Finite Group Databasebeta

← #28373391

Subgroups — #28373391

Order 13952.

404 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 133 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#20295595C211yesmaximal, characteristic
2#20295392C212yesmaximal, complemented
3#20295388C212yesmaximal, complemented
4#20295383C212yesmaximal, complemented
5#20295374C212yesmaximal, complemented
6#20295030C211yesmaximal, characteristic, complemented
7#20295025C211yesmaximal, characteristic, complemented
8#20295020C211yesmaximal, characteristic, complemented
9C2×#16812619C211yesmaximal, characteristic, complemented
10#20294982C211yesmaximal, characteristic, complemented
11#544×C109C211yesmaximal, characteristic, complemented
12C2×#13182029C2211yescharacteristic
13C22×D436C2211yescharacteristic
14C2×#13182013C2211yescharacteristic
15C22×#11724784C2211yescharacteristic
16#16812696C2212yes—
17#16812687C2212yes—
18#16812674—21no—
19#16812674—21no—
20#16812669—22no—
21#16812622C2211yescharacteristic, complemented
22#16812622C2211yescharacteristic, complemented
23#16812622C2212yescomplemented
24#16812619C2212yes—
25#16812619C2212yes—
26#16812619C2212yes—
27#16812619C2212yes—
28#16812619C2211yescharacteristic, complemented
29#16812619C2211yescharacteristic, complemented
30#16812619C2212yescomplemented
31#16812610C2212yescomplemented
32C4×#11724784C2212yescomplemented
33C4⋊Q8×C109—21no—
34C4⋊Q8×C109C2212yescomplemented
35#125×C109—21no—
36#125×C109C2212yescomplemented
37C22⋊Q8×C109—22no—
38#117×C109C2211yescharacteristic
39#117×C109C2211yescharacteristic
40C2×C4⋊C4×C109C2211yescharacteristic
41C23×D109—21no—
42#13182029—22no—
43#13182029—22no—
44C2×Dic218C2311yescharacteristic
45C2×D436—22no—
46C2×D436—22no—
47C2×D436—21no—
48C2×D436—22no—
49C2×D436—22no—
50C2×D436C2312yes—
51C2×D436C2311yescharacteristic
52C2×C22⋊D109—21no—
53C2×C4×D109—21no—
54#13182013—22no—
55C2×#11724784—22no—
56C2×#11724784C2311yescharacteristic
57C2×#11724784C2312yes—
58C2×#11724784C2311yescharacteristic
59#13182006—22no—
60#13182006—22no—
61C2×Q8×C109—22no—
62C22×C4×C109—21no—
63C22×C4×C109C2311yescharacteristic
64C4⋊C4×C109—21no—
65C4⋊C4×C109—22no—
66C4⋊C4×C109—22no—
67C4⋊C4×C109—22no—
68C4⋊C4×C109D411yescharacteristic
69C4⋊C4×C109D411yescharacteristic
70C4⋊C4×C109C2312yes—
71C4⋊C4×C109C2311yescharacteristic
72C4⋊C4×C109C2311yescharacteristic
73C22⋊C4×C109—21no—
74C22⋊C4×C109D412yes—
75C42×C109—21no—
76C42×C109C2312yes—
77C23×C109C2×D411yescharacteristic
78C22×D109—22no—
79C22×D109—21no—
80C22×D109—42no—
81C22×D109—81no—
82C22×D109—81no—
83Q8×C109—82no—
84C2×C4×C109—21no—
85C2×C4×C109—21no—
86C2×C4×C109—22no—
87C2×C4×C109—81no—
88C2×C4×C109C2×D412yes—
89C2×C4×C109C2×D412yes—
90C2×C4×C109C2×D411yescharacteristic
91C2×C4×C109#4112yes—
92C2×C4×C109#4112yes—
93C2×C4×C109C2411yescharacteristic
94C22⋊D109—42no—
95C2×Dic109—21no—
96D436—22no—
97D436—22no—
98D436—21no—
99D436—21no—
100D436—42no—
101D436—82no—
102D436—82no—
103C4×D109—42no—
104Dic218—21no—
105Dic218—21no—
106#11724784—21no—
107#11724784—21no—
108#11724784—22no—
109C22×C109—21no—
110C22×C109—21no—
111C22×C109#12412yes—
112C22×C109C22×D411yescharacteristic
113C2×D109—41no—
114C2×D109—42no—
115C2×D109—41no—
116C2×D109—42no—
117C2×D109—41no—
118C2×D109—42no—
119C2×D109—84no—
120C4×C109—22no—
121C4×C109—22no—
122C4×C109—42no—
123C4×C109—41no—
124C4×C109—41no—
125C4×C109#12412yes—
126C4×C109C2×#4111yescharacteristic
127C4×C109C2×#4111yescharacteristic
128Dic109—41no—
129C2×C109—22no—
130C2×C109C2×#12411yescharacteristic
131C2×C109#57611yescharacteristic
132C2×C109#57711yescharacteristic
133D109—81no—
134D109—82no—
135#3136—1091nomaximal, Sylow
136C109#313611yescharacteristic, complemented, Sylow
