Finite Group Databasebeta

← #28374857

Subgroups — #28374857

Order 13952.

462 conjugacy classes of subgroups; only the 139 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 139 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#20295733C211yesmaximal, characteristic, complemented
2D4×D436C211yesmaximal, characteristic
3#20295373C211yesmaximal, characteristic, complemented
4#20295345C212yesmaximal, complemented
5#20295317C211yesmaximal, characteristic, complemented
6#20295255C212yesmaximal, complemented
7#20295227C211yesmaximal, characteristic, complemented
8#20295179C212yesmaximal, complemented
9C4×#13182018C211yesmaximal, characteristic, complemented
10C2×#16812621C212yesmaximal, complemented
11C8×D4×C109C211yesmaximal, characteristic, complemented
12C22×D436C2212yes—
13C2×D4×D109—21no—
14C2×#13182018C2211yescharacteristic
15C2×#13182018C2212yescomplemented
16#16812699C2211yescharacteristic, complemented
17#16812696C2211yescharacteristic
18#16812690—21no—
19#16812678C2212yescomplemented
20#16812676—22no—
21#16812674—21no—
22#16812673C2212yes—
23#16812669—22no—
24C4×Dic218C2211yescharacteristic, complemented
25C4×D436C2211yescharacteristic
26D4×Dic109—21no—
27C2×C4⋊Dic109C2212yescomplemented
28#16812631C2212yescomplemented
29#16812631C2211yescharacteristic, complemented
30#16812624C2211yescharacteristic, complemented
31#16812621C2212yes—
32#16812621C2211yescharacteristic
33#16812621C2212yescomplemented
34#16812621C2212yescomplemented
35#16812621C2212yescomplemented
36C22×C8×C109C2212yescomplemented
37C4×D4×C109C2211yescharacteristic
38C4⋊C8×C109C2211yescharacteristic, complemented
39C22⋊C8×C109C2212yescomplemented
40C4×C8×C109C2211yescharacteristic, complemented
41C23×D109—22no—
42C2×Dic218—21no—
43C2×Dic218C2311yescharacteristic
44C2×D436—24no—
45C2×D436—22no—
46C2×D436—21no—
47C2×D436—21no—
48C2×D436—42no—
49C2×D436C2312yes—
50C2×D436C2311yescharacteristic
51C2×C22⋊D109—22no—
52D4×D109—44no—
53C22×Dic109—22no—
54C2×C4×D109—21no—
55#13182018—22no—
56#13182018—42no—
57C4⋊Dic109—22no—
58C4⋊Dic109—21no—
59C4⋊Dic109C2311yescharacteristic
60C4⋊Dic109C2312yes—
61C4⋊Dic109C2311yescharacteristic
62#13182008—21no—
63#13182008—22no—
64C22⋊Dic109—22no—
65#13182006—21no—
66#13182006—22no—
67C4×Dic109—21no—
68C2×D4×C109D411yescharacteristic
69C22×C4×C109C2312yes—
70C2×C8×C109—42no—
71C2×C8×C109C2311yescharacteristic
72C2×C8×C109C2311yescharacteristic
73C2×C8×C109C2312yes—
74C4⋊C4×C109D411yescharacteristic
75C22⋊C4×C109D412yes—
76C42×C109C2311yescharacteristic
77C23×C109C2×D412yes—
78C22×D109—21no—
79C22×D109—22no—
80C22×D109—44no—
81C22×D109—44no—
82C22×D109—82no—
83D4×C109SD1614yescomplemented
84C2×C4×C109—22no—
85C2×C4×C109—22no—
86C2×C4×C109C2×D411yescharacteristic
87C2×C4×C109C2×D412yes—
88C2×C4×C109C2×D411yescharacteristic
89C2×C4×C109C2411yescharacteristic
90C22⋊D109—44no—
91C2×Dic109—21no—
92C2×Dic109—21no—
93C2×Dic109—21no—
94C2×Dic109—22no—
95C2×Dic109—82no—
96D436—22no—
97D436—21no—
98D436—22no—
99D436—44no—
100D436—42no—
101D436—81no—
102D436#4112yes—
103C4×D109—42no—
104Dic218—21no—
105Dic218—21no—
106Dic218—81no—
107C8×C109—21no—
108C8×C109—21no—
109C8×C109—22no—
110C22×C109—22no—
111C22×C109—22no—
112C22×C109C2×SD1614yes—
113C22×C109C22×D411yescharacteristic
114C2×D109—22no—
115C2×D109—41no—
116C2×D109—41no—
117C2×D109—42no—
118C2×D109—42no—
119C2×D109—42no—
120C2×D109—88no—
121C4×C109—22no—
122C4×C109—21no—
123C4×C109C2×SD1612yes—
124C4×C109C2×#4111yescharacteristic
125C4×C109H(5,𝔽2)11yescharacteristic
126Dic109—41no—
127Dic109—41no—
128Dic109—42no—
129Dic109—41no—
130C2×C109—24no—
131C2×C109#54611yescharacteristic
132C2×C109C22×SD1611yescharacteristic
133C2×C109#57611yescharacteristic
134D109—42no—
135D109—82no—
136#3295—1091nomaximal, Sylow
137C109#329511yescharacteristic, complemented, Sylow
138#554—1091no—
139D42—1091no—
140#486—1091no—
141#481—1092no—
142#476—1091no—
143#465—1092no—
144#460—1091no—
145#450—1092no—
146C4×SD16—1091no—
147C8×D4D10911yescharacteristic, complemented
148C2×#102—1092no—
149C22×D4—1092no—
150C22×D4—2181no—
151C2×SD16—1092no—
152C2×SD16—1091no—
153C22×C8C2×D10912yescomplemented
154C4⋊Q8—1091no—
155#127—1091no—
156#125—2181no—
157C22⋊Q8—1092no—
158C22⋊Q8—2182no—
159#121—1092no—
160#121—2181no—
161#120—2182no—
162C4×Q8—1091no—
163C4×D4—1091no—
164C4×D4—2181no—
165C4×D4C2×D10911yescharacteristic
166C2×C4⋊C4—1092no—
167#106—1091no—
168#106—1092no—
169C4⋊C8C2×D10911yescharacteristic, complemented
170#103—1091no—
171#102—1091no—
172#102—1092no—
173#102—1092no—
174#102—1092no—
175#102—1092no—
176C22⋊C8C2×D10912yescomplemented
177C4×C8C2×D10911yescharacteristic, complemented
178C24—2182no—
179C2×Q8—1091no—
180C2×Q8—2181no—
181C2×D4—1091no—
182C2×D4—1092no—
183C2×D4—4362no—
184C2×D4—4364no—
185C2×D4—2184no—
186C2×D4—2182no—
187C2×D4—2181no—
188C2×D4—2182no—
189C2×D4—2181no—
190C2×D4D43611yescharacteristic
191C22×C4—2182no—
192C22×C4—2181no—
193C22×C4C22×D10912yes—
194SD16—4362no—
195SD16—2182no—
196C2×C8—42no—
197C2×C8C22×D10911yescharacteristic
198C2×C8C22×D10911yescharacteristic
199C2×C8C22×D10912yes—
200C4⋊C4—1091no—
201C4⋊C4—1092no—
202C4⋊C4—1091no—
203C4⋊C4—2181no—
204C4⋊C4—2182no—
205C4⋊C4—2182no—
206C4⋊C4—2181no—
207C4⋊C4D43611yescharacteristic
208C22⋊C4—2182no—
209C22⋊C4—2181no—
210C22⋊C4—2182no—
211C22⋊C4D43612yes—
212C42—2181no—
213C42C22×D10911yescharacteristic
214C23—8722no—
215C23—4364no—
216C23—4364no—
217C23—2182no—
218C23—2181no—
219C23C2×D43612yes—
220Q8—8721no—
221Q8—2181no—
222Q8—2181no—
223D4—8721no—
224D4—1092no—
225D4—4364no—
226D4—4364no—
227D4—4362no—
228D4—2182no—
229D4—2182no—
230D4—2181no—
231D4#1318201814yescomplemented
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—8722no—
235C2×C4—4362no—
236C2×C4—2181no—
237C2×C4—2181no—
238C2×C4—2181no—
239C2×C4—2182no—
240C2×C4C2×D43611yescharacteristic
241C2×C4C2×D43612yes—
242C2×C4C2×D43611yescharacteristic
243C2×C4C23×D10911yescharacteristic
244C8—21no—
245C8—21no—
246C8—22no—
247C22—22no—
248C22—22no—
249C22—8728no—
250C22—4362no—
251C22—4362no—
252C22—4362no—
253C22—4361no—
254C22—4361no—
255C22—2182no—
256C22C2×#1318201814yes—
257C22C22×D43611yescharacteristic
258C4—21no—
259C4—22no—
260C4—4361no—
261C4—4361no—
262C4—4362no—
263C4—4361no—
264C4C2×#1318201812yes—
265C4#41×D10911yescharacteristic
266C4#1681274411yescharacteristic
267C2—24no—
268C2—8722no—
269C2—4362no—
270C2#2029567111yescharacteristic
271C2C22×#1318201811yescharacteristic
272C2#2029583111yescharacteristic
subgroupG/H
centerC22C22×D436
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC8×D4×C109C2
socleC22×C109C22×D4
prime radicalD4×D436C2
hypercenterC8×D4D109
hypocenterC109#3295
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4

