Subgroups — #35313449

Order 21248.

250 conjugacy classes of subgroups.

The subgroup lattice has 174 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127477	C₂	1	1	yes	maximal, characteristic
2	#24905752	C₂	1	1	yes	maximal, characteristic
3	#24905385	C₂	1	2	yes	maximal, complemented
4	#24905383	C₂	1	2	yes	maximal, complemented
5	C₈₃×#2163	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈₃⋊C₁₆	C₄	1	2	yes	—
7	#19127754	C₄	1	1	yes	characteristic
8	C₄×#13050574	C₄	1	1	yes	characteristic
9	#19127477	C₄	1	2	yes	—
10	#19127477	C₂²	1	1	yes	characteristic
11	#19127477	C₂²	1	1	yes	characteristic
12	#19127423	C₄	1	2	yes	—
13	#19127422	C₂²	1	2	yes	—
14	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₄⋊C₁₆×C₈₃	C₄	1	2	yes	—
16	C₂²⋊C₁₆×C₈₃	C₄	1	2	yes	—
17	#346×C₈₃	C₂²	1	2	yes	—
18	C₂×#13050574	—	2	1	no	—
19	C₂×C₈₃⋊C₁₆	—	2	4	no	—
20	C₂×C₈₃⋊C₁₆	C₂×C₄	1	2	yes	—
21	C₂×C₈₃⋊C₁₆	C₂×C₄	1	2	yes	—
22	C₄⋊(C₈₃⋊C₈)	—	2	1	no	—
23	C₂²⋊(C₈₃⋊C₈)	—	2	1	no	—
24	#14581703	—	2	1	no	—
25	C₄×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
26	C₄×C₈₃⋊C₈	C₂×C₄	1	2	yes	—
27	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
28	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
29	C₂×C₄²×C₈₃	C₂×C₄	1	1	yes	characteristic
30	C₂×C₁₆×C₈₃	C₂×C₄	1	4	yes	—
31	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
32	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
33	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
34	C₄×C₈×C₈₃	C₂³	1	1	yes	characteristic
35	#13050574	—	4	2	no	—
36	C₂×C₈₃⋊C₈	—	2	2	no	—
37	C₂×C₈₃⋊C₈	—	2	2	no	—
38	C₈₃⋊C₁₆	—	2	2	no	—
39	C₈₃⋊C₁₆	M₁₆	1	4	yes	complemented
40	C₂²×C₄×C₈₃	C₄²	1	1	yes	characteristic
41	C₂²×C₄×C₈₃	M₁₆	1	1	yes	characteristic
42	C₂²×C₄×C₈₃	M₁₆	1	1	yes	characteristic
43	C₂×C₈×C₈₃	—	2	2	no	—
44	C₂×C₈×C₈₃	—	2	2	no	—
45	C₂×C₈×C₈₃	C₄²	1	1	yes	characteristic
46	C₂×C₈×C₈₃	C₄²	1	1	yes	characteristic
47	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
49	C₄²×C₈₃	C₄²	1	1	yes	characteristic
50	C₄²×C₈₃	M₁₆	1	1	yes	characteristic
51	C₄²×C₈₃	M₁₆	1	1	yes	characteristic
52	C₄²×C₈₃	C₂²×C₄	1	1	yes	characteristic
53	C₁₆×C₈₃	—	2	4	no	—
54	C₂³×C₈₃	#97	1	1	yes	characteristic
55	C₂×C₄×C₈₃	—	2	2	no	—
56	C₂×C₄×C₈₃	—	2	1	no	—
57	C₂×C₄×C₈₃	—	2	1	no	—
58	C₂×C₄×C₈₃	—	4	1	no	—
59	C₂×C₄×C₈₃	#97	1	1	yes	characteristic
60	C₂×C₄×C₈₃	#97	1	1	yes	characteristic
61	C₂×C₄×C₈₃	#97	1	1	yes	characteristic
62	C₂×C₄×C₈₃	C₂×C₄²	1	1	yes	characteristic
63	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
64	C₂×C₄×C₈₃	C₂×M₁₆	1	1	yes	characteristic
65	C₈×C₈₃	—	2	1	no	—
66	C₈×C₈₃	—	2	1	no	—
67	C₈×C₈₃	C₂×M₁₆	1	2	yes	—
68	C₈×C₈₃	#131	1	2	yes	—
69	C₈₃⋊C₈	—	4	2	no	—
70	C₈₃⋊C₈	—	4	2	no	—
71	C₂²×C₈₃	—	2	1	no	—
72	C₂²×C₈₃	—	2	1	no	—
73	C₂²×C₈₃	#347	1	2	yes	—
74	C₂²×C₈₃	C₂×#97	1	1	yes	characteristic
75	C₄×C₈₃	—	2	1	no	—
76	C₄×C₈₃	—	2	1	no	—
77	C₄×C₈₃	—	2	1	no	—
78	C₄×C₈₃	—	2	1	no	—
79	C₄×C₈₃	—	2	1	no	—
80	C₄×C₈₃	C₄×M₁₆	1	1	yes	characteristic
81	C₄×C₈₃	#405	1	1	yes	characteristic
82	#11413	—	83	1	no	maximal, Sylow
83	C₂×C₈₃	—	2	2	no	—
84	C₂×C₈₃	#1451	1	1	yes	characteristic
85	C₂×C₈₃	C₂×#347	1	1	yes	characteristic
86	C₂×C₈₃	#2110	1	1	yes	characteristic
87	#2163	D₈₃	1	1	yes	characteristic
88	C₂×#346	—	83	1	no	—
89	#1451	—	83	1	no	—
90	#1313	—	83	2	no	—
