Finite Group Databasebeta

← #24909410

Subgroups — #24909410

Order 10624.

226 conjugacy classes of subgroups; only the 47 normal subgroups are shown.

normal subgroups only

This group has 226 conjugacy classes of subgroups — too many to draw. Only its 47 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#19128498C211yesmaximal, characteristic, complemented
2#19128139C211yesmaximal, characteristic, complemented
3#19128136C211yesmaximal, characteristic, complemented
4#19127955C211yesmaximal, characteristic, complemented
5#19127510C211yesmaximal, characteristic, complemented
6#19127482C211yesmaximal, characteristic, complemented
7#360×C83C211yesmaximal, characteristic, complemented
8#41×D83—21no—
9#14581824—21no—
10#14581821—21no—
11#14581819—21no—
12#14581816—21no—
13#14581814C2211yescharacteristic
14#14581812—21no—
15C2×D664C2211yescharacteristic, complemented
16C2×#13050573C2211yescharacteristic
17#14581767—21no—
18#14581737—21no—
19#14581735—21no—
20D1328—21no—
21#14581733—21no—
22#14581732C2211yescharacteristic, complemented
23#14581712—21no—
24#135×C83C2211yescharacteristic
25M32×C83C2211yescharacteristic, complemented
26#106×C83C2211yescharacteristic, complemented
27#13050591—21no—
28#13050590—21no—
29#13050590—41no—
30#13050589—21no—
31#13050589—41no—
32Q8×D83—21no—
33C2×D332—21no—
34C2×D332—41no—
35D4×D83—21no—
36D4×D83—41no—
37C2×C4×D83—41no—
38C2×C4×D83D411yescharacteristic
39Dic332D411yescharacteristic, complemented
40#13050580—21no—
41#13050579—21no—
42#13050578—21no—
43#13050577—21no—
44D664—21no—
45D664—21no—
46D664D411yescharacteristic, complemented
47#13050575—21no—
48#13050573—21no—
49#13050573—21no—
50C2×C83⋊C8D411yescharacteristic
51(C2×D83)⋊C4—41no—
52C83⋊C16—21no—
53#41×C83—21no—
54Q16×C83D411yescharacteristic, complemented
55SD16×C83—21no—
56D8×C83D411yescharacteristic, complemented
57C2×C8×C83C2311yescharacteristic
58C4⋊C4×C83—21no—
59C16×C83—21no—
60C22×D83—41no—
61C22×D83—41no—
62C22×D83D811yescharacteristic
63Q8×C83—21no—
64D4×C83—21no—
65D4×C83—41no—
66C2×C4×C83—41no—
67C2×C4×C83—41no—
68C2×C4×C83C2×D411yescharacteristic
69C22⋊D83—41no—
70C22⋊D83—41no—
71C22⋊D83—41no—
72C2×Dic83—41no—
73C2×Dic83D811yescharacteristic
74D332—21no—
75D332—41no—
76D332—41no—
77D332—41no—
78D332—82no—
79C4×D83—21no—
80C4×D83—21no—
81C4×D83—41no—
82C4×D83—41no—
83C4×D83—41no—
84Dic166—21no—
85Dic166—41no—
86C8×C83C2×D411yescharacteristic
87C8×C83C2×D411yescharacteristic
88C83⋊C8—21no—
89C22×C83—41no—
90C22×C83C2×D811yescharacteristic
91C2×D83—21no—
92C2×D83—41no—
93C2×D83—41no—
94C2×D83—82no—
95C2×D83—81no—
96C2×D83—81no—
97C2×D83—81no—
98C4×C83—41no—
99C4×C83—81no—
100C4×C83#12011yescharacteristic
101C4×C83AGL(1,ℤ/8)11yescharacteristic
102Dic83—21no—
103Dic83—41no—
104C2×C83—21no—
105C2×C83—81no—
106C2×C83#44711yescharacteristic
107D83—41no—
108D83—81no—
109D83—161no—
110#2190—831nomaximal, Sylow
111C83#219011yescharacteristic, complemented, Sylow
112#575—831no—