Complements

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

NH ≅ G/Ntype
1. #20295733C2semidirect
3. #20295373C2semidirect
4. #20295345C2semidirect
5. #20295317C2semidirect
6. #20295255C2semidirect
7. #20295227C2semidirect
8. #20295179C2semidirect
9. C4×#13182018C2semidirect
10. C2×#16812621C2semidirect
11. C8×D4×C109C2semidirect
15. C2×#13182018C22semidirect
16. #16812699C22semidirect
19. #16812678C22semidirect
24. C4×Dic218C22semidirect
27. C2×C4⋊Dic109C22semidirect
28. #16812631C22semidirect
29. #16812631C22semidirect
30. #16812624C22semidirect
33. #16812621C22semidirect
34. #16812621C22semidirect
35. #16812621C22semidirect
36. C22×C8×C109C22semidirect
38. C4⋊C8×C109C22semidirect
39. C22⋊C8×C109C22semidirect
40. C4×C8×C109C22semidirect
83. D4×C109SD16semidirect
137. C109#3295semidirect
147. C8×D4D109semidirect
153. C22×C8C2×D109semidirect
169. C4⋊C8C2×D109semidirect
176. C22⋊C8C2×D109semidirect
177. C4×C8C2×D109semidirect
231. D4#13182018semidirect

Maximal subgroups

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

subgroupindex
#202957332
D4×D4362
#202953732
#202953452
#202953172
#202952552
#202952272
#202951792
C4×#131820182
C2×#168126212
C8×D4×C1092
#3295109