Finite Group Databasebeta

← #28389532

Subgroups — #28389532

Order 14464.

256 conjugacy classes of subgroups; only the 51 normal subgroups are shown.

normal subgroups only

This group has 256 conjugacy classes of subgroups — too many to draw. Only its 51 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#20299223C211yesmaximal, characteristic
2#20298756C211yesmaximal, characteristic, complemented
3#20298753C211yesmaximal, characteristic, complemented
4#20298511C211yesmaximal, characteristic
5#20298334C211yesmaximal, characteristic, complemented
6#20298321C211yesmaximal, characteristic
7#328×C113C211yesmaximal, characteristic, complemented
8C22×Dic226C2211yescharacteristic
9C2×Dic452—21no—
10C2×#14268637—21no—
11#16813602—21no—
12#16813601C2211yescharacteristic
13#16813600—21no—
14#16813600—21no—
15#16813599—21no—
16#16813598—21no—
17C2×#14268624—21no—
18C2×#14268624C2211yescharacteristic
19C22⋊C4×D113—21no—
20#16813547—21no—
21#16813545—21no—
22C22⋊#11744751C2211yescharacteristic
23#16813529—21no—
24C22⋊Q8×C113C2211yescharacteristic
25C22⋊C8×C113C2211yescharacteristic
26#95×C113C2211yescharacteristic
27C23×D113D411yescharacteristic
28C2×Dic226—21no—
29C2×Dic226—21no—
30C2×Dic226—21no—
31C2×Dic226—42no—
32C2×Dic226D411yescharacteristic
33C22×Dic113—21no—
34C22×Dic113D411yescharacteristic
35C2×C4×D113—21no—
36C2×C4×D113—21no—
37Dic452—42no—
38#14268637—42no—
39C2×#11744751—21no—
40C4⋊Dic113—21no—
41C4⋊Dic113D411yescharacteristic
42#14268626—21no—
43#14268626—21no—
44#14268626—21no—
45C22⋊Dic113—21no—
46#14268624—21no—
47#14268624—21no—
48#14268624—21no—
49#14268624—21no—
50#14268624—42no—
51C2×Q8×C113D411yescharacteristic
52C22×C4×C113—21no—
53C22×C4×C113C2311yescharacteristic
54C2×C8×C113—21no—
55C4⋊C4×C113—21no—
56C4⋊C4×C113D411yescharacteristic
57C22⋊C4×C113—21no—
58C23×C113C2×D411yescharacteristic
59C22×D113—21no—
60C22×D113—21no—
61C22×D113—41no—
62C22×D113—41no—
63C22×D113Q1612yescomplemented
64Q8×C113—41no—
65C2×C4×C113—21no—
66C2×C4×C113—21no—
67C2×C4×C113—41no—
68C2×C4×C113—41no—
69C2×C4×C113—41no—
70C2×C4×C113—41no—
71C2×C4×C113C2×D411yescharacteristic
72C2×C4×C113C2×D411yescharacteristic
73C2×Dic113—22no—
74C2×Dic113—21no—
75C2×Dic113—21no—
76C2×Dic113—21no—
77C2×Dic113—41no—
78C2×Dic113—41no—
79C2×Dic113—41no—
80C4×D113—81no—
81C4×D113—81no—
82Dic226—22no—
83Dic226—22no—
84Dic226—41no—
85Dic226—41no—
86Dic226—41no—
87Dic226—81no—
88C8×C113—41no—
89#11744751—41no—
90C22×C113—21no—
91C22×C113—21no—
92C22×C113#12011yescharacteristic
93C22×C113C2×Q1611yescharacteristic
94C22×C113#13711yescharacteristic
95C2×D113—22no—
96C2×D113—41no—
97C2×D113—41no—
98C2×D113—42no—
99C2×D113—81no—
100C4×C113—21no—
101C4×C113—21no—
102C4×C113—41no—
103C4×C113—41no—
104C4×C113—81no—
105Dic113—41no—
106Dic113—41no—
107Dic113—42no—
108Dic113—41no—
109C2×C113—21no—
110C2×C113—21no—
111C2×C113#45111yescharacteristic
