Finite Group Databasebeta

← C2×#20292803 #28345789

Subgroups — C2×#20292803 #28345789

Order 13696.

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

normal subgroups only

This group has 760 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×#16812155C211yesmaximal, characteristic
2C2×#16812140C211yesmaximal, characteristic, complemented
3C2×#16812098C211yesmaximal, characteristic, complemented
4#20292803C218yesmaximal, direct product factor
5C2×#16812036C211yesmaximal, characteristic, complemented
6C2×#16812035C211yesmaximal, characteristic, complemented
7C2×#16812025C211yesmaximal, characteristic, complemented
8C2×#137×C107C211yesmaximal, characteristic, complemented
9#16812155—22no—
10#16812155—22no—
11#16812155—22no—
12#16812155C2214yes—
13C2×#13133819—21no—
14C2×#13133818—21no—
15C2×#13133818C2211yescharacteristic
16C22×D428—21no—
17C22×D428C2211yescharacteristic
18C2×D4×D107—22no—
19C2×D4×D107—21no—
20#16812140—22no—
21#16812140C2214yescomplemented
22C2×#13133808—21no—
23C2×#13133805—21no—
24C2×#13133802C2211yescharacteristic, complemented
25#16812098—22no—
26#16812098C2214yescomplemented
27C2×(C2×D107)⋊C4—22no—
28C2×C4×Dic107C2211yescharacteristic, complemented
29#16812036C2214yescomplemented
30#16812035C2214yescomplemented
31#16812025C2214yescomplemented
32C2×#41×C107C2211yescharacteristic
33C22×Q8×C107C2211yescharacteristic, complemented
34#137×C107—22no—
35#137×C107C2214yescomplemented
36C2×Q16×C107—21no—
37C2×SD16×C107—21no—
38C2×M16×C107C2211yescharacteristic, complemented
39C23×D107—22no—
40C23×D107D412yes—
41#13133819—24no—
42#13133819—24no—
43#13133818—24no—
44#13133818—21no—
45#13133818—21no—
46#13133818—24no—
47#13133818D412yes—
48#13133818C2312yes—
49C2×Dic214D411yescharacteristic
50C2×D428—22no—
51C2×D428—24no—
52C2×D428—21no—
53C2×D428—21no—
54C2×D428—21no—
55C2×D428—21no—
56C2×D428—24no—
57C2×D428D412yes—
58C2×D428D411yescharacteristic
59C2×D428C2312yes—
60C2×C22⋊D107—22no—
61C2×C22⋊D107—21no—
62D4×D107—28no—
63D4×D107—24no—
64D4×D107—28no—
65D4×D107—24no—
66C22×Dic107—21no—
67C2×C4×D107—22no—
68C2×C4×D107—21no—
69#13133808—24no—
70#13133805—24no—
71#13133802—21no—
72#13133802C2312yescomplemented
73C2×C107⋊C8—21no—
74(C2×D107)⋊C4—28no—
75C4×Dic107—21no—
76C4×Dic107C2312yescomplemented
77#41×C107—21no—
78#41×C107—21no—
79#41×C107D412yes—
80#41×C107C2312yes—
81C2×Q8×C107—21no—
82C2×Q8×C107—21no—
83C2×Q8×C107—21no—
84C2×Q8×C107—21no—
85C2×Q8×C107—21no—
86C2×Q8×C107D411yescharacteristic
87C2×Q8×C107C2312yescomplemented
88C2×D4×C107—21no—
89C2×D4×C107D411yescharacteristic
90C22×C4×C107—21no—
91C22×C4×C107—21no—
92C22×C4×C107C2311yescharacteristic
93Q16×C107—24no—
94SD16×C107—24no—
95M16×C107—21no—
96M16×C107C2312yescomplemented
97C2×C8×C107—21no—
98C23×C107—21no—
99C23×C107C2×D411yescharacteristic
100C22×D107—22no—
101C22×D107—21no—
102C22×D107—28no—
103C22×D107—22no—
104C22×D107—44no—
105C22×D107—84no—
106C22×D107C2×D414yes—
107Q8×C107—21no—
108Q8×C107—22no—
109Q8×C107—22no—
110Q8×C107—41no—
111Q8×C107—41no—
