Subgroups — C₂×#21447657 #29833242

Order 15616.

662 conjugacy classes of subgroups.

The subgroup lattice has 272 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	#21447657	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×C₄×C₁₆×C₆₁	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	C₄×C₁₆×C₆₁	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	1	yes	characteristic
50	C₂×C₁₆×C₆₁	D₄	1	2	yes	—
51	C₂×C₁₆×C₆₁	D₄	1	1	yes	characteristic
52	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
53	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
54	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
55	C₄×C₈×C₆₁	C₂³	1	2	yes	—
56	C₂²×C₄×C₆₁	—	2	1	no	—
57	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
58	C₂×C₂²⋊D₆₁	—	2	2	no	—
59	#11758222	—	2	2	no	—
60	#11758222	—	2	2	no	—
61	#11758222	C₂×C₈	1	4	yes	—
62	#11758222	C₂²×C₄	1	4	yes	—
63	C₂×D₂₄₄	C₂×C₈	1	2	yes	—
64	C₂×C₄×D₆₁	—	2	2	no	—
65	C₂×Dic₁₂₂	C₂×C₈	1	2	yes	—
66	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
67	C₂×C₈×C₆₁	C₂×C₈	1	1	yes	characteristic
68	C₂×C₈×C₆₁	C₂×C₈	1	2	yes	—
69	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
71	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
72	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
73	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
74	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
75	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
76	C₄²×C₆₁	—	2	1	no	—
77	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
78	#11758194	—	2	2	no	—
79	#11758194	C₂²×C₄	1	4	yes	—
80	C₂×#92418	—	2	2	no	—
81	Dic₂₄₄	C₂×C₈	1	4	yes	—
82	D₄₈₈	C₂×C₈	1	4	yes	—
83	#11758190	C₂×C₈	1	8	yes	—
84	#11758189	—	2	4	no	—
85	#11758189	—	2	4	no	—
86	C₈×D₆₁	—	2	4	no	—
87	C₈×D₆₁	—	2	4	no	—
88	C₁₆×C₆₁	—	2	2	no	—
89	C₁₆×C₆₁	C₂×D₄	1	2	yes	—
90	C₁₆×C₆₁	C₂×D₄	1	2	yes	—
91	#11758185	—	2	4	no	—
92	C₂³×C₆₁	#131	1	1	yes	characteristic
93	C₂²×D₆₁	—	2	2	no	—
94	C₂×C₄×C₆₁	—	2	4	no	—
95	C₂×C₄×C₆₁	—	2	2	no	—
96	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
98	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
99	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
100	C₂×C₄×C₆₁	#131	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
102	C₂²⋊D₆₁	—	2	4	no	—
103	C₂²⋊D₆₁	—	2	4	no	—
104	C₂×Dic₆₁	—	2	2	no	—
105	D₂₄₄	—	2	2	no	—
106	D₂₄₄	C₂²×C₈	1	4	yes	—
107	C₄×D₆₁	—	2	4	no	—
108	C₄×D₆₁	—	2	4	no	—
109	Dic₁₂₂	—	2	2	no	—
110	Dic₁₂₂	C₂²×C₈	1	4	yes	—
111	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
112	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
113	C₈×C₆₁	C₂²×C₈	1	2	yes	—
114	C₈×C₆₁	C₂²×C₈	1	2	yes	—
115	C₈×C₆₁	C₂²×D₄	1	1	yes	characteristic
116	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
117	#92418	—	2	4	no	—
118	C₂×#2170	—	61	1	no	maximal, Sylow
119	C₂²×C₆₁	—	2	2	no	—
120	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
121	C₂²×C₆₁	C₈×D₄	1	1	yes	characteristic
