Subgroups — #35298063

Order 21248.

422 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	#24907614	C₂	1	2	yes	maximal
2	C₄×C₈×Dic₈₃	C₂	1	1	yes	maximal, characteristic
3	#24905298	C₂	1	1	yes	maximal, characteristic
4	#24905297	C₂	1	2	yes	maximal
5	C₈₃×#1277	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
7	C₂×Dic₈₃⋊C₈	—	2	2	no	—
8	C₂×Dic₈₃⋊C₈	—	2	2	no	—
9	C₂×C₄×C₈₃⋊C₈	C₂²	1	1	yes	characteristic
10	C₂×C₈×Dic₈₃	C₄	1	1	yes	characteristic
11	C₂×C₈×Dic₈₃	C₄	1	1	yes	characteristic
12	C₂×C₈×Dic₈₃	C₄	1	1	yes	characteristic
13	C₂×C₈×Dic₈₃	C₄	1	1	yes	characteristic
14	C₄²×Dic₈₃	C₂²	1	1	yes	characteristic
15	C₂×C₄⋊C₈×C₈₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	C₂²×C₈₃⋊C₈	—	2	2	no	—
18	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
19	C₂²×C₈₃⋊C₈	C₂×C₄	1	1	yes	characteristic
20	C₂×C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
21	C₂×C₄×Dic₈₃	C₂×C₄	1	1	yes	characteristic
22	C₂×C₄×Dic₈₃	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₈₃	D₄	1	1	yes	characteristic
24	C₂×C₄×Dic₈₃	Q₈	1	1	yes	characteristic
25	C₂×C₄×Dic₈₃	Q₈	1	1	yes	characteristic
26	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
27	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
28	Dic₈₃⋊C₈	—	4	4	no	—
29	Dic₈₃⋊C₈	—	4	4	no	—
30	C₄×C₈₃⋊C₈	—	2	2	no	—
31	C₈×Dic₈₃	—	2	2	no	—
32	C₈×Dic₈₃	—	2	2	no	—
33	C₈×Dic₈₃	—	2	2	no	—
34	C₈×Dic₈₃	C₈	1	4	yes	complemented
35	C₂²×C₈×C₈₃	—	2	2	no	—
36	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
37	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
38	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
39	C₄⋊C₈×C₈₃	—	2	2	no	—
40	C₄⋊C₈×C₈₃	—	2	2	no	—
41	C₄×C₈×C₈₃	—	2	2	no	—
42	C₂²×Dic₈₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂²×Dic₈₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂²×Dic₈₃	C₄⋊C₄	1	1	yes	characteristic
45	C₂²×Dic₈₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈₃⋊C₈	—	2	1	no	—
47	C₂×C₈₃⋊C₈	—	2	1	no	—
48	C₂×C₈₃⋊C₈	—	2	1	no	—
49	C₂×C₈₃⋊C₈	—	2	1	no	—
50	C₂×C₈₃⋊C₈	—	4	2	no	—
51	C₂×C₈₃⋊C₈	—	4	2	no	—
52	C₂×C₈₃⋊C₈	—	4	2	no	—
53	C₂×C₈₃⋊C₈	C₂×C₈	1	2	yes	—
54	C₂×C₈₃⋊C₈	M₁₆	1	2	yes	—
55	C₄×Dic₈₃	—	2	1	no	—
56	C₄×Dic₈₃	—	2	1	no	—
57	C₄×Dic₈₃	—	2	2	no	—
58	C₄×Dic₈₃	—	2	2	no	—
59	C₄×Dic₈₃	C₂×C₈	1	2	yes	—
60	C₄×Dic₈₃	M₁₆	1	2	yes	—
61	C₄×Dic₈₃	D₈	1	2	yes	—
62	C₄×Dic₈₃	SD₁₆	1	4	yes	—
