Finite Group Databasebeta

← #28345924

Subgroups — #28345924

Order 13696.

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×#16812137C211yesmaximal, characteristic, complemented
2C2×#16812134C211yesmaximal, characteristic, complemented
3#20293173C211yesmaximal, characteristic, complemented
4C2×#16812071C211yesmaximal, characteristic, complemented
5#20292845C212yesmaximal, complemented
6#20292841C212yesmaximal, complemented
7#20292831C212yesmaximal, complemented
8#20292821C212yesmaximal, complemented
9#20292650C211yesmaximal, characteristic, complemented
10#20292626C211yesmaximal, characteristic, complemented
11#417×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C22×D428C2211yescharacteristic
14C22×C4×D107—21no—
15#16812137—24no—
16#16812134—24no—
17C2×#13133807C2212yes—
18C2×#13133805C2212yes—
19C2×D856C2212yes—
20C2×#13133803C2212yes—
21C2×#13133802C2211yescharacteristic
22#16812107C2211yescharacteristic, complemented
23#16812105C2211yescharacteristic, complemented
24#16812098C2211yescharacteristic, complemented
25#16812096C2211yescharacteristic, complemented
26#16812081—21no—
27#16812081—21no—
28#16812071—21no—
29#16812071—21no—
30#16812071C2212yes—
31#16812071C2212yes—
32C4×C22⋊D107—21no—
33C4×C22⋊D107—21no—
34C2×(C2×D107)⋊C4—21no—
35C2×C4×Dic107C2211yescharacteristic
36Dic107⋊C8C2212yescomplemented
37Dic107⋊C8C2212yescomplemented
38#16812032C2212yescomplemented
39#16812029C2212yescomplemented
40C2×#41×C107C2211yescharacteristic
41C2×M16×C107C2211yescharacteristic
42C2×C4⋊C4×C107C2211yescharacteristic
43#103×C107C2212yescomplemented
44#102×C107C2212yescomplemented
45C23×D107—21no—
46#13133818—22no—
47#13133818—22no—
48C2×Dic214C2311yescharacteristic
49C2×D428—21no—
50C2×D428—22no—
51C2×D428D414yescomplemented
52C2×D428C2311yescharacteristic
53C2×D428C2312yes—
54C2×C22⋊D107—21no—
55C2×C22⋊D107—21no—
56C2×C22⋊D107—21no—
57C22×Dic107D411yescharacteristic
58C22×Dic107D411yescharacteristic
59C2×C4×D107—21no—
60C2×C4×D107—22no—
61C2×C4×D107—22no—
62C2×C4×D107—44no—
63#13133807—24no—
64#13133805—24no—
65D856—24no—
66#13133803—24no—
67#13133802—22no—
68C2×C107⋊C8C2312yes—
69Dic107⋊C4—21no—
70Dic107⋊C4—21no—
71C22⋊Dic107—21no—
72C22⋊Dic107—21no—
73(C2×D107)⋊C4—22no—
74(C2×D107)⋊C4—22no—
75(C2×D107)⋊C4—21no—
76(C2×D107)⋊C4—21no—
77C4×Dic107D411yescharacteristic
78C4×Dic107D411yescharacteristic
79C4×Dic107C2311yescharacteristic
80C4×Dic107C2311yescharacteristic
81#41×C107—22no—
82#41×C107—22no—
83C2×Q8×C107C2311yescharacteristic
84C2×D4×C107—21no—
85C2×D4×C107C2311yescharacteristic
86C22×C4×C107—21no—
87C22×C4×C107—21no—
88C22×C4×C107C2311yescharacteristic
89M16×C107—22no—
90C2×C8×C107C2312yes—
91C4⋊C4×C107—21no—
92C4⋊C4×C107C2312yes—
93C23×C107—21no—
94C23×C107C2×D411yescharacteristic
95C22×D107—24no—
96C22×D107—22no—
97C22×D107—21no—
98C22×D107—42no—
