Subgroups — C₈₃×#3330 #24905015

Order 10624.

420 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	#547×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
2	#547×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
3	#487×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
4	#484×C₈₃	C₂	1	2	yes	maximal, complemented
5	#474×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
6	#468×C₈₃	C₂	1	2	yes	maximal, complemented
7	#461×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
8	#449×C₈₃	C₂	1	2	yes	maximal, complemented
9	C₄×D₈×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
10	#436×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
11	#417×C₈₃	C₂	1	2	yes	maximal, complemented
12	C₂×#41×C₈₃	—	2	1	no	—
13	C₂×#41×C₈₃	C₂²	1	2	yes	—
14	C₂×SD₁₆×C₈₃	C₂²	1	2	yes	—
15	C₂×D₈×C₈₃	C₂²	1	1	yes	characteristic
16	C₂×M₁₆×C₈₃	C₂²	1	2	yes	—
17	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
18	C₄⋊Q₈×C₈₃	C₂²	1	1	yes	characteristic
19	#123×C₈₃	—	2	2	no	—
20	#123×C₈₃	—	2	2	no	—
21	C₂²⋊Q₈×C₈₃	—	2	1	no	—
22	C₂²⋊Q₈×C₈₃	C₂²	1	2	yes	—
23	#121×C₈₃	—	2	1	no	—
24	#121×C₈₃	C₂²	1	2	yes	—
25	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
26	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
27	C₄×D₄×C₈₃	C₂²	1	1	yes	characteristic
28	C₂×C₄⋊C₄×C₈₃	—	2	1	no	—
29	C₂×C₄⋊C₄×C₈₃	C₂²	1	2	yes	—
30	#107×C₈₃	C₂²	1	2	yes	—
31	#107×C₈₃	C₂²	1	1	yes	characteristic
32	C₄⋊C₈×C₈₃	C₂²	1	1	yes	characteristic
33	#103×C₈₃	C₂²	1	2	yes	—
34	#103×C₈₃	C₂²	1	2	yes	—
35	#103×C₈₃	C₂²	1	2	yes	—
36	#102×C₈₃	C₂²	1	1	yes	characteristic
37	#102×C₈₃	C₂²	1	2	yes	—
38	#102×C₈₃	C₂²	1	1	yes	characteristic
39	C₂²⋊C₈×C₈₃	C₂²	1	2	yes	—
40	C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
41	#41×C₈₃	—	2	4	no	—
42	#41×C₈₃	—	4	2	no	—
43	#41×C₈₃	—	4	4	no	—
44	C₂×Q₈×C₈₃	—	2	1	no	—
45	C₂×Q₈×C₈₃	C₂³	1	2	yes	—
46	C₂×D₄×C₈₃	—	2	2	no	—
47	C₂×D₄×C₈₃	—	2	2	no	—
48	C₂×D₄×C₈₃	D₄	1	1	yes	characteristic
49	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
50	C₂×D₄×C₈₃	C₂³	1	1	yes	characteristic
51	C₂²×C₄×C₈₃	—	2	2	no	—
52	C₂²×C₄×C₈₃	—	2	1	no	—
53	C₂²×C₄×C₈₃	—	2	2	no	—
54	C₂²×C₄×C₈₃	—	2	1	no	—
55	C₂²×C₄×C₈₃	C₂³	1	2	yes	—
56	SD₁₆×C₈₃	—	4	2	no	—
57	D₈×C₈₃	—	2	2	no	—
58	M₁₆×C₈₃	—	4	2	no	—
59	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
60	C₂×C₈×C₈₃	C₂³	1	1	yes	characteristic
61	C₂×C₈×C₈₃	C₂³	1	2	yes	—
62	C₄⋊C₄×C₈₃	—	2	1	no	—
63	C₄⋊C₄×C₈₃	—	2	2	no	—
64	C₄⋊C₄×C₈₃	—	2	2	no	—
65	C₄⋊C₄×C₈₃	—	2	1	no	—
66	C₄⋊C₄×C₈₃	—	2	1	no	—
67	C₄⋊C₄×C₈₃	D₄	1	1	yes	characteristic
68	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
69	C₄⋊C₄×C₈₃	C₂³	1	2	yes	—
70	C₄⋊C₄×C₈₃	C₂³	1	1	yes	characteristic
71	C₂²⋊C₄×C₈₃	—	2	2	no	—
72	C₂²⋊C₄×C₈₃	—	2	1	no	—
73	C₂²⋊C₄×C₈₃	—	2	1	no	—
74	C₂²⋊C₄×C₈₃	—	2	2	no	—
75	C₂²⋊C₄×C₈₃	D₄	1	2	yes	—
