Finite Group Databasebeta

← C2×#20292811 #28345804

Subgroups — C2×#20292811 #28345804

Order 13696.

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

normal subgroups only

This group has 708 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×#16812156C211yesmaximal, characteristic
2C2×#16812142C211yesmaximal, characteristic, complemented
3C2×#16812139C211yesmaximal, characteristic, complemented
4C2×#16812109C211yesmaximal, characteristic, complemented
5#20292811C218yesmaximal, direct product factor
6C2×#16812037C211yesmaximal, characteristic, complemented
7C2×#16812027C211yesmaximal, characteristic, complemented
8C2×#104×C107C211yesmaximal, characteristic, complemented
9#16812156—22no—
10#16812156—22no—
11#16812156—22no—
12#16812156C2214yes—
13C2×#13133818—21no—
14C2×#13133818C2211yescharacteristic
15C2×#13133817—22no—
16C2×#13133817—21no—
17C22×Dic214—21no—
18C22×Dic214C2211yescharacteristic
19C22×Dic214C2211yescharacteristic
20C2×Q8×D107—21no—
21#16812142—22no—
22#16812142C2214yescomplemented
23#16812139—22no—
24#16812139C2214yescomplemented
25C2×Dic428—21no—
26C2×#13133808—21no—
27C2×#13133807—21no—
28C2×#13133806—21no—
29C2×#13133802C2211yescharacteristic
30#16812109—22no—
31#16812109C2214yescomplemented
32C2×C4⋊Dic107—22no—
33#16812037C2214yescomplemented
34#16812027C2214yescomplemented
35C2×#41×C107C2211yescharacteristic
36C2×M16×C107C2211yescharacteristic
37C2×C42×C107C2211yescharacteristic
38#104×C107C2214yescomplemented
39#13133818—24no—
40#13133818—21no—
41#13133818—21no—
42#13133818—24no—
43#13133818D412yes—
44#13133818C2312yes—
45#13133817—28no—
46#13133817—24no—
47#13133817—24no—
48#13133817—28no—
49C2×Dic214—22no—
50C2×Dic214—24no—
51C2×Dic214—21no—
52C2×Dic214—21no—
53C2×Dic214—21no—
54C2×Dic214—21no—
55C2×Dic214—24no—
56C2×Dic214—21no—
57C2×Dic214—21no—
58C2×Dic214—21no—
59C2×Dic214—21no—
60C2×Dic214—21no—
61C2×Dic214D412yes—
62C2×Dic214D411yescharacteristic
63C2×Dic214C2312yes—
64C2×Dic214C2312yes—
65Q8×D107—24no—
66Q8×D107—24no—
67C2×D428D411yescharacteristic
68C2×C22⋊D107—22no—
69C2×C22⋊D107—21no—
70C22×Dic107—22no—
71C22×Dic107—21no—
72C22×Dic107D412yes—
73C2×C4×D107—22no—
74C2×C4×D107—21no—
75Dic428—24no—
76#13133808—24no—
77#13133807—24no—
78#13133806—24no—
79#13133802—21no—
80#13133802C2312yes—
81C2×C107⋊C8—21no—
82C4⋊Dic107—28no—
83#41×C107—21no—
84#41×C107—21no—
85#41×C107D412yes—
86#41×C107C2312yes—
87C2×Q8×C107D411yescharacteristic
88C2×D4×C107—21no—
89C2×D4×C107D411yescharacteristic
90C22×C4×C107—21no—
91C22×C4×C107—21no—
92C22×C4×C107C2311yescharacteristic
93M16×C107—21no—
94M16×C107C2312yes—
95C2×C8×C107—21no—
96C42×C107—21no—
97C42×C107C2312yes—
98C23×C107—21no—
99C23×C107C2×D411yescharacteristic
100C22×D107—21no—
101Q8×C107—21no—
102Q8×C107C2×D412yes—
103D4×C107—22no—
104D4×C107—22no—
105D4×C107—21no—
