Subgroups — #35322959

Order 21248.

332 conjugacy classes of subgroups.

The subgroup lattice has 234 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127522	C₂	1	1	yes	maximal, characteristic
2	#24907342	C₂	1	1	yes	maximal, characteristic
3	#24905579	C₂	1	2	yes	maximal
4	#24905564	C₂	1	2	yes	maximal
5	C₈₃×#3160	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581732	C₂²	1	2	yes	—
7	C₂×#14581728	C₂²	1	1	yes	characteristic
8	#19127739	C₂²	1	1	yes	characteristic
9	#19127656	—	2	1	no	—
10	#19127522	—	2	1	no	—
11	#19127522	—	2	1	no	—
12	#19127461	—	2	1	no	—
13	#533×C₈₃	—	2	1	no	—
14	#500×C₈₃	—	2	1	no	—
15	#499×C₈₃	—	2	1	no	—
16	#499×C₈₃	C₄	1	2	yes	complemented
17	#498×C₈₃	C₄	1	2	yes	complemented
18	#427×C₈₃	—	2	1	no	—
19	#427×C₈₃	C₂²	1	2	yes	—
20	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
21	C₂×#13050574	—	2	1	no	—
22	#14581751	—	2	1	no	—
23	#14581732	—	2	2	no	—
24	#14581732	—	2	2	no	—
25	C₂×C₈₃⋊C₁₆	—	2	2	no	—
26	#14581728	—	2	1	no	—
27	#14581728	Q₈	1	2	yes	complemented
28	#14581726	—	2	1	no	—
29	#14581724	Q₈	1	2	yes	complemented
30	#14581702	—	4	1	no	—
31	C₈×Dic₈₃	—	4	1	no	—
32	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
33	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
34	C₄⋊Q₈×C₈₃	—	2	2	no	—
35	#125×C₈₃	—	2	2	no	—
36	C₂²⋊Q₈×C₈₃	—	4	2	no	—
37	C₄×Q₈×C₈₃	—	4	2	no	—
38	#117×C₈₃	—	2	2	no	—
39	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
40	#107×C₈₃	—	2	2	no	—
41	#107×C₈₃	—	2	2	no	—
42	#106×C₈₃	—	2	2	no	—
43	#106×C₈₃	C₂×C₄	1	4	yes	—
44	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
45	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
46	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
47	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
48	#13050574	—	2	2	no	—
49	#13050574	—	4	1	no	—
50	C₂×C₈₃⋊C₈	—	4	1	no	—
51	C₄⋊Dic₈₃	—	2	2	no	—
52	C₂²⋊Dic₈₃	—	4	1	no	—
53	C₄×Dic₈₃	—	4	1	no	—
54	C₈₃⋊C₁₆	—	4	2	no	—
55	C₂×Q₈×C₈₃	—	4	2	no	—
56	C₂²×C₄×C₈₃	—	2	1	no	—
57	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
58	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₈₃	SD₁₆	1	1	yes	characteristic
61	C₂×C₈×C₈₃	SD₁₆	1	1	yes	characteristic
62	C₂×C₈×C₈₃	Q₁₆	1	1	yes	characteristic
63	C₂×C₈×C₈₃	Q₁₆	1	1	yes	characteristic
64	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
65	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
66	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
67	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
68	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
69	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
70	C₄⋊C₄×C₈₃	—	2	4	no	—
71	C₄⋊C₄×C₈₃	—	4	4	no	—
72	C₄⋊C₄×C₈₃	—	4	2	no	—
73	C₂²⋊C₄×C₈₃	—	4	2	no	—
74	C₄²×C₈₃	—	2	1	no	—
75	C₄²×C₈₃	—	4	2	no	—
76	C₄²×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
77	C₄²×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
78	C₂³×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
79	Q₈×C₈₃	—	8	4	no	—
