Finite Group Databasebeta

← #13508597

Subgroups — #13508597

Order 1792.

590 conjugacy classes of subgroups.

The subgroup lattice has 424 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11741187C211yesmaximal, characteristic
2C2×#89878C211yesmaximal, characteristic, complemented
3C2×C7×C22⋊C16C211yesmaximal, characteristic, complemented
4#11729166C212yesmaximal, complemented
5#11729158C212yesmaximal, complemented
6#11726854C212yesmaximal, complemented
7#11726853C211yesmaximal, characteristic, complemented
8#11726852C211yesmaximal, characteristic, complemented
9#11726851C212yesmaximal, complemented
10#11726850C211yesmaximal, characteristic, complemented
11#11726849C211yesmaximal, characteristic, complemented
12C23×C7×C8C2211yescharacteristic
13#90482—21no—
14C2×#6489—21no—
15C2×#6467C2211yescharacteristic
16C22×C7×C16C2212yes—
17#89888C2211yescharacteristic
18#89888C2211yescharacteristic
19#89887C2211yescharacteristic
20#89887C2211yescharacteristic
21#89886—21no—
22#89886—21no—
23#89885—21no—
24#89878—21no—
25#89878—21no—
26#89878C2211yescharacteristic
27#89878C2212yes—
28#89878C2212yes—
29C2×#6392C2211yescharacteristic
30C2×#6392C2211yescharacteristic
31C2×#6391—21no—
32C7×C22⋊C16C2211yescharacteristic
33C7×C22⋊C16C2211yescharacteristic
34C7×C22⋊C16C2212yes—
35#89304C2212yescomplemented
36#89304C2212yescomplemented
37#89300C2212yescomplemented
38#89300C2212yescomplemented
39#89299C2212yescomplemented
40#89299C2212yescomplemented
41#89298C2212yescomplemented
42#89298C2212yescomplemented
43#33645—71nomaximal, Sylow
44C23×C4×C7D411yescharacteristic
45C2×#1157—21no—
46C22×C7×C8—24no—
47C22×C7×C8D411yescharacteristic
48C22×C7×C8D411yescharacteristic
49C22×C7×C8D411yescharacteristic
50C22×C7×C8C2311yescharacteristic
51C22×C7×C8C2311yescharacteristic
52C22×C7×C8C2311yescharacteristic
53#6516—41no—
54C2×C22⋊Dic7—41no—
55#6493—21no—
56#6493—21no—
57#6492—41no—
58C4×C22⋊D7—41no—
59C4×C22⋊D7—41no—
60#6489—22no—
61#6489—22no—
62#6489—21no—
63#6489—41no—
64#6489—41no—
65C2×C4⋊Dic7—21no—
66C2×C4⋊Dic7—21no—
67#6487—21no—
68#6487—21no—
69C2×C4×Dic7—41no—
70C2×Dic28C2311yescharacteristic
71#6467—24no—
72C2×D56C2311yescharacteristic
73C2×#1132—21no—
74C2×C7×C16—24no—
75C2×C7×C16C2312yes—
76C2×C7×C16C2312yes—
77#6395—21no—
78#6395—21no—
79#6392—21no—
80#6392—21no—
81#6392C2312yes—
82#6392C2312yes—
83#6392C2311yescharacteristic
84#6392C2311yescharacteristic
85#6391—22no—
86#6391—22no—
87#6391—21no—
88#6390—21no—
89#6390—21no—
90#3050—71no—
91C2×#427—71no—
92#2237—72no—
93#2236—71no—
94#2235—71no—
95#2234—72no—
96#2233—71no—
97#2232—71no—
98#2158—72no—
99#2138—72no—
100C2×C22⋊C16D711yescharacteristic, complemented
101C24×C7D811yescharacteristic
102C22×C4×C7—22no—
103C22×C4×C7—22no—
104C22×C4×C7D811yescharacteristic
105C22×C4×C7D811yescharacteristic
106C22×C4×C7D811yescharacteristic
107C22×C4×C7C2×D411yescharacteristic
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7C2×D411yescharacteristic
110C2×C22⋊D7—41no—
111C2×C22⋊D7—41no—
112C2×C22⋊D7—41no—
113C22×Dic7—41no—
114C22×Dic7—41no—
115#1157—42no—
116#1157—42no—
117C2×D28—21no—
118C2×C4×D7—41no—
119C2×Dic14—21no—
120C2×C7×C8—22no—
121C2×C7×C8—22no—
122C2×C7×C8—22no—
123C2×C7×C8—22no—
124C2×C7×C8—22no—
125C2×C7×C8C2×D411yescharacteristic
126C2×C7×C8C2×D411yescharacteristic
