Finite Group Databasebeta

← #28400101

Subgroups — #28400101

Order 14464.

458 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 141 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
1C2×#16813653C211yesmaximal, characteristic
2#20299172C211yesmaximal, characteristic
3#20299076C211yesmaximal, characteristic, complemented
4#20298822C214yesmaximal, complemented
5#20298806C214yesmaximal, complemented
6#20298624C211yesmaximal, characteristic, complemented
7#20298607C211yesmaximal, characteristic, complemented
8#20298595C211yesmaximal, characteristic
9C22×Q16×C113C211yesmaximal, characteristic, complemented
10C2×#14268647C2211yescharacteristic
11#16813653—24no—
12C2×#14268637C2212yes—
13C2×#14268634C2212yes—
14C2×#14268631C2211yescharacteristic
15#16813620C2212yes—
16#16813611C2212yes—
17#16813605—22no—
18#16813598—22no—
19Q8×Dic113—22no—
20Q8×Dic113C2214yes—
21#16813581—22no—
22#16813581C2211yescharacteristic
23#16813581C2211yescharacteristic
24#16813551C2214yes—
25#16813547C2212yes—
26#16813546C2214yes—
27#16813544C2212yes—
28C22×Q8×C113C2211yescharacteristic
29C22×Q8×C113C2211yescharacteristic
30C2×Q16×C113—24no—
31C2×Q16×C113C2214yes—
32C22×C8×C113C2211yescharacteristic
33#14268647—22no—
34#14268647—22no—
35C2×Dic226C2311yescharacteristic
36C2×D452C2311yescharacteristic
37C2×C22⋊D113—21no—
38C2×C4×D113—21no—
39#14268637—24no—
40#14268634—24no—
41#14268631—22no—
42C2×#11744751C2312yes—
43C4⋊Dic113—24no—
44C4⋊Dic113D412yes—
45C4⋊Dic113C2312yes—
46#14268626—24no—
47C22⋊Dic113—21no—
48C22⋊Dic113D412yes—
49#14268624—24no—
50C4×Dic113—21no—
51C4×Dic113—24no—
52C4×Dic113C2312yes—
53C2×Q8×C113—21no—
54C2×Q8×C113—21no—
55C2×Q8×C113—22no—
56C2×Q8×C113—22no—
57C2×Q8×C113—22no—
58C2×Q8×C113D414yes—
59C2×Q8×C113C2312yes—
60C2×Q8×C113C2312yes—
61C22×C4×C113—21no—
62C22×C4×C113—21no—
63C22×C4×C113C2311yescharacteristic
64Q16×C113—28no—
65C2×C8×C113—22no—
66C2×C8×C113C2312yes—
67C23×C113C2×D411yescharacteristic
68C22×D113—21no—
69Q8×C113—24no—
70Q8×C113—22no—
71Q8×C113—22no—
72Q8×C113—22no—
73Q8×C113—42no—
74Q8×C113C2×D414yes—
75C2×C4×C113—24no—
76C2×C4×C113—22no—
77C2×C4×C113—22no—
78C2×C4×C113—42no—
79C2×C4×C113C2×D411yescharacteristic
80C2×C4×C113C2×D412yes—
81C2×C4×C113#4112yes—
82C2×C4×C113C2411yescharacteristic
83C22⋊D113—42no—
84C2×Dic113—22no—
85C2×Dic113—21no—
86C2×Dic113C2×D414yes—
87D452—21no—
88D452—21no—
89C4×D113—42no—
90Dic226—21no—
91Dic226—21no—
92C8×C113—22no—
93#11744751—22no—
94C22×C113—22no—
95C22×C113#12112yes—
96C22×C113C22×D411yescharacteristic
97C2×D113—41no—
98C2×D113—42no—
99C4×C113—24no—
100C4×C113—42no—
101C4×C113#12112yes—
102C4×C113C22×D411yescharacteristic
103C4×C113C2×#4111yescharacteristic
104Dic113—24no—
105Dic113—41no—
106Dic113—42no—
107C2×C113—22no—
108C2×C113C2×#12111yescharacteristic
109C2×C113#57812yes—
110D113—81no—
