Finite Group Databasebeta

← #13508595

Subgroups — #13508595

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
1C2×C7×#427C211yesmaximal, characteristic, complemented
2#11741187C211yesmaximal, characteristic
3C2×C22⋊(C7⋊C16)C211yesmaximal, characteristic, complemented
4#11728409C212yesmaximal, complemented
5#11728407C212yesmaximal, complemented
6#11728405C212yesmaximal, complemented
7#11728402C212yesmaximal, complemented
8#11728357C211yesmaximal, characteristic, complemented
9#11728356C211yesmaximal, characteristic, complemented
10#11728354C211yesmaximal, characteristic, complemented
11#11728353C211yesmaximal, characteristic, complemented
12C23×C7×C8C2211yescharacteristic
13C2×C7×#117—21no—
14#90482—21no—
15C2×#6467C2211yescharacteristic
16C7×#427—21no—
17C7×#427—21no—
18C7×#427C2212yes—
19C7×#427C2212yes—
20C2×C7×#107C2211yescharacteristic
21C2×C7×#107C2211yescharacteristic
22C2×C7×#106—21no—
23#89888C2211yescharacteristic
24#89888C2211yescharacteristic
25#89887C2211yescharacteristic
26#89887C2211yescharacteristic
27#89886—21no—
28#89886—21no—
29#89885—21no—
30#89878C2211yescharacteristic
31C22×C7⋊C16C2212yes—
32C22⋊(C7⋊C16)C2212yes—
33C22⋊(C7⋊C16)C2211yescharacteristic
34C22⋊(C7⋊C16)C2211yescharacteristic
35#89287C2212yescomplemented
36#89287C2212yescomplemented
37#89286C2212yescomplemented
38#89286C2212yescomplemented
39#89285C2212yescomplemented
40#89285C2212yescomplemented
41#89284C2212yescomplemented
42#89284C2212yescomplemented
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
53C7×#117—22no—
54C7×#117—22no—
55C7×#117—41no—
56C7×#117—41no—
57C2×C7×C4⋊C4—21no—
58C2×C7×C4⋊C4—21no—
59C2×C7×C22⋊C4—41no—
60C2×C42×C7—41no—
61#6516—41no—
62#6493—21no—
63#6493—21no—
64#6492—41no—
65C4×C22⋊D7—41no—
66C4×C22⋊D7—41no—
67#6489—21no—
68#6487—21no—
69#6487—21no—
70C2×Dic28C2311yescharacteristic
71#6467—24no—
72C2×D56C2311yescharacteristic
73C2×#1132—21no—
74C7×#107—21no—
75C7×#107—21no—
76C7×#107C2312yes—
77C7×#107C2312yes—
78C7×#106—22no—
79C7×#106—22no—
80#6395—21no—
81#6395—21no—
82#6392C2311yescharacteristic
83#6392C2311yescharacteristic
84#6391—21no—
85#6390—21no—
86#6390—21no—
87C2×C7⋊C16—24no—
88C2×C7⋊C16C2312yes—
89C2×C7⋊C16C2312yes—
90#3050—71no—
91C2×#427D711yescharacteristic, complemented
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⋊C16—71no—
101C24×C7D811yescharacteristic
102C22×C4×C7—22no—
103C22×C4×C7—22no—
104C22×C4×C7—41no—
105C22×C4×C7—41no—
106C22×C4×C7D811yescharacteristic
107C22×C4×C7D811yescharacteristic
108C22×C4×C7D811yescharacteristic
109C22×C4×C7C2×D411yescharacteristic
110C22×C4×C7C2×D411yescharacteristic
111C22×C4×C7C2×D411yescharacteristic
112C2×C22⋊D7—41no—
113C2×C22⋊D7—41no—
114C2×C22⋊D7—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
130C7×C4⋊C4—22no—
131C7×C4⋊C4—22no—
132C7×C4⋊C4—41no—
133C7×C4⋊C4—41no—
134C7×C22⋊C4—42no—
135C7×C22⋊C4—42no—
136C42×C7—42no—
137C42×C7—42no—
138C22⋊Dic7—41no—
139C22⋊Dic7—41no—
140C22⋊Dic7—41no—
141(C2×D7)⋊C4—41no—
142(C2×D7)⋊C4—41no—
143C4⋊Dic7—21no—
144C4⋊Dic7—21no—
145Dic7⋊C4—41no—
146Dic7⋊C4—41no—
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—21no—
157C22×C16—72no—
158#467—71no—
159#467—71no—
160#466—71no—
161#466—71no—
162#465—141no—
163#465—141no—
164#427—21no—
165#427—21no—
166#427—71no—
167#427C2×D712yes—
168#427C2×D712yes—
169C2×#107C2×D711yescharacteristic
170C2×#107C2×D711yescharacteristic
171C2×#106—21no—
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⋊C16—71no—
182C22⋊C16—71no—
183C22⋊C16—72no—
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—42no—
199C2×C4×C7—42no—
200C2×C4×C7—82no—
201C2×C4×C7—82no—
202C2×C4×C7#12312yes—
203C2×C4×C7C2×D811yescharacteristic
204C2×C4×C7C2×D811yescharacteristic
205C2×C4×C7C2×D811yescharacteristic
206C2×C4×C7#13712yes—
207C2×C4×C7C22×D411yescharacteristic
208C22⋊D7—82no—
209C22⋊D7—82no—
210C22⋊D7—82no—
211C2×Dic7—41no—
