Subgroups — #35303072

Order 21248.

370 conjugacy classes of subgroups.

The subgroup lattice has 230 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24908224	C₂	1	2	yes	maximal
2	#24907330	C₂	1	1	yes	maximal, characteristic
3	C₂×#19127454	C₂	1	1	yes	maximal, characteristic
4	#24905346	C₂	1	2	yes	maximal
5	C₈₃×#1844	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581723	—	2	2	no	—
7	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	2	yes	—
8	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
9	C₂×#14581702	—	2	1	no	—
10	C₄×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
11	#19127463	—	2	2	no	—
12	#19127455	—	2	1	no	—
13	#19127454	C₄	1	4	yes	complemented
14	C₂×#106×C₈₃	—	2	1	no	—
15	C₂×#106×C₈₃	—	2	1	no	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	#384×C₈₃	C₂²	1	2	yes	—
18	C₂²×C₈₃⋊C₈	—	2	1	no	—
19	C₂²×C₈₃⋊C₈	—	2	2	no	—
20	C₂×C₄⋊Dic₈₃	D₄	1	2	yes	—
21	C₂×C₄×Dic₈₃	—	2	1	no	—
22	#14581723	—	4	4	no	—
23	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
24	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
25	C₄⋊(C₈₃⋊C₈)	C₂×C₄	1	4	yes	—
26	C₄⋊(C₈₃⋊C₈)	Q₈	1	2	yes	—
27	#14581702	—	4	2	no	—
28	C₈₃⋊#95	—	2	1	no	—
29	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
31	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
32	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
33	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
34	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
35	#106×C₈₃	—	4	2	no	—
36	#106×C₈₃	—	4	2	no	—
37	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
38	C₄×C₈×C₈₃	Q₈	1	2	yes	—
39	C₂²×Dic₈₃	—	2	2	no	—
40	C₂×C₈₃⋊C₈	—	2	4	no	—
41	C₂×C₈₃⋊C₈	—	2	2	no	—
42	C₂×C₈₃⋊C₈	—	4	1	no	—
43	C₂×C₈₃⋊C₈	—	4	1	no	—
44	C₂×C₈₃⋊C₈	—	4	2	no	—
45	C₂×C₈₃⋊C₈	—	4	2	no	—
46	C₄⋊Dic₈₃	—	2	2	no	—
47	C₄⋊Dic₈₃	—	2	2	no	—
48	C₄×Dic₈₃	—	4	1	no	—
49	C₄×Dic₈₃	—	4	1	no	—
50	C₂²×C₄×C₈₃	—	4	2	no	—
51	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
52	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
53	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
54	C₂×C₈×C₈₃	—	2	1	no	—
55	C₂×C₈×C₈₃	—	2	1	no	—
56	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
57	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
58	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
59	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
60	C₄⋊C₄×C₈₃	—	4	2	no	—
61	C₄⋊C₄×C₈₃	—	4	2	no	—
62	C₄⋊C₄×C₈₃	—	4	2	no	—
63	C₄⋊C₄×C₈₃	—	4	2	no	—
64	C₄⋊C₄×C₈₃	—	8	2	no	—
65	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
66	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
67	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
68	C₄²×C₈₃	#41	1	1	yes	characteristic
69	C₂³×C₈₃	#121	1	1	yes	characteristic
70	C₂×C₄×C₈₃	—	2	2	no	—
71	C₂×C₄×C₈₃	—	8	2	no	—
72	C₂×C₄×C₈₃	—	8	2	no	—
73	C₂×C₄×C₈₃	—	8	2	no	—
74	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
