Subgroups — C₂×#21447652 #29833178

Order 15616.

742 conjugacy classes of subgroups.

The subgroup lattice has 240 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂²×#14519716	C₂	1	2	yes	maximal, complemented
2	C₂×C₈×D₂₄₄	C₂	1	1	yes	maximal, characteristic
3	#21447652	C₂	1	8	yes	maximal, direct product factor
4	C₂×C₄⋊#11758185	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16839541	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×C₈×D₆₁	C₂²	1	2	yes	—
8	C₂×C₄×D₂₄₄	C₄	1	1	yes	characteristic
9	C₂×#14519716	—	2	8	no	—
10	C₂×#14519716	C₂²	1	8	yes	complemented
11	C₂²×#11758185	C₂²	1	2	yes	—
12	C₈×D₂₄₄	C₄	1	4	yes	—
13	C₈×D₂₄₄	C₂²	1	4	yes	—
14	C₂×C₄⋊#92418	C₄	1	1	yes	characteristic
15	C₂×#14519689	C₄	1	2	yes	—
16	C₄⋊#11758185	C₂²	1	4	yes	complemented
17	#16839541	C₂²	1	8	yes	complemented
18	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
19	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
20	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
21	C₄×C₁₆×C₆₁	C₂²	1	4	yes	complemented
22	C₂²×D₂₄₄	C₈	1	1	yes	characteristic
23	C₂²×C₄×D₆₁	C₂×C₄	1	2	yes	—
24	C₂²×#92418	C₂×C₄	1	2	yes	—
25	C₂×C₈×D₆₁	—	2	4	no	—
26	C₂×C₈×D₆₁	—	2	4	no	—
27	C₂×C₈×D₆₁	C₂×C₄	1	4	yes	—
28	C₂×C₈×D₆₁	C₂³	1	4	yes	—
29	C₄×D₂₄₄	C₈	1	4	yes	—
30	C₄×D₂₄₄	C₂×C₄	1	4	yes	—
31	C₂×C₄⋊Dic₆₁	C₈	1	1	yes	characteristic
32	C₂×#11758198	C₈	1	2	yes	—
33	#14519716	—	2	16	no	—
34	C₂×#11758185	—	2	4	no	—
35	C₂×#11758185	C₂³	1	4	yes	—
36	C₄⋊#92418	C₂×C₄	1	4	yes	—
37	#14519689	C₂×C₄	1	8	yes	—
38	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
39	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
41	C₂×C₁₆×C₆₁	—	2	2	no	—
42	C₂×C₁₆×C₆₁	D₄	1	4	yes	complemented
43	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
44	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
45	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
46	C₄×C₈×C₆₁	C₂³	1	2	yes	—
47	C₂³×D₆₁	C₂×C₈	1	2	yes	—
48	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
49	C₂²×C₄×C₆₁	C₂×C₈	1	1	yes	characteristic
50	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
51	C₂²×Dic₆₁	C₂×C₈	1	2	yes	—
52	C₂×D₂₄₄	—	2	4	no	—
53	C₂×D₂₄₄	C₂×C₈	1	4	yes	—
54	C₂×C₄×D₆₁	—	2	4	no	—
55	C₂×C₄×D₆₁	—	2	4	no	—
56	C₂×C₄×D₆₁	C₂×C₈	1	4	yes	—
57	C₂×C₄×D₆₁	C₂²×C₄	1	4	yes	—
58	C₂×C₈×C₆₁	—	2	2	no	—
59	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
60	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
61	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
62	C₂×C₈×C₆₁	#41	1	2	yes	—
63	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
64	C₄²×C₆₁	C₂×C₈	1	2	yes	—
65	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
66	#11758198	C₂×C₈	1	8	yes	—
67	C₄⋊Dic₆₁	C₂×C₈	1	4	yes	—
68	C₂×#92418	—	2	4	no	—
69	C₂×#92418	C₂²×C₄	1	4	yes	—
70	C₈×D₆₁	—	2	8	no	—
71	C₈×D₆₁	—	2	4	no	—
72	C₈×D₆₁	—	2	4	no	—
73	C₁₆×C₆₁	—	2	2	no	—
74	C₁₆×C₆₁	C₂×D₄	1	4	yes	complemented
75	#11758185	—	2	4	no	—
76	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
77	C₂²×D₆₁	—	2	4	no	—
78	C₂²×D₆₁	—	2	4	no	—
79	C₂²×D₆₁	—	2	4	no	—
80	C₂²×D₆₁	C₂²×C₈	1	4	yes	—
81	C₂×C₄×C₆₁	—	2	2	no	—
82	C₂×C₄×C₆₁	M₃₂	1	4	yes	complemented
83	C₂×C₄×C₆₁	C₄×D₄	1	2	yes	—
