Finite Group Databasebeta

← #28346114

Subgroups — #28346114

Order 13696.

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

normal subgroups only

This group has 460 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×Q16×D107C211yesmaximal, characteristic, complemented
2C2×#16812117C211yesmaximal, characteristic
3#20293200C212yesmaximal
4#20292886C211yesmaximal, characteristic, complemented
5#20292885C211yesmaximal, characteristic, complemented
6#20292882C211yesmaximal, characteristic, complemented
7#20292867C212yesmaximal, complemented
8#20292866C212yesmaximal, complemented
9#20292662C211yesmaximal, characteristic, complemented
10#20292634C212yesmaximal, complemented
11#470×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C2×Q8×D107C2212yes—
14C2×Dic428C2211yescharacteristic
15C2×#13133808C2212yes—
16Q16×D107—24no—
17#16812117—24no—
18C2×#13133801C2211yescharacteristic
19C2×#13133801C2211yescharacteristic
20C22×C107⋊C8C2211yescharacteristic
21C2×C8×D107C2211yescharacteristic
22C2×C8×D107C2211yescharacteristic
23#16812108—22no—
24#16812095—22no—
25#16812094—22no—
26#16812091—22no—
27#16812089C2212yes—
28#16812088C2212yes—
29#16812085C2212yes—
30#16812067C2212yes—
31#16812042C2212yes—
32#16812041C2211yescharacteristic
33#16812034C2212yes—
34#16812033C2212yes—
35#16812032C2212yes—
36C2×Q16×C107C2211yescharacteristic
37C2×M16×C107C2211yescharacteristic
38C22⋊Q8×C107C2212yes—
39#106×C107C2211yescharacteristic
40#103×C107C2212yes—
41#13133818—22no—
42#13133818—22no—
43C2×Dic214—22no—
44C2×Dic214D411yescharacteristic
45C2×Dic214C2312yes—
46Q8×D107—24no—
47Q8×D107—24no—
48C2×D428D411yescharacteristic
49C2×C22⋊D107D411yescharacteristic
50C2×C22⋊D107D411yescharacteristic
51C2×C4×D107—22no—
52C2×C4×D107C2311yescharacteristic
53C2×C4×D107C2311yescharacteristic
54Dic428—22no—
55#13133808—24no—
56#13133801—22no—
57#13133801—22no—
58C2×C107⋊C8—22no—
59C2×C107⋊C8C2311yescharacteristic
60C2×C107⋊C8C2311yescharacteristic
61C8×D107—22no—
62C8×D107D414yes—
63C4⋊Dic107—22no—
64C4⋊Dic107C2312yes—
65Dic107⋊C4—22no—
66Dic107⋊C4—22no—
67C22⋊Dic107—22no—
68(C2×D107)⋊C4—22no—
69(C2×D107)⋊C4—22no—
70(C2×D107)⋊C4—22no—
71C4×Dic107—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×D107C2×D411yescharacteristic
83C22×D107C2×D411yescharacteristic
84Q8×C107—22no—
85Q8×C107—22no—
86C2×C4×C107—21no—
87C2×C4×C107—21no—
88C2×C4×C107—22no—
89C2×C4×C107—22no—
90C2×C4×C107C2×D411yescharacteristic
91C2×C4×C107C2411yescharacteristic
92C22⋊D107—22no—
93C22⋊D107—22no—
94C2×Dic107—22no—
95C2×Dic107—22no—
96C2×Dic107C2×D411yescharacteristic
97C2×Dic107C2×D411yescharacteristic
98D428—22no—
99C4×D107—21no—
100C4×D107—21no—
101C4×D107—44no—
102C4×D107C2×D412yes—
103C4×D107#4112yes—
104Dic214—22no—
105Dic214—22no—
106Dic214—22no—
107Dic214—44no—
108C8×C107—21no—
109C8×C107C2×D412yes—
110C107⋊C8—21no—
111C107⋊C8C2×D412yes—
112C22×C107—21no—
113C22×C107—22no—
114C22×C107C22×D411yescharacteristic
115C2×D107—22no—
116C2×D107—21no—
117C2×D107—22no—
118C2×D107#12112yes—
119C4×C107—21no—
120C4×C107—42no—
121C4×C107—42no—
122C4×C107C22×D411yescharacteristic
123C4×C107C2×#4111yescharacteristic
124Dic107—21no—
125Dic107—42no—
126Dic107—42no—
127Dic107#12112yes—
128C2×C107—41no—
129C2×C107C2×#12111yescharacteristic
130C2×C107#57812yes—
