Finite Group Databasebeta

← #28372493

Subgroups — #28372493

Order 13952.

548 conjugacy classes of subgroups; only the 143 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 143 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×#16812725C211yesmaximal, characteristic, complemented
2#20295572C211yesmaximal, characteristic, complemented
3C2×D4×Dic109C211yesmaximal, characteristic, complemented
4#20295243C212yesmaximal, complemented
5#20295239C212yesmaximal, complemented
6#20295229C212yesmaximal, complemented
7#20295219C212yesmaximal, complemented
8#20295049C211yesmaximal, characteristic, complemented
9#20295028C211yesmaximal, characteristic, complemented
10#20295027C211yesmaximal, characteristic, complemented
11C2×AGL(1,ℤ/8)×C109C211yesmaximal, characteristic, complemented
12C2×#13182029C2211yescharacteristic
13C23×Dic109—21no—
14#16812725—24no—
15C2×#13182017C2212yes—
16C2×#13182014C2212yes—
17C2×#13182013C2211yescharacteristic
18#16812697C2211yescharacteristic, complemented
19#16812695C2211yescharacteristic, complemented
20#16812688C2211yescharacteristic, complemented
21#16812686C2211yescharacteristic, complemented
22#16812671—21no—
23#16812671—21no—
24D4×Dic109—21no—
25D4×Dic109—21no—
26D4×Dic109C2212yes—
27D4×Dic109C2212yes—
28C4×C22⋊D109—21no—
29C4×C22⋊D109—21no—
30C2×C4⋊Dic109C2211yescharacteristic
31C2×C22⋊Dic109—21no—
32C2×C4×Dic109C2211yescharacteristic
33#16812628C2212yescomplemented
34#16812628C2212yescomplemented
35#16812624C2212yescomplemented
36#16812623C2212yescomplemented
37#16812621C2212yescomplemented
38#16812620C2212yescomplemented
39C2×#41×C109C2211yescharacteristic
40C22×D4×C109C2211yescharacteristic
41AGL(1,ℤ/8)×C109—24no—
42C2×SD16×C109C2212yes—
43C2×D8×C109C2212yes—
44C2×M16×C109C2211yescharacteristic
45#13182029—22no—
46#13182029—22no—
47C2×Dic218C2311yescharacteristic
48C2×D436C2311yescharacteristic
49C2×C22⋊D109—21no—
50C2×C22⋊D109—21no—
51C2×C22⋊D109—21no—
52C22×Dic109—21no—
53C22×Dic109—22no—
54C22×Dic109—22no—
55C22×Dic109—44no—
56C22×Dic109D411yescharacteristic
57C22×Dic109D411yescharacteristic
58C2×C4×D109—21no—
59#13182017—24no—
60#13182014—24no—
61#13182013—22no—
62C2×#11724784C2312yes—
63C4⋊Dic109—21no—
64C4⋊Dic109C2312yes—
65#13182008—21no—
66#13182008—21no—
67C22⋊Dic109—22no—
68C22⋊Dic109—22no—
69C22⋊Dic109—21no—
70C22⋊Dic109—21no—
71#13182006—21no—
72#13182006—21no—
73C4×Dic109D411yescharacteristic
74C4×Dic109D411yescharacteristic
75C4×Dic109C2311yescharacteristic
76C4×Dic109C2311yescharacteristic
77C24×C109—21no—
78#41×C109—22no—
79#41×C109—22no—
80C2×Q8×C109C2311yescharacteristic
81C2×D4×C109—21no—
82C2×D4×C109—21no—
83C2×D4×C109—22no—
84C2×D4×C109D414yescomplemented
85C2×D4×C109C2311yescharacteristic
86C2×D4×C109C2312yes—
87C22×C4×C109—21no—
88C22×C4×C109C2311yescharacteristic
89SD16×C109—24no—
90D8×C109—24no—
91M16×C109—22no—