80	C₂×C₄×C₈₃	—	2	1	no	—
81	C₂×C₄×C₈₃	—	2	1	no	—
82	C₂×C₄×C₈₃	—	4	2	no	—
83	C₂×C₄×C₈₃	—	4	4	no	—
84	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
85	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
86	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
87	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
88	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
89	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
90	C₂×Dic₈₃	—	4	2	no	—
91	C₈×C₈₃	#103	1	1	yes	characteristic
92	C₈×C₈₃	#103	1	1	yes	characteristic
93	C₈×C₈₃	#103	1	1	yes	characteristic
94	C₈×C₈₃	#103	1	1	yes	characteristic
95	C₈×C₈₃	C₄×Q₈	1	1	yes	characteristic
96	C₈×C₈₃	C₄×Q₈	1	1	yes	characteristic
97	C₈×C₈₃	#124	1	1	yes	characteristic
98	C₈×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
99	C₈₃⋊C₈	—	4	2	no	—
100	C₂²×C₈₃	—	4	1	no	—
101	C₂²×C₈₃	#418	1	1	yes	characteristic
102	C₂²×C₈₃	#476	1	1	yes	characteristic
103	C₂²×C₈₃	#477	1	1	yes	characteristic
104	C₄×C₈₃	—	2	2	no	—
105	C₄×C₈₃	—	4	1	no	—
106	C₄×C₈₃	—	8	4	no	—
107	C₄×C₈₃	#391	1	1	yes	characteristic
108	C₄×C₈₃	C₂×#103	1	1	yes	characteristic
109	C₄×C₈₃	#476	1	1	yes	characteristic
110	C₄×C₈₃	#477	1	1	yes	characteristic
111	Dic₈₃	—	8	2	no	—
112	#13271	—	83	1	no	maximal, Sylow
113	C₂×C₈₃	—	2	1	no	—
114	C₂×C₈₃	—	2	1	no	—
115	C₂×C₈₃	—	2	1	no	—
116	C₂×C₈₃	#1987	1	1	yes	characteristic
117	#3160	D₈₃	1	1	yes	characteristic
118	C₂×#364	—	83	1	no	—
119	#1772	—	83	1	no	—
120	#1389	—	83	2	no	—
121	#1384	—	83	2	no	—
122	C₈₃	#13271	1	1	yes	characteristic, complemented, Sylow
123	#533	—	2	1	no	—
124	C₂×M₃₂	—	83	2	no	—
125	#500	—	2	1	no	—
126	#499	—	2	1	no	—
127	#499	Dic₈₃	1	2	yes	complemented
128	#498	Dic₈₃	1	2	yes	complemented
129	C₂×#108	—	83	1	no	—
130	#427	—	2	1	no	—
131	#427	—	83	1	no	—
132	#427	C₂×D₈₃	1	2	yes	—
133	#405	—	166	1	no	—
134	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
135	#364	—	166	1	no	—
136	#364	—	166	1	no	—
137	#339	—	166	1	no	—
138	C₂×M₁₆	—	166	1	no	—
139	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
140	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
141	C₄⋊Q₈	—	2	2	no	—
142	#125	—	2	2	no	—
143	C₂²⋊Q₈	—	4	2	no	—
144	C₄×Q₈	—	4	2	no	—
145	#117	—	2	2	no	—
146	#117	—	166	1	no	—
147	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
148	M₃₂	—	166	2	no	—
149	M₃₂	—	166	2	no	—
150	C₂×C₁₆	—	166	2	no	—
151	#108	—	83	2	no	—
152	#108	—	166	1	no	—
153	#107	—	2	2	no	—
154	#107	—	2	2	no	—
155	#107	—	166	1	no	—
156	#106	—	2	2	no	—
157	#106	—	83	2	no	—
158	#106	C₂×Dic₈₃	1	4	yes	—
159	#97	—	332	1	no	—
160	C₄×C₈	—	332	1	no	—
161	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
162	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
163	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
164	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
165	C₂×Q₈	—	4	2	no	—
166	C₂²×C₄	—	2	1	no	—
167	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
168	M₁₆	—	332	1	no	—
