Finite Group Databasebeta

← #27186557

Subgroups — #27186557

Order 12928.

284 conjugacy classes of subgroups; only the 58 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 58 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
1D101⋊(C22⋊C8)C211yesmaximal, characteristic, complemented
2#20266120C211yesmaximal, characteristic, complemented
3#20266030C212yesmaximal, complemented
4#20266028C212yesmaximal, complemented
5#20265687C211yesmaximal, characteristic
6C2×D101⋊C8—21no—
7C2×C22⋊#87938—21no—
8#15692312C2212yes—
9C22⋊#11717581—21no—
10(C2×#87938)⋊C4—21no—
11(C2×#87938)⋊C4C2212yes—
12C2×#13108073C411yescharacteristic, complemented
13C22×C4×D101C2211yescharacteristic
14#15692237—21no—
15#15692235C2211yescharacteristic
16#15692234C411yescharacteristic, complemented
17#15692226C2211yescharacteristic
18#15692223—21no—
19D4×Dic101—21no—
20#15692215—21no—
21#15692213C411yescharacteristic, complemented
22#121×C101C411yescharacteristic, complemented
23C22×#87938—22no—
24D101⋊C8—42no—
25C2×#11717581—22no—
26C22⋊#87938—42no—
27C23×D101D411yescharacteristic
28#13108073—42no—
29C2×Dic202C2×C411yescharacteristic
30C2×C22⋊D101—21no—
31C2×C22⋊D101—21no—
32C22×Dic101—21no—
33C22×Dic101D411yescharacteristic
34C2×C4×D101—42no—
35C2×C4×D101C2×C411yescharacteristic
36C2×C4×D101D411yescharacteristic
37C2×C4×D101D411yescharacteristic
38C2×C4×D101C2311yescharacteristic
39C4⋊Dic101—21no—
40C4⋊Dic101C2×C411yescharacteristic
41#13108053—21no—
42C22⋊Dic101—21no—
43C22⋊Dic101—21no—
44#13108051—21no—
45#13108051—21no—
46C4×Dic101—21no—
47C2×D4×C101—21no—
48C2×D4×C101C2×C411yescharacteristic
49C22×C4×C101C2×C411yescharacteristic
50C4⋊C4×C101C2×C411yescharacteristic
51C22⋊C4×C101—21no—
52C23×C101—21no—
53C23×C101C22⋊C411yescharacteristic
54C22×D101—22no—
55C22×D101—22no—
56C22×D101—41no—
57C22×D101C22⋊C411yescharacteristic
58C22×D101C2×D411yescharacteristic
59C2×#87938—42no—
60C2×#87938—42no—
61C2×#87938—42no—
62D4×C101—41no—
63D4×C101—42no—
64C2×C4×C101—21no—
65C2×C4×C101—41no—
66C2×C4×C101C22⋊C411yescharacteristic
67C2×C4×C101C22×C411yescharacteristic
68C22⋊D101—42no—
69C22⋊D101—42no—
70C2×Dic101—21no—
71C2×Dic101—21no—
72C2×Dic101—41no—
73C2×Dic101—42no—
74C2×Dic101C22⋊C411yescharacteristic
75C2×Dic101C2×D411yescharacteristic
76C4×D101—22no—
77C4×D101—22no—
78C4×D101—41no—
79C4×D101—41no—
80Dic202—41no—
81#11717581—42no—
82C22×C101—21no—
83C22×C101—21no—
84C22×C101—21no—
85C22×C101—41no—
86C22×C101—41no—
87C22×C101—41no—
88C22×C101C2×C22⋊C411yescharacteristic
89C2×D101—21no—
90C2×D101—21no—
91C2×D101—21no—
92C2×D101—21no—
93C2×D101—44no—
94C2×D101#9912yes—
95C2×D101#13512yes—
96#87938—82no—
97C4×C101—21no—
98C4×C101—21no—
99C4×C101—41no—
100Dic101—21no—
101Dic101—21no—
102Dic101—41no—
103Dic101—41no—
104C2×C101—41no—
105C2×C101—81no—
106C2×C101C2×#9911yescharacteristic
107C2×C101#41611yescharacteristic