113#509—831no—
114#509—831no—
115#469—831no—
116#360—831no—
117#360D8311yescharacteristic, complemented
118#349—831no—
119C2×#41—1661no—
120#137—1661no—
121#137—1661no—
122AGL(1,ℤ/8)—1661no—
123AGL(1,ℤ/8)—1661no—
124#135—831no—
125#135—1661no—
126#135C2×D8311yescharacteristic
127C2×D8—831no—
128C2×M16—831no—
129#121—1661no—
130SD32—1661no—
131SD32—1661no—
132D16—1661no—
133D16—1661no—
134M32—831no—
135M32C2×D8311yescharacteristic, complemented
136#106C2×D8311yescharacteristic, complemented
137#102—1661no—
138#41—21no—
139#41—3321no—
140#41—3321no—
141#41—1661no—
142#41—1661no—
143#41—1661no—
144C2×Q8—1661no—
145C2×D4—3321no—
146C2×D4—3321no—
147C2×D4—1661no—
148C2×D4—1661no—
149C22×C4—3321no—
150C22×C4—831no—
151Q16—831no—
152Q16—1661no—
153Q16D33211yescharacteristic, complemented
154SD16—21no—
155SD16—1661no—
156SD16—1661no—
157SD16—1661no—
158D8—831no—
159D8—1661no—
160D8—1661no—
161D8—1661no—
162D8D33211yescharacteristic, complemented
163M16—1661no—
164M16—1661no—
165C2×C8—831no—
166C2×C8C22×D8311yescharacteristic
167C4⋊C4—21no—
168C22⋊C4—3321no—
169C16—21no—
170C16—1661no—
171C23—3321no—
172C23—3321no—
173C23—831no—
174Q8—21no—
175Q8—3321no—
176Q8—1661no—
177D4—21no—
178D4—41no—
179D4—3321no—
180D4—3321no—
181D4—3321no—
182D4—3321no—
183D4—3321no—
184D4—3321no—
185D4—6642no—
186D4—1661no—
187C2×C4—41no—
188C2×C4—41no—
189C2×C4—3321no—
190C2×C4—3321no—
191C2×C4—3321no—
192C2×C4—3321no—
193C2×C4—831no—
194C2×C4—1661no—
195C2×C4—1661no—
196C2×C4D4×D8311yescharacteristic
197C8—1661no—
198C8D4×D8311yescharacteristic
199C8C2×D33211yescharacteristic
200C22—41no—
201C22—3321no—
202C22—3321no—
203C22—6641no—
204C22—6641no—
205C22—6642no—
206C22—6641no—
207C22—1661no—
208C22D8×D8311yescharacteristic
209C4—41no—
210C4—81no—
211C4—3321no—
212C4—1661no—
213C4#1458176211yescharacteristic
214C4#1458181911yescharacteristic
215C2—21no—
216C2—81no—
217C2—13281no—
218C2—3321no—
219C2—6641no—
220C2#1912783611yescharacteristic
subgroupG/H
centerC2#19127836
commutator, cosocleC2×C8×C83C23
FrattiniC2×C8C22×D83
Fitting#360×C83C2
socleC2×C83#447
hypercenter#360D83
hypocenterC83#2190
LCSG ▷ C2×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #360

Complements

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

NH ≅ G/Ntype
1. #19128498C2semidirect
2. #19128139C2semidirect
3. #19128136C2semidirect
4. #19127955C2semidirect
5. #19127510C2semidirect
6. #19127482C2semidirect
7. #360×C83C2semidirect
15. C2×D664C22semidirect
22. #14581732C22semidirect
25. M32×C83C22semidirect
26. #106×C83C22semidirect
39. Dic332D4semidirect
46. D664D4semidirect
54. Q16×C83D4semidirect
56. D8×C83D4semidirect
111. C83#2190semidirect
117. #360D83semidirect
135. M32C2×D83semidirect
136. #106C2×D83semidirect
153. Q16D332semidirect
162. D8D332semidirect

Maximal subgroups

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

subgroupindex
#191284982
#191281392
#191281362
#191279552
#191275102
#191274822
#360×C832
#219083