Finite Group Databasebeta

← #28336416

Subgroups — #28336416

Order 13696.

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

normal subgroups only

This group has 620 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×#16812155C211yesmaximal, characteristic
2C2×#16812037C211yesmaximal, characteristic, complemented
3#20292629C211yesmaximal, characteristic, complemented
4#20292628C211yesmaximal, characteristic, complemented
5C2×#16812025C211yesmaximal, characteristic, complemented
6#20292529C211yesmaximal, characteristic, complemented
7C2×#104×C107C211yesmaximal, characteristic, complemented
8#16812155—22no—
9#16812155—22no—
10#16812155—22no—
11#16812155C414yescomplemented
12C2×#13133819—21no—
13C2×#13133818—21no—
14C2×#13133818C2211yescharacteristic
15C22×D428—21no—
16C22×D428C2211yescharacteristic
17C2×D4×D107—22no—
18C2×D4×D107—21no—
19C2×#13133802C2211yescharacteristic, complemented
20C2×C4⋊Dic107C2211yescharacteristic, complemented
21C2×(C2×D107)⋊C4—22no—
22#16812037—22no—
23#16812034—21no—
24#16812033—21no—
25#16812031—21no—
26#16812030—21no—
27#16812025—22no—
28C2×#41×C107C2211yescharacteristic
29C2×M16×C107C2211yescharacteristic, complemented
30C2×C42×C107C2211yescharacteristic, complemented
31#104×C107—22no—
32C23×D107—22no—
33C23×D107D412yes—
34#13133819—24no—
35#13133819—24no—
36#13133818—24no—
37#13133818—21no—
38#13133818—21no—
39#13133818—24no—
40#13133818C2×C412yes—
41#13133818D412yes—
42C2×Dic214D411yescharacteristic
43C2×D428—22no—
44C2×D428—24no—
45C2×D428—21no—
46C2×D428—21no—
47C2×D428—21no—
48C2×D428—21no—
49C2×D428—24no—
50C2×D428C2×C412yes—
51C2×D428D412yes—
52C2×D428D411yescharacteristic
53C2×C22⋊D107—22no—
54C2×C22⋊D107—21no—
55D4×D107—28no—
56D4×D107—24no—
57D4×D107—28no—
58D4×D107—24no—
59C22×Dic107—21no—
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(C2×D107)⋊C4—44no—
68#41×C107—21no—
69#41×C107—21no—
70#41×C107C2×C412yes—
71#41×C107D412yes—
72C2×Q8×C107D411yescharacteristic
73C2×D4×C107—21no—
74C2×D4×C107D411yescharacteristic
75C22×C4×C107—21no—
76C22×C4×C107—21no—
77C22×C4×C107C2311yescharacteristic
78M16×C107—21no—
79M16×C107—21no—
80C2×C8×C107—21no—
81C42×C107—21no—
82C42×C107—21no—
83C23×C107—21no—
84C23×C107C2×D411yescharacteristic
85C22×D107—22no—
86C22×D107—21no—
87C22×D107—28no—
88C22×D107—22no—
89C22×D107—42no—
90C22×D107—42no—
91C22×D107—82no—
92C22×D107—82no—
93C22×D107C22⋊C414yes—
94Q8×C107—21no—
95Q8×C107C22⋊C412yescomplemented
96D4×C107—22no—
97D4×C107—22no—
98D4×C107—21no—
99D4×C107C22⋊C412yescomplemented
100C2×C4×C107—21no—
101C2×C4×C107—22no—
102C2×C4×C107—22no—
103C2×C4×C107—22no—
104C2×C4×C107—41no—
105C2×C4×C107—41no—
106C2×C4×C107C22×C411yescharacteristic
107C2×C4×C107C2×D411yescharacteristic
