Subgroups — #35320839

Order 21248.

374 conjugacy classes of subgroups.

The subgroup lattice has 184 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127529	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127528	C₂	1	2	yes	maximal
3	C₂×C₁₆×Dic₈₃	C₂	1	1	yes	maximal, characteristic
4	#24907621	C₂	1	2	yes	maximal
5	C₂×#367×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	#19127529	—	2	2	no	—
12	#19127528	—	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	#367×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	SD₃₂	1	8	yes	—
71	C₈×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
72	C₈×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
73	C₈×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
74	C₈×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
75	C₈₃⋊C₈	#107	1	4	yes	—
76	C₂²×C₈₃	C₂×#107	1	1	yes	characteristic
77	C₂²×C₈₃	#493	1	1	yes	characteristic
78	C₂²×C₈₃	#500	1	1	yes	characteristic
79	C₂²×C₈₃	C₂×SD₃₂	1	4	yes	—
80	C₄×C₈₃	—	16	2	no	—
81	C₄×C₈₃	#384	1	1	yes	characteristic
82	C₄×C₈₃	C₂×#107	1	1	yes	characteristic
83	C₄×C₈₃	#494	1	1	yes	characteristic
84	C₄×C₈₃	#500	1	1	yes	characteristic
85	Dic₈₃	—	16	2	no	—
86	Dic₈₃	#367	1	8	yes	complemented
87	#12906	—	83	1	no	maximal, Sylow
88	C₂×C₈₃	#1845	1	1	yes	characteristic
89	C₂×C₈₃	C₂×#367	1	2	yes	—
90	C₂×C₈₃	#2248	1	2	yes	—
91	C₂×C₈₃	#2254	1	2	yes	—
92	C₂×#367	—	83	1	no	—
93	C₂×#367	—	83	2	no	—
94	C₂×#367	D₈₃	1	1	yes	characteristic
95	C₂×C₄×C₁₆	—	83	1	no	—
96	#1845	—	83	2	no	—
97	C₈₃	#12906	1	1	yes	characteristic, complemented, Sylow
98	C₂²×C₁₆	—	83	1	no	—
99	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
100	C₂×#107	—	83	2	no	—
101	C₂×#107	—	166	2	no	—
102	C₂×#107	C₂×D₈₃	1	2	yes	—
103	C₂×C₄×C₈	—	83	1	no	—
104	#384	—	166	2	no	—
105	#367	—	2	2	no	—
106	#367	—	166	4	no	—
107	#367	—	166	2	no	—
108	C₄×C₁₆	—	83	4	no	—
109	C₂²×C₈	—	83	1	no	—
110	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
111	C₂×C₄⋊C₄	—	2	2	no	—
112	C₂×C₄⋊C₄	—	332	4	no	—
113	C₂×C₄⋊C₄	—	166	2	no	—
114	C₂×C₄²	—	83	1	no	—
115	C₂×C₁₆	—	83	2	no	—
116	C₂×C₁₆	—	83	2	no	—
117	C₂×C₁₆	—	83	2	no	—
118	C₂×C₁₆	Dic₁₆₆	1	2	yes	—
119	C₂×C₁₆	C₄×D₈₃	1	2	yes	—
120	C₂×C₁₆	C₂²⋊D₈₃	1	2	yes	—
121	#107	—	2	2	no	—
122	#107	—	2	2	no	—
123	#107	—	332	4	no	—
124	#107	—	166	2	no	—
125	#107	—	166	2	no	—
126	C₄×C₈	—	83	2	no	—
127	C₄×C₈	—	83	2	no	—
128	C₂²×C₄	—	4	2	no	—
129	C₂²×C₄	—	332	2	no	—
130	C₂²×C₄	—	83	2	no	—
131	C₂²×C₄	D₄×D₈₃	1	1	yes	characteristic
132	C₂×C₈	—	83	2	no	—
133	C₂×C₈	—	83	2	no	—
134	C₂×C₈	—	83	2	no	—
135	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
136	C₂×C₈	D₄×D₈₃	1	1	yes	characteristic
137	C₂×C₈	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
138	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
139	C₂×C₈	Q₈×D₈₃	1	1	yes	characteristic
140	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
141	C₄⋊C₄	—	4	2	no	—
142	C₄⋊C₄	—	4	2	no	—
143	C₄⋊C₄	—	332	2	no	—
144	C₄⋊C₄	—	332	2	no	—
145	C₄⋊C₄	—	664	8	no	—
146	C₄²	—	83	2	no	—
147	C₄²	—	83	2	no	—
148	C₁₆	—	83	4	no	—
149	C₁₆	Dic₈₃⋊C₄	1	4	yes	complemented
150	C₂³	D₈×D₈₃	1	1	yes	characteristic
151	C₂×C₄	—	8	2	no	—
152	C₂×C₄	—	8	4	no	—
153	C₂×C₄	—	83	8	no	—
154	C₂×C₄	—	83	4	no	—
155	C₂×C₄	—	664	2	no	—
156	C₂×C₄	—	664	4	no	—
157	C₂×C₄	C₄⋊C₄×D₈₃	1	1	yes	characteristic
158	C₂×C₄	#14581788	1	1	yes	characteristic
159	C₂×C₄	#14581791	1	1	yes	characteristic
160	C₂×C₄	D₈×D₈₃	1	1	yes	characteristic
161	C₂×C₄	Q₁₆×D₈₃	1	1	yes	characteristic
162	C₂×C₄	Q₁₆×D₈₃	1	1	yes	characteristic
163	C₈	—	83	4	no	—
164	C₈	C₄⋊C₄×D₈₃	1	1	yes	characteristic
165	C₈	C₂×Dic₈₃⋊C₄	1	1	yes	characteristic
166	C₈	#14581790	1	1	yes	characteristic
167	C₈	#14581791	1	1	yes	characteristic
168	C₂²	#107×D₈₃	1	1	yes	characteristic
169	C₂²	#19128076	1	1	yes	characteristic
170	C₂²	#19128105	1	1	yes	characteristic
171	C₂²	SD₃₂×D₈₃	1	4	yes	—
172	C₄	—	16	2	no	—
173	C₄	—	1328	2	no	—
174	C₄	—	83	8	no	—
175	C₄	#19127570	1	1	yes	characteristic
176	C₄	#107×D₈₃	1	1	yes	characteristic
177	C₄	#19128079	1	1	yes	characteristic
178	C₄	#19128105	1	1	yes	characteristic
179	C₂	#24907621	1	1	yes	characteristic
180	C₂	#367×D₈₃	1	2	yes	—
181	C₂	#24909717	1	2	yes	—
182	C₂	#24909744	1	2	yes	—

	subgroup	G/H
center	C₂³	D₈×D₈₃
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₂×#367×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂×#367	D₈₃
hypocenter	C₈₃	#12906
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄ ◁ C₂²×C₈ ◁ C₂×#367

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
86. Dic₈₃	#367	semidirect
97. C₈₃	#12906	semidirect
149. 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×#19127529	2
C₂×#19127528	2
C₂×C₁₆×Dic₈₃	2
#24907621	2
C₂×#367×C₈₃	2
#12906	83