75	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
76	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
77	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
78	C₂×C₄×C₈₃	#125	1	1	yes	characteristic
79	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
80	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
81	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
82	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
83	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	2	yes	—
84	C₂×C₄×C₈₃	#137	1	2	yes	—
85	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
86	C₂×Dic₈₃	—	4	2	no	—
87	C₂×Dic₈₃	—	4	2	no	—
88	C₂×Dic₈₃	—	4	2	no	—
89	C₈×C₈₃	—	2	2	no	—
90	C₈×C₈₃	#106	1	4	yes	complemented
91	C₈₃⋊C₈	—	4	4	no	—
92	C₈₃⋊C₈	—	4	2	no	—
93	C₂²×C₈₃	#389	1	1	yes	characteristic
94	C₂²×C₈₃	#460	1	1	yes	characteristic
95	C₂²×C₈₃	#461	1	1	yes	characteristic
96	C₂²×C₈₃	#468	1	2	yes	—
97	C₂²×C₈₃	#471	1	2	yes	—
98	C₄×C₈₃	—	2	1	no	—
99	C₄×C₈₃	—	2	1	no	—
100	C₄×C₈₃	—	16	2	no	—
101	C₄×C₈₃	C₂×#106	1	1	yes	characteristic
102	C₄×C₈₃	#428	1	1	yes	characteristic
103	C₄×C₈₃	#441	1	1	yes	characteristic
104	C₄×C₈₃	#442	1	1	yes	characteristic
105	C₄×C₈₃	#475	1	1	yes	characteristic
106	C₄×C₈₃	#476	1	1	yes	characteristic
107	C₄×C₈₃	#501	1	2	yes	—
108	Dic₈₃	—	8	2	no	—
109	#9618	—	83	1	no	maximal, Sylow
110	C₂×C₈₃	#1686	1	2	yes	—
111	C₂×C₈₃	#1691	1	2	yes	—
112	C₂×C₈₃	#1943	1	1	yes	characteristic
113	C₂×C₈₃	#1948	1	1	yes	characteristic
114	C₂×C₈₃	#1951	1	1	yes	characteristic
115	#1948	—	83	2	no	—
116	#1844	D₈₃	1	1	yes	characteristic
117	#1770	—	83	1	no	—
118	C₂×#335	—	83	1	no	—
119	#1296	—	83	2	no	—
120	C₈₃	#9618	1	1	yes	characteristic, complemented, Sylow
121	C₂×#106	—	2	1	no	—
122	C₂×#106	—	2	1	no	—
123	C₂×#106	—	166	2	no	—
124	C₂×C₄⋊C₈	—	83	1	no	—
125	C₂×C₄⋊C₈	—	83	2	no	—
126	C₂×#97	—	166	1	no	—
127	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
128	#384	C₂×D₈₃	1	2	yes	—
129	C₄×C₄⋊C₄	—	83	1	no	—
130	#340	—	166	2	no	—
131	#336	—	166	1	no	—
132	#335	—	83	4	no	—
133	C₂²×C₈	—	166	2	no	—
134	C₂²×C₈	—	166	1	no	—
135	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
136	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
137	C₂×C₄⋊C₄	—	2	2	no	—
138	C₂×C₄⋊C₄	—	2	2	no	—
139	C₂×C₄⋊C₄	—	2	2	no	—
140	C₂×C₄⋊C₄	—	83	2	no	—
141	C₂×C₄²	—	166	1	no	—
142	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
143	#106	—	4	2	no	—
144	#106	—	4	2	no	—
145	#106	—	332	4	no	—
146	C₄⋊C₈	—	83	4	no	—
147	C₄⋊C₈	—	83	2	no	—
148	C₄⋊C₈	—	166	1	no	—
149	C₄⋊C₈	—	166	2	no	—
150	#97	—	332	2	no	—
151	C₄×C₈	Dic₁₆₆	1	2	yes	—
152	C₄×C₈	C₄×D₈₃	1	2	yes	—
153	#95	—	166	1	no	—
154	C₂²×C₄	—	4	2	no	—
155	C₂²×C₄	—	166	2	no	—
156	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
157	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
158	C₂²×C₄	#13050589	1	1	yes	characteristic
159	C₂×C₈	—	2	1	no	—
