Subgroups — #35322968

Order 21248.

334 conjugacy classes of subgroups.

The subgroup lattice has 202 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#	subgroup	G/H	class size	multiplicity	normal?	properties
1	C₂×#19128103	C₂	1	1	yes	maximal, characteristic
2	C₂×#19127522	C₂	1	1	yes	maximal, characteristic
3	#24905568	C₂	1	2	yes	maximal
4	#24905563	C₂	1	2	yes	maximal
5	C₈₃×#1772	C₂	1	1	yes	maximal, characteristic
6	C₂×#14581784	—	2	1	no	—
7	C₂×#14581732	C₂²	1	2	yes	—
8	#19128103	—	2	2	no	—
9	#19128103	C₄	1	4	yes	complemented
10	C₂×#14581726	—	2	1	no	—
11	C₂×#14581724	C₂²	1	2	yes	—
12	#19127522	—	2	2	no	—
13	C₂×#108×C₈₃	C₂²	1	1	yes	characteristic
14	#427×C₈₃	C₂²	1	1	yes	characteristic
15	#405×C₈₃	—	2	1	no	—
16	C₂×C₄×C₈×C₈₃	C₂²	1	1	yes	characteristic
17	#339×C₈₃	—	2	1	no	—
18	#14581784	—	2	4	no	—
19	#14581784	—	4	2	no	—
20	C₂×C₄⋊Dic₈₃	—	2	2	no	—
21	C₂×Dic₈₃⋊C₄	—	4	2	no	—
22	#14581732	—	2	2	no	—
23	#14581732	—	2	2	no	—
24	C₂×C₈₃⋊C₁₆	—	2	2	no	—
25	#14581726	—	2	4	no	—
26	#14581724	—	2	2	no	—
27	#14581724	C₂×C₄	1	4	yes	—
28	C₂×M₁₆×C₈₃	—	2	1	no	—
29	C₂²×C₈×C₈₃	D₄	1	1	yes	characteristic
30	C₂²×C₈×C₈₃	C₂³	1	1	yes	characteristic
31	#117×C₈₃	—	2	1	no	—
32	C₂×C₄²×C₈₃	D₄	1	1	yes	characteristic
33	#108×C₈₃	—	2	1	no	—
34	#108×C₈₃	Q₈	1	2	yes	—
35	#107×C₈₃	Q₈	1	2	yes	—
36	#106×C₈₃	—	2	1	no	—
37	#97×C₈₃	—	4	1	no	—
38	C₄×C₈×C₈₃	—	4	1	no	—
39	C₄×C₈×C₈₃	C₂×C₄	1	2	yes	—
40	C₄×C₈×C₈₃	D₄	1	2	yes	—
41	C₂²×Dic₈₃	—	4	2	no	—
42	C₄⋊Dic₈₃	—	2	4	no	—
43	C₄⋊Dic₈₃	—	4	2	no	—
44	Dic₈₃⋊C₄	—	4	8	no	—
45	C₈₃⋊C₁₆	—	4	2	no	—
46	C₂²×C₄×C₈₃	—	2	1	no	—
47	C₂²×C₄×C₈₃	C₂×D₄	1	1	yes	characteristic
48	M₁₆×C₈₃	—	2	2	no	—
49	M₁₆×C₈₃	—	4	1	no	—
50	C₂×C₈×C₈₃	—	4	1	no	—
51	C₂×C₈×C₈₃	C₂²⋊C₄	1	2	yes	—
52	C₂×C₈×C₈₃	SD₁₆	1	1	yes	characteristic
53	C₂×C₈×C₈₃	SD₁₆	1	1	yes	characteristic
54	C₂×C₈×C₈₃	Q₁₆	1	2	yes	—
55	C₂×C₈×C₈₃	C₂²×C₄	1	1	yes	characteristic
56	C₂×C₈×C₈₃	C₂×D₄	1	1	yes	characteristic
57	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
58	C₂×C₈×C₈₃	C₂×Q₈	1	1	yes	characteristic
59	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
60	C₂×C₈×C₈₃	#41	1	1	yes	characteristic
61	C₄⋊C₄×C₈₃	—	2	2	no	—
62	C₂²⋊C₄×C₈₃	—	4	1	no	—
63	C₄²×C₈₃	—	2	1	no	—
64	C₄²×C₈₃	—	4	1	no	—
65	C₄²×C₈₃	C₂²⋊C₄	1	2	yes	—
66	C₂³×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
67	C₂×C₄×C₈₃	—	2	2	no	—
68	C₂×C₄×C₈₃	—	4	2	no	—
69	C₂×C₄×C₈₃	—	4	2	no	—
