Finite Group Databasebeta

← #36363114

Subgroups — #36363114

Order 22144.

472 conjugacy classes of subgroups.

The subgroup lattice has 384 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149708C211yesmaximal, characteristic, complemented
2C2×#19149698C211yesmaximal, characteristic
3#26012061C211yesmaximal, characteristic, complemented
4#26011970C211yesmaximal, characteristic, complemented
5#26011712C211yesmaximal, characteristic, complemented
6#26011710C211yesmaximal, characteristic, complemented
7#26011704C211yesmaximal, characteristic, complemented
8#26011702C211yesmaximal, characteristic, complemented
9#26011700C211yesmaximal, characteristic, complemented
10#26011696C211yesmaximal, characteristic, complemented
11#26011694C211yesmaximal, characteristic, complemented
12#26011516C211yesmaximal, characteristic, complemented
13#26011489C211yesmaximal, characteristic, complemented
14#103×D173C211yesmaximal, characteristic, complemented
15#463×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18C2×Q8×D173C2211yescharacteristic
19#19149708—24no—
20#19149698—24no—
21C2×Dic692C2211yescharacteristic
22C2×Dic692C2211yescharacteristic
23C2×#14583816C2211yescharacteristic
24C2×#14583814C2211yescharacteristic
25C2×#14583814C2211yescharacteristic
26C2×#14583814C2211yescharacteristic
27C2×#14583809C2211yescharacteristic
28C2×C8×D173C2211yescharacteristic
29#19149679C2211yescharacteristic
30#19149677C2211yescharacteristic
31#19149666C2211yescharacteristic
32#19149661—21no—
33#19149661—21no—
34#19149656—21no—
35#19149656—21no—
36C4×Dic346C2211yescharacteristic
37#19149644C2211yescharacteristic
38D4×Dic173C2211yescharacteristic
39C4×C22⋊D173—21no—
40#19149639—21no—
41#19149636C2211yescharacteristic
42#19149635—21no—
43C2×#14583805—21no—
44C4⋊C4×D173C2211yescharacteristic
45C4⋊#13063436C2211yescharacteristic
46#19149608C2211yescharacteristic
47#19149608C2211yescharacteristic
48#19149604C2211yescharacteristic
49#19149604C2211yescharacteristic
50#19149603C2211yescharacteristic
51#19149602C2211yescharacteristic
52#19149602C2211yescharacteristic
53#19149600C2211yescharacteristic
54C2×SD16×C173C2211yescharacteristic
55C2×D8×C173C2211yescharacteristic
56#124×C173C2211yescharacteristic
57C4×D4×C173C2211yescharacteristic
58C4⋊C8×C173C2211yescharacteristic
59#103×C173C2211yescharacteristic
60#102×C173C2211yescharacteristic
61#14583825—22no—
62#14583825—22no—
63#14583825—22no—
64#14583825D414yes—
65C2×Dic346—21no—
66C2×Dic346C2311yescharacteristic
67C2×Dic346C2311yescharacteristic
68C2×Dic346C2311yescharacteristic
69Q8×D173—22no—
70Q8×D173—22no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C22×Dic173—21no—
74C22×Dic173—21no—
75C22×Dic173—21no—
76C2×C4×D173—21no—
77C2×C4×D173—21no—
78C2×C4×D173C2311yescharacteristic
79Dic692—22no—
80Dic692—22no—
81#14583816—22no—
82#14583814—22no—
83#14583814—22no—
84#14583814—22no—
85#14583809—22no—
86C2×#13063436C2311yescharacteristic
87C2×#13063436C2311yescharacteristic
88C8×D173—22no—
89C4⋊Dic173—21no—
90C4⋊Dic173—21no—
91C4⋊Dic173C2311yescharacteristic
92#14583805—21no—
93#14583805—21no—
94#14583805—21no—
95#14583805—21no—
96#14583805—21no—
97#14583805D411yescharacteristic
98#14583805D411yescharacteristic
99C22⋊Dic173—21no—
100C22⋊Dic173—21no—
101C22⋊Dic173—21no—
