Finite Group Databasebeta

← #24918925

Subgroups — #24918925

Order 10624.

346 conjugacy classes of subgroups; only the 109 normal subgroups are shown.

normal subgroups only

This group has 346 conjugacy classes of subgroups — too many to draw. Only its 109 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#19128324C212yesmaximal, complemented
2#19127994C212yesmaximal, complemented
3#19127980C212yesmaximal, complemented
4#19127949C212yesmaximal, complemented
5#19127937C211yesmaximal, characteristic
6#19127925C212yesmaximal, complemented
7#19127919C212yesmaximal, complemented
8#19127727C211yesmaximal, characteristic, complemented
9#467×C83C211yesmaximal, characteristic, complemented
10C2×#13050577C2212yes—
11C2×D664C2211yescharacteristic
12C2×#13050574C2211yescharacteristic
13#14581791C2212yes—
14#14581787—22no—
15#14581781C2212yes—
16#14581778—22no—
17#14581774—22no—
18#14581772—22no—
19#14581766C2212yes—
20Q8×Dic83C2212yes—
21#14581754C2212yes—
22#14581751C2211yescharacteristic
23#14581726C2211yescharacteristic
24#14581725C2212yes—
25Dic83⋊C8C2212yes—
26Dic83⋊C8C2212yes—
27#14581716C2212yes—
28#14581714C2212yes—
29#14581712C2212yes—
30C2×Q16×C83C2211yescharacteristic
31C22×C8×C83C2211yescharacteristic
32C22⋊Q8×C83C2212yes—
33#107×C83C2211yescharacteristic
34#103×C83C2212yes—
35C2×Dic166—22no—
36C2×D332C2312yes—
37C2×C22⋊D83—22no—
38C2×C4×D83—22no—
39#13050577—42no—
40D664—41no—
41#13050574—41no—
42C2×C83⋊C8C2312yes—
43C4⋊Dic83—22no—
44C4⋊Dic83D411yescharacteristic
45C4⋊Dic83D411yescharacteristic
46C4⋊Dic83C2312yes—
47Dic83⋊C4—22no—
48C22⋊Dic83D412yes—
49(C2×D83)⋊C4—22no—
50(C2×D83)⋊C4—22no—
51(C2×D83)⋊C4—22no—
52(C2×D83)⋊C4—22no—
53C4×Dic83—22no—
54C4×Dic83C2312yes—
55C2×Q8×C83C2312yes—
56C22×C4×C83C2311yescharacteristic
57Q16×C83—41no—
58C2×C8×C83—41no—
59C2×C8×C83C2311yescharacteristic
60C2×C8×C83C2311yescharacteristic
61C4⋊C4×C83—22no—
62C4⋊C4×C83C2312yes—
63C22⋊C4×C83—22no—
64C23×C83C2×D411yescharacteristic
65C22×D83—22no—
66Q8×C83—22no—
67Q8×C83—22no—
68C2×C4×C83—21no—
69C2×C4×C83—21no—
70C2×C4×C83—22no—
71C2×C4×C83—22no—
72C2×C4×C83C2×D411yescharacteristic
73C2×C4×C83C2411yescharacteristic
74C22⋊D83—82no—
75C2×Dic83—22no—
76C2×Dic83C2×D412yes—
77C2×Dic83C2×D412yes—
78D332—22no—
79D332—22no—
80C4×D83—82no—
81Dic166—82no—
82C8×C83—21no—
83C8×C83—21no—
84C83⋊C8—22no—
85C22×C83—21no—
86C22×C83—21no—
87C22×C83—21no—
88C22×C83C22×D411yescharacteristic
89C2×D83—42no—
90C2×D83—42no—
91C2×D83—42no—
92C4×C83—21no—
93C4×C83—42no—
94C4×C83—42no—
95C4×C83H(5,𝔽2)11yescharacteristic
96C4×C832−1+411yescharacteristic
97Dic83—22no—
98Dic83—22no—
99Dic83—42no—
100C2×C83—41no—
101C2×C83#53711yescharacteristic
102C2×C83#57611yescharacteristic
103C2×C83#57811yescharacteristic
