Finite Group Databasebeta

← #28379289

Subgroups — #28379289

Order 13952.

280 conjugacy classes of subgroups; only the 54 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 54 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×#16812759C211yesmaximal, characteristic, complemented
2C2×#16812757C211yesmaximal, characteristic, complemented
3#20295929C212yesmaximal, complemented
4#20295927C212yesmaximal, complemented
5#20295588C211yesmaximal, characteristic
6C2×#13182038—21no—
7C2×C22⋊#89137—21no—
8#16812759—21no—
9#16812759C2212yes—
10#16812757—21no—
11#16812757C2212yes—
12C22×C22⋊D109C411yescharacteristic, complemented
13C22×C22⋊D109C2211yescharacteristic
14#16812683—21no—
15#16812683C411yescharacteristic, complemented
16#16812683C2211yescharacteristic
17#16812673—21no—
18#16812673—21no—
19#16812669—21no—
20#16812669C411yescharacteristic, complemented
21#16812669C2211yescharacteristic
22C2×C22⋊C4×C109C411yescharacteristic, complemented
23C22×#89137—41no—
24#13182038—22no—
25#13182038—41no—
26C2×#11724786—41no—
27C22⋊#89137—22no—
28C22⋊#89137—41no—
29C23×D109C2×C411yescharacteristic, complemented
30C23×D109D411yescharacteristic
31C2×D436—21no—
32C2×D436—21no—
33C2×C22⋊D109—21no—
34C2×C22⋊D109—21no—
35C2×C22⋊D109—41no—
36C2×C22⋊D109—41no—
37C2×C22⋊D109—41no—
38C2×C22⋊D109—41no—
39C2×C22⋊D109C2×C411yescharacteristic
40C2×C22⋊D109D411yescharacteristic
41C2×C22⋊D109D411yescharacteristic
42C2×C22⋊D109C2311yescharacteristic
43C22×Dic109C2×C411yescharacteristic, complemented
44C22×Dic109D411yescharacteristic
45C4⋊Dic109—21no—
46C4⋊Dic109—21no—
47#13182006—21no—
48#13182006—21no—
49#13182006—21no—
50#13182006—21no—
51C24×C109C2×C411yescharacteristic
52C22×C4×C109—21no—
53C22⋊C4×C109—21no—
54C22⋊C4×C109C2×C411yescharacteristic, complemented
55C22⋊C4×C109C2×C411yescharacteristic, complemented
56C23×C109—21no—
57C23×C109—21no—
58C23×C109—81no—
59C23×C109C22⋊C411yescharacteristic
60C23×C109C22⋊C411yescharacteristic
61C23×C109C22×C411yescharacteristic
62C22×D109—21no—
63C22×D109—21no—
64C22×D109—21no—
65C22×D109—41no—
66C22×D109—81no—
67C22×D109—81no—
68C22×D109C22⋊C411yescharacteristic
69C22×D109C2×D411yescharacteristic
70C2×#89137—42no—
71C2×#89137—82no—
72C2×C4×C109—21no—
73C2×C4×C109—21no—
74C2×C4×C109—81no—
75C22⋊D109—41no—
76C22⋊D109—41no—
77C22⋊D109—41no—
78C22⋊D109—41no—
79C22⋊D109—42no—
80C22⋊D109—41no—
81C22⋊D109—41no—
82C2×Dic109—21no—
83C2×Dic109—21no—
84C2×Dic109—21no—
85C2×Dic109—41no—
86C2×Dic109C22⋊C411yescharacteristic
87C2×Dic109C2×D411yescharacteristic
88D436—81no—
89D436—81no—
90#11724786—42no—
91C22×C109—21no—
92C22×C109—21no—
93C22×C109—41no—
94C22×C109—41no—
95C22×C109—41no—
96C22×C109—81no—
97C22×C109—81no—
98C22×C109#9911yescharacteristic
99C22×C109#9911yescharacteristic
100C22×C109C2×C22⋊C411yescharacteristic
101C2×D109—21no—
102C2×D109—21no—
103C2×D109—42no—
104C2×D109—41no—
105C2×D109—41no—
106C2×D109—41no—
107C2×D109—41no—
108C2×D109—41no—
109C2×D109—81no—
110C2×D109—82no—
111#89137—82no—
112C4×C109—41no—
113C4×C109—41no—
114Dic109—21no—
115Dic109—21no—
116Dic109—41no—
117C2×C109—21no—
118C2×C109—21no—
119C2×C109—21no—
120C2×C109—41no—
121C2×C109—41no—
122C2×C109C2×#9911yescharacteristic
123D109—41no—
124D109—41no—
125D109—81no—
126#2121—1091nomaximal, Sylow
