Finite Group Databasebeta

← #27180366

Subgroups — #27180366

Order 12928.

738 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 738 conjugacy classes of subgroups — too many to draw. Only its 173 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
1C2×#15692286C212yesmaximal, complemented
2#20265688C211yesmaximal, characteristic
3#20265333C214yesmaximal, complemented
4#20265316C214yesmaximal, complemented
5#20265186C212yesmaximal, complemented
6#20265141C211yesmaximal, characteristic, complemented
7C23×Q8×C101C211yesmaximal, characteristic, complemented
8C2×#13108074C2211yescharacteristic
9#15692286—28no—
10C2×#13108064C2214yescomplemented
11C2×#13108061C2214yescomplemented
12C2×#13108058C2212yes—
13#15692238—21no—
14#15692234C2212yes—
15#15692229C2212yes—
16#15692224—21no—
17C4×C22⋊D101—22no—
18#15692214C2211yescharacteristic
19#15692179C2218yescomplemented
20C22⋊#11717579C2214yes—
21C22×Q8×C101—28no—
22C22×Q8×C101—22no—
23C22×Q8×C101—21no—
24C22×Q8×C101C2214yes—
25C22×Q8×C101C2212yes—
26C23×C4×C101—21no—
27C23×C4×C101C2211yescharacteristic
28#13108074—24no—
29C2×Dic202C2311yescharacteristic
30C2×D404C2311yescharacteristic
31C2×C22⋊D101—21no—
32C2×C22⋊D101—22no—
33C2×C4×D101—21no—
34#13108064—28no—
35#13108061—28no—
36#13108058—24no—
37C2×#11717579C2314yes—
38C4⋊Dic101C2312yes—
39#13108053—22no—
40C22⋊Dic101—21no—
41C22⋊Dic101—22no—
42#13108051—22no—
43C4×Dic101—21no—
44C24×C101D411yescharacteristic
45C2×Q8×C101—216no—
46C2×Q8×C101—28no—
47C2×Q8×C101—216no—
48C2×Q8×C101—24no—
49C2×Q8×C101—24no—
50C2×Q8×C101—22no—
51C2×Q8×C101D418yescomplemented
52C2×Q8×C101C2314yes—
53C22×C4×C101—28no—
54C22×C4×C101—24no—
55C22×C4×C101—22no—
56C22×C4×C101—24no—
57C22×C4×C101D412yes—
58C22×C4×C101D411yescharacteristic
59C22×C4×C101C2312yes—
60C22×C4×C101C2311yescharacteristic
61C23×C101—22no—
62C23×C101—24no—
63C23×C101C2×D411yescharacteristic
64C23×C101C2×D412yes—
65C22×D101—21no—
66Q8×C101—216no—
67Q8×C101—28no—
68Q8×C101—24no—
69Q8×C101C2×D418yes—
70C2×C4×C101—216no—
71C2×C4×C101—28no—
72C2×C4×C101—22no—
73C2×C4×C101—24no—
74C2×C4×C101—24no—
75C2×C4×C101—24no—
76C2×C4×C101C2×D412yes—
77C2×C4×C101C2×D411yescharacteristic
78C2×C4×C101C2×D414yes—
79C2×C4×C101C2411yescharacteristic
80C22⋊D101—42no—
81C22⋊D101—44no—
82C2×Dic101—22no—
83C2×Dic101—21no—
84D404—22no—
85C4×D101—42no—
86Dic202—22no—
87#11717579—24no—
88C22×C101—28no—
89C22×C101—22no—
90C22×C101—22no—
91C22×C101—22no—
92C22×C101#12012yes—
93C22×C101#13714yescomplemented
94C22×C101C22×D411yescharacteristic
95C2×D101—41no—
96C2×D101—42no—
97C4×C101—28no—
98C4×C101—22no—
99C4×C101#12012yes—
100C4×C101C22×D412yes—
101Dic101—41no—
102Dic101—42no—
103C2×C101—22no—
104C2×C101—24no—
105C2×C101C2×#12011yescharacteristic
106C2×C101C2×#13712yes—
107D101—81no—
108#3004—1011nomaximal, Sylow
109C101#300411yescharacteristic, complemented, Sylow
110C23×Q8D10111yescharacteristic, complemented
