Finite Group Databasebeta

← #27171578

Subgroups — #27171578

Order 12928.

358 conjugacy classes of subgroups; only the 89 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 89 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
2C2×#15692214C211yesmaximal, characteristic, complemented
3C2×#15692181C212yesmaximal, complemented
4#20264938C212yesmaximal, complemented
5#407×C101C211yesmaximal, characteristic, complemented
6C2×#13108074C2211yescharacteristic
7#15692238—21no—
8#15692234C412yescomplemented
9#15692229C412yescomplemented
10#15692224—21no—
11C4×C22⋊D101—22no—
12#15692214—24no—
13#15692214C2211yescharacteristic
14C2×C4⋊Dic101—21no—
15C2×C22⋊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
24C2×D404C2×C411yescharacteristic
25C2×C22⋊D101—22no—
26C2×C22⋊D101—21no—
27C22×Dic101—22no—
28C2×C4×D101—21no—
29C4⋊Dic101—24no—
30C4⋊Dic101C2×C412yes—
31#13108053—22no—
32C22⋊Dic101—24no—
33C22⋊Dic101—22no—
34C22⋊Dic101—21no—
35#13108051—22no—
36C4×Dic101—22no—
37C4×Dic101—21no—
38C4×Dic101D414yescomplemented
39C24×C101D411yescharacteristic
40C22×C4×C101—22no—
41C22×C4×C101—22no—
42C22×C4×C101C2×C412yes—
43C22×C4×C101D411yescharacteristic
44C22×C4×C101D412yes—
45C22×C4×C101C2311yescharacteristic
46M16×C101—22no—
47M16×C101—22no—
48C2×C8×C101—22no—
49C23×C101—21no—
50C23×C101—21no—
51C23×C101—42no—
52C23×C101C22⋊C412yes—
53C23×C101C2×D411yescharacteristic
54C22×D101—21no—
55C2×C4×C101—21no—
56C2×C4×C101—21no—
57C2×C4×C101—24no—
58C2×C4×C101—24no—
59C2×C4×C101C22⋊C412yes—
60C2×C4×C101C22×C411yescharacteristic
61C2×C4×C101C2×D412yes—
62C2×C4×C101C2×D411yescharacteristic
63C2×C4×C101#4112yes—
64C22⋊D101—44no—
65C22⋊D101—41no—
66C22⋊D101—41no—
67C2×Dic101—22no—
68C2×Dic101—21no—
69C2×Dic101—24no—
70C2×Dic101—44no—
71D404—22no—
72C4×D101—42no—
73Dic202—22no—
74C8×C101—42no—
75C22×C101—22no—
76C22×C101—22no—
77C22×C101—22no—
78C22×C101—22no—
79C22×C101—42no—
80C22×C101—42no—
81C22×C101C2×C22⋊C411yescharacteristic
82C22×C101#12011yescharacteristic
83C22×C101#12311yescharacteristic
84C2×D101—41no—
85C2×D101—42no—
86C4×C101—22no—
87C4×C101C4×D412yes—
88C4×C101#12112yes—
89Dic101—44no—
90Dic101—41no—
91Dic101—42no—
92C2×C101—21no—
93C2×C101—21no—
94C2×C101—42no—
95C2×C101#38611yescharacteristic
96C2×C101#42112yes—
97D101—81no—
98#1897—1011nomaximal, Sylow
99C101#189711yescharacteristic, complemented, Sylow
100#525—1011no—
101C2×#117—1011no—
102C2×#104—1012no—
103#407D10111yescharacteristic, complemented
104#339—1012no—
105C2×#41—1011no—
106C23×C4C2×D10111yescharacteristic
107C2×M16C2×D10112yes—
108#123—2021no—
109C22⋊Q8—1012no—
110#121—1012no—
111#120—2021no—
112C4×D4—2022no—
113#117—1011no—
114#117—2024no—
115C2×C4⋊C4—2021no—
116C2×C22⋊C4—2021no—
117C2×C42—1012no—
118#104—2024no—
119C22⋊C8—22no—
120C24D40411yescharacteristic
121#41—2022no—
122#41—2022no—
123C2×Q8—1011no—
124C2×D4—1011no—
125C2×D4—2022no—
126C2×D4—2021no—
127C22×C4—22no—
128C22×C4—22no—
129C22×C4—2022no—
130C22×C4—2021no—
131C22×C4C4×D10112yes—
132C22×C4D40411yescharacteristic
133C22×C4C22⋊D10112yes—
134C22×C4C22×D10111yescharacteristic
135M16—22no—
136M16—22no—
137C2×C8—22no—
138C4⋊C4—1012no—
139C4⋊C4—2024no—
140C4⋊C4—2022no—
141C22⋊C4—2024no—
142C22⋊C4—2022no—
143C22⋊C4—2021no—
144C22⋊C4—2022no—
145C42—1014no—
146C42—2022no—
147C42—2021no—
148C23—21no—
149C23—21no—
150C23—42no—
151C23—2021no—
152C23#1310805112yes—
153C23C2×D40411yescharacteristic
154Q8—2022no—
155D4—4044no—
156D4—4041no—
157D4—4041no—
158D4—2022no—
159C2×C4—21no—
160C2×C4—21no—
161C2×C4—24no—
162C2×C4—24no—
163C2×C4—4044no—
164C2×C4—4042no—
165C2×C4—2022no—
166C2×C4—2021no—
167C2×C4—2024no—
168C2×C4#1310805112yes—
169C2×C4C2×C4×D10111yescharacteristic
170C2×C4D4×D10112yes—
171C2×C4C2×C22⋊D10111yescharacteristic
172C2×C4#1310807312yes—
173C8—42no—
174C22—22no—
175C22—22no—
176C22—22no—
177C22—22no—
178C22—42no—
179C22—42no—
180C22—4041no—
181C22—4042no—
182C22C2×#1310805111yescharacteristic
183C22#1569222511yescharacteristic
184C22#1569223911yescharacteristic
185C4—22no—
186C4—4044no—
187C4—4041no—
188C4—4042no—
189C4#1569221512yes—
190C4#1569222712yes—
191C2—21no—
192C2—21no—
193C2—42no—
194C2—8081no—
195C2#2026505411yescharacteristic
196C2#2026519312yes—
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#1897
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. C2×#15692214C2semidirect
3. C2×#15692181C2semidirect
4. #20264938C2semidirect
5. #407×C101C2semidirect
8. #15692234C4semidirect
9. #15692229C4semidirect
38. C4×Dic101D4semidirect
99. C101#1897semidirect
103. #407D101semidirect

Maximal subgroups

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

subgroupindex
#202656882
C2×#156922142
C2×#156921812
#202649382
#407×C1012
#1897101