70	C₂×C₄×C₈₃	C₂×C₂²⋊C₄	1	1	yes	characteristic
71	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
72	C₂×C₄×C₈₃	C₂²⋊Q₈	1	1	yes	characteristic
73	C₂×C₄×C₈₃	C₂×SD₁₆	1	1	yes	characteristic
74	C₂×C₄×C₈₃	C₂×Q₁₆	1	1	yes	characteristic
75	C₂×C₄×C₈₃	AGL(1,ℤ/8)	1	1	yes	characteristic
76	C₂×Dic₈₃	—	4	4	no	—
77	C₂×Dic₈₃	—	8	4	no	—
78	C₈×C₈₃	—	4	2	no	—
79	C₈×C₈₃	#103	1	2	yes	—
80	C₈×C₈₃	#103	1	2	yes	—
81	C₈×C₈₃	C₄×Q₈	1	1	yes	characteristic
82	C₈×C₈₃	C₄×Q₈	1	1	yes	characteristic
83	C₈×C₈₃	#124	1	1	yes	characteristic
84	C₈×C₈₃	C₄⋊Q₈	1	1	yes	characteristic
85	C₂²×C₈₃	—	2	2	no	—
86	C₂²×C₈₃	#418	1	1	yes	characteristic
87	C₂²×C₈₃	#476	1	1	yes	characteristic
88	C₂²×C₈₃	#477	1	1	yes	characteristic
89	C₄×C₈₃	—	4	2	no	—
90	C₄×C₈₃	—	8	2	no	—
91	C₄×C₈₃	#391	1	1	yes	characteristic
92	C₄×C₈₃	C₂×#103	1	1	yes	characteristic
93	C₄×C₈₃	#476	1	1	yes	characteristic
94	C₄×C₈₃	#477	1	1	yes	characteristic
95	Dic₈₃	—	8	4	no	—
96	#13273	—	83	1	no	maximal, Sylow
97	C₂×C₈₃	—	2	1	no	—
98	C₂×C₈₃	—	2	1	no	—
99	C₂×C₈₃	#1987	1	1	yes	characteristic
100	C₂×C₈₃	#2231	1	2	yes	—
101	C₂×#499	—	83	1	no	—
102	C₂×#364	—	83	1	no	—
103	#1772	D₈₃	1	1	yes	characteristic
104	#1386	—	83	2	no	—
105	#1383	—	83	2	no	—
106	C₈₃	#13273	1	1	yes	characteristic, complemented, Sylow
107	C₂×#125	—	166	1	no	—
108	C₂×M₃₂	—	83	2	no	—
109	#499	—	83	4	no	—
110	#499	—	166	2	no	—
111	C₂×#108	C₂×D₈₃	1	1	yes	characteristic
112	#427	C₂×D₈₃	1	1	yes	characteristic
113	C₂×#107	—	166	1	no	—
114	C₂×#106	—	83	2	no	—
115	#405	—	2	1	no	—
116	C₂×C₄×C₈	C₂×D₈₃	1	1	yes	characteristic
117	#364	—	166	2	no	—
118	#339	—	2	1	no	—
119	C₂×M₁₆	—	2	1	no	—
120	C₂²×C₈	C₂²⋊D₈₃	1	1	yes	characteristic
121	C₂²×C₈	C₂²×D₈₃	1	1	yes	characteristic
122	#125	—	332	2	no	—
123	#125	—	166	4	no	—
124	#117	—	2	1	no	—
125	C₂×C₄⋊C₄	—	332	2	no	—
126	C₂×C₄⋊C₄	—	166	2	no	—
127	C₂×C₄²	C₂²⋊D₈₃	1	1	yes	characteristic
128	M₃₂	—	166	2	no	—
129	M₃₂	—	166	2	no	—
130	C₂×C₁₆	—	166	2	no	—
131	#108	—	2	1	no	—
132	#108	Dic₁₆₆	1	2	yes	—
133	#107	—	166	4	no	—
134	#107	Dic₁₆₆	1	2	yes	—
135	#106	—	2	1	no	—
136	#106	—	83	4	no	—
137	#106	—	166	2	no	—
138	#97	—	4	1	no	—
139	C₄×C₈	—	4	1	no	—
140	C₄×C₈	C₄×D₈₃	1	2	yes	—
141	C₄×C₈	D₃₃₂	1	2	yes	—
142	C₂²×C₄	—	2	1	no	—
143	C₂²×C₄	—	332	2	no	—
144	C₂²×C₄	C₂×C₂²⋊D₈₃	1	1	yes	characteristic
145	M₁₆	—	2	2	no	—
146	M₁₆	—	4	1	no	—
147	C₂×C₈	—	4	1	no	—
148	C₂×C₈	(C₂×D₈₃)⋊C₄	1	2	yes	—