106D4×C107C2×D412yes—
107C2×C4×C107—24no—
108C2×C4×C107—21no—
109C2×C4×C107—22no—
110C2×C4×C107—22no—
111C2×C4×C107—22no—
112C2×C4×C107C2×D411yescharacteristic
113C2×C4×C107C2×D411yescharacteristic
114C2×C4×C107C2×D411yescharacteristic
115C2×C4×C107C2×D411yescharacteristic
116C2×C4×C107C2×D411yescharacteristic
117C2×C4×C107C2411yescharacteristic
118C22⋊D107—28no—
119C22⋊D107—22no—
120C22⋊D107—22no—
121C2×Dic107—28no—
122C2×Dic107—21no—
123C2×Dic107—22no—
124C2×Dic107—22no—
125C2×Dic107—22no—
126C2×Dic107—41no—
127C2×Dic107—41no—
128C2×Dic107C2×D414yes—
129D428—21no—
130D428C2×D412yes—
131C4×D107—28no—
132C4×D107—22no—
133C4×D107—22no—
134Dic214—28no—
135Dic214—22no—
136Dic214—22no—
137Dic214—21no—
138Dic214—22no—
139Dic214—22no—
140Dic214—22no—
141Dic214—22no—
142Dic214—41no—
143Dic214—41no—
144Dic214C2×D412yes—
145C8×C107—22no—
146C107⋊C8—22no—
147C22×C107—22no—
148C22×C107—22no—
149C22×C107—42no—
150C22×C107#12011yescharacteristic
151C22×C107#12011yescharacteristic
152C22×C107C22×D411yescharacteristic
153C2×D107—22no—
154C2×D107—42no—
155C4×C107—24no—
156C4×C107—22no—
157C4×C107#12011yescharacteristic
158C4×C107#12011yescharacteristic
159C4×C107C22×D411yescharacteristic
160C4×C107C22×D411yescharacteristic
161Dic107—24no—
162Dic107—22no—
163Dic107—42no—
164C2×C107—21no—
165C2×C107—21no—
166C2×C107—42no—
167C2×C107#45612yescomplemented
168C2×C107C2×#12011yescharacteristic
169D107—42no—
170C2×#456—1071nomaximal, Sylow
171C107C2×#45611yescharacteristic, complemented, Sylow
172C2×2−1+4—1071no—
173C2×#137—1071no—
174C2×#137—1071no—
175C2×C4⋊Q8—1071no—
176#456—1078no—
177C2×#104—1071no—
178C2×#104D10711yescharacteristic, complemented
179C2×#101—1071no—
1802−1+4—1074no—
1812−1+4—2142no—
1822−1+4—2142no—
1832−1+4—2142no—
184C2×#41—1071no—
185C2×#41—2141no—
186C2×#41—2142no—
187C2×#41—2141no—
188C2×#41C2×D10711yescharacteristic
189C22×Q8—1071no—
190C22×Q8—1071no—
191C22×Q8—2141no—
192C22×Q8—2141no—
193#137—1074no—
194#137—1074no—
195#137—2142no—
196#137—2142no—
197C2×Q16—2141no—
198C2×Q16—2141no—
199C2×SD16—2141no—
200C2×SD16—2141no—
201C2×M16—1071no—
202C2×M16C2×D10711yescharacteristic
203C4⋊Q8—1074no—
204C4⋊Q8—2142no—
205C2×C4⋊C4—2142no—
206C2×C42C2×D10711yescharacteristic
207#104—1074no—
208#104C2×D10714yescomplemented
209#101—1074no—
210#41—21no—
211#41—21no—
212#41—1072no—
213#41—1072no—
214#41—2144no—
215#41—2144no—
216#41—2148no—
217#41—2148no—
218#41—2144no—
219#41—2144no—
220#41—2141no—
221#41—2141no—
222#41D42812yes—
223#41C22×D10712yes—
224C2×Q8—1072no—
225C2×Q8—1072no—
226C2×Q8—1071no—
227C2×Q8—1072no—
228C2×Q8—2144no—
229C2×Q8—2142no—
230C2×Q8—2144no—
231C2×Q8—2141no—
232C2×Q8—2141no—
233C2×Q8—2141no—
234C2×Q8—2141no—
235C2×Q8—2144no—
236C2×Q8—2144no—
237C2×Q8—2141no—
238C2×Q8—2141no—
239C2×Q8—2141no—
240C2×Q8—2141no—
241C2×Q8—2141no—
