Subgroups — #29589060

Order 15616.

248 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21447622	C₂	1	1	yes	maximal, characteristic
2	C₄×#14519717	C₂	1	1	yes	maximal, characteristic
3	#21443740	C₂	1	2	yes	maximal, complemented
4	#21443740	C₂	1	2	yes	maximal, complemented
5	C₆₁×#1451	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519717	C₄	1	2	yes	—
7	C₄⋊#11758185	C₄	1	2	yes	—
8	C₂²⋊#11758185	C₄	1	2	yes	—
9	#16839538	C₂²	1	2	yes	—
10	#16839538	C₂²	1	1	yes	characteristic
11	#16839538	C₂²	1	1	yes	characteristic
12	C₄×#11758185	C₄	1	2	yes	—
13	C₆₁×#431	C₄	1	1	yes	characteristic
14	C₄×M₁₆×C₆₁	C₄	1	1	yes	characteristic
15	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
16	C₆₁×#322	C₄	1	2	yes	—
17	C₆₁×#322	C₂²	1	2	yes	—
18	#14519717	—	2	4	no	—
19	C₂×#11758185	C₂×C₄	1	2	yes	—
20	C₂×#11758185	C₂×C₄	1	4	yes	—
21	C₂×#11758185	C₂×C₄	1	2	yes	—
22	C₂×M₁₆×C₆₁	—	2	1	no	—
23	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
25	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
26	C₄⋊C₈×C₆₁	—	2	1	no	—
27	C₂²⋊C₈×C₆₁	—	2	1	no	—
28	#97×C₆₁	—	2	1	no	—
29	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
32	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
33	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
35	C₂²×C₄×C₆₁	C₄²	1	1	yes	characteristic
36	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
37	C₂²×C₄×C₆₁	M₁₆	1	1	yes	characteristic
38	M₁₆×C₆₁	—	4	2	no	—
39	C₂×C₈×C₆₁	—	2	2	no	—
40	C₂×C₈×C₆₁	—	2	2	no	—
41	C₂×C₈×C₆₁	—	2	2	no	—
42	C₂×C₈×C₆₁	—	2	2	no	—
43	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
45	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
46	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
47	C₄²×C₆₁	C₄²	1	1	yes	characteristic
48	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
49	C₄²×C₆₁	M₁₆	1	1	yes	characteristic
50	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
51	#11758185	—	2	2	no	—
52	#11758185	—	2	4	no	—
53	#11758185	M₁₆	1	4	yes	complemented
54	C₂³×C₆₁	#97	1	1	yes	characteristic
55	C₂×C₄×C₆₁	—	2	2	no	—
56	C₂×C₄×C₆₁	—	2	1	no	—
57	C₂×C₄×C₆₁	—	2	1	no	—
58	C₂×C₄×C₆₁	—	4	1	no	—
59	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
60	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
61	C₂×C₄×C₆₁	#97	1	1	yes	characteristic
62	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
64	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
65	C₈×C₆₁	—	2	1	no	—
66	C₈×C₆₁	—	2	1	no	—
67	C₈×C₆₁	—	4	2	no	—
68	C₈×C₆₁	—	4	2	no	—
69	C₈×C₆₁	C₂×M₁₆	1	2	yes	—
70	C₈×C₆₁	#131	1	2	yes	—
71	#11414	—	61	1	no	maximal, Sylow
72	C₂²×C₆₁	—	2	1	no	—
73	C₂²×C₆₁	—	2	1	no	—
74	C₂²×C₆₁	—	2	1	no	—
75	C₂²×C₆₁	C₂×#97	1	1	yes	characteristic
76	C₄×C₆₁	—	2	1	no	—
77	C₄×C₆₁	—	2	1	no	—
78	C₄×C₆₁	—	2	1	no	—
79	C₄×C₆₁	—	2	1	no	—
80	C₄×C₆₁	#347	1	2	yes	—
81	C₄×C₆₁	C₄×M₁₆	1	1	yes	characteristic
82	C₄×C₆₁	#405	1	1	yes	characteristic
83	#2163	—	61	1	no	—
84	C₄×M₃₂	—	61	1	no	—
85	#1451	D₆₁	1	1	yes	characteristic
86	#1313	—	61	2	no	—
87	#1313	—	61	2	no	—
88	C₂×C₆₁	—	4	1	no	—
89	C₂×C₆₁	#1451	1	1	yes	characteristic
90	C₂×C₆₁	C₂×#347	1	1	yes	characteristic
91	C₂×C₆₁	#2110	1	1	yes	characteristic
92	C₂×M₃₂	—	61	2	no	—
93	#431	Dic₆₁	1	1	yes	characteristic
94	C₄×M₁₆	Dic₆₁	1	1	yes	characteristic