63	C₄×Dic₈₃	Q₁₆	1	2	yes	—
64	C₂²×C₄×C₈₃	C₂²×C₄	1	1	yes	characteristic
65	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
66	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₈₃	—	2	1	no	—
68	C₂×C₈×C₈₃	—	2	1	no	—
69	C₂×C₈×C₈₃	—	2	1	no	—
70	C₂×C₈×C₈₃	—	2	1	no	—
71	C₂×C₈×C₈₃	—	4	2	no	—
72	C₂×C₈×C₈₃	—	4	2	no	—
73	C₂×C₈×C₈₃	—	4	2	no	—
74	C₂×C₈×C₈₃	C₂×C₈	1	2	yes	—
75	C₂×C₈×C₈₃	M₁₆	1	2	yes	—
76	C₄²×C₈₃	#41	1	2	yes	—
77	C₄²×C₈₃	#41	1	2	yes	—
78	C₂³×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
79	C₂×C₄×C₈₃	—	2	1	no	—
80	C₂×C₄×C₈₃	—	2	1	no	—
81	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
82	C₂×C₄×C₈₃	#117	1	1	yes	characteristic
83	C₂×C₄×C₈₃	#124	1	2	yes	—
84	C₂×C₄×C₈₃	#125	1	2	yes	—
85	C₂×C₄×C₈₃	C₂²×C₈	1	1	yes	characteristic
86	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
87	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
88	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
89	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
90	C₂×C₄×C₈₃	C₂×SD₁₆	1	2	yes	—
91	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
92	C₂×Dic₈₃	—	2	1	no	—
93	C₂×Dic₈₃	—	2	1	no	—
94	C₂×Dic₈₃	—	2	1	no	—
95	C₂×Dic₈₃	—	2	1	no	—
96	C₂×Dic₈₃	C₄⋊C₈	1	2	yes	—
97	C₂×Dic₈₃	C₄⋊C₈	1	2	yes	—
98	C₂×Dic₈₃	#106	1	2	yes	—
99	C₂×Dic₈₃	#106	1	2	yes	—
100	C₂×Dic₈₃	#107	1	2	yes	—
101	C₂×Dic₈₃	#107	1	2	yes	—
102	C₂×Dic₈₃	#108	1	2	yes	—
103	C₂×Dic₈₃	#108	1	2	yes	—
104	C₈×C₈₃	—	2	2	no	—
105	C₈×C₈₃	—	2	2	no	—
106	C₈×C₈₃	—	8	2	no	—
107	C₈₃⋊C₈	—	2	2	no	—
108	C₈₃⋊C₈	—	2	2	no	—
109	C₈₃⋊C₈	—	8	2	no	—
110	C₂²×C₈₃	#382	1	1	yes	characteristic
111	C₂²×C₈₃	C₂×C₄⋊C₈	1	1	yes	characteristic
112	C₂²×C₈₃	#423	1	1	yes	characteristic
113	C₂²×C₈₃	C₂×#106	1	1	yes	characteristic
114	C₂²×C₈₃	C₂×#107	1	1	yes	characteristic
115	C₂²×C₈₃	C₂×#108	1	1	yes	characteristic
116	C₂²×C₈₃	C₂×#108	1	1	yes	characteristic
117	C₄×C₈₃	—	2	2	no	—
118	C₄×C₈₃	#431	1	1	yes	characteristic
119	C₄×C₈₃	#431	1	1	yes	characteristic
120	C₄×C₈₃	#432	1	1	yes	characteristic
121	C₄×C₈₃	#432	1	1	yes	characteristic
122	C₄×C₈₃	#486	1	2	yes	—
123	C₄×C₈₃	#487	1	2	yes	—
124	Dic₈₃	—	2	2	no	—
125	Dic₈₃	—	2	2	no	—
126	Dic₈₃	#334	1	4	yes	—
127	Dic₈₃	#335	1	4	yes	—
128	#8784	—	83	1	no	maximal, Sylow
129	C₂×C₈₃	C₂×#334	1	1	yes	characteristic
130	C₂×C₈₃	C₂×#335	1	1	yes	characteristic
131	C₂×C₈₃	#1566	1	1	yes	characteristic
