Finite Group Databasebeta

← #27170059

Subgroups — #27170059

Order 12928.

272 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 272 conjugacy classes of subgroups — too many to draw. Only its 49 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#20265774C211yesmaximal, characteristic, complemented
2#20265328C211yesmaximal, characteristic, complemented
3#20265311C211yesmaximal, characteristic
4#20264913C211yesmaximal, characteristic, complemented
5#20264910C211yesmaximal, characteristic, complemented
6#20264903C211yesmaximal, characteristic, complemented
7#394×C101C211yesmaximal, characteristic, complemented
8C2×#13108062—21no—
9C2×#13108059—21no—
10#15692250—21no—
11#15692245—21no—
12#15692240—21no—
13#15692238—21no—
14#15692234C2211yescharacteristic
15#15692233—21no—
16#15692229C2211yescharacteristic
17C4×C22⋊D101—21no—
18#15692176—21no—
19#15692176—21no—
20C22⋊#11717579C2211yescharacteristic
21C22⋊#11717579C2211yescharacteristic
22C22×D4×C101C2211yescharacteristic
23C2×C4⋊C4×C101C2211yescharacteristic
24C2×C22⋊C4×C101—21no—
25C2×C22⋊C4×C101—21no—
26#95×C101C2211yescharacteristic
27C2×Dic202D411yescharacteristic
28C2×D404D411yescharacteristic
29C2×C22⋊D101—21no—
30C2×C4×D101—21no—
31#13108062—42no—
32#13108059—42no—
33C2×#11717579—21no—
34C2×#11717579—21no—
35C4⋊Dic101D411yescharacteristic
36C4⋊Dic101D411yescharacteristic
37#13108053—21no—
38#13108053—21no—
39#13108053—21no—
40#13108053—21no—
41C22⋊Dic101—21no—
42#13108051—21no—
43#13108051—21no—
44#13108051—21no—
45#13108051—21no—
46C4×Dic101—21no—
47C24×C101—21no—
48C2×D4×C101—21no—
49C2×D4×C101—21no—
50C2×D4×C101—42no—
51C22×C4×C101—21no—
52C22×C4×C101D411yescharacteristic
53C22×C4×C101D411yescharacteristic
54C22×C4×C101C2311yescharacteristic
55C4⋊C4×C101—21no—
56C4⋊C4×C101—21no—
57C22⋊C4×C101—42no—
58C22⋊C4×C101—42no—
59C23×C101—22no—
60C23×C101—41no—
61C23×C101—41no—
62C23×C101—41no—
63C23×C101—41no—
64C23×C101—41no—
65C23×C101—41no—
66C23×C101C2×D411yescharacteristic
67C22×D101—21no—
68D4×C101—22no—
69D4×C101—22no—
70D4×C101—41no—
71D4×C101—41no—
72C2×C4×C101—21no—
73C2×C4×C101—21no—
74C2×C4×C101—21no—
75C2×C4×C101—21no—
76C2×C4×C101—41no—
77C2×C4×C101—41no—
78C2×C4×C101—41no—
79C2×C4×C101—41no—
80C2×C4×C101—41no—
81C2×C4×C101C2×D411yescharacteristic
82C2×C4×C101C2×D411yescharacteristic
83C22⋊D101—41no—
84C22⋊D101—41no—
85C2×Dic101—21no—
86C2×Dic101—21no—
87C2×Dic101—21no—
88D404—41no—
89C4×D101—81no—
90Dic202—41no—
91#11717579—41no—
92#11717579—41no—
93C22×C101—21no—
94C22×C101—21no—
95C22×C101—42no—
96C22×C101—42no—
97C22×C101—42no—
98C22×C101—81no—
99C22×C101—81no—
100C22×C101—81no—
101C22×C101—81no—
102C22×C101#12011yescharacteristic
103C22×C101#13511yescharacteristic
104C22×C101AGL(1,ℤ/8)11yescharacteristic
105C2×D101—41no—
106C2×D101—41no—
107C2×D101—41no—
108C4×C101—21no—
109C4×C101—21no—
110C4×C101—41no—
111C4×C101—41no—
112C4×C101—81no—
113Dic101—41no—
114Dic101—41no—
115Dic101—41no—
116C2×C101—21no—
117C2×C101—21no—
118C2×C101—82no—
119C2×C101#44911yescharacteristic
120C2×C101#45411yescharacteristic
