Finite Group Databasebeta

← #28335404

Subgroups — #28335404

Order 13696.

320 conjugacy classes of subgroups; only the 49 normal subgroups are shown.

normal subgroups only

This group has 320 conjugacy classes of subgroups — too many to draw. Only its 49 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#20293214C211yesmaximal, characteristic, complemented
2#20292778C211yesmaximal, characteristic, complemented
3#20292535C211yesmaximal, characteristic, complemented
4#20292365C211yesmaximal, characteristic, complemented
5#20292361C211yesmaximal, characteristic, complemented
6#20292350C211yesmaximal, characteristic, complemented
7#447×C107C211yesmaximal, characteristic, complemented
8C22×C22⋊D107—21no—
9C2×#13133806—21no—
10#16812104—21no—
11#16812095—21no—
12#16812093—21no—
13#16812088C2211yescharacteristic
14#16812081—21no—
15#16812077—21no—
16D4×Dic107—21no—
17C2×C22⋊Dic107—21no—
18C2×(C2×D107)⋊C4C2211yescharacteristic
19#16812030—21no—
20C22⋊(C107⋊C8)C2211yescharacteristic
21C107⋊#95C2211yescharacteristic
22C22×D4×C107C2211yescharacteristic
23C2×D8×C107—21no—
24#121×C107C2211yescharacteristic
25#102×C107—21no—
26C22⋊C8×C107C2211yescharacteristic
27C23×D107D411yescharacteristic
28C2×Dic214D411yescharacteristic, complemented
29C2×C22⋊D107—21no—
30C2×C22⋊D107—21no—
31C2×C22⋊D107—42no—
32C2×C22⋊D107—42no—
33C2×C22⋊D107—42no—
34C22×Dic107—21no—
35C22×Dic107—21no—
36C22×Dic107D411yescharacteristic
37#13133806—42no—
38C2×C107⋊C8—21no—
39C4⋊Dic107—21no—
40C4⋊Dic107D411yescharacteristic, complemented
41Dic107⋊C4—21no—
42C22⋊Dic107—21no—
43C22⋊Dic107—21no—
44C22⋊Dic107—42no—
45(C2×D107)⋊C4—21no—
46(C2×D107)⋊C4—21no—
47(C2×D107)⋊C4—21no—
48(C2×D107)⋊C4—21no—
49C4×Dic107—21no—
50C24×C107—21no—
51C2×D4×C107—21no—
52C2×D4×C107—21no—
53C2×D4×C107—21no—
54C2×D4×C107—42no—
55C2×D4×C107D411yescharacteristic, complemented
56C22×C4×C107C2311yescharacteristic
57D8×C107—42no—
58C2×C8×C107—21no—
59C4⋊C4×C107D411yescharacteristic, complemented
60C22⋊C4×C107—21no—
61C23×C107—22no—
62C23×C107—21no—
63C23×C107—41no—
64C23×C107—41no—
65C23×C107—41no—
66C23×C107—41no—
67C23×C107—41no—
68C23×C107—41no—
69C23×C107C2×D411yescharacteristic
70C22×D107—21no—
71C22×D107—21no—
72C22×D107—21no—
73C22×D107—41no—
74C22×D107—41no—
75D4×C107—22no—
76D4×C107—22no—
77D4×C107—41no—
78D4×C107—41no—
79D4×C107—41no—
80D4×C107—41no—
81D4×C107—41no—
82C2×C4×C107—21no—
83C2×C4×C107—41no—
84C2×C4×C107C2×D411yescharacteristic
85C2×C4×C107C2×D411yescharacteristic
86C22⋊D107—48no—
87C22⋊D107—81no—
88C22⋊D107—81no—
89C2×Dic107—22no—
90C2×Dic107—21no—
91C2×Dic107—21no—
92C2×Dic107—21no—
93C2×Dic107—41no—
94C2×Dic107—41no—
95C2×Dic107—41no—
96C2×Dic107—81no—
97Dic214—41no—
98C8×C107—41no—
99C107⋊C8—41no—
100C22×C107—21no—
101C22×C107—21no—
102C22×C107—42no—
103C22×C107—42no—
104C22×C107—41no—
105C22×C107—41no—
106C22×C107—41no—
107C22×C107—42no—
108C22×C107—81no—
109C22×C107—81no—
110C22×C107—81no—
111C22×C107—81no—
112C22×C107#12011yescharacteristic
113C22×C107#13511yescharacteristic
114C22×C107AGL(1,ℤ/8)11yescharacteristic
115C2×D107—42no—
116C2×D107—41no—
117C2×D107—41no—
118C2×D107—41no—
119C2×D107—81no—
120C4×C107—21no—
121C4×C107—21no—
122C4×C107—41no—
123Dic107—42no—
124Dic107—41no—
125Dic107—41no—
126Dic107—41no—
127C2×C107—21no—
128C2×C107—21no—
129C2×C107—81no—
130C2×C107—82no—
131C2×C107#44911yescharacteristic
132C2×C107#45311yescharacteristic