122	C₂²×C₆₁	C₂×#131	1	1	yes	characteristic
123	C₂×D₆₁	—	2	4	no	—
124	C₂×D₆₁	—	4	4	no	—
125	C₄×C₆₁	—	2	4	no	—
126	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
127	C₄×C₆₁	C₈×D₄	1	1	yes	characteristic
128	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
129	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
130	Dic₆₁	—	2	4	no	—
131	C₂×#504	—	61	2	no	—
132	C₂×#443	—	61	1	no	—
133	#2170	—	61	8	no	—
134	C₂×#364	—	61	1	no	—
135	C₂×#350	—	61	2	no	—
136	C₂×C₄×C₁₆	D₆₁	1	1	yes	characteristic, complemented
137	C₂×C₆₁	—	2	1	no	—
138	C₂×C₆₁	—	2	1	no	—
139	C₂×C₆₁	#2170	1	2	yes	complemented
140	C₂×C₆₁	C₂×C₈×D₄	1	1	yes	characteristic
141	D₆₁	—	4	4	no	—
142	C₂×#135	—	61	1	no	—
143	C₂×#131	—	61	2	no	—
144	#504	—	61	8	no	—
145	#504	—	122	4	no	—
146	C₂×M₃₂	—	61	2	no	—
147	C₂×M₃₂	—	122	2	no	—
148	C₂²×C₁₆	—	122	2	no	—
149	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
150	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
151	#443	—	61	4	no	—
152	#443	—	61	4	no	—
153	C₂×#108	—	61	1	no	—
154	C₂×#104	—	61	2	no	—
155	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
156	#364	—	61	4	no	—
157	#350	—	61	8	no	—
158	C₄×C₁₆	C₂×D₆₁	1	4	yes	complemented
159	C₆₁	C₂×#2170	1	1	yes	characteristic, complemented, Sylow
160	C₂×#41	—	61	2	no	—
161	#135	—	61	4	no	—
162	#135	—	61	4	no	—
163	C₂×Q₁₆	—	61	1	no	—
164	C₂×SD₁₆	—	61	2	no	—
165	C₂×D₈	—	61	1	no	—
166	#131	—	61	4	no	—
167	#131	—	61	4	no	—
168	#131	—	122	2	no	—
169	#131	—	122	2	no	—
170	C₂×M₁₆	—	61	2	no	—
171	C₂×M₁₆	—	122	2	no	—
172	C₂²×C₈	—	122	2	no	—
173	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
174	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
175	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
176	M₃₂	—	61	4	no	—
177	M₃₂	—	122	2	no	—
178	M₃₂	—	122	8	no	—
179	C₂×C₁₆	—	2	1	no	—
180	C₂×C₁₆	—	2	1	no	—
181	C₂×C₁₆	—	122	2	no	—
182	C₂×C₁₆	—	122	8	no	—
183	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
184	C₂×C₁₆	D₂₄₄	1	2	yes	—
185	C₂×C₁₆	D₂₄₄	1	1	yes	characteristic
186	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
187	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
188	#108	—	61	4	no	—
189	#104	—	61	8	no	—
190	C₄×C₈	C₄×D₆₁	1	2	yes	—
191	C₄×C₈	C₂²×D₆₁	1	2	yes	—
192	#41	—	61	4	no	—
193	#41	—	61	4	no	—
194	#41	—	122	2	no	—
195	#41	—	122	2	no	—
196	C₂×Q₈	—	61	2	no	—
197	C₂×D₄	—	61	2	no	—
198	C₂×D₄	—	122	2	no	—
199	C₂²×C₄	—	2	1	no	—
200	C₂²×C₄	—	122	2	no	—
201	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
202	Q₁₆	—	61	4	no	—
203	SD₁₆	—	61	8	no	—
204	D₈	—	61	4	no	—
205	M₁₆	—	61	4	no	—
206	M₁₆	—	122	2	no	—
207	M₁₆	—	122	4	no	—
208	M₁₆	—	122	4	no	—
209	C₂×C₈	—	122	2	no	—
210	C₂×C₈	—	122	4	no	—
211	C₂×C₈	—	122	4	no	—
212	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
213	C₂×C₈	C₈×D₆₁	1	1	yes	characteristic
214	C₂×C₈	C₈×D₆₁	1	2	yes	—
215	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
