Finite Group Databasebeta

← #36365118

Subgroups — #36365118

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×#19149708C211yesmaximal, characteristic, complemented
2#26012148C211yesmaximal, characteristic, complemented
3#26012140C211yesmaximal, characteristic, complemented
4#26011882C211yesmaximal, characteristic, complemented
5#26011734C211yesmaximal, characteristic, complemented
6#26011732C211yesmaximal, characteristic, complemented
7#26011704C211yesmaximal, characteristic, complemented
8#26011699C211yesmaximal, characteristic, complemented
9#26011653C211yesmaximal, characteristic, complemented
10#26011653C211yesmaximal, characteristic, complemented
11#26011645C211yesmaximal, characteristic, complemented
12#26011633C211yesmaximal, characteristic, complemented
13#26011592C211yesmaximal, characteristic, complemented
14#26011562C211yesmaximal, characteristic, complemented
15C2×#135×C173C211yesmaximal, characteristic, complemented
16C2×#14583825C2211yescharacteristic
17C2×Q8×D173C2211yescharacteristic
18#19149708—24no—
19C2×Dic692C2211yescharacteristic
20C2×#14583816C2211yescharacteristic
21C2×#14583816C2211yescharacteristic
22C2×#14583814C2211yescharacteristic
23C2×#14583814C2211yescharacteristic
24C2×#14583809C2211yescharacteristic
25#19149677C2211yescharacteristic
26#19149666—21no—
27#19149666C2211yescharacteristic
28#19149662—21no—
29#19149660—21no—
30#19149658—21no—
31#19149658C2211yescharacteristic
32#19149658C2211yescharacteristic
33#19149655C2211yescharacteristic
34#19149651—21no—
35#19149650—21no—
36#19149650C2211yescharacteristic
37Q8×Dic173C2211yescharacteristic
38D4×Dic173C2211yescharacteristic
39C4×C22⋊D173—21no—
40C4×C22⋊D173C2211yescharacteristic
41C2×#14583805—21no—
42#19149613C2211yescharacteristic
43#19149608C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149604C2211yescharacteristic
46#19149603C2211yescharacteristic
47#19149603C2211yescharacteristic
48#19149600C2211yescharacteristic
49#19149600C2211yescharacteristic
50C22⋊#13063436C2211yescharacteristic
51#19149591C2211yescharacteristic
52#19149589C2211yescharacteristic
53C2×#41×C173C2211yescharacteristic
54C2×#41×C173C2211yescharacteristic
55#135×C173—24no—
56C2×Q16×C173C2211yescharacteristic
57C2×SD16×C173C2211yescharacteristic
58C2×SD16×C173C2211yescharacteristic
59C2×D8×C173C2211yescharacteristic
60C22×C8×C173C2211yescharacteristic
61#14583825—22no—
62#14583825—41no—
63C2×Dic346—21no—
64C2×Dic346C2311yescharacteristic
65C2×Dic346C2311yescharacteristic
66Q8×D173—22no—
67Q8×D173—41no—
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173D411yescharacteristic
71C22×Dic173—21no—
72C22×Dic173—21no—
73C2×C4×D173—21no—
74C2×C4×D173C2311yescharacteristic
75Dic692—22no—
76#14583816—22no—
77#14583816—41no—
78#14583814—22no—
79#14583814—41no—
80#14583809—22no—
81C2×#13063436C2311yescharacteristic
82C2×#13063436C2311yescharacteristic
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⋊Dic173—21no—
95C22⋊Dic173—21no—
96C22⋊Dic173—21no—
97C22⋊Dic173—21no—
98C22⋊Dic173D411yescharacteristic
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×D173C2×D411yescharacteristic
124Dic346—21no—
125Dic346—21no—
126Dic346—21no—
127Dic346—21no—
128Dic346—42no—
129C22⋊D173—41no—
130C22⋊D173—42no—
131C22⋊D173—42no—
132C2×Dic173—21no—
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×C173—21no—
148C23×C173C2×D411yescharacteristic
149Q8×C173—21no—
150Q8×C173—21no—
151Q8×C173—21no—
152Q8×C173C2×D412yes—
153D4×C173—21no—
154D4×C173—21no—
155D4×C173—21no—
156D4×C173—42no—
157D4×C173—42no—
158D4×C173C2×D412yes—
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×C173—41no—
176C22×C173—41no—
177C22×C173—41no—
