Finite Group Databasebeta

← #35315921

Subgroups — #35315921

Order 21248.

292 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 87 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#24909279C211yesmaximal, characteristic, complemented
2#24909229C211yesmaximal, characteristic, complemented
3#24905524C212yesmaximal, complemented
4#24905500C212yesmaximal, complemented
5C83×#3176C211yesmaximal, characteristic
6C2×#14581732—21no—
7#19127522—21no—
8#19127522C2212yes—
9#19127522C2212yes—
10#19127483—21no—
11#19127479—21no—
12#540×C83C412yes—
13#499×C83C2211yescharacteristic
14#493×C83C2211yescharacteristic
15#490×C83C412yes—
16#469×C83—21no—
17#466×C83—21no—
18#442×C83—21no—
19C4×D8×C83—21no—
20#405×C83C2211yescharacteristic
21#14581732—22no—
22#14581732—22no—
23#14581732—41no—
24C2×C83⋊C16—41no—
25C2×D8×C83—21no—
26C2×D8×C83—21no—
27C2×M16×C83D411yescharacteristic
28C22×C8×C83D411yescharacteristic
29#127×C83C2×C412yes—
30#125×C83C2×C412yes—
31#123×C83—42no—
32#121×C83—22no—
33#121×C83—42no—
34C4×D4×C83—22no—
35#117×C83C2×C411yescharacteristic
36#107×C83—21no—
37#106×C83—21no—
38#102×C83—22no—
39#102×C83—22no—
40#97×C83C2×C411yescharacteristic
41#97×C83D411yescharacteristic
42C4×C8×C83D411yescharacteristic
43C4×C8×C83C2311yescharacteristic
44C83⋊C16—42no—
45C83⋊C16—42no—
46C2×D4×C83—22no—
47C2×D4×C83—42no—
48C2×D4×C83—42no—
49C22×C4×C83—42no—
50C22×C4×C83C22⋊C411yescharacteristic
51D8×C83—42no—
52D8×C83—41no—
53D8×C83—41no—
54M16×C83—22no—
55C2×C8×C83—21no—
56C2×C8×C83—21no—
57C2×C8×C83C22⋊C411yescharacteristic
58C2×C8×C83C22⋊C411yescharacteristic
59C2×C8×C83C2×D411yescharacteristic
60C2×C8×C83C2×D411yescharacteristic
61C4⋊C4×C83—21no—
62C4⋊C4×C83—22no—
63C4⋊C4×C83—42no—
64C22⋊C4×C83—21no—
65C22⋊C4×C83—42no—
66C22⋊C4×C83—42no—
67C42×C83C22⋊C411yescharacteristic
68C42×C83C22×C411yescharacteristic
69C23×C83—42no—
70C23×C83#10011yescharacteristic
71D4×C83—42no—
72D4×C83—42no—
73D4×C83—162no—
74D4×C83—162no—
75C2×C4×C83—21no—
76C2×C4×C83—21no—
77C2×C4×C83—21no—
78C2×C4×C83—21no—
79C2×C4×C83—42no—
80C2×C4×C83—162no—
81C2×C4×C83#10011yescharacteristic
82C2×C4×C83C2×C22⋊C411yescharacteristic
83C8×C83—21no—
84C8×C83—21no—
85C8×C83#13512yes—
86C8×C83AGL(1,ℤ/8)11yescharacteristic
87C8×C83#13711yescharacteristic
88C22×C83—21no—
89C22×C83—41no—
90C22×C83—82no—
91C22×C83—82no—
92C22×C83—82no—
93C22×C83C2×#10011yescharacteristic
94C4×C83—21no—
95C4×C83—41no—
96C4×C83—41no—
97C4×C83—82no—
98C4×C83#41611yescharacteristic
99C4×C83#41711yescharacteristic
100#11989—831nomaximal, Sylow
101C2×C83—21no—
102C2×C83—41no—
103C2×C83—162no—
104C2×C83#155111yescharacteristic
105#3176D8311yescharacteristic
106#2166—831no—
107#2153—831no—
108#1366—832no—
109#1357—832no—
110C83#1198911yescharacteristic, complemented, Sylow
111#540Dic8312yes—
112C2×M32—1661no—
113#499C2×D8311yescharacteristic
114#493C2×D8311yescharacteristic
115#490Dic8312yes—
116#469—21no—
117#466—21no—
118#442—21no—
119C4×D8—21no—
120#405C2×D8311yescharacteristic
121#364—832no—
122#364—832no—
123#364—1661no—
124#349—1661no—
125#347—1661no—
126C2×D8—21no—
127C2×D8—21no—
128C2×M16C22⋊D8311yescharacteristic
129C22×C8C22⋊D8311yescharacteristic
130#127C2×Dic8312yes—
131#125C2×Dic8312yes—
132#123—42no—
133#121—22no—
134#121—42no—
135C4×D4—22no—
136#117C2×Dic8311yescharacteristic
137M32—3321no—
138M32—1662no—
139M32—1662no—
140C2×C16—3321no—
141#107—21no—
142#106—21no—
143#102—22no—
144#102—22no—
145#97C2×Dic8311yescharacteristic
146#97C22⋊D8311yescharacteristic
147C4×C8C22⋊D8311yescharacteristic
148C4×C8C22×D8311yescharacteristic
149C2×D4—22no—
150C2×D4—42no—
151C2×D4—42no—
152C22×C4—42no—
153C22×C4C22⋊Dic8311yescharacteristic
154D8—42no—
155D8—41no—
156D8—41no—
157M16—22no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8C22⋊Dic8311yescharacteristic
161C2×C8C22⋊Dic8311yescharacteristic
162C2×C8C2×C22⋊D8311yescharacteristic
163C2×C8C2×C22⋊D8311yescharacteristic
164C4⋊C4—21no—
165C4⋊C4—22no—
166C4⋊C4—42no—
167C22⋊C4—21no—
168C22⋊C4—42no—
169C22⋊C4—42no—
170C42C22⋊Dic8311yescharacteristic
171C42C22×Dic8311yescharacteristic
172C16—3322no—
173C16—3322no—
174C23—42no—
175C23#1458170911yescharacteristic
176D4—42no—
177D4—42no—
178D4—162no—
179D4—162no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—42no—
185C2×C4—162no—
186C2×C4#1458170911yescharacteristic
187C2×C4C2×C22⋊Dic8311yescharacteristic
188C8—21no—
189C8—21no—
190C8#1458181212yes—
191C8#1458181711yescharacteristic
192C8#1458182211yescharacteristic
193C22—21no—
194C22—41no—
195C22—82no—
196C22—82no—
197C22—82no—
198C22C2×#1458170911yescharacteristic
199C4—21no—
200C4—41no—
201C4—41no—
202C4—82no—
203C4#1912769711yescharacteristic
204C4#1912770211yescharacteristic
205C2—21no—
206C2—41no—
207C2—162no—
208C2#2490615811yescharacteristic
subgroupG/H
centerC2#24906158
commutatorC42×C83C22×C4
FrattiniC4×C8C22×D83
Fitting, prime radicalC83×#3176C2
socleC2×C83#1551
cosocleC4×C8×C83C23
hypercenter#3176D83
hypocenterC83#11989
LCSG ▷ C42×C83 ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #3176

Complements

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

NH ≅ G/Ntype
1. #24909279C2semidirect
2. #24909229C2semidirect
3. #24905524C2semidirect
4. #24905500C2semidirect
110. C83#11989semidirect

Maximal subgroups

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

subgroupindex
#249092792
#249092292
#249055242
#249055002
C83×#31762
#1198983