Finite Group Databasebeta

← #24920499

Subgroups — #24920499

Order 10624.

350 conjugacy classes of subgroups; only the 117 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 117 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#19128426C211yesmaximal, characteristic, complemented
2#19128425C211yesmaximal, characteristic
3#19128094C212yesmaximal
4#19127925C212yesmaximal
5#19127909C212yesmaximal, complemented
6#19127908C211yesmaximal, characteristic, complemented
7#19127903C211yesmaximal, characteristic
8#19127831C211yesmaximal, characteristic, complemented
9#19127808C212yesmaximal, complemented
10C4×#13050579C211yesmaximal, characteristic
11#494×C83C211yesmaximal, characteristic, complemented
12C2×#13050580C2212yes—
13C2×#13050579C2211yescharacteristic
14C2×#13050577C2212yes—
15#14581790C2212yes—
16#14581787—22no—
17#14581783—21no—
18#14581782—22no—
19#14581781C2212yes—
20#14581772—21no—
21Q8×Dic83C2212yes—
22Q8×Dic83C2212yes—
23C4×Dic166C2211yescharacteristic
24C4×Dic166C2211yescharacteristic
25#14581757—21no—
26C4×D332C2211yescharacteristic
27#14581750—21no—
28#14581724C2211yescharacteristic
29C4⋊(C83⋊C8)C2211yescharacteristic
30Dic83⋊C8C2212yes—
31#14581717C2211yescharacteristic
32#14581716C2212yes—
33#14581716C2212yes—
34#14581714C2211yescharacteristic
35#14581702C2212yes—
36C2×Q16×C83C2212yes—
37C2×Q16×C83C2212yes—
38C4⋊Q8×C83C2211yescharacteristic
39C4⋊Q8×C83C2211yescharacteristic
40C4×C8×C83C2211yescharacteristic
41C2×Dic166D411yescharacteristic
42C2×Dic166C2311yescharacteristic
43C2×D332C2311yescharacteristic
44C2×C4×D83—21no—
45#13050580—42no—
46#13050579—22no—
47#13050577—42no—
48C2×C83⋊C8C2312yes—
49C4⋊Dic83—22no—
50C4⋊Dic83—22no—
51C4⋊Dic83D411yescharacteristic
52C4⋊Dic83C2311yescharacteristic
53C4⋊Dic83C2311yescharacteristic
54Dic83⋊C4—21no—
55Dic83⋊C4—22no—
56Dic83⋊C4—22no—
57Dic83⋊C4D412yes—
58(C2×D83)⋊C4—21no—
59(C2×D83)⋊C4—22no—
60(C2×D83)⋊C4—22no—
61C4×Dic83—21no—
62C4×Dic83—22no—
63C4×Dic83—22no—
64C4×Dic83C2312yes—
65C2×Q8×C83C2312yes—
66C2×Q8×C83C2312yes—
67Q16×C83—24no—
68Q16×C83—42no—
69C2×C8×C83C2311yescharacteristic
70C2×C8×C83C2311yescharacteristic
71C4⋊C4×C83—21no—
72C4⋊C4×C83—21no—
73C4⋊C4×C83—21no—
74C4⋊C4×C83—21no—
75C42×C83C2311yescharacteristic
76C22×D83—21no—
77Q8×C83—22no—
78Q8×C83—22no—
79Q8×C83—22no—
80Q8×C83—22no—
81C2×C4×C83—22no—
82C2×C4×C83—22no—
83C2×C4×C83C2×D411yescharacteristic
84C2×C4×C83C2×D411yescharacteristic
85C2×C4×C83C2411yescharacteristic
86C2×Dic83—21no—
87C2×Dic83—21no—
88C2×Dic83—21no—
89C2×Dic83C2×D412yes—
90C2×Dic83C2×D412yes—
91D332—21no—
92D332—21no—
93C4×D83—42no—
94Dic166—22no—
95Dic166—21no—
96Dic166—21no—
97C8×C83—21no—
98C8×C83#4112yes—
99C83⋊C8—22no—
100C22×C83C22×D411yescharacteristic
101C2×D83—41no—
102C2×D83—41no—
103C2×D83—41no—
104C4×C83—21no—
105C4×C83—42no—
106C4×C83—42no—
107C4×C83#13712yes—
108C4×C83C2×#4111yescharacteristic
109C4×C83H(5,𝔽2)11yescharacteristic
110Dic83—22no—
111Dic83—22no—
112Dic83—41no—
113Dic83—41no—
114Dic83—41no—
