Finite Group Databasebeta

← #28337470

Subgroups — #28337470

Order 13696.

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

normal subgroups only

This group has 392 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×#13133808C211yesmaximal, characteristic, complemented
2C2×#16812089C211yesmaximal, characteristic
3C2×#16812085C211yesmaximal, characteristic
4C2×#16812034C211yesmaximal, characteristic, complemented
5C2×(C2×D107)⋊C8C211yesmaximal, characteristic
6#20292389C211yesmaximal, characteristic, complemented
7C2×#103×C107C211yesmaximal, characteristic, complemented
8C22×Dic214C2211yescharacteristic
9C22×C4×D107C2211yescharacteristic
10C2×#13133808—22no—
11C2×#13133808—22no—
12C2×#13133808—22no—
13C22×C107⋊C8C2211yescharacteristic
14#16812089—22no—
15#16812089—22no—
16#16812085—22no—
17#16812085—22no—
18C2×C4⋊Dic107—21no—
19C2×C4⋊Dic107C2211yescharacteristic
20C2×Dic107⋊C4—21no—
21C2×(C2×D107)⋊C4—21no—
22C2×(C2×D107)⋊C4—21no—
23#16812034—22no—
24(C2×D107)⋊C8—22no—
25C22×Q8×C107C2211yescharacteristic
26C22×C8×C107C2211yescharacteristic
27C2×C4⋊C4×C107C2211yescharacteristic
28#103×C107—22no—
29C23×D107D411yescharacteristic
30C2×Dic214—21no—
31C2×Dic214—21no—
32C2×Dic214—21no—
33C2×Dic214—21no—
34C2×Dic214—21no—
35C2×Dic214D412yes—
36C22×Dic107—21no—
37C22×Dic107—21no—
38C22×Dic107D411yescharacteristic
39C2×C4×D107—21no—
40C2×C4×D107—21no—
41C2×C4×D107—21no—
42C2×C4×D107—21no—
43C2×C4×D107—21no—
44C2×C4×D107—21no—
45#13133808—28no—
46C2×C107⋊C8—21no—
47C2×C107⋊C8—21no—
48C2×C107⋊C8D412yes—
49C4⋊Dic107—24no—
50C4⋊Dic107—21no—
51C4⋊Dic107D412yes—
52Dic107⋊C4—24no—
53(C2×D107)⋊C4—24no—
54(C2×D107)⋊C4—24no—
55C2×Q8×C107—21no—
56C2×Q8×C107—21no—
57C2×Q8×C107—21no—
58C2×Q8×C107—21no—
59C2×Q8×C107—21no—
60C2×Q8×C107D412yes—
61C22×C4×C107—21no—
62C22×C4×C107—21no—
63C22×C4×C107C2311yescharacteristic
64C2×C8×C107—21no—
65C2×C8×C107—21no—
66C2×C8×C107—21no—
67C4⋊C4×C107—21no—
68C4⋊C4×C107D412yes—
69C23×C107C2×D411yescharacteristic
70C22×D107—21no—
71C22×D107—21no—
72C22×D107—21no—
73C22×D107—21no—
74C22×D107—21no—
75C22×D107—21no—
76C22×D107Q1612yescomplemented
77Q8×C107—22no—
78Q8×C107—22no—
79Q8×C107—41no—
80Q8×C107—41no—
81C2×C4×C107—22no—
82C2×C4×C107—22no—
83C2×C4×C107—41no—
84C2×C4×C107—41no—
85C2×C4×C107—41no—
86C2×C4×C107—41no—
87C2×C4×C107C2×D411yescharacteristic
88C2×C4×C107C2×D411yescharacteristic
89C2×C4×C107C2×D411yescharacteristic
90C2×C4×C107C2×D411yescharacteristic
91C2×C4×C107C2×D411yescharacteristic
92C2×C4×C107#4111yescharacteristic
93C2×Dic107—22no—
94C2×Dic107—22no—
95C2×Dic107—21no—
96C2×Dic107—21no—
97C2×Dic107—21no—
98C2×Dic107—41no—
99C2×Dic107—41no—
100C2×Dic107—41no—
101C2×Dic107—41no—
102C4×D107—41no—
103C4×D107—41no—
104C4×D107—41no—
105C4×D107—41no—
106Dic214—22no—
107Dic214—22no—
108Dic214—41no—
109Dic214—41no—
110C8×C107—41no—
111C107⋊C8—22no—
112C22×C107#12011yescharacteristic
113C22×C107#12011yescharacteristic
114C22×C107#12111yescharacteristic
115C22×C107C2×Q1611yescharacteristic
116C22×C107#13511yescharacteristic
117C22×C107#13711yescharacteristic
118C22×C107#13711yescharacteristic
119C2×D107—22no—
120C2×D107—22no—
121C2×D107—22no—
122C2×D107—41no—
123C2×D107—41no—
124C2×D107—41no—
125C2×D107—41no—
126C4×C107—42no—
127C4×C107—42no—
128C4×C107#12011yescharacteristic
129C4×C107#12111yescharacteristic
130C4×C107#12111yescharacteristic
131C4×C107#12711yescharacteristic
132Dic107—42no—
133Dic107—41no—
134Dic107—42no—
135C2×C107#39211yescharacteristic
136C2×C107#45111yescharacteristic
137C2×C107#45111yescharacteristic
138C2×C107#45211yescharacteristic
139C2×C107#45211yescharacteristic
140C2×C107#46711yescharacteristic
141C2×C107#47011yescharacteristic
