Finite Group Databasebeta

← #353×D83 #24909119

Subgroups — #353×D83 #24909119

Order 10624.

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

normal subgroups only

This group has 368 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#127×D83C211yesmaximal, characteristic
2#99×D83C212yesmaximal, complemented
3#19127494C211yesmaximal, characteristic, complemented
4#19127493C212yesmaximal, complemented
5#353×C83C211yesmaximal, characteristic, complemented
6C2×D4×D83—22no—
7C2×D4×D83C411yescharacteristic, complemented
8C2×D4×D83C2211yescharacteristic
9#14581789C411yescharacteristic, complemented
10#14581788—22no—
11#14581788C411yescharacteristic, complemented
12#14581788C2211yescharacteristic
13C22⋊C4×D83—22no—
14C42×D83C411yescharacteristic, complemented
15#14581707C2212yescomplemented
16#14581706—22no—
17#127×C83C2211yescharacteristic
18#99×C83C2212yescomplemented
19C23×D83—21no—
20C23×D83D412yes—
21C2×D332—22no—
22C2×D332C2×C411yescharacteristic
23C2×D332C2×C411yescharacteristic, complemented
24C2×C22⋊D83—22no—
25C2×C22⋊D83—22no—
26C2×C22⋊D83—21no—
27C2×C22⋊D83D412yes—
28D4×D83—41no—
29D4×D83—41no—
30D4×D83—41no—
31D4×D83—41no—
32D4×D83—48no—
33C2×C4×D83—21no—
34C2×C4×D83—42no—
35C2×C4×D83C2×C411yescharacteristic
36C22⋊Dic83—22no—
37(C2×D83)⋊C4—42no—
38C4×Dic83—21no—
39C4×Dic83C2×C411yescharacteristic, complemented
40C2×D4×C83—22no—
41C2×D4×C83C2×C411yescharacteristic, complemented
42C2×D4×C83C2311yescharacteristic
43C22⋊C4×C83—22no—
44C42×C83C2×C411yescharacteristic, complemented
45C23×C83—21no—
46C23×C83C2×D412yes—
47C22×D83—22no—
48C22×D83—22no—
49C22×D83—21no—
50C22×D83—44no—
51C22×D83—44no—
52C22×D83—41no—
53C22×D83—41no—
54C22×D83C22⋊C411yescharacteristic
55C22×D83C22⋊C412yes—
56D4×C83—41no—
57D4×C83—41no—
58D4×C83—44no—
59C2×C4×C83—21no—
60C2×C4×C83—42no—
61C2×C4×C83C22×C411yescharacteristic
62C22⋊D83—42no—
63C22⋊D83—44no—
64C22⋊D83—44no—
65C22⋊D83—42no—
66C2×Dic83—21no—
67C2×Dic83—42no—
68C2×Dic83C22⋊C411yescharacteristic
69D332—41no—
70D332—41no—
71D332—44no—
72C4×D83—24no—
73C4×D83—21no—
74C4×D83—21no—
75C4×D83—84no—
76C22×C83—22no—
77C22×C83—42no—
78C22×C83—42no—
79C22×C83—41no—
80C22×C83C2×C22⋊C411yescharacteristic
81C2×D83—22no—
82C2×D83—22no—
83C2×D83—44no—
84C2×D83—42no—
85C2×D83—42no—
86C2×D83—41no—
87C2×D83—82no—
88C2×D83—82no—
89C2×D83—82no—
90C2×D83#9911yescharacteristic
91C2×D83#9911yescharacteristic
92C4×C83—22no—
93C4×C83—21no—
94C4×C83—82no—
95Dic83—22no—
96Dic83—21no—
97Dic83—82no—
98C2×C83—21no—
99C2×C83—42no—
100C2×C83—81no—
101C2×C83C2×#9911yescharacteristic
102D83—21no—
103D83—42no—
104D83—81no—
105D83#35312yesdirect product factor
106C2×#353—831nomaximal, Sylow
107C83C2×#35311yescharacteristic, complemented, Sylow
108C2×#127—831no—
109C2×#99—832no—
110#353—831no—
