Finite Group Databasebeta

← #36365142

Subgroups — #36365142

Order 22144.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149702C211yesmaximal, characteristic, complemented
2#26012160C211yesmaximal, characteristic, complemented
3#26012152C211yesmaximal, characteristic, complemented
4#26011877C211yesmaximal, characteristic, complemented
5#26011744C211yesmaximal, characteristic, complemented
6#26011717C211yesmaximal, characteristic, complemented
7#26011715C211yesmaximal, characteristic, complemented
8#26011689C211yesmaximal, characteristic, complemented
9#26011652C211yesmaximal, characteristic, complemented
10#26011636C211yesmaximal, characteristic, complemented
11#26011635C211yesmaximal, characteristic, complemented
12#26011630C211yesmaximal, characteristic, complemented
13#26011598C211yesmaximal, characteristic, complemented
14#26011560C211yesmaximal, characteristic, complemented
15C2×#137×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583827C2211yescharacteristic
18#19149702—24no—
19C2×#14583813C2211yescharacteristic
20C2×#14583813C2211yescharacteristic
21C2×#14583813C2211yescharacteristic
22C2×D1384C2211yescharacteristic
23C2×#14583811C2211yescharacteristic
24C2×C8×D173C2211yescharacteristic
25#19149675—21no—
26#19149675C2211yescharacteristic
27#19149668—21no—
28#19149668C2211yescharacteristic
29#19149655—21no—
30#19149655C2211yescharacteristic
31#19149653—21no—
32#19149653—21no—
33#19149653C2211yescharacteristic
34#19149653C2211yescharacteristic
35#19149650—21no—
36#19149650C2211yescharacteristic
37Q8×Dic173C2211yescharacteristic
38Q8×Dic173C2211yescharacteristic
39C4×C22⋊D173—21no—
40C4×C22⋊D173—21no—
41C4×C22⋊D173C2211yescharacteristic
42#19149611C2211yescharacteristic
43#19149608C2211yescharacteristic
44#19149603C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149603C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149601C2211yescharacteristic
49#19149600C2211yescharacteristic
50C22⋊#13063436C2211yescharacteristic
51#19149591C2211yescharacteristic
52#19149589C2211yescharacteristic
53C2×#41×C173C2211yescharacteristic
54C22×Q8×C173C2211yescharacteristic
55#137×C173—24no—
56C2×Q16×C173C2211yescharacteristic
57C2×Q16×C173C2211yescharacteristic
58C2×SD16×C173C2211yescharacteristic
59C2×SD16×C173C2211yescharacteristic
60C2×M16×C173C2211yescharacteristic
61#14583827—22no—
62#14583827—22no—
63#14583827—41no—
64#14583827—41no—
65C2×D692—21no—
66C2×D692—21no—
67C2×D692C2311yescharacteristic
68C2×D692C2311yescharacteristic
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C2×C22⋊D173—21no—
72C2×C22⋊D173—21no—
73C2×C22⋊D173—21no—
74C2×C22⋊D173D411yescharacteristic
75C2×C4×D173—21no—
76C2×C4×D173—21no—
77C2×C4×D173C2311yescharacteristic
78#14583813—22no—
79#14583813—22no—
80#14583813—41no—
81D1384—22no—
82#14583811—41no—
83C2×#13063436C2311yescharacteristic
84C2×#13063436C2311yescharacteristic
85C8×D173—22no—
86C4⋊Dic173—21no—
87C4⋊Dic173—21no—
88C4⋊Dic173C2311yescharacteristic
89C4⋊Dic173C2311yescharacteristic
90#14583805—21no—
91#14583805D411yescharacteristic
92C22⋊Dic173—21no—
93C22⋊Dic173D411yescharacteristic
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803—21no—
98#14583803—21no—
99#14583803D411yescharacteristic
100C4×Dic173—21no—
101C4×Dic173—21no—
102C4×Dic173C2311yescharacteristic
103#41×C173—22no—
104#41×C173—41no—
105C2×Q8×C173—21no—
106C2×Q8×C173—22no—
107C2×Q8×C173—41no—
108C2×Q8×C173C2311yescharacteristic
