Finite Group Databasebeta

← #28336765

Subgroups — #28336765

Order 13696.

404 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 404 conjugacy classes of subgroups — too many to draw. Only its 107 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×#16812085C211yesmaximal, characteristic
2C2×C4⋊C4×D107C211yesmaximal, characteristic
3C2×#16812033C211yesmaximal, characteristic, complemented
4C2×(C2×D107)⋊C8C211yesmaximal, characteristic
5#20292390C211yesmaximal, characteristic, complemented
6#20292389C211yesmaximal, characteristic, complemented
7C2×#103×C107C211yesmaximal, characteristic, complemented
8C22×C4×D107—21no—
9C22×C4×D107C2211yescharacteristic
10C22×C107⋊C8C2211yescharacteristic
11#16812085—22no—
12#16812085C414yescomplemented
13C2×C4⋊Dic107C2211yescharacteristic
14C2×C4⋊Dic107C2211yescharacteristic
15C2×Dic107⋊C4—21no—
16C2×Dic107⋊C4—21no—
17C4⋊C4×D107—22no—
18C4⋊C4×D107—22no—
19C2×(C2×D107)⋊C4—21no—
20#16812033—22no—
21(C2×D107)⋊C8—22no—
22C22×Q8×C107C2211yescharacteristic
23C22×C8×C107C2211yescharacteristic
24C2×C4⋊C4×C107C2211yescharacteristic
25#103×C107—22no—
26C23×D107D411yescharacteristic
27C22×Dic107—21no—
28C22×Dic107—21no—
29C22×Dic107D411yescharacteristic
30C2×C4×D107—21no—
31C2×C4×D107—21no—
32C2×C4×D107—21no—
33C2×C4×D107—21no—
34C2×C4×D107—24no—
35C2×C4×D107—42no—
36C2×C4×D107—42no—
37C2×C4×D107C2×C412yes—
38C2×C4×D107D412yes—
39C2×C107⋊C8—21no—
40C2×C107⋊C8—21no—
41C2×C107⋊C8—21no—
42C4⋊Dic107—21no—
43C4⋊Dic107—21no—
44C4⋊Dic107C2×C412yescomplemented
45C4⋊Dic107D412yes—
46Dic107⋊C4—24no—
47Dic107⋊C4—24no—
48(C2×D107)⋊C4—24no—
49C2×Q8×C107—21no—
50C2×Q8×C107—21no—
51C2×Q8×C107—21no—
52C2×Q8×C107—21no—
53C2×Q8×C107—21no—
54C2×Q8×C107C2×C412yescomplemented
55C22×C4×C107—21no—
56C22×C4×C107—21no—
57C22×C4×C107C2311yescharacteristic
58C2×C8×C107—21no—
59C2×C8×C107—21no—
60C2×C8×C107—21no—
61C4⋊C4×C107—21no—
62C4⋊C4×C107D412yes—
63C23×C107C2×D411yescharacteristic
64C22×D107—21no—
65C22×D107—21no—
66C22×D107—21no—
67C22×D107—21no—
68C22×D107C22⋊C412yes—
69C22×D107SD1612yes—
70C22×D107Q1612yes—
71Q8×C107—22no—
72Q8×C107—22no—
73Q8×C107—41no—
74Q8×C107—41no—
75C2×C4×C107—22no—
76C2×C4×C107—22no—
77C2×C4×C107—41no—
78C2×C4×C107—41no—
79C2×C4×C107—41no—
80C2×C4×C107—41no—
81C2×C4×C107C22×C411yescharacteristic
82C2×C4×C107C2×D411yescharacteristic
83C2×C4×C107C2×D411yescharacteristic
84C2×C4×C107C2×D411yescharacteristic
85C2×C4×C107#4111yescharacteristic
86C2×C4×C107#4111yescharacteristic
87C2×Dic107—21no—
88C2×Dic107—21no—
89C2×Dic107—22no—
90C2×Dic107—22no—
91C2×Dic107—41no—
92C2×Dic107—41no—
93C2×Dic107—41no—
94C2×Dic107—41no—
95C2×Dic107C22⋊C412yes—
96C4×D107—22no—
97C4×D107—22no—
98C4×D107—22no—
99C4×D107—22no—
100C4×D107—48no—
101C8×C107—41no—
102C107⋊C8—41no—
103C22×C107C2×C22⋊C411yescharacteristic
104C22×C107#12011yescharacteristic
105C22×C107#12311yescharacteristic
106C22×C107C2×SD1611yescharacteristic
107C22×C107C2×Q1611yescharacteristic
108C22×C107AGL(1,ℤ/8)11yescharacteristic
109C22×C107#13711yescharacteristic
110C2×D107—22no—
111C2×D107—22no—
112C2×D107—22no—
113C2×D107—22no—
114C2×D107—22no—
115C2×D107—22no—
116C2×D107#10314yescomplemented
117C4×C107—42no—
118C4×C107—42no—
119C4×C107C4×D411yescharacteristic
120C4×C107C4×D411yescharacteristic
121C4×C107#12111yescharacteristic
122C4×C107#12111yescharacteristic
123Dic107—22no—
124Dic107—42no—
125Dic107—42no—
126C2×C107#38611yescharacteristic
127C2×C107C2×#10311yescharacteristic
128C2×C107#41711yescharacteristic
129C2×C107#45011yescharacteristic
130C2×C107#45111yescharacteristic
131C2×C107#48311yescharacteristic
132C2×C107#48411yescharacteristic
133D107—24no—
