Subgroups — #35320790

Order 21248.

318 conjugacy classes of subgroups.

The subgroup lattice has 170 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	#24907627	C₂	1	1	yes	maximal, characteristic
2	#24907623	C₂	1	1	yes	maximal, characteristic
3	#24905555	C₂	1	4	yes	maximal
4	C₂×#346×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14581726	—	2	1	no	—
6	C₂×#14581726	C₂²	1	2	yes	—
7	C₂×#14581726	C₂²	1	2	yes	—
8	C₂×#14581724	—	2	1	no	—
9	#19127571	—	2	1	no	—
10	#19127567	—	2	1	no	—
11	C₂²×C₁₆×C₈₃	C₂²	1	2	yes	—
12	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
13	#346×C₈₃	C₄	1	4	yes	complemented
14	C₂×C₄⋊Dic₈₃	—	2	2	no	—
15	C₂×C₄⋊Dic₈₃	—	2	2	no	—
16	C₂×C₄⋊Dic₈₃	—	4	2	no	—
17	#14581726	—	2	4	no	—
18	#14581726	—	2	4	no	—
19	#14581726	—	4	2	no	—
20	#14581724	—	4	2	no	—
21	C₈₃⋊#95	—	4	2	no	—
22	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
23	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
24	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
25	C₂×C₁₆×C₈₃	—	2	2	no	—
26	C₂×C₁₆×C₈₃	—	2	2	no	—
27	C₂×C₁₆×C₈₃	C₂×C₄	1	4	yes	—
28	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
29	C₄×C₈×C₈₃	Q₈	1	2	yes	—
30	C₂²×Dic₈₃	—	4	2	no	—
31	C₂²×Dic₈₃	—	4	2	no	—
32	C₄⋊Dic₈₃	—	4	2	no	—
33	C₄⋊Dic₈₃	—	4	2	no	—
34	C₄⋊Dic₈₃	—	4	2	no	—
35	C₄⋊Dic₈₃	—	4	2	no	—
36	C₄⋊Dic₈₃	—	8	4	no	—
37	C₂²×C₄×C₈₃	D₈	1	2	yes	—
38	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
39	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
40	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
41	C₂×C₈×C₈₃	SD₁₆	1	2	yes	—
42	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
43	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
44	C₂×C₈×C₈₃	#41	1	2	yes	—
45	C₂×C₈×C₈₃	#41	1	2	yes	—
46	C₄²×C₈₃	C₄⋊C₄	1	2	yes	—
47	C₄²×C₈₃	Q₁₆	1	2	yes	—
48	C₁₆×C₈₃	—	2	4	no	—
49	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
50	C₂×C₄×C₈₃	—	2	2	no	—
51	C₂×C₄×C₈₃	—	2	2	no	—
52	C₂×C₄×C₈₃	#107	1	4	yes	—
53	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
54	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
55	C₂×C₄×C₈₃	#124	1	1	yes	characteristic
56	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
57	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
58	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
59	C₂×Dic₈₃	—	8	2	no	—
60	C₂×Dic₈₃	—	8	2	no	—
61	C₂×Dic₈₃	—	8	4	no	—
62	C₂×Dic₈₃	—	8	2	no	—
63	C₂×Dic₈₃	—	8	2	no	—
64	C₈×C₈₃	#106	1	4	yes	—
65	C₈×C₈₃	#117	1	2	yes	—
66	C₈×C₈₃	#125	1	2	yes	—
67	C₂²×C₈₃	C₂×#107	1	1	yes	characteristic
68	C₂²×C₈₃	#486	1	1	yes	characteristic
69	C₂²×C₈₃	#486	1	1	yes	characteristic
70	C₂²×C₈₃	#509	1	2	yes	—
71	C₂²×C₈₃	#510	1	2	yes	—
72	C₄×C₈₃	—	2	4	no	—
73	C₄×C₈₃	#382	1	1	yes	characteristic
74	C₄×C₈₃	C₂×#106	1	1	yes	characteristic
75	C₄×C₈₃	#487	1	1	yes	characteristic
76	C₄×C₈₃	#487	1	1	yes	characteristic
77	Dic₈₃	—	16	2	no	—
78	Dic₈₃	—	16	2	no	—
79	#12890	—	83	1	no	maximal, Sylow
80	C₂×C₈₃	#1834	1	1	yes	characteristic
81	C₂×C₈₃	#2159	1	2	yes	—
82	C₂×C₈₃	#2243	1	2	yes	—
83	C₂×C₈₃	#2244	1	1	yes	characteristic
84	C₂×C₈₃	#2245	1	1	yes	characteristic
85	C₂×#346	D₈₃	1	1	yes	characteristic
86	#1846	—	83	1	no	—
87	#1845	—	83	1	no	—
88	#1380	—	83	4	no	—
89	C₈₃	#12890	1	1	yes	characteristic, complemented, Sylow
