Subgroups — #29600216

Order 15616.

290 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	C₂×#16840178	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839596	C₂	1	1	yes	maximal, characteristic, complemented
3	#21443958	C₂	1	4	yes	maximal, complemented
4	C₂×C₆₁×M₆₄	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#14519783	C₄	1	1	yes	characteristic
6	#16840178	—	2	1	no	—
7	#16840178	—	2	1	no	—
8	#16840178	—	2	1	no	—
9	#16840178	—	2	1	no	—
10	C₂×#14519717	C₄	1	1	yes	characteristic
11	C₂×#14519716	C₂²	1	2	yes	—
12	C₂×C₁₆×D₆₁	C₄	1	2	yes	—
13	#16839596	—	2	1	no	—
14	#16839596	C₂²	1	2	yes	—
15	C₂×#14519683	—	2	1	no	—
16	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
17	C₆₁×M₆₄	—	2	1	no	—
18	C₆₁×M₆₄	C₂²	1	2	yes	—
19	C₂×C₃₂×C₆₁	—	2	1	no	—
20	C₂×#11758222	C₈	1	1	yes	characteristic
21	#14519783	—	2	1	no	—
22	#14519783	—	2	1	no	—
23	#14519783	—	2	1	no	—
24	#14519783	—	2	1	no	—
25	C₂×#11758194	C₈	1	1	yes	characteristic
26	C₂×#11758189	C₈	1	2	yes	—
27	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
28	#14519717	—	2	1	no	—
29	#14519717	—	2	1	no	—
30	#14519716	—	2	2	no	—
31	#14519716	—	2	2	no	—
32	C₂×#11758185	C₂×C₄	1	2	yes	—
33	C₁₆×D₆₁	—	2	2	no	—
34	C₁₆×D₆₁	—	2	2	no	—
35	#14519683	—	2	2	no	—
36	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
37	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
38	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
39	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
40	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
41	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
42	C₂×C₁₆×C₆₁	C₂³	1	1	yes	characteristic
43	C₃₂×C₆₁	—	2	2	no	—
44	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
45	C₂×C₂²⋊D₆₁	C₁₆	1	2	yes	—
46	#11758222	—	2	1	no	—
47	#11758222	—	2	1	no	—
48	#11758222	—	2	1	no	—
49	#11758222	—	2	1	no	—
50	C₂×D₂₄₄	C₁₆	1	1	yes	characteristic
51	C₂×C₄×D₆₁	C₂×C₈	1	2	yes	—
52	C₂×Dic₁₂₂	C₁₆	1	1	yes	characteristic
53	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
54	C₂×C₈×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
55	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
56	C₂×C₈×C₆₁	M₁₆	1	1	yes	characteristic
57	C₂×C₈×C₆₁	M₁₆	1	1	yes	characteristic
58	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
59	#11758194	—	2	1	no	—
60	#11758194	—	2	1	no	—
61	C₂×#92418	C₂×C₈	1	2	yes	—
62	#11758189	—	2	2	no	—
63	#11758189	—	2	2	no	—
64	C₈×D₆₁	—	2	2	no	—
65	C₈×D₆₁	—	2	2	no	—
66	C₁₆×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
67	C₁₆×C₆₁	C₂²⋊C₄	1	1	yes	characteristic
68	C₁₆×C₆₁	C₂×D₄	1	1	yes	characteristic
69	C₁₆×C₆₁	C₂×D₄	1	1	yes	characteristic
70	#11758185	—	2	2	no	—
71	C₂³×C₆₁	C₂×C₁₆	1	1	yes	characteristic
72	C₂²×D₆₁	C₂×C₁₆	1	2	yes	—
