Subgroups — C₂×#21447701 #29833414

Order 15616.

662 conjugacy classes of subgroups.

The subgroup lattice has 270 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#16840194	C₂	1	1	yes	maximal, characteristic
2	C₂×#16839880	C₂	1	2	yes	maximal, complemented
3	#21447701	C₂	1	8	yes	maximal, direct product factor
4	C₂×#16839592	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#16839570	C₂	1	2	yes	maximal, complemented
6	C₂×C₄×C₁₆×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14519799	C₂²	1	2	yes	—
8	C₂×#14519783	—	2	2	no	—
9	#16840194	C₄	1	4	yes	complemented
10	#16840194	C₂²	1	4	yes	—
11	C₂×Dic₄₈₈	C₄	1	1	yes	characteristic, complemented
12	C₂×#14519722	C₄	1	2	yes	complemented
13	C₂×D₉₇₆	C₄	1	1	yes	characteristic, complemented
14	#16839880	—	2	4	no	—
15	#16839880	C₂²	1	8	yes	complemented
16	C₂×#14519713	C₂²	1	2	yes	—
17	C₂×#14519705	—	2	2	no	—
18	#16839592	C₂²	1	4	yes	complemented
19	#16839570	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	—	2	2	no	—
25	#14519799	—	2	2	no	—
26	#14519799	—	2	2	no	—
27	#14519799	C₂×C₄	1	4	yes	—
28	#14519799	C₂³	1	4	yes	—
29	C₂×Dic₂₄₄	C₂×C₄	1	2	yes	—
30	C₂×#11758190	—	2	2	no	—
31	C₂×D₄₈₈	C₂×C₄	1	2	yes	—
32	#14519783	—	2	8	no	—
33	#14519783	—	4	4	no	—
34	C₂×#11758194	—	2	2	no	—
35	C₂×#11758189	—	4	2	no	—
36	C₂×C₈×D₆₁	—	4	2	no	—
37	Dic₄₈₈	C₂×C₄	1	4	yes	complemented
38	#14519722	C₂×C₄	1	8	yes	complemented
39	D₉₇₆	C₂×C₄	1	4	yes	complemented
40	#14519713	—	2	2	no	—
41	#14519713	C₂³	1	4	yes	—
42	#14519705	—	2	8	no	—
43	C₂²×C₈×C₆₁	D₄	1	1	yes	characteristic
44	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
45	C₂×C₄²×C₆₁	D₄	1	1	yes	characteristic
46	C₂×C₁₆×C₆₁	—	2	1	no	—
47	C₂×C₁₆×C₆₁	—	2	1	no	—
48	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
49	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
50	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
51	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
52	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
53	C₄×C₈×C₆₁	D₄	1	2	yes	—
54	C₄×C₈×C₆₁	C₂³	1	2	yes	—
55	C₂²×C₄×C₆₁	—	2	1	no	—
56	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₂×C₂²⋊D₆₁	—	4	2	no	—
58	#11758222	—	2	4	no	—
59	#11758222	—	2	4	no	—
60	#11758222	—	4	2	no	—
61	#11758222	—	4	2	no	—
62	C₂×D₂₄₄	—	2	2	no	—
63	C₂×C₄×D₆₁	—	4	2	no	—
64	C₂×Dic₁₂₂	—	2	2	no	—
65	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
66	C₂×C₈×C₆₁	D₈	1	2	yes	—
67	C₂×C₈×C₆₁	D₈	1	1	yes	characteristic
68	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
69	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
70	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
71	C₂×C₈×C₆₁	C₂×D₄	1	2	yes	—
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	C₂×D₄	1	2	yes	—
77	#11758194	—	2	4	no	—
78	#11758194	—	4	2	no	—
79	C₂×#92418	—	4	2	no	—
80	Dic₂₄₄	—	2	2	no	—
81	Dic₂₄₄	C₂²×C₄	1	4	yes	—
82	D₄₈₈	—	2	2	no	—
83	D₄₈₈	C₂²×C₄	1	4	yes	—
84	#11758190	—	2	4	no	—
85	#11758190	—	2	4	no	—
86	#11758189	—	4	8	no	—
87	C₈×D₆₁	—	4	8	no	—
88	C₁₆×C₆₁	—	2	2	no	—
89	C₁₆×C₆₁	C₂²×C₄	1	2	yes	—
90	C₁₆×C₆₁	C₂²×C₄	1	2	yes	—
91	C₂³×C₆₁	#135	1	1	yes	characteristic
92	C₂²×D₆₁	—	4	2	no	—
93	C₂×C₄×C₆₁	—	2	4	no	—
94	C₂×C₄×C₆₁	—	2	2	no	—
95	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
97	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
99	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
101	C₂²⋊D₆₁	—	4	4	no	—
102	C₂²⋊D₆₁	—	4	4	no	—
103	C₂×Dic₆₁	—	4	2	no	—
104	D₂₄₄	—	2	4	no	—
105	D₂₄₄	—	4	2	no	—
106	C₄×D₆₁	—	4	4	no	—
107	C₄×D₆₁	—	4	4	no	—
108	Dic₁₂₂	—	2	4	no	—
109	Dic₁₂₂	—	4	2	no	—
110	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
111	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
112	C₈×C₆₁	C₂×D₈	1	2	yes	—
113	C₈×C₆₁	C₂×D₈	1	2	yes	—
114	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
115	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
116	#92418	—	4	4	no	—
117	C₂×#2178	—	61	1	no	maximal, Sylow
118	C₂²×C₆₁	—	2	2	no	—
119	C₂²×C₆₁	C₄×D₈	1	1	yes	characteristic
120	C₂²×C₆₁	C₄×D₈	1	1	yes	characteristic
121	C₂²×C₆₁	C₂×#135	1	1	yes	characteristic
122	C₂×D₆₁	—	4	4	no	—
123	C₂×D₆₁	—	8	4	no	—
124	C₄×C₆₁	—	2	4	no	—
125	C₄×C₆₁	C₄×D₈	1	1	yes	characteristic
126	C₄×C₆₁	C₄×D₈	1	1	yes	characteristic
127	C₄×C₆₁	C₂×C₄×D₄	1	1	yes	characteristic
128	C₄×C₆₁	C₂²×D₈	1	1	yes	characteristic
129	Dic₆₁	—	4	4	no	—
130	C₂×#508	—	61	1	no	—
131	C₂×#443	—	61	2	no	—
132	#2178	—	61	8	no	—
133	C₂×#368	—	61	1	no	—
