Subgroups — C₂×#19127820 #24916356

Order 10624.

464 conjugacy classes of subgroups.

The subgroup lattice has 266 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#14581812	C₂	1	1	yes	maximal, characteristic
2	C₂×#14581800	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂×#14581799	C₂	1	1	yes	maximal, characteristic, complemented
4	#19127820	C₂	1	8	yes	maximal, direct product factor
5	C₂×#14581727	C₂	1	1	yes	maximal, characteristic, complemented
6	C₂×#14581720	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#14581703	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×#104×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#13050590	C₂²	1	1	yes	characteristic
10	#14581812	C₄	1	4	yes	complemented
11	#14581812	C₂²	1	4	yes	—
12	C₂×#13050580	C₄	1	1	yes	characteristic, complemented
13	C₂×#13050578	C₄	1	1	yes	characteristic, complemented
14	C₂×#13050577	C₄	1	1	yes	characteristic, complemented
15	C₂×#13050575	C₄	1	1	yes	characteristic, complemented
16	#14581800	—	2	2	no	—
17	#14581800	C₂²	1	4	yes	complemented
18	#14581799	—	2	2	no	—
19	#14581799	C₂²	1	4	yes	complemented
20	C₂×#13050574	—	2	1	no	—
21	C₂×#13050574	C₂²	1	1	yes	characteristic
22	C₂×#13050573	—	2	1	no	—
23	C₂²×C₈₃⋊C₈	—	2	1	no	—
24	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
25	C₂²×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
26	C₂×C₈×D₈₃	—	2	1	no	—
27	#14581727	C₂²	1	4	yes	complemented
28	#14581720	C₂²	1	4	yes	complemented
29	#14581703	C₂²	1	4	yes	complemented
30	C₂×#41×C₈₃	C₂²	1	1	yes	characteristic
31	C₂×M₁₆×C₈₃	C₂²	1	1	yes	characteristic
32	C₂×C₄²×C₈₃	C₂²	1	1	yes	characteristic
33	#104×C₈₃	C₂²	1	4	yes	complemented
34	#13050590	—	2	1	no	—
35	#13050590	—	2	1	no	—
36	#13050590	C₂×C₄	1	2	yes	—
37	#13050590	C₂³	1	2	yes	—
38	C₂×Dic₁₆₆	C₂×C₄	1	1	yes	characteristic
39	C₂×D₃₃₂	C₂×C₄	1	1	yes	characteristic
40	C₂×C₂²⋊D₈₃	—	2	1	no	—
41	C₂×C₄×D₈₃	—	2	1	no	—
42	#13050580	C₂×C₄	1	4	yes	complemented
43	#13050578	C₂×C₄	1	4	yes	complemented
44	#13050577	C₂×C₄	1	4	yes	complemented
45	#13050575	C₂×C₄	1	4	yes	complemented
46	#13050574	—	2	1	no	—
47	#13050574	—	2	4	no	—
48	#13050574	C₂³	1	2	yes	—
49	#13050573	—	2	4	no	—
50	C₂×C₈₃⋊C₈	—	2	1	no	—
51	C₂×C₈₃⋊C₈	—	2	4	no	—
52	C₂×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
53	C₂×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
54	C₂×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
55	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
56	C₂×C₈₃⋊C₈	D₄	1	1	yes	characteristic
57	C₂×C₈₃⋊C₈	D₄	1	2	yes	—
58	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
59	C₂×C₈₃⋊C₈	C₂³	1	2	yes	—
60	C₈×D₈₃	—	2	4	no	—
61	#41×C₈₃	—	2	1	no	—
62	#41×C₈₃	—	2	1	no	—
63	#41×C₈₃	C₂×C₄	1	2	yes	—
64	#41×C₈₃	C₂³	1	2	yes	—
65	C₂×Q₈×C₈₃	C₂×C₄	1	1	yes	characteristic
66	C₂×D₄×C₈₃	—	2	1	no	—
67	C₂×D₄×C₈₃	C₂×C₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₃	—	2	1	no	—
69	C₂²×C₄×C₈₃	—	2	1	no	—
