Subgroups — #29595387

Order 15616.

296 conjugacy classes of subgroups.

The subgroup lattice has 210 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16839583	C₂	1	1	yes	maximal, characteristic
2	#21447423	C₂	1	2	yes	maximal
3	#21445698	C₂	1	1	yes	maximal, characteristic
4	#21443904	C₂	1	2	yes	maximal
5	C₄×M₃₂×C₆₁	C₂	1	1	yes	maximal, characteristic
6	C₂×#14519717	C₂²	1	2	yes	—
7	C₂×#14519713	C₄	1	1	yes	characteristic
8	#16839800	C₄	1	1	yes	characteristic
9	#16839717	C₂²	1	2	yes	—
10	#16839583	C₄	1	2	yes	complemented
11	#16839583	C₄	1	2	yes	complemented
12	#16839539	—	2	4	no	—
13	#16839522	C₄	1	2	yes	—
14	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
15	C₂×M₃₂×C₆₁	C₂²	1	1	yes	characteristic
16	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×#346	C₄	1	2	yes	complemented
18	C₄×C₁₆×C₆₁	C₄	1	2	yes	complemented
19	C₂×#11758194	C₂×C₄	1	2	yes	—
20	#14519736	C₂×C₄	1	2	yes	—
21	#14519717	—	2	2	no	—
22	#14519717	C₂×C₄	1	4	yes	—
23	C₂×#11758185	—	2	2	no	—
24	#14519713	—	2	1	no	—
25	#14519713	—	2	1	no	—
26	#14519711	—	2	1	no	—
27	#14519709	—	2	1	no	—
28	#14519687	—	2	2	no	—
29	C₈×Dic₆₁	—	2	2	no	—
30	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
32	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
33	M₃₂×C₆₁	—	2	1	no	—
34	M₃₂×C₆₁	—	2	1	no	—
35	M₃₂×C₆₁	D₄	1	2	yes	—
36	M₃₂×C₆₁	Q₈	1	2	yes	—
37	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
38	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
39	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
40	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
41	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
42	C₄×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
43	C₂²×C₄×C₆₁	—	2	1	no	—
44	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
45	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
46	C₂×C₈×C₆₁	C₄²	1	1	yes	characteristic
47	C₂×C₈×C₆₁	M₁₆	1	1	yes	characteristic
48	C₂×C₈×C₆₁	M₁₆	1	1	yes	characteristic
49	C₂×C₈×C₆₁	M₁₆	1	1	yes	characteristic
50	C₂×C₈×C₆₁	M₁₆	1	1	yes	characteristic
51	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
52	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
54	C₂×C₈×C₆₁	C₂×Q₈	1	1	yes	characteristic
55	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
56	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
57	C₄²×C₆₁	—	2	1	no	—
58	C₄²×C₆₁	C₄²	1	1	yes	characteristic
59	C₄²×C₆₁	C₄²	1	1	yes	characteristic
60	C₂²⋊Dic₆₁	—	2	2	no	—
61	C₄⋊Dic₆₁	—	2	2	no	—
62	C₄×Dic₆₁	—	2	2	no	—
63	#11758194	—	2	2	no	—
64	#11758194	—	2	2	no	—
65	C₂×#92418	—	2	2	no	—
66	C₁₆×C₆₁	—	2	2	no	—
67	C₁₆×C₆₁	C₄⋊C₄	1	4	yes	—
68	#11758185	—	2	4	no	—
69	C₂³×C₆₁	#131	1	1	yes	characteristic
70	C₂×C₄×C₆₁	—	2	1	no	—
71	C₂×C₄×C₆₁	—	2	1	no	—
