Subgroups — #30624133

Order 15616.

622 conjugacy classes of subgroups.

The subgroup lattice has 204 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21462481	C₂	1	2	yes	maximal
2	#21462175	C₂	1	6	yes	maximal, complemented
3	C₂×#16840691	C₂	1	3	yes	maximal, complemented
4	D₆₁⋊#368	C₂	1	3	yes	maximal, complemented
5	#504×D₆₁	C₂	1	1	yes	maximal, characteristic
6	#16840841	—	2	2	no	—
7	#16840777	—	2	6	no	—
8	C₂×#14519848	C₂²	1	6	yes	—
9	D₆₁⋊#108	C₂²	1	6	yes	—
10	#16840691	C₂²	1	1	yes	characteristic
11	#16840691	C₂²	1	3	yes	—
12	#16840691	C₂²	1	6	yes	—
13	#16840690	C₂²	1	6	yes	—
14	#131×D₆₁	C₂²	1	1	yes	characteristic
15	#16840180	C₄	1	3	yes	—
16	#16840179	C₄	1	1	yes	characteristic
17	#16840178	C₄	1	3	yes	—
18	M₃₂×D₆₁	C₂²	1	3	yes	—
19	C₂×C₁₆×D₆₁	C₂²	1	3	yes	—
20	C₆₁×#504	C₄	1	1	yes	characteristic
21	#14519865	—	4	2	no	—
22	#14519864	—	4	6	no	—
23	C₂×#11758213	—	2	6	no	—
24	D₆₁⋊M₁₆	—	2	6	no	—
25	#14519848	—	2	6	no	—
26	#14519848	C₂³	1	2	yes	—
27	#14519848	C₂³	1	6	yes	—
28	#14519847	—	2	6	no	—
29	#41×D₆₁	D₄	1	1	yes	characteristic
30	#14519785	Q₈	1	3	yes	—
31	#14519784	Q₈	1	1	yes	characteristic
32	#14519783	C₂×C₄	1	3	yes	—
33	M₁₆×D₆₁	D₄	1	3	yes	—
34	C₂×C₈×D₆₁	C₂³	1	3	yes	—
35	#14519717	C₂×C₄	1	3	yes	—
36	#14519716	C₂×C₄	1	6	yes	—
37	C₂×#11758185	C₂×C₄	1	3	yes	—
38	C₁₆×D₆₁	C₂×C₄	1	3	yes	—
39	C₁₆×D₆₁	C₂×C₄	1	3	yes	—
40	C₁₆×D₆₁	C₂³	1	1	yes	characteristic
41	C₁₆×D₆₁	C₂³	1	3	yes	—
42	#131×C₆₁	C₂×C₄	1	1	yes	characteristic
43	M₃₂×C₆₁	C₂×C₄	1	3	yes	—
44	C₂×C₁₆×C₆₁	C₂×C₄	1	3	yes	—
45	#41×C₆₁	C₄⋊C₄	1	1	yes	characteristic
46	#11758226	D₈	1	1	yes	characteristic
47	Q₈×D₆₁	Q₁₆	1	1	yes	characteristic
48	#11758224	D₈	1	3	yes	—
49	D₄×D₆₁	Q₁₆	1	3	yes	—
50	#11758222	C₄⋊C₄	1	3	yes	—
51	C₂×C₄×D₆₁	C₂×D₄	1	3	yes	—
52	#11758217	—	4	6	no	—
53	C₂×#92420	—	4	6	no	—
54	#11758213	—	2	2	no	—
55	#11758213	—	2	6	no	—
56	#11758213	—	4	6	no	—
57	D₆₁⋊C₈	—	4	6	no	—
58	M₁₆×C₆₁	C₄⋊C₄	1	3	yes	—
59	C₂×C₈×C₆₁	C₂²×C₄	1	3	yes	—
60	#11758194	C₄⋊C₄	1	3	yes	—
61	C₂×#92418	C₂×Q₈	1	3	yes	—
62	#11758189	C₄⋊C₄	1	6	yes	—
63	C₈×D₆₁	C₂²×C₄	1	3	yes	—
64	C₈×D₆₁	C₂×D₄	1	3	yes	—
65	C₈×D₆₁	C₂×Q₈	1	3	yes	—
66	C₈×D₆₁	C₂⁴	1	1	yes	characteristic
