Finite Group Databasebeta

← #24906398

Subgroups — #24906398

Order 10624.

310 conjugacy classes of subgroups; only the 57 normal subgroups are shown.

normal subgroups only

This group has 310 conjugacy classes of subgroups — too many to draw. Only its 57 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#19127834C212yesmaximal, complemented
2#19127618C211yesmaximal, characteristic, complemented
3#19127435C212yesmaximal, complemented
4#19127427C211yesmaximal, characteristic, complemented
5#537×C83C211yesmaximal, characteristic, complemented
6C22×D332C2211yescharacteristic
7C2×#13050575—22no—
8C2×(C2×D83)⋊C4—21no—
9C2×(C2×D83)⋊C4—21no—
10#14581712—22no—
11C22⋊(C83⋊C8)C2212yes—
12C83⋊#95C2211yescharacteristic
13C2×#41×C83—21no—
14C22⋊Q8×C83—22no—
15#121×C83—22no—
16#121×C83—21no—
17#121×C83C2212yes—
18C2×C4⋊C4×C83C2211yescharacteristic
19C23×D83—21no—
20C2×D332—22no—
21C2×D332—42no—
22C22×Dic83—21no—
23#13050575—44no—
24C2×C83⋊C8—22no—
25(C2×D83)⋊C4—42no—
26(C2×D83)⋊C4—42no—
27#41×C83—42no—
28#41×C83—42no—
29C2×Q8×C83—21no—
30C2×D4×C83—21no—
31C2×D4×C83—22no—
32C2×D4×C83—21no—
33C2×D4×C83D412yes—
34C22×C4×C83—22no—
35C22×C4×C83D411yescharacteristic
36C22×C4×C83D411yescharacteristic
37C22×C4×C83C2311yescharacteristic
38C4⋊C4×C83—21no—
39C4⋊C4×C83—22no—
40C4⋊C4×C83D412yes—
41C22⋊C4×C83—22no—
42C22⋊C4×C83—22no—
43C23×C83—22no—
44C23×C83C2×D411yescharacteristic
45C22×D83—22no—
46C22×D83—41no—
47C22×D83—41no—
48C22×D83—41no—
49C22×D83—41no—
50C22×D83—41no—
51C22×D83—41no—
52Q8×C83—42no—
53D4×C83—42no—
54D4×C83—42no—
55D4×C83—42no—
56D4×C83—42no—
57D4×C83—81no—
58C2×C4×C83—21no—
59C2×C4×C83—21no—
60C2×C4×C83—21no—
61C2×C4×C83—22no—
62C2×C4×C83—41no—
63C2×C4×C83—44no—
64C2×C4×C83—41no—
65C2×C4×C83D812yescomplemented
66C2×C4×C83C2×D412yes—
67C2×Dic83—41no—
68C2×Dic83—41no—
69C2×Dic83—41no—
70D332—24no—
71D332—42no—
72C83⋊C8—42no—
73C22×C83—21no—
74C22×C83—21no—
75C22×C83—42no—
76C22×C83—42no—
77C22×C83—42no—
78C22×C83#12011yescharacteristic
79C22×C83C2×D811yescharacteristic
80C22×C83AGL(1,ℤ/8)11yescharacteristic
81C2×D83—42no—
82C2×D83—42no—
83C2×D83—42no—
84C2×D83—81no—
85C2×D83—81no—
86C2×D83—81no—
87C2×D83—81no—
88C4×C83—22no—
89C4×C83—22no—
90C4×C83—42no—
91C4×C83—41no—
92Dic83—81no—
93C2×C83—21no—
94C2×C83—21no—
95C2×C83—82no—
96C2×C83#44711yescharacteristic
97C2×C83#45411yescharacteristic
98C2×C83B(4,𝔽2)11yescharacteristic
99D83—82no—
100#1598—831nomaximal, Sylow
101C83#159811yescharacteristic, complemented, Sylow
102#537D8311yescharacteristic, complemented
103#447—832no—
104#394—831no—
105#327—832no—
106#324—831no—
107C2×#41—21no—
108C22×D4—831no—
109C2×D8—1662no—
110C22⋊Q8—22no—
111#121—22no—
112#121—21no—
113#121C2×D8312yes—
114C2×C4⋊C4C2×D8311yescharacteristic
115C2×C22⋊C4—1661no—
116C2×C22⋊C4—1661no—
117#102—1662no—
118C22⋊C8—832no—
119#95—831no—
120C24—1661no—
121#41—42no—
122#41—42no—
123C2×Q8—21no—
124C2×D4—21no—
125C2×D4—22no—
126C2×D4—21no—
127C2×D4—3322no—
128C2×D4—1662no—
129C2×D4C22⋊D8312yes—
130C22×C4—22no—
131C22×C4—1661no—
132C22×C4C22⋊D8311yescharacteristic
133C22×C4C22⋊D8311yescharacteristic
134C22×C4C22×D8311yescharacteristic
135D8—3324no—
136C2×C8—1662no—
137C4⋊C4—21no—
138C4⋊C4—22no—
139C4⋊C4C22⋊D8312yes—
140C22⋊C4—22no—
141C22⋊C4—22no—
142C22⋊C4—3322no—
143C22⋊C4—3322no—
144C23—22no—
145C23—3321no—
146C23—3321no—
147C23—3321no—
148C23—3321no—
149C23—3321no—
150C23—3321no—
151C23—1662no—
152C23C2×C22⋊D8311yescharacteristic
153Q8—42no—
154D4—42no—
155D4—42no—
156D4—42no—
157D4—42no—
158D4—81no—
159D4—3322no—
160D4—1664no—
161C2×C4—21no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—22no—
165C2×C4—41no—
166C2×C4—44no—
167C2×C4—41no—
168C2×C4—3321no—
169C2×C4—3321no—
170C2×C4—3321no—
171C2×C4#1305057512yescomplemented
172C2×C4C2×C22⋊D8312yes—
173C8—3322no—
174C22—21no—
175C22—21no—
176C22—42no—
177C22—42no—
178C22—42no—
179C22—3322no—
180C22—3322no—
181C22—3322no—
182C22—6641no—
183C22—6641no—
184C22—6641no—
185C22—6641no—
186C22#1458176111yescharacteristic
187C22C2×#1305057511yescharacteristic
188C22#1458181811yescharacteristic
189C4—22no—
190C4—22no—
191C4—42no—
192C4—41no—
193C4—6641no—
194C2—21no—
195C2—21no—
196C2—82no—
197C2—6642no—
198C2#1912783511yescharacteristic
199C2#1912787411yescharacteristic
200C2#1912788811yescharacteristic
subgroupG/H
centerC22#14581761
commutator, cosocleC22×C4×C83C23
FrattiniC22×C4C22×D83
Fitting#537×C83C2
socleC22×C83#120
hypercenter#537D83
hypocenterC83#1598
LCSG ▷ C22×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #537

Complements

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

NH ≅ G/Ntype
1. #19127834C2semidirect
2. #19127618C2semidirect
3. #19127435C2semidirect
4. #19127427C2semidirect
5. #537×C83C2semidirect
65. C2×C4×C83D8semidirect
101. C83#1598semidirect
102. #537D83semidirect
171. C2×C4#13050575semidirect

Maximal subgroups

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

subgroupindex
#191278342
#191276182
#191274352
#191274272
#537×C832
#159883