Finite Group Databasebeta

← #35323518

Subgroups — #35323518

Order 21248.

370 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 370 conjugacy classes of subgroups — too many to draw. Only its 97 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×#19128103C211yesmaximal, characteristic, complemented
2C2×#19127512C212yesmaximal
3#24908351C211yesmaximal, characteristic
4#24905562C212yesmaximal, complemented
5C83×#1376C211yesmaximal, characteristic, complemented
6C2×#14581814C2211yescharacteristic
7C2×#14581784—21no—
8#19128103—24no—
9C2×#14581728C2211yescharacteristic
10C2×#14581726—21no—
11C2×#14581724C2212yes—
12C2×#14581720—21no—
13#19127667—21no—
14#19127512—24no—
15C2×M32×C83C2212yes—
16C2×C4×C8×C83C2211yescharacteristic
17C2×#13050590—21no—
18#14581814—22no—
19#14581814—22no—
20C2×Dic332Q811yescharacteristic
21C2×#13050579—21no—
22C2×D664Q811yescharacteristic
23C2×#13050574—21no—
24#14581784—24no—
25#14581784—42no—
26C2×C4⋊Dic83—22no—
27C2×Dic83⋊C4—42no—
28#14581728—22no—
29#14581726—24no—
30#14581724—22no—
31#14581724D414yes—
32#14581720—42no—
33(C2×D83)⋊C8—42no—
34C22×C8×C83D411yescharacteristic
35C22×C8×C83C2311yescharacteristic
36C2×C42×C83D411yescharacteristic
37M32×C83—22no—
38M32×C83—22no—
39C2×C16×C83—22no—
40C4×C8×C83—22no—
41#13050590—41no—
42#13050590—41no—
43#13050590—41no—
44#13050590—41no—
45C2×Dic166—21no—
46C2×D332—21no—
47C2×C22⋊D83—41no—
48C22×Dic83—42no—
49C2×C4×D83—41no—
50Dic332—22no—
51#13050579—42no—
52D664—22no—
53#13050574—41no—
54#13050574—41no—
55C2×C83⋊C8—41no—
56C4⋊Dic83—24no—
57C4⋊Dic83—42no—
58Dic83⋊C4—48no—
59C22×C4×C83—21no—
60C22×C4×C83C2×D411yescharacteristic
61C2×C8×C83—21no—
62C2×C8×C83—21no—
63C2×C8×C83SD1612yes—
64C2×C8×C83C2×D412yes—
65C2×C8×C83C2×Q811yescharacteristic
66C2×C8×C83#4111yescharacteristic
67C2×C8×C83#4112yes—
68C42×C83—21no—
69C42×C83D812yes—
70C16×C83—42no—
71C23×C83AGL(1,ℤ/8)11yescharacteristic
72C22×D83—41no—
73C2×C4×C83—22no—
74C2×C4×C83—42no—
75C2×C4×C83#12011yescharacteristic
76C2×C4×C83C22⋊Q811yescharacteristic
77C2×C4×C83#12411yescharacteristic
78C2×C4×C83C2×D811yescharacteristic
79C2×C4×C83C2×SD1611yescharacteristic
80C2×C4×C83#13711yescharacteristic
81C22⋊D83—81no—
82C22⋊D83—81no—
83C2×Dic83—44no—
84C2×Dic83—41no—
85C2×Dic83—84no—
86D332—41no—
87D332—41no—
88C4×D83—81no—
89C4×D83—81no—
90Dic166—41no—
91Dic166—41no—
92C8×C83—22no—
93C8×C83C22⋊Q812yes—
94C8×C83#12412yes—
95C83⋊C8—81no—
96C22×C83—22no—
97C22×C83#44711yescharacteristic
98C22×C83#47611yescharacteristic
99C22×C83#48911yescharacteristic
100C2×D83—82no—
101C2×D83—81no—
102C4×C83—42no—
103C4×C83#39311yescharacteristic
104C4×C83#44811yescharacteristic
105C4×C83#47411yescharacteristic
106C4×C83#48611yescharacteristic
107Dic83—84no—
108Dic83—81no—
109#13372—831nomaximal, Sylow
110C2×C83—21no—
111C2×C83—21no—
112C2×C83#202311yescharacteristic
113C2×C83#223112yes—
114D83—161no—
115C2×#499—831no—
116C2×#360—832no—
117#1974—831no—
118#1383—832no—