112C2×C113#45511yescharacteristic
113C2×C113B(4,𝔽2)11yescharacteristic
114D113—42no—
115#1602—1131nomaximal, Sylow
116C113#160211yescharacteristic, complemented, Sylow
117#543—1131no—
118#451—1131no—
119#451—1131no—
120#393—1131no—
121#328—1131no—
122#328D11311yescharacteristic, complemented
123#323—1131no—
124C22×Q8—1131no—
125C2×Q16—2261no—
126C2×Q16—2261no—
127C22⋊Q8—1131no—
128C22⋊Q8—2261no—
129C22⋊Q8—2261no—
130C22⋊Q8—2261no—
131C22⋊Q8—2261no—
132C22⋊Q8—2261no—
133C22⋊Q8C2×D11311yescharacteristic
134C2×C22⋊C4—1131no—
135C2×C22⋊C4—2261no—
136C2×C22⋊C4—2261no—
137#103—2261no—
138#103—2261no—
139C22⋊C8—1131no—
140C22⋊C8C2×D11311yescharacteristic
141#95—2261no—
142#95C2×D11311yescharacteristic
143C24—1131no—
144C2×Q8—1131no—
145C2×Q8—4522no—
146C2×Q8—2261no—
147C2×Q8—2261no—
148C2×Q8—2261no—
149C2×Q8D45211yescharacteristic
150C22×C4—21no—
151C22×C4—1131no—
152C22×C4—2261no—
153C22×C4—2261no—
154C22×C4—2261no—
155C22×C4C22×D11311yescharacteristic
156Q16—4522no—
157Q16—4522no—
158C2×C8—21no—
159C2×C8—2261no—
160C4⋊C4—21no—
161C4⋊C4—1131no—
162C4⋊C4—2261no—
163C4⋊C4—2261no—
164C4⋊C4—2261no—
165C4⋊C4—2261no—
166C4⋊C4D45211yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—4522no—
169C22⋊C4—2261no—
170C22⋊C4—2261no—
171C22⋊C4—2261no—
172C22⋊C4—2261no—
173C22⋊C4—2261no—
174C23—1132no—
175C23—4521no—
176C23—4521no—
177C23—2261no—
178C23—2261no—
179C23D4×D11311yescharacteristic
180Q8—41no—
181Q8—9041no—
182Q8—4521no—
183Q8—4521no—
184Q8—4521no—
185Q8—2262no—
186Q8—2262no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—9041no—
194C2×C4—9041no—
195C2×C4—4521no—
196C2×C4—4521no—
197C2×C4—4521no—
198C2×C4—2262no—
199C2×C4—2261no—
200C2×C4—2261no—
201C2×C4—2261no—
202C2×C4D4×D11311yescharacteristic
203C2×C4C2×D45211yescharacteristic
204C8—41no—
205C8—4521no—
206C22—21no—
207C22—21no—
208C22—9041no—
209C22—4522no—
210C22—4521no—
211C22—4521no—
212C22—2262no—
213C22#1681359211yescharacteristic
214C22Q16×D11311yescharacteristic
215C22#1681365111yescharacteristic
216C4—21no—
217C4—21no—
218C4—41no—
219C4—41no—
220C4—81no—
221C4—4521no—
222C4—4521no—
223C4—4521no—
224C4—4522no—
225C2—21no—
226C2—21no—
227C2—4522no—
228C2#2029875511yescharacteristic
229C2#2029877711yescharacteristic
230C2#2029878611yescharacteristic
subgroupG/H
centerC22#16813592
commutator, cosocleC22×C4×C113C23
FrattiniC22×C4C22×D113
Fitting#328×C113C2
socleC22×C113#120
prime radicalC23×D113D4
hypercenter#328D113
hypocenterC113#1602
LCSG ▷ C22×C4×C113 ▷ C22×C113 ▷ C113
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #20298756C2semidirect
3. #20298753C2semidirect
5. #20298334C2semidirect
7. #328×C113C2semidirect
63. C22×D113Q16semidirect
116. C113#1602semidirect
122. #328D113semidirect

Maximal subgroups

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

subgroupindex
#202992232
#202987562
#202987532
#202985112
#202983342
#202983212
#328×C1132
#1602113