127C109#212111yescharacteristic, complemented, Sylow
128#534—1091no—
129C2×#100—1091no—
130C2×#99—1091no—
131#352—1092no—
132AΣL(1,𝔽16)—1092no—
133C22×D4—1091no—
134C22×D4—1091no—
135C2×M16—2181no—
136#123—1091no—
137#123—1091no—
138#123—2181no—
139#121—2181no—
140#121—2181no—
141#120—1091no—
142#120—1091no—
143#120—2181no—
144C2×C22⋊C4—2181no—
145C2×C22⋊C4#8913711yescharacteristic, complemented
146#100—1092no—
147#100—2181no—
148#99—1092no—
149#99—2181no—
150C24—1091no—
151C24—1091no—
152C24C2×#8913711yescharacteristic
153C2×D4—1091no—
154C2×D4—1091no—
155C2×D4—1091no—
156C2×D4—1091no—
157C2×D4—4361no—
158C2×D4—4361no—
159C2×D4—4361no—
160C2×D4—4361no—
161C2×D4—2181no—
162C2×D4—2181no—
163C2×D4—2181no—
164C2×D4—2181no—
165C22×C4—21no—
166C22×C4—1091no—
167C22×C4—1091no—
168C22×C4—4361no—
169M16—4361no—
170M16—2182no—
171C2×C8—4361no—
172C4⋊C4—2181no—
173C4⋊C4—2181no—
174C22⋊C4—21no—
175C22⋊C4—4361no—
176C22⋊C4—2181no—
177C22⋊C4—2181no—
178C22⋊C4—2181no—
179C22⋊C4—2181no—
180C22⋊C4—2182no—
181C22⋊C4C2×#8913711yescharacteristic, complemented
182C22⋊C4C2×#8913711yescharacteristic, complemented
183C23—21no—
184C23—21no—
185C23—81no—
186C23—8721no—
187C23—8721no—
188C23—1091no—
189C23—1091no—
190C23—4361no—
191C23—2181no—
192C23—2181no—
193C23—2181no—
194C23C22⋊#8913711yescharacteristic
195C23C22⋊#8913711yescharacteristic
196C23C22×#8913711yescharacteristic
197D4—8721no—
198D4—8721no—
199D4—4362no—
200D4—4361no—
201D4—4361no—
202D4—4361no—
203D4—4361no—
204D4—4361no—
205D4—4361no—
206C2×C4—21no—
207C2×C4—21no—
208C2×C4—81no—
209C2×C4—8722no—
210C2×C4—1091no—
211C2×C4—1091no—
212C2×C4—4362no—
213C2×C4—4361no—
214C2×C4—2181no—
215C2×C4—2181no—
216C2×C4—2181no—
217C8—4362no—
218C22—21no—
219C22—21no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—81no—
224C22—81no—
225C22—8721no—
226C22—8722no—
227C22—4361no—
228C22—4361no—
229C22—4361no—
230C22—4361no—
231C22—4362no—
232C22—4361no—
233C22—2181no—
234C22—2181no—
235C22#1681275811yescharacteristic
236C22#1681275811yescharacteristic
237C22C2×C22⋊#8913711yescharacteristic
238C4—41no—
239C4—41no—
240C4—8722no—
241C4—4361no—
242C4—2181no—
243C4—2181no—
244C2—21no—
245C2—21no—
246C2—21no—
247C2—41no—
248C2—41no—
249C2—8721no—
250C2—4361no—
251C2—4361no—
252C2C2×#1681275811yescharacteristic
subgroupG/H
centerC2C2×#16812758
commutatorC23×C109C22×C4
FrattiniC23C22×#89137
FittingC2×C22⋊C4×C109C4
socleC2×C109C2×#99
prime radical#20295588C2
cosocleC2×C22⋊D109C23
hypercenterC2×C22⋊C4#89137
hypocenterC109#2121
LCSG ▷ C23×C109 ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ C2×C22⋊C4

Complements

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

NH ≅ G/Ntype
1. C2×#16812759C2semidirect
2. C2×#16812757C2semidirect
3. #20295929C2semidirect
4. #20295927C2semidirect
12. C22×C22⋊D109C4semidirect
15. #16812683C4semidirect
20. #16812669C4semidirect
22. C2×C22⋊C4×C109C4semidirect
29. C23×D109C2×C4semidirect
43. C22×Dic109C2×C4semidirect
54. C22⋊C4×C109C2×C4semidirect
55. C22⋊C4×C109C2×C4semidirect
127. C109#2121semidirect
145. C2×C22⋊C4#89137semidirect
181. C22⋊C4C2×#89137semidirect
182. C22⋊C4C2×#89137semidirect

Maximal subgroups

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

subgroupindex
C2×#168127592
C2×#168127572
#202959292
#202959272
#202955882
#2121109