Finite Group Databasebeta

← #24920496

Subgroups — #24920496

Order 10624.

350 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 350 conjugacy classes of subgroups — too many to draw. Only its 125 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#19128430C211yesmaximal, characteristic, complemented
2#19128424C211yesmaximal, characteristic
3#19128095C212yesmaximal
4#19128081C211yesmaximal, characteristic, complemented
5#19127924C212yesmaximal
6#19127912C211yesmaximal, characteristic, complemented
7#19127912C212yesmaximal, complemented
8#19127907C211yesmaximal, characteristic
9#19127809C212yesmaximal, complemented
10C4×#13050579C211yesmaximal, characteristic
11#446×C83C211yesmaximal, characteristic, complemented
12C2×Dic332C2212yes—
13C2×Dic332C2212yes—
14C2×Dic332C2212yes—
15C2×#13050579C2211yescharacteristic
16C2×#13050579C2212yes—
17#14581792C2211yescharacteristic
18#14581792C2211yescharacteristic
19#14581792C2212yes—
20#14581787—22no—
21#14581783—22no—
22#14581783—21no—
23#14581780C2212yes—
24#14581768—21no—
25Q8×Dic83—21no—
26C4×Dic166C2212yes—
27C4×Dic166C2212yes—
28C4×Dic166C2211yescharacteristic
29C4×D332C2211yescharacteristic
30#14581750—21no—
31#14581724C2211yescharacteristic
32#14581717C2212yes—
33#14581717C2212yes—
34#14581717C2211yescharacteristic
35#14581714C2211yescharacteristic
36C4×Q8×C83C2211yescharacteristic
37C4⋊C8×C83C2211yescharacteristic
38C4⋊C8×C83C2212yes—
39#97×C83C2212yes—
40C4×C8×C83C2211yescharacteristic
41C2×Dic166C2312yes—
42C2×Dic166C2311yescharacteristic
43C2×Dic166C2312yes—
44C2×D332C2311yescharacteristic
45C2×C4×D83—21no—
46Dic332—24no—
47Dic332—42no—
48Dic332—42no—
49#13050579—22no—
50#13050579—42no—
51C4⋊Dic83—21no—
52C4⋊Dic83—22no—
53C4⋊Dic83—21no—
54C4⋊Dic83—22no—
55C4⋊Dic83—21no—
56C4⋊Dic83—21no—
57C4⋊Dic83C2311yescharacteristic
58C4⋊Dic83C2311yescharacteristic
59Dic83⋊C4—22no—
60Dic83⋊C4—22no—
61Dic83⋊C4—21no—
62(C2×D83)⋊C4—22no—
63(C2×D83)⋊C4—22no—
64(C2×D83)⋊C4—21no—
65C4×Dic83—22no—
66C4×Dic83—21no—
67C4×Dic83—22no—
68C2×Q8×C83D411yescharacteristic
69C2×C8×C83C2312yes—
70C2×C8×C83C2311yescharacteristic
71C2×C8×C83C2311yescharacteristic
72C4⋊C4×C83D412yes—
73C4⋊C4×C83D411yescharacteristic
74C42×C83C2312yes—
75C42×C83C2311yescharacteristic
76C22×D83—21no—
77Q8×C83—22no—
78C2×C4×C83C2×D412yes—
79C2×C4×C83C2×D411yescharacteristic
80C2×C4×C83C2×D412yes—
81C2×C4×C83C2×D411yescharacteristic
82C2×C4×C83C2411yescharacteristic
83C2×Dic83—21no—
84C2×Dic83—21no—
85C2×Dic83—22no—
86C2×Dic83—21no—
87C2×Dic83—22no—
88D332—21no—
89D332—21no—
90C4×D83—42no—
91Dic166—21no—
92Dic166—22no—
93Dic166—22no—
94Dic166—22no—
95Dic166—22no—
96Dic166—21no—
97C8×C83—22no—
98C8×C83—21no—
99C8×C83#4112yes—
100C22×C83C22×D411yescharacteristic
101C2×D83—41no—
102C2×D83—41no—
103C2×D83—41no—
104C4×C83—22no—
105C4×C83—22no—
106C4×C83—21no—
107C4×C83#13712yes—
108C4×C83C2×#4111yescharacteristic
109C4×C83H(5,𝔽2)11yescharacteristic
110Dic83—41no—
111Dic83—42no—
112Dic83—42no—