84	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
85	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
87	C₂×C₄×C₆₁	#131	1	2	yes	—
88	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
89	C₂×Dic₆₁	—	2	4	no	—
90	C₂×Dic₆₁	C₂²×C₈	1	4	yes	—
91	D₂₄₄	—	2	8	no	—
92	C₄×D₆₁	—	2	4	no	—
93	C₄×D₆₁	—	2	4	no	—
94	C₄×D₆₁	—	2	8	no	—
95	C₈×C₆₁	—	2	2	no	—
96	C₈×C₆₁	C₄×D₄	1	2	yes	—
97	C₈×C₆₁	C₂²×D₄	1	1	yes	characteristic
98	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
99	#92418	—	2	4	no	—
100	C₂×#2169	—	61	1	no	maximal, Sylow
101	C₂²×C₆₁	C₈×D₄	1	2	yes	—
102	C₂²×C₆₁	C₂×M₃₂	1	2	yes	—
103	C₂²×C₆₁	#504	1	2	yes	—
104	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
105	C₂×D₆₁	—	2	4	no	—
106	C₂×D₆₁	—	2	4	no	—
107	C₂×D₆₁	—	2	4	no	—
108	C₂×D₆₁	—	4	8	no	—
109	C₄×C₆₁	—	2	2	no	—
110	C₄×C₆₁	C₈×D₄	1	2	yes	—
111	C₄×C₆₁	C₂×M₃₂	1	4	yes	complemented
112	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
113	C₄×C₆₁	C₂×#131	1	1	yes	characteristic
114	Dic₆₁	—	2	4	no	—
115	C₂²×M₃₂	—	61	2	no	—
116	C₂×C₈×D₄	—	61	1	no	—
117	#2169	—	61	8	no	—
118	C₂×C₄⋊C₁₆	—	61	1	no	—
119	C₂×C₂²⋊C₁₆	—	61	2	no	—
120	C₂×C₄×C₁₆	D₆₁	1	1	yes	characteristic, complemented
121	C₂×C₆₁	#2169	1	4	yes	complemented
122	C₂×C₆₁	C₂×C₈×D₄	1	1	yes	characteristic
123	C₂×C₆₁	C₂²×M₃₂	1	1	yes	characteristic
124	C₂×C₆₁	C₂×#504	1	1	yes	characteristic
125	D₆₁	—	4	4	no	—
126	C₂³×C₈	—	61	2	no	—
127	C₂×C₄×D₄	—	61	1	no	—
128	C₂×M₃₂	—	61	8	no	—
129	C₂×M₃₂	—	122	8	no	—
130	C₂²×C₁₆	—	61	2	no	—
131	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
132	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
133	C₈×D₄	—	61	4	no	—
134	C₈×D₄	—	61	4	no	—
135	C₂×C₄⋊C₈	—	61	1	no	—
136	C₂×C₂²⋊C₈	—	61	2	no	—
137	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
138	C₄⋊C₁₆	—	61	4	no	—
139	C₂²⋊C₁₆	—	61	8	no	—
140	C₄×C₁₆	C₂×D₆₁	1	4	yes	complemented
141	C₆₁	C₂×#2169	1	1	yes	characteristic, complemented, Sylow
142	C₂²×D₄	—	61	1	no	—
143	C₂³×C₄	—	61	2	no	—
144	C₂²×C₈	—	61	2	no	—
145	C₂²×C₈	—	61	4	no	—
146	C₂²×C₈	—	61	4	no	—
147	C₂²×C₈	—	122	4	no	—
148	C₂²×C₈	—	122	4	no	—
149	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
150	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
151	C₄×D₄	—	61	4	no	—
152	C₄×D₄	—	61	4	no	—
153	C₂×C₄⋊C₄	—	61	1	no	—
154	C₂×C₂²⋊C₄	—	61	2	no	—
155	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
156	M₃₂	—	122	16	no	—
157	C₂×C₁₆	—	2	2	no	—
158	C₂×C₁₆	—	61	4	no	—
159	C₂×C₁₆	—	122	4	no	—
160	C₂×C₁₆	D₂₄₄	1	4	yes	complemented
161	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
162	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
163	C₄⋊C₈	—	61	4	no	—
164	C₂²⋊C₈	—	61	8	no	—
165	C₄×C₈	C₄×D₆₁	1	2	yes	—
166	C₄×C₈	C₂²×D₆₁	1	2	yes	—
167	C₂⁴	—	61	2	no	—
168	C₂×D₄	—	61	4	no	—
169	C₂×D₄	—	122	4	no	—
170	C₂²×C₄	—	61	2	no	—
171	C₂²×C₄	—	61	4	no	—
172	C₂²×C₄	—	61	4	no	—
173	C₂²×C₄	—	122	4	no	—
174	C₂²×C₄	—	122	4	no	—
175	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
176	C₂²×C₄	C₈×D₆₁	1	1	yes	characteristic
177	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
178	C₂×C₈	—	2	2	no	—
179	C₂×C₈	—	61	4	no	—
180	C₂×C₈	—	122	4	no	—
181	C₂×C₈	—	122	4	no	—
182	C₂×C₈	—	122	4	no	—
