Finite Group Databasebeta

← #27172776

Subgroups — #27172776

Order 12928.

280 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 65 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#20265736C211yesmaximal, characteristic
2C2×#15692172C211yesmaximal, characteristic, complemented
3#20264966C212yesmaximal, complemented
4#20264963C212yesmaximal, complemented
5C2×#99×C101C211yesmaximal, characteristic, complemented
6C22×C22⋊D101C411yescharacteristic, complemented
7C2×#13108058—21no—
8#15692238—21no—
9#15692236C411yescharacteristic, complemented
10#15692236C2211yescharacteristic
11#15692227—21no—
12#15692227—21no—
13#15692224—21no—
14#15692223C411yescharacteristic, complemented
15#15692223C2211yescharacteristic
16C2×C22⋊Dic101C411yescharacteristic, complemented
17#15692172—21no—
18#15692172C2212yes—
19C22×D4×C101C2211yescharacteristic
20C2×C22⋊C4×C101—21no—
21#99×C101—21no—
22#99×C101C2212yes—
23C23×D101C2×C411yescharacteristic, complemented
24C2×C22⋊D101—21no—
25C2×C22⋊D101—21no—
26C2×C22⋊D101—21no—
27C2×C22⋊D101—21no—
28C2×C22⋊D101—41no—
29C2×C22⋊D101—41no—
30C22×Dic101—21no—
31C22×Dic101C2×C411yescharacteristic, complemented
32#13108058—22no—
33#13108058—41no—
34C2×#11717579—41no—
35#13108053—21no—
36#13108053—21no—
37C22⋊Dic101—21no—
38C22⋊Dic101—21no—
39C22⋊Dic101—21no—
40C22⋊Dic101C2×C411yescharacteristic, complemented
41C22⋊Dic101C2×C411yescharacteristic, complemented
42#13108051—21no—
43#13108051—21no—
44C24×C101C2×C411yescharacteristic
45C24×C101D411yescharacteristic
46C2×D4×C101—41no—
47C2×D4×C101—41no—
48C2×D4×C101C2×C411yescharacteristic
49C2×D4×C101D411yescharacteristic
50C2×D4×C101D411yescharacteristic
51C2×D4×C101C2311yescharacteristic
52C22×C4×C101—41no—
53C22×C4×C101D411yescharacteristic
54C22⋊C4×C101—22no—
55C22⋊C4×C101—41no—
56C23×C101—21no—
57C23×C101—21no—
58C23×C101—21no—
59C23×C101—21no—
60C23×C101—81no—
61C23×C101—81no—
62C23×C101C22⋊C411yescharacteristic
63C23×C101C22⋊C411yescharacteristic
64C23×C101C22⋊C411yescharacteristic
65C23×C101C22×C411yescharacteristic
66C23×C101C2×D411yescharacteristic
67C22×D101—21no—
68C22×D101—41no—
69C22×D101—81no—
70D4×C101—41no—
71D4×C101—41no—
72D4×C101—41no—
73D4×C101—41no—
74C2×C4×C101—21no—
75C2×C4×C101—21no—
76C2×C4×C101—42no—
77C2×C4×C101—82no—
78C2×C4×C101C22⋊C411yescharacteristic
79C2×C4×C101C2×D411yescharacteristic
80C22⋊D101—42no—
81C22⋊D101—41no—
82C22⋊D101—41no—
83C22⋊D101—81no—
84C22⋊D101—81no—
85C2×Dic101—21no—
86C2×Dic101—21no—
87C2×Dic101—21no—
88C2×Dic101—41no—
89C2×Dic101—81no—
90#11717579—42no—
91C22×C101—21no—
92C22×C101—21no—
93C22×C101—21no—
94C22×C101—21no—
95C22×C101—41no—
96C22×C101—41no—
97C22×C101—41no—
98C22×C101—41no—
99C22×C101—41no—
100C22×C101—42no—
101C22×C101—81no—
102C22×C101—81no—
103C22×C101—81no—
104C22×C101#9911yescharacteristic
105C22×C101#9911yescharacteristic
106C22×C101C2×C22⋊C411yescharacteristic
107C2×D101—41no—
108C2×D101—41no—
109C2×D101—41no—
110C2×D101—82no—
111C4×C101—21no—
112C4×C101—21no—
113C4×C101—82no—
114Dic101—41no—
115Dic101—41no—
116Dic101—41no—
117C2×C101—21no—
118C2×C101—21no—
119C2×C101—21no—
120C2×C101—41no—
121C2×C101—41no—
122C2×C101—41no—
123C2×C101—41no—
124C2×C101C2×#9911yescharacteristic
125D101—81no—
126#2121—1011nomaximal, Sylow
127C101#212111yescharacteristic, complemented, Sylow
