Finite Group Databasebeta

← #36365103

Subgroups — #36365103

Order 22144.

510 conjugacy classes of subgroups.

The subgroup lattice has 430 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149706C211yesmaximal, characteristic, complemented
2#26012160C211yesmaximal, characteristic, complemented
3#26012129C211yesmaximal, characteristic, complemented
4#26011877C211yesmaximal, characteristic, complemented
5#26011734C211yesmaximal, characteristic, complemented
6#26011727C211yesmaximal, characteristic, complemented
7#26011715C211yesmaximal, characteristic, complemented
8#26011684C211yesmaximal, characteristic, complemented
9#26011652C211yesmaximal, characteristic, complemented
10#26011636C211yesmaximal, characteristic, complemented
11#26011636C211yesmaximal, characteristic, complemented
12#26011623C211yesmaximal, characteristic, complemented
13#26011592C211yesmaximal, characteristic, complemented
14#26011562C211yesmaximal, characteristic, complemented
15C2×#135×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×D4×D173C2211yescharacteristic
18#19149706—24no—
19C2×#14583813C2211yescharacteristic
20C2×#14583813C2211yescharacteristic, complemented
21C2×D1384C2211yescharacteristic
22C2×#14583811C2211yescharacteristic
23C2×#14583811C2211yescharacteristic
24C2×#14583809C2211yescharacteristic
25#19149675—21no—
26#19149675C2211yescharacteristic, complemented
27#19149668C2211yescharacteristic
28#19149662—21no—
29#19149655C2211yescharacteristic
30#19149653—21no—
31#19149653C2211yescharacteristic
32#19149653C2211yescharacteristic, complemented
33#19149651—21no—
34#19149650—21no—
35#19149650C2211yescharacteristic
36#19149647—21no—
37Q8×Dic173C2211yescharacteristic, complemented
38D4×Dic173C2211yescharacteristic
39C4×C22⋊D173—21no—
40C4×C22⋊D173C2211yescharacteristic
41C2×#14583803—21no—
42#19149613C2211yescharacteristic, complemented
43#19149608C2211yescharacteristic, complemented
44#19149603C2211yescharacteristic, complemented
45#19149603C2211yescharacteristic, complemented
46#19149601C2211yescharacteristic
47#19149601C2211yescharacteristic, complemented
48#19149600C2211yescharacteristic
49#19149600C2211yescharacteristic, complemented
50C22⋊#13063436C2211yescharacteristic, complemented
51#19149591C2211yescharacteristic
52#19149589C2211yescharacteristic, complemented
53C2×#41×C173C2211yescharacteristic
54C2×#41×C173C2211yescharacteristic, complemented
55#135×C173—24no—
56C2×Q16×C173C2211yescharacteristic, complemented
57C2×SD16×C173C2211yescharacteristic
58C2×SD16×C173C2211yescharacteristic, complemented
59C2×D8×C173C2211yescharacteristic
60C22×C8×C173C2211yescharacteristic, complemented
61C23×D173—21no—
62#14583827—22no—
63#14583827—41no—
64C2×D692—21no—
65C2×D692C2311yescharacteristic
66C2×D692C2311yescharacteristic