112Q8×C107C2×D412yes—
113D4×C107—22no—
114D4×C107—22no—
115D4×C107—21no—
116D4×C107C2×D412yes—
117C2×C4×C107—21no—
118C2×C4×C107—22no—
119C2×C4×C107—22no—
120C2×C4×C107—22no—
121C2×C4×C107—41no—
122C2×C4×C107—41no—
123C2×C4×C107C2×D411yescharacteristic
124C2×C4×C107C2×D411yescharacteristic
125C2×C4×C107C2×D411yescharacteristic
126C2×C4×C107C2×D411yescharacteristic
127C2×C4×C107C2×D411yescharacteristic
128C2×C4×C107C2411yescharacteristic
129C22⋊D107—28no—
130C22⋊D107—22no—
131C22⋊D107—22no—
132C2×Dic107—21no—
133C2×Dic107—22no—
134C2×Dic107—42no—
135D428—28no—
136D428—22no—
137D428—22no—
138D428—21no—
139D428—22no—
140D428—22no—
141D428C2×D412yes—
142C4×D107—28no—
143C4×D107—22no—
144C4×D107—22no—
145Dic214—21no—
146Dic214C2×D412yes—
147C8×C107—22no—
148C107⋊C8—22no—
149C22×C107—22no—
150C22×C107—22no—
151C22×C107—42no—
152C22×C107#12011yescharacteristic
153C22×C107#12011yescharacteristic
154C22×C107C22×D411yescharacteristic
155C2×D107—24no—
156C2×D107—22no—
157C2×D107—44no—
158C2×D107—42no—
159C2×D107—44no—
160C2×D107—82no—
161C2×D107—88no—
162C4×C107—22no—
163C4×C107—42no—
164C4×C107#12011yescharacteristic
165C4×C107#12011yescharacteristic
166C4×C107C22×D411yescharacteristic
167C4×C107C22×D411yescharacteristic
168Dic107—22no—
169Dic107—42no—
170C2×C107—21no—
171C2×C107—21no—
172C2×C107—42no—
173C2×C107#45512yescomplemented
174C2×C107C2×#12011yescharacteristic
175D107—44no—
176D107—42no—
177C2×#455—1071nomaximal, Sylow
178C107C2×#45511yescharacteristic, complemented, Sylow
179C2×H(5,𝔽2)—1071no—
180C2×#137—1071no—
181C2×#137D10711yescharacteristic, complemented
182C2×#124—1071no—
183#455—1078no—
184C2×#104—1071no—
185C2×#104—1071no—
186C2×#100—1071no—
187H(5,𝔽2)—1074no—
188H(5,𝔽2)—2142no—
189H(5,𝔽2)—2142no—
190H(5,𝔽2)—2142no—
191C2×#41—1071no—
192C2×#41—2141no—
193C2×#41—2141no—
194C2×#41C2×D10711yescharacteristic
195C22×Q8C2×D10711yescharacteristic, complemented
196C22×D4—1071no—
197C22×D4—2141no—
198C22×D4—2142no—
199C22×D4—2141no—
200#137—22no—
201#137—1074no—
202#137—2142no—
203#137C2×D10714yescomplemented
204C2×Q16—21no—
205C2×Q16—2141no—
206C2×SD16—21no—
207C2×SD16—2141no—
208C2×M16—1071no—
209C2×M16C2×D10711yescharacteristic, complemented
210#124—1074no—
211#124—2142no—
212C2×C22⋊C4—2142no—
213C2×C42—1071no—
214#104—1074no—
215#104—1074no—
216#100—1074no—
217C24—1072no—
218C24—2142no—
219#41—21no—
220#41—21no—
221#41—1072no—
222#41—1072no—
223#41—2144no—
224#41—2144no—
225#41—2144no—
226#41—2144no—
227#41—2141no—
228#41—2141no—
229#41C22⋊D10712yes—
230#41C22×D10712yes—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—1071no—
237C2×Q8C22⋊D10711yescharacteristic
238C2×Q8C22×D10712yescomplemented
239C2×D4—21no—
240C2×D4—1071no—
241C2×D4—1072no—
242C2×D4—1072no—
243C2×D4—2144no—
244C2×D4—2142no—
245C2×D4—2144no—
246C2×D4—2148no—
247C2×D4—2148no—
248C2×D4—2144no—
249C2×D4—2142no—
250C2×D4—2141no—
251C2×D4—2144no—
252C2×D4—2141no—
253C2×D4—2141no—
254C2×D4—2141no—
255C2×D4—2141no—
256C2×D4C22⋊D10711yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—2142no—
