Subgroups — #29591585

Order 15616.

242 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#21447573	C₂	1	1	yes	maximal, characteristic
2	C₄⋊C₁₆×D₆₁	C₂	1	1	yes	maximal, characteristic
3	#21444101	C₂	1	1	yes	maximal, characteristic
4	#21443889	C₂	1	1	yes	maximal, characteristic, complemented
5	#21443888	C₂	1	1	yes	maximal, characteristic, complemented
6	#21443888	C₂	1	1	yes	maximal, characteristic, complemented
7	C₆₁×#1370	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#14519716	—	2	1	no	—
9	C₂×C₁₆×D₆₁	—	2	1	no	—
10	#16839813	C₄	1	1	yes	characteristic
11	#16839813	C₄	1	1	yes	characteristic
12	C₄×C₈×D₆₁	C₂²	1	1	yes	characteristic
13	C₄⋊#11758185	C₂²	1	1	yes	characteristic
14	C₄⋊#11758185	C₂²	1	1	yes	characteristic
15	#16839580	—	2	1	no	—
16	#16839580	—	2	1	no	—
17	#16839484	C₄	1	1	yes	characteristic
18	#16839484	C₄	1	1	yes	characteristic
19	C₈×#92418	C₂²	1	1	yes	characteristic
20	C₆₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
21	C₆₁×C₄⋊C₁₆	C₂²	1	1	yes	characteristic
22	C₈²×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×C₈×D₆₁	D₄	1	1	yes	characteristic
24	C₂×C₈×D₆₁	Q₈	1	1	yes	characteristic
25	C₄²×D₆₁	C₂×C₄	1	1	yes	characteristic
26	#14519716	—	4	2	no	—
27	C₂×#11758185	—	2	1	no	—
28	C₂×#11758185	—	2	1	no	—
29	C₁₆×D₆₁	—	4	2	no	—
30	#14519706	—	2	1	no	—
31	#14519706	C₈	1	1	yes	characteristic
32	#14519706	C₈	1	1	yes	characteristic
33	#14519689	—	2	1	no	—
34	#14519689	C₈	1	1	yes	characteristic
35	#14519689	C₈	1	1	yes	characteristic
36	C₄×#92418	C₂×C₄	1	1	yes	characteristic
37	C₄×#92418	C₂×C₄	1	1	yes	characteristic
38	C₄×#92418	C₂×C₄	1	1	yes	characteristic
39	C₈×Dic₆₁	D₄	1	1	yes	characteristic
40	C₈×Dic₆₁	Q₈	1	1	yes	characteristic
41	C₂×C₁₆×C₆₁	—	2	1	no	—
42	C₂×C₁₆×C₆₁	—	2	1	no	—
43	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
44	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₆₁	C₂³	1	1	yes	characteristic
46	C₂×C₄×D₆₁	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₄×D₆₁	C₂×C₈	1	1	yes	characteristic
48	C₂×C₄×D₆₁	M₁₆	1	1	yes	characteristic
49	C₂×C₈×C₆₁	—	2	1	no	—
50	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
51	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
52	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
53	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
54	C₄²×C₆₁	C₂²×C₄	1	1	yes	characteristic
55	C₄×Dic₆₁	C₄⋊C₄	1	1	yes	characteristic
56	C₄×Dic₆₁	C₂×C₈	1	1	yes	characteristic
57	C₄×Dic₆₁	M₁₆	1	1	yes	characteristic
58	C₂×#92418	—	2	1	no	—
59	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
60	C₂×#92418	C₄⋊C₄	1	1	yes	characteristic
