Finite Group Databasebeta

← #35372158

Subgroups — #35372158

Order 21248.

938 conjugacy classes of subgroups; only the 355 normal subgroups are shown.

normal subgroups only

This group has 938 conjugacy classes of subgroups — too many to draw. Only its 355 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#24921273C211yesmaximal, characteristic
2#24910384C211yesmaximal, characteristic
3#24906028C216yesmaximal
4#24905963C216yesmaximal
5C83×#2982C211yesmaximal, characteristic
6Q8×Dic166C414yescomplemented
7#19128430C414yescomplemented
8#19128427—24no—
9#19128332—23no—
10#19128258C2216yes—
11#19128179—21no—
12C2×Q8×Dic83—21no—
13#19127600—23no—
14#19127567C2213yes—
15C4×C4⋊Dic83C2213yes—
16#19127439C22112yes—
17#19127439C2214yes—
18C2×C4×Q8×C83C2211yescharacteristic
19C8×Q8×C83—24no—
20#431×C83C2216yes—
21#14581792C2×C416yes—
22#14581792C2×C416yes—
23#14581791—212no—
24#14581784—26no—
25#14581783—212no—
26#14581771—26no—
27#14581771C2314yes—
28Q8×Dic83—24no—
29Q8×Dic83—24no—
30Q8×Dic83—44no—
31C4×Dic166—26no—
32C4×Dic166C2×C4112yes—
33#14581751—26no—
34C2×C4⋊Dic83—23no—
35C2×C4⋊Dic83—23no—
36C2×C4×Dic83—23no—
37C83⋊#95—23no—
38C83⋊#95C2311yescharacteristic
39C83⋊#95C2313yes—
40C22×Q8×C83D411yescharacteristic
41C4×Q8×C83C2×C414yes—
42C4×Q8×C83D414yes—
43C2×C4⋊C4×C83D413yes—
44C2×C42×C83C2313yes—
45C4⋊C8×C83—26no—
46C22⋊C8×C83C2314yes—
47C4×C8×C83—26no—
48C2×Dic166—26no—
49C2×Dic166C22×C414yes—
50C22×Dic83—21no—
51C22×Dic83—23no—
52C4⋊Dic83—26no—
53C4⋊Dic83—26no—
54C4⋊Dic83—26no—
55C4⋊Dic83—46no—
56C4⋊Dic83—46no—
57C4⋊Dic83C22×C414yes—
58Dic83⋊C4—212no—
59Dic83⋊C4—24no—
60C22⋊Dic83—24no—
61C4×Dic83—212no—
62C4×Dic83—46no—
63C2×Q8×C83C22⋊C414yes—
64C2×Q8×C83SD1614yes—
65C2×Q8×C83Q1614yes—
66C22×C4×C83C2×D413yes—
67C22×C4×C83C2×D413yes—
68C22×C4×C83C2411yescharacteristic
69C2×C8×C83—24no—
70C4⋊C4×C83C22⋊C4112yes—
71C42×C83C22×C416yes—
72C42×C83C2×D416yes—
73C23×C83C22×D411yescharacteristic
74Q8×C83#103116yes—
75C2×C4×C83—43no—
76C2×C4×C83—41no—
77C2×C4×C83C2×C22⋊C416yes—
78C2×C4×C83C2×C22⋊C416yes—
79C2×C4×C83C2×SD1616yes—
80C2×C4×C83C2×Q1616yes—
81C2×C4×C83C23×C411yescharacteristic
82C2×C4×C83C22×D411yescharacteristic
83C2×Dic83—22no—
84C2×Dic83—24no—
85C2×Dic83—24no—
86C2×Dic83—41no—
87C2×Dic83—41no—
88C2×Dic83—43no—
89C2×Dic83—43no—
90C2×Dic83—43no—
91Dic166—44no—
92Dic166—86no—
93C8×C83—44no—
94C22×C83—21no—
95C22×C83—21no—
96C22×C83C22×C22⋊C411yescharacteristic
97C22×C83C22×SD1611yescharacteristic
98C22×C83C22×Q1611yescharacteristic
99C4×C83—21no—
100C4×C83—21no—
101C4×C83—26no—
102C4×C83C2×#103112yes—
103Dic83—42no—
104Dic83—44no—
105Dic83—44no—
106Dic83—83no—
107#17619—831nomaximal, Sylow
108C2×C83—21no—
109C2×C83—21no—
110C2×C83#288411yescharacteristic
111C2×C83C22×#10311yescharacteristic
112C2×C83#290111yescharacteristic
113#3502—831no—
114#2982D8311yescharacteristic
