Finite Group Databasebeta

← #28372708

Subgroups — #28372708

Order 13952.

504 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16812725C211yesmaximal, characteristic, complemented
2C2×#16812706C211yesmaximal, characteristic, complemented
3#20295604C211yesmaximal, characteristic, complemented
4#20295277C212yesmaximal, complemented
5#20295270C212yesmaximal, complemented
6#20295260C212yesmaximal, complemented
7#20295253C212yesmaximal, complemented
8#20295069C211yesmaximal, characteristic, complemented
9#20295020C211yesmaximal, characteristic, complemented
10C2×#16812620C211yesmaximal, characteristic, complemented
11#535×C109C211yesmaximal, characteristic, complemented
12C2×#13182029C2211yescharacteristic
13#16812725—24no—
14C2×#13182017C2212yes—
15C2×#13182014C2212yes—
16#16812706—24no—
17C2×#13182013C2211yescharacteristic
18C2×#13182013C2212yescomplemented
19C22×#11724784C2211yescharacteristic
20C22×#11724784C2212yescomplemented
21#16812678C2212yescomplemented
22#16812675—21no—
23#16812673C2212yescomplemented
24#16812671—22no—
25#16812670—21no—
26C2×C4⋊Dic109C2211yescharacteristic
27#16812632C2212yescomplemented
28#16812630C2212yescomplemented
29#16812622C2211yescharacteristic, complemented
30#16812622C2211yescharacteristic, complemented
31#16812620C2212yes—
32#16812620C2212yes—
33#16812619C2211yescharacteristic, complemented
34#16812619C2211yescharacteristic, complemented
35C2×#41×C109C2211yescharacteristic
36C22×D4×C109C2211yescharacteristic
37#127×C109—21no—
38#124×C109—21no—
39#121×C109C2212yes—
40#121×C109C2212yes—
41#120×C109—22no—
42#117×C109C2211yescharacteristic
43#13182029—22no—
44#13182029—22no—
45C2×Dic218C2311yescharacteristic
46C2×D436C2311yescharacteristic
47C2×C22⋊D109—21no—
48C2×C22⋊D109—22no—
49C22×Dic109—21no—
50C2×C4×D109—21no—
51#13182017—24no—
52#13182014—24no—
53#13182013—24no—
54#13182013D414yescomplemented
55C2×#11724784—22no—
56C2×#11724784—24no—
57C2×#11724784C2311yescharacteristic
58C2×#11724784C2311yescharacteristic
59C2×#11724784C2312yes—
60C4⋊Dic109—21no—
61C4⋊Dic109C2312yes—
62#13182008—22no—
63C22⋊Dic109—22no—
64#13182006—22no—
65C24×C109—21no—
66#41×C109—22no—
67#41×C109—22no—
68C2×Q8×C109D411yescharacteristic
69C2×D4×C109—22no—
70C2×D4×C109—22no—
71C2×D4×C109—22no—
72C2×D4×C109—21no—
73C2×D4×C109—22no—
74C2×D4×C109D412yes—
75C2×D4×C109D411yescharacteristic
76C2×D4×C109C2312yes—
77C22×C4×C109C2312yes—
78C22×C4×C109C2311yescharacteristic
79C4⋊C4×C109C2311yescharacteristic
80C4⋊C4×C109C2311yescharacteristic
81C22⋊C4×C109—21no—
82C22⋊C4×C109—22no—
83C22⋊C4×C109—22no—
84C42×C109—21no—
85C23×C109—22no—
86C23×C109—42no—
87C23×C109—81no—
88C23×C109—81no—
89C23×C109C2×D412yes—
90C23×C109C2×D411yescharacteristic
91C22×D109—21no—
92Q8×C109—22no—
93D4×C109—24no—
94D4×C109—22no—
95D4×C109—22no—
96D4×C109—22no—
97D4×C109—42no—
98D4×C109—82no—
99D4×C109—82no—
100C2×C4×C109—22no—
101C2×C4×C109—22no—
102C2×C4×C109—21no—
103C2×C4×C109—21no—
104C2×C4×C109C2×D412yes—
105C2×C4×C109C2×D412yes—
106C2×C4×C109C2×D411yescharacteristic
107C2×C4×C109#4112yes—
108C2×C4×C109C2411yescharacteristic
109C22⋊D109—42no—
110C22⋊D109—82no—
111C2×Dic109—22no—
112C2×Dic109—21no—
113C2×Dic109—81no—
114D436—21no—
115D436—21no—
116C4×D109—42no—
117Dic218—21no—
118Dic218—21no—
119#11724784—21no—
120#11724784—21no—
121#11724784C2×D414yes—
122C22×C109—21no—
123C22×C109—21no—
124C22×C109—22no—
125C22×C109—22no—
126C22×C109—22no—
127C22×C109—42no—
128C22×C109—42no—
129C22×C109—42no—
130C22×C109—84no—
131C22×C109#12112yes—
132C22×C109C22×D411yescharacteristic
133C2×D109—41no—
134C2×D109—41no—
135C2×D109—41no—
136C4×C109—22no—
137C4×C109—41no—
138C4×C109—41no—
139C4×C109#12112yes—
140C4×C109C22×D411yescharacteristic
141C4×C109C2×#4111yescharacteristic
142Dic109—41no—
143Dic109—42no—
144C2×C109—22no—
145C2×C109—42no—
146C2×C109—82no—
147C2×C109C2×#12111yescharacteristic
148C2×C109#57611yescharacteristic
149C2×C109#57711yescharacteristic
