Subgroups — C₂×#21447700 #29833413

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×#16839879	C₂	1	2	yes	maximal, complemented
2	#21447700	C₂	1	8	yes	maximal, direct product factor
3	C₂×#16839592	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×#16839569	C₂	1	2	yes	maximal, complemented
5	C₂×C₁₆×Dic₆₁	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₆₁×#508	C₂	1	1	yes	maximal, characteristic
7	C₂×#14519784	—	2	2	no	—
8	C₂²×#11758185	C₂²	1	1	yes	characteristic
9	#16839879	—	2	4	no	—
10	#16839879	C₂²	1	8	yes	complemented
11	C₂×#14519713	C₂²	1	2	yes	—
12	C₂×#14519704	—	2	2	no	—
13	C₂×C₈×Dic₆₁	C₂²	1	1	yes	characteristic
14	#16839592	C₂²	1	4	yes	complemented
15	#16839569	C₂²	1	8	yes	complemented
16	C₁₆×Dic₆₁	C₂²	1	4	yes	complemented
17	C₂×#135×C₆₁	C₂²	1	2	yes	—
18	C₆₁×#508	C₄	1	4	yes	complemented
19	C₆₁×#508	C₂²	1	4	yes	—
20	C₂×Q₃₂×C₆₁	C₄	1	1	yes	characteristic, complemented
21	C₂×SD₃₂×C₆₁	C₄	1	2	yes	complemented
22	C₂×D₁₆×C₆₁	C₄	1	1	yes	characteristic, complemented
23	C₂²×C₁₆×C₆₁	C₂²	1	1	yes	characteristic
24	#14519784	—	2	8	no	—
25	#14519784	—	4	4	no	—
26	C₂×#11758194	—	2	2	no	—
27	C₂×#11758194	—	4	2	no	—
28	C₂²×#92418	—	4	2	no	—
29	C₂²×#92418	D₄	1	1	yes	characteristic
30	C₂×C₄×Dic₆₁	D₄	1	1	yes	characteristic
31	C₂×#11758185	—	2	1	no	—
32	C₂×#11758185	—	2	1	no	—
33	C₂×#11758185	C₂³	1	2	yes	—
34	#14519713	—	2	2	no	—
35	#14519713	C₂³	1	4	yes	—
36	#14519704	—	2	8	no	—
37	C₈×Dic₆₁	D₄	1	2	yes	—
38	C₈×Dic₆₁	C₂³	1	2	yes	—
39	C₂×#41×C₆₁	—	2	2	no	—
40	#135×C₆₁	—	2	2	no	—
41	#135×C₆₁	—	2	2	no	—
42	#135×C₆₁	C₂×C₄	1	4	yes	—
43	#135×C₆₁	C₂³	1	4	yes	—
44	C₂×Q₁₆×C₆₁	C₂×C₄	1	2	yes	—
45	C₂×SD₁₆×C₆₁	—	2	2	no	—
46	C₂×D₈×C₆₁	C₂×C₄	1	2	yes	—
47	C₂²×C₈×C₆₁	C₂³	1	1	yes	characteristic
48	Q₃₂×C₆₁	C₂×C₄	1	4	yes	complemented
49	SD₃₂×C₆₁	C₂×C₄	1	8	yes	complemented
50	D₁₆×C₆₁	C₂×C₄	1	4	yes	complemented
51	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
52	C₂×C₁₆×C₆₁	C₂×C₄	1	1	yes	characteristic
53	C₂×C₁₆×C₆₁	C₂×C₄	1	2	yes	—
54	C₂×C₁₆×C₆₁	C₂³	1	2	yes	—
55	#41×C₆₁	—	2	4	no	—
56	#41×C₆₁	—	2	4	no	—
57	#41×C₆₁	—	4	2	no	—
58	#41×C₆₁	—	4	2	no	—
59	C₂×Q₈×C₆₁	—	2	2	no	—
60	C₂×D₄×C₆₁	—	2	2	no	—
61	C₂×D₄×C₆₁	—	4	2	no	—
62	C₂²×C₄×C₆₁	—	4	2	no	—
63	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
64	C₂²×Dic₆₁	—	2	1	no	—
65	Q₁₆×C₆₁	—	2	2	no	—
66	Q₁₆×C₆₁	C₂²×C₄	1	4	yes	—
67	SD₁₆×C₆₁	—	2	4	no	—
68	SD₁₆×C₆₁	—	2	4	no	—