132	C₂×C₈₃	#1567	1	1	yes	characteristic
133	C₂×C₈₃	#1831	1	1	yes	characteristic
134	C₂×C₈₃	#1834	1	1	yes	characteristic
135	C₂×C₈₃	#1837	1	1	yes	characteristic
136	#1842	—	83	2	no	—
137	C₄²×C₈	—	83	1	no	—
138	#1277	—	83	2	no	—
139	#1277	—	83	1	no	—
140	#1277	D₈₃	1	1	yes	characteristic
141	C₈₃	#8784	1	1	yes	characteristic, complemented, Sylow
142	C₂×C₄⋊C₈	—	83	2	no	—
143	C₂×C₄⋊C₈	—	166	2	no	—
144	C₂×C₄⋊C₈	—	166	2	no	—
145	C₂×C₄⋊C₈	C₂×D₈₃	1	2	yes	—
146	C₂×C₄×C₈	—	83	1	no	—
147	C₂×C₄×C₈	—	83	1	no	—
148	C₂×C₄×C₈	—	83	1	no	—
149	C₂×C₄×C₈	—	83	1	no	—
150	C₂×C₄×C₈	—	83	1	no	—
151	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
152	C₄³	—	83	1	no	—
153	C₂²×C₈	—	2	2	no	—
154	C₂²×C₈	—	83	1	no	—
155	C₂²×C₈	—	83	1	no	—
156	C₂²×C₈	—	166	2	no	—
157	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
158	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
159	C₂×C₄²	—	83	1	no	—
160	C₂×C₄²	—	83	1	no	—
161	C₂×C₄²	—	83	1	no	—
162	C₂×C₄²	—	83	1	no	—
163	C₂×C₄²	—	83	1	no	—
164	C₂×C₄²	—	83	1	no	—
165	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
166	C₄⋊C₈	—	2	2	no	—
167	C₄⋊C₈	—	2	2	no	—
168	C₄⋊C₈	—	332	4	no	—
169	C₄⋊C₈	—	332	4	no	—
170	C₄⋊C₈	—	166	2	no	—
171	C₄⋊C₈	—	166	2	no	—
172	C₄×C₈	—	2	2	no	—
173	C₄×C₈	—	83	4	no	—
174	C₄×C₈	—	166	2	no	—
175	C₄×C₈	—	166	2	no	—
176	C₄×C₈	—	166	2	no	—
177	C₄×C₈	—	166	2	no	—
178	C₂²×C₄	—	83	1	no	—
179	C₂²×C₄	—	83	1	no	—
180	C₂²×C₄	—	83	1	no	—
181	C₂²×C₄	—	83	1	no	—
182	C₂²×C₄	C₂×C₄×D₈₃	1	1	yes	characteristic
183	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
184	C₂²×C₄	Q₈×D₈₃	1	1	yes	characteristic
185	C₂×C₈	—	2	1	no	—
186	C₂×C₈	—	2	1	no	—
187	C₂×C₈	—	2	1	no	—
188	C₂×C₈	—	2	1	no	—
189	C₂×C₈	—	4	2	no	—
190	C₂×C₈	—	4	2	no	—
191	C₂×C₈	—	4	2	no	—
192	C₂×C₈	—	332	2	no	—
193	C₂×C₈	—	332	2	no	—
194	C₂×C₈	—	332	2	no	—
195	C₂×C₈	—	83	2	no	—
196	C₂×C₈	—	83	2	no	—
197	C₂×C₈	—	166	1	no	—
198	C₂×C₈	—	166	1	no	—
199	C₂×C₈	—	166	1	no	—
200	C₂×C₈	—	166	1	no	—
201	C₂×C₈	C₈×D₈₃	1	2	yes	—
202	C₂×C₈	#13050573	1	2	yes	—
203	C₄²	—	83	2	no	—
204	C₄²	—	83	2	no	—
205	C₄²	—	83	2	no	—
206	C₄²	—	83	4	no	—
207	C₄²	—	83	2	no	—
208	C₄²	—	166	2	no	—
209	C₄²	—	166	2	no	—
210	C₄²	—	166	1	no	—
211	C₄²	—	166	1	no	—
212	C₄²	#13050589	1	2	yes	—
213	C₄²	#13050590	1	2	yes	—
214	C₂³	C₄⋊C₄×D₈₃	1	1	yes	characteristic
215	C₂×C₄	—	2	1	no	—
216	C₂×C₄	—	2	1	no	—
