Finite Group Databasebeta

← #72277028

Subgroups — #72277028

Order 2688.

1240 conjugacy classes of subgroups; only the 169 normal subgroups are shown.

normal subgroups only

This group has 1240 conjugacy classes of subgroups — too many to draw. Only its 169 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6548⋊C3C211yesmaximal, characteristic
2C2×ΓL(1,𝔽8)×#100C211yesmaximal, characteristic, complemented
3#89993⋊C3C211yesmaximal, characteristic, complemented
4C2×#6399⋊C3C212yesmaximal, complemented
5#89905⋊C3C212yesmaximal, complemented
6#11727840C311yesmaximal, characteristic, complemented, Hall
7C22×D4×ΓL(1,𝔽8)C2211yescharacteristic
8C22×(C22⋊D7)⋊C3—22no—
9#6548⋊C3—22no—
10#6548⋊C3—24no—
11#6548⋊C3C414yescomplemented
12C2×#1159⋊C3—22no—
13C2×D4×AGL(1,𝔽7)—22no—
14C2×#1157⋊C3C2212yes—
15C2×M16×ΓL(1,𝔽8)C2212yescomplemented
16C2×(C22⋊Dic7)⋊C3—22no—
17C2×C4⋊(ΓL(1,𝔽8)⋊C4)C2211yescharacteristic, complemented
18C2×C4×ΓL(1,𝔽8)⋊C4C2211yescharacteristic, complemented
19ΓL(1,𝔽8)×#100—22no—
20#6399⋊C3—24no—
21#6395⋊C3—22no—
22#6390⋊C3—22no—
23C2×#6548C2×C311yescharacteristic
24C2×C7×#100C2×C311yescharacteristic, complemented
25#89993C2×C311yescharacteristic, complemented
26C2×#6399C2×C312yescomplemented
27#89905C2×C312yescomplemented
28C3×#1792—71nomaximal, Hall
29C24×ΓL(1,𝔽8)D412yes—
30C23×AGL(1,𝔽7)—22no—
31C2×D4×ΓL(1,𝔽8)—22no—
32C2×D4×ΓL(1,𝔽8)—22no—
33C2×D4×ΓL(1,𝔽8)C2×C412yes—
34C2×D4×ΓL(1,𝔽8)D412yes—
35C22×C4×ΓL(1,𝔽8)C2311yescharacteristic
36C2×(C22⋊D7)⋊C3—24no—
37C2×(C22⋊D7)⋊C3—28no—
38C2×(C22⋊D7)⋊C3—28no—
39C2×(C22⋊D7)⋊C3—22no—
40C22×ΓL(1,𝔽8)⋊C4—22no—
41C22×ΓL(1,𝔽8)⋊C4—21no—
42C22×ΓL(1,𝔽8)⋊C4—21no—
43#1159⋊C3—28no—
44#1159⋊C3—28no—
45D4×AGL(1,𝔽7)—28no—
46D4×AGL(1,𝔽7)—28no—
47#1157⋊C3—22no—
48#1157⋊C3—22no—
49#1157⋊C3C2×C414yes—
50#1157⋊C3D414yes—
51C2×C4⋊AGL(1,𝔽7)D412yes—
52C2×C4×AGL(1,𝔽7)—22no—
53C2×ΓL(1,𝔽8)⋊Q8D412yes—
54M16×ΓL(1,𝔽8)—22no—
55M16×ΓL(1,𝔽8)—22no—
56C2×C8×ΓL(1,𝔽8)—22no—
57(C22⋊Dic7)⋊C3—44no—
58C4⋊(ΓL(1,𝔽8)⋊C4)—22no—
59C4×ΓL(1,𝔽8)⋊C4—22no—
60C22×C7×D4C22×C311yescharacteristic
61C22×C22⋊D7—22no—
62#6548—22no—
63#6548—24no—
64#6548C3×C414yescomplemented
65C2×#1159—22no—
66C2×D4×D7—22no—
67C2×#1157C22×C312yes—
68C2×C7×M16C22×C312yescomplemented
69C2×C22⋊Dic7—22no—
70C2×C4⋊Dic7C22×C311yescharacteristic, complemented
71C2×C4×Dic7C22×C311yescharacteristic, complemented
72C7×#100—22no—
73#6399—24no—
74#6395—22no—
75#6390—22no—
76C2×C3×H(5,𝔽2)—71no—
77C2×C3×#104—72no—
78C3×#417—72no—
79C2×C3×#100—71no—
80C3×#390—71no—
81C23×ΓL(1,𝔽8)—22no—
82C23×ΓL(1,𝔽8)—22no—
83C23×ΓL(1,𝔽8)—42no—
