Finite Group Databasebeta

← #27186592

Subgroups — #27186592

Order 12928.

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

normal subgroups only

This group has 264 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
1#20266200C211yesmaximal, characteristic, complemented
2C4×#13108084C211yesmaximal, characteristic, complemented
3#20266033C212yesmaximal, complemented
4#20266032C212yesmaximal, complemented
5#124×D101C211yesmaximal, characteristic
6C2×#13108084—21no—
7#15692322—21no—
8#15692312—21no—
9#15692310C2212yescomplemented
10#15692310C2212yescomplemented
11C4×#11717581—21no—
12C2×Q8×D101C411yescharacteristic
13C2×D4×D101C411yescharacteristic
14#15692244C2211yescharacteristic
15#15692243C411yescharacteristic
16#15692242C2211yescharacteristic
17#15692241—21no—
18#15692241—21no—
19C22⋊C4×D101—21no—
20C22⋊C4×D101—21no—
21C42×D101C2211yescharacteristic
22#124×C101C411yescharacteristic
23#13108084—24no—
24C2×#11717581—22no—
25C2×#11717581—22no—
26C23×D101—21no—
27C2×Dic202—21no—
28C2×Dic202C2×C411yescharacteristic
29Q8×D101—42no—
30C2×D404C2×C411yescharacteristic
31C2×C22⋊D101—21no—
32D4×D101—42no—
33C2×C4×D101—21no—
34C2×C4×D101C2×C411yescharacteristic
35C2×C4×D101D411yescharacteristic
36C2×C4×D101D411yescharacteristic
37C22⋊Dic101—21no—
38C22⋊Dic101—21no—
39#13108051—21no—
40#13108051—21no—
41#13108051—21no—
42#13108051—21no—
43C4×Dic101D411yescharacteristic
44C4×Dic101D411yescharacteristic
45C4×Dic101C2311yescharacteristic
46C2×Q8×C101C2×C411yescharacteristic
47C2×D4×C101C2×C411yescharacteristic
48C22⋊C4×C101—21no—
49C22⋊C4×C101—21no—
50C42×C101C2×C411yescharacteristic
51C23×C101—21no—
52C22×D101—21no—
53C22×D101—42no—
54C22×D101—42no—
55C22×D101—42no—
56C22×D101C22⋊C411yescharacteristic
57Q8×C101—41no—
58D4×C101—41no—
59C2×C4×C101—21no—
60C2×C4×C101C22⋊C411yescharacteristic
61C2×C4×C101C22⋊C411yescharacteristic
62C2×C4×C101C22×C411yescharacteristic
63C22⋊D101—42no—
64C2×Dic101—21no—
65C2×Dic101C22⋊C411yescharacteristic
66C2×Dic101C2×D411yescharacteristic
67C2×Dic101C2×D411yescharacteristic
68D404—41no—
69C4×D101—22no—
70C4×D101—22no—
71C4×D101—41no—
72C4×D101—42no—
73Dic202—42no—
74Dic202—41no—
75#11717581—24no—
76#11717581—42no—
77C22×C101—41no—
78C22×C101—41no—
79C22×C101—41no—
80C22×C101C2×C22⋊C411yescharacteristic
81C2×D101—21no—
82C2×D101—21no—
83C2×D101—41no—
84C2×D101—41no—
85C2×D101—41no—
86C2×D101—84no—
87C2×D101#10412yes—
88C4×C101—21no—
89C4×C101—21no—
90C4×C101—21no—
91C4×C101—41no—
92Dic101—21no—
93Dic101—21no—
94Dic101—41no—
95Dic101#10012yes—
96C2×C101—81no—
97C2×C101C2×#10011yescharacteristic
98C2×C101C2×#10411yescharacteristic
99C2×C101#42111yescharacteristic
