Finite Group Databasebeta

← #28337463

Subgroups — #28337463

Order 13696.

444 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 444 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C22×#13133805C211yesmaximal, characteristic, complemented
2C2×#16812085C211yesmaximal, characteristic, complemented
3C2×#16812084C211yesmaximal, characteristic
4C2×#16812031C211yesmaximal, characteristic, complemented
5C2×(C2×D107)⋊C8C211yesmaximal, characteristic, complemented
6#20292389C211yesmaximal, characteristic, complemented
7C2×#103×C107C211yesmaximal, characteristic, complemented
8C22×D428—21no—
9C22×D428C2211yescharacteristic
10C22×C4×D107C2211yescharacteristic
11C2×#13133805—22no—
12C2×#13133805—22no—
13C2×#13133805—22no—
14C22×C107⋊C8C2211yescharacteristic
15#16812085—22no—
16#16812085—22no—
17#16812084—22no—
18#16812084—22no—
19C2×C4⋊Dic107C2211yescharacteristic
20C2×Dic107⋊C4—21no—
21C2×(C2×D107)⋊C4—21no—
22C2×(C2×D107)⋊C4—21no—
23#16812031—22no—
24(C2×D107)⋊C8—22no—
25C22×Q8×C107C2211yescharacteristic
26C22×C8×C107C2211yescharacteristic
27C2×C4⋊C4×C107C2211yescharacteristic
28#103×C107—22no—
29C23×D107—21no—
30C23×D107D411yescharacteristic
31C2×D428—21no—
32C2×D428—21no—
33C2×D428—21no—
34C2×D428—21no—
35C2×D428—24no—
36C2×D428—21no—
37C2×D428—42no—
38C2×D428—42no—
39C2×D428D412yes—
40C22×Dic107—21no—
41C22×Dic107D411yescharacteristic
42C2×C4×D107—21no—
43C2×C4×D107—21no—
44C2×C4×D107—21no—
45C2×C4×D107—21no—
46C2×C4×D107—21no—
47C2×C4×D107—21no—
48#13133805—28no—
49C2×C107⋊C8—21no—
50C2×C107⋊C8—21no—
51C2×C107⋊C8D412yescomplemented
52C4⋊Dic107—21no—
53C4⋊Dic107D412yescomplemented
54Dic107⋊C4—24no—
55(C2×D107)⋊C4—24no—
56(C2×D107)⋊C4—24no—
57C2×Q8×C107—21no—
58C2×Q8×C107—21no—
59C2×Q8×C107—21no—
60C2×Q8×C107—21no—
61C2×Q8×C107—21no—
62C2×Q8×C107D412yescomplemented
63C22×C4×C107—21no—
64C22×C4×C107—21no—
65C22×C4×C107C2311yescharacteristic
66C2×C8×C107—21no—
67C2×C8×C107—21no—
68C2×C8×C107—21no—
69C4⋊C4×C107—21no—
70C4⋊C4×C107D412yes—
71C23×C107C2×D411yescharacteristic
72C22×D107—21no—
73C22×D107—21no—
74C22×D107—21no—
75C22×D107—21no—
76C22×D107—21no—
77C22×D107—21no—
78C22×D107—22no—
79C22×D107—41no—
80C22×D107—41no—
81C22×D107—41no—
82C22×D107—41no—
83C22×D107—41no—
84C22×D107—41no—
85C22×D107SD1612yescomplemented
86Q8×C107—22no—
87Q8×C107—22no—
88Q8×C107—41no—
89Q8×C107—41no—
90C2×C4×C107—22no—
91C2×C4×C107—22no—
92C2×C4×C107—41no—
93C2×C4×C107—41no—
94C2×C4×C107—41no—
95C2×C4×C107—41no—
96C2×C4×C107C2×D411yescharacteristic
97C2×C4×C107C2×D411yescharacteristic
98C2×C4×C107C2×D411yescharacteristic
99C2×C4×C107C2×D411yescharacteristic
100C2×C4×C107C2×D411yescharacteristic