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C2×C22⋊D173D411yescharacteristic
74D4×D173—22no—
75D4×D173—41no—
76C22×Dic173—21no—
77C2×C4×D173—21no—
78C2×C4×D173C2311yescharacteristic
79#14583813—22no—
80#14583813—41no—
81D1384—22no—
82#14583811—22no—
83#14583811—41no—
84#14583809—22no—
85C2×#13063436C2311yescharacteristic
86C2×#13063436C2311yescharacteristic
87C4⋊Dic173—21no—
88C4⋊Dic173C2311yescharacteristic
89C4⋊Dic173C2311yescharacteristic
90#14583805—21no—
91#14583805D411yescharacteristic
92C22⋊Dic173—21no—
93C22⋊Dic173—21no—
94C22⋊Dic173D411yescharacteristic
95#14583803—21no—
96#14583803—21no—
97#14583803—21no—
98#14583803—21no—
99#14583803—21no—
100#14583803D411yescharacteristic
101C4×Dic173—21no—
102C4×Dic173C2311yescharacteristic
103#41×C173—22no—
104#41×C173—22no—
105#41×C173—41no—
106#41×C173—41no—
107C2×Q8×C173C2311yescharacteristic
108C2×Q8×C173C2311yescharacteristic
109C2×D4×C173—21no—
110C2×D4×C173—21no—
111C2×D4×C173C2311yescharacteristic
112C2×D4×C173C2311yescharacteristic
113C22×C4×C173—21no—
114C22×C4×C173—21no—
115C22×C4×C173C2311yescharacteristic
116Q16×C173—22no—
117SD16×C173—22no—
118SD16×C173D414yescomplemented
119D8×C173—22no—
120C2×C8×C173—22no—
121C2×C8×C173C2311yescharacteristic
122C2×C8×C173C2311yescharacteristic
123C22×D173—21no—
124C22×D173—21no—
125C22×D173—42no—
126C22×D173—42no—
127C22×D173—81no—
128C22×D173C2×D411yescharacteristic
129D692—21no—
130D692—21no—
131D692—21no—
132D692—21no—
133D692—42no—
134C22⋊D173—41no—
135C22⋊D173—42no—
136C22⋊D173—42no—
137C22⋊D173—81no—
138C22⋊D173—81no—
139C22⋊D173—81no—
140C22⋊D173—81no—
141C2×Dic173—21no—
142C2×Dic173—21no—
143C2×Dic173—81no—
144C2×Dic173C2×D411yescharacteristic
145C2×Dic173C2×D411yescharacteristic
146C2×Dic173C2×D411yescharacteristic
147C4×D173—21no—
148C4×D173—21no—
149C4×D173—42no—
150#13063436—21no—
151#13063436—21no—
152C23×C173—21no—
153C23×C173—21no—
154C23×C173C2×D411yescharacteristic
155Q8×C173—21no—
156Q8×C173—21no—
157Q8×C173—21no—
158Q8×C173C2×D412yes—
159D4×C173—21no—
160D4×C173—21no—
161D4×C173—21no—
162D4×C173—42no—
163D4×C173—42no—
164D4×C173C2×D412yes—
165C2×C4×C173—21no—
166C2×C4×C173—21no—
167C2×C4×C173—21no—
168C2×C4×C173—21no—
169C2×C4×C173—42no—
170C2×C4×C173—42no—
171C2×C4×C173C2×D411yescharacteristic
172C2×C4×C173C2411yescharacteristic
173C8×C173—21no—
174C8×C173C2×D412yes—
175C22×C173—21no—
176C22×C173—22no—
177C22×C173—21no—
178C22×C173—21no—
179C22×C173—41no—
180C22×C173—41no—
181C22×C173—41no—
182C22×C173—41no—
183C22×C173—41no—
184C22×C173C22×D411yescharacteristic
185C2×D173—21no—
186C2×D173—21no—
187C2×D173—21no—
188C2×D173—41no—
189C2×D173—41no—
190C2×D173—41no—
191C2×D173—41no—
192C2×D173—41no—
193C2×D173—41no—
194C2×D173—84no—
195C4×C173—21no—
196C4×C173—22no—
197C4×C173—41no—
198C4×C173C22×D411yescharacteristic
199C4×C173H(5,𝔽2)11yescharacteristic
200Dic173—21no—
