Finite Group Databasebeta

← #28346632

Subgroups — #28346632

Order 13696.

386 conjugacy classes of subgroups; only the 129 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 129 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#20293161C211yesmaximal, characteristic
2#20292962C211yesmaximal, characteristic, complemented
3#20292945C211yesmaximal, characteristic
4#20292934C212yesmaximal, complemented
5#20292849C212yesmaximal, complemented
6#20292783C211yesmaximal, characteristic, complemented
7#20292766C211yesmaximal, characteristic
8#20292732C212yesmaximal, complemented
9#20292725C212yesmaximal, complemented
10#20292684C211yesmaximal, characteristic, complemented
11C2×C4×Q8×C107C211yesmaximal, characteristic, complemented
12C2×#13133808C2212yes—
13C2×#13133805C2212yes—
14#16812103—21no—
15#16812101C2211yescharacteristic
16#16812097C2211yescharacteristic
17#16812092—21no—
18#16812088C2211yescharacteristic
19#16812083C2211yescharacteristic
20C4×Dic214C2211yescharacteristic
21C4×D428C2211yescharacteristic
22C4×C22⋊D107—21no—
23#16812066—21no—
24#16812042C2212yes—
25#16812040C2212yes—
26C4⋊(C107⋊C8)C2212yes—
27#16812033C2212yes—
28#16812033C2212yes—
29#16812032C2212yes—
30#16812029C2212yes—
31C22⋊(C107⋊C8)C2211yescharacteristic
32C22⋊(C107⋊C8)C2211yescharacteristic
33#16812020C2212yes—
34C22×Q8×C107C2211yescharacteristic
35C4×Q8×C107—21no—
36C4×Q8×C107—21no—
37C4×Q8×C107C2212yes—
38C4×Q8×C107C2212yes—
39C2×C4⋊C4×C107—21no—
40C2×C4⋊C4×C107C2211yescharacteristic
41C2×C42×C107—21no—
42C2×C42×C107C2211yescharacteristic
43C2×Dic214C2311yescharacteristic
44C2×D428C2311yescharacteristic
45C2×C22⋊D107—21no—
46C2×C4×D107—21no—
47#13133808—24no—
48#13133805—24no—
49C2×C107⋊C8C2312yes—
50C2×C107⋊C8C2312yes—
51C4⋊Dic107C2311yescharacteristic
52C4⋊Dic107C2311yescharacteristic
53Dic107⋊C4—21no—
54Dic107⋊C4—21no—
55Dic107⋊C4—21no—
56Dic107⋊C4—21no—
57C22⋊Dic107—21no—
58(C2×D107)⋊C4—21no—
59(C2×D107)⋊C4—21no—
60(C2×D107)⋊C4—21no—
61(C2×D107)⋊C4—21no—
62C4×Dic107—21no—
63C2×Q8×C107—24no—
64C2×Q8×C107—21no—
65C2×Q8×C107C2312yes—
66C22×C4×C107—21no—
67C22×C4×C107—21no—
68C22×C4×C107D411yescharacteristic
69C22×C4×C107D411yescharacteristic
70C22×C4×C107C2311yescharacteristic
71C4⋊C4×C107—22no—
72C4⋊C4×C107—22no—
73C4⋊C4×C107—21no—
74C4⋊C4×C107C2312yes—
75C42×C107—22no—
76C42×C107—22no—
77C42×C107D411yescharacteristic
78C42×C107D411yescharacteristic
79C42×C107C2311yescharacteristic
80C42×C107C2311yescharacteristic
81C23×C107C2×D411yescharacteristic
82C22×D107—21no—
83Q8×C107—24no—
84Q8×C107—22no—
85Q8×C107#4114yes—
86C2×C4×C107—24no—
87C2×C4×C107—22no—
88C2×C4×C107—22no—
89C2×C4×C107—22no—
90C2×C4×C107—22no—
91C2×C4×C107—41no—
92C2×C4×C107—42no—
93C2×C4×C107C2×D411yescharacteristic
94C2×C4×C107C2×D411yescharacteristic
95C2×C4×C107C2×D411yescharacteristic
96C2×C4×C107C2×D411yescharacteristic
97C2×C4×C107C2×D411yescharacteristic
98C2×C4×C107C2411yescharacteristic
99C22⋊D107—42no—
100C2×Dic107—21no—
101C2×Dic107—21no—
102C2×Dic107—21no—
103D428—22no—
104C4×D107—42no—
105Dic214—22no—
106C107⋊C8—22no—
107C107⋊C8—22no—
108C22×C107—21no—
109C22×C107—21no—
110C22×C107#13712yes—
111C22×C107C22×D411yescharacteristic
112C2×D107—41no—
113C2×D107—41no—
114C2×D107—41no—
115C4×C107—24no—
116C4×C107—21no—
117C4×C107—21no—
118C4×C107—21no—
119C4×C107—21no—
120C4×C107—21no—
121C4×C107—42no—
122C4×C107C2×#4112yes—
123Dic107—41no—
124Dic107—41no—
125Dic107—41no—
126C2×C107—22no—
127C2×C107#52511yescharacteristic
128C2×C107C2×#13711yescharacteristic