108C2×C101#42111yescharacteristic
109D101—22no—
110D101—22no—
111#1516—1011nomaximal, Sylow
112C101#151611yescharacteristic, complemented, Sylow
113#525—1011no—
114C2×C22⋊C8—1011no—
115#381—1011no—
116#328—1012no—
117#327—1012no—
118C2×#41—1011no—
119C23×C4—1011no—
120C22×C8—2021no—
121#123—2021no—
122C22⋊Q8—1011no—
123C22⋊Q8—1011no—
124#121—1011no—
125#121#8793811yescharacteristic, complemented
126#120—2021no—
127C4×D4—2021no—
128C4×D4—2021no—
129#117—1011no—
130C2×C22⋊C4—2021no—
131C22⋊C8—1012no—
132C22⋊C8—2021no—
133#95—1012no—
134#95—2021no—
135C24—1011no—
136#41—4042no—
137C2×Q8—1011no—
138C2×D4—21no—
139C2×D4—2021no—
140C2×D4—2021no—
141C2×D4C2×#8793811yescharacteristic
142C22×C4—1011no—
143C22×C4—1011no—
144C22×C4—1011no—
145C22×C4—1011no—
146C22×C4—1011no—
147C22×C4—4042no—
148C22×C4—2021no—
149C22×C4—2022no—
150C22×C4C2×#8793811yescharacteristic
151C2×C8—4042no—
152C2×C8—2022no—
153C4⋊C4—1011no—
154C4⋊C4—2021no—
155C4⋊C4—2021no—
156C4⋊C4C2×#8793811yescharacteristic
157C22⋊C4—21no—
158C22⋊C4—4042no—
159C22⋊C4—2021no—
160C22⋊C4—2021no—
161C22⋊C4—2021no—
162C22⋊C4—2021no—
163C42—2021no—
164C23—21no—
165C23—1011no—
166C23—1011no—
167C23—4041no—
168C23—2022no—
169C23—2022no—
170C23C22⋊#8793811yescharacteristic
171Q8—4041no—
172D4—41no—
173D4—42no—
174D4—4042no—
175D4—4042no—
176C2×C4—21no—
177C2×C4—41no—
178C2×C4—1011no—
179C2×C4—1011no—
180C2×C4—4041no—
181C2×C4—4041no—
182C2×C4—4042no—
183C2×C4—4042no—
184C2×C4—4042no—
185C2×C4—4042no—
186C2×C4—4041no—
187C2×C4—2022no—
188C2×C4—2022no—
189C2×C4—2021no—
190C2×C4—2021no—
191C2×C4C22⋊#8793811yescharacteristic
192C2×C4C22×#8793811yescharacteristic
193C8—4042no—
194C22—21no—
195C22—21no—
196C22—21no—
197C22—41no—
198C22—41no—
199C22—41no—
200C22—1012no—
201C22—1012no—
202C22—4044no—
203C22—2021no—
204C22—2021no—
205C22—2021no—
206C22—2021no—
207C22C2×C22⋊#8793811yescharacteristic
208C4—21no—
209C4—21no—
210C4—41no—
211C4—8082no—
212C4—4041no—
213C4—4041no—
214C4—2021no—
215C4—2021no—
216C2—41no—
217C2—81no—
218C2—2022no—
219C2—2022no—
220C2D101⋊#9911yescharacteristic
221C2#2026616711yescharacteristic
222C2#2026617511yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#121×C101C4
socleC22×C101C2×C22⋊C4
prime radical#20265687C2
cosocleC2×C4×D101C23
hypercenter#121#87938
hypocenterC101#1516
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #121

Complements

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

NH ≅ G/Ntype
1. D101⋊(C22⋊C8)C2semidirect
2. #20266120C2semidirect
3. #20266030C2semidirect
4. #20266028C2semidirect
12. C2×#13108073C4semidirect
16. #15692234C4semidirect
21. #15692213C4semidirect
22. #121×C101C4semidirect
112. C101#1516semidirect
125. #121#87938semidirect

Maximal subgroups

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

subgroupindex
D101⋊(C22⋊C8)2
#202661202
#202660302
#202660282
#202656872
#1516101