Finite Group Databasebeta

← #28346016

Subgroups — #28346016

Order 13696.

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

normal subgroups only

This group has 472 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×#16812140C211yesmaximal, characteristic, complemented
2C2×#16812131C211yesmaximal, characteristic
3#20293201C211yesmaximal, characteristic, complemented
4#20293108C211yesmaximal, characteristic, complemented
5#20292852C212yesmaximal, complemented
6#20292845C212yesmaximal, complemented
7#20292836C212yesmaximal, complemented
8#20292831C212yesmaximal, complemented
9#20292653C211yesmaximal, characteristic, complemented
10#20292626C211yesmaximal, characteristic, complemented
11C2×#103×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C2×#13133818C2211yescharacteristic
14#16812140—24no—
15#16812131—24no—
16C2×Dic428C2211yescharacteristic
17C2×#13133808C2212yes—
18C2×#13133807C2211yescharacteristic
19C2×#13133807C2211yescharacteristic
20C2×#13133805C2212yes—
21C2×D856C2211yescharacteristic
22C2×#13133802C2211yescharacteristic
23#16812107C2212yes—
24#16812098C2212yes—
25#16812092—22no—
26#16812085—21no—
27#16812085—21no—
28#16812074C2212yes—
29#16812071C2212yes—
30C4×C22⋊D107—22no—
31#16812068C2211yescharacteristic
32#16812067—21no—
33C4⋊C4×D107—21no—
34Dic107⋊C8C2212yes—
35Dic107⋊C8C2212yes—
36#16812032C2212yes—
37#16812029C2212yes—
38C22×Q8×C107C2211yescharacteristic
39C22×C8×C107C2211yescharacteristic
40C2×C4⋊C4×C107C2211yescharacteristic
41#103×C107C2212yes—
42#103×C107C2212yes—
43#13133818—22no—
44#13133818—22no—
45#13133818—22no—
46#13133818D414yes—
47C2×Dic214C2311yescharacteristic
48C2×Dic214C2311yescharacteristic
49C2×D428C2311yescharacteristic
50C2×D428C2311yescharacteristic
51C2×C22⋊D107—21no—
52C2×C22⋊D107—21no—
53C2×C4×D107—21no—
54C2×C4×D107—21no—
55C2×C4×D107—22no—
56Dic428—22no—
57#13133808—24no—
58#13133807—22no—
59#13133807—22no—
60#13133805—24no—
61D856—22no—
62#13133802—22no—
63C2×C107⋊C8C2312yes—
64C4⋊Dic107D411yescharacteristic
65C4⋊Dic107D411yescharacteristic
66Dic107⋊C4—22no—
67Dic107⋊C4—22no—
68Dic107⋊C4—22no—
69C22⋊Dic107D412yes—
70(C2×D107)⋊C4—22no—
71(C2×D107)⋊C4—22no—
72(C2×D107)⋊C4—22no—
73C4×Dic107—22no—
74C4×Dic107C2312yes—
75C2×Q8×C107—21no—
76C2×Q8×C107—22no—
77C2×Q8×C107—22no—
78C2×Q8×C107C2312yes—
79C22×C4×C107—21no—
80C22×C4×C107—21no—
81C22×C4×C107C2311yescharacteristic
82C2×C8×C107—22no—
83C2×C8×C107C2311yescharacteristic
84C2×C8×C107C2311yescharacteristic
85C4⋊C4×C107—21no—
86C4⋊C4×C107C2312yes—
87C23×C107C2×D411yescharacteristic
88C22×D107—21no—
89C22×D107—21no—
90Q8×C107—22no—
91Q8×C107—22no—
92Q8×C107—42no—
93C2×C4×C107—22no—
94C2×C4×C107—22no—
95C2×C4×C107—42no—
96C2×C4×C107—81no—
97C2×C4×C107C2×D411yescharacteristic
98C2×C4×C107C2×D412yes—
99C2×C4×C107#4112yes—
100C2×C4×C107C2411yescharacteristic
101C22⋊D107—24no—
102C22⋊D107—42no—
103C2×Dic107—21no—
104C2×Dic107—21no—
105C2×Dic107C2×D412yes—
106C2×Dic107C2×D412yes—
107D428—21no—
108D428—21no—
109D428—21no—
110D428C2×D412yes—
111C4×D107—24no—
112C4×D107—42no—
113C4×D107—82no—
114Dic214—21no—
115Dic214—21no—
116Dic214—21no—
117Dic214C2×D412yes—
118C8×C107—21no—
119C8×C107—21no—
120C107⋊C8—22no—
121C22×C107—21no—
122C22×C107—21no—
123C22×C107#12112yes—
124C22×C107C22×D411yescharacteristic
125C2×D107—22no—
126C2×D107—41no—
127C2×D107—41no—
128C2×D107—41no—
129C2×D107—41no—
130C2×D107—41no—
131C4×C107—42no—
132C4×C107—42no—
133C4×C107#12112yes—
134C4×C107C22×D411yescharacteristic
135C4×C107C2×#4111yescharacteristic
