Finite Group Databasebeta

← #36365112

Subgroups — #36365112

Order 22144.

478 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#19149700C211yesmaximal, characteristic, complemented
2#26012152C211yesmaximal, characteristic, complemented
3#26012140C211yesmaximal, characteristic, complemented
4#26011884C211yesmaximal, characteristic, complemented
5#26011737C211yesmaximal, characteristic, complemented
6#26011729C211yesmaximal, characteristic, complemented
7#26011708C211yesmaximal, characteristic, complemented
8#26011699C211yesmaximal, characteristic, complemented
9#26011652C211yesmaximal, characteristic, complemented
10#26011650C211yesmaximal, characteristic, complemented
11#26011647C211yesmaximal, characteristic, complemented
12#26011636C211yesmaximal, characteristic, complemented
13#26011592C211yesmaximal, characteristic, complemented
14#26011560C211yesmaximal, characteristic, complemented
15C2×#137×C173C211yesmaximal, characteristic, complemented
16C2×#14583827C2211yescharacteristic
17C2×#14583825C2211yescharacteristic
18#19149700—24no—
19C2×#14583816C2211yescharacteristic
20C2×#14583816C2211yescharacteristic
21C2×#14583815C2211yescharacteristic
22C2×#14583813C2211yescharacteristic
23C2×#14583811C2211yescharacteristic
24C2×C8×D173C2211yescharacteristic
25#19149677C2211yescharacteristic
26#19149668—21no—
27#19149668C2211yescharacteristic
28#19149662—21no—
29#19149660—21no—
30#19149658C2211yescharacteristic
31#19149655—21no—
32#19149655C2211yescharacteristic
33#19149655C2211yescharacteristic
34#19149653—21no—
35#19149653C2211yescharacteristic
36#19149651—21no—
37Q8×Dic173C2211yescharacteristic
38D4×Dic173C2211yescharacteristic
39C4×C22⋊D173—21no—
40C4×C22⋊D173C2211yescharacteristic
41C2×#14583805—21no—
42#19149613C2211yescharacteristic
43#19149608C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149603C2211yescharacteristic
46#19149603C2211yescharacteristic
47#19149603C2211yescharacteristic
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—41no—
63#14583825—22no—
64#14583825—41no—
65C2×Dic346C2311yescharacteristic
66C2×D692—21no—
67C2×D692C2311yescharacteristic
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173D411yescharacteristic
71C22×Dic173—21no—
72C22×Dic173—21no—
73C2×C4×D173—21no—
74C2×C4×D173C2311yescharacteristic
75#14583816—22no—
76#14583816—41no—
77#14583815—22no—
78#14583813—41no—
79#14583811—22no—
80C2×#13063436C2311yescharacteristic
81C2×#13063436C2311yescharacteristic
82C8×D173—22no—
83C4⋊Dic173—21no—
84C4⋊Dic173C2311yescharacteristic
85C4⋊Dic173C2311yescharacteristic
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89#14583805—21no—
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—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×C173—22no—
118SD16×C173—22no—
119SD16×C173D414yescomplemented
120M16×C173—22no—
121C2×C8×C173C2311yescharacteristic
122C2×C8×C173C2311yescharacteristic
123C22×D173—21no—
124C22×D173C2×D411yescharacteristic
125Dic346—21no—
126Dic346—21no—
127D692—21no—
128D692—21no—
129D692—42no—
130C22⋊D173—41no—
