Finite Group Databasebeta

← #28348217

Subgroups — #28348217

Order 13696.

504 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 504 conjugacy classes of subgroups — too many to draw. Only its 133 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×#16812129C212yesmaximal, complemented
2#20293298C211yesmaximal, characteristic, complemented
3#20293003C211yesmaximal, characteristic, complemented
4#20292861C212yesmaximal, complemented
5#20292827C211yesmaximal, characteristic, complemented
6#20292824C211yesmaximal, characteristic, complemented
7#20292772C212yesmaximal, complemented
8#20292770C212yesmaximal, complemented
9C4×#13133808C211yesmaximal, characteristic, complemented
10D4×C107⋊C8C211yesmaximal, characteristic, complemented
11#550×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2212yes—
13#16812129—28no—
14C2×#13133808C2211yescharacteristic
15C2×#13133806C2212yes—
16C2×#13133805C2212yes—
17C2×#13133803C2212yes—
18C2×#13133803C2212yes—
19C22×C107⋊C8C2212yes—
20#16812106C2211yescharacteristic
21#16812105—21no—
22#16812083C2212yes—
23#16812082—22no—
24C4×Dic214C2211yescharacteristic
25#16812071—21no—
26#16812042C2211yescharacteristic
27C4⋊(C107⋊C8)C2211yescharacteristic
28#16812033C2211yescharacteristic
29#16812032C2211yescharacteristic
30#16812030C2212yes—
31#16812029C2212yes—
32C22⋊(C107⋊C8)C2212yes—
33C4×C107⋊C8C2211yescharacteristic
34C2×#41×C107C2212yes—
35#127×C107—21no—
36#127×C107C2211yescharacteristic
37#121×C107—22no—
38#121×C107C2212yes—
39C4×Q8×C107C2211yescharacteristic
40C4×D4×C107—21no—
41C4×D4×C107C2211yescharacteristic
42#13133818—24no—
43#13133818—42no—
44C2×Dic214C2311yescharacteristic
45C2×D428—21no—
46C2×D428—21no—
47C2×D428C2312yes—
48C2×C22⋊D107—22no—
49C2×C22⋊D107—22no—
50C2×C4×D107—22no—
51#13133808D414yescomplemented
52#13133806—24no—
53#13133805—24no—
54#13133803—24no—
55#13133803—42no—
56C2×C107⋊C8—24no—
57C2×C107⋊C8C2311yescharacteristic
58C2×C107⋊C8C2312yes—
59C2×C107⋊C8C2311yescharacteristic
60C4⋊Dic107C2311yescharacteristic
61Dic107⋊C4—21no—
62(C2×D107)⋊C4—22no—
63C4×Dic107—21no—
64#41×C107—24no—
65#41×C107—42no—
66C2×Q8×C107C2311yescharacteristic
67C2×D4×C107—22no—
68C2×D4×C107—22no—
69C2×D4×C107—21no—
70C2×D4×C107—21no—
71C2×D4×C107D411yescharacteristic
72C2×D4×C107C2312yes—
73C22×C4×C107—22no—
74C22×C4×C107C2312yes—
75C4⋊C4×C107—21no—
76C4⋊C4×C107D411yescharacteristic
77C4⋊C4×C107C2311yescharacteristic
78C22⋊C4×C107—22no—
79C22⋊C4×C107D412yes—
80C42×C107—21no—
81C42×C107C2311yescharacteristic
82C23×C107—22no—
83C23×C107C2×D412yes—
84C22×D107—22no—
85Q8×C107—21no—
86Q8×C107C2×D412yes—
87D4×C107—22no—
88D4×C107—22no—
89D4×C107—22no—
90D4×C107—44no—
91D4×C107—42no—
92D4×C107—81no—
93D4×C107—82no—
94C2×C4×C107—21no—
95C2×C4×C107—22no—
96C2×C4×C107—22no—
97C2×C4×C107—21no—
98C2×C4×C107—44no—
99C2×C4×C107C2×D411yescharacteristic
100C2×C4×C107C2×D412yes—
101C2×C4×C107C2×D411yescharacteristic
102C2×C4×C107C2411yescharacteristic
103C22⋊D107—44no—
104C22⋊D107—82no—
105C2×Dic107—21no—
106C2×Dic107—21no—
107D428—22no—
108D428—22no—
109D428—42no—
110D428—81no—
111C4×D107—44no—
112Dic214—21no—
113Dic214C2×D412yes—
114C107⋊C8—22no—
115C107⋊C8—21no—
116C107⋊C8C2×D412yes—
117C22×C107—22no—
118C22×C107—22no—
119C22×C107—22no—
120C22×C107—42no—
121C22×C107—42no—
122C22×C107—42no—
123C22×C107C22×D411yescharacteristic
124C2×D107—42no—
125C2×D107—42no—
126C2×D107—42no—
127C4×C107—22no—
128C4×C107—22no—
129C4×C107—21no—
130C4×C107—41no—
131C4×C107#13512yescomplemented
132C4×C107C22×D411yescharacteristic
133C4×C107H(5,𝔽2)11yescharacteristic
134Dic107—22no—
135Dic107—41no—
136C2×C107—42no—
137C2×C107—82no—
138C2×C107D4211yescharacteristic