70	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
71	M₁₆×C₈₃	—	2	1	no	—
72	M₁₆×C₈₃	C₂³	1	2	yes	—
73	C₂×C₈×C₈₃	—	2	1	no	—
74	C₄²×C₈₃	—	2	1	no	—
75	C₄²×C₈₃	C₂³	1	2	yes	—
76	C₂³×C₈₃	—	2	1	no	—
77	C₂³×C₈₃	#41	1	1	yes	characteristic
78	C₂²×D₈₃	—	2	1	no	—
79	Q₈×C₈₃	—	2	1	no	—
80	Q₈×C₈₃	C₂²×C₄	1	2	yes	—
81	D₄×C₈₃	—	2	2	no	—
82	D₄×C₈₃	—	2	2	no	—
83	D₄×C₈₃	—	2	1	no	—
84	D₄×C₈₃	C₂²×C₄	1	2	yes	—
85	C₂×C₄×C₈₃	—	2	1	no	—
86	C₂×C₄×C₈₃	—	2	2	no	—
87	C₂×C₄×C₈₃	—	2	2	no	—
88	C₂×C₄×C₈₃	—	2	2	no	—
89	C₂×C₄×C₈₃	—	4	2	no	—
90	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
91	C₂×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
92	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
93	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
94	C₂×C₄×C₈₃	#41	1	1	yes	characteristic
95	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
96	C₂²⋊D₈₃	—	2	2	no	—
97	C₂²⋊D₈₃	—	2	2	no	—
98	C₂×Dic₈₃	—	2	1	no	—
99	D₃₃₂	—	2	1	no	—
100	D₃₃₂	C₂²×C₄	1	2	yes	—
101	C₄×D₈₃	—	2	2	no	—
102	C₄×D₈₃	—	2	2	no	—
103	Dic₁₆₆	—	2	1	no	—
104	Dic₁₆₆	C₂²×C₄	1	2	yes	—
105	C₈×C₈₃	—	2	2	no	—
106	C₈₃⋊C₈	—	2	2	no	—
107	C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
108	C₈₃⋊C₈	C₂²×C₄	1	2	yes	—
109	C₈₃⋊C₈	C₂×D₄	1	2	yes	—
110	C₈₃⋊C₈	C₂×D₄	1	2	yes	—
111	C₂²×C₈₃	—	2	2	no	—
112	C₂²×C₈₃	—	2	2	no	—
113	C₂²×C₈₃	—	4	2	no	—
114	C₂²×C₈₃	C₄×D₄	1	1	yes	characteristic
115	C₂²×C₈₃	C₄×D₄	1	1	yes	characteristic
116	C₂²×C₈₃	C₂×#41	1	1	yes	characteristic
117	C₂×D₈₃	—	2	2	no	—
118	C₂×D₈₃	—	4	2	no	—
119	C₄×C₈₃	—	2	2	no	—
120	C₄×C₈₃	—	4	2	no	—
121	C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
122	C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
123	C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
124	C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
125	Dic₈₃	—	2	2	no	—
126	C₂×C₈₃	—	2	1	no	—
127	C₂×C₈₃	—	2	1	no	—
128	C₂×C₈₃	—	4	2	no	—
129	C₂×C₈₃	#444	1	2	yes	complemented
130	C₂×C₈₃	C₂×C₄×D₄	1	1	yes	characteristic
131	D₈₃	—	4	2	no	—
132	C₂×#444	—	83	1	no	maximal, Sylow
133	C₈₃	C₂×#444	1	1	yes	characteristic, complemented, Sylow
134	C₂×#135	—	83	1	no	—
135	C₂×#131	—	83	1	no	—
136	C₂×#131	—	83	1	no	—
137	#444	—	83	8	no	—
138	C₂×#108	—	83	1	no	—
139	C₂×#104	—	83	1	no	—
140	C₂×#104	D₈₃	1	1	yes	characteristic, complemented
141	C₂×#97	—	83	1	no	—
142	C₂×#41	—	83	1	no	—
143	C₂×#41	C₂×D₈₃	1	1	yes	characteristic
144	#135	—	83	4	no	—
145	#135	—	83	4	no	—
146	C₂×Q₁₆	—	83	1	no	—
147	C₂×SD₁₆	—	83	1	no	—
148	C₂×SD₁₆	—	83	1	no	—
149	C₂×D₈	—	83	1	no	—
150	#131	—	83	4	no	—
151	#131	—	83	4	no	—
152	#131	—	166	2	no	—
153	#131	—	166	2	no	—
154	C₂×M₁₆	—	83	1	no	—
155	C₂×M₁₆	—	166	1	no	—
156	C₂×M₁₆	—	166	1	no	—
157	C₂×M₁₆	C₂×D₈₃	1	1	yes	characteristic
158	C₂²×C₈	—	83	1	no	—
159	C₂²×C₈	—	83	1	no	—
160	C₂²×C₈	—	166	1	no	—
161	C₂²×C₈	—	166	1	no	—
162	C₂×C₄²	C₂×D₈₃	1	1	yes	characteristic
163	#108	—	83	4	no	—
164	#104	—	83	4	no	—
165	#104	C₂×D₈₃	1	4	yes	complemented
166	#97	—	83	4	no	—
167	#41	—	2	1	no	—