212C2×Dic7—41no—
213C2×Dic7—41no—
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×C4C22⋊D711yescharacteristic
225#135—144no—
226C2×Q16—71no—
227C2×SD16—141no—
228C2×D8—71no—
229C22×C8—24no—
230C22×C8C22⋊D711yescharacteristic
231C22×C8C22⋊D711yescharacteristic
232C22×C8C22⋊D711yescharacteristic
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—22no—
245#117—22no—
246#117—41no—
247#117—41no—
248#117—141no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—21no—
251C2×C22⋊C4—41no—
252C2×C42—41no—
253C2×C16—72no—
254C2×C16—72no—
255C2×C16—144no—
256#107—21no—
257#107—21no—
258#107—71no—
259#107—71no—
260#107C22×D712yes—
261#107C22×D712yes—
262#106—22no—
263#106—22no—
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—82no—
286C4×C7—82no—
287C4×C7#48012yes—
288C4×C7C2×#12311yescharacteristic
289C4×C7C2×#13711yescharacteristic
290Dic7—81no—
291Dic7—81no—
292Dic7—81no—
293C24#114111yescharacteristic
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—41no—
304C22×C4—41no—
305C22×C4—281no—
306C22×C4#114111yescharacteristic
307C22×C4#114111yescharacteristic
308C22×C4#114111yescharacteristic
309C22×C4C2×C22⋊D711yescharacteristic
310C22×C4C2×C22⋊D711yescharacteristic
311C22×C4C2×C22⋊D711yescharacteristic
312Q16—142no—
313SD16—282no—
314D8—142no—
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8#115714yes—
321C2×C8C2×C22⋊D711yescharacteristic
322C2×C8C2×C22⋊D711yescharacteristic
323C2×C8C2×C22⋊D711yescharacteristic
324C2×C8C23×D711yescharacteristic
325C4⋊C4—22no—
326C4⋊C4—22no—
327C4⋊C4—41no—
328C4⋊C4—41no—
329C4⋊C4—141no—
330C4⋊C4—141no—
331C4⋊C4—281no—
332C4⋊C4—281no—
333C22⋊C4—42no—
334C22⋊C4—42no—
335C22⋊C4—281no—
336C22⋊C4—281no—
337C22⋊C4—281no—
338C22⋊C4—281no—
339C22⋊C4—281no—
340C42—42no—
341C42—42no—
342C42—281no—
343C16—142no—
344C16—142no—
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×#114111yescharacteristic
357C23C2×#114111yescharacteristic
358C23C2×#114111yescharacteristic
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—42no—
374C2×C4—42no—
375C2×C4—82no—
376C2×C4—82no—
377C2×C4—281no—
378C2×C4—281no—
379C2×C4—281no—
380C2×C4—562no—
381C2×C4#649212yes—
382C2×C4C2×#114111yescharacteristic
383C2×C4C2×#114111yescharacteristic
384C2×C4C2×#114111yescharacteristic
385C2×C4#650512yes—
386C2×C4C22×C22⋊D711yescharacteristic
387C8—21no—
388C8—21no—
389C8#649212yes—
390C8C2×#115712yes—
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#8973912yes—
404C22#8996612yes—
405C22C22×#114111yescharacteristic
406C4—21no—
407C4—21no—
408C4—82no—
409C4—82no—
410C4—561no—
411C4—561no—
412C4—561no—
413C4#8973912yes—
414C4C2×#649211yescharacteristic
415C4C2×#650511yescharacteristic
416C2—22no—
417C2—22no—
418C2—41no—
419C2—1121no—
420C2C2×#8973911yescharacteristic
421C2C2×#8996611yescharacteristic
422C2#1174133911yescharacteristic
subgroupG/H
centerC2×C4C2×#1141
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
FittingC2×C7×#427C2
socleC22×C7C22×D8
prime radical#11741187C2
hypercenterC2×#427D7
hypocenterC7#33645
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C23×C4 ◁ C23×C8 ◁ C2×#427

Complements

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

NH ≅ G/Ntype
1. C2×C7×#427C2semidirect
3. C2×C22⋊(C7⋊C16)C2semidirect
4. #11728409C2semidirect
5. #11728407C2semidirect
6. #11728405C2semidirect
7. #11728402C2semidirect
8. #11728357C2semidirect
9. #11728356C2semidirect
10. #11728354C2semidirect
11. #11728353C2semidirect
35. #89287C22semidirect
36. #89287C22semidirect
37. #89286C22semidirect
38. #89286C22semidirect
39. #89285C22semidirect
40. #89285C22semidirect
41. #89284C22semidirect
42. #89284C22semidirect
91. C2×#427D7semidirect
391. C7#33645semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4272
#117411872
C2×C22⋊(C7⋊C16)2
#117284092
#117284072
#117284052
#117284022
#117283572
#117283562
#117283542
#117283532
#336457