Subgroups — #35308619

Order 21248.

486 conjugacy classes of subgroups.

The subgroup lattice has 278 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24908416	C₂	1	2	yes	maximal
2	#24908181	C₂	1	1	yes	maximal, characteristic
3	C₂×#19127447	C₂	1	2	yes	maximal
4	#24905298	C₂	1	1	yes	maximal, characteristic
5	C₂×#494×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
7	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
8	C₂×Dic₈₃⋊C₈	—	2	1	no	—
9	C₄×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
10	#19127463	—	2	2	no	—
11	#19127463	—	2	2	no	—
12	#19127455	—	2	1	no	—
13	#19127447	—	2	4	no	—
14	C₂²×Q₁₆×C₈₃	—	2	1	no	—
15	C₂²×Q₁₆×C₈₃	—	2	1	no	—
16	C₂×C₄⋊Q₈×C₈₃	C₂²	1	2	yes	—
17	#494×C₈₃	—	2	4	no	—
18	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
19	C₂²×C₈₃⋊C₈	—	2	1	no	—
20	C₂²×C₈₃⋊C₈	—	2	2	no	—
21	C₂×C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
22	C₂×C₄⋊Dic₈₃	D₄	1	1	yes	characteristic
23	C₂×C₄×Dic₈₃	—	2	1	no	—
24	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
25	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
26	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
27	Dic₈₃⋊C₈	—	4	2	no	—
28	C₈₃⋊#95	—	2	1	no	—
29	C₂²×Q₈×C₈₃	—	2	2	no	—
30	C₂×Q₁₆×C₈₃	—	2	4	no	—
31	C₂×Q₁₆×C₈₃	—	2	4	no	—
32	C₂×Q₁₆×C₈₃	—	4	2	no	—
33	C₂×Q₁₆×C₈₃	—	4	2	no	—
34	C₂×Q₁₆×C₈₃	—	4	2	no	—
35	C₂×Q₁₆×C₈₃	—	4	2	no	—
36	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
37	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
38	C₄⋊Q₈×C₈₃	—	2	2	no	—
39	C₄⋊Q₈×C₈₃	—	2	2	no	—
40	C₄⋊Q₈×C₈₃	—	2	2	no	—
41	C₄⋊Q₈×C₈₃	D₄	1	4	yes	—
42	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
43	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
44	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
45	C₄×C₈×C₈₃	—	2	2	no	—
46	C₂²×Dic₈₃	—	2	1	no	—
47	C₂²×Dic₈₃	—	2	1	no	—
48	C₂×C₈₃⋊C₈	—	4	1	no	—
49	C₂×C₈₃⋊C₈	—	4	2	no	—
50	C₂×C₈₃⋊C₈	—	4	1	no	—
51	C₂×C₈₃⋊C₈	—	4	2	no	—
52	C₂×C₈₃⋊C₈	—	4	1	no	—
53	C₂×C₈₃⋊C₈	—	4	2	no	—
54	C₄⋊Dic₈₃	—	2	1	no	—
55	C₄⋊Dic₈₃	—	2	2	no	—
56	C₄⋊Dic₈₃	Q₁₆	1	2	yes	complemented
57	C₄×Dic₈₃	—	4	1	no	—
58	C₄×Dic₈₃	—	4	1	no	—
59	C₂×Q₈×C₈₃	—	2	4	no	—
60	C₂×Q₈×C₈₃	—	4	2	no	—
61	C₂×Q₈×C₈₃	—	4	2	no	—
62	C₂×Q₈×C₈₃	—	4	2	no	—
63	C₂×Q₈×C₈₃	—	4	2	no	—
64	C₂×Q₈×C₈₃	—	4	2	no	—
65	C₂²×C₄×C₈₃	—	4	2	no	—
66	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
