Finite Group Databasebeta

← #28346015

Subgroups — #28346015

Order 13696.

472 conjugacy classes of subgroups; only the 141 normal subgroups are shown.

normal subgroups only

This group has 472 conjugacy classes of subgroups — too many to draw. Only its 141 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×#16812129C211yesmaximal, characteristic
2#20293201C211yesmaximal, characteristic, complemented
3#20293110C211yesmaximal, characteristic, complemented
4#20292851C212yesmaximal, complemented
5#20292844C212yesmaximal, complemented
6#20292835C212yesmaximal, complemented
7#20292830C212yesmaximal, complemented
8#20292655C211yesmaximal, characteristic, complemented
9#20292629C211yesmaximal, characteristic, complemented
10C2×#16812033C211yesmaximal, characteristic, complemented
11C2×#137×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13#16812129—24no—
14C2×#13133808C2211yescharacteristic
15C2×#13133806C2211yescharacteristic
16C2×#13133805C2211yescharacteristic
17C2×#13133803C2211yescharacteristic
18C22×C107⋊C8C2211yescharacteristic
19#16812107C2212yes—
20#16812098C2212yes—
21#16812092—22no—
22#16812085—21no—
23#16812085—21no—
24Q8×Dic107C2212yes—
25D4×Dic107—22no—
26D4×Dic107C2212yes—
27#16812068—21no—
28#16812068C2211yescharacteristic
29C2×C4⋊Dic107—21no—
30C2×C4⋊Dic107C2211yescharacteristic
31Dic107⋊C8C2212yes—
32Dic107⋊C8C2212yes—
33#16812034C2212yes—
34#16812033C2212yes—
35#16812033C2212yes—
36#16812031C2212yes—
37C2×#41×C107C2211yescharacteristic
38C22×Q8×C107C2211yescharacteristic
39#137×C107—24no—
40C2×Q16×C107C2212yes—
41C2×SD16×C107C2212yes—
42C2×M16×C107C2211yescharacteristic
43#13133818—22no—
44#13133818—22no—
45C2×Dic214C2311yescharacteristic
46C2×D428C2311yescharacteristic
47C2×C22⋊D107—21no—
48C22×Dic107—21no—
49C22×Dic107—22no—
50C2×C4×D107—21no—
51#13133808—22no—
52#13133806—22no—
53#13133805—22no—
54#13133803—22no—
55C2×C107⋊C8—22no—
56C2×C107⋊C8C2311yescharacteristic
57C2×C107⋊C8C2311yescharacteristic
58C4⋊Dic107—21no—
59C4⋊Dic107—22no—
60C4⋊Dic107D411yescharacteristic
61C4⋊Dic107D411yescharacteristic
62C4⋊Dic107C2312yes—
63Dic107⋊C4—22no—
64Dic107⋊C4—22no—
65C22⋊Dic107—22no—
66C22⋊Dic107D412yes—
67(C2×D107)⋊C4—22no—
68(C2×D107)⋊C4—22no—
69C4×Dic107—22no—
70C4×Dic107C2312yes—
71#41×C107—22no—
72#41×C107D414yes—
73C2×Q8×C107—21no—
74C2×Q8×C107—22no—
75C2×Q8×C107—22no—
76C2×Q8×C107C2312yes—
77C2×Q8×C107C2311yescharacteristic
78C2×D4×C107—21no—
79C2×D4×C107C2311yescharacteristic
80C22×C4×C107—21no—
81C22×C4×C107—21no—
82C22×C4×C107C2311yescharacteristic
83Q16×C107—24no—
84SD16×C107—24no—
85M16×C107—22no—
86C2×C8×C107C2312yes—
87C23×C107—21no—
88C23×C107C2×D411yescharacteristic
89C22×D107—21no—
90Q8×C107—21no—
91Q8×C107—22no—
92Q8×C107—22no—
93Q8×C107—42no—
94Q8×C107C2×D412yes—
95D4×C107—24no—
96D4×C107—21no—
97D4×C107C2×D412yes—
98C2×C4×C107—21no—
99C2×C4×C107—24no—
100C2×C4×C107—22no—
101C2×C4×C107—42no—
102C2×C4×C107C2×D411yescharacteristic
103C2×C4×C107C2×D412yes—
104C2×C4×C107#4112yes—
105C2×C4×C107C2411yescharacteristic
106C22⋊D107—42no—
107C2×Dic107—22no—
108C2×Dic107—21no—
109C2×Dic107—82no—
110C2×Dic107—81no—
111C2×Dic107C2×D412yes—
112C2×Dic107C2×D412yes—
113D428—21no—
114D428—21no—
115C4×D107—42no—
116Dic214—21no—
117Dic214—21no—
118C8×C107—22no—
119C107⋊C8—21no—
120C107⋊C8—21no—
121C22×C107—21no—
122C22×C107—21no—
123C22×C107—22no—
124C22×C107—41no—
125C22×C107—41no—
126C22×C107#12112yes—
127C22×C107C22×D411yescharacteristic
128C2×D107—41no—
129C2×D107—41no—
130C2×D107—41no—
131C4×C107—22no—
132C4×C107—42no—
133C4×C107#12112yes—
134C4×C107C22×D411yescharacteristic
135C4×C107C2×#4111yescharacteristic
136Dic107—22no—
137Dic107—22no—