102#14583803D411yescharacteristic
103#14583803D411yescharacteristic
104C4×Dic173—21no—
105C4×Dic173—21no—
106C4×Dic173C2311yescharacteristic
107C2×Q8×C173C2311yescharacteristic
108C2×D4×C173C2311yescharacteristic
109C2×D4×C173C2311yescharacteristic
110C22×C4×C173—21no—
111SD16×C173—22no—
112D8×C173—22no—
113C2×C8×C173C2311yescharacteristic
114C2×C8×C173C2311yescharacteristic
115C4⋊C4×C173C2311yescharacteristic
116C22⋊C4×C173—21no—
117C22⋊C4×C173—21no—
118C22⋊C4×C173—21no—
119C42×C173C2311yescharacteristic
120C22×D173C2×D411yescharacteristic
121Dic346—21no—
122Dic346—21no—
123Dic346—21no—
124Dic346—21no—
125Dic346—21no—
126Dic346—42no—
127Dic346C2×D412yes—
128C22⋊D173—24no—
129C22⋊D173—42no—
130C2×Dic173—21no—
131C2×Dic173—24no—
132C2×Dic173—21no—
133C2×Dic173—21no—
134C2×Dic173—21no—
135C2×Dic173—42no—
136C2×Dic173—81no—
137C2×Dic173C2×D411yescharacteristic
138C2×Dic173C2×D411yescharacteristic
139C2×Dic173C2×D411yescharacteristic
140C4×D173—42no—
141C4×D173—81no—
142C4×D173C2×D412yes—
143C4×D173#4112yes—
144#13063436—21no—
145#13063436—21no—
146C23×C173—21no—
147C23×C173—21no—
148Q8×C173—21no—
149Q8×C173—21no—
150D4×C173—21no—
151D4×C173—21no—
152D4×C173—21no—
153D4×C173C2×D412yes—
154C2×C4×C173—21no—
155C2×C4×C173—21no—
156C2×C4×C173—81no—
157C2×C4×C173C2×D411yescharacteristic
158C2×C4×C173C2×D411yescharacteristic
159C2×C4×C173C2411yescharacteristic
160C8×C173—21no—
161C8×C173—21no—
162C22×C173—22no—
163C22×C173—41no—
164C22×C173—41no—
165C22×C173—41no—
166C22×C173—41no—
167C22×C173—41no—
168C22×C173C22×D411yescharacteristic
169C2×D173—21no—
170C2×D173—21no—
171C2×D173#12112yes—
172C4×C173—21no—
173C4×C173—21no—
174C4×C173—41no—
175C4×C173—41no—
176C4×C173C22×D411yescharacteristic
177C4×C173C2×#4111yescharacteristic
178Dic173—22no—
179Dic173—21no—
180Dic173—21no—
181Dic173—41no—
182Dic173—41no—
183Dic173—41no—
184Dic173#12112yes—
185C2×C173—42no—
186C2×C173—81no—
187C2×C173C2×#12111yescharacteristic
188C2×C173#57711yescharacteristic
189C2×C173#57811yescharacteristic
190D173—22no—
191C173#304611yescharacteristic, complemented, Sylow
192#3046—1731nomaximal, Sylow
193C2×#137—1731no—
194C2×#135—1731no—
195#536—1731no—
196#520—1731no—
197#464—1731no—
198#464—1731no—
199#463—1731no—
200#463D17311yescharacteristic, complemented
201#462—1731no—
202#462—1731no—
203#461—1731no—
204#461—1731no—
205#424—1731no—
206#417—1731no—
207C2×#103—1731no—
208C2×#41—1731no—
209C2×#41—1731no—
210C22×Q8—1731no—
211#137—3464no—
212#135—3464no—
213C2×Q16—1731no—
214C2×Q16—1731no—
215C2×Q16—1731no—
216C2×SD16—1731no—
217C2×SD16—1731no—
218C2×SD16—1731no—
219C2×SD16C2×D17311yescharacteristic
220C2×D8C2×D17311yescharacteristic
221C2×M16—1731no—
222C22×C8—1731no—
223C4⋊Q8—1731no—
224C4⋊Q8—1731no—
225#124—1731no—
226#124C2×D17311yescharacteristic
227#123—3461no—
228#123—3461no—
229C22⋊Q8—3461no—
230C22⋊Q8—3461no—
231C4×Q8—1731no—
232C4×Q8—1731no—
233C4×D4—3461no—
234C4×D4—3461no—
235C4×D4—1731no—
236C4×D4C2×D17311yescharacteristic
237#117—3461no—
238#117—1731no—
239C2×C4⋊C4—3461no—
240C2×C4⋊C4—1731no—
241C4⋊C8—1731no—
242C4⋊C8—1731no—
243C4⋊C8—1731no—
244C4⋊C8C2×D17311yescharacteristic
245#103—1731no—
246#103—1731no—
247#103—1731no—
248#103—1731no—
249#103—1731no—
250#103C2×D17311yescharacteristic
251#102—1731no—
