Subgroups — C₂×#21447417 #29832649

Order 15616.

854 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	C₂²×#14519716	C₂	1	1	yes	maximal, characteristic
2	C₂²×C₁₆×D₆₁	C₂	1	1	yes	maximal, characteristic
3	C₂×#16839716	C₂	1	1	yes	maximal, characteristic
4	#21447417	C₂	1	8	yes	maximal, direct product factor
5	C₂×#16839537	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₄×#11758185	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×C₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₆₁×#346	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂²×#11758189	C₄	1	1	yes	characteristic
10	C₂²×C₈×D₆₁	C₂²	1	1	yes	characteristic
11	C₂×#14519734	C₄	1	1	yes	characteristic
12	C₂×#14519716	C₄	1	8	yes	complemented
13	C₂×#14519716	C₂²	1	4	yes	—
14	C₂²×#11758185	C₂²	1	1	yes	characteristic
15	C₂²×#11758185	C₂²	1	1	yes	characteristic
16	C₂×C₁₆×D₆₁	C₄	1	8	yes	complemented
17	C₂×C₁₆×D₆₁	C₂²	1	4	yes	—
18	#16839716	C₄	1	4	yes	—
19	#16839716	C₂²	1	4	yes	—
20	C₂×#14519688	C₄	1	1	yes	characteristic
21	C₂×#14519687	C₄	1	1	yes	characteristic
22	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
23	#16839537	C₂²	1	4	yes	complemented
24	C₄×#11758185	C₂²	1	4	yes	complemented
25	C₁₆×Dic₆₁	C₂²	1	4	yes	complemented
26	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
27	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
28	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
29	C₆₁×#346	C₂²	1	4	yes	complemented
30	C₂²×C₄×D₆₁	C₂×C₄	1	1	yes	characteristic
31	C₂×#11758189	C₈	1	8	yes	—
32	C₂×#11758189	C₂×C₄	1	4	yes	—
33	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
34	C₂²×#92418	C₂×C₄	1	1	yes	characteristic
35	C₂×C₈×D₆₁	C₂×C₄	1	2	yes	—
36	C₂×C₈×D₆₁	C₂×C₄	1	4	yes	—
37	C₂×C₈×D₆₁	C₂×C₄	1	4	yes	—
38	C₂×C₈×D₆₁	C₂³	1	2	yes	—
39	#14519734	C₈	1	4	yes	—
40	#14519734	C₂×C₄	1	4	yes	—
41	C₂×#11758196	C₈	1	2	yes	—
42	C₂×#11758198	C₈	1	2	yes	—
43	C₂×C₄×Dic₆₁	C₂×C₄	1	1	yes	characteristic
44	#14519716	C₂×C₄	1	16	yes	complemented
45	C₂×#11758185	C₂×C₄	1	4	yes	—
46	C₂×#11758185	C₂×C₄	1	4	yes	—
47	C₂×#11758185	C₂³	1	2	yes	—
48	C₂×#11758185	C₂³	1	2	yes	—
49	C₁₆×D₆₁	C₂×C₄	1	16	yes	complemented
50	#14519688	C₂×C₄	1	4	yes	—
51	#14519687	C₂×C₄	1	4	yes	—
52	C₈×Dic₆₁	C₂×C₄	1	2	yes	—
53	C₈×Dic₆₁	C₂³	1	2	yes	—
54	C₂²×C₈×C₆₁	C₂×C₄	1	1	yes	characteristic
55	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
56	C₂×C₄²×C₆₁	C₂×C₄	1	1	yes	characteristic
57	C₂×C₁₆×C₆₁	C₂×C₄	1	4	yes	—
58	C₂×C₁₆×C₆₁	C₂×C₄	1	4	yes	—
59	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
