Finite Group Databasebeta

← C2×#26041455 #37543049

Subgroups — C2×#26041455 #37543049

Order 23168.

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×#19153862C211yesmaximal, characteristic
2C2×#19153847C211yesmaximal, characteristic, complemented
3C2×#19153805C211yesmaximal, characteristic, complemented
4#26041455C218yesmaximal, direct product factor
5C2×#19153743C211yesmaximal, characteristic, complemented
6C2×#19153742C211yesmaximal, characteristic, complemented
7C2×#19153732C211yesmaximal, characteristic, complemented
8C2×#137×C181C211yesmaximal, characteristic, complemented
9#19153862—22no—
10#19153862—22no—
11#19153862—22no—
12#19153862C2214yes—
13C2×#15668450—21no—
14C2×#15668449—21no—
15C2×#15668449C2211yescharacteristic
16C22×D724—21no—
17C22×D724C2211yescharacteristic
18C2×D4×D181—22no—
19C2×D4×D181—21no—
20#19153847—22no—
21#19153847C2214yescomplemented
22C2×#15668439—21no—
23C2×#15668436—21no—
24C2×#15668433C2211yescharacteristic, complemented
25#19153805—22no—
26#19153805C2214yescomplemented
27C2×#15668426—22no—
28C2×C4×Dic181C2211yescharacteristic, complemented
29#19153743C2214yescomplemented
30#19153742C2214yescomplemented
31#19153732C2214yescomplemented
32C2×#41×C181C2211yescharacteristic
33C22×Q8×C181C2211yescharacteristic, complemented
34#137×C181—22no—
35#137×C181C2214yescomplemented
36C2×Q16×C181—21no—
37C2×SD16×C181—21no—
38C2×M16×C181C2211yescharacteristic, complemented
39C23×D181—22no—
40C23×D181D412yes—
41#15668450—24no—
42#15668450—24no—
43#15668449—24no—
44#15668449—24no—
45#15668449—21no—
46#15668449—21no—
47#15668449D412yes—
48#15668449C2312yes—
49C2×Dic362D411yescharacteristic
50C2×D724—22no—
51C2×D724—24no—
52C2×D724—24no—
53C2×D724—21no—
54C2×D724—21no—
55C2×D724—21no—
56C2×D724—21no—
57C2×D724D412yes—
58C2×D724D411yescharacteristic
59C2×D724C2312yes—
60C2×C22⋊D181—22no—
61C2×C22⋊D181—21no—
62D4×D181—28no—
63D4×D181—24no—
64D4×D181—28no—
65D4×D181—24no—
66C22×Dic181—21no—
67C2×C4×D181—22no—
68C2×C4×D181—21no—
69#15668439—24no—
70#15668436—24no—
71#15668433—21no—
72#15668433C2312yescomplemented
73C2×#13089678—21no—
74#15668426—28no—
75C4×Dic181—21no—
76C4×Dic181C2312yescomplemented
77#41×C181—21no—
78#41×C181—21no—
79#41×C181D412yes—
80#41×C181C2312yes—
81C2×Q8×C181—21no—
82C2×Q8×C181—21no—
83C2×Q8×C181—21no—
84C2×Q8×C181—21no—
85C2×Q8×C181—21no—
86C2×Q8×C181D411yescharacteristic
87C2×Q8×C181C2312yescomplemented
88C2×D4×C181—21no—
89C2×D4×C181D411yescharacteristic
90C22×C4×C181—21no—
91C22×C4×C181—21no—
92C22×C4×C181C2311yescharacteristic
93Q16×C181—24no—
94SD16×C181—24no—
95M16×C181—21no—
96M16×C181C2312yescomplemented
97C2×C8×C181—21no—
98C22×D181—22no—
99C22×D181—21no—
100C22×D181—28no—
101C22×D181—22no—
102C22×D181—44no—
103C22×D181—84no—
104C22×D181C2×D414yes—
105Dic362—21no—
106Dic362C2×D412yes—
107D724—28no—
108D724—22no—
109D724—22no—
110D724—21no—
111D724—22no—
112D724—22no—
113D724C2×D412yes—
114C22⋊D181—28no—
115C22⋊D181—22no—
116C22⋊D181—22no—
117C2×Dic181—21no—
118C2×Dic181—22no—
