Subgroups — #35320576

Order 21248.

438 conjugacy classes of subgroups.

The subgroup lattice has 186 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, complemented
2	#24907492	C₂	1	2	yes	maximal
3	#24905555	C₂	1	2	yes	maximal, complemented
4	C₂³×C₁₆×C₈₃	C₂	1	1	yes	maximal, characteristic
5	C₂×#14581726	C₂²	1	4	yes	—
6	#19127561	—	2	2	no	—
7	#19127529	—	2	4	no	—
8	#19127464	—	2	2	no	—
9	C₂³×C₈×C₈₃	C₂²	1	1	yes	characteristic
10	C₂²×C₁₆×C₈₃	—	2	4	no	—
11	C₂²×C₁₆×C₈₃	C₄	1	4	yes	complemented
12	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
13	C₂²×C₁₆×C₈₃	C₂²	1	1	yes	characteristic
14	C₂×C₄⋊Dic₈₃	—	2	4	no	—
15	C₂×C₂²⋊Dic₈₃	—	4	2	no	—
16	#14581726	—	2	4	no	—
17	#14581726	—	2	4	no	—
18	C₈₃⋊#95	—	4	2	no	—
19	C₂³×C₄×C₈₃	D₄	1	1	yes	characteristic
20	C₂²×C₈×C₈₃	—	2	2	no	—
21	C₂²×C₈×C₈₃	—	2	2	no	—
22	C₂²×C₈×C₈₃	C₂×C₄	1	2	yes	—
23	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
24	C₂²×C₈×C₈₃	Q₈	1	2	yes	—
25	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
26	C₂×C₁₆×C₈₃	—	2	2	no	—
27	C₂×C₁₆×C₈₃	—	2	8	no	—
28	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	—
29	C₂×C₁₆×C₈₃	C₂×C₄	1	2	yes	—
30	C₂×C₁₆×C₈₃	D₄	1	4	yes	—
31	C₂²×Dic₈₃	—	4	4	no	—
32	C₄⋊Dic₈₃	—	4	4	no	—
33	C₄⋊Dic₈₃	—	4	4	no	—
34	C₂²⋊Dic₈₃	—	8	4	no	—
35	C₂⁴×C₈₃	D₈	1	1	yes	characteristic
36	C₂²×C₄×C₈₃	—	2	2	no	—
37	C₂²×C₄×C₈₃	—	2	2	no	—
38	C₂²×C₄×C₈₃	C₄⋊C₄	1	2	yes	—
39	C₂²×C₄×C₈₃	D₈	1	1	yes	characteristic
40	C₂²×C₄×C₈₃	Q₁₆	1	2	yes	—
41	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
42	C₂×C₈×C₈₃	—	2	2	no	—
43	C₂×C₈×C₈₃	—	2	2	no	—
44	C₂×C₈×C₈₃	—	2	2	no	—
45	C₂×C₈×C₈₃	—	2	4	no	—
46	C₂×C₈×C₈₃	C₄⋊C₄	1	2	yes	—
47	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
48	C₂×C₈×C₈₃	C₂×D₄	1	2	yes	—
49	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
50	C₂×C₈×C₈₃	#41	1	2	yes	—
51	C₁₆×C₈₃	—	2	2	no	—
52	C₁₆×C₈₃	C₂²⋊C₄	1	4	yes	complemented
53	C₂³×C₈₃	—	2	2	no	—
54	C₂³×C₈₃	—	2	2	no	—
55	C₂³×C₈₃	#107	1	2	yes	—
56	C₂³×C₈₃	SD₃₂	1	4	yes	—
57	C₂³×C₈₃	C₂×D₈	1	1	yes	characteristic
58	C₂×C₄×C₈₃	—	2	2	no	—
59	C₂×C₄×C₈₃	—	2	2	no	—
60	C₂×C₄×C₈₃	—	2	2	no	—
61	C₂×C₄×C₈₃	—	2	4	no	—
62	C₂×C₄×C₈₃	#107	1	2	yes	—
63	C₂×C₄×C₈₃	C₂×C₄⋊C₄	1	1	yes	characteristic
64	C₂×C₄×C₈₃	#121	1	2	yes	—
