Finite Group Databasebeta

← #36363282

Subgroups — #36363282

Order 22144.

452 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012291C211yesmaximal, characteristic, complemented
2#26012289C211yesmaximal, characteristic, complemented
3C2×#19149699C211yesmaximal, characteristic, complemented
4C2×#19149687C211yesmaximal, characteristic, complemented
5#26011765C212yesmaximal, complemented
6#26011760C212yesmaximal, complemented
7#26011757C212yesmaximal, complemented
8#26011753C212yesmaximal, complemented
9C2×#19149614C211yesmaximal, characteristic, complemented
10#26011473C211yesmaximal, characteristic, complemented
11#413×C173C211yesmaximal, characteristic, complemented
12C2×#14583826C2211yescharacteristic
13C2×#14583826C2211yescharacteristic
14#19149712—21no—
15#19149712C2212yes—
16#19149710C2212yes—
17#19149709—21no—
18#19149707—21no—
19#19149707C2212yes—
20#19149705C2212yes—
21#19149704—21no—
22#19149701—22no—
23#19149699—22no—
24#19149699—21no—
25#19149699—21no—
26#19149699—21no—
27#19149699—21no—
28C2×#14583816C2211yescharacteristic, complemented
29C2×#14583814C2211yescharacteristic, complemented
30C2×#14583813C2211yescharacteristic, complemented
31C2×#14583811C2211yescharacteristic, complemented
32#19149687—21no—
33#19149687—21no—
34#19149687C2212yes—
35#19149687C2212yes—
36C2×#14583810C2211yescharacteristic
37C2×#14583809—21no—
38C22×#13063436C2211yescharacteristic
39C2×C8×D173—21no—
40#19149614C2212yescomplemented
41#19149614C2212yescomplemented
42#19149597C2212yescomplemented
43#19149595C2212yescomplemented
44C2×#41×C173C2211yescharacteristic
45C2×M16×C173C2211yescharacteristic
46C2×M16×C173C2211yescharacteristic
47#101×C173C2212yescomplemented
48#100×C173C2212yescomplemented
49#14583826—21no—
50#14583826—21no—
51#14583826—21no—
52#14583826—21no—
53#14583826—21no—
54#14583826—21no—
55#14583826D412yes—
56#14583826C2312yes—
57C2×Dic346D411yescharacteristic
58C2×Dic346C2311yescharacteristic
59C2×D692D411yescharacteristic
60C2×D692C2311yescharacteristic
61C2×C22⋊D173—21no—
62C2×C22⋊D173—21no—
63C2×C4×D173—21no—
64C2×C4×D173—21no—
65Dic692—22no—
66#14583816—22no—
67#14583816—21no—
68#14583816—21no—
69#14583815—22no—
70#14583815—22no—
71#14583814—22no—
72#14583814—21no—
73#14583814—21no—
74#14583813—22no—
75#14583813—21no—
76#14583813—21no—
77D1384—22no—
78#14583811—22no—
79#14583811—21no—
80#14583811—21no—
81#14583810—21no—
82#14583810C2312yes—
83#14583809—22no—
84#14583809—22no—
85C2×#13063436—21no—
86C2×#13063436D411yescharacteristic
87C2×#13063436D411yescharacteristic
88C2×#13063436D411yescharacteristic
89C2×#13063436D411yescharacteristic
90C2×#13063436C2311yescharacteristic
91C2×#13063436C2311yescharacteristic
92C8×D173—22no—
93C8×D173—22no—
94#41×C173—21no—
95#41×C173—21no—
96#41×C173—21no—
97#41×C173—21no—
98C2×Q8×C173C2311yescharacteristic
99C2×D4×C173—21no—
100C2×D4×C173C2311yescharacteristic
101C22×C4×C173—21no—
102C22×C4×C173C2311yescharacteristic
103M16×C173—21no—
104M16×C173—21no—
105M16×C173C2312yes—
106M16×C173C2312yes—
107C2×C8×C173—21no—
