Finite Group Databasebeta

← #28346125

Subgroups — #28346125

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×#16812117C211yesmaximal, characteristic, complemented
3#20293207C212yesmaximal, complemented
4#20292882C211yesmaximal, characteristic, complemented
5#20292881C211yesmaximal, characteristic, complemented
6#20292880C211yesmaximal, characteristic, complemented
7#20292859C212yesmaximal, complemented
8#20292858C212yesmaximal, complemented
9#106×D107C211yesmaximal, characteristic, complemented
10#20292627C212yesmaximal, complemented
11#470×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2212yes—
13C2×#13133818C2211yescharacteristic
14#16812132—24no—
15C2×#13133805C2212yes—
16C2×D856C2211yescharacteristic
17#16812117—24no—
18C2×#13133801C2211yescharacteristic
19C2×#13133801C2211yescharacteristic
20C22×C107⋊C8C2211yescharacteristic
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#16812041C2211yescharacteristic
32#16812040C2212yes—
33#16812033C2212yes—
34#16812031C2212yes—
35#16812029C2212yes—
36C2×Q16×C107C2211yescharacteristic
37C2×M16×C107C2211yescharacteristic
38C22⋊Q8×C107C2212yes—
39#106×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#13133801—22no—
60#13133801—22no—
61C2×C107⋊C8—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—
75M16×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—22no—
117C22×C107C22×D411yescharacteristic
118C2×D107—22no—
119C2×D107—21no—
120C2×D107—22no—
121C2×D107—42no—
122C2×D107—42no—
123C2×D107—42no—
124C2×D107#12112yes—
125C4×C107—21no—
126C4×C107—42no—
127C4×C107—42no—
128C4×C107C22×D411yescharacteristic
129C4×C107C2×#4111yescharacteristic
130Dic107—21no—
131Dic107—42no—
132Dic107#12112yes—
133C2×C107—41no—
134C2×C107C2×#12111yescharacteristic
135C2×C107#57712yes—
136D107—22no—
137D107—41no—
138D107—82no—
139#3060—1071nomaximal, Sylow
140C107#306011yescharacteristic, complemented, Sylow
141C2×#135—1071no—
142C2×#131—1071no—
143#537—1072no—
144#470—1071no—
145#470D10711yescharacteristic, complemented
146#469—1071no—
147#469—1071no—
148#465—1072no—
149#465—1072no—
150C2×#106—1071no—
151#417—1072no—
152C2×#41—1071no—
153C2×#41—1072no—
154#135—2144no—
155C2×Q16C2×D10711yescharacteristic
156C2×SD16—1072no—
157C2×D8—1071no—
158#131—2144no—
159C2×M16—1071no—
160C2×M16—1071no—
161C2×M16C2×D10711yescharacteristic
162C22×C8—1071no—
163C22×C8—1071no—
164C22×C8—1071no—
165C22⋊Q8—1072no—
166C22⋊Q8—2142no—
167C22⋊Q8C2×D10712yes—
168#121—1072no—
169#121—1072no—
170#121—2142no—
171#121—2142no—
172#121—2142no—
173C2×C4⋊C4—1072no—
174#106—1071no—
175#106—1072no—
176#106C2×D10711yescharacteristic
177#103—1072no—
178#103C2×D10712yes—
179#102—1072no—
180#102—1072no—
181#41—2142no—
182#41—2142no—
183#41—2144no—
184#41—2144no—
185C2×Q8—1071no—
186C2×Q8C22×D10712yes—
187C2×D4—1071no—
188C2×D4—1071no—
189C2×D4—1071no—
190C2×D4—1072no—
191C2×D4—2142no—
192C2×D4—2142no—
193C2×D4—2142no—
194C22×C4—1071no—
195C22×C4—1071no—
196C22×C4—2142no—
197C22×C4—2142no—
198C22×C4C22×D10711yescharacteristic
199Q16—22no—
200SD16—2144no—
201D8—2142no—
202M16—22no—
203M16—2142no—
204M16—2142no—
205C2×C8—1071no—
206C2×C8—1074no—
207C2×C8—1071no—
208C2×C8—2142no—
209C2×C8—2142no—
210C2×C8C22×D10711yescharacteristic
211C2×C8C22×D10711yescharacteristic
212C4⋊C4—22no—
213C4⋊C4—1072no—
214C4⋊C4—2142no—
215C4⋊C4—2142no—
216C4⋊C4C22×D10712yes—
217C22⋊C4—22no—
218C22⋊C4—2142no—
219C22⋊C4—2142no—
220C22⋊C4—2142no—
221C23—1071no—
222C23—1071no—
223C23—2142no—
224C23D4×D10711yescharacteristic
225Q8—22no—
226Q8—22no—
227Q8—2142no—
228D4—8562no—
229D4—8562no—
230D4—4284no—
231D4—2142no—
232D4—2142no—
233D4—2142no—
234D4—2142no—
235D4—2142no—
236C2×C4—21no—
237C2×C4—21no—
238C2×C4—22no—
239C2×C4—22no—
240C2×C4—8562no—
241C2×C4—1071no—
242C2×C4—1072no—
243C2×C4—1072no—
244C2×C4—1071no—
245C2×C4—4284no—
246C2×C4—2142no—
247C2×C4—2141no—
248C2×C4—2141no—
249C2×C4D4×D10711yescharacteristic
250C2×C4C23×D10711yescharacteristic
251C8—21no—
252C8—1072no—
253C8—2141no—
254C8D4×D10712yes—
255C22—22no—
256C22—21no—
257C22—1072no—
258C22—4282no—
259C22—4282no—
260C22—4282no—
261C22—2142no—
262C22—2141no—
263C22—2142no—
264C22C2×D4×D10711yescharacteristic
265C4—21no—
266C4—42no—
267C4—42no—
268C4—1072no—
269C4—4282no—
270C4—2141no—
271C4C2×D4×D10711yescharacteristic
272C4#41×D10711yescharacteristic
273C2—41no—
274C2—8562no—
275C2—4281no—
276C2—2142no—
277C2#121×D10711yescharacteristic
278C2#2029343612yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#470×C107C2
socleC22×C107C22×D4
hypercenter#470D107
hypocenterC107#3060
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #470

Complements

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

NH ≅ G/Ntype
1. C2×#16812132C2semidirect
2. C2×#16812117C2semidirect
3. #20293207C2semidirect
4. #20292882C2semidirect
5. #20292881C2semidirect
6. #20292880C2semidirect
7. #20292859C2semidirect
8. #20292858C2semidirect
9. #106×D107C2semidirect
10. #20292627C2semidirect
11. #470×C107C2semidirect
65. C8×D107D4semidirect
140. C107#3060semidirect
145. #470D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121322
C2×#168121172
#202932072
#202928822
#202928812
#202928802
#202928592
#202928582
#106×D1072
#202926272
#470×C1072
#3060107