Finite Group Databasebeta

← #27180950

Subgroups — #27180950

Order 12928.

384 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 384 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C4×#13108074C211yesmaximal, characteristic
2#20265481C214yesmaximal, complemented
3#20265474C214yesmaximal, complemented
4#20265217C212yesmaximal, complemented
5#20265201C211yesmaximal, characteristic, complemented
6#20265176C212yesmaximal
7C2×#125×C101C211yesmaximal, characteristic, complemented
8C2×#13108074C2211yescharacteristic
9C2×#13108058C2212yes—
10C4×Dic202C2212yes—
11C4×D404C2212yes—
12C4×C22⋊D101—22no—
13#15692214C2211yescharacteristic
14#15692212—21no—
15C42×D101—21no—
16#15692188C2214yes—
17#15692186C2214yes—
18C4⋊#11717579C2214yes—
19#15692178C2214yes—
20#15692175C2214yes—
21#125×C101—21no—
22#125×C101—21no—
23#125×C101C2214yes—
24C2×C4⋊C4×C101—22no—
25C2×C4⋊C4×C101C2212yes—
26C2×C42×C101C2211yescharacteristic
27#13108074—22no—
28#13108074Q814yes—
29C2×Dic202C2311yescharacteristic
30C2×D404C2311yescharacteristic
31C2×C22⋊D101—21no—
32C2×C4×D101—21no—
33C2×C4×D101—22no—
34#13108058—24no—
35C2×#11717579C2314yes—
36C4⋊Dic101C2312yes—
37#13108053—22no—
38C22⋊Dic101—21no—
39#13108051—22no—
40C4×Dic101—21no—
41C4×Dic101—22no—
42C22×C4×C101—22no—
43C22×C4×C101D412yes—
44C22×C4×C101C2311yescharacteristic
45C4⋊C4×C101—24no—
46C4⋊C4×C101—24no—
47C4⋊C4×C101—22no—
48C4⋊C4×C101C2314yes—
49C42×C101D411yescharacteristic
50C42×C101D411yescharacteristic
51C42×C101C2312yes—
52C23×C101C2×D411yescharacteristic
53C22×D101—21no—
54C2×C4×C101—24no—
55C2×C4×C101—42no—
56C2×C4×C101—44no—
57C2×C4×C101C2×D412yes—
58C2×C4×C101C2×D412yes—
59C2×C4×C101C2×D411yescharacteristic
60C2×C4×C101C2×Q812yes—
61C2×C4×C101#4112yes—
62C2×C4×C101C2411yescharacteristic
63C22⋊D101—24no—
64C2×Dic101—21no—
65C2×Dic101—22no—
66D404—21no—
67D404C2×Q812yes—
68C4×D101—24no—
69C4×D101—44no—
70Dic202—21no—
71Dic202C2×Q812yes—
72#11717579—24no—
73C22×C101—22no—
74C22×C101C22⋊Q812yes—
75C22×C101C22×D411yescharacteristic
76C2×D101—22no—
77C2×D101—42no—
78C4×C101—22no—
79C4×C101—22no—
80C4×C101—44no—
81C4×C101C22⋊Q812yes—
82C4×C101C22×Q811yescharacteristic
83C4×C101C2×#4111yescharacteristic
84Dic101—22no—
85Dic101—42no—
86C2×C101—22no—
87C2×C101C2×C22⋊Q811yescharacteristic
88C2×C101#57512yes—
89D101—42no—
90#3080—1011nomaximal, Sylow
91C101#308011yescharacteristic, complemented, Sylow
92C2×#125D10111yescharacteristic, complemented
93C4×#41—1011no—
94#479—1014no—
95#478—1014no—
96#428—1012no—
97#423—1011no—
98#417—1012no—
99C2×#41—1011no—
100C2×M16—1012no—
101#125—21no—
102#125—21no—
103#125C2×D10114yes—
104C4×Q8—1012no—
105C4×D4—1012no—
106C4×D4—2022no—
107#117—1011no—
108#117—2021no—
109C2×C4⋊C4—22no—
110C2×C4⋊C4C2×D10112yes—
111C2×C42—2021no—
112C2×C42C2×D10111yescharacteristic
113#107—1014no—
114#106—1014no—
115C4⋊C8—1014no—
116#103—1014no—
117#102—1014no—
118#41—1014no—
119#41—2022no—
120C2×Q8—1011no—
121C2×D4—1011no—
122C2×D4—2021no—
123C22×C4—22no—
124C22×C4—2022no—
125C22×C4—2021no—
126C22×C4C22⋊D10112yes—
127C22×C4C22×D10111yescharacteristic
128M16—2024no—
129C2×C8—1014no—
130C4⋊C4—24no—
131C4⋊C4—24no—
132C4⋊C4—22no—
133C4⋊C4—1012no—
134C4⋊C4—2022no—
135C4⋊C4C22×D10114yes—
136C22⋊C4—2021no—
137C22⋊C4—2022no—
138C42—2021no—
139C42—2022no—
140C42C22⋊D10111yescharacteristic
141C42C22⋊D10111yescharacteristic
142C42C22×D10112yes—
143C23—2021no—
144C23C2×C22⋊D10111yescharacteristic
145Q8—1012no—
146Q8—2021no—
147D4—1012no—
148D4—2024no—
149D4—2021no—
150C2×C4—24no—
151C2×C4—42no—
152C2×C4—44no—
153C2×C4—4044no—
154C2×C4—2022no—
155C2×C4—2021no—
156C2×C4—2024no—
157C2×C4C2×C22⋊D10112yes—
158C2×C4C2×C22⋊D10112yes—
159C2×C4C2×C22⋊D10111yescharacteristic
160C2×C4Q8×D10112yes—
161C2×C4#1310807512yes—
162C2×C4C23×D10111yescharacteristic
163C8—2024no—
164C22—22no—
165C22—4042no—
166C22—2022no—
167C22#1569223112yes—
168C22C22×C22⋊D10111yescharacteristic
169C4—22no—
170C4—22no—
171C4—44no—
172C4—4042no—
173C4—2022no—
174C4#1569223112yes—
175C4C2×Q8×D10111yescharacteristic
176C4C2×#1310807511yescharacteristic
177C2—22no—
178C2—4042no—
179C2C2×#1569223111yescharacteristic
180C2#2026597512yes—
subgroupG/H
centerC22C22×C22⋊D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×#125×C101C2
socleC22×C101C22×D4
prime radicalC2×#13108074C22
hypercenterC2×#125D101
hypocenterC101#3080
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#125

Complements

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

NH ≅ G/Ntype
2. #20265481C2semidirect
3. #20265474C2semidirect
4. #20265217C2semidirect
5. #20265201C2semidirect
7. C2×#125×C101C2semidirect
91. C101#3080semidirect
92. C2×#125D101semidirect

Maximal subgroups

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

subgroupindex
C4×#131080742
#202654812
#202654742
#202652172
#202652012
#202651762
C2×#125×C1012
#3080101