61	C₂×#92418	C₂×C₈	1	1	yes	characteristic
62	C₂×#92418	C₂×C₈	1	1	yes	characteristic
63	C₈×D₆₁	—	2	2	no	—
64	C₈×D₆₁	D₈	1	2	yes	—
65	C₈×D₆₁	Q₁₆	1	2	yes	—
66	C₁₆×C₆₁	—	4	1	no	—
67	C₁₆×C₆₁	—	4	1	no	—
68	#11758185	—	4	1	no	—
69	#11758185	—	4	1	no	—
70	C₂²×D₆₁	C₄⋊C₈	1	1	yes	characteristic
71	C₂×C₄×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
73	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
74	C₂×Dic₆₁	C₄⋊C₈	1	1	yes	characteristic
75	C₂×Dic₆₁	C₄⋊C₈	1	1	yes	characteristic
76	C₂×Dic₆₁	C₄⋊C₈	1	1	yes	characteristic
77	C₄×D₆₁	—	2	1	no	—
78	C₄×D₆₁	—	2	1	no	—
79	C₄×D₆₁	—	4	1	no	—
80	C₄×D₆₁	#107	1	2	yes	—
81	C₄×D₆₁	#108	1	2	yes	—
82	C₈×C₆₁	—	2	1	no	—
83	C₈×C₆₁	—	2	1	no	—
84	C₈×C₆₁	—	2	1	no	—
85	C₈×C₆₁	—	4	1	no	—
86	C₈×C₆₁	C₂×D₈	1	1	yes	characteristic
87	C₈×C₆₁	C₂×Q₁₆	1	1	yes	characteristic
88	#92418	—	2	1	no	—
89	#92418	—	2	1	no	—
90	#92418	—	2	1	no	—
91	#92418	—	4	1	no	—
92	#92418	#107	1	2	yes	—
93	#12005	—	61	1	no	maximal, Sylow
94	C₂²×C₆₁	C₂×C₄⋊C₈	1	1	yes	characteristic
95	C₂×D₆₁	—	2	1	no	—
96	C₂×D₆₁	—	2	1	no	—
97	C₂×D₆₁	#335	1	2	yes	—
98	C₄×C₆₁	—	2	1	no	—
99	C₄×C₆₁	C₂×#107	1	1	yes	characteristic
100	C₄×C₆₁	C₂×#108	1	1	yes	characteristic
101	C₄×C₆₁	#504	1	1	yes	characteristic
102	C₄×C₆₁	#504	1	1	yes	characteristic
103	Dic₆₁	—	2	1	no	—
104	Dic₆₁	—	2	1	no	—
105	Dic₆₁	#335	1	2	yes	—
106	#2150	—	61	1	no	—
107	C₂×C₄⋊C₁₆	—	61	1	no	—
108	#1450	—	61	1	no	—
109	#1370	—	61	1	no	—
110	#1370	—	61	1	no	—
111	#1370	—	61	1	no	—
112	#1370	D₆₁	1	1	yes	characteristic, complemented
113	C₂×C₆₁	C₂×#335	1	1	yes	characteristic
114	C₂×C₆₁	#2151	1	1	yes	characteristic
115	C₂×C₆₁	#2153	1	1	yes	characteristic
116	D₆₁	—	2	2	no	—
117	C₂×M₃₂	—	122	1	no	—
118	C₂²×C₁₆	—	122	1	no	—
119	#431	—	61	1	no	—
120	#431	—	61	1	no	—
121	C₂×C₄×C₈	—	61	1	no	—
122	C₄⋊C₁₆	—	61	1	no	—
123	C₄⋊C₁₆	—	61	1	no	—
124	C₄⋊C₁₆	—	122	1	no	—
125	C₄⋊C₁₆	—	122	1	no	—
126	C₄⋊C₁₆	C₂×D₆₁	1	1	yes	characteristic
127	C₄⋊C₁₆	C₂×D₆₁	1	1	yes	characteristic
128	#322	—	61	1	no	—
129	#322	—	61	1	no	—
130	C₈²	—	61	1	no	—
131	C₈²	C₂×D₆₁	1	1	yes	characteristic
132	C₆₁	#12005	1	1	yes	characteristic, complemented, Sylow
133	C₂²×C₈	—	61	1	no	—
134	C₂²×C₈	—	61	1	no	—
135	C₂×C₄²	—	61	1	no	—
136	M₃₂	—	244	2	no	—
137	C₂×C₁₆	—	2	1	no	—
138	C₂×C₁₆	—	2	1	no	—
139	C₂×C₁₆	—	122	1	no	—
140	C₂×C₁₆	—	122	1	no	—
141	C₂×C₁₆	—	244	2	no	—
142	C₄⋊C₈	—	61	1	no	—
143	C₄⋊C₈	—	61	1	no	—
144	C₄⋊C₈	—	122	1	no	—
145	C₂²⋊C₈	—	61	1	no	—
146	C₂²⋊C₈	—	61	1	no	—
147	C₂²⋊C₈	—	122	1	no	—
148	C₄×C₈	—	61	1	no	—
149	C₄×C₈	—	61	1	no	—
150	C₄×C₈	—	61	1	no	—
151	C₄×C₈	—	61	1	no	—
152	C₄×C₈	—	61	1	no	—
153	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
