Finite Group Databasebeta

← #27187037

Subgroups — #27187037

Order 12928.

300 conjugacy classes of subgroups; only the 74 normal subgroups are shown.

normal subgroups only

This group has 300 conjugacy classes of subgroups — too many to draw. Only its 74 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
2#20266119C211yesmaximal, characteristic
3#20266050C212yesmaximal, complemented
4#20266050C212yesmaximal, complemented
5#117×D101C211yesmaximal, characteristic
6C2×D101⋊C8C2212yes—
7C2×C4⋊#87938C2212yes—
8C2×C22⋊#87938—21no—
9#15692312—21no—
10C22⋊#11717581—21no—
11(C2×#87938)⋊C4—21no—
12C22×C4×D101C2211yescharacteristic
13#15692214C411yescharacteristic, complemented
14#15692213C411yescharacteristic, complemented
15#15692213C411yescharacteristic, complemented
16#15692212—21no—
17#15692211—21no—
18C4⋊C4×D101C2211yescharacteristic
19C4⋊C4×D101C2211yescharacteristic
20C22⋊C4×D101—21no—
21C42×D101—21no—
22#117×C101C411yescharacteristic, complemented
23C22×#87938—22no—
24D101⋊C8—24no—
25C2×#11717581—22no—
26C4⋊#87938—24no—
27C22⋊#87938—42no—
28C23×D101D411yescharacteristic
29C22×Dic101D411yescharacteristic
30C2×C4×D101—24no—
31C2×C4×D101—21no—
32C2×C4×D101—21no—
33C2×C4×D101C2×C411yescharacteristic
34C2×C4×D101D411yescharacteristic
35C2×C4×D101D411yescharacteristic
36C2×C4×D101C2311yescharacteristic
37C4⋊Dic101C2×C411yescharacteristic
38C4⋊Dic101C2×C411yescharacteristic
39#13108053—21no—
40#13108053—21no—
41C22⋊Dic101—21no—
42#13108051—21no—
43#13108051—21no—
44C4×Dic101—21no—
45C4×Dic101—21no—
46C4×Dic101—21no—
47C22×C4×C101C2×C411yescharacteristic
48C4⋊C4×C101C2×C411yescharacteristic
49C4⋊C4×C101C2×C411yescharacteristic
50C22⋊C4×C101—21no—
51C42×C101—21no—
52C23×C101C22⋊C411yescharacteristic
53C22×D101—22no—
54C22×D101—22no—
55C22×D101—21no—
56C22×D101—21no—
57C22×D101C22⋊C411yescharacteristic
58C22×D101C2×D411yescharacteristic
59C2×#87938—42no—
60C2×#87938—42no—
61C2×#87938—42no—
62C2×C4×C101—22no—
63C2×C4×C101—21no—
64C2×C4×C101—21no—
65C2×C4×C101C22⋊C411yescharacteristic
66C2×C4×C101C22×C411yescharacteristic
67C2×Dic101—22no—
68C2×Dic101—21no—
69C2×Dic101—21no—
70C2×Dic101C22⋊C411yescharacteristic
71C2×Dic101C2×D411yescharacteristic
72C4×D101—22no—
73C4×D101—22no—
74C4×D101—22no—
75C4×D101—42no—
76C4×D101—41no—
77C4×D101—41no—
78C4×D101#4114yes—
79#11717581—42no—
80C22×C101—21no—
81C22×C101—21no—
82C22×C101—21no—
83C22×C101C2×C22⋊C411yescharacteristic
84C2×D101—21no—
85C2×D101—21no—
86C2×D101—21no—
87C2×D101—42no—
88C2×D101—42no—
89C2×D101#12312yes—
90C2×D101#13512yes—
91C2×D101AGL(1,ℤ/8)11yescharacteristic
92C2×D101#13711yescharacteristic
93#87938—82no—
94C4×C101—21no—
95C4×C101—41no—
96C4×C101—41no—
97C4×C101#11712yes—
98Dic101—21no—
99Dic101—41no—
100Dic101—41no—
101Dic101#12312yes—
102C2×C101—41no—
103C2×C101#38111yescharacteristic
