Finite Group Databasebeta

← #36362777

Subgroups — #36362777

Order 22144.

764 conjugacy classes of subgroups.

The subgroup lattice has 476 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#41×D173C211yesmaximal, characteristic, complemented
2C2×#19149699C211yesmaximal, characteristic, complemented
3#26011998C211yesmaximal, characteristic, complemented
4#26011652C211yesmaximal, characteristic, complemented
5#26011650C211yesmaximal, characteristic, complemented
6#26011645C211yesmaximal, characteristic, complemented
7#26011640C211yesmaximal, characteristic, complemented
8#26011635C211yesmaximal, characteristic, complemented
9#26011633C211yesmaximal, characteristic, complemented
10#26011628C211yesmaximal, characteristic, complemented
11#26011623C211yesmaximal, characteristic, complemented
12#26011488C211yesmaximal, characteristic, complemented
13#26011487C211yesmaximal, characteristic, complemented
14C2×#19149591C211yesmaximal, characteristic, complemented
15C2×#135×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583826—21no—
18C2×#14583826C2211yescharacteristic
19C2×#14583825—21no—
20C2×#14583825C2211yescharacteristic
21#41×D173—24no—
22#41×D173—24no—
23C2×Q8×D173C2211yescharacteristic
24C2×D4×D173—21no—
25C2×D4×D173C2211yescharacteristic
26C22×C4×D173—21no—
27C22×C4×D173C2211yescharacteristic
28#19149699—24no—
29C2×#14583816C2211yescharacteristic, complemented
30C2×#14583814C2211yescharacteristic, complemented
31C2×#14583813C2211yescharacteristic, complemented
32C2×#14583811C2211yescharacteristic, complemented
33C22×#13063436C2211yescharacteristic, complemented
34#19149660—21no—
35#19149655C2211yescharacteristic
36#19149655C2211yescharacteristic, complemented
37#19149650C2211yescharacteristic
38#19149650C2211yescharacteristic, complemented
39#19149647—21no—
40C4×C22⋊D173—21no—
41C4×C22⋊D173—21no—
42#19149638C2211yescharacteristic, complemented
43#19149604C2211yescharacteristic, complemented
44#19149604C2211yescharacteristic, complemented
45#19149603C2211yescharacteristic, complemented
46#19149603C2211yescharacteristic, complemented
47#19149601C2211yescharacteristic, complemented
48#19149601C2211yescharacteristic, complemented
49#19149600C2211yescharacteristic, complemented
50#19149600C2211yescharacteristic, complemented
51#19149591C2211yescharacteristic
52#19149591C2211yescharacteristic
53#19149591C2211yescharacteristic, complemented
54#19149591C2211yescharacteristic, complemented
55C2×#41×C173C2211yescharacteristic
56C2×#41×C173C2211yescharacteristic, complemented
57#135×C173—24no—
58C2×Q16×C173C2211yescharacteristic, complemented
59C2×SD16×C173C2211yescharacteristic, complemented
60C2×SD16×C173C2211yescharacteristic, complemented
61C2×D8×C173C2211yescharacteristic, complemented
62C22×C8×C173C2211yescharacteristic, complemented
63C23×D173—21no—
64C23×D173D411yescharacteristic
65#14583827—22no—
66#14583827—22no—
67#14583826—28no—
68#14583826—22no—
69#14583826D414yescomplemented
70#14583825—28no—
71#14583825—22no—
72#14583825—22no—
73C2×Dic346—21no—
74C2×Dic346C2311yescharacteristic
75Q8×D173—22no—
76Q8×D173—22no—
77C2×D692—21no—
78C2×D692C2311yescharacteristic
79C2×C22⋊D173—21no—
80C2×C22⋊D173—21no—
81C2×C22⋊D173—21no—
82C2×C22⋊D173—21no—
