Finite Group Databasebeta

← #28336421

Subgroups — #28336421

Order 13696.

552 conjugacy classes of subgroups; only the 101 normal subgroups are shown.

normal subgroups only

This group has 552 conjugacy classes of subgroups — too many to draw. Only its 101 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×#16812156C211yesmaximal, characteristic
2C2×#16812037C211yesmaximal, characteristic, complemented
3#20292629C211yesmaximal, characteristic, complemented
4#20292628C211yesmaximal, characteristic, complemented
5C2×#16812027C211yesmaximal, characteristic, complemented
6#20292533C211yesmaximal, characteristic, complemented
7C2×#104×C107C211yesmaximal, characteristic, complemented
8#16812156—22no—
9#16812156—22no—
10#16812156—22no—
11#16812156C414yescomplemented
12C2×#13133818—21no—
13C2×#13133818C2211yescharacteristic
14C2×#13133817—22no—
15C2×#13133817—21no—
16C22×Dic214—21no—
17C22×Dic214C2211yescharacteristic
18C2×Q8×D107—21no—
19C2×#13133802C2211yescharacteristic
20C2×C4⋊Dic107C2211yescharacteristic
21#16812037—22no—
22#16812034—21no—
23#16812033—21no—
24#16812031—21no—
25#16812030—21no—
26#16812027—22no—
27C107⋊#95—22no—
28C2×#41×C107C2211yescharacteristic
29C2×M16×C107C2211yescharacteristic
30C2×C42×C107C2211yescharacteristic
31#104×C107—22no—
32#13133818—24no—
33#13133818—21no—
34#13133818—21no—
35#13133818—24no—
36#13133818C2×C412yes—
37#13133818D412yes—
38#13133817—28no—
39#13133817—24no—
40#13133817—24no—
41#13133817—28no—
42C2×Dic214—22no—
43C2×Dic214—24no—
44C2×Dic214—21no—
45C2×Dic214—21no—
46C2×Dic214—21no—
47C2×Dic214—21no—
48C2×Dic214—24no—
49C2×Dic214C2×C412yes—
50C2×Dic214D412yes—
51C2×Dic214D411yescharacteristic
52Q8×D107—24no—
53Q8×D107—24no—
54C2×D428D411yescharacteristic
55C2×C22⋊D107—22no—
56C2×C22⋊D107—21no—
57C22×Dic107—22no—
58C22×Dic107—21no—
59C22×Dic107D412yes—
60C2×C4×D107—22no—
61C2×C4×D107—21no—
62#13133802—21no—
63#13133802—21no—
64C2×C107⋊C8—21no—
65C4⋊Dic107—21no—
66C4⋊Dic107—21no—
67#41×C107—21no—
68#41×C107—21no—
69#41×C107C2×C412yes—
70#41×C107D412yes—
71C2×Q8×C107D411yescharacteristic
72C2×D4×C107—21no—
73C2×D4×C107D411yescharacteristic
74C22×C4×C107—21no—
75C22×C4×C107—21no—
76C22×C4×C107C2311yescharacteristic
77M16×C107—21no—
78M16×C107—21no—
79C2×C8×C107—21no—
80C42×C107—21no—
81C42×C107—21no—
82C23×C107—21no—
83C23×C107C2×D411yescharacteristic
84C22×D107—21no—
85Q8×C107—21no—
86Q8×C107C22⋊C412yes—
87D4×C107—22no—
88D4×C107—22no—
89D4×C107—21no—
90D4×C107C22⋊C412yes—
91C2×C4×C107—21no—
92C2×C4×C107—22no—
93C2×C4×C107—22no—
94C2×C4×C107—22no—
95C2×C4×C107—41no—
96C2×C4×C107—41no—
97C2×C4×C107C22×C411yescharacteristic
98C2×C4×C107C2×D411yescharacteristic
99C2×C4×C107C2×D411yescharacteristic
100C2×C4×C107C2×D411yescharacteristic
101C2×C4×C107C2×D411yescharacteristic
102C2×C4×C107C2×D411yescharacteristic
103C22⋊D107—28no—
104C22⋊D107—22no—
105C22⋊D107—22no—
106C2×Dic107—28no—
107C2×Dic107—21no—
108C2×Dic107—22no—
109C2×Dic107—22no—
110C2×Dic107—42no—
111C2×Dic107—41no—
112C2×Dic107C22⋊C414yes—
113D428—21no—
114D428C22⋊C412yes—
115C4×D107—28no—
116C4×D107—22no—
117C4×D107—22no—
118Dic214—28no—
119Dic214—22no—
120Dic214—22no—
121Dic214—22no—
122Dic214—22no—
123Dic214—21no—
124Dic214C22⋊C412yes—
125C8×C107—41no—
126C107⋊C8—41no—
127C22×C107—21no—
128C22×C107—21no—
129C22×C107—22no—
130C22×C107—41no—
131C22×C107—41no—
132C22×C107C2×C22⋊C411yescharacteristic
133C22×C107#12011yescharacteristic
134C22×C107#12011yescharacteristic
135C2×D107—22no—
136C2×D107—41no—
137C2×D107—41no—
138C4×C107—22no—
139C4×C107—42no—
140C4×C107C2×C22⋊C411yescharacteristic
141C4×C107C2×C22⋊C411yescharacteristic
142C4×C107#12011yescharacteristic
143C4×C107#12011yescharacteristic
144Dic107—24no—
145Dic107—22no—
146Dic107—81no—
147C2×C107—21no—
148C2×C107—21no—
149C2×C107—42no—
