Finite Group Databasebeta

← #13338962

Subgroups — #13338962

Order 1792.

708 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#474C211yesmaximal, characteristic, complemented
2#11740658C211yesmaximal, characteristic, complemented
3#11739752C211yesmaximal, characteristic
4#11739746C211yesmaximal, characteristic, complemented
5#11729529C211yesmaximal, characteristic, complemented
6#11728856C214yesmaximal, complemented
7#11728854C214yesmaximal, complemented
8#11728185C211yesmaximal, characteristic, complemented
9#11728175C211yesmaximal, characteristic, complemented
10C2×C7×C4⋊Q8C2211yescharacteristic
11C2×C4×C7×D4C2211yescharacteristic
12C2×#6495—21no—
13#90168C2211yescharacteristic
14C7×#474—22no—
15C7×#474C2214yes—
16C2×C7×#107—21no—
17C2×C7×C4⋊C8C2211yescharacteristic
18C2×C7×#102—21no—
19#89999—22no—
20C2×C22⋊(C7⋊C8)—22no—
21#89857C2212yescomplemented
22#89855—22no—
23#89853C2212yescomplemented
24#89791C2212yes—
25#89788—22no—
26#89787C2212yes—
27D4×C7⋊C8—22no—
28D4×C7⋊C8C2214yes—
29#89738—21no—
30#89697C2211yescharacteristic
31#89697C2211yescharacteristic
32#89696C2211yescharacteristic
33#89342C2214yescomplemented
34#89280C2212yescomplemented
35#89277C2212yescomplemented
36#89275C2214yescomplemented
37#19385—71nomaximal, Sylow
38C22×C7×Q8—21no—
39C22×C7×D4D411yescharacteristic
40C23×C4×C7—21no—
41C2×#1157—21no—
42C22×C7×C8—21no—
43C7×C4⋊Q8—21no—
44C7×C4⋊Q8—21no—
45C7×C4⋊Q8—21no—
46C7×C4⋊Q8C2312yes—
47C4×C7×D4—21no—
48C4×C7×D4—21no—
49C4×C7×D4D412yes—
50C4×C7×D4C2312yes—
51C2×C7×C4⋊C4—21no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4D411yescharacteristic
54C2×C7×C22⋊C4—21no—
55C2×C42×C7C2311yescharacteristic
56C2×#1143—24no—
57C2×#1142—22no—
58#6495—44no—
59C2×#1141—22no—
60#6493—21no—
61#6493—21no—
62C2×#1136—21no—
63C22×C7⋊C8—24no—
64#6438—21no—
65#6437C2311yescharacteristic
66C4×D28—21no—
67C4×D28—21no—
68#6432C2311yescharacteristic
69C7×#107—24no—
70C7×C4⋊C8—21no—
71C7×C4⋊C8C2312yes—
72C7×#102—24no—
73#6409—24no—
74C22⋊(C7⋊C8)—21no—
75C22⋊(C7⋊C8)—24no—
76C22⋊(C7⋊C8)D412yes—
77#6384—22no—
78#6383—22no—
79C4⋊(C7⋊C8)—21no—
80C4⋊(C7⋊C8)D412yes—
81C4⋊(C7⋊C8)C2312yes—
82C4⋊(C7⋊C8)C2312yes—
83C4×C7⋊C8—21no—
84C4×C7⋊C8C2312yes—
85#3206—71no—
86C2×#474D711yescharacteristic, complemented
87#3036—71no—
88#2972—71no—
89#1679—74no—
90#1665—74no—
91#1570—71no—
92#1549—71no—
93#1540—71no—
94C24×C7—21no—
95C2×C7×Q8—22no—
96C2×C7×Q8—41no—
97C2×C7×Q8—82no—
98C2×C7×D4—21no—
99C2×C7×D4—22no—
100C2×C7×D4SD1614yescomplemented
101C2×C7×D4C2×D412yes—
102C22×C4×C7—21no—
103C22×C4×C7—22no—
104C22×C4×C7—22no—
105C22×C4×C7—41no—
106C22×C4×C7—82no—
107C22×C4×C7C2×D411yescharacteristic
108C22×C4×C7C2×D411yescharacteristic
109C22×C4×C7#4111yescharacteristic
