Finite Group Databasebeta

← #28372638

Subgroups — #28372638

Order 13952.

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

normal subgroups only

This group has 500 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×#16812719C211yesmaximal, characteristic, complemented
2C22×#13182013C211yesmaximal, characteristic, complemented
3#20295537C211yesmaximal, characteristic, complemented
4#20295282C212yesmaximal, complemented
5#20295277C212yesmaximal, complemented
6#20295274C212yesmaximal, complemented
7#20295270C212yesmaximal, complemented
8#20295062C211yesmaximal, characteristic, complemented
9#20295022C211yesmaximal, characteristic, complemented
10#20295020C211yesmaximal, characteristic, complemented
11#525×C109C211yesmaximal, characteristic, complemented
12C2×#13182029C2211yescharacteristic
13#16812719—24no—
14C2×#13182019C2211yescharacteristic, complemented
15C2×#13182017C2211yescharacteristic, complemented
16C2×#13182016C2211yescharacteristic, complemented
17C2×#13182014C2211yescharacteristic, complemented
18C2×#13182013—24no—
19C2×#13182013C2212yes—
20C2×#13182013C2212yes—
21C22×#11724784C2211yescharacteristic
22C22×#11724784C2211yescharacteristic
23#16812682—21no—
24#16812678C2212yes—
25#16812673C2212yes—
26#16812668—21no—
27C4×C22⋊D109—22no—
28#16812658C2211yescharacteristic
29#16812632C2212yescomplemented
30#16812630C2212yescomplemented
31#16812623C2212yescomplemented
32#16812622C2212yescomplemented
33#16812620C2212yescomplemented
34#16812619C2212yescomplemented
35C2×#41×C109C2211yescharacteristic
36C23×C4×C109C2211yescharacteristic
37#123×C109—21no—
38C22⋊Q8×C109C2212yes—
39#121×C109C2212yes—
40#120×C109—21no—
41C4×D4×C109—22no—
42#117×C109C2211yescharacteristic
43#13182029—22no—
44#13182029—22no—
45C2×Dic218C2311yescharacteristic
46C2×D436C2311yescharacteristic
47C2×C22⋊D109—22no—
48C2×C22⋊D109—21no—
49C2×C4×D109—21no—
50#13182019—22no—
51#13182017—22no—
52#13182016—22no—
53#13182014—22no—
54#13182013—24no—
55#13182013—24no—
56C2×#11724784—22no—
57C2×#11724784D414yescomplemented
58C2×#11724784C2312yes—
59C2×#11724784C2311yescharacteristic
60C2×#11724784C2311yescharacteristic
61C4⋊Dic109C2312yes—
62#13182008—22no—
63C22⋊Dic109—21no—
64C22⋊Dic109—22no—
65#13182006—22no—
66C4×Dic109—21no—
67C24×C109D411yescharacteristic
68#41×C109—22no—
69#41×C109—22no—
70C2×Q8×C109C2311yescharacteristic
71C2×D4×C109—22no—
72C2×D4×C109—21no—
73C2×D4×C109C2311yescharacteristic
74C22×C4×C109—22no—
75C22×C4×C109—22no—
76C22×C4×C109—21no—
77C22×C4×C109D412yes—
78C22×C4×C109D411yescharacteristic
79C22×C4×C109C2312yes—
80C22×C4×C109C2311yescharacteristic
81C4⋊C4×C109—22no—
82C4⋊C4×C109C2312yes—
83C22⋊C4×C109—21no—
84C22⋊C4×C109—22no—
85C22⋊C4×C109—22no—
86C42×C109—21no—
87C23×C109—22no—
88C23×C109—21no—
89C23×C109—42no—
90C23×C109C2×D412yes—
91C23×C109C2×D411yescharacteristic
92C22×D109—21no—
93Q8×C109—21no—
94Q8×C109—21no—
95D4×C109—21no—
96D4×C109—21no—
97D4×C109—44no—
98D4×C109—42no—
99C2×C4×C109—22no—
100C2×C4×C109—24no—
101C2×C4×C109—22no—
102C2×C4×C109—22no—
103C2×C4×C109—22no—
104C2×C4×C109—21no—
105C2×C4×C109—42no—
106C2×C4×C109C2×D412yes—
107C2×C4×C109C2×D412yes—
108C2×C4×C109C2×D411yescharacteristic
109C2×C4×C109#4112yes—
110C2×C4×C109C2411yescharacteristic
111C22⋊D109—44no—
112C22⋊D109—42no—
113C2×Dic109—22no—
114C2×Dic109—21no—
115D436—21no—
116D436—21no—
117C4×D109—42no—
118Dic218—21no—
119Dic218—21no—
120#11724784—22no—
121#11724784C2×D412yes—
122#11724784C2×D412yes—
123C22×C109—22no—
124C22×C109—22no—
125C22×C109—22no—
126C22×C109—22no—
127C22×C109—44no—
128C22×C109—41no—
129C22×C109—42no—
130C22×C109#12112yes—
131C22×C109C22×D411yescharacteristic
132C2×D109—41no—
133C2×D109—42no—
134C4×C109—22no—
135C4×C109—41no—
136C4×C109—42no—
137C4×C109#12112yes—
138C4×C109C22×D411yescharacteristic
139C4×C109C2×#4111yescharacteristic
140Dic109—41no—
141Dic109—42no—
142C2×C109—22no—
143C2×C109—42no—
144C2×C109—81no—
145C2×C109C2×#12111yescharacteristic
