Finite Group Databasebeta

← #28374858

Subgroups — #28374858

Order 13952.

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

normal subgroups only

This group has 462 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#20295736C211yesmaximal, characteristic, complemented
2D4×D436C211yesmaximal, characteristic
3#20295373C211yesmaximal, characteristic, complemented
4#20295346C212yesmaximal, complemented
5#20295316C211yesmaximal, characteristic, complemented
6#20295258C212yesmaximal, complemented
7#20295226C211yesmaximal, characteristic, complemented
8#20295175C212yesmaximal, complemented
9C8×D436C211yesmaximal, characteristic, complemented
10#102×D109C212yesmaximal, complemented
11C4×SD16×C109C211yesmaximal, characteristic, complemented
12C22×D436—21no—
13C2×D4×D109C2212yes—
14C2×#13182016C2212yescomplemented
15C2×C8×D109C2212yescomplemented
16#16812699C2211yescharacteristic, complemented
17#16812696C2211yescharacteristic
18#16812691—21no—
19#16812677—22no—
20#16812675C2212yescomplemented
21#16812674C2212yes—
22#16812673—21no—
23#16812669—22no—
24C4×D436—21no—
25C4×D436C2211yescharacteristic
26C4⋊C4×D109C2212yescomplemented
27#16812630C2212yescomplemented
28C4⋊#11724784C2211yescharacteristic, complemented
29#16812621C2212yescomplemented
30#16812621C2212yescomplemented
31#16812620C2212yes—
32#16812619C2212yescomplemented
33#16812611C2212yescomplemented
34C2×SD16×C109C2211yescharacteristic
35C4×Q8×C109C2211yescharacteristic, complemented
36C4×D4×C109C2211yescharacteristic
37#106×C109C2211yescharacteristic, complemented
38#103×C109C2211yescharacteristic, complemented
39#102×C109C2211yescharacteristic
40C4×C8×C109C2211yescharacteristic, complemented
41C23×D109—22no—
42C2×Dic218—21no—
43C2×D436—22no—
44C2×D436—21no—
45C2×D436—21no—
46C2×D436—44no—
47C2×D436D411yescharacteristic
48C2×D436C2312yes—
49C2×C22⋊D109—22no—
50D4×D109—24no—
51D4×D109—42no—
52C2×C4×D109—22no—
53C2×C4×D109C2312yes—
54#13182016—42no—
55C2×#11724784C2312yes—
56C8×D109—42no—
57C4⋊Dic109—21no—
58C4⋊Dic109D411yescharacteristic
59C4⋊Dic109C2312yes—
60#13182008—22no—
61#13182008—22no—
62#13182006—22no—
63#13182006—22no—
64#13182006D412yes—
65C2×Q8×C109C2311yescharacteristic
66C2×D4×C109C2311yescharacteristic
67C22×C4×C109—21no—
68SD16×C109—22no—
69C2×C8×C109C2311yescharacteristic
70C2×C8×C109C2311yescharacteristic
71C4⋊C4×C109—21no—
72C4⋊C4×C109C2311yescharacteristic
73C4⋊C4×C109C2311yescharacteristic
74C22⋊C4×C109—21no—
75C42×C109—21no—
76C42×C109C2311yescharacteristic
77C23×C109—21no—
78C22×D109—22no—
79C22×D109—44no—
80C22×D109—44no—
81C22×D109—82no—
82C22×D109C2×D412yes—
83Q8×C109—21no—
84Q8×C109—21no—
85D4×C109—21no—
86D4×C109#4112yes—
87C2×C4×C109—21no—
88C2×C4×C109—21no—
89C2×C4×C109—21no—
90C2×C4×C109—42no—
91C2×C4×C109C2×D411yescharacteristic
92C2×C4×C109C2×D411yescharacteristic
93C2×C4×C109C2411yescharacteristic
94C22⋊D109—44no—
95C2×Dic109—22no—
96C2×Dic109C2×D412yes—
97D436—22no—
98D436—22no—
99D436—44no—
100D436—42no—
101D436—81no—
102D436SD1614yescomplemented
103C4×D109—22no—
104C4×D109—22no—
105C4×D109—82no—
106Dic218—81no—
107C8×C109—21no—
108C8×C109—21no—
109#11724784—22no—
110C22×C109—22no—
111C22×C109—41no—
112C22×C109—41no—
113C22×C109C22×D411yescharacteristic
114C2×D109—22no—
115C2×D109—22no—
116C2×D109—42no—
117C2×D109—42no—
118C2×D109—42no—
119C2×D109—88no—
120C2×D109C2×SD1614yes—
121C4×C109—21no—
122C4×C109—41no—
123C4×C109—41no—
124C4×C109—41no—
125C4×C109C2×SD1612yes—
126C4×C109C2×#4111yescharacteristic
127C4×C109H(5,𝔽2)11yescharacteristic
128Dic109—22no—
129Dic109—42no—
130C2×C109—42no—
131C2×C109#54611yescharacteristic
132C2×C109C22×SD1611yescharacteristic
133C2×C109#57611yescharacteristic
134D109—24no—
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×SD16D10911yescharacteristic, complemented
147C8×D4—1091no—
148C2×#102—1092no—
149C22×D4—1092no—
150C22×D4—2181no—
151C2×SD16—1092no—
152C2×SD16C2×D10911yescharacteristic
153C22×C8—1092no—
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×Q8C2×D10911yescharacteristic, complemented
163C4×D4—1091no—
164C4×D4—2181no—
165C4×D4C2×D10911yescharacteristic
166C2×C4⋊C4—1092no—
167#106—1092no—
168#106C2×D10911yescharacteristic, complemented
169C4⋊C8—1091no—
170#103C2×D10911yescharacteristic, complemented
171#102—1092no—
172#102—1092no—
173#102—1092no—
174#102—1092no—
175#102C2×D10911yescharacteristic
176C22⋊C8—1092no—
177C4×C8C2×D10911yescharacteristic, complemented
178C24—2182no—
179C2×Q8—2181no—
180C2×Q8C22×D10911yescharacteristic
181C2×D4—1091no—
182C2×D4—1092no—
183C2×D4—4364no—
184C2×D4—4362no—
185C2×D4—2182no—
186C2×D4—2182no—
187C2×D4—2184no—
188C2×D4—2181no—
189C2×D4—2181no—
190C2×D4C22×D10911yescharacteristic
191C22×C4—21no—
192C22×C4—1092no—
193C22×C4—2182no—
194SD16—22no—
195SD16—4362no—
196C2×C8—1092no—
197C2×C8—4362no—
198C2×C8C22×D10911yescharacteristic
199C2×C8C22×D10911yescharacteristic
200C4⋊C4—21no—
201C4⋊C4—1091no—
202C4⋊C4—1092no—
203C4⋊C4—2182no—
204C4⋊C4—2182no—
205C4⋊C4—2181no—
206C4⋊C4C22×D10911yescharacteristic
207C4⋊C4C22×D10911yescharacteristic
208C22⋊C4—21no—
209C22⋊C4—1092no—
210C22⋊C4—2182no—
211C22⋊C4—2182no—
212C42—21no—
213C42C22×D10911yescharacteristic
214C23—21no—
215C23—8722no—
216C23—1092no—
217C23—4364no—
218C23—4364no—
219C23—2182no—
220Q8—21no—
221Q8—21no—
222Q8—8721no—
223D4—21no—
224D4—8721no—
225D4—1094no—
226D4—4364no—
227D4—4364no—
228D4—4362no—
229D4—2182no—
230D4—2182no—
231D4#1318202912yes—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—42no—
236C2×C4—8722no—
237C2×C4—1092no—
238C2×C4—2182no—
239C2×C4—2182no—
240C2×C4—2182no—
241C2×C4D4×D10911yescharacteristic
242C2×C4D4×D10911yescharacteristic
243C2×C4C23×D10911yescharacteristic
244C8—21no—
245C8—21no—
246C8—2182no—
247C22—22no—
248C22—41no—
249C22—41no—
250C22—8728no—
251C22—1094no—
252C22—4362no—
253C22—4362no—
254C22—4362no—
255C22—2182no—
256C22—2182no—
257C22C2×D4×D10911yescharacteristic
258C4—21no—
259C4—41no—
260C4—41no—
261C4—41no—
262C4—4362no—
263C4—2182no—
264C4SD16×D10912yes—
265C4C2×#1318202911yescharacteristic
266C4#1681274511yescharacteristic
267C2—42no—
268C2—8722no—
269C2—2184no—
270C2#2029567211yescharacteristic
271C2C2×SD16×D10911yescharacteristic
272C2#2029582811yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC4×SD16×C109C2
socleC22×C109C22×D4
prime radicalD4×D436C2
hypercenterC4×SD16D109
hypocenterC109#3295
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ C4×SD16