83C2×C22⋊D173—21no—
84D4×D173—28no—
85D4×D173—22no—
86D4×D173—22no—
87C22×Dic173—21no—
88C22×Dic173D411yescharacteristic
89C2×C4×D173—28no—
90C2×C4×D173—21no—
91C2×C4×D173—21no—
92C2×C4×D173—21no—
93C2×C4×D173—22no—
94C2×C4×D173—22no—
95C2×C4×D173D411yescharacteristic
96C2×C4×D173D411yescharacteristic
97C2×C4×D173C2311yescharacteristic
98C2×C4×D173C2311yescharacteristic
99#14583816—22no—
100#14583814—22no—
101#14583813—22no—
102#14583811—22no—
103C2×#13063436—22no—
104C2×#13063436C2311yescharacteristic
105C2×#13063436C2311yescharacteristic
106C4⋊Dic173C2311yescharacteristic
107C4⋊Dic173C2311yescharacteristic
108#14583805—21no—
109#14583805—21no—
110C22⋊Dic173—21no—
111C22⋊Dic173—21no—
112C22⋊Dic173—21no—
113#14583803—21no—
114#14583803—21no—
115C4×Dic173—21no—
116#41×C173—22no—
117#41×C173—22no—
118#41×C173—22no—
119#41×C173D414yescomplemented
120C2×Q8×C173C2311yescharacteristic
121C2×Q8×C173C2311yescharacteristic
122C2×D4×C173—21no—
123C2×D4×C173—21no—
124C2×D4×C173C2311yescharacteristic
125C2×D4×C173C2311yescharacteristic
126C22×C4×C173—21no—
127C22×C4×C173—21no—
128C22×C4×C173C2311yescharacteristic
129Q16×C173—22no—
130SD16×C173—22no—
131SD16×C173—22no—
132D8×C173—22no—
133C2×C8×C173—22no—
134C2×C8×C173C2311yescharacteristic
135C2×C8×C173C2311yescharacteristic
136C22×D173—24no—
137C22×D173—22no—
138C22×D173—21no—
139C22×D173—22no—
140C22×D173—41no—
141C22×D173—42no—
142C22×D173—42no—
143C22×D173C2×D411yescharacteristic
144C22×D173C2×D411yescharacteristic
145Dic346—24no—
146Dic346—21no—
147Dic346C2×D412yes—
148D692—24no—
149D692—21no—
150D692C2×D412yes—
151C22⋊D173—24no—
152C22⋊D173—24no—
153C22⋊D173—24no—
154C22⋊D173—42no—
155C22⋊D173—42no—
156C2×Dic173—24no—
157C2×Dic173—22no—
158C2×Dic173—21no—
159C2×Dic173—21no—
160C2×Dic173—21no—
161C2×Dic173C2×D411yescharacteristic
162C2×Dic173C2×D411yescharacteristic
163C4×D173—24no—
164C4×D173—24no—
165C4×D173—22no—
166C4×D173—24no—
167C4×D173—22no—
168C4×D173—21no—
169C4×D173—21no—
170C4×D173—21no—
171C4×D173—21no—
172#13063436—21no—
173#13063436—21no—
174C23×C173—21no—
175C23×C173—21no—
176C23×C173C2×D411yescharacteristic
177Q8×C173—21no—
178Q8×C173—21no—
179Q8×C173—21no—
180Q8×C173C2×D412yes—
181D4×C173—24no—
182D4×C173—21no—
183D4×C173—21no—
184D4×C173—21no—
185D4×C173—42no—
186D4×C173C2×D412yes—
187C2×C4×C173—21no—
188C2×C4×C173—24no—
189C2×C4×C173—21no—
190C2×C4×C173—42no—
191C2×C4×C173C2×D412yes—
192C2×C4×C173C2×D411yescharacteristic
193C2×C4×C173C2×D412yes—
194C2×C4×C173C2411yescharacteristic
195C8×C173—21no—
196C8×C173—21no—
197C22×C173—21no—
198C22×C173—21no—
199C22×C173—22no—
200C22×C173—41no—
201C22×C173—41no—
202C22×C173—41no—
203C22×C173—41no—
204C22×C173—41no—
205C22×C173#12012yes—
206C22×C173C22×D411yescharacteristic
207C2×D173—21no—
208C2×D173—21no—
209C2×D173—21no—
210C2×D173—22no—
211C2×D173—21no—
212C2×D173—21no—
213C2×D173—44no—
214C2×D173—48no—
215C2×D173—41no—
216C2×D173—41no—
217C2×D173#13512yescomplemented
218C4×C173—22no—
219C4×C173—41no—
220C4×C173#12012yes—
221C4×C173C22×D411yescharacteristic
222C4×C173C22×D411yescharacteristic
223Dic173—21no—
224Dic173—22no—
225Dic173—21no—
226Dic173—41no—
227Dic173—41no—
228C2×C173—22no—
229C2×C173—42no—
230C2×C173—81no—
