Finite Group Databasebeta

← #24919852

Subgroups — #24919852

Order 10624.

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

normal subgroups only

This group has 356 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#19128409C212yesmaximal
2#19128105C211yesmaximal, characteristic, complemented
3#19128035C212yesmaximal, complemented
4#19128034C212yesmaximal, complemented
5#19127994C212yesmaximal, complemented
6#19127939C211yesmaximal, characteristic, complemented
7Q16×Dic83C211yesmaximal, characteristic, complemented
8C8×C22⋊D83C211yesmaximal, characteristic
9C2×#14581726C211yesmaximal, characteristic, complemented
10#107×D83C211yesmaximal, characteristic, complemented
11#467×C83C211yesmaximal, characteristic, complemented
12C2×C8×D83C2211yescharacteristic
13#14581791C2212yes—
14#14581782—22no—
15#14581772—22no—
16#14581769—22no—
17#14581768C2212yes—
18Q8×Dic83C2212yes—
19C4×C22⋊D83C2211yescharacteristic
20(C22⋊D83)⋊C4—22no—
21C2×C4⋊Dic83C2212yes—
22C4⋊C4×D83C2212yes—
23#14581726C2211yescharacteristic
24#14581726C2211yescharacteristic
25#14581726C2212yes—
26#14581725C2212yes—
27#14581725C2212yes—
28Dic83⋊C8C2211yescharacteristic
29#14581716C2212yes—
30#14581716C2212yes—
31C22⋊(C83⋊C8)C2211yescharacteristic
32(C2×D83)⋊C8C2211yescharacteristic
33C8×Dic83C2211yescharacteristic
34C2×Q16×C83C2211yescharacteristic
35C22×C8×C83C2211yescharacteristic
36C22⋊Q8×C83C2212yes—
37#107×C83C2211yescharacteristic
38#103×C83C2212yes—
39C2×Dic166—22no—
40C2×C22⋊D83D411yescharacteristic
41C22×Dic83—22no—
42C2×C4×D83—22no—
43C2×C4×D83C2311yescharacteristic
44C2×C83⋊C8C2311yescharacteristic
45C2×C83⋊C8C2311yescharacteristic
46C8×D83—22no—
47C4⋊Dic83—22no—
48C4⋊Dic83—22no—
49C4⋊Dic83C2312yes—
50C4⋊Dic83C2312yes—
51Dic83⋊C4—22no—
52Dic83⋊C4—22no—
53Dic83⋊C4—22no—
54Dic83⋊C4D411yescharacteristic
55C22⋊Dic83D411yescharacteristic
56(C2×D83)⋊C4—22no—
57(C2×D83)⋊C4D411yescharacteristic
58C4×Dic83—22no—
59C4×Dic83C2311yescharacteristic
60C2×Q8×C83C2312yes—
61C22×C4×C83C2311yescharacteristic
62Q16×C83—22no—
63C2×C8×C83—22no—
64C2×C8×C83C2311yescharacteristic
65C2×C8×C83C2311yescharacteristic
66C4⋊C4×C83—22no—
67C4⋊C4×C83C2312yes—
68C22⋊C4×C83—22no—
69C23×C83C2×D411yescharacteristic
70C22×D83C2×D411yescharacteristic
71Q8×C83—22no—
72Q8×C83—22no—
73C2×C4×C83—21no—
74C2×C4×C83—21no—
75C2×C4×C83—22no—
76C2×C4×C83—22no—
77C2×C4×C83C2×D411yescharacteristic
78C2×C4×C83C2411yescharacteristic
79C22⋊D83Q1614yescomplemented
80C2×Dic83—22no—
81C2×Dic83—22no—
82C2×Dic83—82no—
83C2×Dic83C2×D411yescharacteristic
84C2×Dic83C2×D411yescharacteristic
85C2×Dic83C2×D411yescharacteristic
86C4×D83—21no—
87C4×D83—21no—
88C4×D83—82no—
89Dic166—82no—
90C8×C83—21no—
91C8×C83#4112yes—
92C83⋊C8—21no—
93C83⋊C8—21no—
94C22×C83—21no—
95C22×C83—21no—
96C22×C83C2×Q1612yes—
97C22×C83C22×D411yescharacteristic
98C2×D83—21no—
99C2×D83—21no—
100C2×D83C2×Q1612yes—
101C4×C83—21no—
102C4×C83—42no—
103C4×C83—42no—
104C4×C83C2×#4111yescharacteristic
105C4×C832−1+411yescharacteristic
106Dic83—21no—
107Dic83—21no—
108Dic83—42no—
109Dic83—42no—
110Dic83C2×Q1612yes—