72	C₂×C₄×C₆₁	—	4	2	no	—
73	C₂×C₄×C₆₁	C₂×C₄²	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₄×Q₈	1	1	yes	characteristic
76	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
77	C₂×C₄×C₆₁	C₂×M₁₆	1	1	yes	characteristic
78	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
79	C₂×Dic₆₁	—	4	2	no	—
80	C₈×C₆₁	#97	1	1	yes	characteristic
81	C₈×C₆₁	#97	1	1	yes	characteristic
82	C₈×C₆₁	#97	1	1	yes	characteristic
83	C₈×C₆₁	#97	1	1	yes	characteristic
84	C₈×C₆₁	C₂×C₄⋊C₄	1	1	yes	characteristic
85	C₈×C₆₁	#117	1	1	yes	characteristic
86	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
87	C₈×C₆₁	C₄×Q₈	1	1	yes	characteristic
88	#92418	—	4	2	no	—
89	#12667	—	61	1	no	maximal, Sylow
90	C₂²×C₆₁	—	4	1	no	—
91	C₂²×C₆₁	#405	1	1	yes	characteristic
92	C₂²×C₆₁	#436	1	1	yes	characteristic
93	C₂²×C₆₁	#446	1	1	yes	characteristic
94	C₄×C₆₁	—	2	2	no	—
95	C₄×C₆₁	—	4	1	no	—
96	C₄×C₆₁	C₄×C₄⋊C₄	1	1	yes	characteristic
97	C₄×C₆₁	C₂×#97	1	1	yes	characteristic
98	C₄×C₆₁	#436	1	1	yes	characteristic
99	C₄×C₆₁	#446	1	1	yes	characteristic
100	Dic₆₁	—	8	2	no	—
101	C₂×#364	—	61	1	no	—
102	#2110	—	61	2	no	—
103	C₄×M₃₂	D₆₁	1	1	yes	characteristic
104	#1772	—	61	1	no	—
105	#1378	—	61	2	no	—
106	C₂×C₆₁	—	2	1	no	—
107	C₂×C₆₁	—	2	1	no	—
108	C₂×C₆₁	—	2	1	no	—
109	C₂×C₆₁	#1770	1	1	yes	characteristic
110	C₂×M₃₂	—	61	2	no	—
111	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
112	C₂×M₃₂	C₂×D₆₁	1	1	yes	characteristic
113	C₂×#108	—	61	1	no	—
114	#427	—	61	1	no	—
115	#405	—	61	2	no	—
116	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
117	#364	—	61	2	no	—
118	#364	—	61	2	no	—
119	#347	—	122	4	no	—
120	#346	Dic₆₁	1	2	yes	complemented
121	C₄×C₁₆	Dic₆₁	1	2	yes	complemented
122	#339	—	61	2	no	—
123	C₆₁	#12667	1	1	yes	characteristic, complemented, Sylow
124	C₂×M₁₆	—	61	2	no	—
125	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
126	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
127	#117	—	61	2	no	—
128	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
129	M₃₂	—	2	1	no	—
130	M₃₂	—	2	1	no	—
131	M₃₂	—	61	4	no	—
132	M₃₂	—	122	2	no	—
133	M₃₂	Dic₁₂₂	1	2	yes	—
134	M₃₂	D₂₄₄	1	2	yes	—
135	C₂×C₁₆	—	122	2	no	—
136	C₂×C₁₆	C₂×Dic₆₁	1	2	yes	—
137	C₂×C₁₆	C₂×Dic₆₁	1	2	yes	—
138	#108	—	122	1	no	—
139	#108	—	122	1	no	—
140	#107	—	122	1	no	—
141	#106	—	122	1	no	—
142	#97	—	122	2	no	—
143	C₄×C₈	—	122	2	no	—
144	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
145	C₄×C₈	C₄×D₆₁	1	1	yes	characteristic
146	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
147	C₄×C₈	C₂×Dic₆₁	1	1	yes	characteristic
148	C₂²×C₄	—	2	1	no	—
149	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
150	M₁₆	—	122	2	no	—
151	M₁₆	—	122	2	no	—
152	C₂×C₈	—	122	2	no	—
153	C₂×C₈	#11758189	1	1	yes	characteristic
