Subgroups — #35557110

Order 21248.

1114 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	#24920566	C₂	1	1	yes	maximal, characteristic, complemented
2	C₂×#19128118	C₂	1	1	yes	maximal, characteristic, complemented
3	#24909091	C₂	1	6	yes	maximal, complemented
4	C₂×#19127485	C₂	1	6	yes	maximal, complemented
5	C₈₃×#3573	C₂	1	1	yes	maximal, characteristic
6	#19128118	—	2	4	no	—
7	#19128118	—	2	4	no	—
8	C₂×#14581732	C₂²	1	6	yes	complemented
9	C₂×#14581732	C₂²	1	3	yes	complemented
10	C₂²×C₈₃⋊C₁₆	C₂²	1	3	yes	complemented
11	#19127485	C₂²	1	4	yes	complemented
12	#19127485	C₂²	1	12	yes	complemented
13	#585×C₈₃	C₄	1	1	yes	characteristic
14	#568×C₈₃	—	2	4	no	—
15	#568×C₈₃	C₄	1	6	yes	—
16	#568×C₈₃	C₄	1	1	yes	characteristic
17	C₂×#131×C₈₃	—	2	4	no	—
18	C₂×#131×C₈₃	C₂²	1	6	yes	—
19	C₂×#131×C₈₃	C₂²	1	1	yes	characteristic
20	#14581732	—	2	6	no	—
21	#14581732	—	2	6	no	—
22	#14581732	C₂³	1	4	yes	—
23	C₂×C₈₃⋊C₁₆	—	2	6	no	—
24	C₂×C₈₃⋊C₁₆	—	2	6	no	—
25	C₂×C₈₃⋊C₁₆	C₂³	1	1	yes	characteristic
26	C₂×C₈₃⋊C₁₆	C₂³	1	3	yes	—
27	2₋¹⁺⁴×C₈₃	—	2	1	no	—
28	2₋¹⁺⁴×C₈₃	C₈	1	4	yes	—
29	H(5,𝔽₂)×C₈₃	—	2	3	no	—
30	H(5,𝔽₂)×C₈₃	C₈	1	4	yes	—
31	C₂×#41×C₈₃	—	2	4	no	—
32	C₂×#41×C₈₃	C₂×C₄	1	6	yes	—
33	C₂×#41×C₈₃	C₂×C₄	1	1	yes	characteristic
34	#131×C₈₃	—	2	4	no	—
35	#131×C₈₃	—	2	12	no	—
36	#131×C₈₃	—	2	6	no	—
37	#131×C₈₃	—	2	6	no	—
38	#131×C₈₃	C₂×C₄	1	12	yes	—
39	#131×C₈₃	D₄	1	4	yes	—
40	#131×C₈₃	D₄	1	4	yes	—
41	#131×C₈₃	C₂³	1	4	yes	—
42	C₂×M₁₆×C₈₃	—	2	12	no	—
43	C₂×M₁₆×C₈₃	—	2	6	no	—
44	C₂×M₁₆×C₈₃	C₂×C₄	1	6	yes	—
45	C₂×M₁₆×C₈₃	C₂×C₄	1	3	yes	—
46	C₂²×C₈×C₈₃	—	2	6	no	—
47	C₂²×C₈×C₈₃	C₂³	1	3	yes	—
48	C₈₃⋊C₁₆	—	2	1	no	—
49	C₈₃⋊C₁₆	—	2	4	no	—
50	C₈₃⋊C₁₆	—	2	3	no	—
51	#41×C₈₃	—	2	4	no	—
52	#41×C₈₃	—	2	12	no	—
53	#41×C₈₃	—	2	6	no	—
54	#41×C₈₃	—	2	6	no	—
55	#41×C₈₃	C₂²⋊C₄	1	4	yes	—
56	#41×C₈₃	C₂×C₈	1	12	yes	—
57	#41×C₈₃	C₂×C₈	1	4	yes	—
58	#41×C₈₃	C₂²×C₄	1	4	yes	—
59	C₂×Q₈×C₈₃	—	2	4	no	—
60	C₂×Q₈×C₈₃	C₂×C₈	1	6	yes	—
61	C₂×Q₈×C₈₃	M₁₆	1	1	yes	characteristic
62	C₂×D₄×C₈₃	—	2	6	no	—
63	C₂×D₄×C₈₃	—	2	12	no	—
64	C₂×D₄×C₈₃	C₂×C₈	1	6	yes	—
65	C₂×D₄×C₈₃	M₁₆	1	3	yes	—
66	C₂²×C₄×C₈₃	—	2	6	no	—
67	C₂²×C₄×C₈₃	C₂²×C₄	1	3	yes	—
68	M₁₆×C₈₃	—	2	6	no	—
69	M₁₆×C₈₃	—	2	12	no	—
70	M₁₆×C₈₃	—	2	4	no	—
