Finite Group Databasebeta

← #28373390

Subgroups — #28373390

Order 13952.

404 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 125 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#20295654C211yesmaximal, characteristic, complemented
2#20295590C211yesmaximal, characteristic
3#20295389C212yesmaximal, complemented
4#20295387C212yesmaximal, complemented
5#20295382C212yesmaximal, complemented
6#20295371C212yesmaximal, complemented
7#20295030C211yesmaximal, characteristic, complemented
8#20295027C211yesmaximal, characteristic, complemented
9C2×#16812621C211yesmaximal, characteristic, complemented
10#20294981C211yesmaximal, characteristic, complemented
11#416×C109C211yesmaximal, characteristic, complemented
12C22×D436C2211yescharacteristic
13C2×#13182013C2211yescharacteristic
14#16812699—21no—
15#16812698C2212yescomplemented
16#16812695C2212yes—
17#16812691—21no—
18#16812690C2212yescomplemented
19#16812688C2212yes—
20#16812676—22no—
21#16812673—21no—
22#16812673—21no—
23#16812667—22no—
24#16812658C2211yescharacteristic
25C2×C4⋊Dic109C2211yescharacteristic
26#16812623C2212yescomplemented
27#16812621C2212yes—
28#16812621C2212yes—
29#16812620C2212yescomplemented
30#16812619C2212yes—
31#16812619C2212yes—
32#16812609C2212yescomplemented
33C8×Dic109C2212yescomplemented
34C2×#41×C109C2211yescharacteristic
35C22×C8×C109C2211yescharacteristic
36#117×C109C2211yescharacteristic
37#103×C109C2211yescharacteristic, complemented
38#103×C109C2211yescharacteristic, complemented
39#102×C109C2211yescharacteristic, complemented
40#102×C109C2211yescharacteristic, complemented
41C23×D109—21no—
42C2×Dic218—22no—
43C2×D436—22no—
44C2×D436—22no—
45C2×D436—21no—
46C2×D436C2312yes—
47C2×C22⋊D109—22no—
48C2×C22⋊D109—22no—
49C22×Dic109—21no—
50#13182013—22no—
51C2×#11724784C2312yes—
52C4⋊Dic109—21no—
53C4⋊Dic109—22no—
54C4⋊Dic109D411yescharacteristic
55C4⋊Dic109D411yescharacteristic
56C4⋊Dic109C2312yes—
57#13182008—22no—
58#13182008—22no—
59C22⋊Dic109D412yes—
60#13182006—22no—
61#13182006—22no—
62C4×Dic109C2312yes—
63#41×C109—22no—
64#41×C109—22no—
65C2×Q8×C109C2311yescharacteristic
66C2×D4×C109—21no—
67C2×D4×C109C2311yescharacteristic
68C22×C4×C109—21no—
69C22×C4×C109C2311yescharacteristic
70C2×C8×C109—22no—
71C2×C8×C109C2311yescharacteristic
72C2×C8×C109C2311yescharacteristic
73C4⋊C4×C109C2311yescharacteristic
74C4⋊C4×C109C2311yescharacteristic
75C22⋊C4×C109—21no—
76C42×C109—21no—
77C23×C109—21no—
78C23×C109C2×D411yescharacteristic
79C22×D109—22no—
80C22×D109—42no—
81C22×D109—81no—
82C22×D109—81no—
83Q8×C109—21no—
84Q8×C109—21no—
85D4×C109—21no—
86D4×C109—21no—
87D4×C109—42no—
88C2×C4×C109—21no—
89C2×C4×C109—21no—
90C2×C4×C109—21no—
91C2×C4×C109—42no—
92C2×C4×C109C2×D411yescharacteristic
93C2×C4×C109#4112yes—
94C2×C4×C109#4112yes—
95C2×C4×C109C2411yescharacteristic
96C22⋊D109—82no—
97C22⋊D109—82no—
98C2×Dic109—22no—
99C2×Dic109—81no—
100C2×Dic109C2×D412yes—
101C2×Dic109C2×D412yes—
102D436—22no—
103D436—22no—
104D436—42no—
105Dic218—82no—
106C8×C109—21no—
107C8×C109—21no—
108#11724784—22no—
109C22×C109—21no—
110C22×C109—21no—
111C22×C109—41no—
112C22×C109—41no—
113C22×C109—41no—
114C22×C109#12412yes—
115C22×C109C22×D411yescharacteristic
116C2×D109—42no—
117C2×D109—42no—
118C2×D109—42no—
119C2×D109—84no—
120C4×C109—41no—
121C4×C109—41no—
122C4×C109—41no—
123C4×C109#12412yes—
124C4×C109C2×#4111yescharacteristic
125C4×C109C2×#4111yescharacteristic
126Dic109—22no—
127Dic109—22no—
128Dic109—42no—
129C2×C109—22no—
130C2×C109—81no—
131C2×C109C2×#12411yescharacteristic
132C2×C109#57611yescharacteristic
133C2×C109#57711yescharacteristic
134D109—82no—
135#3136—1091nomaximal, Sylow
136C109#313611yescharacteristic, complemented, Sylow
137#544—1091no—
138#535—1091no—
139#490—1092no—
