Finite Group Databasebeta

← #27181520

Subgroups — #27181520

Order 12928.

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

normal subgroups only

This group has 400 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#20265730C211yesmaximal, characteristic, complemented
2C2×#15692244C211yesmaximal, characteristic
3#20265538C214yesmaximal, complemented
4#20265533C214yesmaximal, complemented
5#20265186C211yesmaximal, characteristic, complemented
6#20265183C211yesmaximal, characteristic
7#20265181C211yesmaximal, characteristic
8M16×Dic101C211yesmaximal, characteristic, complemented
9#419×C101C211yesmaximal, characteristic, complemented
10C22×D404C2211yescharacteristic
11C2×#13108058C2211yescharacteristic
12#15692254C2212yescomplemented
13#15692246C2212yescomplemented
14#15692244—22no—
15#15692244C2212yes—
16#15692244C2212yes—
17#15692232—22no—
18C4×Dic202—22no—
19#15692214C2211yescharacteristic
20C2×#13108051—22no—
21C2×C4×Dic101C2211yescharacteristic
22#15692179C2214yescomplemented
23#15692177C2214yes—
24#15692175C2214yes—
25#15692165C2212yescomplemented
26C8×Dic101C2212yescomplemented
27C22×Q8×C101C2211yescharacteristic
28C2×M16×C101C2211yescharacteristic
29#117×C101C2211yescharacteristic
30#103×C101C2214yescomplemented
31C23×D101—21no—
32C2×Dic202—22no—
33C2×D404—22no—
34C2×D404—22no—
35C2×D404—21no—
36C2×D404C2312yes—
37C22×Dic101D412yes—
38#13108058—22no—
39C2×#11717579C2312yes—
40C4⋊Dic101—22no—
41C4⋊Dic101C2312yes—
42#13108053—22no—
43#13108053—22no—
44C22⋊Dic101—21no—
45#13108051—24no—
46#13108051—24no—
47C4×Dic101—21no—
48C4×Dic101D412yes—
49C4×Dic101C2311yescharacteristic
50C4×Dic101C2311yescharacteristic
51C2×Q8×C101—21no—
52C2×Q8×C101—22no—
53C2×Q8×C101—22no—
54C2×Q8×C101C2312yes—
55C22×C4×C101—21no—
56C22×C4×C101C2311yescharacteristic
57M16×C101—22no—
58C2×C8×C101C2312yes—
59C4⋊C4×C101C2312yes—
60C22⋊C4×C101—21no—
61C42×C101—21no—
62C23×C101C2×D411yescharacteristic
63C22×D101—22no—
64C22×D101—44no—
65C22×D101—42no—
66Q8×C101—22no—
67Q8×C101—22no—
68Q8×C101—42no—
69C2×C4×C101—22no—
70C2×C4×C101—22no—
71C2×C4×C101—42no—
72C2×C4×C101C2×D411yescharacteristic
73C2×C4×C101#4112yes—
74C2×C4×C101#4112yes—
75C2×C4×C101C2411yescharacteristic
76C2×Dic101—22no—
77C2×Dic101—42no—
78C2×Dic101C2×D412yes—
79C2×Dic101C2×D412yes—
80D404—22no—
81D404—22no—
82D404—42no—
83Dic202—44no—
84C8×C101—22no—
85#11717579—22no—
86C22×C101—22no—
87C22×C101#12412yes—
88C22×C101C22×D411yescharacteristic
89C2×D101—42no—
90C2×D101—44no—
91C2×D101—84no—
92C4×C101—42no—
93C4×C101—42no—
94C4×C101#12412yes—
95C4×C101C2×#4111yescharacteristic
96C4×C101C2×#4111yescharacteristic
97Dic101—22no—
98Dic101—22no—
99Dic101—42no—
100C2×C101—22no—
101C2×C101C2×#12411yescharacteristic
102C2×C101#57512yes—
103D101—82no—
104#3140—1011nomaximal, Sylow