108C2×C8×C173—21no—
109C22×D173—21no—
110C22×D173—21no—
111Dic346—21no—
112Dic346—21no—
113Dic346—21no—
114Dic346C2×D412yes—
115D692—21no—
116D692—21no—
117D692—21no—
118D692C2×D412yes—
119C22⋊D173—22no—
120C22⋊D173—22no—
121C22⋊D173—41no—
122C22⋊D173—41no—
123C2×Dic173—21no—
124C2×Dic173—21no—
125C4×D173—22no—
126C4×D173—22no—
127C4×D173—41no—
128C4×D173—41no—
129#13063436—22no—
130#13063436C2×D411yescharacteristic
131#13063436C2×D411yescharacteristic
132#13063436C2×D411yescharacteristic
133#13063436C2×D411yescharacteristic
134C23×C173—21no—
135C23×C173#4111yescharacteristic
136Q8×C173—21no—
137Q8×C173—21no—
138D4×C173—21no—
139D4×C173—21no—
140D4×C173—41no—
141D4×C173—41no—
142C2×C4×C173—21no—
143C2×C4×C173—41no—
144C2×C4×C173—41no—
145C2×C4×C173C2×D411yescharacteristic
146C2×C4×C173C2×D411yescharacteristic
147C2×C4×C173C2×D411yescharacteristic
148C2×C4×C173C2×D411yescharacteristic
149C2×C4×C173#4111yescharacteristic
150C2×C4×C173C2411yescharacteristic
151C8×C173—22no—
152C8×C173—22no—
153C22×C173—41no—
154C22×C173—41no—
155C22×C173—81no—
156C22×C173#12111yescharacteristic
157C22×C173#12111yescharacteristic
158C22×C173C2×#4111yescharacteristic
159C2×D173—22no—
160C2×D173—42no—
161C2×D173—41no—
162C2×D173—81no—
163C4×C173—41no—
164C4×C173#12111yescharacteristic
165C4×C173#12111yescharacteristic
166C4×C173C22×D411yescharacteristic
167C4×C173C22×D411yescharacteristic
168Dic173—22no—
169Dic173—41no—
170C2×C173—21no—
171C2×C173—21no—
172C2×C173—21no—
173C2×C173—81no—
174C2×C173C2×#12111yescharacteristic
175D173—42no—
176D173—81no—
177C173#306711yescharacteristic, complemented, Sylow
178#3067—1731nomaximal, Sylow
179#575—1731no—
180#575—1731no—
181C2×#135—1731no—
182C2×#131—1731no—
183#473—1732no—
184#472—1732no—
185#471—1732no—
186#471—1732no—
187C2×#108—1731no—
188#413—1731no—
189#413D17311yescharacteristic, complemented
190C2×#41—1731no—
191C2×#41—1731no—
192C2×#41C2×D17311yescharacteristic
193#137—3461no—
194#137—3461no—
195#137—1732no—
196#137—1732no—
197AGL(1,ℤ/8)—3461no—
198AGL(1,ℤ/8)—3461no—
199AGL(1,ℤ/8)—1732no—
200AGL(1,ℤ/8)—1732no—
201#135—3462no—
202#135—3462no—
203#135—3461no—
204#135—3461no—
205#135—3461no—
206#135—3461no—
207C2×Q16—1731no—
208C2×SD16—1731no—
209C2×SD16—1731no—
210C2×D8—1731no—
211#131—3461no—
212#131—3461no—
213#131—1732no—
214#131—1732no—
215C2×M16—3461no—
216C2×M16—1731no—
217C2×M16C2×D17311yescharacteristic
218C2×M16C2×D17311yescharacteristic
219C22×C8—3461no—
220C22×C8—1731no—
221#108—1732no—
222#108—1732no—
223#101—1732no—
224#101C2×D17312yescomplemented
225#100—1732no—
226#100C2×D17312yescomplemented
227#41—21no—
228#41—21no—
229#41—21no—
230#41—21no—
231#41—3461no—
232#41—3461no—
233#41—3461no—
234#41—3461no—
235#41—3461no—
236#41—3461no—
237#41—1732no—
238#41—1732no—
239C2×Q8—1731no—
240C2×Q8—1731no—
241C2×Q8C22×D17311yescharacteristic
242C2×D4—21no—
243C2×D4—3461no—
244C2×D4—3461no—
245C2×D4—1731no—
246C2×D4—1731no—
247C2×D4C22×D17311yescharacteristic
248C22×C4—21no—
249C22×C4—3461no—
250C22×C4—3461no—
251C22×C4C22×D17311yescharacteristic