95	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
96	C₄⋊C₁₆	—	61	2	no	—
97	C₂²⋊C₁₆	—	61	2	no	—
98	#346	—	61	1	no	—
99	#346	—	61	1	no	—
100	#346	—	61	2	no	—
101	C₄×C₁₆	—	61	2	no	—
102	#322	Dic₆₁	1	2	yes	—
103	#322	C₂×D₆₁	1	2	yes	—
104	C₆₁	#11414	1	1	yes	characteristic, complemented, Sylow
105	C₂×M₁₆	—	2	1	no	—
106	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
107	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
108	C₂×C₄²	C₂×Dic₆₁	1	1	yes	characteristic
109	M₃₂	—	122	4	no	—
110	C₂×C₁₆	—	61	2	no	—
111	C₂×C₁₆	—	61	2	no	—
112	C₂×C₁₆	—	61	4	no	—
113	C₄⋊C₈	—	2	1	no	—
114	C₂²⋊C₈	—	2	1	no	—
115	#97	—	2	1	no	—
116	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
117	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
118	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
119	C₄×C₈	C₂×Dic₆₁	1	2	yes	—
120	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
121	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
122	C₂²×C₄	#11758194	1	1	yes	characteristic
123	C₂²×C₄	#11758194	1	1	yes	characteristic
124	C₂²×C₄	C₄×Dic₆₁	1	1	yes	characteristic
125	M₁₆	—	4	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	2	2	no	—
129	C₂×C₈	—	2	2	no	—
130	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
131	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
132	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
133	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
134	C₄²	#11758194	1	1	yes	characteristic
135	C₄²	#11758194	1	1	yes	characteristic
136	C₄²	C₄×Dic₆₁	1	1	yes	characteristic
137	C₄²	C₂²×Dic₆₁	1	1	yes	characteristic
138	C₁₆	—	61	4	no	—
139	C₁₆	—	122	2	no	—
140	C₁₆	—	122	4	no	—
141	C₂³	#14519688	1	1	yes	characteristic
142	C₂×C₄	—	2	1	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	2	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	#14519688	1	1	yes	characteristic
147	C₂×C₄	#14519688	1	1	yes	characteristic
148	C₂×C₄	#14519688	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
150	C₂×C₄	C₂×#11758194	1	1	yes	characteristic
151	C₂×C₄	C₂×#11758194	1	1	yes	characteristic
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	4	2	no	—
155	C₈	—	4	2	no	—
156	C₈	M₁₆×D₆₁	1	2	yes	—
157	C₈	#14519785	1	2	yes	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	—	2	1	no	—
161	C₂²	C₂×#14519688	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	#16839540	1	2	yes	—
167	C₄	M₁₆×Dic₆₁	1	1	yes	characteristic
168	C₄	#16839720	1	1	yes	characteristic
169	C₂	—	4	1	no	—
170	C₂	#21444106	1	1	yes	characteristic
171	C₂	C₂×#16839540	1	1	yes	characteristic
172	C₂	#21447424	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₆₁
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1451	C₂
socle	C₂²×C₆₁	C₂×#97
prime radical	C₂³×C₆₁	#97
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	#1451	D₆₁
hypocenter	C₆₁	#11414
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ #1451

Complements

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

N	H ≅ G/N	type
3. #21443740	C₂	semidirect
4. #21443740	C₂	semidirect
53. #11758185	M₁₆	semidirect
104. C₆₁	#11414	semidirect

Maximal subgroups

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

subgroup	index
#21447622	2
C₄×#14519717	2
#21443740	2
C₆₁×#1451	2
#11414	61