201Dic173—21no—
202Dic173—21no—
203Dic173—41no—
204Dic173—41no—
205C2×C173—42no—
206C2×C173—41no—
207C2×C173—81no—
208C2×C173D4211yescharacteristic
209C2×C173#57511yescharacteristic
210C2×C173#57611yescharacteristic
211D173—41no—
212D173—81no—
213D173—81no—
214C173#326811yescharacteristic, complemented, Sylow
215#3268—1731nomaximal, Sylow
216C2×AGL(1,ℤ/8)—1731no—
217C2×#135D17311yescharacteristic, complemented
218#550—1731no—
219#546—1731no—
220#496—1731no—
221#468—1731no—
222#466—1731no—
223#464—1731no—
224#459—1731no—
225#452—1731no—
226#449—1731no—
227#449—1731no—
228#447—1731no—
229#440—1731no—
230#435—1731no—
231C2×#41—1731no—
232C2×#41C2×D17311yescharacteristic
233C2×#41C2×D17311yescharacteristic, complemented
234C22×D4—1731no—
235AGL(1,ℤ/8)—3464no—
236#135—24no—
237C2×Q16C2×D17311yescharacteristic, complemented
238C2×SD16—1731no—
239C2×SD16—1731no—
240C2×SD16C2×D17311yescharacteristic
241C2×SD16C2×D17311yescharacteristic, complemented
242C2×D8—1731no—
243C2×D8—1731no—
244C2×D8—1731no—
245C2×D8C2×D17311yescharacteristic
246C2×M16—1731no—
247C22×C8C2×D17311yescharacteristic, complemented
248#127—3461no—
249#127—1731no—
250#124—1731no—
251#123—3461no—
252C22⋊Q8—1731no—
253#121—3461no—
254#121—3461no—
255#121—3461no—
256#121—1731no—
257#121—1731no—
258#121—1731no—
259#120—3461no—
260C4×Q8—1731no—
261C4×D4—3461no—
262C4×D4—1731no—
263C4×D4—1731no—
264C2×C22⋊C4—3461no—
265#107—1731no—
266C4⋊C8—1731no—
267#103—1731no—
268#103—1731no—
269#102—1731no—
270#102—1731no—
271#102—1731no—
272#102—1731no—
273C22⋊C8—1731no—
274C22⋊C8—1731no—
275#97—1731no—
276C24—3461no—
277#41—22no—
278#41—22no—
279#41—41no—
280#41—41no—
281#41—3462no—
282#41—6921no—
283C2×Q8C22×D17311yescharacteristic
284C2×Q8C22×D17311yescharacteristic
285C2×D4—21no—
286C2×D4—21no—
287C2×D4—3461no—
288C2×D4—3462no—
289C2×D4—3461no—
290C2×D4—3461no—
291C2×D4—3461no—
292C2×D4—3461no—
293C2×D4—3461no—
294C2×D4—3461no—
295C2×D4—1731no—
296C2×D4—1731no—
297C2×D4—1731no—
298C2×D4—6921no—
299C2×D4C22×D17311yescharacteristic
300C2×D4C22×D17311yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—3461no—
304C22×C4—3461no—
305C22×C4—1731no—
306C22×C4C22×D17311yescharacteristic
307Q16—22no—
308SD16—22no—
309SD16—3462no—
310SD16—6921no—
311SD16C22⋊D17314yescomplemented
312D8—22no—
313D8—3462no—
314D8—3462no—
315D8—6921no—
316M16—3462no—
317C2×C8—22no—
318C2×C8—1731no—
319C2×C8—1731no—
320C2×C8C22×D17311yescharacteristic
321C2×C8C22×D17311yescharacteristic
322C4⋊C4—3461no—
323C4⋊C4—3461no—
324C4⋊C4—1731no—
325C4⋊C4—1731no—
326C4⋊C4—1731no—
327C22⋊C4—3461no—
328C22⋊C4—3461no—
329C22⋊C4—3461no—
330C22⋊C4—3461no—
331C22⋊C4—3461no—
332C22⋊C4—3461no—
333C22⋊C4—3461no—
334C22⋊C4—1731no—
335C22⋊C4—1731no—
336C42—3461no—
337C42—1731no—
338C23—21no—
339C23—21no—
340C23—3461no—
341C23—3461no—
342C23—13841no—
343C23—1731no—