231C2×C173C2×#12011yescharacteristic
232C2×C173C2×#13511yescharacteristic
233C2×C173#57511yescharacteristic
234D173—22no—
235D173—41no—
236D173—42no—
237C173#300211yescharacteristic, complemented, Sylow
238#3002—1731nomaximal, Sylow
239C22×#41—1731no—
240C2×#135—1731no—
241C2×#135D17311yescharacteristic, complemented
242#525—1731no—
243#452—1731no—
244#452—1731no—
245#451—1731no—
246#450—1731no—
247#449—1731no—
248#449—1731no—
249#448—1731no—
250#447—1731no—
251#416—1731no—
252#416—1731no—
253C2×C22⋊C8—1731no—
254C2×#41—3461no—
255C2×#41—3464no—
256C2×#41—3464no—
257C2×#41—3461no—
258C2×#41—1731no—
259C2×#41—1731no—
260C2×#41—1731no—
261C2×#41C2×D17311yescharacteristic
262C2×#41C2×D17311yescharacteristic, complemented
263C22×Q8—1731no—
264C22×D4—3461no—
265C22×D4—1731no—
266C23×C4—3461no—
267C23×C4—1731no—
268#135—24no—
269#135—3464no—
270C2×Q16—1731no—
271C2×Q16C2×D17311yescharacteristic, complemented
272C2×SD16—1731no—
273C2×SD16—1731no—
274C2×SD16C2×D17311yescharacteristic, complemented
275C2×SD16C2×D17311yescharacteristic, complemented
276C2×D8—1731no—
277C2×D8C2×D17311yescharacteristic, complemented
278C22×C8—1731no—
279C22×C8C2×D17311yescharacteristic, complemented
280#123—3461no—
281C22⋊Q8—1731no—
282C22⋊Q8—1731no—
283#121—1731no—
284#121—1731no—
285#120—3461no—
286C4×D4—3461no—
287C4×D4—3461no—
288#117—1731no—
289#103—1731no—
290#103—1731no—
291#103—1731no—
292#103—1731no—
293#102—1731no—
294#102—1731no—
295#102—1731no—
296#102—1731no—
297C22⋊C8—1731no—
298C22⋊C8—1731no—
299C22⋊C8—1731no—
300C22⋊C8—1731no—
301C24—3461no—
302C24—1731no—
303#41—22no—
304#41—22no—
305#41—22no—
306#41—3468no—
307#41—3468no—
308#41—3462no—
309#41—3462no—
310#41—3462no—
311#41—3462no—
312#41—3462no—
313#41—1734no—
314#41C22⋊D17314yescomplemented
315C2×Q8—3461no—
316C2×Q8—3462no—
317C2×Q8—3462no—
318C2×Q8—1731no—
319C2×Q8C22×D17311yescharacteristic
320C2×Q8C22×D17311yescharacteristic
321C2×D4—21no—
322C2×D4—21no—
323C2×D4—3461no—
324C2×D4—3468no—
325C2×D4—3462no—
326C2×D4—3461no—
327C2×D4—3461no—
328C2×D4—3462no—
329C2×D4—3461no—
330C2×D4—3461no—
331C2×D4—3461no—
332C2×D4—1731no—
333C2×D4C22×D17311yescharacteristic
334C2×D4C22×D17311yescharacteristic
335C22×C4—21no—
336C22×C4—21no—
337C22×C4—3468no—
338C22×C4—3461no—
339C22×C4—3461no—
340C22×C4—3461no—
341C22×C4—3462no—
342C22×C4—3461no—
343C22×C4—3462no—
344C22×C4—1731no—
345C22×C4—1731no—
346C22×C4—1731no—
347C22×C4—1731no—
348C22×C4—1731no—
349C22×C4C22×D17311yescharacteristic
350Q16—22no—
351Q16—3462no—
352SD16—22no—
353SD16—22no—
354SD16—3462no—
355SD16—3462no—
356D8—22no—
357D8—3462no—
358C2×C8—22no—
359C2×C8—3462no—
360C2×C8—1731no—
361C2×C8—1731no—
362C2×C8C22×D17311yescharacteristic
363C2×C8C22×D17311yescharacteristic
364C4⋊C4—3461no—
365C4⋊C4—3461no—
366C4⋊C4—1731no—
367C4⋊C4—1731no—
368C22⋊C4—3461no—
369C22⋊C4—3461no—
370C22⋊C4—3461no—
371C22⋊C4—3461no—
372C22⋊C4—3461no—
373C42—3461no—
374C23—21no—
375C23—21no—
376C23—3464no—
377C23—3462no—
378C23—3462no—
379C23—3461no—
380C23—1731no—
381C23—1731no—
382C23—6921no—
383C23—6922no—
384C23—6922no—
385C23D4×D17311yescharacteristic
386Q8—21no—
387Q8—21no—
388Q8—21no—
389Q8—3464no—
390Q8—3461no—
391Q8—1732no—
392Q8C2×C22⋊D17312yes—
393D4—24no—
394D4—21no—
395D4—21no—
396D4—21no—