111C2×#137—1012no—
112#525—1011no—
113#451—1014no—
114#448—1014no—
115#419—1012no—
116#407—1011no—
117C2×#41—1011no—
118C22×Q8—28no—
119C22×Q8—22no—
120C22×Q8—21no—
121C22×Q8C2×D10114yes—
122C22×Q8C2×D10112yes—
123C23×C4—21no—
124C23×C4C2×D10111yescharacteristic
125#137—2028no—
126C2×Q16—1014no—
127C2×SD16—1014no—
128C2×M16—1012no—
129#123—2021no—
130C22⋊Q8—1012no—
131#121—1012no—
132#120—2021no—
133C4×D4—2022no—
134#117—1011no—
135#103—1018no—
136C22⋊C8—1014no—
137C24C22⋊D10111yescharacteristic
138#41—2024no—
139C2×Q8—216no—
140C2×Q8—28no—
141C2×Q8—216no—
142C2×Q8—24no—
143C2×Q8—22no—
144C2×Q8—24no—
145C2×Q8—1011no—
146C2×Q8C22⋊D10118yescomplemented
147C2×Q8C22×D10114yes—
148C2×D4—1011no—
149C2×D4—2022no—
150C2×D4—2021no—
151C22×C4—28no—
152C22×C4—24no—
153C22×C4—22no—
154C22×C4—24no—
155C22×C4—2021no—
156C22×C4C22⋊D10112yes—
157C22×C4C22⋊D10111yescharacteristic
158C22×C4C22×D10112yes—
159C22×C4C22×D10111yescharacteristic
160Q16—2028no—
161SD16—2028no—
162M16—2024no—
163C2×C8—1014no—
164C4⋊C4—1012no—
165C4⋊C4—2022no—
166C22⋊C4—2021no—
167C22⋊C4—2022no—
168C22⋊C4—2022no—
169C42—2021no—
170C23—22no—
171C23—24no—
172C23—2021no—
173C23C2×C22⋊D10111yescharacteristic
174C23C2×C22⋊D10112yes—
175Q8—216no—
176Q8—24no—
177Q8—28no—
178Q8—2022no—
179Q8C2×C22⋊D10118yes—
180D4—4042no—
181D4—4044no—
182D4—2022no—
183C2×C4—216no—
184C2×C4—24no—
185C2×C4—28no—
186C2×C4—22no—
187C2×C4—24no—
188C2×C4—24no—
189C2×C4—4042no—
190C2×C4—2022no—
191C2×C4—2021no—
192C2×C4C2×C22⋊D10112yes—
193C2×C4C2×C22⋊D10114yes—
194C2×C4C2×C22⋊D10111yescharacteristic
195C2×C4C23×D10111yescharacteristic
196C8—2024no—
197C22—28no—
198C22—22no—
199C22—22no—
200C22—22no—
201C22—4041no—
202C22—4042no—
203C22#1569222412yes—
204C22#1569228614yescomplemented
205C22C22×C22⋊D10111yescharacteristic
206C4—28no—
207C4—22no—
208C4—4041no—
209C4—4042no—
210C4#1569222412yes—
211C4C22×C22⋊D10112yes—
212C2—22no—
213C2—24no—
214C2—8081no—
215C2C2×#1569222411yescharacteristic
216C2C2×#1569228612yes—
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC23×Q8×C101C2
socleC22×C101C22×D4
prime radical#20265688C2
hypercenterC23×Q8D101
hypocenterC101#3004
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C23×C4 ◁ C23×Q8

Complements

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

NH ≅ G/Ntype
1. C2×#15692286C2semidirect
3. #20265333C2semidirect
4. #20265316C2semidirect
5. #20265186C2semidirect
6. #20265141C2semidirect
7. C23×Q8×C101C2semidirect
10. C2×#13108064C22semidirect
11. C2×#13108061C22semidirect
19. #15692179C22semidirect
51. C2×Q8×C101D4semidirect
93. C22×C101#137semidirect
109. C101#3004semidirect
110. C23×Q8D101semidirect
146. C2×Q8C22⋊D101semidirect
204. C22#15692286semidirect

Maximal subgroups

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

subgroupindex
C2×#156922862
#202656882
#202653332
#202653162
#202651862
#202651412
C23×Q8×C1012
#3004101