Finite Group Databasebeta

← #28373512

Subgroups — #28373512

Order 13952.

596 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 137 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×#16812723C212yesmaximal, complemented
2C2×#16812718C211yesmaximal, characteristic, complemented
3C2×D8×D109C211yesmaximal, characteristic, complemented
4#20295589C212yesmaximal, complemented
5#20295414C212yesmaximal, complemented
6#20295413C212yesmaximal, complemented
7#20295410C211yesmaximal, characteristic, complemented
8#20295409C211yesmaximal, characteristic, complemented
9#20295401C211yesmaximal, characteristic, complemented
10#20294980C211yesmaximal, characteristic, complemented
11#493×C109C211yesmaximal, characteristic, complemented
12C2×#13182028C2212yes—
13C2×D4×D109C2212yes—
14#16812723—28no—
15#16812718—24no—
16C2×Dic436C2211yescharacteristic, complemented
17C2×#13182018C2212yescomplemented
18C2×#13182017C2212yescomplemented
19C2×#13182017C2212yescomplemented
20C2×D872C2211yescharacteristic
21C2×#13182014C2212yes—
22C2×#13182014C2212yescomplemented
23D8×D109—24no—
24C2×#13182012C2211yescharacteristic
25C2×C8×D109C2211yescharacteristic
26#16812695C2212yescomplemented
27#16812687C2212yescomplemented
28#16812686C2212yescomplemented
29#16812671—24no—
30#16812667—24no—
31#16812656C2211yescharacteristic
32#16812610C2211yescharacteristic, complemented
33#16812609C2211yescharacteristic, complemented
34C8×Dic109C2211yescharacteristic, complemented
35C2×D8×C109C2211yescharacteristic
36C2×D8×C109C2211yescharacteristic, complemented
37C2×D8×C109C2212yescomplemented
38#127×C109C2212yescomplemented
39C4×C8×C109C2211yescharacteristic, complemented
40C23×D109—22no—
41#13182028—24no—
42#13182028—24no—
43C2×Dic218C2312yes—
44C2×D436C2312yes—
45C2×C22⋊D109—22no—
46C2×C22⋊D109—22no—
47C2×C22⋊D109—24no—
48D4×D109—24no—
49D4×D109—24no—
50C22×Dic109—22no—
51C2×C4×D109C2311yescharacteristic
52Dic436—22no—
53#13182018—24no—
54#13182017—24no—
55#13182017—24no—
56D872—22no—
57#13182014—24no—
58#13182014—24no—
59#13182012D414yescomplemented
60C2×#11724784C2311yescharacteristic
61C2×#11724784C2311yescharacteristic
62C8×D109D414yescomplemented
63#13182008D412yes—
64C22⋊Dic109—24no—
65#13182006—24no—
66#13182006D412yes—
67C4×Dic109C2311yescharacteristic
68C2×D4×C109—24no—
69C2×D4×C109C2312yes—
70C2×D4×C109C2312yes—
71D8×C109—22no—
72D8×C109—22no—
73D8×C109—24no—
74C2×C8×C109C2311yescharacteristic
75C2×C8×C109C2311yescharacteristic
76C42×C109C2311yescharacteristic
77C23×C109—22no—
78C23×C109—22no—
79C22×D109—22no—
80C22×D109—44no—
81C22×D109—84no—
82C22×D109C2×D411yescharacteristic
83D4×C109—22no—
84D4×C109—22no—
85D4×C109—22no—
86D4×C109—22no—
87D4×C109—84no—
88C2×C4×C109C2×D412yes—
89C2×C4×C109C2411yescharacteristic
90C22⋊D109—44no—
91C22⋊D109—44no—
92C22⋊D109—84no—
93C2×Dic109—22no—
94C2×Dic109—44no—
95C2×Dic109C2×D412yes—
96C2×Dic109C2×D411yescharacteristic
97D436—22no—
98D436—22no—
99C4×D109C2×D412yes—
100C4×D109C2×D412yes—
101Dic218—22no—
102Dic218—22no—
103C8×C109C2×D412yes—
104C8×C109C2×D412yes—
105#11724784C2×D412yes—
106#11724784C2×D412yes—
107C22×C109—42no—
108C22×C109—42no—
109C22×C109—44no—
110C22×C109—44no—
111C22×C109C22×D411yescharacteristic
112C2×D109—22no—
113C2×D109—42no—
114C2×D109—44no—
115C2×D109—88no—
116C2×D109#12712yes—
117C4×C109—22no—
118C4×C109C22×D411yescharacteristic
119C4×C109C22×D411yescharacteristic
120Dic109—22no—
121Dic109—42no—
122Dic109#12712yes—
123C2×C109—82no—
124C2×C109—82no—
125C2×C109C2×#12711yescharacteristic
126C2×C109#57612yes—
127D109—22no—
128D109—82no—
129#3155—1091nomaximal, Sylow
130C109#315511yescharacteristic, complemented, Sylow
131C2×AGL(1,ℤ/8)—1092no—
132C2×#135—1091no—
133C22×D8—1091no—
134#535—1092no—
135#496—1092no—
136#496—1092no—
137#495—1091no—
138#495—1091no—
139#493—1091no—
140#493D10911yescharacteristic, complemented
141#405—1091no—
142C2×#41—1092no—
143C22×D4—1092no—
144AGL(1,ℤ/8)—2188no—
145#135—2184no—
146C2×Q16—1091no—
147C2×SD16—1092no—
148C2×SD16—1092no—
149C2×SD16—1092no—
150C2×D8—1092no—
151C2×D8—1092no—
152C2×D8—1091no—
153C2×D8—2184no—
154C2×D8C2×D10911yescharacteristic
155C2×D8C2×D10911yescharacteristic, complemented
156C2×D8C2×D10912yescomplemented
157C2×M16—1091no—
158C22×C8—1091no—
159#127—1092no—
160#127C2×D10912yescomplemented
161#124—1092no—
162#124—1092no—
163#121—2184no—
164#120—2184no—
165#117—1091no—
166#97—1091no—
167#97—1091no—
168C4×C8—1091no—
169C4×C8C2×D10911yescharacteristic, complemented
170C24—2182no—
171#41—2184no—
172#41—2184no—
173C2×Q8—1092no—
174C2×D4—24no—
175C2×D4—1092no—
176C2×D4—2184no—
177C2×D4—2184no—
178C2×D4—2184no—
179C2×D4—2182no—
180C2×D4—2182no—
181C2×D4C22×D10912yes—
182C2×D4C22×D10912yes—
183C22×C4—1091no—
184C22×C4—2182no—
185Q16—2182no—
186SD16—2184no—
187SD16—2184no—
188SD16—2184no—
189D8—22no—
190D8—22no—
191D8—24no—
192D8—2184no—
193D8—2184no—
194D8—2182no—
195M16—1094no—
196C2×C8—1091no—
197C2×C8—1091no—
198C2×C8—1094no—
199C2×C8C22×D10911yescharacteristic
200C2×C8C22×D10911yescharacteristic
201C4⋊C4—1092no—
202C22⋊C4—1092no—
203C22⋊C4—2184no—
204C22⋊C4—2184no—
205C42—1091no—
206C42C22×D10911yescharacteristic
207C23—22no—
208C23—22no—
209C23—8724no—
210C23—1091no—
211C23—4364no—
212C23—2182no—
213Q8—2182no—
214Q8—2182no—
215D4—22no—
216D4—22no—
217D4—22no—
218D4—22no—
219D4—84no—
220D4—8724no—
221D4—4364no—
222D4—4364no—
223D4—2182no—
224D4—2182no—
225C2×C4—1091no—
226C2×C4—1092no—
227C2×C4—1092no—
228C2×C4—1092no—
229C2×C4—4364no—
230C2×C4—2182no—
231C2×C4D4×D10912yes—
232C2×C4C23×D10911yescharacteristic
233C8—1092no—
234C8—1092no—
235C8D4×D10912yes—
236C8D4×D10912yes—
237C22—42no—
238C22—42no—
239C22—44no—
240C22—44no—
241C22—8728no—
242C22—1092no—
243C22—4362no—
244C22—4364no—
245C22—2182no—
246C22C2×D4×D10911yescharacteristic
247C4—22no—
248C4—1092no—
249C4—4362no—
250C4—2182no—
251C4C2×D4×D10911yescharacteristic
252C4C2×D4×D10911yescharacteristic
253C2—82no—
254C2—82no—
255C2—8722no—
256C2—2182no—
257C2#127×D10911yescharacteristic
258C2#2029582712yes—
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#493×C109C2
socleC22×C109C22×D4
hypercenter#493D109
hypocenterC109#3155
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #493

