Finite Group Databasebeta

← #28372411

Subgroups — #28372411

Order 13952.

704 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 161 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
1C2×#16812729C212yesmaximal, complemented
2#20295597C211yesmaximal, characteristic, complemented
3#20295209C214yesmaximal, complemented
4#20295204C214yesmaximal, complemented
5#20295043C212yesmaximal, complemented
6#20295010C211yesmaximal, characteristic, complemented
7#585×C109C211yesmaximal, characteristic
8C22×Dic218C2211yescharacteristic, complemented
9#16812729—24no—
10#16812729C2218yescomplemented
11C2×#13182019—22no—
12C2×#13182017—22no—
13C2×#13182013C2212yescomplemented
14#16812697C2212yescomplemented
15#16812686C2212yescomplemented
16#16812680—22no—
17#16812678—22no—
18#16812658C2211yescharacteristic, complemented
19#16812626C2218yescomplemented
20#16812618C2212yescomplemented
21#16812616C2212yescomplemented
222−1+4×C109—22no—
232−1+4×C109C2212yes—
24H(5,𝔽2)×C109—22no—
25H(5,𝔽2)×C109—22no—
26H(5,𝔽2)×C109C2212yes—
27C2×#41×C109—24no—
28C2×#41×C109—22no—
29C2×#41×C109C2212yes—
30C2×#41×C109C2211yescharacteristic
31C2×Dic218—22no—
32C2×Dic218—22no—
33C2×Dic218—21no—
34C2×Dic218C2312yes—
35C22×Dic109—21no—
36#13182019—28no—
37#13182017—28no—
38#13182013—22no—
39#13182013C2314yes—
40C2×#11724784—22no—
41C4⋊Dic109—21no—
42#13182008—24no—
43C22⋊Dic109—21no—
44C22⋊Dic109—24no—
45C4×Dic109C2312yes—
46#41×C109—24no—
47#41×C109—28no—
48#41×C109—24no—
49#41×C109—24no—
50#41×C109—24no—
51#41×C109—22no—
52#41×C109—22no—
53#41×C109—24no—
54#41×C109—22no—
55#41×C109—22no—
56#41×C109D414yes—
57#41×C109C2314yes—
58C2×Q8×C109—22no—
59C2×Q8×C109—24no—
60C2×Q8×C109D412yes—
61C2×Q8×C109C2311yescharacteristic
62C2×D4×C109—28no—
63C2×D4×C109—22no—
64C2×D4×C109—24no—
65C2×D4×C109—22no—
66C2×D4×C109—24no—
67C2×D4×C109D412yes—
68C2×D4×C109D412yes—
69C2×D4×C109C2311yescharacteristic
70C22×C4×C109—24no—
71C22×C4×C109—22no—
72C22×C4×C109D412yes—
73C22×C4×C109C2311yescharacteristic
74C23×C109—24no—
75C23×C109—22no—
76C23×C109C2×D412yes—
77C23×C109C2×D411yescharacteristic
78Q8×C109—24no—
79Q8×C109—22no—
80Q8×C109—22no—
81Q8×C109C2×D414yes—
82D4×C109—28no—
83D4×C109—24no—
84D4×C109—24no—
85D4×C109—24no—
86D4×C109—24no—
87D4×C109—22no—
88D4×C109—22no—
89D4×C109C2×D414yes—
90C2×C4×C109—24no—
91C2×C4×C109—28no—
92C2×C4×C109—22no—
93C2×C4×C109—24no—
94C2×C4×C109—22no—
95C2×C4×C109—22no—
96C2×C4×C109—24no—
97C2×C4×C109C2×D412yes—
98C2×C4×C109C2×D411yescharacteristic
99C2×C4×C109C2×D412yes—
100C2×C4×C109C2×D412yes—
101C2×C4×C109C2411yescharacteristic
102C2×Dic109—22no—
103C2×Dic109—22no—
104C2×Dic109—42no—
105Dic218—24no—
106Dic218—41no—
107Dic218—41no—
108#11724784—24no—
109C22×C109—22no—
110C22×C109—24no—
111C22×C109—41no—
112C22×C109—42no—
113C22×C109—82no—
114C22×C109—84no—
115C22×C109#12012yes—
116C22×C109C22×D411yescharacteristic
117C4×C109—24no—
118C4×C109—22no—
119C4×C109#12011yescharacteristic
120C4×C109#12011yescharacteristic
121C4×C109C22×D412yes—
122Dic109—42no—
123Dic109—42no—
124C2×C109—21no—
125C2×C109—22no—
126C2×C109—42no—
127C2×C109—44no—
128C2×C109C2×#12011yescharacteristic
129#3022—1091nomaximal, Sylow
130C109#302211yescharacteristic, complemented, Sylow
131#585D10911yescharacteristic