397D4—42no—
398D4—3464no—
399D4—3464no—
400D4—3464no—
401D4—3464no—
402D4—3461no—
403D4—1732no—
404D4—6922no—
405D4—6922no—
406D4C2×C22⋊D17312yes—
407C2×C4—21no—
408C2×C4—24no—
409C2×C4—21no—
410C2×C4—42no—
411C2×C4—3464no—
412C2×C4—3464no—
413C2×C4—3464no—
414C2×C4—3462no—
415C2×C4—3461no—
416C2×C4—3462no—
417C2×C4—3464no—
418C2×C4—3462no—
419C2×C4—3461no—
420C2×C4—3461no—
421C2×C4—3461no—
422C2×C4—3461no—
423C2×C4—3461no—
424C2×C4—3461no—
425C2×C4—1731no—
426C2×C4—1731no—
427C2×C4D4×D17312yes—
428C2×C4D4×D17311yescharacteristic
429C2×C4C2×C22⋊D17312yes—
430C2×C4C23×D17311yescharacteristic
431C8—21no—
432C8—21no—
433C8—3461no—
434C8—3461no—
435C22—21no—
436C22—21no—
437C22—22no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—3461no—
444C22—3461no—
445C22—3461no—
446C22—3462no—
447C22—3461no—
448C22—3461no—
449C22—1732no—
450C22—6924no—
451C22—6928no—
452C22—6921no—
453C22—6921no—
454C22#1914964712yes—
455C22C2×D4×D17311yescharacteristic
456C4—22no—
457C4—41no—
458C4—3461no—
459C4—3462no—
460C4—3461no—
461C4—6921no—
462C4—6921no—
463C4#1914964712yes—
464C4C2×D4×D17311yescharacteristic
465C4C22×C22⋊D17311yescharacteristic
466C2—22no—
467C2—42no—
468C2—81no—
469C2—3462no—
470C2—6921no—
471C2—6922no—
472C2C2×#1914964711yescharacteristic
473C2#135×D17311yescharacteristic
474C2#2601228811yescharacteristic
subgroupG/H
centerC2×C4D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#135×C173C2
socleC22×C173C22×D4
hypercenterC2×#135D173
hypocenterC173#3002
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C2×C4 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#41×D173C2semidirect
2. C2×#19149699C2semidirect
3. #26011998C2semidirect
4. #26011652C2semidirect
5. #26011650C2semidirect
6. #26011645C2semidirect
7. #26011640C2semidirect
8. #26011635C2semidirect
9. #26011633C2semidirect
10. #26011628C2semidirect
11. #26011623C2semidirect
12. #26011488C2semidirect
13. #26011487C2semidirect
14. C2×#19149591C2semidirect
15. C2×#135×C173C2semidirect
29. C2×#14583816C22semidirect
30. C2×#14583814C22semidirect
31. C2×#14583813C22semidirect
32. C2×#14583811C22semidirect
33. C22×#13063436C22semidirect
36. #19149655C22semidirect
38. #19149650C22semidirect
42. #19149638C22semidirect
43. #19149604C22semidirect
44. #19149604C22semidirect
45. #19149603C22semidirect
46. #19149603C22semidirect
47. #19149601C22semidirect
48. #19149601C22semidirect
49. #19149600C22semidirect
50. #19149600C22semidirect
53. #19149591C22semidirect
54. #19149591C22semidirect
56. C2×#41×C173C22semidirect
58. C2×Q16×C173C22semidirect
59. C2×SD16×C173C22semidirect
60. C2×SD16×C173C22semidirect
61. C2×D8×C173C22semidirect
62. C22×C8×C173C22semidirect
69. #14583826D4semidirect
119. #41×C173D4semidirect
217. C2×D173#135semidirect
237. C173#3002semidirect
241. C2×#135D173semidirect
262. C2×#41C2×D173semidirect
271. C2×Q16C2×D173semidirect
274. C2×SD16C2×D173semidirect
275. C2×SD16C2×D173semidirect
277. C2×D8C2×D173semidirect
279. C22×C8C2×D173semidirect
314. #41C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×#41×D1732
C2×#191496992
#260119982
#260116522
#260116502
#260116452
#260116402
#260116352
#260116332
#260116282
#260116232
#260114882
#260114872
C2×#191495912
C2×#135×C1732
#3002173