Finite Group Databasebeta

← #36362922

Subgroups — #36362922

Order 22144.

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

normal subgroups only

This group has 762 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
1C2×#19149725C211yesmaximal, characteristic, complemented
2#26012291C211yesmaximal, characteristic, complemented
3#26012289C211yesmaximal, characteristic, complemented
4#26012060C211yesmaximal, characteristic, complemented
5#26011670C214yesmaximal, complemented
6#26011660C214yesmaximal, complemented
7#26011509C211yesmaximal, characteristic, complemented
8C2×#19149595C211yesmaximal, characteristic, complemented
9#421×C173C211yesmaximal, characteristic, complemented
10#19149725—22no—
11#19149725—22no—
12#19149725—22no—
13#19149725C2214yes—
14C2×#14583827—21no—
15C2×#14583826—21no—
16C2×#14583826C2211yescharacteristic
17C2×#14583826C2211yescharacteristic, complemented
18C22×D692—21no—
19C22×D692C2211yescharacteristic
20C2×D4×D173—21no—
21C2×D4×D173—22no—
22#19149712—21no—
23#19149712C2212yescomplemented
24#19149710—21no—
25#19149709C2212yescomplemented
26#19149707—21no—
27#19149707C2212yescomplemented
28#19149705—21no—
29#19149704C2212yescomplemented
30#19149701—22no—
31#19149699—22no—
32C2×#14583810C2211yescharacteristic, complemented
33#19149676C2212yescomplemented
34#19149667C2212yescomplemented
35#19149654—22no—
36#19149649—22no—
37#19149607C2214yescomplemented
38#19149595C2214yescomplemented
39C2×#41×C173C2211yescharacteristic
40C2×M16×C173C2211yescharacteristic, complemented
41#117×C173C2211yescharacteristic, complemented
42#104×C173C2214yescomplemented
43C23×D173—22no—
44C23×D173D412yes—
45#14583827—24no—
46#14583827—24no—
47#14583826—24no—
48#14583826—24no—
49#14583826—21no—
50#14583826—21no—
51#14583826—21no—
52#14583826—21no—
53#14583826—21no—
54#14583826—21no—
55#14583826D412yescomplemented
56#14583826C2312yes—
57C2×Dic346D411yescharacteristic, complemented
58C2×Dic346C2311yescharacteristic, complemented
59C2×D692—22no—
60C2×D692—24no—
61C2×D692—24no—
62C2×D692—24no—
63C2×D692—21no—
64C2×D692—21no—
65C2×D692—21no—
66C2×D692—21no—
67C2×D692D412yes—
68C2×D692D411yescharacteristic, complemented
69C2×D692C2312yes—
70C2×D692C2311yescharacteristic, complemented
71C2×C22⋊D173—22no—
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74D4×D173—28no—
75D4×D173—24no—
76D4×D173—28no—
77D4×D173—24no—
78C2×C4×D173—22no—
79C2×C4×D173—21no—
80C2×C4×D173—21no—
81Dic692—22no—
82#14583816—22no—
83#14583815—22no—
84#14583815—22no—
85#14583814—22no—
86#14583813—22no—
87D1384—22no—
88#14583811—22no—
89#14583810—21no—
90#14583810C2312yescomplemented
91C2×#13063436—21no—
92#14583803—24no—
93#41×C173—21no—
94#41×C173—21no—
95#41×C173D412yescomplemented
96#41×C173C2312yes—
97C2×Q8×C173D411yescharacteristic, complemented
98C2×D4×C173—21no—
99C2×D4×C173D411yescharacteristic, complemented
100C22×C4×C173—21no—
101C22×C4×C173C2311yescharacteristic
102M16×C173—21no—
103M16×C173C2312yescomplemented
104C2×C8×C173—21no—
105C4⋊C4×C173—21no—
106C22⋊C4×C173—21no—
107C42×C173C2312yescomplemented
108C22×D173—22no—
109C22×D173—21no—
110C22×D173—28no—
111C22×D173—22no—
112C22×D173—21no—
113C22×D173—82no—
114C22×D173—84no—
115C22×D173C2×D414yes—
116Dic346—21no—
117Dic346—21no—
118Dic346—21no—
119Dic346C2×D412yes—
120D692—28no—
121D692—22no—
122D692—22no—
123D692—21no—
