Finite Group Databasebeta

← #35318614

Subgroups — #35318614

Order 21248.

236 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 236 conjugacy classes of subgroups — too many to draw. Only its 65 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
1#24909715C211yesmaximal, characteristic, complemented
2#24909282C211yesmaximal, characteristic
3#24906829C211yesmaximal, characteristic
4#24906825C211yesmaximal, characteristic
5#24905534C211yesmaximal, characteristic, complemented
6#24905513C211yesmaximal, characteristic, complemented
7C83×#1362C211yesmaximal, characteristic, complemented
8C2×#14581738—21no—
9C2×#14581738—21no—
10C2×C16×D83—21no—
11#19128081C2211yescharacteristic
12#19127982—21no—
13#19127978—21no—
14C8×D332C2211yescharacteristic
15(C2×D83)⋊C16—21no—
16C4×C83⋊C16C2211yescharacteristic
17#19127454C2211yescharacteristic
18#19127447—21no—
19#19127446—21no—
20#494×C83C2211yescharacteristic
21C4⋊C16×C83C2211yescharacteristic
22#335×C83C2211yescharacteristic
23C2×Dic332—21no—
24C2×Dic332—21no—
25C2×C8×D83—21no—
26#14581792—21no—
27C4×D332D411yescharacteristic
28#14581738—42no—
29#14581738—42no—
30C2×C83⋊C16D411yescharacteristic
31C2×C83⋊C16D411yescharacteristic
32C16×D83—42no—
33#14581726—41no—
34#14581725—41no—
35C4⋊(C83⋊C8)—21no—
36C4⋊(C83⋊C8)D411yescharacteristic
37#14581714—41no—
38#14581712—41no—
39(C2×D83)⋊C8—21no—
40C2×Q16×C83—21no—
41C2×Q16×C83—21no—
42C4⋊Q8×C83—21no—
43C2×C16×C83—21no—
44C4⋊C8×C83—21no—
45C4×C8×C83C2311yescharacteristic
46C2×Dic166—41no—
47C2×D332D811yescharacteristic
48C2×C4×D83—21no—
49Dic332—41no—
50Dic332—41no—
51Dic332—81no—
52C2×C83⋊C8—21no—
53C2×C83⋊C8—41no—
54C8×D83—41no—
55C8×D83—41no—
56C4⋊Dic83—41no—
57C4⋊Dic83—41no—
58C4⋊Dic83D811yescharacteristic
59(C2×D83)⋊C4—21no—
60C83⋊C16—21no—
61C83⋊C16D812yes—
62C2×Q8×C83—41no—
63Q16×C83—41no—
64Q16×C83—41no—
65Q16×C83—81no—
66C2×C8×C83—41no—
67C2×C8×C83C2×D411yescharacteristic
68C2×C8×C83#4111yescharacteristic
69C4⋊C4×C83—41no—
70C4⋊C4×C83—41no—
71C42×C83C2×D411yescharacteristic
72C16×C83—41no—
73C22×D83—21no—
74Q8×C83—81no—
75Q8×C83—81no—
76C2×C4×C83—81no—
77C2×C4×C83#12111yescharacteristic
78C2×C4×C83C2×D811yescharacteristic
79C2×C4×C83#13511yescharacteristic
80C2×Dic83—21no—
81C2×Dic83—81no—
82D332—21no—
83D332Q3212yescomplemented
84C4×D83—41no—
85C4×D83—41no—
86Dic166—81no—
87Dic166—81no—
88C8×C83—21no—
89C8×C83—81no—
90C8×C83C2×D811yescharacteristic
91C8×C83AGL(1,ℤ/8)11yescharacteristic
92C83⋊C8—41no—
93C83⋊C8—81no—
94C22×C83#46611yescharacteristic
95C2×D83—22no—
96C2×D83—41no—
97C2×D83—41no—
98C4×C83—21no—
99C4×C83—161no—
100C4×C83#45911yescharacteristic
101C4×C83#47211yescharacteristic
102C4×C83C2×Q3211yescharacteristic
103C4×C83#50811yescharacteristic
104Dic83—41no—
105Dic83—161no—
106#12431—831nomaximal, Sylow
107C2×C83#166711yescharacteristic
108C2×C83#221611yescharacteristic
