Finite Group Databasebeta

← #28348190

Subgroups — #28348190

Order 13696.

494 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 494 conjugacy classes of subgroups — too many to draw. Only its 125 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×#16812130C212yesmaximal, complemented
2#20293297C211yesmaximal, characteristic, complemented
3#20293280C211yesmaximal, characteristic, complemented
4#20292853C212yesmaximal, complemented
5#20292834C211yesmaximal, characteristic, complemented
6#20292819C211yesmaximal, characteristic, complemented
7#20292786C212yesmaximal, complemented
8#20292771C212yesmaximal, complemented
9C4×#13133807C211yesmaximal, characteristic, complemented
10C8×D428C211yesmaximal, characteristic, complemented
11#492×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2212yes—
13C2×#13133817C2212yes—
14#16812130—28no—
15C2×#13133808C2212yes—
16C2×#13133807C2211yescharacteristic
17C2×#13133803C2212yes—
18C2×C8×D107C2212yes—
19#16812105—21no—
20#16812094—22no—
21#16812085C2212yes—
22#16812084—21no—
23#16812081—22no—
24#16812080C2212yes—
25C4×Dic214C2211yescharacteristic
26C4×D428C2211yescharacteristic
27C4×D428C2211yescharacteristic
28D4×Dic107—21no—
29C4⋊C4×D107—21no—
30#16812041C2211yescharacteristic
31C4⋊(C107⋊C8)C2211yescharacteristic
32#16812034C2211yescharacteristic
33#16812033C2212yes—
34#16812031C2211yescharacteristic
35#16812030C2212yes—
36(C2×D107)⋊C8C2212yes—
37C2×SD16×C107C2212yes—
38C2×SD16×C107C2212yes—
39C4⋊Q8×C107C2211yescharacteristic
40#127×C107C2211yescharacteristic
41C4×C8×C107C2211yescharacteristic
42#13133819—24no—
43#13133819—42no—
44#13133817—24no—
45#13133817—42no—
46C2×Dic214C2311yescharacteristic
47C2×D428—22no—
48C2×D428D411yescharacteristic
49C2×D428C2311yescharacteristic
50C2×C22⋊D107—22no—
51C2×C22⋊D107—22no—
52C22×Dic107—22no—
53C2×C4×D107—21no—
54C2×C4×D107—22no—
55C2×C4×D107C2312yes—
56#13133808—24no—
57#13133807D414yescomplemented
58#13133803—24no—
59C2×C107⋊C8C2312yes—
60C8×D107—24no—
61C4⋊Dic107D411yescharacteristic
62C4⋊Dic107C2311yescharacteristic
63C4⋊Dic107C2311yescharacteristic
64Dic107⋊C4—21no—
65Dic107⋊C4—22no—
66C22⋊Dic107—22no—
67(C2×D107)⋊C4—22no—
68(C2×D107)⋊C4—21no—
69(C2×D107)⋊C4D412yes—
70C4×Dic107—21no—
71C2×Q8×C107C2312yes—
72C2×D4×C107—21no—
73C2×D4×C107—21no—
74C2×D4×C107C2312yes—
75SD16×C107—24no—
76SD16×C107—42no—
77C2×C8×C107C2311yescharacteristic
78C2×C8×C107C2311yescharacteristic
79C4⋊C4×C107—21no—
80C4⋊C4×C107—21no—
81C42×C107C2311yescharacteristic
82C23×C107—22no—
83C22×D107—21no—
84C22×D107C2×D412yes—
85Q8×C107—22no—
86Q8×C107—22no—
87D4×C107—22no—
88D4×C107—22no—
89D4×C107—42no—
90D4×C107—81no—
91C2×C4×C107—22no—
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2411yescharacteristic
95C22⋊D107—44no—
96C22⋊D107—82no—
97C2×Dic107—21no—
98C2×Dic107—21no—
99C2×Dic107—21no—
100C2×Dic107—44no—
101C2×Dic107C2×D412yes—
102D428—22no—
103D428—21no—
104D428—44no—
105D428C2×D412yes—
106C4×D107—22no—
107C4×D107—22no—
108C4×D107—44no—
109C4×D107—42no—
110Dic214—21no—
111Dic214C2×D412yes—
112C8×C107—21no—
113C8×C107C2×D412yes—
114C107⋊C8—22no—
115C22×C107—42no—
116C22×C107—42no—
117C22×C107—42no—
118C22×C107C22×D411yescharacteristic
119C2×D107—22no—
120C2×D107—22no—
121C2×D107—22no—
122C2×D107—22no—
123C2×D107—41no—
124C2×D107—41no—
125C4×C107—21no—
126C4×C107—42no—
127C4×C107#13512yescomplemented
128C4×C107C22×D411yescharacteristic
129C4×C107H(5,𝔽2)11yescharacteristic
130Dic107—22no—
131Dic107—22no—
132Dic107—41no—
133Dic107—41no—
134C2×C107—82no—
135C2×C107D4211yescharacteristic