129C2×C107#57511yescharacteristic
130D107—81no—
131#3117—1071nomaximal, Sylow
132C107#311711yescharacteristic, complemented, Sylow
133#529—1071no—
134C2×C4×Q8D10711yescharacteristic, complemented
135#484—1071no—
136#481—1071no—
137#479—1072no—
138#464—1072no—
139#451—1071no—
140#448—1071no—
141#441—1072no—
142#440—1072no—
143#432—1071no—
144C22×Q8C2×D10711yescharacteristic
145C2×Q16—1072no—
146C2×SD16—1072no—
147#126—2141no—
148#125—1071no—
149#124—1071no—
150#123—2141no—
151C22⋊Q8—1071no—
152#121—1071no—
153C4×Q8—21no—
154C4×Q8—21no—
155C4×Q8—1071no—
156C4×Q8C2×D10712yes—
157C4×Q8C2×D10712yes—
158C4×D4—1071no—
159C4×D4—2141no—
160#117—2141no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4C2×D10711yescharacteristic
163C2×C42—21no—
164C2×C42C2×D10711yescharacteristic
165#107—1072no—
166#106—1072no—
167C4⋊C8—1072no—
168#103—1072no—
169#103—1072no—
170#103—1072no—
171#102—1072no—
172C22⋊C8—1071no—
173C22⋊C8—1071no—
174#97—1072no—
175C2×Q8—24no—
176C2×Q8—21no—
177C2×Q8—1071no—
178C2×Q8C22×D10712yes—
179C2×D4—1071no—
180C2×D4—2141no—
181C22×C4—21no—
182C22×C4—21no—
183C22×C4—2141no—
184C22×C4C22⋊D10711yescharacteristic
185C22×C4C22⋊D10711yescharacteristic
186C22×C4C22×D10711yescharacteristic
187Q16—2144no—
188SD16—2144no—
189C2×C8—1072no—
190C2×C8—1072no—
191C4⋊C4—22no—
192C4⋊C4—22no—
193C4⋊C4—21no—
194C4⋊C4—1071no—
195C4⋊C4—1071no—
196C4⋊C4—2141no—
197C4⋊C4—2141no—
198C4⋊C4—2141no—
199C4⋊C4—2141no—
200C4⋊C4C22×D10712yes—
201C22⋊C4—2141no—
202C22⋊C4—2141no—
203C22⋊C4—2141no—
204C22⋊C4—2141no—
205C22⋊C4—2141no—
206C42—22no—
207C42—22no—
208C42—2141no—
209C42C22⋊D10711yescharacteristic
210C42C22⋊D10711yescharacteristic
211C42C22×D10711yescharacteristic
212C42C22×D10711yescharacteristic
213C23—2141no—
214C23C2×C22⋊D10711yescharacteristic
215Q8—24no—
216Q8—22no—
217Q8—2142no—
218Q8#1313381814yes—
219D4—4282no—
220D4—2142no—
221C2×C4—24no—
222C2×C4—22no—
223C2×C4—22no—
224C2×C4—22no—
225C2×C4—22no—
226C2×C4—41no—
227C2×C4—42no—
228C2×C4—4282no—
229C2×C4—2141no—
230C2×C4—2141no—
231C2×C4—2141no—
232C2×C4C2×C22⋊D10711yescharacteristic
233C2×C4C2×C22⋊D10711yescharacteristic
234C2×C4C2×C22⋊D10711yescharacteristic
235C2×C4C2×C22⋊D10711yescharacteristic
236C2×C4C2×C22⋊D10711yescharacteristic
237C2×C4C23×D10711yescharacteristic
238C8—2142no—
239C8—2142no—
240C22—21no—
241C22—21no—
242C22—4281no—
243C22—4281no—
244C22—4281no—
245C22#1681214012yes—
246C22C22×C22⋊D10711yescharacteristic
247C4—24no—
248C4—21no—
249C4—21no—
250C4—21no—
251C4—21no—
252C4—21no—
253C4—42no—
254C4—4281no—
255C4—4281no—
256C4—4281no—
257C4C2×#1313381812yes—
258C2—22no—
259C2—8561no—
260C2#2029313811yescharacteristic
261C2C2×#1681214011yescharacteristic
262C2#2029342511yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×C4×Q8×C107C2
socleC22×C107C22×D4
prime radical#16812083C22
hypercenterC2×C4×Q8D107
hypocenterC107#3117
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×C4×Q8

Complements

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

NH ≅ G/Ntype
2. #20292962C2semidirect
4. #20292934C2semidirect
5. #20292849C2semidirect
6. #20292783C2semidirect
8. #20292732C2semidirect
9. #20292725C2semidirect
10. #20292684C2semidirect
11. C2×C4×Q8×C107C2semidirect
132. C107#3117semidirect
134. C2×C4×Q8D107semidirect

Maximal subgroups

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

subgroupindex
#202931612
#202929622
#202929452
#202929342
#202928492
#202927832
#202927662
#202927322
#202927252
#202926842
C2×C4×Q8×C1072
#3117107