Subgroups — #35291600

Order 21248.

222 conjugacy classes of subgroups.

The subgroup lattice has 190 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×C₄⋊(C₈₃⋊C₁₆)	C₂	1	1	yes	maximal, characteristic
2	#24907635	C₂	1	1	yes	maximal, characteristic
3	C₈₃×#1564	C₂	1	1	yes	maximal, characteristic
4	C₂²×C₈₃⋊C₁₆	—	2	1	no	—
5	C₂×#14581728	—	2	1	no	—
6	C₂×#14581728	C₄	1	2	yes	—
7	#19127725	—	2	1	no	—
8	C₄⋊(C₈₃⋊C₁₆)	—	2	1	no	—
9	C₄⋊(C₈₃⋊C₁₆)	C₄	1	2	yes	—
10	#431×C₈₃	—	2	1	no	—
11	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
12	#335×C₈₃	C₄	1	2	yes	—
13	#334×C₈₃	—	2	1	no	—
14	C₂×#13050574	—	2	2	no	—
15	C₂×C₈₃⋊C₁₆	—	2	2	no	—
16	C₂×C₈₃⋊C₁₆	—	4	1	no	—
17	C₂×C₈₃⋊C₁₆	—	4	1	no	—
18	#14581728	—	4	2	no	—
19	#14581728	—	4	1	no	—
20	#14581728	—	4	1	no	—
21	C₄⋊(C₈₃⋊C₈)	—	4	1	no	—
22	C₄⋊(C₈₃⋊C₈)	—	4	1	no	—
23	C₂²×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
24	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
25	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
26	C₄⋊C₈×C₈₃	—	2	2	no	—
27	C₂²⋊C₈×C₈₃	—	4	1	no	—
28	C₄×C₈×C₈₃	—	4	1	no	—
29	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
30	C₄×C₈×C₈₃	C₂×C₄	1	1	yes	characteristic
31	C₄×C₈×C₈₃	D₄	1	1	yes	characteristic
32	C₄×C₈×C₈₃	Q₈	1	1	yes	characteristic
33	#13050574	—	4	2	no	—
34	#13050574	—	4	2	no	—
35	C₂×C₈₃⋊C₈	—	4	2	no	—
36	C₈₃⋊C₁₆	—	4	2	no	—
37	C₂²×C₄×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
38	C₂²×C₄×C₈₃	SD₁₆	1	1	yes	characteristic
39	C₂²×C₄×C₈₃	Q₁₆	1	1	yes	characteristic
40	C₂×C₈×C₈₃	—	2	1	no	—
41	C₂×C₈×C₈₃	—	2	1	no	—
42	C₂×C₈×C₈₃	—	4	2	no	—
43	C₂×C₈×C₈₃	C₄²	1	1	yes	characteristic
44	C₂×C₈×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
45	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
46	C₂×C₈×C₈₃	C₄⋊C₄	1	1	yes	characteristic
47	C₂×C₈×C₈₃	D₈	1	1	yes	characteristic
48	C₂×C₈×C₈₃	SD₁₆	1	1	yes	characteristic
49	C₂×C₈×C₈₃	SD₁₆	1	1	yes	characteristic
50	C₂×C₈×C₈₃	Q₁₆	1	1	yes	characteristic
51	C₄²×C₈₃	C₂²⋊C₄	1	1	yes	characteristic
52	C₄²×C₈₃	C₄⋊C₄	1	1	yes	characteristic
53	C₄²×C₈₃	D₈	1	1	yes	characteristic
54	C₄²×C₈₃	SD₁₆	1	1	yes	characteristic
55	C₂³×C₈₃	#103	1	1	yes	characteristic
56	C₂×C₄×C₈₃	—	2	2	no	—
57	C₂×C₄×C₈₃	—	4	1	no	—
58	C₂×C₄×C₈₃	#95	1	1	yes	characteristic
59	C₂×C₄×C₈₃	#100	1	1	yes	characteristic
60	C₂×C₄×C₈₃	#101	1	1	yes	characteristic
61	C₂×C₄×C₈₃	#102	1	1	yes	characteristic
62	C₂×C₄×C₈₃	#102	1	1	yes	characteristic
