Subgroups — C₂×#452×C₈₃ #24904686

Order 10624.

760 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	C₂²×#41×C₈₃	C₂	1	1	yes	maximal, characteristic
2	C₂²×Q₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₂²×SD₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
4	C₂×C₂²⋊Q₈×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
5	#452×C₈₃	C₂	1	8	yes	maximal, direct product factor
6	C₂×#103×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
7	C₂×#102×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂×C₂²⋊C₈×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
9	C₂×#41×C₈₃	—	2	4	no	—
10	C₂×#41×C₈₃	—	2	4	no	—
11	C₂×#41×C₈₃	—	2	2	no	—
12	C₂×#41×C₈₃	C₂²	1	4	yes	—
13	C₂²×Q₈×C₈₃	C₂²	1	1	yes	characteristic
14	C₂²×Q₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₂²×D₄×C₈₃	—	2	1	no	—
16	C₂²×D₄×C₈₃	C₂²	1	1	yes	characteristic
17	C₂³×C₄×C₈₃	—	2	1	no	—
18	C₂³×C₄×C₈₃	C₂²	1	1	yes	characteristic
19	C₂×Q₁₆×C₈₃	—	2	4	no	—
20	C₂×Q₁₆×C₈₃	C₂²	1	4	yes	complemented
21	C₂×SD₁₆×C₈₃	—	2	4	no	—
22	C₂×SD₁₆×C₈₃	C₂²	1	4	yes	complemented
23	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
24	C₂²×C₈×C₈₃	C₂²	1	1	yes	characteristic
25	C₂²⋊Q₈×C₈₃	—	2	2	no	—
26	C₂²⋊Q₈×C₈₃	C₂²	1	4	yes	complemented
27	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
28	C₂×C₄⋊C₄×C₈₃	C₂²	1	1	yes	characteristic
29	C₂×C₂²⋊C₄×C₈₃	—	2	1	no	—
30	#103×C₈₃	C₂²	1	4	yes	complemented
31	#102×C₈₃	C₂²	1	4	yes	complemented
32	C₂²⋊C₈×C₈₃	C₂²	1	4	yes	complemented
33	C₂⁴×C₈₃	—	2	1	no	—
34	C₂⁴×C₈₃	D₄	1	1	yes	characteristic
35	#41×C₈₃	—	2	16	no	—
36	#41×C₈₃	—	2	8	no	—
37	#41×C₈₃	—	2	8	no	—
38	C₂×Q₈×C₈₃	—	2	2	no	—
39	C₂×Q₈×C₈₃	—	2	1	no	—
40	C₂×Q₈×C₈₃	—	2	1	no	—
41	C₂×Q₈×C₈₃	—	2	2	no	—
42	C₂×Q₈×C₈₃	—	2	2	no	—
43	C₂×Q₈×C₈₃	D₄	1	4	yes	—
44	C₂×Q₈×C₈₃	C₂³	1	2	yes	—
45	C₂×Q₈×C₈₃	C₂³	1	2	yes	—
46	C₂×D₄×C₈₃	—	2	8	no	—
47	C₂×D₄×C₈₃	—	2	4	no	—
48	C₂×D₄×C₈₃	—	2	1	no	—
49	C₂×D₄×C₈₃	—	2	2	no	—
50	C₂×D₄×C₈₃	D₄	1	4	yes	—
51	C₂×D₄×C₈₃	C₂³	1	2	yes	—