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#494—831no—
125#464—832no—
126#462—832no—
127#462—831no—
128#461—831no—
129#446D8311yescharacteristic, complemented
130#441—832no—
131C4×SD16—831no—
132C2×Q16—832no—
133C2×Q16—832no—
134C2×Q16—832no—
135C2×SD16—832no—
136C2×SD16—831no—
137C4⋊Q8—831no—
138C4⋊Q8—831no—
139C4⋊Q8—832no—
140#126—1662no—
141#125—1661no—
142#125—1662no—
143#124—832no—
144C22⋊Q8—1661no—
145C4×Q8—832no—
146C4×Q8—831no—
147C4×Q8—832no—
148C4×Q8—1661no—
149C4×Q8C2×D8311yescharacteristic
150C4×D4—831no—
151#117—1661no—
152#106—831no—
153C4⋊C8C2×D8311yescharacteristic
154C4⋊C8C2×D8312yes—
155#103—831no—
156#103—832no—
157#103—832no—
158#102—831no—
159#97C2×D8312yes—
160C4×C8C2×D8311yescharacteristic
161C2×Q8—832no—
162C2×Q8—831no—
163C2×Q8—832no—
164C2×Q8D33211yescharacteristic
165C2×D4—831no—
166C22×C4—1661no—
167Q16—3322no—
168Q16—3322no—
169Q16—1664no—
170SD16—3322no—
171SD16—1662no—
172C2×C8C22×D8311yescharacteristic
173C2×C8C22×D8312yes—
174C2×C8C22×D8311yescharacteristic
175C4⋊C4—831no—
176C4⋊C4—831no—
177C4⋊C4—1661no—
178C4⋊C4—1662no—
179C4⋊C4—1662no—
180C4⋊C4—1661no—
181C4⋊C4—1662no—
182C4⋊C4—1661no—
183C4⋊C4—1662no—
184C4⋊C4—1661no—
185C4⋊C4—1661no—
186C4⋊C4D33211yescharacteristic
187C4⋊C4D33212yes—
188C22⋊C4—1662no—
189C22⋊C4—1661no—
190C22⋊C4—1662no—
191C42—1662no—
192C42—1662no—
193C42—1661no—
194C42C22×D8311yescharacteristic
195C42C22×D8312yes—
196C23—1661no—
197Q8—22no—
198Q8—1661no—
199Q8—1662no—
200Q8—1662no—
201Q8—1662no—
202Q8—1662no—
203Q8—1661no—
204D4—1661no—
205D4—1661no—
206C2×C4—3322no—
207C2×C4—1661no—
208C2×C4—1661no—
209C2×C4—1662no—
210C2×C4—1661no—
211C2×C4—1662no—
212C2×C4C2×D33212yes—
213C2×C4C2×D33212yes—
214C2×C4C2×D33211yescharacteristic
215C2×C4C2×D33211yescharacteristic
216C2×C4C23×D8311yescharacteristic
217C8—22no—
218C8—21no—
219C8#1305059112yes—
220C22—3321no—
221C22—3321no—
222C22—3321no—
223C22C22×D33211yescharacteristic
224C4—22no—
225C4—22no—
226C4—21no—
227C4—3322no—
228C4—3321no—
229C4—3322no—
230C4—3321no—
231C4—3321no—
232C4#1458182512yes—
233C4C2×#1305059111yescharacteristic
234C4#1458183811yescharacteristic
235C2—6641no—
236C2#1912837311yescharacteristic
237C2C2×#1458182511yescharacteristic
238C2#1912851811yescharacteristic
subgroupG/H
centerC22C22×D332
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#446×C83C2
socleC22×C83C22×D4
prime radicalC2×D332C23
hypercenter#446D83
hypocenterC83#3398
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C4×Q8 ◁ #446

Complements

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

NH ≅ G/Ntype
1. #19128430C2semidirect
4. #19128081C2semidirect
6. #19127912C2semidirect
7. #19127912C2semidirect
9. #19127809C2semidirect
11. #446×C83C2semidirect
120. C83#3398semidirect
129. #446D83semidirect

Maximal subgroups

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

subgroupindex
#191284302
#191284242
#191280952
#191280812
#191279242
#191279122
#191279072
#191278092
C4×#130505792
#446×C832
#339883