146C2×C109#57512yes—
147D109—81no—
148#3054—1091nomaximal, Sylow
149C109#305411yescharacteristic, complemented, Sylow
150C2×#135—1091no—
151C22×M16—1091no—
152#525—1091no—
153#525D10911yescharacteristic, complemented
154#470—1092no—
155#469—1092no—
156#468—1092no—
157#468—1092no—
158#427—1091no—
159#416—1091no—
160#416—1091no—
161C2×#41—1091no—
162C2×#41C2×D10911yescharacteristic
163C23×C4C2×D10911yescharacteristic
164#135—2184no—
165C2×Q16—1091no—
166C2×SD16—1091no—
167C2×SD16—1091no—
168C2×D8—1091no—
169C2×M16—1092no—
170C2×M16—1092no—
171C2×M16—2184no—
172C22×C8—1091no—
173C22×C8—1091no—
174#123—21no—
175#123—2181no—
176C22⋊Q8—1092no—
177C22⋊Q8C2×D10912yes—
178#121—1092no—
179#121C2×D10912yes—
180#120—21no—
181#120—2181no—
182C4×D4—22no—
183C4×D4—2182no—
184#117—1091no—
185#117C2×D10911yescharacteristic
186#107—1092no—
187#106—1092no—
188#103—1092no—
189#103—1092no—
190#102—1092no—
191#102—1092no—
192C24C22⋊D10911yescharacteristic
193#41—22no—
194#41—22no—
195#41—2182no—
196#41—2182no—
197C2×Q8—1091no—
198C2×Q8C22×D10911yescharacteristic
199C2×D4—22no—
200C2×D4—21no—
201C2×D4—1091no—
202C2×D4—2182no—
203C2×D4—2181no—
204C2×D4C22×D10911yescharacteristic
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—21no—
208C22×C4—2181no—
209C22×C4C22⋊D10911yescharacteristic
210C22×C4C22⋊D10912yes—
211C22×C4C22×D10912yes—
212C22×C4C22×D10911yescharacteristic
213Q16—2182no—
214SD16—2182no—
215SD16—2182no—
216D8—2182no—
217M16—2184no—
218M16—2184no—
219C2×C8—1094no—
220C2×C8—1091no—
221C2×C8—1091no—
222C2×C8—1092no—
223C2×C8—2182no—
224C4⋊C4—22no—
225C4⋊C4—1092no—
226C4⋊C4—2182no—
227C4⋊C4C22×D10912yes—
228C22⋊C4—21no—
229C22⋊C4—22no—
230C22⋊C4—22no—
231C22⋊C4—2181no—
232C22⋊C4—2182no—
233C22⋊C4—2182no—
234C42—21no—
235C42—2181no—
236C23—22no—
237C23—21no—
238C23—42no—
239C23—2181no—
240C23C2×C22⋊D10912yes—
241C23C2×C22⋊D10911yescharacteristic
242Q8—21no—
243Q8—21no—
244Q8—2181no—
245Q8—2181no—
246D4—21no—
247D4—21no—
248D4—44no—
249D4—42no—
250D4—4364no—
251D4—4362no—
252D4—2181no—
253D4—2181no—
254C2×C4—22no—
255C2×C4—24no—
256C2×C4—22no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—21no—
260C2×C4—42no—
261C2×C4—4362no—
262C2×C4—2182no—
263C2×C4—2181no—
264C2×C4D4×D10912yes—
265C2×C4C2×C22⋊D10912yes—
266C2×C4C2×C22⋊D10911yescharacteristic
267C2×C4#1318202812yes—
268C2×C4C23×D10911yescharacteristic
269C8—1092no—
270C8—1092no—
271C8—2182no—
272C22—22no—
273C22—22no—
274C22—22no—
275C22—22no—
276C22—44no—
277C22—41no—
278C22—42no—
279C22—4361no—
280C22—4362no—
281C22#1681267112yes—
282C22C22×C22⋊D10911yescharacteristic
283C4—22no—
284C4—41no—
285C4—42no—
286C4—4361no—
287C4—4362no—
288C4#1681267112yes—
289C4C2×D4×D10911yescharacteristic
290C4C2×#1318202811yescharacteristic
291C2—22no—
292C2—42no—
293C2—81no—
294C2—8721no—
295C2C2×#1681267111yescharacteristic
296C2#2029582412yes—
subgroupG/H
centerC2×C4C2×C22⋊D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#525×C109C2
socleC22×C109C22×D4
hypercenter#525D109
hypocenterC109#3054
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ #525

Complements

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

NH ≅ G/Ntype
1. C2×#16812719C2semidirect
2. C22×#13182013C2semidirect
3. #20295537C2semidirect
4. #20295282C2semidirect
5. #20295277C2semidirect
6. #20295274C2semidirect
7. #20295270C2semidirect
8. #20295062C2semidirect
9. #20295022C2semidirect
10. #20295020C2semidirect
11. #525×C109C2semidirect
14. C2×#13182019C22semidirect
15. C2×#13182017C22semidirect
16. C2×#13182016C22semidirect
17. C2×#13182014C22semidirect
29. #16812632C22semidirect
30. #16812630C22semidirect
31. #16812623C22semidirect
32. #16812622C22semidirect
33. #16812620C22semidirect
34. #16812619C22semidirect
57. C2×#11724784D4semidirect
149. C109#3054semidirect
153. #525D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127192
C22×#131820132
#202955372
#202952822
#202952772
#202952742
#202952702
#202950622
#202950222
#202950202
#525×C1092
#3054109