119C2×Dic181—42no—
120C4×D181—28no—
121C4×D181—22no—
122C4×D181—22no—
123#13089678—22no—
124C23×C181—21no—
125C23×C181C2×D411yescharacteristic
126Q8×C181—21no—
127Q8×C181—22no—
128Q8×C181—22no—
129Q8×C181—41no—
130Q8×C181—41no—
131Q8×C181C2×D412yes—
132D4×C181—22no—
133D4×C181—22no—
134D4×C181—21no—
135D4×C181C2×D412yes—
136C2×C4×C181—21no—
137C2×C4×C181—22no—
138C2×C4×C181—22no—
139C2×C4×C181—22no—
140C2×C4×C181—41no—
141C2×C4×C181—41no—
142C2×C4×C181C2×D411yescharacteristic
143C2×C4×C181C2×D411yescharacteristic
144C2×C4×C181C2×D411yescharacteristic
145C2×C4×C181C2×D411yescharacteristic
146C2×C4×C181C2×D411yescharacteristic
147C2×C4×C181C2411yescharacteristic
148C8×C181—22no—
149C22×C181—22no—
150C22×C181—22no—
151C22×C181—42no—
152C22×C181#12011yescharacteristic
153C22×C181#12011yescharacteristic
154C22×C181C22×D411yescharacteristic
155C2×D181—24no—
156C2×D181—22no—
157C2×D181—44no—
158C2×D181—42no—
159C2×D181—44no—
160C2×D181—82no—
161C2×D181—88no—
162C4×C181—22no—
163C4×C181—42no—
164C4×C181#12011yescharacteristic
165C4×C181#12011yescharacteristic
166C4×C181C22×D411yescharacteristic
167C4×C181C22×D411yescharacteristic
168Dic181—22no—
169Dic181—42no—
170C2×C181—21no—
171C2×C181—21no—
172C2×C181—42no—
173C2×C181#45512yescomplemented
174C2×C181C2×#12011yescharacteristic
175D181—44no—
176D181—42no—
177C181C2×#45511yescharacteristic, complemented, Sylow
178C2×#455—1811nomaximal, Sylow
179C2×H(5,𝔽2)—1811no—
180C2×#137—1811no—
181C2×#137D18111yescharacteristic, complemented
182C2×#124—1811no—
183#455—1818no—
184C2×#104—1811no—
185C2×#104—1811no—
186C2×#100—1811no—
187H(5,𝔽2)—3622no—
188H(5,𝔽2)—3622no—
189H(5,𝔽2)—3622no—
190H(5,𝔽2)—1814no—
191C2×#41—3621no—
192C2×#41—3621no—
193C2×#41—1811no—
194C2×#41C2×D18111yescharacteristic
195C22×Q8C2×D18111yescharacteristic, complemented
196C22×D4—3621no—
197C22×D4—3622no—
198C22×D4—3621no—
199C22×D4—1811no—
200#137—22no—
201#137—3622no—
202#137—1814no—
203#137C2×D18114yescomplemented
204C2×Q16—21no—
205C2×Q16—3621no—
206C2×SD16—21no—
207C2×SD16—3621no—
208C2×M16—1811no—
209C2×M16C2×D18111yescharacteristic, complemented
210#124—3622no—
211#124—1814no—
212C2×C22⋊C4—3622no—
213C2×C42—1811no—
214#104—1814no—
215#104—1814no—
216#100—1814no—
217C24—3622no—
218C24—1812no—
219#41—21no—
220#41—21no—
221#41—3624no—
222#41—3624no—
223#41—3624no—
224#41—3624no—
225#41—3621no—
226#41—3621no—
227#41—1812no—
228#41—1812no—
229#41C22⋊D18112yes—
230#41C22×D18112yes—
231C2×Q8—21no—
232C2×Q8—21no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—21no—
236C2×Q8—1811no—
237C2×Q8C22⋊D18111yescharacteristic
238C2×Q8C22×D18112yescomplemented
239C2×D4—21no—
240C2×D4—3622no—
241C2×D4—3624no—
242C2×D4—3624no—
243C2×D4—3628no—
244C2×D4—3624no—
245C2×D4—3628no—
246C2×D4—3622no—
247C2×D4—3621no—
248C2×D4—3624no—
249C2×D4—3621no—
250C2×D4—3621no—
251C2×D4—3621no—
252C2×D4—3621no—
253C2×D4—1812no—
254C2×D4—1811no—
255C2×D4—1812no—
256C2×D4C22⋊D18111yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—3622no—
260C22×C4—3621no—
