Finite Group Databasebeta

← #28346831

Subgroups — #28346831

Order 13696.

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

normal subgroups only

This group has 372 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
1#20293203C211yesmaximal, characteristic
2#20292997C212yesmaximal, complemented
3#20292989C212yesmaximal, complemented
4#20292984C212yesmaximal, complemented
5#20292979C212yesmaximal, complemented
6#20292635C211yesmaximal, characteristic, complemented
7#20292626C211yesmaximal, characteristic
8#20292621C211yesmaximal, characteristic
9C2×#16812032C211yesmaximal, characteristic, complemented
10#20292583C211yesmaximal, characteristic, complemented
11#544×C107C211yesmaximal, characteristic, complemented
12C2×#13133818C2211yescharacteristic
13C22×Dic214C2211yescharacteristic
14C2×#13133802C2211yescharacteristic
15C22×C107⋊C8C2211yescharacteristic
16#16812109C2212yes—
17#16812097C2212yes—
18#16812093—22no—
19#16812089—21no—
20#16812089—21no—
21#16812032C2212yes—
22#16812032C2212yes—
23#16812032C2211yescharacteristic
24#16812032C2211yescharacteristic
25#16812032C2212yes—
26#16812032C2212yes—
27#16812032C2212yes—
28#16812029C2211yescharacteristic
29#16812029C2211yescharacteristic
30#16812029C2212yes—
31#16812020C2212yes—
32C4×C107⋊C8C2212yes—
33C4⋊Q8×C107—21no—
34C4⋊Q8×C107C2212yes—
35#125×C107—21no—
36#125×C107C2212yes—
37C22⋊Q8×C107—22no—
38#117×C107C2211yescharacteristic
39#117×C107C2211yescharacteristic
40C2×C4⋊C4×C107C2211yescharacteristic
41#13133818—22no—
42#13133818—22no—
43C2×Dic214—22no—
44C2×Dic214—22no—
45C2×Dic214—21no—
46C2×Dic214C2312yes—
47C2×Dic214C2311yescharacteristic
48C2×D428C2311yescharacteristic
49C2×C22⋊D107—21no—
50C22×Dic107—21no—
51C2×C4×D107—21no—
52#13133802—22no—
53C2×C107⋊C8—22no—
54C2×C107⋊C8C2311yescharacteristic
55C2×C107⋊C8C2311yescharacteristic
56C2×C107⋊C8C2312yes—
57C4⋊Dic107—22no—
58C4⋊Dic107—22no—
59(C2×D107)⋊C4—22no—
60(C2×D107)⋊C4—22no—
61C2×Q8×C107—22no—
62C22×C4×C107—21no—
63C22×C4×C107C2311yescharacteristic
64C4⋊C4×C107—21no—
65C4⋊C4×C107—22no—
66C4⋊C4×C107—22no—
67C4⋊C4×C107—22no—
68C4⋊C4×C107D411yescharacteristic
69C4⋊C4×C107D411yescharacteristic
70C4⋊C4×C107C2312yes—
71C4⋊C4×C107C2311yescharacteristic
72C4⋊C4×C107C2311yescharacteristic
73C22⋊C4×C107—21no—
74C22⋊C4×C107D412yes—
75C42×C107—21no—
76C42×C107C2312yes—
77C23×C107C2×D411yescharacteristic
78C22×D107—21no—
79Q8×C107—82no—
80C2×C4×C107—21no—
81C2×C4×C107—21no—
82C2×C4×C107—22no—
83C2×C4×C107—81no—
84C2×C4×C107C2×D412yes—
85C2×C4×C107C2×D412yes—
86C2×C4×C107C2×D411yescharacteristic
87C2×C4×C107#4112yes—
88C2×C4×C107#4112yes—
89C2×C4×C107C2411yescharacteristic
90C22⋊D107—42no—
91C2×Dic107—22no—
92C2×Dic107—21no—
93C2×Dic107—42no—
94D428—21no—
95D428—21no—
96C4×D107—42no—
97Dic214—22no—
98Dic214—22no—
99Dic214—21no—
100Dic214—21no—
101Dic214—42no—
102C107⋊C8—21no—
103C107⋊C8—21no—
104C107⋊C8—22no—
105C22×C107—21no—
106C22×C107—21no—
107C22×C107#12412yes—
108C22×C107C22×D411yescharacteristic
109C2×D107—41no—
110C2×D107—41no—
111C2×D107—41no—
112C4×C107—22no—
113C4×C107—22no—
114C4×C107—42no—
115C4×C107—41no—
116C4×C107—41no—
117C4×C107#12412yes—
118C4×C107C2×#4111yescharacteristic
119C4×C107C2×#4111yescharacteristic
120Dic107—42no—
121Dic107—41no—
122C2×C107—22no—
123C2×C107C2×#12411yescharacteristic
124C2×C107#57711yescharacteristic
125C2×C107#57811yescharacteristic
