Finite Group Databasebeta

← C2×#27132842 #37581851

Subgroups — C2×#27132842 #37581851

Order 24448.

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×#20259368C211yesmaximal, characteristic
2C2×#20259355C211yesmaximal, characteristic, complemented
3C2×#20259352C211yesmaximal, characteristic, complemented
4C2×#20259310C211yesmaximal, characteristic, complemented
5#27132842C218yesmaximal, direct product factor
6C2×#20259250C211yesmaximal, characteristic, complemented
7C2×#20259238C211yesmaximal, characteristic, complemented
8C2×#104×C191C211yesmaximal, characteristic, complemented
9#20259368—22no—
10#20259368—22no—
11#20259368—22no—
12#20259368C2214yes—
13C2×#15690288—21no—
14C2×#15690287—21no—
15C2×#15690287C2211yescharacteristic
16C22×Dic382C2211yescharacteristic, complemented
17C22×D764—21no—
18C22×D764C2211yescharacteristic
19C2×D4×D191—22no—
20C2×D4×D191—21no—
21#20259355—22no—
22#20259355C2214yescomplemented
23#20259352—22no—
24#20259352C2214yescomplemented
25C2×Dic764—21no—
26C2×#15690277—21no—
27C2×#15690276—21no—
28C2×#15690275—21no—
29C2×#15690271C2211yescharacteristic, complemented
30#20259310—22no—
31#20259310C2214yescomplemented
32C2×(C2×D191)⋊C4—22no—
33#20259250C2214yescomplemented
34#20259238C2214yescomplemented
35C2×#41×C191C2211yescharacteristic
36C2×M16×C191C2211yescharacteristic, complemented
37C2×C42×C191C2211yescharacteristic, complemented
38#104×C191C2214yescomplemented
39C23×D191—22no—
40C23×D191D412yes—
41#15690288—24no—
42#15690288—24no—
43#15690287—24no—
44#15690287—24no—
45#15690287—21no—
46#15690287—21no—
47#15690287D412yes—
48#15690287C2312yes—
49C2×Dic382—21no—
50C2×Dic382—21no—
51C2×Dic382—21no—
52C2×Dic382—21no—
53C2×Dic382—21no—
54C2×Dic382D411yescharacteristic
55C2×Dic382C2312yescomplemented
56C2×D764—22no—
57C2×D764—24no—
58C2×D764—24no—
59C2×D764—21no—
60C2×D764—21no—
61C2×D764—21no—
62C2×D764—21no—
63C2×D764D412yes—
64C2×D764D411yescharacteristic
65C2×D764C2312yes—
66C2×C22⋊D191—22no—
67C2×C22⋊D191—21no—
68D4×D191—28no—
69D4×D191—24no—
70D4×D191—28no—
71D4×D191—24no—
72C22×Dic191—21no—
73C2×C4×D191—22no—
74C2×C4×D191—21no—
75Dic764—24no—
76#15690277—24no—
77#15690276—24no—
78#15690275—24no—
79#15690271—21no—
80#15690271C2312yescomplemented
81C2×C191⋊C8—21no—
82(C2×D191)⋊C4—28no—
83#41×C191—21no—
84#41×C191—21no—
85#41×C191D412yes—
86#41×C191C2312yes—
87C2×Q8×C191D411yescharacteristic
88C2×D4×C191—21no—
89C2×D4×C191D411yescharacteristic
90C22×C4×C191—21no—
91C22×C4×C191—21no—
92C22×C4×C191C2311yescharacteristic
93M16×C191—21no—
94M16×C191C2312yescomplemented
95C2×C8×C191—21no—
96C42×C191—21no—
97C42×C191C2312yescomplemented
98C22×D191—22no—
99C22×D191—21no—
100C22×D191—28no—
101C22×D191—22no—
102C22×D191—44no—
103C22×D191—84no—
104C22×D191C2×D414yes—
105Dic382—21no—
106Dic382—22no—
107Dic382—22no—
108Dic382—41no—
109Dic382—41no—
110Dic382C2×D412yes—
111D764—28no—
112D764—22no—
113D764—22no—
114D764—21no—
115D764—22no—
116D764—22no—
117D764C2×D412yes—
118C22⋊D191—28no—
119C22⋊D191—22no—
