Finite Group Databasebeta

← #28348106

Subgroups — #28348106

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
2#20293292C211yesmaximal, characteristic, complemented
3#20293280C211yesmaximal, characteristic
4#20292882C212yesmaximal, complemented
5#20292834C211yesmaximal, characteristic, complemented
6#20292829C211yesmaximal, characteristic, complemented
7#20292788C212yesmaximal, complemented
8#20292781C212yesmaximal, complemented
9C4×#13133807C211yesmaximal, characteristic, complemented
10#20292706C211yesmaximal, characteristic, complemented
11#494×C107C211yesmaximal, characteristic, complemented
12C2×#13133819C2212yes—
13C2×Q8×D107C2212yes—
14#16812141—28no—
15C2×#13133808C2212yes—
16C2×#13133807C2211yescharacteristic
17C2×#13133805C2212yes—
18C2×#13133801C2212yes—
19#16812108—21no—
20#16812094—22no—
21#16812087—22no—
22#16812085C2212yes—
23#16812085C2212yes—
24#16812084—21no—
25C4×Dic214C2211yescharacteristic
26C4×D428C2211yescharacteristic
27C4×D428C2211yescharacteristic
28#16812071—21no—
29C4⋊C4×D107—21no—
30#16812041C2211yescharacteristic
31C4⋊(C107⋊C8)C2211yescharacteristic
32#16812034C2211yescharacteristic
33#16812033C2212yes—
34#16812033C2212yes—
35#16812031C2211yescharacteristic
36(C2×D107)⋊C8C2212yes—
37C2×Q16×C107C2212yes—
38C2×Q16×C107C2212yes—
39C4⋊Q8×C107C2211yescharacteristic
40C4⋊Q8×C107C2211yescharacteristic
41C4×C8×C107C2211yescharacteristic
42#13133819—24no—
43#13133819—42no—
44C2×Dic214—22no—
45C2×Dic214C2311yescharacteristic
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#13133807D414yescomplemented
57#13133805—24no—
58#13133801—24no—
59C2×C107⋊C8C2312yes—
60C4⋊Dic107D411yescharacteristic
61C4⋊Dic107C2311yescharacteristic
62C4⋊Dic107C2311yescharacteristic
63Dic107⋊C4—21no—
64Dic107⋊C4—22no—
65Dic107⋊C4—22no—
66(C2×D107)⋊C4—22no—
67(C2×D107)⋊C4—22no—
68(C2×D107)⋊C4—21no—
69(C2×D107)⋊C4D412yes—
70C4×Dic107—21no—
71C2×Q8×C107C2312yes—
72C2×Q8×C107C2312yes—
73Q16×C107—24no—
74Q16×C107—42no—
75C2×C8×C107C2311yescharacteristic
76C2×C8×C107C2311yescharacteristic
77C4⋊C4×C107—21no—
78C4⋊C4×C107—21no—
79C4⋊C4×C107—21no—
80C4⋊C4×C107—21no—
81C42×C107C2311yescharacteristic
82C22×D107—21no—
83C22×D107C2×D412yes—
84Q8×C107—22no—
85Q8×C107—22no—
86Q8×C107—22no—
87Q8×C107—22no—
88C2×C4×C107—22no—
89C2×C4×C107—22no—
90C2×C4×C107C2×D411yescharacteristic
91C2×C4×C107C2×D411yescharacteristic
92C2×C4×C107C2411yescharacteristic
93C2×Dic107—21no—
94C2×Dic107—21no—
95C2×Dic107—21no—
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—21no—
107Dic214—44no—
108Dic214C2×D412yes—
109C8×C107—21no—
110C8×C107C2×D412yes—
111C107⋊C8—22no—
112C22×C107C22×D411yescharacteristic
113C2×D107—22no—
114C2×D107—22no—
115C2×D107—22no—
116C2×D107—22no—
117C2×D107—41no—
118C2×D107—41no—
119C4×C107—21no—
120C4×C107—42no—
121C4×C107—42no—
122C4×C107#13712yescomplemented
123C4×C107C22×D411yescharacteristic
124C4×C107H(5,𝔽2)11yescharacteristic
125Dic107—22no—
126Dic107—22no—
127Dic107—41no—
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#494D10711yescharacteristic, complemented
140#470—1072no—
141#462—1071no—