109C2×Q8×C173C2311yescharacteristic
110C2×Q8×C173C2311yescharacteristic
111C2×D4×C173—21no—
112C2×D4×C173C2311yescharacteristic
113C22×C4×C173—21no—
114C22×C4×C173—21no—
115C22×C4×C173C2311yescharacteristic
116Q16×C173—22no—
117Q16×C173D414yescomplemented
118SD16×C173—22no—
119SD16×C173—22no—
120M16×C173—22no—
121C2×C8×C173C2311yescharacteristic
122C2×C8×C173C2311yescharacteristic
123C22×D173—21no—
124C22×D173—21no—
125C22×D173C2×D411yescharacteristic
126D692—21no—
127D692—21no—
128D692—21no—
129D692—21no—
130D692—42no—
131D692—42no—
132C22⋊D173—41no—
133C22⋊D173—42no—
134C22⋊D173—42no—
135C22⋊D173—81no—
136C22⋊D173—81no—
137C22⋊D173—81no—
138C2×Dic173—21no—
139C2×Dic173—21no—
140C2×Dic173C2×D411yescharacteristic
141C2×Dic173C2×D411yescharacteristic
142C2×Dic173C2×D411yescharacteristic
143C4×D173—21no—
144C4×D173—21no—
145C4×D173—42no—
146C4×D173—42no—
147#13063436—21no—
148#13063436—21no—
149C23×C173—21no—
150C23×C173C2×D411yescharacteristic
151Q8×C173—21no—
152Q8×C173—21no—
153Q8×C173—21no—
154Q8×C173—21no—
155Q8×C173—42no—
156Q8×C173C2×D412yes—
157Q8×C173C2×D412yes—
158D4×C173—21no—
159D4×C173—21no—
160D4×C173—42no—
161C2×C4×C173—21no—
162C2×C4×C173—21no—
163C2×C4×C173—21no—
164C2×C4×C173—21no—
165C2×C4×C173—21no—
166C2×C4×C173—42no—
167C2×C4×C173—42no—
168C2×C4×C173C2×D411yescharacteristic
169C2×C4×C173C2411yescharacteristic
170C8×C173—21no—
171C8×C173C2×D412yes—
172C22×C173—21no—
173C22×C173—21no—
174C22×C173—21no—
175C22×C173—41no—
176C22×C173—41no—
177C22×C173—41no—
178C22×C173C22×D411yescharacteristic
179C2×D173—21no—
180C2×D173—21no—
181C2×D173—21no—
182C2×D173—41no—
183C2×D173—41no—
184C2×D173—41no—
185C2×D173—41no—
186C2×D173—41no—
187C2×D173—41no—
188C4×C173—21no—
189C4×C173—22no—
190C4×C173—22no—
191C4×C173—41no—
192C4×C173C22×D411yescharacteristic
193C4×C173H(5,𝔽2)11yescharacteristic
194Dic173—21no—
195Dic173—21no—
196Dic173—21no—
197Dic173—41no—
198Dic173—41no—
199C2×C173—41no—
200C2×C173—81no—
201C2×C173D4211yescharacteristic
202C2×C173#57511yescharacteristic
203C2×C173#57711yescharacteristic
204D173—41no—
205D173—81no—
206D173—81no—
207C173#327111yescharacteristic, complemented, Sylow
208#3271—1731nomaximal, Sylow
209C2×#137D17311yescharacteristic, complemented
210C2×#135—1731no—
211#550—1731no—
212#548—1731no—
213#496—1731no—
214#469—1731no—
215#465—1731no—
216#464—1731no—
217#460—1731no—
218#452—1731no—
219#449—1731no—
220#449—1731no—
221#448—1731no—
222#441—1731no—
223#435—1731no—
224C2×#41—1731no—
225C2×#41—1731no—
226C2×#41C2×D17311yescharacteristic
227C22×Q8C2×D17311yescharacteristic
228#137—24no—
229#135—3464no—
230C2×Q16C2×D17311yescharacteristic
231C2×Q16C2×D17311yescharacteristic
232C2×SD16—1731no—
233C2×SD16—1731no—
234C2×SD16—1731no—
235C2×SD16C2×D17311yescharacteristic
236C2×SD16C2×D17311yescharacteristic
237C2×D8—1731no—
238C2×D8—1731no—
239C2×M16C2×D17311yescharacteristic
240C22×C8—1731no—
241#127—3461no—
242#127—1731no—
243#124—3461no—
244#124—1731no—
245C22⋊Q8—3461no—
246C22⋊Q8—1731no—
247#121—3461no—
248#121—3461no—
249#121—3461no—
250#121—1731no—
251#121—1731no—
252#121—1731no—
253C4×Q8—1731no—
254C4×Q8—1731no—
255C4×D4—3461no—
256C4×D4—3461no—
257C4×D4—1731no—
258#106—1731no—
259C4⋊C8—1731no—
260#103—1731no—
261#103—1731no—
262#103—1731no—
263#102—1731no—
264#102—1731no—
265#102—1731no—
266C22⋊C8—1731no—
267C22⋊C8—1731no—
268#97—1731no—
269#41—22no—
270#41—41no—
271#41—3462no—
272#41—3462no—