134#1872—1071nomaximal, Sylow
135C107#187211yescharacteristic, complemented, Sylow
136C2×C22⋊Q8—1071no—
137C22×C4⋊C4—1071no—
138C2×#103—1071no—
139C2×#103D10711yescharacteristic, complemented
140C2×C22⋊C8—1071no—
141#340—1071no—
142#340—1071no—
143C22×Q8C2×D10711yescharacteristic
144C23×C4—1071no—
145C23×C4—2141no—
146C22×C8—1071no—
147C22×C8C2×D10711yescharacteristic
148C22⋊Q8—1074no—
149C22⋊Q8—2142no—
150C2×C4⋊C4—1071no—
151C2×C4⋊C4—1071no—
152C2×C4⋊C4—2142no—
153C2×C4⋊C4—2141no—
154C2×C4⋊C4—2141no—
155C2×C4⋊C4—2142no—
156C2×C4⋊C4C2×D10711yescharacteristic
157C2×C22⋊C4—2141no—
158#103—22no—
159#103—2142no—
160C22⋊C8—2142no—
161C24—1071no—
162C2×Q8—21no—
163C2×Q8—21no—
164C2×Q8—21no—
165C2×Q8—21no—
166C2×Q8—21no—
167C2×Q8C4×D10712yescomplemented
168C22×C4—21no—
169C22×C4—21no—
170C22×C4—1071no—
171C22×C4—1072no—
172C22×C4—1072no—
173C22×C4—4282no—
174C22×C4—4282no—
175C22×C4—2144no—
176C22×C4—2141no—
177C22×C4—2141no—
178C22×C4—2141no—
179C22×C4—2141no—
180C22×C4—2141no—
181C22×C4—2141no—
182C22×C4C22×D10711yescharacteristic
183C2×C8—21no—
184C2×C8—21no—
185C2×C8—21no—
186C2×C8—2141no—
187C2×C8—2141no—
188C2×C8—2141no—
189C4⋊C4—21no—
190C4⋊C4—1072no—
191C4⋊C4—1072no—
192C4⋊C4—2144no—
193C4⋊C4—2144no—
194C4⋊C4—2141no—
195C4⋊C4—2141no—
196C4⋊C4C22⋊D10712yes—
197C22⋊C4—2144no—
198C23—1072no—
199C23—1072no—
200C23—1072no—
201C23—2141no—
202C23—2141no—
203C23—2141no—
204C23—2141no—
205C23D4×D10711yescharacteristic
206Q8—22no—
207Q8—22no—
208Q8—41no—
209Q8—41no—
210C2×C4—22no—
211C2×C4—22no—
212C2×C4—41no—
213C2×C4—41no—
214C2×C4—41no—
215C2×C4—41no—
216C2×C4—1072no—
217C2×C4—4288no—
218C2×C4—4281no—
219C2×C4—4281no—
220C2×C4—4281no—
221C2×C4—4281no—
222C2×C4—2142no—
223C2×C4—2142no—
224C2×C4—2141no—
225C2×C4—2141no—
226C2×C4—2142no—
227C2×C4—2142no—
228C2×C4—2142no—
229C2×C4—2142no—
230C2×C4C2×C4×D10711yescharacteristic
231C2×C4D4×D10711yescharacteristic
232C2×C4D4×D10711yescharacteristic
233C2×C4C2×C22⋊D10711yescharacteristic
234C2×C4#1313381711yescharacteristic
235C2×C4#1313381811yescharacteristic
236C8—41no—
237C8—4281no—
238C22—1074no—
239C22—2142no—
240C22—2142no—
241C22—2142no—
242C22—2142no—
243C22—2142no—
244C22—2142no—
245C22C22⋊C4×D10711yescharacteristic
246C22#1681207711yescharacteristic
247C22#1681209111yescharacteristic
248C22SD16×D10711yescharacteristic
249C22Q16×D10711yescharacteristic
250C22#1681213611yescharacteristic
251C22#1681214111yescharacteristic
252C4—42no—
253C4—42no—
254C4—4282no—
255C4—4282no—
256C4—2142no—
257C4(C22⋊D107)⋊C411yescharacteristic
258C4C4×C22⋊D10711yescharacteristic
259C4#1681208011yescharacteristic
260C4#1681208211yescharacteristic
261C2—2144no—
262C2#2029250311yescharacteristic
263C2#103×D10711yescharacteristic
264C2#2029262711yescharacteristic
265C2#2029277611yescharacteristic
266C2#2029278111yescharacteristic
267C2#2029295611yescharacteristic
268C2#2029296111yescharacteristic
subgroupG/H
centerC23D4×D107
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
FittingC2×#103×C107C2
socleC23×C107C2×D4
prime radicalC23×D107D4
cosocleC22×C4×C107C23
hypercenterC2×#103D107
hypocenterC107#1872
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C23 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
3. C2×#16812033C2semidirect
5. #20292390C2semidirect
6. #20292389C2semidirect
7. C2×#103×C107C2semidirect
12. #16812085C4semidirect
44. C4⋊Dic107C2×C4semidirect
54. C2×Q8×C107C2×C4semidirect
116. C2×D107#103semidirect
135. C107#1872semidirect
139. C2×#103D107semidirect
167. C2×Q8C4×D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168120852
C2×C4⋊C4×D1072
C2×#168120332
C2×(C2×D107)⋊C82
#202923902
#202923892
C2×#103×C1072
#1872107