Finite Group Databasebeta

← #24907787

Subgroups — #24907787

Order 10624.

324 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 83 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×#14581812C211yesmaximal, characteristic
2C2×#14581800C211yesmaximal, characteristic, complemented
3C2×#14581720C211yesmaximal, characteristic, complemented
4#19127668C211yesmaximal, characteristic, complemented
5#19127465C211yesmaximal, characteristic, complemented
6#19127456C211yesmaximal, characteristic, complemented
7C2×#104×C83C211yesmaximal, characteristic, complemented
8C2×#13050590C2211yescharacteristic
9#14581812—22no—
10#14581812—22no—
11C2×#13050580C411yescharacteristic, complemented
12C2×#13050578C411yescharacteristic, complemented
13C2×#13050577C411yescharacteristic, complemented
14C2×#13050575C411yescharacteristic, complemented
15#14581800—22no—
16#14581800—22no—
17C2×#13050574C2211yescharacteristic
18C2×#13050573—21no—
19C22×C83⋊C8C2211yescharacteristic
20C22×C83⋊C8C2211yescharacteristic
21C2×C8×D83—21no—
22#14581720—22no—
23C22⋊(C83⋊C8)—21no—
24C22⋊(C83⋊C8)—21no—
25C2×#41×C83C2211yescharacteristic
26C2×M16×C83C2211yescharacteristic
27C2×C42×C83C2211yescharacteristic
28#104×C83—22no—
29#13050590—21no—
30#13050590—21no—
31#13050590D412yes—
32#13050590D412yes—
33C2×Dic166C2×C411yescharacteristic
34C2×D332C2×C411yescharacteristic
35C2×C22⋊D83—21no—
36C2×C4×D83—21no—
37#13050580—22no—
38#13050578—22no—
39#13050577—22no—
40#13050575—22no—
41#13050574—21no—
42#13050574—21no—
43#13050573—24no—
44C2×C83⋊C8—21no—
45C2×C83⋊C8—21no—
46C2×C83⋊C8—21no—
47C2×C83⋊C8—21no—
48C2×C83⋊C8—21no—
49C2×C83⋊C8C2×C411yescharacteristic
50C2×C83⋊C8C2×C411yescharacteristic
51C2×C83⋊C8D411yescharacteristic
52C2×C83⋊C8D411yescharacteristic
53C8×D83—24no—
54#41×C83—21no—
55#41×C83—21no—
56#41×C83—21no—
57#41×C83—21no—
58C2×Q8×C83C2×C411yescharacteristic
59C2×D4×C83—21no—
60C2×D4×C83C2×C411yescharacteristic
61C22×C4×C83—21no—
62C22×C4×C83—21no—
63C22×C4×C83C2311yescharacteristic
64M16×C83—21no—
65M16×C83D412yes—
66C2×C8×C83—21no—
67C42×C83—21no—
68C42×C83—21no—
69C23×C83—21no—
70C23×C83#4111yescharacteristic
71C22×D83—21no—
72Q8×C83—21no—
73Q8×C83—21no—
74D4×C83—21no—
75D4×C83—21no—
76D4×C83—41no—
77D4×C83—41no—
78C2×C4×C83—22no—
79C2×C4×C83—21no—
80C2×C4×C83—41no—
81C2×C4×C83—41no—
82C2×C4×C83—41no—
83C2×C4×C83—41no—
84C2×C4×C83C22×C411yescharacteristic
85C2×C4×C83C2×D411yescharacteristic
86C2×C4×C83C2×D411yescharacteristic
87C2×C4×C83C2×D411yescharacteristic
88C2×C4×C83C2×D411yescharacteristic
89C2×C4×C83#4111yescharacteristic
90C22⋊D83—22no—
91C22⋊D83—22no—
92C2×Dic83—21no—
93D332—21no—
94D332C22⋊C412yes—
95C4×D83—22no—
96C4×D83—22no—
97Dic166—21no—
98Dic166C22⋊C412yes—
99C8×C83—22no—
100C83⋊C8—21no—
101C83⋊C8—21no—
102C83⋊C8—21no—
103C83⋊C8—21no—
104C83⋊C8—41no—
105C22×C83—21no—
106C22×C83—21no—
107C22×C83—41no—
108C22×C83—41no—
109C22×C83—41no—
110C22×C83C4×D411yescharacteristic
111C22×C83#12111yescharacteristic
112C22×C83#12311yescharacteristic
113C2×D83—22no—
114C2×D83—41no—
115C2×D83—41no—
116C4×C83—42no—
117C4×C83—41no—
118C4×C83C2×C22⋊C411yescharacteristic
119C4×C83C4×D411yescharacteristic
120C4×C83#12011yescharacteristic
121C4×C83#12111yescharacteristic
122Dic83—22no—
123C2×C83—21no—
124C2×C83—21no—
125C2×C83—81no—
126C2×C83#38611yescharacteristic
127C2×C83#44311yescharacteristic
