Finite Group Databasebeta

← #27171573

Subgroups — #27171573

Order 12928.

442 conjugacy classes of subgroups; only the 97 normal subgroups are shown.

normal subgroups only

This group has 442 conjugacy classes of subgroups — too many to draw. Only its 97 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#20265688C211yesmaximal, characteristic
2C22×C4×Dic101C211yesmaximal, characteristic, complemented
3C2×#15692181C212yesmaximal, complemented
4#20264938C212yesmaximal, complemented
5#407×C101C211yesmaximal, characteristic, complemented
6C2×#13108074C2211yescharacteristic
7C23×Dic101—21no—
8#15692238—21no—
9#15692234C412yescomplemented
10#15692229C412yescomplemented
11#15692224—21no—
12C4×C22⋊D101—22no—
13#15692214C2211yescharacteristic
14C2×C4×Dic101—24no—
15C2×C4×Dic101—21no—
16C2×C4×Dic101C2212yes—
17#15692181—24no—
18C23×C4×C101C2211yescharacteristic
19C2×M16×C101C2212yes—
20C22⋊C8×C101—22no—
21#13108074—22no—
22#13108074—22no—
23C2×Dic202C2×C411yescharacteristic, complemented
24C2×D404C2×C411yescharacteristic, complemented
25C2×C22⋊D101—22no—
26C2×C22⋊D101—21no—
27C22×Dic101—28no—
28C22×Dic101—24no—
29C22×Dic101—22no—
30C2×C4×D101—21no—
31C4⋊Dic101C2×C412yescomplemented
32#13108053—22no—
33C22⋊Dic101—22no—
34C22⋊Dic101—21no—
35#13108051—22no—
36C4×Dic101—24no—
37C4×Dic101—22no—
38C4×Dic101—21no—
39C4×Dic101D414yescomplemented
40C24×C101D411yescharacteristic
41C22×C4×C101—22no—
42C22×C4×C101—22no—
43C22×C4×C101C2×C412yes—
44C22×C4×C101D411yescharacteristic
45C22×C4×C101D412yes—
46C22×C4×C101C2311yescharacteristic
47M16×C101—22no—
48M16×C101—22no—
49C2×C8×C101—22no—
50C23×C101—24no—
51C23×C101—21no—
52C23×C101—21no—
53C23×C101C22⋊C412yes—
54C23×C101C2×D411yescharacteristic
55C22×D101—21no—
56C2×C4×C101—21no—
57C2×C4×C101—21no—
58C2×C4×C101—24no—
59C2×C4×C101—24no—
60C2×C4×C101C22⋊C412yes—
61C2×C4×C101C22×C411yescharacteristic
62C2×C4×C101C2×D412yes—
63C2×C4×C101C2×D411yescharacteristic
64C2×C4×C101#4112yes—
65C22⋊D101—44no—
66C22⋊D101—41no—
67C22⋊D101—41no—
68C2×Dic101—216no—
69C2×Dic101—28no—
70C2×Dic101—24no—
71C2×Dic101—22no—
72C2×Dic101—21no—
73D404—22no—
74C4×D101—42no—
75Dic202—22no—
76C8×C101—42no—
77C22×C101—24no—
78C22×C101—24no—
79C22×C101—22no—
80C22×C101—22no—
81C22×C101—22no—
82C22×C101#10414yescomplemented
83C22×C101C2×C22⋊C411yescharacteristic
84C22×C101#12011yescharacteristic
85C22×C101#12311yescharacteristic
86C2×D101—41no—
87C2×D101—42no—
88C4×C101—22no—
89C4×C101C4×D412yes—
90C4×C101#12112yes—
91Dic101—28no—
92Dic101—41no—
93Dic101—42no—
94C2×C101—21no—
95C2×C101—24no—
96C2×C101—21no—
97C2×C101#38611yescharacteristic
98C2×C101C2×#10412yes—
99D101—81no—
100#1896—1011nomaximal, Sylow
101C101#189611yescharacteristic, complemented, Sylow
102#525—1011no—
103C22×C42—1011no—