134	C₂×#359	—	61	2	no	—
135	C₂×C₄×C₁₆	D₆₁	1	1	yes	characteristic, complemented
136	C₂×C₆₁	—	2	1	no	—
137	C₂×C₆₁	—	2	1	no	—
138	C₂×C₆₁	#2178	1	2	yes	complemented
139	C₂×C₆₁	C₂×C₄×D₈	1	1	yes	characteristic
140	D₆₁	—	8	4	no	—
141	C₂×#135	—	61	2	no	—
142	C₂×#131	—	122	2	no	—
143	#508	—	61	4	no	—
144	#508	—	61	4	no	—
145	C₂×Q₃₂	—	61	1	no	—
146	C₂×SD₃₂	—	61	2	no	—
147	C₂×D₁₆	—	61	1	no	—
148	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
149	C₂²×C₁₆	C₂×D₆₁	1	1	yes	characteristic
150	#443	—	61	8	no	—
151	#443	—	122	4	no	—
152	C₂×#108	—	61	2	no	—
153	C₂×#104	—	122	2	no	—
154	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
155	#368	—	61	4	no	—
156	#359	—	61	8	no	—
157	C₄×C₁₆	C₂×D₆₁	1	4	yes	complemented
158	C₆₁	C₂×#2178	1	1	yes	characteristic, complemented, Sylow
159	C₂×#41	—	122	2	no	—
160	#135	—	61	4	no	—
161	#135	—	61	4	no	—
162	#135	—	122	2	no	—
163	#135	—	122	2	no	—
164	C₂×Q₁₆	—	61	2	no	—
165	C₂×SD₁₆	—	122	2	no	—
166	C₂×D₈	—	61	2	no	—
167	#131	—	122	8	no	—
168	#131	—	244	4	no	—
169	C₂×M₁₆	—	122	2	no	—
170	C₂×M₁₆	—	244	2	no	—
171	C₂²×C₈	—	244	2	no	—
172	C₂²×C₈	D₂₄₄	1	1	yes	characteristic
173	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
174	C₂×C₄²	D₂₄₄	1	1	yes	characteristic
175	Q₃₂	—	61	4	no	—
176	SD₃₂	—	61	8	no	—
177	D₁₆	—	61	4	no	—
178	C₂×C₁₆	—	2	1	no	—
179	C₂×C₁₆	—	2	1	no	—
180	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
181	C₂×C₁₆	C₄×D₆₁	1	1	yes	characteristic
182	C₂×C₁₆	C₄×D₆₁	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	#108	—	122	2	no	—
187	#104	—	122	8	no	—
188	C₄×C₈	D₂₄₄	1	2	yes	—
189	C₄×C₈	C₂²×D₆₁	1	2	yes	—
190	#41	—	122	4	no	—
191	#41	—	122	4	no	—
192	#41	—	244	2	no	—
193	#41	—	244	2	no	—
194	C₂×Q₈	—	122	2	no	—
195	C₂×D₄	—	122	2	no	—
196	C₂×D₄	—	244	2	no	—
197	C₂²×C₄	—	2	1	no	—
198	C₂²×C₄	—	244	2	no	—
199	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
200	Q₁₆	—	61	4	no	—
201	Q₁₆	—	122	2	no	—
202	SD₁₆	—	122	4	no	—
203	SD₁₆	—	122	4	no	—
204	D₈	—	61	4	no	—
205	D₈	—	122	2	no	—
206	M₁₆	—	122	4	no	—
207	M₁₆	—	244	2	no	—
208	M₁₆	—	244	8	no	—
209	C₂×C₈	—	244	2	no	—
210	C₂×C₈	—	244	8	no	—
211	C₂×C₈	D₄₈₈	1	1	yes	characteristic
212	C₂×C₈	D₄₈₈	1	2	yes	—
213	C₂×C₈	D₄₈₈	1	1	yes	characteristic
214	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
215	C₂×C₈	C₂×C₄×D₆₁	1	1	yes	characteristic
216	C₂×C₈	C₂×D₂₄₄	1	1	yes	characteristic
217	C₂×C₈	C₂×D₂₄₄	1	2	yes	—
218	C₂×C₈	#11758222	1	1	yes	characteristic
219	C₂×C₈	#11758222	1	1	yes	characteristic
220	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
221	C₄²	—	2	1	no	—
222	C₄²	C₂×D₂₄₄	1	2	yes	—
223	C₁₆	—	2	2	no	—
224	C₁₆	C₂×C₄×D₆₁	1	2	yes	—
225	C₁₆	C₂×C₄×D₆₁	1	2	yes	—
226	C₂³	—	244	2	no	—
227	C₂³	#14519799	1	1	yes	characteristic
228	Q₈	—	122	4	no	—
229	Q₈	—	244	2	no	—
230	D₄	—	122	4	no	—
231	D₄	—	244	4	no	—
232	D₄	—	244	4	no	—
233	D₄	—	244	2	no	—
234	C₂×C₄	—	2	4	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	244	2	no	—
237	C₂×C₄	—	244	4	no	—
238	C₂×C₄	—	244	4	no	—
239	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
240	C₂×C₄	C₄×D₂₄₄	1	1	yes	characteristic
241	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
242	C₂×C₄	C₂×D₄₈₈	1	1	yes	characteristic
243	C₂×C₄	#14519799	1	1	yes	characteristic
244	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
245	C₈	—	244	4	no	—
246	C₈	C₄×D₂₄₄	1	1	yes	characteristic
247	C₈	C₄×D₂₄₄	1	1	yes	characteristic
248	C₈	C₂×D₄₈₈	1	2	yes	—
249	C₈	C₂×D₄₈₈	1	2	yes	—
250	C₈	C₂²×C₄×D₆₁	1	1	yes	characteristic
251	C₈	C₂×#11758222	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	488	4	no	—
254	C₂²	—	244	4	no	—
255	C₂²	C₄×D₄₈₈	1	1	yes	characteristic
256	C₂²	C₄×D₄₈₈	1	1	yes	characteristic
257	C₂²	C₂×#14519799	1	1	yes	characteristic
258	C₄	—	2	4	no	—
259	C₄	—	244	4	no	—
260	C₄	C₄×D₄₈₈	1	1	yes	characteristic
261	C₄	C₄×D₄₈₈	1	1	yes	characteristic
262	C₄	C₂×C₄×D₂₄₄	1	1	yes	characteristic
263	C₄	C₂²×D₄₈₈	1	1	yes	characteristic
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	—	488	4	no	—
267	C₂	#21447701	1	2	yes	direct product factor
268	C₂	C₂×C₄×D₄₈₈	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	D₄₈₈
commutator	C₈×C₆₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₁
Fitting	C₂×C₄×C₁₆×C₆₁	C₂
socle	C₂²×C₆₁	C₄×D₈
prime radical	C₂×#16840194	C₂
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	C₂×C₄×C₁₆	D₆₁
hypocenter	C₆₁	C₂×#2178
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆

Complements

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

N	H ≅ G/N	type
2. C₂×#16839880	C₂	semidirect
3. #21447701	C₂	direct (N × H)
4. C₂×#16839592	C₂	semidirect
5. C₂×#16839570	C₂	semidirect
6. C₂×C₄×C₁₆×C₆₁	C₂	semidirect
9. #16840194	C₄	semidirect
11. C₂×Dic₄₈₈	C₄	semidirect
12. C₂×#14519722	C₄	semidirect
13. C₂×D₉₇₆	C₄	semidirect
15. #16839880	C₂²	semidirect
18. #16839592	C₂²	semidirect
19. #16839570	C₂²	semidirect
23. C₄×C₁₆×C₆₁	C₂²	semidirect
37. Dic₄₈₈	C₂×C₄	semidirect
38. #14519722	C₂×C₄	semidirect
39. D₉₇₆	C₂×C₄	semidirect
135. C₂×C₄×C₁₆	D₆₁	semidirect
138. C₂×C₆₁	#2178	semidirect
157. C₄×C₁₆	C₂×D₆₁	semidirect
158. C₆₁	C₂×#2178	semidirect
267. C₂	#21447701	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×#16840194	2
C₂×#16839880	2
#21447701	2
C₂×#16839592	2
C₂×#16839570	2
C₂×C₄×C₁₆×C₆₁	2
C₂×#2178	61

Subgroups — C2×#21447701 #29833414

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21447701 #29833414