Finite Group Databasebeta

← #28379287

Subgroups — #28379287

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#20295928C212yesmaximal, complemented
4#20295926C212yesmaximal, complemented
5#20295585C211yesmaximal, characteristic
6C2×#13182038—21no—
7C2×C22⋊#89137—21no—
8#16812759—21no—
9#16812759C2212yes—
10#16812757—21no—
11#16812757C2212yes—
12C22×C22⋊D109C2211yescharacteristic
13#16812682C411yescharacteristic, complemented
14#16812682C2211yescharacteristic
15#16812680—21no—
16#16812671—21no—
17#16812671—21no—
18#16812668C411yescharacteristic, complemented
19#16812668C2211yescharacteristic
20#16812667—21no—
21C2×C22⋊Dic109C411yescharacteristic, complemented
22C22×D4×C109C411yescharacteristic, complemented
23C22×#89137—41no—
24#13182038—22no—
25#13182038—41no—
26C2×#11724786—41no—
27C22⋊#89137—22no—
28C22⋊#89137—41no—
29C23×D109D411yescharacteristic
30C2×C22⋊D109—21no—
31C2×C22⋊D109—21no—
32C2×C22⋊D109—41no—
33C2×C22⋊D109—41no—
34C2×C22⋊D109C2×C411yescharacteristic
35C2×C22⋊D109D411yescharacteristic
36C2×C22⋊D109D411yescharacteristic
37C2×C22⋊D109C2311yescharacteristic
38C22×Dic109—21no—
39C22×Dic109D411yescharacteristic
40#13182008—21no—
41#13182008—21no—
42C22⋊Dic109—21no—
43C22⋊Dic109—21no—
44C22⋊Dic109—21no—
45C22⋊Dic109C2×C411yescharacteristic, complemented
46C22⋊Dic109C2×C411yescharacteristic, complemented
47#13182006—21no—
48#13182006—21no—
49C24×C109C2×C411yescharacteristic
50C24×C109C2×C411yescharacteristic, complemented
51C2×D4×C109—21no—
52C2×D4×C109—21no—
53C2×D4×C109—41no—
54C2×D4×C109—41no—
55C22×C4×C109C2×C411yescharacteristic, complemented
56C23×C109—21no—
57C23×C109—21no—
58C23×C109—21no—
59C23×C109—41no—
60C23×C109—81no—
61C23×C109—81no—
62C23×C109C22⋊C411yescharacteristic
63C23×C109C22⋊C411yescharacteristic
64C23×C109C22×C411yescharacteristic
65C22×D109—21no—
66C22×D109—21no—
67C22×D109—81no—
68C22×D109C22⋊C411yescharacteristic
69C22×D109C2×D411yescharacteristic
70C2×#89137—42no—
71C2×#89137—82no—
72D4×C109—42no—
73D4×C109—41no—
74D4×C109—41no—
75C2×C4×C109—21no—
76C2×C4×C109—41no—
77C22⋊D109—41no—
78C22⋊D109—41no—
79C22⋊D109—41no—
80C22⋊D109—41no—
81C22⋊D109—81no—
82C22⋊D109—81no—
83C2×Dic109—21no—
84C2×Dic109—21no—
85C2×Dic109—21no—
86C2×Dic109—21no—
87C2×Dic109—81no—
88C2×Dic109C22⋊C411yescharacteristic
89C2×Dic109C2×D411yescharacteristic
90#11724786—42no—
91C22×C109—21no—
92C22×C109—21no—
93C22×C109—41no—
94C22×C109—41no—
95C22×C109—41no—
96C22×C109—41no—
97C22×C109—41no—
98C22×C109—41no—
99C22×C109—81no—
100C22×C109—81no—
101C22×C109—82no—
102C22×C109#9911yescharacteristic
103C22×C109#9911yescharacteristic
104C22×C109C2×C22⋊C411yescharacteristic
105C2×D109—21no—
106C2×D109—21no—
107C2×D109—42no—
108C2×D109—41no—
109C2×D109—41no—
110C2×D109—81no—
111#89137—82no—
112C4×C109—41no—
113Dic109—21no—
114Dic109—21no—
115Dic109—41no—
116Dic109—41no—
117C2×C109—21no—
118C2×C109—21no—
119C2×C109—21no—
120C2×C109—41no—
121C2×C109—41no—
122C2×C109—81no—
123C2×C109C2×#9911yescharacteristic