344C23—6922no—
345C23—6922no—
346C23D4×D17311yescharacteristic
347Q8—21no—
348Q8—21no—
349Q8—21no—
350Q8C2×C22⋊D17312yes—
351D4—21no—
352D4—21no—
353D4—21no—
354D4—42no—
355D4—42no—
356D4—3461no—
357D4—3461no—
358D4—3461no—
359D4—3461no—
360D4—13841no—
361D4—13841no—
362D4—13841no—
363D4—13841no—
364D4—6922no—
365D4—6921no—
366D4—6922no—
367D4—6922no—
368D4C2×C22⋊D17312yes—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—42no—
374C2×C4—42no—
375C2×C4—3461no—
376C2×C4—3461no—
377C2×C4—3461no—
378C2×C4—3461no—
379C2×C4—13841no—
380C2×C4—1731no—
381C2×C4—1731no—
382C2×C4—1731no—
383C2×C4—6922no—
384C2×C4D4×D17311yescharacteristic
385C2×C4C23×D17311yescharacteristic
386C8—21no—
387C8—3461no—
388C8—3461no—
389C8C2×C22⋊D17312yes—
390C22—21no—
391C22—21no—
392C22—21no—
393C22—22no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—3461no—
400C22—3461no—
401C22—3461no—
402C22—13844no—
403C22—6921no—
404C22—6921no—
405C22—6921no—
406C22—6921no—
407C22—6921no—
408C22—6921no—
409C22C2×D4×D17311yescharacteristic
410C4—21no—
411C4—22no—
412C4—41no—
413C4—3461no—
414C4—3461no—
415C4—3461no—
416C4—6921no—
417C4—6921no—
418C4C22×C22⋊D17311yescharacteristic
419C4#1914972411yescharacteristic
420C2—41no—
421C2—42no—
422C2—81no—
423C2—13841no—
424C2—13841no—
425C2—6921no—
426C2D4×C22⋊D17311yescharacteristic
427C2#2601228811yescharacteristic
428C2#2601229311yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#135×C173C2
socleC22×C173C22×D4
hypercenterC2×#135D173
hypocenterC173#3268
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×#19149706C2semidirect
2. #26012160C2semidirect
3. #26012129C2semidirect
4. #26011877C2semidirect
5. #26011734C2semidirect
6. #26011727C2semidirect
7. #26011715C2semidirect
8. #26011684C2semidirect
9. #26011652C2semidirect
10. #26011636C2semidirect
11. #26011636C2semidirect
12. #26011623C2semidirect
13. #26011592C2semidirect
14. #26011562C2semidirect
15. C2×#135×C173C2semidirect
20. C2×#14583813C22semidirect
26. #19149675C22semidirect
32. #19149653C22semidirect
37. Q8×Dic173C22semidirect
42. #19149613C22semidirect
43. #19149608C22semidirect
44. #19149603C22semidirect
45. #19149603C22semidirect
47. #19149601C22semidirect
49. #19149600C22semidirect
50. C22⋊#13063436C22semidirect
52. #19149589C22semidirect
54. C2×#41×C173C22semidirect
56. C2×Q16×C173C22semidirect
58. C2×SD16×C173C22semidirect
60. C22×C8×C173C22semidirect
118. SD16×C173D4semidirect
214. C173#3268semidirect
217. C2×#135D173semidirect
233. C2×#41C2×D173semidirect
237. C2×Q16C2×D173semidirect
241. C2×SD16C2×D173semidirect
247. C22×C8C2×D173semidirect
311. SD16C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497062
#260121602
#260121292
#260118772
#260117342
#260117272
#260117152
#260116842
#260116522
#260116362
#260116232
#260115922
#260115622
C2×#135×C1732
#3268173