65	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
66	C₂×C₄×C₈₃	#135	1	2	yes	—
67	C₂×Dic₈₃	—	8	4	no	—
68	C₂×Dic₈₃	—	8	4	no	—
69	C₂×Dic₈₃	—	8	4	no	—
70	C₈×C₈₃	—	2	2	no	—
71	C₈×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
72	C₈×C₈₃	#117	1	1	yes	characteristic
73	C₈×C₈₃	C₂²⋊Q₈	1	2	yes	—
74	C₂²×C₈₃	—	2	2	no	—
75	C₂²×C₈₃	—	2	2	no	—
76	C₂²×C₈₃	—	2	8	no	—
77	C₂²×C₈₃	#367	1	4	yes	complemented
78	C₂²×C₈₃	C₂×#107	1	1	yes	characteristic
79	C₂²×C₈₃	#466	1	2	yes	—
80	C₂²×C₈₃	C₂×SD₃₂	1	2	yes	—
81	C₂²×C₈₃	#508	1	2	yes	—
82	C₄×C₈₃	—	2	2	no	—
83	C₄×C₈₃	#380	1	1	yes	characteristic
84	C₄×C₈₃	#427	1	1	yes	characteristic
85	C₄×C₈₃	#467	1	2	yes	—
86	Dic₈₃	—	16	4	no	—
87	#12835	—	83	1	no	maximal, Sylow
88	C₂×C₈₃	—	2	4	no	—
89	C₂×C₈₃	#1808	1	1	yes	characteristic
90	C₂×C₈₃	C₂×#367	1	1	yes	characteristic
91	C₂×C₈₃	#2158	1	1	yes	characteristic
92	C₂×C₈₃	#2217	1	4	yes	—
93	C₂³×C₁₆	D₈₃	1	1	yes	characteristic
94	C₂×#367	—	83	2	no	—
95	#1808	—	83	2	no	—
96	#1380	—	83	2	no	—
97	C₈₃	#12835	1	1	yes	characteristic, complemented, Sylow
98	C₂³×C₈	C₂×D₈₃	1	1	yes	characteristic
99	C₂²×C₁₆	—	2	4	no	—
100	C₂²×C₁₆	Dic₈₃	1	4	yes	complemented
101	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
102	C₂²×C₁₆	C₂×D₈₃	1	1	yes	characteristic
103	C₂×#107	—	83	4	no	—
104	#380	—	166	2	no	—
105	#367	—	166	4	no	—
106	#340	—	166	2	no	—
107	C₂³×C₄	D₃₃₂	1	1	yes	characteristic
108	C₂²×C₈	—	2	2	no	—
109	C₂²×C₈	—	2	2	no	—
110	C₂²×C₈	Dic₁₆₆	1	2	yes	—
111	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
112	C₂²×C₈	C₂×Dic₈₃	1	2	yes	—
113	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
114	C₂×C₄⋊C₄	—	166	4	no	—
115	C₂×C₂²⋊C₄	—	332	2	no	—
116	C₂×C₁₆	—	2	2	no	—
117	C₂×C₁₆	—	2	8	no	—
118	C₂×C₁₆	C₂×Dic₈₃	1	2	yes	—
119	C₂×C₁₆	C₂×Dic₈₃	1	2	yes	—
120	C₂×C₁₆	C₂²⋊D₈₃	1	4	yes	—
121	#107	—	166	4	no	—
122	#107	—	166	4	no	—
123	#95	—	332	2	no	—
124	C₂⁴	D₆₆₄	1	1	yes	characteristic
125	C₂²×C₄	—	2	2	no	—
126	C₂²×C₄	—	2	2	no	—
127	C₂²×C₄	—	332	4	no	—
128	C₂²×C₄	C₄⋊Dic₈₃	1	2	yes	—
129	C₂²×C₄	D₆₆₄	1	1	yes	characteristic
130	C₂²×C₄	Dic₃₃₂	1	2	yes	—
131	C₂²×C₄	C₂×D₃₃₂	1	1	yes	characteristic
132	C₂×C₈	—	2	2	no	—
133	C₂×C₈	—	2	4	no	—
134	C₂×C₈	—	2	2	no	—
135	C₂×C₈	—	2	2	no	—
136	C₂×C₈	C₄⋊Dic₈₃	1	2	yes	—
