Subgroups — C₂×#24909306 #35557639

Order 21248.

654 conjugacy classes of subgroups.

The subgroup lattice has 268 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128117	C₂	1	2	yes	maximal, complemented
2	C₂×#19127819	C₂	1	1	yes	maximal, characteristic
3	#24909306	C₂	1	8	yes	maximal, direct product factor
4	C₂×#19127522	C₂	1	1	yes	maximal, characteristic, complemented
5	C₂×#19127484	C₂	1	2	yes	maximal, complemented
6	C₂×#346×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14581814	C₄	1	1	yes	characteristic
8	C₂×#14581798	C₂²	1	2	yes	—
9	#19128117	—	2	4	no	—
10	#19128117	C₂²	1	8	yes	complemented
11	C₂×#14581732	C₂²	1	2	yes	—
12	C₂×#14581731	—	2	2	no	—
13	C₂×C₁₆×D₈₃	—	2	2	no	—
14	#19127819	C₄	1	4	yes	—
15	#19127819	C₂²	1	4	yes	—
16	C₂×#14581728	C₄	1	1	yes	characteristic
17	C₂×#14581720	C₄	1	2	yes	—
18	#19127522	C₂²	1	4	yes	complemented
19	#19127484	C₂²	1	8	yes	complemented
20	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
21	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
22	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
23	#346×C₈₃	C₂²	1	4	yes	complemented
24	C₂×#13050590	C₂×C₄	1	2	yes	—
25	#14581814	C₈	1	4	yes	—
26	#14581814	C₂×C₄	1	4	yes	—
27	C₂×Dic₃₃₂	C₈	1	1	yes	characteristic
28	C₂×#13050579	C₈	1	2	yes	—
29	C₂×D₆₆₄	C₈	1	1	yes	characteristic
30	#14581798	—	2	2	no	—
31	#14581798	—	2	2	no	—
32	#14581798	C₂×C₄	1	4	yes	—
33	#14581798	C₂³	1	4	yes	—
34	C₂×#13050574	C₂×C₄	1	2	yes	—
35	C₂×#13050573	—	2	2	no	—
36	C₂×C₈×D₈₃	—	2	2	no	—
37	#14581732	—	2	2	no	—
38	#14581732	C₂³	1	4	yes	—
39	#14581731	—	2	8	no	—
40	C₂×C₈₃⋊C₁₆	—	2	2	no	—
41	C₁₆×D₈₃	—	2	8	no	—
42	#14581728	C₂×C₄	1	4	yes	—
43	#14581720	C₂×C₄	1	8	yes	—
44	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
45	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
46	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
47	C₂×C₁₆×C₈₃	—	2	1	no	—
48	C₂×C₁₆×C₈₃	—	2	1	no	—
49	C₂×C₁₆×C₈₃	D₄	1	2	yes	—
50	C₂×C₁₆×C₈₃	D₄	1	2	yes	—
51	C₂×C₁₆×C₈₃	C₂³	1	2	yes	—
52	C₂×C₁₆×C₈₃	C₂³	1	2	yes	—
53	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
54	C₄×C₈×C₈₃	C₂³	1	2	yes	—
55	#13050590	—	2	2	no	—
56	#13050590	—	2	2	no	—
57	#13050590	C₂×C₈	1	4	yes	—
58	#13050590	C₂²×C₄	1	4	yes	—
59	C₂×Dic₁₆₆	C₂×C₈	1	2	yes	—
60	C₂×D₃₃₂	C₂×C₈	1	2	yes	—
61	C₂×C₂²⋊D₈₃	—	2	2	no	—
62	C₂×C₄×D₈₃	—	2	2	no	—
63	Dic₃₃₂	C₂×C₈	1	4	yes	—
64	#13050579	C₂×C₈	1	8	yes	—
65	D₆₆₄	C₂×C₈	1	4	yes	—
66	#13050574	—	2	2	no	—
67	#13050574	C₂²×C₄	1	4	yes	—
