Finite Group Databasebeta

← #27151313

Subgroups — #27151313

Order 12416.

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×D97C211yesmaximal, characteristic, complemented
2#20261242C212yesmaximal, complemented
3#20260973C212yesmaximal, complemented
4#20260968C212yesmaximal, complemented
5#20260961C214yesmaximal, complemented
6#20260773C211yesmaximal, characteristic, complemented
7#20260731C212yesmaximal, complemented
8C2×#135×C97C211yesmaximal, characteristic, complemented
9C22×C4×D97C2211yescharacteristic
10C22×#11715895C2211yescharacteristic
11C2×C8×D97—24no—
12C2×C8×D97C2212yes—
13C2×C8×D97C2212yes—
14#15690791—22no—
15#15690786C2214yes—
16#15690781C2214yes—
17#15690778—22no—
18C4×C22⋊D97—24no—
19#15690769C2212yes—
20#15690744C2212yescomplemented
21#15690742C2212yescomplemented
22#15690734C2214yescomplemented
23#15690731C2214yescomplemented
24C2×#41×C97C2212yes—
25#135×C97—24no—
26C2×Q16×C97C2211yescharacteristic, complemented
27C2×SD16×C97C2212yescomplemented
28C2×D8×C97C2211yescharacteristic, complemented
29C22×C8×C97C2211yescharacteristic
30C23×D97D411yescharacteristic
31C2×C22⋊D97—24no—
32C22×Dic97D411yescharacteristic
33C2×C4×D97—22no—
34C2×C4×D97—22no—
35C2×C4×D97D412yes—
36C2×C4×D97C2312yes—
37C2×#11715895—22no—
38C2×#11715895C2312yes—
39C8×D97—24no—
40C8×D97—24no—
41C4⋊Dic97C2314yes—
42#13095490—24no—
43C22⋊Dic97—22no—
44C22⋊Dic97—24no—
45#13095488—24no—
46C4×Dic97—22no—
47#41×C97—24no—
48#41×C97—24no—
49C2×Q8×C97C2312yes—
50C2×D4×C97—22no—
51C2×D4×C97C2312yes—
52C22×C4×C97—22no—
53C22×C4×C97C2311yescharacteristic
54Q16×C97—22no—
55SD16×C97—24no—
56D8×C97—22no—
57C2×C8×C97D414yescomplemented
58C2×C8×C97C2311yescharacteristic
59C2×C8×C97C2311yescharacteristic
60C23×C97—22no—
61C23×C97C2×D411yescharacteristic
62C22×D97—22no—
63C22×D97—24no—
64C22×D97C2×D412yes—
65Q8×C97—22no—
66Q8×C97—22no—
67D4×C97—22no—
68D4×C97—22no—
69D4×C97—44no—
70C2×C4×C97—22no—
71C2×C4×C97—44no—
72C2×C4×C97C2×D411yescharacteristic
73C2×C4×C97C2×D412yes—
74C2×C4×C97#4112yes—
75C2×C4×C97C2411yescharacteristic
76C22⋊D97—48no—
77C2×Dic97—22no—
78C2×Dic97—24no—
79C2×Dic97C2×D412yes—
80C4×D97—24no—
81C4×D97—22no—
82C4×D97—22no—
83C8×C97C2×D412yes—
84C8×C97C2×D412yes—
85#11715895—22no—
86C22×C97—22no—
87C22×C97—42no—
88C22×C97—44no—
89C22×C97#12112yes—
90C22×C97C22×D411yescharacteristic
91C2×D97—28no—
92C2×D97—22no—
93C2×D97—22no—
94C2×D97#13514yescomplemented
95C4×C97—42no—
96C4×C97#12112yes—
97C4×C97C22×D411yescharacteristic
98C4×C97C2×#4111yescharacteristic
99Dic97—22no—
100Dic97—44no—
101C2×C97—22no—
102C2×C97—82no—
103C2×C97C2×#12111yescharacteristic
104C2×C97C2×#13512yes—
105D97—24no—
106#3050—971nomaximal, Sylow
107C97#305011yescharacteristic, complemented, Sylow
108C2×#135D9711yescharacteristic, complemented
109C23×C8—971no—
110#525—972no—
111#467—972no—
112#466—972no—
113#465—974no—
114#427—971no—
115#416—972no—
116C2×#41C2×D9712yes—