63	C₂×C₄×C₈₃	#102	1	1	yes	characteristic
64	C₂×C₄×C₈₃	#103	1	1	yes	characteristic
65	C₂×C₄×C₈₃	#103	1	1	yes	characteristic
66	C₂×C₄×C₈₃	#106	1	1	yes	characteristic
67	C₂×C₄×C₈₃	#107	1	1	yes	characteristic
68	C₈×C₈₃	—	2	1	no	—
69	C₈×C₈₃	—	2	1	no	—
70	C₈×C₈₃	—	8	2	no	—
71	C₈×C₈₃	#102	1	1	yes	characteristic
72	C₈×C₈₃	#103	1	1	yes	characteristic
73	C₈×C₈₃	#106	1	1	yes	characteristic
74	C₈×C₈₃	#107	1	1	yes	characteristic
75	C₈₃⋊C₈	—	8	2	no	—
76	C₂²×C₈₃	—	2	1	no	—
77	C₂²×C₈₃	—	2	1	no	—
78	C₂²×C₈₃	#332	1	1	yes	characteristic
79	C₂²×C₈₃	#333	1	1	yes	characteristic
80	C₂²×C₈₃	#340	1	1	yes	characteristic
81	C₄×C₈₃	—	2	1	no	—
82	C₄×C₈₃	—	2	1	no	—
83	C₄×C₈₃	—	2	1	no	—
84	C₄×C₈₃	#331	1	1	yes	characteristic
85	C₄×C₈₃	#332	1	1	yes	characteristic
86	C₄×C₈₃	#340	1	1	yes	characteristic
87	C₄×C₈₃	#343	1	1	yes	characteristic
88	C₄×C₈₃	#359	1	2	yes	—
89	#7277	—	83	1	no	maximal, Sylow
90	C₂×C₈₃	—	2	1	no	—
91	C₂×C₈₃	—	2	1	no	—
92	C₂×C₈₃	#1296	1	1	yes	characteristic
93	C₂×C₈₃	#1382	1	1	yes	characteristic
94	C₂×C₈₃	#1388	1	1	yes	characteristic
95	C₂×C₄⋊C₁₆	—	83	1	no	—
96	#1848	—	83	1	no	—
97	#1564	D₈₃	1	1	yes	characteristic
98	C₈₃	#7277	1	1	yes	characteristic, complemented, Sylow
99	C₂²×C₁₆	—	166	1	no	—
100	#431	—	2	1	no	—
101	C₂×#108	—	83	2	no	—
102	C₂×#108	—	166	1	no	—
103	#423	—	166	1	no	—
104	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
105	C₄⋊C₁₆	—	83	2	no	—
106	C₄⋊C₁₆	—	166	1	no	—
107	#335	Dic₈₃	1	2	yes	—
108	#334	—	2	1	no	—
109	C₂×M₁₆	—	166	2	no	—
110	C₂²×C₈	C₄×D₈₃	1	1	yes	characteristic
111	C₂²×C₈	D₃₃₂	1	1	yes	characteristic
112	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
113	C₂×C₁₆	—	332	1	no	—
114	C₂×C₁₆	—	332	1	no	—
115	C₂×C₁₆	—	166	2	no	—
116	#108	—	332	2	no	—
117	#108	—	332	1	no	—
118	#108	—	332	1	no	—
119	C₄⋊C₈	—	2	2	no	—
120	C₄⋊C₈	—	332	1	no	—
121	C₄⋊C₈	—	332	1	no	—
122	C₂²⋊C₈	—	4	1	no	—
123	C₄×C₈	—	4	1	no	—
124	C₄×C₈	Dic₁₆₆	1	1	yes	characteristic
125	C₄×C₈	C₄×D₈₃	1	1	yes	characteristic
126	C₄×C₈	C₂×Dic₈₃	1	1	yes	characteristic
127	C₄×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
128	C₂²×C₄	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
129	C₂²×C₄	#13050578	1	1	yes	characteristic
130	C₂²×C₄	#13050580	1	1	yes	characteristic
131	M₁₆	—	332	2	no	—
132	M₁₆	—	332	2	no	—
133	C₂×C₈	—	2	1	no	—
134	C₂×C₈	—	2	1	no	—
135	C₂×C₈	—	4	2	no	—
136	C₂×C₈	—	332	2	no	—
137	C₂×C₈	C₄×Dic₈₃	1	1	yes	characteristic
