Finite Group Databasebeta

← #36364976

Subgroups — #36364976

Order 22144.

358 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#26012112C211yesmaximal, characteristic, complemented
2#26012096C211yesmaximal, characteristic, complemented
3#26011799C211yesmaximal, characteristic, complemented
4#26011792C211yesmaximal, characteristic, complemented
5#26011782C211yesmaximal, characteristic, complemented
6#26011772C211yesmaximal, characteristic, complemented
7#26011744C211yesmaximal, characteristic, complemented
8#26011737C211yesmaximal, characteristic, complemented
9#26011727C211yesmaximal, characteristic, complemented
10#26011720C211yesmaximal, characteristic, complemented
11#26011707C211yesmaximal, characteristic, complemented
12#26011707C211yesmaximal, characteristic, complemented
13#26011692C211yesmaximal, characteristic, complemented
14#26011687C211yesmaximal, characteristic, complemented
15#424×C173C211yesmaximal, characteristic, complemented
16C2×#14583815C2211yescharacteristic
17C2×#14583815C2211yescharacteristic
18C2×D1384C2211yescharacteristic
19C2×D1384C2211yescharacteristic
20#19149679C2211yescharacteristic
21#19149676C2211yescharacteristic
22#19149674—21no—
23#19149671C2211yescharacteristic
24#19149667C2211yescharacteristic
25#19149665—21no—
26#19149664—21no—
27#19149664—21no—
28#19149661—21no—
29#19149658C2211yescharacteristic
30#19149657—21no—
31#19149653C2211yescharacteristic
32#19149653C2211yescharacteristic
33#19149653C2211yescharacteristic
34#19149652—21no—
35#19149652—21no—
36C4×Dic346C2211yescharacteristic
37C4×D692C2211yescharacteristic
38C4×D692C2211yescharacteristic
39C4×D692C2211yescharacteristic
40#19149613C2211yescharacteristic
41#19149613C2211yescharacteristic
42#19149611C2211yescharacteristic
43#19149611C2211yescharacteristic
44#19149604C2211yescharacteristic
45#19149604C2211yescharacteristic
46#19149601C2211yescharacteristic
47#19149601C2211yescharacteristic
48#19149601C2211yescharacteristic
49#19149601C2211yescharacteristic
50#19149601C2211yescharacteristic
51#19149601C2211yescharacteristic
52C2×M16×C173C2211yescharacteristic
53C22×C8×C173C2211yescharacteristic
54#117×C173C2211yescharacteristic
55C4⋊C8×C173C2211yescharacteristic
56C4⋊C8×C173C2211yescharacteristic
57C4⋊C8×C173C2211yescharacteristic
58C4⋊C8×C173C2211yescharacteristic
59C2×Dic346—21no—
60C2×Dic346C2311yescharacteristic
61C2×D692—21no—
62C2×D692—21no—
63C2×D692C2311yescharacteristic
64C2×D692C2311yescharacteristic
65C2×D692C2311yescharacteristic
66C2×C22⋊D173—21no—
67C2×C22⋊D173—21no—
68C2×C22⋊D173—21no—
69C2×C4×D173—21no—
70C2×C4×D173—21no—
71C2×C4×D173—21no—
72#14583815—41no—
73#14583815—41no—
74D1384—41no—
75D1384—41no—
76C4⋊Dic173—21no—
77C4⋊Dic173C2311yescharacteristic
78C4⋊Dic173C2311yescharacteristic
79C4⋊Dic173C2311yescharacteristic
80C4⋊Dic173C2311yescharacteristic
81#14583805—21no—
82#14583805—21no—
83#14583805—21no—
84#14583805—21no—
85C22⋊Dic173—21no—
86#14583803—21no—
87#14583803—21no—
88#14583803—21no—
89#14583803—21no—
90#14583803—21no—
91#14583803—21no—
92#14583803—21no—
93#14583803—21no—
94C4×Dic173—21no—
95C22×C4×C173C2311yescharacteristic
96M16×C173—41no—
