Finite Group Databasebeta

← #24906709

Subgroups — #24906709

Order 10624.

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

normal subgroups only

This group has 234 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#19128366C211yesmaximal, characteristic
2#19127978C211yesmaximal, characteristic
3#19127908C211yesmaximal, characteristic, complemented
4#19127447C211yesmaximal, characteristic, complemented
5#19127433C211yesmaximal, characteristic, complemented
6#19127430C211yesmaximal, characteristic
7#500×C83C211yesmaximal, characteristic, complemented
8C2×Q8×D83—21no—
9C2×#13050580—21no—
10#14581791—21no—
11#14581770—22no—
12#14581768—21no—
13C4×Dic166C2211yescharacteristic
14C4×D332C2211yescharacteristic
15#14581754—21no—
16#14581725—21no—
17C4⋊(C83⋊C8)C2211yescharacteristic
18C4⋊(C83⋊C8)C2211yescharacteristic
19#14581716—21no—
20#14581712—21no—
21C4⋊Q8×C83C2211yescharacteristic
22C4⋊Q8×C83C2211yescharacteristic
23#107×C83—22no—
24C4×C8×C83C2211yescharacteristic
25C2×Dic166—22no—
26C2×Dic166D411yescharacteristic
27Q8×D83—42no—
28Q8×D83—42no—
29C2×D332D411yescharacteristic
30C2×C4×D83—22no—
31C2×C4×D83—21no—
32#13050580—42no—
33C2×C83⋊C8—21no—
34C2×C83⋊C8—21no—
35C4⋊Dic83D411yescharacteristic
36C4⋊Dic83D411yescharacteristic
37Dic83⋊C4—22no—
38Dic83⋊C4—21no—
39(C2×D83)⋊C4—22no—
40(C2×D83)⋊C4—21no—
41C4×Dic83—21no—
42C2×Q8×C83—21no—
43C2×Q8×C83—21no—
44C2×C8×C83—21no—
45C4⋊C4×C83—21no—
46C4⋊C4×C83—21no—
47C4⋊C4×C83—21no—
48C4⋊C4×C83D412yes—
49C42×C83C2311yescharacteristic
50C22×D83—21no—
51Q8×C83—41no—
52Q8×C83—41no—
53Q8×C83—41no—
54Q8×C83—41no—
55C2×C4×C83—22no—
56C2×C4×C83—41no—
57C2×C4×C83C2×D411yescharacteristic
58C2×C4×C83C2×D411yescharacteristic
59C2×C4×C83C2×D411yescharacteristic
60C2×Dic83—21no—
61C2×Dic83—21no—
62C2×Dic83—21no—
63D332—21no—
64D332Q1612yescomplemented
65C4×D83—44no—
66C4×D83—41no—
67C4×D83—41no—
68Dic166—21no—
69Dic166—44no—
70Dic166D812yes—
71C8×C83—22no—
72C83⋊C8—41no—
73C83⋊C8—41no—
74C22×C83#12011yescharacteristic
75C2×D83—22no—
76C2×D83—41no—
77C2×D83—41no—
78C4×C83—21no—
79C4×C83—42no—
80C4×C83—81no—
81C4×C83C2×D811yescharacteristic
82C4×C83C2×Q1611yescharacteristic
83C4×C83AGL(1,ℤ/8)11yescharacteristic
84C4×C83#13711yescharacteristic
85Dic83—22no—
86Dic83—41no—
87Dic83—41no—
88C2×C83#44711yescharacteristic
89C2×C83#45111yescharacteristic
90C2×C83#45611yescharacteristic
91D83—42no—
92#1649—831nomaximal, Sylow
93C83#164911yescharacteristic, complemented, Sylow
94D4×Q8—831no—
95#500D8311yescharacteristic, complemented
96#474—831no—
97#462—831no—
98#332—831no—
99Q8⋊C8—831no—
100D4⋊C8—831no—
101C22×Q8—1661no—
102C2×Q16—1661no—
103C4⋊Q8—1661no—
104C4⋊Q8C2×D8311yescharacteristic
105C4⋊Q8C2×D8311yescharacteristic
106C22⋊Q8—1661no—
107C22⋊Q8—1662no—
108C4×Q8—831no—
109C4×D4—831no—
110C4×D4—1661no—
111#107—22no—
112#107—1661no—
113C4⋊C8—831no—
114C4⋊C8—831no—
115#103—1661no—
116#102—1661no—
117C4×C8C2×D8311yescharacteristic
118C2×Q8—21no—
119C2×Q8—21no—
120C2×Q8—3322no—
121C2×Q8—3322no—
122C2×Q8—831no—
123C2×Q8—1662no—
124C2×D4—831no—
125C22×C4—1662no—
126C22×C4—1661no—
127Q16—3322no—
128C2×C8—21no—
129C2×C8—1661no—
130C2×C8—1661no—
131C4⋊C4—21no—
132C4⋊C4—21no—
133C4⋊C4—21no—
134C4⋊C4—831no—
135C4⋊C4—831no—
136C4⋊C4—1662no—
137C4⋊C4—1661no—
138C4⋊C4C22⋊D8312yes—
139C22⋊C4—1662no—
140C22⋊C4—1661no—
141C42—1661no—
142C42C22×D8311yescharacteristic
143C23—1661no—
144Q8—41no—
145Q8—41no—
146Q8—41no—
147Q8—41no—
148Q8—3324no—
149Q8—832no—
150Q8—1661no—
151D4—832no—
152D4—1661no—
153C2×C4—22no—
154C2×C4—41no—
155C2×C4—3324no—
156C2×C4—3321no—
157C2×C4—3321no—
158C2×C4—1661no—
159C2×C4—1661no—
160C2×C4—1661no—
161C2×C4D4×D8311yescharacteristic
162C2×C4D4×D8311yescharacteristic
163C2×C4C2×C22⋊D8311yescharacteristic
164C8—22no—
165C8—3321no—
166C8—3321no—
167C22—3321no—
168C22—3321no—
169C22—1662no—
170C22#1458176011yescharacteristic
171C4—21no—
172C4—42no—
173C4—81no—
174C4—3321no—
175C4—3321no—
176C4—1662no—
177C4D8×D8311yescharacteristic
178C4Q16×D8311yescharacteristic
179C4#1458181611yescharacteristic
180C4#1458182411yescharacteristic
181C2—3322no—
182C2#1912783311yescharacteristic
183C2#1912785511yescharacteristic
184C2#1912788311yescharacteristic
subgroupG/H
centerC22#14581760
commutator, cosocleC42×C83C23
FrattiniC42C22×D83
Fitting#500×C83C2
socleC22×C83#120
prime radicalC2×D332D4
hypercenter#500D83
hypocenterC83#1649
LCSG ▷ C42×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #500

Complements

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

NH ≅ G/Ntype
3. #19127908C2semidirect
4. #19127447C2semidirect
5. #19127433C2semidirect
7. #500×C83C2semidirect
64. D332Q16semidirect
93. C83#1649semidirect
95. #500D83semidirect

Maximal subgroups

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

subgroupindex
#191283662
#191279782
#191279082
#191274472
#191274332
#191274302
#500×C832
#164983