Finite Group Databasebeta

← #28372495

Subgroups — #28372495

Order 13952.

548 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 548 conjugacy classes of subgroups — too many to draw. Only its 135 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×#16812724C211yesmaximal, characteristic, complemented
3#20295572C211yesmaximal, characteristic, complemented
4C2×#16812661C211yesmaximal, characteristic, complemented
5#20295244C212yesmaximal, complemented
6#20295240C212yesmaximal, complemented
7#20295230C212yesmaximal, complemented
8#20295220C212yesmaximal, complemented
9#20295049C211yesmaximal, characteristic, complemented
10#20295025C211yesmaximal, characteristic, complemented
11#417×C109C211yesmaximal, characteristic, complemented
12C2×#13182029C2211yescharacteristic
13C22×D436C2211yescharacteristic
14C22×C4×D109—21no—
15#16812727—24no—
16#16812724—24no—
17C2×#13182018C2212yes—
18C2×#13182016C2212yes—
19C2×D872C2212yes—
20C2×#13182014C2212yes—
21C2×#13182013C2211yescharacteristic
22#16812697C2211yescharacteristic, complemented
23#16812695C2211yescharacteristic, complemented
24#16812688C2211yescharacteristic, complemented
25#16812686C2211yescharacteristic, complemented
26#16812671—21no—
27#16812671—21no—
28#16812661—21no—
29#16812661—21no—
30#16812661C2212yes—
31#16812661C2212yes—
32C4×C22⋊D109—21no—
33C4×C22⋊D109—21no—
34C2×#13182006—21no—
35C2×C4×Dic109C2211yescharacteristic
36#16812628C2212yescomplemented
37#16812628C2212yescomplemented
38#16812622C2212yescomplemented
39#16812619C2212yescomplemented
40C2×#41×C109C2211yescharacteristic
41C2×M16×C109C2211yescharacteristic
42C2×C4⋊C4×C109C2211yescharacteristic
43#103×C109C2212yescomplemented
44#102×C109C2212yescomplemented
45C23×D109—21no—
46#13182029—22no—
47#13182029—22no—
48C2×Dic218C2311yescharacteristic
49C2×D436—21no—
50C2×D436—22no—
51C2×D436D414yescomplemented
52C2×D436C2311yescharacteristic
53C2×D436C2312yes—
54C2×C22⋊D109—21no—
55C2×C22⋊D109—21no—
56C2×C22⋊D109—21no—
57C22×Dic109D411yescharacteristic
58C22×Dic109D411yescharacteristic
59C2×C4×D109—21no—
60C2×C4×D109—22no—
61C2×C4×D109—22no—
62C2×C4×D109—44no—
63#13182018—24no—
64#13182016—24no—
65D872—24no—
66#13182014—24no—
67#13182013—22no—
68C2×#11724784C2312yes—
69#13182008—21no—
70#13182008—21no—
71C22⋊Dic109—21no—
72C22⋊Dic109—21no—
73#13182006—22no—
74#13182006—22no—
75#13182006—21no—
76#13182006—21no—
77C4×Dic109D411yescharacteristic
78C4×Dic109D411yescharacteristic
79C4×Dic109C2311yescharacteristic
80C4×Dic109C2311yescharacteristic
81#41×C109—22no—
82#41×C109—22no—
83C2×Q8×C109C2311yescharacteristic
84C2×D4×C109—21no—
85C2×D4×C109C2311yescharacteristic
86C22×C4×C109—21no—
87C22×C4×C109—21no—
88C22×C4×C109C2311yescharacteristic
89M16×C109—22no—
90C2×C8×C109C2312yes—
91C4⋊C4×C109—21no—
92C4⋊C4×C109C2312yes—
