Subgroups — C₂×#21447662 #29833248

Order 15616.

654 conjugacy classes of subgroups.

The subgroup lattice has 268 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	2	yes	maximal, complemented
2	C₂×#16839880	C₂	1	1	yes	maximal, characteristic
3	#21447662	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16839583	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16839545	C₂	1	2	yes	maximal, complemented
6	C₂×C₆₁×#346	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14519799	C₄	1	1	yes	characteristic
8	C₂×#14519783	C₂²	1	2	yes	—
9	#16840178	—	2	4	no	—
10	#16840178	C₂²	1	8	yes	complemented
11	C₂×#14519717	C₂²	1	2	yes	—
12	C₂×#14519716	—	2	2	no	—
13	C₂×C₁₆×D₆₁	—	2	2	no	—
14	#16839880	C₄	1	4	yes	—
15	#16839880	C₂²	1	4	yes	—
16	C₂×#14519713	C₄	1	1	yes	characteristic
17	C₂×#14519705	C₄	1	2	yes	—
18	#16839583	C₂²	1	4	yes	complemented
19	#16839545	C₂²	1	8	yes	complemented
20	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₆₁×#346	C₂²	1	4	yes	complemented
24	C₂×#11758222	C₂×C₄	1	2	yes	—
25	#14519799	C₈	1	4	yes	—
26	#14519799	C₂×C₄	1	4	yes	—
27	C₂×Dic₂₄₄	C₈	1	1	yes	characteristic
28	C₂×#11758190	C₈	1	2	yes	—
29	C₂×D₄₈₈	C₈	1	1	yes	characteristic
30	#14519783	—	2	2	no	—
31	#14519783	—	2	2	no	—
32	#14519783	C₂×C₄	1	4	yes	—
33	#14519783	C₂³	1	4	yes	—
34	C₂×#11758194	C₂×C₄	1	2	yes	—
35	C₂×#11758189	—	2	2	no	—
36	C₂×C₈×D₆₁	—	2	2	no	—
37	#14519717	—	2	2	no	—
38	#14519717	C₂³	1	4	yes	—
39	#14519716	—	2	8	no	—
40	C₂×#11758185	—	2	2	no	—
41	C₁₆×D₆₁	—	2	8	no	—
42	#14519713	C₂×C₄	1	4	yes	—
43	#14519705	C₂×C₄	1	8	yes	—
44	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
47	C₂×C₁₆×C₆₁	—	2	1	no	—
48	C₂×C₁₆×C₆₁	—	2	1	no	—
49	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
50	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
51	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
52	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
53	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₆₁	C₂³	1	2	yes	—
55	C₂²×C₄×C₆₁	—	2	1	no	—
56	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
57	C₂×C₂²⋊D₆₁	—	2	2	no	—
58	#11758222	—	2	2	no	—
59	#11758222	—	2	2	no	—
60	#11758222	C₂×C₈	1	4	yes	—
61	#11758222	C₂²×C₄	1	4	yes	—
62	C₂×D₂₄₄	C₂×C₈	1	2	yes	—
63	C₂×C₄×D₆₁	—	2	2	no	—
64	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
65	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
66	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
67	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
68	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
70	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
73	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
74	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
75	C₄²×C₆₁	—	2	1	no	—
76	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
77	#11758194	—	2	2	no	—
78	#11758194	C₂²×C₄	1	4	yes	—
79	C₂×#92418	—	2	2	no	—
80	Dic₂₄₄	C₂×C₈	1	4	yes	—
81	D₄₈₈	C₂×C₈	1	4	yes	—
82	#11758190	C₂×C₈	1	8	yes	—
83	#11758189	—	2	4	no	—
84	#11758189	—	2	4	no	—
85	C₈×D₆₁	—	2	4	no	—
86	C₈×D₆₁	—	2	4	no	—
87	C₁₆×C₆₁	—	2	2	no	—
88	C₁₆×C₆₁	C₂×D₄	1	4	yes	—
89	#11758185	—	2	4	no	—
90	C₂³×C₆₁	#131	1	1	yes	characteristic
91	C₂²×D₆₁	—	2	2	no	—