69	D₈×C₆₁	—	2	2	no	—
70	D₈×C₆₁	C₂²×C₄	1	4	yes	—
71	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
72	C₂×C₈×C₆₁	C₂²×C₄	1	1	yes	characteristic
73	C₂×C₈×C₆₁	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
75	C₂×C₈×C₆₁	#41	1	1	yes	characteristic
76	C₂×C₈×C₆₁	C₂⁴	1	1	yes	characteristic
77	C₄×Dic₆₁	—	2	1	no	—
78	C₄×Dic₆₁	C₂×D₄	1	2	yes	—
79	#11758194	—	2	4	no	—
80	#11758194	—	4	2	no	—
81	#11758194	—	4	8	no	—
82	C₂×#92418	—	4	2	no	—
83	C₂×#92418	—	4	8	no	—
84	C₂×#92418	D₈	1	1	yes	characteristic
85	C₂×#92418	D₈	1	1	yes	characteristic
86	C₂×#92418	D₈	1	2	yes	—
87	C₂×#92418	C₂×D₄	1	2	yes	—
88	C₁₆×C₆₁	C₂²×C₄	1	2	yes	—
89	C₁₆×C₆₁	C₂²×C₄	1	2	yes	—
90	#11758185	—	2	2	no	—
91	C₂³×C₆₁	—	4	2	no	—
92	C₂³×C₆₁	#135	1	1	yes	characteristic
93	Q₈×C₆₁	—	2	4	no	—
94	Q₈×C₆₁	—	4	2	no	—
95	D₄×C₆₁	—	2	4	no	—
96	D₄×C₆₁	—	4	4	no	—
97	D₄×C₆₁	—	4	4	no	—
98	D₄×C₆₁	—	4	2	no	—
99	C₂×C₄×C₆₁	—	4	2	no	—
100	C₂×C₄×C₆₁	—	4	4	no	—
101	C₂×C₄×C₆₁	—	4	4	no	—
102	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
103	C₂×C₄×C₆₁	C₄×D₄	1	1	yes	characteristic
104	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₂×D₈	1	1	yes	characteristic
106	C₂×C₄×C₆₁	#135	1	1	yes	characteristic
107	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
108	C₂×Dic₆₁	—	2	4	no	—
109	C₂×Dic₆₁	—	2	2	no	—
110	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
111	C₈×C₆₁	C₄×D₄	1	1	yes	characteristic
112	C₈×C₆₁	C₂³×C₄	1	1	yes	characteristic
113	C₈×C₆₁	C₂×#41	1	1	yes	characteristic
114	#92418	—	4	4	no	—
115	#92418	C₂×D₈	1	2	yes	—
116	#92418	C₂×D₈	1	2	yes	—
117	C₂×#2178	—	61	1	no	maximal, Sylow
118	C₂²×C₆₁	—	2	2	no	—
119	C₂²×C₆₁	—	4	4	no	—
120	C₂²×C₆₁	—	8	4	no	—
121	C₂²×C₆₁	C₄×D₈	1	1	yes	characteristic
122	C₂²×C₆₁	C₄×D₈	1	1	yes	characteristic
123	C₂²×C₆₁	C₂×#135	1	1	yes	characteristic
124	C₄×C₆₁	—	4	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	—	2	4	no	—
130	C₂×#508	D₆₁	1	1	yes	characteristic
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	—	61	1	no	—
136	C₂×C₆₁	—	2	1	no	—
137	C₂×C₆₁	—	2	1	no	—
138	C₂×C₆₁	—	8	4	no	—
139	C₂×C₆₁	#2178	1	2	yes	complemented
140	C₂×C₆₁	C₂×C₄×D₈	1	1	yes	characteristic
141	C₂×#135	C₂×D₆₁	1	2	yes	—
142	C₂×#131	—	122	2	no	—
143	#508	Dic₆₁	1	4	yes	complemented
144	#508	C₂×D₆₁	1	4	yes	—
145	C₂×Q₃₂	Dic₆₁	1	1	yes	characteristic, complemented
146	C₂×SD₃₂	Dic₆₁	1	2	yes	complemented