76	C₄²×C₈₃	C₂³	1	1	yes	characteristic
77	C₂³×C₈₃	—	2	1	no	—
78	C₂³×C₈₃	—	2	1	no	—
79	C₂³×C₈₃	C₂×D₄	1	2	yes	—
80	Q₈×C₈₃	—	2	2	no	—
81	Q₈×C₈₃	—	2	2	no	—
82	Q₈×C₈₃	—	4	2	no	—
83	D₄×C₈₃	—	2	2	no	—
84	D₄×C₈₃	—	2	1	no	—
85	D₄×C₈₃	—	2	1	no	—
86	D₄×C₈₃	—	2	1	no	—
87	D₄×C₈₃	—	4	4	no	—
88	D₄×C₈₃	—	4	4	no	—
89	D₄×C₈₃	#41	1	2	yes	complemented
90	C₂×C₄×C₈₃	—	2	2	no	—
91	C₂×C₄×C₈₃	—	2	2	no	—
92	C₂×C₄×C₈₃	—	2	1	no	—
93	C₂×C₄×C₈₃	—	2	2	no	—
94	C₂×C₄×C₈₃	—	2	1	no	—
95	C₂×C₄×C₈₃	—	2	2	no	—
96	C₂×C₄×C₈₃	—	4	4	no	—
97	C₂×C₄×C₈₃	—	4	4	no	—
98	C₂×C₄×C₈₃	—	4	2	no	—
99	C₂×C₄×C₈₃	—	4	2	no	—
100	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
101	C₂×C₄×C₈₃	C₂×D₄	1	2	yes	—
102	C₂×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
103	C₂×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
104	C₈×C₈₃	—	2	1	no	—
105	C₈×C₈₃	—	2	1	no	—
106	C₈×C₈₃	—	2	2	no	—
107	C₂²×C₈₃	—	2	2	no	—
108	C₂²×C₈₃	—	2	2	no	—
109	C₂²×C₈₃	—	2	2	no	—
110	C₂²×C₈₃	—	2	2	no	—
111	C₂²×C₈₃	—	4	1	no	—
112	C₂²×C₈₃	—	4	1	no	—
113	C₂²×C₈₃	—	4	1	no	—
114	C₂²×C₈₃	—	4	1	no	—
115	C₂²×C₈₃	—	4	1	no	—
116	C₂²×C₈₃	C₂²×D₄	1	1	yes	characteristic
117	C₄×C₈₃	—	2	2	no	—
118	C₄×C₈₃	—	2	1	no	—
119	C₄×C₈₃	—	4	2	no	—
120	C₄×C₈₃	—	4	1	no	—
121	C₄×C₈₃	—	4	1	no	—
122	C₄×C₈₃	—	4	2	no	—
123	C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
124	C₄×C₈₃	C₂×#41	1	1	yes	characteristic
125	C₄×C₈₃	H(5,𝔽₂)	1	1	yes	characteristic
126	C₂×C₈₃	—	4	2	no	—
127	C₂×C₈₃	—	4	2	no	—
128	C₂×C₈₃	—	8	1	no	—
129	C₂×C₈₃	#546	1	1	yes	characteristic
130	C₂×C₈₃	C₂×AGL(1,ℤ/8)	1	1	yes	characteristic
131	C₂×C₈₃	#578	1	1	yes	characteristic
132	#3330	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
133	C₈₃	#3330	1	1	yes	characteristic, direct product factor, Sylow
134	#547	C₂×C₈₃	1	1	yes	characteristic, complemented
135	#547	C₂×C₈₃	1	1	yes	characteristic, complemented
136	#487	C₂×C₈₃	1	1	yes	characteristic, complemented
137	#484	C₂×C₈₃	1	2	yes	complemented
138	#474	C₂×C₈₃	1	1	yes	characteristic, complemented
139	#468	C₂×C₈₃	1	2	yes	complemented
140	#461	C₂×C₈₃	1	1	yes	characteristic, complemented
141	#449	C₂×C₈₃	1	2	yes	complemented
142	C₄×D₈	C₂×C₈₃	1	1	yes	characteristic, complemented
143	#436	C₂×C₈₃	1	1	yes	characteristic, complemented
144	#417	C₂×C₈₃	1	2	yes	complemented
145	C₂×#41	—	2	1	no	—
146	C₂×#41	C₂²×C₈₃	1	2	yes	—
147	C₂×SD₁₆	C₂²×C₈₃	1	2	yes	—
148	C₂×D₈	C₂²×C₈₃	1	1	yes	characteristic
149	C₂×M₁₆	C₂²×C₈₃	1	2	yes	—
150	C₄⋊Q₈	C₂²×C₈₃	1	1	yes	characteristic
151	C₄⋊Q₈	C₂²×C₈₃	1	1	yes	characteristic
152	#123	—	2	2	no	—
153	#123	—	2	2	no	—
154	C₂²⋊Q₈	—	2	1	no	—
155	C₂²⋊Q₈	C₂²×C₈₃	1	2	yes	—
156	#121	—	2	1	no	—
157	#121	C₂²×C₈₃	1	2	yes	—
158	C₄×D₄	C₂²×C₈₃	1	1	yes	characteristic
159	C₄×D₄	C₂²×C₈₃	1	1	yes	characteristic
160	C₄×D₄	C₂²×C₈₃	1	1	yes	characteristic
161	C₂×C₄⋊C₄	—	2	1	no	—
