Finite Group Databasebeta

← #23777884

Subgroups — #23777884

Order 10112.

910 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#19099317C214yesmaximal, complemented
2Q8×Dic158C211yesmaximal, characteristic, complemented
3#19099289C212yesmaximal, complemented
4#19099287C212yesmaximal, complemented
5#19099287C211yesmaximal, characteristic, complemented
6#19099286C212yesmaximal, complemented
7#19099273C211yesmaximal, characteristic, complemented
8#19099251C212yesmaximal, complemented
9#19099193C212yesmaximal, complemented
10#19099185C212yesmaximal
11#19099174C212yesmaximal, complemented
12#19099172C214yesmaximal, complemented
13#19099171C212yesmaximal, complemented
14#19099118C211yesmaximal, characteristic, complemented
15#19099117C211yesmaximal, characteristic, complemented
16#19099100C211yesmaximal, characteristic
17#557×C79C211yesmaximal, characteristic, complemented
18#14579161C2212yes—
19#14579161C2211yescharacteristic
20#14579160C2214yes—
21#14579160C2212yes—
22#14579160C2211yescharacteristic
23#14579160C2214yes—
24#14579159C2211yescharacteristic
25#14579156C2214yes—
26#14579156C2212yes—
27#14579156C2214yes—
28#14579156C2214yes—
29#14579156C2214yes—
30#14579156C2214yes—
31#14579155C2214yes—
32#14579153C2214yes—
33#14579153C2212yes—
34#14579152C2212yes—
35#14579152C2214yes—
36#14579152C2211yescharacteristic
37#14579151C2214yes—
38#14579151C2212yes—
39#14579151C2214yes—
40#14579150C2212yes—
41#14579150C2211yescharacteristic
42#14579149C2214yes—
43#14579146C2214yes—
44#14579146C2212yes—
45#14579141C2212yes—
46#14579141C2214yes—
47#14579141C2211yescharacteristic
48#14579141C2212yes—
49#14579139C2214yes—
50#14579139C2214yes—
51#14579139C2212yes—
52#14579137C2212yes—
53#14579137C2211yescharacteristic
54#14579137C2212yes—
55#14579137C2211yescharacteristic
56#14579136C2211yescharacteristic
57Q8×Dic79C2212yes—
58Q8×Dic79C2212yes—
59Q8×Dic79C2212yes—
60C4×Dic158C2212yes—
61C4×Dic158C2214yes—
62C4×Dic158C2212yes—
63C4×Dic158C2211yescharacteristic
64#14579126C2212yes—
65C4×D316C2212yes—
66C4×D316C2212yes—
67C4×D316C2211yescharacteristic
68#14579119C2214yes—
69#14579119C2212yes—
70#14579119C2212yes—
71#14579118C2212yes—
72#14579118C2212yes—
73C4⋊C4×D79C2212yes—
74C42×D79C2211yescharacteristic
75C4⋊Q8×C79C2212yes—
76C4⋊Q8×C79C2211yescharacteristic
77#125×C79C2214yes—
78#125×C79C2212yes—
79C4×Q8×C79C2212yes—
80C4×Q8×C79C2212yes—
81C4×Q8×C79C2211yescharacteristic
82C4×Q8×C79C2211yescharacteristic
83C2×Dic158C2311yescharacteristic
84C2×Dic158C2312yes—
85C2×Dic158C2312yes—
86C2×Dic158C2314yes—
87C2×D316C2311yescharacteristic
88C2×D316C2312yes—
89C2×C4×D79C2311yescharacteristic
90C2×C4×D79C2311yescharacteristic
91C2×C4×D79C2312yes—
92C2×C4×D79C2311yescharacteristic
93C2×C4×D79C2314yes—
94C4⋊Dic79C2312yes—
95C4⋊Dic79C2311yescharacteristic
96C4⋊Dic79C2312yes—
97C4⋊Dic79C2312yes—
98C4⋊Dic79C2311yescharacteristic
99C4⋊Dic79C2312yes—
100C4⋊Dic79C2314yes—
101C4⋊Dic79C2314yes—
102C4⋊Dic79C2314yes—
103Dic79⋊C4C2314yes—
104Dic79⋊C4C2312yes—
105Dic79⋊C4C2312yes—
106Dic79⋊C4C2312yes—
107Dic79⋊C4C2312yes—
108Dic79⋊C4C2314yes—
109Dic79⋊C4C2312yes—
110Dic79⋊C4C2314yes—
111Dic79⋊C4C2312yes—
112Dic79⋊C4C2312yes—
113Dic79⋊C4C2314yes—
114(C2×D79)⋊C4C2312yes—
