Finite Group Databasebeta

← C2×#30631863 #44559693

Subgroups — C2×#30631863 #44559693

Order 32128.

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×#22548761C211yesmaximal, characteristic
2C2×#22548746C211yesmaximal, characteristic, complemented
3C2×#22548704C211yesmaximal, characteristic, complemented
4#30631863C218yesmaximal, direct product factor
5C2×#22548642C211yesmaximal, characteristic, complemented
6C2×#22548641C211yesmaximal, characteristic, complemented
7C2×#22548631C211yesmaximal, characteristic, complemented
8C2×#137×C251C211yesmaximal, characteristic, complemented
9#22548761—22no—
10#22548761—22no—
11#22548761—22no—
12#22548761C2214yes—
13C2×#16860811—21no—
14C2×#16860810—21no—
15C2×#16860810C2211yescharacteristic
16C22×D1004—21no—
17C22×D1004C2211yescharacteristic
18C2×D4×D251—22no—
19C2×D4×D251—21no—
20#22548746—22no—
21#22548746C2214yescomplemented
22C2×#16860800—21no—
23C2×#16860797—21no—
24C2×#16860794C2211yescharacteristic, complemented
25#22548704—22no—
26#22548704C2214yescomplemented
27C2×(C2×D251)⋊C4—22no—
28C2×C4×Dic251C2211yescharacteristic, complemented
29#22548642C2214yescomplemented
30#22548641C2214yescomplemented
31#22548631C2214yescomplemented
32C2×#41×C251C2211yescharacteristic
33C22×Q8×C251C2211yescharacteristic, complemented
34#137×C251—22no—
35#137×C251C2214yescomplemented
36C2×Q16×C251—21no—
37C2×SD16×C251—21no—
38C2×M16×C251C2211yescharacteristic, complemented
39C23×D251—22no—
40C23×D251D412yes—
41#16860811—24no—
42#16860811—24no—
43#16860810—24no—
44#16860810—24no—
45#16860810—21no—
46#16860810—21no—
47#16860810D412yes—
48#16860810C2312yes—
49C2×Dic502D411yescharacteristic
50C2×D1004—22no—
51C2×D1004—24no—
52C2×D1004—24no—
53C2×D1004—21no—
54C2×D1004—21no—
55C2×D1004—21no—
56C2×D1004—21no—
57C2×D1004D412yes—
58C2×D1004D411yescharacteristic
59C2×D1004C2312yes—
60C2×C22⋊D251—22no—
61C2×C22⋊D251—21no—
62D4×D251—28no—
63D4×D251—24no—
64D4×D251—28no—
65D4×D251—24no—
66C22×Dic251—21no—
67C2×C4×D251—22no—
68C2×C4×D251—21no—
69#16860800—24no—
70#16860797—24no—
71#16860794—21no—
72#16860794C2312yescomplemented
73C2×C251⋊C8—21no—
74(C2×D251)⋊C4—28no—
75C4×Dic251—21no—
76C4×Dic251C2312yescomplemented
77#41×C251—21no—
78#41×C251—21no—
79#41×C251D412yes—
80#41×C251C2312yes—
81C2×Q8×C251—21no—
82C2×Q8×C251—21no—
83C2×Q8×C251—21no—
84C2×Q8×C251—21no—
85C2×Q8×C251—21no—
86C2×Q8×C251D411yescharacteristic
87C2×Q8×C251C2312yescomplemented
88C2×D4×C251—21no—
89C2×D4×C251D411yescharacteristic
90C22×C4×C251—21no—
91C22×C4×C251—21no—
92C22×C4×C251C2311yescharacteristic
93Q16×C251—24no—
94SD16×C251—24no—
95M16×C251—21no—
96M16×C251C2312yescomplemented
97C2×C8×C251—21no—
98C22×D251—22no—
99C22×D251—21no—
100C22×D251—28no—
101C22×D251—22no—
102C22×D251—44no—
103C22×D251—84no—
104C22×D251C2×D414yes—
105Dic502—21no—
106Dic502C2×D412yes—
107D1004—28no—
108D1004—22no—
109D1004—22no—
110D1004—21no—
111D1004—22no—
112D1004—22no—
113D1004C2×D412yes—
114C22⋊D251—28no—
115C22⋊D251—22no—
116C22⋊D251—22no—
117C2×Dic251—21no—