150D109—81no—
151#3062—1091nomaximal, Sylow
152C109#306211yescharacteristic, complemented, Sylow
153C2×AGL(1,ℤ/8)—1091no—
154C2×#131—1091no—
155#537—1091no—
156#535D10911yescharacteristic, complemented
157#469—1092no—
158#468—1092no—
159#466—1092no—
160#465—1092no—
161#428—1091no—
162#416—1091no—
163C2×#102—1091no—
164C2×#41—1091no—
165C2×#41C2×D10911yescharacteristic
166C22×D4C2×D10911yescharacteristic
167AGL(1,ℤ/8)—2184no—
168C2×SD16—1092no—
169C2×D8—1092no—
170#131—2184no—
171C2×M16—1091no—
172C2×M16—1092no—
173C22×C8—1091no—
174C22×C8—1092no—
175#127—21no—
176#124—21no—
177C22⋊Q8—1092no—
178C22⋊Q8—2181no—
179#121—1092no—
180#121—2181no—
181#121—2182no—
182#121C2×D10912yes—
183#121C2×D10912yes—
184#120—22no—
185#117C2×D10911yescharacteristic
186C2×C4⋊C4—1091no—
187#107—1092no—
188#106—1092no—
189#103—1091no—
190#103—1091no—
191#102—1092no—
192#102—1092no—
193#102—1091no—
194#102—1091no—
195C24—21no—
196#41—22no—
197#41—22no—
198#41—2182no—
199#41—2182no—
200C2×Q8—1091no—
201C2×Q8C22⋊D10911yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—21no—
206C2×D4—22no—
207C2×D4—1091no—
208C2×D4—2182no—
209C2×D4—2181no—
210C2×D4C22⋊D10912yes—
211C2×D4C22⋊D10911yescharacteristic
212C2×D4C22×D10912yes—
213C22×C4—2181no—
214C22×C4—2181no—
215C22×C4C22×D10912yes—
216C22×C4C22×D10911yescharacteristic
217SD16—2184no—
218D8—2184no—
219M16—1094no—
220M16—2184no—
221C2×C8—1092no—
222C2×C8—1091no—
223C2×C8—1091no—
224C2×C8—2184no—
225C2×C8—2182no—
226C4⋊C4—1092no—
227C4⋊C4—2182no—
228C4⋊C4—2181no—
229C4⋊C4C22×D10911yescharacteristic
230C4⋊C4C22×D10911yescharacteristic
231C22⋊C4—21no—
232C22⋊C4—22no—
233C22⋊C4—22no—
234C22⋊C4—2182no—
235C22⋊C4—2182no—
236C42—21no—
237C23—22no—
238C23—42no—
239C23—81no—
240C23—81no—
241C23—2181no—
242C23C2×C22⋊D10912yes—
243C23C2×C22⋊D10911yescharacteristic
244Q8—22no—
245Q8—2181no—
246Q8—2181no—
247D4—24no—
248D4—22no—
249D4—22no—
250D4—22no—
251D4—42no—
252D4—82no—
253D4—82no—
254D4—8722no—
255D4—4362no—
256D4—2181no—
257D4—2181no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—8721no—
263C2×C4—4362no—
264C2×C4—2182no—
265C2×C4—2181no—
266C2×C4D4×D10912yes—
267C2×C4C2×C22⋊D10912yes—
268C2×C4C2×C22⋊D10911yescharacteristic
269C2×C4#1318202812yes—
270C2×C4C23×D10911yescharacteristic
271C8—1094no—
272C8—2181no—
273C8—2181no—
274C22—21no—
275C22—21no—
276C22—22no—
277C22—22no—
278C22—22no—
279C22—42no—
280C22—42no—
281C22—42no—
282C22—84no—
283C22—4361no—
284C22—4361no—
285C22—4361no—
286C22#1681267112yes—
287C22C22×C22⋊D10911yescharacteristic
288C4—22no—
289C4—41no—
290C4—41no—
291C4—4361no—
292C4—4362no—
293C4#1681267112yes—
294C4C2×D4×D10911yescharacteristic
295C4C2×#1318202811yescharacteristic
296C2—22no—
297C2—42no—
298C2—82no—
299C2—8721no—
300C2C2×#1681267111yescharacteristic
301C2#2029582911yescharacteristic
302C2#2029583711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#535×C109C2
socleC22×C109C22×D4
hypercenter#535D109
hypocenterC109#3062
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C2×#41 ◁ #535

Complements

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

NH ≅ G/Ntype
1. C2×#16812725C2semidirect
2. C2×#16812706C2semidirect
3. #20295604C2semidirect
4. #20295277C2semidirect
5. #20295270C2semidirect
6. #20295260C2semidirect
7. #20295253C2semidirect
8. #20295069C2semidirect
9. #20295020C2semidirect
10. C2×#16812620C2semidirect
11. #535×C109C2semidirect
18. C2×#13182013C22semidirect
20. C22×#11724784C22semidirect
21. #16812678C22semidirect
23. #16812673C22semidirect
27. #16812632C22semidirect
28. #16812630C22semidirect
29. #16812622C22semidirect
30. #16812622C22semidirect
33. #16812619C22semidirect
34. #16812619C22semidirect
54. #13182013D4semidirect
152. C109#3062semidirect
156. #535D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127252
C2×#168127062
#202956042
#202952772
#202952702
#202952602
#202952532
#202950692
#202950202
C2×#168126202
#535×C1092
#3062109