Finite Group Databasebeta

← #36364447

Subgroups — #36364447

Order 22144.

390 conjugacy classes of subgroups.

The subgroup lattice has 382 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012069C211yesmaximal, characteristic
2#26012062C211yesmaximal, characteristic, complemented
3#26011833C211yesmaximal, characteristic, complemented
4#26011827C211yesmaximal, characteristic, complemented
5#26011822C211yesmaximal, characteristic, complemented
6#26011816C211yesmaximal, characteristic, complemented
7#26011749C211yesmaximal, characteristic, complemented
8#26011737C211yesmaximal, characteristic, complemented
9#26011732C211yesmaximal, characteristic, complemented
10#26011720C211yesmaximal, characteristic, complemented
11#26011655C211yesmaximal, characteristic, complemented
12#26011655C211yesmaximal, characteristic, complemented
13#26011649C211yesmaximal, characteristic, complemented
14#26011632C211yesmaximal, characteristic, complemented
15#408×C173C211yesmaximal, characteristic, complemented
16C2×#14583826C2211yescharacteristic
17C22×Dic346C2211yescharacteristic
18C2×Dic692C2211yescharacteristic
19C2×Dic692C2211yescharacteristic
20C2×#14583815C2211yescharacteristic
21C2×#14583815C2211yescharacteristic
22#19149677—21no—
23#19149666—21no—
24#19149660—21no—
25#19149660—21no—
26#19149658C2211yescharacteristic
27#19149658C2211yescharacteristic
28#19149658C2211yescharacteristic
29#19149658C2211yescharacteristic
30#19149658C2211yescharacteristic
31#19149658C2211yescharacteristic
32#19149655—21no—
33#19149653C2211yescharacteristic
34#19149653C2211yescharacteristic
35#19149651—21no—
36#19149651—21no—
37#19149650—21no—
38#19149638C2211yescharacteristic
39C2×C4⋊Dic173C2211yescharacteristic
40#19149613C2211yescharacteristic
41#19149613C2211yescharacteristic
42#19149611C2211yescharacteristic
43#19149611C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149604C2211yescharacteristic
46#19149604C2211yescharacteristic
47#19149604C2211yescharacteristic
48#19149604C2211yescharacteristic
49#19149604C2211yescharacteristic
50#19149601C2211yescharacteristic
51#19149601C2211yescharacteristic
52C2×#41×C173C2211yescharacteristic
53C2×M16×C173C2211yescharacteristic
54C22×C8×C173C2211yescharacteristic
55C22⋊C8×C173C2211yescharacteristic
56C22⋊C8×C173C2211yescharacteristic
57C22⋊C8×C173C2211yescharacteristic
58C22⋊C8×C173C2211yescharacteristic
59#14583826—41no—
60#14583826—41no—
61C2×Dic346—21no—
62C2×Dic346—41no—
63C2×Dic346—41no—
64C2×Dic346C2311yescharacteristic
65C2×Dic346C2311yescharacteristic
66C2×Dic346C2311yescharacteristic
67C2×D692C2311yescharacteristic
68C2×C22⋊D173—21no—
69C2×C22⋊D173—21no—
70C2×C22⋊D173—21no—
71C22×Dic173—21no—
72C22×Dic173—21no—
73C2×C4×D173—21no—
74Dic692—41no—
75Dic692—41no—
76#14583815—41no—
77#14583815—41no—
78C4⋊Dic173—21no—
79C4⋊Dic173C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81C4⋊Dic173C2311yescharacteristic
82C4⋊Dic173C2311yescharacteristic
83#14583805—21no—
84#14583805—21no—
85#14583805—21no—
86#14583805—21no—
87#14583805—21no—
88#14583805—21no—
89C22⋊Dic173—21no—
90C22⋊Dic173—21no—
91C22⋊Dic173—21no—