137	C₂×C₈	C₂²×Dic₈₃	1	1	yes	characteristic
138	C₂×C₈	C₂×C₂²⋊D₈₃	1	2	yes	—
139	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
140	C₂×C₈	#13050590	1	2	yes	—
141	C₄⋊C₄	—	332	4	no	—
142	C₄⋊C₄	—	332	4	no	—
143	C₂²⋊C₄	—	664	4	no	—
144	C₁₆	—	2	2	no	—
145	C₁₆	C₂²⋊Dic₈₃	1	4	yes	complemented
146	C₂³	—	2	2	no	—
147	C₂³	—	2	2	no	—
148	C₂³	#14581726	1	2	yes	—
149	C₂³	#14581737	1	4	yes	—
150	C₂³	C₂×D₆₆₄	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	2	2	no	—
153	C₂×C₄	—	2	4	no	—
154	C₂×C₄	—	2	2	no	—
155	C₂×C₄	—	664	4	no	—
156	C₂×C₄	—	664	4	no	—
157	C₂×C₄	—	664	4	no	—
158	C₂×C₄	#14581726	1	2	yes	—
159	C₂×C₄	C₂×C₄⋊Dic₈₃	1	1	yes	characteristic
160	C₂×C₄	#14581766	1	2	yes	—
161	C₂×C₄	C₂×Dic₃₃₂	1	1	yes	characteristic
162	C₂×C₄	#14581814	1	2	yes	—
163	C₈	—	2	2	no	—
164	C₈	C₂×C₂²⋊Dic₈₃	1	1	yes	characteristic
165	C₈	#14581751	1	1	yes	characteristic
166	C₈	#14581771	1	2	yes	—
167	C₂²	—	2	2	no	—
168	C₂²	—	2	2	no	—
169	C₂²	—	2	8	no	—
170	C₂²	#19127529	1	4	yes	complemented
171	C₂²	C₂×#14581726	1	1	yes	characteristic
172	C₂²	#19127937	1	2	yes	—
173	C₂²	C₂×#14581737	1	2	yes	—
174	C₂²	#19128133	1	2	yes	—
175	C₄	—	2	2	no	—
176	C₄	—	1328	4	no	—
177	C₄	#19127561	1	1	yes	characteristic
178	C₄	#19127739	1	1	yes	characteristic
179	C₄	#19127942	1	2	yes	—
180	C₂	—	2	4	no	—
181	C₂	#24907492	1	1	yes	characteristic
182	C₂	C₂×#19127529	1	1	yes	characteristic
183	C₂	#24909247	1	1	yes	characteristic
184	C₂	#24909559	1	4	yes	—

	subgroup	G/H
center	C₂³	C₂×D₆₆₄
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₂³×C₁₆×C₈₃	C₂
socle	C₂³×C₈₃	C₂×D₈
prime radical	C₂⁴×C₈₃	D₈
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	C₂³×C₁₆	D₈₃
hypocenter	C₈₃	#12835
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ C₂³×C₈ ◁ C₂³×C₁₆

Complements

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

N	H ≅ G/N	type
1. C₂×#19127529	C₂	semidirect
3. #24905555	C₂	semidirect
11. C₂²×C₁₆×C₈₃	C₄	semidirect
52. C₁₆×C₈₃	C₂²⋊C₄	semidirect
77. C₂²×C₈₃	#367	semidirect
97. C₈₃	#12835	semidirect
100. C₂²×C₁₆	Dic₈₃	semidirect
145. C₁₆	C₂²⋊Dic₈₃	semidirect
170. C₂²	#19127529	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
#24907492	2
#24905555	2
C₂³×C₁₆×C₈₃	2
#12835	83