Subgroups — #35320793

Order 21248.

282 conjugacy classes of subgroups.

The subgroup lattice has 168 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄×C₈₃⋊C₁₆	C₂	1	1	yes	maximal, characteristic
2	#24907635	C₂	1	1	yes	maximal, characteristic
3	#24905556	C₂	1	4	yes	maximal
4	C₈₃×#1848	C₂	1	1	yes	maximal, characteristic
5	C₂²×C₈₃⋊C₁₆	C₂²	1	2	yes	—
6	C₂×#14581728	—	2	1	no	—
7	C₂×#14581728	C₂²	1	2	yes	—
8	#19127725	—	2	1	no	—
9	C₄×C₈₃⋊C₁₆	C₄	1	2	yes	—
10	C₄×C₈₃⋊C₁₆	C₄	1	2	yes	—
11	C₂×#108×C₈₃	—	2	1	no	—
12	C₂×#108×C₈₃	C₂²	1	2	yes	—
13	#423×C₈₃	—	2	1	no	—
14	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×#13050574	—	2	2	no	—
16	C₂×C₈₃⋊C₁₆	—	2	2	no	—
17	C₂×C₈₃⋊C₁₆	—	2	2	no	—
18	C₂×C₈₃⋊C₁₆	C₂×C₄	1	4	yes	—
19	#14581728	—	2	4	no	—
20	#14581728	—	4	2	no	—
21	C₄⋊(C₈₃⋊C₈)	—	4	2	no	—
22	C₂×M₁₆×C₈₃	—	2	2	no	—
23	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
25	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
26	#108×C₈₃	—	2	4	no	—
27	#108×C₈₃	—	4	2	no	—
28	C₄⋊C₈×C₈₃	—	4	2	no	—
29	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₈₃	Q₈	1	1	yes	characteristic
32	C₄×C₈×C₈₃	Q₈	1	1	yes	characteristic
33	#13050574	—	4	2	no	—
34	#13050574	—	4	2	no	—
35	C₂×C₈₃⋊C₈	—	4	2	no	—
36	C₈₃⋊C₁₆	—	2	4	no	—
37	C₂²×C₄×C₈₃	Q₁₆	1	2	yes	—
38	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
39	M₁₆×C₈₃	—	4	2	no	—
40	M₁₆×C₈₃	—	4	2	no	—
41	C₂×C₈×C₈₃	—	4	2	no	—
42	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
43	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
44	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
45	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
46	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
47	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
48	C₂×C₈×C₈₃	#41	1	2	yes	—
49	C₂×C₈×C₈₃	#41	1	2	yes	—
50	C₄²×C₈₃	C₄⋊C₄	1	1	yes	characteristic
51	C₄²×C₈₃	C₄⋊C₄	1	1	yes	characteristic
52	C₄²×C₈₃	D₈	1	2	yes	—
53	C₂³×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
54	C₂×C₄×C₈₃	—	2	4	no	—
55	C₂×C₄×C₈₃	#107	1	2	yes	—
56	C₂×C₄×C₈₃	#107	1	2	yes	—
57	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
58	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
59	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
60	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
61	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
62	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
63	C₈×C₈₃	—	8	2	no	—
64	C₈×C₈₃	#106	1	2	yes	—
65	C₈×C₈₃	#106	1	2	yes	—
66	C₈×C₈₃	#117	1	2	yes	—
67	C₈×C₈₃	#125	1	2	yes	—
68	C₈₃⋊C₈	—	8	2	no	—
69	C₂²×C₈₃	—	2	2	no	—
70	C₂²×C₈₃	C₂×#107	1	1	yes	characteristic
71	C₂²×C₈₃	#487	1	1	yes	characteristic
72	C₂²×C₈₃	#487	1	1	yes	characteristic
73	C₄×C₈₃	—	2	2	no	—
74	C₄×C₈₃	#368	1	4	yes	—
75	C₄×C₈₃	#382	1	1	yes	characteristic
76	C₄×C₈₃	C₂×#106	1	1	yes	characteristic
77	C₄×C₈₃	#486	1	1	yes	characteristic
78	C₄×C₈₃	#486	1	1	yes	characteristic
79	#12891	—	83	1	no	maximal, Sylow
80	C₂×C₈₃	—	2	1	no	—
81	C₂×C₈₃	—	2	1	no	—
82	C₂×C₈₃	#1834	1	1	yes	characteristic
83	C₂×C₈₃	C₂×#368	1	2	yes	—
84	C₂×C₄×C₁₆	—	83	1	no	—
85	#1848	—	83	1	no	—
86	#1848	D₈₃	1	1	yes	characteristic
87	#1381	—	83	4	no	—
88	C₈₃	#12891	1	1	yes	characteristic, complemented, Sylow