73	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
74	C₂×C₄×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
75	C₂×C₄×C₆₁	C₂×C₁₆	1	1	yes	characteristic
76	C₂×C₄×C₆₁	M₃₂	1	1	yes	characteristic
77	C₂×C₄×C₆₁	M₃₂	1	1	yes	characteristic
78	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
79	C₂²⋊D₆₁	—	2	2	no	—
80	C₂²⋊D₆₁	—	2	2	no	—
81	C₂×Dic₆₁	C₂×C₁₆	1	2	yes	—
82	D₂₄₄	—	2	1	no	—
83	D₂₄₄	—	2	1	no	—
84	C₄×D₆₁	—	2	2	no	—
85	C₄×D₆₁	—	2	2	no	—
86	Dic₁₂₂	—	2	1	no	—
87	Dic₁₂₂	—	2	1	no	—
88	C₈×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
89	C₈×C₆₁	C₂²⋊C₈	1	1	yes	characteristic
90	C₈×C₆₁	C₂×C₂²⋊C₄	1	1	yes	characteristic
91	C₈×C₆₁	C₂×M₁₆	1	1	yes	characteristic
92	#92418	—	2	2	no	—
93	#13556	—	61	1	no	maximal, Sylow
94	C₂²×C₆₁	—	4	1	no	—
95	C₂²×C₆₁	C₂²⋊C₁₆	1	1	yes	characteristic
96	C₂²×C₆₁	C₂²⋊C₁₆	1	1	yes	characteristic
97	C₂²×C₆₁	C₂²×C₁₆	1	1	yes	characteristic
98	C₂×D₆₁	—	2	2	no	—
99	C₂×D₆₁	—	4	2	no	—
100	C₄×C₆₁	C₂²⋊C₁₆	1	1	yes	characteristic
101	C₄×C₆₁	C₂²⋊C₁₆	1	1	yes	characteristic
102	C₄×C₆₁	C₂×C₂²⋊C₈	1	1	yes	characteristic
103	C₄×C₆₁	C₂×M₃₂	1	1	yes	characteristic
104	Dic₆₁	—	2	2	no	—
105	C₂×#504	—	61	1	no	—
106	C₂×M₆₄	—	61	1	no	—
107	C₂×M₆₄	D₆₁	1	1	yes	characteristic, complemented
108	#1400	—	61	4	no	—
109	C₂×C₆₁	—	2	1	no	—
110	C₂×C₆₁	—	2	1	no	—
111	C₂×C₆₁	—	2	1	no	—
112	C₂×C₆₁	C₂×C₂²⋊C₁₆	1	1	yes	characteristic
113	D₆₁	—	4	2	no	—
114	C₂×#131	—	61	1	no	—
115	#504	—	122	1	no	—
116	#504	—	122	1	no	—
117	#504	—	122	1	no	—
118	#504	—	122	1	no	—
119	C₂×M₃₂	—	61	1	no	—
120	C₂×M₃₂	—	61	2	no	—
121	C₂²×C₁₆	—	61	2	no	—
122	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
123	M₆₄	—	2	1	no	—
124	M₆₄	—	61	2	no	—
125	M₆₄	—	122	1	no	—
126	M₆₄	C₂×D₆₁	1	2	yes	—
127	C₂×C₃₂	—	2	1	no	—
128	C₂×C₃₂	—	122	1	no	—
129	C₆₁	#13556	1	1	yes	characteristic, complemented, Sylow
130	C₂×#41	—	61	1	no	—
131	#131	—	122	1	no	—
132	#131	—	122	1	no	—
133	#131	—	122	1	no	—
134	#131	—	122	1	no	—
135	C₂×M₁₆	—	61	1	no	—
136	C₂×M₁₆	—	61	2	no	—
137	C₂²×C₈	—	61	2	no	—
138	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
139	M₃₂	—	122	1	no	—
140	M₃₂	—	122	1	no	—
141	M₃₂	—	122	2	no	—
142	M₃₂	—	122	2	no	—
143	C₂×C₁₆	—	61	2	no	—
144	C₂×C₁₆	—	122	2	no	—
145	C₂×C₁₆	—	122	2	no	—
146	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
147	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
148	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
149	C₂×C₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
150	C₂×C₁₆	C₂²⋊D₆₁	1	1	yes	characteristic
151	C₂×C₁₆	C₂²×D₆₁	1	1	yes	characteristic
152	C₃₂	—	2	2	no	—
153	C₃₂	—	122	2	no	—
154	#41	—	122	1	no	—
155	#41	—	122	1	no	—
156	#41	—	122	1	no	—
157	#41	—	122	1	no	—
158	C₂×Q₈	—	61	1	no	—