99C22×D107—42no—
100Q8×C107—21no—
101Q8×C107—21no—
102D4×C107—21no—
103D4×C107—21no—
104D4×C107—42no—
105C2×C4×C107—22no—
106C2×C4×C107—21no—
107C2×C4×C107—42no—
108C2×C4×C107—42no—
109C2×C4×C107C2×D411yescharacteristic
110C2×C4×C107C2×D412yes—
111C2×C4×C107#4112yes—
112C2×C4×C107C2411yescharacteristic
113C22⋊D107—42no—
114C22⋊D107—42no—
115C22⋊D107—42no—
116C2×Dic107—22no—
117C2×Dic107—21no—
118C2×Dic107—41no—
119C2×Dic107C2×D411yescharacteristic
120C2×Dic107C2×D411yescharacteristic
121C2×Dic107C2×D411yescharacteristic
122C2×Dic107C2×D411yescharacteristic
123D428—24no—
124D428—21no—
125D428—22no—
126D428—21no—
127D428C2×D414yes—
128C4×D107—44no—
129C4×D107—42no—
130C4×D107—44no—
131Dic214—21no—
132Dic214—21no—
133C8×C107—22no—
134C107⋊C8—22no—
135C22×C107—21no—
136C22×C107—21no—
137C22×C107—41no—
138C22×C107—41no—
139C22×C107—41no—
140C22×C107#12112yes—
141C22×C107C22×D411yescharacteristic
142C2×D107—24no—
143C2×D107—48no—
144C2×D107—42no—
145C2×D107—42no—
146C2×D107—41no—
147C2×D107—41no—
148C2×D107—41no—
149C4×C107—41no—
150C4×C107—42no—
151C4×C107#12112yes—
152C4×C107C22×D411yescharacteristic
153C4×C107C2×#4111yescharacteristic
154Dic107—21no—
155Dic107—21no—
156Dic107—21no—
157Dic107—41no—
158Dic107AGL(1,ℤ/8)12yescomplemented
159C2×C107—22no—
160C2×C107—81no—
161C2×C107C2×#12111yescharacteristic
162C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
163C2×C107#57511yescharacteristic
164D107—44no—
165D107—81no—
166#3036—1071nomaximal, Sylow
167C107#303611yescharacteristic, complemented, Sylow
168C2×AGL(1,ℤ/8)—1071no—
169C2×AGL(1,ℤ/8)—1071no—
170#532—1071no—
171C2×C4×D4—1071no—
172#463—1072no—
173#463—1072no—
174#461—1072no—
175#459—1072no—
176#423—1071no—
177#417—1071no—
178#417D10711yescharacteristic, complemented
179C2×#41—1071no—
180C2×#41C2×D10711yescharacteristic
181C22×D4—1071no—
182C23×C4—2141no—
183AGL(1,ℤ/8)—2144no—
184AGL(1,ℤ/8)—2144no—
185C2×SD16—1072no—
186C2×SD16—1072no—
187C2×D8—1072no—
188C2×D8—1072no—
189C2×M16—1071no—
190C2×M16C2×D10711yescharacteristic
191C4⋊Q8—1071no—
192#127—1071no—
193#124—1071no—
194#124—1071no—
195#121—2141no—
196#121—2141no—
197C4×D4—1072no—
198C4×D4—1072no—
199C4×D4—2141no—
200C4×D4—2141no—
201C4×D4—2141no—
202C4×D4—2141no—
203C2×C4⋊C4C2×D10711yescharacteristic
204C2×C22⋊C4—2141no—
205C2×C42—1071no—
206C4⋊C8—1072no—
207C4⋊C8—1072no—
208#103—1072no—
209#103C2×D10712yescomplemented
210#102—1072no—
211#102C2×D10712yescomplemented
212C24—2141no—
213#41—22no—
214#41—22no—
215#41—2142no—
216#41—2142no—
217C2×Q8—1071no—
218C2×Q8C22×D10711yescharacteristic
219C2×D4—21no—
220C2×D4—1074no—
221C2×D4—1072no—
222C2×D4—1071no—
223C2×D4—2141no—
224C2×D4—2141no—
225C2×D4—2141no—
226C2×D4—2141no—
227C2×D4—2142no—
228C2×D4C22×D10711yescharacteristic
229C22×C4—21no—
230C22×C4—21no—
231C22×C4—1071no—
232C22×C4—1071no—