252Q16—3462no—
253Q16—3462no—
254Q16—3461no—
255Q16—3461no—
256SD16—3462no—
257SD16—3462no—
258SD16—3462no—
259SD16—3462no—
260SD16—3461no—
261SD16—3461no—
262SD16—3461no—
263SD16—3461no—
264D8—3462no—
265D8—3462no—
266D8—3461no—
267D8—3461no—
268M16—21no—
269M16—21no—
270M16—3462no—
271M16—3462no—
272M16—3461no—
273M16—1732no—
274M16C22×D17312yes—
275M16C22×D17312yes—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—3462no—
279C2×C8—3462no—
280C2×C8—3461no—
281C2×C8—1731no—
282C2×C8—1731no—
283C2×C8—1731no—
284C2×C8—1731no—
285C2×C8—1731no—
286C2×C8—1731no—
287C23—21no—
288C23—3461no—
289C23—3461no—
290C23#1458382611yescharacteristic
291Q8—21no—
292Q8—21no—
293Q8—3461no—
294Q8—3461no—
295Q8—3461no—
296Q8—1732no—
297D4—21no—
298D4—21no—
299D4—41no—
300D4—41no—
301D4—3462no—
302D4—3462no—
303D4—3461no—
304D4—3461no—
305D4—3461no—
306D4—1732no—
307D4—6921no—
308D4—6921no—
309C2×C4—21no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—3461no—
313C2×C4—3462no—
314C2×C4—3462no—
315C2×C4—3461no—
316C2×C4—6921no—
317C2×C4—6921no—
318C2×C4D4×D17311yescharacteristic
319C2×C4D4×D17311yescharacteristic
320C2×C4D4×D17311yescharacteristic
321C2×C4D4×D17311yescharacteristic
322C2×C4#1458382611yescharacteristic
323C2×C4C23×D17311yescharacteristic
324C8—22no—
325C8—22no—
326C8—3462no—
327C8—1731no—
328C8—1731no—
329C8—1731no—
330C8—1731no—
331C22—41no—
332C22—41no—
333C22—81no—
334C22—3462no—
335C22—13841no—
336C22—6922no—
337C22—6921no—
338C22#1914965211yescharacteristic
339C22#1914965211yescharacteristic
340C22C2×#1458382611yescharacteristic
341C4—41no—
342C4—3462no—
343C4—6921no—
344C4#1914965211yescharacteristic
345C4#1914965211yescharacteristic
346C4C2×D4×D17311yescharacteristic
347C4C2×D4×D17311yescharacteristic
348C2—21no—
349C2—21no—
350C2—21no—
351C2—81no—
352C2—13841no—
353C2—6922no—
354C2C2×#1914965211yescharacteristic
subgroupG/H
centerC4#19149652
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#413×C173C2
socleC2×C173C2×#121
hypercenter#413D173
hypocenterC173#3067
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C4 ◁ C22×C4 ◁ #413

Complements

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

NH ≅ G/Ntype
1. #26012291C2semidirect
2. #26012289C2semidirect
3. C2×#19149699C2semidirect
4. C2×#19149687C2semidirect
5. #26011765C2semidirect
6. #26011760C2semidirect
7. #26011757C2semidirect
8. #26011753C2semidirect
9. C2×#19149614C2semidirect
10. #26011473C2semidirect
11. #413×C173C2semidirect
28. C2×#14583816C22semidirect
29. C2×#14583814C22semidirect
30. C2×#14583813C22semidirect
31. C2×#14583811C22semidirect
40. #19149614C22semidirect
41. #19149614C22semidirect
42. #19149597C22semidirect
43. #19149595C22semidirect
47. #101×C173C22semidirect
48. #100×C173C22semidirect
177. C173#3067semidirect
189. #413D173semidirect
224. #101C2×D173semidirect
226. #100C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260122912
#260122892
C2×#191496992
C2×#191496872
#260117652
#260117602
#260117572
#260117532
C2×#191496142
#260114732
#413×C1732
#3067173