Finite Group Databasebeta

← #28345932

Subgroups — #28345932

Order 13696.

484 conjugacy classes of subgroups; only the 135 normal subgroups are shown.

normal subgroups only

This group has 484 conjugacy classes of subgroups — too many to draw. Only its 135 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×#16812142C211yesmaximal, characteristic, complemented
2C2×#16812139C211yesmaximal, characteristic, complemented
3#20293173C211yesmaximal, characteristic
4C2×#16812074C211yesmaximal, characteristic, complemented
5#20292852C212yesmaximal, complemented
6#20292848C212yesmaximal, complemented
7#20292836C212yesmaximal, complemented
8#20292826C212yesmaximal, complemented
9#20292650C211yesmaximal, characteristic, complemented
10#20292626C211yesmaximal, characteristic, complemented
11#417×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C22×Dic214C2211yescharacteristic
14#16812142—24no—
15#16812139—24no—
16C2×Dic428C2212yes—
17C2×#13133808C2212yes—
18C2×#13133807C2212yes—
19C2×#13133806C2212yes—
20C2×#13133802C2211yescharacteristic
21#16812107C2211yescharacteristic
22#16812105C2211yescharacteristic
23#16812098C2211yescharacteristic
24#16812096C2211yescharacteristic
25#16812081—21no—
26#16812081—21no—
27#16812074—21no—
28#16812074—21no—
29#16812074C2212yes—
30#16812074C2212yes—
31C4×C22⋊D107—21no—
32C4×C22⋊D107—21no—
33C2×Dic107⋊C4—21no—
34C2×C4×Dic107—21no—
35C2×C4×Dic107C2211yescharacteristic
36Dic107⋊C8C2212yes—
37Dic107⋊C8C2212yes—
38#16812032C2212yes—
39#16812029C2212yes—
40C2×#41×C107C2211yescharacteristic
41C2×M16×C107C2211yescharacteristic
42C2×C4⋊C4×C107C2211yescharacteristic
43#103×C107C2212yes—
44#102×C107C2212yes—
45#13133818—22no—
46#13133818—22no—
47C2×Dic214—21no—
48C2×Dic214—22no—
49C2×Dic214D414yescomplemented
50C2×Dic214C2311yescharacteristic
51C2×Dic214C2312yes—
52C2×D428C2311yescharacteristic
53C2×C22⋊D107—21no—
54C2×C22⋊D107—21no—
55C2×C22⋊D107—21no—
56C22×Dic107—21no—
57C22×Dic107D411yescharacteristic
58C22×Dic107D411yescharacteristic
59C2×C4×D107—21no—
60Dic428—24no—
61#13133808—24no—
62#13133807—24no—
63#13133806—24no—
64#13133802—22no—
65C2×C107⋊C8C2312yes—
66Dic107⋊C4—22no—
67Dic107⋊C4—22no—
68Dic107⋊C4—21no—
69Dic107⋊C4—21no—
70C22⋊Dic107—21no—
71C22⋊Dic107—21no—
72(C2×D107)⋊C4—21no—
73(C2×D107)⋊C4—21no—
74C4×Dic107—22no—
75C4×Dic107—22no—
76C4×Dic107D411yescharacteristic
77C4×Dic107D411yescharacteristic
78C4×Dic107C2311yescharacteristic
79C4×Dic107C2311yescharacteristic
80#41×C107—22no—
81#41×C107—22no—
82C2×Q8×C107C2311yescharacteristic
83C2×D4×C107—21no—
84C2×D4×C107C2311yescharacteristic
85C22×C4×C107—21no—
86C22×C4×C107—21no—
87C22×C4×C107C2311yescharacteristic
88M16×C107—22no—
89C2×C8×C107C2312yes—
90C4⋊C4×C107—21no—
91C4⋊C4×C107C2312yes—
92C23×C107—21no—
93C23×C107C2×D411yescharacteristic
94C22×D107—21no—
95Q8×C107—21no—
96Q8×C107—21no—
97D4×C107—21no—
98D4×C107—21no—
99D4×C107—42no—
100C2×C4×C107—22no—
101C2×C4×C107—21no—
102C2×C4×C107—42no—
103C2×C4×C107—42no—
104C2×C4×C107C2×D411yescharacteristic
105C2×C4×C107C2×D412yes—
106C2×C4×C107#4112yes—
107C2×C4×C107C2411yescharacteristic
108C22⋊D107—42no—
109C22⋊D107—42no—
110C22⋊D107—42no—
111C2×Dic107—24no—
112C2×Dic107—22no—
113C2×Dic107—22no—
114C2×Dic107—21no—
115C2×Dic107—41no—
116C2×Dic107C2×D411yescharacteristic
117C2×Dic107C2×D411yescharacteristic
118C2×Dic107C2×D411yescharacteristic
119C2×Dic107C2×D411yescharacteristic
120D428—21no—
121D428—21no—
122C4×D107—42no—
123Dic214—24no—
124Dic214—22no—
125Dic214—21no—
126Dic214—21no—
127Dic214C2×D414yes—
128C8×C107—22no—
129C107⋊C8—22no—
130C22×C107—21no—
131C22×C107—21no—
132C22×C107—41no—
133C22×C107—41no—
134C22×C107—41no—
135C22×C107#12112yes—
136C22×C107C22×D411yescharacteristic
137C2×D107—41no—
138C2×D107—41no—
139C2×D107—41no—
140C4×C107—41no—
141C4×C107—42no—
142C4×C107#12112yes—
143C4×C107C22×D411yescharacteristic
144C4×C107C2×#4111yescharacteristic
145Dic107—24no—
146Dic107—21no—
147Dic107—21no—
148Dic107—21no—
