Subgroups — C₈₃×#12822 #35240646

Order 21248.

400 conjugacy classes of subgroups.

The subgroup lattice has 296 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#443×C₈₃	C₂	1	1	yes	maximal, characteristic
2	C₂×#364×C₈₃	C₂	1	1	yes	maximal, characteristic, complemented
3	C₈₃×#2143	C₂	1	2	yes	maximal
4	C₈₃×#1848	C₂	1	1	yes	maximal, characteristic, complemented
5	C₈₃×#1388	C₂	1	2	yes	maximal, complemented
6	C₂×#135×C₈₃	C₂²	1	1	yes	characteristic
7	C₂×#131×C₈₃	—	2	1	no	—
8	C₂×M₃₂×C₈₃	C₂²	1	2	yes	—
9	#443×C₈₃	—	2	1	no	—
10	#443×C₈₃	—	2	1	no	—
11	#443×C₈₃	C₄	1	2	yes	—
12	#443×C₈₃	C₄	1	2	yes	—
13	C₂×#108×C₈₃	—	2	1	no	—
14	C₂×#108×C₈₃	C₂²	1	2	yes	—
15	C₂×#108×C₈₃	C₂²	1	1	yes	characteristic
16	#423×C₈₃	—	2	1	no	—
17	C₂×#104×C₈₃	—	2	1	no	—
18	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
19	#364×C₈₃	—	2	1	no	—
20	#364×C₈₃	—	2	1	no	—
21	#359×C₈₃	—	2	4	no	—
22	C₂×#41×C₈₃	—	2	1	no	—
23	#135×C₈₃	—	2	1	no	—
24	#135×C₈₃	—	2	1	no	—
25	#135×C₈₃	C₂×C₄	1	2	yes	—
26	#135×C₈₃	D₄	1	2	yes	—
27	C₂×Q₁₆×C₈₃	Q₈	1	1	yes	characteristic
28	C₂×SD₁₆×C₈₃	—	2	1	no	—
29	C₂×D₈×C₈₃	Q₈	1	1	yes	characteristic
30	#131×C₈₃	—	2	2	no	—
31	#131×C₈₃	—	2	2	no	—
32	#131×C₈₃	—	4	1	no	—
33	#131×C₈₃	—	4	1	no	—
34	C₂×M₁₆×C₈₃	—	2	1	no	—
35	C₂×M₁₆×C₈₃	—	2	2	no	—
36	C₂×M₁₆×C₈₃	—	4	1	no	—
37	C₂²×C₈×C₈₃	—	4	1	no	—
38	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
39	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
40	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
41	M₃₂×C₈₃	—	2	2	no	—
42	M₃₂×C₈₃	—	2	2	no	—
43	C₂×C₁₆×C₈₃	—	2	2	no	—
44	#108×C₈₃	—	2	1	no	—
45	#108×C₈₃	—	2	2	no	—
46	#108×C₈₃	—	2	2	no	—
47	#108×C₈₃	—	4	1	no	—
48	#108×C₈₃	—	4	1	no	—
49	#108×C₈₃	C₂×C₄	1	2	yes	—
50	C₄⋊C₈×C₈₃	—	4	2	no	—
51	#104×C₈₃	—	2	2	no	—
52	#104×C₈₃	—	2	2	no	—
53	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
54	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
55	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
56	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
57	#41×C₈₃	—	2	2	no	—
58	#41×C₈₃	—	2	2	no	—
59	#41×C₈₃	—	4	1	no	—
60	#41×C₈₃	—	4	1	no	—
61	C₂×Q₈×C₈₃	—	2	1	no	—
62	C₂×D₄×C₈₃	—	2	1	no	—
63	C₂×D₄×C₈₃	—	4	1	no	—
64	C₂²×C₄×C₈₃	—	2	1	no	—
65	C₂²×C₄×C₈₃	—	4	1	no	—
66	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
67	Q₁₆×C₈₃	—	2	1	no	—
68	Q₁₆×C₈₃	C₄⋊C₄	1	2	yes	—
69	SD₁₆×C₈₃	—	2	2	no	—
70	SD₁₆×C₈₃	—	2	2	no	—
71	D₈×C₈₃	—	2	1	no	—
72	D₈×C₈₃	C₄⋊C₄	1	2	yes	—
73	M₁₆×C₈₃	—	2	2	no	—
74	M₁₆×C₈₃	—	4	1	no	—
75	M₁₆×C₈₃	—	4	2	no	—
76	M₁₆×C₈₃	—	4	2	no	—
77	M₁₆×C₈₃	—	4	2	no	—
78	M₁₆×C₈₃	—	4	2	no	—
79	C₂×C₈×C₈₃	—	4	1	no	—
