Subgroups — C₂×#21445412 #29667657

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×#16840139	C₂	1	1	yes	maximal, characteristic
2	C₂×#16840136	C₂	1	2	yes	maximal, complemented
3	C₂×#16840110	C₂	1	2	yes	maximal, complemented
4	#21445412	C₂	1	8	yes	maximal, direct product factor
5	C₂×C₈×#92418	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×C₆₁×#500	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂²×#11758201	—	2	4	no	—
8	C₂²×D₄₈₈	—	2	2	no	—
9	C₂×#14519774	C₂²	1	2	yes	—
10	#16840139	—	2	2	no	—
11	#16840139	C₂²	1	4	yes	—
12	#16840136	—	2	4	no	—
13	#16840136	C₂²	1	8	yes	complemented
14	#16840110	—	2	4	no	—
15	#16840110	C₂²	1	8	yes	complemented
16	C₂×#14519697	—	2	4	no	—
17	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
18	C₂×C₄×#92418	C₂²	1	1	yes	characteristic
19	C₈×#92418	C₂²	1	4	yes	complemented
20	C₂×C₄⋊Q₈×C₆₁	C₂²	1	2	yes	—
21	C₆₁×#500	—	2	2	no	—
22	C₆₁×#500	C₂²	1	4	yes	complemented
23	C₂×#107×C₆₁	—	2	2	no	—
24	C₂×C₄×C₈×C₆₁	C₂²	1	1	yes	characteristic
25	C₂²×D₂₄₄	—	2	2	no	—
26	C₂²×D₂₄₄	—	2	2	no	—
27	C₂²×D₂₄₄	—	2	2	no	—
28	C₂×#11758201	—	2	16	no	—
29	C₂×#11758201	—	4	16	no	—
30	C₂×D₄₈₈	—	2	8	no	—
31	C₂×D₄₈₈	—	4	8	no	—
32	C₂²×#92418	D₄	1	2	yes	—
33	C₂²×#92418	D₄	1	2	yes	—
34	#14519774	—	2	2	no	—
35	#14519774	—	2	2	no	—
36	#14519774	—	2	2	no	—
37	#14519774	C₂³	1	4	yes	—
38	#14519697	—	2	16	no	—
39	C₄×#92418	D₄	1	2	yes	—
40	C₄×#92418	D₄	1	2	yes	—
41	C₄×#92418	C₂³	1	2	yes	—
42	C₄×#92418	C₂³	1	2	yes	—
43	C₂²×Q₈×C₆₁	—	2	2	no	—
44	C₂²×C₈×C₆₁	D₄	1	2	yes	—
45	C₄⋊Q₈×C₆₁	—	2	2	no	—
46	C₄⋊Q₈×C₆₁	—	2	2	no	—
47	C₄⋊Q₈×C₆₁	—	2	2	no	—
48	C₄⋊Q₈×C₆₁	C₂³	1	4	yes	—
49	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
50	C₂×C₄⋊C₄×C₆₁	—	2	2	no	—
51	C₂×C₄²×C₆₁	C₂³	1	1	yes	characteristic
52	#107×C₆₁	—	2	8	no	—
53	C₄×C₈×C₆₁	D₄	1	2	yes	—
54	C₄×C₈×C₆₁	C₂³	1	2	yes	—
55	C₂³×D₆₁	—	4	2	no	—
56	C₂×Q₈×C₆₁	—	2	4	no	—
57	C₂×Q₈×C₆₁	—	4	2	no	—
58	C₂×Q₈×C₆₁	—	4	8	no	—
59	C₂²×C₄×C₆₁	—	4	2	no	—
60	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
61	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
62	C₂²×C₄×C₆₁	C₂×D₄	1	1	yes	characteristic
63	C₂×D₂₄₄	—	2	4	no	—
64	C₂×D₂₄₄	—	2	4	no	—
65	C₂×D₂₄₄	—	2	4	no	—
66	C₂×D₂₄₄	—	4	8	no	—
67	C₂×D₂₄₄	—	4	8	no	—
68	C₂×D₂₄₄	—	4	8	no	—
69	C₂×D₂₄₄	—	4	2	no	—
70	C₂×D₂₄₄	—	4	2	no	—
71	C₂×D₂₄₄	—	4	2	no	—
72	C₂×C₈×C₆₁	SD₁₆	1	8	yes	complemented
73	C₂×C₈×C₆₁	C₂×D₄	1	4	yes	—
74	C₄⋊C₄×C₆₁	—	2	4	no	—
75	C₄⋊C₄×C₆₁	—	2	4	no	—
76	C₄⋊C₄×C₆₁	—	4	2	no	—
77	C₄⋊C₄×C₆₁	—	4	2	no	—
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	#11758201	—	4	32	no	—
83	C₂×#92418	—	2	4	no	—
84	C₂×#92418	D₈	1	8	yes	complemented
85	C₂×#92418	C₂×D₄	1	4	yes	—
86	C₂×#92418	C₂×D₄	1	4	yes	—
87	D₄₈₈	—	4	16	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	Q₈×C₆₁	—	4	8	no	—
94	Q₈×C₆₁	—	8	4	no	—
95	C₂×C₄×C₆₁	—	4	4	no	—
96	C₂×C₄×C₆₁	—	8	4	no	—
97	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
98	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
99	C₂×C₄×C₆₁	#127	1	1	yes	characteristic
100	C₂×C₄×C₆₁	C₂×D₈	1	4	yes	—
101	C₂×C₄×C₆₁	C₂×SD₁₆	1	4	yes	—
102	C₂×C₄×C₆₁	#135	1	4	yes	—
103	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
104	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
105	C₂×C₄×C₆₁	C₂²×D₄	1	1	yes	characteristic
106	D₂₄₄	—	4	8	no	—
107	D₂₄₄	—	4	8	no	—
108	D₂₄₄	—	4	8	no	—
109	D₂₄₄	—	8	4	no	—
110	D₂₄₄	—	8	4	no	—
111	D₂₄₄	—	8	4	no	—
112	C₈×C₆₁	C₂×SD₁₆	1	8	yes	complemented
113	#92418	—	2	4	no	—
114	#92418	C₂×D₈	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	—	8	4	no	—