101C2×C4×C107#4111yescharacteristic
102C2×Dic107—21no—
103C2×Dic107—22no—
104C2×Dic107—21no—
105C2×Dic107—21no—
106C2×Dic107—41no—
107C2×Dic107—41no—
108D428—22no—
109D428—22no—
110D428—48no—
111D428—41no—
112D428—41no—
113C4×D107—41no—
114C4×D107—41no—
115C4×D107—41no—
116C4×D107—41no—
117C8×C107—41no—
118C107⋊C8—22no—
119C22×C107#12011yescharacteristic
120C22×C107#12011yescharacteristic
121C22×C107#12111yescharacteristic
122C22×C107C2×SD1611yescharacteristic
123C22×C107#13511yescharacteristic
124C22×C107AGL(1,ℤ/8)11yescharacteristic
125C22×C107#13711yescharacteristic
126C2×D107—22no—
127C2×D107—22no—
128C2×D107—22no—
129C2×D107—41no—
130C2×D107—41no—
131C2×D107—41no—
132C2×D107—41no—
133C2×D107—42no—
134C2×D107—42no—
135C2×D107—42no—
136C2×D107—81no—
137C2×D107—81no—
138C2×D107—81no—
139C2×D107—81no—
140C4×C107—42no—
141C4×C107—42no—
142C4×C107#12011yescharacteristic
143C4×C107#12111yescharacteristic
144C4×C107#12111yescharacteristic
145C4×C107#12711yescharacteristic
146Dic107—41no—
147Dic107—42no—
148C2×C107#39211yescharacteristic
149C2×C107#44811yescharacteristic
150C2×C107#44911yescharacteristic
151C2×C107#45011yescharacteristic
152C2×C107#45211yescharacteristic
153C2×C107#46511yescharacteristic
154C2×C107#46811yescharacteristic
155D107—42no—
156D107—82no—
157#2000—1071nomaximal, Sylow
158C107#200011yescharacteristic, complemented, Sylow
159C22×SD16—1071no—
160C2×C22⋊Q8—1071no—
161C2×#121—1071no—
162C2×#103D10711yescharacteristic, complemented
163C2×#102—1071no—
164C2×C22⋊C8—1071no—
165#340—1071no—
166C22×Q8C2×D10711yescharacteristic
167C22×D4—1071no—
168C22×D4—2141no—
169C23×C4—1071no—
170C2×SD16—2142no—
171C2×SD16—2142no—
172C2×SD16—2142no—
173C22×C8—1071no—
174C22×C8C2×D10711yescharacteristic
175C22⋊Q8—2142no—
176C22⋊Q8—2142no—
177#121—2142no—
178#121—2142no—
179C2×C4⋊C4—1071no—
180C2×C4⋊C4—2141no—
181C2×C4⋊C4C2×D10711yescharacteristic
182C2×C22⋊C4—2141no—
183C2×C22⋊C4—2141no—
184#103—22no—
185#102—2142no—
186C22⋊C8—2142no—
187C24—1071no—
188C24—2141no—
189C2×Q8—21no—
190C2×Q8—21no—
191C2×Q8—21no—
192C2×Q8—21no—
193C2×Q8—21no—
194C2×Q8D42812yescomplemented
195C2×D4—1072no—
196C2×D4—4282no—
197C2×D4—4282no—
198C2×D4—2144no—
199C2×D4—2141no—
200C2×D4—2141no—
201C2×D4—2141no—
202C2×D4—2141no—
203C2×D4—2141no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—1071no—
207C22×C4—2141no—
208C22×C4—2141no—
209C22×C4—2141no—
210C22×C4—2141no—
211C22×C4—2141no—
212C22×C4—2141no—
213C22×C4—2141no—
214C22×C4C22×D10711yescharacteristic
215SD16—2148no—
216C2×C8—21no—
217C2×C8—21no—
218C2×C8—21no—
219C2×C8—1072no—
220C2×C8—2141no—
221C2×C8—2141no—
222C4⋊C4—21no—
223C4⋊C4—1072no—
224C4⋊C4—2144no—
225C4⋊C4—2141no—
