Finite Group Databasebeta

← #24906451

Subgroups — #24906451

Order 10624.

280 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 53 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#19128263C211yesmaximal, characteristic
2#19128015C212yesmaximal, complemented
3#19127626C211yesmaximal, characteristic, complemented
4#19127436C212yesmaximal, complemented
5#323×C83C211yesmaximal, characteristic, complemented
6C22×C22⋊D83—21no—
7#14581776—22no—
8#14581766—21no—
9#14581766C2212yes—
10#14581762—22no—
11C2×C4⋊Dic83C2211yescharacteristic, complemented
12C2×C22⋊Dic83—21no—
13#14581726—22no—
14#14581714—22no—
15C83⋊#95—21no—
16C83⋊#95C2211yescharacteristic, complemented
17C2×C22⋊C4×C83C2211yescharacteristic
18C22⋊C8×C83C2212yescomplemented
19C23×D83—21no—
20C2×D332—21no—
21C2×D332D412yes—
22C2×C22⋊D83—22no—
23C2×C22⋊D83—22no—
24C2×C22⋊D83—42no—
25C2×C22⋊D83—42no—
26C22×Dic83—21no—
27C22×Dic83—21no—
28C22×Dic83—21no—
29C4⋊Dic83—21no—
30C4⋊Dic83—22no—
31C4⋊Dic83D412yes—
32C22⋊Dic83—42no—
33(C2×D83)⋊C4—22no—
34(C2×D83)⋊C4—22no—
35C24×C83D411yescharacteristic
36C22×C4×C83D411yescharacteristic
37C22×C4×C83C2311yescharacteristic
38C2×C8×C83—22no—
39C22⋊C4×C83—22no—
40C23×C83—21no—
41C23×C83—21no—
42C23×C83—41no—
43C23×C83—41no—
44C23×C83D812yes—
45C23×C83C2×D411yescharacteristic
46C22×D83—22no—
47C22×D83—42no—
48C22×D83—81no—
49C22×D83—81no—
50C2×C4×C83—21no—
51C2×C4×C83—41no—
52C2×C4×C83—41no—
53C2×C4×C83C2×D412yes—
54C22⋊D83—44no—
55C22⋊D83—42no—
56C22⋊D83—42no—
57C2×Dic83—22no—
58C2×Dic83—42no—
59C2×Dic83—41no—
60C2×Dic83—41no—
61C2×Dic83—41no—
62C2×Dic83—41no—
63C2×Dic83—41no—
64C2×Dic83—41no—
65D332—42no—
66D332—81no—
67C8×C83—42no—
68C22×C83—21no—
69C22×C83—21no—
70C22×C83—22no—
71C22×C83—42no—
72C22×C83—41no—
73C22×C83—41no—
74C22×C83—81no—
75C22×C83#12011yescharacteristic
76C22×C83C2×D811yescharacteristic
77C22×C83AGL(1,ℤ/8)11yescharacteristic
78C2×D83—42no—
79C2×D83—42no—
80C2×D83—42no—
81C2×D83—84no—
82C4×C83—22no—
83C4×C83—41no—
84Dic83—42no—
85Dic83—81no—
86Dic83—81no—
87C2×C83—21no—
88C2×C83—21no—
89C2×C83—42no—
90C2×C83#44711yescharacteristic
91C2×C83#45511yescharacteristic
92C2×C83#45811yescharacteristic
93D83—82no—
94#1607—831nomaximal, Sylow
95C83#160711yescharacteristic, complemented, Sylow
96#534—831no—
97#480—832no—
98#397—831no—
99#327—832no—
100#323D8311yescharacteristic, complemented
101C22×D4—1661no—
102#123—1662no—
103#121—832no—
104#121—1661no—
105#120—1662no—
106C2×C4⋊C4—831no—
107C2×C22⋊C4—1661no—
108C2×C22⋊C4C2×D8311yescharacteristic
109#107—1662no—
110#102—1662no—
111C22⋊C8C2×D8312yescomplemented
112#95—831no—
113#95—1661no—
114C24—1661no—
115C24D33211yescharacteristic
116C2×D4—3322no—
117C2×D4—3322no—
118C2×D4—832no—
119C2×D4—1661no—
120C2×D4—1662no—
121C2×D4—1662no—
122C22×C4—1661no—
123C22×C4—1661no—
124C22×C4—1661no—
125C22×C4D33211yescharacteristic
126C22×C4C22×D8311yescharacteristic
127C2×C8—22no—
128C4⋊C4—832no—
129C4⋊C4—1661no—
130C4⋊C4—1662no—
131C22⋊C4—22no—
132C22⋊C4—3322no—
133C22⋊C4—1662no—
134C22⋊C4—1662no—
135C23—21no—
136C23—21no—
137C23—41no—
138C23—41no—
139C23—3322no—
140C23—6641no—
141C23—6641no—
142C23—1662no—
143C23D66412yes—
144C23C2×D33211yescharacteristic
145D4—3322no—
146D4—3324no—
147D4—3322no—
148D4—3322no—
149D4—6641no—
150C2×C4—21no—
151C2×C4—41no—
152C2×C4—41no—
153C2×C4—3322no—
154C2×C4—3321no—
155C2×C4—3321no—
156C2×C4—3321no—
157C2×C4—3321no—
158C2×C4—3321no—
159C2×C4—3321no—
160C2×C4—1662no—
161C2×C4D4×D8312yes—
162C8—42no—
163C22—21no—
164C22—21no—
165C22—22no—
166C22—42no—
167C22—41no—
168C22—41no—
169C22—81no—
170C22—3322no—
171C22—3322no—
172C22—3322no—
173C22—6644no—
174C22#1458176211yescharacteristic
175C22C2×D66411yescharacteristic
176C22#1458182011yescharacteristic
177C4—22no—
178C4—41no—
179C4—3322no—
180C4—6641no—
181C4—6641no—
182C2—21no—
183C2—21no—
184C2—42no—
185C2—6642no—
186C2#1912783711yescharacteristic
187C2#1912788111yescharacteristic
188C2#1912789211yescharacteristic
subgroupG/H
centerC22#14581762
commutator, cosocleC22×C4×C83C23
FrattiniC22×C4C22×D83
Fitting#323×C83C2
socleC22×C83#120
prime radical#19128263C2
hypercenter#323D83
hypocenterC83#1607
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #323

Complements

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

NH ≅ G/Ntype
2. #19128015C2semidirect
3. #19127626C2semidirect
4. #19127436C2semidirect
5. #323×C83C2semidirect
11. C2×C4⋊Dic83C22semidirect
16. C83⋊#95C22semidirect
18. C22⋊C8×C83C22semidirect
95. C83#1607semidirect
100. #323D83semidirect
111. C22⋊C8C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282632
#191280152
#191276262
#191274362
#323×C832
#160783