84C23×ΓL(1,𝔽8)—42no—
85C23×ΓL(1,𝔽8)C22⋊C414yes—
86C23×ΓL(1,𝔽8)C2×D411yescharacteristic
87C22×AGL(1,𝔽7)—22no—
88C22×AGL(1,𝔽7)—28no—
89C22×AGL(1,𝔽7)—82no—
90C22×AGL(1,𝔽7)—82no—
91D4×ΓL(1,𝔽8)—24no—
92D4×ΓL(1,𝔽8)—24no—
93C2×C4×ΓL(1,𝔽8)C22×C411yescharacteristic
94C2×C4×ΓL(1,𝔽8)C2×D412yes—
95C2×C4×ΓL(1,𝔽8)C2×D412yes—
96C2×C4×ΓL(1,𝔽8)C2×D411yescharacteristic
97(C22⋊D7)⋊C3—216no—
98(C22⋊D7)⋊C3—24no—
99(C22⋊D7)⋊C3—24no—
100C2×ΓL(1,𝔽8)⋊C4—28no—
101C2×ΓL(1,𝔽8)⋊C4—22no—
102C2×ΓL(1,𝔽8)⋊C4—22no—
103C2×ΓL(1,𝔽8)⋊C4—41no—
104C2×ΓL(1,𝔽8)⋊C4—41no—
105C2×ΓL(1,𝔽8)⋊C4—42no—
106C2×ΓL(1,𝔽8)⋊C4—41no—
107C4⋊AGL(1,𝔽7)—22no—
108C4⋊AGL(1,𝔽7)C22⋊C414yescomplemented
109C4×AGL(1,𝔽7)—24no—
110C4×AGL(1,𝔽7)—24no—
111ΓL(1,𝔽8)⋊Q8—22no—
112ΓL(1,𝔽8)⋊Q8C22⋊C414yescomplemented
113C8×ΓL(1,𝔽8)—42no—
114#1792—71noSylow
115C24×C7C3×D412yes—
116C23×D7—22no—
117C2×C7×D4—22no—
118C2×C7×D4—22no—
119C2×C7×D4C2×C3×C412yes—
120C2×C7×D4C3×D412yes—
121C22×C4×C7C23×C311yescharacteristic
122C2×C22⋊D7—24no—
123C2×C22⋊D7—28no—
124C2×C22⋊D7—28no—
125C2×C22⋊D7—22no—
126C22×Dic7—22no—
127C22×Dic7—21no—
128C22×Dic7—21no—
129#1159—28no—
130#1159—28no—
131D4×D7—28no—
132D4×D7—28no—
133#1157—22no—
134#1157—22no—
135#1157C2×C3×C414yes—
136#1157C3×D414yes—
137C2×D28C3×D412yes—
138C2×C4×D7—22no—
139C2×Dic14C3×D412yes—
140C7×M16—22no—
141C7×M16—22no—
142C2×C7×C8—22no—
143C22⋊Dic7—44no—
144C4⋊Dic7—22no—
145C4×Dic7—22no—
146C3×H(5,𝔽2)—74no—
147C3×H(5,𝔽2)—142no—
148C3×H(5,𝔽2)—144no—
149C2×C3×#41—72no—
150C2×C3×#41—142no—
151C22×C3×D4—71no—
152C22×C3×D4—142no—
153C22×C3×D4—142no—
154C2×C3×M16—72no—
155C2×C3×C4⋊C4—71no—
156C2×C3×C22⋊C4—142no—
157C2×C3×C42—71no—
158C3×#104—144no—
159C3×#103—142no—
160C3×#102—142no—
161C3×#100—142no—
162C22×ΓL(1,𝔽8)—21no—
163C22×ΓL(1,𝔽8)—21no—
164C22×ΓL(1,𝔽8)—24no—
165C22×ΓL(1,𝔽8)—44no—
166C22×ΓL(1,𝔽8)—42no—
167C22×ΓL(1,𝔽8)—42no—
168C22×ΓL(1,𝔽8)—82no—
169C22×ΓL(1,𝔽8)C2×C22⋊C411yescharacteristic
170C22×ΓL(1,𝔽8)#12011yescharacteristic
171C22×ΓL(1,𝔽8)#12011yescharacteristic
172C2×AGL(1,𝔽7)—24no—
173C2×AGL(1,𝔽7)—42no—
174C2×AGL(1,𝔽7)—42no—
175C2×AGL(1,𝔽7)—88no—
176C4×ΓL(1,𝔽8)C2×C22⋊C412yes—
177C4×ΓL(1,𝔽8)#12012yes—
178ΓL(1,𝔽8)⋊C4—24no—
179ΓL(1,𝔽8)⋊C4—42no—
180ΓL(1,𝔽8)⋊C4—81no—
181C2×H(5,𝔽2)—71no—
182C2×#104—72no—
183#417—72no—
184C2×#100AGL(1,𝔽7)11yescharacteristic, complemented
185#390—71no—
186C23×C7—22no—
187C23×C7—22no—
188C23×C7—42no—
189C23×C7—42no—
190C23×C7C3×C22⋊C414yes—
191C23×C7C2×C3×D411yescharacteristic
192C22×D7—22no—
193C22×D7—28no—
194C22×D7—82no—
195C22×D7—82no—
196C7×D4—24no—
197C7×D4—24no—