60	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
61	C₄×C₈×C₆₁	C₂×C₄	1	2	yes	—
62	C₄×C₈×C₆₁	C₂³	1	2	yes	—
63	C₂³×D₆₁	C₂×C₈	1	1	yes	characteristic
64	C₂²×C₄×C₆₁	C₂×C₈	1	2	yes	—
65	C₂²×C₄×C₆₁	C₂²×C₄	1	1	yes	characteristic
66	C₂²×Dic₆₁	C₂×C₈	1	2	yes	—
67	C₂²×Dic₆₁	C₂×C₈	1	1	yes	characteristic
68	C₂×C₄×D₆₁	C₄²	1	4	yes	—
69	C₂×C₄×D₆₁	C₂×C₈	1	2	yes	—
70	C₂×C₄×D₆₁	C₂×C₈	1	4	yes	—
71	C₂×C₄×D₆₁	C₂²×C₄	1	2	yes	—
72	C₂×C₈×C₆₁	C₂×C₈	1	4	yes	—
73	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
74	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
75	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
76	C₂×C₈×C₆₁	C₂²×C₄	1	2	yes	—
77	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
78	C₄²×C₆₁	C₂×C₈	1	2	yes	—
79	C₄²×C₆₁	C₂²×C₄	1	2	yes	—
80	#11758198	C₂×C₈	1	8	yes	—
81	#11758196	C₂×C₈	1	8	yes	—
82	C₄×Dic₆₁	C₂×C₈	1	2	yes	—
83	C₄×Dic₆₁	C₂²×C₄	1	2	yes	—
84	C₂×#92418	C₄²	1	4	yes	—
85	C₂×#92418	C₂×C₈	1	4	yes	—
86	C₂×#92418	C₂²×C₄	1	2	yes	—
87	C₂×#92418	C₂²×C₄	1	2	yes	—
88	#11758189	C₂×C₈	1	16	yes	—
89	C₈×D₆₁	C₄²	1	8	yes	—
90	C₈×D₆₁	C₂²×C₄	1	4	yes	—
91	C₈×D₆₁	C₂²×C₄	1	4	yes	—
92	C₁₆×C₆₁	C₂²×C₄	1	4	yes	—
93	C₁₆×C₆₁	C₂²×C₄	1	4	yes	—
94	#11758185	C₂²×C₄	1	4	yes	—
95	#11758185	C₂²×C₄	1	4	yes	—
96	C₂³×C₆₁	C₂²×C₈	1	1	yes	characteristic
97	C₂²×D₆₁	—	2	4	no	—
98	C₂²×D₆₁	C₄×C₈	1	4	yes	—
99	C₂²×D₆₁	C₂²×C₈	1	2	yes	—
100	C₂×C₄×C₆₁	—	2	4	no	—
101	C₂×C₄×C₆₁	C₂×C₄²	1	2	yes	—
102	C₂×C₄×C₆₁	C₂²×C₈	1	4	yes	—
103	C₂×C₄×C₆₁	C₂²×C₈	1	2	yes	—
104	C₂×C₄×C₆₁	C₂²×C₈	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₂³×C₄	1	1	yes	characteristic
106	C₂×Dic₆₁	—	2	4	no	—
107	C₂×Dic₆₁	C₄×C₈	1	4	yes	—
108	C₂×Dic₆₁	C₂²×C₈	1	4	yes	—
109	C₂×Dic₆₁	C₂²×C₈	1	2	yes	—
110	C₄×D₆₁	C₄×C₈	1	8	yes	—
111	C₄×D₆₁	C₂×C₄²	1	4	yes	—
112	C₄×D₆₁	C₂²×C₈	1	4	yes	—
113	C₈×C₆₁	C₂×C₄²	1	2	yes	—
114	C₈×C₆₁	C₂²×C₈	1	4	yes	—
115	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
116	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
117	#92418	C₂×C₄²	1	4	yes	—
118	#92418	C₂²×C₈	1	4	yes	—
119	C₂×#2108	—	61	1	no	maximal, Sylow
120	C₂²×C₆₁	C₂×C₄×C₈	1	2	yes	—
121	C₂²×C₆₁	#504	1	4	yes	—
122	C₂²×C₆₁	C₂³×C₈	1	1	yes	characteristic
123	C₂×D₆₁	—	2	8	no	—
124	C₂×D₆₁	—	2	4	no	—
125	C₂×D₆₁	C₂×C₄×C₈	1	4	yes	—
126	C₄×C₆₁	—	2	4	no	—
127	C₄×C₆₁	C₂×C₄×C₈	1	2	yes	—
128	C₄×C₆₁	C₂²×C₄²	1	1	yes	characteristic
129	C₄×C₆₁	C₂³×C₈	1	1	yes	characteristic
130	Dic₆₁	—	2	4	no	—
131	Dic₆₁	C₂×C₄×C₈	1	4	yes	—
132	C₂²×M₃₂	—	61	1	no	—
133	C₂³×C₁₆	—	61	1	no	—
134	C₂×#405	—	61	1	no	—
135	#2108	—	61	8	no	—
136	C₂×#346	—	61	1	no	—