104C2×#104—1012no—
105#407D10111yescharacteristic, complemented
106#339—1012no—
107C2×#41—1011no—
108C23×C4—2021no—
109C23×C4C2×D10111yescharacteristic
110C2×M16C2×D10112yes—
111#123—2021no—
112C22⋊Q8—1012no—
113#121—1012no—
114#120—2021no—
115C4×D4—2022no—
116#117—1011no—
117C2×C42—1012no—
118C2×C42—2024no—
119C2×C42—2021no—
120#104—2024no—
121C22⋊C8—22no—
122C24D40411yescharacteristic
123#41—2022no—
124#41—2022no—
125C2×Q8—1011no—
126C2×D4—1011no—
127C2×D4—2022no—
128C2×D4—2021no—
129C22×C4—22no—
130C22×C4—22no—
131C22×C4—2021no—
132C22×C4—2024no—
133C22×C4—2028no—
134C22×C4—2022no—
135C22×C4C4×D10112yes—
136C22×C4D40411yescharacteristic
137C22×C4C22⋊D10112yes—
138C22×C4C22×D10111yescharacteristic
139M16—22no—
140M16—22no—
141C2×C8—22no—
142C4⋊C4—1012no—
143C4⋊C4—2022no—
144C22⋊C4—2022no—
145C22⋊C4—2021no—
146C22⋊C4—2022no—
147C42—1014no—
148C42—2024no—
149C42—2022no—
150C42—2021no—
151C23—24no—
152C23—21no—
153C23—21no—
154C23—2021no—
155C23#1310805112yes—
156C23C2×D40411yescharacteristic
157Q8—2022no—
158D4—4044no—
159D4—4041no—
160D4—4041no—
161D4—2022no—
162C2×C4—21no—
163C2×C4—21no—
164C2×C4—24no—
165C2×C4—24no—
166C2×C4—4042no—
167C2×C4—2022no—
168C2×C4—2021no—
169C2×C4—20216no—
170C2×C4—2028no—
171C2×C4—2024no—
172C2×C4#1310805112yes—
173C2×C4C2×C4×D10111yescharacteristic
174C2×C4D4×D10112yes—
175C2×C4C2×C22⋊D10111yescharacteristic
176C2×C4#1310807312yes—
177C8—42no—
178C22—24no—
179C22—24no—
180C22—22no—
181C22—22no—
182C22—22no—
183C22—4041no—
184C22—4042no—
185C22#1569218114yescomplemented
186C22C2×#1310805111yescharacteristic
187C22#1569222511yescharacteristic
188C22#1569223911yescharacteristic
189C4—22no—
190C4—4041no—
191C4—4042no—
192C4—2028no—
193C4#1569221512yes—
194C4#1569222712yes—
195C2—21no—
196C2—24no—
197C2—21no—
198C2—8081no—
199C2#2026505411yescharacteristic
200C2C2×#1569218112yes—
subgroupG/H
centerC2×C4C2×C22⋊D101
commutatorC2×C4×C101C22×C4
FrattiniC22×C4C22×D101
Fitting#407×C101C2
socleC22×C101C2×C22⋊C4
prime radical#20265688C2
cosocleC22×C4×C101C23
hypercenter#407D101
hypocenterC101#1896
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #407

Complements

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

NH ≅ G/Ntype
2. C22×C4×Dic101C2semidirect
3. C2×#15692181C2semidirect
4. #20264938C2semidirect
5. #407×C101C2semidirect
9. #15692234C4semidirect
10. #15692229C4semidirect
23. C2×Dic202C2×C4semidirect
24. C2×D404C2×C4semidirect
31. C4⋊Dic101C2×C4semidirect
39. C4×Dic101D4semidirect
82. C22×C101#104semidirect
101. C101#1896semidirect
105. #407D101semidirect
185. C22#15692181semidirect

Maximal subgroups

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

subgroupindex
#202656882
C22×C4×Dic1012
C2×#156921812
#202649382
#407×C1012
#1896101