Finite Group Databasebeta

← #28373480

Subgroups — #28373480

Order 13952.

656 conjugacy classes of subgroups; only the 153 normal subgroups are shown.

normal subgroups only

This group has 656 conjugacy classes of subgroups — too many to draw. Only its 153 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×#16812727C211yesmaximal, characteristic, complemented
2C2×#16812725C211yesmaximal, characteristic, complemented
3C2×#16812724C211yesmaximal, characteristic, complemented
4#20295572C211yesmaximal, characteristic, complemented
5C2×#16812695C211yesmaximal, characteristic, complemented
6#20295414C212yesmaximal, complemented
7#20295412C212yesmaximal, complemented
8#20295401C212yesmaximal, complemented
9#20295397C212yesmaximal, complemented
10M16×Dic109C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C109C211yesmaximal, characteristic, complemented
12C2×#13182029C2211yescharacteristic
13C22×D436C2211yescharacteristic
14C22×C22⋊D109—22no—
15#16812727—24no—
16#16812725—24no—
17#16812724—24no—
18C2×#13182018C2212yescomplemented
19C2×#13182017C2212yescomplemented
20C2×#13182016C2212yescomplemented
21C2×D872C2212yescomplemented
22C2×#13182014C2212yescomplemented
23C2×#13182014C2212yescomplemented
24C2×#13182013C2211yescharacteristic
25#16812697C2211yescharacteristic, complemented
26#16812695—22no—
27#16812695C2212yes—
28#16812695C2212yes—
29#16812695C2211yescharacteristic, complemented
30#16812688C2211yescharacteristic, complemented
31#16812686C2211yescharacteristic, complemented
32#16812671—22no—
33C4×C22⋊D109—22no—
34C2×C4×Dic109C2211yescharacteristic
35#16812609C2212yescomplemented
36C8×Dic109C2212yescomplemented
37C2×#41×C109C2211yescharacteristic
38C22×D4×C109C2211yescharacteristic
39AGL(1,ℤ/8)×C109—24no—
40C2×SD16×C109C2212yescomplemented
41C2×D8×C109C2212yescomplemented
42C2×M16×C109C2211yescharacteristic
43C23×D109—21no—
44#13182029—22no—
45#13182029—22no—
46C2×Dic218C2311yescharacteristic
47C2×D436—22no—
48C2×D436—22no—
49C2×D436—21no—
50C2×D436C2311yescharacteristic
51C2×D436C2312yes—
52C2×C22⋊D109—21no—
53C2×C22⋊D109—22no—
54C2×C22⋊D109—24no—
55C2×C22⋊D109—24no—
56C2×C22⋊D109—48no—
57C22×Dic109D412yes—
58C2×C4×D109—21no—
59#13182018—24no—
60#13182017—24no—
61#13182016—24no—
62D872—24no—
63#13182014—24no—
64#13182014—24no—
65#13182013D414yescomplemented
66C2×#11724784C2312yes—
67#13182008—22no—
68C22⋊Dic109—22no—
69#13182006—22no—
70C4×Dic109D412yes—
71C4×Dic109C2311yescharacteristic
72C4×Dic109C2311yescharacteristic
73C24×C109—21no—
74#41×C109—22no—
75#41×C109—22no—
76C2×Q8×C109C2311yescharacteristic
77C2×D4×C109—21no—
78C2×D4×C109—22no—
79C2×D4×C109—22no—
80C2×D4×C109—21no—
81C2×D4×C109C2311yescharacteristic
82C2×D4×C109C2312yes—
83C22×C4×C109—21no—
84C22×C4×C109C2311yescharacteristic
85SD16×C109—24no—
86D8×C109—24no—
87M16×C109D414yescomplemented
88C2×C8×C109C2312yes—
89C23×C109—21no—
90C23×C109—22no—
91C23×C109—44no—
92C23×C109—42no—
93C23×C109C2×D411yescharacteristic
94C22×D109—21no—
95C22×D109—22no—
96C22×D109—44no—
97C22×D109—42no—
98Q8×C109—21no—
99Q8×C109—21no—
100D4×C109—21no—
101D4×C109—22no—
102D4×C109—22no—
103D4×C109—21no—
104D4×C109—42no—
105D4×C109—42no—
106C2×C4×C109—21no—
107C2×C4×C109—42no—
108C2×C4×C109C2×D411yescharacteristic
109C2×C4×C109C2×D412yes—
110C2×C4×C109C2×D412yes—
111C2×C4×C109C2411yescharacteristic
112C22⋊D109—48no—
113C22⋊D109—42no—
114C22⋊D109—44no—
115C22⋊D109—48no—
116C2×Dic109—24no—
117C2×Dic109—21no—
118C2×Dic109C2×D412yes—
119C2×Dic109C2×D412yes—
120D436—21no—
121D436—22no—
122D436—21no—
123D436—22no—
124D436—42no—
125C4×D109—42no—
126Dic218—21no—
127Dic218—21no—
128C8×C109C2×D414yescomplemented
129#11724784C2×D414yescomplemented
130C22×C109—22no—
131C22×C109—41no—
132C22×C109—42no—