136C2×C107C2×#13511yescharacteristic
137C2×C107#57711yescharacteristic
138D107—42no—
139D107—42no—
140#3284—1071nomaximal, Sylow
141C107#328411yescharacteristic, complemented, Sylow
142C2×#135—1072no—
143#550—1071no—
144#547—1071no—
145#492D10711yescharacteristic, complemented
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×#41—1072no—
155#135—2148no—
156C2×Q16—1072no—
157C2×SD16—1071no—
158C2×SD16C2×D10712yes—
159C2×SD16C2×D10712yes—
160C2×D8—1072no—
161C22×C8—1072no—
162C4⋊Q8C2×D10711yescharacteristic
163#127—2141no—
164#127C2×D10711yescharacteristic
165#123—2142no—
166C22⋊Q8—1072no—
167#121—1072no—
168#121—2142no—
169#121—2141no—
170C4×Q8—1071no—
171C4×D4—1071no—
172C4×D4—1071no—
173C4×D4—2141no—
174C2×C4⋊C4—2141no—
175#106—1071no—
176C4⋊C8—1071no—
177#103—1072no—
178#103—1071no—
179#102—1072no—
180#102—1071no—
181C22⋊C8—1072no—
182C4×C8C2×D10711yescharacteristic
183#41—4282no—
184#41—4282no—
185#41—2144no—
186#41—2144no—
187C2×Q8—1071no—
188C2×Q8C22×D10712yes—
189C2×D4—21no—
190C2×D4—21no—
191C2×D4—1071no—
192C2×D4—1071no—
193C2×D4—2142no—
194C2×D4—2142no—
195C2×D4—2142no—
196C2×D4C22×D10712yes—
197C22×C4—1072no—
198C22×C4—2142no—
199C22×C4—2142no—
200C22×C4—2141no—
201Q16—2144no—
202SD16—24no—
203SD16—42no—
204SD16—1074no—
205D8—2144no—
206C2×C8—1072no—
207C2×C8—2144no—
208C2×C8C22×D10711yescharacteristic
209C2×C8C22×D10711yescharacteristic
210C4⋊C4—21no—
211C4⋊C4—21no—
212C4⋊C4—1071no—
213C4⋊C4—1071no—
214C4⋊C4—1071no—
215C4⋊C4—2142no—
216C4⋊C4—2141no—
217C22⋊C4—1072no—
218C22⋊C4—2142no—
219C22⋊C4—2142no—
220C22⋊C4—2141no—
221C42—2141no—
222C42C22×D10711yescharacteristic
223C23—22no—
224C23—1072no—
225C23—2141no—
226Q8—22no—
227Q8—22no—
228Q8—1072no—
229Q8—2141no—
230D4—22no—
231D4—22no—
232D4—42no—
233D4—81no—
234D4—8562no—
235D4—1072no—
236D4—4284no—
237D4—4284no—
238D4—2142no—
239D4—2141no—
240C2×C4—22no—
241C2×C4—1072no—
242C2×C4—4284no—
243C2×C4—4282no—
244C2×C4—4284no—
245C2×C4—2141no—
246C2×C4—2141no—
247C2×C4—2141no—
248C2×C4—2142no—
249C2×C4—2142no—
250C2×C4D4×D10711yescharacteristic
251C2×C4D4×D10711yescharacteristic
252C2×C4C23×D10711yescharacteristic
253C8—21no—
254C8—2142no—
255C8D4×D10712yes—
256C22—42no—
257C22—42no—
258C22—42no—
259C22—4281no—
260C22—4281no—
261C22—2142no—
262C22—2142no—
263C22—2142no—
264C22—2142no—
265C22C2×D4×D10711yescharacteristic
266C4—21no—
267C4—42no—
268C4—4281no—
269C4—4281no—
270C4—2142no—
271C4—2142no—
272C4#1681213012yescomplemented
273C4C2×D4×D10711yescharacteristic
274C4#1681215411yescharacteristic
275C2—82no—
276C2—4282no—
277C2—4282no—
278C2#2029326211yescharacteristic
279C2C2×#1681213011yescharacteristic
280C2#2029343311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#492×C107C2
socleC22×C107C22×D4
hypercenter#492D107
hypocenterC107#3284
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #492

Complements

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

NH ≅ G/Ntype
1. C2×#16812130C2semidirect
2. #20293297C2semidirect
3. #20293280C2semidirect
4. #20292853C2semidirect
5. #20292834C2semidirect
6. #20292819C2semidirect
7. #20292786C2semidirect
8. #20292771C2semidirect
9. C4×#13133807C2semidirect
10. C8×D428C2semidirect
11. #492×C107C2semidirect
57. #13133807D4semidirect
127. C4×C107#135semidirect
141. C107#3284semidirect
145. #492D107semidirect
272. C4#16812130semidirect

Maximal subgroups

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

subgroupindex
C2×#168121302
#202932972
#202932802
#202928532
#202928342
#202928192
#202927862
#202927712
C4×#131338072
C8×D4282
#492×C1072
#3284107