252#102C2×D17311yescharacteristic
253#41—3462no—
254#41—3462no—
255#41—3462no—
256#41—1734no—
257C2×Q8—3461no—
258C2×Q8—3462no—
259C2×Q8—3462no—
260C2×Q8—1731no—
261C2×Q8—1731no—
262C2×Q8—1731no—
263C2×Q8C22×D17311yescharacteristic
264C2×D4—3461no—
265C2×D4—3461no—
266C2×D4C22×D17311yescharacteristic
267C2×D4C22×D17311yescharacteristic
268C22×C4—21no—
269C22×C4—3461no—
270C22×C4—3461no—
271C22×C4—3461no—
272C22×C4—3461no—
273C22×C4—3461no—
274C22×C4—1731no—
275Q16—3462no—
276Q16—3462no—
277Q16—3462no—
278SD16—22no—
279SD16—3462no—
280SD16—3462no—
281SD16—3462no—
282D8—22no—
283M16—3462no—
284C2×C8—3462no—
285C2×C8—1731no—
286C2×C8—1731no—
287C2×C8C22×D17311yescharacteristic
288C2×C8C22×D17311yescharacteristic
289C4⋊C4—3461no—
290C4⋊C4—3461no—
291C4⋊C4—3461no—
292C4⋊C4—3461no—
293C4⋊C4—3461no—
294C4⋊C4—3461no—
295C4⋊C4—3461no—
296C4⋊C4—1731no—
297C4⋊C4—1731no—
298C4⋊C4—1731no—
299C4⋊C4C22×D17311yescharacteristic
300C22⋊C4—21no—
301C22⋊C4—21no—
302C22⋊C4—21no—
303C22⋊C4—3461no—
304C22⋊C4—3461no—
305C22⋊C4—3461no—
306C22⋊C4—1731no—
307C22⋊C4—1731no—
308C42—3461no—
309C42—3461no—
310C42—1731no—
311C42C22×D17311yescharacteristic
312C23—21no—
313C23—21no—
314C23—1731no—
315Q8—21no—
316Q8—21no—
317Q8—3461no—
318Q8—3461no—
319Q8—3461no—
320Q8—3461no—
321Q8—3461no—
322Q8—1732no—
323Q8—6922no—
324D4—21no—
325D4—21no—
326D4—21no—
327D4—3464no—
328D4—6922no—
329D4D4×D17312yes—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—81no—
333C2×C4—3461no—
334C2×C4—3464no—
335C2×C4—3461no—
336C2×C4—3461no—
337C2×C4—3461no—
338C2×C4—13841no—
339C2×C4—13841no—
340C2×C4—1731no—
341C2×C4—1731no—
342C2×C4—1731no—
343C2×C4—1732no—
344C2×C4—1732no—
345C2×C4—6922no—
346C2×C4—6922no—
347C2×C4D4×D17311yescharacteristic
348C2×C4D4×D17311yescharacteristic
349C2×C4C23×D17311yescharacteristic
350C8—21no—
351C8—21no—
352C8—3461no—
353C8—3461no—
354C22—22no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22—41no—
359C22—41no—
360C22—3461no—
361C22—3461no—
362C22—1732no—
363C22C2×D4×D17311yescharacteristic
364C4—21no—
365C4—21no—
366C4—41no—
367C4—41no—
368C4—3462no—
369C4—3461no—
370C4—3461no—
371C4—1732no—
372C4—6921no—
373C4—6921no—
374C4—6921no—
375C4C2×D4×D17311yescharacteristic
376C4#41×D17311yescharacteristic
377C2—42no—
378C2—81no—
379C2—3462no—
380C2#121×D17311yescharacteristic
381C2#2601230111yescharacteristic
382C2#2601230511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#463×C173C2
socleC22×C173C22×D4
prime radicalC2×#19149698C2
hypercenter#463D173
hypocenterC173#3046
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C42 ◁ #463

Complements

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

NH ≅ G/Ntype
1. C2×#19149708C2semidirect
3. #26012061C2semidirect
4. #26011970C2semidirect
5. #26011712C2semidirect
6. #26011710C2semidirect
7. #26011704C2semidirect
8. #26011702C2semidirect
9. #26011700C2semidirect
10. #26011696C2semidirect
11. #26011694C2semidirect
12. #26011516C2semidirect
13. #26011489C2semidirect
14. #103×D173C2semidirect
15. #463×C173C2semidirect
191. C173#3046semidirect
200. #463D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497082
C2×#191496982
#260120612
#260119702
#260117122
#260117102
#260117042
#260117022
#260117002
#260116962
#260116942
#260115162
#260114892
#103×D1732
#463×C1732
#3046173