261C22×C4—3621no—
262C22×C4C22×D18111yescharacteristic
263Q16—24no—
264Q16—3624no—
265SD16—24no—
266SD16—3624no—
267M16—21no—
268M16—3621no—
269M16—1812no—
270M16C22×D18112yescomplemented
271C2×C8—21no—
272C2×C8—3621no—
273C22⋊C4—3628no—
274C42—3621no—
275C42—1812no—
276C23—21no—
277C23—3628no—
278C23—3622no—
279C23—3622no—
280C23—3621no—
281C23—14484no—
282C23—1814no—
283C23—7244no—
284C23D4×D18111yescharacteristic
285Q8—21no—
286Q8—22no—
287Q8—22no—
288Q8—41no—
289Q8—41no—
290Q8—3621no—
291Q8—1812no—
292Q8C2×C22⋊D18112yes—
293D4—22no—
294D4—22no—
295D4—21no—
296D4—3628no—
297D4—3628no—
298D4—3622no—
299D4—3622no—
300D4—3622no—
301D4—3621no—
302D4—3622no—
303D4—3622no—
304D4—3622no—
305D4—1812no—
306D4C2×C22⋊D18112yes—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—3628no—
314C2×C4—3621no—
315C2×C4—3622no—
316C2×C4—3622no—
317C2×C4—3622no—
318C2×C4—7242no—
319C2×C4D4×D18111yescharacteristic
320C2×C4D4×D18111yescharacteristic
321C2×C4D4×D18111yescharacteristic
322C2×C4C2×C22⋊D18111yescharacteristic
323C2×C4C2×C22⋊D18111yescharacteristic
324C2×C4C23×D18111yescharacteristic
325C8—22no—
326C8—3622no—
327C22—22no—
328C22—22no—
329C22—42no—
330C22—3624no—
331C22—3622no—
332C22—14482no—
333C22—14488no—
334C22—7244no—
335C22—7242no—
336C22—7244no—
337C22#1915378411yescharacteristic
338C22#1915378411yescharacteristic
339C22C2×D4×D18111yescharacteristic
340C4—22no—
341C4—42no—
342C4—3622no—
343C4—7242no—
344C4#1915378411yescharacteristic
345C4#1915378411yescharacteristic
346C4C2×D4×D18111yescharacteristic
347C4C22×C22⋊D18111yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—7242no—
352C2—7244no—
353C2#2604145512yesdirect product factor
354C2C2×#1915378411yescharacteristic
subgroupG/H
centerC22#19153784
commutator, cosocleC2×C4×C181C24
FrattiniC2×C4C23×D181
FittingC2×#137×C181C2
socleC22×C181#120
prime radicalC2×#19153862C2
hypercenterC2×#137D181
hypocenterC181C2×#455
LCSG ▷ C2×C4×C181 ▷ C2×C181 ▷ C181
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
2. C2×#19153847C2semidirect
3. C2×#19153805C2semidirect
4. #26041455C2direct (N × H)
5. C2×#19153743C2semidirect
6. C2×#19153742C2semidirect
7. C2×#19153732C2semidirect
8. C2×#137×C181C2semidirect
21. #19153847C22semidirect
24. C2×#15668433C22semidirect
26. #19153805C22semidirect
28. C2×C4×Dic181C22semidirect
29. #19153743C22semidirect
30. #19153742C22semidirect
31. #19153732C22semidirect
33. C22×Q8×C181C22semidirect
35. #137×C181C22semidirect
38. C2×M16×C181C22semidirect
72. #15668433C23semidirect
76. C4×Dic181C23semidirect
87. C2×Q8×C181C23semidirect
96. M16×C181C23semidirect
173. C2×C181#455semidirect
177. C181C2×#455semidirect
181. C2×#137D181semidirect
195. C22×Q8C2×D181semidirect
203. #137C2×D181semidirect
209. C2×M16C2×D181semidirect
238. C2×Q8C22×D181semidirect
270. M16C22×D181semidirect
353. C2#26041455direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191538622
C2×#191538472
C2×#191538052
#260414552
C2×#191537432
C2×#191537422
C2×#191537322
C2×#137×C1812
C2×#455181