Subgroups — C₂×#21445411 #29667656

Order 15616.

1046 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×#16840167	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#16840139	C₂	1	1	yes	maximal, characteristic
3	C₂×#16840135	C₂	1	2	yes	maximal, complemented
4	C₂×#16840109	C₂	1	2	yes	maximal, complemented
5	#21445411	C₂	1	8	yes	maximal, direct product factor
6	C₂×C₈²×C₆₁	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11758190	—	2	4	no	—
8	C₂²×D₄₈₈	—	2	2	no	—
9	C₂×#14519777	C₂²	1	2	yes	—
10	C₂×#14519774	C₂²	1	2	yes	—
11	#16840167	—	2	2	no	—
12	#16840167	C₂²	1	4	yes	complemented
13	#16840139	—	2	2	no	—
14	#16840139	C₂²	1	4	yes	—
15	#16840135	—	2	4	no	—
16	#16840135	C₂²	1	8	yes	complemented
17	#16840109	—	2	4	no	—
18	#16840109	C₂²	1	8	yes	complemented
19	C₂×#14519711	—	2	2	no	—
20	C₂×#14519699	—	2	4	no	—
21	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
23	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
24	C₈²×C₆₁	C₂²	1	4	yes	complemented
25	C₂²×Dic₁₂₂	—	2	2	no	—
26	C₂²×D₂₄₄	—	2	2	no	—
27	C₂²×D₂₄₄	—	2	2	no	—
28	C₂²×D₂₄₄	—	2	2	no	—
29	C₂×#11758190	—	2	16	no	—
30	C₂×#11758190	—	4	16	no	—
31	C₂×D₄₈₈	—	2	8	no	—
32	C₂×D₄₈₈	—	4	8	no	—
33	#14519777	—	2	2	no	—
34	#14519777	—	2	2	no	—
35	#14519777	—	2	2	no	—
36	#14519777	C₂³	1	4	yes	—
37	#14519774	—	2	2	no	—
38	#14519774	—	2	2	no	—
39	#14519774	—	2	2	no	—
40	#14519774	C₂³	1	4	yes	—
41	C₂×C₄⋊Dic₆₁	—	2	2	no	—
42	C₂×C₄⋊Dic₆₁	—	2	2	no	—
43	#14519711	—	2	8	no	—
44	#14519699	—	2	16	no	—
45	C₂²×C₈×C₆₁	D₄	1	2	yes	—
46	C₂²×C₈×C₆₁	D₄	1	2	yes	—
47	C₂²×C₈×C₆₁	D₄	1	2	yes	—
48	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
49	C₄×C₈×C₆₁	D₄	1	2	yes	—
50	C₄×C₈×C₆₁	D₄	1	2	yes	—
51	C₄×C₈×C₆₁	D₄	1	2	yes	—
52	C₄×C₈×C₆₁	C₂³	1	2	yes	—
53	C₄×C₈×C₆₁	C₂³	1	2	yes	—
54	C₄×C₈×C₆₁	C₂³	1	2	yes	—
55	C₂³×D₆₁	—	4	2	no	—
56	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
57	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
58	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
59	C₂²×Dic₆₁	—	4	2	no	—
60	C₂×D₂₄₄	—	2	4	no	—
61	C₂×D₂₄₄	—	2	4	no	—
62	C₂×D₂₄₄	—	2	4	no	—
63	C₂×D₂₄₄	—	4	8	no	—
64	C₂×D₂₄₄	—	4	8	no	—
65	C₂×D₂₄₄	—	4	8	no	—
66	C₂×D₂₄₄	—	4	2	no	—
67	C₂×D₂₄₄	—	4	2	no	—
68	C₂×D₂₄₄	—	4	2	no	—
69	C₂×Dic₁₂₂	—	2	4	no	—
70	C₂×Dic₁₂₂	—	4	8	no	—
71	C₂×Dic₁₂₂	—	4	2	no	—
72	C₂×C₈×C₆₁	—	2	4	no	—
73	C₂×C₈×C₆₁	D₈	1	8	yes	complemented
74	C₂×C₈×C₆₁	SD₁₆	1	8	yes	complemented
75	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
76	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
77	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
78	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
79	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
80	C₄²×C₆₁	C₂×D₄	1	1	yes	characteristic
81	C₄²×C₆₁	C₂⁴	1	1	yes	characteristic
82	C₄⋊Dic₆₁	—	2	4	no	—
83	C₄⋊Dic₆₁	—	2	4	no	—
84	C₄⋊Dic₆₁	—	4	2	no	—
85	C₄⋊Dic₆₁	—	4	2	no	—
86	D₄₈₈	—	4	16	no	—