92C2×C8×C109C2312yes—
93C23×C109—24no—
94C23×C109—22no—
95C23×C109—21no—
96C23×C109—42no—
97C23×C109—42no—
98C23×C109C2×D411yescharacteristic
99C22×D109—21no—
100Q8×C109—21no—
101Q8×C109—21no—
102D4×C109—24no—
103D4×C109—21no—
104D4×C109—22no—
105D4×C109—21no—
106D4×C109—42no—
107D4×C109C2×D414yes—
108C2×C4×C109—21no—
109C2×C4×C109—42no—
110C2×C4×C109C2×D411yescharacteristic
111C2×C4×C109C2×D412yes—
112C2×C4×C109#4112yes—
113C2×C4×C109C2411yescharacteristic
114C22⋊D109—42no—
115C22⋊D109—42no—
116C22⋊D109—42no—
117C2×Dic109—22no—
118C2×Dic109—21no—
119C2×Dic109—22no—
120C2×Dic109—41no—
121C2×Dic109—44no—
122C2×Dic109—42no—
123C2×Dic109—44no—
124C2×Dic109C2×D411yescharacteristic
125C2×Dic109C2×D411yescharacteristic
126C2×Dic109C2×D411yescharacteristic
127C2×Dic109C2×D411yescharacteristic
128D436—21no—
129D436—21no—
130C4×D109—42no—
131Dic218—21no—
132Dic218—21no—
133C8×C109—22no—
134#11724784—22no—
135C22×C109—21no—
136C22×C109—21no—
137C22×C109—24no—
138C22×C109—48no—
139C22×C109—42no—
140C22×C109—42no—
141C22×C109—41no—
142C22×C109—41no—
143C22×C109—41no—
144C22×C109#12112yes—
145C22×C109C22×D411yescharacteristic
146C2×D109—41no—
147C2×D109—41no—
148C2×D109—41no—
149C4×C109—41no—
150C4×C109#12112yes—
151C4×C109C22×D411yescharacteristic
152C4×C109C2×#4111yescharacteristic
153Dic109—21no—
154Dic109—21no—
155Dic109—21no—
156Dic109—41no—
157Dic109—42no—
158Dic109AGL(1,ℤ/8)12yescomplemented
159C2×C109—22no—
160C2×C109—44no—
161C2×C109—81no—
162C2×C109C2×#12111yescharacteristic
163C2×C109C2×AGL(1,ℤ/8)11yescharacteristic
164C2×C109#57511yescharacteristic
165D109—81no—
166#3036—1091nomaximal, Sylow
167C109#303611yescharacteristic, complemented, Sylow
168C2×AGL(1,ℤ/8)—1091no—
169C2×AGL(1,ℤ/8)D10911yescharacteristic, complemented
170#532—1091no—
171C2×C4×D4—1091no—
172#463—1092no—
173#463—1092no—
174#461—1092no—
175#459—1092no—
176#423—1091no—
177#417—1091no—
178#417—1091no—
179C2×#41—1091no—
180C2×#41C2×D10911yescharacteristic
181C22×D4C2×D10911yescharacteristic
182C23×C4—2181no—
183AGL(1,ℤ/8)—24no—
184AGL(1,ℤ/8)—2184no—
185C2×SD16—1092no—
186C2×SD16C2×D10912yes—
187C2×D8—1092no—
188C2×D8C2×D10912yes—
189C2×M16—1091no—
190C2×M16C2×D10911yescharacteristic
191C4⋊Q8—1091no—
192#127—1091no—
193#124—1091no—
194#124—1091no—
195#121—2181no—
196#121—2181no—
197C4×D4—1092no—
198C4×D4—1092no—
199C4×D4—2181no—
200C4×D4—2181no—
201C4×D4—2181no—
202C4×D4—2181no—
203C2×C4⋊C4—1091no—
204C2×C22⋊C4—2181no—
205C2×C42—1091no—
206C4⋊C8—1092no—
207C4⋊C8—1092no—
208#103—1092no—
209#103—1092no—
210#102—1092no—
211#102—1092no—
212C24—21no—
213#41—22no—
214#41—22no—
215#41—2182no—
216#41—2182no—
217C2×Q8—1091no—
218C2×Q8C22×D10911yescharacteristic
219C2×D4—21no—
220C2×D4—21no—
221C2×D4—22no—
222C2×D4—1091no—
223C2×D4—2181no—
224C2×D4—2181no—
225C2×D4—2181no—
226C2×D4C22⋊D10914yescomplemented
227C2×D4C22×D10911yescharacteristic
