Finite Group Databasebeta

← #28373481

Subgroups — #28373481

Order 13952.

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

normal subgroups only

This group has 656 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×#16812726C212yesmaximal, complemented
2C2×#16812723C212yesmaximal, complemented
3#20295575C211yesmaximal, characteristic, complemented
4#127×D109C211yesmaximal, characteristic, complemented
5#20295416C211yesmaximal, characteristic, complemented
6#20295415C211yesmaximal, characteristic, complemented
7#20295413C211yesmaximal, characteristic, complemented
8#20295401C212yesmaximal, complemented
9#20295397C212yesmaximal, complemented
10#20294978C211yesmaximal, characteristic, complemented
11#496×C109C211yesmaximal, characteristic, complemented
12C2×#13182030C2211yescharacteristic
13C2×#13182028C2211yescharacteristic
14C2×D4×D109—22no—
15C2×D4×D109C2212yes—
16#16812726—28no—
17#16812723—28no—
18C2×#13182018C2212yescomplemented
19C2×#13182017C2212yescomplemented
20C2×#13182016C2212yescomplemented
21C2×D872C2212yescomplemented
22C2×#13182014C2212yescomplemented
23C2×#13182014C2212yescomplemented
24C2×#13182012C2212yes—
25#16812697C2211yescharacteristic, complemented
26#16812696C2211yescharacteristic
27#16812695—22no—
28#16812695C2212yes—
29#16812695C2211yescharacteristic, complemented
30#16812687C2211yescharacteristic, complemented
31#16812672—22no—
32#16812659—22no—
33C42×D109C2211yescharacteristic
34#16812610C2211yescharacteristic, complemented
35C8×Dic109C2212yescomplemented
36C2×SD16×C109C2212yescomplemented
37C2×D8×C109C2212yescomplemented
38#127×C109C2211yescharacteristic
39#124×C109C2211yescharacteristic, complemented
40#97×C109C2211yescharacteristic, complemented
41C23×D109—22no—
42#13182030—24no—
43#13182028—24no—
44C2×Dic218C2311yescharacteristic
45C2×D436—21no—
46C2×D436—22no—
47C2×D436C2311yescharacteristic
48C2×D436C2312yes—
49C2×C22⋊D109—21no—
50C2×C22⋊D109—22no—
51C2×C22⋊D109—22no—
52C2×C22⋊D109—24no—
53D4×D109—24no—
54D4×D109—24no—
55D4×D109—48no—
56C22×Dic109—21no—
57C2×C4×D109—21no—
58C2×C4×D109D412yes—
59C2×C4×D109C2311yescharacteristic
60#13182018—24no—
61#13182017—24no—
62#13182016—24no—
63D872—24no—
64#13182014—24no—
65#13182014—24no—
66#13182012D418yescomplemented
67C2×#11724784C2312yes—
68#13182008—22no—
69#13182006—22no—
70C4×Dic109D412yes—
71C4×Dic109C2311yescharacteristic
72C2×Q8×C109C2311yescharacteristic
73C2×D4×C109—22no—
74C2×D4×C109C2311yescharacteristic
75C2×D4×C109C2312yes—
76SD16×C109—24no—
77D8×C109—24no—
78C2×C8×C109C2312yes—
79C22⋊C4×C109—22no—
80C42×C109C2311yescharacteristic
81C23×C109—21no—
82C23×C109—22no—
83C22×D109—21no—
84C22×D109—22no—
85C22×D109—48no—
86C22×D109—44no—
87C22×D109C2×D411yescharacteristic
88Q8×C109—22no—
89D4×C109—22no—
90D4×C109—22no—
91D4×C109—22no—
92D4×C109—44no—
93C2×C4×C109—21no—
94C2×C4×C109C2×D412yes—
95C2×C4×C109C2411yescharacteristic
96C22⋊D109—42no—
97C22⋊D109—44no—
98C22⋊D109—48no—
99C22⋊D109—44no—
100C2×Dic109—21no—
101C2×Dic109—42no—
102C2×Dic109C2×D411yescharacteristic
103C2×Dic109C2×D412yes—
104D436—22no—
105D436—22no—
106D436—22no—
107D436—44no—
108D436—42no—
109C4×D109—24no—
110C4×D109—42no—
111C4×D109C2×D414yes—
112Dic218—22no—
113C8×C109C2×D414yescomplemented
114#11724784C2×D414yescomplemented
115C22×C109—41no—
116C22×C109—42no—
117C22×C109—44no—
118C22×C109—42no—
119C22×C109C22×D411yescharacteristic
120C2×D109—22no—
121C2×D109—41no—
122C2×D109—42no—
123C2×D109—44no—
124C2×D109—42no—
125C2×D109—88no—
126C2×D109#12712yes—
127C4×C109—22no—
128C4×C109—41no—
129C4×C109C22×D412yes—
130Dic109—41no—
131Dic109#12712yes—
132Dic109AGL(1,ℤ/8)14yescomplemented
133C2×C109—82no—
134C2×C109—81no—
135C2×C109C2×#12711yescharacteristic
136C2×C109C2×AGL(1,ℤ/8)12yes—
137D109—22no—
138D109—81no—
139D109—82no—
140#3151—1091nomaximal, Sylow
141C109#315111yescharacteristic, complemented, Sylow
142C2×AGL(1,ℤ/8)—1092no—
143C2×AGL(1,ℤ/8)—1092no—
144#532—1091no—
145C2×#127—1091no—
146#496—1091no—
147#496—1091no—
148#496—1091no—
149#496D10911yescharacteristic, complemented
150#493—1092no—
151#492—1092no—
152C4×M16—1091no—
153C2×#41—1091no—
154C2×#41—1091no—
155C22×D4—1092no—
156C22×D4—2182no—
157AGL(1,ℤ/8)—2188no—
158AGL(1,ℤ/8)—2188no—
159C2×SD16—1092no—
160C2×SD16—1092no—
161C2×SD16—1092no—
162C2×SD16C2×D10912yescomplemented
163C2×D8—1092no—
164C2×D8—1092no—
165C2×D8—1092no—
166C2×D8C2×D10912yescomplemented
167C2×M16—1092no—
168C4⋊Q8—1091no—
169#127—1092no—
170#127—1091no—
171#127—1091no—
172#127—2182no—
173#127C2×D10911yescharacteristic
174#124—1091no—
175#124C2×D10911yescharacteristic, complemented
176#121—2182no—
177C4×D4—2182no—
178C2×C42—1091no—
179#97—1091no—
180#97C2×D10911yescharacteristic, complemented
181C4×C8—1092no—
182C24—2182no—
183#41—2184no—
184#41—2184no—
185C2×Q8—1091no—
186C2×Q8C22×D10911yescharacteristic
187C2×D4—22no—
188C2×D4—1091no—
189C2×D4—1092no—
190C2×D4—4368no—
191C2×D4—2181no—
192C2×D4—2184no—
193C2×D4—2184no—
194C2×D4—2182no—
195C2×D4—2184no—
196C2×D4—2182no—
197C2×D4—2181no—
198C2×D4—2182no—
199C2×D4C22×D10911yescharacteristic
200C2×D4C22×D10912yes—
201C22×C4—1091no—
202C22×C4—1092no—
203C22×C4—2181no—
204C22×C4—2181no—
205SD16—24no—
206SD16—2184no—
207SD16—2184no—
208SD16—2184no—
209D8—24no—
210D8—2184no—
211D8—2184no—
212D8—2184no—
213M16—1098no—
214C2×C8—1092no—
215C2×C8C22×D10912yes—
216C4⋊C4—2182no—
217C22⋊C4—22no—
218C22⋊C4—2182no—
219C42—1092no—
220C42—1091no—
221C42C22×D10911yescharacteristic
222C23—21no—
223C23—22no—
224C23—1091no—
225C23—4364no—
226C23—4368no—
227C23—2181no—
228C23—2182no—
229Q8—22no—
230Q8—2182no—
231D4—22no—
232D4—22no—
233D4—22no—
234D4—44no—
235D4—4364no—
236D4—4362no—
237D4—4368no—
238D4—4364no—
239D4—4364no—
240D4—4362no—
241D4—2182no—
242D4—2182no—
243D4—2182no—
244C2×C4—21no—
245C2×C4—1092no—
246C2×C4—1091no—
247C2×C4—1094no—
248C2×C4—4362no—
249C2×C4—4362no—
250C2×C4—2184no—
251C2×C4—2181no—
252C2×C4D4×D10912yes—
253C2×C4C23×D10911yescharacteristic
254C8—1094no—
255C8D4×D10914yescomplemented
256C22—44no—
257C22—42no—
258C22—41no—
259C22—42no—
260C22—8728no—
261C22—1092no—
262C22—4364no—
263C22—4362no—
264C22—4361no—
265C22—4362no—
266C22—2182no—
267C22C2×D4×D10911yescharacteristic
268C4—22no—
269C4—41no—
270C4—1094no—
271C4—1092no—
272C4—4361no—
273C4C2×D4×D10912yes—
274C2—82no—
275C2—81no—
276C2—8722no—
277C2—8721no—
278C2—2182no—
279C2#127×D10911yescharacteristic
280C2AGL(1,ℤ/8)×D10912yes—
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#496×C109C2
socleC22×C109C22×D4
hypercenter#496D109
hypocenterC109#3151
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C42 ◁ #496