67	C₁₆×C₆₁	C₂²×C₄	1	1	yes	characteristic
68	C₁₆×C₆₁	C₂²×C₄	1	3	yes	—
69	#11758185	C₂²×C₄	1	1	yes	characteristic
70	#11758185	C₂²×C₄	1	3	yes	—
71	C₂²×D₆₁	C₂×Q₁₆	1	3	yes	—
72	Q₈×C₆₁	#107	1	1	yes	characteristic
73	D₄×C₆₁	#107	1	3	yes	—
74	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
75	C₂²⋊D₆₁	#107	1	6	yes	—
76	C₂×Dic₆₁	C₂×D₈	1	3	yes	—
77	D₂₄₄	#107	1	3	yes	—
78	C₄×D₆₁	C₂×C₄⋊C₄	1	3	yes	—
79	C₄×D₆₁	C₂×D₈	1	3	yes	—
80	C₄×D₆₁	C₂×Q₁₆	1	3	yes	—
81	C₄×D₆₁	C₂²×D₄	1	1	yes	characteristic
82	Dic₁₂₂	#107	1	3	yes	—
83	#92420	—	4	2	no	—
84	#92420	—	4	6	no	—
85	C₈×C₆₁	C₂×C₄⋊C₄	1	3	yes	—
86	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
87	#92418	C₂×C₄⋊C₄	1	3	yes	—
88	#92418	C₂²×Q₈	1	1	yes	characteristic
89	#33414	—	61	1	no	maximal, Sylow
90	C₂²×C₆₁	C₂×#107	1	3	yes	—
91	C₂×D₆₁	—	4	3	no	—
92	C₂×D₆₁	C₂×#107	1	3	yes	—
93	C₂×D₆₁	C₂²×Q₁₆	1	1	yes	characteristic
94	C₄×C₆₁	C₂×#107	1	3	yes	—
95	C₄×C₆₁	C₂²×C₄⋊C₄	1	1	yes	characteristic
96	Dic₆₁	C₂×#107	1	3	yes	—
97	Dic₆₁	C₂²×D₈	1	1	yes	characteristic
98	C₂×#504	—	61	1	no	—
99	#2916	—	61	2	no	—
100	#2161	—	61	6	no	—
101	C₂×#368	—	61	3	no	—
102	C₂×#368	—	61	3	no	—
103	C₂×C₆₁	—	2	3	no	—
104	C₂×C₆₁	C₂²×#107	1	1	yes	characteristic
105	D₆₁	—	2	1	no	—
106	D₆₁	—	2	3	no	—
107	#568	—	122	2	no	—
108	C₂×#131	—	61	1	no	—
109	#504	—	61	1	no	—
110	#504	—	61	3	no	—
111	#504	—	61	3	no	—
112	#504	#6923	1	1	yes	characteristic
113	C₂×M₃₂	—	61	3	no	—
114	C₂²×C₁₆	—	61	3	no	—
115	#430	—	122	6	no	—
116	C₂×#108	—	61	6	no	—
117	C₂×#108	—	61	6	no	—
118	#368	—	61	1	no	—
119	#368	—	61	3	no	—
120	#368	—	61	6	no	—
121	#368	—	61	6	no	—
122	C₆₁	#33414	1	1	yes	characteristic, complemented, Sylow
123	C₂×#41	—	61	1	no	—
124	#131	—	61	1	no	—
125	#131	—	61	3	no	—
126	#131	—	61	3	no	—
127	#131	—	244	6	no	—
128	#131	—	244	2	no	—
129	#131	C₂×#6923	1	1	yes	characteristic
130	C₂×M₁₆	—	61	3	no	—
131	C₂×M₁₆	—	122	6	no	—
132	C₂×M₁₆	—	122	6	no	—
133	C₂²×C₈	—	61	3	no	—
134	M₃₂	—	61	3	no	—
135	M₃₂	—	61	6	no	—
136	M₃₂	C₂×#6923	1	3	yes	—
137	C₂×C₁₆	—	61	3	no	—
138	C₂×C₁₆	—	61	1	no	—
139	C₂×C₁₆	—	61	3	no	—
140	C₂×C₁₆	—	61	3	no	—
141	C₂×C₁₆	—	61	3	no	—
