Finite Group Databasebeta

← #24911774

Subgroups — #24911774

Order 10624.

584 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 584 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×C4×C22⋊D83C211yesmaximal, characteristic
2#19127639C212yesmaximal, complemented
3#19127636C212yesmaximal, complemented
4#19127597C212yesmaximal, complemented
5#19127580C212yesmaximal
6#19127574C212yesmaximal, complemented
7#19127561C211yesmaximal, characteristic, complemented
8#19127558C212yesmaximal, complemented
9C22×C4⋊C4×C83C211yesmaximal, characteristic, complemented
10C22×C22⋊D83C2211yescharacteristic
11C22×C4×D83C2211yescharacteristic
12C4×C22⋊D83—24no—
13C2×C4⋊Dic83C2212yes—
14C2×Dic83⋊C4C2211yescharacteristic
15C2×Dic83⋊C4C2214yes—
16C2×C22⋊Dic83C2212yes—
17C2×C22⋊Dic83C2211yescharacteristic
18C2×(C2×D83)⋊C4C2211yescharacteristic
19C2×(C2×D83)⋊C4C2214yes—
20C2×C4×Dic83C2211yescharacteristic
21C83⋊#95C2214yes—
22C83⋊#95C2212yes—
23C83⋊#95C2214yes—
24C23×C4×C83C2211yescharacteristic
25C23×C4×C83C2212yes—
26C2×C4⋊C4×C83—24no—
27C2×C4⋊C4×C83C2212yes—
28C2×C4⋊C4×C83C2212yes—
29C23×D83C2311yescharacteristic
30C2×C22⋊D83—22no—
31C2×C22⋊D83—22no—
32C2×C22⋊D83Q814yes—
33C22×Dic83C2311yescharacteristic
34C22×Dic83C2314yes—
35C22×Dic83C2311yescharacteristic
36C22×Dic83C2311yescharacteristic
37C2×C4×D83—22no—
38C2×C4×D83—24no—
39C4⋊Dic83—24no—
40Dic83⋊C4—22no—
41Dic83⋊C4—44no—
42C22⋊Dic83—22no—
43C22⋊Dic83—24no—
44(C2×D83)⋊C4—22no—
45(C2×D83)⋊C4—44no—
46C4×Dic83—22no—
47C24×C83C2311yescharacteristic
48C22×C4×C83—24no—
49C22×C4×C83—24no—
50C22×C4×C83—44no—
51C22×C4×C83D414yes—
52C22×C4×C83C2314yes—
53C22×C4×C83C2311yescharacteristic
54C22×C4×C83C2311yescharacteristic
55C4⋊C4×C83—24no—
56C4⋊C4×C83—24no—
57C23×C83—22no—
58C23×C83—22no—
59C23×C83C2×D412yes—
60C23×C83C2×Q812yes—
61C23×C83#4112yes—
62C23×C83C2411yescharacteristic
63C22×D83—21no—
64C22×D83—22no—
65C22×D83—21no—
66C22×D83—22no—
67C22×D83C2×Q812yes—
68C2×C4×C83—22no—
69C2×C4×C83—28no—
70C2×C4×C83—24no—
71C2×C4×C83—24no—
72C2×C4×C83—24no—
73C2×C4×C83—48no—
74C2×C4×C83C2×D412yes—
75C2×C4×C83C2×D412yes—
76C2×C4×C83#4114yes—
77C22⋊D83—28no—
78C2×Dic83—21no—
79C2×Dic83—21no—
80C2×Dic83—21no—
81C2×Dic83—21no—
82C2×Dic83—24no—
83C2×Dic83—24no—
84C2×Dic83—22no—
85C2×Dic83—24no—
86C2×Dic83—22no—
87C2×Dic83C2×Q812yes—
88C4×D83—44no—
89C22×C83—22no—
90C22×C83—22no—
91C22×C83—24no—
92C22×C83—24no—
93C22×C83C22⋊Q814yes—
94C22×C83C22×D411yescharacteristic
95C22×C83C22×Q811yescharacteristic
96C22×C83C2×#4112yes—
97C22×C83C2×#4111yescharacteristic
98C22×C83H(5,𝔽2)12yes—
99C2×D83—22no—
100C2×D83—24no—
101C2×D83—41no—
102C2×D83—41no—
103C2×D83—42no—
104C4×C83—22no—
105C4×C83—44no—
106C4×C83#12314yes—
107Dic83—22no—
108Dic83—41no—
109Dic83—41no—
110Dic83—44no—
111C2×C83—24no—
112C2×C83C2×C22⋊Q811yescharacteristic
113C2×C83C2×#12311yescharacteristic
114C2×C83#53511yescharacteristic
115C2×C83#55312yes—
116C2×C83#55412yes—
117D83—42no—