52	C₂²×C₄×C₈₃	—	2	8	no	—
53	C₂²×C₄×C₈₃	—	2	1	no	—
54	C₂²×C₄×C₈₃	—	2	4	no	—
55	C₂²×C₄×C₈₃	—	2	2	no	—
56	C₂²×C₄×C₈₃	—	2	2	no	—
57	C₂²×C₄×C₈₃	—	2	1	no	—
58	C₂²×C₄×C₈₃	—	2	1	no	—
59	C₂²×C₄×C₈₃	D₄	1	2	yes	—
60	C₂²×C₄×C₈₃	D₄	1	1	yes	characteristic
61	C₂²×C₄×C₈₃	C₂³	1	2	yes	—
62	C₂²×C₄×C₈₃	C₂³	1	1	yes	characteristic
63	Q₁₆×C₈₃	—	2	8	no	—
64	SD₁₆×C₈₃	—	2	8	no	—
65	C₂×C₈×C₈₃	—	2	2	no	—
66	C₂×C₈×C₈₃	—	2	2	no	—
67	C₂×C₈×C₈₃	C₂³	1	2	yes	—
68	C₂×C₈×C₈₃	C₂³	1	2	yes	—
69	C₄⋊C₄×C₈₃	—	2	4	no	—
70	C₄⋊C₄×C₈₃	—	2	1	no	—
71	C₄⋊C₄×C₈₃	C₂³	1	2	yes	—
72	C₂²⋊C₄×C₈₃	—	2	4	no	—
73	C₂³×C₈₃	—	2	2	no	—
74	C₂³×C₈₃	—	2	4	no	—
75	C₂³×C₈₃	—	2	2	no	—
76	C₂³×C₈₃	—	2	2	no	—
77	C₂³×C₈₃	—	2	2	no	—
78	C₂³×C₈₃	—	4	2	no	—
79	C₂³×C₈₃	—	4	2	no	—
80	C₂³×C₈₃	C₂×D₄	1	2	yes	—
81	C₂³×C₈₃	C₂×D₄	1	1	yes	characteristic
82	Q₈×C₈₃	—	2	4	no	—
83	Q₈×C₈₃	—	2	2	no	—
84	Q₈×C₈₃	—	2	2	no	—
85	Q₈×C₈₃	—	2	2	no	—
86	Q₈×C₈₃	—	4	2	no	—
87	Q₈×C₈₃	C₂×D₄	1	4	yes	—
88	D₄×C₈₃	—	2	16	no	—
89	D₄×C₈₃	—	2	4	no	—
90	D₄×C₈₃	—	2	2	no	—
91	D₄×C₈₃	C₂×D₄	1	4	yes	—
92	C₂×C₄×C₈₃	—	2	16	no	—
93	C₂×C₄×C₈₃	—	2	2	no	—
94	C₂×C₄×C₈₃	—	2	4	no	—
95	C₂×C₄×C₈₃	—	2	2	no	—
96	C₂×C₄×C₈₃	—	2	4	no	—
97	C₂×C₄×C₈₃	—	2	2	no	—
98	C₂×C₄×C₈₃	—	2	2	no	—
99	C₂×C₄×C₈₃	—	2	2	no	—
100	C₂×C₄×C₈₃	—	2	2	no	—
101	C₂×C₄×C₈₃	—	4	2	no	—
102	C₂×C₄×C₈₃	—	4	2	no	—
103	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
104	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
105	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
106	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
107	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
108	C₈×C₈₃	—	2	2	no	—
109	C₈×C₈₃	—	2	2	no	—
110	C₂²×C₈₃	—	2	2	no	—
111	C₂²×C₈₃	—	2	4	no	—
112	C₂²×C₈₃	—	2	2	no	—
113	C₂²×C₈₃	—	2	2	no	—
114	C₂²×C₈₃	—	4	2	no	—
115	C₂²×C₈₃	—	4	2	no	—
116	C₂²×C₈₃	—	4	4	no	—
117	C₂²×C₈₃	—	4	8	no	—
118	C₂²×C₈₃	#120	1	2	yes	—
119	C₂²×C₈₃	#135	1	2	yes	—
120	C₂²×C₈₃	#137	1	2	yes	—
121	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