104C2×C101#41611yescharacteristic
105C2×C101#41711yescharacteristic
106D101—41no—
107D101#48212yes—
108D101#48512yes—
109#1825—1011nomaximal, Sylow
110C101#182511yescharacteristic, complemented, Sylow
111C2×#117—1011no—
112C2×C22⋊C8—1011no—
113#380—1011no—
114#340—1012no—
115#340—1012no—
116C23×C4—1011no—
117C22×C8—1012no—
118#117—1011no—
119#117—1011no—
120#117—1011no—
121#117—2021no—
122#117—2021no—
123#117#8793811yescharacteristic, complemented
124C2×C4⋊C4—1011no—
125C2×C4⋊C4—1011no—
126C2×C4⋊C4—1012no—
127C2×C22⋊C4—2021no—
128C2×C22⋊C4—2021no—
129C2×C42—2021no—
130C22⋊C8—2021no—
131C22⋊C8—2021no—
132#95—2021no—
133C24—1011no—
134C22×C4—1011no—
135C22×C4—1011no—
136C22×C4—1011no—
137C22×C4—1011no—
138C22×C4—1011no—
139C22×C4—2024no—
140C22×C4—2021no—
141C22×C4—2021no—
142C22×C4—2022no—
143C22×C4C2×#8793811yescharacteristic
144C2×C8—2022no—
145C2×C8—2024no—
146C4⋊C4—1011no—
147C4⋊C4—1011no—
148C4⋊C4—2021no—
149C4⋊C4—2021no—
150C4⋊C4—2024no—
151C4⋊C4C2×#8793811yescharacteristic
152C4⋊C4C2×#8793811yescharacteristic
153C22⋊C4—21no—
154C22⋊C4—4042no—
155C22⋊C4—2021no—
156C22⋊C4—2021no—
157C22⋊C4—2021no—
158C42—21no—
159C42—2021no—
160C42—2021no—
161C42—2021no—
162C23—1011no—
163C23—1011no—
164C23—2021no—
165C23—2022no—
166C23—2022no—
167C23—2021no—
168C23C22⋊#8793811yescharacteristic
169C2×C4—22no—
170C2×C4—21no—
171C2×C4—21no—
172C2×C4—1011no—
173C2×C4—1014no—
174C2×C4—1011no—
175C2×C4—4042no—
176C2×C4—4041no—
177C2×C4—4041no—
178C2×C4—4042no—
179C2×C4—4042no—
180C2×C4—4042no—
181C2×C4—2022no—
182C2×C4—2022no—
183C2×C4—2022no—
184C2×C4—2021no—
185C2×C4—2021no—
186C2×C4—2022no—
187C2×C4C22⋊#8793811yescharacteristic
188C2×C4C22×#8793811yescharacteristic
189C8—4042no—
190C22—21no—
191C22—21no—
192C22—21no—
193C22—1012no—
194C22—1012no—
195C22—1011no—
196C22—1011no—
197C22—4042no—
198C22—4042no—
199C22—2021no—
200C22—2021no—
201C22—2021no—
202C22C2×C22⋊#8793811yescharacteristic
203C4—21no—
204C4—41no—
205C4—41no—
206C4—8082no—
207C4—1012no—
208C4—4041no—
209C4—4041no—
210C4—2021no—
211C4#1569233412yes—
212C2—41no—
213C2—1012no—
214C2—1012no—
215C2—4041no—
216C2#2026612011yescharacteristic
217C2#2026616711yescharacteristic
218C2#2026616811yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#117×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC23×D101D4
cosocleC2×C4×D101C23
hypercenter#117#87938
hypocenterC101#1825
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #117

Complements

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

NH ≅ G/Ntype
3. #20266050C2semidirect
4. #20266050C2semidirect
13. #15692214C4semidirect
14. #15692213C4semidirect
15. #15692213C4semidirect
22. #117×C101C4semidirect
110. C101#1825semidirect
123. #117#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
#202661192
#202660502
#117×D1012
#1825101