Finite Group Databasebeta

← #28345943

Subgroups — #28345943

Order 13696.

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

normal subgroups only

This group has 480 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
1C2×#16812138C212yesmaximal
2C4⋊Q8×D107C211yesmaximal, characteristic, complemented
3C4×#13133817C211yesmaximal, characteristic
4#20292838C211yesmaximal, characteristic, complemented
5#20292836C212yesmaximal, complemented
6#20292834C211yesmaximal, characteristic, complemented
7#20292832C211yesmaximal, characteristic, complemented
8#20292830C212yesmaximal, complemented
9#20292649C211yesmaximal, characteristic, complemented
10#20292634C212yesmaximal, complemented
11#461×C107C211yesmaximal, characteristic, complemented
12C2×#13133817C2211yescharacteristic
13C2×Q8×D107C2212yes—
14#16812138—28no—
15C2×Dic428C2212yes—
16C2×#13133808C2212yes—
17C2×#13133806C2212yes—
18C2×#13133801C2212yes—
19#16812109C2211yescharacteristic
20#16812108—21no—
21#16812108—21no—
22#16812107C2212yes—
23C4×Dic214C2211yescharacteristic
24#16812074C2211yescharacteristic
25D4×Dic107C2211yescharacteristic
26C4×C22⋊D107—21no—
27(C22⋊D107)⋊C4—21no—
28#16812067C2211yescharacteristic
29#16812065—21no—
30C4⋊C4×D107—21no—
31C4⋊C4×D107—21no—
32C2×C4×Dic107—21no—
33C42×D107C2211yescharacteristic
34C4⋊(C107⋊C8)C2211yescharacteristic
35Dic107⋊C8C2212yes—
36#16812034C2212yes—
37#16812033C2212yes—
38#16812032C2212yes—
39C2×SD16×C107C2212yes—
40C4⋊Q8×C107C2211yescharacteristic
41C4×D4×C107C2211yescharacteristic
42C4⋊C8×C107C2211yescharacteristic
43#103×C107C2212yes—
44#13133817—22no—
45#13133817D414yes—
46C2×Dic214—22no—
47C2×Dic214C2312yes—
48C2×Dic214C2311yescharacteristic
49Q8×D107—24no—
50Q8×D107—24no—
51C2×C22⋊D107—21no—
52C22×Dic107—21no—
53C22×Dic107—21no—
54C2×C4×D107—22no—
55C2×C4×D107D411yescharacteristic
56C2×C4×D107D411yescharacteristic
57C2×C4×D107C2311yescharacteristic
58Dic428—24no—
59#13133808—24no—
60#13133806—24no—
61#13133801—24no—
62C2×C107⋊C8C2312yes—
63C4⋊Dic107—21no—
64C4⋊Dic107—21no—
65C4⋊Dic107C2311yescharacteristic
66Dic107⋊C4—21no—
67Dic107⋊C4—22no—
68Dic107⋊C4—22no—
69Dic107⋊C4—21no—
70C22⋊Dic107—21no—
71(C2×D107)⋊C4—21no—
72C4×Dic107—21no—
73C4×Dic107—21no—
74C4×Dic107—21no—
75C4×Dic107D411yescharacteristic
76C4×Dic107D411yescharacteristic
77C4×Dic107C2311yescharacteristic
78C2×Q8×C107C2312yes—
79C2×D4×C107C2311yescharacteristic
80C22×C4×C107—21no—
81SD16×C107—24no—
82C2×C8×C107C2312yes—
83C4⋊C4×C107—21no—
84C4⋊C4×C107—21no—
85C4⋊C4×C107C2311yescharacteristic
86C22⋊C4×C107—21no—
87C42×C107C2311yescharacteristic
88C23×C107—21no—
89C22×D107C2×D411yescharacteristic
90Q8×C107—22no—
91Q8×C107—22no—
92D4×C107—21no—
93D4×C107C2×D412yes—
94C2×C4×C107—21no—
95C2×C4×C107—22no—
96C2×C4×C107—42no—
97C2×C4×C107C2×D411yescharacteristic
98C2×C4×C107C2×D411yescharacteristic
99C2×C4×C107C2411yescharacteristic
100C22⋊D107—24no—
101C2×Dic107—21no—
102C2×Dic107—24no—
103C2×Dic107—21no—
104C2×Dic107—22no—
105C2×Dic107—42no—
106C2×Dic107C2×D411yescharacteristic
107C2×Dic107C2×D411yescharacteristic
108C2×Dic107C2×D411yescharacteristic
109C4×D107—22no—
110C4×D107—22no—
111C4×D107—44no—
112C4×D107C2×D412yes—
113C4×D107#4112yes—
114Dic214—21no—
115Dic214—22no—
116Dic214—22no—
117Dic214—44no—
118Dic214C2×D412yes—
119C8×C107—22no—
120C107⋊C8—22no—
121C22×C107—22no—
122C22×C107—41no—
123C22×C107—41no—
124C22×C107C22×D411yescharacteristic
125C2×D107—21no—
126C2×D107—21no—
127C2×D107#12112yes—
128C4×C107—21no—
129C4×C107—42no—
130C4×C107—41no—
131C4×C107#13712yescomplemented
132C4×C107C22×D411yescharacteristic
133C4×C107C2×#4111yescharacteristic
134Dic107—22no—
135Dic107—21no—
136Dic107—42no—
137Dic107—41no—
138Dic107#12112yes—
139Dic107#13712yescomplemented
140C2×C107—42no—