136Dic107—22no—
137Dic107—22no—
138Dic107—22no—
139Dic107—41no—
140C2×C107—22no—
141C2×C107C2×#12111yescharacteristic
142C2×C107#57711yescharacteristic
143C2×C107#57811yescharacteristic
144D107—42no—
145D107—81no—
146#3046—1071nomaximal, Sylow
147C107#304611yescharacteristic, complemented, Sylow
148C2×#137—1071no—
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×#103D10711yescharacteristic, complemented
159C2×#41—1071no—
160C2×#41—1071no—
161C22×Q8C2×D10711yescharacteristic
162#137—2144no—
163#135—2144no—
164C2×Q16—1072no—
165C2×Q16—1071no—
166C2×SD16—1072no—
167C2×SD16—1071no—
168C2×SD16—1071no—
169C2×D8—1071no—
170C2×M16—1071no—
171C22×C8C2×D10711yescharacteristic
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—2141no—
183C2×C4⋊C4C2×D10711yescharacteristic
184C4⋊C8—1072no—
185C4⋊C8—1072no—
186#103—1072no—
187#103C2×D10712yes—
188#103C2×D10712yes—
189#102—1072no—
190#41—1074no—
191#41—2142no—
192#41—2142no—
193#41—2142no—
194C2×Q8—21no—
195C2×Q8—22no—
196C2×Q8—22no—
197C2×Q8—1071no—
198C2×Q8—1071no—
199C2×Q8C22×D10712yes—
200C2×D4—1071no—
201C2×D4—1071no—
202C2×D4—2141no—
203C2×D4—2141no—
204C22×C4—21no—
205C22×C4—21no—
206C22×C4—2142no—
207C22×C4—2141no—
208C22×C4—2141no—
209C22×C4C22×D10711yescharacteristic
210Q16—2142no—
211Q16—2144no—
212SD16—2142no—
213SD16—2142no—
214SD16—2144no—
215D8—2142no—
216M16—2142no—
217C2×C8—22no—
218C2×C8—1072no—
219C2×C8C22×D10711yescharacteristic
220C2×C8C22×D10711yescharacteristic
221C4⋊C4—21no—
222C4⋊C4—1071no—
223C4⋊C4—1071no—
224C4⋊C4—2142no—
225C4⋊C4—2142no—
226C4⋊C4—2142no—
227C4⋊C4C22×D10712yes—
228C22⋊C4—1072no—
229C22⋊C4—2142no—
230C22⋊C4—2142no—
231C22⋊C4—2142no—
232C42—1072no—
233C42—2142no—
234C23—2141no—
235C23—2141no—
236C23D4×D10711yescharacteristic
237Q8—22no—
238Q8—22no—
239Q8—42no—
240Q8—1072no—
241Q8—2141no—
242Q8—2141no—
243Q8—2141no—
244D4—1072no—
245D4—4282no—
246D4—2144no—
247D4—2141no—
248D4—2141no—
249D4—2141no—
250C2×C4—22no—
251C2×C4—22no—
252C2×C4—42no—
253C2×C4—81no—
254C2×C4—8562no—
255C2×C4—1072no—
256C2×C4—1072no—
257C2×C4—4282no—
258C2×C4—2141no—
259C2×C4—2141no—
260C2×C4—2144no—
261C2×C4D4×D10711yescharacteristic
262C2×C4D4×D10712yes—
263C2×C4#1313381812yes—
264C2×C4C23×D10711yescharacteristic
265C8—21no—
266C8—21no—
267C8—2142no—
268C22—21no—
269C22—21no—
270C22—4281no—
271C22—4281no—
272C22—4281no—
273C22—4281no—
274C22—4281no—
275C22—2142no—
276C22#1681208212yes—
277C22C2×D4×D10711yescharacteristic
278C4—42no—
279C4—42no—
280C4—4281no—
281C4—2142no—
282C4—2142no—
283C4—2142no—
284C4#1681208212yes—
285C4C2×D4×D10711yescharacteristic
286C4C2×#1313381811yescharacteristic
287C2—22no—
288C2—8561no—
289C2—4282no—
290C2C2×#1681208211yescharacteristic
291C2#2029343311yescharacteristic
292C2#2029344011yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#103×C107C2
socleC22×C107C22×D4
prime radicalC2×#16812131C2
hypercenterC2×#103D107
hypocenterC107#3046
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#103

Complements

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

NH ≅ G/Ntype
1. C2×#16812140C2semidirect
3. #20293201C2semidirect
4. #20293108C2semidirect
5. #20292852C2semidirect
6. #20292845C2semidirect
7. #20292836C2semidirect
8. #20292831C2semidirect
9. #20292653C2semidirect
10. #20292626C2semidirect
11. C2×#103×C107C2semidirect
147. C107#3046semidirect
158. C2×#103D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121402
C2×#168121312
#202932012
#202931082
#202928522
#202928452
#202928362
#202928312
#202926532
#202926262
C2×#103×C1072
#3046107