242C2×Q8D42811yescharacteristic
243C2×D4—21no—
244C2×D4—1071no—
245C2×D4—2142no—
246C2×D4—2141no—
247C2×D4D42811yescharacteristic
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—1072no—
251C22×C4—2142no—
252C22×C4—2141no—
253C22×C4—2142no—
254C22×C4—2141no—
255C22×C4C22×D10711yescharacteristic
256Q16—2144no—
257Q16—2144no—
258SD16—2144no—
259SD16—2144no—
260M16—21no—
261M16—1072no—
262M16—2141no—
263M16C22×D10712yes—
264C2×C8—21no—
265C2×C8—2141no—
266C4⋊C4—2148no—
267C42—21no—
268C42C22×D10712yes—
269C23—21no—
270C23—2141no—
271C23D4×D10711yescharacteristic
272Q8—21no—
273Q8—1072no—
274Q8—4281no—
275Q8—4281no—
276Q8—2148no—
277Q8—2142no—
278Q8—2142no—
279Q8—2142no—
280Q8—2142no—
281Q8—2141no—
282Q8—2142no—
283Q8—2142no—
284Q8C2×D42812yes—
285D4—22no—
286D4—22no—
287D4—21no—
288D4—1072no—
289D4—2148no—
290D4—2142no—
291D4—2142no—
292D4—2141no—
293D4C2×D42812yes—
294C2×C4—24no—
295C2×C4—21no—
296C2×C4—22no—
297C2×C4—22no—
298C2×C4—22no—
299C2×C4—1074no—
300C2×C4—4281no—
301C2×C4—4281no—
302C2×C4—2148no—
303C2×C4—2142no—
304C2×C4—2148no—
305C2×C4—2142no—
306C2×C4—2142no—
307C2×C4—2141no—
308C2×C4—2142no—
309C2×C4—2142no—
310C2×C4D4×D10711yescharacteristic
311C2×C4D4×D10711yescharacteristic
312C2×C4D4×D10711yescharacteristic
313C2×C4C2×D42811yescharacteristic
314C2×C4C2×D42811yescharacteristic
315C2×C4C23×D10711yescharacteristic
316C8—22no—
317C8—2142no—
318C22—22no—
319C22—22no—
320C22—42no—
321C22—4282no—
322C22—2142no—
323C22#1681207911yescharacteristic
324C22#1681207911yescharacteristic
325C22C2×D4×D10711yescharacteristic
326C4—24no—
327C4—22no—
328C4—4282no—
329C4—2144no—
330C4—2142no—
331C4#1681207911yescharacteristic
332C4#1681207911yescharacteristic
333C4C2×D4×D10711yescharacteristic
334C4C22×D42811yescharacteristic
335C2—21no—
336C2—21no—
337C2—42no—
338C2—4282no—
339C2#2029281112yesdirect product factor
340C2C2×#1681207911yescharacteristic
subgroupG/H
centerC22#16812079
commutator, cosocleC2×C4×C107C24
FrattiniC2×C4C23×D107
FittingC2×#104×C107C2
socleC22×C107#120
prime radicalC2×#16812156C2
hypercenterC2×#104D107
hypocenterC107C2×#456
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×#16812142C2semidirect
3. C2×#16812139C2semidirect
4. C2×#16812109C2semidirect
5. #20292811C2direct (N × H)
6. C2×#16812037C2semidirect
7. C2×#16812027C2semidirect
8. C2×#104×C107C2semidirect
22. #16812142C22semidirect
24. #16812139C22semidirect
31. #16812109C22semidirect
33. #16812037C22semidirect
34. #16812027C22semidirect
38. #104×C107C22semidirect
167. C2×C107#456semidirect
171. C107C2×#456semidirect
178. C2×#104D107semidirect
208. #104C2×D107semidirect
339. C2#20292811direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#168121562
C2×#168121422
C2×#168121392
C2×#168121092
#202928112
C2×#168120372
C2×#168120272
C2×#104×C1072
C2×#456107