Finite Group Databasebeta

← #28345752

Subgroups — #28345752

Order 13696.

700 conjugacy classes of subgroups; only the 149 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 149 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×#16812157C211yesmaximal, characteristic
2C2×#16812140C211yesmaximal, characteristic, complemented
3#20293201C211yesmaximal, characteristic, complemented
4#20292788C214yesmaximal, complemented
5#20292781C214yesmaximal, complemented
6#20292636C211yesmaximal, characteristic, complemented
7#20292624C211yesmaximal, characteristic, complemented
8#20292593C211yesmaximal, characteristic, complemented
9C22×Q16×C107C211yesmaximal, characteristic, complemented
10#16812157—24no—
11#16812157—24no—
12C2×#13133819—21no—
13C2×#13133819C2212yes—
14C2×#13133818—22no—
15C2×#13133818C2211yescharacteristic
16C2×#13133818C2211yescharacteristic
17C2×Q8×D107—21no—
18C2×Q8×D107C2212yes—
19#16812140—24no—
20C2×#13133808C2212yes—
21C2×#13133805C2212yes—
22C2×#13133802C2211yescharacteristic
23#16812107—21no—
24#16812098—21no—
25#16812092—22no—
26#16812085C2214yes—
27#16812068C2211yescharacteristic
28#16812034C2212yes—
29#16812033C2214yes—
30#16812031C2212yes—
31(C2×D107)⋊C8C2214yes—
32C22×Q8×C107C2211yescharacteristic
33C22×Q8×C107C2211yescharacteristic
34C2×Q16×C107—24no—
35C2×Q16×C107C2214yes—
36C22×C8×C107C2211yescharacteristic
37#13133819—28no—
38#13133819—24no—
39#13133819—24no—
40#13133818—216no—
41#13133818—22no—
42#13133818—22no—
43#13133818—22no—
44#13133818D414yes—
45C2×Dic214—22no—
46C2×Dic214D411yescharacteristic
47C2×Dic214C2311yescharacteristic
48Q8×D107—28no—
49Q8×D107—24no—
50Q8×D107—24no—
51C2×D428—22no—
52C2×D428D411yescharacteristic
53C2×D428C2311yescharacteristic
54C2×C22⋊D107—21no—
55C2×C22⋊D107D412yes—
56C2×C4×D107—22no—
57C2×C4×D107—21no—
58C2×C4×D107—22no—
59C2×C4×D107C2312yes—
60#13133808—24no—
61#13133805—24no—
62#13133802—22no—
63C2×C107⋊C8C2312yes—
64C4⋊Dic107C2312yes—
65Dic107⋊C4—24no—
66C22⋊Dic107—21no—
67(C2×D107)⋊C4—24no—
68C4×Dic107—21no—
69C2×Q8×C107—21no—
70C2×Q8×C107—22no—
71C2×Q8×C107—21no—
72C2×Q8×C107—22no—
73C2×Q8×C107—22no—
74C2×Q8×C107D414yes—
75C2×Q8×C107C2312yes—
76C2×Q8×C107C2312yes—
77C22×C4×C107—21no—
78C22×C4×C107—21no—
79C22×C4×C107C2311yescharacteristic
80Q16×C107—28no—
81C2×C8×C107—22no—
82C2×C8×C107C2312yes—
83C23×C107C2×D411yescharacteristic
84C22×D107—21no—
85C22×D107C2×D412yes—
86Q8×C107—24no—
87Q8×C107—22no—
88Q8×C107—22no—
89Q8×C107—22no—
90Q8×C107—42no—
91Q8×C107C2×D414yes—
92C2×C4×C107—24no—
93C2×C4×C107—22no—
94C2×C4×C107—22no—
95C2×C4×C107—42no—
96C2×C4×C107C2×D411yescharacteristic
97C2×C4×C107C2×D412yes—
98C2×C4×C107C2×D412yes—
99C2×C4×C107C2411yescharacteristic
100C22⋊D107—24no—
101C22⋊D107—24no—
102C2×Dic107—21no—
103C2×Dic107—22no—
104C2×Dic107C2×D412yes—
105D428—28no—
106D428—22no—
107D428—21no—
108D428C2×D412yes—
109C4×D107—28no—
110C4×D107—28no—
111C4×D107—24no—
112C4×D107—22no—
113C4×D107—22no—
114Dic214—28no—
115Dic214—22no—
116Dic214—21no—
117Dic214C2×D412yes—
118C8×C107—22no—
119C107⋊C8—22no—
120C22×C107—22no—
121C22×C107#12012yes—
122C22×C107C22×D411yescharacteristic
123C2×D107—22no—
124C2×D107—22no—
125C2×D107—24no—
126C2×D107—42no—
127C4×C107—24no—
128C4×C107—42no—
129C4×C107#12012yes—
130C4×C107C22×D411yescharacteristic
131C4×C107C22×D411yescharacteristic
132Dic107—22no—
133Dic107—22no—
134Dic107—42no—
135C2×C107—22no—
136C2×C107C2×#12011yescharacteristic