120C22⋊D191—22no—
121C2×Dic191—21no—
122C2×Dic191—22no—
123C2×Dic191—41no—
124C2×Dic191—41no—
125C4×D191—28no—
126C4×D191—22no—
127C4×D191—22no—
128C191⋊C8—22no—
129C23×C191—21no—
130C23×C191C2×D411yescharacteristic
131Q8×C191—21no—
132Q8×C191C2×D412yes—
133D4×C191—22no—
134D4×C191—22no—
135D4×C191—21no—
136D4×C191C2×D412yes—
137C2×C4×C191—21no—
138C2×C4×C191—22no—
139C2×C4×C191—22no—
140C2×C4×C191—22no—
141C2×C4×C191—42no—
142C2×C4×C191C2×D411yescharacteristic
143C2×C4×C191C2×D411yescharacteristic
144C2×C4×C191C2×D411yescharacteristic
145C2×C4×C191C2×D411yescharacteristic
146C2×C4×C191C2×D411yescharacteristic
147C2×C4×C191C2411yescharacteristic
148C8×C191—22no—
149C22×C191—22no—
150C22×C191—22no—
151C22×C191—42no—
152C22×C191#12011yescharacteristic
153C22×C191#12011yescharacteristic
154C22×C191C22×D411yescharacteristic
155C2×D191—24no—
156C2×D191—22no—
157C2×D191—44no—
158C2×D191—42no—
159C2×D191—44no—
160C2×D191—82no—
161C2×D191—88no—
162C4×C191—22no—
163C4×C191—42no—
164C4×C191#12011yescharacteristic
165C4×C191#12011yescharacteristic
166C4×C191C22×D411yescharacteristic
167C4×C191C22×D411yescharacteristic
168Dic191—22no—
169Dic191—42no—
170C2×C191—21no—
171C2×C191—21no—
172C2×C191—42no—
173C2×C191#45512yescomplemented
174C2×C191C2×#12011yescharacteristic
175D191—44no—
176D191—42no—
177C191C2×#45511yescharacteristic, complemented, Sylow
178C2×#455—1911nomaximal, Sylow
179C2×H(5,𝔽2)—1911no—
180C2×#137—1911no—
181C2×#137—1911no—
182C2×#124—1911no—
183#455—1918no—
184C2×#104—1911no—
185C2×#104D19111yescharacteristic, complemented
186C2×#100—1911no—
187H(5,𝔽2)—3822no—
188H(5,𝔽2)—3822no—
189H(5,𝔽2)—3822no—
190H(5,𝔽2)—1914no—
191C2×#41—3821no—
192C2×#41—3821no—
193C2×#41—1911no—
194C2×#41C2×D19111yescharacteristic
195C22×Q8—1911no—
196C22×D4—3821no—
197C22×D4—3822no—
198C22×D4—3821no—
199C22×D4—1911no—
200#137—3822no—
201#137—3822no—
202#137—1914no—
203#137—1914no—
204C2×Q16—3821no—
205C2×Q16—3821no—
206C2×SD16—3821no—
207C2×SD16—3821no—
208C2×M16—1911no—
209C2×M16C2×D19111yescharacteristic, complemented
210#124—3822no—
211#124—1914no—
212C2×C22⋊C4—3822no—
213C2×C42C2×D19111yescharacteristic, complemented
214#104—1914no—
215#104C2×D19114yescomplemented
216#100—1914no—
217C24—3822no—
218C24—1912no—
219#41—21no—
220#41—21no—
221#41—3824no—
222#41—3824no—
223#41—3824no—
224#41—3824no—
225#41—3821no—
226#41—3821no—
227#41—1912no—
228#41—1912no—
229#41D76412yes—
230#41C22×D19112yes—
231C2×Q8—3821no—
232C2×Q8—3821no—
233C2×Q8—3821no—
234C2×Q8—3821no—
235C2×Q8—3821no—
236C2×Q8—1911no—
237C2×Q8—1912no—
238C2×Q8D76411yescharacteristic
239C2×D4—21no—
240C2×D4—3822no—
241C2×D4—3824no—
242C2×D4—3824no—
243C2×D4—3828no—
244C2×D4—3824no—
245C2×D4—3828no—
246C2×D4—3822no—
247C2×D4—3821no—
248C2×D4—3824no—
249C2×D4—3821no—
250C2×D4—3821no—
251C2×D4—3821no—
252C2×D4—3821no—
253C2×D4—1912no—
254C2×D4—1911no—
255C2×D4—1912no—
256C2×D4D76411yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—3822no—
260C22×C4—3821no—
261C22×C4—3821no—