89	C₂²×C₁₆	—	83	2	no	—
90	C₂×#108	—	2	1	no	—
91	C₂×#108	—	83	2	no	—
92	C₂×#108	—	166	1	no	—
93	C₂×#108	C₂×D₈₃	1	2	yes	—
94	#423	—	2	1	no	—
95	#423	—	166	1	no	—
96	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
97	C₄×C₁₆	—	83	2	no	—
98	C₄×C₁₆	—	83	2	no	—
99	C₂×M₁₆	—	2	2	no	—
100	C₂×M₁₆	—	166	2	no	—
101	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
102	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
103	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
104	C₂×C₁₆	—	83	4	no	—
105	C₂×C₁₆	—	166	2	no	—
106	C₂×C₁₆	—	166	2	no	—
107	#108	—	2	4	no	—
108	#108	—	4	2	no	—
109	#108	—	332	2	no	—
110	#108	—	166	4	no	—
111	C₄⋊C₈	—	4	2	no	—
112	C₄⋊C₈	—	332	2	no	—
113	C₄×C₈	Dic₁₆₆	1	1	yes	characteristic
114	C₄×C₈	Dic₁₆₆	1	1	yes	characteristic
115	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
116	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
117	C₂²×C₄	#13050580	1	2	yes	—
118	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
119	M₁₆	—	4	2	no	—
120	M₁₆	—	4	2	no	—
121	M₁₆	—	332	2	no	—
122	M₁₆	—	332	2	no	—
123	C₂×C₈	—	4	2	no	—
124	C₂×C₈	—	332	2	no	—
125	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
126	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
127	C₂×C₈	#13050577	1	2	yes	—
128	C₂×C₈	#13050578	1	2	yes	—
129	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
130	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
131	C₂×C₈	#13050589	1	2	yes	—
132	C₂×C₈	#13050591	1	2	yes	—
133	C₄²	Dic₈₃⋊C₄	1	1	yes	characteristic
134	C₄²	Dic₈₃⋊C₄	1	1	yes	characteristic
135	C₄²	#13050575	1	2	yes	—
136	C₁₆	—	166	4	no	—
137	C₂³	C₂×#13050580	1	1	yes	characteristic
138	C₂×C₄	—	2	4	no	—
139	C₂×C₄	#14581725	1	2	yes	—
140	C₂×C₄	#14581725	1	2	yes	—
141	C₂×C₄	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
142	C₂×C₄	#14581779	1	1	yes	characteristic
143	C₂×C₄	#14581781	1	1	yes	characteristic
144	C₂×C₄	C₂×#13050575	1	1	yes	characteristic
145	C₂×C₄	C₂×#13050577	1	1	yes	characteristic
146	C₂×C₄	C₂×#13050578	1	1	yes	characteristic
147	C₈	—	8	2	no	—
148	C₈	—	664	2	no	—
149	C₈	#14581723	1	2	yes	—
150	C₈	#14581723	1	2	yes	—
151	C₈	#14581750	1	2	yes	—
152	C₈	#14581783	1	2	yes	—
153	C₂²	—	2	2	no	—
154	C₂²	C₂×#14581725	1	1	yes	characteristic
155	C₂²	#19128051	1	1	yes	characteristic
156	C₂²	#19128053	1	1	yes	characteristic
157	C₄	—	2	2	no	—
158	C₄	#19127530	1	4	yes	—
159	C₄	#19127566	1	1	yes	characteristic
160	C₄	C₂×#14581723	1	1	yes	characteristic
161	C₄	#19128047	1	1	yes	characteristic
162	C₄	#19128049	1	1	yes	characteristic
163	C₂	—	2	1	no	—
164	C₂	—	2	1	no	—
165	C₂	#24907590	1	1	yes	characteristic
166	C₂	C₂×#19127530	1	2	yes	—

	subgroup	G/H
center	C₂×C₄	C₂×#13050575
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1848	C₂
socle	C₂²×C₈₃	C₂×#107
prime radical	C₂³×C₈₃	C₂×Q₁₆
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#1848	D₈₃
hypocenter	C₈₃	#12891
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1848

Complements

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

N	H ≅ G/N	type
88. C₈₃	#12891	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄×C₈₃⋊C₁₆	2
#24907635	2
#24905556	2
C₈₃×#1848	2
#12891	83