127C2×C7×C8C2×D411yescharacteristic
128C2×C7×C8#4114yes—
129C2×C7×C8C2411yescharacteristic
130C22⋊Dic7—41no—
131C22⋊Dic7—41no—
132C22⋊Dic7—42no—
133C22⋊Dic7—42no—
134C22⋊Dic7—41no—
135(C2×D7)⋊C4—41no—
136(C2×D7)⋊C4—41no—
137C4⋊Dic7—21no—
138C4⋊Dic7—21no—
139C4⋊Dic7—22no—
140C4⋊Dic7—22no—
141C4⋊Dic7—41no—
142C4⋊Dic7—41no—
143Dic7⋊C4—41no—
144Dic7⋊C4—41no—
145C4×Dic7—42no—
146C4×Dic7—42no—
147C4×Dic7—41no—
148Dic28—22no—
149D56—22no—
150#1132—42no—
151C7×C16—22no—
152C7×C16—22no—
153C2×#135—71no—
154C23×C8C2×D711yescharacteristic
155#525—141no—
156C2×#117—141no—
157C22×C16C2×D712yes—
158#467—71no—
159#467—71no—
160#466—71no—
161#466—71no—
162#465—141no—
163#465—141no—
164#427—72no—
165#427—72no—
166#427—71no—
167#427—141no—
168#427—141no—
169C2×#107—71no—
170C2×#107—71no—
171C2×#106—141no—
172#416—141no—
173#367—72no—
174#367—72no—
175#366—72no—
176#366—72no—
177#358—72no—
178#358—72no—
179#357—72no—
180#357—72no—
181C22⋊C16C2×D711yescharacteristic
182C22⋊C16C2×D711yescharacteristic
183C22⋊C16C2×D712yes—
184C23×C7—22no—
185C23×C7—22no—
186C23×C7—41no—
187C23×C7C2×D811yescharacteristic
188C23×C7C2×D811yescharacteristic
189C23×C7C2×D811yescharacteristic
190C22×D7—41no—
191C2×C4×C7—22no—
192C2×C4×C7—21no—
193C2×C4×C7—21no—
194C2×C4×C7—21no—
195C2×C4×C7—21no—
196C2×C4×C7—22no—
197C2×C4×C7—22no—
198C2×C4×C7#12312yes—
199C2×C4×C7C2×D811yescharacteristic
200C2×C4×C7C2×D811yescharacteristic
201C2×C4×C7C2×D811yescharacteristic
202C2×C4×C7#13712yes—
203C2×C4×C7C22×D411yescharacteristic
204C22⋊D7—82no—
205C22⋊D7—82no—
206C22⋊D7—82no—
207C2×Dic7—42no—
208C2×Dic7—42no—
209C2×Dic7—41no—
210C2×Dic7—41no—
211C2×Dic7—41no—
212C2×Dic7—82no—
213C2×Dic7—82no—
214D28—41no—
215D28—41no—
216C4×D7—82no—
217Dic14—41no—
218Dic14—41no—
219C7×C8—21no—
220C7×C8—21no—
221C7×C8#12312yes—
222C7×C8C2×#4112yes—
223C2×#41—141no—
224C23×C4D2811yescharacteristic
225#135—144no—
226C2×Q16—71no—
227C2×SD16—141no—
228C2×D8—71no—
229C22×C8—24no—
230C22×C8D2811yescharacteristic
231C22×C8D2811yescharacteristic
232C22×C8D2811yescharacteristic
233C22×C8C22×D711yescharacteristic
234C22×C8C22×D711yescharacteristic
235C22×C8C22×D711yescharacteristic
236#123—281no—
237C22⋊Q8—141no—
238C22⋊Q8—141no—
239#121—141no—
240#121—141no—
241#120—281no—
242C4×D4—281no—
243C4×D4—281no—
244#117—142no—
245#117—142no—
246#117—141no—
247#117—281no—
248#117—281no—
249C2×C4⋊C4—141no—
250C2×C4⋊C4—141no—
251C2×C22⋊C4—281no—
252C2×C42—281no—
253C2×C16—24no—
254C2×C16C22×D712yes—
255C2×C16C22×D712yes—
256#107—72no—
257#107—72no—
258#107—71no—
259#107—71no—
260#107—141no—
261#107—141no—
262#106—142no—
263#106—142no—
264#106—141no—
265#103—141no—
266#103—141no—
267#102—141no—
268#102—141no—
269C22×C7—21no—
270C22×C7—21no—
271C22×C7—21no—
272C22×C7—21no—
273C22×C7—21no—
274C22×C7—21no—
275C22×C7—21no—
276C22×C7—44no—
277C22×C7#48012yes—
278C22×C7#50812yes—
279C22×C7C22×D811yescharacteristic
280C2×D7—81no—
281C2×D7—81no—
282C2×D7—81no—
283C4×C7—21no—
284C4×C7—21no—
285C4×C7#48012yes—
286C4×C7C2×#12311yescharacteristic
287C4×C7C2×#13711yescharacteristic
288Dic7—81no—
289Dic7—82no—
290Dic7—81no—
291Dic7—82no—
292Dic7—81no—
293C24D5611yescharacteristic
294#41—282no—
295#41—282no—
296C2×Q8—141no—
297C2×D4—141no—
298C2×D4—281no—
299C2×D4—281no—
300C2×D4—281no—
301C22×C4—22no—
302C22×C4—22no—
303C22×C4—281no—
304C22×C4—281no—
