Subgroups — #35401737

Order 21248.

1502 conjugacy classes of subgroups.

The subgroup lattice has 198 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19127653	C₂	1	3	yes	maximal
2	C₂×C₄×#13050573	C₂	1	3	yes	maximal
3	#376×D₈₃	C₂	1	1	yes	maximal, characteristic
4	#24907151	C₂	1	3	yes	maximal, complemented
5	#24907150	C₂	1	1	yes	maximal, characteristic, complemented
6	#24907149	C₂	1	3	yes	maximal, complemented
7	C₈₃×#1726	C₂	1	1	yes	maximal, characteristic, complemented
8	C₂²×#13050573	C₂²	1	1	yes	characteristic
9	C₂²×#13050573	C₂²	1	3	yes	—
10	C₂×C₄²×D₈₃	C₂²	1	3	yes	—
11	#19127653	C₄	1	24	yes	complemented
12	C₄×#13050573	C₄	1	24	yes	complemented
13	C₂×#14581703	C₂²	1	3	yes	—
14	C₂×#14581702	C₂²	1	3	yes	—
15	C₂×#14581702	C₂²	1	3	yes	—
16	C₂×C₄×C₈₃⋊C₈	C₂²	1	3	yes	—
17	C₂×C₈×Dic₈₃	C₂²	1	6	yes	—
18	#19127547	C₂²	1	3	yes	—
19	#19127546	C₄	1	4	yes	—
20	#95×D₈₃	C₄	1	4	yes	—
21	C₂×#97×C₈₃	C₂²	1	3	yes	—
22	C₂×C₄×C₈×C₈₃	C₂²	1	3	yes	—
23	#376×C₈₃	C₂²	1	1	yes	characteristic
24	C₂²×C₄×D₈₃	C₂×C₄	1	6	yes	—
25	C₂²×C₄×D₈₃	C₂³	1	1	yes	characteristic
26	C₂×#13050573	C₂×C₄	1	12	yes	—
27	C₂×#13050573	C₂×C₄	1	12	yes	—
28	C₂×#13050573	C₂×C₄	1	24	yes	—
29	C₂²×C₈₃⋊C₈	C₂³	1	3	yes	—
30	C₂²×C₈₃⋊C₈	C₂³	1	1	yes	characteristic
31	C₂×C₄×Dic₈₃	C₂×C₄	1	6	yes	—
32	C₂×C₄×Dic₈₃	C₂³	1	3	yes	—
33	C₄²×D₈₃	C₂×C₄	1	12	yes	—
34	C₄²×D₈₃	C₂×C₄	1	12	yes	—
35	#14581703	C₂×C₄	1	12	yes	complemented
36	#14581702	C₂×C₄	1	12	yes	complemented
37	#14581702	C₂×C₄	1	12	yes	complemented
38	C₄×C₈₃⋊C₈	C₂×C₄	1	12	yes	complemented
39	C₈×Dic₈₃	C₂×C₄	1	24	yes	complemented
40	C₈₃⋊#95	C₂×C₄	1	12	yes	—
41	C₂²×C₈×C₈₃	C₂³	1	3	yes	—
42	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
43	C₂×C₄²×C₈₃	C₂³	1	3	yes	—
44	#97×C₈₃	C₂×C₄	1	12	yes	complemented
45	C₄×C₈×C₈₃	C₂×C₄	1	12	yes	complemented
46	#95×C₈₃	C₂×C₄	1	4	yes	—
47	C₂³×D₈₃	C₂²×C₄	1	1	yes	characteristic
48	C₂²×Dic₈₃	C₂²×C₄	1	6	yes	—
49	C₂²×Dic₈₃	C₂²×C₄	1	1	yes	characteristic
50	C₂×C₄×D₈₃	—	4	12	no	—
51	C₂×C₄×D₈₃	C₄²	1	24	yes	—
52	C₂×C₄×D₈₃	C₂²×C₄	1	6	yes	—
53	C₂×C₄×D₈₃	C₂²×C₄	1	6	yes	—
54	#13050573	C₄²	1	16	yes	—
55	#13050573	C₄²	1	48	yes	—
56	C₂×C₈₃⋊C₈	C₂²×C₄	1	12	yes	—
57	C₂×C₈₃⋊C₈	C₂²×C₄	1	6	yes	—
58	C₂×C₈₃⋊C₈	C₂²×C₄	1	6	yes	—
59	C₄×Dic₈₃	C₄²	1	24	yes	—
60	C₄×Dic₈₃	C₂²×C₄	1	6	yes	—
61	C₄×Dic₈₃	C₂²×C₄	1	6	yes	—
62	C₂²×C₄×C₈₃	C₂²×C₄	1	6	yes	—
63	C₂²×C₄×C₈₃	C₂⁴	1	1	yes	characteristic
64	C₂×C₈×C₈₃	C₂²×C₄	1	6	yes	—
65	C₂×C₈×C₈₃	C₂²×C₄	1	6	yes	—
66	C₂×C₈×C₈₃	C₂²×C₄	1	12	yes	—
67	C₄²×C₈₃	C₂²×C₄	1	6	yes	—
68	C₄²×C₈₃	C₂²×C₄	1	6	yes	—
69	C₂³×C₈₃	C₂³×C₄	1	1	yes	characteristic
70	C₂²×D₈₃	—	2	4	no	—
71	C₂²×D₈₃	C₂×C₄²	1	6	yes	—
72	C₂×C₄×C₈₃	—	2	12	no	—
73	C₂×C₄×C₈₃	C₂×C₄²	1	12	yes	—
74	C₂×C₄×C₈₃	C₂³×C₄	1	3	yes	—
75	C₂×C₄×C₈₃	C₂³×C₄	1	3	yes	—
76	C₂×Dic₈₃	—	2	12	no	—
77	C₂×Dic₈₃	C₂×C₄²	1	12	yes	—
78	C₂×Dic₈₃	C₂×C₄²	1	6	yes	—
79	C₄×D₈₃	—	4	24	no	—
80	C₄×D₈₃	C₂×C₄²	1	4	yes	—
81	C₄×D₈₃	C₂×C₄²	1	12	yes	—
82	C₈×C₈₃	C₂×C₄²	1	4	yes	—
83	C₈×C₈₃	C₂×C₄²	1	12	yes	—
84	C₈₃⋊C₈	C₂×C₄²	1	12	yes	—
