Finite Group Databasebeta

← #27180709

Subgroups — #27180709

Order 12928.

520 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 145 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
1C22×C8×D101C211yesmaximal, characteristic, complemented
2#20265684C212yesmaximal, complemented
3#20265415C212yesmaximal, complemented
4#20265410C212yesmaximal, complemented
5#20265403C214yesmaximal, complemented
6#20265215C211yesmaximal, characteristic, complemented
7#20265173C212yesmaximal, complemented
8C2×#135×C101C211yesmaximal, characteristic, complemented
9C22×C4×D101C2211yescharacteristic
10C22×#11717579C2211yescharacteristic
11C2×C8×D101—24no—
12C2×C8×D101C2212yes—
13C2×C8×D101C2212yes—
14#15692236—22no—
15#15692231C2214yes—
16#15692226C2214yes—
17#15692223—22no—
18C4×C22⋊D101—24no—
19#15692214C2212yes—
20#15692189C2212yescomplemented
21#15692187C2212yescomplemented
22#15692179C2214yescomplemented
23#15692176C2214yescomplemented
24C2×#41×C101C2212yes—
25#135×C101—24no—
26C2×Q16×C101C2211yescharacteristic, complemented
27C2×SD16×C101C2212yescomplemented
28C2×D8×C101C2211yescharacteristic, complemented
29C22×C8×C101C2211yescharacteristic
30C23×D101D411yescharacteristic
31C2×C22⋊D101—24no—
32C22×Dic101D411yescharacteristic
33C2×C4×D101—22no—
34C2×C4×D101—22no—
35C2×C4×D101D412yes—
36C2×C4×D101C2312yes—
37C2×#11717579—22no—
38C2×#11717579C2312yes—
39C8×D101—24no—
40C8×D101—24no—
41C4⋊Dic101C2314yes—
42#13108053—24no—
43C22⋊Dic101—22no—
44C22⋊Dic101—24no—
45#13108051—24no—
46C4×Dic101—22no—
47#41×C101—24no—
48#41×C101—24no—
49C2×Q8×C101C2312yes—
50C2×D4×C101—22no—
51C2×D4×C101C2312yes—
52C22×C4×C101—22no—
53C22×C4×C101C2311yescharacteristic
54Q16×C101—22no—
55SD16×C101—24no—
56D8×C101—22no—
57C2×C8×C101D414yescomplemented
58C2×C8×C101C2311yescharacteristic
59C2×C8×C101C2311yescharacteristic
60C23×C101—22no—
61C23×C101C2×D411yescharacteristic
62C22×D101—22no—
63C22×D101—24no—
64C22×D101C2×D412yes—
65Q8×C101—22no—
66Q8×C101—22no—
67D4×C101—22no—
68D4×C101—22no—
69D4×C101—44no—
70C2×C4×C101—22no—
71C2×C4×C101—44no—
72C2×C4×C101C2×D411yescharacteristic
73C2×C4×C101C2×D412yes—
74C2×C4×C101#4112yes—
75C2×C4×C101C2411yescharacteristic
76C22⋊D101—48no—
77C2×Dic101—22no—
78C2×Dic101—24no—
79C2×Dic101C2×D412yes—
80C4×D101—24no—
81C4×D101—22no—
82C4×D101—22no—
83C8×C101C2×D412yes—
84C8×C101C2×D412yes—
85#11717579—22no—
86C22×C101—22no—
87C22×C101—42no—
88C22×C101—44no—
89C22×C101#12112yes—
90C22×C101C22×D411yescharacteristic
91C2×D101—28no—
92C2×D101—22no—
93C2×D101—22no—
94C2×D101#13514yescomplemented
95C4×C101—42no—
96C4×C101#12112yes—
97C4×C101C22×D411yescharacteristic
98C4×C101C2×#4111yescharacteristic
99Dic101—22no—
100Dic101—44no—
101C2×C101—22no—
102C2×C101—82no—
103C2×C101C2×#12111yescharacteristic
104C2×C101C2×#13512yes—
105D101—24no—
106#3050—1011nomaximal, Sylow
107C101#305011yescharacteristic, complemented, Sylow
108C2×#135D10111yescharacteristic, complemented
109C23×C8—1011no—
110#525—1012no—
111#467—1012no—
112#466—1012no—
113#465—1014no—
