Finite Group Databasebeta

← #22602878

Subgroups — #22602878

Order 9088.

1024 conjugacy classes of subgroups.

The subgroup lattice has 402 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#17986749C212yesmaximal, complemented
2#17986743C212yesmaximal, complemented
3#17986708C212yesmaximal, complemented
4#17986701C212yesmaximal, complemented
5#17986699C212yesmaximal, complemented
6#17986649C212yesmaximal, complemented
7#17986648C214yesmaximal, complemented
8#17986626C212yesmaximal, complemented
9#17986603C212yesmaximal, complemented
10#17986594C212yesmaximal, complemented
11#17986591C214yesmaximal, complemented
12#17986576C212yesmaximal, complemented
13#17986533C211yesmaximal, characteristic, complemented
14C2×#14556120C211yesmaximal, characteristic, complemented
15#532×C71C211yesmaximal, characteristic, complemented
16C2×#11765864C2214yes—
17#14556127C2212yes—
18#14556126C2214yes—
19#14556122C2212yes—
20#14556121C2214yes—
21#14556121C2214yes—
22#14556120C2212yes—
23#14556120C2212yes—
24#14556120C2212yes—
25#14556120C2212yes—
26#14556120C2211yescharacteristic
27#14556119C2212yes—
28#14556118C2214yes—
29#14556116C2211yescharacteristic
30#14556112C2214yes—
31#14556111C2214yes—
32#14556111C2212yes—
33#14556110C2214yes—
34#14556109C2214yes—
35#14556109C2214yes—
36#14556109C2214yes—
37#14556107C2214yes—
38#14556107C2211yescharacteristic
39#14556106C2214yes—
40#14556105C2214yes—
41#14556105C2214yes—
42#14556105C2214yes—
43#14556104C2214yes—
44#14556102C2211yescharacteristic
45#14556101C2214yes—
46#14556100C2214yes—
47#14556100C2214yes—
48#14556100C2214yes—
49#14556099C2212yes—
50#14556099C2212yes—
51C4×Dic142C2211yescharacteristic
52C4×D284C2211yescharacteristic
53D4×Dic71C2214yes—
54D4×Dic71C2212yes—
55D4×Dic71C2212yes—
56C4×C22⋊D71C2214yes—
57C4×C22⋊D71C2212yes—
58#14556086C2212yes—
59#14556085C2212yes—
60C2×C4⋊Dic71C2211yescharacteristic
61C2×C4⋊Dic71C2211yescharacteristic
62C2×Dic71⋊C4C2212yes—
63C2×Dic71⋊C4C2214yes—
64C2×Dic71⋊C4C2212yes—
65C4⋊C4×D71C2212yes—
66C2×#41×C71C2212yes—
67C4⋊Q8×C71C2211yescharacteristic
68#127×C71C2211yescharacteristic
69#124×C71C2212yes—
70#121×C71C2212yes—
71#121×C71C2212yes—
72C4×D4×C71C2212yes—
73C4×D4×C71C2212yes—
74C2×C42×C71C2211yescharacteristic
75#11765864—48no—
76C2×Dic142C2314yes—
77C2×Dic142C2311yescharacteristic
78C2×D284C2311yescharacteristic
79C2×C22⋊D71C2312yes—
80C2×C22⋊D71C2314yes—
81C2×C22⋊D71C2314yes—
82C22×Dic71C2312yes—
83C22×Dic71C2314yes—
84C22×Dic71C2312yes—
85C22×Dic71C2314yes—
86C2×C4×D71C2312yes—
87C2×C4×D71C2314yes—
88C4⋊Dic71C2312yes—
89C4⋊Dic71C2312yes—
90C4⋊Dic71C2312yes—
91C4⋊Dic71C2312yes—
92C4⋊Dic71C2314yes—
93C4⋊Dic71C2311yescharacteristic
94C4⋊Dic71C2311yescharacteristic
95Dic71⋊C4C2314yes—
96Dic71⋊C4C2314yes—
97Dic71⋊C4C2312yes—
98Dic71⋊C4C2314yes—
99Dic71⋊C4C2314yes—
100Dic71⋊C4C2314yes—
101Dic71⋊C4C2314yes—
102Dic71⋊C4C2314yes—
103Dic71⋊C4C2312yes—
104Dic71⋊C4C2312yes—
105Dic71⋊C4C2312yes—
106C22⋊Dic71C2312yes—
107C22⋊Dic71C2314yes—
108C22⋊Dic71C2314yes—
109C22⋊Dic71C2314yes—
110C22⋊Dic71C2314yes—
111(C2×D71)⋊C4C2312yes—
112(C2×D71)⋊C4C2312yes—
113(C2×D71)⋊C4C2314yes—
114(C2×D71)⋊C4C2312yes—