233C22×C4—4284no—
234C22×C4—2142no—
235C22×C4—2142no—
236C22×C4—2141no—
237C22×C4C22×D10711yescharacteristic
238SD16—2144no—
239SD16—2144no—
240D8—2144no—
241D8—2144no—
242M16—22no—
243M16—2142no—
244C2×C8—1072no—
245C2×C8C22×D10712yes—
246C4⋊C4—21no—
247C4⋊C4—2141no—
248C4⋊C4—2141no—
249C4⋊C4C22×D10712yes—
250C22⋊C4—2142no—
251C22⋊C4—2142no—
252C22⋊C4—2141no—
253C22⋊C4—2141no—
254C22⋊C4—2141no—
255C22⋊C4—2141no—
256C42—1071no—
257C42—1071no—
258C42—1071no—
259C42—1071no—
260C23—21no—
261C23—4282no—
262C23—4282no—
263C23—2141no—
264C23—2144no—
265C23—2142no—
266C23D4×D10711yescharacteristic
267Q8—21no—
268Q8—21no—
269Q8—2141no—
270Q8—2141no—
271D4—21no—
272D4—21no—
273D4—42no—
274D4—1074no—
275D4—4282no—
276D4—4282no—
277D4—4282no—
278D4—2144no—
279D4—2141no—
280D4—2141no—
281D4—2142no—
282C2×C4—22no—
283C2×C4—21no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—1071no—
287C2×C4—1071no—
288C2×C4—1071no—
289C2×C4—1071no—
290C2×C4—4284no—
291C2×C4—4284no—
292C2×C4—4282no—
293C2×C4—4281no—
294C2×C4—2142no—
295C2×C4—2141no—
296C2×C4D4×D10711yescharacteristic
297C2×C4D4×D10712yes—
298C2×C4#1313381812yes—
299C2×C4C23×D10711yescharacteristic
300C8—22no—
301C8—2142no—
302C22—21no—
303C22—21no—
304C22—41no—
305C22—41no—
306C22—41no—
307C22—4288no—
308C22—4281no—
309C22—4281no—
310C22—4281no—
311C22—4282no—
312C22—4282no—
313C22—2144no—
314C22#1681208212yes—
315C22C2×D4×D10711yescharacteristic
316C4—42no—
317C4—41no—
318C4—1072no—
319C4—4281no—
320C4—2141no—
321C4—2141no—
322C4—2141no—
323C4#1681208212yes—
324C4C2×D4×D10711yescharacteristic
325C4C2×#1313381811yescharacteristic
326C2—22no—
327C2—81no—
328C2—8561no—
329C2—4284no—
330C2C2×#1681208211yescharacteristic
331C2AGL(1,ℤ/8)×D10711yescharacteristic
332C2#2029342611yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#417×C107C2
socleC22×C107C22×D4
hypercenter#417D107
hypocenterC107#3036
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#16812137C2semidirect
2. C2×#16812134C2semidirect
3. #20293173C2semidirect
4. C2×#16812071C2semidirect
5. #20292845C2semidirect
6. #20292841C2semidirect
7. #20292831C2semidirect
8. #20292821C2semidirect
9. #20292650C2semidirect
10. #20292626C2semidirect
11. #417×C107C2semidirect
22. #16812107C22semidirect
23. #16812105C22semidirect
24. #16812098C22semidirect
25. #16812096C22semidirect
36. Dic107⋊C8C22semidirect
37. Dic107⋊C8C22semidirect
38. #16812032C22semidirect
39. #16812029C22semidirect
43. #103×C107C22semidirect
44. #102×C107C22semidirect
51. C2×D428D4semidirect
158. Dic107AGL(1,ℤ/8)semidirect
167. C107#3036semidirect
178. #417D107semidirect
209. #103C2×D107semidirect
211. #102C2×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121372
C2×#168121342
#202931732
C2×#168120712
#202928452
#202928412
#202928312
#202928212
#202926502
#202926262
#417×C1072
#3036107