124D109—41no—
125D109—41no—
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#8913711yescharacteristic, complemented
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—1091no—
145C2×C22⋊C4—2181no—
146#100—1092no—
147#100—2181no—
148#99—1092no—
149#99—2181no—
150C24—1091no—
151C24C2×#8913711yescharacteristic
152C24C2×#8913711yescharacteristic, complemented
153C2×D4—21no—
154C2×D4—21no—
155C2×D4—41no—
156C2×D4—41no—
157C2×D4—1091no—
158C2×D4—1091no—
159C2×D4—1091no—
160C2×D4—1091no—
161C2×D4—4361no—
162C2×D4—4361no—
163C2×D4—2181no—
164C2×D4—2181no—
165C22×C4—1091no—
166C22×C4—4361no—
167C22×C4—2181no—
168C22×C4C2×#8913711yescharacteristic, complemented
169M16—4361no—
170M16—2182no—
171C2×C8—4361no—
172C4⋊C4—2181no—
173C4⋊C4—2181no—
174C22⋊C4—1091no—
175C22⋊C4—1091no—
176C22⋊C4—4361no—
177C22⋊C4—2181no—
178C22⋊C4—2181no—
179C22⋊C4—2181no—
180C22⋊C4—2181no—
181C22⋊C4—2182no—
182C22⋊C4—2181no—
183C23—21no—
184C23—21no—
185C23—21no—
186C23—41no—
187C23—81no—
188C23—81no—
189C23—8721no—
190C23—1091no—
191C23—1091no—
192C23—2181no—
193C23—2181no—
194C23C22⋊#8913711yescharacteristic
195C23C22⋊#8913711yescharacteristic
196C23C22×#8913711yescharacteristic
197D4—42no—
198D4—41no—
199D4—41no—
200D4—8721no—
201D4—8721no—
202D4—4361no—
203D4—4361no—
204D4—4361no—
205D4—4361no—
206C2×C4—21no—
207C2×C4—41no—
208C2×C4—8722no—
209C2×C4—8721no—
210C2×C4—1091no—
211C2×C4—1091no—
212C2×C4—4362no—
213C2×C4—2181no—
214C2×C4—2181no—
215C2×C4—2181no—
216C2×C4—2181no—
217C8—4362no—
218C22—21no—
219C22—21no—
220C22—41no—
221C22—41no—
222C22—41no—
223C22—41no—
224C22—41no—
225C22—41no—
226C22—81no—
227C22—81no—
228C22—82no—
229C22—8721no—
230C22—4361no—
231C22—4361no—
232C22—4362no—
233C22—2181no—
234C22—2181no—
235C22#1681275711yescharacteristic
236C22#1681275711yescharacteristic
237C22C2×C22⋊#8913711yescharacteristic
238C4—41no—
239C4—8722no—
240C4—4361no—
241C4—4361no—
242C4—2181no—
243C4—2181no—
244C2—21no—
245C2—21no—
246C2—21no—
247C2—41no—
248C2—41no—
249C2—81no—
250C2—4361no—
251C2—4361no—
252C2C2×#1681275711yescharacteristic
subgroupG/H
centerC2C2×#16812757
commutatorC23×C109C22×C4
FrattiniC23C22×#89137
FittingC22×D4×C109C4
socleC2×C109C2×#99
prime radical#20295585C2
cosocleC2×C22⋊D109C23
hypercenterC22×D4#89137
hypocenterC109#2121
LCSG ▷ C23×C109 ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C23 ◁ C24 ◁ C22×D4

Complements

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

NH ≅ G/Ntype
1. C2×#16812759C2semidirect
2. C2×#16812757C2semidirect
3. #20295928C2semidirect
4. #20295926C2semidirect
13. #16812682C4semidirect
18. #16812668C4semidirect
21. C2×C22⋊Dic109C4semidirect
22. C22×D4×C109C4semidirect
45. C22⋊Dic109C2×C4semidirect
46. C22⋊Dic109C2×C4semidirect
50. C24×C109C2×C4semidirect
55. C22×C4×C109C2×C4semidirect
127. C109#2121semidirect
134. C22×D4#89137semidirect
152. C24C2×#89137semidirect
168. 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
#202959282
#202959262
#202955852
#2121109