Finite Group Databasebeta

← #24919331

Subgroups — #24919331

Order 10624.

420 conjugacy classes of subgroups; only the 123 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 123 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, complemented
2#19128356C211yesmaximal, characteristic
3#19128051C211yesmaximal, characteristic, complemented
4#19128030C212yesmaximal, complemented
5#19127987C211yesmaximal, characteristic, complemented
6#19127927C212yesmaximal, complemented
7#19127838C212yesmaximal, complemented
8C4×#13050578C211yesmaximal, characteristic, complemented
9#19127766C211yesmaximal, characteristic, complemented
10#103×D83C212yesmaximal, complemented
11#461×C83C211yesmaximal, characteristic, complemented
12C2×#13050590—21no—
13C2×Q8×D83C2212yes—
14C2×#13050578C2211yescharacteristic
15C2×C8×D83C2212yes—
16#14581792C2211yescharacteristic
17#14581791—21no—
18#14581774—22no—
19#14581771—21no—
20#14581770—22no—
21#14581768C2212yes—
22#14581763C2212yes—
23C4×Dic166C2211yescharacteristic
24C4×D332C2211yescharacteristic
25#14581754—21no—
26C4⋊C4×D83C2212yes—
27#14581724C2212yes—
28#14581723C2211yescharacteristic
29#14581717C2212yes—
30#14581716C2212yes—
31#14581716C2212yes—
32#14581715C2211yescharacteristic
33#14581713C2211yescharacteristic
34(C2×D83)⋊C8C2212yes—
35C4×C83⋊C8C2211yescharacteristic
36C2×SD16×C83C2212yes—
37C4⋊Q8×C83C2211yescharacteristic
38C4×D4×C83C2211yescharacteristic
39C4⋊C8×C83C2211yescharacteristic
40#103×C83C2212yes—
41#13050590—44no—
42C2×Dic166—21no—
43C2×Dic166—22no—
44C2×Dic166C2311yescharacteristic
45Q8×D83—24no—
46Q8×D83—42no—
47C2×D332D411yescharacteristic
48C2×C22⋊D83—22no—
49C2×C4×D83—22no—
50C2×C4×D83—22no—
51C2×C4×D83C2312yes—
52#13050578—22no—
53C2×C83⋊C8C2311yescharacteristic
54C2×C83⋊C8C2311yescharacteristic
55C8×D83—42no—
56C4⋊Dic83—21no—
57C4⋊Dic83D411yescharacteristic
58C4⋊Dic83C2312yes—
59C4⋊Dic83C2311yescharacteristic
60Dic83⋊C4—21no—
61Dic83⋊C4—22no—
62Dic83⋊C4—22no—
63C22⋊Dic83—22no—
64(C2×D83)⋊C4—22no—
65(C2×D83)⋊C4D412yes—
66C4×Dic83—21no—
67C2×Q8×C83C2312yes—
68C2×D4×C83C2311yescharacteristic
69C22×C4×C83—21no—
70SD16×C83—42no—
71C2×C8×C83C2312yes—
72C4⋊C4×C83—21no—
73C4⋊C4×C83—21no—
74C4⋊C4×C83C2311yescharacteristic
75C22⋊C4×C83—21no—
76C42×C83C2311yescharacteristic
77C23×C83—21no—
78C22×D83C2×D412yes—
79Q8×C83—22no—
80Q8×C83—22no—
81D4×C83—21no—
82D4×C83—21no—
83C2×C4×C83—21no—
84C2×C4×C83—22no—
85C2×C4×C83—42no—
86C2×C4×C83C2×D411yescharacteristic
87C2×C4×C83C2×D411yescharacteristic
88C2×C4×C83C2411yescharacteristic
89C22⋊D83—44no—
90C2×Dic83—21no—
91C2×Dic83—21no—
92C2×Dic83—22no—
93C2×Dic83C2×D412yes—
94D332SD1614yescomplemented
95C4×D83—22no—
96C4×D83—22no—
97C4×D83—44no—
98C4×D83—44no—
99Dic166—21no—
100Dic166—44no—
101Dic166—42no—
102Dic166#4112yescomplemented
103C8×C83—22no—
104C83⋊C8—21no—
105C83⋊C8—21no—
106C22×C83—41no—
107C22×C83—41no—
108C22×C83—41no—
109C22×C83C22×D411yescharacteristic
110C2×D83—22no—
111C2×D83—22no—
112C2×D83C2×SD1614yes—
113C4×C83—21no—
114C4×C83—42no—
115C4×C83—41no—
116C4×C83C2×SD1612yes—
117C4×C83C2×#4111yescharacteristic
118C4×C83H(5,𝔽2)11yescharacteristic
119Dic83—22no—
120Dic83—22no—
121Dic83—42no—
122Dic83—41no—
123C2×C83—81no—
124C2×C83#54611yescharacteristic