85	C₈₃⋊C₈	C₂×C₄²	1	4	yes	—
86	C₂²×C₈₃	C₂²×C₄²	1	3	yes	—
87	C₂²×C₈₃	#568	1	4	yes	—
88	C₂×D₈₃	—	2	12	no	—
89	C₂×D₈₃	C₄³	1	4	yes	—
90	C₄×C₈₃	—	2	12	no	—
91	C₄×C₈₃	C₂²×C₄²	1	3	yes	—
92	C₄×C₈₃	C₂²×C₄²	1	1	yes	characteristic
93	Dic₈₃	—	2	12	no	—
94	Dic₈₃	C₄³	1	4	yes	—
95	#20329	—	83	1	no	maximal, Sylow
96	C₂×C₈₃	C₂×C₄³	1	1	yes	characteristic
97	C₂×C₈₃	#2873	1	6	yes	—
98	D₈₃	—	2	4	no	—
99	C₂×C₄×M₁₆	—	83	3	no	—
100	C₂×C₄×M₁₆	—	83	3	no	—
101	C₂×#376	—	83	1	no	—
102	#1726	—	83	3	no	—
103	#1726	—	83	3	no	—
104	#1726	—	83	1	no	—
105	#1726	D₈₃	1	1	yes	characteristic, complemented
106	C₈₃	#20329	1	1	yes	characteristic, complemented, Sylow
107	C₂²×M₁₆	—	83	3	no	—
108	C₂²×M₁₆	—	83	1	no	—
109	C₂²×C₄²	—	83	3	no	—
110	C₄×M₁₆	—	83	24	no	—
111	C₄×M₁₆	—	83	24	no	—
112	C₂×#97	—	83	3	no	—
113	C₂×#97	—	83	3	no	—
114	C₂×#97	—	83	3	no	—
115	C₂×#97	C₂×D₈₃	1	3	yes	—
116	C₂×C₄×C₈	—	83	6	no	—
117	C₂×C₄×C₈	—	83	3	no	—
118	C₂×C₄×C₈	C₂×D₈₃	1	3	yes	—
119	#376	—	83	3	no	—
120	#376	—	83	4	no	—
121	#376	C₂×D₈₃	1	1	yes	characteristic
122	C₂×#95	—	83	4	no	—
123	C₂³×C₄	—	83	6	no	—
124	C₂³×C₄	—	83	1	no	—
125	C₂×M₁₆	—	83	24	no	—
126	C₂×M₁₆	—	83	12	no	—
127	C₂×M₁₆	—	83	12	no	—
128	C₂²×C₈	—	83	3	no	—
129	C₂²×C₈	—	83	1	no	—
130	C₂²×C₈	C₂²×D₈₃	1	3	yes	—
131	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
132	C₂×C₄²	—	83	6	no	—
133	C₂×C₄²	—	83	3	no	—
134	C₂×C₄²	—	83	12	no	—
135	C₂×C₄²	—	83	12	no	—
136	C₂×C₄²	C₂²×D₈₃	1	3	yes	—
137	#97	—	83	12	no	—
138	#97	—	83	12	no	—
139	#97	—	83	12	no	—
140	#97	C₄×D₈₃	1	12	yes	complemented
141	C₄×C₈	—	83	24	no	—
142	C₄×C₈	—	83	12	no	—
143	C₄×C₈	C₄×D₈₃	1	12	yes	complemented
144	#95	—	83	12	no	—
145	#95	C₄×D₈₃	1	4	yes	—
146	C₂⁴	—	83	1	no	—
147	C₂²×C₄	—	332	12	no	—
148	C₂²×C₄	—	83	24	no	—
149	C₂²×C₄	—	83	1	no	—
150	C₂²×C₄	—	83	6	no	—
151	C₂²×C₄	—	83	6	no	—
152	C₂²×C₄	—	83	6	no	—
153	C₂²×C₄	C₂×C₄×D₈₃	1	6	yes	—
154	C₂²×C₄	C₂³×D₈₃	1	1	yes	characteristic
155	M₁₆	—	83	48	no	—
156	M₁₆	—	83	16	no	—
157	C₂×C₈	—	83	12	no	—
158	C₂×C₈	—	83	6	no	—
159	C₂×C₈	—	83	6	no	—
160	C₂×C₈	C₂×C₄×D₈₃	1	12	yes	—
161	C₂×C₈	C₂×C₄×D₈₃	1	6	yes	—
162	C₂×C₈	C₂×C₄×D₈₃	1	6	yes	—
163	C₄²	—	83	6	no	—
164	C₄²	—	83	6	no	—
165	C₄²	—	83	24	no	—
166	C₄²	C₂×C₄×D₈₃	1	6	yes	—
167	C₄²	C₂×C₄×D₈₃	1	6	yes	—
168	C₂³	—	83	6	no	—
169	C₂³	—	166	4	no	—
170	C₂³	C₂²×C₄×D₈₃	1	1	yes	characteristic
171	C₂×C₄	—	2	12	no	—
172	C₂×C₄	—	332	24	no	—
173	C₂×C₄	—	83	6	no	—
174	C₂×C₄	—	83	12	no	—
175	C₂×C₄	—	83	4	no	—
176	C₂×C₄	—	83	12	no	—
177	C₂×C₄	—	166	12	no	—
178	C₂×C₄	C₄²×D₈₃	1	12	yes	—
179	C₂×C₄	C₂²×C₄×D₈₃	1	3	yes	—
180	C₂×C₄	C₂²×C₄×D₈₃	1	3	yes	—
181	C₈	—	83	12	no	—
182	C₈	—	83	4	no	—
183	C₈	C₄²×D₈₃	1	12	yes	—
184	C₈	C₄²×D₈₃	1	4	yes	—
185	C₂²	—	83	4	no	—
186	C₂²	—	166	12	no	—
187	C₂²	C₂×C₄²×D₈₃	1	3	yes	—
188	C₂²	#19128466	1	4	yes	—
189	C₄	—	2	12	no	—
190	C₄	—	83	4	no	—
191	C₄	—	166	12	no	—
192	C₄	C₂×C₄²×D₈₃	1	3	yes	—
193	C₄	C₂×C₄²×D₈₃	1	1	yes	characteristic
194	C₂	—	166	4	no	—
195	C₂	C₄³×D₈₃	1	1	yes	characteristic
196	C₂	#24915879	1	6	yes	—

