Subgroups — #35303146

Order 21248.

310 conjugacy classes of subgroups.

The subgroup lattice has 218 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24907635	C₂	1	1	yes	maximal, characteristic
2	#24907330	C₂	1	1	yes	maximal, characteristic
3	#24906234	C₂	1	2	yes	maximal
4	#24905365	C₂	1	2	yes	maximal
5	C₈₃×#1564	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581728	—	2	1	no	—
7	C₂×#14581728	—	2	1	no	—
8	#19127725	C₂²	1	2	yes	—
9	C₂×C₄⋊(C₈₃⋊C₈)	C₂²	1	1	yes	characteristic
10	C₂×#14581702	—	2	1	no	—
11	C₄×C₄⋊Dic₈₃	C₂²	1	1	yes	characteristic
12	#19127455	—	2	1	no	—
13	#19127452	—	2	4	no	—
14	#431×C₈₃	C₂²	1	2	yes	—
15	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
16	#335×C₈₃	C₄	1	2	yes	complemented
17	#334×C₈₃	C₄	1	2	yes	complemented
18	C₂×#13050574	—	2	2	no	—
19	C₂²×C₈₃⋊C₈	—	2	1	no	—
20	C₂×C₄⋊Dic₈₃	—	2	1	no	—
21	C₂×C₄×Dic₈₃	—	2	1	no	—
22	#14581728	—	4	2	no	—
23	#14581728	—	4	2	no	—
24	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
25	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
26	C₄⋊(C₈₃⋊C₈)	—	2	2	no	—
27	C₄⋊(C₈₃⋊C₈)	D₄	1	2	yes	—
28	#14581702	—	4	2	no	—
29	C₈₃⋊#95	Q₈	1	2	yes	—
30	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
31	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
32	C₂×C₄²×C₈₃	C₂³	1	1	yes	characteristic
33	C₄⋊C₈×C₈₃	C₂×C₄	1	4	yes	—
34	C₂²⋊C₈×C₈₃	—	2	2	no	—
35	C₄×C₈×C₈₃	—	2	2	no	—
36	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
37	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
38	C₄×C₈×C₈₃	Q₈	1	1	yes	characteristic
39	C₄×C₈×C₈₃	Q₈	1	1	yes	characteristic
40	C₂²×Dic₈₃	—	2	2	no	—
41	#13050574	—	4	2	no	—
42	#13050574	—	4	2	no	—
43	C₂×C₈₃⋊C₈	—	2	2	no	—
44	C₂×C₈₃⋊C₈	—	4	1	no	—
45	C₂×C₈₃⋊C₈	—	4	1	no	—
46	C₂×C₈₃⋊C₈	—	4	2	no	—
47	C₄⋊Dic₈₃	—	4	2	no	—
48	C₄×Dic₈₃	—	4	1	no	—
49	C₄×Dic₈₃	—	4	1	no	—
50	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
51	C₂²×C₄×C₈₃	C₂×Q₈	1	1	yes	characteristic
52	C₂²×C₄×C₈₃	#41	1	1	yes	characteristic
53	C₂×C₈×C₈₃	—	2	1	no	—
54	C₂×C₈×C₈₃	—	2	1	no	—
55	C₂×C₈×C₈₃	—	2	4	no	—
56	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
57	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
59	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
60	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
61	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
62	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
63	C₄²×C₈₃	C₂×D₄	1	1	yes	characteristic
64	C₄²×C₈₃	C₂×Q₈	1	1	yes	characteristic
65	C₄²×C₈₃	#41	1	1	yes	characteristic
66	C₂³×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
67	C₂×C₄×C₈₃	—	2	2	no	—
68	C₂×C₄×C₈₃	—	4	1	no	—
69	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
70	C₂×C₄×C₈₃	C₄×D₄	1	1	yes	characteristic