125C2×C83C22×SD1611yescharacteristic
126C2×C83#57811yescharacteristic
127D83—24no—
128#3298—831nomaximal, Sylow
129C83#329811yescharacteristic, complemented, Sylow
130D4×Q8—831no—
131#547—831no—
132#487—831no—
133#483—832no—
134#475—831no—
135#465—832no—
136#461D8311yescharacteristic, complemented
137#448—832no—
138C4×SD16—831no—
139C8×D4—831no—
140C2×#103—832no—
141C2×#41—1661no—
142C22×Q8—832no—
143C2×SD16—831no—
144C2×SD16C2×D8312yes—
145C22×C8—832no—
146C4⋊Q8—831no—
147C4⋊Q8—1661no—
148C4⋊Q8C2×D8311yescharacteristic
149#123—1662no—
150C22⋊Q8—832no—
151C22⋊Q8—1662no—
152C22⋊Q8—1661no—
153#121—832no—
154C4×Q8—831no—
155C4×D4—831no—
156C4×D4—1661no—
157C4×D4C2×D8311yescharacteristic
158C2×C4⋊C4—832no—
159#106—832no—
160#106—831no—
161C4⋊C8C2×D8311yescharacteristic
162#103—832no—
163#103—832no—
164#103—832no—
165#103—831no—
166#103C2×D8312yes—
167#102—831no—
168C22⋊C8—832no—
169C4×C8—831no—
170#41—3324no—
171C2×Q8—3322no—
172C2×Q8—831no—
173C2×Q8—1662no—
174C2×Q8—1664no—
175C2×Q8—1661no—
176C2×Q8C22×D8312yes—
177C2×D4—831no—
178C2×D4—1662no—
179C2×D4C22×D8311yescharacteristic
180C22×C4—21no—
181C22×C4—832no—
182C22×C4—1662no—
183C22×C4—1662no—
184SD16—42no—
185SD16—1662no—
186C2×C8—3322no—
187C2×C8—831no—
188C2×C8—831no—
189C2×C8C22×D8312yes—
190C4⋊C4—21no—
191C4⋊C4—21no—
192C4⋊C4—831no—
193C4⋊C4—831no—
194C4⋊C4—832no—
195C4⋊C4—1662no—
196C4⋊C4—1662no—
197C4⋊C4—1661no—
198C4⋊C4—1661no—
199C4⋊C4C22×D8311yescharacteristic
200C22⋊C4—21no—
201C22⋊C4—832no—
202C22⋊C4—1662no—
203C22⋊C4—1662no—
204C42—1661no—
205C42C22×D8311yescharacteristic
206C23—21no—
207C23—832no—
208Q8—22no—
209Q8—22no—
210Q8—3324no—
211Q8—3322no—
212Q8—832no—
213Q8—1661no—
214D4—21no—
215D4—21no—
216D4—3324no—
217D4—834no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—42no—
221C2×C4—3324no—
222C2×C4—3324no—
223C2×C4—832no—
224C2×C4—1661no—
225C2×C4—1662no—
226C2×C4—1661no—
227C2×C4—1662no—
228C2×C4—1662no—
229C2×C4D4×D8311yescharacteristic
230C2×C4D4×D8311yescharacteristic
231C2×C4C23×D8311yescharacteristic
232C8—22no—
233C8—1661no—
234C8—1661no—
235C22—41no—
236C22—41no—
237C22—41no—
238C22—834no—
239C22—1662no—
240C22—1662no—
241C22C2×D4×D8311yescharacteristic
242C4—21no—
243C4—42no—
244C4—41no—
245C4—3322no—
246C4—3321no—
247C4—1662no—
248C4—1662no—
249C4SD16×D8312yes—
250C4#41×D8311yescharacteristic
251C4#1458183711yescharacteristic
252C2—81no—
253C2—1664no—
254C2#1912834811yescharacteristic
255C2C2×SD16×D8311yescharacteristic
256C2#1912851711yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#461×C83C2
socleC22×C83C22×D4
prime radical#19128356C2
hypercenter#461D83
hypocenterC83#3298
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
1. #19128366C2semidirect
3. #19128051C2semidirect
4. #19128030C2semidirect
5. #19127987C2semidirect
6. #19127927C2semidirect
7. #19127838C2semidirect
8. C4×#13050578C2semidirect
9. #19127766C2semidirect
10. #103×D83C2semidirect
11. #461×C83C2semidirect
94. D332SD16semidirect
102. Dic166#41semidirect
129. C83#3298semidirect
136. #461D83semidirect

Maximal subgroups

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

subgroupindex
#191283662
#191283562
#191280512
#191280302
#191279872
#191279272
#191278382
C4×#130505782
#191277662
#103×D832
#461×C832
#329883