132C2×#137—1092no—
133#536—1091no—
134#456—1094no—
135#455—1094no—
136#421—1092no—
137#413—1091no—
1382−1+4—22no—
1392−1+4C2×D10912yes—
140H(5,𝔽2)—22no—
141H(5,𝔽2)—22no—
142H(5,𝔽2)C2×D10912yes—
143C2×#41—24no—
144C2×#41—22no—
145C2×#41C2×D10912yes—
146C2×#41C2×D10911yescharacteristic
147C22×Q8—1091no—
148#137—1098no—
149#137—2184no—
150C2×Q16—2182no—
151C2×SD16—2182no—
152C2×M16—1092no—
153C4⋊Q8—1092no—
154#124—1092no—
155#123—2182no—
156C22⋊Q8—2182no—
157#117—1091no—
158#104—1098no—
159#101—1092no—
160#100—1092no—
161#41—24no—
162#41—28no—
163#41—24no—
164#41—24no—
165#41—24no—
166#41—22no—
167#41—22no—
168#41—24no—
169#41—22no—
170#41—22no—
171#41C22⋊D10914yes—
172#41C22×D10914yes—
173C2×Q8—22no—
174C2×Q8—24no—
175C2×Q8—1092no—
176C2×Q8—2182no—
177C2×Q8—2182no—
178C2×Q8—2181no—
179C2×Q8C22⋊D10912yes—
180C2×Q8C22×D10911yescharacteristic
181C2×D4—28no—
182C2×D4—22no—
183C2×D4—24no—
184C2×D4—22no—
185C2×D4—24no—
186C2×D4C22⋊D10912yes—
187C2×D4C22⋊D10912yes—
188C2×D4C22×D10911yescharacteristic
189C22×C4—24no—
190C22×C4—22no—
191C22×C4—2181no—
192C22×C4C22⋊D10912yes—
193C22×C4C22×D10911yescharacteristic
194Q16—2188no—
195SD16—2188no—
196M16—1094no—
197M16—2182no—
198C2×C8—2182no—
199C4⋊C4—2184no—
200C4⋊C4—2181no—
201C22⋊C4—2181no—
202C22⋊C4—2184no—
203C42—1092no—
204C23—24no—
205C23—22no—
206C23C2×C22⋊D10912yes—
207C23C2×C22⋊D10911yescharacteristic
208Q8—24no—
209Q8—22no—
210Q8—22no—
211Q8—4361no—
212Q8—4361no—
213Q8—2184no—
214Q8C2×C22⋊D10914yes—
215D4—28no—
216D4—24no—
217D4—24no—
218D4—24no—
219D4—24no—
220D4—22no—
221D4—22no—
222D4C2×C22⋊D10914yes—
223C2×C4—24no—
224C2×C4—28no—
225C2×C4—22no—
226C2×C4—24no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—24no—
230C2×C4—4362no—
231C2×C4—2182no—
232C2×C4—2182no—
233C2×C4C2×C22⋊D10912yes—
234C2×C4C2×C22⋊D10911yescharacteristic
235C2×C4C2×C22⋊D10912yes—
236C2×C4C2×C22⋊D10912yes—
237C2×C4C23×D10911yescharacteristic
238C8—2184no—
239C22—22no—
240C22—24no—
241C22—41no—
242C22—42no—
243C22—82no—
244C22—84no—
245C22#1681266812yes—
246C22C22×C22⋊D10911yescharacteristic
247C4—24no—
248C4—22no—
249C4—4362no—
250C4—4362no—
251C4#1681266811yescharacteristic
252C4#1681266811yescharacteristic
253C4C22×C22⋊D10912yes—
254C2—21no—
255C2—22no—
256C2—42no—
257C2—44no—
258C2C2×#1681266811yescharacteristic
subgroupG/H
centerC2C2×#16812668
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting, prime radical#585×C109C2
socleC2×C109C2×#120
hypercenter#585D109
hypocenterC109#3022
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2 ◁ C22×C4 ◁ #585

Complements

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

NH ≅ G/Ntype
1. C2×#16812729C2semidirect
2. #20295597C2semidirect
3. #20295209C2semidirect
4. #20295204C2semidirect
5. #20295043C2semidirect
6. #20295010C2semidirect
8. C22×Dic218C22semidirect
10. #16812729C22semidirect
13. C2×#13182013C22semidirect
14. #16812697C22semidirect
15. #16812686C22semidirect
18. #16812658C22semidirect
19. #16812626C22semidirect
20. #16812618C22semidirect
21. #16812616C22semidirect
130. C109#3022semidirect

Maximal subgroups

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

subgroupindex
C2×#168127292
#202955972
#202952092
#202952042
#202950432
#202950102
#585×C1092
#3022109