68	#13050573	—	2	4	no	—
69	#13050573	—	2	4	no	—
70	C₂×C₈₃⋊C₈	—	2	2	no	—
71	C₈×D₈₃	—	2	4	no	—
72	C₈×D₈₃	—	2	4	no	—
73	C₈₃⋊C₁₆	—	2	4	no	—
74	C₂²×C₄×C₈₃	—	2	1	no	—
75	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
76	C₂×C₈×C₈₃	C₂×C₈	1	1	yes	characteristic
77	C₂×C₈×C₈₃	C₂×C₈	1	1	yes	characteristic
78	C₂×C₈×C₈₃	C₂×C₈	1	2	yes	—
79	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
80	C₂×C₈×C₈₃	C₂²×C₄	1	2	yes	—
81	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
82	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
83	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
84	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
85	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
86	C₄²×C₈₃	—	2	1	no	—
87	C₄²×C₈₃	C₂²×C₄	1	2	yes	—
88	C₁₆×C₈₃	—	2	2	no	—
89	C₁₆×C₈₃	C₂×D₄	1	4	yes	—
90	C₂³×C₈₃	#131	1	1	yes	characteristic
91	C₂²×D₈₃	—	2	2	no	—
92	C₂×C₄×C₈₃	—	2	2	no	—
93	C₂×C₄×C₈₃	—	4	2	no	—
94	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
95	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
96	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
97	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
98	C₂×C₄×C₈₃	#131	1	1	yes	characteristic
99	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
100	C₂²⋊D₈₃	—	2	4	no	—
101	C₂²⋊D₈₃	—	2	4	no	—
102	C₂×Dic₈₃	—	2	2	no	—
103	D₃₃₂	—	2	2	no	—
104	D₃₃₂	C₂²×C₈	1	4	yes	—
105	C₄×D₈₃	—	2	4	no	—
106	C₄×D₈₃	—	2	4	no	—
107	Dic₁₆₆	—	2	2	no	—
108	Dic₁₆₆	C₂²×C₈	1	4	yes	—
109	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
110	C₈×C₈₃	C₄×D₄	1	1	yes	characteristic
111	C₈×C₈₃	C₂²×C₈	1	2	yes	—
112	C₈×C₈₃	C₂²×C₈	1	2	yes	—
113	C₈×C₈₃	C₂²×D₄	1	1	yes	characteristic
114	C₈×C₈₃	C₂×#41	1	1	yes	characteristic
115	C₈₃⋊C₈	—	2	4	no	—
116	C₂²×C₈₃	—	2	2	no	—
117	C₂²×C₈₃	C₈×D₄	1	1	yes	characteristic
118	C₂²×C₈₃	C₈×D₄	1	1	yes	characteristic
119	C₂²×C₈₃	C₂×#131	1	1	yes	characteristic
120	C₂×D₈₃	—	2	4	no	—
121	C₂×D₈₃	—	4	4	no	—
122	C₄×C₈₃	—	4	2	no	—
123	C₄×C₈₃	C₈×D₄	1	1	yes	characteristic
124	C₄×C₈₃	C₈×D₄	1	1	yes	characteristic
125	C₄×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
126	C₄×C₈₃	C₂³×C₈	1	1	yes	characteristic
127	Dic₈₃	—	2	4	no	—
128	C₂×#2171	—	83	1	no	maximal, Sylow
129	C₂×C₈₃	—	2	1	no	—
130	C₂×C₈₃	—	2	1	no	—
131	C₂×C₈₃	#2171	1	2	yes	complemented
132	C₂×C₈₃	C₂×C₈×D₄	1	1	yes	characteristic
133	D₈₃	—	4	4	no	—
134	C₂×#504	—	83	2	no	—
135	C₂×#443	—	83	1	no	—
136	#2171	—	83	8	no	—
137	C₂×#364	—	83	1	no	—
138	C₂×#350	—	83	2	no	—
