Finite Group Databasebeta

← #28379284

Subgroups — #28379284

Order 13952.

264 conjugacy classes of subgroups; only the 52 normal subgroups are shown.

normal subgroups only

This group has 264 conjugacy classes of subgroups — too many to draw. Only its 52 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#20296013C211yesmaximal, characteristic, complemented
2#20296006C211yesmaximal, characteristic, complemented
3#20295929C212yesmaximal, complemented
4#20295926C212yesmaximal, complemented
5#20295532C211yesmaximal, characteristic
6C2×#13182038—21no—
7#16812778—21no—
8#16812760—21no—
9#16812759C2212yes—
10#16812757—21no—
11#16812757C2212yes—
12C2×#13182028C2211yescharacteristic
13C23×Dic109C411yescharacteristic, complemented
14#16812683C2211yescharacteristic
15#16812680C411yescharacteristic, complemented
16#16812676—21no—
17#16812671—21no—
18#16812668C2211yescharacteristic
19D4×Dic109—21no—
20#16812659—21no—
21#16812655C411yescharacteristic, complemented
22#120×C109C411yescharacteristic, complemented
23#13182038—22no—
24#13182038—41no—
25C2×#11724786—41no—
26C4⋊#89137—41no—
27C22⋊#89137—22no—
28C4×#89137—41no—
29#13182028—41no—
30#13182028—41no—
31C2×Dic218D411yescharacteristic
32C2×C22⋊D109—21no—
33C2×C22⋊D109D411yescharacteristic
34C2×C22⋊D109C2311yescharacteristic
35C22×Dic109—21no—
36C22×Dic109—21no—
37C22×Dic109—41no—
38C22×Dic109—41no—
39C22×Dic109C2×C411yescharacteristic
40C22×Dic109C2×C411yescharacteristic
41C22×Dic109D411yescharacteristic
42C2×C4×D109D411yescharacteristic
43C4⋊Dic109—21no—
44#13182008—21no—
45#13182008—21no—
46C22⋊Dic109—21no—
47C22⋊Dic109—21no—
48C22⋊Dic109C2×C411yescharacteristic
49#13182006—21no—
50C4×Dic109—21no—
51C24×C109C2×C411yescharacteristic
52C2×D4×C109—21no—
53C2×D4×C109C2×C411yescharacteristic
54C22⋊C4×C109—21no—
55C22⋊C4×C109C2×C411yescharacteristic
56C23×C109—21no—
57C23×C109—42no—
58C23×C109—41no—
59C23×C109C22⋊C411yescharacteristic
60C23×C109C22×C411yescharacteristic
61C22×D109C2×D411yescharacteristic
62C2×#89137—42no—
63D4×C109—41no—
64D4×C109—42no—
65C2×C4×C109—21no—
66C2×C4×C109C22⋊C411yescharacteristic
67C22⋊D109—41no—
68C22⋊D109—41no—
69C22⋊D109—42no—
70C2×Dic109—21no—
71C2×Dic109—21no—
72C2×Dic109—21no—
73C2×Dic109—21no—
74C2×Dic109—21no—
75C2×Dic109—21no—
76C2×Dic109—42no—
77C2×Dic109—41no—
78C2×Dic109—41no—
79C2×Dic109—41no—
80C2×Dic109C22⋊C411yescharacteristic
81C2×Dic109C22⋊C411yescharacteristic
82C2×Dic109C2×D411yescharacteristic
83C4×D109—21no—
84C4×D109—21no—
85Dic218—41no—
86#11724786—42no—
87C22×C109—22no—
88C22×C109—21no—
89C22×C109—21no—
90C22×C109—41no—
91C22×C109—41no—
92C22×C109—41no—
93C22×C109—42no—
94C22×C109—42no—
95C22×C109—41no—
96C22×C109C2×C22⋊C411yescharacteristic
97C2×D109—21no—
98C2×D109—41no—
99#89137—82no—
100C4×C109—21no—
101C4×C109—41no—
102Dic109—21no—
103Dic109—21no—
104Dic109—41no—
105Dic109—41no—
106Dic109#9911yescharacteristic
107Dic109#9911yescharacteristic
108C2×C109—21no—
109C2×C109—41no—