118C2×Dic251—22no—
119C2×Dic251—42no—
120C4×D251—28no—
121C4×D251—22no—
122C4×D251—22no—
123C251⋊C8—22no—
124C23×C251—21no—
125C23×C251C2×D411yescharacteristic
126Q8×C251—21no—
127Q8×C251—22no—
128Q8×C251—22no—
129Q8×C251—41no—
130Q8×C251—41no—
131Q8×C251C2×D412yes—
132D4×C251—22no—
133D4×C251—22no—
134D4×C251—21no—
135D4×C251C2×D412yes—
136C2×C4×C251—21no—
137C2×C4×C251—22no—
138C2×C4×C251—22no—
139C2×C4×C251—22no—
140C2×C4×C251—41no—
141C2×C4×C251—41no—
142C2×C4×C251C2×D411yescharacteristic
143C2×C4×C251C2×D411yescharacteristic
144C2×C4×C251C2×D411yescharacteristic
145C2×C4×C251C2×D411yescharacteristic
146C2×C4×C251C2×D411yescharacteristic
147C2×C4×C251C2411yescharacteristic
148C8×C251—22no—
149C22×C251—22no—
150C22×C251—22no—
151C22×C251—42no—
152C22×C251#12011yescharacteristic
153C22×C251#12011yescharacteristic
154C22×C251C22×D411yescharacteristic
155C2×D251—24no—
156C2×D251—22no—
157C2×D251—44no—
158C2×D251—42no—
159C2×D251—44no—
160C2×D251—82no—
161C2×D251—88no—
162C4×C251—22no—
163C4×C251—42no—
164C4×C251#12011yescharacteristic
165C4×C251#12011yescharacteristic
166C4×C251C22×D411yescharacteristic
167C4×C251C22×D411yescharacteristic
168Dic251—22no—
169Dic251—42no—
170C2×C251—21no—
171C2×C251—21no—
172C2×C251—42no—
173C2×C251#45512yescomplemented
174C2×C251C2×#12011yescharacteristic
175D251—44no—
176D251—42no—
177C251C2×#45511yescharacteristic, complemented, Sylow
178C2×#455—2511nomaximal, Sylow
179C2×H(5,𝔽2)—2511no—
180C2×#137—2511no—
181C2×#137D25111yescharacteristic, complemented
182C2×#124—2511no—
183#455—2518no—
184C2×#104—2511no—
185C2×#104—2511no—
186C2×#100—2511no—
187H(5,𝔽2)—5022no—
188H(5,𝔽2)—5022no—
189H(5,𝔽2)—5022no—
190H(5,𝔽2)—2514no—
191C2×#41—5021no—
192C2×#41—5021no—
193C2×#41—2511no—
194C2×#41C2×D25111yescharacteristic
195C22×Q8C2×D25111yescharacteristic, complemented
196C22×D4—5021no—
197C22×D4—5022no—
198C22×D4—5021no—
199C22×D4—2511no—
200#137—22no—
201#137—5022no—
202#137—2514no—
203#137C2×D25114yescomplemented
204C2×Q16—21no—
205C2×Q16—5021no—
206C2×SD16—21no—
207C2×SD16—5021no—
208C2×M16—2511no—
209C2×M16C2×D25111yescharacteristic, complemented
210#124—5022no—
211#124—2514no—
212C2×C22⋊C4—5022no—
213C2×C42—2511no—
214#104—2514no—
215#104—2514no—
216#100—2514no—
217C24—5022no—
218C24—2512no—
219#41—21no—
220#41—21no—
221#41—5024no—
222#41—5024no—
223#41—5024no—
224#41—5024no—
225#41—5021no—
226#41—5021no—
227#41—2512no—
228#41—2512no—
229#41C22⋊D25112yes—
230#41C22×D25112yes—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—2511no—
237C2×Q8C22⋊D25111yescharacteristic
238C2×Q8C22×D25112yescomplemented
239C2×D4—21no—
240C2×D4—5022no—
241C2×D4—5024no—
242C2×D4—5024no—
243C2×D4—5028no—
244C2×D4—5024no—
245C2×D4—5028no—
246C2×D4—5022no—
247C2×D4—5021no—
248C2×D4—5024no—
249C2×D4—5021no—
250C2×D4—5021no—
251C2×D4—5021no—
252C2×D4—5021no—
253C2×D4—2512no—
254C2×D4—2511no—
255C2×D4—2512no—
256C2×D4C22⋊D25111yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—5022no—
260C22×C4—5021no—