111C2×C83—22no—
112C2×C83#54711yescharacteristic
113C2×C83C22×Q1611yescharacteristic
114C2×C83#57611yescharacteristic
115D83—22no—
116#3338—831nomaximal, Sylow
117C83#333811yescharacteristic, complemented, Sylow
118#554—832no—
119#500—831no—
120#484—832no—
121#484—832no—
122#477—832no—
123#467—831no—
124#467D8311yescharacteristic, complemented
125C4×Q16—831no—
126C8×D4—831no—
127C2×#107—831no—
128C2×#107—831no—
129C2×Q16C2×D8311yescharacteristic
130C22×C8—831no—
131C22×C8C2×D8311yescharacteristic
132C4⋊Q8—832no—
133#125—1662no—
134C22⋊Q8—832no—
135C22⋊Q8—1662no—
136C22⋊Q8—1662no—
137C22⋊Q8C2×D8312yes—
138C4×Q8—832no—
139C4×D4—831no—
140C4×D4—1662no—
141C2×C4⋊C4—832no—
142C2×C4⋊C4—832no—
143#107—831no—
144#107—831no—
145#107—832no—
146#107—832no—
147#107—832no—
148#107C2×D8311yescharacteristic
149C4⋊C8—831no—
150#103—832no—
151#103—832no—
152#103C2×D8312yes—
153C22⋊C8—831no—
154C22⋊C8—831no—
155C4×C8—831no—
156C2×Q8—1662no—
157C2×Q8C22×D8312yes—
158C2×D4—831no—
159C22×C4—831no—
160C22×C4—1662no—
161C22×C4—1662no—
162C22×C4C22×D8311yescharacteristic
163Q16—22no—
164C2×C8—22no—
165C2×C8—831no—
166C2×C8—831no—
167C2×C8—1662no—
168C2×C8C22×D8311yescharacteristic
169C2×C8C22×D8311yescharacteristic
170C4⋊C4—22no—
171C4⋊C4—831no—
172C4⋊C4—832no—
173C4⋊C4—832no—
174C4⋊C4—1662no—
175C4⋊C4—1662no—
176C4⋊C4—1662no—
177C4⋊C4—1662no—
178C4⋊C4—1662no—
179C4⋊C4C22×D8312yes—
180C22⋊C4—22no—
181C22⋊C4—831no—
182C22⋊C4—831no—
183C22⋊C4—1662no—
184C42—831no—
185C42—1662no—
186C23—831no—
187C23D4×D8311yescharacteristic
188Q8—22no—
189Q8—22no—
190Q8—6642no—
191D4—834no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—831no—
197C2×C4—831no—
198C2×C4—831no—
199C2×C4—6642no—
200C2×C4—6642no—
201C2×C4—1662no—
202C2×C4—1662no—
203C2×C4—1661no—
204C2×C4—1661no—
205C2×C4D4×D8311yescharacteristic
206C2×C4C23×D8311yescharacteristic
207C8—21no—
208C8—1661no—
209C8—1661no—
210C8#1305058912yes—
211C22—21no—
212C22—21no—
213C22—832no—
214C22—1661no—
215C22—1661no—
216C22Q16×D8312yes—
217C22C2×D4×D8311yescharacteristic
218C4—21no—
219C4—42no—
220C4—42no—
221C4—3322no—
222C4—3322no—
223C4—832no—
224C4—1661no—
225C4—1661no—
226C4C2×#1305058911yescharacteristic
227C4#1458183911yescharacteristic
228C2—22no—
229C2—1662no—
230C2#1912835111yescharacteristic
231C2C2×Q16×D8311yescharacteristic
232C2#1912850311yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#467×C83C2
socleC22×C83C22×D4
prime radicalC2×C22⋊D83D4
hypercenter#467D83
hypocenterC83#3338
LCSG ▷ C2×C4×C83 ▷ C2×C83 ▷ C83
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
2. #19128105C2semidirect
3. #19128035C2semidirect
4. #19128034C2semidirect
5. #19127994C2semidirect
6. #19127939C2semidirect
7. Q16×Dic83C2semidirect
9. C2×#14581726C2semidirect
10. #107×D83C2semidirect
11. #467×C83C2semidirect
79. C22⋊D83Q16semidirect
117. C83#3338semidirect
124. #467D83semidirect

Maximal subgroups

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

subgroupindex
#191284092
#191281052
#191280352
#191280342
#191279942
#191279392
Q16×Dic832
C8×C22⋊D832
C2×#145817262
#107×D832
#467×C832
#333883