Subgroups — #35320836

Order 21248.

374 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127527	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127526	C₂	1	2	yes	maximal
3	C₂×C₁₆×Dic₈₃	C₂	1	1	yes	maximal, characteristic
4	#24907621	C₂	1	2	yes	maximal
5	C₂×#366×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂²×C₈₃⋊C₁₆	C₂²	1	1	yes	characteristic
7	C₂×#14581726	C₂²	1	2	yes	—
8	C₂×#14581725	—	2	2	no	—
9	C₂×C₈×Dic₈₃	C₂²	1	1	yes	characteristic
10	#19127570	—	2	2	no	—
11	#19127527	—	2	2	no	—
12	#19127526	—	2	4	no	—
13	C₁₆×Dic₈₃	C₄	1	4	yes	complemented
14	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
15	C₂×#107×C₈₃	C₂²	1	2	yes	—
16	#366×C₈₃	—	2	2	no	—
17	C₂²×C₈₃⋊C₈	D₄	1	1	yes	characteristic
18	C₂×C₄⋊Dic₈₃	—	2	2	no	—
19	C₂×Dic₈₃⋊C₄	—	4	4	no	—
20	C₂×C₄×Dic₈₃	D₄	1	1	yes	characteristic
21	C₂×C₈₃⋊C₁₆	C₂×C₄	1	2	yes	—
22	C₂×C₈₃⋊C₁₆	D₄	1	2	yes	—
23	C₂×C₈₃⋊C₁₆	Q₈	1	2	yes	—
24	#14581726	—	2	2	no	—
25	#14581726	—	2	2	no	—
26	#14581725	—	4	4	no	—
27	C₈×Dic₈₃	C₂×C₄	1	2	yes	—
28	C₈×Dic₈₃	Q₈	1	2	yes	—
29	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
30	C₂×C₄⋊C₄×C₈₃	—	2	2	no	—
31	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	—
32	C₂×C₁₆×C₈₃	D₄	1	2	yes	—
33	C₂×C₁₆×C₈₃	Q₈	1	2	yes	—
34	#107×C₈₃	—	2	2	no	—
35	#107×C₈₃	—	2	2	no	—
36	C₂²×Dic₈₃	—	4	2	no	—
37	C₂²×Dic₈₃	D₈	1	2	yes	—
38	C₂×C₈₃⋊C₈	C₄⋊C₄	1	2	yes	—
39	C₂×C₈₃⋊C₈	D₈	1	2	yes	—
40	C₂×C₈₃⋊C₈	Q₁₆	1	2	yes	—
41	C₄⋊Dic₈₃	—	4	2	no	—
42	C₄⋊Dic₈₃	—	4	2	no	—
43	Dic₈₃⋊C₄	—	8	8	no	—
44	C₄×Dic₈₃	C₄⋊C₄	1	2	yes	—
45	C₄×Dic₈₃	Q₁₆	1	2	yes	—
46	C₈₃⋊C₁₆	C₄⋊C₄	1	4	yes	complemented
47	C₂²×C₄×C₈₃	—	4	2	no	—
48	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
51	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
52	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
53	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
54	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
55	C₄⋊C₄×C₈₃	—	4	2	no	—
56	C₄⋊C₄×C₈₃	—	4	2	no	—
57	C₁₆×C₈₃	C₄⋊C₄	1	4	yes	complemented
58	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
59	C₂×C₄×C₈₃	—	8	2	no	—
60	C₂×C₄×C₈₃	—	8	4	no	—
61	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
62	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
63	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
64	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
65	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
67	C₂×Dic₈₃	—	8	2	no	—
68	C₂×Dic₈₃	—	8	4	no	—
69	C₂×Dic₈₃	#107	1	4	yes	—
70	C₂×Dic₈₃	D₁₆	1	4	yes	—
71	C₂×Dic₈₃	Q₃₂	1	4	yes	—
72	C₈×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₈×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
74	C₈×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
75	C₈×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
76	C₈₃⋊C₈	#107	1	4	yes	—
77	C₂²×C₈₃	C₂×#107	1	1	yes	characteristic
78	C₂²×C₈₃	#493	1	1	yes	characteristic
79	C₂²×C₈₃	#500	1	1	yes	characteristic
80	C₂²×C₈₃	C₂×D₁₆	1	2	yes	—
81	C₂²×C₈₃	C₂×Q₃₂	1	2	yes	—
82	C₄×C₈₃	—	16	2	no	—
83	C₄×C₈₃	#384	1	1	yes	characteristic
84	C₄×C₈₃	C₂×#107	1	1	yes	characteristic
85	C₄×C₈₃	#494	1	1	yes	characteristic
86	C₄×C₈₃	#500	1	1	yes	characteristic
87	Dic₈₃	—	16	2	no	—
88	Dic₈₃	#366	1	8	yes	complemented
89	#12905	—	83	1	no	maximal, Sylow
90	C₂×C₈₃	#1845	1	1	yes	characteristic
91	C₂×C₈₃	C₂×#366	1	2	yes	—
92	C₂×C₈₃	#2246	1	1	yes	characteristic
93	C₂×C₈₃	#2247	1	1	yes	characteristic
94	C₂×C₈₃	#2252	1	2	yes	—
95	C₂×#366	—	83	1	no	—
96	C₂×#366	—	83	2	no	—
97	C₂×#366	D₈₃	1	1	yes	characteristic
98	C₂×C₄×C₁₆	—	83	1	no	—