67	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
68	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
69	Q₁₆×C₈₃	—	4	8	no	—
70	Q₁₆×C₈₃	—	4	8	no	—
71	C₂×C₈×C₈₃	—	2	1	no	—
72	C₂×C₈×C₈₃	—	2	1	no	—
73	C₂×C₈×C₈₃	—	2	1	no	—
74	C₂×C₈×C₈₃	—	2	1	no	—
75	C₂×C₈×C₈₃	D₈	1	2	yes	—
76	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
77	C₄⋊C₄×C₈₃	—	2	4	no	—
78	C₄⋊C₄×C₈₃	—	2	4	no	—
79	C₄⋊C₄×C₈₃	—	4	2	no	—
80	C₄⋊C₄×C₈₃	—	4	2	no	—
81	C₄²×C₈₃	C₂×D₄	1	2	yes	—
82	C₄²×C₈₃	#41	1	2	yes	—
83	C₂³×C₈₃	#121	1	1	yes	characteristic
84	Q₈×C₈₃	—	4	4	no	—
85	Q₈×C₈₃	—	4	4	no	—
86	Q₈×C₈₃	—	8	2	no	—
87	Q₈×C₈₃	—	8	2	no	—
88	C₂×C₄×C₈₃	—	2	1	no	—
89	C₂×C₄×C₈₃	—	2	1	no	—
90	C₂×C₄×C₈₃	—	4	4	no	—
91	C₂×C₄×C₈₃	—	8	2	no	—
92	C₂×C₄×C₈₃	—	8	2	no	—
93	C₂×C₄×C₈₃	#120	1	1	yes	characteristic
94	C₂×C₄×C₈₃	#121	1	2	yes	—
95	C₂×C₄×C₈₃	#123	1	2	yes	—
96	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
97	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
98	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
99	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
100	C₂×C₄×C₈₃	#135	1	1	yes	characteristic
101	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
102	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
103	C₂×C₄×C₈₃	#137	1	2	yes	—
104	C₂×Dic₈₃	—	2	2	no	—
105	C₂×Dic₈₃	—	4	1	no	—
106	C₂×Dic₈₃	—	4	1	no	—
107	C₂×Dic₈₃	—	4	1	no	—
108	C₂×Dic₈₃	—	4	1	no	—
109	C₂×Dic₈₃	—	4	1	no	—
110	C₈×C₈₃	—	2	2	no	—
111	C₈×C₈₃	—	2	2	no	—
112	C₈₃⋊C₈	—	8	1	no	—
113	C₈₃⋊C₈	—	8	2	no	—
114	C₂²×C₈₃	#394	1	1	yes	characteristic
115	C₂²×C₈₃	#459	1	1	yes	characteristic
116	C₂²×C₈₃	#460	1	1	yes	characteristic
117	C₂²×C₈₃	#467	1	1	yes	characteristic
118	C₂²×C₈₃	#470	1	1	yes	characteristic
119	C₂²×C₈₃	#473	1	1	yes	characteristic
120	C₂²×C₈₃	#473	1	1	yes	characteristic
121	C₄×C₈₃	—	2	2	no	—
122	C₄×C₈₃	—	8	4	no	—
123	C₄×C₈₃	#447	1	1	yes	characteristic
124	C₄×C₈₃	#450	1	1	yes	characteristic
125	C₄×C₈₃	#462	1	2	yes	—
126	C₄×C₈₃	#464	1	2	yes	—
127	C₄×C₈₃	#486	1	1	yes	characteristic
128	C₄×C₈₃	#490	1	1	yes	characteristic
129	Dic₈₃	—	4	2	no	—
130	Dic₈₃	—	8	1	no	—
131	#10538	—	83	1	no	maximal, Sylow
132	C₂×C₈₃	#1669	1	1	yes	characteristic
133	C₂×C₈₃	#1670	1	1	yes	characteristic
134	C₂×C₈₃	#1689	1	1	yes	characteristic
135	C₂×C₈₃	#1690	1	1	yes	characteristic
136	C₂×C₈₃	#2041	1	1	yes	characteristic
