Finite Group Databasebeta

← #28346275

Subgroups — #28346275

Order 13696.

416 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 416 conjugacy classes of subgroups — too many to draw. Only its 133 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
1C4⋊Q8×D107C211yesmaximal, characteristic, complemented
2C4×#13133819C211yesmaximal, characteristic
3#20292922C211yesmaximal, characteristic, complemented
4#20292918C211yesmaximal, characteristic, complemented
5#20292914C211yesmaximal, characteristic, complemented
6#20292910C212yesmaximal, complemented
7#20292906C212yesmaximal, complemented
8#20292668C212yesmaximal, complemented
9#20292649C211yesmaximal, characteristic, complemented
10#20292627C212yesmaximal
11#477×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2211yescharacteristic
13C2×Q8×D107—21no—
14C2×#13133801C2212yes—
15#16812109C2211yescharacteristic
16#16812108—21no—
17#16812108C2212yes—
18#16812107—21no—
19Q8×Dic107C2211yescharacteristic
20C4×D428—21no—
21C4×D428C2211yescharacteristic
22#16812071—21no—
23#16812071C2211yescharacteristic
24#16812067—21no—
25#16812067C2211yescharacteristic
26C4⋊C4×D107—21no—
27C4⋊C4×D107C2212yes—
28C42×D107—21no—
29C42×D107C2211yescharacteristic
30#16812042C2212yes—
31#16812041C2212yes—
32#16812040C2212yes—
33C4⋊(C107⋊C8)C2211yescharacteristic
34Dic107⋊C8C2212yes—
35#16812033C2212yes—
36#16812031C2212yes—
37#16812029C2212yes—
38C4⋊Q8×C107C2211yescharacteristic
39C4×Q8×C107C2211yescharacteristic
40#107×C107C2212yes—
41C4⋊C8×C107C2211yescharacteristic
42#103×C107C2212yes—
43#13133819—22no—
44#13133819Q814yescomplemented
45C2×Dic214—21no—
46C2×Dic214—22no—
47Q8×D107—44no—
48C2×D428—21no—
49C2×D428C2311yescharacteristic
50C2×C4×D107—21no—
51C2×C4×D107—22no—
52C2×C4×D107—21no—
53C2×C4×D107—21no—
54C2×C4×D107D411yescharacteristic
55C2×C4×D107D411yescharacteristic
56C2×C4×D107C2311yescharacteristic
57#13133801—24no—
58C2×C107⋊C8C2312yes—
59C4⋊Dic107—21no—
60C4⋊Dic107—21no—
61C4⋊Dic107C2312yes—
62C4⋊Dic107C2311yescharacteristic
63Dic107⋊C4—22no—
64Dic107⋊C4—22no—
65(C2×D107)⋊C4—21no—
66(C2×D107)⋊C4—21no—
67(C2×D107)⋊C4—21no—
68C4×Dic107—21no—
69C4×Dic107—21no—
70C4×Dic107D411yescharacteristic
71C4×Dic107D411yescharacteristic
72C4×Dic107C2311yescharacteristic
73C2×Q8×C107—21no—
74C2×Q8×C107C2311yescharacteristic
75C2×C8×C107C2312yes—
76C4⋊C4×C107—21no—
77C4⋊C4×C107—21no—
78C4⋊C4×C107C2312yes—
79C4⋊C4×C107C2311yescharacteristic
80C42×C107—21no—
81C42×C107C2311yescharacteristic
82C22×D107—21no—
83C22×D107C2×D411yescharacteristic
84Q8×C107—21no—
85Q8×C107—42no—
86Q8×C107C2×Q812yescomplemented
87C2×C4×C107—21no—
88C2×C4×C107—21no—
89C2×C4×C107—22no—
90C2×C4×C107C2×D411yescharacteristic
91C2×C4×C107C2×D411yescharacteristic
92C2×C4×C107C2411yescharacteristic
93C2×Dic107—21no—
94C2×Dic107—22no—
95C2×Dic107C2×D411yescharacteristic
96C2×Dic107C2×D411yescharacteristic
97C2×Dic107C2×D411yescharacteristic
98D428—24no—
99D428—21no—
100D428C2×Q812yescomplemented
101C4×D107—22no—
102C4×D107—24no—
103C4×D107—22no—
104C4×D107—44no—
105C4×D107—42no—
106C4×D107—42no—
107C4×D107C2×Q812yes—
108C4×D107#4112yes—
109Dic214—42no—
110Dic214—44no—
111C8×C107—22no—
112C107⋊C8—22no—
113C22×C107C22×D411yescharacteristic
114C2×D107—21no—
115C2×D107—21no—
116C2×D107—22no—
117C2×D107—41no—
118C2×D107—41no—
119C2×D107C22⋊Q812yes—
120C4×C107—22no—
121C4×C107—21no—
122C4×C107—42no—
123C4×C107—41no—
124C4×C107#13712yes—
125C4×C107C22×Q811yescharacteristic
126C4×C107C2×#4111yescharacteristic
127Dic107—21no—
128Dic107—42no—
129Dic107—41no—
130Dic107C22⋊Q812yes—
131Dic107AGL(1,ℤ/8)12yes—
132C2×C107C2×C22⋊Q811yescharacteristic