71	M₁₆×C₈₃	C₂²⋊C₄	1	12	yes	—
72	M₁₆×C₈₃	C₂²×C₄	1	4	yes	—
73	C₂×C₈×C₈₃	—	2	6	no	—
74	C₂×C₈×C₈₃	—	2	12	no	—
75	C₂×C₈×C₈₃	—	2	4	no	—
76	C₂×C₈×C₈₃	C₂²×C₄	1	3	yes	—
77	C₂×C₈×C₈₃	C₂×D₄	1	6	yes	—
78	C₂×C₈×C₈₃	C₂×D₄	1	6	yes	—
79	C₂×C₈×C₈₃	C₂⁴	1	1	yes	characteristic
80	C₂³×C₈₃	—	2	6	no	—
81	C₂³×C₈₃	C₂×M₁₆	1	3	yes	—
82	Q₈×C₈₃	—	2	6	no	—
83	Q₈×C₈₃	C₂²⋊C₈	1	4	yes	—
84	Q₈×C₈₃	C₂²×C₈	1	4	yes	—
85	D₄×C₈₃	—	2	4	no	—
86	D₄×C₈₃	—	2	12	no	—
87	D₄×C₈₃	—	2	6	no	—
88	D₄×C₈₃	C₂²⋊C₈	1	12	yes	—
89	D₄×C₈₃	C₂²×C₈	1	4	yes	—
90	C₂×C₄×C₈₃	—	2	6	no	—
91	C₂×C₄×C₈₃	—	2	4	no	—
92	C₂×C₄×C₈₃	—	2	12	no	—
93	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	6	yes	—
94	C₂×C₄×C₈₃	C₂²×C₈	1	6	yes	—
95	C₂×C₄×C₈₃	C₂×M₁₆	1	3	yes	—
96	C₂×C₄×C₈₃	C₂³×C₄	1	1	yes	characteristic
97	C₈×C₈₃	—	2	4	no	—
98	C₈×C₈₃	C₂×C₂²⋊C₄	1	6	yes	—
99	C₈×C₈₃	C₂²×D₄	1	1	yes	characteristic
100	C₈×C₈₃	C₂²×D₄	1	1	yes	characteristic
101	C₂²×C₈₃	—	2	4	no	—
102	C₂²×C₈₃	—	4	3	no	—
103	C₂²×C₈₃	—	8	6	no	—
104	C₂²×C₈₃	C₂×C₂²⋊C₈	1	6	yes	—
105	C₂²×C₈₃	C₂²×M₁₆	1	1	yes	characteristic
106	C₄×C₈₃	—	2	4	no	—
107	C₄×C₈₃	C₂×C₂²⋊C₈	1	6	yes	—
108	C₄×C₈₃	C₂²×C₂²⋊C₄	1	1	yes	characteristic
109	C₄×C₈₃	C₂³×C₈	1	1	yes	characteristic
110	#33340	—	83	1	no	maximal, Sylow
111	C₂×C₈₃	—	2	1	no	—
112	C₂×C₈₃	—	2	6	no	—
113	C₂×C₈₃	—	4	4	no	—
114	C₂×C₈₃	C₂²×C₂²⋊C₈	1	1	yes	characteristic
115	#3573	D₈₃	1	1	yes	characteristic
116	#3407	—	83	1	no	—
117	C₂×#504	—	83	1	no	—
118	#2117	—	83	6	no	—
119	C₂×#350	—	83	6	no	—
120	C₈₃	#33340	1	1	yes	characteristic, complemented, Sylow
121	#585	Dic₈₃	1	1	yes	characteristic
122	#568	—	2	4	no	—
123	#568	Dic₈₃	1	6	yes	—
124	#568	Dic₈₃	1	1	yes	characteristic
125	C₂×#131	—	2	4	no	—
126	C₂×#131	C₂×D₈₃	1	6	yes	—
127	C₂×#131	C₂×D₈₃	1	1	yes	characteristic
128	#504	—	166	4	no	—
129	#504	—	166	4	no	—
130	C₂×M₃₂	—	83	3	no	—
131	C₂×M₃₂	—	83	6	no	—
132	C₂²×C₁₆	—	83	3	no	—
133	#350	—	83	4	no	—
134	#350	—	83	12	no	—
135	2₋¹⁺⁴	—	2	1	no	—
136	2₋¹⁺⁴	C₈₃⋊C₈	1	4	yes	—
137	H(5,𝔽₂)	—	2	3	no	—
138	H(5,𝔽₂)	C₈₃⋊C₈	1	4	yes	—
139	C₂×#41	—	2	4	no	—
140	C₂×#41	C₂×Dic₈₃	1	6	yes	—
141	C₂×#41	C₂×Dic₈₃	1	1	yes	characteristic
142	#131	—	2	4	no	—
143	#131	—	2	12	no	—
144	#131	—	2	6	no	—
145	#131	—	2	6	no	—
146	#131	C₂×Dic₈₃	1	12	yes	—