109C2×C83#222211yescharacteristic
110D83—42no—
111#2247—831no—
112D4×C16—831no—
113#1669—831no—
114#1669—831no—
115#1371—831no—
116#1362—831no—
117#1362D8311yescharacteristic, complemented
118C83#1243111yescharacteristic, complemented, Sylow
119C2×Q32—1661no—
120C2×Q32—1661no—
121C22×C16—1661no—
122#494—831no—
123#494C2×D8311yescharacteristic
124#474—1661no—
125#474—1661no—
126C8×D4—831no—
127C4⋊C16C2×D8311yescharacteristic
128C22⋊C16—1661no—
129C4×C16—831no—
130#335—831no—
131#335C2×D8311yescharacteristic
132#332—1661no—
133#332—1661no—
134C2×Q16—21no—
135C2×Q16—21no—
136C2×Q16—1661no—
137C2×Q16—1661no—
138C22×C8—1661no—
139C4⋊Q8—21no—
140C4⋊Q8—1661no—
141C4×D4—831no—
142Q32—3322no—
143Q32—3322no—
144C2×C16—21no—
145C2×C16—3322no—
146C2×C16—831no—
147C2×C16—831no—
148#107—3321no—
149#107—3321no—
150C4⋊C8—21no—
151C4⋊C8—831no—
152C4⋊C8—1661no—
153#102—3321no—
154#102—3321no—
155C22⋊C8—1661no—
156C4×C8C22×D8311yescharacteristic
157C2×Q8—41no—
158C2×Q8—3321no—
159C2×D4—831no—
160C22×C4—1661no—
161Q16—41no—
162Q16—41no—
163Q16—81no—
164Q16—3321no—
165Q16—3321no—
166Q16—6641no—
167C2×C8—41no—
168C2×C8—3321no—
169C2×C8—3321no—
170C2×C8—3321no—
171C2×C8—1661no—
172C2×C8D4×D8311yescharacteristic
173C2×C8#1305059011yescharacteristic
174C4⋊C4—41no—
175C4⋊C4—41no—
176C4⋊C4—3321no—
177C4⋊C4—3321no—
178C4⋊C4—831no—
179C22⋊C4—1661no—
180C42D4×D8311yescharacteristic
181C16—41no—
182C16—832no—
183C16—1661no—
184C23—1661no—
185Q8—81no—
186Q8—81no—
187Q8—6641no—
188Q8—6641no—
189D4—832no—
190D4—1661no—
191C2×C4—81no—
192C2×C4—3321no—
193C2×C4—3321no—
194C2×C4—6641no—
195C2×C4—1661no—
196C2×C4#1458176511yescharacteristic
197C2×C4D8×D8311yescharacteristic
198C2×C4#1458181311yescharacteristic
199C8—21no—
200C8—81no—
201C8—3321no—
202C8—6641no—
203C8D8×D8311yescharacteristic
204C8#1458181911yescharacteristic
205C22—3321no—
206C22—3321no—
207C22—1662no—
208C22#1912793611yescharacteristic
209C4—21no—
210C4—161no—
211C4—13281no—
212C4—3321no—
213C4#1912789511yescharacteristic
214C4#1912797011yescharacteristic
215C4Q32×D8311yescharacteristic
216C4#1912813211yescharacteristic
217C2—3322no—
218C2#2490681711yescharacteristic
219C2#2490955311yescharacteristic
220C2#2490958711yescharacteristic
subgroupG/H
centerC22#19127936
commutator, cosocleC4×C8×C83C23
FrattiniC4×C8C22×D83
FittingC83×#1362C2
socleC22×C83#466
prime radicalC2×D332D8
hypercenter#1362D83
hypocenterC83#12431
LCSG ▷ C4×C8×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
1. #24909715C2semidirect
5. #24905534C2semidirect
6. #24905513C2semidirect
7. C83×#1362C2semidirect
83. D332Q32semidirect
117. #1362D83semidirect
118. C83#12431semidirect

Maximal subgroups

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

subgroupindex
#249097152
#249092822
#249068292
#249068252
#249055342
#249055132
C83×#13622
#1243183