Finite Group Databasebeta

← #28346822

Subgroups — #28346822

Order 13696.

372 conjugacy classes of subgroups; only the 125 normal subgroups are shown.

normal subgroups only

This group has 372 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
1#20293255C211yesmaximal, characteristic, complemented
2#20293201C211yesmaximal, characteristic
3#20292994C212yesmaximal, complemented
4#20292988C212yesmaximal, complemented
5#20292983C212yesmaximal, complemented
6#20292976C212yesmaximal, complemented
7#20292629C211yesmaximal, characteristic
8#20292621C211yesmaximal, characteristic
9C2×#16812033C211yesmaximal, characteristic, complemented
10#20292585C211yesmaximal, characteristic, complemented
11#419×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C22×C107⋊C8C2211yescharacteristic
14#16812109—21no—
15#16812108C2212yes—
16#16812107C2212yes—
17#16812101—21no—
18#16812100C2212yes—
19#16812098C2212yes—
20#16812092—22no—
21#16812086—22no—
22#16812085—21no—
23#16812085—21no—
24#16812068C2211yescharacteristic
25C2×C4⋊Dic107C2211yescharacteristic
26#16812034C2212yes—
27#16812033C2212yes—
28#16812033C2212yes—
29#16812032C2211yescharacteristic
30#16812032C2211yescharacteristic
31#16812031C2212yes—
32#16812029C2211yescharacteristic
33#16812029C2211yescharacteristic
34#16812019C2212yes—
35C8×Dic107C2212yes—
36C22×Q8×C107C2211yescharacteristic
37C2×M16×C107C2211yescharacteristic
38#117×C107C2211yescharacteristic
39#103×C107C2212yes—
40#103×C107C2212yes—
41#13133818—22no—
42#13133818—22no—
43C2×Dic214—22no—
44C2×Dic214C2311yescharacteristic
45C2×D428C2311yescharacteristic
46C2×C22⋊D107—21no—
47C22×Dic107—21no—
48C2×C4×D107—21no—
49C2×C107⋊C8—22no—
50C2×C107⋊C8C2311yescharacteristic
51C2×C107⋊C8C2311yescharacteristic
52C4⋊Dic107—21no—
53C4⋊Dic107—22no—
54C4⋊Dic107D411yescharacteristic
55C4⋊Dic107D411yescharacteristic
56C4⋊Dic107C2312yes—
57Dic107⋊C4—22no—
58Dic107⋊C4—22no—
59Dic107⋊C4—22no—
60Dic107⋊C4—22no—
61C22⋊Dic107D412yes—
62(C2×D107)⋊C4—22no—
63(C2×D107)⋊C4—22no—
64C4×Dic107C2312yes—
65C2×Q8×C107—21no—
66C2×Q8×C107—22no—
67C2×Q8×C107—22no—
68C2×Q8×C107C2312yes—
69C22×C4×C107—21no—
70C22×C4×C107C2311yescharacteristic
71M16×C107—22no—
72C2×C8×C107C2312yes—
73C4⋊C4×C107C2311yescharacteristic
74C4⋊C4×C107C2311yescharacteristic
75C22⋊C4×C107—21no—
76C42×C107—21no—
77C23×C107C2×D411yescharacteristic
78C22×D107—21no—
79Q8×C107—22no—
80Q8×C107—22no—
81Q8×C107—42no—
82C2×C4×C107—21no—
83C2×C4×C107—21no—
84C2×C4×C107—22no—
85C2×C4×C107—42no—
86C2×C4×C107C2×D411yescharacteristic
87C2×C4×C107#4112yes—
88C2×C4×C107#4112yes—
89C2×C4×C107C2411yescharacteristic
90C22⋊D107—42no—
91C2×Dic107—21no—
92C2×Dic107—22no—
93C2×Dic107—81no—
94C2×Dic107C2×D412yes—
95C2×Dic107C2×D412yes—
96D428—21no—
97D428—21no—
98C4×D107—42no—
99Dic214—21no—
100Dic214—21no—
101Dic214—82no—
102C8×C107—22no—
103C107⋊C8—21no—
104C107⋊C8—21no—
105C22×C107—21no—
106C22×C107—21no—
107C22×C107#12412yes—
108C22×C107C22×D411yescharacteristic
109C2×D107—41no—
110C2×D107—41no—
111C2×D107—41no—
112C4×C107—42no—
113C4×C107—41no—
114C4×C107—41no—
115C4×C107#12412yes—
116C4×C107C2×#4111yescharacteristic
117C4×C107C2×#4111yescharacteristic
118Dic107—22no—
119Dic107—22no—
120Dic107—41no—
121Dic107—42no—
122C2×C107—22no—
123C2×C107C2×#12411yescharacteristic
124C2×C107#57711yescharacteristic