122	C₄×C₈₃	—	2	2	no	—
123	C₄×C₈₃	—	2	4	no	—
124	C₄×C₈₃	—	4	2	no	—
125	C₄×C₈₃	—	4	2	no	—
126	C₄×C₈₃	#120	1	2	yes	—
127	C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
128	C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
129	C₂×C₈₃	—	4	2	no	—
130	C₂×C₈₃	—	4	4	no	—
131	C₂×C₈₃	#452	1	4	yes	direct product factor
132	C₂×C₈₃	C₂×#120	1	1	yes	characteristic
133	C₂×C₈₃	C₂×#135	1	1	yes	characteristic
134	C₂×C₈₃	C₂×#137	1	1	yes	characteristic
135	C₂×#452	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
136	C₈₃	C₂×#452	1	1	yes	characteristic, direct product factor, Sylow
137	C₂²×#41	C₂×C₈₃	1	1	yes	characteristic
138	C₂²×Q₁₆	C₂×C₈₃	1	1	yes	characteristic, complemented
139	C₂²×SD₁₆	C₂×C₈₃	1	1	yes	characteristic, complemented
140	C₂×C₂²⋊Q₈	C₂×C₈₃	1	1	yes	characteristic, complemented
141	#452	C₂×C₈₃	1	8	yes	direct product factor
142	C₂×#103	C₂×C₈₃	1	1	yes	characteristic, complemented
143	C₂×#102	C₂×C₈₃	1	1	yes	characteristic, complemented
144	C₂×C₂²⋊C₈	C₂×C₈₃	1	1	yes	characteristic, complemented
145	C₂×#41	—	2	4	no	—
146	C₂×#41	—	2	4	no	—
147	C₂×#41	—	2	2	no	—
148	C₂×#41	C₂²×C₈₃	1	4	yes	—
149	C₂²×Q₈	C₂²×C₈₃	1	1	yes	characteristic
150	C₂²×Q₈	C₂²×C₈₃	1	1	yes	characteristic
151	C₂²×D₄	—	2	1	no	—
152	C₂²×D₄	C₂²×C₈₃	1	1	yes	characteristic
153	C₂³×C₄	—	2	1	no	—
154	C₂³×C₄	C₂²×C₈₃	1	1	yes	characteristic
155	C₂×Q₁₆	—	2	4	no	—
156	C₂×Q₁₆	C₂²×C₈₃	1	4	yes	complemented
157	C₂×SD₁₆	—	2	4	no	—
158	C₂×SD₁₆	C₂²×C₈₃	1	4	yes	complemented
159	C₂²×C₈	C₂²×C₈₃	1	1	yes	characteristic
160	C₂²×C₈	C₂²×C₈₃	1	1	yes	characteristic
161	C₂²⋊Q₈	—	2	2	no	—
162	C₂²⋊Q₈	C₂²×C₈₃	1	4	yes	complemented
163	C₂×C₄⋊C₄	—	2	1	no	—
164	C₂×C₄⋊C₄	C₂²×C₈₃	1	1	yes	characteristic
165	C₂×C₂²⋊C₄	—	2	1	no	—
166	#103	C₂²×C₈₃	1	4	yes	complemented
167	#102	C₂²×C₈₃	1	4	yes	complemented
168	C₂²⋊C₈	C₂²×C₈₃	1	4	yes	complemented
169	C₂⁴	—	2	1	no	—
170	C₂⁴	D₄×C₈₃	1	1	yes	characteristic
171	#41	—	2	16	no	—
172	#41	—	2	8	no	—