198C2×C4×C7C22×C3×C411yescharacteristic
199C2×C4×C7C2×C3×D412yes—
200C2×C4×C7C2×C3×D412yes—
201C2×C4×C7C2×C3×D411yescharacteristic
202C22⋊D7—216no—
203C22⋊D7—24no—
204C22⋊D7—24no—
205C2×Dic7—28no—
206C2×Dic7—22no—
207C2×Dic7—22no—
208C2×Dic7—41no—
209C2×Dic7—41no—
210C2×Dic7—42no—
211C2×Dic7—41no—
212D28—22no—
213D28C3×C22⋊C414yescomplemented
214C4×D7—24no—
215C4×D7—24no—
216Dic14—22no—
217Dic14C3×C22⋊C414yescomplemented
218C7×C8—42no—
219C24×C3—72no—
220C24×C3—142no—
221C3×#41—74no—
222C3×#41—74no—
223C3×#41—142no—
224C3×#41—142no—
225C3×#41—148no—
226C3×#41—148no—
227C2×C3×Q8—72no—
228C2×C3×D4—72no—
229C2×C3×D4—72no—
230C2×C3×D4—72no—
231C2×C3×D4—142no—
232C2×C3×D4—142no—
233C2×C3×D4—148no—
234C2×C3×D4—142no—
235C2×C3×D4—148no—
236C2×C3×D4—148no—
237C2×C3×D4—144no—
238C2×C3×D4—148no—
239C22×C3×C4—71no—
240C22×C3×C4—142no—
241C22×C3×C4—142no—
242C22×C3×C4—141no—
243C22×C3×C4—141no—
244C3×M16—142no—
245C3×M16—142no—
246C2×C3×C8—142no—
247C3×C4⋊C4—142no—
248C3×C22⋊C4—284no—
249C3×C42—142no—
250C2×ΓL(1,𝔽8)—21no—
251C2×ΓL(1,𝔽8)—21no—
252C2×ΓL(1,𝔽8)—44no—
253C2×ΓL(1,𝔽8)#37911yescharacteristic
254C2×ΓL(1,𝔽8)#45311yescharacteristic
255C2×ΓL(1,𝔽8)#45511yescharacteristic
256AGL(1,𝔽7)—44no—
257H(5,𝔽2)—74no—
258H(5,𝔽2)—144no—
259H(5,𝔽2)—142no—
260C2×#41—72no—
261C2×#41—142no—
262C22×D4—142no—
263C22×D4—142no—
264C22×D4C2×AGL(1,𝔽7)11yescharacteristic
265C2×M16C2×AGL(1,𝔽7)12yescomplemented
266C2×C4⋊C4—71no—
267C2×C22⋊C4—142no—
268C2×C42—71no—
269#104—144no—
270#103—142no—
271#102—142no—
272#100—22no—
273C22×C7—21no—
274C22×C7—21no—
275C22×C7—24no—
276C22×C7—44no—
277C22×C7—42no—
278C22×C7—42no—
279C22×C7—82no—
280C22×C7C2×C3×C22⋊C411yescharacteristic
281C22×C7C3×#12011yescharacteristic
282C22×C7C3×#12011yescharacteristic
283C2×D7—24no—
284C2×D7—42no—
285C2×D7—42no—
286C2×D7—88no—
287C4×C7C2×C3×C22⋊C412yes—
288C4×C7C3×#12012yes—
289Dic7—24no—
290Dic7—42no—
291Dic7—81no—
292C23×C3—71no—
293C23×C3—74no—
294C23×C3—142no—
295C23×C3—142no—
296C23×C3—142no—
297C23×C3—148no—
298C23×C3—282no—
299C23×C3—282no—
300C23×C3—562no—
301C23×C3—562no—
302C3×Q8—74no—
303C3×Q8—142no—
304C3×D4—74no—
305C3×D4—144no—
306C3×D4—144no—
307C3×D4—142no—
308C3×D4—144no—
309C3×D4—144no—
310C3×D4—1416no—
311C2×C3×C4—71no—
312C2×C3×C4—72no—
313C2×C3×C4—72no—
314C2×C3×C4—71no—
315C2×C3×C4—144no—
316C2×C3×C4—144no—
317C2×C3×C4—142no—
318C2×C3×C4—148no—
319C2×C3×C4—142no—
320C2×C3×C4—281no—
321C2×C3×C4—281no—
322C2×C3×C4—281no—
323C2×C3×C4—282no—
324C3×C8—282no—
325ΓL(1,𝔽8)#179211yescharacteristic, complemented, Hall
326C24—142no—
327C24C4⋊AGL(1,𝔽7)12yes—