124D692—22no—
125D692—22no—
126D692—21no—
127D692—21no—
128D692—84no—
129D692C2×D412yes—
130C22⋊D173—28no—
131C22⋊D173—22no—
132C22⋊D173—22no—
133C22⋊D173—41no—
134C22⋊D173—41no—
135C2×Dic173—21no—
136C2×Dic173—21no—
137C4×D173—28no—
138C4×D173—22no—
139C4×D173—22no—
140C4×D173—41no—
141C4×D173—41no—
142#13063436—22no—
143C23×C173—21no—
144C23×C173C2×D411yescharacteristic
145Q8×C173—21no—
146Q8×C173C2×D412yes—
147D4×C173—22no—
148D4×C173—22no—
149D4×C173—21no—
150D4×C173C2×D412yes—
151C2×C4×C173—21no—
152C2×C4×C173—22no—
153C2×C4×C173—22no—
154C2×C4×C173—22no—
155C2×C4×C173C2×D411yescharacteristic
156C2×C4×C173C2×D411yescharacteristic
157C2×C4×C173C2×D411yescharacteristic
158C2×C4×C173C2×D411yescharacteristic
159C2×C4×C173C2×D411yescharacteristic
160C2×C4×C173C2411yescharacteristic
161C8×C173—22no—
162C22×C173—22no—
163C22×C173—41no—
164C22×C173—42no—
165C22×C173#12011yescharacteristic
166C22×C173#12011yescharacteristic
167C22×C173C22×D411yescharacteristic
168C2×D173—24no—
169C2×D173—22no—
170C2×D173—44no—
171C2×D173—42no—
172C2×D173—44no—
173C2×D173—42no—
174C2×D173—41no—
175C2×D173—82no—
176C2×D173—88no—
177C4×C173—22no—
178C4×C173—42no—
179C4×C173#12011yescharacteristic
180C4×C173#12011yescharacteristic
181C4×C173C22×D411yescharacteristic
182C4×C173C22×D411yescharacteristic
183Dic173—22no—
184Dic173—41no—
185C2×C173—21no—
186C2×C173—21no—
187C2×C173—21no—
188C2×C173—42no—
189C2×C173C2×#12011yescharacteristic
190D173—44no—
191D173—42no—
192D173—81no—
193C173#301911yescharacteristic, complemented, Sylow
194#3019—1731nomaximal, Sylow
195C2×H(5,𝔽2)—1731no—
196#575—1731no—
197#575—1731no—
198#535—1731no—
199#455—1734no—
200#453—1734no—
201#421—1731no—
202#421D17311yescharacteristic, complemented
203C2×#100—1731no—
204H(5,𝔽2)—3462no—
205H(5,𝔽2)—3462no—
206H(5,𝔽2)—3462no—
207H(5,𝔽2)—1734no—
208C2×#41—3461no—
209C2×#41—3461no—
210C2×#41—1731no—
211C2×#41—1731no—
212C2×#41C2×D17311yescharacteristic
213C22×D4—3461no—
214C22×D4—3462no—
215C22×D4—3461no—
216C22×D4—1731no—
217#137—3461no—
218#137—3461no—
219#137—1732no—
220#137—1732no—
221AGL(1,ℤ/8)—3461no—
222AGL(1,ℤ/8)—3461no—
223AGL(1,ℤ/8)—1732no—
224AGL(1,ℤ/8)—1732no—
225#135—3462no—
226#135—3462no—
227C2×M16—1731no—
228C2×M16C2×D17311yescharacteristic, complemented
229#127—1732no—
230#124—1732no—
231#121—3462no—
232#120—3462no—
233#117C2×D17311yescharacteristic, complemented
234#104—1734no—
235#104C2×D17314yescomplemented
236#100—1734no—
237C24—3462no—
238C24—1732no—
239#41—21no—
240#41—21no—
241#41—3464no—
242#41—3464no—
243#41—3464no—
244#41—3464no—
245#41—3461no—
246#41—3461no—
247#41—3461no—
248#41—3461no—
249#41—3461no—
250#41—3461no—
251#41—1732no—
252#41—1732no—
253#41D69212yescomplemented
254#41C22×D17312yes—
255C2×Q8—1731no—
256C2×Q8—1731no—
257C2×Q8D69211yescharacteristic, complemented
258C2×D4—21no—
259C2×D4—3462no—
260C2×D4—3464no—
261C2×D4—3464no—
262C2×D4—3464no—
263C2×D4—3468no—
264C2×D4—3464no—
265C2×D4—3468no—
266C2×D4—3462no—
267C2×D4—3461no—
268C2×D4—3464no—
269C2×D4—3461no—
270C2×D4—3461no—
271C2×D4—3461no—
272C2×D4—3461no—
273C2×D4—3461no—
274C2×D4—1732no—
275C2×D4—1731no—
276C2×D4—1731no—
277C2×D4—1732no—
278C2×D4D69211yescharacteristic, complemented
279C22×C4—21no—
280C22×C4—3462no—
281C22×C4—3461no—
282C22×C4—3461no—
283C22×C4C22×D17311yescharacteristic
284Q16—3462no—
285Q16—3462no—