168	#41	—	2	1	no	—
169	#41	—	83	2	no	—
170	#41	—	83	2	no	—
171	#41	—	166	1	no	—
172	#41	—	166	1	no	—
173	#41	C₄×D₈₃	1	2	yes	—
174	#41	C₂²×D₈₃	1	2	yes	—
175	C₂×Q₈	—	83	1	no	—
176	C₂×Q₈	C₄×D₈₃	1	1	yes	characteristic
177	C₂×D₄	—	2	1	no	—
178	C₂×D₄	—	83	1	no	—
179	C₂×D₄	—	166	1	no	—
180	C₂×D₄	C₄×D₈₃	1	1	yes	characteristic
181	C₂²×C₄	—	2	1	no	—
182	C₂²×C₄	—	2	1	no	—
183	C₂²×C₄	—	166	1	no	—
184	C₂²×C₄	C₂²×D₈₃	1	1	yes	characteristic
185	Q₁₆	—	83	4	no	—
186	SD₁₆	—	83	4	no	—
187	SD₁₆	—	83	4	no	—
188	D₈	—	83	4	no	—
189	M₁₆	—	2	1	no	—
190	M₁₆	—	83	2	no	—
191	M₁₆	—	166	4	no	—
192	M₁₆	—	166	4	no	—
193	M₁₆	—	166	1	no	—
194	M₁₆	C₂²×D₈₃	1	2	yes	—
195	C₂×C₈	—	2	1	no	—
196	C₂×C₈	—	83	1	no	—
197	C₂×C₈	—	83	1	no	—
198	C₂×C₈	—	83	2	no	—
199	C₂×C₈	—	83	2	no	—
200	C₂×C₈	—	83	1	no	—
201	C₂×C₈	—	83	1	no	—
202	C₂×C₈	—	83	2	no	—
203	C₂×C₈	—	83	2	no	—
204	C₂×C₈	—	166	4	no	—
205	C₂×C₈	—	166	4	no	—
206	C₂×C₈	—	166	1	no	—
207	C₄²	—	2	1	no	—
208	C₄²	C₂²×D₈₃	1	2	yes	—
209	C₂³	—	2	1	no	—
210	C₂³	—	166	1	no	—
211	C₂³	#13050589	1	1	yes	characteristic
212	Q₈	—	2	1	no	—
213	Q₈	—	83	2	no	—
214	Q₈	—	166	1	no	—
215	Q₈	C₂×C₄×D₈₃	1	2	yes	—
216	D₄	—	2	2	no	—
217	D₄	—	2	2	no	—
218	D₄	—	2	1	no	—
219	D₄	—	83	2	no	—
220	D₄	—	166	2	no	—
221	D₄	—	166	2	no	—
222	D₄	—	166	1	no	—
223	D₄	C₂×C₄×D₈₃	1	2	yes	—
224	C₂×C₄	—	2	1	no	—
225	C₂×C₄	—	2	2	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	2	no	—
228	C₂×C₄	—	4	2	no	—
229	C₂×C₄	—	166	1	no	—
230	C₂×C₄	—	166	2	no	—
231	C₂×C₄	—	166	2	no	—
232	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
233	C₂×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
234	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
235	C₂×C₄	D₄×D₈₃	1	1	yes	characteristic
236	C₂×C₄	#13050589	1	1	yes	characteristic
237	C₂×C₄	C₂³×D₈₃	1	1	yes	characteristic
238	C₈	—	2	2	no	—
239	C₈	—	83	2	no	—
240	C₈	—	83	2	no	—
241	C₈	—	83	2	no	—
242	C₈	—	83	2	no	—
243	C₈	—	166	2	no	—
244	C₂²	—	2	2	no	—
245	C₂²	—	2	2	no	—
246	C₂²	—	4	2	no	—
247	C₂²	—	332	2	no	—
248	C₂²	—	166	2	no	—
249	C₂²	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
250	C₂²	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
251	C₂²	C₂×#13050589	1	1	yes	characteristic
252	C₄	—	2	2	no	—
253	C₄	—	4	2	no	—
254	C₄	—	166	2	no	—
255	C₄	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
256	C₄	(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic
257	C₄	C₂²×C₄×D₈₃	1	1	yes	characteristic
258	C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
259	C₂	—	2	1	no	—
260	C₂	—	2	1	no	—
261	C₂	—	4	2	no	—
262	C₂	—	332	2	no	—
263	C₂	#19127820	1	2	yes	direct product factor
264	C₂	C₂×(C₂²⋊D₈₃)⋊C₄	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	D₄×D₈₃
commutator	C₄×C₈₃	C₂³×C₄
Frattini	C₂×C₄	C₂³×D₈₃
Fitting	C₂×#104×C₈₃	C₂
socle	C₂²×C₈₃	C₄×D₄
prime radical	C₂×#14581812	C₂
cosocle	C₂×C₄×C₈₃	C₂⁴
hypercenter	C₂×#104	D₈₃
hypocenter	C₈₃	C₂×#444
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄ ◁ C₂×#104