121C2×C101B(4,𝔽2)11yescharacteristic
122D101—81no—
123#1597—1011nomaximal, Sylow
124C101#159711yescharacteristic, complemented, Sylow
125#539—1011no—
126#450—1011no—
127#447—1011no—
128#394D10111yescharacteristic, complemented
129#328—1011no—
130#327—1011no—
131#324—1011no—
132C22×D4C2×D10111yescharacteristic
133C2×SD16—2021no—
134C2×D8—2021no—
135#126—2021no—
136#125—2021no—
137#123—2021no—
138#123—2021no—
139C22⋊Q8—1011no—
140C22⋊Q8—2021no—
141#121—1011no—
142C4×D4—2021no—
143C2×C4⋊C4C2×D10111yescharacteristic
144C2×C22⋊C4—21no—
145C2×C22⋊C4—21no—
146#102—2021no—
147#102—2021no—
148C22⋊C8—1011no—
149C22⋊C8—1011no—
150#95C2×D10111yescharacteristic
151C24—21no—
152C2×Q8—1011no—
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—42no—
156C2×D4—1011no—
157C2×D4—2021no—
158C22×C4—21no—
159C22×C4—2021no—
160C22×C4C22⋊D10111yescharacteristic
161C22×C4C22⋊D10111yescharacteristic
162C22×C4C22×D10111yescharacteristic
163SD16—4042no—
164D8—4042no—
165C2×C8—2021no—
166C2×C8—2021no—
167C4⋊C4—21no—
168C4⋊C4—21no—
169C4⋊C4—1011no—
170C4⋊C4—1011no—
171C4⋊C4—2021no—
172C4⋊C4—2021no—
173C4⋊C4—2021no—
174C4⋊C4—2021no—
175C22⋊C4—42no—
176C22⋊C4—42no—
177C22⋊C4—2021no—
178C22⋊C4—2021no—
179C22⋊C4—2021no—
180C22⋊C4—2021no—
181C22⋊C4—2021no—
182C42—2021no—
183C23—22no—
184C23—41no—
185C23—41no—
186C23—41no—
187C23—41no—
188C23—41no—
189C23—41no—
190C23—2021no—
191C23C2×C22⋊D10111yescharacteristic
192Q8—4041no—
193D4—22no—
194D4—22no—
195D4—41no—
196D4—41no—
197D4—4041no—
198D4—4041no—
199D4—4041no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—21no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—41no—
208C2×C4—41no—
209C2×C4—8081no—
210C2×C4—2021no—
211C2×C4—2021no—
212C2×C4—2021no—
213C2×C4D4×D10111yescharacteristic
214C2×C4D4×D10111yescharacteristic
215C8—4041no—
216C8—4041no—
217C22—21no—
218C22—21no—
219C22—42no—
220C22—42no—
221C22—42no—
222C22—81no—
223C22—81no—
224C22—81no—
225C22—81no—
226C22—4041no—
227C22—4041no—
228C22—4041no—
229C22#1569222311yescharacteristic
230C22#1569227511yescharacteristic
231C22#1569228111yescharacteristic
232C4—21no—
233C4—21no—
234C4—41no—
235C4—41no—
236C4—81no—
237C4—4041no—
238C4—4041no—
239C4—4041no—
240C2—21no—
241C2—21no—
242C2—82no—
243C2—8081no—
244C2#2026532011yescharacteristic
245C2#2026534911yescharacteristic
246C2#2026536311yescharacteristic
subgroupG/H
centerC22#15692223
commutator, cosocleC22×C4×C101C23
FrattiniC22×C4C22×D101
Fitting#394×C101C2
socleC22×C101#120
prime radical#20265311C2
hypercenter#394D101
hypocenterC101#1597
LCSG ▷ C22×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #394

Complements

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

NH ≅ G/Ntype
1. #20265774C2semidirect
2. #20265328C2semidirect
4. #20264913C2semidirect
5. #20264910C2semidirect
6. #20264903C2semidirect
7. #394×C101C2semidirect
124. C101#1597semidirect
128. #394D101semidirect

Maximal subgroups

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

subgroupindex
#202657742
#202653282
#202653112
#202649132
#202649102
#202649032
#394×C1012
#1597101