133C22×C109—44no—
134C22×C109—42no—
135C22×C109—84no—
136C22×C109#12712yes—
137C22×C109C22×D411yescharacteristic
138C2×D109—41no—
139C2×D109—42no—
140C2×D109—44no—
141C2×D109—42no—
142C2×D109—84no—
143C4×C109—41no—
144C4×C109#12712yes—
145C4×C109C22×D411yescharacteristic
146C4×C109C22×D411yescharacteristic
147Dic109—22no—
148Dic109—41no—
149Dic109AGL(1,ℤ/8)14yescomplemented
150C2×C109—22no—
151C2×C109—81no—
152C2×C109—82no—
153C2×C109C2×#12711yescharacteristic
154C2×C109C2×AGL(1,ℤ/8)12yes—
155D109—81no—
156D109—82no—
157#3151—1091nomaximal, Sylow
158C109#315111yescharacteristic, complemented, Sylow
159C2×AGL(1,ℤ/8)—1091no—
160C2×AGL(1,ℤ/8)—1091no—
161C2×AGL(1,ℤ/8)—1091no—
162C2×AGL(1,ℤ/8)D10911yescharacteristic, complemented
163#532—1091no—
164C2×#127—1091no—
165#496—1092no—
166#496—1092no—
167#493—1092no—
168#492—1092no—
169C4×M16—1091no—
170C2×#41—1091no—
171C2×#41C2×D10911yescharacteristic
172C22×D4—1091no—
173C22×D4—2182no—
174C22×D4C2×D10911yescharacteristic
175AGL(1,ℤ/8)—24no—
176AGL(1,ℤ/8)—2184no—
177AGL(1,ℤ/8)—2184no—
178AGL(1,ℤ/8)—2184no—
179C2×SD16—1092no—
180C2×SD16—1092no—
181C2×SD16—1092no—
182C2×SD16C2×D10912yescomplemented
183C2×D8—1092no—
184C2×D8—1092no—
185C2×D8—1092no—
186C2×D8C2×D10912yescomplemented
187C2×M16—1091no—
188C2×M16C2×D10911yescharacteristic
189C4⋊Q8—1091no—
190#127—1092no—
191#127—1092no—
192#127—1091no—
193#127—2182no—
194#124—1091no—
195#124—1091no—
196#121—2182no—
197C4×D4—2182no—
198C2×C42—1091no—
199#97—1092no—
200C4×C8—1092no—
201C24—21no—
202C24—2181no—
203#41—22no—
204#41—22no—
205#41—2182no—
206#41—2182no—
207C2×Q8—1091no—
208C2×Q8C22×D10911yescharacteristic
209C2×D4—21no—
210C2×D4—22no—
211C2×D4—21no—
212C2×D4—22no—
213C2×D4—1091no—
214C2×D4—1092no—
215C2×D4—4368no—
216C2×D4—2182no—
217C2×D4—2184no—
218C2×D4—2184no—
219C2×D4—2181no—
220C2×D4—2181no—
221C2×D4—2182no—
222C2×D4—2182no—
223C2×D4C22×D10911yescharacteristic
224C2×D4C22×D10912yes—
225C22×C4—21no—
226C22×C4—1092no—
227C22×C4—2181no—
228C22×C4C22×D10911yescharacteristic
229SD16—24no—
230SD16—2184no—
231SD16—2184no—
232SD16—2184no—
233D8—24no—
234D8—2184no—
235D8—2184no—
236D8—2184no—
237M16—1094no—
238M16C22⋊D10914yescomplemented
239C2×C8—1092no—
240C2×C8C22×D10912yes—
241C4⋊C4—2182no—
242C22⋊C4—2182no—
243C22⋊C4—2182no—
244C42—1092no—
245C42—1091no—
246C42—1091no—
247C23—21no—
248C23—22no—
249C23—44no—
250C23—42no—
251C23—4364no—
252C23—4362no—
253C23—2181no—
254C23—2182no—
255C23D4×D10911yescharacteristic
256Q8—21no—
257Q8—21no—
258Q8—2181no—
259Q8—2181no—
260D4—21no—
261D4—22no—
262D4—21no—
263D4—22no—
264D4—42no—
265D4—42no—
266D4—4368no—
267D4—4364no—
268D4—4362no—
269D4—4368no—
270D4—4362no—
271D4—2181no—
272D4—2182no—
273D4—2181no—
274D4—2182no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—1092no—
278C2×C4—1092no—
279C2×C4—4362no—
280C2×C4—2184no—
281C2×C4—2181no—
282C2×C4D4×D10911yescharacteristic
283C2×C4D4×D10912yes—
284C2×C4C2×C22⋊D10912yes—
285C2×C4C23×D10911yescharacteristic
286C8—1094no—
287C8C2×C22⋊D10914yescomplemented
288C22—22no—
289C22—44no—
290C22—42no—
291C22—41no—
292C22—42no—
293C22—84no—
294C22—8724no—
295C22—4364no—
296C22—4362no—
297C22—4361no—
298C22—4362no—
299C22#1681269512yes—
300C22C2×D4×D10911yescharacteristic
301C4—41no—
302C4—1094no—
303C4—4361no—
304C4—2182no—
305C4#1681269512yes—
306C4C2×D4×D10911yescharacteristic
307C4C22×C22⋊D10911yescharacteristic
308C2—22no—
309C2—82no—
310C2—81no—
311C2—8722no—
312C2—8721no—
313C2C2×#1681269511yescharacteristic
314C2AGL(1,ℤ/8)×D10912yes—
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
FittingC2×AGL(1,ℤ/8)×C109C2
socleC22×C109C22×D4
hypercenterC2×AGL(1,ℤ/8)D109
hypocenterC109#3151
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×AGL(1,ℤ/8)

