Finite Group Databasebeta

← #36364417

Subgroups — #36364417

Order 22144.

422 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012069C211yesmaximal, characteristic, complemented
2#26012055C211yesmaximal, characteristic, complemented
3#26011833C211yesmaximal, characteristic, complemented
4#26011816C211yesmaximal, characteristic, complemented
5#26011810C211yesmaximal, characteristic, complemented
6#26011805C211yesmaximal, characteristic, complemented
7#26011744C211yesmaximal, characteristic, complemented
8#26011737C211yesmaximal, characteristic, complemented
9#26011727C211yesmaximal, characteristic, complemented
10#26011720C211yesmaximal, characteristic, complemented
11#26011644C211yesmaximal, characteristic, complemented
12#26011638C211yesmaximal, characteristic, complemented
13#26011638C211yesmaximal, characteristic, complemented
14#26011627C211yesmaximal, characteristic, complemented
15#408×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic, complemented
17C22×D692C2211yescharacteristic
18C2×#14583815C2211yescharacteristic
19C2×#14583815C2211yescharacteristic
20C2×D1384C2211yescharacteristic
21C2×D1384C2211yescharacteristic
22#19149675—21no—
23#19149668—21no—
24#19149658C2211yescharacteristic
25#19149658C2211yescharacteristic, complemented
26#19149655—21no—
27#19149653C2211yescharacteristic
28#19149653C2211yescharacteristic
29#19149653C2211yescharacteristic
30#19149653C2211yescharacteristic
31#19149653C2211yescharacteristic
32#19149653C2211yescharacteristic, complemented
33#19149651—21no—
34#19149651—21no—
35#19149650—21no—
36#19149647—21no—
37#19149647—21no—
38#19149638C2211yescharacteristic
39C2×C4⋊Dic173C2211yescharacteristic, complemented
40#19149613C2211yescharacteristic, complemented
41#19149613C2211yescharacteristic, complemented
42#19149611C2211yescharacteristic, complemented
43#19149611C2211yescharacteristic, complemented
44#19149604C2211yescharacteristic, complemented
45#19149604C2211yescharacteristic, complemented
46#19149601C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149601C2211yescharacteristic
49#19149601C2211yescharacteristic
50#19149601C2211yescharacteristic, complemented
51#19149601C2211yescharacteristic, complemented
52C2×#41×C173C2211yescharacteristic
53C2×M16×C173C2211yescharacteristic, complemented
54C22×C8×C173C2211yescharacteristic, complemented
55C22⋊C8×C173C2211yescharacteristic
56C22⋊C8×C173C2211yescharacteristic
57C22⋊C8×C173C2211yescharacteristic, complemented
58C22⋊C8×C173C2211yescharacteristic, complemented
59C23×D173—21no—
60#14583826—41no—
61#14583826—41no—
62C2×Dic346C2311yescharacteristic
63C2×D692—21no—
64C2×D692—41no—
65C2×D692—41no—
66C2×D692C2311yescharacteristic
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⋊D173—21no—
75C2×C22⋊D173—21no—
76C22×Dic173—21no—
77C2×C4×D173—21no—
78#14583815—41no—
79#14583815—41no—
80D1384—41no—
81D1384—41no—
82C4⋊Dic173—21no—
83C4⋊Dic173C2311yescharacteristic
84C4⋊Dic173C2311yescharacteristic
85C4⋊Dic173C2311yescharacteristic
86C4⋊Dic173C2311yescharacteristic
87#14583805—21no—
88#14583805—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92#14583803—21no—
93#14583803—21no—
94#14583803—21no—
95#14583803—21no—
96#14583803—21no—
97#14583803—21no—
98C4×Dic173—21no—
99#41×C173—41no—
100#41×C173—41no—
101C2×Q8×C173D411yescharacteristic
102C2×D4×C173D411yescharacteristic
103C2×D4×C173D411yescharacteristic
104C2×D4×C173D411yescharacteristic
105C22×C4×C173C2311yescharacteristic
106C22×C4×C173C2311yescharacteristic
107C22×C4×C173C2311yescharacteristic
108M16×C173—41no—
109C2×C8×C173—41no—
110C2×C8×C173C2311yescharacteristic
111C2×C8×C173C2311yescharacteristic
112C2×C8×C173C2311yescharacteristic
113C2×C8×C173C2311yescharacteristic
114C22×D173—21no—
115C22×D173—21no—
116C22×D173—21no—
117C22×D173—42no—
118C22×D173—81no—
119C22×D173—81no—
120Dic346—21no—
121Dic346—21no—
122D692—21no—
123D692—21no—
124D692—21no—
125D692—21no—
126D692—21no—
127D692—21no—
128D692—42no—
129C22⋊D173—42no—
130C22⋊D173—81no—
131C22⋊D173—81no—
132C22⋊D173—81no—
133C22⋊D173—81no—
134C22⋊D173—81no—
135C22⋊D173—81no—
136C2×Dic173—21no—
137C2×Dic173—21no—
138C2×Dic173—21no—
139C2×Dic173—21no—
140C2×Dic173—21no—
141C2×Dic173—81no—
142C4×D173—42no—
143C23×C173C2×D411yescharacteristic
144C23×C173C2×D411yescharacteristic
145C23×C173C2×D411yescharacteristic
146Q8×C173—41no—
147D4×C173—41no—
148D4×C173—41no—
149D4×C173—41no—
150C2×C4×C173—21no—
151C2×C4×C173—21no—
152C2×C4×C173—21no—
153C2×C4×C173—21no—
154C2×C4×C173—21no—
155C2×C4×C173—21no—
156C2×C4×C173C2×D411yescharacteristic
157C2×C4×C173C2×D411yescharacteristic
158C2×C4×C173C2×D411yescharacteristic
159C2×C4×C173C2411yescharacteristic
160C8×C173—21no—
161C8×C173—21no—
162C8×C173—21no—
163C8×C173—21no—
164C22×C173—21no—
165C22×C173—21no—
166C22×C173—21no—
