Subgroups — #35320849

Order 21248.

350 conjugacy classes of subgroups.

The subgroup lattice has 194 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	2	yes	maximal
2	C₂×#19127527	C₂	1	2	yes	maximal
3	#24907627	C₂	1	1	yes	maximal, characteristic
4	#24907623	C₂	1	1	yes	maximal, characteristic
5	C₂×#346×C₈₃	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581726	—	2	1	no	—
7	C₂×#14581726	C₂²	1	2	yes	—
8	C₂×#14581726	C₂²	1	1	yes	characteristic
9	C₂×#14581726	C₂²	1	1	yes	characteristic
10	C₂×#14581724	—	2	1	no	—
11	#19127571	—	2	1	no	—
12	#19127567	—	2	1	no	—
13	#19127529	—	2	4	no	—
14	#19127527	—	2	4	no	—
15	C₂²×C₁₆×C₈₃	C₂²	1	2	yes	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	#346×C₈₃	C₄	1	4	yes	complemented
18	C₂×C₄⋊Dic₈₃	—	2	1	no	—
19	C₂×C₄⋊Dic₈₃	—	2	1	no	—
20	C₂×C₄⋊Dic₈₃	—	2	2	no	—
21	C₂×C₄⋊Dic₈₃	—	4	2	no	—
22	#14581726	—	2	2	no	—
23	#14581726	—	2	2	no	—
24	#14581726	—	2	2	no	—
25	#14581726	—	2	2	no	—
26	#14581726	—	4	2	no	—
27	#14581724	—	4	2	no	—
28	C₈₃⋊#95	—	4	2	no	—
29	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
31	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
32	C₂×C₁₆×C₈₃	C₂×C₄	1	4	yes	—
33	C₂×C₁₆×C₈₃	D₄	1	4	yes	—
34	C₂×C₁₆×C₈₃	Q₈	1	4	yes	—
35	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
36	C₄×C₈×C₈₃	Q₈	1	2	yes	—
37	C₂²×Dic₈₃	—	4	1	no	—
38	C₂²×Dic₈₃	—	4	1	no	—
39	C₂²×Dic₈₃	—	4	2	no	—
40	C₄⋊Dic₈₃	—	4	1	no	—
41	C₄⋊Dic₈₃	—	4	2	no	—
42	C₄⋊Dic₈₃	—	4	1	no	—
43	C₄⋊Dic₈₃	—	4	1	no	—
44	C₄⋊Dic₈₃	—	4	1	no	—
45	C₄⋊Dic₈₃	—	4	2	no	—
46	C₄⋊Dic₈₃	—	8	4	no	—
47	C₂²×C₄×C₈₃	D₈	1	2	yes	—
48	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
49	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
50	C₂×C₈×C₈₃	D₈	1	2	yes	—
51	C₂×C₈×C₈₃	Q₁₆	1	2	yes	—
52	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
53	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
54	C₂×C₈×C₈₃	C₂×Q₈	1	2	yes	—
55	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
56	C₄²×C₈₃	C₄⋊C₄	1	2	yes	—
57	C₄²×C₈₃	Q₁₆	1	2	yes	—
58	C₁₆×C₈₃	C₄⋊C₄	1	8	yes	complemented
59	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
60	C₂×C₄×C₈₃	—	2	2	no	—
61	C₂×C₄×C₈₃	—	2	2	no	—
62	C₂×C₄×C₈₃	#107	1	4	yes	—
63	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₃	#127	1	1	yes	characteristic
65	C₂×C₄×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
66	C₂×C₄×C₈₃	C₂×D₈	1	1	yes	characteristic
67	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
68	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
69	C₂×Dic₈₃	—	8	1	no	—
70	C₂×Dic₈₃	—	8	2	no	—
71	C₂×Dic₈₃	—	8	1	no	—
72	C₂×Dic₈₃	—	8	4	no	—
73	C₂×Dic₈₃	—	8	2	no	—
74	C₂×Dic₈₃	—	8	2	no	—
75	C₈×C₈₃	#107	1	4	yes	—
76	C₈×C₈₃	C₂×C₄⋊C₄	1	2	yes	—
77	C₈×C₈₃	C₄⋊Q₈	1	2	yes	—
78	C₂²×C₈₃	C₂×#107	1	1	yes	characteristic
79	C₂²×C₈₃	#493	1	1	yes	characteristic
80	C₂²×C₈₃	#500	1	1	yes	characteristic
81	C₂²×C₈₃	#509	1	2	yes	—
82	C₂²×C₈₃	#510	1	2	yes	—
83	C₄×C₈₃	—	2	4	no	—
84	C₄×C₈₃	#384	1	1	yes	characteristic
85	C₄×C₈₃	C₂×#107	1	1	yes	characteristic
86	C₄×C₈₃	#494	1	1	yes	characteristic
87	C₄×C₈₃	#500	1	1	yes	characteristic
88	Dic₈₃	—	16	2	no	—
89	Dic₈₃	—	16	1	no	—
90	Dic₈₃	—	16	1	no	—
91	#12908	—	83	1	no	maximal, Sylow
92	C₂×C₈₃	#1845	1	1	yes	characteristic
93	C₂×C₈₃	#2159	1	2	yes	—
94	C₂×C₈₃	#2250	1	1	yes	characteristic
95	C₂×C₈₃	#2251	1	1	yes	characteristic
96	C₂×C₈₃	#2255	1	2	yes	—
97	C₂×#367	—	83	2	no	—