147	C₂×D₁₆	Dic₆₁	1	1	yes	characteristic, complemented
148	C₂²×C₁₆	—	61	1	no	—
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	—	61	1	no	—
155	#368	—	61	4	no	—
156	#359	—	61	8	no	—
157	C₄×C₁₆	—	61	4	no	—
158	C₆₁	C₂×#2178	1	1	yes	characteristic, complemented, Sylow
159	C₂×#41	—	2	2	no	—
160	#135	—	2	2	no	—
161	#135	—	2	2	no	—
162	#135	C₂×Dic₆₁	1	4	yes	—
163	#135	C₂²×D₆₁	1	4	yes	—
164	C₂×Q₁₆	C₂×Dic₆₁	1	2	yes	—
165	C₂×SD₁₆	—	2	2	no	—
166	C₂×D₈	C₂×Dic₆₁	1	2	yes	—
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	—	61	1	no	—
172	C₂²×C₈	—	244	2	no	—
173	C₂²×C₈	C₂²×D₆₁	1	1	yes	characteristic
174	C₂×C₄²	—	61	1	no	—
175	Q₃₂	C₂×Dic₆₁	1	4	yes	complemented
176	SD₃₂	C₂×Dic₆₁	1	8	yes	complemented
177	D₁₆	C₂×Dic₆₁	1	4	yes	complemented
178	C₂×C₁₆	—	61	2	no	—
179	C₂×C₁₆	—	122	1	no	—
180	C₂×C₁₆	—	122	1	no	—
181	C₂×C₁₆	C₂×Dic₆₁	1	1	yes	characteristic
182	C₂×C₁₆	C₂×Dic₆₁	1	1	yes	characteristic
183	C₂×C₁₆	C₂×Dic₆₁	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	—	61	2	no	—
189	C₄×C₈	—	61	2	no	—
190	#41	—	2	4	no	—
191	#41	—	2	4	no	—
192	#41	—	4	2	no	—
193	#41	—	4	2	no	—
194	C₂×Q₈	—	2	2	no	—
195	C₂×D₄	—	2	2	no	—
196	C₂×D₄	—	4	2	no	—
197	C₂²×C₄	—	4	2	no	—
198	C₂²×C₄	—	122	1	no	—
199	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
200	Q₁₆	—	2	2	no	—
201	Q₁₆	C₂²×Dic₆₁	1	4	yes	—
202	SD₁₆	—	2	4	no	—
203	SD₁₆	—	2	4	no	—
204	D₈	—	2	2	no	—
205	D₈	C₂²×Dic₆₁	1	4	yes	—
206	M₁₆	—	122	4	no	—
207	M₁₆	—	244	8	no	—
208	M₁₆	—	244	2	no	—
209	C₂×C₈	—	61	2	no	—
210	C₂×C₈	—	61	2	no	—
211	C₂×C₈	—	61	1	no	—
212	C₂×C₈	—	61	1	no	—
213	C₂×C₈	—	244	8	no	—
214	C₂×C₈	—	244	2	no	—
215	C₂×C₈	D₄×D₆₁	1	1	yes	characteristic
216	C₂×C₈	#11758224	1	1	yes	characteristic
217	C₂×C₈	#11758224	1	1	yes	characteristic
218	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
219	C₂×C₈	C₂²×Dic₆₁	1	1	yes	characteristic
220	C₂×C₈	C₂³×D₆₁	1	1	yes	characteristic
221	C₄²	—	61	2	no	—
222	C₄²	—	122	1	no	—
223	C₁₆	—	122	2	no	—
224	C₁₆	C₂²×Dic₆₁	1	2	yes	—
225	C₁₆	C₂²×Dic₆₁	1	2	yes	—
226	C₂³	—	4	2	no	—
227	C₂³	#14519796	1	1	yes	characteristic
228	Q₈	—	2	4	no	—
229	Q₈	—	4	2	no	—
230	D₄	—	2	4	no	—
231	D₄	—	4	4	no	—
232	D₄	—	4	4	no	—
233	D₄	—	4	2	no	—
234	C₂×C₄	—	4	2	no	—
235	C₂×C₄	—	4	4	no	—
236	C₂×C₄	—	4	4	no	—
237	C₂×C₄	—	122	4	no	—
238	C₂×C₄	—	122	2	no	—
239	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
240	C₂×C₄	D₄×Dic₆₁	1	1	yes	characteristic