119#1376D8311yescharacteristic, complemented
120C83#1337211yescharacteristic, complemented, Sylow
121C2×#135—831no—
122C2×#125—1661no—
123C2×M32C2×D8312yes—
124#499—1664no—
125C2×#108—831no—
126C2×#107—1661no—
127C2×#106—832no—
128C2×#104—1661no—
129#408—1661no—
130C2×C4×C8C2×D8311yescharacteristic
131#360—1664no—
132C2×#41—1661no—
133#135—1662no—
134#135—1662no—
135C2×Q16—831no—
136C2×SD16—1661no—
137C2×D8—831no—
138C2×M16—1661no—
139C22×C8D33211yescharacteristic
140C22×C8C22×D8311yescharacteristic
141#125—3322no—
142#125—1664no—
143C2×C4⋊C4—3322no—
144C2×C4⋊C4—1662no—
145C2×C42D33211yescharacteristic
146M32—22no—
147M32—22no—
148C2×C16—22no—
149#108—1662no—
150#107—1664no—
151#106—834no—
152#106—1662no—
153#104—3322no—
154C22⋊C8—3322no—
155C4×C8—22no—
156#41—3321no—
157#41—3321no—
158#41—3321no—
159#41—3321no—
160C2×Q8—1661no—
161C2×D4—3321no—
162C2×D4—1661no—
163C22×C4—21no—
164C22×C4—3322no—
165C22×C4—3321no—
166C22×C4C2×D33211yescharacteristic
167Q16—1662no—
168SD16—3322no—
169D8—1662no—
170M16—3321no—
171M16—3321no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—3321no—
175C2×C8#1305057912yes—
176C2×C8D4×D8312yes—
177C2×C8Q8×D8311yescharacteristic
178C2×C8#1305058911yescharacteristic
179C2×C8#1305059012yes—
180C4⋊C4—3328no—
181C4⋊C4—3322no—
182C4⋊C4—1664no—
183C42—21no—
184C42D66412yes—
185C16—42no—
186C23—3321no—
187C23#1458182011yescharacteristic
188Q8—3321no—
189Q8—3321no—
190D4—3321no—
191D4—3321no—
192D4—6641no—
193D4—6641no—
194C2×C4—22no—
195C2×C4—42no—
196C2×C4—3324no—
197C2×C4—3321no—
198C2×C4—6644no—
199C2×C4—6641no—
200C2×C4—6641no—
201C2×C4#1458176211yescharacteristic
202C2×C4#1458177211yescharacteristic
203C2×C4#1458178011yescharacteristic
204C2×C4C2×D66411yescharacteristic
205C2×C4C2×#1305057911yescharacteristic
206C2×C4#1458182511yescharacteristic
207C8—22no—
208C8—6641no—
209C8#1458177012yes—
210C8#1458177812yes—
211C22—22no—
212C22—6641no—
213C22—6642no—
214C22#1912783711yescharacteristic
215C22#1912799211yescharacteristic
216C22#1912806111yescharacteristic
217C4—42no—
218C4—6644no—
219C4—6641no—
220C4#1912761411yescharacteristic
221C4#1912784211yescharacteristic
222C4#1912798211yescharacteristic
223C4#1912804811yescharacteristic
224C2—21no—
225C2—21no—
226C2—13281no—
227C2#2490860011yescharacteristic
228C2#2490964012yes—
subgroupG/H
centerC22#19127992
commutator, cosocleC22×C8×C83C23
FrattiniC22×C8C22×D83
FittingC83×#1376C2
socleC22×C83#476
prime radicalC2×#14581814C22
hypercenter#1376D83
hypocenterC83#13372
LCSG ▷ C22×C8×C83 ▷ C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ #1376

Complements

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

NH ≅ G/Ntype
1. C2×#19128103C2semidirect
4. #24905562C2semidirect
5. C83×#1376C2semidirect
119. #1376D83semidirect
120. C83#13372semidirect

Maximal subgroups

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

subgroupindex
C2×#191281032
C2×#191275122
#249083512
#249055622
C83×#13762
#1337283