125C2×C107#57811yescharacteristic
126D107—81no—
127#3137—1071nomaximal, Sylow
128C107#313711yescharacteristic, complemented, Sylow
129#544—1071no—
130#536—1071no—
131#491—1072no—
132#489—1072no—
133#488—1072no—
134#487—1072no—
135#419D10711yescharacteristic, complemented
136#417—1071no—
137#416—1071no—
138C2×#103—1071no—
139#405—1071no—
140C2×#41—1071no—
141C22×Q8C2×D10711yescharacteristic
142C2×M16C2×D10711yescharacteristic
143C22×C8—1071no—
144C4⋊Q8—1072no—
145C4⋊Q8—1072no—
146C4⋊Q8—2141no—
147#125—1072no—
148#125—2141no—
149#124—1072no—
150#123—2142no—
151C22⋊Q8—2142no—
152C22⋊Q8—2141no—
153C22⋊Q8—2141no—
154#117—1071no—
155#117C2×D10711yescharacteristic
156C2×C4⋊C4—1071no—
157#103—1072no—
158#103—1072no—
159#103—1072no—
160#103—1071no—
161#103—1071no—
162#103C2×D10712yes—
163#103C2×D10712yes—
164#102—1072no—
165#102—1071no—
166#102—1071no—
167#97—1072no—
168C4×C8—1072no—
169#41—2142no—
170#41—2142no—
171C2×Q8—21no—
172C2×Q8—22no—
173C2×Q8—22no—
174C2×Q8—1071no—
175C2×Q8—2142no—
176C2×Q8C22×D10712yes—
177C2×D4—1071no—
178C2×D4—2141no—
179C22×C4—21no—
180C22×C4—2141no—
181C22×C4—2141no—
182C22×C4C22×D10711yescharacteristic
183M16—22no—
184C2×C8—1071no—
185C2×C8—1071no—
186C2×C8—2142no—
187C2×C8C22×D10712yes—
188C4⋊C4—1071no—
189C4⋊C4—1071no—
190C4⋊C4—1072no—
191C4⋊C4—2142no—
192C4⋊C4—2142no—
193C4⋊C4—2142no—
194C4⋊C4—2142no—
195C4⋊C4—2142no—
196C4⋊C4—2141no—
197C4⋊C4C22×D10711yescharacteristic
198C4⋊C4C22×D10711yescharacteristic
199C22⋊C4—21no—
200C22⋊C4—1072no—
201C22⋊C4—2142no—
202C22⋊C4—2142no—
203C42—21no—
204C42—1072no—
205C23—2141no—
206C23D4×D10711yescharacteristic
207Q8—22no—
208Q8—22no—
209Q8—42no—
210Q8—8562no—
211Q8—2141no—
212Q8—2141no—
213D4—4282no—
214D4—2141no—
215D4—2141no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—42no—
220C2×C4—8561no—
221C2×C4—1072no—
222C2×C4—1072no—
223C2×C4—4282no—
224C2×C4—2142no—
225C2×C4—2141no—
226C2×C4D4×D10711yescharacteristic
227C2×C4#1313381712yes—
228C2×C4#1313381812yes—
229C2×C4C23×D10711yescharacteristic
230C8—22no—
231C8—2141no—
232C8—2141no—
233C22—21no—
234C22—21no—
235C22—4281no—
236C22—4281no—
237C22—4281no—
238C22#1681209512yes—
239C22C2×D4×D10711yescharacteristic
240C4—42no—
241C4—41no—
242C4—41no—
243C4—4281no—
244C4—4282no—
245C4—2142no—
246C4—2142no—
247C4#1681209512yes—
248C4C2×#1313381711yescharacteristic
249C4C2×#1313381811yescharacteristic
250C2—22no—
251C2—8561no—
252C2C2×#1681209511yescharacteristic
253C2#2029343611yescharacteristic
254C2#2029344311yescharacteristic
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#419×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133818C22
hypercenter#419D107
hypocenterC107#3137
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ #419

Complements

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

NH ≅ G/Ntype
1. #20293255C2semidirect
3. #20292994C2semidirect
4. #20292988C2semidirect
5. #20292983C2semidirect
6. #20292976C2semidirect
9. C2×#16812033C2semidirect
10. #20292585C2semidirect
11. #419×C107C2semidirect
128. C107#3137semidirect
135. #419D107semidirect

Maximal subgroups

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

subgroupindex
#202932552
#202932012
#202929942
#202929882
#202929832
#202929762
#202926292
#202926212
C2×#168120332
#202925852
#419×C1072
#3137107