108C2×C4×C107C2×D411yescharacteristic
109C2×C4×C107C2×D411yescharacteristic
110C2×C4×C107C2×D411yescharacteristic
111C2×C4×C107C2×D411yescharacteristic
112C22⋊D107—28no—
113C22⋊D107—22no—
114C22⋊D107—22no—
115C2×Dic107—21no—
116C2×Dic107—42no—
117C2×Dic107—41no—
118D428—28no—
119D428—22no—
120D428—22no—
121D428—22no—
122D428—22no—
123D428—21no—
124D428C22⋊C412yescomplemented
125C4×D107—28no—
126C4×D107—22no—
127C4×D107—22no—
128Dic214—21no—
129Dic214C22⋊C412yescomplemented
130C8×C107—41no—
131C107⋊C8—41no—
132C22×C107—21no—
133C22×C107—21no—
134C22×C107—22no—
135C22×C107—41no—
136C22×C107—41no—
137C22×C107C2×C22⋊C411yescharacteristic
138C22×C107#12011yescharacteristic
139C22×C107#12011yescharacteristic
140C2×D107—24no—
141C2×D107—22no—
142C2×D107—42no—
143C2×D107—41no—
144C2×D107—42no—
145C2×D107—41no—
146C2×D107—44no—
147C2×D107—82no—
148C2×D107—88no—
149C4×C107—22no—
150C4×C107—42no—
151C4×C107C2×C22⋊C411yescharacteristic
152C4×C107C2×C22⋊C411yescharacteristic
153C4×C107#12011yescharacteristic
154C4×C107#12011yescharacteristic
155Dic107—22no—
156Dic107—81no—
157C2×C107—21no—
158C2×C107—21no—
159C2×C107—42no—
160C2×C107#37911yescharacteristic
161C2×C107#45311yescharacteristic
162C2×C107#45511yescharacteristic
163D107—44no—
164D107—42no—
165#1792—1071nomaximal, Sylow
166C107#179211yescharacteristic, complemented, Sylow
167C2×H(5,𝔽2)—1071no—
168C2×#104—1071no—
169C2×#104D10711yescharacteristic, complemented
170#417—1071no—
171#417—1071no—
172C2×#100—1071no—
173#390—1071no—
174H(5,𝔽2)—1074no—
175H(5,𝔽2)—2142no—
176H(5,𝔽2)—2142no—
177H(5,𝔽2)—2142no—
178C2×#41—1071no—
179C2×#41—2141no—
180C2×#41—2141no—
181C2×#41C2×D10711yescharacteristic
182C22×D4—1071no—
183C22×D4—2141no—
184C22×D4—2142no—
185C22×D4—2141no—
186C2×M16—1071no—
187C2×M16C2×D10711yescharacteristic, complemented
188C2×C4⋊C4—1071no—
189C2×C22⋊C4—2142no—
190C2×C42C2×D10711yescharacteristic, complemented
191#104—22no—
192#104—2142no—
193#103—2141no—
194#103—2141no—
195#102—2141no—
196#102—2141no—
197#100—2142no—
198C24—1072no—
199C24—2142no—
200#41—21no—
201#41—21no—
202#41—1072no—
203#41—1072no—
204#41—2144no—
205#41—2144no—
206#41—2144no—
207#41—2144no—
208#41—2141no—
209#41—2141no—
210#41C4×D10712yes—
211#41C22⋊D10712yes—
212C2×Q8—1071no—
213C2×Q8D42811yescharacteristic
214C2×D4—21no—
215C2×D4—1072no—
216C2×D4—1071no—
217C2×D4—1072no—
218C2×D4—2144no—
219C2×D4—2142no—
220C2×D4—2144no—
221C2×D4—2148no—
222C2×D4—2148no—
223C2×D4—2144no—
224C2×D4—2142no—
225C2×D4—2141no—
226C2×D4—2144no—
227C2×D4—2141no—
228C2×D4—2141no—
229C2×D4—2141no—
230C2×D4—2141no—
231C2×D4D42811yescharacteristic
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—2142no—
235C22×C4—2141no—
