yes	—
66	C₂×C₄×C₆₁	#135	1	2	yes	—
67	C₂×C₄×C₆₁	#135	1	4	yes	—
68	C₂×C₄×C₆₁	#135	1	2	yes	—
69	C₂×Dic₆₁	—	8	2	no	—
70	C₂×Dic₆₁	—	8	2	no	—
71	C₂×Dic₆₁	—	8	2	no	—
72	C₈×C₆₁	—	2	4	no	—
73	#92418	—	2	4	no	—
74	#92418	#106	1	8	yes	—
75	#8815	—	61	1	no	maximal, Sylow
76	C₂²×C₆₁	#384	1	1	yes	characteristic
77	C₂²×C₆₁	#492	1	2	yes	—
78	C₂²×C₆₁	#495	1	2	yes	—
79	C₂²×C₆₁	#495	1	1	yes	characteristic
80	C₂²×C₆₁	#495	1	1	yes	characteristic
81	C₄×C₆₁	—	16	2	no	—
82	C₄×C₆₁	C₂×#106	1	2	yes	—
83	C₄×C₆₁	#427	1	2	yes	—
84	C₄×C₆₁	#427	1	2	yes	—
85	C₄×C₆₁	#498	1	2	yes	—
86	C₄×C₆₁	#499	1	2	yes	—
87	C₄×C₆₁	#499	1	2	yes	—
88	Dic₆₁	—	16	2	no	—
89	#1844	—	61	2	no	—
90	#1834	—	61	1	no	—
91	#1834	—	61	2	no	—
92	#1834	D₆₁	1	1	yes	characteristic
93	C₂×C₈²	—	61	1	no	—
94	C₂×C₆₁	#1708	1	2	yes	—
95	C₂×C₆₁	#1708	1	2	yes	—
96	C₂×C₆₁	#1844	1	1	yes	characteristic
97	C₂×C₆₁	#1846	1	1	yes	characteristic
98	C₂×C₆₁	#1846	1	1	yes	characteristic
99	C₂×#106	—	122	4	no	—
100	C₂×C₄×C₈	—	61	1	no	—
101	C₂×C₄×C₈	—	61	1	no	—
102	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
103	#384	—	61	2	no	—
104	#384	C₂×D₆₁	1	2	yes	—
105	#340	—	2	2	no	—
106	#340	—	122	4	no	—
107	#340	—	122	2	no	—
108	C₈²	—	61	4	no	—
109	C₆₁	#8815	1	1	yes	characteristic, complemented, Sylow
110	C₂²×C₈	—	61	2	no	—
111	C₂²×C₈	—	61	2	no	—
112	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
113	C₂×C₄⋊C₄	—	2	2	no	—
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	2	2	no	—
116	C₂×C₄⋊C₄	—	122	2	no	—
117	C₂×C₄⋊C₄	—	122	2	no	—
118	C₂×C₄⋊C₄	—	122	2	no	—
119	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
120	#106	—	244	8	no	—
121	C₄×C₈	—	61	2	no	—
122	C₄×C₈	—	61	2	no	—
123	C₄×C₈	—	61	2	no	—
124	C₄×C₈	—	61	2	no	—
125	C₄×C₈	Dic₁₂₂	1	2	yes	—
126	C₄×C₈	C₄×D₆₁	1	2	yes	—
127	C₂²×C₄	—	4	2	no	—
128	C₂²×C₄	—	244	2	no	—
129	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
130	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
131	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
132	C₂×C₈	—	2	4	no	—
133	C₂×C₈	—	61	4	no	—
134	C₂×C₈	—	61	8	no	—
135	C₂×C₈	—	61	4	no	—
136	C₂×C₈	—	122	2	no	—
137	C₂×C₈	—	122	2	no	—
138	C₂×C₈	#11758196	1	4	yes	—
139	C₄⋊C₄	—	4	4	no	—
140	C₄⋊C₄	—	4	4	no	—
141	C₄⋊C₄	—	4	4	no	—
142	C₄⋊C₄	—	244	4	no	—
143	C₄⋊C₄	—	244	4	no	—
144	C₄⋊C₄	—	244	4	no	—
145	C₄²	C₂×Dic₁₂₂	1	1	yes	characteristic
146	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
147	C₄²	Q₈×D₆₁	1	1	yes	characteristic
148	C₄²	Q₈×D₆₁	1	1	yes	characteristic
149	C₂³	#14519773	1	1	yes	characteristic
150	C₂×C₄	—	8	2	no	—
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	8	2	no	—
153	C₂×C₄	—	488	2	no	—
154	C₂×C₄	—	488	2	no	—
155	C₂×C₄	—	488	2	no	—
156	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
157	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
158	C₂×C₄	C₂×#11758196	1	1	yes	characteristic
159	C₂×C₄	#14519775	1	1	yes	characteristic
160	C₂×C₄	#14519776	1	1	yes	characteristic
161	C₂×C₄	#14519776	1	1	yes	characteristic
162	C₂×C₄	SD₁₆×D₆₁	1	4	yes	—
163	C₂×C₄	#14519796	1	2	yes	—
164	C₂×C₄	#14519797	1	4	yes	—
165	C₂×C₄	#14519800	1	2	yes	—
166	C₈	—	2	4	no	—
167	C₈	—	61	8	no	—
168	C₈	—	122	4	no	—
169	C₂²	#16839631	1	1	yes	characteristic
170	C₂²	#16840133	1	2	yes	—
171	C₂²	#16840144	1	2	yes	—
172	C₂²	#16840145	1	1	yes	characteristic
173	C₂²	#16840147	1	1	yes	characteristic
174	C₄	—	16	2	no	—
175	C₄	—	976	2	no	—
176	C₄	#106×D₆₁	1	2	yes	—
177	C₄	#16839798	1	2	yes	—
178	C₄	#16839799	1	2	yes	—
179	C₄	#16840159	1	2	yes	—
180	C₄	#16840161	1	2	yes	—
181	C₄	#16840163	1	2	yes	—
182	C₂	#21445419	1	2	yes	—
183	C₂	#21445422	1	2	yes	—
184	C₂	#21445974	1	1	yes	characteristic
185	C₂	#21445981	1	1	yes	characteristic
186	C₂	#21445982	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	#8815
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
109. C₆₁	#8815	semidirect

Maximal subgroups

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

subgroup	index
#21445975	2
#21445947	2
#21445946	2
C₂×C₈×#92418	2
C₆₁×#1834	2
#8815	61