71	C₂×C₄×C₈₃	C₄×Q₈	1	1	yes	characteristic
72	C₂×C₄×C₈₃	#121	1	1	yes	characteristic
73	C₂×C₄×C₈₃	#125	1	1	yes	characteristic
74	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
75	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
76	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
77	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
78	C₂×C₄×C₈₃	#137	1	1	yes	characteristic
79	C₂×Dic₈₃	—	4	2	no	—
80	C₂×Dic₈₃	—	4	2	no	—
81	C₂×Dic₈₃	—	4	2	no	—
82	C₈×C₈₃	—	2	1	no	—
83	C₈×C₈₃	—	2	1	no	—
84	C₈×C₈₃	—	4	4	no	—
85	C₈×C₈₃	#106	1	2	yes	—
86	C₈×C₈₃	#106	1	2	yes	—
87	C₈₃⋊C₈	—	4	2	no	—
88	C₈₃⋊C₈	—	8	2	no	—
89	C₂²×C₈₃	—	2	1	no	—
90	C₂²×C₈₃	—	2	1	no	—
91	C₂²×C₈₃	#389	1	1	yes	characteristic
92	C₂²×C₈₃	#475	1	1	yes	characteristic
93	C₂²×C₈₃	#476	1	1	yes	characteristic
94	C₄×C₈₃	—	2	1	no	—
95	C₄×C₈₃	—	2	1	no	—
96	C₄×C₈₃	—	2	1	no	—
97	C₄×C₈₃	C₂×#106	1	1	yes	characteristic
98	C₄×C₈₃	#428	1	1	yes	characteristic
99	C₄×C₈₃	#444	1	2	yes	—
100	C₄×C₈₃	#460	1	1	yes	characteristic
101	C₄×C₈₃	#461	1	1	yes	characteristic
102	Dic₈₃	—	8	2	no	—
103	#9631	—	83	1	no	maximal, Sylow
104	C₂×C₈₃	—	2	1	no	—
105	C₂×C₈₃	—	2	1	no	—
106	C₂×C₈₃	#1946	1	1	yes	characteristic
107	C₂×C₈₃	#1948	1	1	yes	characteristic
108	C₂×C₈₃	#1953	1	1	yes	characteristic
109	#1848	—	83	1	no	—
110	#1770	—	83	1	no	—
111	#1570	—	83	2	no	—
112	#1564	D₈₃	1	1	yes	characteristic
113	#1303	—	83	2	no	—
114	C₈₃	#9631	1	1	yes	characteristic, complemented, Sylow
115	#431	C₂×D₈₃	1	2	yes	—
116	C₂×#108	—	166	1	no	—
117	C₂×#108	—	166	1	no	—
118	#423	—	83	2	no	—
119	C₂×C₄⋊C₈	—	83	1	no	—
120	C₂×#97	—	166	1	no	—
121	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
122	C₄×C₄⋊C₄	—	83	1	no	—
123	#336	—	166	1	no	—
124	#335	Dic₈₃	1	2	yes	complemented
125	#334	—	166	4	no	—
126	#334	Dic₈₃	1	2	yes	complemented
127	C₂×M₁₆	—	166	2	no	—
128	C₂²×C₈	—	166	1	no	—
129	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
130	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
131	C₂×C₄⋊C₄	—	166	1	no	—
132	C₂×C₄²	—	166	1	no	—
133	C₂×C₄²	C₂²×D₈₃	1	1	yes	characteristic
134	#108	—	332	2	no	—
135	#108	—	332	2	no	—
136	C₄⋊C₈	—	83	2	no	—
137	C₄⋊C₈	—	166	2	no	—
138	C₄⋊C₈	—	166	2	no	—
139	C₄⋊C₈	—	166	1	no	—
140	C₄⋊C₈	C₂×Dic₈₃	1	4	yes	—
141	C₂²⋊C₈	—	2	2	no	—
142	#97	—	332	2	no	—
143	C₄×C₈	—	2	2	no	—
144	C₄×C₈	Dic₁₆₆	1	1	yes	characteristic
145	C₄×C₈	Dic₁₆₆	1	1	yes	characteristic
146	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
147	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
148	#95	—	83	2	no	—
149	C₂²×C₄	—	166	2	no	—
150	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
151	C₂²×C₄	Q₈×D₈₃	1	1	yes	characteristic
152	C₂²×C₄	#13050589	1	1	yes	characteristic
153	M₁₆	—	332	2	no	—
154	M₁₆	—	332	2	no	—
155	C₂×C₈	—	2	1	no	—