80	C₂×C₈×C₈₃	—	4	2	no	—
81	C₂×C₈×C₈₃	—	4	2	no	—
82	C₂×C₈×C₈₃	—	4	2	no	—
83	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
84	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₈×C₈₃	D₈	1	1	yes	characteristic
86	C₂×C₈×C₈₃	D₈	1	1	yes	characteristic
87	C₂×C₈×C₈₃	SD₁₆	1	1	yes	characteristic
88	C₂×C₈×C₈₃	SD₁₆	1	1	yes	characteristic
89	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
90	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
91	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
92	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
93	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
94	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
95	C₄²×C₈₃	—	2	1	no	—
96	C₄²×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
97	C₄²×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
98	C₁₆×C₈₃	—	4	2	no	—
99	C₂³×C₈₃	—	4	1	no	—
100	C₂³×C₈₃	#137	1	1	yes	characteristic
101	Q₈×C₈₃	—	2	2	no	—
102	Q₈×C₈₃	—	4	1	no	—
103	D₄×C₈₃	—	2	2	no	—
104	D₄×C₈₃	—	4	2	no	—
105	D₄×C₈₃	—	4	2	no	—
106	D₄×C₈₃	—	4	1	no	—
107	C₂×C₄×C₈₃	—	2	1	no	—
108	C₂×C₄×C₈₃	—	2	1	no	—
109	C₂×C₄×C₈₃	—	4	2	no	—
110	C₂×C₄×C₈₃	—	4	1	no	—
111	C₂×C₄×C₈₃	—	4	2	no	—
112	C₂×C₄×C₈₃	—	4	2	no	—
113	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
114	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
115	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
116	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
117	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
118	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
119	C₈×C₈₃	—	4	2	no	—
120	C₈×C₈₃	—	8	2	no	—
121	C₈×C₈₃	#102	1	1	yes	characteristic
122	C₈×C₈₃	#102	1	1	yes	characteristic
123	C₈×C₈₃	#102	1	1	yes	characteristic
124	C₈×C₈₃	#102	1	1	yes	characteristic
125	C₈×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
126	C₈×C₈₃	#117	1	1	yes	characteristic
127	C₈×C₈₃	#121	1	1	yes	characteristic
128	C₈×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
129	C₂²×C₈₃	—	4	1	no	—
130	C₂²×C₈₃	—	4	2	no	—
131	C₂²×C₈₃	—	8	2	no	—
132	C₂²×C₈₃	#417	1	1	yes	characteristic
133	C₂²×C₈₃	#461	1	1	yes	characteristic
134	C₂²×C₈₃	#474	1	1	yes	characteristic
135	C₄×C₈₃	—	2	2	no	—
136	C₄×C₈₃	—	4	1	no	—
137	C₄×C₈₃	—	4	2	no	—
138	C₄×C₈₃	#380	1	1	yes	characteristic
139	C₄×C₈₃	C₂×#102	1	1	yes	characteristic
140	C₄×C₈₃	#459	1	1	yes	characteristic
141	C₄×C₈₃	#476	1	1	yes	characteristic
142	#12822	C₈₃	1	1	yes	maximal, characteristic, direct product factor, Sylow
143	C₂×C₈₃	—	2	1	no	—
144	C₂×C₈₃	—	2	1	no	—
145	C₂×C₈₃	—	2	1	no	—
146	C₂×C₈₃	—	8	2	no	—