99	#1845	—	83	2	no	—
100	C₈₃	#12905	1	1	yes	characteristic, complemented, Sylow
101	C₂²×C₁₆	—	83	1	no	—
102	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
103	C₂×#107	—	83	2	no	—
104	C₂×#107	—	166	2	no	—
105	C₂×#107	C₂×D₈₃	1	2	yes	—
106	C₂×C₄×C₈	—	83	1	no	—
107	#384	—	166	2	no	—
108	#366	—	2	2	no	—
109	#366	—	166	4	no	—
110	#366	—	166	2	no	—
111	C₄×C₁₆	—	83	4	no	—
112	C₂²×C₈	—	83	1	no	—
113	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	2	2	no	—
115	C₂×C₄⋊C₄	—	332	4	no	—
116	C₂×C₄⋊C₄	—	166	2	no	—
117	C₂×C₄²	—	83	1	no	—
118	C₂×C₁₆	—	83	2	no	—
119	C₂×C₁₆	—	83	2	no	—
120	C₂×C₁₆	—	83	2	no	—
121	C₂×C₁₆	Dic₁₆₆	1	2	yes	—
122	C₂×C₁₆	C₄×D₈₃	1	2	yes	—
123	C₂×C₁₆	C₂²⋊D₈₃	1	2	yes	—
124	#107	—	2	2	no	—
125	#107	—	2	2	no	—
126	#107	—	332	4	no	—
127	#107	—	166	2	no	—
128	#107	—	166	2	no	—
129	C₄×C₈	—	83	2	no	—
130	C₄×C₈	—	83	2	no	—
131	C₂²×C₄	—	4	2	no	—
132	C₂²×C₄	—	332	2	no	—
133	C₂²×C₄	—	83	2	no	—
134	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
135	C₂×C₈	—	83	2	no	—
136	C₂×C₈	—	83	2	no	—
137	C₂×C₈	—	83	2	no	—
138	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
139	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
140	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
141	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
142	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
143	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
144	C₄⋊C₄	—	4	2	no	—
145	C₄⋊C₄	—	4	2	no	—
146	C₄⋊C₄	—	332	2	no	—
147	C₄⋊C₄	—	332	2	no	—
148	C₄⋊C₄	—	664	8	no	—
149	C₄²	—	83	2	no	—
150	C₄²	—	83	2	no	—
151	C₁₆	—	83	4	no	—
152	C₁₆	Dic₈₃⋊C₄	1	4	yes	complemented
153	C₂³	D₈×D₈₃	1	1	yes	characteristic
154	C₂×C₄	—	8	2	no	—
155	C₂×C₄	—	8	4	no	—
156	C₂×C₄	—	83	4	no	—
157	C₂×C₄	—	83	4	no	—
158	C₂×C₄	—	83	4	no	—
159	C₂×C₄	—	664	2	no	—
160	C₂×C₄	—	664	4	no	—
161	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
162	C₂×C₄	#14581788	1	1	yes	characteristic
163	C₂×C₄	#14581791	1	1	yes	characteristic
164	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
165	C₂×C₄	Q₁₆×D₈₃	1	1	yes	characteristic
166	C₂×C₄	Q₁₆×D₈₃	1	1	yes	characteristic
167	C₈	—	83	4	no	—
168	C₈	C₄⋊C₄×D₈₃	1	1	yes	characteristic
169	C₈	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
170	C₈	#14581790	1	1	yes	characteristic
171	C₈	#14581791	1	1	yes	characteristic
172	C₂²	#107×D₈₃	1	1	yes	characteristic
173	C₂²	#19128076	1	1	yes	characteristic
174	C₂²	#19128105	1	1	yes	characteristic
175	C₂²	D₁₆×D₈₃	1	2	yes	—
176	C₂²	Q₃₂×D₈₃	1	2	yes	—
177	C₄	—	16	2	no	—
178	C₄	—	1328	2	no	—
179	C₄	—	83	8	no	—
180	C₄	#19127570	1	1	yes	characteristic
181	C₄	#107×D₈₃	1	1	yes	characteristic
182	C₄	#19128079	1	1	yes	characteristic
183	C₄	#19128105	1	1	yes	characteristic
184	C₂	#24907621	1	1	yes	characteristic
185	C₂	#366×D₈₃	1	2	yes	—
186	C₂	#24909711	1	1	yes	characteristic
187	C₂	#24909714	1	1	yes	characteristic
188	C₂	#24909737	1	2	yes	—

	subgroup	G/H
center	C₂³	D₈×D₈₃
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₂×#366×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂×#366	D₈₃
hypocenter	C₈₃	#12905
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#366

Complements

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

N	H ≅ G/N	type
13. C₁₆×Dic₈₃	C₄	semidirect
46. C₈₃⋊C₁₆	C₄⋊C₄	semidirect
57. C₁₆×C₈₃	C₄⋊C₄	semidirect
88. Dic₈₃	#366	semidirect
100. C₈₃	#12905	semidirect
152. C₁₆	Dic₈₃⋊C₄	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19127527	2
C₂×#19127526	2
C₂×C₁₆×Dic₈₃	2
#24907621	2
C₂×#366×C₈₃	2
#12905	83