131C22⋊D173—42no—
132C22⋊D173—42no—
133C2×Dic173—21no—
134C2×Dic173—21no—
135C2×Dic173—21no—
136C2×Dic173—42no—
137C2×Dic173—42no—
138C2×Dic173C2×D411yescharacteristic
139C2×Dic173C2×D411yescharacteristic
140C2×Dic173C2×D411yescharacteristic
141C4×D173—21no—
142C4×D173—21no—
143C4×D173—42no—
144#13063436—21no—
145#13063436—21no—
146C23×C173—21no—
147C23×C173C2×D411yescharacteristic
148Q8×C173—21no—
149Q8×C173—21no—
150Q8×C173—21no—
151Q8×C173—21no—
152Q8×C173—21no—
153Q8×C173—42no—
154Q8×C173C2×D412yes—
155D4×C173—21no—
156D4×C173—42no—
157D4×C173C2×D412yes—
158C2×C4×C173—21no—
159C2×C4×C173—21no—
160C2×C4×C173—21no—
161C2×C4×C173—21no—
162C2×C4×C173—21no—
163C2×C4×C173—42no—
164C2×C4×C173—42no—
165C2×C4×C173C2×D411yescharacteristic
166C2×C4×C173C2411yescharacteristic
167C8×C173—21no—
168C8×C173C2×D412yes—
169C22×C173—21no—
170C22×C173—22no—
171C22×C173—21no—
172C22×C173—21no—
173C22×C173—41no—
174C22×C173—41no—
175C22×C173C22×D411yescharacteristic
176C2×D173—21no—
177C2×D173—21no—
178C2×D173—21no—
179C2×D173—41no—
180C2×D173—41no—
181C2×D173—41no—
182C4×C173—21no—
183C4×C173—22no—
184C4×C173—41no—
185C4×C173—41no—
186C4×C173C22×D411yescharacteristic
187C4×C173H(5,𝔽2)11yescharacteristic
188Dic173—21no—
189Dic173—21no—
190Dic173—21no—
191Dic173—41no—
192Dic173—41no—
193Dic173—41no—
194C2×C173—41no—
195C2×C173—42no—
196C2×C173D4211yescharacteristic
197C2×C173#57511yescharacteristic
198C2×C173#57811yescharacteristic
199D173—41no—
200D173—81no—
201C173#326911yescharacteristic, complemented, Sylow
202#3269—1731nomaximal, Sylow
203C2×#137D17311yescharacteristic, complemented
204C2×#135—1731no—
205#548—1731no—
206#547—1731no—
207#497—1731no—
208#468—1731no—
209#467—1731no—
210#463—1731no—
211#462—1731no—
212#452—1731no—
213#452—1731no—
214#451—1731no—
215#449—1731no—
216#440—1731no—
217#435—1731no—
218C2×#41—1731no—
219C2×#41—1731no—
220C2×#41C2×D17311yescharacteristic
221C22×Q8C2×D17311yescharacteristic
222#137—24no—
223#135—3464no—
224C2×Q16—1731no—
225C2×Q16—1731no—
226C2×Q16C2×D17311yescharacteristic
227C2×Q16C2×D17311yescharacteristic
228C2×SD16—1731no—
229C2×SD16—1731no—
230C2×SD16C2×D17311yescharacteristic
231C2×SD16C2×D17311yescharacteristic
232C2×D8—1731no—
233C2×M16C2×D17311yescharacteristic
234C22×C8—1731no—
235C4⋊Q8—1731no—
236#124—3461no—
237#124—1731no—
238#123—3461no—
239#123—3461no—
240C22⋊Q8—3461no—
241C22⋊Q8—1731no—
242C22⋊Q8—1731no—
243C22⋊Q8—1731no—
244#121—3461no—
245#121—3461no—
246#121—1731no—
247C4×Q8—1731no—
248C4×D4—3461no—
249C4×D4—1731no—
250C4×D4—1731no—
251C2×C4⋊C4—3461no—
252#107—1731no—
253C4⋊C8—1731no—
254#103—1731no—
255#103—1731no—
256#103—1731no—
257#103—1731no—
258#102—1731no—
259#102—1731no—
260C22⋊C8—1731no—
261C22⋊C8—1731no—
262#97—1731no—
263#41—22no—
264#41—41no—
265#41—3462no—
266#41—3462no—
267#41—6921no—
268#41—6921no—
269C2×Q8—21no—
270C2×Q8—22no—
271C2×Q8—41no—
272C2×Q8—1731no—
273C2×Q8C22×D17311yescharacteristic
274C2×Q8C22×D17311yescharacteristic
275C2×Q8C22×D17311yescharacteristic
276C2×D4—21no—
277C2×D4—3461no—