216	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
217	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
218	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
219	C₂×C₈	#11758222	1	1	yes	characteristic
220	C₂×C₈	#11758222	1	1	yes	characteristic
221	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
222	C₄²	—	2	1	no	—
223	C₄²	C₂×C₄×D₆₁	1	2	yes	—
224	C₁₆	—	2	2	no	—
225	C₁₆	—	122	4	no	—
226	C₁₆	C₂×D₂₄₄	1	2	yes	—
227	C₁₆	C₂×D₂₄₄	1	2	yes	—
228	C₂³	—	122	2	no	—
229	C₂³	#14519783	1	1	yes	characteristic
230	Q₈	—	61	4	no	—
231	Q₈	—	122	2	no	—
232	D₄	—	61	4	no	—
233	D₄	—	122	4	no	—
234	D₄	—	122	4	no	—
235	D₄	—	122	2	no	—
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	2	2	no	—
238	C₂×C₄	—	122	2	no	—
239	C₂×C₄	—	122	4	no	—
240	C₂×C₄	—	122	4	no	—
241	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
242	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
243	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
244	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
245	C₂×C₄	#14519783	1	1	yes	characteristic
246	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
247	C₈	—	122	4	no	—
248	C₈	C₄×D₂₄₄	1	1	yes	characteristic
249	C₈	C₄×D₂₄₄	1	1	yes	characteristic
250	C₈	C₂×C₈×D₆₁	1	2	yes	—
251	C₈	C₂×C₈×D₆₁	1	2	yes	—
252	C₈	C₂²×D₂₄₄	1	1	yes	characteristic
253	C₈	C₂×#11758222	1	1	yes	characteristic
254	C₂²	—	2	2	no	—
255	C₂²	—	122	4	no	—
256	C₂²	—	244	4	no	—
257	C₂²	C₈×D₂₄₄	1	1	yes	characteristic
258	C₂²	C₈×D₂₄₄	1	1	yes	characteristic
259	C₂²	C₂×#14519783	1	1	yes	characteristic
260	C₄	—	2	4	no	—
261	C₄	—	122	4	no	—
262	C₄	C₈×D₂₄₄	1	1	yes	characteristic
263	C₄	C₈×D₂₄₄	1	1	yes	characteristic
264	C₄	C₂×C₄×D₂₄₄	1	1	yes	characteristic
265	C₄	C₂²×C₈×D₆₁	1	1	yes	characteristic
266	C₂	—	2	1	no	—
267	C₂	—	2	1	no	—
268	C₂	—	244	4	no	—
269	C₂	#21447657	1	2	yes	direct product factor
270	C₂	C₂×C₈×D₂₄₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₁₆	D₂₄₄
commutator	C₄×C₆₁	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₂×C₄×C₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₈×D₄
prime radical	C₂×#14519799	C₄
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₆₁
hypocenter	C₆₁	C₂×#2170
LCS	G ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₁₆ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#16840178	C₂	semidirect
3. #21447657	C₂	direct (N × H)
4. C₂×#16839583	C₂	semidirect
5. C₂×#16839545	C₂	semidirect
6. C₂×C₄×C₁₆×C₆₁	C₂	semidirect
10. #16840178	C₂²	semidirect
18. #16839583	C₂²	semidirect
19. #16839545	C₂²	semidirect
23. C₄×C₁₆×C₆₁	C₂²	semidirect
136. C₂×C₄×C₁₆	D₆₁	semidirect
139. C₂×C₆₁	#2170	semidirect
158. C₄×C₁₆	C₂×D₆₁	semidirect
159. C₆₁	C₂×#2170	semidirect
269. C₂	#21447657	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
#21447657	2
C₂×#16839583	2
C₂×#16839545	2
C₂×C₄×C₁₆×C₆₁	2
C₂×#2170	61

Subgroups — C2×#21447657 #29833242

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21447657 #29833242