226C4⋊C4C22⋊D10712yes—
227C22⋊C4—2144no—
228C22⋊C4—2144no—
229C23—1072no—
230C23—4281no—
231C23—4281no—
232C23—4281no—
233C23—4281no—
234C23—4281no—
235C23—4281no—
236C23—2142no—
237C23—2141no—
238C23—2141no—
239C23—2141no—
240C23—2141no—
241C23—2141no—
242C23—2141no—
243C23D4×D10711yescharacteristic
244Q8—22no—
245Q8—22no—
246Q8—41no—
247Q8—41no—
248D4—4288no—
249D4—4281no—
250D4—4281no—
251D4—2142no—
252D4—2142no—
253C2×C4—22no—
254C2×C4—22no—
255C2×C4—41no—
256C2×C4—41no—
257C2×C4—41no—
258C2×C4—41no—
259C2×C4—4281no—
260C2×C4—4281no—
261C2×C4—4281no—
262C2×C4—4281no—
263C2×C4—4281no—
264C2×C4—4281no—
265C2×C4—2141no—
266C2×C4—2142no—
267C2×C4—2141no—
268C2×C4—2141no—
269C2×C4D4×D10711yescharacteristic
270C2×C4D4×D10711yescharacteristic
271C2×C4D4×D10711yescharacteristic
272C2×C4C2×C22⋊D10711yescharacteristic
273C2×C4C2×D42811yescharacteristic
274C2×C4#1313381811yescharacteristic
275C8—41no—
276C8—2142no—
277C22—8561no—
278C22—8561no—
279C22—8561no—
280C22—8561no—
281C22—4282no—
282C22—4282no—
283C22—4282no—
284C22—4281no—
285C22—4281no—
286C22—4281no—
287C22—4281no—
288C22—2142no—
289C22—2142no—
290C22—2142no—
291C22#1681207711yescharacteristic
292C22#1681207911yescharacteristic
293C22#1681208211yescharacteristic
294C22SD16×D10711yescharacteristic
295C22#1681213211yescharacteristic
296C22#1681213611yescharacteristic
297C22#1681214111yescharacteristic
298C4—42no—
299C4—42no—
300C4—4282no—
301C4—4281no—
302C4#1681207911yescharacteristic
303C4#1681208211yescharacteristic
304C4#1681208311yescharacteristic
305C4#1681210511yescharacteristic
306C2—8562no—
307C2—4282no—
308C2#2029253611yescharacteristic
309C2#2029276711yescharacteristic
310C2#2029277511yescharacteristic
311C2#2029277611yescharacteristic
312C2#2029278811yescharacteristic
313C2#2029285911yescharacteristic
314C2#2029287611yescharacteristic
subgroupG/H
centerC23D4×D107
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
FittingC2×#103×C107C2
socleC23×C107C2×D4
prime radicalC2×#16812084C2
hypercenterC2×#103D107
hypocenterC107#2000
LCSG ▷ C22×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
1. C22×#13133805C2semidirect
2. C2×#16812085C2semidirect
4. C2×#16812031C2semidirect
5. C2×(C2×D107)⋊C8C2semidirect
6. #20292389C2semidirect
7. C2×#103×C107C2semidirect
51. C2×C107⋊C8D4semidirect
53. C4⋊Dic107D4semidirect
62. C2×Q8×C107D4semidirect
85. C22×D107SD16semidirect
158. C107#2000semidirect
162. C2×#103D107semidirect
194. C2×Q8D428semidirect

Maximal subgroups

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

subgroupindex
C22×#131338052
C2×#168120852
C2×#168120842
C2×#168120312
C2×(C2×D107)⋊C82
#202923892
C2×#103×C1072
#2000107