Complements

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

NH ≅ G/Ntype
1. C2×#16812723C2semidirect
2. C2×#16812718C2semidirect
3. C2×D8×D109C2semidirect
4. #20295589C2semidirect
5. #20295414C2semidirect
6. #20295413C2semidirect
7. #20295410C2semidirect
8. #20295409C2semidirect
9. #20295401C2semidirect
10. #20294980C2semidirect
11. #493×C109C2semidirect
16. C2×Dic436C22semidirect
17. C2×#13182018C22semidirect
18. C2×#13182017C22semidirect
19. C2×#13182017C22semidirect
22. C2×#13182014C22semidirect
26. #16812695C22semidirect
27. #16812687C22semidirect
28. #16812686C22semidirect
32. #16812610C22semidirect
33. #16812609C22semidirect
34. C8×Dic109C22semidirect
36. C2×D8×C109C22semidirect
37. C2×D8×C109C22semidirect
38. #127×C109C22semidirect
39. C4×C8×C109C22semidirect
59. #13182012D4semidirect
62. C8×D109D4semidirect
130. C109#3155semidirect
140. #493D109semidirect
155. C2×D8C2×D109semidirect
156. C2×D8C2×D109semidirect
160. #127C2×D109semidirect
169. C4×C8C2×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127232
C2×#168127182
C2×D8×D1092
#202955892
#202954142
#202954132
#202954102
#202954092
#202954012
#202949802
#493×C1092
#3155109