133C2×C107B(4,𝔽2)11yescharacteristic
134D107—81no—
135#1600—1071nomaximal, Sylow
136C107#160011yescharacteristic, complemented, Sylow
137#538—1071no—
138#450—1071no—
139#447D10711yescharacteristic, complemented
140#392—1071no—
141#328—1071no—
142#327—1071no—
143#323—1071no—
144C22×D4—2141no—
145C22×D4C2×D10711yescharacteristic
146C2×SD16—2141no—
147C2×D8—21no—
148#126—2141no—
149#124—2141no—
150#123—2141no—
151C22⋊Q8—1071no—
152#121—2141no—
153#121C2×D10711yescharacteristic
154#120—2141no—
155C4×D4—2141no—
156C2×C22⋊C4—1071no—
157C2×C22⋊C4—2141no—
158#102—21no—
159#102—2141no—
160C22⋊C8—1071no—
161C22⋊C8C2×D10711yescharacteristic
162#95—1071no—
163C24—21no—
164C24—1071no—
165C2×Q8—1071no—
166C2×D4—21no—
167C2×D4—21no—
168C2×D4—21no—
169C2×D4—42no—
170C2×D4—4282no—
171C2×D4—4282no—
172C2×D4—4282no—
173C2×D4—2141no—
174C2×D4—2141no—
175C2×D4C22⋊D10711yescharacteristic, complemented
176C22×C4—1071no—
177C22×C4—2141no—
178C22×C4—2141no—
179C22×C4C22×D10711yescharacteristic
180SD16—4282no—
181D8—42no—
182C2×C8—21no—
183C2×C8—2141no—
184C4⋊C4—1071no—
185C4⋊C4—2141no—
186C4⋊C4—2141no—
187C4⋊C4C22⋊D10711yescharacteristic, complemented
188C22⋊C4—21no—
189C22⋊C4—4282no—
190C22⋊C4—2141no—
191C22⋊C4—2141no—
192C22⋊C4—2141no—
193C22⋊C4—2141no—
194C22⋊C4—2141no—
195C22⋊C4—2141no—
196C42—2141no—
197C23—22no—
198C23—21no—
199C23—41no—
200C23—41no—
201C23—41no—
202C23—41no—
203C23—41no—
204C23—41no—
205C23—4281no—
206C23—4281no—
207C23—2141no—
208C23—2141no—
209C23—2141no—
210C23D4×D10711yescharacteristic
211Q8—4281no—
212D4—22no—
213D4—22no—
214D4—41no—
215D4—41no—
216D4—41no—
217D4—41no—
218D4—41no—
219D4—8561no—
220D4—8561no—
221D4—4288no—
222C2×C4—21no—
223C2×C4—41no—
224C2×C4—8561no—
225C2×C4—4281no—
226C2×C4—4281no—
227C2×C4—4281no—
228C2×C4—2142no—
229C2×C4—2141no—
230C2×C4—2141no—
231C2×C4—2141no—
232C2×C4D4×D10711yescharacteristic
233C2×C4C2×C22⋊D10711yescharacteristic
234C8—41no—
235C8—4281no—
236C22—21no—
237C22—21no—
238C22—42no—
239C22—42no—
240C22—41no—
241C22—41no—
242C22—42no—
243C22—41no—
244C22—81no—
245C22—81no—
246C22—81no—
247C22—81no—
248C22—8561no—
249C22—4282no—
250C22—4281no—
251C22—4281no—
252C22—4281no—
253C22#1681207711yescharacteristic
254C22#1681212811yescharacteristic
255C22#1681213311yescharacteristic
256C4—21no—
257C4—21no—
258C4—41no—
259C4—4282no—
260C4—4281no—
261C4—4281no—
262C4—4281no—
263C2—21no—
264C2—21no—
265C2—82no—
266C2—81no—
267C2—8561no—
268C2#2029276911yescharacteristic
269C2#2029279311yescharacteristic
270C2#2029281311yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
Fitting#447×C107C2
socleC22×C107#120
hypercenter#447D107
hypocenterC107#1600
LCSG ▷ C22×C4×C107 ▷ C22×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #447

Complements

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

NH ≅ G/Ntype
1. #20293214C2semidirect
2. #20292778C2semidirect
3. #20292535C2semidirect
4. #20292365C2semidirect
5. #20292361C2semidirect
6. #20292350C2semidirect
7. #447×C107C2semidirect
28. C2×Dic214D4semidirect
40. C4⋊Dic107D4semidirect
55. C2×D4×C107D4semidirect
59. C4⋊C4×C107D4semidirect
136. C107#1600semidirect
139. #447D107semidirect
175. C2×D4C22⋊D107semidirect
187. C4⋊C4C22⋊D107semidirect

Maximal subgroups

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

subgroupindex
#202932142
#202927782
#202925352
#202923652
#202923612
#202923502
#447×C1072
#1600107