92C22⋊Dic173—21no—
93C22⋊Dic173—21no—
94C22⋊Dic173—21no—
95C22⋊Dic173—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—
115Dic346—21no—
116Dic346—21no—
117Dic346—21no—
118Dic346—21no—
119Dic346—21no—
120Dic346—21no—
121Dic346—42no—
122D692—21no—
123D692—21no—
124C22⋊D173—42no—
125C22⋊D173—81no—
126C22⋊D173—81no—
127C2×Dic173—21no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C2×Dic173—21no—
131C2×Dic173—21no—
132C2×Dic173—21no—
133C2×Dic173—21no—
134C2×Dic173—42no—
135C2×Dic173—81no—
136C4×D173—42no—
137C23×C173C2×D411yescharacteristic
138C23×C173C2×D411yescharacteristic
139C23×C173C2×D411yescharacteristic
140Q8×C173—41no—
141D4×C173—41no—
142D4×C173—41no—
143D4×C173—41no—
144C2×C4×C173—21no—
145C2×C4×C173—21no—
146C2×C4×C173—21no—
147C2×C4×C173—21no—
148C2×C4×C173—21no—
149C2×C4×C173—21no—
150C2×C4×C173C2×D411yescharacteristic
151C2×C4×C173C2×D411yescharacteristic
152C2×C4×C173C2×D411yescharacteristic
153C2×C4×C173C2411yescharacteristic
154C8×C173—21no—
155C8×C173—21no—
156C8×C173—21no—
157C8×C173—21no—
158C22×C173—21no—
159C22×C173—21no—
160C22×C173—21no—
161C22×C173—21no—
162C22×C173—21no—
163C22×C173—21no—
164C22×C173—21no—
165C22×C173—21no—
166C22×C173—21no—
167C22×C173C22×D411yescharacteristic
168C2×D173—41no—
169C2×D173—41no—
170C2×D173—41no—
171C4×C173—21no—
172C4×C173—21no—
173C4×C173—21no—
174C4×C173H(5,𝔽2)11yescharacteristic
175C4×C173H(5,𝔽2)11yescharacteristic
176Dic173—41no—
177Dic173—41no—
178Dic173—41no—
179Dic173—41no—
180Dic173—41no—
181Dic173—41no—
182Dic173—41no—
183C2×C173—41no—
184C2×C173—41no—
185C2×C173—41no—
186C2×C173#53411yescharacteristic
187C2×C173#57711yescharacteristic
188C2×C173#57811yescharacteristic
189D173—81no—
190C173#320111yescharacteristic, complemented, Sylow
191#3201—1731nomaximal, Sylow
192#537—1731no—
193#536—1731no—
194#485—1731no—
195#484—1731no—
196#483—1731no—
197#482—1731no—
198#470—1731no—
199#468—1731no—
200#467—1731no—
201#465—1731no—
202#452—1731no—
203#452—1731no—
204#451—1731no—
205#448—1731no—
206#408D17311yescharacteristic, complemented
207C2×#41—1731no—
208C2×#41C2×D17311yescharacteristic
209C22×Q8—1731no—
210C2×Q16—1731no—
211C2×Q16—1731no—
212C2×SD16—1731no—
213C2×SD16—1731no—
214C2×M16C2×D17311yescharacteristic
215C22×C8C2×D17311yescharacteristic
216C4⋊Q8—3461no—
217#124—3461no—
218#123—3461no—
219#123—3461no—
220C22⋊Q8—3461no—
221C22⋊Q8—1731no—
222C22⋊Q8—1731no—
223C22⋊Q8—1731no—
224C22⋊Q8—1731no—
225C22⋊Q8—1731no—
226C22⋊Q8—1731no—
227#121—3461no—
228#121—3461no—
229#121—3461no—
230#121—1731no—
231#121—1731no—
232#117—1731no—
233C2×C4⋊C4—1731no—
234#107—1731no—
235#107—1731no—
236#106—1731no—
237#106—1731no—
238#103—1731no—
239#103—1731no—
240#103—1731no—
241#103—1731no—
242#103—1731no—
243#103—1731no—
244#102—1731no—
245#102—1731no—
246C22⋊C8C2×D17311yescharacteristic
247C22⋊C8C2×D17311yescharacteristic
248C22⋊C8C2×D17311yescharacteristic
249C22⋊C8C2×D17311yescharacteristic
250#41—41no—
251#41—41no—
252#41—6921no—
253#41—6921no—
254C2×Q8—3461no—
255C2×Q8—1731no—
256C2×Q8—1731no—
257C2×Q8—1731no—
258C2×Q8—6921no—
259C2×Q8—6921no—
260C2×Q8D69211yescharacteristic