160	C₂×C₈	—	2	1	no	—
161	C₂×C₈	—	332	1	no	—
162	C₂×C₈	—	332	1	no	—
163	C₂×C₈	—	332	2	no	—
164	C₂×C₈	—	332	2	no	—
165	C₂×C₈	—	166	2	no	—
166	C₂×C₈	—	166	4	no	—
167	C₂×C₈	Dic₈₃⋊C₄	1	2	yes	—
168	C₂×C₈	Dic₈₃⋊C₄	1	2	yes	—
169	C₂×C₈	#13050577	1	2	yes	—
170	C₂×C₈	#13050578	1	2	yes	—
171	C₄⋊C₄	—	4	2	no	—
172	C₄⋊C₄	—	4	2	no	—
173	C₄⋊C₄	—	4	2	no	—
174	C₄⋊C₄	—	4	2	no	—
175	C₄⋊C₄	—	8	2	no	—
176	C₄⋊C₄	—	166	2	no	—
177	C₄⋊C₄	—	166	2	no	—
178	C₄²	—	332	1	no	—
179	C₄²	—	332	1	no	—
180	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
181	C₄²	Q₈×D₈₃	1	1	yes	characteristic
182	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
183	C₄²	#13050591	1	1	yes	characteristic
184	C₂³	#14581763	1	1	yes	characteristic
185	C₂×C₄	—	2	2	no	—
186	C₂×C₄	—	8	2	no	—
187	C₂×C₄	—	8	2	no	—
188	C₂×C₄	—	8	2	no	—
189	C₂×C₄	—	332	2	no	—
190	C₂×C₄	—	332	2	no	—
191	C₂×C₄	—	332	2	no	—
192	C₂×C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
193	C₂×C₄	#14581754	1	1	yes	characteristic
194	C₂×C₄	#14581757	1	1	yes	characteristic
195	C₂×C₄	#14581768	1	1	yes	characteristic
196	C₂×C₄	#14581783	1	1	yes	characteristic
197	C₂×C₄	#14581791	1	1	yes	characteristic
198	C₂×C₄	C₂×#13050577	1	1	yes	characteristic
199	C₂×C₄	C₂×#13050578	1	1	yes	characteristic
200	C₂×C₄	#14581818	1	1	yes	characteristic
201	C₂×C₄	#14581819	1	2	yes	—
202	C₂×C₄	#14581821	1	2	yes	—
203	C₂×C₄	#14581823	1	1	yes	characteristic
204	C₈	—	2	2	no	—
205	C₈	—	332	4	no	—
206	C₈	—	332	2	no	—
207	C₈	#14581723	1	4	yes	complemented
208	C₂²	#19127597	1	1	yes	characteristic
209	C₂²	#19127898	1	1	yes	characteristic
210	C₂²	#19127903	1	1	yes	characteristic
211	C₂²	#19127944	1	2	yes	—
212	C₂²	#19127961	1	2	yes	—
213	C₄	—	2	1	no	—
214	C₄	—	2	1	no	—
215	C₄	—	16	2	no	—
216	C₄	—	664	2	no	—
217	C₄	C₂×#14581723	1	1	yes	characteristic
218	C₄	#19127741	1	1	yes	characteristic
219	C₄	#19127807	1	1	yes	characteristic
220	C₄	#19127812	1	1	yes	characteristic
221	C₄	#19127983	1	1	yes	characteristic
222	C₄	#19127988	1	1	yes	characteristic
223	C₄	#19128108	1	2	yes	—
224	C₂	#24906932	1	2	yes	—
225	C₂	#24906959	1	2	yes	—
226	C₂	#24908199	1	1	yes	characteristic
227	C₂	#24908224	1	1	yes	characteristic
228	C₂	#24908241	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	#14581763
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1844	C₂
socle, prime radical	C₂³×C₈₃	#121
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1844	D₈₃
hypocenter	C₈₃	#9618
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ #1844

Complements

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

N	H ≅ G/N	type
13. #19127454	C₄	semidirect
90. C₈×C₈₃	#106	semidirect
120. C₈₃	#9618	semidirect
207. C₈	#14581723	semidirect

Maximal subgroups

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

subgroup	index
#24908224	2
#24907330	2
C₂×#19127454	2
#24905346	2
C₈₃×#1844	2
#9618	83