141C2×C107C2×#12111yescharacteristic
142C2×C107C2×#13711yescharacteristic
143C2×C107C2×#13711yescharacteristic
144D107—22no—
145#3039—1071nomaximal, Sylow
146C107#303911yescharacteristic, complemented, Sylow
147C2×#137—1072no—
148C2×C4⋊Q8—1071no—
149C4×#41—1071no—
150#462—1071no—
151#462—1071no—
152#462—1072no—
153#461—1072no—
154#461—1071no—
155#461D10711yescharacteristic, complemented
156#423—1071no—
157#419—1072no—
158C2×#41—1071no—
159C22×Q8—1072no—
160#137—2148no—
161C2×Q16—1072no—
162C2×Q16—1072no—
163C2×SD16—1072no—
164C2×SD16C2×D10712yes—
165C2×M16—1072no—
166C4⋊Q8—1072no—
167C4⋊Q8—1071no—
168C4⋊Q8—2141no—
169C4⋊Q8—2141no—
170C4⋊Q8C2×D10711yescharacteristic
171C4×Q8—1071no—
172C4×Q8—1071no—
173C4×D4—1071no—
174C4×D4—2141no—
175C4×D4—2141no—
176C4×D4C2×D10711yescharacteristic
177#117—1071no—
178#117—2141no—
179C2×C4⋊C4—2141no—
180C2×C4⋊C4—2141no—
181C2×C42—1071no—
182C2×C42—2141no—
183C4⋊C8—1072no—
184C4⋊C8—1071no—
185C4⋊C8C2×D10711yescharacteristic
186#103—1072no—
187#103—1072no—
188#103—1072no—
189#103C2×D10712yes—
190#41—1074no—
191#41—2142no—
192C2×Q8—1071no—
193C2×Q8—1072no—
194C2×Q8—2142no—
195C2×Q8—2144no—
196C2×Q8—2144no—
197C2×Q8C22×D10712yes—
198C2×D4—2141no—
199C2×D4C22×D10711yescharacteristic
200C22×C4—21no—
201C22×C4—1071no—
202C22×C4—1071no—
203C22×C4—1071no—
204C22×C4—2142no—
205C22×C4—2141no—
206C22×C4—2141no—
207Q16—2144no—
208Q16—2144no—
209SD16—24no—
210SD16—2144no—
211M16—2144no—
212C2×C8—1072no—
213C2×C8C22×D10712yes—
214C4⋊C4—21no—
215C4⋊C4—21no—
216C4⋊C4—1071no—
217C4⋊C4—2142no—
218C4⋊C4—2142no—
219C4⋊C4—2141no—
220C4⋊C4—2141no—
221C4⋊C4—2141no—
222C4⋊C4—2141no—
223C4⋊C4C22×D10711yescharacteristic
224C22⋊C4—21no—
225C22⋊C4—2141no—
226C22⋊C4—2141no—
227C42—1071no—
228C42—1071no—
229C42—1071no—
230C42—2141no—
231C42—2141no—
232C42—2141no—
233C42C22×D10711yescharacteristic
234C23—21no—
235C23—1071no—
236Q8—22no—
237Q8—22no—
238Q8—1072no—
239Q8—4284no—
240Q8—2142no—
241Q8—2142no—
242Q8—2141no—
243D4—21no—
244D4—2144no—
245D4D4×D10712yes—
246C2×C4—21no—
247C2×C4—22no—
248C2×C4—42no—
249C2×C4—1071no—
250C2×C4—1071no—
251C2×C4—1072no—
252C2×C4—1072no—
253C2×C4—1071no—
254C2×C4—4284no—
255C2×C4—4282no—
256C2×C4—2142no—
257C2×C4—2144no—
258C2×C4—2141no—
259C2×C4—2142no—
260C2×C4—2141no—
261C2×C4—2142no—
262C2×C4D4×D10711yescharacteristic
263C2×C4D4×D10711yescharacteristic
264C2×C4C23×D10711yescharacteristic
265C8—22no—
266C8—2142no—
267C22—22no—
268C22—41no—
269C22—41no—
270C22—1072no—
271C22—2141no—
272C22—2141no—
273C22C2×D4×D10711yescharacteristic
274C4—21no—
275C4—41no—
276C4—42no—
277C4—1072no—
278C4—1072no—
279C4—4281no—
280C4—4282no—
281C4—2141no—
282C4—2142no—
283C4#1681213812yescomplemented
284C4C2×D4×D10711yescharacteristic
285C4#41×D10711yescharacteristic
286C2—42no—
287C2—2142no—
288C2#121×D10711yescharacteristic
289C2#137×D10711yescharacteristic
290C2C2×#1681213811yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#461×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133817C22
hypercenter#461D107
hypocenterC107#3039
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C42 ◁ #461

Complements

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

NH ≅ G/Ntype
2. C4⋊Q8×D107C2semidirect
4. #20292838C2semidirect
5. #20292836C2semidirect
6. #20292834C2semidirect
7. #20292832C2semidirect
8. #20292830C2semidirect
9. #20292649C2semidirect
10. #20292634C2semidirect
11. #461×C107C2semidirect
131. C4×C107#137semidirect
139. Dic107#137semidirect
146. C107#3039semidirect
155. #461D107semidirect
283. C4#16812138semidirect

Maximal subgroups

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

subgroupindex
C2×#168121382
C4⋊Q8×D1072
C4×#131338172
#202928382
#202928362
#202928342
#202928322
#202928302
#202926492
#202926342
#461×C1072
#3039107