262C22×C4C22×D19111yescharacteristic
263Q16—3824no—
264Q16—3824no—
265SD16—3824no—
266SD16—3824no—
267M16—21no—
268M16—3821no—
269M16—1912no—
270M16C22×D19112yescomplemented
271C2×C8—21no—
272C2×C8—3821no—
273C22⋊C4—3828no—
274C42—21no—
275C42C22×D19112yescomplemented
276C23—21no—
277C23—3828no—
278C23—3822no—
279C23—3822no—
280C23—3821no—
281C23—1914no—
282C23—15284no—
283C23—7644no—
284C23D4×D19111yescharacteristic
285Q8—21no—
286Q8—3821no—
287Q8—3822no—
288Q8—3822no—
289Q8—1912no—
290Q8—7641no—
291Q8—7641no—
292Q8C2×D76412yes—
293D4—22no—
294D4—22no—
295D4—21no—
296D4—3828no—
297D4—3828no—
298D4—3822no—
299D4—3822no—
300D4—3822no—
301D4—3822no—
302D4—3822no—
303D4—3821no—
304D4—3822no—
305D4—1912no—
306D4C2×D76412yes—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—42no—
312C2×C4—3828no—
313C2×C4—3821no—
314C2×C4—3822no—
315C2×C4—3822no—
316C2×C4—3822no—
317C2×C4—7641no—
318C2×C4—7641no—
319C2×C4D4×D19111yescharacteristic
320C2×C4D4×D19111yescharacteristic
321C2×C4D4×D19111yescharacteristic
322C2×C4C2×D76411yescharacteristic
323C2×C4C2×D76411yescharacteristic
324C2×C4C23×D19111yescharacteristic
325C8—22no—
326C8—3822no—
327C22—22no—
328C22—22no—
329C22—42no—
330C22—3824no—
331C22—3822no—
332C22—15282no—
333C22—15288no—
334C22—7644no—
335C22—7642no—
336C22—7644no—
337C22#2025929211yescharacteristic
338C22#2025929211yescharacteristic
339C22C2×D4×D19111yescharacteristic
340C4—22no—
341C4—42no—
342C4—3822no—
343C4—7642no—
344C4#2025929211yescharacteristic
345C4#2025929211yescharacteristic
346C4C2×D4×D19111yescharacteristic
347C4C22×D76411yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—7642no—
352C2—7644no—
353C2#2713284212yesdirect product factor
354C2C2×#2025929211yescharacteristic
subgroupG/H
centerC22#20259292
commutator, cosocleC2×C4×C191C24
FrattiniC2×C4C23×D191
FittingC2×#104×C191C2
socleC22×C191#120
prime radicalC2×#20259368C2
hypercenterC2×#104D191
hypocenterC191C2×#455
LCSG ▷ C2×C4×C191 ▷ C2×C191 ▷ C191
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#20259355C2semidirect
3. C2×#20259352C2semidirect
4. C2×#20259310C2semidirect
5. #27132842C2direct (N × H)
6. C2×#20259250C2semidirect
7. C2×#20259238C2semidirect
8. C2×#104×C191C2semidirect
16. C22×Dic382C22semidirect
22. #20259355C22semidirect
24. #20259352C22semidirect
29. C2×#15690271C22semidirect
31. #20259310C22semidirect
33. #20259250C22semidirect
34. #20259238C22semidirect
36. C2×M16×C191C22semidirect
37. C2×C42×C191C22semidirect
38. #104×C191C22semidirect
55. C2×Dic382C23semidirect
80. #15690271C23semidirect
94. M16×C191C23semidirect
97. C42×C191C23semidirect
173. C2×C191#455semidirect
177. C191C2×#455semidirect
185. C2×#104D191semidirect
209. C2×M16C2×D191semidirect
213. C2×C42C2×D191semidirect
215. #104C2×D191semidirect
270. M16C22×D191semidirect
275. C42C22×D191semidirect
353. C2#27132842direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#202593682
C2×#202593552
C2×#202593522
C2×#202593102
#271328422
C2×#202592502
C2×#202592382
C2×#104×C1912
C2×#455191