Subgroups — #35559895

Order 21248.

490 conjugacy classes of subgroups.

The subgroup lattice has 220 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24917976	C₂	1	2	yes	maximal
2	#24909744	C₂	1	2	yes	maximal, complemented
3	#24909742	C₂	1	1	yes	maximal, characteristic, complemented
4	#24909739	C₂	1	2	yes	maximal, complemented
5	#24909737	C₂	1	2	yes	maximal, complemented
6	#24909246	C₂	1	2	yes	maximal
7	#24909244	C₂	1	2	yes	maximal
8	C₄×C₁₆×D₈₃	C₂	1	1	yes	maximal, characteristic
9	C₈₃×#2253	C₂	1	1	yes	maximal, characteristic, complemented
10	#19128257	—	2	2	no	—
11	C₂×C₁₆×D₈₃	C₂²	1	1	yes	characteristic
12	C₂×C₁₆×D₈₃	C₂²	1	1	yes	characteristic
13	#19128105	C₂²	1	4	yes	—
14	#19128103	C₂²	1	2	yes	—
15	#19128100	—	2	2	no	—
16	#19128098	—	2	2	no	—
17	#19127738	C₂²	1	4	yes	—
18	#19127736	—	2	2	no	—
19	C₄×C₈×D₈₃	C₂²	1	1	yes	characteristic
20	#19127529	C₂²	1	2	yes	—
21	#19127528	C₂²	1	4	yes	—
22	#19127527	C₂²	1	2	yes	—
23	#19127526	C₂²	1	4	yes	—
24	C₄×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
25	C₁₆×Dic₈₃	C₂²	1	1	yes	characteristic
26	C₁₆×Dic₈₃	C₂²	1	1	yes	characteristic
27	#499×C₈₃	C₂²	1	2	yes	—
28	#367×C₈₃	C₂²	1	2	yes	—
29	#366×C₈₃	C₂²	1	2	yes	—
30	C₄×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
31	C₂×C₈×D₈₃	D₄	1	1	yes	characteristic
32	C₂×C₈×D₈₃	C₂³	1	1	yes	characteristic
33	#14581791	—	2	4	no	—
34	#14581784	—	2	2	no	—
35	#14581770	—	4	4	no	—
36	#14581757	—	4	4	no	—
37	#14581750	—	2	4	no	—
38	C₄²×D₈₃	D₄	1	1	yes	characteristic
39	C₂×C₈₃⋊C₁₆	C₂³	1	1	yes	characteristic
40	C₂×C₈₃⋊C₁₆	C₂³	1	1	yes	characteristic
41	C₁₆×D₈₃	Q₈	1	4	yes	complemented
42	C₁₆×D₈₃	Q₈	1	4	yes	complemented
43	#14581726	C₂³	1	4	yes	—
44	#14581725	—	2	4	no	—
45	#14581724	—	2	2	no	—
46	#14581723	—	2	4	no	—
47	C₄×C₈₃⋊C₈	D₄	1	1	yes	characteristic
48	C₈×Dic₈₃	D₄	1	1	yes	characteristic
49	C₈×Dic₈₃	C₂³	1	1	yes	characteristic
50	#125×C₈₃	—	2	2	no	—
51	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
52	C₂×C₁₆×C₈₃	C₂³	1	1	yes	characteristic
53	#107×C₈₃	C₂³	1	4	yes	—
54	#106×C₈₃	—	2	2	no	—
55	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
56	C₂×Dic₁₆₆	—	4	4	no	—
57	C₂×C₄×D₈₃	D₈	1	2	yes	—
58	C₂×C₄×D₈₃	C₂×D₄	1	1	yes	characteristic
59	C₂×C₈₃⋊C₈	C₂×D₄	1	1	yes	characteristic
60	C₂×C₈₃⋊C₈	C₂×D₄	1	1	yes	characteristic
61	C₈×D₈₃	Q₁₆	1	4	yes	—
62	C₈×D₈₃	C₂×Q₈	1	2	yes	—
63	C₈×D₈₃	C₂×Q₈	1	2	yes	—
64	C₄⋊Dic₈₃	—	2	4	no	—
65	Dic₈₃⋊C₄	—	4	4	no	—
66	Dic₈₃⋊C₄	—	4	4	no	—
67	(C₂×D₈₃)⋊C₄	—	4	4	no	—
68	C₄×Dic₈₃	—	4	4	no	—
69	C₄×Dic₈₃	D₈	1	2	yes	—