261C2×D4—3461no—
262C2×D4—3461no—
263C2×D4—3461no—
264C2×D4—1731no—
265C2×D4D69211yescharacteristic
266C2×D4D69211yescharacteristic
267C2×D4D69211yescharacteristic
268C22×C4—3461no—
269C22×C4—3461no—
270C22×C4—3461no—
271C22×C4C22×D17311yescharacteristic
272C22×C4C22×D17311yescharacteristic
273C22×C4C22×D17311yescharacteristic
274Q16—6921no—
275Q16—6921no—
276SD16—6921no—
277SD16—6921no—
278M16—41no—
279C2×C8—41no—
280C2×C8C22×D17311yescharacteristic
281C2×C8C22×D17311yescharacteristic
282C2×C8C22×D17311yescharacteristic
283C2×C8C22×D17311yescharacteristic
284C4⋊C4—3461no—
285C4⋊C4—3461no—
286C4⋊C4—3461no—
287C4⋊C4—3461no—
288C4⋊C4—3461no—
289C4⋊C4—3461no—
290C4⋊C4—3461no—
291C4⋊C4—1731no—
292C4⋊C4—1731no—
293C4⋊C4—1731no—
294C4⋊C4—1731no—
295C22⋊C4—3461no—
296C22⋊C4—3461no—
297C22⋊C4—3461no—
298C22⋊C4—3461no—
299C22⋊C4—3461no—
300C22⋊C4—3461no—
301C22⋊C4—3461no—
302C22⋊C4—3461no—
303C22⋊C4—3461no—
304C42—3461no—
305C23—3461no—
306C23C2×D69211yescharacteristic
307C23C2×D69211yescharacteristic
308C23C2×D69211yescharacteristic
309Q8—41no—
310Q8—3461no—
311Q8—3461no—
312Q8—3461no—
313Q8—3461no—
314Q8—3461no—
315Q8—3461no—
316Q8—6922no—
317D4—41no—
318D4—41no—
319D4—41no—
320D4—3461no—
321D4—3461no—
322D4—13841no—
323D4—13841no—
324D4—6922no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—3461no—
332C2×C4—3461no—
333C2×C4—3461no—
334C2×C4—3461no—
335C2×C4—3461no—
336C2×C4—3461no—
337C2×C4—3461no—
338C2×C4—13841no—
339C2×C4—6922no—
340C2×C4—6922no—
341C2×C4C2×D69211yescharacteristic
342C2×C4C2×D69211yescharacteristic
343C2×C4C2×D69211yescharacteristic
344C2×C4C23×D17311yescharacteristic
345C8—21no—
346C8—21no—
347C8—21no—
348C8—21no—
349C22—21no—
350C22—21no—
351C22—21no—
352C22—21no—
353C22—21no—
354C22—21no—
355C22—21no—
356C22—21no—
357C22—21no—
358C22—6921no—
359C22—6921no—
360C22—6921no—
361C22C22×D69211yescharacteristic
362C4—21no—
363C4—21no—
364C4—21no—
365C4—6921no—
366C4—6921no—
367C4—6921no—
368C4—6921no—
369C4—6921no—
370C4—6921no—
371C4—6921no—
372C4#1914972411yescharacteristic
373C4#1914972411yescharacteristic
374C2—41no—
375C2—41no—
376C2—41no—
377C2—13841no—
378C2#2601205311yescharacteristic
379C2#2601230311yescharacteristic
380C2#2601230811yescharacteristic
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#408×C173C2
socleC22×C173C22×D4
prime radical#26012069C2
hypercenter#408D173
hypocenterC173#3201
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
2. #26012062C2semidirect
3. #26011833C2semidirect
4. #26011827C2semidirect
5. #26011822C2semidirect
6. #26011816C2semidirect
7. #26011749C2semidirect
8. #26011737C2semidirect
9. #26011732C2semidirect
10. #26011720C2semidirect
11. #26011655C2semidirect
12. #26011655C2semidirect
13. #26011649C2semidirect
14. #26011632C2semidirect
15. #408×C173C2semidirect
190. C173#3201semidirect
206. #408D173semidirect

Maximal subgroups

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

subgroupindex
#260120692
#260120622
#260118332
#260118272
#260118222
#260118162
#260117492
#260117372
#260117322
#260117202
#260116552
#260116492
#260116322
#408×C1732
#3201173