167C22×C173—21no—
168C22×C173—21no—
169C22×C173—21no—
170C22×C173—21no—
171C22×C173—21no—
172C22×C173—21no—
173C22×C173C22×D411yescharacteristic
174C2×D173—41no—
175C2×D173—41no—
176C2×D173—41no—
177C2×D173—41no—
178C2×D173—41no—
179C2×D173—41no—
180C2×D173—41no—
181C2×D173—41no—
182C2×D173—41no—
183C2×D173—84no—
184C4×C173—21no—
185C4×C173—21no—
186C4×C173—21no—
187C4×C173H(5,𝔽2)11yescharacteristic
188C4×C173H(5,𝔽2)11yescharacteristic
189Dic173—41no—
190Dic173—41no—
191Dic173—41no—
192Dic173—41no—
193Dic173—41no—
194C2×C173—41no—
195C2×C173—41no—
196C2×C173—41no—
197C2×C173#53411yescharacteristic
198C2×C173#57611yescharacteristic
199C2×C173#57711yescharacteristic
200D173—81no—
201D173—81no—
202D173—81no—
203C173#319811yescharacteristic, complemented, Sylow
204#3198—1731nomaximal, Sylow
205#537—1731no—
206#535—1731no—
207#485—1731no—
208#482—1731no—
209#481—1731no—
210#480—1731no—
211#469—1731no—
212#468—1731no—
213#466—1731no—
214#465—1731no—
215#450—1731no—
216#449—1731no—
217#449—1731no—
218#447—1731no—
219#408D17311yescharacteristic, complemented
220C2×#41—1731no—
221C2×#41C2×D17311yescharacteristic
222C22×D4—1731no—
223C2×SD16—1731no—
224C2×SD16—1731no—
225C2×D8—1731no—
226C2×D8—1731no—
227C2×M16C2×D17311yescharacteristic, complemented
228C22×C8C2×D17311yescharacteristic, complemented
229#127—3461no—
230#124—3461no—
231C22⋊Q8—3461no—
232C22⋊Q8—1731no—
233C22⋊Q8—1731no—
234#121—3461no—
235#121—3461no—
236#121—3461no—
237#121—1731no—
238#121—1731no—
239#121—1731no—
240#121—1731no—
241#121—1731no—
242#121—1731no—
243#120—3461no—
244#120—3461no—
245#117—1731no—
246C2×C4⋊C4—1731no—
247#107—1731no—
248#107—1731no—
249#106—1731no—
250#106—1731no—
251#103—1731no—
252#103—1731no—
253#102—1731no—
254#102—1731no—
255#102—1731no—
256#102—1731no—
257#102—1731no—
258#102—1731no—
259C22⋊C8C2×D17311yescharacteristic
260C22⋊C8C2×D17311yescharacteristic
261C22⋊C8C2×D17311yescharacteristic, complemented
262C22⋊C8C2×D17311yescharacteristic, complemented
263C24—3461no—
264#41—41no—
265#41—41no—
266#41—6921no—
267#41—6921no—
268C2×Q8—1731no—
269C2×Q8D69211yescharacteristic
270C2×D4—3461no—
271C2×D4—3461no—
272C2×D4—3461no—
273C2×D4—3461no—
274C2×D4—3461no—
275C2×D4—3461no—
276C2×D4—3461no—
277C2×D4—3461no—
278C2×D4—1731no—
279C2×D4—1731no—
280C2×D4—1731no—
281C2×D4—6921no—
282C2×D4—6921no—
283C2×D4D69211yescharacteristic
284C2×D4D69211yescharacteristic
285C2×D4D69211yescharacteristic
286C22×C4—3461no—
287C22×C4—3461no—
288C22×C4C22×D17311yescharacteristic
289C22×C4C22×D17311yescharacteristic
290C22×C4C22×D17311yescharacteristic
291SD16—6921no—
292SD16—6921no—
293D8—6921no—
294D8—6921no—
295M16—41no—
296C2×C8—41no—
297C2×C8C22×D17311yescharacteristic
298C2×C8C22×D17311yescharacteristic
299C2×C8C22×D17311yescharacteristic
300C2×C8C22×D17311yescharacteristic
301C4⋊C4—3461no—
302C4⋊C4—3461no—
303C4⋊C4—3461no—
304C4⋊C4—1731no—
305C4⋊C4—1731no—
306C4⋊C4—1731no—
307C4⋊C4—1731no—
308C22⋊C4—3461no—
309C22⋊C4—3461no—
310C22⋊C4—3461no—
311C22⋊C4—3461no—
312C22⋊C4—3461no—
313C22⋊C4—3461no—
314C22⋊C4—3461no—
315C22⋊C4—3461no—
316C22⋊C4—3461no—
317C42—3461no—
318C23—3461no—
319C23—3461no—
320C23—3461no—
321C23—13841no—
322C23—13841no—
323C23—6922no—
324C23C2×D69211yescharacteristic
325C23C2×D69211yescharacteristic
326C23C2×D69211yescharacteristic
327Q8—41no—
328Q8—3461no—
329Q8—3461no—
330D4—41no—
331D4—41no—
332D4—41no—
333D4—3461no—
334D4—3461no—
335D4—3461no—
336D4—3461no—
337D4—3461no—
338D4—3461no—
339D4—13841no—
340D4—13841no—
341D4—13841no—
342D4—13841no—
343D4—13841no—
344D4—13841no—
345D4—6922no—
346D4—6922no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—3461no—
354C2×C4—3461no—
355C2×C4—3461no—
356C2×C4—3461no—
357C2×C4—3461no—
358C2×C4—13841no—
359C2×C4—6922no—
360C2×C4C2×D69211yescharacteristic
361C2×C4C2×D69211yescharacteristic
362C2×C4C2×D69211yescharacteristic
363C2×C4C23×D17311yescharacteristic
364C8—21no—
365C8—21no—
366C8—21no—
367C8—21no—
368C22—21no—
369C22—21no—
370C22—21no—
371C22—21no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—21no—
376C22—21no—
377C22—13844no—
378C22—6921no—
379C22—6921no—
380C22—6921no—
381C22—6921no—
382C22—6921no—
383C22—6921no—
384C22—6921no—
385C22—6921no—
386C22—6921no—
387C22C22×D69211yescharacteristic
388C4—21no—
389C4—21no—
390C4—21no—
391C4—6921no—
392C4—6921no—
393C4—6921no—
394C4—6921no—
395C4—6921no—
396C4#1914972411yescharacteristic
397C4#1914972411yescharacteristic
398C2—41no—
399C2—41no—
400C2—41no—
401C2—13841no—
402C2—13841no—
403C2—13841no—
404C2#2601205311yescharacteristic
405C2#2601229611yescharacteristic
406C2#2601230311yescharacteristic
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#408×C173C2
socleC22×C173C22×D4
hypercenter#408D173
hypocenterC173#3198
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ C2×#41 ◁ #408