70	C₄×Dic₈₃	C₂×D₄	1	1	yes	characteristic
71	C₈₃⋊C₁₆	C₂×Q₈	1	2	yes	—
72	C₈₃⋊C₁₆	C₂×Q₈	1	2	yes	—
73	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
74	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
75	C₄⋊C₄×C₈₃	—	2	4	no	—
76	C₄⋊C₄×C₈₃	—	4	4	no	—
77	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
78	C₁₆×C₈₃	C₂×Q₈	1	2	yes	—
79	C₁₆×C₈₃	C₂×Q₈	1	2	yes	—
80	C₂²×D₈₃	C₂×D₈	1	1	yes	characteristic
81	C₂×C₄×C₈₃	—	4	4	no	—
82	C₂×C₄×C₈₃	C₂×D₈	1	2	yes	—
83	C₂×C₄×C₈₃	C₂²×D₄	1	1	yes	characteristic
84	C₂×Dic₈₃	—	4	4	no	—
85	C₂×Dic₈₃	C₂×D₈	1	1	yes	characteristic
86	C₂×Dic₈₃	C₂×D₈	1	2	yes	—
87	C₄×D₈₃	—	2	4	no	—
88	C₄×D₈₃	C₄⋊Q₈	1	2	yes	—
89	C₄×D₈₃	C₂×Q₁₆	1	2	yes	—
90	Dic₁₆₆	—	8	8	no	—
91	C₈×C₈₃	C₂×Q₁₆	1	2	yes	—
92	C₈×C₈₃	C₂²×Q₈	1	1	yes	characteristic
93	C₈×C₈₃	C₂²×Q₈	1	1	yes	characteristic
94	C₈₃⋊C₈	C₄⋊Q₈	1	2	yes	—
95	C₈₃⋊C₈	C₂×Q₁₆	1	2	yes	—
96	C₂²×C₈₃	C₂²×D₈	1	1	yes	characteristic
97	C₂×D₈₃	—	2	2	no	—
98	C₂×D₈₃	#500	1	2	yes	—
99	C₄×C₈₃	—	2	2	no	—
100	C₄×C₈₃	—	8	4	no	—
101	C₄×C₈₃	C₂×C₄⋊Q₈	1	1	yes	characteristic
102	C₄×C₈₃	C₂²×Q₁₆	1	1	yes	characteristic
103	Dic₈₃	—	8	4	no	—
104	Dic₈₃	#500	1	2	yes	—
105	Dic₈₃	#508	1	4	yes	—
106	#33726	—	83	1	no	maximal, Sylow
107	C₂×C₈₃	C₂×#500	1	1	yes	characteristic
108	C₂×C₈₃	C₂×#508	1	2	yes	—
109	D₈₃	—	2	2	no	—
110	#3160	—	83	2	no	—
111	#2254	—	83	2	no	—
112	#2253	—	83	2	no	—
113	#2253	—	83	1	no	—
114	#2253	D₈₃	1	1	yes	characteristic, complemented
115	#2252	—	83	2	no	—
116	#2158	—	83	2	no	—
117	#2158	—	83	2	no	—
118	C₂×C₄×C₁₆	—	83	1	no	—
119	C₈₃	#33726	1	1	yes	characteristic, complemented, Sylow
120	#533	—	166	2	no	—
121	C₂²×C₁₆	—	83	1	no	—
122	C₂²×C₁₆	—	83	1	no	—
123	#500	—	83	4	no	—
124	#499	—	83	2	no	—
125	#499	—	166	2	no	—
126	#499	C₂×D₈₃	1	2	yes	—
127	#498	—	166	2	no	—
128	#427	—	83	4	no	—
129	#427	—	166	2	no	—
130	C₂×C₄×C₈	—	83	1	no	—
131	#367	—	83	2	no	—
132	#367	—	83	4	no	—
133	#367	C₂×D₈₃	1	2	yes	—
134	#366	—	83	2	no	—
135	#366	—	83	4	no	—
136	#366	C₂×D₈₃	1	2	yes	—
137	C₄×C₁₆	—	83	1	no	—
138	C₄×C₁₆	—	83	1	no	—
139	C₄×C₁₆	—	83	1	no	—
140	C₄×C₁₆	C₂×D₈₃	1	1	yes	characteristic
141	C₂²×C₈	—	83	1	no	—
142	C₂²×C₈	—	83	1	no	—
143	C₄⋊Q₈	—	166	4	no	—
144	#125	—	2	2	no	—
145	#125	—	166	2	no	—
146	C₂²⋊Q₈	—	332	4	no	—
147	C₄×Q₈	—	332	4	no	—
148	#117	—	166	4	no	—
149	C₂×C₄²	—	83	1	no	—
150	C₂×C₁₆	—	83	1	no	—
151	C₂×C₁₆	—	83	1	no	—
152	C₂×C₁₆	—	83	4	no	—
153	C₂×C₁₆	—	83	4	no	—
154	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
155	C₂×C₁₆	C₂²×D₈₃	1	1	yes	characteristic