147	#131	C₂²⋊D₈₃	1	4	yes	—
148	#131	C₂²⋊D₈₃	1	4	yes	—
149	#131	C₂²×D₈₃	1	4	yes	—
150	C₂×M₁₆	—	2	12	no	—
151	C₂×M₁₆	—	2	6	no	—
152	C₂×M₁₆	C₂×Dic₈₃	1	6	yes	—
153	C₂×M₁₆	C₂×Dic₈₃	1	3	yes	—
154	C₂²×C₈	—	2	6	no	—
155	C₂²×C₈	C₂²×D₈₃	1	3	yes	—
156	M₃₂	—	83	4	no	—
157	M₃₂	—	166	6	no	—
158	M₃₂	—	166	6	no	—
159	C₂×C₁₆	—	83	1	no	—
160	C₂×C₁₆	—	83	3	no	—
161	C₂×C₁₆	—	166	6	no	—
162	C₂×C₁₆	—	166	6	no	—
163	#41	—	2	4	no	—
164	#41	—	2	12	no	—
165	#41	—	2	6	no	—
166	#41	—	2	6	no	—
167	#41	C₂²⋊Dic₈₃	1	4	yes	—
168	#41	C₂×C₈₃⋊C₈	1	12	yes	—
169	#41	C₂×C₈₃⋊C₈	1	4	yes	—
170	#41	C₂²×Dic₈₃	1	4	yes	—
171	C₂×Q₈	—	2	4	no	—
172	C₂×Q₈	C₂×C₈₃⋊C₈	1	6	yes	—
173	C₂×Q₈	#13050574	1	1	yes	characteristic
174	C₂×D₄	—	2	6	no	—
175	C₂×D₄	—	2	12	no	—
176	C₂×D₄	C₂×C₈₃⋊C₈	1	6	yes	—
177	C₂×D₄	#13050574	1	3	yes	—
178	C₂²×C₄	—	2	6	no	—
179	C₂²×C₄	C₂²×Dic₈₃	1	3	yes	—
180	M₁₆	—	2	6	no	—
181	M₁₆	—	2	4	no	—
182	M₁₆	—	2	12	no	—
183	M₁₆	C₂²⋊Dic₈₃	1	12	yes	—
184	M₁₆	C₂²×Dic₈₃	1	4	yes	—
185	C₂×C₈	—	2	6	no	—
186	C₂×C₈	—	2	4	no	—
187	C₂×C₈	—	2	12	no	—
188	C₂×C₈	C₂²×Dic₈₃	1	3	yes	—
189	C₂×C₈	C₂×C₂²⋊D₈₃	1	6	yes	—
190	C₂×C₈	C₂×C₂²⋊D₈₃	1	6	yes	—
191	C₂×C₈	C₂³×D₈₃	1	1	yes	characteristic
192	C₁₆	—	166	1	no	—
193	C₁₆	—	166	3	no	—
194	C₁₆	—	166	4	no	—
195	C₂³	—	2	6	no	—
196	C₂³	C₂×#13050574	1	3	yes	—
197	Q₈	—	2	6	no	—
198	Q₈	C₂²⋊(C₈₃⋊C₈)	1	4	yes	—
199	Q₈	C₂²×C₈₃⋊C₈	1	4	yes	—
200	D₄	—	2	12	no	—
201	D₄	—	2	4	no	—
202	D₄	—	2	6	no	—
203	D₄	C₂²⋊(C₈₃⋊C₈)	1	12	yes	—
204	D₄	C₂²×C₈₃⋊C₈	1	4	yes	—
205	C₂×C₄	—	2	6	no	—
206	C₂×C₄	—	2	4	no	—
207	C₂×C₄	—	2	12	no	—
208	C₂×C₄	C₂×C₂²⋊Dic₈₃	1	6	yes	—
209	C₂×C₄	C₂²×C₈₃⋊C₈	1	6	yes	—
210	C₂×C₄	C₂×#13050574	1	3	yes	—
211	C₂×C₄	C₂³×Dic₈₃	1	1	yes	characteristic
212	C₈	—	2	4	no	—
213	C₈	C₂×C₂²⋊Dic₈₃	1	6	yes	—
214	C₈	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
215	C₈	C₂²×C₂²⋊D₈₃	1	1	yes	characteristic
216	C₂²	—	2	4	no	—
217	C₂²	—	4	3	no	—
218	C₂²	—	8	6	no	—
219	C₂²	C₂×C₂²⋊(C₈₃⋊C₈)	1	6	yes	—
220	C₂²	C₂²×#13050574	1	1	yes	characteristic
221	C₄	—	2	4	no	—
222	C₄	C₂×C₂²⋊(C₈₃⋊C₈)	1	6	yes	—
223	C₄	C₂²×C₂²⋊Dic₈₃	1	1	yes	characteristic
224	C₄	C₂³×C₈₃⋊C₈	1	1	yes	characteristic
225	C₂	—	2	1	no	—
226	C₂	—	2	6	no	—
227	C₂	—	4	4	no	—
228	C₂	C₂²×C₂²⋊(C₈₃⋊C₈)	1	1	yes	characteristic

