Finite Group Databasebeta

← #28349507

Subgroups — #28349507

Order 13696.

418 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 418 conjugacy classes of subgroups — too many to draw. Only its 115 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
1#20293363C211yesmaximal, characteristic, complemented
2#20293298C211yesmaximal, characteristic, complemented
3#20293015C212yesmaximal, complemented
4#20293003C211yesmaximal, characteristic, complemented
5#20292841C212yesmaximal, complemented
6#20292828C211yesmaximal, characteristic, complemented
7#20292820C212yesmaximal, complemented
8#20292754C211yesmaximal, characteristic, complemented
9#20292736C212yesmaximal, complemented
10C4×#13133806C211yesmaximal, characteristic, complemented
11#459×C107C211yesmaximal, characteristic, complemented
12C2×#13133818—21no—
13C2×#13133807C2212yes—
14C2×#13133806C2211yescharacteristic
15C2×#13133803C2212yes—
16C2×#13133803C2212yes—
17#16812106C2211yescharacteristic
18#16812105C2212yes—
19#16812096—21no—
20#16812096C2212yes—
21#16812090—22no—
22#16812083—21no—
23#16812082—22no—
24C4×Dic214C2211yescharacteristic
25C4×Dic214C2211yescharacteristic
26D4×Dic107—21no—
27D4×Dic107C2212yes—
28#16812071C2212yes—
29#16812040C2211yescharacteristic
30Dic107⋊C8C2212yes—
31#16812032C2211yescharacteristic
32#16812031C2212yes—
33#16812030C2211yescharacteristic
34#16812019C2212yes—
35C4×C107⋊C8C2211yescharacteristic
36C2×D8×C107C2212yes—
37#127×C107C2211yescharacteristic
38C4×D4×C107C2211yescharacteristic
39C4⋊C8×C107C2211yescharacteristic
40#102×C107C2212yes—
41#13133818—44no—
42C2×Dic214D411yescharacteristic
43C2×Dic214C2311yescharacteristic
44C2×D428—21no—
45C2×D428—21no—
46C2×D428C2312yes—
47C2×C22⋊D107—22no—
48C2×C22⋊D107—22no—
49C22×Dic107—22no—
50C2×C4×D107—22no—
51#13133807—42no—
52#13133806—22no—
53#13133803—24no—
54#13133803—42no—
55C2×C107⋊C8C2311yescharacteristic
56C2×C107⋊C8C2311yescharacteristic
57C4⋊Dic107D411yescharacteristic
58C4⋊Dic107C2311yescharacteristic
59Dic107⋊C4—21no—
60Dic107⋊C4D412yes—
61C22⋊Dic107—22no—
62C22⋊Dic107—22no—
63C22⋊Dic107—22no—
64(C2×D107)⋊C4—22no—
65C4×Dic107—21no—
66C4×Dic107C2312yes—
67C2×D4×C107—21no—
68C2×D4×C107—21no—
69C2×D4×C107C2312yes—
70C2×D4×C107C2311yescharacteristic
71C22×C4×C107—21no—
72D8×C107—42no—
73C2×C8×C107C2312yes—
74C4⋊C4×C107C2311yescharacteristic
75C22⋊C4×C107—21no—
76C42×C107C2311yescharacteristic
77C23×C107—21no—
78C23×C107—22no—
79C22×D107—22no—
80D4×C107—22no—
81D4×C107—21no—
82D4×C107—21no—
83D4×C107—22no—
84D4×C107—42no—
85D4×C107—81no—
86C2×C4×C107—21no—
87C2×C4×C107—42no—
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107C2×D411yescharacteristic
90C2×C4×C107C2411yescharacteristic
91C22⋊D107—44no—
92C22⋊D107—82no—
93C2×Dic107—21no—
94C2×Dic107—21no—
95C2×Dic107—82no—
96C2×Dic107C2×D412yes—
97C2×Dic107C2×D412yes—
98D428—22no—
99D428—22no—
100D428—42no—
101D428—81no—
102C4×D107—44no—
103Dic214—22no—
104Dic214—21no—
105Dic214—21no—
106C8×C107—22no—
107C107⋊C8—21no—
108C107⋊C8#4112yescomplemented
109C22×C107—42no—
110C22×C107—41no—
111C22×C107—41no—
112C22×C107—42no—
113C22×C107—42no—
114C22×C107—41no—
115C22×C107C22×D411yescharacteristic
116C2×D107—42no—
117C2×D107—42no—
118C2×D107—42no—
119C4×C107—21no—
120C4×C107—41no—
121C4×C107AGL(1,ℤ/8)12yes—
122C4×C107C2×#4111yescharacteristic
123C4×C107H(5,𝔽2)11yescharacteristic
124Dic107—22no—
125Dic107—22no—
126Dic107—41no—
127Dic107—41no—
128C2×C107—81no—
129C2×C107—82no—