154	C₂×C₈	#11758189	1	1	yes	characteristic
155	C₂×C₈	#11758189	1	1	yes	characteristic
156	C₂×C₈	#11758189	1	1	yes	characteristic
157	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
158	C₂×C₈	C₄×Dic₆₁	1	1	yes	characteristic
159	C₂×C₈	C₂×Dic₁₂₂	1	1	yes	characteristic
160	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
161	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
162	C₂×C₈	#11758222	1	1	yes	characteristic
163	C₂×C₈	#11758222	1	1	yes	characteristic
164	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
165	C₄⋊C₄	—	122	2	no	—
166	C₂²⋊C₄	—	122	2	no	—
167	C₄²	—	2	1	no	—
168	C₄²	—	122	2	no	—
169	C₄²	C₄×Dic₆₁	1	1	yes	characteristic
170	C₄²	C₄×Dic₆₁	1	1	yes	characteristic
171	C₁₆	—	2	2	no	—
172	C₁₆	—	122	4	no	—
173	C₁₆	C₄⋊Dic₆₁	1	4	yes	—
174	C₂³	#14519783	1	1	yes	characteristic
175	C₂×C₄	—	2	1	no	—
176	C₂×C₄	—	2	1	no	—
177	C₂×C₄	—	4	2	no	—
178	C₂×C₄	—	244	2	no	—
179	C₂×C₄	C₂×C₄×Dic₆₁	1	1	yes	characteristic
180	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
181	C₂×C₄	C₄×Dic₁₂₂	1	1	yes	characteristic
182	C₂×C₄	C₂×#11758189	1	1	yes	characteristic
183	C₂×C₄	C₂×#11758189	1	1	yes	characteristic
184	C₂×C₄	#14519783	1	1	yes	characteristic
185	C₈	—	244	2	no	—
186	C₈	#14519687	1	1	yes	characteristic
187	C₈	#14519687	1	1	yes	characteristic
188	C₈	#14519687	1	1	yes	characteristic
189	C₈	#14519687	1	1	yes	characteristic
190	C₈	C₂×C₄⋊Dic₆₁	1	1	yes	characteristic
191	C₈	#14519736	1	1	yes	characteristic
192	C₈	C₄×D₂₄₄	1	1	yes	characteristic
193	C₈	C₄×Dic₁₂₂	1	1	yes	characteristic
194	C₂²	—	4	1	no	—
195	C₂²	#16839717	1	1	yes	characteristic
196	C₂²	#16839841	1	1	yes	characteristic
197	C₂²	#16839892	1	1	yes	characteristic
198	C₄	—	2	2	no	—
199	C₄	—	4	1	no	—
200	C₄	—	488	2	no	—
201	C₄	C₄×C₄⋊Dic₆₁	1	1	yes	characteristic
202	C₄	C₂×#14519687	1	1	yes	characteristic
203	C₄	#16839841	1	1	yes	characteristic
204	C₄	#16839892	1	1	yes	characteristic
205	C₂	—	2	1	no	—
206	C₂	—	2	1	no	—
207	C₂	—	2	1	no	—
208	C₂	#21445686	1	1	yes	characteristic

	subgroup	G/H
center	C₈	#14519736
commutator	C₂×C₄×C₆₁	C₂×C₄²
Frattini	C₂²×C₈	C₂²×D₆₁
Fitting	C₄×M₃₂×C₆₁	C₂
socle	C₂×C₆₁	#1770
prime radical	C₂³×C₆₁	#131
cosocle	C₂²×C₈×C₆₁	C₂³
hypercenter	C₄×M₃₂	D₆₁
hypocenter	C₆₁	#12667
LCS	G ▷ C₂×C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₈ ◁ C₂×M₃₂ ◁ C₄×M₃₂

Complements

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

N	H ≅ G/N	type
10. #16839583	C₄	semidirect
11. #16839583	C₄	semidirect
17. C₆₁×#346	C₄	semidirect
18. C₄×C₁₆×C₆₁	C₄	semidirect
120. #346	Dic₆₁	semidirect
121. C₄×C₁₆	Dic₆₁	semidirect
123. C₆₁	#12667	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16839583	2
#21447423	2
#21445698	2
#21443904	2
C₄×M₃₂×C₆₁	2
#12667	61