91	#1313	—	83	2	no	—
92	C₈₃	#11413	1	1	yes	characteristic, complemented, Sylow
93	C₂²×C₁₆	—	83	2	no	—
94	#431	—	83	1	no	—
95	C₄×M₁₆	—	83	1	no	—
96	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
97	C₄⋊C₁₆	Dic₈₃	1	2	yes	—
98	C₂²⋊C₁₆	Dic₈₃	1	2	yes	—
99	#346	—	83	2	no	—
100	#346	—	83	1	no	—
101	#346	—	83	1	no	—
102	#346	C₂×D₈₃	1	2	yes	—
103	#322	—	83	2	no	—
104	#322	—	83	2	no	—
105	C₂×M₁₆	—	166	1	no	—
106	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
107	C₂²×C₈	C₂×Dic₈₃	1	1	yes	characteristic
108	C₂×C₄²	C₄×D₈₃	1	1	yes	characteristic
109	C₂×C₁₆	—	83	2	no	—
110	C₂×C₁₆	—	83	2	no	—
111	C₂×C₁₆	—	166	4	no	—
112	C₂×C₁₆	C₂×Dic₈₃	1	4	yes	—
113	C₄⋊C₈	—	166	1	no	—
114	C₂²⋊C₈	—	166	1	no	—
115	#97	—	166	1	no	—
116	C₄×C₈	—	83	2	no	—
117	C₄×C₈	—	83	2	no	—
118	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
119	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
120	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
121	C₄×C₈	C₂²×D₈₃	1	1	yes	characteristic
122	C₂²×C₄	C₄×Dic₈₃	1	1	yes	characteristic
123	C₂²×C₄	#13050573	1	1	yes	characteristic
124	C₂²×C₄	#13050573	1	1	yes	characteristic
125	M₁₆	—	332	2	no	—
126	C₂×C₈	—	2	2	no	—
127	C₂×C₈	—	2	2	no	—
128	C₂×C₈	—	166	2	no	—
129	C₂×C₈	—	166	2	no	—
130	C₂×C₈	C₄×Dic₈₃	1	1	yes	characteristic
131	C₂×C₈	C₄×Dic₈₃	1	1	yes	characteristic
132	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
133	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
134	C₄²	C₄×Dic₈₃	1	1	yes	characteristic
135	C₄²	#13050573	1	1	yes	characteristic
136	C₄²	#13050573	1	1	yes	characteristic
137	C₄²	C₂×C₄×D₈₃	1	1	yes	characteristic
138	C₁₆	—	2	4	no	—
139	C₁₆	—	83	4	no	—
140	C₁₆	—	166	2	no	—
141	C₂³	#14581702	1	1	yes	characteristic
142	C₂×C₄	—	2	2	no	—
143	C₂×C₄	—	2	1	no	—
144	C₂×C₄	—	2	1	no	—
145	C₂×C₄	—	4	1	no	—
146	C₂×C₄	#14581702	1	1	yes	characteristic
147	C₂×C₄	#14581702	1	1	yes	characteristic
148	C₂×C₄	#14581702	1	1	yes	characteristic
149	C₂×C₄	C₂×C₄×Dic₈₃	1	1	yes	characteristic
150	C₂×C₄	C₂×#13050573	1	1	yes	characteristic
151	C₂×C₄	C₂×#13050573	1	1	yes	characteristic
152	C₈	—	2	1	no	—
153	C₈	—	2	1	no	—
154	C₈	—	332	2	no	—
155	C₈	—	332	2	no	—
156	C₈	M₁₆×D₈₃	1	2	yes	—
157	C₈	#14581799	1	2	yes	—
158	C₂²	—	2	1	no	—
159	C₂²	—	2	1	no	—
160	C₂²	#19127478	1	2	yes	—
161	C₂²	C₂×#14581702	1	1	yes	characteristic
162	C₄	—	2	1	no	—
163	C₄	—	2	1	no	—
164	C₄	—	2	1	no	—
165	C₄	—	2	1	no	—
166	C₄	—	2	1	no	—
167	C₄	M₁₆×Dic₈₃	1	1	yes	characteristic
168	C₄	#19127657	1	1	yes	characteristic
169	C₂	—	2	2	no	—
170	C₂	#24905753	1	1	yes	characteristic
171	C₂	C₂×#19127478	1	1	yes	characteristic
172	C₂	#24909067	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	C₂×C₄×Dic₈₃
commutator	C₂×C₄×C₈₃	C₂×C₄²
Frattini	C₄×C₈	C₂²×D₈₃
Fitting	C₈₃×#2163	C₂
socle	C₂²×C₈₃	C₂×#97
prime radical	C₂³×C₈₃	#97
cosocle	C₄×C₈×C₈₃	C₂³
hypercenter	#2163	D₈₃
hypocenter	C₈₃	#11413
LCS	G ▷ C₂×C₄×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ #2163

Complements

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

N	H ≅ G/N	type
3. #24905385	C₂	semidirect
4. #24905383	C₂	semidirect
39. C₈₃⋊C₁₆	M₁₆	semidirect
92. C₈₃	#11413	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19127477	2
#24905752	2
#24905385	2
#24905383	2
C₈₃×#2163	2
#11413	83