183	C₂×C₈	—	122	8	no	—
184	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
185	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
186	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
187	C₂×C₈	#11758222	1	2	yes	—
188	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
189	C₄⋊C₄	—	61	4	no	—
190	C₂²⋊C₄	—	61	8	no	—
191	C₄²	C₈×D₆₁	1	2	yes	—
192	C₄²	C₂×C₄×D₆₁	1	2	yes	—
193	C₁₆	—	2	2	no	—
194	C₁₆	—	122	4	no	—
195	C₁₆	C₂×D₂₄₄	1	4	yes	complemented
196	C₂³	—	61	4	no	—
197	C₂³	—	122	4	no	—
198	C₂³	—	122	4	no	—
199	C₂³	—	122	4	no	—
200	C₂³	C₂×C₈×D₆₁	1	1	yes	characteristic
201	D₄	—	122	8	no	—
202	C₂×C₄	—	2	2	no	—
203	C₂×C₄	—	61	4	no	—
204	C₂×C₄	—	122	4	no	—
205	C₂×C₄	—	122	4	no	—
206	C₂×C₄	—	122	4	no	—
207	C₂×C₄	—	122	8	no	—
208	C₂×C₄	#14519716	1	4	yes	complemented
209	C₂×C₄	C₄×D₂₄₄	1	2	yes	—
210	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
211	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
212	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
213	C₂×C₄	#14519783	1	2	yes	—
214	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
215	C₈	—	2	2	no	—
216	C₈	—	122	4	no	—
217	C₈	C₄×D₂₄₄	1	2	yes	—
218	C₈	C₂²×D₂₄₄	1	1	yes	characteristic
219	C₈	C₂×#11758222	1	1	yes	characteristic
220	C₂²	—	122	4	no	—
221	C₂²	—	122	4	no	—
222	C₂²	—	122	4	no	—
223	C₂²	—	244	8	no	—
224	C₂²	C₈×D₂₄₄	1	2	yes	—
225	C₂²	C₂×#14519716	1	2	yes	—
226	C₂²	#16840178	1	2	yes	—
227	C₂²	C₂²×C₈×D₆₁	1	1	yes	characteristic
228	C₄	—	2	2	no	—
229	C₄	—	122	4	no	—
230	C₄	C₈×D₂₄₄	1	2	yes	—
231	C₄	C₂×#14519716	1	4	yes	complemented
232	C₄	C₂×C₄×D₂₄₄	1	1	yes	characteristic
233	C₄	C₂×#14519783	1	1	yes	characteristic
234	C₂	—	244	4	no	—
235	C₂	#21447652	1	4	yes	direct product factor
236	C₂	C₂×C₈×D₂₄₄	1	1	yes	characteristic
237	C₂	C₂²×#14519716	1	1	yes	characteristic
238	C₂	C₂×#16840178	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₈	C₂²×D₆₁
commutator	C₂²×C₆₁	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₂×C₄×C₁₆×C₆₁	C₂
socle	C₂³×C₆₁	C₂²×C₈
prime radical	C₂²×D₂₄₄	C₈
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₆₁
hypocenter	C₆₁	C₂×#2169
LCS	G ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂²×#14519716	C₂	semidirect
3. #21447652	C₂	direct (N × H)
4. C₂×C₄⋊#11758185	C₂	semidirect
5. C₂×#16839541	C₂	semidirect
6. C₂×C₄×C₁₆×C₆₁	C₂	semidirect
10. C₂×#14519716	C₂²	semidirect
16. C₄⋊#11758185	C₂²	semidirect
17. #16839541	C₂²	semidirect
21. C₄×C₁₆×C₆₁	C₂²	semidirect
42. C₂×C₁₆×C₆₁	D₄	semidirect
74. C₁₆×C₆₁	C₂×D₄	semidirect
82. C₂×C₄×C₆₁	M₃₂	semidirect
111. C₄×C₆₁	C₂×M₃₂	semidirect
120. C₂×C₄×C₁₆	D₆₁	semidirect
121. C₂×C₆₁	#2169	semidirect
140. C₄×C₁₆	C₂×D₆₁	semidirect
141. C₆₁	C₂×#2169	semidirect
160. C₂×C₁₆	D₂₄₄	semidirect
195. C₁₆	C₂×D₂₄₄	semidirect
208. C₂×C₄	#14519716	semidirect
231. C₄	C₂×#14519716	semidirect
235. C₂	#21447652	direct (N × H)

Maximal subgroups

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

subgroup	index
C₂²×#14519716	2
C₂×C₈×D₂₄₄	2
#21447652	2
C₂×C₄⋊#11758185	2
C₂×#16839541	2
C₂×C₄×C₁₆×C₆₁	2
C₂×#2169	61

Subgroups — C2×#21447652 #29833178

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21447652 #29833178