92	C₂×C₄×C₆₁	—	2	2	no	—
93	C₂×C₄×C₆₁	—	4	2	no	—
94	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
100	C₂²⋊D₆₁	—	2	4	no	—
101	C₂²⋊D₆₁	—	2	4	no	—
102	C₂×Dic₆₁	—	2	2	no	—
103	D₂₄₄	—	2	2	no	—
104	D₂₄₄	C₂²×C₈	1	4	yes	—
105	C₄×D₆₁	—	2	4	no	—
106	C₄×D₆₁	—	2	4	no	—
107	Dic₁₂₂	—	2	2	no	—
108	Dic₁₂₂	C₂²×C₈	1	4	yes	—
109	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
110	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
111	C₈×C₆₁	C₂²×C₈	1	2	yes	—
112	C₈×C₆₁	C₂²×C₈	1	2	yes	—
113	C₈×C₆₁	C₂²×D₄	1	1	yes	characteristic
114	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
115	#92418	—	2	4	no	—
116	C₂×#2171	—	61	1	no	maximal, Sylow
117	C₂²×C₆₁	—	2	2	no	—
118	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
119	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
120	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
121	C₂×D₆₁	—	2	4	no	—
122	C₂×D₆₁	—	4	4	no	—
123	C₄×C₆₁	—	4	2	no	—
124	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
125	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
126	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
127	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
128	Dic₆₁	—	2	4	no	—
129	C₂×#504	—	61	2	no	—
130	C₂×#443	—	61	1	no	—
131	#2171	—	61	8	no	—
132	C₂×#364	—	61	1	no	—
133	C₂×#350	—	61	2	no	—
134	C₂×#346	D₆₁	1	1	yes	characteristic, complemented
135	C₂×C₆₁	—	2	1	no	—
136	C₂×C₆₁	—	2	1	no	—
137	C₂×C₆₁	#2171	1	2	yes	complemented
138	C₂×C₆₁	C₂×C₈×D₄	1	1	yes	characteristic
139	D₆₁	—	4	4	no	—
140	C₂×#135	—	61	1	no	—
141	C₂×#131	—	61	2	no	—
142	#504	—	61	8	no	—
143	#504	—	122	4	no	—
144	C₂×M₃₂	—	61	2	no	—
145	C₂×M₃₂	—	122	2	no	—
146	C₂²×C₁₆	—	122	2	no	—
147	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
148	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
149	#443	—	61	4	no	—
150	#443	—	61	4	no	—
151	C₂×#108	—	61	1	no	—
152	C₂×#104	—	61	2	no	—
153	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
154	#364	—	61	4	no	—
155	#350	—	61	8	no	—
156	#346	C₂×D₆₁	1	4	yes	complemented
157	C₆₁	C₂×#2171	1	1	yes	characteristic, complemented, Sylow
158	C₂×#41	—	61	2	no	—
159	#135	—	61	4	no	—
160	#135	—	61	4	no	—
161	C₂×Q₁₆	—	61	1	no	—
162	C₂×SD₁₆	—	61	2	no	—
163	C₂×D₈	—	61	1	no	—
164	#131	—	61	4	no	—
165	#131	—	61	4	no	—
166	#131	—	122	2	no	—
167	#131	—	122	2	no	—
168	C₂×M₁₆	—	61	2	no	—
169	C₂×M₁₆	—	122	2	no	—
170	C₂²×C₈	—	122	2	no	—
171	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
173	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
174	M₃₂	—	61	4	no	—
175	M₃₂	—	122	2	no	—
176	M₃₂	—	122	8	no	—
177	C₂×C₁₆	—	2	1	no	—
178	C₂×C₁₆	—	2	1	no	—
179	C₂×C₁₆	—	122	2	no	—
180	C₂×C₁₆	—	122	8	no	—
181	C₂×C₁₆	D₂₄₄	1	2	yes	—
182	C₂×C₁₆	D₂₄₄	1	2	yes	—
183	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
184	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
185	#108	—	61	4	no	—
186	#104	—	61	8	no	—
187	C₄×C₈	C₄×D₆₁	1	2	yes	—
188	C₄×C₈	C₂²×D₆₁	1	2	yes	—
189	#41	—	61	4	no	—
190	#41	—	61	4	no	—
191	#41	—	122	2	no	—
192	#41	—	122	2	no	—
193	C₂×Q₈	—	61	2	no	—
194	C₂×D₄	—	61	2	no	—
195	C₂×D₄	—	122	2	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	122	2	no	—
198	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