115(C2×D79)⋊C4C2312yes—
116(C2×D79)⋊C4C2312yes—
117(C2×D79)⋊C4C2312yes—
118(C2×D79)⋊C4C2312yes—
119(C2×D79)⋊C4C2314yes—
120(C2×D79)⋊C4C2314yes—
121(C2×D79)⋊C4C2314yes—
122(C2×D79)⋊C4C2314yes—
123(C2×D79)⋊C4C2312yes—
124(C2×D79)⋊C4C2312yes—
125C4×Dic79C2311yescharacteristic
126C4×Dic79C2311yescharacteristic
127C4×Dic79C2311yescharacteristic
128C4×Dic79C2312yes—
129C4×Dic79C2314yes—
130C4×Dic79C2314yes—
131C4×Dic79C2314yes—
132C2×Q8×C79C2311yescharacteristic
133C2×Q8×C79C2312yes—
134C2×Q8×C79C2311yescharacteristic
135C4⋊C4×C79C2311yescharacteristic
136C4⋊C4×C79C2311yescharacteristic
137C4⋊C4×C79C2312yes—
138C4⋊C4×C79C2314yes—
139C4⋊C4×C79C2312yes—
140C4⋊C4×C79C2312yes—
141C4⋊C4×C79C2314yes—
142C4⋊C4×C79C2314yes—
143C4⋊C4×C79C2312yes—
144C42×C79C2312yes—
145C42×C79C2314yes—
146C42×C79C2311yescharacteristic
147C42×C79C2312yes—
148C22×D79C2411yescharacteristic
149C22×D79C2412yes—
150Q8×C79—24no—
151Q8×C79—22no—
152Q8×C79—22no—
153C2×C4×C79C2414yes—
154C2×C4×C79C2412yes—
155C2×C4×C79C2411yescharacteristic
156C2×C4×C79C2412yes—
157C2×C4×C79C2412yes—
158C2×C4×C79C2412yes—
159C2×C4×C79C2411yescharacteristic
160C2×C4×C79C2411yescharacteristic
161C2×Dic79C2411yescharacteristic
162C2×Dic79C2412yes—
163C2×Dic79C2412yes—
164C2×Dic79C2411yescharacteristic
165C2×Dic79C2412yes—
166C2×Dic79C2411yescharacteristic
167C2×Dic79C2414yes—
168D316—22no—
169D316—42no—
170C4×D79—22no—
171C4×D79—22no—
172C4×D79—24no—
173C4×D79—41no—
174C4×D79—44no—
175Dic158—22no—
176Dic158—44no—
177Dic158—42no—
178Dic158—42no—
179C22×C79C2511yescharacteristic
180C2×D79—21no—
181C2×D79—21no—
182C2×D79—22no—
183C2×D79—22no—
184C2×D79—22no—
185C2×D79—21no—
186C4×C79—24no—
187C4×C79—22no—
188C4×C79—22no—
189C4×C79—21no—
190C4×C79—22no—
191C4×C79—22no—
192C4×C792−1+412yes—
193C4×C792−1+412yes—
194Dic79—22no—
195Dic79—21no—
196Dic79—21no—
197Dic79—22no—
198Dic79—22no—
199Dic79—24no—
200Dic79—21no—
201C2×C79C2×2−1+411yescharacteristic
202C2×C79C2×2−1+411yescharacteristic
203C2×C79#58511yescharacteristic
204D79—41no—
205D79—42no—
206#3555—791nomaximal, Sylow
207C79#355511yescharacteristic, complemented, Sylow
208#563—794no—
209Q82—791no—
210#557—792no—
211#557D7911yescharacteristic, complemented
212#556—792no—
213#556—791no—
214#556—792no—
215#554—791no—
216#552—792no—
217#544—792no—
218#542—792no—
219#541—792no—
220#541—794no—
221#541—792no—
222#533—791no—
223#533—791no—
224#529—791no—
225C4⋊Q8—791no—
226C4⋊Q8—791no—
227C4⋊Q8—794no—
228C4⋊Q8—792no—
229C4⋊Q8—794no—
230C4⋊Q8—791no—
231C4⋊Q8—792no—
232C4⋊Q8C2×D7911yescharacteristic
233C4⋊Q8C2×D7912yes—
234#126—792no—
235#126—794no—
236#126—794no—
237#126—794no—
238#126—794no—
239#126—794no—
240#126—794no—
241#125—791no—
242#125—792no—
243#125—792no—
244#125—794no—
245#125—794no—
246#125—794no—
247#125—792no—
248#125—794no—
249#125C2×D7914yes—
250#125C2×D7912yes—
251#124—791no—
252#124—792no—
253#124—794no—
254#123—792no—
255#123—794no—
256C22⋊Q8—791no—
257C22⋊Q8—792no—
258C22⋊Q8—791no—
259C22⋊Q8—792no—
260C22⋊Q8—791no—
261C22⋊Q8—792no—
262C22⋊Q8—792no—
263C22⋊Q8—794no—
264C22⋊Q8—792no—
265C22⋊Q8—794no—
266C22⋊Q8—794no—
267#121—791no—
268C4×Q8—792no—
269C4×Q8—792no—
270C4×Q8—792no—
271C4×Q8—792no—
272C4×Q8—792no—
273C4×Q8—791no—
274C4×Q8—792no—
275C4×Q8—794no—