137C2×C107#57812yes—
138D107—42no—
139D107—42no—
140#3011—1071nomaximal, Sylow
141C107#301111yescharacteristic, complemented, Sylow
142C2×2−1+4—1071no—
143C2×#137—1071no—
144C22×Q16D10711yescharacteristic, complemented
145#536—1071no—
146#452—1074no—
147#451—1074no—
148#419—1071no—
149#416—1071no—
150#408—1071no—
1512−1+4—2144no—
1522−1+4—2144no—
153C2×#41—1071no—
154C2×#41—1072no—
155C2×#41—1071no—
156C2×#41—2142no—
157C2×#41—2141no—
158C22×Q8—1072no—
159C22×Q8—2141no—
160C22×Q8C2×D10711yescharacteristic
161C22×Q8C2×D10711yescharacteristic
162#137—2144no—
163C2×Q16—24no—
164C2×Q16—1072no—
165C2×Q16C2×D10714yes—
166C2×SD16—1072no—
167C2×M16—1071no—
168C22×C8C2×D10711yescharacteristic
169C4⋊Q8—2141no—
170#124—2141no—
171#123—2142no—
172C22⋊Q8—1074no—
173#117—1071no—
174#103—1074no—
175#103—1072no—
176#102—1072no—
177C22⋊C8—1074no—
178#41—1074no—
179#41—21416no—
180#41—2144no—
181#41—2148no—
182#41—2144no—
183#41—2142no—
184#41—2142no—
185#41—2142no—
186C2×Q8—21no—
187C2×Q8—22no—
188C2×Q8—21no—
189C2×Q8—22no—
190C2×Q8—22no—
191C2×Q8—1071no—
192C2×Q8—1071no—
193C2×Q8—2142no—
194C2×Q8—2144no—
195C2×Q8—2148no—
196C2×Q8—2144no—
197C2×Q8C22⋊D10714yes—
198C2×Q8C22×D10712yes—
199C2×Q8C22×D10712yes—
200C2×D4—1072no—
201C2×D4—1071no—
202C2×D4—1071no—
203C2×D4—2142no—
204C2×D4—2141no—
205C22×C4—21no—
206C22×C4—21no—
207C22×C4—1072no—
208C22×C4—2142no—
209C22×C4—2142no—
210C22×C4—2141no—
211C22×C4C22×D10711yescharacteristic
212Q16—28no—
213Q16—2144no—
214SD16—2144no—
215M16—2142no—
216C2×C8—22no—
217C2×C8—1072no—
218C2×C8C22×D10712yes—
219C4⋊C4—1072no—
220C4⋊C4—2144no—
221C22⋊C4—2144no—
222C22⋊C4—2141no—
223C42—2141no—
224C23—1072no—
225C23—2141no—
226C23D4×D10711yescharacteristic
227Q8—24no—
228Q8—22no—
229Q8—22no—
230Q8—22no—
231Q8—42no—
232Q8—1072no—
233Q8—2148no—
234Q8—2142no—
235Q8—2141no—
236Q8C2×C22⋊D10714yes—
237D4—1072no—
238D4—2148no—
239D4—2144no—
240D4—2144no—
241D4—2142no—
242D4—2141no—
243C2×C4—24no—
244C2×C4—22no—
245C2×C4—22no—
246C2×C4—42no—
247C2×C4—1072no—
248C2×C4—2148no—
249C2×C4—2148no—
250C2×C4—2142no—
251C2×C4—2141no—
252C2×C4—2144no—
253C2×C4—2142no—
254C2×C4—2142no—
255C2×C4D4×D10711yescharacteristic
256C2×C4D4×D10712yes—
257C2×C4C2×C22⋊D10712yes—
258C2×C4C23×D10711yescharacteristic
259C8—22no—
260C8—2142no—
261C22—22no—
262C22—4282no—
263C22—2144no—
264C22—2142no—
265C22—2142no—
266C22#1681207712yes—
267C22C2×D4×D10711yescharacteristic
268C4—24no—
269C4—42no—
270C4—4282no—
271C4—2142no—
272C4—2142no—
273C4#1681207712yes—
274C4C2×D4×D10711yescharacteristic
275C4C22×C22⋊D10711yescharacteristic
276C2—22no—
277C2—4282no—
278C2—4282no—
279C2C2×#1681207711yescharacteristic
280C2#2029344012yes—
subgroupG/H
centerC22C2×D4×D107
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC22×Q16×C107C2
socleC22×C107C22×D4
prime radicalC2×#16812157C2
hypercenterC22×Q16D107
hypocenterC107#3011
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×Q16

Complements

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

NH ≅ G/Ntype
2. C2×#16812140C2semidirect
3. #20293201C2semidirect
4. #20292788C2semidirect
5. #20292781C2semidirect
6. #20292636C2semidirect
7. #20292624C2semidirect
8. #20292593C2semidirect
9. C22×Q16×C107C2semidirect
141. C107#3011semidirect
144. C22×Q16D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121572
C2×#168121402
#202932012
#202927882
#202927812
#202926362
#202926242
#202925932
C22×Q16×C1072
#3011107