260C22×C4—2141no—
261C22×C4—2141no—
262C22×C4C22×D10711yescharacteristic
263Q16—24no—
264Q16—2144no—
265SD16—24no—
266SD16—2144no—
267M16—21no—
268M16—1072no—
269M16—2141no—
270M16C22×D10712yescomplemented
271C2×C8—21no—
272C2×C8—2141no—
273C22⋊C4—2148no—
274C42—1072no—
275C42—2141no—
276C23—21no—
277C23—8564no—
278C23—1074no—
279C23—4284no—
280C23—2148no—
281C23—2142no—
282C23—2141no—
283C23—2142no—
284C23D4×D10711yescharacteristic
285Q8—21no—
286Q8—22no—
287Q8—22no—
288Q8—41no—
289Q8—41no—
290Q8—1072no—
291Q8—2141no—
292Q8C2×C22⋊D10712yes—
293D4—22no—
294D4—22no—
295D4—21no—
296D4—1072no—
297D4—2148no—
298D4—2148no—
299D4—2142no—
300D4—2142no—
301D4—2141no—
302D4—2142no—
303D4—2142no—
304D4—2142no—
305D4—2142no—
306D4C2×C22⋊D10712yes—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—4282no—
314C2×C4—2148no—
315C2×C4—2142no—
316C2×C4—2141no—
317C2×C4—2142no—
318C2×C4—2142no—
319C2×C4D4×D10711yescharacteristic
320C2×C4D4×D10711yescharacteristic
321C2×C4D4×D10711yescharacteristic
322C2×C4C2×C22⋊D10711yescharacteristic
323C2×C4C2×C22⋊D10711yescharacteristic
324C2×C4C23×D10711yescharacteristic
325C8—22no—
326C8—2142no—
327C22—22no—
328C22—22no—
329C22—42no—
330C22—8562no—
331C22—8568no—
332C22—4284no—
333C22—4282no—
334C22—4284no—
335C22—2142no—
336C22—2144no—
337C22#1681207711yescharacteristic
338C22#1681207711yescharacteristic
339C22C2×D4×D10711yescharacteristic
340C4—22no—
341C4—42no—
342C4—4282no—
343C4—2142no—
344C4#1681207711yescharacteristic
345C4#1681207711yescharacteristic
346C4C2×D4×D10711yescharacteristic
347C4C22×C22⋊D10711yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—4282no—
352C2—4284no—
353C2#2029280312yesdirect product factor
354C2C2×#1681207711yescharacteristic
subgroupG/H
centerC22#16812077
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#137×C107C2
socleC22×C107#120
prime radicalC2×#16812155C2
hypercenterC2×#137D107
hypocenterC107C2×#455
LCSG ▷ C2×C4×C107 ▷ C2×C107 ▷ C107
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#16812140C2semidirect
3. C2×#16812098C2semidirect
4. #20292803C2direct (N × H)
5. C2×#16812036C2semidirect
6. C2×#16812035C2semidirect
7. C2×#16812025C2semidirect
8. C2×#137×C107C2semidirect
21. #16812140C22semidirect
24. C2×#13133802C22semidirect
26. #16812098C22semidirect
28. C2×C4×Dic107C22semidirect
29. #16812036C22semidirect
30. #16812035C22semidirect
31. #16812025C22semidirect
33. C22×Q8×C107C22semidirect
35. #137×C107C22semidirect
38. C2×M16×C107C22semidirect
72. #13133802C23semidirect
76. C4×Dic107C23semidirect
87. C2×Q8×C107C23semidirect
96. M16×C107C23semidirect
173. C2×C107#455semidirect
178. C107C2×#455semidirect
181. C2×#137D107semidirect
195. C22×Q8C2×D107semidirect
203. #137C2×D107semidirect
209. C2×M16C2×D107semidirect
238. C2×Q8C22×D107semidirect
270. M16C22×D107semidirect
353. C2#20292803direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168121552
C2×#168121402
C2×#168120982
#202928032
C2×#168120362
C2×#168120352
C2×#168120252
C2×#137×C1072
C2×#455107