147	C₂×C₈₃	#1802	1	1	yes	characteristic
148	C₂×#443	C₂×C₈₃	1	1	yes	characteristic
149	C₂×#364	C₂×C₈₃	1	1	yes	characteristic, complemented
150	#2143	C₂×C₈₃	1	2	yes	—
151	#1848	C₂×C₈₃	1	1	yes	characteristic, complemented
152	#1388	C₂×C₈₃	1	2	yes	complemented
153	C₈₃	#12822	1	1	yes	characteristic, direct product factor, Sylow
154	C₂×#135	C₂²×C₈₃	1	1	yes	characteristic
155	C₂×#131	—	2	1	no	—
156	C₂×M₃₂	C₂²×C₈₃	1	2	yes	—
157	#443	—	2	1	no	—
158	#443	—	2	1	no	—
159	#443	C₄×C₈₃	1	2	yes	—
160	#443	C₄×C₈₃	1	2	yes	—
161	C₂×#108	—	2	1	no	—
162	C₂×#108	C₂²×C₈₃	1	2	yes	—
163	C₂×#108	C₂²×C₈₃	1	1	yes	characteristic
164	#423	—	2	1	no	—
165	C₂×#104	—	2	1	no	—
166	C₂×C₄×C₈	C₂²×C₈₃	1	1	yes	characteristic
167	#364	—	2	1	no	—
168	#364	—	2	1	no	—
169	#359	—	2	4	no	—
170	C₂×#41	—	2	1	no	—
171	#135	—	2	1	no	—
172	#135	—	2	1	no	—
173	#135	C₂×C₄×C₈₃	1	2	yes	—
174	#135	D₄×C₈₃	1	2	yes	—
175	C₂×Q₁₆	Q₈×C₈₃	1	1	yes	characteristic
176	C₂×SD₁₆	—	2	1	no	—
177	C₂×D₈	Q₈×C₈₃	1	1	yes	characteristic
178	#131	—	2	2	no	—
179	#131	—	2	2	no	—
180	#131	—	4	1	no	—
181	#131	—	4	1	no	—
182	C₂×M₁₆	—	2	1	no	—
183	C₂×M₁₆	—	2	2	no	—
184	C₂×M₁₆	—	4	1	no	—
185	C₂²×C₈	—	4	1	no	—
186	C₂²×C₈	D₄×C₈₃	1	1	yes	characteristic
187	C₂²×C₈	C₂³×C₈₃	1	1	yes	characteristic
188	C₂×C₄²	D₄×C₈₃	1	1	yes	characteristic
189	M₃₂	—	2	2	no	—
190	M₃₂	—	2	2	no	—
191	C₂×C₁₆	—	2	2	no	—
192	#108	—	2	1	no	—
193	#108	—	2	2	no	—
194	#108	—	2	2	no	—
195	#108	—	4	1	no	—
196	#108	—	4	1	no	—
197	#108	C₂×C₄×C₈₃	1	2	yes	—
198	C₄⋊C₈	—	4	2	no	—
199	#104	—	2	2	no	—
200	#104	—	2	2	no	—
201	C₄×C₈	C₂×C₄×C₈₃	1	1	yes	characteristic
202	C₄×C₈	C₂×C₄×C₈₃	1	1	yes	characteristic
203	C₄×C₈	D₄×C₈₃	1	1	yes	characteristic
204	C₄×C₈	D₄×C₈₃	1	1	yes	characteristic
205	#41	—	2	2	no	—
206	#41	—	2	2	no	—
207	#41	—	4	1	no	—
208	#41	—	4	1	no	—
209	C₂×Q₈	—	2	1	no	—
210	C₂×D₄	—	2	1	no	—
211	C₂×D₄	—	4	1	no	—
212	C₂²×C₄	—	2	1	no	—
213	C₂²×C₄	—	4	1	no	—
214	C₂²×C₄	C₂×D₄×C₈₃	1	1	yes	characteristic
215	Q₁₆	—	2	1	no	—
216	Q₁₆	C₄⋊C₄×C₈₃	1	2	yes	—
217	SD₁₆	—	2	2	no	—
218	SD₁₆	—	2	2	no	—
219	D₈	—	2	1	no	—
220	D₈	C₄⋊C₄×C₈₃	1	2	yes	—
221	M₁₆	—	2	2	no	—
222	M₁₆	—	4	1	no	—
223	M₁₆	—	4	2	no	—
224	M₁₆	—	4	2	no	—
225	M₁₆	—	4	2	no	—
226	M₁₆	—	4	2	no	—
227	C₂×C₈	—	4	1	no	—
228	C₂×C₈	—	4	2	no	—
229	C₂×C₈	—	4	2	no	—
230	C₂×C₈	—	4	2	no	—
231	C₂×C₈	C₂²⋊C₄×C₈₃	1	1	yes	characteristic
232	C₂×C₈	C₂²⋊C₄×C₈₃	1	1	yes	characteristic
233	C₂×C₈	D₈×C₈₃	1	1	yes	characteristic
234	C₂×C₈	D₈×C₈₃	1	1	yes	characteristic
235	C₂×C₈	SD₁₆×C₈₃	1	1	yes	characteristic
236	C₂×C₈	SD₁₆×C₈₃	1	1	yes	characteristic
237	C₂×C₈	C₂²×C₄×C₈₃	1	1	yes	characteristic