111#353—832no—
112#353D8311yescharacteristic, direct product factor
113C22×D4—831no—
114C22×D4—831no—
115C22×D4—1662no—
116#127—831no—
117#127—831no—
118#127—831no—
119#127—1662no—
120#127C2×D8311yescharacteristic
121C2×C22⋊C4—1662no—
122C2×C42—831no—
123#99—832no—
124#99—1662no—
125#99C2×D8312yescomplemented
126C24—832no—
127C24—1661no—
128C2×D4—22no—
129C2×D4—3321no—
130C2×D4—3321no—
131C2×D4—3328no—
132C2×D4—3321no—
133C2×D4—3321no—
134C2×D4—832no—
135C2×D4—831no—
136C2×D4—831no—
137C2×D4—1662no—
138C2×D4—1661no—
139C2×D4—1662no—
140C2×D4—1662no—
141C2×D4C4×D8311yescharacteristic, complemented
142C2×D4C22×D8311yescharacteristic
143C22×C4—3322no—
144C22×C4—831no—
145C22×C4—1661no—
146C22⋊C4—22no—
147C22⋊C4—3322no—
148C22⋊C4—1662no—
149C42—831no—
150C42—1661no—
151C42C4×D8311yescharacteristic, complemented
152C23—21no—
153C23—3324no—
154C23—3321no—
155C23—3321no—
156C23—3324no—
157C23—832no—
158C23—831no—
159C23—1662no—
160C23—1662no—
161C23—1661no—
162C23D4×D8312yes—
163D4—41no—
164D4—41no—
165D4—44no—
166D4—3321no—
167D4—3324no—
168D4—3321no—
169D4—3322no—
170D4—3324no—
171D4—3324no—
172D4—3322no—
173C2×C4—21no—
174C2×C4—42no—
175C2×C4—3322no—
176C2×C4—831no—
177C2×C4—6644no—
178C2×C4—1661no—
179C2×C4—1661no—
180C2×C4—1664no—
181C2×C4—1661no—
182C2×C4C2×C4×D8311yescharacteristic
183C22—22no—
184C22—42no—
185C22—42no—
186C22—41no—
187C22—3322no—
188C22—3322no—
189C22—3324no—
190C22—3321no—
191C22—831no—
192C22—831no—
193C22—6642no—
194C22—6642no—
195C22—6642no—
196C22—1662no—
197C22—1662no—
198C22C22⋊C4×D8311yescharacteristic
199C4—22no—
200C4—21no—
201C4—82no—
202C4—6642no—
203C4—1662no—
204C4—1661no—
205C2—21no—
206C2—42no—
207C2—81no—
208C2—3322no—
209C2—832no—
210C2—6641no—
211C2—1661no—
212C2#99×D8311yescharacteristic
subgroupG/H
centerC2#99×D83
commutatorC2×C4×C83C22×C4
FrattiniC2×D4C22×D83
Fitting#353×C83C2
socleC2×C83C2×#99
prime radical#127×D83C2
cosocleC2×D4×C83C23
hypercenter#353D83
hypocenterC83C2×#353
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #353

Complements

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

NH ≅ G/Ntype
2. #99×D83C2semidirect
3. #19127494C2semidirect
4. #19127493C2semidirect
5. #353×C83C2semidirect
7. C2×D4×D83C4semidirect
9. #14581789C4semidirect
11. #14581788C4semidirect
14. C42×D83C4semidirect
15. #14581707C22semidirect
18. #99×C83C22semidirect
23. C2×D332C2×C4semidirect
39. C4×Dic83C2×C4semidirect
41. C2×D4×C83C2×C4semidirect
44. C42×C83C2×C4semidirect
105. D83#353direct (N × H)
107. C83C2×#353semidirect
112. #353D83direct (N × H)
125. #99C2×D83semidirect
141. C2×D4C4×D83semidirect
151. C42C4×D83semidirect

Maximal subgroups

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

subgroupindex
#127×D832
#99×D832
#191274942
#191274932
#353×C832
C2×#35383