97C2×C8×C173—41no—
98C2×C8×C173C2311yescharacteristic
99C2×C8×C173C2311yescharacteristic
100C2×C8×C173C2311yescharacteristic
101C2×C8×C173C2311yescharacteristic
102C4⋊C4×C173D411yescharacteristic
103C4⋊C4×C173D411yescharacteristic
104C22⋊C4×C173D411yescharacteristic
105C22⋊C4×C173D411yescharacteristic
106C42×C173C2311yescharacteristic
107C42×C173C2311yescharacteristic
108C22×D173—21no—
109C22×D173—21no—
110C22×D173—21no—
111Dic346—21no—
112Dic346—21no—
113Dic346—81no—
114D692—21no—
115D692—21no—
116D692—21no—
117D692—21no—
118D692—21no—
119D692—21no—
120D692—81no—
121D692—81no—
122C22⋊D173—81no—
123C22⋊D173—81no—
124C22⋊D173—81no—
125C2×Dic173—21no—
126C2×Dic173—21no—
127C2×Dic173—21no—
128C2×Dic173—21no—
129C2×Dic173—21no—
130C4×D173—81no—
131C4×D173—81no—
132C4×D173—81no—
133C23×C173C2×D411yescharacteristic
134C2×C4×C173—21no—
135C2×C4×C173—21no—
136C2×C4×C173C2×D411yescharacteristic
137C2×C4×C173C2×D411yescharacteristic
138C2×C4×C173C2×D411yescharacteristic
139C2×C4×C173C2×D411yescharacteristic
140C2×C4×C173C2×D411yescharacteristic
141C2×C4×C173C2411yescharacteristic
142C8×C173—21no—
143C8×C173—21no—
144C8×C173—21no—
145C8×C173—21no—
146C22×C173—21no—
147C22×C173—21no—
148C22×C173—21no—
149C22×C173C22×D411yescharacteristic
150C2×D173—41no—
151C2×D173—41no—
152C2×D173—41no—
153C2×D173—41no—
154C2×D173—41no—
155C2×D173—41no—
156C2×D173—41no—
157C2×D173—41no—
158C2×D173—41no—
159C4×C173—21no—
160C4×C173—21no—
161C4×C173—21no—
162C4×C173—21no—
163C4×C173—21no—
164C4×C173H(5,𝔽2)11yescharacteristic
165C4×C1732−1+411yescharacteristic
166Dic173—41no—
167Dic173—41no—
168Dic173—41no—
169Dic173—41no—
170Dic173—41no—
171C2×C173—41no—
172C2×C173#53711yescharacteristic
173C2×C173#57611yescharacteristic
174C2×C173#57711yescharacteristic
175D173—81no—
176D173—81no—
177D173—81no—
178C173#325711yescharacteristic, complemented, Sylow
179#3257—1731nomaximal, Sylow
180#542—1731no—
181#540—1731no—
182#479—1731no—
183#478—1731no—
184#476—1731no—
185#474—1731no—
186#469—1731no—
187#468—1731no—
188#466—1731no—
189#465—1731no—
190#463—1731no—
191#463—1731no—
192#460—1731no—
193#459—1731no—
194#424D17311yescharacteristic, complemented
195C2×SD16—1731no—
196C2×SD16—1731no—
197C2×D8—1731no—
198C2×D8—1731no—
199C2×M16C2×D17311yescharacteristic
200C22×C8C2×D17311yescharacteristic
201C4⋊Q8—1731no—
202#127—1731no—
203#126—3461no—
204#125—1731no—
205#124—3461no—
206#124—1731no—
207#123—3461no—
208#123—3461no—
209#123—3461no—
210C22⋊Q8—3461no—
211C22⋊Q8—1731no—
212#121—3461no—
213#121—3461no—
214#121—1731no—
215#121—1731no—
216#121—1731no—
217C4×Q8—1731no—
218C4×D4—1731no—
219C4×D4—1731no—
220C4×D4—1731no—
221#117C2×D17311yescharacteristic
222#107—1731no—
223#107—1731no—
224#106—1731no—
225#106—1731no—
226C4⋊C8C2×D17311yescharacteristic
227C4⋊C8C2×D17311yescharacteristic
228C4⋊C8C2×D17311yescharacteristic
229C4⋊C8C2×D17311yescharacteristic
230#103—1731no—
231#103—1731no—
232#102—1731no—
233#102—1731no—
234#102—1731no—
235#102—1731no—
236#102—1731no—
237#102—1731no—
238C2×Q8—3461no—
239C2×Q8—1731no—
240C2×D4—3461no—
241C2×D4—3461no—
242C2×D4—3461no—
243C2×D4—3461no—
244C2×D4—3461no—
245C2×D4—1731no—
