Finite Group Databasebeta

← #28346132

Subgroups — #28346132

Order 13696.

484 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 127 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×#16812132C211yesmaximal, characteristic, complemented
2C2×#16812115C211yesmaximal, characteristic, complemented
3#20293207C212yesmaximal, complemented
4#20292876C212yesmaximal, complemented
5#20292875C212yesmaximal, complemented
6#20292865C211yesmaximal, characteristic, complemented
7#20292864C211yesmaximal, characteristic, complemented
8#20292863C211yesmaximal, characteristic, complemented
9#107×D107C211yesmaximal, characteristic, complemented
10#20292627C212yesmaximal, complemented
11#467×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2212yes—
13C2×#13133818C2211yescharacteristic
14#16812132—24no—
15C2×#13133805C2212yes—
16C2×D856C2211yescharacteristic
17#16812115—24no—
18C2×#13133802C2211yescharacteristic
19C2×#13133801C2211yescharacteristic
20C2×#13133801C2211yescharacteristic
21C2×C8×D107C2211yescharacteristic
22C2×C8×D107C2211yescharacteristic
23#16812087—22no—
24#16812085C2212yes—
25#16812084—22no—
26#16812084C2212yes—
27#16812083C2212yes—
28#16812082—22no—
29#16812082—22no—
30C4⋊C4×D107C2212yes—
31#16812043C2211yescharacteristic
32#16812042C2212yes—
33#16812033C2212yes—
34#16812031C2212yes—
35#16812029C2212yes—
36C2×Q16×C107C2211yescharacteristic
37C22×C8×C107C2211yescharacteristic
38C22⋊Q8×C107C2212yes—
39#107×C107C2211yescharacteristic
40#103×C107C2212yes—
41#13133819—24no—
42#13133819—24no—
43#13133818—22no—
44#13133818—22no—
45C2×Dic214D411yescharacteristic
46C2×D428—22no—
47C2×D428—22no—
48C2×D428D411yescharacteristic
49C2×D428C2312yes—
50C2×C22⋊D107—22no—
51C2×C22⋊D107D411yescharacteristic
52C2×C22⋊D107D411yescharacteristic
53C2×C4×D107—22no—
54C2×C4×D107—22no—
55C2×C4×D107C2311yescharacteristic
56C2×C4×D107C2311yescharacteristic
57#13133805—24no—
58D856—22no—
59#13133802—22no—
60#13133801—22no—
61#13133801—22no—
62C2×C107⋊C8C2311yescharacteristic
63C2×C107⋊C8C2311yescharacteristic
64C8×D107—22no—
65C8×D107D414yescomplemented
66C4⋊Dic107C2312yes—
67Dic107⋊C4—22no—
68Dic107⋊C4—22no—
69(C2×D107)⋊C4—22no—
70(C2×D107)⋊C4—22no—
71(C2×D107)⋊C4—22no—
72C2×Q8×C107C2312yes—
73C22×C4×C107C2311yescharacteristic
74Q16×C107—22no—
75C2×C8×C107—22no—
76C2×C8×C107C2311yescharacteristic
77C2×C8×C107C2311yescharacteristic
78C4⋊C4×C107—22no—
79C4⋊C4×C107C2312yes—
80C22⋊C4×C107—22no—
81C23×C107C2×D411yescharacteristic
82C22×D107—22no—
83C22×D107C2×D411yescharacteristic
84C22×D107C2×D411yescharacteristic
85Q8×C107—22no—
86Q8×C107—22no—
87C2×C4×C107—21no—
88C2×C4×C107—21no—
89C2×C4×C107—22no—
90C2×C4×C107—22no—
91C2×C4×C107C2×D411yescharacteristic
92C2×C4×C107C2411yescharacteristic
93C22⋊D107—22no—
94C22⋊D107—22no—
95C22⋊D107—82no—
96C2×Dic107—22no—
97C2×Dic107C2×D411yescharacteristic
98C2×Dic107C2×D411yescharacteristic
99D428—22no—
100D428—22no—
101D428—22no—
102D428—44no—
103D428—82no—
104C4×D107—21no—
105C4×D107—21no—
106C4×D107—44no—
107C4×D107—82no—
108C4×D107C2×D412yes—
109C4×D107#4112yes—
110Dic214—22no—
111C8×C107—21no—
112C8×C107C2×D412yes—
113C107⋊C8—21no—
114C107⋊C8C2×D412yes—
115C22×C107—21no—
116C22×C107—21no—
117C22×C107—21no—
118C22×C107C22×D411yescharacteristic
119C2×D107—21no—
120C2×D107—21no—
121C2×D107—21no—
122C2×D107—21no—
123C2×D107—21no—
124C2×D107—42no—
125C2×D107—42no—
126C2×D107—42no—
127C2×D107#12112yes—
128C4×C107—21no—
129C4×C107—42no—
130C4×C107—42no—
131C4×C107C22×D411yescharacteristic
132C4×C107C2×#4111yescharacteristic
133Dic107—21no—
134Dic107—42no—
