Finite Group Databasebeta

← #28372678

Subgroups — #28372678

Order 13952.

524 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 524 conjugacy classes of subgroups — too many to draw. Only its 127 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×D8×D109C211yesmaximal, characteristic, complemented
2C2×#16812707C211yesmaximal, characteristic, complemented
3#20295592C212yesmaximal, complemented
4#20295280C211yesmaximal, characteristic, complemented
5#20295279C211yesmaximal, characteristic, complemented
6#20295276C211yesmaximal, characteristic, complemented
7#20295261C212yesmaximal, complemented
8#20295260C212yesmaximal, complemented
9#20295061C211yesmaximal, characteristic, complemented
10#20295029C212yesmaximal, complemented
11#469×C109C211yesmaximal, characteristic, complemented
12C2×#13182029C2211yescharacteristic
13C2×D4×D109C2212yes—
14C2×D872C2211yescharacteristic
15C2×#13182014C2212yes—
16D8×D109—24no—
17#16812707—24no—
18C2×#13182012C2211yescharacteristic
19C2×#13182012C2211yescharacteristic, complemented
20C22×#11724784C2211yescharacteristic, complemented
21C2×C8×D109C2211yescharacteristic
22C2×C8×D109C2211yescharacteristic, complemented
23#16812695—22no—
24#16812685—22no—
25#16812674C2212yescomplemented
26#16812673C2212yescomplemented
27#16812670C2212yescomplemented
28#16812669—22no—
29#16812667—22no—
30#16812657C2212yes—
31#16812632C2212yescomplemented
32#16812631C2211yescharacteristic, complemented
33#16812621C2212yescomplemented
34#16812620C2212yescomplemented
35#16812619C2212yescomplemented
36C2×D8×C109C2211yescharacteristic
37C2×M16×C109C2211yescharacteristic, complemented
38#121×C109C2212yescomplemented
39#106×C109C2211yescharacteristic, complemented
40#102×C109C2212yescomplemented
41C23×D109—22no—
42#13182029—22no—
43#13182029—22no—
44C2×Dic218D411yescharacteristic
45C2×D436—22no—
46C2×D436D411yescharacteristic
47C2×D436C2312yes—
48C2×C22⋊D109—22no—
49C2×C22⋊D109—22no—
50C2×C22⋊D109—22no—
51C2×C22⋊D109D411yescharacteristic
52C2×C22⋊D109D411yescharacteristic
53D4×D109—24no—
54D4×D109—24no—
55C2×C4×D109C2311yescharacteristic
56C2×C4×D109C2311yescharacteristic
57D872—22no—
58#13182014—24no—
59#13182012—22no—
60#13182012—22no—
61C2×#11724784—22no—
62C2×#11724784C2311yescharacteristic
63C2×#11724784C2311yescharacteristic
64C8×D109—22no—
65C8×D109D414yescomplemented
66C4⋊Dic109C2312yes—
67C22⋊Dic109—22no—
68#13182006—22no—
69#13182006—22no—
70#13182006—22no—
71C4×Dic109—22no—
72C2×D4×C109—22no—
73C2×D4×C109C2312yes—
74C22×C4×C109C2311yescharacteristic
75D8×C109—22no—
76M16×C109—22no—
77C2×C8×C109C2311yescharacteristic
78C2×C8×C109C2311yescharacteristic
79C4⋊C4×C109C2312yes—
80C22⋊C4×C109—22no—
81C23×C109—22no—
82C23×C109C2×D411yescharacteristic
83C22×D109—22no—
84C22×D109—44no—
85C22×D109—82no—
86C22×D109—82no—
87C22×D109C2×D411yescharacteristic
88C22×D109C2×D411yescharacteristic
89D4×C109—22no—
90D4×C109—22no—
91D4×C109—82no—
92C2×C4×C109—21no—
93C2×C4×C109—21no—
94C2×C4×C109—22no—
95C2×C4×C109C2×D411yescharacteristic
96C2×C4×C109C2411yescharacteristic
97C22⋊D109—22no—
98C22⋊D109—22no—
99C22⋊D109—44no—
100C22⋊D109—82no—
101C22⋊D109—82no—
102C2×Dic109—22no—
103C2×Dic109C2×D411yescharacteristic
104C2×Dic109C2×D411yescharacteristic
105D436—22no—
106D436—22no—
107D436—22no—
108D436—82no—
109C4×D109—21no—
110C4×D109—21no—
111C4×D109C2×D412yes—
112C4×D109#4112yes—
113Dic218—22no—
114C8×C109—21no—
115C8×C109C2×D412yes—
116#11724784—21no—
117#11724784C2×D412yes—
118C22×C109—21no—
119C22×C109—22no—
120C22×C109—42no—
121C22×C109—42no—
122C22×C109—42no—
123C22×C109C22×D411yescharacteristic
124C2×D109—22no—
125C2×D109—21no—
126C2×D109—22no—
127C2×D109—42no—
128C2×D109—42no—
129C2×D109—42no—
130C2×D109—88no—
131C2×D109#12112yes—
132C4×C109—21no—
133C4×C109—42no—
134C4×C109C22×D411yescharacteristic
135C4×C109C2×#4111yescharacteristic
136Dic109—21no—
137Dic109—42no—
138Dic109#12112yes—
139C2×C109—41no—
140C2×C109—82no—
141C2×C109C2×#12111yescharacteristic
142C2×C109#57612yes—
143D109—22no—
144D109—41no—
145D109—82no—
146#3058—1091nomaximal, Sylow
147C109#305811yescharacteristic, complemented, Sylow
148C22×D8—1091no—
149C2×#131—1091no—
150#535—1092no—
151#469—1091no—
152#469—1091no—
153#469—1091no—
154#469D10911yescharacteristic, complemented
155#466—1092no—
156#466—1092no—
157#427—1091no—
158#418—1092no—
159C2×#41—1091no—
160C22×D4—1092no—
161C2×D8—1092no—
162C2×D8—1091no—
163C2×D8—2184no—
164C2×D8C2×D10911yescharacteristic
165#131—2184no—
166C2×M16—1091no—
167C2×M16—1091no—
168C2×M16C2×D10911yescharacteristic, complemented
169C22×C8—1091no—
170C22×C8—1091no—
171C22×C8—1091no—
172#127—2182no—
173#124—2182no—
174#121—1092no—
175#121—1092no—
176#121—1092no—
177#121C2×D10912yescomplemented
178#120—2182no—
179#120—2182no—
180#117—1092no—
181#107—1092no—
182#106—1091no—
183#106C2×D10911yescharacteristic, complemented
184#102—1092no—
185#102—1092no—
186#102—1092no—
187#102C2×D10912yescomplemented
188C24—2182no—
189#41—2182no—
190#41—2182no—
191C2×Q8—1091no—
192C2×D4—22no—
193C2×D4—1091no—
194C2×D4—1091no—
195C2×D4—1091no—
196C2×D4—1092no—
197C2×D4—2182no—
198C2×D4—2184no—
199C2×D4—2184no—
200C2×D4—2182no—
201C2×D4—2182no—
202C2×D4—2182no—
203C2×D4C22×D10912yes—
204C22×C4—1091no—
205C22×C4—1091no—
206C22×C4C22×D10911yescharacteristic
207D8—22no—
208D8—2184no—
209D8—2182no—
210M16—22no—
211M16—2182no—
212M16—2182no—
213C2×C8—1091no—
214C2×C8—1094no—
215C2×C8—1091no—
216C2×C8—2182no—
217C2×C8—2182no—
218C2×C8C22×D10911yescharacteristic
219C2×C8C22×D10911yescharacteristic
220C4⋊C4—1092no—
221C4⋊C4C22×D10912yes—
222C22⋊C4—22no—
223C22⋊C4—2182no—
224C22⋊C4—2182no—
225C22⋊C4—2182no—
226C22⋊C4—2182no—
227C42—2182no—
228C23—22no—
229C23—8722no—
230C23—8722no—
231C23—1091no—
232C23—1091no—
233C23—4364no—
234C23—2182no—
235C23D4×D10911yescharacteristic
236Q8—2182no—
237D4—22no—
238D4—22no—
239D4—82no—
240D4—8722no—
241D4—8722no—
242D4—8722no—
243D4—4364no—
244D4—2182no—
245D4—2182no—
246D4—2182no—
247D4—2182no—
248D4—2182no—
249C2×C4—21no—
250C2×C4—21no—
251C2×C4—22no—
252C2×C4—1091no—
253C2×C4—1092no—
254C2×C4—1092no—
255C2×C4—1091no—
256C2×C4—2182no—
257C2×C4—2181no—
258C2×C4—2181no—
259C2×C4D4×D10911yescharacteristic
260C2×C4C23×D10911yescharacteristic
261C8—21no—
262C8—1092no—
263C8—2181no—
264C8D4×D10912yes—
265C22—21no—
266C22—22no—
267C22—42no—
268C22—42no—
269C22—42no—
270C22—8728no—
271C22—1092no—
272C22—4362no—
273C22—4362no—
274C22—4362no—
275C22—2181no—
276C22—2182no—
277C22—2182no—
278C22C2×D4×D10911yescharacteristic
279C4—21no—
280C4—42no—
281C4—1092no—
282C4—4362no—
283C4—2181no—
284C4C2×D4×D10911yescharacteristic
285C4#41×D10911yescharacteristic
286C2—41no—
287C2—82no—
288C2—8722no—
289C2—4361no—
290C2—2182no—
291C2#121×D10911yescharacteristic
292C2#2029583012yes—
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#469×C109C2
socleC22×C109C22×D4
hypercenter#469D109
hypocenterC109#3058
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #469