115(C2×D71)⋊C4C2312yes—
116C4×Dic71C2312yes—
117C4×Dic71C2314yes—
118#41×C71—44no—
119C2×Q8×C71C2312yes—
120C2×D4×C71C2314yes—
121C2×D4×C71C2312yes—
122C2×D4×C71C2312yes—
123C2×D4×C71C2312yes—
124C22×C4×C71C2311yescharacteristic
125C22×C4×C71C2311yescharacteristic
126C22×C4×C71C2311yescharacteristic
127C22×C4×C71C2314yes—
128C4⋊C4×C71C2312yes—
129C4⋊C4×C71C2312yes—
130C22⋊C4×C71C2314yes—
131C22⋊C4×C71C2314yes—
132C42×C71C2311yescharacteristic
133C42×C71C2311yescharacteristic
134C42×C71C2311yescharacteristic
135C42×C71C2311yescharacteristic
136C23×C71C2414yes—
137C23×C71C2411yescharacteristic
138C22×D71C2412yes—
139Q8×C71—42no—
140D4×C71—28no—
141D4×C71—24no—
142D4×C71—42no—
143D4×C71—42no—
144C2×C4×C71—22no—
145C2×C4×C71—22no—
146C2×C4×C71—41no—
147C2×C4×C71—44no—
148C2×C4×C71C2411yescharacteristic
149C2×C4×C71C2411yescharacteristic
150C2×C4×C71C2411yescharacteristic
151C2×C4×C71C2411yescharacteristic
152C2×C4×C71C2411yescharacteristic
153C2×C4×C71C2414yes—
154C2×C4×C71C2411yescharacteristic
155C22⋊D71—24no—
156C22⋊D71—44no—
157C22⋊D71—44no—
158C2×Dic71—24no—
159C2×Dic71—44no—
160C2×Dic71—42no—
161C2×Dic71—44no—
162C2×Dic71C2412yes—
163C2×Dic71C2412yes—
164C2×Dic71C2414yes—
165C2×Dic71C2412yes—
166C2×Dic71C2414yes—
167D284—41no—
168C4×D71—24no—
169C4×D71—44no—
170Dic142—41no—
171Dic142—44no—
172C22×C71—21no—
173C22×C71—21no—
174C22×C71—24no—
175C22×C71—24no—
176C22×C71—24no—
177C22×C71H(5,𝔽2)12yes—
178C22×C71C2511yescharacteristic
179C2×D71—22no—
180C2×D71—22no—
181C2×D71—22no—
182C4×C71—24no—
183C4×C71—21no—
184C4×C71—21no—
185C4×C71—21no—
186C4×C71—21no—
187C4×C71—21no—
188C4×C712−1+412yes—
189Dic71—22no—
190Dic71—24no—
191Dic71—22no—
192Dic71—22no—
193Dic71—24no—
194C2×C71—22no—
195C2×C71—44no—
196C2×C71C2×H(5,𝔽2)11yescharacteristic
197C2×C71C2×2−1+411yescharacteristic
198C2×C71#58511yescharacteristic
199D71—42no—
200#3559—711nomaximal, Sylow
201C71#355911yescharacteristic, complemented, Sylow
202#563—712no—
203#562—712no—
204#554—712no—
205#553—712no—
206#553—712no—
207#547—714no—
208#547—712no—
209#544—712no—
210#540—712no—
211#539—712no—
212#539—714no—
213#537—712no—
214#532D7111yescharacteristic, complemented
215#529—711no—
216C2×#125—711no—
217C2×#41—714no—
218C2×#41C2×D7112yes—
219C4⋊Q8—714no—
220C4⋊Q8—712no—
221C4⋊Q8C2×D7111yescharacteristic
222#127C2×D7111yescharacteristic
223#126—714no—
224#126—714no—
225#126—712no—
226#125—714no—
227#125—712no—
228#125—712no—
229#125—712no—
230#125—712no—
231#125—712no—
232#125—711no—
233#124—711no—
234#124C2×D7112yes—
235#123—714no—
236#123—714no—
237#123—714no—
238#123—712no—
239#123—714no—
240#123—714no—
241#123—714no—
242C22⋊Q8—714no—
243C22⋊Q8—711no—
244C22⋊Q8—714no—
245C22⋊Q8—714no—
246C22⋊Q8—714no—
247C22⋊Q8—714no—
248C22⋊Q8—714no—
249#121—714no—
250#121—714no—
251#121—712no—
252#121—714no—
253#121—712no—
254#121—711no—
255#121—714no—
256#121C2×D7112yes—
257#121C2×D7112yes—
258C4×Q8—711no—
259C4×D4—714no—
260C4×D4—712no—
261C4×D4—712no—
262C4×D4—712no—
263C4×D4—714no—
264C4×D4—711no—
265C4×D4C2×D7112yes—
266C4×D4C2×D7112yes—
267#117—712no—
268#117—712no—
269C2×C4⋊C4—712no—
270C2×C4⋊C4—712no—