162	C₂×C₄⋊C₄	C₂²×C₈₃	1	2	yes	—
163	#107	C₂²×C₈₃	1	1	yes	characteristic
164	#107	C₂²×C₈₃	1	2	yes	—
165	C₄⋊C₈	C₂²×C₈₃	1	1	yes	characteristic
166	#103	C₂²×C₈₃	1	2	yes	—
167	#103	C₂²×C₈₃	1	2	yes	—
168	#103	C₂²×C₈₃	1	2	yes	—
169	#102	C₂²×C₈₃	1	1	yes	characteristic
170	#102	C₂²×C₈₃	1	2	yes	—
171	#102	C₂²×C₈₃	1	1	yes	characteristic
172	C₂²⋊C₈	C₂²×C₈₃	1	2	yes	—
173	C₄×C₈	C₂²×C₈₃	1	1	yes	characteristic
174	#41	—	2	4	no	—
175	#41	—	4	2	no	—
176	#41	—	4	4	no	—
177	C₂×Q₈	—	2	1	no	—
178	C₂×Q₈	C₂³×C₈₃	1	2	yes	—
179	C₂×D₄	—	2	2	no	—
180	C₂×D₄	—	2	2	no	—
181	C₂×D₄	D₄×C₈₃	1	1	yes	characteristic
182	C₂×D₄	C₂³×C₈₃	1	1	yes	characteristic
183	C₂×D₄	C₂³×C₈₃	1	1	yes	characteristic
184	C₂²×C₄	—	2	2	no	—
185	C₂²×C₄	—	2	1	no	—
186	C₂²×C₄	—	2	2	no	—
187	C₂²×C₄	—	2	1	no	—
188	C₂²×C₄	C₂³×C₈₃	1	2	yes	—
189	SD₁₆	—	4	2	no	—
190	D₈	—	2	2	no	—
191	M₁₆	—	4	2	no	—
192	C₂×C₈	C₂³×C₈₃	1	1	yes	characteristic
193	C₂×C₈	C₂³×C₈₃	1	1	yes	characteristic
194	C₂×C₈	C₂³×C₈₃	1	2	yes	—
195	C₄⋊C₄	—	2	1	no	—
196	C₄⋊C₄	—	2	2	no	—
197	C₄⋊C₄	—	2	2	no	—
198	C₄⋊C₄	—	2	1	no	—
199	C₄⋊C₄	—	2	1	no	—
200	C₄⋊C₄	D₄×C₈₃	1	1	yes	characteristic
201	C₄⋊C₄	C₂³×C₈₃	1	1	yes	characteristic
202	C₄⋊C₄	C₂³×C₈₃	1	2	yes	—
203	C₄⋊C₄	C₂³×C₈₃	1	1	yes	characteristic
204	C₂²⋊C₄	—	2	2	no	—
205	C₂²⋊C₄	—	2	1	no	—
206	C₂²⋊C₄	—	2	1	no	—
207	C₂²⋊C₄	—	2	2	no	—
208	C₂²⋊C₄	D₄×C₈₃	1	2	yes	—
209	C₄²	C₂³×C₈₃	1	1	yes	characteristic
210	C₂³	—	2	1	no	—
211	C₂³	—	2	1	no	—
212	C₂³	C₂×D₄×C₈₃	1	2	yes	—
213	Q₈	—	2	2	no	—
214	Q₈	—	2	2	no	—
215	Q₈	—	4	2	no	—
216	D₄	—	2	2	no	—
217	D₄	—	2	1	no	—
218	D₄	—	2	1	no	—
219	D₄	—	2	1	no	—
220	D₄	—	4	4	no	—
221	D₄	—	4	4	no	—
222	D₄	#41×C₈₃	1	2	yes	complemented
223	C₂×C₄	—	2	2	no	—
224	C₂×C₄	—	2	2	no	—
225	C₂×C₄	—	2	1	no	—
226	C₂×C₄	—	2	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	2	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	4	4	no	—
231	C₂×C₄	—	4	2	no	—
232	C₂×C₄	—	4	2	no	—
233	C₂×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
234	C₂×C₄	C₂×D₄×C₈₃	1	2	yes	—
235	C₂×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
236	C₂×C₄	C₂⁴×C₈₃	1	1	yes	characteristic
237	C₈	—	2	1	no	—
238	C₈	—	2	1	no	—
239	C₈	—	2	2	no	—
240	C₂²	—	2	2	no	—
241	C₂²	—	2	2	no	—
242	C₂²	—	2	2	no	—
243	C₂²	—	2	2	no	—
244	C₂²	—	4	1	no	—
245	C₂²	—	4	1	no	—
246	C₂²	—	4	1	no	—
247	C₂²	—	4	1	no	—
248	C₂²	—	4	1	no	—
249	C₂²	C₂²×D₄×C₈₃	1	1	yes	characteristic
250	C₄	—	2	2	no	—
251	C₄	—	2	1	no	—
252	C₄	—	4	2	no	—
253	C₄	—	4	1	no	—
254	C₄	—	4	2	no	—
255	C₄	—	4	1	no	—
256	C₄	AGL(1,ℤ/8)×C₈₃	1	2	yes	—
257	C₄	C₂×#41×C₈₃	1	1	yes	characteristic
258	C₄	H(5,𝔽₂)×C₈₃	1	1	yes	characteristic
259	C₂	—	4	2	no	—
260	C₂	—	4	2	no	—
261	C₂	—	8	1	no	—
262	C₂	#546×C₈₃	1	1	yes	characteristic
263	C₂	C₂×AGL(1,ℤ/8)×C₈₃	1	1	yes	characteristic
264	C₂	#578×C₈₃	1	1	yes	characteristic