137	C₂×#346	D₆₁	1	1	yes	characteristic, complemented
138	C₂×C₄×C₁₆	—	61	1	no	—
139	C₂×C₄×C₁₆	—	61	1	no	—
140	C₂×C₆₁	#2108	1	4	yes	complemented
141	C₂×C₆₁	C₂²×C₄×C₈	1	1	yes	characteristic
142	C₂×C₆₁	C₂×#504	1	2	yes	—
143	D₆₁	—	2	4	no	—
144	C₂²×M₁₆	—	61	1	no	—
145	C₂³×C₈	—	61	1	no	—
146	C₂×#117	—	61	1	no	—
147	C₂×M₃₂	—	61	8	no	—
148	C₂×M₃₂	—	61	4	no	—
149	C₂²×C₁₆	—	61	1	no	—
150	C₂²×C₁₆	—	61	1	no	—
151	C₂²×C₁₆	—	61	8	no	—
152	C₂²×C₁₆	—	61	4	no	—
153	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
154	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
155	#405	—	61	4	no	—
156	#405	—	61	4	no	—
157	C₂×#97	—	61	1	no	—
158	C₂×#97	—	61	1	no	—
159	C₂×C₄×C₈	—	61	1	no	—
160	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
161	#346	—	61	4	no	—
162	#346	C₂×D₆₁	1	4	yes	complemented
163	C₄×C₁₆	—	61	4	no	—
164	C₄×C₁₆	—	61	4	no	—
165	C₆₁	C₂×#2108	1	1	yes	characteristic, complemented, Sylow
166	C₂³×C₄	—	61	1	no	—
167	C₂×M₁₆	—	61	8	no	—
168	C₂×M₁₆	—	61	4	no	—
169	C₂²×C₈	—	61	1	no	—
170	C₂²×C₈	—	61	1	no	—
171	C₂²×C₈	—	61	2	no	—
172	C₂²×C₈	—	61	4	no	—
173	C₂²×C₈	—	61	4	no	—
174	C₂²×C₈	—	61	2	no	—
175	C₂²×C₈	C₄×D₆₁	1	1	yes	characteristic
176	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
177	#117	—	61	4	no	—
178	#117	—	61	4	no	—
179	C₂×C₄⋊C₄	—	61	2	no	—
180	C₂×C₂²⋊C₄	—	61	2	no	—
181	C₂×C₄²	—	61	1	no	—
182	C₂×C₄²	C₄×D₆₁	1	1	yes	characteristic
183	M₃₂	—	61	16	no	—
184	C₂×C₁₆	—	61	4	no	—
185	C₂×C₁₆	—	61	4	no	—
186	C₂×C₁₆	—	61	2	no	—
187	C₂×C₁₆	—	61	2	no	—
188	C₂×C₁₆	—	61	16	no	—
189	C₂×C₁₆	C₄×D₆₁	1	4	yes	—
190	C₂×C₁₆	C₄×D₆₁	1	4	yes	—
191	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
192	C₂×C₁₆	C₂²×D₆₁	1	2	yes	—
193	#97	—	61	4	no	—
194	#97	—	61	4	no	—
195	C₄×C₈	—	61	2	no	—
196	C₄×C₈	—	61	2	no	—
197	C₄×C₈	C₄×D₆₁	1	2	yes	—
198	C₄×C₈	C₂²×D₆₁	1	2	yes	—
199	C₂⁴	—	61	1	no	—
200	C₂²×C₄	—	61	1	no	—
201	C₂²×C₄	—	61	2	no	—
202	C₂²×C₄	—	61	4	no	—
203	C₂²×C₄	—	61	2	no	—
204	C₂²×C₄	—	61	4	no	—
205	C₂²×C₄	—	61	2	no	—
206	C₂²×C₄	C₈×D₆₁	1	2	yes	—
207	C₂²×C₄	C₂×C₄×D₆₁	1	1	yes	characteristic
208	M₁₆	—	61	16	no	—
209	C₂×C₈	—	61	2	no	—
210	C₂×C₈	—	61	2	no	—
211	C₂×C₈	—	61	4	no	—
212	C₂×C₈	—	61	4	no	—
213	C₂×C₈	—	61	4	no	—
214	C₂×C₈	—	61	4	no	—
215	C₂×C₈	—	61	8	no	—
216	C₂×C₈	C₈×D₆₁	1	4	yes	—
217	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
218	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
219	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
220	C₂×C₈	C₂×C₄×D₆₁	1	2	yes	—
221	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
222	C₄⋊C₄	—	61	8	no	—
