Finite Group Databasebeta

← #27187392

Subgroups — #27187392

Order 12928.

266 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 52 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#20266164C211yesmaximal, characteristic, complemented
2#20266157C211yesmaximal, characteristic, complemented
3#20266078C212yesmaximal, complemented
4#20266077C212yesmaximal, complemented
5#20265742C211yesmaximal, characteristic
6C2×#13108083—21no—
7#15692334—21no—
8#15692316—21no—
9#15692315C2212yes—
10#15692313—21no—
11#15692313C2212yes—
12C2×#13108073C2211yescharacteristic
13C22×C22⋊D101C411yescharacteristic, complemented
14#15692251—21no—
15#15692241—21no—
16#15692228—21no—
17#15692227—21no—
18#15692225C2211yescharacteristic
19#15692224C2211yescharacteristic
20#15692223C411yescharacteristic, complemented
21#15692211C411yescharacteristic, complemented
22#120×C101C411yescharacteristic, complemented
23#13108083—22no—
24#13108083—41no—
25C2×#11717581—41no—
26C4⋊#87938—41no—
27C22⋊#87938—22no—
28C4×#87938—41no—
29C23×D101C2×C411yescharacteristic
30#13108073—41no—
31#13108073—41no—
32C2×Dic202D411yescharacteristic
33C2×D404—21no—
34C2×C22⋊D101—21no—
35C2×C22⋊D101—21no—
36C2×C22⋊D101—21no—
37C2×C22⋊D101—21no—
38C2×C22⋊D101—41no—
39C2×C22⋊D101—41no—
40C2×C22⋊D101D411yescharacteristic
41C2×C22⋊D101C2311yescharacteristic
42C22×Dic101C2×C411yescharacteristic
43C22×Dic101D411yescharacteristic
44C2×C4×D101D411yescharacteristic
45#13108053—21no—
46C22⋊Dic101—21no—
47C22⋊Dic101C2×C411yescharacteristic
48#13108051—21no—
49#13108051—21no—
50C4×Dic101—21no—
51C24×C101C2×C411yescharacteristic
52C2×D4×C101—21no—
53C2×D4×C101C2×C411yescharacteristic
54C22⋊C4×C101—21no—
55C22⋊C4×C101C2×C411yescharacteristic
56C23×C101—21no—
57C23×C101—41no—
58C23×C101—81no—
59C23×C101C22⋊C411yescharacteristic
60C23×C101C22×C411yescharacteristic
61C22×D101—21no—
62C22×D101—41no—
63C22×D101—81no—
64C22×D101C2×D411yescharacteristic
65C2×#87938—42no—
66D4×C101—41no—
67D4×C101—81no—
68C2×C4×C101—21no—
69C2×C4×C101C22⋊C411yescharacteristic
70C22⋊D101—41no—
71C22⋊D101—41no—
72C22⋊D101—42no—
73C22⋊D101—41no—
74C22⋊D101—41no—
75C22⋊D101—81no—
76C22⋊D101—81no—
77C2×Dic101—21no—
78C2×Dic101—21no—
79C2×Dic101—21no—
80C2×Dic101—21no—
81C2×Dic101—21no—
82C2×Dic101C22⋊C411yescharacteristic
83C2×Dic101C22⋊C411yescharacteristic
84C2×Dic101C2×D411yescharacteristic
85D404—81no—
86C4×D101—21no—
87C4×D101—21no—
88Dic202—41no—
89#11717581—42no—
90C22×C101—21no—
91C22×C101—21no—
92C22×C101—41no—
93C22×C101—41no—
94C22×C101—41no—
95C22×C101—41no—
96C22×C101—41no—
97C22×C101—82no—
98C22×C101C2×C22⋊C411yescharacteristic
99C2×D101—21no—
100C2×D101—41no—
101C2×D101—41no—
102C2×D101—41no—
103C2×D101—41no—
104C2×D101—82no—
105#87938—82no—
106C4×C101—21no—
107C4×C101—41no—
108Dic101—21no—
109Dic101—21no—
110Dic101—41no—
111Dic101#9911yescharacteristic
112Dic101#9911yescharacteristic
113C2×C101—21no—
114C2×C101—41no—