Complements

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

NH ≅ G/Ntype
1. C2×#16812727C2semidirect
2. C2×#16812725C2semidirect
3. C2×#16812724C2semidirect
4. #20295572C2semidirect
5. C2×#16812695C2semidirect
6. #20295414C2semidirect
7. #20295412C2semidirect
8. #20295401C2semidirect
9. #20295397C2semidirect
10. M16×Dic109C2semidirect
11. C2×AGL(1,ℤ/8)×C109C2semidirect
18. C2×#13182018C22semidirect
19. C2×#13182017C22semidirect
20. C2×#13182016C22semidirect
21. C2×D872C22semidirect
22. C2×#13182014C22semidirect
23. C2×#13182014C22semidirect
25. #16812697C22semidirect
29. #16812695C22semidirect
30. #16812688C22semidirect
31. #16812686C22semidirect
35. #16812609C22semidirect
36. C8×Dic109C22semidirect
40. C2×SD16×C109C22semidirect
41. C2×D8×C109C22semidirect
65. #13182013D4semidirect
87. M16×C109D4semidirect
128. C8×C109C2×D4semidirect
129. #11724784C2×D4semidirect
149. Dic109AGL(1,ℤ/8)semidirect
158. C109#3151semidirect
162. C2×AGL(1,ℤ/8)D109semidirect
182. C2×SD16C2×D109semidirect
186. C2×D8C2×D109semidirect
238. M16C22⋊D109semidirect
287. C8C2×C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127272
C2×#168127252
C2×#168127242
#202955722
C2×#168126952
#202954142
#202954122
#202954012
#202953972
M16×Dic1092
C2×AGL(1,ℤ/8)×C1092
#3151109