139C2×C107C2×#13511yescharacteristic
140C2×C107#57611yescharacteristic
141D107—82no—
142#3287—1071nomaximal, Sylow
143C107#328711yescharacteristic, complemented, Sylow
144C2×#135—1072no—
145#550—1071no—
146#550D10711yescharacteristic, complemented
147#493—1071no—
148#466—1072no—
149#460—1071no—
150#460—1071no—
151#449—1072no—
152#449—1072no—
153C4×Q16—1071no—
154C8×D4—1071no—
155C2×#41—1072no—
156C2×#41C2×D10712yes—
157#135—2148no—
158C2×Q16—1071no—
159C2×SD16—1072no—
160C2×SD16—1072no—
161C2×D8—1072no—
162C2×D8—1072no—
163C22×C8—1072no—
164#127—21no—
165#127—1071no—
166#127—2141no—
167#127C2×D10711yescharacteristic
168#121—22no—
169#121—1072no—
170#121—2142no—
171#121C2×D10712yes—
172C4×Q8—1071no—
173C4×Q8C2×D10711yescharacteristic
174C4×D4—21no—
175C4×D4—2141no—
176C4×D4C2×D10711yescharacteristic
177#107—1071no—
178C4⋊C8—1071no—
179#103—1071no—
180#103—1071no—
181#102—1072no—
182#102—1072no—
183C22⋊C8—1072no—
184C4×C8—1071no—
185#41—24no—
186#41—42no—
187#41—4282no—
188#41—2144no—
189C2×Q8—1071no—
190C2×Q8C22×D10711yescharacteristic
191C2×D4—22no—
192C2×D4—22no—
193C2×D4—21no—
194C2×D4—21no—
195C2×D4—1072no—
196C2×D4—2142no—
197C2×D4—2142no—
198C2×D4—2141no—
199C2×D4—2141no—
200C2×D4C22⋊D10711yescharacteristic
201C2×D4C22×D10712yes—
202C22×C4—22no—
203C22×C4—2142no—
204C22×C4C22×D10712yes—
205Q16—1074no—
206SD16—2144no—
207SD16—2144no—
208D8—4282no—
209D8—2144no—
210C2×C8—1071no—
211C2×C8—1071no—
212C2×C8—1072no—
213C2×C8—2144no—
214C4⋊C4—21no—
215C4⋊C4—1071no—
216C4⋊C4—2141no—
217C4⋊C4C22⋊D10711yescharacteristic
218C4⋊C4C22×D10711yescharacteristic
219C22⋊C4—22no—
220C22⋊C4—2142no—
221C22⋊C4C22⋊D10712yes—
222C42—21no—
223C42—2141no—
224C42C22×D10711yescharacteristic
225C23—22no—
226C23—2142no—
227C23C2×C22⋊D10712yes—
228Q8—21no—
229Q8—1072no—
230Q8—2141no—
231Q8D4×D10712yes—
232D4—22no—
233D4—22no—
234D4—22no—
235D4—44no—
236D4—42no—
237D4—81no—
238D4—82no—
239D4—8561no—
240D4—8562no—
241D4—4284no—
242D4—4282no—
243D4—2142no—
244D4—2142no—
245C2×C4—21no—
246C2×C4—22no—
247C2×C4—22no—
248C2×C4—21no—
249C2×C4—44no—
250C2×C4—4284no—
251C2×C4—2141no—
252C2×C4—2141no—
253C2×C4C2×C22⋊D10711yescharacteristic
254C2×C4C2×C22⋊D10712yes—
255C2×C4C2×C22⋊D10711yescharacteristic
256C2×C4C23×D10711yescharacteristic
257C8—1072no—
258C8—2142no—
259C8—2141no—
260C22—22no—
261C22—22no—
262C22—22no—
263C22—42no—
264C22—42no—
265C22—42no—
266C22—4282no—
267C22—4282no—
268C22—4282no—
269C22C22×C22⋊D10711yescharacteristic
270C4—21no—
271C4—22no—
272C4—22no—
273C4—41no—
274C4—4281no—
275C4—2142no—
276C4#1681212912yescomplemented
277C4C2×D4×D10711yescharacteristic
278C4#1681215411yescharacteristic
279C2—42no—
280C2—82no—
281C2—8562no—
282C2D4×C22⋊D10711yescharacteristic
283C2C2×#1681212911yescharacteristic
284C2#2029343011yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#550×C107C2
socleC22×C107C22×D4
hypercenter#550D107
hypocenterC107#3287
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C4×D4 ◁ #550

Complements

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

NH ≅ G/Ntype
1. C2×#16812129C2semidirect
2. #20293298C2semidirect
3. #20293003C2semidirect
4. #20292861C2semidirect
5. #20292827C2semidirect
6. #20292824C2semidirect
7. #20292772C2semidirect
8. #20292770C2semidirect
9. C4×#13133808C2semidirect
10. D4×C107⋊C8C2semidirect
11. #550×C107C2semidirect
51. #13133808D4semidirect
131. C4×C107#135semidirect
143. C107#3287semidirect
146. #550D107semidirect
276. C4#16812129semidirect

Maximal subgroups

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

subgroupindex
C2×#168121292
#202932982
#202930032
#202928612
#202928272
#202928242
#202927722
#202927702
C4×#131338082
D4×C107⋊C82
#550×C1072
#3287107