Finite Group Databasebeta

← #28348191

Subgroups — #28348191

Order 13696.

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

normal subgroups only

This group has 494 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#20293282C211yesmaximal, characteristic, complemented
3#20292999C211yesmaximal, characteristic, complemented
4#20292854C212yesmaximal, complemented
5#20292837C211yesmaximal, characteristic, complemented
6#20292819C211yesmaximal, characteristic, complemented
7#20292789C212yesmaximal, complemented
8#20292770C212yesmaximal, complemented
9C4×#13133805C211yesmaximal, characteristic, complemented
10D4×C107⋊C8C211yesmaximal, characteristic, complemented
11#550×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2212yes—
13#16812129—28no—
14C2×#13133808C2212yes—
15C2×#13133806C2212yes—
16C2×#13133806C2212yes—
17C2×#13133805C2211yescharacteristic
18C2×#13133803C2212yes—
19C22×C107⋊C8C2212yes—
20#16812109C2211yescharacteristic
21#16812091—22no—
22#16812088C2212yes—
23#16812080—21no—
24C4×D428C2211yescharacteristic
25C4⋊C4×D107—21no—
26#16812040C2211yescharacteristic
27C4⋊(C107⋊C8)C2211yescharacteristic
28#16812033C2211yescharacteristic
29#16812032C2212yes—
30#16812030C2212yes—
31#16812029C2211yescharacteristic
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×Dic214C2312yes—
45C2×D428C2311yescharacteristic
46C2×C22⋊D107—22no—
47C2×C4×D107—21no—
48C2×C4×D107—22no—
49#13133808—24no—
50#13133806—24no—
51#13133806—42no—
52#13133805D414yescomplemented
53#13133803—24no—
54C2×C107⋊C8—24no—
55C2×C107⋊C8C2312yes—
56C2×C107⋊C8C2311yescharacteristic
57C2×C107⋊C8C2311yescharacteristic
58C4⋊Dic107—21no—
59C4⋊Dic107—21no—
60C4⋊Dic107C2311yescharacteristic
61Dic107⋊C4—22no—
62C22⋊Dic107—22no—
63(C2×D107)⋊C4—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—21no—
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—
104C2×Dic107—21no—
105C2×Dic107—22no—
106D428—21no—
107D428C2×D412yes—
108C4×D107—44no—
109C4×D107—42no—
110Dic214—22no—
111Dic214—22no—
112C107⋊C8—22no—
113C107⋊C8—21no—
114C107⋊C8C2×D412yes—
115C22×C107—22no—
116C22×C107—22no—
117C22×C107—22no—
118C22×C107—42no—
119C22×C107—42no—
120C22×C107—42no—
121C22×C107C22×D411yescharacteristic
122C2×D107—22no—
123C2×D107—41no—
124C2×D107—41no—
125C4×C107—22no—
126C4×C107—22no—
127C4×C107—21no—
128C4×C107—41no—
129C4×C107#13512yescomplemented
130C4×C107C22×D411yescharacteristic
131C4×C107H(5,𝔽2)11yescharacteristic
132Dic107—42no—
133Dic107—41no—
134C2×C107—42no—
135C2×C107—82no—
136C2×C107D4211yescharacteristic
137C2×C107C2×#13511yescharacteristic
138C2×C107#57711yescharacteristic
139D107—42no—
140#3284—1071nomaximal, Sylow
141C107#328411yescharacteristic, complemented, Sylow
142C2×#135—1072no—
143#550D10711yescharacteristic, complemented
144#547—1071no—
145#492—1071no—
146#465—1072no—
147#462—1071no—
148#459—1071no—
149#452—1072no—
150#449—1072no—
151C4×SD16—1071no—
152C8×D4—1071no—
153C2×#41—1072no—
154C2×#41C2×D10712yes—
155#135—2148no—
156C2×Q16—1072no—
157C2×SD16—1071no—
158C2×SD16—1072no—
159C2×SD16—1072no—