105C101#314011yescharacteristic, complemented, Sylow
106#533—1011no—
107C2×#124—1011no—
108#489—1014no—
109#488—1014no—
110#419—1011no—
111#419D10111yescharacteristic, complemented
112#418—1011no—
113#418—1011no—
114C4×M16—1011no—
115C22×Q8C2×D10111yescharacteristic
116C22×D4—1011no—
117C2×M16—1011no—
118C2×M16C2×D10111yescharacteristic
119C4⋊Q8—1012no—
120#125—1012no—
121#124—1012no—
122#124—1012no—
123#124—2022no—
124C22⋊Q8—2022no—
125C4×Q8—2022no—
126#117—1011no—
127#117C2×D10111yescharacteristic
128C2×C22⋊C4—2022no—
129C2×C42—1011no—
130#103—1014no—
131#103C2×D10114yescomplemented
132#102—1014no—
133#102—1014no—
134#97—1012no—
135C4×C8—1012no—
136C24—2021no—
137C2×Q8—21no—
138C2×Q8—22no—
139C2×Q8—22no—
140C2×Q8—2022no—
141C2×Q8C22×D10112yes—
142C2×D4—1012no—
143C2×D4—2021no—
144C2×D4—2022no—
145C2×D4—2022no—
146C22×C4—21no—
147C22×C4—1012no—
148C22×C4C22×D10111yescharacteristic
149M16—22no—
150M16—2022no—
151C2×C8—1012no—
152C2×C8C22×D10112yes—
153C4⋊C4—1012no—
154C4⋊C4—2022no—
155C4⋊C4—2022no—
156C4⋊C4—2022no—
157C4⋊C4C22×D10112yes—
158C22⋊C4—21no—
159C22⋊C4—2024no—
160C22⋊C4—2024no—
161C22⋊C4—2021no—
162C42—21no—
163C42—1012no—
164C42—1011no—
165C42—1011no—
166C42—2021no—
167C23—4044no—
168C23—4042no—
169C23—2022no—
170C23D4×D10111yescharacteristic
171Q8—22no—
172Q8—22no—
173Q8—42no—
174Q8—4044no—
175D4—4042no—
176D4—2022no—
177D4—2022no—
178C2×C4—22no—
179C2×C4—22no—
180C2×C4—42no—
181C2×C4—1012no—
182C2×C4—1012no—
183C2×C4—4042no—
184C2×C4—2022no—
185C2×C4D4×D10111yescharacteristic
186C2×C4#1310807312yes—
187C2×C4#1310807412yes—
188C2×C4C23×D10111yescharacteristic
189C8—22no—
190C8—2022no—
191C22—22no—
192C22—8084no—
193C22—4042no—
194C22—4044no—
195C22#1569224112yes—
196C22C2×D4×D10111yescharacteristic
197C4—42no—
198C4—42no—
199C4—4042no—
200C4—2022no—
201C4—2022no—
202C4#1569224112yes—
203C4C2×#1310807311yescharacteristic
204C4C2×#1310807411yescharacteristic
205C2—22no—
206C2—8082no—
207C2C2×#1569224111yescharacteristic
208C2#2026597612yes—
subgroupG/H
centerC22C2×D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
Fitting#419×C101C2
socleC22×C101C22×D4
prime radicalC22×D404C22
hypercenter#419D101
hypocenterC101#3140
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. #20265730C2semidirect
3. #20265538C2semidirect
4. #20265533C2semidirect
5. #20265186C2semidirect
8. M16×Dic101C2semidirect
9. #419×C101C2semidirect
12. #15692254C22semidirect
13. #15692246C22semidirect
22. #15692179C22semidirect
25. #15692165C22semidirect
26. C8×Dic101C22semidirect
30. #103×C101C22semidirect
105. C101#3140semidirect
111. #419D101semidirect
131. #103C2×D101semidirect

Maximal subgroups

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

subgroupindex
#202657302
C2×#156922442
#202655382
#202655332
#202651862
#202651832
#202651812
M16×Dic1012
#419×C1012
#3140101