156	#107	—	83	4	no	—
157	#107	—	166	4	no	—
158	#107	C₂²×D₈₃	1	4	yes	—
159	#106	—	2	2	no	—
160	#106	—	166	4	no	—
161	#106	—	166	2	no	—
162	C₄×C₈	—	83	1	no	—
163	C₄×C₈	—	83	1	no	—
164	C₄×C₈	—	83	1	no	—
165	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
166	C₂×Q₈	—	332	4	no	—
167	C₂²×C₄	—	83	1	no	—
168	C₂²×C₄	—	83	2	no	—
169	C₂×C₈	—	83	1	no	—
170	C₂×C₈	—	83	1	no	—
171	C₂×C₈	—	83	4	no	—
172	C₂×C₈	—	83	2	no	—
173	C₂×C₈	—	83	2	no	—
174	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
175	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
176	C₄⋊C₄	—	2	4	no	—
177	C₄⋊C₄	—	4	4	no	—
178	C₄⋊C₄	—	332	4	no	—
179	C₄⋊C₄	—	332	4	no	—
180	C₄⋊C₄	—	166	4	no	—
181	C₂²⋊C₄	—	332	4	no	—
182	C₄²	—	332	4	no	—
183	C₄²	—	83	2	no	—
184	C₄²	—	83	1	no	—
185	C₄²	D₄×D₈₃	1	1	yes	characteristic
186	C₁₆	—	83	2	no	—
187	C₁₆	—	83	2	no	—
188	C₁₆	Q₈×D₈₃	1	2	yes	—
189	C₁₆	Q₈×D₈₃	1	2	yes	—
190	C₂³	—	83	1	no	—
191	Q₈	—	664	8	no	—
192	C₂×C₄	—	4	4	no	—
193	C₂×C₄	—	332	4	no	—
194	C₂×C₄	—	83	1	no	—
195	C₂×C₄	—	83	2	no	—
196	C₂×C₄	—	83	2	no	—
197	C₂×C₄	—	83	2	no	—
198	C₂×C₄	—	166	4	no	—
199	C₂×C₄	D₈×D₈₃	1	2	yes	—
200	C₂×C₄	C₂×D₄×D₈₃	1	1	yes	characteristic
201	C₈	—	83	2	no	—
202	C₈	—	83	2	no	—
203	C₈	Q₁₆×D₈₃	1	2	yes	—
204	C₈	C₂×Q₈×D₈₃	1	1	yes	characteristic
205	C₈	C₂×Q₈×D₈₃	1	1	yes	characteristic
206	C₂²	—	83	2	no	—
207	C₂²	—	166	2	no	—
208	C₂²	C₂×D₈×D₈₃	1	1	yes	characteristic
209	C₄	—	2	2	no	—
210	C₄	—	8	4	no	—
211	C₄	—	83	4	no	—
212	C₄	—	83	2	no	—
213	C₄	—	664	4	no	—
214	C₄	C₄⋊Q₈×D₈₃	1	1	yes	characteristic
215	C₄	C₂×Q₁₆×D₈₃	1	1	yes	characteristic
216	C₂	—	166	2	no	—
217	C₂	#500×D₈₃	1	1	yes	characteristic
218	C₂	#508×D₈₃	1	2	yes	—

	subgroup	G/H
center	C₂²	C₂×D₈×D₈₃
commutator, cosocle	C₂×C₈×C₈₃	C₂⁴
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₈₃×#2253	C₂
socle	C₂²×C₈₃	C₂²×D₈
prime radical	C₂²×D₈₃	C₂×D₈
hypercenter	#2253	D₈₃
hypocenter	C₈₃	#33726
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₄² ◁ C₄×C₈ ◁ #2253

Complements

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

N	H ≅ G/N	type
2. #24909744	C₂	semidirect
3. #24909742	C₂	semidirect
4. #24909739	C₂	semidirect
5. #24909737	C₂	semidirect
9. C₈₃×#2253	C₂	semidirect
41. C₁₆×D₈₃	Q₈	semidirect
42. C₁₆×D₈₃	Q₈	semidirect
114. #2253	D₈₃	semidirect
119. C₈₃	#33726	semidirect

Maximal subgroups

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

subgroup	index
#24917976	2
#24909744	2
#24909742	2
#24909739	2
#24909737	2
#24909246	2
#24909244	2
C₄×C₁₆×D₈₃	2
C₈₃×#2253	2
#33726	83