137	C₂×C₈₃	#2047	1	1	yes	characteristic
138	C₂×C₈₃	#2056	1	1	yes	characteristic
139	C₂×#494	D₈₃	1	1	yes	characteristic
140	#1988	—	83	2	no	—
141	#1939	—	83	1	no	—
142	C₂×#332	—	83	2	no	—
143	#1277	—	83	1	no	—
144	C₈₃	#10538	1	1	yes	characteristic, complemented, Sylow
145	C₂²×Q₁₆	—	2	1	no	—
146	C₂²×Q₁₆	—	2	1	no	—
147	C₂×C₄⋊Q₈	C₂×D₈₃	1	2	yes	—
148	#494	—	2	4	no	—
149	C₂×C₄⋊C₈	—	83	1	no	—
150	C₂×C₄⋊C₈	—	83	2	no	—
151	C₂×C₄⋊C₈	—	166	1	no	—
152	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
153	C₄×C₄⋊C₄	—	83	1	no	—
154	#340	—	166	2	no	—
155	#340	—	166	2	no	—
156	#336	—	166	1	no	—
157	#332	—	166	4	no	—
158	C₂²×Q₈	—	2	2	no	—
159	C₂×Q₁₆	—	2	4	no	—
160	C₂×Q₁₆	—	2	4	no	—
161	C₂×Q₁₆	—	4	2	no	—
162	C₂×Q₁₆	—	4	2	no	—
163	C₂×Q₁₆	—	4	2	no	—
164	C₂×Q₁₆	—	4	2	no	—
165	C₂²×C₈	—	166	2	no	—
166	C₂²×C₈	—	166	1	no	—
167	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
168	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
169	C₄⋊Q₈	—	2	2	no	—
170	C₄⋊Q₈	—	2	2	no	—
171	C₄⋊Q₈	—	2	2	no	—
172	C₄⋊Q₈	C₂²⋊D₈₃	1	4	yes	—
173	C₂×C₄⋊C₄	—	2	2	no	—
174	C₂×C₄⋊C₄	—	2	2	no	—
175	C₂×C₄⋊C₄	—	83	1	no	—
176	C₂×C₄⋊C₄	—	83	1	no	—
177	C₂×C₄²	—	166	1	no	—
178	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
179	C₄⋊C₈	—	332	2	no	—
180	C₄⋊C₈	—	166	2	no	—
181	C₄⋊C₈	—	166	2	no	—
182	C₄⋊C₈	—	166	2	no	—
183	C₄×C₈	—	2	2	no	—
184	#95	—	166	1	no	—
185	C₂×Q₈	—	2	4	no	—
186	C₂×Q₈	—	4	2	no	—
187	C₂×Q₈	—	4	2	no	—
188	C₂×Q₈	—	4	2	no	—
189	C₂×Q₈	—	4	2	no	—
190	C₂×Q₈	—	4	2	no	—
191	C₂²×C₄	—	4	2	no	—
192	C₂²×C₄	—	166	1	no	—
193	C₂²×C₄	—	166	1	no	—
194	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
195	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
196	C₂²×C₄	#13050589	1	1	yes	characteristic
197	Q₁₆	—	4	8	no	—
198	Q₁₆	—	4	8	no	—
199	C₂×C₈	—	2	1	no	—
200	C₂×C₈	—	2	1	no	—
201	C₂×C₈	—	2	1	no	—
202	C₂×C₈	—	2	1	no	—
203	C₂×C₈	—	332	1	no	—
204	C₂×C₈	—	332	2	no	—
205	C₂×C₈	—	332	1	no	—
206	C₂×C₈	—	332	2	no	—
207	C₂×C₈	—	332	1	no	—
208	C₂×C₈	—	332	2	no	—
209	C₂×C₈	#13050575	1	2	yes	—
210	C₂×C₈	#13050578	1	2	yes	—
211	C₄⋊C₄	—	2	4	no	—
212	C₄⋊C₄	—	2	4	no	—
213	C₄⋊C₄	—	4	2	no	—
214	C₄⋊C₄	—	4	2	no	—
215	C₄⋊C₄	—	83	2	no	—
216	C₄⋊C₄	—	166	2	no	—
217	C₄⋊C₄	—	166	1	no	—
218	C₄²	—	332	1	no	—
219	C₄²	—	332	1	no	—
220	C₄²	C₂×C₂²⋊D₈₃	1	2	yes	—
221	C₄²	#13050589	1	2	yes	—
222	C₂³	#14581763	1	1	yes	characteristic
223	Q₈	—	4	4	no	—
