Finite Group Databasebeta

← #24907657

Subgroups — #24907657

Order 10624.

360 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 360 conjugacy classes of subgroups — too many to draw. Only its 107 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⋊C4×D83C211yesmaximal, characteristic
2C2×#14581724C211yesmaximal, characteristic, complemented
3C2×#14581723C211yesmaximal, characteristic, complemented
4C2×(C2×D83)⋊C8C211yesmaximal, characteristic
5#19127558C211yesmaximal, characteristic
6#19127463C211yesmaximal, characteristic, complemented
7#340×C83C211yesmaximal, characteristic, complemented
8C22×C4×D83—21no—
9C22×C4×D83C2211yescharacteristic
10C22×C83⋊C8C2211yescharacteristic
11C2×C4⋊Dic83C2211yescharacteristic
12C2×C4⋊Dic83C2211yescharacteristic
13C2×Dic83⋊C4—21no—
14C4⋊C4×D83—22no—
15C4⋊C4×D83—22no—
16C2×(C2×D83)⋊C4—21no—
17#14581724—22no—
18#14581723—22no—
19(C2×D83)⋊C8C414yescomplemented
20C83⋊#95—21no—
21C22×C8×C83C2211yescharacteristic
22C2×C4⋊C4×C83C2211yescharacteristic
23C2×C4⋊C4×C83C2211yescharacteristic
24C23×D83D411yescharacteristic
25C22×Dic83—21no—
26C22×Dic83—21no—
27C22×Dic83D411yescharacteristic
28C2×C4×D83—21no—
29C2×C4×D83—21no—
30C2×C4×D83—21no—
31C2×C4×D83—21no—
32C2×C4×D83—24no—
33C2×C4×D83—42no—
34C2×C4×D83—42no—
35C2×C4×D83C2×C412yes—
36C2×C4×D83Q812yes—
37C2×C83⋊C8—21no—
38C2×C83⋊C8—21no—
39C2×C83⋊C8C2×C412yescomplemented
40C4⋊Dic83—21no—
41C4⋊Dic83—21no—
42C4⋊Dic83—21no—
43C4⋊Dic83D412yes—
44Dic83⋊C4—24no—
45(C2×D83)⋊C4—42no—
46C22×C4×C83—21no—
47C22×C4×C83—21no—
48C22×C4×C83C2311yescharacteristic
49C2×C8×C83—21no—
50C2×C8×C83—21no—
51C2×C8×C83C2×C412yescomplemented
52C4⋊C4×C83—21no—
53C4⋊C4×C83—21no—
54C4⋊C4×C83—21no—
55C4⋊C4×C83D412yes—
56C23×C83C2×D411yescharacteristic
57C22×D83—21no—
58C22×D83—21no—
59C22×D83—21no—
60C22×D83—21no—
61C22×D83C4⋊C412yes—
62C22×D83SD1612yes—
63C22×D83SD1612yes—
64C2×C4×C83—22no—
65C2×C4×C83—41no—
66C2×C4×C83—41no—
67C2×C4×C83—41no—
68C2×C4×C83—41no—
69C2×C4×C83—41no—
70C2×C4×C83C22×C411yescharacteristic
71C2×C4×C83C2×D411yescharacteristic
72C2×C4×C83C2×D411yescharacteristic
73C2×C4×C83C2×Q811yescharacteristic
74C2×C4×C83#4111yescharacteristic
75C2×C4×C83#4111yescharacteristic
76C2×Dic83—21no—
77C2×Dic83—21no—
78C2×Dic83—22no—
79C2×Dic83—41no—
80C2×Dic83—41no—
81C2×Dic83—41no—
82C2×Dic83—41no—
83C2×Dic83—41no—
84C2×Dic83C4⋊C412yes—
85C4×D83—22no—
86C4×D83—22no—
87C4×D83—22no—
88C4×D83—22no—
89C4×D83—48no—
90C8×C83—22no—
91C83⋊C8—22no—
92C22×C83C2×C4⋊C411yescharacteristic
93C22×C83#12011yescharacteristic
94C22×C83#12311yescharacteristic
95C22×C83C2×SD1611yescharacteristic
96C22×C83C2×SD1611yescharacteristic
97C22×C83AGL(1,ℤ/8)11yescharacteristic
98C22×C83#13711yescharacteristic
99C2×D83—22no—
100C2×D83—22no—
101C2×D83—22no—
102C2×D83—22no—
103C2×D83—22no—
104C2×D83—22no—
105C2×D83#10614yescomplemented
106C4×C83—42no—
107C4×C83—81no—
108C4×C83C4×D411yescharacteristic
109C4×C83C4×D411yescharacteristic
110C4×C83C22⋊Q811yescharacteristic
111C4×C83C22⋊Q811yescharacteristic
112Dic83—22no—
113Dic83—42no—
114Dic83—81no—
115C2×C83#38511yescharacteristic
116C2×C83C2×#10611yescharacteristic
117C2×C83#42811yescharacteristic
118C2×C83#44811yescharacteristic
119C2×C83#45011yescharacteristic
120C2×C83#48111yescharacteristic
121C2×C83#48311yescharacteristic
122D83—24no—
123#1853—831nomaximal, Sylow
124C83#185311yescharacteristic, complemented, Sylow