93C23×C109—21no—
94C23×C109C2×D411yescharacteristic
95C22×D109—24no—
96C22×D109—22no—
97C22×D109—21no—
98C22×D109—42no—
99C22×D109—42no—
100Q8×C109—21no—
101Q8×C109—21no—
102D4×C109—21no—
103D4×C109—21no—
104D4×C109—42no—
105C2×C4×C109—22no—
106C2×C4×C109—21no—
107C2×C4×C109—42no—
108C2×C4×C109—42no—
109C2×C4×C109C2×D411yescharacteristic
110C2×C4×C109C2×D412yes—
111C2×C4×C109#4112yes—
112C2×C4×C109C2411yescharacteristic
113C22⋊D109—42no—
114C22⋊D109—42no—
115C22⋊D109—42no—
116C2×Dic109—22no—
117C2×Dic109—21no—
118C2×Dic109—41no—
119C2×Dic109C2×D411yescharacteristic
120C2×Dic109C2×D411yescharacteristic
121C2×Dic109C2×D411yescharacteristic
122C2×Dic109C2×D411yescharacteristic
123D436—24no—
124D436—21no—
125D436—22no—
126D436—21no—
127D436C2×D414yes—
128C4×D109—44no—
129C4×D109—42no—
130C4×D109—44no—
131Dic218—21no—
132Dic218—21no—
133C8×C109—22no—
134#11724784—22no—
135C22×C109—21no—
136C22×C109—21no—
137C22×C109—41no—
138C22×C109—41no—
139C22×C109—41no—
140C22×C109#12112yes—
141C22×C109C22×D411yescharacteristic
142C2×D109—24no—
143C2×D109—48no—
144C2×D109—42no—
145C2×D109—42no—
146C2×D109—41no—
147C2×D109—41no—
148C2×D109—41no—
149C4×C109—41no—
150C4×C109—42no—
151C4×C109#12112yes—
152C4×C109C22×D411yescharacteristic
153C4×C109C2×#4111yescharacteristic
154Dic109—21no—
155Dic109—21no—
156Dic109—21no—
157Dic109—41no—
158Dic109AGL(1,ℤ/8)12yescomplemented
159C2×C109—22no—
160C2×C109—81no—
161C2×C109C2×#12111yescharacteristic
162C2×C109C2×AGL(1,ℤ/8)11yescharacteristic
163C2×C109#57511yescharacteristic
164D109—44no—
165D109—81no—
166#3036—1091nomaximal, Sylow
167C109#303611yescharacteristic, complemented, Sylow
168C2×AGL(1,ℤ/8)—1091no—
169C2×AGL(1,ℤ/8)—1091no—
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#417D10911yescharacteristic, complemented
179C2×#41—1091no—
180C2×#41C2×D10911yescharacteristic
181C22×D4—1091no—
182C23×C4—2181no—
183AGL(1,ℤ/8)—2184no—
184AGL(1,ℤ/8)—2184no—
185C2×SD16—1092no—
186C2×SD16—1092no—
187C2×D8—1092no—
188C2×D8—1092no—
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⋊C4C2×D10911yescharacteristic
204C2×C22⋊C4—2181no—
205C2×C42—1091no—
206C4⋊C8—1092no—
207C4⋊C8—1092no—
208#103—1092no—
209#103C2×D10912yescomplemented
210#102—1092no—
211#102C2×D10912yescomplemented
212C24—2181no—
213#41—22no—
214#41—22no—
215#41—2182no—
216#41—2182no—
217C2×Q8—1091no—
218C2×Q8C22×D10911yescharacteristic
219C2×D4—21no—
220C2×D4—1094no—
221C2×D4—1092no—
222C2×D4—1091no—
223C2×D4—2181no—
224C2×D4—2181no—
225C2×D4—2181no—
226C2×D4—2181no—
227C2×D4—2182no—
228C2×D4C22×D10911yescharacteristic
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—1091no—
232C22×C4—1091no—
233C22×C4—4364no—
234C22×C4—2182no—
235C22×C4—2182no—
236C22×C4—2181no—
237C22×C4C22×D10911yescharacteristic
238SD16—2184no—
239SD16—2184no—
240D8—2184no—