241	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
242	C₂×C₄	D₈×D₆₁	1	1	yes	characteristic
243	C₂×C₄	#14519796	1	1	yes	characteristic
244	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
245	C₈	—	61	2	no	—
246	C₈	—	61	2	no	—
247	C₈	—	244	4	no	—
248	C₈	D₄×Dic₆₁	1	1	yes	characteristic
249	C₈	D₄×Dic₆₁	1	1	yes	characteristic
250	C₈	C₂³×Dic₆₁	1	1	yes	characteristic
251	C₈	C₂×#11758224	1	1	yes	characteristic
252	C₂²	—	2	2	no	—
253	C₂²	—	4	4	no	—
254	C₂²	—	8	4	no	—
255	C₂²	D₈×Dic₆₁	1	1	yes	characteristic
256	C₂²	D₈×Dic₆₁	1	1	yes	characteristic
257	C₂²	C₂×#14519796	1	1	yes	characteristic
258	C₄	—	4	4	no	—
259	C₄	—	122	4	no	—
260	C₄	D₈×Dic₆₁	1	1	yes	characteristic
261	C₄	D₈×Dic₆₁	1	1	yes	characteristic
262	C₄	C₂×D₄×Dic₆₁	1	1	yes	characteristic
263	C₄	C₂×D₈×D₆₁	1	1	yes	characteristic
264	C₂	—	2	1	no	—
265	C₂	—	2	1	no	—
266	C₂	—	8	4	no	—
267	C₂	#21447700	1	2	yes	direct product factor
268	C₂	C₂×D₈×Dic₆₁	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₈×D₆₁
commutator	C₈×C₆₁	C₂³×C₄
Frattini	C₂×C₈	C₂³×D₆₁
Fitting, prime radical	C₂×C₆₁×#508	C₂
socle	C₂²×C₆₁	C₄×D₈
cosocle	C₂×C₈×C₆₁	C₂⁴
hypercenter	C₂×#508	D₆₁
hypocenter	C₆₁	C₂×#2178
LCS	G ▷ C₈×C₆₁ ▷ C₄×C₆₁ ▷ C₂×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#508

Complements

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

N	H ≅ G/N	type
1. C₂×#16839879	C₂	semidirect
2. #21447700	C₂	direct (N × H)
3. C₂×#16839592	C₂	semidirect
4. C₂×#16839569	C₂	semidirect
5. C₂×C₁₆×Dic₆₁	C₂	semidirect
10. #16839879	C₂²	semidirect
14. #16839592	C₂²	semidirect
15. #16839569	C₂²	semidirect
16. C₁₆×Dic₆₁	C₂²	semidirect
18. C₆₁×#508	C₄	semidirect
20. C₂×Q₃₂×C₆₁	C₄	semidirect
21. C₂×SD₃₂×C₆₁	C₄	semidirect
22. C₂×D₁₆×C₆₁	C₄	semidirect
48. Q₃₂×C₆₁	C₂×C₄	semidirect
49. SD₃₂×C₆₁	C₂×C₄	semidirect
50. D₁₆×C₆₁	C₂×C₄	semidirect
139. C₂×C₆₁	#2178	semidirect
143. #508	Dic₆₁	semidirect
145. C₂×Q₃₂	Dic₆₁	semidirect
146. C₂×SD₃₂	Dic₆₁	semidirect
147. C₂×D₁₆	Dic₆₁	semidirect
158. C₆₁	C₂×#2178	semidirect
175. Q₃₂	C₂×Dic₆₁	semidirect
176. SD₃₂	C₂×Dic₆₁	semidirect
177. D₁₆	C₂×Dic₆₁	semidirect
267. C₂	#21447700	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×#16839879	2
#21447700	2
C₂×#16839592	2
C₂×#16839569	2
C₂×C₁₆×Dic₆₁	2
C₂×C₆₁×#508	2
C₂×#2178	61

Subgroups — C2×#21447700 #29833413

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21447700 #29833413