Finite Group Databasebeta

← #35318617

Subgroups — #35318617

Order 21248.

252 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 252 conjugacy classes of subgroups — too many to draw. Only its 69 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C16×D332C211yesmaximal, characteristic, complemented
2#24906825C211yesmaximal, characteristic, complemented
3#24906815C211yesmaximal, characteristic
4#24905537C211yesmaximal, characteristic, complemented
5#24905512C211yesmaximal, characteristic, complemented
6#24905509C211yesmaximal, characteristic, complemented
7C83×#2248C211yesmaximal, characteristic, complemented
8C2×C16×D83—21no—
9#19127978—21no—
10#19127893—21no—
11C8×D332C2211yescharacteristic
12C4⋊(C83⋊C16)C2211yescharacteristic
13(C2×D83)⋊C16—21no—
14#19127454C2211yescharacteristic
15#19127454C2211yescharacteristic
16#19127447—21no—
17#19127445—21no—
18C2×SD32×C83—21no—
19C2×SD32×C83—21no—
20#494×C83C2211yescharacteristic
21#493×C83C2211yescharacteristic
22C4×C16×C83C2211yescharacteristic
23C2×#13050575—41no—
24C2×C8×D83—21no—
25C4×D332D411yescharacteristic
26C2×C83⋊C16—21no—
27C16×D83—42no—
28#14581725—41no—
29C4⋊(C83⋊C8)—21no—
30C4⋊(C83⋊C8)—21no—
31C4⋊(C83⋊C8)D411yescharacteristic
32#14581713—41no—
33#14581712—41no—
34(C2×D83)⋊C8—21no—
35C2×Q16×C83—21no—
36C2×Q16×C83—21no—
37C2×D8×C83—21no—
38C2×D8×C83—21no—
39C4⋊Q8×C83—21no—
40#127×C83—21no—
41SD32×C83—42no—
42SD32×C83—42no—
43C2×C16×C83D411yescharacteristic
44C2×C16×C83D411yescharacteristic
45C4×C8×C83C2311yescharacteristic
46C2×D332D811yescharacteristic
47C2×C4×D83—21no—
48#13050575—82no—
49C2×C83⋊C8—21no—
50C2×C83⋊C8—41no—
51C2×C83⋊C8—41no—
52C8×D83—41no—
53C8×D83—41no—
54C4⋊Dic83D811yescharacteristic
55(C2×D83)⋊C4—21no—
56C83⋊C16—41no—
57C2×Q8×C83—41no—
58C2×D4×C83—41no—
59C2×D4×C83—41no—
60C2×D4×C83—41no—
61Q16×C83—41no—
62Q16×C83—41no—
63Q16×C83—81no—
64D8×C83—41no—
65D8×C83—41no—
66D8×C83—81no—
67C2×C8×C83C2×D411yescharacteristic
68C2×C8×C83#4111yescharacteristic
69C4⋊C4×C83—41no—
70C4⋊C4×C83—41no—
71C42×C83C2×D411yescharacteristic
72C16×C83—21no—
73C16×C83D812yescomplemented
74C23×C83—81no—
75C22×D83—21no—
76Q8×C83—81no—
77Q8×C83—81no—
78D4×C83—81no—
79D4×C83—81no—
80D4×C83—81no—
81D4×C83—81no—
82D4×C83—161no—
83C2×C4×C83—81no—
84C2×C4×C83#12111yescharacteristic
85C2×C4×C83C2×D811yescharacteristic
86C2×C4×C83#13511yescharacteristic
87C2×Dic83—21no—
88D332—21no—
89D332SD3212yescomplemented
90C4×D83—41no—
91C4×D83—41no—
92C8×C83—21no—
93C8×C83C2×D811yescharacteristic
94C8×C83AGL(1,ℤ/8)11yescharacteristic
95C83⋊C8—41no—
96C83⋊C8—81no—
97C83⋊C8—81no—
98C22×C83—161no—
99C22×C83—161no—
100C22×C83—161no—
101C22×C83#46611yescharacteristic
102C2×D83—22no—
103C2×D83—41no—
104C2×D83—41no—
105C4×C83—21no—
106C4×C83—161no—
107C4×C83#45911yescharacteristic
108C4×C83#47211yescharacteristic
109C4×C83C2×SD3211yescharacteristic
110C4×C83#50811yescharacteristic
111Dic83—41no—
112#12432—831nomaximal, Sylow
113C2×C83—321no—
114C2×C83#166711yescharacteristic
115C2×C83#221711yescharacteristic
116C2×C83#222311yescharacteristic
117D83—42no—
