Finite Group Databasebeta

← C2×#26011670 #36362894

Subgroups — C2×#26011670 #36362894

Order 22144.

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×#19149725C211yesmaximal, characteristic
2C2×#19149712C211yesmaximal, characteristic, complemented
3C2×#19149709C211yesmaximal, characteristic, complemented
4C2×#19149667C211yesmaximal, characteristic, complemented
5#26011670C218yesmaximal, direct product factor
6C2×#19149607C211yesmaximal, characteristic, complemented
7C2×#19149595C211yesmaximal, characteristic, complemented
8C2×#104×C173C211yesmaximal, characteristic, complemented
9#19149725—22no—
10#19149725—22no—
11#19149725—22no—
12#19149725C2214yes—
13C2×#14583827—21no—
14C2×#14583826—21no—
15C2×#14583826C2211yescharacteristic
16C22×Dic346C2211yescharacteristic, complemented
17C22×D692—21no—
18C22×D692C2211yescharacteristic
19C2×D4×D173—21no—
20C2×D4×D173—22no—
21#19149712—22no—
22#19149712C2214yescomplemented
23#19149709—22no—
24#19149709C2214yescomplemented
25C2×Dic692—21no—
26C2×#14583816—21no—
27C2×#14583815—21no—
28C2×#14583814—21no—
29C2×#14583810C2211yescharacteristic, complemented
30#19149667—22no—
31#19149667C2214yescomplemented
32C2×#14583803—22no—
33#19149607C2214yescomplemented
34#19149595C2214yescomplemented
35C2×#41×C173C2211yescharacteristic
36C2×M16×C173C2211yescharacteristic, complemented
37C2×C42×C173C2211yescharacteristic, complemented
38#104×C173C2214yescomplemented
39C23×D173—22no—
40C23×D173D412yes—
41#14583827—24no—
42#14583827—24no—
43#14583826—24no—
44#14583826—24no—
45#14583826—21no—
46#14583826—21no—
47#14583826D412yes—
48#14583826C2312yes—
49C2×Dic346—21no—
50C2×Dic346—21no—
51C2×Dic346—21no—
52C2×Dic346—21no—
53C2×Dic346—21no—
54C2×Dic346D411yescharacteristic
55C2×Dic346C2312yescomplemented
56C2×D692—22no—
57C2×D692—24no—
58C2×D692—24no—
59C2×D692—21no—
60C2×D692—21no—
61C2×D692—21no—
62C2×D692—21no—
63C2×D692D412yes—
64C2×D692D411yescharacteristic
65C2×D692C2312yes—
66C2×C22⋊D173—22no—
67C2×C22⋊D173—21no—
68D4×D173—28no—
69D4×D173—24no—
70D4×D173—28no—
71D4×D173—24no—
72C22×Dic173—21no—
73C2×C4×D173—22no—
74C2×C4×D173—21no—
75Dic692—24no—
76#14583816—24no—
77#14583815—24no—
78#14583814—24no—
79#14583810—21no—
80#14583810C2312yescomplemented
81C2×#13063436—21no—
82#14583803—28no—
83#41×C173—21no—
84#41×C173—21no—
85#41×C173D412yes—
86#41×C173C2312yes—
87C2×Q8×C173D411yescharacteristic
88C2×D4×C173—21no—
89C2×D4×C173D411yescharacteristic
90C22×C4×C173—21no—
91C22×C4×C173—21no—
92C22×C4×C173C2311yescharacteristic
93M16×C173—21no—
94M16×C173C2312yescomplemented
95C2×C8×C173—21no—
96C42×C173—21no—
97C42×C173C2312yescomplemented
98C22×D173—22no—
99C22×D173—21no—
100C22×D173—28no—
101C22×D173—22no—
102C22×D173—44no—
103C22×D173—84no—
104C22×D173C2×D414yes—
105Dic346—21no—
106Dic346—22no—
107Dic346—22no—
108Dic346—41no—
109Dic346—41no—
110Dic346C2×D412yes—
111D692—28no—
112D692—22no—
113D692—22no—
114D692—21no—
115D692—22no—
116D692—22no—
117D692C2×D412yes—
118C22⋊D173—28no—
119C22⋊D173—22no—
120C22⋊D173—22no—
121C2×Dic173—21no—
122C2×Dic173—22no—
123C2×Dic173—41no—
124C2×Dic173—41no—
125C4×D173—28no—
126C4×D173—22no—
127C4×D173—22no—
128#13063436—22no—
129C23×C173—21no—
130C23×C173C2×D411yescharacteristic
131Q8×C173—21no—
132Q8×C173C2×D412yes—
133D4×C173—22no—
134D4×C173—22no—
135D4×C173—21no—
136D4×C173C2×D412yes—
137C2×C4×C173—21no—
138C2×C4×C173—22no—
139C2×C4×C173—22no—
140C2×C4×C173—22no—
141C2×C4×C173—42no—
142C2×C4×C173C2×D411yescharacteristic
143C2×C4×C173C2×D411yescharacteristic
144C2×C4×C173C2×D411yescharacteristic
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2×D411yescharacteristic
147C2×C4×C173C2411yescharacteristic
148C8×C173—22no—
149C22×C173—22no—
150C22×C173—22no—
151C22×C173—42no—
152C22×C173#12011yescharacteristic
153C22×C173#12011yescharacteristic
154C22×C173C22×D411yescharacteristic
155C2×D173—24no—
156C2×D173—22no—
157C2×D173—44no—
158C2×D173—42no—
159C2×D173—44no—
160C2×D173—82no—
161C2×D173—88no—
162C4×C173—22no—
163C4×C173—42no—