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×Q16C2×D10712yes—
152C2×Q16C2×D10712yes—
153C2×SD16—1072no—
154C2×SD16—1071no—
155C2×M16—1072no—
156C4⋊Q8—2141no—
157C4⋊Q8C2×D10711yescharacteristic
158C4⋊Q8C2×D10711yescharacteristic
159#123—2142no—
160C22⋊Q8—1072no—
161C22⋊Q8—1072no—
162C22⋊Q8—2142no—
163#121—2141no—
164C4×Q8—1071no—
165C4×D4—1071no—
166C4×D4—1071no—
167C4×D4—2141no—
168C2×C4⋊C4—2141no—
169#106—1071no—
170C4⋊C8—1071no—
171#103—1072no—
172#103—1072no—
173#103—1071no—
174#102—1071no—
175C22⋊C8—1072no—
176C4×C8C2×D10711yescharacteristic
177#41—4282no—
178#41—2144no—
179C2×Q8—1071no—
180C2×Q8—4282no—
181C2×Q8—2142no—
182C2×Q8—2144no—
183C2×Q8C22×D10712yes—
184C2×Q8C22×D10712yes—
185C2×D4—1071no—
186C2×D4—1071no—
187C2×D4—2142no—
188C22×C4—1072no—
189C22×C4—2142no—
190C22×C4—2142no—
191C22×C4—2141no—
192Q16—24no—
193Q16—42no—
194Q16—2144no—
195SD16—1074no—
196SD16—2144no—
197M16—2144no—
198C2×C8—1072no—
199C2×C8C22×D10711yescharacteristic
200C2×C8C22×D10711yescharacteristic
201C4⋊C4—21no—
202C4⋊C4—21no—
203C4⋊C4—21no—
204C4⋊C4—21no—
205C4⋊C4—1071no—
206C4⋊C4—1071no—
207C4⋊C4—1071no—
208C4⋊C4—2142no—
209C4⋊C4—2142no—
210C4⋊C4—2141no—
211C22⋊C4—1072no—
212C22⋊C4—2142no—
213C22⋊C4—2142no—
214C22⋊C4—2141no—
215C42—2141no—
216C42C22×D10711yescharacteristic
217C23—1072no—
218C23—2141no—
219Q8—22no—
220Q8—22no—
221Q8—22no—
222Q8—22no—
223Q8—1072no—
224Q8—4284no—
225Q8—2141no—
226D4—1072no—
227D4—4284no—
228D4—2142no—
229D4—2141no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—1072no—
233C2×C4—4284no—
234C2×C4—4284no—
235C2×C4—4282no—
236C2×C4—2141no—
237C2×C4—2141no—
238C2×C4—2141no—
239C2×C4—2142no—
240C2×C4—2142no—
241C2×C4D4×D10711yescharacteristic
242C2×C4D4×D10711yescharacteristic
243C2×C4C23×D10711yescharacteristic
244C8—21no—
245C8—2142no—
246C8D4×D10712yes—
247C22—4281no—
248C22—4281no—
249C22—2142no—
250C22—2142no—
251C22—2142no—
252C22—2142no—
253C22C2×D4×D10711yescharacteristic
254C4—21no—
255C4—42no—
256C4—42no—
257C4—4281no—
258C4—4281no—
259C4—2142no—
260C4—2142no—
261C4#1681214112yescomplemented
262C4C2×D4×D10711yescharacteristic
263C4#1681215411yescharacteristic
264C2—4282no—
265C2—4282no—
266C2#2029326211yescharacteristic
267C2C2×#1681214111yescharacteristic
268C2#2029344011yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#494×C107C2
socleC22×C107C22×D4
prime radical#20293280C2
hypercenter#494D107
hypocenterC107#3276
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #494

Complements

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

NH ≅ G/Ntype
1. C2×#16812141C2semidirect
2. #20293292C2semidirect
4. #20292882C2semidirect
5. #20292834C2semidirect
6. #20292829C2semidirect
7. #20292788C2semidirect
8. #20292781C2semidirect
9. C4×#13133807C2semidirect
10. #20292706C2semidirect
11. #494×C107C2semidirect
56. #13133807D4semidirect
122. C4×C107#137semidirect
135. C107#3276semidirect
139. #494D107semidirect
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
#202932922
#202932802
#202928822
#202928342
#202928292
#202927882
#202927812
C4×#131338072
#202927062
#494×C1072
#3276107