140#489—1092no—
141#488—1092no—
142#486—1092no—
143#418—1091no—
144#417—1091no—
145#416D10911yescharacteristic, complemented
146C2×#102—1091no—
147#405—1091no—
148C2×#41C2×D10911yescharacteristic
149C22×D4—1091no—
150C2×M16—1091no—
151C22×C8C2×D10911yescharacteristic
152C4⋊Q8—1092no—
153C4⋊Q8—2181no—
154#127—1092no—
155#125—1092no—
156#125—2181no—
157#124—1092no—
158C22⋊Q8—2182no—
159#121—2181no—
160#121—2181no—
161#120—2182no—
162#117—1091no—
163#117C2×D10911yescharacteristic
164C2×C4⋊C4—1091no—
165#103—1092no—
166#103C2×D10911yescharacteristic, complemented
167#103C2×D10911yescharacteristic, complemented
168#102—1092no—
169#102—1092no—
170#102—1092no—
171#102—1092no—
172#102—1092no—
173#102C2×D10911yescharacteristic, complemented
174#102C2×D10911yescharacteristic, complemented
175#97—1092no—
176C4×C8—1092no—
177C24—2181no—
178#41—22no—
179#41—22no—
180C2×Q8—2182no—
181C2×Q8C22×D10911yescharacteristic
182C2×D4—21no—
183C2×D4—1092no—
184C2×D4—2182no—
185C2×D4—2182no—
186C2×D4—2181no—
187C2×D4—2182no—
188C2×D4—2182no—
189C2×D4C22×D10911yescharacteristic
190C22×C4—21no—
191C22×C4—2181no—
192C22×C4C22×D10911yescharacteristic
193M16—2182no—
194C2×C8—22no—
195C2×C8—1092no—
196C2×C8C22×D10911yescharacteristic
197C2×C8C22×D10911yescharacteristic
198C4⋊C4—1091no—
199C4⋊C4—1091no—
200C4⋊C4—1092no—
201C4⋊C4—2182no—
202C4⋊C4—2182no—
203C4⋊C4—2182no—
204C4⋊C4—2181no—
205C4⋊C4C22×D10911yescharacteristic
206C4⋊C4C22×D10911yescharacteristic
207C22⋊C4—21no—
208C22⋊C4—1092no—
209C22⋊C4—2182no—
210C22⋊C4—2182no—
211C42—21no—
212C42—1092no—
213C23—21no—
214C23—8721no—
215C23—8721no—
216C23—4362no—
217C23—2182no—
218C23D4×D10911yescharacteristic
219Q8—21no—
220Q8—21no—
221Q8—8722no—
222D4—21no—
223D4—21no—
224D4—42no—
225D4—8722no—
226D4—8722no—
227D4—4362no—
228D4—2182no—
229D4—2182no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—42no—
234C2×C4—8721no—
235C2×C4—1092no—
236C2×C4—1092no—
237C2×C4—2182no—
238C2×C4D4×D10911yescharacteristic
239C2×C4#1318202812yes—
240C2×C4#1318202912yes—
241C2×C4C23×D10911yescharacteristic
242C8—21no—
243C8—21no—
244C8—2182no—
245C22—21no—
246C22—21no—
247C22—41no—
248C22—41no—
249C22—41no—
250C22—8724no—
251C22—4362no—
252C22—4362no—
253C22—4362no—
254C22#1681268512yes—
255C22C2×D4×D10911yescharacteristic
256C4—41no—
257C4—41no—
258C4—41no—
259C4—4362no—
260C4—2182no—
261C4—2182no—
262C4#1681268512yes—
263C4C2×#1318202811yescharacteristic
264C4C2×#1318202911yescharacteristic
265C2—22no—
266C2—81no—
267C2—8722no—
268C2C2×#1681268511yescharacteristic
269C2#2029582811yescharacteristic
270C2#2029583311yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#416×C109C2
socleC22×C109C22×D4
prime radical#20295590C2
hypercenter#416D109
hypocenterC109#3136
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #416

Complements

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

NH ≅ G/Ntype
1. #20295654C2semidirect
3. #20295389C2semidirect
4. #20295387C2semidirect
5. #20295382C2semidirect
6. #20295371C2semidirect
7. #20295030C2semidirect
8. #20295027C2semidirect
9. C2×#16812621C2semidirect
10. #20294981C2semidirect
11. #416×C109C2semidirect
15. #16812698C22semidirect
18. #16812690C22semidirect
26. #16812623C22semidirect
29. #16812620C22semidirect
32. #16812609C22semidirect
33. C8×Dic109C22semidirect
37. #103×C109C22semidirect
38. #103×C109C22semidirect
39. #102×C109C22semidirect
40. #102×C109C22semidirect
136. C109#3136semidirect
145. #416D109semidirect
166. #103C2×D109semidirect
167. #103C2×D109semidirect
173. #102C2×D109semidirect
174. #102C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202956542
#202955902
#202953892
#202953872
#202953822
#202953712
#202950302
#202950272
C2×#168126212
#202949812
#416×C1092
#3136109