217	C₂×C₄	—	83	2	no	—
218	C₂×C₄	—	83	2	no	—
219	C₂×C₄	—	83	2	no	—
220	C₂×C₄	—	83	2	no	—
221	C₂×C₄	—	83	2	no	—
222	C₂×C₄	—	83	2	no	—
223	C₂×C₄	—	83	2	no	—
224	C₂×C₄	—	83	2	no	—
225	C₂×C₄	—	166	1	no	—
226	C₂×C₄	—	166	1	no	—
227	C₂×C₄	—	166	1	no	—
228	C₂×C₄	—	166	1	no	—
229	C₂×C₄	#14581748	1	1	yes	characteristic
230	C₂×C₄	#14581749	1	1	yes	characteristic
231	C₂×C₄	#14581778	1	2	yes	—
232	C₂×C₄	#14581782	1	2	yes	—
233	C₂×C₄	C₂×C₈×D₈₃	1	1	yes	characteristic
234	C₂×C₄	M₁₆×D₈₃	1	1	yes	characteristic
235	C₂×C₄	M₁₆×D₈₃	1	1	yes	characteristic
236	C₂×C₄	C₂×#13050573	1	1	yes	characteristic
237	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
238	C₂×C₄	SD₁₆×D₈₃	1	2	yes	—
239	C₂×C₄	Q₁₆×D₈₃	1	1	yes	characteristic
240	C₈	—	2	2	no	—
241	C₈	—	2	2	no	—
242	C₈	—	8	2	no	—
243	C₈	—	664	2	no	—
244	C₈	—	166	2	no	—
245	C₈	—	166	2	no	—
246	C₂²	#19127565	1	1	yes	characteristic
247	C₂²	C₄⋊C₈×D₈₃	1	1	yes	characteristic
248	C₂²	#19127723	1	1	yes	characteristic
249	C₂²	#106×D₈₃	1	1	yes	characteristic
250	C₂²	#107×D₈₃	1	1	yes	characteristic
251	C₂²	#108×D₈₃	1	1	yes	characteristic
252	C₂²	#108×D₈₃	1	1	yes	characteristic
253	C₄	—	2	2	no	—
254	C₄	—	83	4	no	—
255	C₄	—	83	4	no	—
256	C₄	—	166	2	no	—
257	C₄	—	166	2	no	—
258	C₄	#19127751	1	1	yes	characteristic
259	C₄	#19127752	1	1	yes	characteristic
260	C₄	#19127755	1	1	yes	characteristic
261	C₄	#19127756	1	1	yes	characteristic
262	C₄	#19128046	1	2	yes	—
263	C₄	#19128050	1	2	yes	—
264	C₂	#334×D₈₃	1	1	yes	characteristic
265	C₂	#335×D₈₃	1	1	yes	characteristic
266	C₂	#24906217	1	1	yes	characteristic
267	C₂	#24906220	1	1	yes	characteristic
268	C₂	#24907578	1	1	yes	characteristic
269	C₂	#24907589	1	1	yes	characteristic
270	C₂	#24907599	1	1	yes	characteristic

	subgroup	G/H
center	C₂²×C₄	D₄×D₈₃
commutator	C₂×C₄×C₈₃	C₂²×C₈
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1277	C₂
socle, prime radical	C₂³×C₈₃	C₂×C₄⋊C₄
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1277	D₈₃
hypocenter	C₈₃	#8784
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄² ◁ #1277

Complements

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

N	H ≅ G/N	type
34. C₈×Dic₈₃	C₈	semidirect
141. C₈₃	#8784	semidirect

Maximal subgroups

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

subgroup	index
#24907614	2
C₄×C₈×Dic₈₃	2
#24905298	2
#24905297	2
C₈₃×#1277	2
#8784	83