Finite Group Databasebeta

← #28361990

Subgroups — #28361990

Order 13952.

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#20295651C211yesmaximal, characteristic
2#20295184C211yesmaximal, characteristic, complemented
3#20295181C211yesmaximal, characteristic, complemented
4#20294939C211yesmaximal, characteristic
5#20294762C211yesmaximal, characteristic, complemented
6#20294749C211yesmaximal, characteristic
7#328×C109C211yesmaximal, characteristic, complemented
8C22×Dic218C2211yescharacteristic
9C2×Dic436—21no—
10C2×#13182019—21no—
11#16812679—21no—
12#16812678C2211yescharacteristic
13#16812677—21no—
14#16812677—21no—
15#16812676—21no—
16#16812675—21no—
17C2×#13182006—21no—
18C2×#13182006C2211yescharacteristic
19C22⋊C4×D109—21no—
20#16812624—21no—
21#16812622—21no—
22C22⋊#11724784C2211yescharacteristic
23#16812606—21no—
24C22⋊Q8×C109C2211yescharacteristic
25C22⋊C8×C109C2211yescharacteristic
26#95×C109C2211yescharacteristic
27C23×D109D411yescharacteristic
28C2×Dic218—21no—
29C2×Dic218—21no—
30C2×Dic218—21no—
31C2×Dic218—42no—
32C2×Dic218D411yescharacteristic
33C22×Dic109—21no—
34C22×Dic109D411yescharacteristic
35C2×C4×D109—21no—
36C2×C4×D109—21no—
37Dic436—42no—
38#13182019—42no—
39C2×#11724784—21no—
40C4⋊Dic109—21no—
41C4⋊Dic109D411yescharacteristic
42#13182008—21no—
43#13182008—21no—
44#13182008—21no—
45C22⋊Dic109—21no—
46#13182006—21no—
47#13182006—21no—
48#13182006—21no—
49#13182006—21no—
50#13182006—42no—
51C2×Q8×C109D411yescharacteristic
52C22×C4×C109—21no—
53C22×C4×C109C2311yescharacteristic
54C2×C8×C109—21no—
55C4⋊C4×C109—21no—
56C4⋊C4×C109D411yescharacteristic
57C22⋊C4×C109—21no—
58C23×C109C2×D411yescharacteristic
59C22×D109—21no—
60C22×D109—21no—
61C22×D109—41no—
62C22×D109—41no—
63C22×D109Q1612yescomplemented
64Q8×C109—41no—
65C2×C4×C109—21no—
66C2×C4×C109—21no—
67C2×C4×C109—41no—
68C2×C4×C109—41no—
69C2×C4×C109—41no—
70C2×C4×C109—41no—
71C2×C4×C109C2×D411yescharacteristic
72C2×C4×C109C2×D411yescharacteristic
73C2×Dic109—22no—
74C2×Dic109—21no—
75C2×Dic109—21no—
76C2×Dic109—21no—
77C2×Dic109—41no—
78C2×Dic109—41no—
79C2×Dic109—41no—
80C4×D109—81no—
81C4×D109—81no—
82Dic218—22no—
83Dic218—22no—
84Dic218—41no—
85Dic218—41no—
86Dic218—41no—
87Dic218—81no—
88C8×C109—41no—
89#11724784—41no—
90C22×C109—21no—
91C22×C109—21no—
92C22×C109#12011yescharacteristic
93C22×C109C2×Q1611yescharacteristic
94C22×C109#13711yescharacteristic
95C2×D109—22no—
96C2×D109—41no—
97C2×D109—41no—
98C2×D109—42no—
99C2×D109—81no—
100C4×C109—21no—
101C4×C109—21no—
102C4×C109—41no—
103C4×C109—41no—
104C4×C109—81no—
105Dic109—41no—
106Dic109—41no—
107Dic109—42no—
108Dic109—41no—
109C2×C109—21no—
110C2×C109—21no—
111C2×C109#45111yescharacteristic
112C2×C109#45511yescharacteristic
113C2×C109B(4,𝔽2)11yescharacteristic
114D109—42no—
115#1602—1091nomaximal, Sylow
116C109#160211yescharacteristic, complemented, Sylow