286SD16—3462no—
287SD16—3462no—
288SD16—3462no—
289SD16—3462no—
290D8—3462no—
291D8—3462no—
292M16—21no—
293M16—3461no—
294M16—1732no—
295M16C22×D17312yescomplemented
296C2×C8—21no—
297C2×C8—3461no—
298C4⋊C4—21no—
299C22⋊C4—21no—
300C22⋊C4—3464no—
301C42C22×D17312yescomplemented
302C23—21no—
303C23—3468no—
304C23—3462no—
305C23—3462no—
306C23—3461no—
307C23—3461no—
308C23—13844no—
309C23—13842no—
310C23—1734no—
311C23D4×D17311yescharacteristic
312Q8—21no—
313Q8—3461no—
314Q8—3461no—
315Q8—3461no—
316Q8—1732no—
317Q8C2×D69212yes—
318D4—22no—
319D4—22no—
320D4—21no—
321D4—3468no—
322D4—3468no—
323D4—3462no—
324D4—3462no—
325D4—3462no—
326D4—3461no—
327D4—3462no—
328D4—3462no—
329D4—3461no—
330D4—3462no—
331D4—3461no—
332D4—13844no—
333D4—1732no—
334D4—6921no—
335D4—6921no—
336D4C2×D69212yes—
337C2×C4—21no—
338C2×C4—22no—
339C2×C4—22no—
340C2×C4—22no—
341C2×C4—3468no—
342C2×C4—3461no—
343C2×C4—3462no—
344C2×C4—3462no—
345C2×C4—3461no—
346C2×C4—6921no—
347C2×C4—6921no—
348C2×C4D4×D17311yescharacteristic
349C2×C4D4×D17311yescharacteristic
350C2×C4D4×D17311yescharacteristic
351C2×C4C2×D69211yescharacteristic
352C2×C4C2×D69211yescharacteristic
353C2×C4C23×D17311yescharacteristic
354C8—22no—
355C8—3462no—
356C22—22no—
357C22—41no—
358C22—42no—
359C22—3464no—
360C22—3462no—
361C22—13842no—
362C22—13848no—
363C22—6922no—
364C22—6924no—
365C22—6922no—
366C22—6924no—
367C22—6921no—
368C22#1914964911yescharacteristic
369C22#1914964911yescharacteristic
370C22C2×D4×D17311yescharacteristic
371C4—22no—
372C4—42no—
373C4—3462no—
374C4—6921no—
375C4#1914964911yescharacteristic
376C4#1914964911yescharacteristic
377C4C2×D4×D17311yescharacteristic
378C4C22×D69211yescharacteristic
379C2—21no—
380C2—21no—
381C2—21no—
382C2—42no—
383C2—13841no—
384C2—6922no—
385C2—6924no—
386C2C2×#1914964911yescharacteristic
subgroupG/H
centerC2C2×#19149649
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#421×C173C2
socleC2×C173C2×#120
hypercenter#421D173
hypocenterC173#3019
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
1. C2×#19149725C2semidirect
2. #26012291C2semidirect
3. #26012289C2semidirect
4. #26012060C2semidirect
5. #26011670C2semidirect
6. #26011660C2semidirect
7. #26011509C2semidirect
8. C2×#19149595C2semidirect
9. #421×C173C2semidirect
17. C2×#14583826C22semidirect
23. #19149712C22semidirect
25. #19149709C22semidirect
27. #19149707C22semidirect
29. #19149704C22semidirect
32. C2×#14583810C22semidirect
33. #19149676C22semidirect
34. #19149667C22semidirect
37. #19149607C22semidirect
38. #19149595C22semidirect
40. C2×M16×C173C22semidirect
41. #117×C173C22semidirect
42. #104×C173C22semidirect
55. #14583826D4semidirect
57. C2×Dic346D4semidirect
58. C2×Dic346C23semidirect
68. C2×D692D4semidirect
70. C2×D692C23semidirect
90. #14583810C23semidirect
95. #41×C173D4semidirect
97. C2×Q8×C173D4semidirect
99. C2×D4×C173D4semidirect
103. M16×C173C23semidirect
107. C42×C173C23semidirect
193. C173#3019semidirect
202. #421D173semidirect
228. C2×M16C2×D173semidirect
233. #117C2×D173semidirect
235. #104C2×D173semidirect
253. #41D692semidirect
257. C2×Q8D692semidirect
278. C2×D4D692semidirect
295. M16C22×D173semidirect
301. C42C22×D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497252
#260122912
#260122892
#260120602
#260116702
#260116602
#260115092
C2×#191495952
#421×C1732
#3019173