139	C₂×#346	D₈₃	1	1	yes	characteristic, complemented
140	C₈₃	C₂×#2171	1	1	yes	characteristic, complemented, Sylow
141	C₂×#135	—	83	1	no	—
142	C₂×#131	—	83	2	no	—
143	#504	—	83	8	no	—
144	#504	—	166	4	no	—
145	C₂×M₃₂	—	83	2	no	—
146	C₂×M₃₂	—	166	2	no	—
147	C₂²×C₁₆	—	166	2	no	—
148	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
149	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
150	#443	—	83	4	no	—
151	#443	—	83	4	no	—
152	C₂×#108	—	83	1	no	—
153	C₂×#104	—	83	2	no	—
154	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
155	#364	—	83	4	no	—
156	#350	—	83	8	no	—
157	#346	C₂×D₈₃	1	4	yes	complemented
158	C₂×#41	—	83	2	no	—
159	#135	—	83	4	no	—
160	#135	—	83	4	no	—
161	C₂×Q₁₆	—	83	1	no	—
162	C₂×SD₁₆	—	83	2	no	—
163	C₂×D₈	—	83	1	no	—
164	#131	—	83	4	no	—
165	#131	—	83	4	no	—
166	#131	—	166	2	no	—
167	#131	—	166	2	no	—
168	C₂×M₁₆	—	83	2	no	—
169	C₂×M₁₆	—	166	2	no	—
170	C₂²×C₈	—	166	2	no	—
171	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
172	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
173	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
174	M₃₂	—	83	4	no	—
175	M₃₂	—	166	2	no	—
176	M₃₂	—	166	8	no	—
177	C₂×C₁₆	—	2	1	no	—
178	C₂×C₁₆	—	2	1	no	—
179	C₂×C₁₆	—	166	2	no	—
180	C₂×C₁₆	—	166	8	no	—
181	C₂×C₁₆	D₃₃₂	1	2	yes	—
182	C₂×C₁₆	D₃₃₂	1	2	yes	—
183	C₂×C₁₆	C₂²×D₈₃	1	2	yes	—
184	C₂×C₁₆	C₂²×D₈₃	1	2	yes	—
185	#108	—	83	4	no	—
186	#104	—	83	8	no	—
187	C₄×C₈	C₄×D₈₃	1	2	yes	—
188	C₄×C₈	C₂²×D₈₃	1	2	yes	—
189	#41	—	83	4	no	—
190	#41	—	83	4	no	—
191	#41	—	166	2	no	—
192	#41	—	166	2	no	—
193	C₂×Q₈	—	83	2	no	—
194	C₂×D₄	—	83	2	no	—
195	C₂×D₄	—	166	2	no	—
196	C₂²×C₄	—	2	1	no	—
197	C₂²×C₄	—	166	2	no	—
198	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
199	Q₁₆	—	83	4	no	—
200	SD₁₆	—	83	8	no	—
201	D₈	—	83	4	no	—
202	M₁₆	—	83	4	no	—
203	M₁₆	—	166	2	no	—
204	M₁₆	—	166	4	no	—
205	M₁₆	—	166	4	no	—
206	C₂×C₈	—	166	2	no	—
207	C₂×C₈	—	166	4	no	—
208	C₂×C₈	—	166	4	no	—
209	C₂×C₈	C₈×D₈₃	1	1	yes	characteristic
210	C₂×C₈	C₈×D₈₃	1	1	yes	characteristic
211	C₂×C₈	C₈×D₈₃	1	2	yes	—
212	C₂×C₈	C₂×C₄×D₈₃	1	2	yes	—
213	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
214	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
215	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
216	C₂×C₈	#13050590	1	1	yes	characteristic
217	C₂×C₈	#13050590	1	1	yes	characteristic
218	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
219	C₄²	—	2	1	no	—
220	C₄²	C₂×C₄×D₈₃	1	2	yes	—
221	C₁₆	—	2	2	no	—