Complements

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

N	H ≅ G/N	type
2. C₂×#14581800	C₂	semidirect
3. C₂×#14581799	C₂	semidirect
4. #19127820	C₂	direct (N × H)
5. C₂×#14581727	C₂	semidirect
6. C₂×#14581720	C₂	semidirect
7. C₂×#14581703	C₂	semidirect
8. C₂×#104×C₈₃	C₂	semidirect
10. #14581812	C₄	semidirect
12. C₂×#13050580	C₄	semidirect
13. C₂×#13050578	C₄	semidirect
14. C₂×#13050577	C₄	semidirect
15. C₂×#13050575	C₄	semidirect
17. #14581800	C₂²	semidirect
19. #14581799	C₂²	semidirect
27. #14581727	C₂²	semidirect
28. #14581720	C₂²	semidirect
29. #14581703	C₂²	semidirect
33. #104×C₈₃	C₂²	semidirect
42. #13050580	C₂×C₄	semidirect
43. #13050578	C₂×C₄	semidirect
44. #13050577	C₂×C₄	semidirect
45. #13050575	C₂×C₄	semidirect
129. C₂×C₈₃	#444	semidirect
133. C₈₃	C₂×#444	semidirect
140. C₂×#104	D₈₃	semidirect
165. #104	C₂×D₈₃	semidirect
263. C₂	#19127820	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×#14581812	2
C₂×#14581800	2
C₂×#14581799	2
#19127820	2
C₂×#14581727	2
C₂×#14581720	2
C₂×#14581703	2
C₂×#104×C₈₃	2
C₂×#444	83

Subgroups — C2×#19127820 #24916356

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#19127820 #24916356