328#41—74no—
329#41—74no—
330#41—148no—
331#41—148no—
332#41—142no—
333#41—142no—
334C2×Q8—72no—
335C2×D4—22no—
336C2×D4—22no—
337C2×D4—72no—
338C2×D4—148no—
339C2×D4—148no—
340C2×D4—144no—
341C2×D4—148no—
342C2×D4—142no—
343C2×D4—148no—
344C2×D4C4×AGL(1,𝔽7)12yes—
345C2×D4(C22⋊D7)⋊C312yes—
346C22×C4—142no—
347C22×C4—142no—
348C22×C4—141no—
349C22×C4—141no—
350C22×C4C22×AGL(1,𝔽7)11yescharacteristic
351M16—22no—
352M16—22no—
353C2×C8—22no—
354C4⋊C4—142no—
355C22⋊C4—284no—
356C42—142no—
357C2×C7—21no—
358C2×C7—21no—
359C2×C7—44no—
360C2×C7C3×#37911yescharacteristic
361C2×C7C3×#45311yescharacteristic
362C2×C7C3×#45511yescharacteristic
363D7—44no—
364C22×C3—71no—
365C22×C3—71no—
366C22×C3—71no—
367C22×C3—141no—
368C22×C3—141no—
369C22×C3—144no—
370C22×C3—144no—
371C22×C3—282no—
372C22×C3—282no—
373C22×C3—284no—
374C22×C3—282no—
375C22×C3—282no—
376C22×C3—562no—
377C22×C3—568no—
378C3×C4—72no—
379C3×C4—72no—
380C3×C4—144no—
381C3×C4—282no—
382C3×C4—561no—
383C23—22no—
384C23—22no—
385C23—42no—
386C23—42no—
387C23—148no—
388C23—142no—
389C23—562no—
390C23—562no—
391C23(C2×AGL(1,𝔽7))⋊C414yes—
392C23C2×C4⋊AGL(1,𝔽7)11yescharacteristic
393Q8—74no—
394Q8—142no—
395D4—24no—
396D4—24no—
397D4—74no—
398D4—1416no—
399D4—144no—
400D4—144no—
401D4—142no—
402C2×C4—148no—
403C2×C4—142no—
404C2×C4—144no—
405C2×C4—144no—
406C2×C4—142no—
407C2×C4—281no—
408C2×C4—281no—
409C2×C4—282no—
410C2×C4—281no—
411C2×C4C2×C4×AGL(1,𝔽7)11yescharacteristic
412C2×C4D4×AGL(1,𝔽7)12yes—
413C2×C4D4×AGL(1,𝔽7)12yes—
414C2×C4C2×(C22⋊D7)⋊C311yescharacteristic
415C8—42no—
416C7C3×#179211yescharacteristic, complemented, Sylow
417C2×C3—71no—
418C2×C3—71no—
419C2×C3—71no—
420C2×C3—141no—
421C2×C3—141no—
422C2×C3—284no—
423C2×C3—284no—
424C22—21no—
425C22—21no—
426C22—24no—
427C22—44no—
428C22—42no—
429C22—42no—
430C22—82no—
431C22—144no—
432C22—282no—
433C22—282no—
434C22—568no—
435C22#6445⋊C311yescharacteristic
436C22#6445⋊C311yescharacteristic
437C22C2×(C2×AGL(1,𝔽7))⋊C411yescharacteristic
438C4—144no—
439C4—282no—
440C4—561no—
441C4C22⋊C4×AGL(1,𝔽7)12yes—
442C4#6500⋊C312yes—
443C3—71noSylow
444C2—21no—
445C2—21no—
446C2—44no—
447C2—284no—
448C2#89518⋊C311yescharacteristic
449C2#89519⋊C311yescharacteristic
450C2#89727⋊C311yescharacteristic
subgroupG/H
centerC22#6445⋊C3
commutatorC2×C4×C7C22×C3×C4
FrattiniC22×C4C22×AGL(1,𝔽7)
FittingC2×C7×#100C2×C3
socleC22×C7C3×#120
prime radicalC2×#6548⋊C3C2
cosocleC22×C4×C7C23×C3
hypercenterC2×#100AGL(1,𝔽7)
hypocenterC7C3×#1792
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#100

