Finite Group Databasebeta

← #27187391

Subgroups — #27187391

Order 12928.

266 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 266 conjugacy classes of subgroups — too many to draw. Only its 54 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#20266163C211yesmaximal, characteristic, complemented
2#20266156C211yesmaximal, characteristic, complemented
3#20266077C214yesmaximal, complemented
4#20265744C211yesmaximal, characteristic
5D101⋊M16—21no—
6#15692334—21no—
7#15692316—21no—
8#15692315C2212yes—
9#15692314—21no—
10#15692313C2212yes—
11C2×#13108074C2211yescharacteristic
12#15692251—21no—
13#15692242—21no—
14#15692226—21no—
15#15692226—21no—
16#15692224C412yescomplemented
17#15692224C2212yes—
18#15692214C411yescharacteristic, complemented
19C22×D4×C101C411yescharacteristic, complemented
20#13108084—41no—
21#13108083—22no—
22D101⋊C8—41no—
23C4⋊#87938—41no—
24C22⋊#87938—22no—
25C4×#87938—41no—
26#13108074—41no—
27#13108074—41no—
28C2×Dic202D411yescharacteristic
29C2×D404D411yescharacteristic
30C2×C22⋊D101—22no—
31C2×C22⋊D101—22no—
32C2×C22⋊D101C2×C411yescharacteristic
33C2×C22⋊D101C2311yescharacteristic
34C2×C4×D101D411yescharacteristic
35C2×C4×D101D411yescharacteristic
36C4⋊Dic101—21no—
37C22⋊Dic101—22no—
38C22⋊Dic101—22no—
39C22⋊Dic101C2×C412yes—
40C4×Dic101—21no—
41C24×C101C2×C412yes—
42C2×D4×C101—21no—
43C2×D4×C101—21no—
44C2×D4×C101—41no—
45C2×D4×C101—41no—
46C22×C4×C101C2×C411yescharacteristic
47C23×C101—22no—
48C23×C101—42no—
49C23×C101—82no—
50C23×C101C22×C411yescharacteristic
51C22×D101C22⋊C411yescharacteristic
52C22×D101C2×D411yescharacteristic
53C2×#87938—42no—
54D4×C101—42no—
55D4×C101—41no—
56D4×C101—41no—
57C2×C4×C101—21no—
58C2×C4×C101—21no—
59C2×C4×C101C22⋊C411yescharacteristic
60C2×C4×C101C22⋊C411yescharacteristic
61C22⋊D101—41no—
62C22⋊D101—41no—
63C22⋊D101—82no—
64C22⋊D101—82no—
65C2×Dic101—22no—
66C2×Dic101C22⋊C411yescharacteristic
67C2×Dic101C2×D411yescharacteristic
68D404—41no—
69C4×D101—21no—
70C4×D101—21no—
71C4×D101—21no—
72C4×D101—21no—
73Dic202—41no—
74#11717581—42no—
75C22×C101—21no—
76C22×C101—41no—
77C22×C101—42no—
78C22×C101—42no—
79C22×C101—42no—
80C22×C101—84no—
81C22×C101C2×C22⋊C411yescharacteristic
82C2×D101—21no—
83C2×D101—21no—
84C2×D101—41no—
85C2×D101—41no—
86#87938—82no—
87C4×C101—21no—
88C4×C101#9911yescharacteristic
89C4×C101#9911yescharacteristic
90Dic101—21no—
91Dic101—21no—
92Dic101—42no—
93C2×C101—21no—
94C2×C101—41no—
95C2×C101—82no—
96C2×C101C2×#9911yescharacteristic
97D101—41no—
98D101—41no—
99#2122—1011nomaximal, Sylow
100C101#212211yescharacteristic, complemented, Sylow
101#535—1011no—
102#413—1011no—
103#410—1011no—
104AΣL(1,𝔽16)—1014no—
105C2×#41—1011no—
106C22×D4#8793811yescharacteristic, complemented
107C2×M16—2021no—
108#127—2021no—
109#124—2021no—
110#121—2021no—
111#121—2021no—
112#120—1012no—
113#120—1012no—
114#117—1011no—
115#117—2021no—
116#101—2021no—
117#100—1012no—
118#99—1012no—
119#99—2021no—
120C24C2×#8793812yes—
121#41—4041no—
122#41—4041no—
123C2×Q8—1011no—
124C2×D4—21no—
125C2×D4—21no—
126C2×D4—41no—
127C2×D4—41no—
128C2×D4—1011no—
129C2×D4—1011no—
130C2×D4—1011no—
131C2×D4—2022no—
132C2×D4—2022no—
133C22×C4—1011no—
134C22×C4—1011no—
135C22×C4C2×#8793811yescharacteristic
136M16—4041no—
137M16—2022no—
138C2×C8—4041no—
139C4⋊C4—4041no—
140C4⋊C4—2021no—
141C22⋊C4—1012no—
142C22⋊C4—2022no—
143C22⋊C4—2022no—
144C22⋊C4—2022no—
145C42—4041no—
146C42—2021no—
147C23—22no—
148C23—42no—
149C23—82no—
150C23—1011no—
151C23—1011no—
152C23C22×#8793811yescharacteristic
153Q8—4041no—
154D4—42no—
155D4—41no—
156D4—41no—
157D4—8082no—
158D4—8082no—
159D4—4041no—
160D4—4041no—
161D4—4041no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—1011no—
165C2×C4—1011no—
166C2×C4—4042no—
167C2×C4—2022no—
168C2×C4—2021no—
169C2×C4—2021no—
170C2×C4—2021no—
171C2×C4—2021no—
172C2×C4C22⋊#8793811yescharacteristic
173C2×C4C22⋊#8793811yescharacteristic
174C8—4042no—
175C22—21no—
176C22—41no—
177C22—42no—
178C22—42no—
179C22—42no—
180C22—84no—
181C22—4041no—
182C22—4041no—
183C22—2021no—
184C22—2021no—
185C22C2×C22⋊#8793811yescharacteristic
186C4—21no—
187C4—8082no—
188C4—4042no—
189C4—2021no—
190C4—2021no—
191C4#1569231311yescharacteristic
192C4#1569231311yescharacteristic
193C2—21no—
194C2—41no—
195C2—82no—
196C2—4041no—
197C2—4041no—
198C2C2×#1569231311yescharacteristic
subgroupG/H
centerC2C2×#15692313
commutatorC23×C101C22×C4
FrattiniC23C22×#87938
FittingC22×D4×C101C4
socleC2×C101C2×#99
prime radical#20265744C2
cosocleC2×C22⋊D101C23
hypercenterC22×D4#87938
hypocenterC101#2122
LCSG ▷ C23×C101 ▷ C22×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. #20266163C2semidirect
2. #20266156C2semidirect
3. #20266077C2semidirect
16. #15692224C4semidirect
18. #15692214C4semidirect
19. C22×D4×C101C4semidirect
100. C101#2122semidirect
106. C22×D4#87938semidirect

Maximal subgroups

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

subgroupindex
#202661632
#202661562
#202660772
#202657442
#2122101