Finite Group Databasebeta

← #28348102

Subgroups — #28348102

Order 13696.

472 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 125 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×#16812141C212yesmaximal, complemented
2Q8×D428C211yesmaximal, characteristic, complemented
3#20293280C211yesmaximal, characteristic
4#20293006C211yesmaximal, characteristic, complemented
5#20292884C212yesmaximal, complemented
6#20292833C211yesmaximal, characteristic, complemented
7#20292792C212yesmaximal, complemented
8#20292781C212yesmaximal, complemented
9C4×#13133805C211yesmaximal, characteristic, complemented
10#20292704C211yesmaximal, characteristic, complemented
11#462×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2212yes—
13C2×Q8×D107C2212yes—
14#16812141—28no—
15C2×#13133808C2212yes—
16C2×#13133808C2212yes—
17C2×#13133807C2212yes—
18C2×#13133805C2211yescharacteristic
19C2×#13133801C2212yes—
20#16812109—21no—
21#16812109C2211yescharacteristic
22#16812094—22no—
23#16812089—22no—
24#16812089C2212yes—
25#16812085C2212yes—
26#16812084—21no—
27C4×D428—21no—
28C4×D428C2211yescharacteristic
29C4×D428C2211yescharacteristic
30C4⋊C4×D107—21no—
31#16812040C2211yescharacteristic
32#16812034C2212yes—
33#16812033C2211yescharacteristic
34#16812029C2211yescharacteristic
35(C2×D107)⋊C8C2212yes—
36C4×C107⋊C8C2211yescharacteristic
37C2×Q16×C107C2212yes—
38C4⋊Q8×C107C2211yescharacteristic
39C4×Q8×C107C2211yescharacteristic
40C4⋊C8×C107C2211yescharacteristic
41#103×C107C2212yes—
42#13133819—24no—
43#13133819—42no—
44C2×Dic214—22no—
45C2×Dic214C2312yes—
46Q8×D107—24no—
47Q8×D107—42no—
48C2×D428—22no—
49C2×D428D411yescharacteristic
50C2×D428C2311yescharacteristic
51C2×C4×D107—22no—
52C2×C4×D107—21no—
53C2×C4×D107—22no—
54C2×C4×D107C2312yes—
55#13133808—24no—
56#13133808—42no—
57#13133807—24no—
58#13133805D414yescomplemented
59#13133801—24no—
60C2×C107⋊C8C2311yescharacteristic
61C2×C107⋊C8C2311yescharacteristic
62C4⋊Dic107—21no—
63C4⋊Dic107—21no—
64C4⋊Dic107—22no—
65C4⋊Dic107D411yescharacteristic
66C4⋊Dic107C2311yescharacteristic
67Dic107⋊C4—22no—
68(C2×D107)⋊C4—22no—
69(C2×D107)⋊C4—22no—
70(C2×D107)⋊C4—21no—
71(C2×D107)⋊C4D412yes—
72C2×Q8×C107C2312yes—
73C2×Q8×C107C2311yescharacteristic
74Q16×C107—24no—
75C2×C8×C107C2312yes—
76C4⋊C4×C107—21no—
77C4⋊C4×C107—21no—
78C4⋊C4×C107—21no—
79C4⋊C4×C107C2311yescharacteristic
80C42×C107—21no—
81C42×C107C2311yescharacteristic
82C22×D107—21no—
83C22×D107C2×D412yes—
84Q8×C107—21no—
85Q8×C107—22no—
86Q8×C107—22no—
87Q8×C107C2×D412yes—
88C2×C4×C107—21no—
89C2×C4×C107—21no—
90C2×C4×C107—22no—
91C2×C4×C107C2×D411yescharacteristic
92C2×C4×C107C2×D411yescharacteristic
93C2×C4×C107C2411yescharacteristic
94C2×Dic107—21no—
95C2×Dic107—22no—
96C2×Dic107C2×D412yes—
97D428—22no—
98D428—21no—
99D428—44no—
100D428C2×D412yes—
101C4×D107—22no—
102C4×D107—22no—
103C4×D107—44no—
104C4×D107—44no—
105C4×D107—42no—
106Dic214—22no—
107Dic214—22no—
108Dic214—44no—
109C8×C107—22no—
110C107⋊C8—21no—
111C107⋊C8C2×D412yes—
112C22×C107C22×D411yescharacteristic
113C2×D107—22no—
114C2×D107—22no—
115C2×D107—22no—
116C2×D107—22no—
117C2×D107—41no—
118C2×D107—41no—
119C4×C107—22no—
120C4×C107—21no—
121C4×C107—42no—
122C4×C107—41no—
123C4×C107#13712yescomplemented
124C4×C107C22×D411yescharacteristic
125C4×C107H(5,𝔽2)11yescharacteristic
126Dic107—22no—
127Dic107—42no—
128Dic107—41no—
129C2×C107D4211yescharacteristic
130C2×C107C2×#13711yescharacteristic
131C2×C107#57811yescharacteristic
132D107—42no—
133D107—42no—