Complements

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

NH ≅ G/Ntype
1. C2×#16812726C2semidirect
2. C2×#16812723C2semidirect
3. #20295575C2semidirect
4. #127×D109C2semidirect
5. #20295416C2semidirect
6. #20295415C2semidirect
7. #20295413C2semidirect
8. #20295401C2semidirect
9. #20295397C2semidirect
10. #20294978C2semidirect
11. #496×C109C2semidirect
18. C2×#13182018C22semidirect
19. C2×#13182017C22semidirect
20. C2×#13182016C22semidirect
21. C2×D872C22semidirect
22. C2×#13182014C22semidirect
23. C2×#13182014C22semidirect
25. #16812697C22semidirect
29. #16812695C22semidirect
30. #16812687C22semidirect
34. #16812610C22semidirect
35. C8×Dic109C22semidirect
36. C2×SD16×C109C22semidirect
37. C2×D8×C109C22semidirect
39. #124×C109C22semidirect
40. #97×C109C22semidirect
66. #13182012D4semidirect
113. C8×C109C2×D4semidirect
114. #11724784C2×D4semidirect
132. Dic109AGL(1,ℤ/8)semidirect
141. C109#3151semidirect
149. #496D109semidirect
162. C2×SD16C2×D109semidirect
166. C2×D8C2×D109semidirect
175. #124C2×D109semidirect
180. #97C2×D109semidirect
255. C8D4×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127262
C2×#168127232
#202955752
#127×D1092
#202954162
#202954152
#202954132
#202954012
#202953972
#202949782
#496×C1092
#3151109