126D107—81no—
127#3137—1071nomaximal, Sylow
128C107#313711yescharacteristic, complemented, Sylow
129#544D10711yescharacteristic, complemented
130#536—1071no—
131#491—1072no—
132#489—1072no—
133#488—1072no—
134#487—1072no—
135#419—1071no—
136#417—1071no—
137#416—1071no—
138C2×#103—1071no—
139#405—1071no—
140C2×#41—1071no—
141C22×Q8—1071no—
142C2×M16—1071no—
143C22×C8—1071no—
144C4⋊Q8—21no—
145C4⋊Q8—1072no—
146C4⋊Q8C2×D10712yes—
147#125—21no—
148#125C2×D10712yes—
149#124—1072no—
150#123—2142no—
151C22⋊Q8—22no—
152C22⋊Q8—2141no—
153C22⋊Q8—2141no—
154#117C2×D10711yescharacteristic
155#117C2×D10711yescharacteristic
156C2×C4⋊C4C2×D10711yescharacteristic
157#103—1072no—
158#103—1072no—
159#103—1071no—
160#103—1071no—
161#103—1072no—
162#103—1072no—
163#103—1072no—
164#102—1071no—
165#102—1071no—
166#102—1072no—
167#97—1072no—
168C4×C8—1072no—
169#41—2142no—
170#41—2142no—
171C2×Q8—22no—
172C2×Q8—1072no—
173C2×Q8—1071no—
174C2×Q8—2142no—
175C2×Q8—2142no—
176C2×Q8—2141no—
177C2×D4—1071no—
178C2×D4—2141no—
179C22×C4—21no—
180C22×C4—2141no—
181C22×C4—2141no—
182C22×C4C22×D10711yescharacteristic
183M16—2142no—
184C2×C8—1071no—
185C2×C8—1072no—
186C2×C8—1071no—
187C2×C8—2142no—
188C4⋊C4—21no—
189C4⋊C4—22no—
190C4⋊C4—22no—
191C4⋊C4—22no—
192C4⋊C4—2142no—
193C4⋊C4—2142no—
194C4⋊C4C22⋊D10711yescharacteristic
195C4⋊C4C22⋊D10711yescharacteristic
196C4⋊C4C22×D10712yes—
197C4⋊C4C22×D10711yescharacteristic
198C4⋊C4C22×D10711yescharacteristic
199C22⋊C4—21no—
200C22⋊C4—2142no—
201C22⋊C4—2142no—
202C22⋊C4C22⋊D10712yes—
203C42—21no—
204C42C22×D10712yes—
205C23—2141no—
206C23C2×C22⋊D10711yescharacteristic
207Q8—82no—
208Q8—4282no—
209Q8—2142no—
210Q8—2142no—
211Q8—2141no—
212Q8—2141no—
213D4—4282no—
214D4—2141no—
215D4—2141no—
216C2×C4—21no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—81no—
220C2×C4—4282no—
221C2×C4—4282no—
222C2×C4—2142no—
223C2×C4—2141no—
224C2×C4C2×C22⋊D10712yes—
225C2×C4C2×C22⋊D10712yes—
226C2×C4C2×C22⋊D10711yescharacteristic
227C2×C4#1313381912yes—
228C2×C4#1313381912yes—
229C2×C4C23×D10711yescharacteristic
230C8—2141no—
231C8—2141no—
232C8—2142no—
233C22—21no—
234C22—21no—
235C22—4281no—
236C22—4281no—
237C22—4281no—
238C22#1681209812yes—
239C22C22×C22⋊D10711yescharacteristic
240C4—22no—
241C4—22no—
242C4—42no—
243C4—41no—
244C4—41no—
245C4—4281no—
246C4—4282no—
247C4#1681209812yes—
248C4C2×#1313381911yescharacteristic
249C4C2×#1313381911yescharacteristic
250C2—22no—
251C2—8561no—
252C2C2×#1681209811yescharacteristic
253C2#2029343811yescharacteristic
254C2#2029344211yescharacteristic
subgroupG/H
centerC22C22×C22⋊D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
Fitting#544×C107C2
socleC22×C107C22×D4
prime radicalC2×#13133818C22
hypercenter#544D107
hypocenterC107#3137
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ #117 ◁ #544

Complements

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

NH ≅ G/Ntype
2. #20292997C2semidirect
3. #20292989C2semidirect
4. #20292984C2semidirect
5. #20292979C2semidirect
6. #20292635C2semidirect
9. C2×#16812032C2semidirect
10. #20292583C2semidirect
11. #544×C107C2semidirect
128. C107#3137semidirect
129. #544D107semidirect

Maximal subgroups

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

subgroupindex
#202932032
#202929972
#202929892
#202929842
#202929792
#202926352
#202926262
#202926212
C2×#168120322
#202925832
#544×C1072
#3137107