142D107—42no—
143#2001—1071nomaximal, Sylow
144C107#200111yescharacteristic, complemented, Sylow
145C22×Q16—1071no—
146C2×C22⋊Q8—1071no—
147C2×C22⋊Q8—1071no—
148C2×#103—1071no—
149C2×#103D10711yescharacteristic, complemented
150C2×C22⋊C8—1071no—
151#340—1071no—
152C22×Q8—1071no—
153C22×Q8C2×D10711yescharacteristic
154C23×C4—1071no—
155C2×Q16—2142no—
156C2×Q16—2142no—
157C2×Q16—2142no—
158C22×C8—1071no—
159C22×C8C2×D10711yescharacteristic
160C22⋊Q8—2142no—
161C22⋊Q8—2142no—
162C22⋊Q8—2142no—
163C22⋊Q8—2142no—
164C2×C4⋊C4—1071no—
165C2×C4⋊C4—2141no—
166C2×C4⋊C4—2141no—
167C2×C4⋊C4C2×D10711yescharacteristic
168C2×C22⋊C4—2141no—
169C2×C22⋊C4—2141no—
170#103—22no—
171#103—2142no—
172C22⋊C8—2142no—
173C24—1071no—
174C2×Q8—21no—
175C2×Q8—21no—
176C2×Q8—21no—
177C2×Q8—21no—
178C2×Q8—21no—
179C2×Q8—1072no—
180C2×Q8—2141no—
181C2×Q8—2141no—
182C2×Q8—2141no—
183C2×Q8—2141no—
184C2×Q8—2141no—
185C2×Q8D42812yes—
186C22×C4—21no—
187C22×C4—21no—
188C22×C4—1071no—
189C22×C4—2141no—
190C22×C4—2141no—
191C22×C4—2141no—
192C22×C4—2141no—
193C22×C4—2141no—
194C22×C4—2141no—
195C22×C4—2141no—
196C22×C4—2141no—
197C22×C4C22×D10711yescharacteristic
198Q16—2148no—
199C2×C8—21no—
200C2×C8—21no—
201C2×C8—21no—
202C2×C8—1072no—
203C2×C8—2141no—
204C2×C8—2141no—
205C4⋊C4—21no—
206C4⋊C4—1072no—
207C4⋊C4—2144no—
208C4⋊C4—2144no—
209C4⋊C4—2141no—
210C4⋊C4C22⋊D10712yes—
211C22⋊C4—2144no—
212C22⋊C4—2144no—
213C23—1072no—
214C23—2141no—
215C23—2141no—
216C23—2141no—
217C23—2141no—
218C23—2141no—
219C23—2141no—
220C23D4×D10711yescharacteristic
221Q8—22no—
222Q8—22no—
223Q8—41no—
224Q8—41no—
225Q8—4281no—
226Q8—4281no—
227Q8—2142no—
228Q8—2142no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—41no—
232C2×C4—41no—
233C2×C4—41no—
234C2×C4—41no—
235C2×C4—4281no—
236C2×C4—4281no—
237C2×C4—4281no—
238C2×C4—4281no—
239C2×C4—4281no—
240C2×C4—4281no—
241C2×C4—4281no—
242C2×C4—4281no—
243C2×C4—2142no—
244C2×C4—2141no—
245C2×C4—2142no—
246C2×C4—2141no—
247C2×C4—2141no—
248C2×C4D4×D10711yescharacteristic
249C2×C4D4×D10711yescharacteristic
250C2×C4D4×D10711yescharacteristic
251C2×C4C2×C22⋊D10711yescharacteristic
252C2×C4C2×D42811yescharacteristic
253C2×C4#1313381811yescharacteristic
254C8—41no—
255C8—2142no—
256C22—4281no—
257C22—4281no—
258C22—4281no—
259C22—4281no—
260C22—2142no—
261C22—2142no—
262C22—2142no—
263C22#1681207711yescharacteristic
264C22#1681207911yescharacteristic
265C22#1681208211yescharacteristic
266C22Q16×D10711yescharacteristic
267C22#1681213011yescharacteristic
268C22#1681213811yescharacteristic
269C22#1681214111yescharacteristic
270C4—42no—
271C4—42no—
272C4—4281no—
273C4—4282no—
274C4—4282no—
275C4#1681207911yescharacteristic
276C4#1681208211yescharacteristic
277C4#1681208311yescharacteristic
278C4#1681210511yescharacteristic
279C2—4282no—
280C2#2029253611yescharacteristic
281C2#2029278111yescharacteristic
282C2#2029278411yescharacteristic
283C2#2029278611yescharacteristic
284C2#2029279211yescharacteristic
285C2#2029286711yescharacteristic
286C2#2029288411yescharacteristic
subgroupG/H
centerC23D4×D107
commutator, cosocleC22×C4×C107C23
FrattiniC22×C4C22×D107
FittingC2×#103×C107C2
socleC23×C107C2×D4
prime radicalC23×D107D4
hypercenterC2×#103D107
hypocenterC107#2001
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×#13133808C2semidirect
4. C2×#16812034C2semidirect
6. #20292389C2semidirect
7. C2×#103×C107C2semidirect
76. C22×D107Q16semidirect
144. C107#2001semidirect
149. C2×#103D107semidirect

Maximal subgroups

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

subgroupindex
C22×#131338082
C2×#168120892
C2×#168120852
C2×#168120342
C2×(C2×D107)⋊C82
#202923892
C2×#103×C1072
#2001107