Finite Group Databasebeta

← #28372643

Subgroups — #28372643

Order 13952.

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

normal subgroups only

This group has 500 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×#16812722C211yesmaximal, characteristic, complemented
2C2×M16×D109C211yesmaximal, characteristic, complemented
3#20295539C212yesmaximal, complemented
4#20295281C211yesmaximal, characteristic, complemented
5#20295280C211yesmaximal, characteristic, complemented
6#20295279C211yesmaximal, characteristic, complemented
7#20295275C212yesmaximal, complemented
8#20295274C212yesmaximal, complemented
9#20295061C211yesmaximal, characteristic, complemented
10#20295021C212yesmaximal, complemented
11#470×C109C211yesmaximal, characteristic, complemented
12C2×#13182030C2212yes—
13C22×C4×D109C2211yescharacteristic
14#16812722—24no—
15C2×#13182016C2212yescomplemented
16C2×D872C2211yescharacteristic, complemented
17C2×#13182013C2211yescharacteristic
18C2×#13182012C2211yescharacteristic
19C2×#13182012C2211yescharacteristic
20M16×D109—24no—
21C2×C8×D109C2211yescharacteristic
22C2×C8×D109C2211yescharacteristic
23#16812684—22no—
24#16812675C2212yes—
25#16812674C2212yes—
26#16812673C2212yes—
27#16812669—22no—
28#16812661—22no—
29#16812659—22no—
30#16812657C2212yes—
31#16812632C2212yescomplemented
32#16812631C2211yescharacteristic, complemented
33#16812623C2212yescomplemented
34#16812621C2212yescomplemented
35#16812619C2212yescomplemented
36C2×Q16×C109C2211yescharacteristic, complemented
37C2×M16×C109C2211yescharacteristic
38C22⋊Q8×C109C2212yes—
39#106×C109C2211yescharacteristic, complemented
40#103×C109C2212yescomplemented
41C23×D109D411yescharacteristic
42#13182030—24no—
43#13182030—24no—
44C2×D436—22no—
45C2×D436—22no—
46C2×D436C2312yes—
47C2×C22⋊D109—22no—
48C22×Dic109D411yescharacteristic
49C2×C4×D109—22no—
50C2×C4×D109—22no—
51C2×C4×D109—22no—
52C2×C4×D109D411yescharacteristic
53C2×C4×D109D411yescharacteristic
54C2×C4×D109C2311yescharacteristic
55C2×C4×D109C2311yescharacteristic
56#13182016—24no—
57D872—22no—
58#13182013—22no—
59#13182012—22no—
60#13182012—22no—
61C2×#11724784C2311yescharacteristic
62C2×#11724784C2311yescharacteristic
63C8×D109—22no—
64C8×D109D414yescomplemented
65C4⋊Dic109C2312yes—
66#13182008—22no—
67#13182006—22no—
68#13182006—22no—
69#13182006—22no—
70#13182006—22no—
71C4×Dic109—22no—
72C2×Q8×C109C2312yes—
73C22×C4×C109C2311yescharacteristic
74Q16×C109—22no—
75M16×C109—22no—
76C2×C8×C109C2311yescharacteristic
77C2×C8×C109C2311yescharacteristic
78C4⋊C4×C109—22no—
79C4⋊C4×C109C2312yes—
80C22⋊C4×C109—22no—
81C23×C109C2×D411yescharacteristic
82C22×D109—22no—
83C22×D109—22no—
84C22×D109—42no—
85C22×D109C2×D411yescharacteristic
86C22×D109C2×D411yescharacteristic
87Q8×C109—22no—
88Q8×C109—22no—
89C2×C4×C109—21no—
90C2×C4×C109—21no—
91C2×C4×C109—22no—
92C2×C4×C109—22no—
93C2×C4×C109C2×D411yescharacteristic
94C2×C4×C109C2411yescharacteristic
95C22⋊D109—44no—
96C2×Dic109—22no—
97C2×Dic109—22no—
98C2×Dic109C2×D411yescharacteristic
99C2×Dic109C2×D411yescharacteristic
100D436—22no—
101D436—22no—
102D436—44no—
103D436—44no—
104C4×D109—22no—
105C4×D109—21no—
106C4×D109—21no—
107C4×D109—22no—
108C4×D109—44no—
109C4×D109C2×D412yes—
110C4×D109#4112yes—
111C8×C109—21no—
112C8×C109C2×D412yes—
113#11724784—21no—
114#11724784C2×D412yes—
115C22×C109—21no—
116C22×C109—22no—
117C22×C109C22×D411yescharacteristic
118C2×D109—22no—
119C2×D109—21no—
120C2×D109—22no—
121C2×D109—44no—
122C2×D109—42no—
123C2×D109—42no—
124C2×D109—42no—
125C2×D109#12112yes—
126C4×C109—21no—
127C4×C109—42no—
128C4×C109—42no—
129C4×C109C22×D411yescharacteristic
130C4×C109C2×#4111yescharacteristic
131Dic109—21no—
132Dic109—42no—
133Dic109#12112yes—
134C2×C109—41no—
135C2×C109C2×#12111yescharacteristic
