Finite Group Databasebeta

← #27140687

Subgroups — #27140687

Order 12416.

308 conjugacy classes of subgroups; only the 53 normal subgroups are shown.

normal subgroups only

This group has 308 conjugacy classes of subgroups — too many to draw. Only its 53 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#20261297C211yesmaximal, characteristic
2#20260874C212yesmaximal, complemented
3#20260468C212yesmaximal, complemented
4#20260459C211yesmaximal, characteristic, complemented
5#393×C97C211yesmaximal, characteristic, complemented
6C22×C22⋊D97—21no—
7C2×#13095498—22no—
8#15690794—22no—
9#15690784—21no—
10#15690784C2212yes—
11#15690780—22no—
12#15690734—22no—
13C22⋊#11715895C2212yescomplemented
14C22×Q8×C97C2211yescharacteristic, complemented
15C2×C22⋊C4×C97—21no—
16C2×C22⋊C4×C97C2211yescharacteristic
17#95×C97—21no—
18#95×C97C2211yescharacteristic, complemented
19C23×D97—21no—
20C2×D388—21no—
21C2×D388D412yes—
22C2×C22⋊D97—22no—
23C2×C22⋊D97—22no—
24C2×C22⋊D97—42no—
25C2×C22⋊D97—42no—
26C22×Dic97—21no—
27#13095498—44no—
28C2×#11715895—22no—
29C4⋊Dic97—21no—
30C4⋊Dic97D412yes—
31#13095488—22no—
32#13095488—22no—
33C24×C97D411yescharacteristic
34C2×Q8×C97—22no—
35C2×Q8×C97—42no—
36C22×C4×C97—21no—
37C22×C4×C97—21no—
38C22×C4×C97D411yescharacteristic
39C22×C4×C97C2311yescharacteristic
40C22⋊C4×C97—22no—
41C22⋊C4×C97—42no—
42C23×C97—21no—
43C23×C97—21no—
44C23×C97—41no—
45C23×C97—41no—
46C23×C97SD1612yescomplemented
47C23×C97C2×D411yescharacteristic
48C22×D97—22no—
49C22×D97—42no—
50C22×D97—81no—
51C22×D97—81no—
52Q8×C97—24no—
53Q8×C97—42no—
54C2×C4×C97—22no—
55C2×C4×C97—21no—
56C2×C4×C97—41no—
57C2×C4×C97—41no—
58C2×C4×C97—41no—
59C2×C4×C97—41no—
60C2×C4×C97—41no—
61C2×C4×C97—41no—
62C2×C4×C97—41no—
63C2×C4×C97C2×D412yes—
64C22⋊D97—44no—
65C22⋊D97—42no—
66C22⋊D97—42no—
67C2×Dic97—22no—
68C2×Dic97—42no—
69D388—42no—
70D388—81no—
71#11715895—42no—
72C22×C97—21no—
73C22×C97—21no—
74C22×C97—22no—
75C22×C97—42no—
76C22×C97—41no—
77C22×C97—41no—
78C22×C97—81no—
79C22×C97#12011yescharacteristic
80C22×C97C2×SD1611yescharacteristic
81C22×C97AGL(1,ℤ/8)11yescharacteristic
82C2×D97—42no—
83C2×D97—42no—
84C2×D97—42no—
85C2×D97—84no—
86C4×C97—22no—
87C4×C97—41no—
88C4×C97—42no—
89C4×C97—81no—
90Dic97—42no—
91C2×C97—21no—
92C2×C97—21no—
93C2×C97—42no—
94C2×C97#45011yescharacteristic
95C2×C97#45511yescharacteristic
96C2×C97B(4,𝔽2)11yescharacteristic
97D97—82no—
98#1601—971nomaximal, Sylow
99C97#160111yescharacteristic, complemented, Sylow
100#534—971no—
101#448—972no—
102#393D9711yescharacteristic, complemented
103#327—972no—
104#323—971no—
105C22×Q8C2×D9711yescharacteristic, complemented
106C22×D4—1941no—
107C2×SD16—1942no—
108#123—1942no—
109#121—972no—
110#121—1941no—
111#120—1942no—
112C2×C22⋊C4—21no—
113C2×C22⋊C4C2×D9711yescharacteristic
114#103—1942no—
115C22⋊C8—972no—
116#95—21no—
117#95C2×D9711yescharacteristic, complemented
118C24—1941no—
119C24C22⋊D9711yescharacteristic
120C2×Q8—22no—
121C2×Q8—42no—
122C2×D4—972no—
123C2×D4—3882no—
124C2×D4—3882no—
125C2×D4—1941no—
126C2×D4—1942no—
127C2×D4—1942no—
128C22×C4—21no—
129C22×C4—21no—
130C22×C4—1941no—
131C22×C4C22⋊D9711yescharacteristic
132C22×C4C22×D9711yescharacteristic
133SD16—3884no—
134C2×C8—1942no—
135C4⋊C4—972no—
136C4⋊C4—1941no—
137C22⋊C4—22no—
138C22⋊C4—42no—
139C22⋊C4—1942no—
140C22⋊C4—1942no—
141C23—21no—
142C23—21no—
143C23—41no—
144C23—41no—
145C23—7761no—
146C23—7761no—
147C23—3882no—
148C23—1942no—
149C23#1309549812yescomplemented
150C23C2×C22⋊D9711yescharacteristic
151Q8—24no—
152Q8—42no—
153D4—7761no—
154D4—3882no—
155D4—3884no—
156D4—3882no—
157D4—3882no—
158C2×C4—22no—
159C2×C4—21no—
160C2×C4—41no—
161C2×C4—41no—
162C2×C4—41no—
163C2×C4—41no—
164C2×C4—41no—
165C2×C4—41no—
166C2×C4—41no—
167C2×C4—3882no—
168C2×C4—1942no—
169C2×C4D4×D9712yes—
170C8—3882no—
171C22—21no—
172C22—21no—
173C22—22no—
174C22—42no—
175C22—41no—
176C22—41no—
177C22—81no—
178C22—7764no—
179C22—3882no—
180C22—3882no—
181C22—3882no—
182C22#1569077811yescharacteristic
183C22C2×#1309549811yescharacteristic
184C22#1569083511yescharacteristic
185C4—22no—
186C4—41no—
187C4—42no—
188C4—81no—
189C4—3882no—
190C2—21no—
191C2—21no—
192C2—42no—
193C2—7762no—
194C2#2026088511yescharacteristic
195C2#2026091211yescharacteristic
196C2#2026092111yescharacteristic
subgroupG/H
centerC22#15690778
commutator, cosocleC22×C4×C97C23
FrattiniC22×C4C22×D97
Fitting#393×C97C2
socleC22×C97#120
prime radical#20261297C2
hypercenter#393D97
hypocenterC97#1601
LCSG ▷ C22×C4×C97 ▷ C22×C97 ▷ C97
UCSC1 ◁ C22 ◁ C22×C4 ◁ #393

Complements

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

NH ≅ G/Ntype
2. #20260874C2semidirect
3. #20260468C2semidirect
4. #20260459C2semidirect
5. #393×C97C2semidirect
13. C22⋊#11715895C22semidirect
14. C22×Q8×C97C22semidirect
18. #95×C97C22semidirect
46. C23×C97SD16semidirect
99. C97#1601semidirect
102. #393D97semidirect
105. C22×Q8C2×D97semidirect
117. #95C2×D97semidirect
149. C23#13095498semidirect

Maximal subgroups

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

subgroupindex
#202612972
#202608742
#202604682
#202604592
#393×C972
#160197