Finite Group Databasebeta

← #24909134

Subgroups — #24909134

Order 10624.

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

normal subgroups only

This group has 270 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#19128265C211yesmaximal, characteristic
2#19127675C211yesmaximal, characteristic, complemented
3#19127493C214yesmaximal, complemented
4#410×C83C211yesmaximal, characteristic, complemented
5C22×D332C411yescharacteristic, complemented
6#14581788C2212yes—
7#14581781C412yescomplemented
8#14581766—21no—
9#14581766—21no—
10#14581760—21no—
11#14581760—21no—
12#14581751—21no—
13#14581751C411yescharacteristic, complemented
14#14581707—21no—
15#14581707C2212yescomplemented
16C2×#41×C83C2211yescharacteristic
17#117×C83—21no—
18#99×C83—21no—
19#99×C83C2212yescomplemented
20C23×D83—21no—
21C2×D332—21no—
22C2×D332—41no—
23C2×D332—41no—
24C2×D332C2×C412yes—
25C2×C22⋊D83—22no—
26C2×C22⋊D83—22no—
27C4⋊Dic83—21no—
28C4⋊Dic83—41no—
29C22⋊Dic83—21no—
30C22⋊Dic83—22no—
31(C2×D83)⋊C4—22no—
32(C2×D83)⋊C4—22no—
33C4×Dic83—41no—
34C4×Dic83C2×C412yes—
35#41×C83—41no—
36#41×C83—41no—
37C2×Q8×C83C2×C411yescharacteristic
38C2×D4×C83D411yescharacteristic
39C2×D4×C83D411yescharacteristic
40C2×D4×C83C2311yescharacteristic
41C22×C4×C83C2×C411yescharacteristic
42C22×C4×C83D411yescharacteristic
43C22×C4×C83D411yescharacteristic
44C4⋊C4×C83—41no—
45C22⋊C4×C83—22no—
46C42×C83—41no—
47C23×C83C22⋊C411yescharacteristic
48C23×C83C2×D411yescharacteristic
49C23×C83C2×D411yescharacteristic
50C22×D83—22no—
51C22×D83—42no—
52C22×D83—82no—
53Q8×C83—41no—
54D4×C83—41no—
55D4×C83—41no—
56D4×C83—41no—
57C2×C4×C83—21no—
58C2×C4×C83—21no—
59C2×C4×C83—21no—
60C2×C4×C83—21no—
61C2×C4×C83—21no—
62C2×C4×C83—21no—
63C2×C4×C83—42no—
64C2×C4×C83C22⋊C411yescharacteristic
65C2×C4×C83C22⋊C411yescharacteristic
66C2×C4×C83C22⋊C411yescharacteristic
67C2×C4×C83C22×C411yescharacteristic
68C22⋊D83—82no—
69C22⋊D83—82no—
70C2×Dic83—22no—
71C2×Dic83—42no—
72D332—42no—
73D332—41no—
74D332—41no—
75C22×C83—21no—
76C22×C83—21no—
77C22×C83—21no—
78C22×C83—41no—
79C22×C83—41no—
80C22×C83—41no—
81C22×C83C2×C22⋊C411yescharacteristic
82C2×D83—42no—
83C2×D83—42no—
84C2×D83—42no—
85C2×D83—84no—
86C4×C83—21no—
87C4×C83—21no—
88C4×C83—21no—
89C4×C83—82no—
90C4×C83#9911yescharacteristic
91C4×C83#9911yescharacteristic
92Dic83—42no—
93Dic83—82no—
94C2×C83—21no—
95C2×C83—41no—
96C2×C83—41no—
97C2×C83—41no—
98C2×C83C2×#9911yescharacteristic
99D83—82no—
100#2128—831nomaximal, Sylow
101C83#212811yescharacteristic, complemented, Sylow
102#535—831no—
103#410—831no—
104#410D8311yescharacteristic, complemented
105#353—834no—
106C2×#41C2×D8311yescharacteristic
107C22×D4—831no—
108#127—832no—
109#124—832no—
110#121—1661no—
111#121—1661no—
112#120—1661no—
113#120—1661no—
114#117—21no—
115#117—831no—
116#117—1661no—
117#99—21no—
118#99—832no—
119#99—1661no—
120#99C2×D8312yescomplemented
121C24—1661no—
122#41—41no—
123#41—41no—
124C2×Q8C4×D8311yescharacteristic
125C2×D4—3321no—
126C2×D4—3321no—
127C2×D4—832no—
128C2×D4—1662no—
129C2×D4—1662no—
130C2×D4—1661no—
131C2×D4D33211yescharacteristic
132C2×D4C22⋊D8311yescharacteristic
133C2×D4C22×D8311yescharacteristic
134C22×C4C4×D8311yescharacteristic
135C22×C4D33211yescharacteristic
136C22×C4C22⋊D8311yescharacteristic
137C4⋊C4—41no—
138C4⋊C4—3321no—
139C4⋊C4—1661no—
140C22⋊C4—22no—
141C22⋊C4—1662no—
142C22⋊C4—1661no—
143C22⋊C4—1662no—
144C22⋊C4—1662no—
145C42—41no—
146C42—3321no—
147C42—832no—
148C23—3322no—
149C23—6642no—
150C23—1662no—
151C23(C2×D83)⋊C411yescharacteristic
152C23C2×C22⋊D8311yescharacteristic
153C23C2×D33211yescharacteristic
154Q8—41no—
155D4—41no—
156D4—41no—
157D4—41no—
158D4—3322no—
159D4—3321no—
160D4—3321no—
161D4—6642no—
162D4—6642no—
163C2×C4—21no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—42no—
170C2×C4—3322no—
171C2×C4—1662no—
172C2×C4(C2×D83)⋊C411yescharacteristic
173C2×C4(C2×D83)⋊C411yescharacteristic
174C2×C4(C2×D83)⋊C411yescharacteristic
175C2×C4C2×C4×D8311yescharacteristic
176C22—21no—
177C22—21no—
178C22—21no—
179C22—41no—
180C22—41no—
181C22—41no—
182C22—3322no—
183C22—3322no—
184C22—3322no—
185C22—6644no—
186C22C2×(C2×D83)⋊C411yescharacteristic
187C4—21no—
188C4—21no—
189C4—21no—
190C4—82no—
191C4—3322no—
192C4—6642no—
193C4#1458170611yescharacteristic
194C4#1458170611yescharacteristic
195C2—21no—
196C2—41no—
197C2—41no—
198C2—41no—
199C2—6642no—
200C2C2×#1458170611yescharacteristic
subgroupG/H
centerC2C2×#14581706
commutatorC2×C4×C83C22×C4
FrattiniC2×D4C22×D83
Fitting#410×C83C2
socleC2×C83C2×#99
prime radical#19128265C2
cosocleC2×D4×C83C23
hypercenter#410D83
hypocenterC83#2128
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #410

Complements

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

NH ≅ G/Ntype
2. #19127675C2semidirect
3. #19127493C2semidirect
4. #410×C83C2semidirect
5. C22×D332C4semidirect
7. #14581781C4semidirect
13. #14581751C4semidirect
15. #14581707C22semidirect
19. #99×C83C22semidirect
101. C83#2128semidirect
104. #410D83semidirect
120. #99C2×D83semidirect

Maximal subgroups

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

subgroupindex
#191282652
#191276752
#191274932
#410×C832
#212883