276C4×Q8C2×D7911yescharacteristic
277C4×Q8C2×D7911yescharacteristic
278C4×Q8C2×D7912yes—
279C4×Q8C2×D7912yes—
280C4×D4—792no—
281C4×D4—791no—
282C4×D4—792no—
283#117—792no—
284#117—794no—
285#117—792no—
286#117—792no—
287#117—792no—
288C2×C4⋊C4—792no—
289C2×C42—791no—
290C2×Q8—792no—
291C2×Q8—792no—
292C2×Q8—794no—
293C2×Q8—791no—
294C2×Q8C22×D7912yes—
295C2×Q8C22×D7911yescharacteristic
296C2×Q8C22×D7911yescharacteristic
297C2×D4—792no—
298C2×D4—791no—
299C22×C4—791no—
300C22×C4—794no—
301C22×C4—791no—
302C22×C4—791no—
303C22×C4—792no—
304C4⋊C4—794no—
305C4⋊C4—792no—
306C4⋊C4—792no—
307C4⋊C4—792no—
308C4⋊C4—792no—
309C4⋊C4—792no—
310C4⋊C4—794no—
311C4⋊C4—794no—
312C4⋊C4—792no—
313C4⋊C4—792no—
314C4⋊C4—794no—
315C4⋊C4—792no—
316C4⋊C4—792no—
317C4⋊C4—791no—
318C4⋊C4—791no—
319C4⋊C4—792no—
320C4⋊C4—792no—
321C4⋊C4—794no—
322C4⋊C4—794no—
323C4⋊C4—794no—
324C4⋊C4C22×D7911yescharacteristic
325C4⋊C4C22×D7912yes—
326C4⋊C4C22×D7914yes—
327C4⋊C4C22×D7912yes—
328C4⋊C4C22×D7911yescharacteristic
329C4⋊C4C22×D7912yes—
330C4⋊C4C22×D7914yes—
331C4⋊C4C22×D7912yes—
332C4⋊C4C22×D7914yes—
333C22⋊C4—792no—
334C22⋊C4—792no—
335C22⋊C4—792no—
336C22⋊C4—792no—
337C22⋊C4—794no—
338C22⋊C4—794no—
339C22⋊C4—792no—
340C22⋊C4—794no—
341C22⋊C4—792no—
342C22⋊C4—794no—
343C22⋊C4—792no—
344C42—791no—
345C42—791no—
346C42—791no—
347C42—794no—
348C42—792no—
349C42—794no—
350C42—794no—
351C42C22×D7911yescharacteristic
352C42C22×D7912yes—
353C42C22×D7912yes—
354C42C22×D7914yes—
355C23—791no—
356C23—792no—
357Q8—24no—
358Q8—22no—
359Q8—22no—
360Q8—3162no—
361Q8—3162no—
362Q8—3164no—
363Q8—1582no—
364D4—3162no—
365D4—1582no—
366C2×C4—3164no—
367C2×C4—3161no—
368C2×C4—791no—
369C2×C4—792no—
370C2×C4—792no—
371C2×C4—791no—
372C2×C4—792no—
373C2×C4—791no—
374C2×C4—794no—
375C2×C4—1582no—
376C2×C4—1582no—
377C2×C4—1584no—
378C2×C4C23×D7914yes—
379C2×C4C23×D7912yes—
380C2×C4C23×D7911yescharacteristic
381C2×C4C23×D7912yes—
382C2×C4C23×D7912yes—
383C2×C4C23×D7912yes—
384C2×C4C23×D7911yescharacteristic
385C2×C4C23×D7911yescharacteristic
386C22—1581no—
387C22—1582no—
388C22—1581no—
389C22—1582no—
390C22—1582no—
391C22—1581no—
392C22C24×D7911yescharacteristic
393C4—24no—
394C4—22no—
395C4—22no—
396C4—22no—
397C4—21no—
398C4—22no—
399C4—1581no—
400C4—1581no—
401C4—1582no—
402C4—1582no—
403C4—1584no—
404C4—1581no—
405C4—1582no—
406C4#1457920812yes—
407C4#1457920912yes—
408C2—3161no—
409C2—3162no—
410C2C2×#1457920811yescharacteristic
411C2C2×#1457920911yescharacteristic
412C2#1909940011yescharacteristic
subgroupG/H
center, FrattiniC22C24×D79
commutator, socle, cosocleC22×C79C25
Fitting#557×C79C2
prime radical#14579136C22
hypercenter#557D79
hypocenterC79#3555
LCSG ▷ C22×C79 ▷ C79
UCSC1 ◁ C22 ◁ #557

Complements

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

NH ≅ G/Ntype
1. #19099317C2semidirect
2. Q8×Dic158C2semidirect
3. #19099289C2semidirect
4. #19099287C2semidirect
5. #19099287C2semidirect
6. #19099286C2semidirect
7. #19099273C2semidirect
8. #19099251C2semidirect
9. #19099193C2semidirect
11. #19099174C2semidirect
12. #19099172C2semidirect
13. #19099171C2semidirect
14. #19099118C2semidirect
15. #19099117C2semidirect
17. #557×C79C2semidirect
207. C79#3555semidirect
211. #557D79semidirect