Finite Group Databasebeta

← #28366202

Subgroups — #28366202

Order 13952.

520 conjugacy classes of subgroups; only the 173 normal subgroups are shown.

normal subgroups only

This group has 520 conjugacy classes of subgroups — too many to draw. Only its 173 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×#16812686C211yesmaximal, characteristic
2#20294930C212yesmaximal, complemented
3#20294920C212yesmaximal, complemented
4#20294914C212yesmaximal, complemented
5#20294906C211yesmaximal, characteristic
6#20294899C212yesmaximal, complemented
7#20294898C212yesmaximal, complemented
8#20294890C212yesmaximal, complemented
9#386×C109C211yesmaximal, characteristic
10C22×Dic218C2211yescharacteristic
11C23×Dic109C2211yescharacteristic
12#16812686C418yescomplemented
13C2×#13182008C2212yes—
14C2×#13182008C2212yes—
15C2×#13182008C2212yes—
16C2×C22⋊Dic109C2212yes—
17C2×C22⋊Dic109C2211yescharacteristic
18C2×C22⋊Dic109C2212yes—
19C2×C22⋊Dic109C2212yes—
20C2×C4×Dic109C2212yes—
21C2×C4×Dic109C2211yescharacteristic
22#16812606C2212yes—
23#16812606C2212yes—
24#16812606C2212yes—
25#16812606C2212yes—
26#16812606C2212yes—
27C22×D4×C109C2211yescharacteristic
28C23×C4×C109C2211yescharacteristic
29C2×C22⋊C4×C109C2211yescharacteristic
30C2×C22⋊C4×C109C2212yes—
31#95×C109C2212yes—
32C2×Dic218—41no—
33C2×Dic218—41no—
34C2×Dic218C2×C414yescomplemented
35C22×Dic109—42no—
36C22×Dic109—41no—
37C22×Dic109C2312yes—
38C22×Dic109C2312yes—
39C22×Dic109C2312yes—
40C22×Dic109C2312yes—
41#13182008—42no—
42#13182008—42no—
43#13182008—42no—
44C22⋊Dic109—41no—
45C22⋊Dic109—42no—
46C22⋊Dic109C2×C418yescomplemented
47C22⋊Dic109C2×C418yescomplemented
48C4×Dic109—42no—
49C4×Dic109C2×C414yescomplemented
50C24×C109C2311yescharacteristic
51C24×C109C2311yescharacteristic
52C2×D4×C109—41no—
53C2×D4×C109—41no—
54C2×D4×C109C2×C414yes—
55C22×C4×C109—42no—
56C22×C4×C109—41no—
57C22×C4×C109C2312yes—
58C22×C4×C109C2312yes—
59C22×C4×C109C2311yescharacteristic
60C22⋊C4×C109—42no—
61C22⋊C4×C109—41no—
62C23×C109—21no—
63C23×C109—22no—
64C23×C109—21no—
65C23×C109—22no—
66C23×C109—21no—
67C23×C109—21no—
68C23×C109—22no—
69C23×C109—22no—
70C23×C109C22×C412yes—
71C23×C109C22×C412yes—
72C23×C109C2411yescharacteristic
73D4×C109—44no—
74C2×C4×C109—21no—
75C2×C4×C109—21no—
76C2×C4×C109—22no—
77C2×C4×C109—22no—
78C2×C4×C109—22no—
79C2×C4×C109—22no—
80C2×C4×C109—22no—
81C2×C4×C109—22no—
82C2×C4×C109—44no—
83C2×C4×C109C22×C412yes—
84C2×Dic109—22no—
85C2×Dic109—22no—
86C2×Dic109—22no—
87C2×Dic109—22no—
88C2×Dic109—22no—
89C2×Dic109—22no—
90C2×Dic109—22no—
91C2×Dic109—22no—
92C2×Dic109—22no—
93C2×Dic109—22no—
94C2×Dic109—44no—
95C2×Dic109C22×C414yes—
96C2×Dic109C22×C414yes—
97Dic218—44no—
98C22×C109—22no—
99C22×C109—22no—
100C22×C109—24no—
101C22×C109—24no—
102C22×C109—41no—
103C22×C109—41no—
104C22×C109—42no—
105C22×C109—42no—
106C22×C109—41no—
107C22×C109—41no—
108C22×C109C23×C411yescharacteristic
109C22×C109H(5,𝔽2)11yescharacteristic
110C22×C109H(5,𝔽2)11yescharacteristic
111C22×C109H(5,𝔽2)12yes—
112C22×C1092−1+412yes—
113C4×C109—22no—
114C4×C109—42no—
115C4×C109—42no—
116Dic109—24no—
117Dic109—24no—
118Dic109—42no—
119Dic109—42no—
120C2×C109—42no—
121C2×C109—42no—
122C2×C109#51811yescharacteristic
123C2×C109#52012yes—
124C2×C109H(3,𝔽4)11yescharacteristic
125C2×C109#56212yes—