133C2×C107C2×AGL(1,ℤ/8)11yescharacteristic
134C2×C107C2×#13711yescharacteristic
135D107—22no—
136D107—42no—
137#3079—1071nomaximal, Sylow
138C107#307911yescharacteristic, complemented, Sylow
139C2×C4⋊Q8—1071no—
140C4×#41—1071no—
141#477—1071no—
142#477D10711yescharacteristic, complemented
143#476—1071no—
144#476—1071no—
145#475—1072no—
146#474—1072no—
147#428—1072no—
148#423—1071no—
149#417—1072no—
150C2×#41—1071no—
151C22×Q8—2141no—
152C2×M16—1072no—
153C4⋊Q8—1072no—
154C4⋊Q8—1071no—
155C4⋊Q8—2141no—
156C4⋊Q8—2141no—
157C4⋊Q8C2×D10711yescharacteristic
158C4×Q8—1071no—
159C4×Q8C2×D10711yescharacteristic
160C4×D4—1071no—
161C4×D4—1071no—
162C4×D4—2141no—
163C4×D4—2141no—
164#117—1071no—
165#117—2141no—
166C2×C4⋊C4—1072no—
167C2×C4⋊C4—2141no—
168C2×C42—1071no—
169C2×C42—2141no—
170#107—1072no—
171#107C2×D10712yes—
172#106—1072no—
173#106—1072no—
174C4⋊C8—1072no—
175C4⋊C8—1071no—
176C4⋊C8C2×D10711yescharacteristic
177#103—1072no—
178#103C2×D10712yes—
179#102—1072no—
180#102—1072no—
181#41—1074no—
182#41—2142no—
183C2×Q8—21no—
184C2×Q8—4284no—
185C2×Q8—2142no—
186C2×Q8—2141no—
187C2×Q8C22×D10711yescharacteristic
188C2×D4—1071no—
189C2×D4—2141no—
190C22×C4—1071no—
191C22×C4—1071no—
192C22×C4—1071no—
193C22×C4—2141no—
194C22×C4—2141no—
195C22×C4—2142no—
196C22×C4—2141no—
197M16—2144no—
198C2×C8—1072no—
199C2×C8C22×D10712yes—
200C4⋊C4—21no—
201C4⋊C4—21no—
202C4⋊C4—1072no—
203C4⋊C4—1071no—
204C4⋊C4—2142no—
205C4⋊C4—2142no—
206C4⋊C4—2141no—
207C4⋊C4—2141no—
208C4⋊C4C22×D10712yes—
209C4⋊C4C22×D10711yescharacteristic
210C22⋊C4—2141no—
211C22⋊C4—2141no—
212C22⋊C4—2141no—
213C42—21no—
214C42—1071no—
215C42—1071no—
216C42—1071no—
217C42—2141no—
218C42—2141no—
219C42C22×D10711yescharacteristic
220C23—1071no—
221C23—2141no—
222Q8—21no—
223Q8—42no—
224Q8—4282no—
225Q8—4284no—
226Q8Q8×D10712yescomplemented
227D4—1072no—
228D4—2144no—
229D4—2141no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—1071no—
234C2×C4—1071no—
235C2×C4—1072no—
236C2×C4—1072no—
237C2×C4—1071no—
238C2×C4—4282no—
239C2×C4—4284no—
240C2×C4—4282no—
241C2×C4—2142no—
242C2×C4—2144no—
243C2×C4—2141no—
244C2×C4—2142no—
245C2×C4—2142no—
246C2×C4D4×D10711yescharacteristic
247C2×C4D4×D10711yescharacteristic
248C2×C4C23×D10711yescharacteristic
249C8—22no—
250C8—2142no—
251C22—1072no—
252C22—4281no—
253C22—4281no—
254C22—2142no—
255C22—2141no—
256C22—2141no—
257C22C2×D4×D10711yescharacteristic
258C4—22no—
259C4—21no—
260C4—42no—
261C4—41no—
262C4—1072no—
263C4—1072no—
264C4—4282no—
265C4—4281no—
266C4—2141no—
267C4#1681214112yes—
268C4C2×Q8×D10711yescharacteristic
269C4#41×D10711yescharacteristic
270C2—4282no—
271C2—2142no—
272C2C22⋊Q8×D10711yescharacteristic
273C2AGL(1,ℤ/8)×D10711yescharacteristic
274C2C2×#1681214111yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#477×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133819C22
hypercenter#477D107
hypocenterC107#3079
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #477

Complements

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

NH ≅ G/Ntype
1. C4⋊Q8×D107C2semidirect
3. #20292922C2semidirect
4. #20292918C2semidirect
5. #20292914C2semidirect
6. #20292910C2semidirect
7. #20292906C2semidirect
8. #20292668C2semidirect
9. #20292649C2semidirect
11. #477×C107C2semidirect
44. #13133819Q8semidirect
86. Q8×C107C2×Q8semidirect
100. D428C2×Q8semidirect
138. C107#3079semidirect
142. #477D107semidirect
226. Q8Q8×D107semidirect

Maximal subgroups

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

subgroupindex
C4⋊Q8×D1072
C4×#131338192
#202929222
#202929182
#202929142
#202929102
#202929062
#202926682
#202926492
#202926272
#477×C1072
#3079107