222	C₁₆	—	166	4	no	—
223	C₁₆	C₂×D₃₃₂	1	4	yes	—
224	C₂³	—	166	2	no	—
225	C₂³	#14581798	1	1	yes	characteristic
226	Q₈	—	83	4	no	—
227	Q₈	—	166	2	no	—
228	D₄	—	83	4	no	—
229	D₄	—	166	4	no	—
230	D₄	—	166	4	no	—
231	D₄	—	166	2	no	—
232	C₂×C₄	—	2	2	no	—
233	C₂×C₄	—	4	2	no	—
234	C₂×C₄	—	166	2	no	—
235	C₂×C₄	—	166	4	no	—
236	C₂×C₄	—	166	4	no	—
237	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
238	C₂×C₄	C₄×D₃₃₂	1	1	yes	characteristic
239	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
240	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
241	C₂×C₄	#14581798	1	1	yes	characteristic
242	C₂×C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
243	C₈	—	166	4	no	—
244	C₈	C₄×D₃₃₂	1	1	yes	characteristic
245	C₈	C₄×D₃₃₂	1	1	yes	characteristic
246	C₈	C₂×C₈×D₈₃	1	2	yes	—
247	C₈	C₂×C₈×D₈₃	1	2	yes	—
248	C₈	C₂²×D₃₃₂	1	1	yes	characteristic
249	C₈	C₂×#13050590	1	1	yes	characteristic
250	C₂²	—	2	2	no	—
251	C₂²	—	332	4	no	—
252	C₂²	—	166	4	no	—
253	C₂²	C₈×D₃₃₂	1	1	yes	characteristic
254	C₂²	C₈×D₃₃₂	1	1	yes	characteristic
255	C₂²	C₂×#14581798	1	1	yes	characteristic
256	C₄	—	4	2	no	—
257	C₄	—	166	4	no	—
258	C₄	C₈×D₃₃₂	1	1	yes	characteristic
259	C₄	C₈×D₃₃₂	1	1	yes	characteristic
260	C₄	C₂×C₄×D₃₃₂	1	1	yes	characteristic
261	C₄	C₂²×C₈×D₈₃	1	1	yes	characteristic
262	C₂	—	2	1	no	—
263	C₂	—	2	1	no	—
264	C₂	—	332	4	no	—
265	C₂	#24909306	1	2	yes	direct product factor
266	C₂	C₂×C₈×D₃₃₂	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₈	C₂×D₃₃₂
commutator	C₄×C₈₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₂×#346×C₈₃	C₂
socle	C₂²×C₈₃	C₈×D₄
prime radical	C₂×#14581814	C₄
cosocle	C₂×C₈×C₈₃	C₂⁴
hypercenter	C₂×#346	D₈₃
hypocenter	C₈₃	C₂×#2171
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×C₁₆ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
1. C₂×#19128117	C₂	semidirect
3. #24909306	C₂	direct (N × H)
4. C₂×#19127522	C₂	semidirect
5. C₂×#19127484	C₂	semidirect
6. C₂×#346×C₈₃	C₂	semidirect
10. #19128117	C₂²	semidirect
18. #19127522	C₂²	semidirect
19. #19127484	C₂²	semidirect
23. #346×C₈₃	C₂²	semidirect
131. C₂×C₈₃	#2171	semidirect
139. C₂×#346	D₈₃	semidirect
140. C₈₃	C₂×#2171	semidirect
157. #346	C₂×D₈₃	semidirect
265. C₂	#24909306	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×#19128117	2
C₂×#19127819	2
#24909306	2
C₂×#19127522	2
C₂×#19127484	2
C₂×#346×C₈₃	2
C₂×#2171	83

Subgroups — C2×#24909306 #35557639

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#24909306 #35557639