87	#11758190	—	4	32	no	—
88	C₂³×C₆₁	#127	1	1	yes	characteristic
89	C₂²×D₆₁	—	4	4	no	—
90	C₂²×D₆₁	—	8	4	no	—
91	C₂²×D₆₁	—	8	4	no	—
92	C₂²×D₆₁	—	8	4	no	—
93	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
94	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
95	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
96	C₂×C₄×C₆₁	C₂×D₈	1	4	yes	—
97	C₂×C₄×C₆₁	C₂×SD₁₆	1	4	yes	—
98	C₂×C₄×C₆₁	#135	1	4	yes	—
99	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
101	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
102	C₂×Dic₆₁	—	4	4	no	—
103	C₂×Dic₆₁	—	8	4	no	—
104	D₂₄₄	—	4	8	no	—
105	D₂₄₄	—	4	8	no	—
106	D₂₄₄	—	4	8	no	—
107	D₂₄₄	—	8	4	no	—
108	D₂₄₄	—	8	4	no	—
109	D₂₄₄	—	8	4	no	—
110	Dic₁₂₂	—	4	8	no	—
111	Dic₁₂₂	—	8	4	no	—
112	C₈×C₆₁	—	2	4	no	—
113	C₈×C₆₁	C₂×D₈	1	8	yes	complemented
114	C₈×C₆₁	C₂×SD₁₆	1	8	yes	complemented
115	C₂×#1706	—	61	1	no	maximal, Sylow
116	C₂²×C₆₁	#492	1	2	yes	—
117	C₂²×C₆₁	#493	1	2	yes	—
118	C₂²×C₆₁	#495	1	2	yes	—
119	C₂²×C₆₁	C₂×#127	1	1	yes	characteristic
120	C₂×D₆₁	—	8	4	no	—
121	C₂×D₆₁	—	8	4	no	—
122	C₂×D₆₁	—	8	4	no	—
123	C₂×D₆₁	—	16	8	no	—
124	C₄×C₆₁	#492	1	2	yes	—
125	C₄×C₆₁	#493	1	2	yes	—
126	C₄×C₆₁	#495	1	2	yes	—
127	C₄×C₆₁	C₂²×D₈	1	2	yes	—
128	C₄×C₆₁	C₂²×SD₁₆	1	2	yes	—
129	C₄×C₆₁	C₂×#135	1	2	yes	—
130	Dic₆₁	—	8	4	no	—
131	C₂×#500	—	61	1	no	—
132	C₂×#493	—	61	1	no	—
133	C₂×#492	—	61	2	no	—
134	C₂×#486	—	61	2	no	—
135	#1706	—	61	8	no	—
136	C₂×C₈²	D₆₁	1	1	yes	characteristic, complemented
137	C₂×C₆₁	#1706	1	4	yes	complemented
138	C₂×C₆₁	C₂×#492	1	1	yes	characteristic
139	C₂×C₆₁	C₂×#493	1	1	yes	characteristic
140	C₂×C₆₁	C₂×#495	1	1	yes	characteristic
141	D₆₁	—	16	4	no	—
142	C₂²×SD₁₆	—	122	4	no	—
143	C₂²×D₈	—	122	2	no	—
144	C₂×C₄⋊Q₈	—	61	2	no	—
145	C₂×#127	—	61	2	no	—
146	#500	—	61	4	no	—
147	#500	—	122	2	no	—
148	#493	—	61	4	no	—
149	#493	—	122	2	no	—
150	#492	—	61	8	no	—
151	#492	—	122	4	no	—
152	#486	—	61	8	no	—
153	#486	—	122	4	no	—
154	C₂×#107	—	122	2	no	—
155	C₂×#102	—	122	4	no	—
156	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
157	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
158	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
159	C₈²	C₂×D₆₁	1	4	yes	complemented
160	C₆₁	C₂×#1706	1	1	yes	characteristic, complemented, Sylow
161	C₂²×Q₈	—	122	2	no	—
162	C₂²×D₄	—	122	2	no	—
163	C₂²×D₄	—	122	2	no	—
164	C₂²×D₄	—	122	2	no	—
165	C₂×SD₁₆	—	122	16	no	—
166	C₂×SD₁₆	—	244	16	no	—
167	C₂×D₈	—	122	8	no	—
168	C₂×D₈	—	244	8	no	—
169	C₂²×C₈	D₂₄₄	1	2	yes	—
170	C₂²×C₈	D₂₄₄	1	2	yes	—
171	C₂²×C₈	D₂₄₄	1	2	yes	—
172	C₄⋊Q₈	—	61	4	no	—
173	C₄⋊Q₈	—	122	2	no	—
174	C₄⋊Q₈	—	122	2	no	—
175	C₄⋊Q₈	—	122	2	no	—
176	#127	—	61	4	no	—
177	#127	—	122	2	no	—
178	#127	—	122	2	no	—
179	#127	—	122	2	no	—
180	C₂×C₄⋊C₄	—	122	2	no	—
181	C₂×C₄⋊C₄	—	122	2	no	—
182	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
183	#107	—	122	8	no	—
184	#102	—	122	16	no	—