115C2×C83#55011yescharacteristic
116C2×C83C2×#13711yescharacteristic
117C2×C83#57811yescharacteristic
118D83—81no—
119#3398—831nomaximal, Sylow
120C83#339811yescharacteristic, complemented, Sylow
121#557—831no—
122#556—831no—
123#497—832no—
124#494D8311yescharacteristic, complemented
125#464—832no—
126#462—832no—
127#462—831no—
128#461—831no—
129#446—831no—
130#441—832no—
131C4×SD16—831no—
132C2×Q16—832no—
133C2×Q16C2×D8312yes—
134C2×Q16C2×D8312yes—
135C2×SD16—832no—
136C2×SD16—831no—
137C4⋊Q8—832no—
138C4⋊Q8C2×D8311yescharacteristic
139C4⋊Q8C2×D8311yescharacteristic
140#126—1662no—
141#125—1662no—
142#125—1661no—
143#124—832no—
144C22⋊Q8—1661no—
145C4×Q8—831no—
146C4×Q8—832no—
147C4×Q8—832no—
148C4×Q8—831no—
149C4×Q8—1661no—
150C4×D4—831no—
151#117—1661no—
152#106—831no—
153C4⋊C8—832no—
154C4⋊C8—831no—
155#103—832no—
156#103—832no—
157#103—831no—
158#102—831no—
159#97—832no—
160C4×C8C2×D8311yescharacteristic
161C2×Q8—831no—
162C2×Q8—831no—
163C2×Q8C22×D8312yes—
164C2×Q8C22×D8312yes—
165C2×D4—831no—
166C22×C4—1661no—
167Q16—24no—
168Q16—42no—
169Q16—3322no—
170SD16—3322no—
171SD16—1662no—
172C2×C8—832no—
173C2×C8C22×D8311yescharacteristic
174C2×C8C22×D8311yescharacteristic
175C4⋊C4—21no—
176C4⋊C4—21no—
177C4⋊C4—21no—
178C4⋊C4—21no—
179C4⋊C4—831no—
180C4⋊C4—832no—
181C4⋊C4—831no—
182C4⋊C4—831no—
183C4⋊C4—1662no—
184C4⋊C4—1662no—
185C4⋊C4—1662no—
186C4⋊C4—1662no—
187C4⋊C4—1661no—
188C22⋊C4—1662no—
189C22⋊C4—1662no—
190C22⋊C4—1661no—
191C42—832no—
192C42—1662no—
193C42—1662no—
194C42—1661no—
195C42C22×D8311yescharacteristic
196C23—1661no—
197Q8—22no—
198Q8—22no—
199Q8—22no—
200Q8—22no—
201Q8—1661no—
202Q8—1661no—
203Q8—1662no—
204D4—1661no—
205D4—1661no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—3322no—
209C2×C4—832no—
210C2×C4—832no—
211C2×C4—1661no—
212C2×C4—1661no—
213C2×C4—1661no—
214C2×C4D4×D8311yescharacteristic
215C2×C4D4×D8311yescharacteristic
216C2×C4C23×D8311yescharacteristic
217C8—21no—
218C8—1662no—
219C8#1305058912yes—
220C22—3321no—
221C22—3321no—
222C22—3321no—
223C22C2×D4×D8311yescharacteristic
224C4—21no—
225C4—42no—
226C4—42no—
227C4—3321no—
228C4—3321no—
229C4—3321no—
230C4—1662no—
231C4—1662no—
232C4#1458182412yes—
233C4C2×#1305058911yescharacteristic
234C4#1458183711yescharacteristic
235C2—6641no—
236C2#1912837111yescharacteristic
237C2C2×#1458182411yescharacteristic
238C2#1912851411yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#494×C83C2
socleC22×C83C22×D4
prime radicalC2×D332C23
hypercenter#494D83
hypocenterC83#3398
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
1. #19128426C2semidirect
5. #19127909C2semidirect
6. #19127908C2semidirect
8. #19127831C2semidirect
9. #19127808C2semidirect
11. #494×C83C2semidirect
120. C83#3398semidirect
124. #494D83semidirect

Maximal subgroups

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

subgroupindex
#191284262
#191284252
#191280942
#191279252
#191279092
#191279082
#191279032
#191278312
#191278082
C4×#130505792
#494×C832
#339883