	subgroup	G/H
center	C₈	C₂²×C₂²⋊D₈₃
commutator	C₄×C₈₃	C₂³×C₈
Frattini	C₂×C₈	C₂³×D₈₃
Fitting	C₈₃×#3573	C₂
socle	C₂×C₈₃	C₂²×C₂²⋊C₈
prime radical	#585×C₈₃	C₄
cosocle	C₂×C₈×C₈₃	C₂⁴
hypercenter	#3573	D₈₃
hypocenter	C₈₃	#33340
LCS	G ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₈ ◁ C₂×#131 ◁ #3573

Complements

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

N	H ≅ G/N	type
1. #24920566	C₂	semidirect
2. C₂×#19128118	C₂	semidirect
3. #24909091	C₂	semidirect
4. C₂×#19127485	C₂	semidirect
8. C₂×#14581732	C₂²	semidirect
9. C₂×#14581732	C₂²	semidirect
10. C₂²×C₈₃⋊C₁₆	C₂²	semidirect
11. #19127485	C₂²	semidirect
12. #19127485	C₂²	semidirect
120. C₈₃	#33340	semidirect

Maximal subgroups

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

subgroup	index
#24920566	2
C₂×#19128118	2
#24909091	2
C₂×#19127485	2
C₈₃×#3573	2
#33340	83