199	Q₁₆	—	61	4	no	—
200	SD₁₆	—	61	8	no	—
201	D₈	—	61	4	no	—
202	M₁₆	—	61	4	no	—
203	M₁₆	—	122	2	no	—
204	M₁₆	—	122	4	no	—
205	M₁₆	—	122	4	no	—
206	C₂×C₈	—	122	2	no	—
207	C₂×C₈	—	122	4	no	—
208	C₂×C₈	—	122	4	no	—
209	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
210	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
211	C₂×C₈	C₈×D₆₁	1	2	yes	—
212	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
213	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
214	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
215	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
216	C₂×C₈	#11758222	1	1	yes	characteristic
217	C₂×C₈	#11758222	1	1	yes	characteristic
218	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
219	C₄²	—	2	1	no	—
220	C₄²	C₂×C₄×D₆₁	1	2	yes	—
221	C₁₆	—	2	2	no	—
222	C₁₆	—	122	4	no	—
223	C₁₆	C₂×D₂₄₄	1	4	yes	—
224	C₂³	—	122	2	no	—
225	C₂³	#14519783	1	1	yes	characteristic
226	Q₈	—	61	4	no	—
227	Q₈	—	122	2	no	—
228	D₄	—	61	4	no	—
229	D₄	—	122	4	no	—
230	D₄	—	122	4	no	—
231	D₄	—	122	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	122	2	no	—
235	C₂×C₄	—	122	4	no	—
236	C₂×C₄	—	122	4	no	—
237	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
238	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
239	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
240	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
241	C₂×C₄	#14519783	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
243	C₈	—	122	4	no	—
244	C₈	C₄×D₂₄₄	1	1	yes	characteristic
245	C₈	C₄×D₂₄₄	1	1	yes	characteristic
246	C₈	C₂×C₈×D₆₁	1	2	yes	—
247	C₈	C₂×C₈×D₆₁	1	2	yes	—
248	C₈	C₂²×D₂₄₄	1	1	yes	characteristic
249	C₈	C₂×#11758222	1	1	yes	characteristic
250	C₂²	—	2	2	no	—
251	C₂²	—	122	4	no	—
252	C₂²	—	244	4	no	—
253	C₂²	C₈×D₂₄₄	1	1	yes	characteristic
254	C₂²	C₈×D₂₄₄	1	1	yes	characteristic
255	C₂²	C₂×#14519783	1	1	yes	characteristic
256	C₄	—	4	2	no	—
257	C₄	—	122	4	no	—
258	C₄	C₈×D₂₄₄	1	1	yes	characteristic
259	C₄	C₈×D₂₄₄	1	1	yes	characteristic
260	C₄	C₂×C₄×D₂₄₄	1	1	yes	characteristic
261	C₄	C₂²×C₈×D₆₁	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	244	4	no	—
265	C₂	#21447662	1	2	yes	direct product factor
266	C₂	C₂×C₈×D₂₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₂₄₄
commutator	C₄×C₆₁	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₂×C₆₁×#346	C₂
socle	C₂²×C₆₁	C₈×D₄
prime radical	C₂×#14519799	C₄
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	C₂×#346	D₆₁
hypocenter	C₆₁	C₂×#2171
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
1. C₂×#16840178	C₂	semidirect
3. #21447662	C₂	direct (N × H)
4. C₂×#16839583	C₂	semidirect
5. C₂×#16839545	C₂	semidirect
6. C₂×C₆₁×#346	C₂	semidirect
10. #16840178	C₂²	semidirect
18. #16839583	C₂²	semidirect
19. #16839545	C₂²	semidirect
23. C₆₁×#346	C₂²	semidirect
134. C₂×#346	D₆₁	semidirect
137. C₂×C₆₁	#2171	semidirect
156. #346	C₂×D₆₁	semidirect
157. C₆₁	C₂×#2171	semidirect
265. C₂	#21447662	direct (N × H)

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×#16839880	2
#21447662	2
C₂×#16839583	2
C₂×#16839545	2
C₂×C₆₁×#346	2
C₂×#2171	61

Subgroups — C2×#21447662 #29833248

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21447662 #29833248