Complements

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

NH ≅ G/Ntype
1. C2×D8×D109C2semidirect
2. C2×#16812707C2semidirect
3. #20295592C2semidirect
4. #20295280C2semidirect
5. #20295279C2semidirect
6. #20295276C2semidirect
7. #20295261C2semidirect
8. #20295260C2semidirect
9. #20295061C2semidirect
10. #20295029C2semidirect
11. #469×C109C2semidirect
19. C2×#13182012C22semidirect
20. C22×#11724784C22semidirect
22. C2×C8×D109C22semidirect
25. #16812674C22semidirect
26. #16812673C22semidirect
27. #16812670C22semidirect
31. #16812632C22semidirect
32. #16812631C22semidirect
33. #16812621C22semidirect
34. #16812620C22semidirect
35. #16812619C22semidirect
37. C2×M16×C109C22semidirect
38. #121×C109C22semidirect
39. #106×C109C22semidirect
40. #102×C109C22semidirect
65. C8×D109D4semidirect
147. C109#3058semidirect
154. #469D109semidirect
168. C2×M16C2×D109semidirect
177. #121C2×D109semidirect
183. #106C2×D109semidirect
187. #102C2×D109semidirect

Maximal subgroups

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

subgroupindex
C2×D8×D1092
C2×#168127072
#202955922
#202952802
#202952792
#202952762
#202952612
#202952602
#202950612
#202950292
#469×C1092
#3058109