185	C₄×C₈	D₂₄₄	1	2	yes	—
186	C₄×C₈	D₂₄₄	1	2	yes	—
187	C₄×C₈	D₂₄₄	1	2	yes	—
188	C₄×C₈	C₂²×D₆₁	1	2	yes	—
189	C₄×C₈	C₂²×D₆₁	1	2	yes	—
190	C₄×C₈	C₂²×D₆₁	1	2	yes	—
191	C₂⁴	—	244	2	no	—
192	C₂×Q₈	—	122	4	no	—
193	C₂×Q₈	—	244	8	no	—
194	C₂×Q₈	—	244	2	no	—
195	C₂×D₄	—	122	4	no	—
196	C₂×D₄	—	122	4	no	—
197	C₂×D₄	—	122	4	no	—
198	C₂×D₄	—	244	2	no	—
199	C₂×D₄	—	244	2	no	—
200	C₂×D₄	—	244	2	no	—
201	C₂×D₄	—	244	8	no	—
202	C₂×D₄	—	244	8	no	—
203	C₂×D₄	—	244	8	no	—
204	C₂²×C₄	—	244	2	no	—
205	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
206	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
207	C₂²×C₄	C₂×D₂₄₄	1	1	yes	characteristic
208	SD₁₆	—	244	32	no	—
209	D₈	—	244	16	no	—
210	C₂×C₈	—	2	4	no	—
211	C₂×C₈	#11758190	1	8	yes	complemented
212	C₂×C₈	D₄₈₈	1	8	yes	complemented
213	C₂×C₈	C₂×D₂₄₄	1	4	yes	—
214	C₂×C₈	C₂×D₂₄₄	1	4	yes	—
215	C₂×C₈	C₂×D₂₄₄	1	4	yes	—
216	C₄⋊C₄	—	122	4	no	—
217	C₄⋊C₄	—	122	4	no	—
218	C₄⋊C₄	—	244	2	no	—
219	C₄⋊C₄	—	244	2	no	—
220	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
221	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
222	C₄²	C₂×D₂₄₄	1	1	yes	characteristic
223	C₄²	C₂³×D₆₁	1	1	yes	characteristic
224	C₂³	—	488	4	no	—
225	C₂³	—	488	4	no	—
226	C₂³	—	488	4	no	—
227	C₂³	—	244	4	no	—
228	C₂³	#14519774	1	1	yes	characteristic
229	Q₈	—	488	4	no	—
230	Q₈	—	244	8	no	—
231	D₄	—	488	4	no	—
232	D₄	—	488	4	no	—
233	D₄	—	488	4	no	—
234	D₄	—	244	8	no	—
235	D₄	—	244	8	no	—
236	D₄	—	244	8	no	—
237	C₂×C₄	—	488	4	no	—
238	C₂×C₄	—	244	4	no	—
239	C₂×C₄	#14519774	1	1	yes	characteristic
240	C₂×C₄	#14519774	1	1	yes	characteristic
241	C₂×C₄	#14519774	1	1	yes	characteristic
242	C₂×C₄	C₂×D₄₈₈	1	4	yes	—
243	C₂×C₄	C₂×#11758190	1	4	yes	—
244	C₂×C₄	#14519799	1	4	yes	—
245	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
246	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
247	C₂×C₄	C₂²×D₂₄₄	1	1	yes	characteristic
248	C₈	—	2	4	no	—
249	C₈	C₂×D₄₈₈	1	8	yes	complemented
250	C₈	C₂×#11758190	1	8	yes	complemented
251	C₂²	—	976	8	no	—
252	C₂²	—	488	4	no	—
253	C₂²	—	488	4	no	—
254	C₂²	—	488	4	no	—
255	C₂²	#16840135	1	2	yes	—
256	C₂²	#16840139	1	2	yes	—
257	C₂²	#16840146	1	2	yes	—
258	C₂²	C₂×#14519774	1	1	yes	characteristic
259	C₄	—	488	4	no	—
260	C₄	#16840135	1	2	yes	—
261	C₄	#16840139	1	2	yes	—
262	C₄	#16840146	1	2	yes	—
263	C₄	C₂²×D₄₈₈	1	2	yes	—
264	C₄	C₂²×#11758190	1	2	yes	—
265	C₄	C₂×#14519799	1	2	yes	—
266	C₂	—	976	4	no	—
267	C₂	#21445411	1	4	yes	direct product factor
268	C₂	C₂×#16840135	1	1	yes	characteristic
269	C₂	C₂×#16840139	1	1	yes	characteristic
270	C₂	C₂×#16840146	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519774
commutator, cosocle	C₄²×C₆₁	C₂⁴
Frattini	C₄²	C₂³×D₆₁
Fitting	C₂×C₈²×C₆₁	C₂
socle	C₂³×C₆₁	#127
prime radical	C₂×#16840139	C₂
hypercenter	C₂×C₈²	D₆₁
hypocenter	C₆₁	C₂×#1706
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈²

