Finite Group Databasebeta

← #36362943

Subgroups — #36362943

Order 22144.

704 conjugacy classes of subgroups; only the 145 normal subgroups are shown.

normal subgroups only

This group has 704 conjugacy classes of subgroups — too many to draw. Only its 145 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
1#26012327C211yesmaximal, characteristic
2C2×#19149712C211yesmaximal, characteristic, complemented
3C2×#19149709C211yesmaximal, characteristic, complemented
4#26012067C211yesmaximal, characteristic, complemented
5#26011675C214yesmaximal, complemented
6#26011670C214yesmaximal, complemented
7#26011509C211yesmaximal, characteristic, complemented
8#26011473C211yesmaximal, characteristic, complemented
9#421×C173C211yesmaximal, characteristic, complemented
10#19149727—21no—
11#19149726—21no—
12#19149726C2212yes—
13#19149725—21no—
14#19149725C2212yes—
15#19149724—21no—
16#19149724—22no—
17C2×#14583826—22no—
18C2×#14583826C2211yescharacteristic
19C2×#14583826C2211yescharacteristic
20#41×D173—22no—
21#41×D173—22no—
22C22×Dic346C2211yescharacteristic, complemented
23#19149712—22no—
24#19149712C2214yescomplemented
25#19149709—22no—
26#19149709C2214yescomplemented
27C2×Dic692—21no—
28C2×#14583816—21no—
29C2×#14583815—21no—
30C2×#14583814—21no—
31C2×#14583810C2211yescharacteristic, complemented
32#19149679C2212yescomplemented
33#19149667C2212yescomplemented
34#19149663—22no—
35#19149659—22no—
36#19149607C2214yescomplemented
37#19149597C2212yescomplemented
38#19149595C2212yescomplemented
39C2×#41×C173C2211yescharacteristic
40C2×M16×C173C2211yescharacteristic, complemented
41#117×C173C2211yescharacteristic, complemented
42#104×C173C2214yescomplemented
43#14583827—22no—
44#14583827—22no—
45#14583826—24no—
46#14583826—24no—
47#14583826—22no—
48#14583826—22no—
49#14583826—21no—
50#14583826—21no—
51#14583826—21no—
52#14583826—21no—
53#14583826—21no—
54#14583826—21no—
55#14583826D412yes—
56#14583826C2312yes—
57#14583825—24no—
58#14583825—22no—
59#14583825—24no—
60#14583825—22no—
61C2×Dic346—22no—
62C2×Dic346—21no—
63C2×Dic346—21no—
64C2×Dic346—21no—
65C2×Dic346—21no—
66C2×Dic346—21no—
67C2×Dic346D411yescharacteristic
68C2×Dic346C2312yes—
69C2×Dic346C2311yescharacteristic
70Q8×D173—22no—
71Q8×D173—22no—
72C2×D692—22no—
73C2×D692D411yescharacteristic
74C2×D692C2311yescharacteristic
75C2×C22⋊D173—24no—
76C2×C22⋊D173—21no—
77C2×C22⋊D173D412yes—
78D4×D173—24no—
79D4×D173—22no—
80D4×D173—24no—
81D4×D173—22no—
82C22×Dic173—21no—
83C2×C4×D173—24no—
84C2×C4×D173—21no—
85C2×C4×D173D412yes—
86Dic692—24no—
87#14583816—24no—
88#14583815—24no—
89#14583814—24no—
90#14583810—21no—
91#14583810C2312yes—
92C2×#13063436—21no—
93C4⋊Dic173—24no—
94#14583803—24no—
95#41×C173—21no—
96#41×C173—21no—
97#41×C173D412yes—
98#41×C173C2312yes—
99C2×Q8×C173D411yescharacteristic
100C2×D4×C173—21no—
101C2×D4×C173D411yescharacteristic
102C22×C4×C173—21no—
103C22×C4×C173C2311yescharacteristic
104M16×C173—21no—
105M16×C173C2312yes—
106C2×C8×C173—21no—
107C4⋊C4×C173—21no—
108C22⋊C4×C173—21no—
109C42×C173C2312yes—
110C22×D173—24no—
111C22×D173—21no—
112C22×D173C2×D412yes—
113Dic346—24no—
114Dic346—21no—
115Dic346—21no—
116Dic346—21no—
117Dic346—22no—