159	C₂×D₄	—	61	2	no	—
160	C₂×D₄	—	61	1	no	—
161	C₂²×C₄	—	61	2	no	—
162	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
163	M₁₆	—	122	1	no	—
164	M₁₆	—	122	1	no	—
165	M₁₆	—	122	2	no	—
166	M₁₆	—	122	2	no	—
167	C₂×C₈	—	61	2	no	—
168	C₂×C₈	—	122	2	no	—
169	C₂×C₈	—	122	2	no	—
170	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
171	C₂×C₈	#11758189	1	1	yes	characteristic
172	C₂×C₈	#11758189	1	1	yes	characteristic
173	C₂×C₈	#11758198	1	1	yes	characteristic
174	C₂×C₈	#11758198	1	1	yes	characteristic
175	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
176	C₁₆	—	122	2	no	—
177	C₁₆	#11758198	1	1	yes	characteristic
178	C₁₆	#11758198	1	1	yes	characteristic
179	C₁₆	C₂×D₂₄₄	1	1	yes	characteristic
180	C₁₆	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
181	C₂³	—	61	2	no	—
182	C₂³	C₁₆×D₆₁	1	1	yes	characteristic
183	Q₈	—	122	1	no	—
184	Q₈	—	122	1	no	—
185	D₄	—	122	2	no	—
186	D₄	—	122	2	no	—
187	D₄	—	122	1	no	—
188	D₄	—	122	1	no	—
189	C₂×C₄	—	61	2	no	—
190	C₂×C₄	—	122	2	no	—
191	C₂×C₄	—	122	2	no	—
192	C₂×C₄	#14519689	1	1	yes	characteristic
193	C₂×C₄	#14519689	1	1	yes	characteristic
194	C₂×C₄	C₁₆×D₆₁	1	1	yes	characteristic
195	C₂×C₄	#14519716	1	1	yes	characteristic
196	C₂×C₄	#14519716	1	1	yes	characteristic
197	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
198	C₈	—	122	2	no	—
199	C₈	#14519689	1	1	yes	characteristic
200	C₈	#14519689	1	1	yes	characteristic
201	C₈	C₂×#11758198	1	1	yes	characteristic
202	C₈	C₂×#11758189	1	1	yes	characteristic
203	C₂²	—	4	1	no	—
204	C₂²	—	122	2	no	—
205	C₂²	—	244	2	no	—
206	C₂²	#16839541	1	1	yes	characteristic
207	C₂²	#16839541	1	1	yes	characteristic
208	C₂²	C₂×C₁₆×D₆₁	1	1	yes	characteristic
209	C₄	—	122	2	no	—
210	C₄	#16839541	1	1	yes	characteristic
211	C₄	#16839541	1	1	yes	characteristic
212	C₄	C₂×#14519689	1	1	yes	characteristic
213	C₄	C₂×#14519716	1	1	yes	characteristic
214	C₂	—	2	1	no	—
215	C₂	—	2	1	no	—
216	C₂	—	2	1	no	—
217	C₂	—	244	2	no	—
218	C₂	C₂×#16839541	1	1	yes	characteristic

	subgroup	G/H
center	C₁₆	#11758198
commutator	C₂²×C₆₁	C₂²×C₁₆
Frattini	C₂×C₁₆	C₂²×D₆₁
Fitting	C₂×C₆₁×M₆₄	C₂
socle	C₂×C₆₁	C₂×C₂²⋊C₁₆
prime radical	C₂×#11758222	C₈
cosocle	C₂×C₁₆×C₆₁	C₂³
hypercenter	C₂×M₆₄	D₆₁
hypocenter	C₆₁	#13556
LCS	G ▷ C₂²×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₁₆ ◁ C₂²×C₁₆ ◁ C₂×M₆₄

Complements

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

N	H ≅ G/N	type
2. C₂×#16839596	C₂	semidirect
3. #21443958	C₂	semidirect
4. C₂×C₆₁×M₆₄	C₂	semidirect
107. C₂×M₆₄	D₆₁	semidirect
129. C₆₁	#13556	semidirect

Maximal subgroups

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

subgroup	index
C₂×#16840178	2
C₂×#16839596	2
#21443958	2
C₂×C₆₁×M₆₄	2
#13556	61