118#2248D8311yescharacteristic, complemented
119D4×C16—831no—
120#1669—831no—
121#1667—831no—
122#1372—831no—
123#1362—831no—
124#1361—831no—
125C83#1243211yescharacteristic, complemented, Sylow
126C2×SD32—21no—
127C2×SD32—21no—
128C22×C16—1661no—
129#494C2×D8311yescharacteristic
130#493C2×D8311yescharacteristic
131#474—1661no—
132#459—1661no—
133C8×D4—831no—
134C4⋊C16—831no—
135C22⋊C16—1661no—
136C4×C16C2×D8311yescharacteristic
137#335—831no—
138#335—831no—
139#332—1661no—
140#331—1661no—
141C2×Q16—21no—
142C2×Q16—21no—
143C2×D8—21no—
144C2×D8—21no—
145C2×D8—3321no—
146C22×C8—1661no—
147C4⋊Q8—21no—
148#127—21no—
149C4×D4—831no—
150SD32—42no—
151SD32—42no—
152C2×C16—3322no—
153C2×C16—1661no—
154C2×C16C22⋊D8311yescharacteristic
155C2×C16C22⋊D8311yescharacteristic
156#107—3321no—
157C4⋊C8—831no—
158C4⋊C8—1661no—
159C4⋊C8—1661no—
160#102—3321no—
161#102—3321no—
162C22⋊C8—1661no—
163C4×C8C22×D8311yescharacteristic
164C2×Q8—41no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4—41no—
168C2×D4—831no—
169C22×C4—1661no—
170Q16—41no—
171Q16—41no—
172Q16—81no—
173D8—41no—
174D8—41no—
175D8—81no—
176D8—6642no—
177C2×C8—3321no—
178C2×C8—3321no—
179C2×C8—3321no—
180C2×C8—3321no—
181C2×C8—1661no—
182C2×C8C2×C22⋊D8311yescharacteristic
183C2×C8#1305058911yescharacteristic
184C4⋊C4—41no—
185C4⋊C4—41no—
186C4⋊C4—831no—
187C22⋊C4—1661no—
188C42D4×D8311yescharacteristic
189C16—21no—
190C16—3321no—
191C16#1305057512yescomplemented
192C23—81no—
193C23—1661no—
194Q8—81no—
195Q8—81no—
196D4—81no—
197D4—81no—
198D4—81no—
199D4—81no—
200D4—161no—
201D4—832no—
202D4—1661no—
203C2×C4—81no—
204C2×C4—3321no—
205C2×C4—3321no—
206C2×C4—1661no—
207C2×C4#1458176311yescharacteristic
208C2×C4D8×D8311yescharacteristic
209C2×C4#1458181111yescharacteristic
210C8—21no—
211C8—3321no—
212C8—6641no—
213C8—6641no—
214C8C2×#1305057511yescharacteristic
215C8#1458181811yescharacteristic
216C22—161no—
217C22—161no—
218C22—161no—
219C22—3321no—
220C22—3321no—
221C22—1662no—
222C22#1912793411yescharacteristic
223C4—21no—
224C4—161no—
225C4—3321no—
226C4#1912789311yescharacteristic
227C4#1912796811yescharacteristic
228C4SD32×D8311yescharacteristic
229C4#1912813211yescharacteristic
230C2—321no—
231C2—3322no—
232C2#2490681511yescharacteristic
233C2#2490955611yescharacteristic
234C2#2490959011yescharacteristic
subgroupG/H
centerC22#19127934
commutator, cosocleC4×C8×C83C23
FrattiniC4×C8C22×D83
FittingC83×#2248C2
socleC22×C83#466
prime radical#24906815C2
hypercenter#2248D83
hypocenterC83#12432
LCSG ▷ C4×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2248

Complements

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

NH ≅ G/Ntype
1. C16×D332C2semidirect
2. #24906825C2semidirect
4. #24905537C2semidirect
5. #24905512C2semidirect
6. #24905509C2semidirect
7. C83×#2248C2semidirect
73. C16×C83D8semidirect
89. D332SD32semidirect
118. #2248D83semidirect
125. C83#12432semidirect
191. C16#13050575semidirect

Maximal subgroups

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

subgroupindex
C16×D3322
#249068252
#249068152
#249055372
#249055122
#249055092
C83×#22482
#1243283