118Dic346—22no—
119Dic346—41no—
120Dic346—41no—
121Dic346C2×D412yes—
122D692—24no—
123D692—21no—
124D692—21no—
125D692—21no—
126D692C2×D412yes—
127C22⋊D173—24no—
128C22⋊D173—24no—
129C22⋊D173—22no—
130C22⋊D173—22no—
131C22⋊D173—22no—
132C22⋊D173—22no—
133C2×Dic173—24no—
134C2×Dic173—21no—
135C2×Dic173—22no—
136C2×Dic173—41no—
137C2×Dic173—41no—
138C2×Dic173C2×D412yes—
139C4×D173—24no—
140C4×D173—24no—
141C4×D173—22no—
142C4×D173—22no—
143C4×D173—22no—
144C4×D173—22no—
145#13063436—22no—
146C23×C173—21no—
147C23×C173C2×D411yescharacteristic
148Q8×C173—21no—
149Q8×C173C2×D412yes—
150D4×C173—22no—
151D4×C173—22no—
152D4×C173—21no—
153D4×C173C2×D412yes—
154C2×C4×C173—21no—
155C2×C4×C173—22no—
156C2×C4×C173—22no—
157C2×C4×C173—22no—
158C2×C4×C173C2×D411yescharacteristic
159C2×C4×C173C2×D411yescharacteristic
160C2×C4×C173C2×D411yescharacteristic
161C2×C4×C173C2×D411yescharacteristic
162C2×C4×C173C2×D411yescharacteristic
163C2×C4×C173C2411yescharacteristic
164C8×C173—22no—
165C22×C173—22no—
166C22×C173—41no—
167C22×C173—81no—
168C22×C173#12011yescharacteristic
169C22×C173#12011yescharacteristic
170C22×C173C22×D411yescharacteristic
171C2×D173—22no—
172C2×D173—22no—
173C2×D173—42no—
174C2×D173—81no—
175C2×D173—84no—
176C4×C173—22no—
177C4×C173—42no—
178C4×C173#12011yescharacteristic
179C4×C173#12011yescharacteristic
180C4×C173C22×D411yescharacteristic
181C4×C173C22×D411yescharacteristic
182Dic173—22no—
183Dic173—22no—
184Dic173—42no—
185C2×C173—21no—
186C2×C173—21no—
187C2×C173—21no—
188C2×C173—42no—
189C2×C173C2×#12011yescharacteristic
190D173—42no—
191D173—42no—
192C173#302211yescharacteristic, complemented, Sylow
193#3022—1731nomaximal, Sylow
194#585—1731no—
195C2×#137—1731no—
196C2×#137—1731no—
197#536—1731no—
198#456—1734no—
199#455—1734no—
200#421—1731no—
201#421D17311yescharacteristic, complemented
202#413—1731no—
2032−1+4—3461no—
2042−1+4—3461no—
2052−1+4—1732no—
206H(5,𝔽2)—3462no—
207H(5,𝔽2)—3461no—
208H(5,𝔽2)—3461no—
209H(5,𝔽2)—1732no—
210C2×#41—3462no—
211C2×#41—3462no—
212C2×#41—3462no—
213C2×#41—1731no—
214C2×#41—1731no—
215C2×#41C2×D17311yescharacteristic
216C22×Q8—1731no—
217#137—3462no—
218#137—3462no—
219#137—1734no—
220#137—1734no—
221C2×Q16—3461no—
222C2×Q16—3461no—
223C2×SD16—3461no—
224C2×SD16—3461no—
225C2×M16—1731no—
226C2×M16C2×D17311yescharacteristic, complemented
227C4⋊Q8—1732no—
228#124—1732no—
229#123—3462no—
230C22⋊Q8—3462no—
231#117C2×D17311yescharacteristic, complemented
232#104—1734no—
233#104C2×D17314yescomplemented
234#101—1732no—
235#100—1732no—
236#41—21no—
237#41—21no—
238#41—3464no—
239#41—3464no—
240#41—3462no—
241#41—3462no—
242#41—3464no—
243#41—3462no—
244#41—3462no—
245#41—3462no—
246#41—3462no—
247#41—3464no—
248#41—3461no—
249#41—3461no—
250#41—3461no—
251#41—3461no—
252#41—3461no—
253#41—3461no—
254#41—1732no—
255#41—1732no—
256#41D69212yes—
257#41C22×D17312yes—
258C2×Q8—3462no—
259C2×Q8—3462no—
260C2×Q8—3462no—
261C2×Q8—3461no—
262C2×Q8—3461no—
263C2×Q8—3461no—
264C2×Q8—3461no—
265C2×Q8—3461no—
266C2×Q8—1731no—
267C2×Q8—1732no—
268C2×Q8—1731no—
269C2×Q8D69211yescharacteristic
270C2×D4—21no—
271C2×D4—3462no—
272C2×D4—3464no—
273C2×D4—3462no—
274C2×D4—3464no—