117C23×C4—971no—
118#135—24no—
119C2×Q16C2×D9711yescharacteristic, complemented
120C2×SD16C2×D9712yescomplemented
121C2×D8C2×D9711yescharacteristic, complemented
122C22×C8—971no—
123C22×C8—972no—
124C22×C8—972no—
125C22×C8—1944no—
126C22×C8C2×D9711yescharacteristic
127#123—1942no—
128C22⋊Q8—974no—
129#121—974no—
130#120—1942no—
131C4×D4—1944no—
132#117—972no—
133#107—972no—
134#106—972no—
135#103—974no—
136#102—974no—
137C24—971no—
138#41—24no—
139#41—24no—
140C2×Q8C22×D9712yes—
141C2×D4—22no—
142C2×D4—1944no—
143C2×D4C22×D9712yes—
144C22×C4—22no—
145C22×C4—971no—
146C22×C4—972no—
147C22×C4—972no—
148C22×C4—1942no—
149C22×C4—1942no—
150C22×C4C22×D9711yescharacteristic
151Q16—22no—
152SD16—24no—
153D8—22no—
154C2×C8—972no—
155C2×C8—1942no—
156C2×C8—1944no—
157C2×C8—1944no—
158C2×C8C22⋊D9714yescomplemented
159C2×C8C22×D9711yescharacteristic
160C2×C8C22×D9711yescharacteristic
161C4⋊C4—974no—
162C4⋊C4—1944no—
163C22⋊C4—1944no—
164C22⋊C4—1944no—
165C22⋊C4—1942no—
166C42—1942no—
167C23—22no—
168C23—972no—
169C23—1944no—
170C23—1942no—
171C23D4×D9711yescharacteristic
172Q8—22no—
173Q8—22no—
174D4—22no—
175D4—22no—
176D4—44no—
177D4—3888no—
178C2×C4—22no—
179C2×C4—44no—
180C2×C4—972no—
181C2×C4—1944no—
182C2×C4—1942no—
183C2×C4—1944no—
184C2×C4—1942no—
185C2×C4—1942no—
186C2×C4D4×D9711yescharacteristic
187C2×C4C2×C22⋊D9712yes—
188C2×C4#1309551012yes—
189C2×C4C23×D9711yescharacteristic
190C8—1942no—
191C8C2×C22⋊D9712yes—
192C8C2×C22⋊D9712yes—
193C22—22no—
194C22—42no—
195C22—44no—
196C22—974no—
197C22—1948no—
198C22—1942no—
199C22—1942no—
200C22#1569078112yes—
201C22C2×D4×D9711yescharacteristic
202C4—42no—
203C4—3884no—
204C4—1942no—
205C4#1569078112yes—
206C4C22×C22⋊D9711yescharacteristic
207C4C2×#1309551011yescharacteristic
208C2—22no—
209C2—82no—
210C2—1944no—
211C2C2×#1569078111yescharacteristic
212C2#135×D9712yes—
subgroupG/H
centerC2×C4D4×D97
commutator, cosocleC2×C4×C97C24
FrattiniC2×C4C23×D97
FittingC2×#135×C97C2
socleC22×C97C22×D4
hypercenterC2×#135D97
hypocenterC97#3050
LCSG ▷ C2×C4×C97 ▷ C2×C97 ▷ C97
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C22×C8×D97C2semidirect
2. #20261242C2semidirect
3. #20260973C2semidirect
4. #20260968C2semidirect
5. #20260961C2semidirect
6. #20260773C2semidirect
7. #20260731C2semidirect
8. C2×#135×C97C2semidirect
20. #15690744C22semidirect
21. #15690742C22semidirect
22. #15690734C22semidirect
23. #15690731C22semidirect
26. C2×Q16×C97C22semidirect
27. C2×SD16×C97C22semidirect
28. C2×D8×C97C22semidirect
57. C2×C8×C97D4semidirect
94. C2×D97#135semidirect
107. C97#3050semidirect
108. C2×#135D97semidirect
119. C2×Q16C2×D97semidirect
120. C2×SD16C2×D97semidirect
121. C2×D8C2×D97semidirect
158. C2×C8C22⋊D97semidirect

Maximal subgroups

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

subgroupindex
C22×C8×D972
#202612422
#202609732
#202609682
#202609612
#202607732
#202607312
C2×#135×C972
#305097