164C4×C173#12011yescharacteristic
165C4×C173#12011yescharacteristic
166C4×C173C22×D411yescharacteristic
167C4×C173C22×D411yescharacteristic
168Dic173—22no—
169Dic173—42no—
170C2×C173—21no—
171C2×C173—21no—
172C2×C173—42no—
173C2×C173#45512yescomplemented
174C2×C173C2×#12011yescharacteristic
175D173—44no—
176D173—42no—
177C173C2×#45511yescharacteristic, complemented, Sylow
178C2×#455—1731nomaximal, Sylow
179C2×H(5,𝔽2)—1731no—
180C2×#137—1731no—
181C2×#137—1731no—
182C2×#124—1731no—
183#455—1738no—
184C2×#104—1731no—
185C2×#104D17311yescharacteristic, complemented
186C2×#100—1731no—
187H(5,𝔽2)—3462no—
188H(5,𝔽2)—3462no—
189H(5,𝔽2)—3462no—
190H(5,𝔽2)—1734no—
191C2×#41—3461no—
192C2×#41—3461no—
193C2×#41—1731no—
194C2×#41C2×D17311yescharacteristic
195C22×Q8—1731no—
196C22×D4—3461no—
197C22×D4—3462no—
198C22×D4—3461no—
199C22×D4—1731no—
200#137—3462no—
201#137—3462no—
202#137—1734no—
203#137—1734no—
204C2×Q16—3461no—
205C2×Q16—3461no—
206C2×SD16—3461no—
207C2×SD16—3461no—
208C2×M16—1731no—
209C2×M16C2×D17311yescharacteristic, complemented
210#124—3462no—
211#124—1734no—
212C2×C22⋊C4—3462no—
213C2×C42C2×D17311yescharacteristic, complemented
214#104—1734no—
215#104C2×D17314yescomplemented
216#100—1734no—
217C24—3462no—
218C24—1732no—
219#41—21no—
220#41—21no—
221#41—3464no—
222#41—3464no—
223#41—3464no—
224#41—3464no—
225#41—3461no—
226#41—3461no—
227#41—1732no—
228#41—1732no—
229#41D69212yes—
230#41C22×D17312yes—
231C2×Q8—3461no—
232C2×Q8—3461no—
233C2×Q8—3461no—
234C2×Q8—3461no—
235C2×Q8—3461no—
236C2×Q8—1731no—
237C2×Q8—1732no—
238C2×Q8D69211yescharacteristic
239C2×D4—21no—
240C2×D4—3462no—
241C2×D4—3464no—
242C2×D4—3464no—
243C2×D4—3468no—
244C2×D4—3464no—
245C2×D4—3468no—
246C2×D4—3462no—
247C2×D4—3461no—
248C2×D4—3464no—
249C2×D4—3461no—
250C2×D4—3461no—
251C2×D4—3461no—
252C2×D4—3461no—
253C2×D4—1732no—
254C2×D4—1731no—
255C2×D4—1732no—
256C2×D4D69211yescharacteristic
257C22×C4—21no—
258C22×C4—21no—
259C22×C4—3462no—
260C22×C4—3461no—
261C22×C4—3461no—
262C22×C4C22×D17311yescharacteristic
263Q16—3464no—
264Q16—3464no—
265SD16—3464no—
266SD16—3464no—
267M16—21no—
268M16—3461no—
269M16—1732no—
270M16C22×D17312yescomplemented
271C2×C8—21no—
272C2×C8—3461no—
273C22⋊C4—3468no—
274C42—21no—
275C42C22×D17312yescomplemented
276C23—21no—
277C23—3468no—
278C23—3462no—
279C23—3462no—
280C23—3461no—
281C23—13844no—
282C23—1734no—
283C23—6924no—
284C23D4×D17311yescharacteristic
285Q8—21no—
286Q8—3461no—
287Q8—3462no—
288Q8—3462no—
289Q8—1732no—
290Q8—6921no—
291Q8—6921no—
292Q8C2×D69212yes—
293D4—22no—
294D4—22no—
295D4—21no—
296D4—3468no—
297D4—3468no—
298D4—3462no—
299D4—3462no—
300D4—3462no—
301D4—3461no—
302D4—3462no—
303D4—3462no—
304D4—3462no—
305D4—1732no—
306D4C2×D69212yes—
307C2×C4—21no—
308C2×C4—22no—
309C2×C4—22no—
310C2×C4—22no—
311C2×C4—42no—
312C2×C4—3468no—
313C2×C4—3461no—
314C2×C4—3462no—
315C2×C4—3462no—
316C2×C4—3462no—
317C2×C4—6921no—
318C2×C4—6921no—
319C2×C4D4×D17311yescharacteristic
320C2×C4D4×D17311yescharacteristic
321C2×C4D4×D17311yescharacteristic
322C2×C4C2×D69211yescharacteristic
323C2×C4C2×D69211yescharacteristic
324C2×C4C23×D17311yescharacteristic
325C8—22no—
326C8—3462no—
327C22—22no—
328C22—22no—
329C22—42no—
330C22—3464no—
331C22—3462no—
332C22—13842no—
333C22—13848no—
334C22—6924no—
335C22—6922no—
336C22—6924no—
337C22#1914964911yescharacteristic
338C22#1914964911yescharacteristic
339C22C2×D4×D17311yescharacteristic
340C4—22no—
341C4—42no—
342C4—3462no—
343C4—6922no—
344C4#1914964911yescharacteristic
345C4#1914964911yescharacteristic
346C4C2×D4×D17311yescharacteristic
347C4C22×D69211yescharacteristic
348C2—21no—
349C2—21no—
350C2—42no—
351C2—6922no—
352C2—6924no—
353C2#2601167012yesdirect product factor
354C2C2×#1914964911yescharacteristic
subgroupG/H
centerC22#19149649
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#104×C173C2
socleC22×C173#120
prime radicalC2×#19149725C2
hypercenterC2×#104D173
hypocenterC173C2×#455
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#104