228C2×D4C22×D10912yes—
229C22×C4—21no—
230C22×C4—1091no—
231C22×C4—1091no—
232C22×C4—4364no—
233C22×C4—2182no—
234C22×C4—2182no—
235C22×C4—2181no—
236C22×C4—2181no—
237C22×C4C22×D10911yescharacteristic
238SD16—24no—
239SD16—2184no—
240D8—24no—
241D8—2184no—
242M16—22no—
243M16—2182no—
244C2×C8—1092no—
245C2×C8C22×D10912yes—
246C4⋊C4—1092no—
247C4⋊C4—2181no—
248C4⋊C4—2181no—
249C4⋊C4—2181no—
250C22⋊C4—2181no—
251C22⋊C4—2181no—
252C22⋊C4—2182no—
253C22⋊C4—2182no—
254C22⋊C4—2181no—
255C22⋊C4—2181no—
256C42—1091no—
257C42—1091no—
258C42—1091no—
259C42—1091no—
260C23—24no—
261C23—22no—
262C23—21no—
263C23—42no—
264C23—42no—
265C23—2181no—
266C23D4×D10911yescharacteristic
267Q8—21no—
268Q8—21no—
269Q8—2181no—
270Q8—2181no—
271D4—24no—
272D4—21no—
273D4—22no—
274D4—21no—
275D4—42no—
276D4—4362no—
277D4—4362no—
278D4—4362no—
279D4—2181no—
280D4—2181no—
281D4C2×C22⋊D10914yes—
282C2×C4—21no—
283C2×C4—42no—
284C2×C4—1091no—
285C2×C4—1091no—
286C2×C4—1091no—
287C2×C4—1091no—
288C2×C4—4364no—
289C2×C4—4362no—
290C2×C4—4364no—
291C2×C4—4362no—
292C2×C4—4361no—
293C2×C4—2182no—
294C2×C4—2182no—
295C2×C4—2181no—
296C2×C4D4×D10911yescharacteristic
297C2×C4C2×C22⋊D10912yes—
298C2×C4#1318202812yes—
299C2×C4C23×D10911yescharacteristic
300C8—22no—
301C8—2182no—
302C22—21no—
303C22—21no—
304C22—24no—
305C22—48no—
306C22—42no—
307C22—42no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—4361no—
312C22—4361no—
313C22—4361no—
314C22#1681267112yes—
315C22C2×D4×D10911yescharacteristic
316C4—41no—
317C4—1092no—
318C4—4361no—
319C4—4362no—
320C4—2181no—
321C4—2181no—
322C4—2181no—
323C4#1681267112yes—
324C4C22×C22⋊D10911yescharacteristic
325C4C2×#1318202811yescharacteristic
326C2—22no—
327C2—44no—
328C2—81no—
329C2—8721no—
330C2C2×#1681267111yescharacteristic
331C2AGL(1,ℤ/8)×D10911yescharacteristic
332C2#2029582211yescharacteristic
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#3036
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×#16812725C2semidirect
2. #20295572C2semidirect
3. C2×D4×Dic109C2semidirect
4. #20295243C2semidirect
5. #20295239C2semidirect
6. #20295229C2semidirect
7. #20295219C2semidirect
8. #20295049C2semidirect
9. #20295028C2semidirect
10. #20295027C2semidirect
11. C2×AGL(1,ℤ/8)×C109C2semidirect
18. #16812697C22semidirect
19. #16812695C22semidirect
20. #16812688C22semidirect
21. #16812686C22semidirect
33. #16812628C22semidirect
34. #16812628C22semidirect
35. #16812624C22semidirect
36. #16812623C22semidirect
37. #16812621C22semidirect
38. #16812620C22semidirect
84. C2×D4×C109D4semidirect
158. Dic109AGL(1,ℤ/8)semidirect
167. C109#3036semidirect
169. C2×AGL(1,ℤ/8)D109semidirect
226. C2×D4C22⋊D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127252
#202955722
C2×D4×Dic1092
#202952432
#202952392
#202952292
#202952192
#202950492
#202950282
#202950272
C2×AGL(1,ℤ/8)×C1092
#3036109