128C2×C83#44411yescharacteristic
129D83—42no—
130#1876—831nomaximal, Sylow
131C83#187611yescharacteristic, complemented, Sylow
132C2×#135—831no—
133C2×#131—831no—
134C2×#104—831no—
135C2×#104D8311yescharacteristic, complemented
136#408—831no—
137#341—831no—
138#336—831no—
139C2×#41—831no—
140C2×#41C2×D8311yescharacteristic
141#135—1662no—
142#135—1662no—
143C2×Q16—831no—
144C2×SD16—831no—
145C2×SD16—831no—
146C2×D8—831no—
147#131—1662no—
148#131—1662no—
149C2×M16—831no—
150C2×M16—1661no—
151C2×M16C2×D8311yescharacteristic
152C22×C8—831no—
153C22×C8—831no—
154C22×C8—1661no—
155C2×C42C2×D8311yescharacteristic
156#104—22no—
157#104—1662no—
158C22⋊C8—1661no—
159C22⋊C8—1661no—
160#41—21no—
161#41—21no—
162#41—21no—
163#41—21no—
164#41—832no—
165#41—832no—
166#41—1661no—
167#41—1661no—
168C2×Q8—831no—
169C2×Q8C4×D8311yescharacteristic
170C2×D4—21no—
171C2×D4—831no—
172C2×D4—1661no—
173C2×D4C4×D8311yescharacteristic
174C22×C4—21no—
175C22×C4—21no—
176C22×C4—1661no—
177C22×C4C22×D8311yescharacteristic
178Q16—1662no—
179SD16—1662no—
180SD16—1662no—
181D8—1662no—
182M16—21no—
183M16—1664no—
184M16—1661no—
185M16—1661no—
186M16C22⋊D8312yes—
187C2×C8—21no—
188C2×C8—831no—
189C2×C8—831no—
190C2×C8—831no—
191C2×C8—831no—
192C2×C8—1664no—
193C2×C8—1661no—
194C2×C8—1661no—
195C2×C8—1661no—
196C2×C8—1661no—
197C2×C8—1661no—
198C42—21no—
199C42—21no—
200C23—21no—
201C23—1661no—
202C23#1305058911yescharacteristic
203Q8—21no—
204Q8—21no—
205Q8—832no—
206Q8—1661no—
207D4—21no—
208D4—21no—
209D4—41no—
210D4—41no—
211D4—832no—
212D4—1662no—
213D4—1662no—
214D4—1661no—
215C2×C4—22no—
216C2×C4—21no—
217C2×C4—41no—
218C2×C4—41no—
219C2×C4—41no—
220C2×C4—41no—
221C2×C4—1661no—
222C2×C4—1662no—
223C2×C4—1662no—
224C2×C4C2×C4×D8311yescharacteristic
225C2×C4D4×D8311yescharacteristic
226C2×C4D4×D8311yescharacteristic
227C2×C4D4×D8311yescharacteristic
228C2×C4C2×C22⋊D8311yescharacteristic
229C2×C4#1305059011yescharacteristic
230C8—22no—
231C8—3321no—
232C8—1661no—
233C8—1661no—
234C8—1661no—
235C8—1661no—
236C22—21no—
237C22—21no—
238C22—41no—
239C22—41no—
240C22—41no—
241C22—3321no—
242C22—3321no—
243C22—1662no—
244C22(C22⋊D83)⋊C411yescharacteristic
245C22#1458176311yescharacteristic
246C22#1458177411yescharacteristic
247C4—42no—
248C4—41no—
249C4—1662no—
250C4C22⋊C4×D8311yescharacteristic
251C4C4×C22⋊D8311yescharacteristic
252C4#1458176011yescharacteristic
253C4#1458176511yescharacteristic
254C2—21no—
255C2—21no—
256C2—81no—
257C2—3322no—
258C2#1912757711yescharacteristic
259C2#1912781511yescharacteristic
260C2#1912782011yescharacteristic
subgroupG/H
centerC2×C4D4×D83
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
FittingC2×#104×C83C2
socleC22×C83C4×D4
prime radicalC2×#14581812C2
cosocleC22×C4×C83C23
hypercenterC2×#104D83
hypocenterC83#1876
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#14581800C2semidirect
3. C2×#14581720C2semidirect
4. #19127668C2semidirect
5. #19127465C2semidirect
6. #19127456C2semidirect
7. C2×#104×C83C2semidirect
11. C2×#13050580C4semidirect
12. C2×#13050578C4semidirect
13. C2×#13050577C4semidirect
14. C2×#13050575C4semidirect
131. C83#1876semidirect
135. C2×#104D83semidirect

Maximal subgroups

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

subgroupindex
C2×#145818122
C2×#145818002
C2×#145817202
#191276682
#191274652
#191274562
C2×#104×C832
#187683