Finite Group Databasebeta

← #27187384

Subgroups — #27187384

Order 12928.

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

normal subgroups only

This group has 264 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#20266080C212yesmaximal, complemented
4#20266077C212yesmaximal, complemented
5#20265683C211yesmaximal, characteristic
6C2×#13108083—21no—
7#15692334—21no—
8#15692316—21no—
9#15692315C2212yes—
10#15692313—21no—
11#15692313C2212yes—
12C2×#13108073C2211yescharacteristic
13C23×Dic101C411yescharacteristic, complemented
14#15692239C2211yescharacteristic
15#15692236C411yescharacteristic, complemented
16#15692232—21no—
17#15692227—21no—
18#15692224C2211yescharacteristic
19D4×Dic101—21no—
20#15692215—21no—
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—
29#13108073—41no—
30#13108073—41no—
31C2×Dic202D411yescharacteristic
32C2×C22⋊D101—21no—
33C2×C22⋊D101D411yescharacteristic
34C2×C22⋊D101C2311yescharacteristic
35C22×Dic101—21no—
36C22×Dic101—21no—
37C22×Dic101—41no—
38C22×Dic101—41no—
39C22×Dic101C2×C411yescharacteristic
40C22×Dic101C2×C411yescharacteristic
41C22×Dic101D411yescharacteristic
42C2×C4×D101D411yescharacteristic
43C4⋊Dic101—21no—
44#13108053—21no—
45#13108053—21no—
46C22⋊Dic101—21no—
47C22⋊Dic101—21no—
48C22⋊Dic101C2×C411yescharacteristic
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—42no—
58C23×C101—41no—
59C23×C101C22⋊C411yescharacteristic
60C23×C101C22×C411yescharacteristic
61C22×D101C2×D411yescharacteristic
62C2×#87938—42no—
63D4×C101—41no—
64D4×C101—42no—
65C2×C4×C101—21no—
66C2×C4×C101C22⋊C411yescharacteristic
67C22⋊D101—41no—
68C22⋊D101—41no—
69C22⋊D101—42no—
70C2×Dic101—21no—
71C2×Dic101—21no—
72C2×Dic101—21no—
73C2×Dic101—21no—
74C2×Dic101—21no—
75C2×Dic101—21no—
76C2×Dic101—42no—
77C2×Dic101—41no—
78C2×Dic101—41no—
79C2×Dic101—41no—
80C2×Dic101C22⋊C411yescharacteristic
81C2×Dic101C22⋊C411yescharacteristic
82C2×Dic101C2×D411yescharacteristic
83C4×D101—21no—
84C4×D101—21no—
85Dic202—41no—
86#11717581—42no—
87C22×C101—22no—
88C22×C101—21no—
89C22×C101—21no—
90C22×C101—41no—
91C22×C101—41no—
92C22×C101—41no—
93C22×C101—42no—
94C22×C101—42no—
95C22×C101—41no—
96C22×C101C2×C22⋊C411yescharacteristic
97C2×D101—21no—
98C2×D101—41no—
99#87938—82no—
100C4×C101—21no—
101C4×C101—41no—
102Dic101—21no—
103Dic101—21no—
104Dic101—41no—
105Dic101—41no—
106Dic101#9911yescharacteristic
107Dic101#9911yescharacteristic
108C2×C101—21no—
109C2×C101—41no—
110C2×C101—42no—
111C2×C101—41no—
112C2×C101C2×#9911yescharacteristic
113D101—41no—
114#2120—1011nomaximal, Sylow
115C101#212011yescharacteristic, complemented, Sylow
116#525—1011no—
117#413—1011no—
118#410—1011no—
119#352—1012no—
120AΣL(1,𝔽16)—1012no—
121C2×#41—1011no—
122C23×C4—1011no—
123C2×M16—2021no—
124#123—1011no—
125#123—1011no—
126C22⋊Q8—2021no—
127#121—2021no—
128#120—1011no—
129#120#8793811yescharacteristic, complemented
130C4×D4—2021no—
131C4×D4—2021no—
132#117—1011no—
133#117—2021no—
134#101—2021no—
135#100—1012no—
136#99—1012no—
137#99—2021no—
138C24C2×#8793811yescharacteristic
139#41—4041no—
140#41—4041no—
141C2×Q8—1011no—
142C2×D4—21no—
143C2×D4—1011no—
144C2×D4—1011no—
145C2×D4—2021no—
146C2×D4C2×#8793811yescharacteristic
147C22×C4—1011no—
148C22×C4—1011no—
149C22×C4—1011no—
150C22×C4—1011no—
151C22×C4—4041no—
152C22×C4—4041no—
153C22×C4—2021no—
154C22×C4—2021no—
155M16—4041no—
156M16—2022no—
157C2×C8—4041no—
158C4⋊C4—4041no—
159C4⋊C4—2021no—
160C4⋊C4—2021no—
161C4⋊C4—2021no—
162C22⋊C4—21no—
163C22⋊C4—1011no—
164C22⋊C4—2021no—
165C22⋊C4—2021no—
166C22⋊C4—2022no—
167C22⋊C4—2021no—
168C22⋊C4C2×#8793811yescharacteristic
169C42—4041no—
170C42—2021no—
171C23—21no—
172C23—41no—
173C23—42no—
174C23—1011no—
175C23C22⋊#8793811yescharacteristic
176C23C22×#8793811yescharacteristic
177Q8—4041no—
178D4—41no—
179D4—42no—
180D4—4042no—
181D4—4041no—
182D4—4041no—
183C2×C4—21no—
184C2×C4—1011no—
185C2×C4—1011no—
186C2×C4—1011no—
187C2×C4—4042no—
188C2×C4—4041no—
189C2×C4—4041no—
190C2×C4—4042no—
191C2×C4—4041no—
192C2×C4—2021no—
193C2×C4—2021no—
194C2×C4—2021no—
195C2×C4—2021no—
196C2×C4—2021no—
197C2×C4—2021no—
198C2×C4—2021no—
199C2×C4—2021no—
200C2×C4C22⋊#8793811yescharacteristic
201C8—4042no—
202C22—22no—
203C22—21no—
204C22—21no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—42no—
209C22—42no—
210C22—41no—
211C22—4041no—
212C22—2021no—
213C22C2×C22⋊#8793811yescharacteristic
214C4—21no—
215C4—41no—
216C4—8082no—
217C4—1011no—
218C4—1011no—
219C4—4041no—
220C4—4041no—
221C4—2021no—
222C4—2021no—
223C2—21no—
224C2—42no—
225C2—41no—
226C2—41no—
227C2—4041no—
228C2D101⋊#9911yescharacteristic
subgroupG/H
centerC2D101⋊#99
commutatorC23×C101C22×C4
FrattiniC23C22×#87938
Fitting#120×C101C4
socleC2×C101C2×#99
prime radical#20265683C2
cosocleC2×C22⋊D101C23
hypercenter#120#87938
hypocenterC101#2120
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. #20266080C2semidirect
4. #20266077C2semidirect
13. C23×Dic101C4semidirect
15. #15692236C4semidirect
21. #15692211C4semidirect
22. #120×C101C4semidirect
115. C101#2120semidirect
129. #120#87938semidirect

Maximal subgroups

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

subgroupindex
#202661642
#202661572
#202660802
#202660772
#202656832
#2120101