173	#41	—	2	8	no	—
174	C₂×Q₈	—	2	2	no	—
175	C₂×Q₈	—	2	1	no	—
176	C₂×Q₈	—	2	1	no	—
177	C₂×Q₈	—	2	2	no	—
178	C₂×Q₈	—	2	2	no	—
179	C₂×Q₈	D₄×C₈₃	1	4	yes	—
180	C₂×Q₈	C₂³×C₈₃	1	2	yes	—
181	C₂×Q₈	C₂³×C₈₃	1	2	yes	—
182	C₂×D₄	—	2	8	no	—
183	C₂×D₄	—	2	4	no	—
184	C₂×D₄	—	2	1	no	—
185	C₂×D₄	—	2	2	no	—
186	C₂×D₄	D₄×C₈₃	1	4	yes	—
187	C₂×D₄	C₂³×C₈₃	1	2	yes	—
188	C₂²×C₄	—	2	8	no	—
189	C₂²×C₄	—	2	1	no	—
190	C₂²×C₄	—	2	4	no	—
191	C₂²×C₄	—	2	2	no	—
192	C₂²×C₄	—	2	2	no	—
193	C₂²×C₄	—	2	1	no	—
194	C₂²×C₄	—	2	1	no	—
195	C₂²×C₄	D₄×C₈₃	1	1	yes	characteristic
196	C₂²×C₄	D₄×C₈₃	1	2	yes	—
197	C₂²×C₄	C₂³×C₈₃	1	2	yes	—
198	C₂²×C₄	C₂³×C₈₃	1	1	yes	characteristic
199	Q₁₆	—	2	8	no	—
200	SD₁₆	—	2	8	no	—
201	C₂×C₈	—	2	2	no	—
202	C₂×C₈	—	2	2	no	—
203	C₂×C₈	C₂³×C₈₃	1	2	yes	—
204	C₂×C₈	C₂³×C₈₃	1	2	yes	—
205	C₄⋊C₄	—	2	4	no	—
206	C₄⋊C₄	—	2	1	no	—
207	C₄⋊C₄	C₂³×C₈₃	1	2	yes	—
208	C₂²⋊C₄	—	2	4	no	—
209	C₂³	—	2	2	no	—
210	C₂³	—	2	4	no	—
211	C₂³	—	2	2	no	—
212	C₂³	—	2	2	no	—
213	C₂³	—	2	2	no	—
214	C₂³	—	4	2	no	—
215	C₂³	—	4	2	no	—
216	C₂³	C₂×D₄×C₈₃	1	2	yes	—
217	C₂³	C₂×D₄×C₈₃	1	1	yes	characteristic
218	Q₈	—	2	4	no	—
219	Q₈	—	2	2	no	—
220	Q₈	—	2	2	no	—
221	Q₈	—	2	2	no	—
222	Q₈	—	4	2	no	—
223	Q₈	C₂×D₄×C₈₃	1	4	yes	—
224	D₄	—	2	16	no	—
225	D₄	—	2	4	no	—
226	D₄	—	2	2	no	—
227	D₄	C₂×D₄×C₈₃	1	4	yes	—
228	C₂×C₄	—	2	16	no	—
229	C₂×C₄	—	2	2	no	—
230	C₂×C₄	—	2	4	no	—
231	C₂×C₄	—	2	2	no	—
232	C₂×C₄	—	2	4	no	—
233	C₂×C₄	—	2	2	no	—
234	C₂×C₄	—	2	2	no	—
235	C₂×C₄	—	2	2	no	—
236	C₂×C₄	—	2	2	no	—
237	C₂×C₄	—	4	2	no	—
238	C₂×C₄	—	4	2	no	—
239	C₂×C₄	C₂×D₄×C₈₃	1	2	yes	—
240	C₂×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
241	C₂×C₄	C₂×D₄×C₈₃	1	2	yes	—
242	C₂×C₄	C₂×D₄×C₈₃	1	2	yes	—
243	C₂×C₄	C₂⁴×C₈₃	1	1	yes	characteristic
244	C₈	—	2	2	no	—
245	C₈	—	2	2	no	—
246	C₂²	—	2	2	no	—
247	C₂²	—	2	4	no	—
248	C₂²	—	2	2	no	—
249	C₂²	—	2	2	no	—
250	C₂²	—	4	2	no	—
251	C₂²	—	4	2	no	—
252	C₂²	—	4	4	no	—
253	C₂²	—	4	8	no	—
254	C₂²	#120×C₈₃	1	2	yes	—
255	C₂²	#135×C₈₃	1	2	yes	—