261C22×C4—5021no—
262C22×C4C22×D25111yescharacteristic
263Q16—24no—
264Q16—5024no—
265SD16—24no—
266SD16—5024no—
267M16—21no—
268M16—5021no—
269M16—2512no—
270M16C22×D25112yescomplemented
271C2×C8—21no—
272C2×C8—5021no—
273C22⋊C4—5028no—
274C42—5021no—
275C42—2512no—
276C23—21no—
277C23—20084no—
278C23—10044no—
279C23—5028no—
280C23—5022no—
281C23—5022no—
282C23—5021no—
283C23—2514no—
284C23D4×D25111yescharacteristic
285Q8—21no—
286Q8—22no—
287Q8—22no—
288Q8—41no—
289Q8—41no—
290Q8—5021no—
291Q8—2512no—
292Q8C2×C22⋊D25112yes—
293D4—22no—
294D4—22no—
295D4—21no—
296D4—5028no—
297D4—5028no—
298D4—5022no—
299D4—5022no—
300D4—5022no—
301D4—5022no—
302D4—5022no—
303D4—5021no—
304D4—5022no—
305D4—2512no—
306D4C2×C22⋊D25112yes—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—10042no—
314C2×C4—5028no—
315C2×C4—5022no—
316C2×C4—5021no—
317C2×C4—5022no—
318C2×C4—5022no—
319C2×C4D4×D25111yescharacteristic
320C2×C4D4×D25111yescharacteristic
321C2×C4D4×D25111yescharacteristic
322C2×C4C2×C22⋊D25111yescharacteristic
323C2×C4C2×C22⋊D25111yescharacteristic
324C2×C4C23×D25111yescharacteristic
325C8—22no—
326C8—5022no—
327C22—22no—
328C22—22no—
329C22—42no—
330C22—20082no—
331C22—20088no—
332C22—10044no—
333C22—10042no—
334C22—10044no—
335C22—5024no—
336C22—5022no—
337C22#2254868311yescharacteristic
338C22#2254868311yescharacteristic
339C22C2×D4×D25111yescharacteristic
340C4—22no—
341C4—42no—
342C4—10042no—
343C4—5022no—
344C4#2254868311yescharacteristic
345C4#2254868311yescharacteristic
346C4C2×D4×D25111yescharacteristic
347C4C22×C22⋊D25111yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—10042no—
352C2—10044no—
353C2#3063186312yesdirect product factor
354C2C2×#2254868311yescharacteristic
subgroupG/H
centerC22#22548683
commutator, cosocleC2×C4×C251C24
FrattiniC2×C4C23×D251
FittingC2×#137×C251C2
socleC22×C251#120
prime radicalC2×#22548761C2
hypercenterC2×#137D251
hypocenterC251C2×#455
LCSG ▷ C2×C4×C251 ▷ C2×C251 ▷ C251
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#22548746C2semidirect
3. C2×#22548704C2semidirect
4. #30631863C2direct (N × H)
5. C2×#22548642C2semidirect
6. C2×#22548641C2semidirect
7. C2×#22548631C2semidirect
8. C2×#137×C251C2semidirect
21. #22548746C22semidirect
24. C2×#16860794C22semidirect
26. #22548704C22semidirect
28. C2×C4×Dic251C22semidirect
29. #22548642C22semidirect
30. #22548641C22semidirect
31. #22548631C22semidirect
33. C22×Q8×C251C22semidirect
35. #137×C251C22semidirect
38. C2×M16×C251C22semidirect
72. #16860794C23semidirect
76. C4×Dic251C23semidirect
87. C2×Q8×C251C23semidirect
96. M16×C251C23semidirect
173. C2×C251#455semidirect
177. C251C2×#455semidirect
181. C2×#137D251semidirect
195. C22×Q8C2×D251semidirect
203. #137C2×D251semidirect
209. C2×M16C2×D251semidirect
238. C2×Q8C22×D251semidirect
270. M16C22×D251semidirect
353. C2#30631863direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#225487612
C2×#225487462
C2×#225487042
#306318632
C2×#225486422
C2×#225486412
C2×#225486312
C2×#137×C2512
C2×#455251