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

Complements

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

N	H ≅ G/N	type
1. #547×C₈₃	C₂	semidirect
2. #547×C₈₃	C₂	semidirect
3. #487×C₈₃	C₂	semidirect
4. #484×C₈₃	C₂	semidirect
5. #474×C₈₃	C₂	semidirect
6. #468×C₈₃	C₂	semidirect
7. #461×C₈₃	C₂	semidirect
8. #449×C₈₃	C₂	semidirect
9. C₄×D₈×C₈₃	C₂	semidirect
10. #436×C₈₃	C₂	semidirect
11. #417×C₈₃	C₂	semidirect
89. D₄×C₈₃	#41	semidirect
132. #3330	C₈₃	direct (N × H)
133. C₈₃	#3330	direct (N × H)
134. #547	C₂×C₈₃	semidirect
135. #547	C₂×C₈₃	semidirect
136. #487	C₂×C₈₃	semidirect
137. #484	C₂×C₈₃	semidirect
138. #474	C₂×C₈₃	semidirect
139. #468	C₂×C₈₃	semidirect
140. #461	C₂×C₈₃	semidirect
141. #449	C₂×C₈₃	semidirect
142. C₄×D₈	C₂×C₈₃	semidirect
143. #436	C₂×C₈₃	semidirect
144. #417	C₂×C₈₃	semidirect
222. D₄	#41×C₈₃	semidirect

Maximal subgroups

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

subgroup	index
#547×C₈₃	2
#487×C₈₃	2
#484×C₈₃	2
#474×C₈₃	2
#468×C₈₃	2
#461×C₈₃	2
#449×C₈₃	2
C₄×D₈×C₈₃	2
#436×C₈₃	2
#417×C₈₃	2
#3330	83

Subgroups — C83×#3330 #24905015

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₃×#3330 #24905015