125	C₄×C₆₁	#492	1	2	yes	—
126	C₄×C₆₁	#493	1	2	yes	—
127	C₄×C₆₁	#495	1	2	yes	—
128	C₄×C₆₁	C₂²×D₈	1	2	yes	—
129	C₄×C₆₁	C₂²×SD₁₆	1	2	yes	—
130	C₄×C₆₁	C₂×#135	1	2	yes	—
131	C₂×#500	D₆₁	1	1	yes	characteristic, complemented
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	—	61	1	no	—
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	C₂×D₆₁	1	2	yes	—
145	C₂×#127	—	61	2	no	—
146	#500	—	2	2	no	—
147	#500	C₂×D₆₁	1	4	yes	complemented
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	—	2	2	no	—
155	C₂×#102	—	122	4	no	—
156	C₂×C₄×C₈	—	61	1	no	—
157	C₂×C₄×C₈	—	61	1	no	—
158	C₂×C₄×C₈	C₂×D₆₁	1	1	yes	characteristic
159	C₈²	—	61	4	no	—
160	C₆₁	C₂×#1706	1	1	yes	characteristic, complemented, Sylow
161	C₂²×Q₈	—	2	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	—	61	2	no	—
170	C₂²×C₈	—	61	2	no	—
171	C₂²×C₈	C₂²⋊D₆₁	1	2	yes	—
172	C₄⋊Q₈	—	2	2	no	—
173	C₄⋊Q₈	—	2	2	no	—
174	C₄⋊Q₈	—	2	2	no	—
175	C₄⋊Q₈	C₂²×D₆₁	1	4	yes	—
176	#127	—	61	4	no	—
177	#127	—	122	2	no	—
178	#127	—	122	2	no	—
179	#127	—	122	2	no	—
180	C₂×C₄⋊C₄	—	2	2	no	—
181	C₂×C₄⋊C₄	—	2	2	no	—
182	C₂×C₄²	C₂²×D₆₁	1	1	yes	characteristic
183	#107	—	2	8	no	—
184	#102	—	122	16	no	—
185	C₄×C₈	—	61	2	no	—
186	C₄×C₈	—	61	2	no	—
187	C₄×C₈	—	61	2	no	—
188	C₄×C₈	—	61	2	no	—
189	C₄×C₈	C₂²⋊D₆₁	1	2	yes	—
190	C₄×C₈	C₂²×D₆₁	1	2	yes	—
191	C₂⁴	—	244	2	no	—
192	C₂×Q₈	—	2	4	no	—
193	C₂×Q₈	—	4	2	no	—
194	C₂×Q₈	—	4	8	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	—	4	2	no	—
205	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
206	C₂²×C₄	D₄×D₆₁	1	1	yes	characteristic
207	C₂²×C₄	C₂×C₂²⋊D₆₁	1	1	yes	characteristic
208	SD₁₆	—	244	32	no	—
209	D₈	—	244	16	no	—
210	C₂×C₈	—	61	4	no	—
211	C₂×C₈	—	61	8	no	—
212	C₂×C₈	—	61	4	no	—
213	C₂×C₈	—	122	4	no	—
214	C₂×C₈	#11758201	1	8	yes	complemented
215	C₂×C₈	C₂×C₂²⋊D₆₁	1	4	yes	—
216	C₄⋊C₄	—	2	4	no	—
217	C₄⋊C₄	—	2	4	no	—
218	C₄⋊C₄	—	4	2	no	—
219	C₄⋊C₄	—	4	2	no	—
220	C₄²	D₄×D₆₁	1	1	yes	characteristic
221	C₄²	D₄×D₆₁	1	1	yes	characteristic
222	C₄²	C₂×C₂²⋊D₆₁	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	#14519773	1	1	yes	characteristic
229	Q₈	—	4	8	no	—
230	Q₈	—	8	4	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	—	4	4	no	—
238	C₂×C₄	—	8	4	no	—
239	C₂×C₄	#14519773	1	1	yes	characteristic
240	C₂×C₄	#14519773	1	1	yes	characteristic
241	C₂×C₄	#14519773	1	1	yes	characteristic
242	C₂×C₄	D₈×D₆₁	1	4	yes	—
243	C₂×C₄	C₂×#11758201	1	4	yes	—
244	C₂×C₄	#14519800	1	4	yes	—
245	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
246	C₂×C₄	C₂×D₄×D₆₁	1	1	yes	characteristic
247	C₂×C₄	C₂²×C₂²⋊D₆₁	1	1	yes	characteristic
248	C₈	—	61	8	no	—
249	C₈	—	122	4	no	—
250	C₈	C₂×#11758201	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	#16840136	1	2	yes	—
256	C₂²	#16840137	1	2	yes	—
257	C₂²	#16840147	1	2	yes	—
258	C₂²	C₂×#14519773	1	1	yes	characteristic
259	C₄	—	8	4	no	—
260	C₄	#16840136	1	2	yes	—
261	C₄	#16840137	1	2	yes	—
262	C₄	#16840147	1	2	yes	—
263	C₄	C₂×D₈×D₆₁	1	2	yes	—
264	C₄	C₂²×#11758201	1	2	yes	—
265	C₄	C₂×#14519800	1	2	yes	—
266	C₂	—	976	4	no	—
267	C₂	#21445412	1	4	yes	direct product factor
268	C₂	C₂×#16840136	1	1	yes	characteristic
269	C₂	C₂×#16840137	1	1	yes	characteristic
270	C₂	C₂×#16840147	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14519773
commutator, cosocle	C₄²×C₆₁	C₂⁴
Frattini	C₄²	C₂³×D₆₁
Fitting	C₂×C₆₁×#500	C₂
socle	C₂³×C₆₁	#127
prime radical	C₂×#16840139	C₂
hypercenter	C₂×#500	D₆₁
hypocenter	C₆₁	C₂×#1706
LCS	G ▷ C₄²×C₆₁ ▷ C₂²×C₆₁ ▷ C₆₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#500