278C2×D4—3461no—
279C2×D4—3461no—
280C2×D4—1731no—
281C2×D4—1731no—
282C2×D4C22×D17311yescharacteristic
283C22×C4—21no—
284C22×C4—21no—
285C22×C4—3461no—
286C22×C4—3461no—
287C22×C4—3461no—
288C22×C4—1731no—
289C22×C4C22×D17311yescharacteristic
290Q16—22no—
291Q16—22no—
292Q16—3462no—
293Q16—6921no—
294SD16—22no—
295SD16—3462no—
296SD16—6921no—
297SD16C22⋊D17314yescomplemented
298D8—3462no—
299M16—22no—
300C2×C8—3462no—
301C2×C8—1731no—
302C2×C8—1731no—
303C2×C8C22×D17311yescharacteristic
304C2×C8C22×D17311yescharacteristic
305C4⋊C4—3461no—
306C4⋊C4—3461no—
307C4⋊C4—3461no—
308C4⋊C4—3461no—
309C4⋊C4—3461no—
310C4⋊C4—3461no—
311C4⋊C4—1731no—
312C4⋊C4—1731no—
313C4⋊C4—1731no—
314C22⋊C4—3461no—
315C22⋊C4—3461no—
316C22⋊C4—3461no—
317C22⋊C4—3461no—
318C22⋊C4—3461no—
319C22⋊C4—3461no—
320C22⋊C4—3461no—
321C22⋊C4—1731no—
322C22⋊C4—1731no—
323C42—3461no—
324C42—1731no—
325C23—21no—
326C23—3461no—
327C23—1731no—
328C23D4×D17311yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—21no—
333Q8—21no—
334Q8—42no—
335Q8—3461no—
336Q8—3461no—
337Q8C2×C22⋊D17312yes—
338D4—21no—
339D4—42no—
340D4—3461no—
341D4—3461no—
342D4—6922no—
343D4—6921no—
344D4—6922no—
345D4—6922no—
346D4C2×C22⋊D17312yes—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—42no—
353C2×C4—42no—
354C2×C4—3461no—
355C2×C4—3461no—
356C2×C4—3461no—
357C2×C4—3461no—
358C2×C4—3461no—
359C2×C4—1731no—
360C2×C4—1731no—
361C2×C4—1731no—
362C2×C4—6922no—
363C2×C4—6922no—
364C2×C4—6922no—
365C2×C4D4×D17311yescharacteristic
366C2×C4C23×D17311yescharacteristic
367C8—21no—
368C8—3461no—
369C8—3461no—
370C8C2×C22⋊D17312yes—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—22no—
375C22—41no—
376C22—41no—
377C22—3461no—
378C22—3461no—
379C22—3461no—
380C22—6921no—
381C22—6921no—
382C22—6921no—
383C22C2×D4×D17311yescharacteristic
384C4—21no—
385C4—22no—
386C4—41no—
387C4—41no—
388C4—3461no—
389C4—3461no—
390C4—3461no—
391C4—6921no—
392C4—6921no—
393C4—6921no—
394C4C22×C22⋊D17311yescharacteristic
395C4#1914972411yescharacteristic
396C2—41no—
397C2—42no—
398C2—13841no—
399C2—6921no—
400C2D4×C22⋊D17311yescharacteristic
401C2#2601229011yescharacteristic
402C2#2601230711yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#137×C173C2
socleC22×C173C22×D4
hypercenterC2×#137D173
hypocenterC173#3269
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×#19149700C2semidirect
2. #26012152C2semidirect
3. #26012140C2semidirect
4. #26011884C2semidirect
5. #26011737C2semidirect
6. #26011729C2semidirect
7. #26011708C2semidirect
8. #26011699C2semidirect
9. #26011652C2semidirect
10. #26011650C2semidirect
11. #26011647C2semidirect
12. #26011636C2semidirect
13. #26011592C2semidirect
14. #26011560C2semidirect
15. C2×#137×C173C2semidirect
119. SD16×C173D4semidirect
201. C173#3269semidirect
203. C2×#137D173semidirect
297. SD16C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497002
#260121522
#260121402
#260118842
#260117372
#260117292
#260117082
#260116992
#260116522
#260116502
#260116472
#260116362
#260115922
#260115602
C2×#137×C1732
#3269173