115C2×C101—41no—
116C2×C101—81no—
117C2×C101C2×#9911yescharacteristic
118D101—41no—
119D101—81no—
120#2122—1011nomaximal, Sylow
121C101#212211yescharacteristic, complemented, Sylow
122#535—1011no—
123#413—1011no—
124#410—1011no—
125AΣL(1,𝔽16)—1012no—
126AΣL(1,𝔽16)—1012no—
127C2×#41—1011no—
128C22×D4—1011no—
129C2×M16—2021no—
130#127—2021no—
131#124—2021no—
132#121—2021no—
133#121—2021no—
134#120—1011no—
135#120—1011no—
136#120—1011no—
137#120#8793811yescharacteristic, complemented
138#117—1011no—
139#117—2021no—
140#101—2021no—
141#100—1012no—
142#99—1012no—
143#99—2021no—
144C24—1011no—
145C24C2×#8793811yescharacteristic
146#41—4041no—
147#41—4041no—
148C2×Q8—1011no—
149C2×D4—21no—
150C2×D4—1011no—
151C2×D4—1011no—
152C2×D4—4041no—
153C2×D4—4041no—
154C2×D4—2021no—
155C2×D4—2021no—
156C2×D4—2021no—
157C2×D4—2021no—
158C2×D4—2021no—
159C2×D4C2×#8793811yescharacteristic
160C22×C4—1011no—
161C22×C4—1011no—
162C22×C4—1011no—
163M16—4041no—
164M16—2022no—
165C2×C8—4041no—
166C4⋊C4—4041no—
167C4⋊C4—2021no—
168C22⋊C4—21no—
169C22⋊C4—1011no—
170C22⋊C4—2021no—
171C22⋊C4—2021no—
172C22⋊C4—2022no—
173C22⋊C4—2021no—
174C22⋊C4C2×#8793811yescharacteristic
175C42—4041no—
176C42—2021no—
177C23—21no—
178C23—41no—
179C23—81no—
180C23—8081no—
181C23—1011no—
182C23—4041no—
183C23—2021no—
184C23C22⋊#8793811yescharacteristic
185C23C22×#8793811yescharacteristic
186Q8—4041no—
187D4—41no—
188D4—81no—
189D4—8081no—
190D4—8081no—
191D4—8081no—
192D4—4041no—
193D4—4041no—
194D4—4041no—
195D4—4041no—
196D4—4042no—
197C2×C4—21no—
198C2×C4—1011no—
199C2×C4—1011no—
200C2×C4—1011no—
201C2×C4—4042no—
202C2×C4—2021no—
203C2×C4—2021no—
204C2×C4—2021no—
205C2×C4—2021no—
206C2×C4—2021no—
207C2×C4—2021no—
208C2×C4—2021no—
209C2×C4C22⋊#8793811yescharacteristic
210C8—4042no—
211C22—21no—
212C22—21no—
213C22—41no—
214C22—41no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—82no—
219C22—8082no—
220C22—4041no—
221C22—4041no—
222C22—4041no—
223C22—4041no—
224C22—2021no—
225C22C2×C22⋊#8793811yescharacteristic
226C4—21no—
227C4—41no—
228C4—8082no—
229C4—1011no—
230C4—1011no—
231C4—4041no—
232C4—2021no—
233C4—2021no—
234C2—21no—
235C2—41no—
236C2—41no—
237C2—81no—
238C2—8081no—
239C2—4041no—
240C2D101⋊#9911yescharacteristic
subgroupG/H
centerC2D101⋊#99
commutatorC23×C101C22×C4
FrattiniC23C22×#87938
Fitting#120×C101C4
socleC2×C101C2×#99
prime radical#20265742C2
cosocleC2×C22⋊D101C23
hypercenter#120#87938
hypocenterC101#2122
LCSG ▷ C23×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #20266164C2semidirect
2. #20266157C2semidirect
3. #20266078C2semidirect
4. #20266077C2semidirect
13. C22×C22⋊D101C4semidirect
20. #15692223C4semidirect
21. #15692211C4semidirect
22. #120×C101C4semidirect
121. C101#2122semidirect
137. #120#87938semidirect

Maximal subgroups

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

subgroupindex
#202661642
#202661572
#202660782
#202660772
#202657422
#2122101