114#427—1011no—
115#416—1012no—
116C2×#41C2×D10112yes—
117C23×C4—1011no—
118#135—24no—
119C2×Q16C2×D10111yescharacteristic, complemented
120C2×SD16C2×D10112yescomplemented
121C2×D8C2×D10111yescharacteristic, complemented
122C22×C8—1011no—
123C22×C8—1012no—
124C22×C8—1012no—
125C22×C8—2024no—
126C22×C8C2×D10111yescharacteristic
127#123—2022no—
128C22⋊Q8—1014no—
129#121—1014no—
130#120—2022no—
131C4×D4—2024no—
132#117—1012no—
133#107—1012no—
134#106—1012no—
135#103—1014no—
136#102—1014no—
137C24—1011no—
138#41—24no—
139#41—24no—
140C2×Q8C22×D10112yes—
141C2×D4—22no—
142C2×D4—2024no—
143C2×D4C22×D10112yes—
144C22×C4—22no—
145C22×C4—1011no—
146C22×C4—1012no—
147C22×C4—1012no—
148C22×C4—2022no—
149C22×C4—2022no—
150C22×C4C22×D10111yescharacteristic
151Q16—22no—
152SD16—24no—
153D8—22no—
154C2×C8—1012no—
155C2×C8—2022no—
156C2×C8—2024no—
157C2×C8—2024no—
158C2×C8C22⋊D10114yescomplemented
159C2×C8C22×D10111yescharacteristic
160C2×C8C22×D10111yescharacteristic
161C4⋊C4—1014no—
162C4⋊C4—2024no—
163C22⋊C4—2024no—
164C22⋊C4—2024no—
165C22⋊C4—2022no—
166C42—2022no—
167C23—22no—
168C23—1012no—
169C23—2024no—
170C23—2022no—
171C23D4×D10111yescharacteristic
172Q8—22no—
173Q8—22no—
174D4—22no—
175D4—22no—
176D4—44no—
177D4—4048no—
178C2×C4—22no—
179C2×C4—44no—
180C2×C4—1012no—
181C2×C4—2024no—
182C2×C4—2022no—
183C2×C4—2022no—
184C2×C4—2022no—
185C2×C4—2024no—
186C2×C4D4×D10111yescharacteristic
187C2×C4C2×C22⋊D10112yes—
188C2×C4#1310807312yes—
189C2×C4C23×D10111yescharacteristic
190C8—2022no—
191C8C2×C22⋊D10112yes—
192C8C2×C22⋊D10112yes—
193C22—22no—
194C22—42no—
195C22—44no—
196C22—1014no—
197C22—2028no—
198C22—2022no—
199C22—2022no—
200C22#1569222612yes—
201C22C2×D4×D10111yescharacteristic
202C4—42no—
203C4—4044no—
204C4—2022no—
205C4#1569222612yes—
206C4C22×C22⋊D10111yescharacteristic
207C4C2×#1310807311yescharacteristic
208C2—22no—
209C2—82no—
210C2—2024no—
211C2C2×#1569222611yescharacteristic
212C2#135×D10112yes—
subgroupG/H
centerC2×C4D4×D101
commutator, cosocleC2×C4×C101C24
FrattiniC2×C4C23×D101
FittingC2×#135×C101C2
socleC22×C101C22×D4
hypercenterC2×#135D101
hypocenterC101#3050
LCSG ▷ C2×C4×C101 ▷ C2×C101 ▷ C101
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C22×C8×D101C2semidirect
2. #20265684C2semidirect
3. #20265415C2semidirect
4. #20265410C2semidirect
5. #20265403C2semidirect
6. #20265215C2semidirect
7. #20265173C2semidirect
8. C2×#135×C101C2semidirect
20. #15692189C22semidirect
21. #15692187C22semidirect
22. #15692179C22semidirect
23. #15692176C22semidirect
26. C2×Q16×C101C22semidirect
27. C2×SD16×C101C22semidirect
28. C2×D8×C101C22semidirect
57. C2×C8×C101D4semidirect
94. C2×D101#135semidirect
107. C101#3050semidirect
108. C2×#135D101semidirect
119. C2×Q16C2×D101semidirect
120. C2×SD16C2×D101semidirect
121. C2×D8C2×D101semidirect
158. C2×C8C22⋊D101semidirect

Maximal subgroups

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

subgroupindex
C22×C8×D1012
#202656842
#202654152
#202654102
#202654032
#202652152
#202651732
C2×#135×C1012
#3050101