178C22×C173C22×D411yescharacteristic
179C2×D173—21no—
180C2×D173—21no—
181C2×D173—21no—
182C4×C173—21no—
183C4×C173—22no—
184C4×C173—41no—
185C4×C173C22×D411yescharacteristic
186C4×C173H(5,𝔽2)11yescharacteristic
187Dic173—21no—
188Dic173—21no—
189Dic173—21no—
190Dic173—41no—
191Dic173—41no—
192Dic173—41no—
193Dic173—41no—
194C2×C173—41no—
195C2×C173—42no—
196C2×C173—81no—
197C2×C173D4211yescharacteristic
198C2×C173#57511yescharacteristic
199C2×C173#57811yescharacteristic
200D173—41no—
201C173#326911yescharacteristic, complemented, Sylow
202#3269—1731nomaximal, Sylow
203C2×#137—1731no—
204C2×#135D17311yescharacteristic, complemented
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×#41C2×D17311yescharacteristic
220C2×#41C2×D17311yescharacteristic
221C22×Q8—1731no—
222#137—3464no—
223#135—24no—
224C2×Q16—1731no—
225C2×Q16—1731no—
226C2×Q16—1731no—
227C2×Q16C2×D17311yescharacteristic
228C2×SD16—1731no—
229C2×SD16—1731no—
230C2×SD16C2×D17311yescharacteristic
231C2×SD16C2×D17311yescharacteristic
232C2×D8C2×D17311yescharacteristic
233C2×M16—1731no—
234C22×C8C2×D17311yescharacteristic
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—22no—
265#41—41no—
266#41—41no—
267#41—3462no—
268#41—6921no—
269C2×Q8—3461no—
270C2×Q8—3462no—
271C2×Q8—1731no—
272C2×Q8—1731no—
273C2×Q8—6921no—
274C2×Q8C22×D17311yescharacteristic
275C2×Q8C22×D17311yescharacteristic
276C2×D4—21no—
277C2×D4—21no—
278C2×D4—3461no—
279C2×D4—3461no—
280C2×D4—1731no—
281C2×D4C22×D17311yescharacteristic
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—3462no—
292Q16—3462no—
293Q16—6921no—
294SD16—22no—
295SD16—3462no—
296SD16—6921no—
297SD16C22⋊D17314yescomplemented
298D8—22no—
299M16—3462no—
300C2×C8—22no—
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—21no—
327C23—1731no—
328C23D4×D17311yescharacteristic
329Q8—21no—
330Q8—21no—
331Q8—21no—
332Q8—3461no—
333Q8—3461no—
334Q8—3461no—
335Q8—3461no—
336Q8—6922no—
337Q8C2×C22⋊D17312yes—
338D4—21no—
339D4—21no—
340D4—21no—
341D4—42no—
342D4—42no—
343D4—6921no—
344D4—6922no—
345D4—6922no—
346D4C2×C22⋊D17312yes—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—42no—
352C2×C4—42no—
353C2×C4—3461no—
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—41no—
378C22—41no—
379C22—41no—
380C22—3461no—
381C22—3461no—
382C22—3461no—
383C22C2×D4×D17311yescharacteristic
384C4—21no—
385C4—22no—
386C4—41no—
387C4—3461no—
388C4—3461no—
389C4—3461no—
390C4—6921no—
391C4—6921no—
392C4—6921no—
393C4—6921no—
394C4C22×C22⋊D17311yescharacteristic
395C4#1914972411yescharacteristic
396C2—41no—
397C2—42no—
398C2—81no—
399C2—6921no—
400C2D4×C22⋊D17311yescharacteristic
401C2#2601228811yescharacteristic
402C2#2601230511yescharacteristic
subgroupG/H
centerC22C2×D4×D173
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
FittingC2×#135×C173C2
socleC22×C173C22×D4
hypercenterC2×#135D173
hypocenterC173#3269
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×#19149708C2semidirect
2. #26012148C2semidirect
3. #26012140C2semidirect
4. #26011882C2semidirect
5. #26011734C2semidirect
6. #26011732C2semidirect
7. #26011704C2semidirect
8. #26011699C2semidirect
9. #26011653C2semidirect
10. #26011653C2semidirect
11. #26011645C2semidirect
12. #26011633C2semidirect
13. #26011592C2semidirect
14. #26011562C2semidirect
15. C2×#135×C173C2semidirect
118. SD16×C173D4semidirect
201. C173#3269semidirect
204. C2×#135D173semidirect
297. SD16C22⋊D173semidirect

Maximal subgroups

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

subgroupindex
C2×#191497082
#260121482
#260121402
#260118822
#260117342
#260117322
#260117042
#260116992
#260116532
#260116452
#260116332
#260115922
#260115622
C2×#135×C1732
#3269173