Complements

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

NH ≅ G/Ntype
2. C2×#19149712C2semidirect
3. C2×#19149709C2semidirect
4. C2×#19149667C2semidirect
5. #26011670C2direct (N × H)
6. C2×#19149607C2semidirect
7. C2×#19149595C2semidirect
8. C2×#104×C173C2semidirect
16. C22×Dic346C22semidirect
22. #19149712C22semidirect
24. #19149709C22semidirect
29. C2×#14583810C22semidirect
31. #19149667C22semidirect
33. #19149607C22semidirect
34. #19149595C22semidirect
36. C2×M16×C173C22semidirect
37. C2×C42×C173C22semidirect
38. #104×C173C22semidirect
55. C2×Dic346C23semidirect
80. #14583810C23semidirect
94. M16×C173C23semidirect
97. C42×C173C23semidirect
173. C2×C173#455semidirect
177. C173C2×#455semidirect
185. C2×#104D173semidirect
209. C2×M16C2×D173semidirect
213. C2×C42C2×D173semidirect
215. #104C2×D173semidirect
270. M16C22×D173semidirect
275. C42C22×D173semidirect
353. C2#26011670direct (N × H)

Maximal subgroups

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

subgroupindex
C2×#191497252
C2×#191497122
C2×#191497092
C2×#191496672
#260116702
C2×#191496072
C2×#191495952
C2×#104×C1732
C2×#455173