Finite Group Databasebeta

← #28363479

Subgroups — #28363479

Order 13952.

358 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 358 conjugacy classes of subgroups — too many to draw. Only its 83 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#20295538C211yesmaximal, characteristic
2C2×#16812656C211yesmaximal, characteristic, complemented
3C2×#16812627C211yesmaximal, characteristic, complemented
4C2×#16812625C211yesmaximal, characteristic, complemented
5#20294989C211yesmaximal, characteristic, complemented
6#20294785C211yesmaximal, characteristic, complemented
7#339×C109C211yesmaximal, characteristic, complemented
8C2×#13182029C2211yescharacteristic
9C22×C4×D109C2211yescharacteristic
10C2×#13182013C2211yescharacteristic
11#16812683—21no—
12#16812679C412yescomplemented
13#16812674C412yescomplemented
14#16812669—21no—
15C4×D436—22no—
16#16812656—22no—
17#16812656—22no—
18C2×#13182008—21no—
19C2×#13182006—21no—
20C2×C4×Dic109C2211yescharacteristic
21#16812627—22no—
22#16812625—22no—
23#16812611—22no—
24C2×M16×C109C2211yescharacteristic
25#117×C109C2211yescharacteristic
26C2×C42×C109C2211yescharacteristic
27C23×D109D411yescharacteristic
28#13182029—21no—
29#13182029—21no—
30#13182029—21no—
31#13182029—21no—
32C2×Dic218C2×C411yescharacteristic
33C2×D436—22no—
34C2×D436C2×C411yescharacteristic
35C2×C22⋊D109—21no—
36C22×Dic109—21no—
37C22×Dic109D411yescharacteristic
38C2×C4×D109—21no—
39C2×C4×D109—21no—
40C2×C4×D109—21no—
41C2×C4×D109—21no—
42C2×C4×D109—21no—
43C2×C4×D109C2×C412yes—
44C2×C4×D109D412yes—
45#13182013—21no—
46#13182013—21no—
47C2×#11724784—21no—
48C4⋊Dic109—22no—
49#13182008—24no—
50#13182006—24no—
51#13182006—22no—
52#13182006—22no—
53C4×Dic109—21no—
54C4×Dic109D412yescomplemented
55C22×C4×C109—21no—
56C22×C4×C109C2311yescharacteristic
57M16×C109—21no—
58M16×C109—21no—
59C2×C8×C109—21no—
60C4⋊C4×C109C2×C412yes—
61C22⋊C4×C109—21no—
62C42×C109—21no—
63C42×C109—21no—
64C42×C109D412yescomplemented
65C23×C109C2×D411yescharacteristic
66C22×D109—21no—
67C22×D109—21no—
68C22×D109—21no—
69C22×D109—42no—
70C22×D109C22⋊C412yes—
71C2×C4×C109—22no—
72C2×C4×C109—22no—
73C2×C4×C109—42no—
74C2×C4×C109C22×C411yescharacteristic
75C2×C4×C109C2×D411yescharacteristic
76C2×C4×C109C2×D411yescharacteristic
77C2×C4×C109C2×D411yescharacteristic
78C2×C4×C109#4111yescharacteristic
79C2×C4×C109#4111yescharacteristic
80C22⋊D109—41no—
81C22⋊D109—41no—
82C2×Dic109—21no—
83C2×Dic109—21no—
84C2×Dic109—21no—
85C2×Dic109—22no—
86C2×Dic109—42no—
87C2×Dic109C22⋊C412yes—
88D436—21no—
89D436—21no—
90D436—44no—
91C4×D109—22no—
92C4×D109—22no—
93C4×D109—22no—
94C4×D109—22no—
95C4×D109—41no—
96C4×D109—41no—
97Dic218—21no—
98Dic218—21no—
99C8×C109—41no—
100#11724784—41no—
101C22×C109—22no—
102C22×C109C2×C22⋊C411yescharacteristic
103C22×C109#12011yescharacteristic
104C22×C109#12311yescharacteristic
105C2×D109—22no—
106C2×D109—22no—
107C2×D109—22no—
108C2×D109—42no—
109C2×D109—42no—
110C2×D109—41no—
111C2×D109—42no—
112C4×C109—42no—
113C4×C109—42no—
114C4×C109C4×D411yescharacteristic
115C4×C109C4×D411yescharacteristic
116C4×C109#12111yescharacteristic
117C4×C109#12111yescharacteristic
118Dic109—22no—
119Dic109—42no—
120Dic109—41no—
121C2×C109—21no—
122C2×C109—21no—
123C2×C109#38611yescharacteristic
124C2×C109#42112yes—
125D109—42no—
126D109—81no—
127#1897—1091nomaximal, Sylow
