Finite Group Databasebeta

← #27171522

Subgroups — #27171522

Order 12928.

324 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 324 conjugacy classes of subgroups — too many to draw. Only its 83 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
1C2×#15692253C211yesmaximal, characteristic
2#20265654C211yesmaximal, characteristic
3#20265186C211yesmaximal, characteristic
4C2×#15692173C211yesmaximal, characteristic
5#20264938C211yesmaximal, characteristic
6#20264936C211yesmaximal, characteristic
7#419×C101C211yesmaximal, characteristic
8C22×Dic202C2211yescharacteristic
9C2×#13108058C2211yescharacteristic
10#15692253—22no—
11#15692253C414yescomplemented
12C4×Dic202—22no—
13#15692220—22no—
14#15692214C2211yescharacteristic
15#15692211—22no—
16C2×#13108053—22no—
17C2×C4×Dic101C2211yescharacteristic
18#15692179—22no—
19#15692173—22no—
20C22×Q8×C101C2211yescharacteristic
21C2×M16×C101C2211yescharacteristic
22#117×C101C2211yescharacteristic
23#103×C101—22no—
24C2×Dic202—21no—
25C2×Dic202—21no—
26C2×Dic202—21no—
27C2×Dic202—21no—
28C2×Dic202C2×C412yes—
29C2×Dic202D412yes—
30C22×Dic101—21no—
31C22×Dic101D412yes—
32#13108058—21no—
33#13108058—21no—
34C2×#11717579—21no—
35C4⋊Dic101D412yes—
36#13108053—24no—
37#13108053—28no—
38C22⋊Dic101—21no—
39C4×Dic101—24no—
40C4×Dic101—21no—
41C4×Dic101—21no—
42C4×Dic101C2×C412yes—
43C2×Q8×C101—21no—
44C2×Q8×C101—21no—
45C2×Q8×C101—21no—
46C2×Q8×C101—21no—
47C2×Q8×C101—21no—
48C2×Q8×C101C2×C412yes—
49C22×C4×C101—21no—
50C22×C4×C101C2311yescharacteristic
51M16×C101—21no—
52M16×C101—21no—
53C2×C8×C101—21no—
54C4⋊C4×C101D412yes—
55C22⋊C4×C101—21no—
56C42×C101—21no—
57C23×C101C2×D411yescharacteristic
58Q8×C101—22no—
59Q8×C101—22no—
60Q8×C101—41no—
61Q8×C101—41no—
62C2×C4×C101—22no—
63C2×C4×C101—22no—
64C2×C4×C101—41no—
65C2×C4×C101—41no—
66C2×C4×C101C22×C411yescharacteristic
67C2×C4×C101C2×D411yescharacteristic
68C2×C4×C101C2×D411yescharacteristic
69C2×C4×C101C2×D411yescharacteristic
70C2×C4×C101#4111yescharacteristic
71C2×C4×C101#4111yescharacteristic
72C2×Dic101—24no—
73C2×Dic101—22no—
74C2×Dic101—22no—
75C2×Dic101—22no—
76C2×Dic101—22no—
77C2×Dic101C22⋊C414yes—
78Dic202—22no—
79Dic202—22no—
80Dic202—22no—
81Dic202—22no—
82C8×C101—41no—
83#11717579—41no—
84C22×C101—22no—
85C22×C101C2×C22⋊C411yescharacteristic
86C22×C101#12011yescharacteristic
87C22×C101#12311yescharacteristic
88C4×C101—42no—
89C4×C101—42no—
90C4×C101C4×D411yescharacteristic
91C4×C101C4×D411yescharacteristic
92C4×C101#12111yescharacteristic
93C4×C101#12111yescharacteristic
94Dic101—24no—
95Dic101—24no—
96Dic101—42no—
97C2×C101—21no—
98C2×C101—21no—
99C2×C101#38611yescharacteristic
100C2×C101#45612yes—
101#1886—1011nomaximal, Sylow
102C101#188611yescharacteristic, complemented, Sylow
103C2×C4⋊Q8—1011no—
104#519—1011no—
105#419—1011no—
106#419D10111yescharacteristic
107C2×#101—1011no—
108#339—1011no—
109#339—1011no—
110C22×Q8—1011no—
111C22×Q8C2×D10111yescharacteristic
112C2×M16—1011no—
113C2×M16C2×D10111yescharacteristic
114C4⋊Q8—1014no—
115C4⋊Q8—2022no—
116C4×Q8—2022no—
117C4×Q8—2022no—
118#117—1011no—
119#117—2022no—
120#117C2×D10111yescharacteristic
121C2×C4⋊C4—2022no—
122C2×C42—1011no—
123#103—22no—
124#103—2022no—
125#101—2022no—
126C2×Q8—21no—
127C2×Q8—21no—
128C2×Q8—21no—
129C2×Q8—21no—
130C2×Q8—21no—
131C2×Q8—1012no—
132C2×Q8—1012no—
133C2×Q8—2021no—
134C2×Q8—2021no—
135C2×Q8—2021no—
136C2×Q8—2021no—
137C2×Q8C4×D10112yes—
138C22×C4—21no—
139C22×C4—1012no—
140C22×C4—2021no—
141C22×C4C22×D10111yescharacteristic
142M16—21no—
143M16—21no—
144M16—2021no—
145M16—2021no—
146C2×C8—21no—
147C2×C8—2021no—
148C4⋊C4—1012no—
149C4⋊C4—2028no—
150C4⋊C4—2024no—
151C4⋊C4C22⋊D10112yes—
152C22⋊C4—21no—
153C22⋊C4—2021no—
154C42—21no—
155C42—1012no—
156C42—2024no—
157C42—2021no—
158C42—2021no—
159C23D4×D10111yescharacteristic
160Q8—22no—
161Q8—22no—
162Q8—41no—
163Q8—41no—
164Q8—2022no—
165Q8—2022no—
166Q8—2022no—
167Q8—2022no—
168C2×C4—22no—
169C2×C4—22no—
170C2×C4—41no—
171C2×C4—41no—
172C2×C4—1014no—
173C2×C4—2022no—
174C2×C4—2024no—
175C2×C4—2022no—
176C2×C4—2022no—
177C2×C4—2022no—
178C2×C4C2×C4×D10111yescharacteristic
179C2×C4D4×D10111yescharacteristic
180C2×C4D4×D10111yescharacteristic
181C2×C4C2×C22⋊D10111yescharacteristic
182C2×C4#1310807311yescharacteristic
183C2×C4#1310807411yescharacteristic
184C8—41no—
185C8—4041no—
186C22—22no—
187C22C22⋊C4×D10111yescharacteristic
188C22#1569222311yescharacteristic
189C22#1569223711yescharacteristic
190C4—42no—
191C4—42no—
192C4—4042no—
193C4—2024no—
194C4—2024no—
195C4#1569221511yescharacteristic
196C4C4×C22⋊D10111yescharacteristic
197C4#1569222611yescharacteristic
198C4#1569222811yescharacteristic
199C2—21no—
200C2—21no—
201C2#2026505311yescharacteristic
202C2#2026535812yes—
subgroupG/H
centerC22#15692223
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#419×C101C2
socleC22×C101#120
prime radicalC23×C101C2×D4
cosocleC22×C4×C101C23
hypercenter#419D101
hypocenterC101#1886
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
11. #15692253C4semidirect
102. C101#1886semidirect

Maximal subgroups

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

subgroupindex
C2×#156922532
#202656542
#202651862
C2×#156921732
#202649382
#202649362
#419×C1012
#1886101