Complements

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

NH ≅ G/Ntype
1. #20295736C2semidirect
3. #20295373C2semidirect
4. #20295346C2semidirect
5. #20295316C2semidirect
6. #20295258C2semidirect
7. #20295226C2semidirect
8. #20295175C2semidirect
9. C8×D436C2semidirect
10. #102×D109C2semidirect
11. C4×SD16×C109C2semidirect
14. C2×#13182016C22semidirect
15. C2×C8×D109C22semidirect
16. #16812699C22semidirect
20. #16812675C22semidirect
26. C4⋊C4×D109C22semidirect
27. #16812630C22semidirect
28. C4⋊#11724784C22semidirect
29. #16812621C22semidirect
30. #16812621C22semidirect
32. #16812619C22semidirect
33. #16812611C22semidirect
35. C4×Q8×C109C22semidirect
37. #106×C109C22semidirect
38. #103×C109C22semidirect
40. C4×C8×C109C22semidirect
102. D436SD16semidirect
137. C109#3295semidirect
146. C4×SD16D109semidirect
162. C4×Q8C2×D109semidirect
168. #106C2×D109semidirect
170. #103C2×D109semidirect
177. C4×C8C2×D109semidirect

Maximal subgroups

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

subgroupindex
#202957362
D4×D4362
#202953732
#202953462
#202953162
#202952582
#202952262
#202951752
C8×D4362
#102×D1092
C4×SD16×C1092
#3295109