Finite Group Databasebeta

← #27170053

Subgroups — #27170053

Order 12928.

316 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 51 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#20265802C211yesmaximal, characteristic, complemented
2#20265328C211yesmaximal, characteristic, complemented
3#20265085C211yesmaximal, characteristic
4#20264915C211yesmaximal, characteristic, complemented
5#20264914C211yesmaximal, characteristic, complemented
6#20264900C211yesmaximal, characteristic, complemented
7#450×C101C211yesmaximal, characteristic, complemented
8C22×C22⋊D101—21no—
9C2×#13108062—21no—
10#15692235—21no—
11#15692234C2211yescharacteristic
12#15692233—21no—
13#15692233—21no—
14#15692232—21no—
15#15692231—21no—
16C2×C22⋊Dic101—21no—
17C2×#13108051C2211yescharacteristic
18C22⋊C4×D101—21no—
19#15692176—21no—
20C22⋊#11717579C2211yescharacteristic
21#15692162C2211yescharacteristic
22C22×D4×C101C2211yescharacteristic
23C2×SD16×C101—21no—
24C22⋊Q8×C101C2211yescharacteristic
25#102×C101—21no—
26C22⋊C8×C101C2211yescharacteristic
27C23×D101D411yescharacteristic
28C2×Dic202—21no—
29C2×Dic202D411yescharacteristic, complemented
30C2×C22⋊D101—42no—
31C2×C22⋊D101—42no—
32C2×C22⋊D101—42no—
33C22×Dic101—21no—
34C22×Dic101D411yescharacteristic
35C2×C4×D101—21no—
36C2×C4×D101—21no—
37#13108062—42no—
38C2×#11717579—21no—
39C4⋊Dic101—21no—
40C4⋊Dic101D411yescharacteristic, complemented
41#13108053—21no—
42#13108053—21no—
43#13108053—21no—
44C22⋊Dic101—21no—
45C22⋊Dic101—42no—
46#13108051—21no—
47#13108051—21no—
48#13108051—21no—
49#13108051—21no—
50C24×C101—21no—
51C2×Q8×C101D411yescharacteristic, complemented
52C2×D4×C101—21no—
53C2×D4×C101—21no—
54C2×D4×C101—42no—
55C22×C4×C101C2311yescharacteristic
56SD16×C101—42no—
57C2×C8×C101—21no—
58C4⋊C4×C101—21no—
59C4⋊C4×C101D411yescharacteristic, complemented
60C22⋊C4×C101—21no—
61C23×C101—22no—
62C23×C101—41no—
63C23×C101—41no—
64C23×C101—41no—
65C23×C101—41no—
66C23×C101—41no—
67C23×C101—41no—
68C23×C101C2×D411yescharacteristic
69C22×D101—21no—
70C22×D101—21no—
71C22×D101—41no—
72C22×D101—41no—
73C22×D101SD1612yescomplemented
74Q8×C101—41no—
75D4×C101—22no—
76D4×C101—22no—
77D4×C101—41no—
78D4×C101—41no—
79C2×C4×C101—21no—
80C2×C4×C101—21no—
81C2×C4×C101—41no—
82C2×C4×C101C2×D411yescharacteristic
83C2×C4×C101C2×D411yescharacteristic
84C22⋊D101—48no—
85C2×Dic101—22no—
86C2×Dic101—21no—
87C2×Dic101—21no—
88C2×Dic101—21no—
89C2×Dic101—41no—
90C2×Dic101—41no—
91C2×Dic101—41no—
92C4×D101—81no—
93C4×D101—81no—
94Dic202—41no—
95Dic202—81no—
96C8×C101—41no—
97#11717579—41no—
98C22×C101—21no—
99C22×C101—21no—
100C22×C101—42no—
101C22×C101—42no—
102C22×C101—42no—
103C22×C101—81no—
104C22×C101—81no—
105C22×C101—81no—
106C22×C101—81no—
107C22×C101#12011yescharacteristic
108C22×C101C2×SD1611yescharacteristic
109C22×C101#13711yescharacteristic
110C2×D101—22no—
111C2×D101—41no—
112C2×D101—41no—
113C2×D101—42no—
114C2×D101—81no—
115C4×C101—21no—
116C4×C101—21no—
117C4×C101—41no—
118C4×C101—41no—
119Dic101—42no—
120Dic101—41no—
121Dic101—41no—
122Dic101—41no—
123C2×C101—21no—
124C2×C101—21no—
125C2×C101—82no—
126C2×C101#44811yescharacteristic
127C2×C101#45311yescharacteristic
128C2×C101B(4,𝔽2)11yescharacteristic
129D101—42no—