275C2×D4—3464no—
276C2×D4—3461no—
277C2×D4—3462no—
278C2×D4—1732no—
279C2×D4—1731no—
280C2×D4—1731no—
281C2×D4D69211yescharacteristic
282C22×C4—21no—
283C22×C4—3464no—
284C22×C4—3461no—
285C22×C4—3461no—
286C22×C4—1732no—
287C22×C4C22×D17311yescharacteristic
288Q16—3464no—
289Q16—3464no—
290SD16—3464no—
291SD16—3464no—
292M16—21no—
293M16—3461no—
294M16—1732no—
295M16C22×D17312yes—
296C2×C8—21no—
297C2×C8—3461no—
298C4⋊C4—21no—
299C4⋊C4—3464no—
300C22⋊C4—21no—
301C22⋊C4—3464no—
302C42C22×D17312yes—
303C23—21no—
304C23—3464no—
305C23—3461no—
306C23—1732no—
307C23D4×D17311yescharacteristic
308Q8—21no—
309Q8—3464no—
310Q8—3461no—
311Q8—3461no—
312Q8—3461no—
313Q8—3462no—
314Q8—3462no—
315Q8—1732no—
316Q8—6921no—
317Q8—6921no—
318Q8C2×D69212yes—
319D4—22no—
320D4—22no—
321D4—21no—
322D4—3464no—
323D4—3464no—
324D4—3464no—
325D4—3462no—
326D4—3462no—
327D4—3462no—
328D4—3462no—
329D4—3461no—
330D4—3461no—
331D4—3461no—
332D4—1732no—
333D4C2×D69212yes—
334C2×C4—21no—
335C2×C4—22no—
336C2×C4—22no—
337C2×C4—22no—
338C2×C4—3464no—
339C2×C4—3464no—
340C2×C4—3464no—
341C2×C4—3461no—
342C2×C4—3462no—
343C2×C4—3462no—
344C2×C4—3462no—
345C2×C4—3462no—
346C2×C4—3462no—
347C2×C4—1732no—
348C2×C4—6921no—
349C2×C4—6921no—
350C2×C4D4×D17311yescharacteristic
351C2×C4D4×D17311yescharacteristic
352C2×C4D4×D17311yescharacteristic
353C2×C4C2×D69211yescharacteristic
354C2×C4C2×D69211yescharacteristic
355C2×C4C23×D17311yescharacteristic
356C8—22no—
357C8—3462no—
358C22—22no—
359C22—41no—
360C22—81no—
361C22—3462no—
362C22—3462no—
363C22—13841no—
364C22—13844no—
365C22—6922no—
366C22#1914964911yescharacteristic
367C22#1914964911yescharacteristic
368C22C2×D4×D17311yescharacteristic
369C4—22no—
370C4—42no—
371C4—3462no—
372C4—3462no—
373C4—6922no—
374C4#1914964911yescharacteristic
375C4#1914964911yescharacteristic
376C4C2×D4×D17311yescharacteristic
377C4C22×D69211yescharacteristic
378C2—21no—
379C2—21no—
380C2—21no—
381C2—42no—
382C2—6922no—
383C2—6922no—
384C2C2×#1914964911yescharacteristic
subgroupG/H
centerC2C2×#19149649
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#421×C173C2
socleC2×C173C2×#120
prime radical#26012327C2
hypercenter#421D173
hypocenterC173#3022
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2 ◁ C22×C4 ◁ #421

Complements

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

NH ≅ G/Ntype
2. C2×#19149712C2semidirect
3. C2×#19149709C2semidirect
4. #26012067C2semidirect
5. #26011675C2semidirect
6. #26011670C2semidirect
7. #26011509C2semidirect
8. #26011473C2semidirect
9. #421×C173C2semidirect
22. C22×Dic346C22semidirect
24. #19149712C22semidirect
26. #19149709C22semidirect
31. C2×#14583810C22semidirect
32. #19149679C22semidirect
33. #19149667C22semidirect
36. #19149607C22semidirect
37. #19149597C22semidirect
38. #19149595C22semidirect
40. C2×M16×C173C22semidirect
41. #117×C173C22semidirect
42. #104×C173C22semidirect
192. C173#3022semidirect
201. #421D173semidirect
226. C2×M16C2×D173semidirect
231. #117C2×D173semidirect
233. #104C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260123272
C2×#191497122
C2×#191497092
#260120672
#260116752
#260116702
#260115092
#260114732
#421×C1732
#3022173