138	C₂×C₈	(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
139	C₂×C₈	Dic₈₃⋊C₄	1	1	yes	characteristic
140	C₂×C₈	C₄⋊Dic₈₃	1	1	yes	characteristic
141	C₂×C₈	D₆₆₄	1	1	yes	characteristic
142	C₂×C₈	#13050579	1	1	yes	characteristic
143	C₂×C₈	#13050579	1	1	yes	characteristic
144	C₂×C₈	Dic₃₃₂	1	1	yes	characteristic
145	C₄²	C₂²⋊Dic₈₃	1	1	yes	characteristic
146	C₄²	Dic₈₃⋊C₄	1	1	yes	characteristic
147	C₄²	#13050575	1	1	yes	characteristic
148	C₄²	#13050577	1	1	yes	characteristic
149	C₁₆	—	332	2	no	—
150	C₂³	#14581715	1	1	yes	characteristic
151	C₂×C₄	—	2	2	no	—
152	C₂×C₄	—	4	1	no	—
153	C₂×C₄	C₈₃⋊#95	1	1	yes	characteristic
154	C₂×C₄	#14581708	1	1	yes	characteristic
155	C₂×C₄	#14581710	1	1	yes	characteristic
156	C₂×C₄	#14581712	1	1	yes	characteristic
157	C₂×C₄	#14581713	1	1	yes	characteristic
158	C₂×C₄	#14581714	1	1	yes	characteristic
159	C₂×C₄	#14581716	1	1	yes	characteristic
160	C₂×C₄	#14581717	1	1	yes	characteristic
161	C₂×C₄	#14581723	1	1	yes	characteristic
162	C₂×C₄	#14581725	1	1	yes	characteristic
163	C₈	—	2	1	no	—
164	C₈	—	2	1	no	—
165	C₈	—	8	2	no	—
166	C₈	—	664	2	no	—
167	C₈	#14581714	1	1	yes	characteristic
168	C₈	#14581717	1	1	yes	characteristic
169	C₈	#14581724	1	1	yes	characteristic
170	C₈	#14581726	1	1	yes	characteristic
171	C₂²	—	2	1	no	—
172	C₂²	—	2	1	no	—
173	C₂²	#19127446	1	1	yes	characteristic
174	C₂²	#19127449	1	1	yes	characteristic
175	C₂²	#19127463	1	1	yes	characteristic
176	C₄	—	2	1	no	—
177	C₄	—	2	1	no	—
178	C₄	—	2	1	no	—
179	C₄	#19127444	1	1	yes	characteristic
180	C₄	#19127448	1	1	yes	characteristic
181	C₄	#19127464	1	1	yes	characteristic
182	C₄	#19127469	1	1	yes	characteristic
183	C₄	#19127508	1	2	yes	—
184	C₂	—	2	1	no	—
185	C₂	—	2	1	no	—
186	C₂	#24905347	1	1	yes	characteristic
187	C₂	#24905558	1	1	yes	characteristic
188	C₂	#24905576	1	1	yes	characteristic

	subgroup	G/H
center	C₂×C₄	#14581712
commutator	C₂×C₈×C₈₃	C₄²
Frattini	C₂×C₄×C₈	C₂×D₈₃
Fitting	C₈₃×#1564	C₂
socle	C₂²×C₈₃	#340
prime radical	C₂³×C₈₃	#103
cosocle	C₂×C₄×C₈×C₈₃	C₂²
hypercenter	#1564	D₈₃
hypocenter	C₈₃	#7277
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₂×C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ #1564

Complements

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

N	H ≅ G/N	type
98. C₈₃	#7277	semidirect

Maximal subgroups

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

subgroup	index
C₂×C₄⋊(C₈₃⋊C₁₆)	2
#24907635	2
C₈₃×#1564	2
#7277	83