111#3044—1131nomaximal, Sylow
112C113#304411yescharacteristic, complemented, Sylow
113C2×#137—1131no—
114C22×Q16D11311yescharacteristic, complemented
115#536—1131no—
116#519—1131no—
117#464—1134no—
118#462—1134no—
119#424—1131no—
120#419—1131no—
121#416—1131no—
122C2×#41—1131no—
123C22×Q8C2×D11311yescharacteristic
124C22×Q8C2×D11311yescharacteristic
125#137—2264no—
126C2×Q16—24no—
127C2×Q16—1132no—
128C2×Q16C2×D11314yes—
129C2×SD16—1132no—
130C2×M16—1131no—
131C22×C8C2×D11311yescharacteristic
132C4⋊Q8—1132no—
133#124—1132no—
134#123—2262no—
135C22⋊Q8—2262no—
136C4×Q8—1134no—
137C4×Q8—2262no—
138#117—1131no—
139#117—1131no—
140#117—2262no—
141C4⋊C8—1134no—
142#103—1134no—
143#103—1132no—
144#102—1132no—
145#41—2262no—
146#41—2262no—
147C2×Q8—21no—
148C2×Q8—21no—
149C2×Q8—22no—
150C2×Q8—22no—
151C2×Q8—22no—
152C2×Q8—1131no—
153C2×Q8C22⋊D11314yes—
154C2×Q8C22×D11312yes—
155C2×Q8C22×D11312yes—
156C2×D4—1131no—
157C2×D4—2261no—
158C22×C4—21no—
159C22×C4—21no—
160C22×C4—2261no—
161C22×C4C22×D11311yescharacteristic
162Q16—28no—
163Q16—2264no—
164SD16—2264no—
165M16—2262no—
166C2×C8—22no—
167C2×C8—1132no—
168C2×C8C22×D11312yes—
169C4⋊C4—1132no—
170C4⋊C4—1132no—
171C4⋊C4—2264no—
172C4⋊C4—2264no—
173C22⋊C4—1132no—
174C22⋊C4—2264no—
175C22⋊C4—2261no—
176C42—1132no—
177C42—2264no—
178C42—2261no—
179C23—2261no—
180C23D4×D11311yescharacteristic
181Q8—24no—
182Q8—22no—
183Q8—22no—
184Q8—22no—
185Q8—42no—
186Q8—2261no—
187Q8—2261no—
188Q8C2×C22⋊D11314yes—
189D4—4522no—
190D4—2261no—
191D4—2261no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—42no—
196C2×C4—1134no—
197C2×C4—4522no—
198C2×C4—2262no—
199C2×C4—2261no—
200C2×C4D4×D11311yescharacteristic
201C2×C4C2×C22⋊D11312yes—
202C2×C4#1426864612yes—
203C2×C4C23×D11311yescharacteristic
204C8—22no—
205C8—2262no—
206C22—22no—
207C22—4521no—
208C22—4522no—
209C22#1681359412yes—
210C22C2×D4×D11311yescharacteristic
211C4—24no—
212C4—42no—
213C4—4521no—
214C4—4522no—
215C4—2264no—
216C4#1681359412yes—
217C4C22×C22⋊D11311yescharacteristic
218C4C2×#1426864611yescharacteristic
219C2—22no—
220C2—9041no—
221C2C2×#1681359411yescharacteristic
222C2#2029941112yes—
subgroupG/H
centerC22C2×D4×D113
commutator, cosocleC2×C4×C113C24
FrattiniC2×C4C23×D113
FittingC22×Q16×C113C2
socleC22×C113C22×D4
prime radicalC2×#14268647C22
hypercenterC22×Q16D113
hypocenterC113#3044
LCSG ▷ C2×C4×C113 ▷ C2×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
3. #20299076C2semidirect
4. #20298822C2semidirect
5. #20298806C2semidirect
6. #20298624C2semidirect
7. #20298607C2semidirect
9. C22×Q16×C113C2semidirect
112. C113#3044semidirect
114. C22×Q16D113semidirect

Maximal subgroups

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

subgroupindex
C2×#168136532
#202991722
#202990762
#202988222
#202988062
#202986242
#202986072
#202985952
C22×Q16×C1132
#3044113