131D107—22no—
132D107—41no—
133#3059—1071nomaximal, Sylow
134C107#305911yescharacteristic, complemented, Sylow
135C22×Q16—1071no—
136C2×#131—1071no—
137#536—1072no—
138#470—1071no—
139#470—1071no—
140#470—1071no—
141#470D10711yescharacteristic, complemented
142#467—1072no—
143#467—1072no—
144#427—1071no—
145#419—1072no—
146C2×#41—1071no—
147C22×Q8—1072no—
148C2×Q16—1072no—
149C2×Q16—1071no—
150C2×Q16—2144no—
151C2×Q16C2×D10711yescharacteristic
152#131—2144no—
153C2×M16—1071no—
154C2×M16—1071no—
155C2×M16C2×D10711yescharacteristic
156C22×C8—1071no—
157C22×C8—1071no—
158C22×C8—1071no—
159C4⋊Q8—2142no—
160#124—2142no—
161#123—2142no—
162#123—2142no—
163C22⋊Q8—1072no—
164C22⋊Q8—1072no—
165C22⋊Q8—1072no—
166C22⋊Q8C2×D10712yes—
167#117—1072no—
168#107—1072no—
169#106—1071no—
170#106C2×D10711yescharacteristic
171#103—1072no—
172#103—1072no—
173#103—1072no—
174#103C2×D10712yes—
175#41—2142no—
176#41—2142no—
177C2×Q8—1071no—
178C2×Q8—1072no—
179C2×Q8—2142no—
180C2×Q8—2144no—
181C2×Q8—2144no—
182C2×Q8C22×D10712yes—
183C2×D4—1071no—
184C2×D4—1071no—
185C2×D4—1071no—
186C22×C4—1071no—
187C22×C4—1071no—
188C22×C4—2142no—
189C22×C4C22×D10711yescharacteristic
190Q16—22no—
191Q16—2144no—
192Q16—2142no—
193M16—22no—
194M16—2142no—
195M16—2142no—
196C2×C8—1071no—
197C2×C8—1074no—
198C2×C8—1071no—
199C2×C8—2142no—
200C2×C8—2142no—
201C2×C8C22×D10711yescharacteristic
202C2×C8C22×D10711yescharacteristic
203C4⋊C4—22no—
204C4⋊C4—1072no—
205C4⋊C4—2142no—
206C4⋊C4—2142no—
207C4⋊C4—2142no—
208C4⋊C4C22×D10712yes—
209C22⋊C4—22no—
210C22⋊C4—2142no—
211C22⋊C4—2142no—
212C22⋊C4—2142no—
213C22⋊C4—2142no—
214C42—2142no—
215C23—1071no—
216C23—1071no—
217C23D4×D10711yescharacteristic
218Q8—22no—
219Q8—22no—
220Q8—4284no—
221Q8—2142no—
222Q8—2142no—
223Q8—2142no—
224D4—2142no—
225D4—2142no—
226D4—2142no—
227C2×C4—21no—
228C2×C4—21no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—1071no—
232C2×C4—1072no—
233C2×C4—1072no—
234C2×C4—1071no—
235C2×C4—4284no—
236C2×C4—2142no—
237C2×C4—2142no—
238C2×C4—2141no—
239C2×C4—2141no—
240C2×C4D4×D10711yescharacteristic
241C2×C4C23×D10711yescharacteristic
242C8—21no—
243C8—1072no—
244C8—2141no—
245C8D4×D10712yes—
246C22—22no—
247C22—21no—
248C22—1072no—
249C22—2142no—
250C22—2141no—
251C22—2142no—
252C22C2×D4×D10711yescharacteristic
253C4—21no—
254C4—42no—
255C4—42no—
256C4—1072no—
257C4—4282no—
258C4—4282no—
259C4—2141no—
260C4C2×D4×D10711yescharacteristic
261C4#41×D10711yescharacteristic
262C2—41no—
263C2—4281no—
264C2—2142no—
265C2#121×D10711yescharacteristic
266C2#2029344312yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#470×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133818C22
hypercenter#470D107
hypocenterC107#3059
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×Q16×D107C2semidirect
4. #20292886C2semidirect
5. #20292885C2semidirect
6. #20292882C2semidirect
7. #20292867C2semidirect
8. #20292866C2semidirect
9. #20292662C2semidirect
10. #20292634C2semidirect
11. #470×C107C2semidirect
134. C107#3059semidirect
141. #470D107semidirect

Maximal subgroups

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

subgroupindex
C2×Q16×D1072
C2×#168121172
#202932002
#202928862
#202928852
#202928822
#202928672
#202928662
#202926622
#202926342
#470×C1072
#3059107