256	C₂²	#137×C₈₃	1	2	yes	—
257	C₂²	C₂²×D₄×C₈₃	1	1	yes	characteristic
258	C₄	—	2	2	no	—
259	C₄	—	2	4	no	—
260	C₄	—	4	2	no	—
261	C₄	—	4	2	no	—
262	C₄	#120×C₈₃	1	2	yes	—
263	C₄	C₂²×D₄×C₈₃	1	1	yes	characteristic
264	C₄	C₂²×D₄×C₈₃	1	1	yes	characteristic
265	C₂	—	4	2	no	—
266	C₂	—	4	4	no	—
267	C₂	#452×C₈₃	1	4	yes	direct product factor
268	C₂	C₂×#120×C₈₃	1	1	yes	characteristic
269	C₂	C₂×#135×C₈₃	1	1	yes	characteristic
270	C₂	C₂×#137×C₈₃	1	1	yes	characteristic

	subgroup	G/H
center, socle	C₂³×C₈₃	C₂×D₄
commutator, Frattini, cosocle	C₂×C₄	C₂⁴×C₈₃
prime radical	C₂²×#41×C₈₃	C₂
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₈₃ ◁ C₂³×C₄×C₈₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂²×Q₁₆×C₈₃	C₂	semidirect
3. C₂²×SD₁₆×C₈₃	C₂	semidirect
4. C₂×C₂²⋊Q₈×C₈₃	C₂	semidirect
5. #452×C₈₃	C₂	direct (N × H)
6. C₂×#103×C₈₃	C₂	semidirect
7. C₂×#102×C₈₃	C₂	semidirect
8. C₂×C₂²⋊C₈×C₈₃	C₂	semidirect
20. C₂×Q₁₆×C₈₃	C₂²	semidirect
22. C₂×SD₁₆×C₈₃	C₂²	semidirect
26. C₂²⋊Q₈×C₈₃	C₂²	semidirect
30. #103×C₈₃	C₂²	semidirect
31. #102×C₈₃	C₂²	semidirect
32. C₂²⋊C₈×C₈₃	C₂²	semidirect
131. C₂×C₈₃	#452	direct (N × H)
135. C₂×#452	C₈₃	direct (N × H)
136. C₈₃	C₂×#452	direct (N × H)
138. C₂²×Q₁₆	C₂×C₈₃	semidirect
139. C₂²×SD₁₆	C₂×C₈₃	semidirect
140. C₂×C₂²⋊Q₈	C₂×C₈₃	semidirect
141. #452	C₂×C₈₃	direct (N × H)
142. C₂×#103	C₂×C₈₃	semidirect
143. C₂×#102	C₂×C₈₃	semidirect
144. C₂×C₂²⋊C₈	C₂×C₈₃	semidirect
156. C₂×Q₁₆	C₂²×C₈₃	semidirect
158. C₂×SD₁₆	C₂²×C₈₃	semidirect
162. C₂²⋊Q₈	C₂²×C₈₃	semidirect
166. #103	C₂²×C₈₃	semidirect
167. #102	C₂²×C₈₃	semidirect
168. C₂²⋊C₈	C₂²×C₈₃	semidirect
267. C₂	#452×C₈₃	direct (N × H)

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroup	index
C₂²×#41×C₈₃	2
C₂²×Q₁₆×C₈₃	2
C₂²×SD₁₆×C₈₃	2
C₂×C₂²⋊Q₈×C₈₃	2
#452×C₈₃	2
C₂×#103×C₈₃	2
C₂×#102×C₈₃	2
C₂×C₂²⋊C₈×C₈₃	2
C₂×#452	83

Subgroups — C2×#452×C83 #24904686

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₂×#452×C₈₃ #24904686