224	Q₈	—	4	4	no	—
225	Q₈	—	8	2	no	—
226	Q₈	—	8	2	no	—
227	C₂×C₄	—	2	1	no	—
228	C₂×C₄	—	2	1	no	—
229	C₂×C₄	—	4	4	no	—
230	C₂×C₄	—	8	2	no	—
231	C₂×C₄	—	8	2	no	—
232	C₂×C₄	—	332	1	no	—
233	C₂×C₄	—	332	1	no	—
234	C₂×C₄	—	332	1	no	—
235	C₂×C₄	—	332	1	no	—
236	C₂×C₄	—	332	1	no	—
237	C₂×C₄	—	166	2	no	—
238	C₂×C₄	#14581761	1	1	yes	characteristic
239	C₂×C₄	#14581764	1	2	yes	—
240	C₂×C₄	#14581773	1	2	yes	—
241	C₂×C₄	#14581779	1	1	yes	characteristic
242	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
243	C₂×C₄	C₂×#13050578	1	1	yes	characteristic
244	C₂×C₄	Q₁₆×D₈₃	1	1	yes	characteristic
245	C₂×C₄	#14581815	1	1	yes	characteristic
246	C₂×C₄	#14581818	1	1	yes	characteristic
247	C₂×C₄	#14581818	1	1	yes	characteristic
248	C₂×C₄	#14581824	1	2	yes	—
249	C₈	—	2	2	no	—
250	C₈	—	2	2	no	—
251	C₈	—	664	1	no	—
252	C₈	—	664	2	no	—
253	C₂²	#19127615	1	1	yes	characteristic
254	C₂²	#19127893	1	1	yes	characteristic
255	C₂²	#19127898	1	1	yes	characteristic
256	C₂²	#19127939	1	1	yes	characteristic
257	C₂²	#19127956	1	1	yes	characteristic
258	C₂²	#19127973	1	1	yes	characteristic
259	C₂²	#19127973	1	1	yes	characteristic
260	C₄	—	2	2	no	—
261	C₄	—	8	4	no	—
262	C₄	—	332	2	no	—
263	C₄	—	664	1	no	—
264	C₄	#19127835	1	1	yes	characteristic
265	C₄	#19127852	1	1	yes	characteristic
266	C₄	#19127909	1	2	yes	—
267	C₄	#19127925	1	2	yes	—
268	C₄	#19128047	1	1	yes	characteristic
269	C₄	#19128065	1	1	yes	characteristic
270	C₂	#24906825	1	1	yes	characteristic
271	C₂	#24906830	1	1	yes	characteristic
272	C₂	#24906949	1	1	yes	characteristic
273	C₂	#24906954	1	1	yes	characteristic
274	C₂	#24908694	1	1	yes	characteristic
275	C₂	#24908728	1	1	yes	characteristic
276	C₂	#24908783	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14581763
commutator, cosocle	C₂×C₄²×C₈₃	C₂³
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₂×#494×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	#121
hypercenter	C₂×#494	D₈₃
hypocenter	C₈₃	#10538
LCS	G ▷ C₂×C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×#494

Complements

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

N	H ≅ G/N	type
56. C₄⋊Dic₈₃	Q₁₆	semidirect
144. C₈₃	#10538	semidirect

Maximal subgroups

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

subgroup	index
#24908416	2
#24908181	2
C₂×#19127447	2
#24905298	2
C₂×#494×C₈₃	2
#10538	83