118#2487—831nomaximal, Sylow
119C83#248711yescharacteristic, complemented, Sylow
120C2×C4×D4—831no—
121C22×C4⋊C4D8311yescharacteristic, complemented
122#400—832no—
123#399—832no—
124#389—832no—
125#386—832no—
126#385—832no—
127#380—831no—
128#380—832no—
129C22×D4—831no—
130C23×C4—831no—
131C23×C4C2×D8311yescharacteristic
132C23×C4C2×D8312yes—
133C4×D4—1664no—
134C2×C4⋊C4—24no—
135C2×C4⋊C4—834no—
136C2×C4⋊C4—831no—
137C2×C4⋊C4—832no—
138C2×C4⋊C4C2×D8312yes—
139C2×C4⋊C4C2×D8312yes—
140C2×C22⋊C4—834no—
141C2×C22⋊C4—831no—
142C2×C22⋊C4—832no—
143C2×C22⋊C4—831no—
144C2×C42—831no—
145#95—834no—
146#95—832no—
147#95—834no—
148C24—831no—
149C24C22×D8311yescharacteristic
150C2×D4—834no—
151C2×D4—1662no—
152C2×D4—1662no—
153C22×C4—24no—
154C22×C4—24no—
155C22×C4—44no—
156C22×C4—831no—
157C22×C4—831no—
158C22×C4—834no—
159C22×C4—831no—
160C22×C4—1662no—
161C22×C4—1664no—
162C22×C4C22⋊D8314yes—
163C22×C4C22×D8314yes—
164C22×C4C22×D8311yescharacteristic
165C22×C4C22×D8311yescharacteristic
166C4⋊C4—24no—
167C4⋊C4—24no—
168C4⋊C4—3324no—
169C4⋊C4—1662no—
170C4⋊C4—1664no—
171C22⋊C4—3324no—
172C22⋊C4—1662no—
173C22⋊C4—1664no—
174C22⋊C4—1662no—
175C42—1662no—
176C23—22no—
177C23—22no—
178C23—832no—
179C23—1661no—
180C23—1662no—
181C23—1661no—
182C23—1662no—
183C23C2×C22⋊D8312yes—
184C23Q8×D8312yes—
185C23#1305059112yes—
186C23C23×D8311yescharacteristic
187D4—1668no—
188C2×C4—22no—
189C2×C4—28no—
190C2×C4—24no—
191C2×C4—24no—
192C2×C4—24no—
193C2×C4—48no—
194C2×C4—3324no—
195C2×C4—832no—
196C2×C4—1661no—
197C2×C4—1661no—
198C2×C4—1664no—
199C2×C4—1662no—
200C2×C4—1664no—
201C2×C4—1664no—
202C2×C4—1662no—
203C2×C4—1661no—
204C2×C4—1661no—
205C2×C4C2×C22⋊D8312yes—
206C2×C4C2×C22⋊D8312yes—
207C2×C4#1305059014yes—
208C22—24no—
209C22—24no—
210C22—22no—
211C22—22no—
212C22—3322no—
213C22—3321no—
214C22—3321no—
215C22—1664no—
216C22—1662no—
217C22#1458176814yes—
218C22C22×C22⋊D8311yescharacteristic
219C22C2×Q8×D8311yescharacteristic
220C22C2×#1305059012yes—
221C22C2×#1305059111yescharacteristic
222C22#1458183712yes—
223C4—22no—
224C4—44no—
225C4—3324no—
226C4—3321no—
227C4—3321no—
228C4—1662no—
229C4#1458177514yes—
230C2—24no—
231C2—3322no—
232C2C2×#1458176811yescharacteristic
233C2C2×#1458177511yescharacteristic
234C2#1912826811yescharacteristic
235C2#1912839912yes—
236C2#1912840612yes—
subgroupG/H
center, FrattiniC23C23×D83
commutator, socle, cosocleC23×C83C24
FittingC22×C4⋊C4×C83C2
prime radicalC22×C22⋊D83C22
hypercenterC22×C4⋊C4D83
hypocenterC83#2487
LCSG ▷ C23×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
2. #19127639C2semidirect
3. #19127636C2semidirect
4. #19127597C2semidirect
6. #19127574C2semidirect
7. #19127561C2semidirect
8. #19127558C2semidirect
9. C22×C4⋊C4×C83C2semidirect
119. C83#2487semidirect
121. C22×C4⋊C4D83semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D832
#191276392
#191276362
#191275972
#191275802
#191275742
#191275612
#191275582
C22×C4⋊C4×C832
#248783