160C2×D8—1072no—
161C22×C8—1072no—
162C4⋊Q8—1071no—
163#127—21no—
164#127C2×D10711yescharacteristic
165#123—2142no—
166C22⋊Q8—1072no—
167#121—22no—
168#121—2141no—
169#121C2×D10712yes—
170C4×Q8C2×D10711yescharacteristic
171C4×D4—21no—
172C4×D4—1071no—
173C4×D4C2×D10711yescharacteristic
174C2×C4⋊C4—2141no—
175#106—1071no—
176C4⋊C8—1071no—
177#103—1071no—
178#103—1072no—
179#102—1072no—
180#102—1071no—
181C22⋊C8—1072no—
182C4×C8—1071no—
183#41—24no—
184#41—42no—
185#41—4282no—
186#41—2144no—
187C2×Q8—1072no—
188C2×Q8C22×D10711yescharacteristic
189C2×D4—22no—
190C2×D4—22no—
191C2×D4—21no—
192C2×D4—21no—
193C2×D4—1071no—
194C2×D4—2142no—
195C2×D4C22⋊D10711yescharacteristic
196C2×D4C22×D10712yes—
197C22×C4—22no—
198C22×C4—2141no—
199C22×C4—2142no—
200C22×C4C22×D10712yes—
201Q16—2144no—
202SD16—1074no—
203SD16—4282no—
204SD16—2144no—
205D8—2144no—
206C2×C8—1071no—
207C2×C8—1071no—
208C2×C8—1072no—
209C2×C8—2144no—
210C4⋊C4—21no—
211C4⋊C4—1071no—
212C4⋊C4—2141no—
213C4⋊C4—2142no—
214C4⋊C4—2141no—
215C4⋊C4C22⋊D10711yescharacteristic
216C4⋊C4C22×D10711yescharacteristic
217C22⋊C4—22no—
218C22⋊C4—2142no—
219C22⋊C4—2141no—
220C22⋊C4C22⋊D10712yes—
221C42—21no—
222C42C22×D10711yescharacteristic
223C23—22no—
224C23—2141no—
225C23C2×C22⋊D10712yes—
226Q8—21no—
227Q8—2142no—
228Q8—2142no—
229Q8D4×D10712yes—
230D4—22no—
231D4—22no—
232D4—22no—
233D4—44no—
234D4—42no—
235D4—81no—
236D4—82no—
237D4—1072no—
238D4—4284no—
239D4—2141no—
240C2×C4—21no—
241C2×C4—22no—
242C2×C4—22no—
243C2×C4—21no—
244C2×C4—44no—
245C2×C4—4284no—
246C2×C4—4282no—
247C2×C4—2141no—
248C2×C4—2142no—
249C2×C4C2×C22⋊D10711yescharacteristic
250C2×C4C2×C22⋊D10712yes—
251C2×C4C2×C22⋊D10711yescharacteristic
252C2×C4C23×D10711yescharacteristic
253C8—1072no—
254C8—2142no—
255C8—2141no—
256C22—22no—
257C22—22no—
258C22—22no—
259C22—42no—
260C22—42no—
261C22—42no—
262C22—4281no—
263C22—4281no—
264C22—2142no—
265C22C22×C22⋊D10711yescharacteristic
266C4—21no—
267C4—22no—
268C4—22no—
269C4—41no—
270C4—4281no—
271C4—4282no—
272C4#1681212912yescomplemented
273C4C2×D4×D10711yescharacteristic
274C4#1681215411yescharacteristic
275C2—42no—
276C2—82no—
277C2—4282no—
278C2D4×C22⋊D10711yescharacteristic
279C2C2×#1681212911yescharacteristic
280C2#2029343811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#550×C107C2
socleC22×C107C22×D4
hypercenter#550D107
hypocenterC107#3284
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. #20293282C2semidirect
3. #20292999C2semidirect
4. #20292854C2semidirect
5. #20292837C2semidirect
6. #20292819C2semidirect
7. #20292789C2semidirect
8. #20292770C2semidirect
9. C4×#13133805C2semidirect
10. D4×C107⋊C8C2semidirect
11. #550×C107C2semidirect
52. #13133805D4semidirect
129. C4×C107#135semidirect
141. C107#3284semidirect
143. #550D107semidirect
272. C4#16812129semidirect

Maximal subgroups

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

subgroupindex
C2×#168121292
#202932822
#202929992
#202928542
#202928372
#202928192
#202927892
#202927702
C4×#131338052
D4×C107⋊C82
#550×C1072
#3284107