Complements

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

N	H ≅ G/N	type
2. C₂×#16840136	C₂	semidirect
3. C₂×#16840110	C₂	semidirect
4. #21445412	C₂	direct (N × H)
5. C₂×C₈×#92418	C₂	semidirect
6. C₂×C₆₁×#500	C₂	semidirect
13. #16840136	C₂²	semidirect
15. #16840110	C₂²	semidirect
19. C₈×#92418	C₂²	semidirect
22. C₆₁×#500	C₂²	semidirect
72. C₂×C₈×C₆₁	SD₁₆	semidirect
84. C₂×#92418	D₈	semidirect
112. C₈×C₆₁	C₂×SD₁₆	semidirect
114. #92418	C₂×D₈	semidirect
131. C₂×#500	D₆₁	semidirect
137. C₂×C₆₁	#1706	semidirect
147. #500	C₂×D₆₁	semidirect
160. C₆₁	C₂×#1706	semidirect
214. C₂×C₈	#11758201	semidirect
250. C₈	C₂×#11758201	semidirect
267. C₂	#21445412	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×#16840139	2
C₂×#16840136	2
C₂×#16840110	2
#21445412	2
C₂×C₈×#92418	2
C₂×C₆₁×#500	2
C₂×#1706	61

Subgroups — C2×#21445412 #29667657

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#21445412 #29667657