Complements

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

N	H ≅ G/N	type
1. C₂×#16840167	C₂	semidirect
3. C₂×#16840135	C₂	semidirect
4. C₂×#16840109	C₂	semidirect
5. #21445411	C₂	direct (N × H)
6. C₂×C₈²×C₆₁	C₂	semidirect
12. #16840167	C₂²	semidirect
16. #16840135	C₂²	semidirect
18. #16840109	C₂²	semidirect
24. C₈²×C₆₁	C₂²	semidirect
73. C₂×C₈×C₆₁	D₈	semidirect
74. C₂×C₈×C₆₁	SD₁₆	semidirect
113. C₈×C₆₁	C₂×D₈	semidirect
114. C₈×C₆₁	C₂×SD₁₆	semidirect
136. C₂×C₈²	D₆₁	semidirect
137. C₂×C₆₁	#1706	semidirect
159. C₈²	C₂×D₆₁	semidirect
160. C₆₁	C₂×#1706	semidirect
211. C₂×C₈	#11758190	semidirect
212. C₂×C₈	D₄₈₈	semidirect
249. C₈	C₂×D₄₈₈	semidirect
250. C₈	C₂×#11758190	semidirect
267. C₂	#21445411	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×#16840167	2
C₂×#16840139	2
C₂×#16840135	2
C₂×#16840109	2
#21445411	2
C₂×C₈²×C₆₁	2
C₂×#1706	61

Subgroups — C2×#21445411 #29667656

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21445411 #29667656