135Dic107#12112yes—
136C2×C107—41no—
137C2×C107C2×#12111yescharacteristic
138C2×C107#57611yescharacteristic
139C2×C107#57811yescharacteristic
140D107—22no—
141D107—41no—
142D107—82no—
143#3061—1071nomaximal, Sylow
144C107#306111yescharacteristic, complemented, Sylow
145C2×#135—1071no—
146C2×#131—1071no—
147#537—1072no—
148#468—1072no—
149#468—1072no—
150#467—1071no—
151#467D10711yescharacteristic, complemented
152#466—1071no—
153#466—1071no—
154C2×#107—1071no—
155#417—1072no—
156C2×#41—1071no—
157C2×#41—1072no—
158#135—2144no—
159C2×Q16C2×D10711yescharacteristic
160C2×SD16—1072no—
161C2×D8—1071no—
162#131—2144no—
163C2×M16—1071no—
164C2×M16—1071no—
165C2×M16—1071no—
166C22×C8—1071no—
167C22×C8—1071no—
168C22×C8C2×D10711yescharacteristic
169C22⋊Q8—1072no—
170C22⋊Q8—2142no—
171C22⋊Q8C2×D10712yes—
172#121—1072no—
173#121—1072no—
174#121—2142no—
175#121—2142no—
176#121—2142no—
177C2×C4⋊C4—1072no—
178#107—1071no—
179#107—1072no—
180#107C2×D10711yescharacteristic
181#103—1072no—
182#103C2×D10712yes—
183#102—1072no—
184#102—1072no—
185#41—2142no—
186#41—2142no—
187#41—2144no—
188#41—2144no—
189C2×Q8—1071no—
190C2×Q8C22×D10712yes—
191C2×D4—1071no—
192C2×D4—1071no—
193C2×D4—1071no—
194C2×D4—1072no—
195C2×D4—2142no—
196C2×D4—2142no—
197C2×D4—2142no—
198C22×C4—1071no—
199C22×C4—1071no—
200C22×C4—2142no—
201C22×C4—2142no—
202C22×C4C22×D10711yescharacteristic
203Q16—22no—
204SD16—2144no—
205D8—2142no—
206M16—2142no—
207M16—2142no—
208M16—2142no—
209C2×C8—22no—
210C2×C8—1071no—
211C2×C8—1074no—
212C2×C8—1071no—
213C2×C8—2142no—
214C2×C8C22×D10711yescharacteristic
215C2×C8C22×D10711yescharacteristic
216C4⋊C4—22no—
217C4⋊C4—1072no—
218C4⋊C4—2142no—
219C4⋊C4—2142no—
220C4⋊C4C22×D10712yes—
221C22⋊C4—22no—
222C22⋊C4—2142no—
223C22⋊C4—2142no—
224C22⋊C4—2142no—
225C23—1071no—
226C23—1071no—
227C23—2142no—
228C23D4×D10711yescharacteristic
229Q8—22no—
230Q8—22no—
231Q8—2142no—
232D4—8562no—
233D4—8562no—
234D4—4284no—
235D4—2142no—
236D4—2142no—
237D4—2142no—
238D4—2142no—
239D4—2142no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—22no—
243C2×C4—22no—
244C2×C4—8562no—
245C2×C4—1071no—
246C2×C4—1072no—
247C2×C4—1072no—
248C2×C4—1071no—
249C2×C4—4284no—
250C2×C4—2142no—
251C2×C4—2141no—
252C2×C4—2141no—
253C2×C4D4×D10711yescharacteristic
254C2×C4C23×D10711yescharacteristic
255C8—21no—
256C8—1072no—
257C8—2141no—
258C8D4×D10712yes—
259C22—21no—
260C22—21no—
261C22—21no—
262C22—1072no—
263C22—4282no—
264C22—4282no—
265C22—4282no—
266C22—2141no—
267C22—2141no—
268C22—2141no—
269C22—2141no—
270C22—2141no—
271C22C2×D4×D10711yescharacteristic
272C4—21no—
273C4—42no—
274C4—42no—
275C4—1072no—
276C4—4282no—
277C4—2141no—
278C4C2×D4×D10711yescharacteristic
279C4#41×D10711yescharacteristic
280C2—41no—
281C2—8562no—
282C2—4281no—
283C2—2142no—
284C2#121×D10711yescharacteristic
285C2#2029342911yescharacteristic
286C2#2029344011yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#467×C107C2
socleC22×C107C22×D4
hypercenter#467D107
hypocenterC107#3061
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
1. C2×#16812132C2semidirect
2. C2×#16812115C2semidirect
3. #20293207C2semidirect
4. #20292876C2semidirect
5. #20292875C2semidirect
6. #20292865C2semidirect
7. #20292864C2semidirect
8. #20292863C2semidirect
9. #107×D107C2semidirect
10. #20292627C2semidirect
11. #467×C107C2semidirect
65. C8×D107D4semidirect
144. C107#3061semidirect
151. #467D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121322
C2×#168121152
#202932072
#202928762
#202928752
#202928652
#202928642
#202928632
#107×D1072
#202926272
#467×C1072
#3061107