154	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
155	C₄×C₈	C₂²×D₆₁	1	1	yes	characteristic
156	C₂²×C₄	—	61	1	no	—
157	C₂²×C₄	—	61	1	no	—
158	C₂²×C₄	—	61	1	no	—
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	61	1	no	—
161	C₂×C₈	—	61	2	no	—
162	C₂×C₈	—	61	2	no	—
163	C₂×C₈	—	61	1	no	—
164	C₂×C₈	—	61	1	no	—
165	C₂×C₈	—	61	1	no	—
166	C₂×C₈	—	122	2	no	—
167	C₂×C₈	—	122	1	no	—
168	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
169	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
170	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
171	C₂×C₈	Q₈×D₆₁	1	1	yes	characteristic
172	C₄²	—	61	1	no	—
173	C₄²	—	61	1	no	—
174	C₄²	—	61	1	no	—
175	C₄²	C₂×C₄×D₆₁	1	1	yes	characteristic
176	C₁₆	—	4	1	no	—
177	C₁₆	—	4	1	no	—
178	C₁₆	—	244	1	no	—
179	C₁₆	—	244	1	no	—
180	C₂³	—	61	1	no	—
181	C₂×C₄	—	61	1	no	—
182	C₂×C₄	—	61	2	no	—
183	C₂×C₄	—	61	2	no	—
184	C₂×C₄	—	61	1	no	—
185	C₂×C₄	—	61	1	no	—
186	C₂×C₄	—	122	1	no	—
187	C₂×C₄	—	122	1	no	—
188	C₂×C₄	—	244	1	no	—
189	C₂×C₄	C₄⋊C₄×D₆₁	1	1	yes	characteristic
190	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
191	C₂×C₄	M₁₆×D₆₁	1	1	yes	characteristic
192	C₈	—	2	1	no	—
193	C₈	—	2	1	no	—
194	C₈	—	2	1	no	—
195	C₈	—	4	1	no	—
196	C₈	—	61	2	no	—
197	C₈	—	122	1	no	—
198	C₈	—	122	1	no	—
199	C₈	—	122	1	no	—
200	C₈	—	244	1	no	—
201	C₈	D₈×D₆₁	1	1	yes	characteristic
202	C₈	Q₁₆×D₆₁	1	1	yes	characteristic
203	C₂²	—	61	2	no	—
204	C₂²	—	122	1	no	—
205	C₂²	—	122	1	no	—
206	C₂²	C₄⋊C₈×D₆₁	1	1	yes	characteristic
207	C₄	—	2	1	no	—
208	C₄	—	61	2	no	—
209	C₄	—	122	1	no	—
210	C₄	—	122	1	no	—
211	C₄	#107×D₆₁	1	1	yes	characteristic
212	C₄	#108×D₆₁	1	1	yes	characteristic
213	C₄	#16840178	1	1	yes	characteristic
214	C₄	#16840180	1	1	yes	characteristic
215	C₂	—	122	2	no	—
216	C₂	#335×D₆₁	1	1	yes	characteristic
217	C₂	#21447576	1	1	yes	characteristic
218	C₂	#21447583	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄×D₆₁
commutator	C₂×C₄×C₆₁	C₂²×C₈
Frattini	C₄×C₈	C₂²×D₆₁
Fitting	C₆₁×#1370	C₂
socle	C₂²×C₆₁	C₂×C₄⋊C₈
prime radical	C₂²×D₆₁	C₄⋊C₈
cosocle	C₄×C₈×C₆₁	C₂³
hypercenter	#1370	D₆₁
hypocenter	C₆₁	#12005
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₄×C₈ ◁ #1370

Complements

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

N	H ≅ G/N	type
4. #21443889	C₂	semidirect
5. #21443888	C₂	semidirect
6. #21443888	C₂	semidirect
7. C₆₁×#1370	C₂	semidirect
112. #1370	D₆₁	semidirect
132. C₆₁	#12005	semidirect

Maximal subgroups

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

subgroup	index
#21447573	2
C₄⋊C₁₆×D₆₁	2
#21444101	2
#21443889	2
#21443888	2
C₆₁×#1370	2
#12005	61