Complements

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

NH ≅ G/Ntype
1. #26012069C2semidirect
2. #26012055C2semidirect
3. #26011833C2semidirect
4. #26011816C2semidirect
5. #26011810C2semidirect
6. #26011805C2semidirect
7. #26011744C2semidirect
8. #26011737C2semidirect
9. #26011727C2semidirect
10. #26011720C2semidirect
11. #26011644C2semidirect
12. #26011638C2semidirect
13. #26011638C2semidirect
14. #26011627C2semidirect
15. #408×C173C2semidirect
16. C2×#14583826C22semidirect
25. #19149658C22semidirect
32. #19149653C22semidirect
39. C2×C4⋊Dic173C22semidirect
40. #19149613C22semidirect
41. #19149613C22semidirect
42. #19149611C22semidirect
43. #19149611C22semidirect
44. #19149604C22semidirect
45. #19149604C22semidirect
50. #19149601C22semidirect
51. #19149601C22semidirect
53. C2×M16×C173C22semidirect
54. C22×C8×C173C22semidirect
57. C22⋊C8×C173C22semidirect
58. C22⋊C8×C173C22semidirect
203. C173#3198semidirect
219. #408D173semidirect
227. C2×M16C2×D173semidirect
228. C22×C8C2×D173semidirect
261. C22⋊C8C2×D173semidirect
262. C22⋊C8C2×D173semidirect

Maximal subgroups

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

subgroupindex
#260120692
#260120552
#260118332
#260118162
#260118102
#260118052
#260117442
#260117372
#260117272
#260117202
#260116442
#260116382
#260116272
#408×C1732
#3198173