238	C₂×C₈	C₂×D₄×C₈₃	1	1	yes	characteristic
239	C₂×C₈	C₂×D₄×C₈₃	1	1	yes	characteristic
240	C₂×C₈	C₂×Q₈×C₈₃	1	1	yes	characteristic
241	C₂×C₈	#41×C₈₃	1	1	yes	characteristic
242	C₂×C₈	#41×C₈₃	1	1	yes	characteristic
243	C₄²	—	2	1	no	—
244	C₄²	C₂²⋊C₄×C₈₃	1	1	yes	characteristic
245	C₄²	C₂²⋊C₄×C₈₃	1	1	yes	characteristic
246	C₁₆	—	4	2	no	—
247	C₂³	—	4	1	no	—
248	C₂³	#137×C₈₃	1	1	yes	characteristic
249	Q₈	—	2	2	no	—
250	Q₈	—	4	1	no	—
251	D₄	—	2	2	no	—
252	D₄	—	4	2	no	—
253	D₄	—	4	2	no	—
254	D₄	—	4	1	no	—
255	C₂×C₄	—	2	1	no	—
256	C₂×C₄	—	2	1	no	—
257	C₂×C₄	—	4	2	no	—
258	C₂×C₄	—	4	1	no	—
259	C₂×C₄	—	4	2	no	—
260	C₂×C₄	—	4	2	no	—
261	C₂×C₄	C₂×C₂²⋊C₄×C₈₃	1	1	yes	characteristic
262	C₂×C₄	#121×C₈₃	1	1	yes	characteristic
263	C₂×C₄	C₂²⋊Q₈×C₈₃	1	1	yes	characteristic
264	C₂×C₄	C₂×D₈×C₈₃	1	1	yes	characteristic
265	C₂×C₄	C₂×SD₁₆×C₈₃	1	1	yes	characteristic
266	C₂×C₄	AGL(1,ℤ/8)×C₈₃	1	1	yes	characteristic
267	C₈	—	4	2	no	—
268	C₈	—	8	2	no	—
269	C₈	#102×C₈₃	1	1	yes	characteristic
270	C₈	#102×C₈₃	1	1	yes	characteristic
271	C₈	#102×C₈₃	1	1	yes	characteristic
272	C₈	#102×C₈₃	1	1	yes	characteristic
273	C₈	C₂×C₄⋊C₄×C₈₃	1	1	yes	characteristic
274	C₈	#117×C₈₃	1	1	yes	characteristic
275	C₈	#121×C₈₃	1	1	yes	characteristic
276	C₈	C₂²⋊Q₈×C₈₃	1	1	yes	characteristic
277	C₂²	—	4	1	no	—
278	C₂²	—	4	2	no	—
279	C₂²	—	8	2	no	—
280	C₂²	#417×C₈₃	1	1	yes	characteristic
281	C₂²	#461×C₈₃	1	1	yes	characteristic
282	C₂²	#474×C₈₃	1	1	yes	characteristic
283	C₄	—	2	2	no	—
284	C₄	—	4	1	no	—
285	C₄	—	4	2	no	—
286	C₄	#380×C₈₃	1	1	yes	characteristic
287	C₄	C₂×#102×C₈₃	1	1	yes	characteristic
288	C₄	#459×C₈₃	1	1	yes	characteristic
289	C₄	#476×C₈₃	1	1	yes	characteristic
290	C₂	—	2	1	no	—
291	C₂	—	2	1	no	—
292	C₂	—	2	1	no	—
293	C₂	—	8	2	no	—
294	C₂	C₈₃×#1802	1	1	yes	characteristic

	subgroup	G/H
center	C₄×C₈₃	#459
commutator	C₂×C₈	C₂²×C₄×C₈₃
Frattini, cosocle	C₂²×C₈	C₂³×C₈₃
socle	C₂×C₈₃	#1802
prime radical	C₂×#135×C₈₃	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄×C₈₃ ◁ C₂²×C₄×C₈₃ ◁ C₂×C₄×C₈×C₈₃ ◁ G

Complements

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

N	H ≅ G/N	type
2. C₂×#364×C₈₃	C₂	semidirect
4. C₈₃×#1848	C₂	semidirect
5. C₈₃×#1388	C₂	semidirect
142. #12822	C₈₃	direct (N × H)
149. C₂×#364	C₂×C₈₃	semidirect
151. #1848	C₂×C₈₃	semidirect
152. #1388	C₂×C₈₃	semidirect
153. C₈₃	#12822	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×#443×C₈₃	2
C₂×#364×C₈₃	2
C₈₃×#2143	2
C₈₃×#1848	2
C₈₃×#1388	2
#12822	83

Subgroups — C83×#12822 #35240646

Conjugacy classes of subgroups

Complements

Maximal subgroups

Subgroups — C₈₃×#12822 #35240646