137#544D10911yescharacteristic, complemented
138#535—1091no—
139#490—1092no—
140#489—1092no—
141#488—1092no—
142#486—1092no—
143#418—1091no—
144#417—1091no—
145#416—1091no—
146C2×#102—1091no—
147#405—1091no—
148C2×#41—1091no—
149C22×D4—1091no—
150C2×M16—1091no—
151C22×C8—1091no—
152C4⋊Q8—21no—
153C4⋊Q8C2×D10912yescomplemented
154#127—1092no—
155#125—21no—
156#125C2×D10912yescomplemented
157#124—1092no—
158C22⋊Q8—22no—
159#121—2181no—
160#121—2181no—
161#120—2182no—
162#117C2×D10911yescharacteristic
163#117C2×D10911yescharacteristic
164C2×C4⋊C4C2×D10911yescharacteristic
165#103—1091no—
166#103—1091no—
167#103—1092no—
168#102—1092no—
169#102—1092no—
170#102—1092no—
171#102—1092no—
172#102—1091no—
173#102—1091no—
174#102—1092no—
175#97—1092no—
176C4×C8—1092no—
177C24—2181no—
178#41—2182no—
179#41—2182no—
180C2×Q8—22no—
181C2×Q8—1091no—
182C2×D4—1092no—
183C2×D4—1091no—
184C2×D4—2181no—
185C2×D4—2182no—
186C2×D4—2182no—
187C2×D4—2181no—
188C2×D4—2182no—
189C2×D4—2182no—
190C22×C4—21no—
191C22×C4—2181no—
192C22×C4C22×D10911yescharacteristic
193M16—2182no—
194C2×C8—1091no—
195C2×C8—1092no—
196C2×C8—1091no—
197C2×C8—2182no—
198C4⋊C4—21no—
199C4⋊C4—22no—
200C4⋊C4—22no—
201C4⋊C4—22no—
202C4⋊C4C22⋊D10911yescharacteristic
203C4⋊C4C22⋊D10911yescharacteristic
204C4⋊C4C22×D10912yes—
205C4⋊C4C22×D10911yescharacteristic
206C4⋊C4C22×D10911yescharacteristic
207C22⋊C4—21no—
208C22⋊C4—2182no—
209C22⋊C4—2182no—
210C22⋊C4C22⋊D10912yes—
211C42—21no—
212C42C22×D10912yes—
213C23—8721no—
214C23—8721no—
215C23—4362no—
216C23—2182no—
217C23—2181no—
218C23C2×C22⋊D10911yescharacteristic
219Q8—82no—
220Q8—2181no—
221Q8—2181no—
222D4—8722no—
223D4—8722no—
224D4—4362no—
225D4—4362no—
226D4—2182no—
227D4—2182no—
228D4—2181no—
229D4—2181no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—81no—
234C2×C4—4362no—
235C2×C4—2181no—
236C2×C4C2×C22⋊D10912yes—
237C2×C4C2×C22⋊D10912yes—
238C2×C4C2×C22⋊D10911yescharacteristic
239C2×C4#1318203012yes—
240C2×C4#1318203012yes—
241C2×C4C23×D10911yescharacteristic
242C8—2181no—
243C8—2181no—
244C8—2182no—
245C22—21no—
246C22—21no—
247C22—8724no—
248C22—4361no—
249C22—4362no—
250C22—4361no—
251C22—4362no—
252C22—4361no—
253C22—4362no—
254C22#1681268812yes—
255C22C22×C22⋊D10911yescharacteristic
256C4—22no—
257C4—22no—
258C4—42no—
259C4—41no—
260C4—41no—
261C4—4361no—
262C4#1681268812yes—
263C4C2×#1318203011yescharacteristic
264C4C2×#1318203011yescharacteristic
265C2—22no—
266C2—8722no—
267C2—8721no—
268C2C2×#1681268811yescharacteristic
269C2#2029582911yescharacteristic
270C2#2029583411yescharacteristic
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#544×C109C2
socleC22×C109C22×D4
prime radical#20295595C2
hypercenter#544D109
hypocenterC109#3136
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ #117 ◁ #544

Complements

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

NH ≅ G/Ntype
2. #20295392C2semidirect
3. #20295388C2semidirect
4. #20295383C2semidirect
5. #20295374C2semidirect
6. #20295030C2semidirect
7. #20295025C2semidirect
8. #20295020C2semidirect
9. C2×#16812619C2semidirect
10. #20294982C2semidirect
11. #544×C109C2semidirect
21. #16812622C22semidirect
22. #16812622C22semidirect
23. #16812622C22semidirect
28. #16812619C22semidirect
29. #16812619C22semidirect
30. #16812619C22semidirect
31. #16812610C22semidirect
32. C4×#11724784C22semidirect
34. C4⋊Q8×C109C22semidirect
36. #125×C109C22semidirect
136. C109#3136semidirect
137. #544D109semidirect
153. C4⋊Q8C2×D109semidirect
156. #125C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202955952
#202953922
#202953882
#202953832
#202953742
#202950302
#202950252
#202950202
C2×#168126192
#202949822
#544×C1092
#3136109