130C2×C107#55011yescharacteristic
131C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
132C2×C107#57611yescharacteristic
133D107—82no—
134#3399—1071nomaximal, Sylow
135C107#339911yescharacteristic, complemented, Sylow
136#559—1071no—
137#550—1071no—
138#496—1072no—
139#493—1071no—
140#463—1072no—
141#460—1071no—
142#459—1072no—
143#459D10711yescharacteristic, complemented
144#446—1071no—
145#442—1072no—
146C4×SD16—1071no—
147C2×#41—2141no—
148C2×SD16—1071no—
149C2×SD16—1072no—
150C2×D8—1072no—
151C2×D8—1072no—
152C2×D8C2×D10712yes—
153#127—1072no—
154#127—1071no—
155#127C2×D10711yescharacteristic
156#124—1072no—
157#124—2141no—
158#123—2142no—
159#121—2142no—
160#121—2141no—
161C4×Q8—1071no—
162C4×Q8—1071no—
163C4×D4—1072no—
164C4×D4—1072no—
165C4×D4—2141no—
166C4×D4C2×D10711yescharacteristic
167#106—1071no—
168C4⋊C8—1072no—
169C4⋊C8C2×D10711yescharacteristic
170#103—1071no—
171#102—1071no—
172#102—1072no—
173#102C2×D10712yes—
174#97—1072no—
175C4×C8—1071no—
176#41—4284no—
177C2×Q8—1071no—
178C2×Q8—1071no—
179C2×D4—21no—
180C2×D4—21no—
181C2×D4—1072no—
182C2×D4—2142no—
183C2×D4—2142no—
184C2×D4—2141no—
185C2×D4—2141no—
186C2×D4C22×D10711yescharacteristic
187C2×D4C22×D10712yes—
188C22×C4—21no—
189C22×C4—2142no—
190C22×C4—2142no—
191SD16—4282no—
192SD16—2142no—
193D8—42no—
194D8—4282no—
195D8—2144no—
196C2×C8—1071no—
197C2×C8—1071no—
198C2×C8C22×D10712yes—
199C4⋊C4—1071no—
200C4⋊C4—1072no—
201C4⋊C4—1071no—
202C4⋊C4—2141no—
203C4⋊C4C22×D10711yescharacteristic
204C22⋊C4—21no—
205C22⋊C4—2142no—
206C22⋊C4—2142no—
207C22⋊C4—2142no—
208C22⋊C4—2142no—
209C42—1072no—
210C42—2141no—
211C42C22×D10711yescharacteristic
212C23—21no—
213C23—22no—
214C23—2142no—
215Q8—2141no—
216Q8—2141no—
217Q8—2142no—
218D4—22no—
219D4—21no—
220D4—21no—
221D4—22no—
222D4—42no—
223D4—81no—
224D4—8561no—
225D4—8562no—
226D4—4284no—
227D4—4282no—
228D4—2142no—
229D4—2142no—
230C2×C4—21no—
231C2×C4—42no—
232C2×C4—8562no—
233C2×C4—1072no—
234C2×C4—1072no—
235C2×C4—4284no—
236C2×C4—2141no—
237C2×C4—2141no—
238C2×C4D4×D10711yescharacteristic
239C2×C4D4×D10711yescharacteristic
240C2×C4C23×D10711yescharacteristic
241C8—22no—
242C8—1072no—
243C8—2141no—
244C22—41no—
245C22—42no—
246C22—42no—
247C22—41no—
248C22—42no—
249C22—41no—
250C22—4282no—
251C22—4282no—
252C22—4282no—
253C22C2×D4×D10711yescharacteristic
254C4—21no—
255C4—41no—
256C4—4281no—
257C4—4281no—
258C4—2142no—
259C4—2142no—
260C4#1681213312yes—
261C4#41×D10711yescharacteristic
262C4#1681215411yescharacteristic
263C2—81no—
264C2—82no—
265C2—8562no—
266C2#2029329511yescharacteristic
267C2C2×#1681213311yescharacteristic
268C2#2029343111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#459×C107C2
socleC22×C107C22×D4
hypercenter#459D107
hypocenterC107#3399
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #459

Complements

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

NH ≅ G/Ntype
1. #20293363C2semidirect
2. #20293298C2semidirect
3. #20293015C2semidirect
4. #20293003C2semidirect
5. #20292841C2semidirect
6. #20292828C2semidirect
7. #20292820C2semidirect
8. #20292754C2semidirect
9. #20292736C2semidirect
10. C4×#13133806C2semidirect
11. #459×C107C2semidirect
108. C107⋊C8#41semidirect
135. C107#3399semidirect
143. #459D107semidirect

Maximal subgroups

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

subgroupindex
#202933632
#202932982
#202930152
#202930032
#202928412
#202928282
#202928202
#202927542
#202927362
C4×#131338062
#459×C1072
#3399107