130#1596—1011nomaximal, Sylow
131C101#159611yescharacteristic, complemented, Sylow
132#543—1011no—
133#450—1011no—
134#450D10111yescharacteristic, complemented
135#392—1011no—
136#328—1011no—
137#328—1011no—
138#323—1011no—
139C22×D4—2021no—
140C22×D4C2×D10111yescharacteristic
141C2×SD16—21no—
142C2×SD16—2021no—
143C22⋊Q8—1011no—
144C22⋊Q8—2021no—
145C22⋊Q8—2021no—
146C22⋊Q8—2021no—
147C22⋊Q8—2021no—
148C22⋊Q8—2021no—
149C22⋊Q8C2×D10111yescharacteristic
150C2×C22⋊C4—1011no—
151C2×C22⋊C4—2021no—
152C2×C22⋊C4—2021no—
153#102—21no—
154#102—2021no—
155C22⋊C8—1011no—
156C22⋊C8C2×D10111yescharacteristic
157#95—1011no—
158C24—21no—
159C24—1011no—
160C2×Q8—1011no—
161C2×Q8—2021no—
162C2×Q8C22⋊D10111yescharacteristic, complemented
163C2×D4—21no—
164C2×D4—21no—
165C2×D4—42no—
166C2×D4—4042no—
167C2×D4—4042no—
168C2×D4—4042no—
169C22×C4—1011no—
170C22×C4—2021no—
171C22×C4—2021no—
172C22×C4—2021no—
173C22×C4C22×D10111yescharacteristic
174SD16—42no—
175SD16—4042no—
176C2×C8—21no—
177C2×C8—2021no—
178C4⋊C4—21no—
179C4⋊C4—1011no—
180C4⋊C4—2021no—
181C4⋊C4—2021no—
182C4⋊C4—2021no—
183C4⋊C4—2021no—
184C4⋊C4C22⋊D10111yescharacteristic, complemented
185C22⋊C4—21no—
186C22⋊C4—4042no—
187C22⋊C4—2021no—
188C22⋊C4—2021no—
189C22⋊C4—2021no—
190C22⋊C4—2021no—
191C22⋊C4—2021no—
192C23—22no—
193C23—41no—
194C23—41no—
195C23—41no—
196C23—41no—
197C23—41no—
198C23—41no—
199C23—1012no—
200C23—4041no—
201C23—4041no—
202C23—2021no—
203C23—2021no—
204C23D4×D10111yescharacteristic
205Q8—41no—
206Q8—8081no—
207Q8—4041no—
208D4—22no—
209D4—22no—
210D4—41no—
211D4—41no—
212D4—4048no—
213C2×C4—21no—
214C2×C4—21no—
215C2×C4—41no—
216C2×C4—8081no—
217C2×C4—8081no—
218C2×C4—4041no—
219C2×C4—4041no—
220C2×C4—4041no—
221C2×C4—2022no—
222C2×C4—2021no—
223C2×C4—2021no—
224C2×C4—2021no—
225C2×C4D4×D10111yescharacteristic
226C2×C4C2×C22⋊D10111yescharacteristic
227C8—41no—
228C8—4041no—
229C22—21no—
230C22—21no—
231C22—42no—
232C22—42no—
233C22—42no—
234C22—81no—
235C22—81no—
236C22—81no—
237C22—81no—
238C22—8081no—
239C22—4042no—
240C22—4041no—
241C22—4041no—
242C22—2022no—
243C22#1569222311yescharacteristic
244C22SD16×D10111yescharacteristic
245C22#1569228411yescharacteristic
246C4—21no—
247C4—21no—
248C4—41no—
249C4—41no—
250C4—4042no—
251C4—4041no—
252C4—4041no—
253C4—4041no—
254C2—21no—
255C2—21no—
256C2—82no—
257C2—4042no—
258C2#2026531411yescharacteristic
259C2#2026534311yescharacteristic
260C2#2026536311yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#450×C101C2
socleC22×C101#120
prime radical#20265085C2
hypercenter#450D101
hypocenterC101#1596
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #450

Complements

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

NH ≅ G/Ntype
1. #20265802C2semidirect
2. #20265328C2semidirect
4. #20264915C2semidirect
5. #20264914C2semidirect
6. #20264900C2semidirect
7. #450×C101C2semidirect
29. C2×Dic202D4semidirect
40. C4⋊Dic101D4semidirect
51. C2×Q8×C101D4semidirect
59. C4⋊C4×C101D4semidirect
73. C22×D101SD16semidirect
131. C101#1596semidirect
134. #450D101semidirect
162. C2×Q8C22⋊D101semidirect
184. C4⋊C4C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
#202658022
#202653282
#202650852
#202649152
#202649142
#202649002
#450×C1012
#1596101