246C2×D4—1731no—
247C2×D4—1731no—
248C22×C4—3461no—
249C22×C4—3461no—
250C22×C4—3461no—
251C22×C4C22×D17311yescharacteristic
252SD16—6921no—
253SD16—6921no—
254D8—6921no—
255D8—6921no—
256M16—41no—
257C2×C8—41no—
258C2×C8C22×D17311yescharacteristic
259C2×C8C22×D17311yescharacteristic
260C2×C8C22×D17311yescharacteristic
261C2×C8C22×D17311yescharacteristic
262C4⋊C4—3461no—
263C4⋊C4—3461no—
264C4⋊C4—3461no—
265C4⋊C4—3461no—
266C4⋊C4—3461no—
267C4⋊C4—1731no—
268C4⋊C4—1731no—
269C4⋊C4—1731no—
270C4⋊C4—1731no—
271C4⋊C4D69211yescharacteristic
272C4⋊C4D69211yescharacteristic
273C22⋊C4—3461no—
274C22⋊C4—3461no—
275C22⋊C4—3461no—
276C22⋊C4—3461no—
277C22⋊C4—3461no—
278C22⋊C4—3461no—
279C22⋊C4—3461no—
280C22⋊C4—3461no—
281C22⋊C4—3461no—
282C22⋊C4D69211yescharacteristic
283C22⋊C4D69211yescharacteristic
284C42—3461no—
285C42C22×D17311yescharacteristic
286C42C22×D17311yescharacteristic
287C23—3461no—
288C23—3461no—
289C23—3461no—
290C23C2×D69211yescharacteristic
291Q8—3461no—
292Q8—3461no—
293Q8—13841no—
294D4—3461no—
295D4—3461no—
296D4—3461no—
297D4—3461no—
298D4—3461no—
299D4—3461no—
300D4—13841no—
301D4—13841no—
302D4—13841no—
303D4—13841no—
304D4—13841no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—3461no—
308C2×C4—3461no—
309C2×C4—3461no—
310C2×C4—3461no—
311C2×C4—3461no—
312C2×C4—13841no—
313C2×C4—13841no—
314C2×C4—13841no—
315C2×C4C2×D69211yescharacteristic
316C2×C4C2×D69211yescharacteristic
317C2×C4C2×D69211yescharacteristic
318C2×C4C2×D69211yescharacteristic
319C2×C4C2×D69211yescharacteristic
320C2×C4C23×D17311yescharacteristic
321C8—21no—
322C8—21no—
323C8—21no—
324C8—21no—
325C22—21no—
326C22—21no—
327C22—21no—
328C22—6921no—
329C22—6921no—
330C22—6921no—
331C22—6921no—
332C22—6921no—
333C22—6921no—
334C22—6921no—
335C22—6921no—
336C22—6921no—
337C22C22×D69211yescharacteristic
338C4—21no—
339C4—21no—
340C4—21no—
341C4—21no—
342C4—21no—
343C4—6921no—
344C4—6921no—
345C4—6921no—
346C4—6921no—
347C4—6921no—
348C4#1914972411yescharacteristic
349C4#1914972711yescharacteristic
350C2—41no—
351C2—13841no—
352C2—13841no—
353C2—13841no—
354C2#2601207211yescharacteristic
355C2#2601229611yescharacteristic
356C2#2601230311yescharacteristic
subgroupG/H
centerC22C22×D692
commutator, cosocleC2×C4×C173C24
FrattiniC2×C4C23×D173
Fitting#424×C173C2
socleC22×C173C22×D4
hypercenter#424D173
hypocenterC173#3257
LCSG ▷ C2×C4×C173 ▷ C2×C173 ▷ C173
UCSC1 ◁ C22 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
1. #26012112C2semidirect
2. #26012096C2semidirect
3. #26011799C2semidirect
4. #26011792C2semidirect
5. #26011782C2semidirect
6. #26011772C2semidirect
7. #26011744C2semidirect
8. #26011737C2semidirect
9. #26011727C2semidirect
10. #26011720C2semidirect
11. #26011707C2semidirect
12. #26011707C2semidirect
13. #26011692C2semidirect
14. #26011687C2semidirect
15. #424×C173C2semidirect
178. C173#3257semidirect
194. #424D173semidirect

Maximal subgroups

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

subgroupindex
#260121122
#260120962
#260117992
#260117922
#260117822
#260117722
#260117442
#260117372
#260117272
#260117202
#260117072
#260116922
#260116872
#424×C1732
#3257173