273#41—6921no—
274#41—6921no—
275C2×Q8—21no—
276C2×Q8—22no—
277C2×Q8—41no—
278C2×Q8C22×D17311yescharacteristic
279C2×Q8C22×D17311yescharacteristic
280C2×Q8C22×D17311yescharacteristic
281C2×D4—21no—
282C2×D4—3461no—
283C2×D4—3461no—
284C2×D4—3461no—
285C2×D4—3461no—
286C2×D4—3461no—
287C2×D4—3461no—
288C2×D4—3461no—
289C2×D4—1731no—
290C2×D4—1731no—
291C2×D4—1731no—
292C2×D4C22×D17311yescharacteristic
293C22×C4—21no—
294C22×C4—21no—
295C22×C4—3461no—
296C22×C4—3461no—
297C22×C4—1731no—
298C22×C4C22×D17311yescharacteristic
299Q16—22no—
300Q16C22⋊D17314yescomplemented
301SD16—22no—
302SD16—22no—
303SD16—3462no—
304SD16—3462no—
305SD16—6921no—
306D8—3462no—
307D8—6921no—
308M16—22no—
309C2×C8—3462no—
310C2×C8—1731no—
311C2×C8—1731no—
312C2×C8C22×D17311yescharacteristic
313C2×C8C22×D17311yescharacteristic
314C4⋊C4—3461no—
315C4⋊C4—3461no—
316C4⋊C4—3461no—
317C4⋊C4—1731no—
318C4⋊C4—1731no—
319C4⋊C4—1731no—
320C22⋊C4—3461no—
321C22⋊C4—3461no—
322C22⋊C4—3461no—
323C22⋊C4—3461no—
324C22⋊C4—3461no—
325C22⋊C4—3461no—
326C22⋊C4—1731no—
327C22⋊C4—1731no—
328C42—3461no—
329C42—3461no—
330C42—1731no—
331C23—21no—
332C23—3461no—
333C23—3461no—
334C23—1731no—
335C23D4×D17311yescharacteristic
336Q8—21no—
337Q8—21no—
338Q8—21no—
339Q8—21no—
340Q8—42no—
341Q8C2×C22⋊D17312yes—
342Q8C2×C22⋊D17312yes—
343D4—21no—
344D4—21no—
345D4—42no—
346D4—3461no—
347D4—3461no—
348D4—3461no—
349D4—3461no—
350D4—13841no—
351D4—13841no—
352D4—13841no—
353D4—6922no—
354D4—6922no—
355D4—6921no—
356D4—6922no—
357D4—6922no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—42no—
364C2×C4—42no—
365C2×C4—3461no—
366C2×C4—3461no—
367C2×C4—3461no—
368C2×C4—3461no—
369C2×C4—1731no—
370C2×C4—1731no—
371C2×C4—1731no—
372C2×C4—6922no—
373C2×C4—6922no—
374C2×C4D4×D17311yescharacteristic
375C2×C4C23×D17311yescharacteristic
376C8—21no—
377C8—3461no—
378C8—3461no—
379C8C2×C22⋊D17312yes—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—41no—
386C22—3461no—
387C22—3461no—
388C22—3461no—
389C22—6921no—
390C22—6921no—
391C22—6921no—
392C22—6921no—
393C22—6921no—
394C22—6921no—
395C22C2×D4×D17311yescharacteristic
396C4—21no—
397C4—22no—
398C4—22no—
399C4—41no—
400C4—3461no—
401C4—3461no—
402C4—3461no—
403C4—6921no—
404C4—6921no—
405C4C22×C22⋊D17311yescharacteristic
406C4#1914972411yescharacteristic
407C2—41no—
408C2—81no—
409C2—13841no—
410C2—13841no—
411C2—6921no—
412C2D4×C22⋊D17311yescharacteristic
413C2#2601229011yescharacteristic
414C2#2601230011yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#137×C173C2
socleC22×C173C22×D4
hypercenterC2×#137D173
hypocenterC173#3271
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#137

Complements

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

NH ≅ G/Ntype
1. C2×#19149702C2semidirect
2. #26012160C2semidirect
3. #26012152C2semidirect
4. #26011877C2semidirect
5. #26011744C2semidirect
6. #26011717C2semidirect
7. #26011715C2semidirect
8. #26011689C2semidirect
9. #26011652C2semidirect
10. #26011636C2semidirect
11. #26011635C2semidirect
12. #26011630C2semidirect
13. #26011598C2semidirect
14. #26011560C2semidirect
15. C2×#137×C173C2semidirect
117. Q16×C173D4semidirect
207. C173#3271semidirect
209. C2×#137D173semidirect
300. Q16C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497022
#260121602
#260121522
#260118772
#260117442
#260117172
#260117152
#260116892
#260116522
#260116362
#260116352
#260116302
#260115982
#260115602
C2×#137×C1732
#3271173