169	M₁₆	—	166	2	no	—
170	C₂×C₈	—	332	1	no	—
171	C₂×C₈	C₂²⋊Dic₈₃	1	1	yes	characteristic
172	C₂×C₈	C₂²⋊Dic₈₃	1	1	yes	characteristic
173	C₂×C₈	#13050577	1	1	yes	characteristic
174	C₂×C₈	#13050577	1	1	yes	characteristic
175	C₂×C₈	#13050580	1	1	yes	characteristic
176	C₂×C₈	#13050580	1	1	yes	characteristic
177	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
178	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
179	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
180	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
181	C₂×C₈	#13050591	1	1	yes	characteristic
182	C₂×C₈	#13050591	1	1	yes	characteristic
183	C₄⋊C₄	—	2	4	no	—
184	C₄⋊C₄	—	4	4	no	—
185	C₄⋊C₄	—	4	2	no	—
186	C₄⋊C₄	—	166	2	no	—
187	C₂²⋊C₄	—	4	2	no	—
188	C₂²⋊C₄	—	332	1	no	—
189	C₄²	—	2	1	no	—
190	C₄²	—	4	2	no	—
191	C₄²	—	332	1	no	—
192	C₄²	C₂²⋊Dic₈₃	1	1	yes	characteristic
193	C₄²	C₂²⋊Dic₈₃	1	1	yes	characteristic
194	C₁₆	—	332	2	no	—
195	C₂³	#14581818	1	1	yes	characteristic
196	Q₈	—	8	4	no	—
197	C₂×C₄	—	2	1	no	—
198	C₂×C₄	—	2	1	no	—
199	C₂×C₄	—	4	2	no	—
200	C₂×C₄	—	4	4	no	—
201	C₂×C₄	—	332	2	no	—
202	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
203	C₂×C₄	#14581768	1	1	yes	characteristic
204	C₂×C₄	#14581768	1	1	yes	characteristic
205	C₂×C₄	C₂×#13050577	1	1	yes	characteristic
206	C₂×C₄	C₂×#13050580	1	1	yes	characteristic
207	C₂×C₄	#14581818	1	1	yes	characteristic
208	C₈	—	332	2	no	—
209	C₈	#14581716	1	1	yes	characteristic
210	C₈	#14581716	1	1	yes	characteristic
211	C₈	#14581716	1	1	yes	characteristic
212	C₈	#14581716	1	1	yes	characteristic
213	C₈	Q₈×Dic₈₃	1	1	yes	characteristic
214	C₈	Q₈×Dic₈₃	1	1	yes	characteristic
215	C₈	#14581781	1	1	yes	characteristic
216	C₈	#14581790	1	1	yes	characteristic
217	C₂²	—	4	1	no	—
218	C₂²	#19127706	1	1	yes	characteristic
219	C₂²	#19127988	1	1	yes	characteristic
220	C₂²	#19127993	1	1	yes	characteristic
221	C₄	—	2	2	no	—
222	C₄	—	4	1	no	—
223	C₄	—	8	4	no	—
224	C₄	—	664	2	no	—
225	C₄	#19127608	1	1	yes	characteristic
226	C₄	C₂×#14581716	1	1	yes	characteristic
227	C₄	#19127988	1	1	yes	characteristic
228	C₄	#19127993	1	1	yes	characteristic
229	C₂	—	2	1	no	—
230	C₂	—	2	1	no	—
231	C₂	—	2	1	no	—
232	C₂	#24908412	1	1	yes	characteristic

	subgroup	G/H
center	C₄	#19127988
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#3160	C₂
socle	C₂×C₈₃	#1987
prime radical	C₂³×C₈₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#3160	D₈₃
hypocenter	C₈₃	#13271
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₄ ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #3160

Complements

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

N	H ≅ G/N	type
16. #499×C₈₃	C₄	semidirect
17. #498×C₈₃	C₄	semidirect
27. #14581728	Q₈	semidirect
29. #14581724	Q₈	semidirect
122. C₈₃	#13271	semidirect
127. #499	Dic₈₃	semidirect
128. #498	Dic₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19127522	2
#24907342	2
#24905579	2
#24905564	2
C₈₃×#3160	2
#13271	83