241D8—2184no—
242M16—22no—
243M16—2182no—
244C2×C8—1092no—
245C2×C8C22×D10912yes—
246C4⋊C4—21no—
247C4⋊C4—2181no—
248C4⋊C4—2181no—
249C4⋊C4C22×D10912yes—
250C22⋊C4—2182no—
251C22⋊C4—2182no—
252C22⋊C4—2181no—
253C22⋊C4—2181no—
254C22⋊C4—2181no—
255C22⋊C4—2181no—
256C42—1091no—
257C42—1091no—
258C42—1091no—
259C42—1091no—
260C23—21no—
261C23—4362no—
262C23—4362no—
263C23—2181no—
264C23—2184no—
265C23—2182no—
266C23D4×D10911yescharacteristic
267Q8—21no—
268Q8—21no—
269Q8—2181no—
270Q8—2181no—
271D4—21no—
272D4—21no—
273D4—42no—
274D4—1094no—
275D4—4362no—
276D4—4362no—
277D4—4362no—
278D4—2184no—
279D4—2181no—
280D4—2181no—
281D4—2182no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—1091no—
287C2×C4—1091no—
288C2×C4—1091no—
289C2×C4—1091no—
290C2×C4—4364no—
291C2×C4—4364no—
292C2×C4—4362no—
293C2×C4—4361no—
294C2×C4—2182no—
295C2×C4—2181no—
296C2×C4D4×D10911yescharacteristic
297C2×C4D4×D10912yes—
298C2×C4#1318202912yes—
299C2×C4C23×D10911yescharacteristic
300C8—22no—
301C8—2182no—
302C22—21no—
303C22—21no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—4368no—
308C22—4361no—
309C22—4361no—
310C22—4361no—
311C22—4362no—
312C22—4362no—
313C22—2184no—
314C22#1681267212yes—
315C22C2×D4×D10911yescharacteristic
316C4—41no—
317C4—42no—
318C4—1092no—
319C4—4361no—
320C4—2181no—
321C4—2181no—
322C4—2181no—
323C4#1681267212yes—
324C4C2×D4×D10911yescharacteristic
325C4C2×#1318202911yescharacteristic
326C2—22no—
327C2—81no—
328C2—8721no—
329C2—4364no—
330C2C2×#1681267211yescharacteristic
331C2AGL(1,ℤ/8)×D10911yescharacteristic
332C2#2029582511yescharacteristic
subgroupG/H
centerC22C2×D4×D109
commutator, cosocleC2×C4×C109C24
FrattiniC2×C4C23×D109
Fitting#417×C109C2
socleC22×C109C22×D4
hypercenter#417D109
hypocenterC109#3036
LCSG ▷ C2×C4×C109 ▷ C2×C109 ▷ C109
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#16812727C2semidirect
2. C2×#16812724C2semidirect
3. #20295572C2semidirect
4. C2×#16812661C2semidirect
5. #20295244C2semidirect
6. #20295240C2semidirect
7. #20295230C2semidirect
8. #20295220C2semidirect
9. #20295049C2semidirect
10. #20295025C2semidirect
11. #417×C109C2semidirect
22. #16812697C22semidirect
23. #16812695C22semidirect
24. #16812688C22semidirect
25. #16812686C22semidirect
36. #16812628C22semidirect
37. #16812628C22semidirect
38. #16812622C22semidirect
39. #16812619C22semidirect
43. #103×C109C22semidirect
44. #102×C109C22semidirect
51. C2×D436D4semidirect
158. Dic109AGL(1,ℤ/8)semidirect
167. C109#3036semidirect
178. #417D109semidirect
209. #103C2×D109semidirect
211. #102C2×D109semidirect

Maximal subgroups

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

subgroupindex
C2×#168127272
C2×#168127242
#202955722
C2×#168126612
#202952442
#202952402
#202952302
#202952202
#202950492
#202950252
#417×C1092
#3036109