128C109#189711yescharacteristic, complemented, Sylow
129#525—1091no—
130C2×#117—1091no—
131C2×#104—1091no—
132C2×#104—1091no—
133#407—1091no—
134#339—1091no—
135#339D10911yescharacteristic, complemented
136C2×#41—1091no—
137C23×C4—1091no—
138C2×M16—1091no—
139C2×M16C2×D10911yescharacteristic
140#123—2181no—
141C22⋊Q8—1092no—
142#121—1092no—
143#120—2181no—
144C4×D4—2182no—
145#117—2182no—
146#117—2182no—
147#117C2×D10911yescharacteristic
148C2×C4⋊C4—2181no—
149C2×C22⋊C4—2181no—
150C2×C42—1091no—
151C2×C42C2×D10911yescharacteristic
152#104—2182no—
153#104—2182no—
154C22⋊C8—2182no—
155C24—1091no—
156#41—2181no—
157#41—2181no—
158#41—2181no—
159#41—2181no—
160C2×Q8—1091no—
161C2×D4—1091no—
162C2×D4—2182no—
163C2×D4—2181no—
164C22×C4—21no—
165C22×C4—1091no—
166C22×C4—1092no—
167C22×C4—1092no—
168C22×C4—2181no—
169C22×C4—2181no—
170C22×C4—2181no—
171C22×C4—2181no—
172C22×C4—2181no—
173C22×C4—2181no—
174C22×C4C22×D10911yescharacteristic
175M16—21no—
176M16—21no—
177M16—2181no—
178M16—2181no—
179C2×C8—21no—
180C2×C8—2181no—
181C4⋊C4—2182no—
182C4⋊C4—2184no—
183C4⋊C4C4×D10912yes—
184C22⋊C4—21no—
185C22⋊C4—2182no—
186C22⋊C4—2182no—
187C22⋊C4—2184no—
188C42—21no—
189C42—21no—
190C42—1092no—
191C42—2181no—
192C42D43612yescomplemented
193C23—1092no—
194C23—4362no—
195C23—2181no—
196C23—2181no—
197C23—2181no—
198C23D4×D10911yescharacteristic
199Q8—2181no—
200Q8—2181no—
201D4—4364no—
202D4—4361no—
203D4—4361no—
204D4—2181no—
205D4—2181no—
206C2×C4—22no—
207C2×C4—22no—
208C2×C4—42no—
209C2×C4—1092no—
210C2×C4—4362no—
211C2×C4—4361no—
212C2×C4—4361no—
213C2×C4—2181no—
214C2×C4—2182no—
215C2×C4—2181no—
216C2×C4—2181no—
217C2×C4—2182no—
218C2×C4—2182no—
219C2×C4—2182no—
220C2×C4—2182no—
221C2×C4C2×C4×D10911yescharacteristic
222C2×C4D4×D10911yescharacteristic
223C2×C4D4×D10911yescharacteristic
224C2×C4C2×D43611yescharacteristic
225C2×C4#1318202911yescharacteristic
226C2×C4#1318203011yescharacteristic
227C8—41no—
228C8—4361no—
229C22—22no—
230C22—4362no—
231C22—4362no—
232C22—4362no—
233C22—4361no—
234C22—2182no—
235C22—2182no—
236C22—2182no—
237C22C22⋊C4×D10911yescharacteristic
238C22#1681266911yescharacteristic
239C22#1681268411yescharacteristic
240C4—42no—
241C4—42no—
242C4—4362no—
243C4—4361no—
244C4—2182no—
245C4#1681266111yescharacteristic
246C4C4×D43611yescharacteristic
247C4#1681267211yescharacteristic
248C4#1681267411yescharacteristic
249C2—21no—
250C2—21no—
251C2—8721no—
252C2—4362no—
253C2#2029490411yescharacteristic
254C2#2029504112yes—
subgroupG/H
centerC2×C4D4×D109
commutatorC2×C4×C109C22×C4
FrattiniC22×C4C22×D109
Fitting#339×C109C2
socleC22×C109C2×C22⋊C4
prime radical#20295538C2
cosocleC22×C4×C109C23
hypercenter#339D109
hypocenterC109#1897
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ #339

Complements

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

NH ≅ G/Ntype
2. C2×#16812656C2semidirect
3. C2×#16812627C2semidirect
4. C2×#16812625C2semidirect
5. #20294989C2semidirect
6. #20294785C2semidirect
7. #339×C109C2semidirect
12. #16812679C4semidirect
13. #16812674C4semidirect
54. C4×Dic109D4semidirect
64. C42×C109D4semidirect
128. C109#1897semidirect
135. #339D109semidirect
192. C42D436semidirect

Maximal subgroups

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

subgroupindex
#202955382
C2×#168126562
C2×#168126272
C2×#168126252
#202949892
#202947852
#339×C1092
#1897109