223	C₂²⋊C₄	—	61	8	no	—
224	C₄²	—	61	2	no	—
225	C₄²	—	61	2	no	—
226	C₄²	C₈×D₆₁	1	2	yes	—
227	C₄²	C₂×C₄×D₆₁	1	2	yes	—
228	C₁₆	—	61	4	no	—
229	C₁₆	—	61	4	no	—
230	C₁₆	C₂×C₄×D₆₁	1	4	yes	—
231	C₁₆	C₂×C₄×D₆₁	1	4	yes	—
232	C₂³	—	61	4	no	—
233	C₂³	—	61	2	no	—
234	C₂³	—	122	4	no	—
235	C₂³	C₂×C₈×D₆₁	1	1	yes	characteristic
236	C₂×C₄	—	2	4	no	—
237	C₂×C₄	—	61	4	no	—
238	C₂×C₄	—	61	2	no	—
239	C₂×C₄	—	61	4	no	—
240	C₂×C₄	—	61	8	no	—
241	C₂×C₄	—	61	4	no	—
242	C₂×C₄	—	61	4	no	—
243	C₂×C₄	—	122	4	no	—
244	C₂×C₄	C₄²×D₆₁	1	2	yes	—
245	C₂×C₄	C₂×C₈×D₆₁	1	4	yes	—
246	C₂×C₄	C₂×C₈×D₆₁	1	1	yes	characteristic
247	C₂×C₄	C₂×C₈×D₆₁	1	2	yes	—
248	C₂×C₄	C₂²×C₄×D₆₁	1	1	yes	characteristic
249	C₈	—	61	4	no	—
250	C₈	—	61	4	no	—
251	C₈	C₄²×D₆₁	1	2	yes	—
252	C₈	C₂×C₈×D₆₁	1	4	yes	—
253	C₈	C₂²×C₄×D₆₁	1	1	yes	characteristic
254	C₈	C₂²×C₄×D₆₁	1	1	yes	characteristic
255	C₂²	—	61	4	no	—
256	C₂²	—	122	4	no	—
257	C₂²	—	122	8	no	—
258	C₂²	C₄×C₈×D₆₁	1	2	yes	—
259	C₂²	#16840180	1	4	yes	—
260	C₂²	C₂²×C₈×D₆₁	1	1	yes	characteristic
261	C₄	—	2	4	no	—
262	C₄	—	61	4	no	—
263	C₄	—	122	4	no	—
264	C₄	C₄×C₈×D₆₁	1	2	yes	—
265	C₄	C₂×C₄²×D₆₁	1	1	yes	characteristic
266	C₄	C₂²×C₈×D₆₁	1	1	yes	characteristic
267	C₂	—	122	4	no	—
268	C₂	#21447417	1	4	yes	direct product factor
269	C₂	C₂×C₄×C₈×D₆₁	1	1	yes	characteristic
270	C₂	C₂×#16840180	1	2	yes	—

	subgroup	G/H
center	C₂²×C₈	C₂²×D₆₁
commutator	C₂×C₆₁	C₂²×C₄×C₈
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₂×C₆₁×#346	C₂
socle	C₂³×C₆₁	C₂²×C₈
prime radical	C₂³×D₆₁	C₂×C₈
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	C₂×#346	D₆₁
hypocenter	C₆₁	C₂×#2108
LCS	G ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂²×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
4. #21447417	C₂	direct (N × H)
5. C₂×#16839537	C₂	semidirect
6. C₂×C₄×#11758185	C₂	semidirect
7. C₂×C₁₆×Dic₆₁	C₂	semidirect
8. C₂×C₆₁×#346	C₂	semidirect
12. C₂×#14519716	C₄	semidirect
16. C₂×C₁₆×D₆₁	C₄	semidirect
23. #16839537	C₂²	semidirect
24. C₄×#11758185	C₂²	semidirect
25. C₁₆×Dic₆₁	C₂²	semidirect
29. C₆₁×#346	C₂²	semidirect
44. #14519716	C₂×C₄	semidirect
49. C₁₆×D₆₁	C₂×C₄	semidirect
137. C₂×#346	D₆₁	semidirect
140. C₂×C₆₁	#2108	semidirect
162. #346	C₂×D₆₁	semidirect
165. C₆₁	C₂×#2108	semidirect
268. C₂	#21447417	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
C₂²×C₁₆×D₆₁	2
C₂×#16839716	2
#21447417	2
C₂×#16839537	2
C₂×C₄×#11758185	2
C₂×C₁₆×Dic₆₁	2
C₂×C₆₁×#346	2
C₂×#2108	61

Subgroups — C2×#21447417 #29832649

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21447417 #29832649