305C22×C4—281no—
306C22×C4D5611yescharacteristic
307C22×C4D5611yescharacteristic
308C22×C4D5611yescharacteristic
309C22×C4C2×D2811yescharacteristic
310C22×C4C2×D2811yescharacteristic
311C22×C4C2×D2811yescharacteristic
312Q16—142no—
313SD16—282no—
314D8—142no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8C2×D2811yescharacteristic
321C2×C8C2×D2811yescharacteristic
322C2×C8C2×D2811yescharacteristic
323C2×C8#115914yes—
324C2×C8C23×D711yescharacteristic
325C4⋊C4—141no—
326C4⋊C4—141no—
327C4⋊C4—142no—
328C4⋊C4—142no—
329C4⋊C4—281no—
330C4⋊C4—281no—
331C4⋊C4—281no—
332C4⋊C4—281no—
333C22⋊C4—282no—
334C22⋊C4—282no—
335C22⋊C4—281no—
336C22⋊C4—281no—
337C22⋊C4—281no—
338C22⋊C4—281no—
339C22⋊C4—281no—
340C42—282no—
341C42—282no—
342C42—281no—
343C16—22no—
344C16—22no—
345C2×C7—22no—
346C2×C7—22no—
347C2×C7—41no—
348C2×C7C2×#48011yescharacteristic
349C2×C7C2×#50811yescharacteristic
350C2×C7#341411yescharacteristic
351D7—161no—
352C23—22no—
353C23—22no—
354C23—41no—
355C23—281no—
356C23C2×D5611yescharacteristic
357C23C2×D5611yescharacteristic
358C23C2×D5611yescharacteristic
359Q8—281no—
360Q8—281no—
361D4—281no—
362D4—281no—
363D4—562no—
364D4—562no—
365D4—562no—
366C2×C4—22no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—22no—
372C2×C4—22no—
373C2×C4—282no—
374C2×C4—282no—
375C2×C4—281no—
376C2×C4—281no—
377C2×C4—281no—
378C2×C4—562no—
379C2×C4—562no—
380C2×C4—562no—
381C2×C4#644912yes—
382C2×C4C2×D5611yescharacteristic
383C2×C4C2×D5611yescharacteristic
384C2×C4C2×D5611yescharacteristic
385C2×C4#647212yes—
386C2×C4C22×D2811yescharacteristic
387C8—21no—
388C8—21no—
389C8#644912yes—
390C8C2×#115912yes—
391C7#3364511yescharacteristic, complemented, Sylow
392C22—21no—
393C22—21no—
394C22—21no—
395C22—21no—
396C22—21no—
397C22—21no—
398C22—21no—
399C22—44no—
400C22—561no—
401C22—561no—
402C22—561no—
403C22#8950812yes—
404C22#8967612yes—
405C22C22×D5611yescharacteristic
406C4—21no—
407C4—21no—
408C4—562no—
409C4—561no—
410C4—562no—
411C4—561no—
412C4—561no—
413C4#8950812yes—
414C4C2×#644911yescharacteristic
415C4C2×#647211yescharacteristic
416C2—22no—
417C2—22no—
418C2—41no—
419C2—1121no—
420C2C2×#8950811yescharacteristic
421C2C2×#8967611yescharacteristic
422C2#1173945411yescharacteristic
subgroupG/H
centerC2×C4C2×D56
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
FittingC2×C7×C22⋊C16C2
socleC22×C7C22×D8
prime radical#11741187C2
hypercenterC2×C22⋊C16D7
hypocenterC7#33645
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ C2×C22⋊C16

Complements

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

NH ≅ G/Ntype
2. C2×#89878C2semidirect
3. C2×C7×C22⋊C16C2semidirect
4. #11729166C2semidirect
5. #11729158C2semidirect
6. #11726854C2semidirect
7. #11726853C2semidirect
8. #11726852C2semidirect
9. #11726851C2semidirect
10. #11726850C2semidirect
11. #11726849C2semidirect
35. #89304C22semidirect
36. #89304C22semidirect
37. #89300C22semidirect
38. #89300C22semidirect
39. #89299C22semidirect
40. #89299C22semidirect
41. #89298C22semidirect
42. #89298C22semidirect
100. C2×C22⋊C16D7semidirect
391. C7#33645semidirect

Maximal subgroups

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

subgroupindex
#117411872
C2×#898782
C2×C7×C22⋊C162
#117291662
#117291582
#117268542
#117268532
#117268522
#117268512
#117268502
#117268492
#336457