98	C₂×#366	—	83	2	no	—
99	C₂×#346	D₈₃	1	1	yes	characteristic
100	#1846	—	83	1	no	—
101	#1845	—	83	1	no	—
102	C₈₃	#12908	1	1	yes	characteristic, complemented, Sylow
103	C₂²×C₁₆	C₂×D₈₃	1	2	yes	—
104	C₂×#107	—	83	2	no	—
105	C₂×#107	—	83	1	no	—
106	C₂×#107	—	83	1	no	—
107	C₂×#107	—	166	1	no	—
108	C₂×#106	—	166	1	no	—
109	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
110	#384	—	166	1	no	—
111	#382	—	166	1	no	—
112	#367	—	166	4	no	—
113	#366	—	166	4	no	—
114	#346	Dic₈₃	1	4	yes	complemented
115	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
116	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
117	C₂×C₄⋊C₄	—	332	2	no	—
118	C₂×C₄⋊C₄	—	166	1	no	—
119	C₂×C₄⋊C₄	—	166	1	no	—
120	C₂×C₄⋊C₄	—	166	2	no	—
121	C₂×C₄²	D₃₃₂	1	1	yes	characteristic
122	C₂×C₁₆	Dic₁₆₆	1	4	yes	—
123	C₂×C₁₆	D₃₃₂	1	4	yes	—
124	C₂×C₁₆	C₂×Dic₈₃	1	4	yes	—
125	#107	—	332	2	no	—
126	#107	—	166	2	no	—
127	#107	—	166	2	no	—
128	#107	—	166	2	no	—
129	#107	—	166	2	no	—
130	#106	—	332	2	no	—
131	C₄×C₈	Dic₁₆₆	1	2	yes	—
132	C₄×C₈	C₂×Dic₈₃	1	2	yes	—
133	#95	—	332	2	no	—
134	C₂²×C₄	—	332	1	no	—
135	C₂²×C₄	—	332	1	no	—
136	C₂²×C₄	—	332	2	no	—
137	C₂²×C₄	D₆₆₄	1	2	yes	—
138	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
139	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
140	C₂×C₈	D₆₆₄	1	2	yes	—
141	C₂×C₈	Dic₃₃₂	1	2	yes	—
142	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
143	C₂×C₈	C₂×D₃₃₂	1	2	yes	—
144	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
145	C₂×C₈	C₂×Dic₁₆₆	1	2	yes	—
146	C₄⋊C₄	—	332	1	no	—
147	C₄⋊C₄	—	332	2	no	—
148	C₄⋊C₄	—	332	1	no	—
149	C₄⋊C₄	—	332	1	no	—
150	C₄⋊C₄	—	332	1	no	—
151	C₄⋊C₄	—	332	2	no	—
152	C₄⋊C₄	—	664	4	no	—
153	C₄²	C₄⋊Dic₈₃	1	2	yes	—
154	C₄²	Dic₃₃₂	1	2	yes	—
155	C₁₆	C₄⋊Dic₈₃	1	8	yes	complemented
156	C₂³	C₂×D₆₆₄	1	1	yes	characteristic
157	C₂×C₄	—	2	2	no	—
158	C₂×C₄	—	2	2	no	—
159	C₂×C₄	—	664	1	no	—
160	C₂×C₄	—	664	2	no	—
161	C₂×C₄	—	664	1	no	—
162	C₂×C₄	—	664	4	no	—
163	C₂×C₄	—	664	2	no	—
164	C₂×C₄	—	664	2	no	—
165	C₂×C₄	#14581726	1	4	yes	—
166	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
167	C₂×C₄	#14581789	1	1	yes	characteristic
168	C₂×C₄	#14581792	1	1	yes	characteristic
169	C₂×C₄	C₂×D₆₆₄	1	1	yes	characteristic
170	C₂×C₄	C₂×Dic₃₃₂	1	1	yes	characteristic
171	C₂×C₄	C₂×Dic₃₃₂	1	1	yes	characteristic
172	C₈	#14581726	1	4	yes	—
173	C₈	C₂×C₄⋊Dic₈₃	1	2	yes	—
174	C₈	#14581792	1	2	yes	—
175	C₂²	C₂×#14581726	1	1	yes	characteristic
176	C₂²	#19128078	1	1	yes	characteristic
177	C₂²	#19128106	1	1	yes	characteristic
178	C₂²	#19128139	1	2	yes	—
179	C₂²	#19128144	1	2	yes	—
180	C₄	—	2	4	no	—
181	C₄	—	1328	2	no	—
182	C₄	—	1328	1	no	—
183	C₄	—	1328	1	no	—
184	C₄	#19127571	1	1	yes	characteristic
185	C₄	C₂×#14581726	1	1	yes	characteristic
186	C₄	#19128081	1	1	yes	characteristic
187	C₄	#19128106	1	1	yes	characteristic
188	C₂	#24907623	1	1	yes	characteristic
189	C₂	#24909251	1	2	yes	—
190	C₂	#24909729	1	1	yes	characteristic
191	C₂	#24909734	1	1	yes	characteristic
192	C₂	#24909749	1	2	yes	—

	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	#12908
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
17. #346×C₈₃	C₄	semidirect
58. C₁₆×C₈₃	C₄⋊C₄	semidirect
102. C₈₃	#12908	semidirect
114. #346	Dic₈₃	semidirect
155. C₁₆	C₄⋊Dic₈₃	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×#19127527	2
#24907627	2
#24907623	2
C₂×#346×C₈₃	2
#12908	83