	subgroup	G/H
center, Frattini	C₂²×C₄	C₂³×D₈₃
commutator	C₂×C₈₃	C₂×C₄³
Fitting	C₈₃×#1726	C₂
socle	C₂³×C₈₃	C₂³×C₄
prime radical	C₂³×D₈₃	C₂²×C₄
cosocle	C₂²×C₄×C₈₃	C₂⁴
hypercenter	#1726	D₈₃
hypocenter	C₈₃	#20329
LCS	G ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂²×C₄ ◁ #1726

Complements

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

N	H ≅ G/N	type
4. #24907151	C₂	semidirect
5. #24907150	C₂	semidirect
6. #24907149	C₂	semidirect
7. C₈₃×#1726	C₂	semidirect
11. #19127653	C₄	semidirect
12. C₄×#13050573	C₄	semidirect
35. #14581703	C₂×C₄	semidirect
36. #14581702	C₂×C₄	semidirect
37. #14581702	C₂×C₄	semidirect
38. C₄×C₈₃⋊C₈	C₂×C₄	semidirect
39. C₈×Dic₈₃	C₂×C₄	semidirect
44. #97×C₈₃	C₂×C₄	semidirect
45. C₄×C₈×C₈₃	C₂×C₄	semidirect
105. #1726	D₈₃	semidirect
106. C₈₃	#20329	semidirect
140. #97	C₄×D₈₃	semidirect
143. C₄×C₈	C₄×D₈₃	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19127653	2
C₂×C₄×#13050573	2
#376×D₈₃	2
#24907151	2
#24907150	2
#24907149	2
C₈₃×#1726	2
#20329	83