110C2×C109—42no—
111C2×C109—41no—
112C2×C109C2×#9911yescharacteristic
113D109—41no—
114#2120—1091nomaximal, Sylow
115C109#212011yescharacteristic, complemented, Sylow
116#525—1091no—
117#413—1091no—
118#410—1091no—
119#352—1092no—
120AΣL(1,𝔽16)—1092no—
121C2×#41—1091no—
122C23×C4—1091no—
123C2×M16—2181no—
124#123—1091no—
125#123—1091no—
126C22⋊Q8—2181no—
127#121—2181no—
128#120—1091no—
129#120#8913711yescharacteristic, complemented
130C4×D4—2181no—
131C4×D4—2181no—
132#117—1091no—
133#117—2181no—
134#101—2181no—
135#100—1092no—
136#99—1092no—
137#99—2181no—
138C24C2×#8913711yescharacteristic
139#41—4361no—
140#41—4361no—
141C2×Q8—1091no—
142C2×D4—21no—
143C2×D4—1091no—
144C2×D4—1091no—
145C2×D4—2181no—
146C2×D4C2×#8913711yescharacteristic
147C22×C4—1091no—
148C22×C4—1091no—
149C22×C4—1091no—
150C22×C4—1091no—
151C22×C4—4361no—
152C22×C4—4361no—
153C22×C4—2181no—
154C22×C4—2181no—
155M16—4361no—
156M16—2182no—
157C2×C8—4361no—
158C4⋊C4—4361no—
159C4⋊C4—2181no—
160C4⋊C4—2181no—
161C4⋊C4—2181no—
162C22⋊C4—21no—
163C22⋊C4—1091no—
164C22⋊C4—2181no—
165C22⋊C4—2181no—
166C22⋊C4—2182no—
167C22⋊C4—2181no—
168C22⋊C4C2×#8913711yescharacteristic
169C42—4361no—
170C42—2181no—
171C23—21no—
172C23—41no—
173C23—42no—
174C23—1091no—
175C23C22⋊#8913711yescharacteristic
176C23C22×#8913711yescharacteristic
177Q8—4361no—
178D4—41no—
179D4—42no—
180D4—4362no—
181D4—4361no—
182D4—4361no—
183C2×C4—21no—
184C2×C4—1091no—
185C2×C4—1091no—
186C2×C4—1091no—
187C2×C4—4362no—
188C2×C4—4361no—
189C2×C4—4361no—
190C2×C4—4362no—
191C2×C4—4361no—
192C2×C4—2181no—
193C2×C4—2181no—
194C2×C4—2181no—
195C2×C4—2181no—
196C2×C4—2181no—
197C2×C4—2181no—
198C2×C4—2181no—
199C2×C4—2181no—
200C2×C4C22⋊#8913711yescharacteristic
201C8—4362no—
202C22—22no—
203C22—21no—
204C22—21no—
205C22—41no—
206C22—41no—
207C22—41no—
208C22—42no—
209C22—42no—
210C22—41no—
211C22—4361no—
212C22—2181no—
213C22C2×C22⋊#8913711yescharacteristic
214C4—21no—
215C4—41no—
216C4—8722no—
217C4—1091no—
218C4—1091no—
219C4—4361no—
220C4—4361no—
221C4—2181no—
222C4—2181no—
223C2—21no—
224C2—42no—
225C2—41no—
226C2—41no—
227C2—4361no—
228C2D109⋊#9911yescharacteristic
subgroupG/H
centerC2D109⋊#99
commutatorC23×C109C22×C4
FrattiniC23C22×#89137
Fitting#120×C109C4
socleC2×C109C2×#99
prime radical#20295532C2
cosocleC2×C22⋊D109C23
hypercenter#120#89137
hypocenterC109#2120
LCSG ▷ C23×C109 ▷ C22×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22 ◁ C2×D4 ◁ #120

Complements

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

NH ≅ G/Ntype
1. #20296013C2semidirect
2. #20296006C2semidirect
3. #20295929C2semidirect
4. #20295926C2semidirect
13. C23×Dic109C4semidirect
15. #16812680C4semidirect
21. #16812655C4semidirect
22. #120×C109C4semidirect
115. C109#2120semidirect
129. #120#89137semidirect

Maximal subgroups

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

subgroupindex
#202960132
#202960062
#202959292
#202959262
#202955322
#2120109