149Dic107—41no—
150Dic107#13712yescomplemented
151C2×C107—22no—
152C2×C107—81no—
153C2×C107C2×#12111yescharacteristic
154C2×C107C2×#13711yescharacteristic
155C2×C107#57511yescharacteristic
156D107—81no—
157#3037—1071nomaximal, Sylow
158C107#303711yescharacteristic, complemented, Sylow
159C2×#137—1071no—
160C2×#137—1071no—
161#532—1071no—
162C2×C4×Q8—1071no—
163#464—1072no—
164#464—1072no—
165#462—1072no—
166#460—1072no—
167#423—1071no—
168#417—1071no—
169#417D10711yescharacteristic, complemented
170C2×#41—1071no—
171C2×#41C2×D10711yescharacteristic
172C22×Q8—1071no—
173#137—2144no—
174#137—2144no—
175C2×Q16—1072no—
176C2×Q16—1072no—
177C2×SD16—1072no—
178C2×SD16—1072no—
179C2×M16—1071no—
180C2×M16C2×D10711yescharacteristic
181C4⋊Q8—1071no—
182#127—1071no—
183#124—1071no—
184#124—1071no—
185#121—2141no—
186#121—2141no—
187C4×Q8—1072no—
188C4×Q8—1072no—
189C4×Q8—2141no—
190C4×Q8—2141no—
191C4×D4—2141no—
192C4×D4—2141no—
193C2×C4⋊C4—2141no—
194C2×C4⋊C4C2×D10711yescharacteristic
195C2×C42—1071no—
196C2×C42—2141no—
197C4⋊C8—1072no—
198C4⋊C8—1072no—
199#103—1072no—
200#103C2×D10712yes—
201#102—1072no—
202#102C2×D10712yes—
203#41—22no—
204#41—22no—
205#41—2142no—
206#41—2142no—
207C2×Q8—1074no—
208C2×Q8—1072no—
209C2×Q8—1071no—
210C2×Q8—2141no—
211C2×Q8—2142no—
212C2×Q8C22×D10711yescharacteristic
213C2×D4—21no—
214C2×D4—1071no—
215C2×D4—2141no—
216C2×D4—2141no—
217C2×D4—2141no—
218C2×D4C22×D10711yescharacteristic
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—1071no—
222C22×C4—1071no—
223C22×C4—2141no—
224C22×C4—2141no—
225C22×C4C22×D10711yescharacteristic
226Q16—2144no—
227Q16—2144no—
228SD16—2144no—
229SD16—2144no—
230M16—22no—
231M16—2142no—
232C2×C8—1072no—
233C2×C8C22×D10712yes—
234C4⋊C4—21no—
235C4⋊C4—2142no—
236C4⋊C4—2142no—
237C4⋊C4—2141no—
238C4⋊C4—2141no—
239C4⋊C4C22×D10712yes—
240C22⋊C4—2141no—
241C22⋊C4—2141no—
242C22⋊C4—2141no—
243C22⋊C4—2141no—
244C42—1071no—
245C42—1071no—
246C42—1071no—
247C42—1071no—
248C42—2142no—
249C42—2142no—
250C23—21no—
251C23—2141no—
252C23D4×D10711yescharacteristic
253Q8—21no—
254Q8—21no—
255Q8—1074no—
256Q8—2141no—
257Q8—2141no—
258Q8—2144no—
259Q8—2142no—
260D4—21no—
261D4—21no—
262D4—42no—
263D4—4282no—
264D4—4282no—
265D4—4282no—
266D4—2141no—
267D4—2141no—
268C2×C4—22no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—1071no—
273C2×C4—1071no—
274C2×C4—1071no—
275C2×C4—1071no—
276C2×C4—4282no—
277C2×C4—4281no—
278C2×C4—2142no—
279C2×C4—2141no—
280C2×C4—2144no—
281C2×C4—2142no—
282C2×C4D4×D10711yescharacteristic
283C2×C4D4×D10712yes—
284C2×C4#1313381812yes—
285C2×C4C23×D10711yescharacteristic
286C8—22no—
287C8—2142no—
288C22—21no—
289C22—21no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22—4281no—
294C22—4281no—
295C22—4281no—
296C22#1681208212yes—
297C22C2×D4×D10711yescharacteristic
298C4—42no—
299C4—41no—
300C4—1072no—
301C4—4281no—
302C4—2141no—
303C4—2141no—
304C4—2141no—
305C4—2144no—
306C4#1681208212yes—
307C4C2×D4×D10711yescharacteristic
308C4C2×#1313381811yescharacteristic
309C2—22no—
310C2—81no—
311C2—8561no—
312C2C2×#1681208211yescharacteristic
313C2#137×D10711yescharacteristic
314C2#2029342311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#417×C107C2
socleC22×C107C22×D4
prime radical#20293173C2
hypercenter#417D107
hypocenterC107#3037
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #417

Complements

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

NH ≅ G/Ntype
1. C2×#16812142C2semidirect
2. C2×#16812139C2semidirect
4. C2×#16812074C2semidirect
5. #20292852C2semidirect
6. #20292848C2semidirect
7. #20292836C2semidirect
8. #20292826C2semidirect
9. #20292650C2semidirect
10. #20292626C2semidirect
11. #417×C107C2semidirect
49. C2×Dic214D4semidirect
150. Dic107#137semidirect
158. C107#3037semidirect
169. #417D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121422
C2×#168121392
#202931732
C2×#168120742
#202928522
#202928482
#202928362
#202928262
#202926502
#202926262
#417×C1072
#3037107