271C2×C4⋊C4—711no—
272C2×C4⋊C4—711no—
273C2×C4⋊C4—714no—
274C2×C4⋊C4—712no—
275C2×C42C2×D7111yescharacteristic
276#41—44no—
277#41—2848no—
278C2×Q8—711no—
279C2×Q8—714no—
280C2×Q8C22×D7112yes—
281C2×D4—714no—
282C2×D4—714no—
283C2×D4—712no—
284C2×D4—711no—
285C2×D4C22×D7112yes—
286C2×D4C22×D7114yes—
287C2×D4C22×D7112yes—
288C2×D4C22×D7112yes—
289C22×C4—712no—
290C22×C4—714no—
291C22×C4—714no—
292C22×C4—714no—
293C22×C4—712no—
294C22×C4—712no—
295C22×C4C22×D7111yescharacteristic
296C22×C4C22×D7111yescharacteristic
297C22×C4C22×D7111yescharacteristic
298C22×C4C22×D7114yes—
299C4⋊C4—714no—
300C4⋊C4—712no—
301C4⋊C4—714no—
302C4⋊C4—714no—
303C4⋊C4—714no—
304C4⋊C4—714no—
305C4⋊C4—714no—
306C4⋊C4—714no—
307C4⋊C4—712no—
308C4⋊C4—712no—
309C4⋊C4—712no—
310C4⋊C4—712no—
311C4⋊C4—712no—
312C4⋊C4—712no—
313C4⋊C4—712no—
314C4⋊C4—714no—
315C4⋊C4—711no—
316C4⋊C4—711no—
317C4⋊C4C22×D7112yes—
318C4⋊C4C22×D7112yes—
319C22⋊C4—712no—
320C22⋊C4—714no—
321C22⋊C4—714no—
322C22⋊C4—714no—
323C22⋊C4—714no—
324C22⋊C4—712no—
325C22⋊C4—714no—
326C22⋊C4—712no—
327C22⋊C4—712no—
328C22⋊C4—712no—
329C22⋊C4C22×D7114yes—
330C22⋊C4C22×D7114yes—
331C42—712no—
332C42—714no—
333C42C22×D7111yescharacteristic
334C42C22×D7111yescharacteristic
335C42C22×D7111yescharacteristic
336C42C22×D7111yescharacteristic
337C23—712no—
338C23C23×D7114yes—
339C23C23×D7111yescharacteristic
340Q8—42no—
341Q8—2844no—
342Q8—2841no—
343D4—28no—
344D4—24no—
345D4—42no—
346D4—42no—
347D4—2841no—
348D4—2844no—
349D4—2844no—
350D4—1424no—
351C2×C4—22no—
352C2×C4—22no—
353C2×C4—44no—
354C2×C4—41no—
355C2×C4—2844no—
356C2×C4—2844no—
357C2×C4—2842no—
358C2×C4—2844no—
359C2×C4—712no—
360C2×C4—712no—
361C2×C4—714no—
362C2×C4—712no—
363C2×C4—714no—
364C2×C4—1424no—
365C2×C4—1424no—
366C2×C4C23×D7114yes—
367C2×C4C23×D7111yescharacteristic
368C2×C4C23×D7111yescharacteristic
369C2×C4C23×D7111yescharacteristic
370C2×C4C23×D7111yescharacteristic
371C2×C4C23×D7111yescharacteristic
372C2×C4C23×D7111yescharacteristic
373C22—21no—
374C22—21no—
375C22—24no—
376C22—24no—
377C22—24no—
378C22—1422no—
379C22—1422no—
380C22—1422no—
381C22#1455617312yes—
382C22C24×D7111yescharacteristic
383C4—21no—
384C4—21no—
385C4—24no—
386C4—21no—
387C4—21no—
388C4—21no—
389C4—1424no—
390C4—1422no—
391C4—1422no—
392C4—1424no—
393C4—1422no—
394C4#1455617512yes—
395C2—22no—
396C2—44no—
397C2—2842no—
398C2C2×#1455617311yescharacteristic
399C2C2×#1455617511yescharacteristic
400C2#1798683511yescharacteristic
subgroupG/H
center, FrattiniC22C24×D71
commutator, socleC22×C71C25
Fitting#532×C71C2
hypercenter#532D71
hypocenterC71#3559
LCSG ▷ C22×C71 ▷ C71
UCSC1 ◁ C22 ◁ #532

Complements

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

NH ≅ G/Ntype
1. #17986749C2semidirect
2. #17986743C2semidirect
3. #17986708C2semidirect
4. #17986701C2semidirect
5. #17986699C2semidirect
6. #17986649C2semidirect
7. #17986648C2semidirect
8. #17986626C2semidirect
9. #17986603C2semidirect
10. #17986594C2semidirect
11. #17986591C2semidirect
12. #17986576C2semidirect
13. #17986533C2semidirect
14. C2×#14556120C2semidirect
15. #532×C71C2semidirect
201. C71#3559semidirect
214. #532D71semidirect