128#534—1011no—
129C2×#100—1011no—
130C2×#99D10111yescharacteristic, complemented
131#352—1012no—
132AΣL(1,𝔽16)—1012no—
133C22×D4—1011no—
134C22×D4C2×D10111yescharacteristic
135C2×M16—2021no—
136#123—1011no—
137#123—1011no—
138#123—2021no—
139#121—2021no—
140#121—2021no—
141#120—1011no—
142#120—1011no—
143#120—2021no—
144C2×C22⋊C4—21no—
145C2×C22⋊C4—1011no—
146#100—1012no—
147#100—2021no—
148#99—21no—
149#99C2×D10112yes—
150C24—1011no—
151C24C4×D10111yescharacteristic
152C24C22⋊D10111yescharacteristic
153C2×D4—41no—
154C2×D4—41no—
155C2×D4—4041no—
156C2×D4—4041no—
157C2×D4—2021no—
158C2×D4—2021no—
159C2×D4—2021no—
160C2×D4—2021no—
161C2×D4C4×D10111yescharacteristic
162C2×D4D40411yescharacteristic
163C2×D4C22⋊D10111yescharacteristic
164C2×D4C22×D10111yescharacteristic
165C22×C4—41no—
166C22×C4—1011no—
167C22×C4—2021no—
168C22×C4D40411yescharacteristic
169M16—4041no—
170M16—2022no—
171C2×C8—4041no—
172C4⋊C4—2021no—
173C4⋊C4—2021no—
174C22⋊C4—22no—
175C22⋊C4—41no—
176C22⋊C4—1011no—
177C22⋊C4—1011no—
178C22⋊C4—2021no—
179C22⋊C4—2021no—
180C22⋊C4—2021no—
181C22⋊C4—2021no—
182C22⋊C4—2021no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—21no—
187C23—81no—
188C23—81no—
189C23—8081no—
190C23—4041no—
191C23—2021no—
192C23#1310805111yescharacteristic
193C23#1310805111yescharacteristic
194C23#1310805111yescharacteristic
195C23C2×C4×D10111yescharacteristic
196C23C2×C22⋊D10111yescharacteristic
197D4—41no—
198D4—41no—
199D4—41no—
200D4—41no—
201D4—8081no—
202D4—8081no—
203D4—4042no—
204D4—4041no—
205D4—4041no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—42no—
209C2×C4—82no—
210C2×C4—8081no—
211C2×C4—4041no—
212C2×C4—2021no—
213C2×C4—2021no—
214C2×C4—2021no—
215C2×C4#1310805111yescharacteristic
216C2×C4C2×D40411yescharacteristic
217C8—4042no—
218C22—21no—
219C22—21no—
220C22—21no—
221C22—21no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—41no—
227C22—42no—
228C22—81no—
229C22—81no—
230C22—81no—
231C22—8082no—
232C22—4041no—
233C22—4041no—
234C22—4041no—
235C22#1569216911yescharacteristic
236C22#1569216911yescharacteristic
237C22C2×#1310805111yescharacteristic
238C4—21no—
239C4—21no—
240C4—82no—
241C4—4041no—
242C4—4041no—
243C4—4041no—
244C2—21no—
245C2—21no—
246C2—21no—
247C2—41no—
248C2—41no—
249C2—41no—
250C2—41no—
251C2—8081no—
252C2C2×#1569216911yescharacteristic
subgroupG/H
centerC2C2×#15692169
commutatorC23×C101C22×C4
FrattiniC2×D4C22×D101
FittingC2×#99×C101C2
socleC2×C101C2×#99
prime radical#20265736C2
cosocleC2×D4×C101C23
hypercenterC2×#99D101
hypocenterC101#2121
LCSG ▷ C23×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C23 ◁ C22×D4 ◁ C2×#99

Complements

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

NH ≅ G/Ntype
2. C2×#15692172C2semidirect
3. #20264966C2semidirect
4. #20264963C2semidirect
5. C2×#99×C101C2semidirect
6. C22×C22⋊D101C4semidirect
9. #15692236C4semidirect
14. #15692223C4semidirect
16. C2×C22⋊Dic101C4semidirect
23. C23×D101C2×C4semidirect
31. C22×Dic101C2×C4semidirect
40. C22⋊Dic101C2×C4semidirect
41. C22⋊Dic101C2×C4semidirect
127. C101#2121semidirect
130. C2×#99D101semidirect

Maximal subgroups

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

subgroupindex
#202657362
C2×#156921722
#202649662
#202649632
C2×#99×C1012
#2121101