104D83—82no—
105#3262—831nomaximal, Sylow
106C83#326211yescharacteristic, complemented, Sylow
107#542—832no—
108#477—832no—
109#474—832no—
110#468—832no—
111#467D8311yescharacteristic, complemented
112#466—831no—
113#464—832no—
114#463—832no—
115#424—831no—
116C2×Q16C2×D8311yescharacteristic
117C2×SD16—832no—
118C2×D8—831no—
119C2×M16—831no—
120C22×C8C2×D8311yescharacteristic
121C4⋊Q8—832no—
122#126—1662no—
123#124—832no—
124#124—1662no—
125#123—1662no—
126C22⋊Q8—1662no—
127C22⋊Q8C2×D8312yes—
128#121—832no—
129C4×Q8—832no—
130C4×D4—832no—
131#117—831no—
132#107—831no—
133#107—832no—
134#107C2×D8311yescharacteristic
135C4⋊C8—832no—
136C4⋊C8—832no—
137#103—832no—
138#103C2×D8312yes—
139#102—832no—
140#102—832no—
141C2×Q8—1662no—
142C2×Q8C22×D8312yes—
143C2×D4—832no—
144C2×D4—1662no—
145C22×C4—1662no—
146C22×C4C22×D8311yescharacteristic
147Q16—41no—
148SD16—3322no—
149D8—3321no—
150M16—3321no—
151C2×C8—41no—
152C2×C8—832no—
153C2×C8C22×D8311yescharacteristic
154C2×C8C22×D8311yescharacteristic
155C4⋊C4—22no—
156C4⋊C4—831no—
157C4⋊C4—831no—
158C4⋊C4—832no—
159C4⋊C4—1662no—
160C4⋊C4—1662no—
161C4⋊C4C22×D8312yes—
162C22⋊C4—22no—
163C22⋊C4—832no—
164C22⋊C4—1662no—
165C22⋊C4—1662no—
166C22⋊C4—1662no—
167C22⋊C4—1662no—
168C42—832no—
169C42—1662no—
170C23—1662no—
171C23D4×D8311yescharacteristic
172Q8—22no—
173Q8—22no—
174Q8—6642no—
175D4—6642no—
176D4—1662no—
177D4—1662no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—22no—
181C2×C4—22no—
182C2×C4—832no—
183C2×C4—832no—
184C2×C4—6642no—
185C2×C4—1662no—
186C2×C4D4×D8311yescharacteristic
187C2×C4C23×D8311yescharacteristic
188C8—21no—
189C8—21no—
190C8—1662no—
191C22—21no—
192C22—21no—
193C22—21no—
194C22—3322no—
195C22—3322no—
196C22—3322no—
197C22C2×D4×D8311yescharacteristic
198C4—21no—
199C4—42no—
200C4—42no—
201C4—3322no—
202C4—1662no—
203C4—1662no—
204C4#1458183711yescharacteristic
205C4#1458183911yescharacteristic
206C2—41no—
207C2—6642no—
208C2#1912827811yescharacteristic
209C2#1912850311yescharacteristic
210C2#1912851411yescharacteristic
subgroupG/H
centerC22C2×D4×D83
commutator, cosocleC2×C4×C83C24
FrattiniC2×C4C23×D83
Fitting#467×C83C2
socleC22×C83C22×D4
prime radical#19127937C2
hypercenter#467D83
hypocenterC83#3262
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
1. #19128324C2semidirect
2. #19127994C2semidirect
3. #19127980C2semidirect
4. #19127949C2semidirect
6. #19127925C2semidirect
7. #19127919C2semidirect
8. #19127727C2semidirect
9. #467×C83C2semidirect
106. C83#3262semidirect
111. #467D83semidirect

Maximal subgroups

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

subgroupindex
#191283242
#191279942
#191279802
#191279492
#191279372
#191279252
#191279192
#191277272
#467×C832
#326283