136C2×C109#57512yes—
137D109—22no—
138D109—41no—
139D109—82no—
140#3054—1091nomaximal, Sylow
141C109#305411yescharacteristic, complemented, Sylow
142C2×#135—1091no—
143C22×M16—1091no—
144#525—1092no—
145#470—1091no—
146#470D10911yescharacteristic, complemented
147#469—1091no—
148#469—1091no—
149#468—1092no—
150#468—1092no—
151#427—1091no—
152#416—1092no—
153C2×#41—1092no—
154C23×C4—1091no—
155#135—2184no—
156C2×Q16C2×D10911yescharacteristic, complemented
157C2×SD16—1092no—
158C2×D8—1091no—
159C2×M16—1091no—
160C2×M16—1091no—
161C2×M16—1091no—
162C2×M16—2184no—
163C2×M16C2×D10911yescharacteristic
164C22×C8—1091no—
165C22×C8—1091no—
166#123—2182no—
167C22⋊Q8—1092no—
168C22⋊Q8C2×D10912yes—
169#121—1092no—
170#121—1092no—
171#120—2182no—
172C4×D4—2182no—
173C4×D4—2182no—
174#117—1092no—
175#107—1092no—
176#106—1091no—
177#106C2×D10911yescharacteristic, complemented
178#103—1092no—
179#103C2×D10912yescomplemented
180#102—1092no—
181#102—1092no—
182C24—1091no—
183#41—2184no—
184#41—2184no—
185C2×Q8C22×D10912yes—
186C2×D4—1092no—
187C2×D4—2182no—
188C2×D4—2182no—
189C2×D4—2182no—
190C22×C4—1091no—
191C22×C4—1091no—
192C22×C4—1091no—
193C22×C4—1091no—
194C22×C4—1091no—
195C22×C4—2182no—
196C22×C4—2182no—
197C22×C4—2182no—
198C22×C4C22×D10911yescharacteristic
199Q16—22no—
200SD16—2184no—
201D8—2182no—
202M16—22no—
203M16—2182no—
204M16—2182no—
205M16—2182no—
206C2×C8—1091no—
207C2×C8—1094no—
208C2×C8—1091no—
209C2×C8—2182no—
210C2×C8C22×D10911yescharacteristic
211C2×C8C22×D10911yescharacteristic
212C4⋊C4—22no—
213C4⋊C4—1092no—
214C4⋊C4—2182no—
215C4⋊C4C22×D10912yes—
216C22⋊C4—22no—
217C22⋊C4—2182no—
218C22⋊C4—2182no—
219C22⋊C4—2182no—
220C22⋊C4—2182no—
221C42—2182no—
222C23—1091no—
223C23—1091no—
224C23—4362no—
225C23—2182no—
226C23—2182no—
227C23D4×D10911yescharacteristic
228Q8—22no—
229Q8—22no—
230D4—4364no—
231D4—4364no—
232D4—4364no—
233D4—2182no—
234D4—2182no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—1091no—
240C2×C4—1092no—
241C2×C4—1092no—
242C2×C4—1091no—
243C2×C4—4364no—
244C2×C4—2182no—
245C2×C4—2182no—
246C2×C4—2182no—
247C2×C4—2182no—
248C2×C4—2181no—
249C2×C4—2181no—
250C2×C4D4×D10911yescharacteristic
251C2×C4C23×D10911yescharacteristic
252C8—21no—
253C8—1092no—
254C8—2181no—
255C8D4×D10912yes—
256C22—21no—
257C22—22no—
258C22—1092no—
259C22—4364no—
260C22—4362no—
261C22—4362no—
262C22—4362no—
263C22—2181no—
264C22—2182no—
265C22—2182no—
266C22C2×D4×D10911yescharacteristic
267C4—21no—
268C4—42no—
269C4—42no—
270C4—1092no—
271C4—4362no—
272C4—2181no—
273C4C2×D4×D10911yescharacteristic
274C4#41×D10911yescharacteristic
275C2—41no—
276C2—8722no—
277C2—4361no—
278C2—2182no—
279C2#121×D10911yescharacteristic
280C2#2029582512yes—
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#470×C109C2
socleC22×C109C22×D4
hypercenter#470D109
hypocenterC109#3054
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#16812722C2semidirect
2. C2×M16×D109C2semidirect
3. #20295539C2semidirect
4. #20295281C2semidirect
5. #20295280C2semidirect
6. #20295279C2semidirect
7. #20295275C2semidirect
8. #20295274C2semidirect
9. #20295061C2semidirect
10. #20295021C2semidirect
11. #470×C109C2semidirect
15. C2×#13182016C22semidirect
16. C2×D872C22semidirect
31. #16812632C22semidirect
32. #16812631C22semidirect
33. #16812623C22semidirect
34. #16812621C22semidirect
35. #16812619C22semidirect
36. C2×Q16×C109C22semidirect
39. #106×C109C22semidirect
40. #103×C109C22semidirect
64. C8×D109D4semidirect
141. C109#3054semidirect
146. #470D109semidirect
156. C2×Q16C2×D109semidirect
177. #106C2×D109semidirect
179. #103C2×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127222
C2×M16×D1092
#202955392
#202952812
#202952802
#202952792
#202952752
#202952742
#202950612
#202950212
#470×C1092
#3054109