Finite Group Databasebeta

← #27181423

Subgroups — #27181423

Order 12928.

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

normal subgroups only

This group has 382 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#20265782C211yesmaximal, characteristic
2#20265501C212yesmaximal, complemented
3#20265484C212yesmaximal, complemented
4#20265377C212yesmaximal, complemented
5#20265316C212yesmaximal, complemented
6#20265275C212yesmaximal, complemented
7C4×#13108061C212yesmaximal, complemented
8#20265238C211yesmaximal, characteristic, complemented
9#519×C101C211yesmaximal, characteristic, complemented
10C2×#13108061C2212yes—
11C2×#13108061C2212yes—
12#15692252C2211yescharacteristic
13#15692247C2211yescharacteristic
14#15692240—22no—
15#15692229C2212yes—
16#15692228—22no—
17C4×D404C2212yes—
18#15692188C2212yes—
19#15692186C2212yes—
20C4⋊#11717579C2212yes—
21#15692179C2212yes—
22#15692179C2212yes—
23#15692175C2212yes—
24#15692175C2212yes—
25C22⋊#11717579C2212yes—
26#15692166C2211yescharacteristic
27C4×#11717579C2211yescharacteristic
28C22×Q8×C101C2211yescharacteristic
29C4×Q8×C101—22no—
30C4×Q8×C101C2212yes—
31C4×Q8×C101C2212yes—
32#117×C101—22no—
33#117×C101C2211yescharacteristic
34#117×C101C2211yescharacteristic
35C2×D404—22no—
36C2×D404C2312yes—
37C2×C22⋊D101—22no—
38C2×C4×D101—22no—
39#13108061—24no—
40#13108061—24no—
41C2×#11717579C2312yes—
42C2×#11717579C2312yes—
43C4⋊Dic101C2312yes—
44#13108053—22no—
45#13108051—22no—
46#13108051—22no—
47C2×Q8×C101—24no—
48C2×Q8×C101—21no—
49C2×Q8×C101C2312yes—
50C22×C4×C101—21no—
51C22×C4×C101C2311yescharacteristic
52C4⋊C4×C101—22no—
53C4⋊C4×C101—22no—
54C4⋊C4×C101D412yes—
55C4⋊C4×C101C2312yes—
56C22⋊C4×C101—21no—
57C22⋊C4×C101D412yes—
58C42×C101—22no—
59C42×C101—22no—
60C42×C101—21no—
61C42×C101C2311yescharacteristic
62C42×C101C2311yescharacteristic
63C23×C101C2×D411yescharacteristic
64C22×D101—22no—
65Q8×C101—24no—
66Q8×C101—22no—
67Q8×C101#4114yes—
68C2×C4×C101—24no—
69C2×C4×C101—22no—
70C2×C4×C101—22no—
71C2×C4×C101—22no—
72C2×C4×C101C2×D412yes—
73C2×C4×C101C2×D412yes—
74C2×C4×C101C2×D411yescharacteristic
75C2×C4×C101C2411yescharacteristic
76C22⋊D101—82no—
77C2×Dic101—22no—
78D404—22no—
79D404—22no—
80D404—82no—
81C4×D101—82no—
82#11717579—22no—
83#11717579—22no—
84C22×C101—21no—
85C22×C101—22no—
86C22×C101C22×D411yescharacteristic
87C2×D101—42no—
88C2×D101—42no—
89C2×D101—42no—
90C4×C101—24no—
91C4×C101—21no—
92C4×C101—22no—
93C4×C101—22no—
94C4×C101—42no—
95C4×C101C2×#4112yes—
96Dic101—42no—
97C2×C101—41no—
98C2×C101#52511yescharacteristic
99C2×C101#57712yes—
100D101—82no—
101#3126—1011nomaximal, Sylow
102C101#312611yescharacteristic, complemented, Sylow
103#540—1011no—
104#519D10111yescharacteristic, complemented
105#482—1012no—
106#479—1012no—
107#460—1012no—
108#448—1012no—
109#440—1012no—
110C4×SD16—1012no—
111#433—1011no—
112C22×Q8C2×D10111yescharacteristic
113C2×SD16—1012no—
114C2×SD16—1012no—
115#127—1011no—
116#125—1011no—
117#123—2022no—
118#121—1012no—
119#121—2022no—
120C4×Q8—22no—
121C4×Q8C2×D10112yes—
122C4×Q8C2×D10112yes—
123C4×D4—1012no—
124#117—22no—
125#117C2×D10111yescharacteristic
126#117C2×D10111yescharacteristic
127#107—1012no—
128#106—1012no—
129C4⋊C8—1012no—
130#103—1012no—
131#103—1012no—
132#102—1012no—
133#102—1012no—
134C22⋊C8—1012no—
135#97—1011no—
136C4×C8—1011no—
137C2×Q8—24no—
138C2×Q8—21no—
139C2×Q8C22×D10112yes—
140C2×D4—1012no—
141C2×D4—2022no—
142C2×D4—2022no—
143C22×C4—21no—
144C22×C4—2022no—
145C22×C4C22×D10111yescharacteristic
146SD16—2024no—
147SD16—2024no—
148C2×C8—1012no—
149C2×C8—1012no—
150C4⋊C4—22no—
151C4⋊C4—22no—
152C4⋊C4—1012no—
153C4⋊C4—2022no—
154C4⋊C4C22⋊D10112yes—
155C4⋊C4C22×D10112yes—
156C22⋊C4—21no—
157C22⋊C4—2022no—
158C22⋊C4—2022no—
159C22⋊C4C22⋊D10112yes—
160C42—22no—
161C42—22no—
162C42—21no—
163C42C22×D10111yescharacteristic
164C42C22×D10111yescharacteristic
165C23—2022no—
166C23C2×C22⋊D10111yescharacteristic
167Q8—24no—
168Q8—22no—
169Q8#1310807414yes—
170D4—8082no—
171D4—8082no—
172D4—2022no—
173D4—2022no—
174C2×C4—24no—
175C2×C4—22no—
176C2×C4—22no—
177C2×C4—22no—
178C2×C4—8082no—
179C2×C4—2022no—
180C2×C4C2×C22⋊D10112yes—
181C2×C4C2×C22⋊D10112yes—
182C2×C4C2×C22⋊D10111yescharacteristic
183C2×C4C23×D10111yescharacteristic
184C8—2022no—
185C8—2022no—
186C22—22no—
187C22—21no—
188C22—4042no—
189C22—4042no—
190C22—4042no—
191C22C22×C22⋊D10111yescharacteristic
192C4—22no—
193C4—22no—
194C4—24no—
195C4—21no—
196C4—42no—
197C4—4042no—
198C4C2×#1310807412yes—
199C2—41no—
200C2—8082no—
201C2#2026568811yescharacteristic
202C2#2026598512yes—
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#519×C101C2
socleC22×C101C22×D4
prime radical#20265782C2
hypercenter#519D101
hypocenterC101#3126
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ #117 ◁ #519

Complements

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

NH ≅ G/Ntype
2. #20265501C2semidirect
3. #20265484C2semidirect
4. #20265377C2semidirect
5. #20265316C2semidirect
6. #20265275C2semidirect
7. C4×#13108061C2semidirect
8. #20265238C2semidirect
9. #519×C101C2semidirect
102. C101#3126semidirect
104. #519D101semidirect

Maximal subgroups

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

subgroupindex
#202657822
#202655012
#202654842
#202653772
#202653162
#202652752
C4×#131080612
#202652382
#519×C1012
#3126101