Complements

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

NH ≅ G/Ntype
2. C2×ΓL(1,𝔽8)×#100C2semidirect
3. #89993⋊C3C2semidirect
4. C2×#6399⋊C3C2semidirect
5. #89905⋊C3C2semidirect
6. #11727840C3semidirect
11. #6548⋊C3C4semidirect
15. C2×M16×ΓL(1,𝔽8)C22semidirect
17. C2×C4⋊(ΓL(1,𝔽8)⋊C4)C22semidirect
18. C2×C4×ΓL(1,𝔽8)⋊C4C22semidirect
24. C2×C7×#100C2×C3semidirect
25. #89993C2×C3semidirect
26. C2×#6399C2×C3semidirect
27. #89905C2×C3semidirect
64. #6548C3×C4semidirect
68. C2×C7×M16C22×C3semidirect
70. C2×C4⋊Dic7C22×C3semidirect
71. C2×C4×Dic7C22×C3semidirect
108. C4⋊AGL(1,𝔽7)C22⋊C4semidirect
112. ΓL(1,𝔽8)⋊Q8C22⋊C4semidirect
184. C2×#100AGL(1,𝔽7)semidirect
213. D28C3×C22⋊C4semidirect
217. Dic14C3×C22⋊C4semidirect
265. C2×M16C2×AGL(1,𝔽7)semidirect
325. ΓL(1,𝔽8)#1792semidirect
416. C7C3×#1792semidirect

Maximal subgroups

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

subgroupindex
C2×#6548⋊C32
C2×ΓL(1,𝔽8)×#1002
#89993⋊C32
C2×#6399⋊C32
#89905⋊C32
#117278403
C3×#17927