134#3276—1071nomaximal, Sylow
135C107#327611yescharacteristic, complemented, Sylow
136C2×#137—1072no—
137D4×Q8—1071no—
138#547—1071no—
139#494—1071no—
140#470—1072no—
141#462D10711yescharacteristic, complemented
142#461—1071no—
143#452—1072no—
144#451—1072no—
145C4×SD16—1071no—
146#436—1071no—
147C2×#41—1072no—
148C22×Q8—1072no—
149#137—2148no—
150C2×Q16—1072no—
151C2×Q16—1072no—
152C2×Q16C2×D10712yes—
153C2×SD16—1071no—
154C2×SD16—1072no—
155C2×M16—1072no—
156C4⋊Q8—1071no—
157C4⋊Q8—2141no—
158C4⋊Q8C2×D10711yescharacteristic
159#123—2142no—
160C22⋊Q8—1072no—
161C22⋊Q8—1072no—
162C22⋊Q8—2142no—
163#121—2141no—
164C4×Q8C2×D10711yescharacteristic
165C4×D4—1071no—
166C4×D4—1071no—
167C4×D4—2141no—
168C2×C4⋊C4—2141no—
169#106—1071no—
170C4⋊C8C2×D10711yescharacteristic
171#103—1071no—
172#103—1072no—
173#103C2×D10712yes—
174#102—1071no—
175C22⋊C8—1072no—
176C4×C8—1071no—
177#41—4282no—
178#41—2144no—
179C2×Q8—1072no—
180C2×Q8—4282no—
181C2×Q8—2142no—
182C2×Q8—2144no—
183C2×Q8C22×D10712yes—
184C2×Q8C22×D10711yescharacteristic
185C2×D4—1071no—
186C2×D4—1071no—
187C2×D4—2142no—
188C22×C4—1072no—
189C22×C4—2142no—
190C22×C4—2142no—
191C22×C4—2141no—
192Q16—24no—
193Q16—4282no—
194Q16—2144no—
195SD16—1074no—
196SD16—2144no—
197M16—2144no—
198C2×C8—1071no—
199C2×C8—1071no—
200C2×C8C22×D10712yes—
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—1071no—
205C4⋊C4—1071no—
206C4⋊C4—2142no—
207C4⋊C4—2142no—
208C4⋊C4—2141no—
209C4⋊C4—2141no—
210C4⋊C4C22×D10711yescharacteristic
211C22⋊C4—1072no—
212C22⋊C4—2142no—
213C22⋊C4—2142no—
214C22⋊C4—2141no—
215C42—21no—
216C42C22×D10711yescharacteristic
217C23—1072no—
218C23—2141no—
219Q8—21no—
220Q8—22no—
221Q8—22no—
222Q8—4284no—
223Q8—2142no—
224Q8—2142no—
225Q8D4×D10712yes—
226D4—1072no—
227D4—4284no—
228D4—2142no—
229D4—2141no—
230C2×C4—21no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—1072no—
234C2×C4—4284no—
235C2×C4—4284no—
236C2×C4—4282no—
237C2×C4—2141no—
238C2×C4—2142no—
239C2×C4—2142no—
240C2×C4—2142no—
241C2×C4D4×D10711yescharacteristic
242C2×C4D4×D10711yescharacteristic
243C2×C4C23×D10711yescharacteristic
244C8—22no—
245C8—1072no—
246C8—2141no—
247C22—4281no—
248C22—4281no—
249C22—2142no—
250C22—2142no—
251C22—2142no—
252C22—2142no—
253C22C2×D4×D10711yescharacteristic
254C4—21no—
255C4—22no—
256C4—41no—
257C4—42no—
258C4—4281no—
259C4—4282no—
260C4—2142no—
261C4#1681214112yescomplemented
262C4C2×D4×D10711yescharacteristic
263C4#1681215511yescharacteristic
264C2—4282no—
265C2—4282no—
266C2#2029326111yescharacteristic
267C2C2×#1681214111yescharacteristic
268C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#462×C107C2
socleC22×C107C22×D4
prime radical#20293280C2
hypercenter#462D107
hypocenterC107#3276
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #462

Complements

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

NH ≅ G/Ntype
1. C2×#16812141C2semidirect
2. Q8×D428C2semidirect
4. #20293006C2semidirect
5. #20292884C2semidirect
6. #20292833C2semidirect
7. #20292792C2semidirect
8. #20292781C2semidirect
9. C4×#13133805C2semidirect
10. #20292704C2semidirect
11. #462×C107C2semidirect
58. #13133805D4semidirect
123. C4×C107#137semidirect
135. C107#3276semidirect
141. #462D107semidirect
261. C4#16812141semidirect

Maximal subgroups

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

subgroupindex
C2×#168121412
Q8×D4282
#202932802
#202930062
#202928842
#202928332
#202927922
#202927812
C4×#131338052
#202927042
#462×C1072
#3276107