100D101—22no—
101D101—81no—
102#1527—1011nomaximal, Sylow
103C101#152711yescharacteristic, complemented, Sylow
104C2×#124—1011no—
105#432—1011no—
106C4×M16—1011no—
107#329—1012no—
108#329—1012no—
109C22×Q8—1011no—
110C22×D4—1011no—
111C2×M16—2021no—
112#124—1011no—
113#124—1011no—
114#124—1011no—
115#124—2021no—
116#124—2021no—
117#124#8793811yescharacteristic
118C2×C22⋊C4—2021no—
119C2×C22⋊C4—2021no—
120C2×C42—1011no—
121C4⋊C8—2021no—
122C22⋊C8—2021no—
123#97—1012no—
124#97—1012no—
125C4×C8—2021no—
126C24—2021no—
127C2×Q8—1011no—
128C2×Q8—4042no—
129C2×Q8—2021no—
130C2×Q8C2×#8793811yescharacteristic
131C2×D4—1011no—
132C2×D4—4042no—
133C2×D4—2021no—
134C2×D4C2×#8793811yescharacteristic
135C22×C4—1011no—
136C22×C4—1011no—
137C22×C4—1011no—
138C22×C4—2021no—
139M16—2024no—
140C2×C8—2022no—
141C2×C8—2022no—
142C22⋊C4—21no—
143C22⋊C4—21no—
144C22⋊C4—2021no—
145C22⋊C4—2021no—
146C22⋊C4—2021no—
147C22⋊C4—2021no—
148C22⋊C4—2021no—
149C22⋊C4—2021no—
150C42—1011no—
151C42—1011no—
152C42—1011no—
153C42C2×#8793811yescharacteristic
154C23—21no—
155C23—1011no—
156C23—4042no—
157C23—4042no—
158C23—4042no—
159C23—2021no—
160Q8—41no—
161Q8—4041no—
162Q8—4042no—
163D4—41no—
164D4—4041no—
165D4—4042no—
166C2×C4—21no—
167C2×C4—1011no—
168C2×C4—1011no—
169C2×C4—1011no—
170C2×C4—4041no—
171C2×C4—4042no—
172C2×C4—2022no—
173C2×C4—2022no—
174C2×C4—2021no—
175C2×C4C22⋊#8793811yescharacteristic
176C2×C4C22⋊#8793811yescharacteristic
177C2×C4C22×#8793811yescharacteristic
178C8—4042no—
179C8—2024no—
180C22—41no—
181C22—41no—
182C22—41no—
183C22—8084no—
184C22—1012no—
185C22—4041no—
186C22—4041no—
187C22—4041no—
188C22—2021no—
189C22—2021no—
190C22C2×C22⋊#8793811yescharacteristic
191C4—21no—
192C4—21no—
193C4—21no—
194C4—41no—
195C4—1012no—
196C4—4041no—
197C4—2021no—
198C4—2021no—
199C2—81no—
200C2—8081no—
201C2—2022no—
202C2D101⋊#10011yescharacteristic
203C2D101⋊#10411yescharacteristic
204C2#2026617411yescharacteristic
subgroupG/H
centerC22C2×C22⋊#87938
commutatorC2×C4×C101C22×C4
FrattiniC2×C4C22×#87938
Fitting#124×C101C4
socleC22×C101C2×C22⋊C4
prime radicalC2×D4×D101C4
cosocleC4×Dic101C23
hypercenter#124#87938
hypocenterC101#1527
LCSG ▷ C2×C4×C101 ▷ C22×C101 ▷ C101
UCSC1 ◁ C22 ◁ C42 ◁ #124

Complements

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

NH ≅ G/Ntype
1. #20266200C2semidirect
2. C4×#13108084C2semidirect
3. #20266033C2semidirect
4. #20266032C2semidirect
9. #15692310C22semidirect
10. #15692310C22semidirect
103. C101#1527semidirect

Maximal subgroups

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

subgroupindex
#202662002
C4×#131080842
#202660332
#202660322
#124×D1012
#1527101