90	C₂²×C₁₆	C₂×D₈₃	1	2	yes	—
91	C₂×#107	—	83	2	no	—
92	C₂×#107	—	83	2	no	—
93	C₂×#107	—	166	1	no	—
94	C₂×#106	—	166	1	no	—
95	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
96	#384	—	166	1	no	—
97	#382	—	166	1	no	—
98	#346	Dic₈₃	1	4	yes	complemented
99	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
100	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
101	C₂×C₄⋊C₄	—	332	2	no	—
102	C₂×C₄⋊C₄	—	166	2	no	—
103	C₂×C₄⋊C₄	—	166	2	no	—
104	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
105	C₂×C₁₆	—	2	2	no	—
106	C₂×C₁₆	—	2	2	no	—
107	C₂×C₁₆	C₂×Dic₈₃	1	4	yes	—
108	#107	—	332	2	no	—
109	#107	—	166	4	no	—
110	#107	—	166	4	no	—
111	#106	—	332	2	no	—
112	C₄×C₈	Dic₁₆₆	1	2	yes	—
113	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
114	#95	—	332	2	no	—
115	C₂²×C₄	—	332	2	no	—
116	C₂²×C₄	—	332	2	no	—
117	C₂²×C₄	D₆₆₄	1	2	yes	—
118	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
119	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
120	C₂×C₈	#13050579	1	2	yes	—
121	C₂×C₈	#13050579	1	2	yes	—
122	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
123	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
124	C₂×C₈	#13050590	1	2	yes	—
125	C₂×C₈	#13050590	1	2	yes	—
126	C₄⋊C₄	—	332	2	no	—
127	C₄⋊C₄	—	332	2	no	—
128	C₄⋊C₄	—	332	2	no	—
129	C₄⋊C₄	—	332	2	no	—
130	C₄⋊C₄	—	664	4	no	—
131	C₄²	C₄⋊Dic₈₃	1	2	yes	—
132	C₄²	Dic₃₃₂	1	2	yes	—
133	C₁₆	—	2	4	no	—
134	C₂³	C₂×D₆₆₄	1	1	yes	characteristic
135	C₂×C₄	—	2	2	no	—
136	C₂×C₄	—	2	2	no	—
137	C₂×C₄	—	664	4	no	—
138	C₂×C₄	—	664	2	no	—
139	C₂×C₄	—	664	2	no	—
140	C₂×C₄	—	664	2	no	—
141	C₂×C₄	—	664	2	no	—
142	C₂×C₄	#14581726	1	4	yes	—
143	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
144	C₂×C₄	#14581780	1	1	yes	characteristic
145	C₂×C₄	#14581780	1	1	yes	characteristic
146	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
147	C₂×C₄	C₂×#13050579	1	1	yes	characteristic
148	C₂×C₄	C₂×Dic₃₃₂	1	1	yes	characteristic
149	C₈	#14581724	1	4	yes	—
150	C₈	#14581751	1	2	yes	—
151	C₈	#14581784	1	2	yes	—
152	C₂²	C₂×#14581726	1	1	yes	characteristic
153	C₂²	#19128048	1	1	yes	characteristic
154	C₂²	#19128048	1	1	yes	characteristic
155	C₂²	#19128139	1	2	yes	—
156	C₂²	#19128144	1	2	yes	—
157	C₄	—	2	4	no	—
158	C₄	—	1328	2	no	—
159	C₄	—	1328	2	no	—
160	C₄	#19127567	1	1	yes	characteristic
161	C₄	C₂×#14581724	1	1	yes	characteristic
162	C₄	#19128052	1	1	yes	characteristic
163	C₄	#19128052	1	1	yes	characteristic
164	C₂	#24907591	1	1	yes	characteristic
165	C₂	#24909251	1	2	yes	—
166	C₂	#24909700	1	2	yes	—
167	C₂	#24909705	1	1	yes	characteristic
168	C₂	#24909709	1	1	yes	characteristic

	subgroup	G/H
center	C₂³	C₂×D₆₆₄
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₂×#346×C₈₃	C₂
socle, prime radical	C₂³×C₈₃	C₂×D₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂×#346	D₈₃
hypocenter	C₈₃	#12890
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ C₂×#346

Complements

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

N	H ≅ G/N	type
13. #346×C₈₃	C₄	semidirect
89. C₈₃	#12890	semidirect
98. #346	Dic₈₃	semidirect

Maximal subgroups

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

subgroup	index
#24907627	2
#24907623	2
#24905555	2
C₂×#346×C₈₃	2
#12890	83