149	C₂×C₈	#13050578	1	1	yes	characteristic
150	C₂×C₈	#13050578	1	1	yes	characteristic
151	C₂×C₈	#13050580	1	2	yes	—
152	C₂×C₈	C₂×C₄×D₈₃	1	1	yes	characteristic
153	C₂×C₈	C₂×D₃₃₂	1	1	yes	characteristic
154	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
155	C₂×C₈	C₂×Dic₁₆₆	1	1	yes	characteristic
156	C₂×C₈	#13050590	1	1	yes	characteristic
157	C₂×C₈	#13050590	1	1	yes	characteristic
158	C₄⋊C₄	—	2	2	no	—
159	C₄⋊C₄	—	332	8	no	—
160	C₄⋊C₄	—	332	2	no	—
161	C₄⋊C₄	—	166	4	no	—
162	C₂²⋊C₄	—	4	1	no	—
163	C₄²	—	2	1	no	—
164	C₄²	—	4	1	no	—
165	C₄²	(C₂×D₈₃)⋊C₄	1	2	yes	—
166	C₁₆	—	332	2	no	—
167	C₂³	#14581817	1	1	yes	characteristic
168	C₂×C₄	—	2	2	no	—
169	C₂×C₄	—	4	2	no	—
170	C₂×C₄	—	4	2	no	—
171	C₂×C₄	—	332	4	no	—
172	C₂×C₄	—	664	4	no	—
173	C₂×C₄	C₂×(C₂×D₈₃)⋊C₄	1	1	yes	characteristic
174	C₂×C₄	#14581771	1	1	yes	characteristic
175	C₂×C₄	#14581771	1	1	yes	characteristic
176	C₂×C₄	C₂×#13050578	1	1	yes	characteristic
177	C₂×C₄	C₂×#13050580	1	1	yes	characteristic
178	C₂×C₄	#14581817	1	1	yes	characteristic
179	C₈	—	4	2	no	—
180	C₈	#14581715	1	2	yes	—
181	C₈	#14581715	1	2	yes	—
182	C₈	C₄×Dic₁₆₆	1	1	yes	characteristic
183	C₈	C₄×Dic₁₆₆	1	1	yes	characteristic
184	C₈	#14581780	1	1	yes	characteristic
185	C₈	#14581792	1	1	yes	characteristic
186	C₂²	—	2	2	no	—
187	C₂²	#19127705	1	1	yes	characteristic
188	C₂²	#19127991	1	1	yes	characteristic
189	C₂²	#19127996	1	1	yes	characteristic
190	C₄	—	4	2	no	—
191	C₄	—	8	2	no	—
192	C₄	—	664	4	no	—
193	C₄	#19127607	1	1	yes	characteristic
194	C₄	C₂×#14581715	1	1	yes	characteristic
195	C₄	#19127991	1	1	yes	characteristic
196	C₄	#19127996	1	1	yes	characteristic
197	C₂	—	2	1	no	—
198	C₂	—	2	1	no	—
199	C₂	#24908411	1	1	yes	characteristic
200	C₂	#24909639	1	2	yes	—

	subgroup	G/H
center	C₂²	#19127991
commutator	C₂×C₈×C₈₃	C₂²×C₄
Frattini	C₂²×C₈	C₂²×D₈₃
Fitting	C₈₃×#1772	C₂
socle	C₂²×C₈₃	#476
prime radical	C₂³×C₈₃	AGL(1,ℤ/8)
cosocle	C₂²×C₈×C₈₃	C₂³
hypercenter	#1772	D₈₃
hypocenter	C₈₃	#13273
LCS	G ▷ C₂×C₈×C₈₃ ▷ C₄×C₈₃ ▷ C₂×C₈₃ ▷ C₈₃
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₄×C₈ ◁ #1772

Complements

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

N	H ≅ G/N	type
9. #19128103	C₄	semidirect
106. C₈₃	#13273	semidirect

Maximal subgroups

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

subgroup	index
C₂×#19128103	2
C₂×#19127522	2
#24905568	2
#24905563	2
C₈₃×#1772	2
#13273	83