142	C₂×C₁₆	C₂×#6923	1	3	yes	—
143	#108	—	61	2	no	—
144	#108	—	61	6	no	—
145	#108	—	122	6	no	—
146	#108	—	122	6	no	—
147	#41	—	61	1	no	—
148	#41	—	61	3	no	—
149	#41	—	61	3	no	—
150	#41	C₄⋊#6923	1	1	yes	characteristic
151	C₂×Q₈	—	61	1	no	—
152	C₂×D₄	—	61	3	no	—
153	C₂²×C₄	—	61	3	no	—
154	M₁₆	—	61	3	no	—
155	M₁₆	—	61	6	no	—
156	M₁₆	—	122	2	no	—
157	M₁₆	—	122	6	no	—
158	M₁₆	—	244	6	no	—
159	M₁₆	—	244	6	no	—
160	M₁₆	C₄⋊#6923	1	3	yes	—
161	C₂×C₈	—	61	3	no	—
162	C₂×C₈	—	61	3	no	—
163	C₂×C₈	—	61	3	no	—
164	C₂×C₈	—	61	1	no	—
165	C₂×C₈	—	61	3	no	—
166	C₂×C₈	—	244	6	no	—
167	C₂×C₈	—	244	6	no	—
168	C₂×C₈	C₂²×#6923	1	3	yes	—
169	C₁₆	—	61	1	no	—
170	C₁₆	—	61	3	no	—
171	C₁₆	C₂²×#6923	1	1	yes	characteristic
172	C₁₆	C₂²×#6923	1	3	yes	—
173	C₂³	—	61	3	no	—
174	Q₈	—	61	3	no	—
175	Q₈	#14519846	1	1	yes	characteristic
176	D₄	—	61	6	no	—
177	D₄	—	61	3	no	—
178	D₄	#14519846	1	3	yes	—
179	C₂×C₄	—	61	3	no	—
180	C₂×C₄	—	61	3	no	—
181	C₂×C₄	—	61	3	no	—
182	C₂×C₄	—	61	3	no	—
183	C₂×C₄	—	61	1	no	—
184	C₂×C₄	C₂×C₄⋊#6923	1	3	yes	—
185	C₈	—	61	3	no	—
186	C₈	—	61	1	no	—
187	C₈	—	244	2	no	—
188	C₈	—	244	6	no	—
189	C₈	C₂×C₄⋊#6923	1	3	yes	—
190	C₈	C₂³×#6923	1	1	yes	characteristic
191	C₂²	—	61	1	no	—
192	C₂²	—	61	3	no	—
193	C₂²	—	244	3	no	—
194	C₂²	C₂×#14519846	1	3	yes	—
195	C₄	—	61	1	no	—
196	C₄	—	61	3	no	—
197	C₄	C₂×#14519846	1	3	yes	—
198	C₄	C₂²×C₄⋊#6923	1	1	yes	characteristic
199	C₂	—	2	3	no	—
200	C₂	—	122	1	no	—
201	C₂	—	122	3	no	—
202	C₂	C₂²×#14519846	1	1	yes	characteristic

	subgroup	G/H
center	C₂	C₂²×#14519846
commutator	C₈×C₆₁	C₂³×C₄
Frattini	C₈	C₂³×#6923
Fitting	C₆₁×#504	C₄
socle	C₂×C₆₁	C₂²×#107
prime radical	#41×D₆₁	D₄
cosocle	C₈×D₆₁	C₂⁴
hypercenter	#504	#6923
hypocenter	C₆₁	#33414
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂ ◁ #41 ◁ #131 ◁ #504

Complements

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

N	H ≅ G/N	type
2. #21462175	C₂	semidirect
3. C₂×#16840691	C₂	semidirect
4. D₆₁⋊#368	C₂	semidirect
122. C₆₁	#33414	semidirect

Maximal subgroups

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

subgroup	index
#21462481	2
#21462175	2
C₂×#16840691	2
D₆₁⋊#368	2
#504×D₆₁	2
#33414	61