126C2×C109#56311yescharacteristic
127#2379—1091nomaximal, Sylow
128C109#237911yescharacteristic, complemented, Sylow
129C2×#124—1091no—
130#391—1092no—
131#388—1092no—
132#388—1092no—
133#386—1091no—
134#386D10911yescharacteristic
135#385—1092no—
136#385—1092no—
137#382—1092no—
138C22×Q8—1091no—
139C22×D4C2×D10911yescharacteristic
140C23×C4—1091no—
141C23×C4C2×D10911yescharacteristic
142#124—1098no—
143C2×C4⋊C4—1092no—
144C2×C4⋊C4—1092no—
145C2×C4⋊C4—1092no—
146C2×C22⋊C4—1092no—
147C2×C22⋊C4—1092no—
148C2×C22⋊C4—1092no—
149C2×C22⋊C4—1091no—
150C2×C22⋊C4C2×D10912yes—
151C2×C22⋊C4C2×D10911yescharacteristic
152C2×C42—1091no—
153C2×C42—1092no—
154#95—1092no—
155#95—1092no—
156#95—1092no—
157#95—1092no—
158#95—1092no—
159#95C2×D10912yes—
160C24C22×D10911yescharacteristic
161C24C22×D10911yescharacteristic
162C2×Q8—1094no—
163C2×Q8—4361no—
164C2×Q8—4361no—
165C2×D4—41no—
166C2×D4—41no—
167C2×D4C4×D10914yes—
168C22×C4—42no—
169C22×C4—41no—
170C22×C4—1092no—
171C22×C4—1092no—
172C22×C4—1092no—
173C22×C4—1092no—
174C22×C4—4362no—
175C22×C4—4361no—
176C22×C4C22×D10912yes—
177C22×C4C22×D10912yes—
178C22×C4C22×D10911yescharacteristic
179C4⋊C4—4362no—
180C4⋊C4—4362no—
181C4⋊C4—4362no—
182C22⋊C4—41no—
183C22⋊C4—42no—
184C22⋊C4—1098no—
185C22⋊C4—1098no—
186C22⋊C4—4361no—
187C22⋊C4—4362no—
188C42—1094no—
189C42—4362no—
190C23—22no—
191C23—21no—
192C23—22no—
193C23—21no—
194C23—21no—
195C23—21no—
196C23—22no—
197C23—22no—
198C23C2×C4×D10912yes—
199C23C2×C4×D10912yes—
200C23C23×D10911yescharacteristic
201Q8—4364no—
202D4—44no—
203C2×C4—21no—
204C2×C4—21no—
205C2×C4—22no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—22no—
209C2×C4—22no—
210C2×C4—22no—
211C2×C4—44no—
212C2×C4—1094no—
213C2×C4—1094no—
214C2×C4—4364no—
215C2×C4—2182no—
216C2×C4—2182no—
217C2×C4—2182no—
218C2×C4—2182no—
219C2×C4—2182no—
220C2×C4—2182no—
221C2×C4—2182no—
222C2×C4—2182no—
223C2×C4—2182no—
224C2×C4—2182no—
225C2×C4C2×C4×D10912yes—
226C22—22no—
227C22—22no—
228C22—24no—
229C22—24no—
230C22—41no—
231C22—41no—
232C22—42no—
233C22—42no—
234C22—41no—
235C22—41no—
236C22C22×C4×D10911yescharacteristic
237C22#1681274411yescharacteristic
238C22#1681274411yescharacteristic
239C22#1681274412yes—
240C22#1681274612yes—
241C4—22no—
242C4—42no—
243C4—42no—
244C4—4362no—
245C4—4362no—
246C4—2184no—
247C4—2184no—
248C2—42no—
249C2—42no—
250C2#2029549911yescharacteristic
251C2#2029550512yes—
252C2#2029576911yescharacteristic
253C2#2029577112yes—
254C2#2029577911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D109
commutatorC22×C109C23×C4
Fitting#386×C109C2
socle, cosocleC23×C109C24
prime radicalC22×D4×C109C22
hypercenter#386D109
hypocenterC109#2379
LCSG ▷ C22×C109 ▷ C109
UCSC1 ◁ C23 ◁ #386

Complements

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

NH ≅ G/Ntype
2. #20294930C2semidirect
3. #20294920C2semidirect
4. #20294914C2semidirect
6. #20294899C2semidirect
7. #20294898C2semidirect
8. #20294890C2semidirect
12. #16812686C4semidirect
34. C2×Dic218C2×C4semidirect
46. C22⋊Dic109C2×C4semidirect
47. C22⋊Dic109C2×C4semidirect
49. C4×Dic109C2×C4semidirect
128. C109#2379semidirect

Maximal subgroups

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

subgroupindex
C2×#168126862
#202949302
#202949202
#202949142
#202949062
#202948992
#202948982
#202948902
#386×C1092
#2379109