150C2×C107#37911yescharacteristic
151C2×C107#45411yescharacteristic
152C2×C107#45611yescharacteristic
153D107—42no—
154#1793—1071nomaximal, Sylow
155C107#179311yescharacteristic, complemented, Sylow
156C2×2−1+4—1071no—
157C2×#104—1071no—
158C2×#104D10711yescharacteristic, complemented
159#417—1071no—
160#417—1071no—
161C2×#101—1071no—
162#391—1071no—
1632−1+4—1074no—
1642−1+4—2142no—
1652−1+4—2142no—
1662−1+4—2142no—
167C2×#41—1071no—
168C2×#41—2141no—
169C2×#41—2142no—
170C2×#41—2141no—
171C2×#41C2×D10711yescharacteristic
172C22×Q8—1071no—
173C22×Q8—2141no—
174C22×Q8—2141no—
175C2×M16—1071no—
176C2×M16C2×D10711yescharacteristic
177C2×C4⋊C4—1071no—
178C2×C42C2×D10711yescharacteristic
179#104—22no—
180#104—2142no—
181#103—2141no—
182#103—2141no—
183#102—2141no—
184#102—2141no—
185#101—2142no—
186#95—2142no—
187#41—21no—
188#41—21no—
189#41—1072no—
190#41—1072no—
191#41—2144no—
192#41—2144no—
193#41—2148no—
194#41—2148no—
195#41—2144no—
196#41—2144no—
197#41—2141no—
198#41—2141no—
199#41C4×D10712yes—
200#41C22⋊D10712yes—
201C2×Q8—1072no—
202C2×Q8—1072no—
203C2×Q8—1071no—
204C2×Q8—2144no—
205C2×Q8—2142no—
206C2×Q8—2144no—
207C2×Q8—2144no—
208C2×Q8—2144no—
209C2×Q8—2141no—
210C2×Q8—2141no—
211C2×Q8—2141no—
212C2×Q8—2141no—
213C2×Q8D42811yescharacteristic
214C2×D4—21no—
215C2×D4—1071no—
216C2×D4—2142no—
217C2×D4—2141no—
218C2×D4D42811yescharacteristic
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—1072no—
222C22×C4—2142no—
223C22×C4—2141no—
224C22×C4—2142no—
225C22×C4—2141no—
226C22×C4C22×D10711yescharacteristic
227M16—21no—
228M16—21no—
229M16—2141no—
230M16—2141no—
231C2×C8—21no—
232C2×C8—2141no—
233C4⋊C4—2141no—
234C4⋊C4—2141no—
235C42—21no—
236C42—21no—
237C23—21no—
238C23—2141no—
239C23D4×D10711yescharacteristic
240Q8—21no—
241Q8—1072no—
242Q8—2148no—
243Q8—2142no—
244Q8—2142no—
245Q8—2142no—
246Q8—2141no—
247Q8—2142no—
248Q8(C2×D107)⋊C412yes—
249D4—22no—
250D4—22no—
251D4—21no—
252D4—1072no—
253D4—2148no—
254D4—2142no—
255D4—2142no—
256D4—2141no—
257D4(C2×D107)⋊C412yes—
258C2×C4—21no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—41no—
263C2×C4—41no—
264C2×C4—1074no—
265C2×C4—4282no—
266C2×C4—4281no—
267C2×C4—2148no—
268C2×C4—2148no—
269C2×C4—2142no—
270C2×C4—2142no—
271C2×C4—2141no—
272C2×C4—2142no—
273C2×C4—2142no—
274C2×C4C2×C4×D10711yescharacteristic
275C2×C4D4×D10711yescharacteristic
276C2×C4D4×D10711yescharacteristic
277C2×C4D4×D10711yescharacteristic
278C2×C4C2×C22⋊D10711yescharacteristic
279C2×C4C2×D42811yescharacteristic
280C8—41no—
281C8—4281no—
282C22—21no—
283C22—21no—
284C22—22no—
285C22—41no—
286C22—41no—
287C22—4281no—
288C22—4281no—
289C22—2142no—
290C22C22⋊C4×D10711yescharacteristic
291C22#1681207711yescharacteristic
292C22#1681207911yescharacteristic
293C4—22no—
294C4—42no—
295C4—8561no—
296C4—2144no—
297C4—2142no—
298C4C22⋊C4×D10711yescharacteristic
299C4C2×(C2×D107)⋊C411yescharacteristic
300C4#1681207711yescharacteristic
301C4#1681207911yescharacteristic
302C2—21no—
303C2—21no—
304C2—42no—
305C2—4282no—
306C2#2029248211yescharacteristic
307C2#2029280111yescharacteristic
308C2#2029281111yescharacteristic
subgroupG/H
centerC22#16812079
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
FittingC2×#104×C107C2
socleC22×C107#120
prime radicalC2×#16812156C2
cosocleC22×C4×C107C23
hypercenterC2×#104D107
hypocenterC107#1793
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#16812037C2semidirect
3. #20292629C2semidirect
4. #20292628C2semidirect
5. C2×#16812027C2semidirect
6. #20292533C2semidirect
7. C2×#104×C107C2semidirect
11. #16812156C4semidirect
155. C107#1793semidirect
158. C2×#104D107semidirect

Maximal subgroups

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

subgroupindex
C2×#168121562
C2×#168120372
#202926292
#202926282
C2×#168120272
#202925332
C2×#104×C1072
#1793107