236C22×C4—2141no—
237C22×C4C22×D10711yescharacteristic
238M16—21no—
239M16—21no—
240M16—2141no—
241M16—2141no—
242C2×C8—21no—
243C2×C8—2141no—
244C4⋊C4—2141no—
245C4⋊C4—2141no—
246C22⋊C4—4284no—
247C42—21no—
248C42—21no—
249C23—21no—
250C23—8562no—
251C23—8562no—
252C23—1074no—
253C23—4282no—
254C23—4282no—
255C23—2148no—
256C23—2142no—
257C23—2141no—
258C23—2142no—
259C23D4×D10711yescharacteristic
260Q8—21no—
261Q8—1072no—
262Q8—2141no—
263Q8(C2×D107)⋊C412yescomplemented
264D4—22no—
265D4—22no—
266D4—21no—
267D4—1072no—
268D4—2148no—
269D4—2148no—
270D4—2142no—
271D4—2142no—
272D4—2142no—
273D4—2142no—
274D4—2141no—
275D4—2142no—
276D4—2142no—
277D4(C2×D107)⋊C412yescomplemented
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—4282no—
285C2×C4—4281no—
286C2×C4—2148no—
287C2×C4—2141no—
288C2×C4—2142no—
289C2×C4—2142no—
290C2×C4C2×C4×D10711yescharacteristic
291C2×C4D4×D10711yescharacteristic
292C2×C4D4×D10711yescharacteristic
293C2×C4D4×D10711yescharacteristic
294C2×C4C2×C22⋊D10711yescharacteristic
295C2×C4C2×D42811yescharacteristic
296C8—41no—
297C8—4281no—
298C22—21no—
299C22—21no—
300C22—22no—
301C22—41no—
302C22—41no—
303C22—8562no—
304C22—8568no—
305C22—4284no—
306C22—4281no—
307C22—4282no—
308C22—4281no—
309C22—4282no—
310C22—2142no—
311C22—2144no—
312C22C22⋊C4×D10711yescharacteristic
313C22#1681207711yescharacteristic
314C22#1681207911yescharacteristic
315C4—22no—
316C4—42no—
317C4—8561no—
318C4—2142no—
319C4C22⋊C4×D10711yescharacteristic
320C4C2×(C2×D107)⋊C411yescharacteristic
321C4#1681207711yescharacteristic
322C4#1681207911yescharacteristic
323C2—21no—
324C2—21no—
325C2—42no—
326C2—4282no—
327C2—4284no—
328C2#2029248211yescharacteristic
329C2#2029279611yescharacteristic
330C2#2029280611yescharacteristic
subgroupG/H
centerC22#16812079
commutatorC2×C4×C107C22×C4
FrattiniC22×C4C22×D107
FittingC2×#104×C107C2
socleC22×C107#120
prime radicalC2×#16812155C2
cosocleC22×C4×C107C23
hypercenterC2×#104D107
hypocenterC107#1792
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×#16812025C2semidirect
6. #20292529C2semidirect
7. C2×#104×C107C2semidirect
11. #16812155C4semidirect
19. C2×#13133802C22semidirect
20. C2×C4⋊Dic107C22semidirect
29. C2×M16×C107C22semidirect
30. C2×C42×C107C22semidirect
95. Q8×C107C22⋊C4semidirect
99. D4×C107C22⋊C4semidirect
124. D428C22⋊C4semidirect
129. Dic214C22⋊C4semidirect
166. C107#1792semidirect
169. C2×#104D107semidirect
187. C2×M16C2×D107semidirect
190. C2×C42C2×D107semidirect
263. Q8(C2×D107)⋊C4semidirect
277. D4(C2×D107)⋊C4semidirect

Maximal subgroups

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

subgroupindex
C2×#168121552
C2×#168120372
#202926292
#202926282
C2×#168120252
#202925292
C2×#104×C1072
#1792107