Finite Group Databasebeta

← #24915347

Subgroups — #24915347

Order 10624.

376 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 376 conjugacy classes of subgroups — too many to draw. Only its 137 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#19127643C212yesmaximal
2#19127640C211yesmaximal, characteristic
3#19127635C212yesmaximal
4#19127621C211yesmaximal, characteristic
5#19127608C212yesmaximal
6#19127605C212yesmaximal
7#19127588C212yesmaximal
8#19127585C212yesmaximal
9#391×C83C211yesmaximal, characteristic
10C22×Dic166C2211yescharacteristic
11C2×C4⋊Dic83C2211yescharacteristic
12C2×C4⋊Dic83C2212yes—
13C2×Dic83⋊C4C2212yes—
14C2×Dic83⋊C4C2212yes—
15C2×Dic83⋊C4C2212yes—
16C2×C4×Dic83C2212yes—
17C83⋊#95C2212yes—
18C83⋊#95C2212yes—
19C83⋊#95C2212yes—
20C83⋊#95C2212yes—
21C83⋊#95C2212yes—
22C83⋊#95C2212yes—
23C83⋊#95C2212yes—
24C83⋊#95C2211yescharacteristic
25C83⋊#95C2211yescharacteristic
26C22×Q8×C83C2211yescharacteristic
27C2×C4⋊C4×C83C2211yescharacteristic
28C2×C42×C83C2211yescharacteristic
29#95×C83C2212yes—
30#95×C83C2212yes—
31C2×Dic166—24no—
32C2×Dic166—41no—
33C22×Dic83C2312yes—
34C22×Dic83C2312yes—
35C22×Dic83C2312yes—
36C22×Dic83C2312yes—
37C4⋊Dic83—41no—
38C4⋊Dic83—42no—
39Dic83⋊C4—24no—
40Dic83⋊C4—24no—
41Dic83⋊C4—42no—
42C4×Dic83—42no—
43C2×Q8×C83—41no—
44C2×Q8×C83—41no—
45C2×Q8×C83Q814yescomplemented
46C22×C4×C83C2311yescharacteristic
47C22×C4×C83C2312yes—
48C22×C4×C83C2311yescharacteristic
49C22×C4×C83C2312yes—
50C22×C4×C83C2311yescharacteristic
51C4⋊C4×C83—41no—
52C42×C83—41no—
53C23×C83C2411yescharacteristic
54Q8×C83—44no—
55C2×C4×C83—21no—
56C2×C4×C83—21no—
57C2×C4×C83—22no—
58C2×C4×C83—22no—
59C2×C4×C83—22no—
60C2×C4×C83—22no—
61C2×C4×C83—22no—
62C2×C4×C83—22no—
63C2×C4×C83—21no—
64C2×C4×C83—21no—
65C2×C4×C83—22no—
66C2×C4×C83C2×Q814yes—
67C2×C4×C83C2×Q812yes—
68C2×Dic83—22no—
69C2×Dic83—22no—
70C2×Dic83—22no—
71C2×Dic83—22no—
72C2×Dic83—22no—
73C2×Dic83—22no—
74C2×Dic83—22no—
75C2×Dic83—22no—
76C2×Dic83—22no—
77C2×Dic83—24no—
78C2×Dic83#4114yes—
79Dic166—44no—
80C22×C83C22×Q811yescharacteristic
81C22×C83C2×#4112yes—
82C22×C83H(5,𝔽2)11yescharacteristic
83C22×C83H(5,𝔽2)11yescharacteristic
84C22×C832−1+412yes—
85C4×C83—22no—
86C4×C83—24no—
87C4×C83—42no—
88C4×C83—41no—
89C4×C83—41no—
90Dic83—24no—
91Dic83—42no—
92Dic83—42no—
93Dic83—42no—
94C2×C83#54212yes—
95C2×C83#54311yescharacteristic
96C2×C83#55111yescharacteristic
97C2×C83#55712yes—
98C2×C83#56311yescharacteristic
99#2799—831nomaximal, Sylow
100C83#279911yescharacteristic, complemented, Sylow
101#400—831no—
102#400—832no—
103#398—832no—
104#395—831no—
105#391—832no—
106#391—832no—
107#391D8311yescharacteristic
108#387—832no—
109#387—832no—
110C22×Q8—831no—
111C22×Q8C2×D8311yescharacteristic
112C2×C4⋊C4—831no—
113C2×C4⋊C4—832no—
114C2×C4⋊C4—832no—
115C2×C4⋊C4—832no—
116C2×C4⋊C4—832no—
117C2×C4⋊C4C2×D8311yescharacteristic
118C2×C42—832no—
119C2×C42C2×D8311yescharacteristic
120#95—832no—
121#95—832no—
122#95—832no—
123#95—832no—
124#95—832no—
125#95—832no—
126#95—832no—
127#95—831no—
128#95—831no—
129#95C2×D8312yes—
130#95C2×D8312yes—
131C2×Q8—41no—
132C2×Q8—41no—
133C2×Q8—3321no—
134C2×Q8—1664no—
135C2×Q8Dic16614yescomplemented
136C22×C4—832no—
137C22×C4—832no—
138C22×C4—832no—
139C22×C4—832no—
140C22×C4C22×D8311yescharacteristic
141C22×C4C22×D8312yes—
142C22×C4C22×D8311yescharacteristic
143C22×C4C22×D8311yescharacteristic
144C22×C4C22×D8312yes—
145C4⋊C4—41no—
146C4⋊C4—3322no—
147C4⋊C4—3322no—
148C4⋊C4—3321no—
149C4⋊C4—1664no—
150C4⋊C4—1664no—
151C42—41no—
152C42—3322no—
153C23C23×D8311yescharacteristic
154Q8—44no—
155Q8—3324no—
156C2×C4—21no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—22no—
160C2×C4—22no—
161C2×C4—22no—
162C2×C4—22no—
163C2×C4—22no—
164C2×C4—21no—
165C2×C4—21no—
166C2×C4—22no—
167C2×C4—834no—
168C2×C4—1662no—
169C2×C4—1662no—
170C2×C4—1662no—
171C2×C4—1662no—
172C2×C4—1662no—
173C2×C4—1662no—
174C2×C4—1662no—
175C2×C4—1664no—
176C2×C4—1662no—
177C2×C4—1662no—
178C2×C4C2×Dic16612yes—
179C2×C4C2×Dic16614yes—
180C22C22×Dic16611yescharacteristic
181C22#41×D8312yes—
182C22#1458183711yescharacteristic
183C22#1458183711yescharacteristic
184C22#1458184012yes—
185C4—22no—
186C4—24no—
187C4—42no—
188C4—41no—
189C4—41no—
190C4—3322no—
191C4—3322no—
192C4—3322no—
193C4—1664no—
194C2#1912832112yes—
195C2#1912832711yescharacteristic
196C2#1912838111yescharacteristic
197C2#1912842712yes—
198C2#1912845011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, prime radical, cosocleC23×C83C24
Fitting#391×C83C2
hypercenter#391D83
hypocenterC83#2799
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
45. C2×Q8×C83Q8semidirect
100. C83#2799semidirect
135. C2×Q8Dic166semidirect

Maximal subgroups

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

subgroupindex
#191276432
#191276402
#191276352
#191276212
#191276082
#191276052
#191275882
#191275852
#391×C832
#279983