117#543—1091no—
118#451—1091no—
119#451—1091no—
120#393—1091no—
121#328—1091no—
122#328D10911yescharacteristic, complemented
123#323—1091no—
124C22×Q8—1091no—
125C2×Q16—2181no—
126C2×Q16—2181no—
127C22⋊Q8—1091no—
128C22⋊Q8—2181no—
129C22⋊Q8—2181no—
130C22⋊Q8—2181no—
131C22⋊Q8—2181no—
132C22⋊Q8—2181no—
133C22⋊Q8C2×D10911yescharacteristic
134C2×C22⋊C4—1091no—
135C2×C22⋊C4—2181no—
136C2×C22⋊C4—2181no—
137#103—2181no—
138#103—2181no—
139C22⋊C8—1091no—
140C22⋊C8C2×D10911yescharacteristic
141#95—2181no—
142#95C2×D10911yescharacteristic
143C24—1091no—
144C2×Q8—1091no—
145C2×Q8—4362no—
146C2×Q8—2181no—
147C2×Q8—2181no—
148C2×Q8—2181no—
149C2×Q8D43611yescharacteristic
150C22×C4—21no—
151C22×C4—1091no—
152C22×C4—2181no—
153C22×C4—2181no—
154C22×C4—2181no—
155C22×C4C22×D10911yescharacteristic
156Q16—4362no—
157Q16—4362no—
158C2×C8—21no—
159C2×C8—2181no—
160C4⋊C4—21no—
161C4⋊C4—1091no—
162C4⋊C4—2181no—
163C4⋊C4—2181no—
164C4⋊C4—2181no—
165C4⋊C4—2181no—
166C4⋊C4D43611yescharacteristic
167C22⋊C4—21no—
168C22⋊C4—4362no—
169C22⋊C4—2181no—
170C22⋊C4—2181no—
171C22⋊C4—2181no—
172C22⋊C4—2181no—
173C22⋊C4—2181no—
174C23—1092no—
175C23—4361no—
176C23—4361no—
177C23—2181no—
178C23—2181no—
179C23D4×D10911yescharacteristic
180Q8—41no—
181Q8—8721no—
182Q8—4361no—
183Q8—4361no—
184Q8—4361no—
185Q8—2182no—
186Q8—2182no—
187C2×C4—21no—
188C2×C4—21no—
189C2×C4—41no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—8721no—
194C2×C4—8721no—
195C2×C4—4361no—
196C2×C4—4361no—
197C2×C4—4361no—
198C2×C4—2182no—
199C2×C4—2181no—
200C2×C4—2181no—
201C2×C4—2181no—
202C2×C4D4×D10911yescharacteristic
203C2×C4C2×D43611yescharacteristic
204C8—41no—
205C8—4361no—
206C22—21no—
207C22—21no—
208C22—8721no—
209C22—4362no—
210C22—4361no—
211C22—4361no—
212C22—2182no—
213C22#1681266911yescharacteristic
214C22Q16×D10911yescharacteristic
215C22#1681272811yescharacteristic
216C4—21no—
217C4—21no—
218C4—41no—
219C4—41no—
220C4—81no—
221C4—4361no—
222C4—4361no—
223C4—4361no—
224C4—4362no—
225C2—21no—
226C2—21no—
227C2—4362no—
228C2#2029518311yescharacteristic
229C2#2029520511yescharacteristic
230C2#2029521411yescharacteristic
subgroupG/H
centerC22#16812669
commutator, cosocleC22×C4×C109C23
FrattiniC22×C4C22×D109
Fitting#328×C109C2
socleC22×C109#120
prime radicalC23×D109D4
hypercenter#328D109
hypocenterC109#1602
LCSG ▷ C22×C4×C109 ▷ C22×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #328

Complements

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

NH ≅ G/Ntype
2. #20295184C2semidirect
3. #20295181C2semidirect
5. #20294762C2semidirect
7. #328×C109C2semidirect
63. C22×D109Q16semidirect
116. C109#1602semidirect
122. #328D109semidirect

Maximal subgroups

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

subgroupindex
#202956512
#202951842
#202951812
#202949392
#202947622
#202947492
#328×C1092
#1602109