115#2351—831no—
116#1518—836no—
117#1503—836no—
118C83#1761911yescharacteristic, complemented, Sylow
119Q82—834no—
120#557—834no—
121#557—1664no—
122#544—1663no—
123#533—836no—
124#519—1661no—
125C2×C4×Q8—1661no—
126C2×C4×Q8C2×D8311yescharacteristic
127C8×Q8—24no—
128#431C2×D8316yes—
129#389—1663no—
130#382—833no—
131C4×C4⋊C4—833no—
132#328—834no—
133#328—8312no—
134C22×Q8D33211yescharacteristic
135C4⋊Q8—836no—
136C4⋊Q8—836no—
137C4⋊Q8—16612no—
138#125—16612no—
139#125—1666no—
140C22⋊Q8—834no—
141C22⋊Q8—1666no—
142C4×Q8—3324no—
143C4×Q8—8312no—
144C4×Q8—1664no—
145C4×Q8—1664no—
146C4×Q8—1666no—
147C4×Q8C4×D8314yes—
148C4×Q8C22⋊D8314yes—
149#117—1666no—
150C2×C4⋊C4—1663no—
151C2×C4⋊C4—1663no—
152C2×C4⋊C4D33213yes—
153C2×C42—1663no—
154C2×C42C22×D8313yes—
155C4⋊C8—26no—
156C22⋊C8C22×D8314yes—
157C4×C8—26no—
158#95—831no—
159#95—833no—
160#95—1663no—
161C2×Q8—834no—
162C2×Q8—1666no—
163C2×Q8(C2×D83)⋊C414yes—
164C2×Q8#1305057914yes—
165C2×Q8Dic33214yes—
166C22×C4—1663no—
167C22×C4—1661no—
168C22×C4C2×D33213yes—
169C22×C4C2×D33213yes—
170C22×C4C23×D8311yescharacteristic
171C2×C8—24no—
172C4⋊C4—3326no—
173C4⋊C4—3326no—
174C4⋊C4—834no—
175C4⋊C4—1664no—
176C4⋊C4—1666no—
177C4⋊C4—1666no—
178C4⋊C4—16612no—
179C4⋊C4—1666no—
180C4⋊C4(C2×D83)⋊C4112yes—
181C22⋊C4—1664no—
182C42—3326no—
183C42—16612no—
184C42C2×C4×D8316yes—
185C42C2×C22⋊D8316yes—
186C23C22×D33211yescharacteristic
187Q8—3324no—
188Q8—6646no—
189Q8#14581717116yes—
190C2×C4—43no—
191C2×C4—41no—
192C2×C4—3321no—
193C2×C4—3321no—
194C2×C4—3323no—
195C2×C4—3323no—
196C2×C4—3323no—
197C2×C4—1662no—
198C2×C4—1664no—
199C2×C4—1664no—
200C2×C4C2×(C2×D83)⋊C416yes—
201C2×C4C2×(C2×D83)⋊C416yes—
202C2×C4C2×#1305057916yes—
203C2×C4C2×Dic33216yes—
204C2×C4C22×C4×D8311yescharacteristic
205C2×C4C22×C22⋊D8311yescharacteristic
206C8—44no—
207C22—21no—
208C22—21no—
209C22C22×(C2×D83)⋊C411yescharacteristic
210C22C22×#1305057911yescharacteristic
211C22C22×Dic33211yescharacteristic
212C4—26no—
213C4—21no—
214C4—21no—
215C4—3322no—
216C4—3324no—
217C4—3324no—
218C4—6643no—
219C4C2×#14581717112yes—
220C2—21no—
221C2—21no—
222C2#2491592611yescharacteristic
223C2C22×#1458171711yescharacteristic
224C2#2491600811yescharacteristic
subgroupG/H
centerC22C22×(C2×D83)⋊C4
commutatorC2×C4×C83C23×C4
FrattiniC22×C4C23×D83
FittingC83×#2982C2
socleC22×C83C22×C22⋊C4
prime radicalC23×C83C22×D4
cosocleC22×C4×C83C24
hypercenter#2982D83
hypocenterC83#17619
LCSG ▷ C2×C4×C83 ▷ C22×C83 ▷ C83
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ #2982

Complements

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

NH ≅ G/Ntype
6. Q8×Dic166C4semidirect
7. #19128430C4semidirect
118. C83#17619semidirect

Maximal subgroups

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

subgroupindex
#249212732
#249103842
#249060282
#249059632
C83×#29822
#1761983