125C22×C4⋊C4—831no—
126C2×#106—831no—
127C2×#106—831no—
128C2×C22⋊C8—831no—
129#380—831no—
130#340—831no—
131#340D8311yescharacteristic, complemented
132C23×C4—831no—
133C23×C4—1661no—
134C22×C8—831no—
135C22×C8C2×D8311yescharacteristic
136C2×C4⋊C4—831no—
137C2×C4⋊C4—831no—
138C2×C4⋊C4—1662no—
139C2×C4⋊C4—1661no—
140C2×C4⋊C4—1662no—
141C2×C4⋊C4C2×D8311yescharacteristic
142C2×C4⋊C4C2×D8311yescharacteristic
143C2×C22⋊C4—1661no—
144#106—1662no—
145#106—1662no—
146C22⋊C8—834no—
147#95—1661no—
148C24—831no—
149C22×C4—21no—
150C22×C4—21no—
151C22×C4—3322no—
152C22×C4—3322no—
153C22×C4—831no—
154C22×C4—832no—
155C22×C4—832no—
156C22×C4—1664no—
157C22×C4—1661no—
158C22×C4—1661no—
159C22×C4—1661no—
160C22×C4—1661no—
161C22×C4—1661no—
162C22×C4—1661no—
163C22×C4C22×D8311yescharacteristic
164C2×C8—21no—
165C2×C8—21no—
166C2×C8—832no—
167C2×C8—1661no—
168C2×C8—1661no—
169C2×C8C4×D8312yescomplemented
170C4⋊C4—21no—
171C4⋊C4—21no—
172C4⋊C4—21no—
173C4⋊C4—832no—
174C4⋊C4—1664no—
175C4⋊C4—1661no—
176C4⋊C4—1661no—
177C4⋊C4—1661no—
178C4⋊C4D33212yes—
179C22⋊C4—3322no—
180C23—832no—
181C23—832no—
182C23—832no—
183C23—1661no—
184C23—1661no—
185C23—1661no—
186C23—1661no—
187C23D4×D8311yescharacteristic
188C2×C4—22no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4—3328no—
195C2×C4—3321no—
196C2×C4—3321no—
197C2×C4—3321no—
198C2×C4—3321no—
199C2×C4—3321no—
200C2×C4—832no—
201C2×C4—1662no—
202C2×C4—1661no—
203C2×C4—1661no—
204C2×C4—1662no—
205C2×C4—1662no—
206C2×C4—1662no—
207C2×C4—1662no—
208C2×C4C2×C4×D8311yescharacteristic
209C2×C4D4×D8311yescharacteristic
210C2×C4C2×D33211yescharacteristic
211C2×C4Q8×D8311yescharacteristic
212C2×C4#1305058911yescharacteristic
213C2×C4#1305059011yescharacteristic
214C8—22no—
215C8—1662no—
216C22—834no—
217C22—1662no—
218C22—1662no—
219C22—1662no—
220C22—1662no—
221C22—1662no—
222C22—1662no—
223C22C4⋊C4×D8311yescharacteristic
224C22#1458176211yescharacteristic
225C22#1458177411yescharacteristic
226C22SD16×D8311yescharacteristic
227C22SD16×D8311yescharacteristic
228C22#1458181611yescharacteristic
229C22#1458182411yescharacteristic
230C4—42no—
231C4—81no—
232C4—3322no—
233C4—6641no—
234C4—1662no—
235C4(C22⋊D83)⋊C411yescharacteristic
236C4C4×D33211yescharacteristic
237C4#1458177011yescharacteristic
238C4#1458177211yescharacteristic
239C2—1664no—
240C2#1912757511yescharacteristic
241C2#106×D8311yescharacteristic
242C2#1912774211yescharacteristic
243C2#1912784111yescharacteristic
244C2#1912785311yescharacteristic
245C2#1912801811yescharacteristic
246C2#1912803011yescharacteristic
subgroupG/H
centerC23D4×D83
commutatorC2×C4×C83C22×C4
FrattiniC22×C4C22×D83
Fitting#340×C83C2
socleC23×C83C2×D4
prime radicalC23×D83D4
cosocleC22×C4×C83C23
hypercenter#340D83
hypocenterC83#1853
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C23 ◁ C22×C4 ◁ #340

Complements

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

NH ≅ G/Ntype
2. C2×#14581724C2semidirect
3. C2×#14581723C2semidirect
6. #19127463C2semidirect
7. #340×C83C2semidirect
19. (C2×D83)⋊C8C4semidirect
39. C2×C83⋊C8C2×C4semidirect
51. C2×C8×C83C2×C4semidirect
105. C2×D83#106semidirect
124. C83#1853semidirect
131. #340D83semidirect
169. C2×C8C4×D83semidirect

Maximal subgroups

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

subgroupindex
C2×C4⋊C4×D832
C2×#145817242
C2×#145817232
C2×(C2×D83)⋊C82
#191275582
#191274632
#340×C832
#185383