156	C₂×C₈	—	2	1	no	—
157	C₂×C₈	—	2	4	no	—
158	C₂×C₈	—	332	1	no	—
159	C₂×C₈	—	332	1	no	—
160	C₂×C₈	—	332	2	no	—
161	C₂×C₈	—	166	2	no	—
162	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
163	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
164	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
165	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
166	C₂×C₈	#13050579	1	2	yes	—
167	C₂×C₈	#13050579	1	2	yes	—
168	C₄⋊C₄	—	332	2	no	—
169	C₄²	—	332	1	no	—
170	C₄²	—	332	1	no	—
171	C₄²	C₂²×Dic₈₃	1	1	yes	characteristic
172	C₄²	D₄×D₈₃	1	1	yes	characteristic
173	C₄²	C₂×Dic₁₆₆	1	1	yes	characteristic
174	C₄²	#13050591	1	1	yes	characteristic
175	C₂³	#14581772	1	1	yes	characteristic
176	C₂×C₄	—	2	2	no	—
177	C₂×C₄	—	4	1	no	—
178	C₂×C₄	—	332	2	no	—
179	C₂×C₄	—	332	2	no	—
180	C₂×C₄	—	332	2	no	—
181	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
182	C₂×C₄	D₄×Dic₈₃	1	1	yes	characteristic
183	C₂×C₄	Q₈×Dic₈₃	1	1	yes	characteristic
184	C₂×C₄	#14581767	1	1	yes	characteristic
185	C₂×C₄	#14581783	1	1	yes	characteristic
186	C₂×C₄	#14581791	1	1	yes	characteristic
187	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
188	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
189	C₂×C₄	#14581820	1	1	yes	characteristic
190	C₂×C₄	#14581825	1	1	yes	characteristic
191	C₈	—	2	1	no	—
192	C₈	—	2	1	no	—
193	C₈	—	4	4	no	—
194	C₈	—	332	2	no	—
195	C₈	—	664	2	no	—
196	C₈	#14581724	1	2	yes	—
197	C₈	#14581724	1	2	yes	—
198	C₂²	—	2	1	no	—
199	C₂²	—	2	1	no	—
200	C₂²	#19127600	1	1	yes	characteristic
201	C₂²	#19127987	1	1	yes	characteristic
202	C₂²	#19127992	1	1	yes	characteristic
203	C₄	—	2	1	no	—
204	C₄	—	2	1	no	—
205	C₄	—	2	1	no	—
206	C₄	—	664	2	no	—
207	C₄	C₂×#14581724	1	1	yes	characteristic
208	C₄	#19127743	1	1	yes	characteristic
209	C₄	#19127823	1	2	yes	—
210	C₄	#19127902	1	1	yes	characteristic
211	C₄	#19127907	1	1	yes	characteristic
212	C₂	—	2	1	no	—
213	C₂	—	2	1	no	—
214	C₂	#24908217	1	1	yes	characteristic
215	C₂	#24908227	1	1	yes	characteristic
216	C₂	#24908254	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14581767
commutator	C₄²×C₈₃	C₂²×C₄
Frattini	C₂×C₄²	C₂²×D₈₃
Fitting	C₈₃×#1564	C₂
socle	C₂²×C₈₃	#389
prime radical	C₂³×C₈₃	C₂²⋊Q₈
cosocle	C₂×C₄²×C₈₃	C₂³
hypercenter	#1564	D₈₃
hypocenter	C₈₃	#9631
LCS	G ▷ C₄²×C₈₃ ▷ C₂²×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ #1564

Complements

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

N	H ≅ G/N	type
16. #335×C₈₃	C₄	semidirect
17. #334×C₈₃	C₄	semidirect
114. C₈₃	#9631	semidirect
124. #335	Dic₈₃	semidirect
126. #334	Dic₈₃	semidirect

Maximal subgroups

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

subgroup	index
#24907635	2
#24907330	2
#24906234	2
#24905365	2
C₈₃×#1564	2
#9631	83