138Dic107—41no—
139Dic107—42no—
140C2×C107—22no—
141C2×C107—42no—
142C2×C107C2×#12111yescharacteristic
143C2×C107#57711yescharacteristic
144C2×C107#57811yescharacteristic
145D107—81no—
146#3046—1071nomaximal, Sylow
147C107#304611yescharacteristic, complemented, Sylow
148C2×#137D10711yescharacteristic, complemented
149C2×#135—1071no—
150#536—1071no—
151#520—1071no—
152#464—1072no—
153#463—1072no—
154#462—1072no—
155#461—1072no—
156#424—1071no—
157#417—1071no—
158C2×#103—1071no—
159C2×#41—1071no—
160C2×#41C2×D10711yescharacteristic
161C22×Q8C2×D10711yescharacteristic
162#137—24no—
163#135—2144no—
164C2×Q16—1071no—
165C2×Q16C2×D10712yes—
166C2×SD16—1071no—
167C2×SD16—1071no—
168C2×SD16C2×D10712yes—
169C2×D8—1071no—
170C2×M16C2×D10711yescharacteristic
171C22×C8—1071no—
172C4⋊Q8—1072no—
173#124—1072no—
174#123—2142no—
175C22⋊Q8—2141no—
176C22⋊Q8—2141no—
177C4×Q8—1072no—
178C4×D4—1072no—
179C4×D4—2142no—
180#117—1071no—
181#117—2141no—
182C2×C4⋊C4—1071no—
183C2×C4⋊C4—2141no—
184C4⋊C8—1072no—
185C4⋊C8—1072no—
186#103—1072no—
187#103—1072no—
188#103—1072no—
189#102—1072no—
190#41—22no—
191#41—2142no—
192#41—2142no—
193#41C22⋊D10714yes—
194C2×Q8—21no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—1071no—
198C2×Q8C22×D10712yes—
199C2×Q8C22×D10711yescharacteristic
200C2×D4—21no—
201C2×D4—1071no—
202C2×D4—2141no—
203C2×D4C22×D10711yescharacteristic
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—2142no—
207C22×C4—2141no—
208C22×C4—2141no—
209C22×C4C22×D10711yescharacteristic
210Q16—24no—
211Q16—2142no—
212SD16—24no—
213SD16—2142no—
214SD16—2142no—
215D8—2142no—
216M16—22no—
217C2×C8—1071no—
218C2×C8—1071no—
219C2×C8—2142no—
220C2×C8C22×D10712yes—
221C4⋊C4—1071no—
222C4⋊C4—1071no—
223C4⋊C4—1072no—
224C4⋊C4—2142no—
225C4⋊C4—2142no—
226C4⋊C4—2142no—
227C4⋊C4—2141no—
228C22⋊C4—1072no—
229C22⋊C4—2142no—
230C22⋊C4—2142no—
231C22⋊C4—2142no—
232C42—1072no—
233C42—2142no—
234C23—21no—
235C23—2141no—
236C23D4×D10711yescharacteristic
237Q8—21no—
238Q8—22no—
239Q8—22no—
240Q8—42no—
241Q8—2141no—
242Q8—2141no—
243Q8C2×C22⋊D10712yes—
244D4—24no—
245D4—21no—
246D4—4282no—
247D4—2141no—
248D4—2141no—
249D4C2×C22⋊D10712yes—
250C2×C4—21no—
251C2×C4—24no—
252C2×C4—22no—
253C2×C4—42no—
254C2×C4—8561no—
255C2×C4—8562no—
256C2×C4—1072no—
257C2×C4—1072no—
258C2×C4—4282no—
259C2×C4—2142no—
260C2×C4—2141no—
261C2×C4D4×D10711yescharacteristic
262C2×C4C2×C22⋊D10712yes—
263C2×C4#1313381712yes—
264C2×C4C23×D10711yescharacteristic
265C8—22no—
266C8—2141no—
267C8—2141no—
268C22—21no—
269C22—21no—
270C22—22no—
271C22—41no—
272C22—41no—
273C22—4281no—
274C22—4281no—
275C22—4281no—
276C22#1681208112yes—
277C22C2×D4×D10711yescharacteristic
278C4—22no—
279C4—42no—
280C4—4282no—
281C4—4281no—
282C4—2142no—
283C4—2142no—
284C4#1681208112yes—
285C4C22×C22⋊D10711yescharacteristic
286C4C2×#1313381711yescharacteristic
287C2—22no—
288C2—42no—
289C2—8561no—
290C2C2×#1681208111yescharacteristic
291C2#2029343611yescharacteristic
292C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#137×C107C2
socleC22×C107C22×D4
prime radicalC2×#16812129C2
hypercenterC2×#137D107
hypocenterC107#3046
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. #20293201C2semidirect
3. #20293110C2semidirect
4. #20292851C2semidirect
5. #20292844C2semidirect
6. #20292835C2semidirect
7. #20292830C2semidirect
8. #20292655C2semidirect
9. #20292629C2semidirect
10. C2×#16812033C2semidirect
11. C2×#137×C107C2semidirect
147. C107#3046semidirect
148. C2×#137D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121292
#202932012
#202931102
#202928512
#202928442
#202928352
#202928302
#202926552
#202926292
C2×#168120332
C2×#137×C1072
#3046107