110C2×C22⋊D7—41no—
111#1157—42no—
112#1157—42no—
113C2×D28—21no—
114C2×D28—21no—
115C2×D28—21no—
116C2×C4×D7—41no—
117C2×Dic14—21no—
118C2×C7×C8—22no—
119C2×C7×C8—42no—
120C7×C4⋊C4—21no—
121C7×C4⋊C4—22no—
122C7×C4⋊C4—22no—
123C7×C4⋊C4—41no—
124C7×C4⋊C4—41no—
125C7×C4⋊C4C2×D412yes—
126C7×C22⋊C4—22no—
127C7×C22⋊C4—22no—
128C42×C7C2×D411yescharacteristic
129C42×C7C2×D411yescharacteristic
130C42×C7#4111yescharacteristic
131C42×C7C2411yescharacteristic
132#1143—84no—
133#1142—44no—
134#1141—44no—
135(C2×D7)⋊C4—41no—
136(C2×D7)⋊C4—41no—
137C4⋊Dic7—21no—
138C4⋊Dic7—21no—
139#1136—42no—
140C2×C7⋊C8—22no—
141C2×C7⋊C8—22no—
142C2×C7⋊C8—84no—
143C2×C7⋊C8C2×D414yes—
144C2×AGL(1,ℤ/8)—141no—
145#532—71no—
146C2×C4⋊Q8C2×D711yescharacteristic
147C2×C4×D4C2×D711yescharacteristic
148#492—72no—
149#487—72no—
150#474—22no—
151#474C2×D714yes—
152#463—142no—
153#461—72no—
154#461—142no—
155#459—72no—
156#448—142no—
157C8×D4—74no—
158C8×D4—142no—
159#431—71no—
160#431—71no—
161C2×#107—21no—
162#423—71no—
163C2×C4⋊C8C2×D711yescharacteristic
164#417—141no—
165C2×#102—21no—
166C2×C22⋊C8—142no—
167#334—74no—
168#332—74no—
169#332—72no—
170#331—72no—
171C23×C7—22no—
172C23×C7—24no—
173C23×C7—81no—
174C23×C7—81no—
175C23×C7#12111yescharacteristic
176C22×D7—41no—
177C7×Q8—44no—
178C7×Q8—82no—
179C7×D4—22no—
180C7×D4—24no—
181C7×D4C2×SD1614yes—
182C2×C4×C7—22no—
183C2×C4×C7—22no—
184C2×C4×C7—41no—
185C2×C4×C7—44no—
186C2×C4×C7—42no—
187C2×C4×C7—82no—
188C2×C4×C7—81no—
189C2×C4×C7—82no—
190C2×C4×C7#12111yescharacteristic
191C2×C4×C7#12111yescharacteristic
192C2×C4×C7#12111yescharacteristic
193C2×C4×C7C2×SD1612yes—
194C2×C4×C7#13512yes—
195C2×C4×C7C22×D411yescharacteristic
196C2×C4×C7C22×D411yescharacteristic
197C2×C4×C7C2×#4111yescharacteristic
198C22⋊D7—82no—
199C2×Dic7—41no—
200D28—41no—
201D28—41no—
202D28—42no—
203D28—81no—
204C4×D7—82no—
205Dic14—41no—
206Dic14—41no—
207C7×C8—42no—
208C7⋊C8—24no—
209C7⋊C8—42no—
210C7⋊C8—42no—
211C2×#41—141no—
212C22×Q8—21no—
213C22×D4C22⋊D711yescharacteristic
214C23×C4—21no—
215AGL(1,ℤ/8)—284no—
216C2×SD16—144no—
217C2×SD16—142no—
218C2×D8—142no—
219C2×M16—141no—
220C22×C8—21no—
221C22×C8—144no—
222C4⋊Q8—21no—
223C4⋊Q8—21no—
224C4⋊Q8—21no—
225C4⋊Q8—71no—
226C4⋊Q8C22×D712yes—
227#127—71no—
228#124—141no—
229#121—141no—
230#121—141no—
231C4×D4—21no—
232C4×D4—21no—
233C4×D4—141no—
234C4×D4—141no—
235C4×D4C22⋊D712yes—
236C4×D4C22×D712yes—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—21no—
239C2×C4⋊C4C22⋊D711yescharacteristic
240C2×C22⋊C4—21no—
241C2×C42C22×D711yescharacteristic
242#107—24no—
243C4⋊C8—21no—
244C4⋊C8—72no—
245C4⋊C8—72no—
246C4⋊C8—72no—
247C4⋊C8—141no—
248C4⋊C8C22×D712yes—
249#103—144no—
250#103—142no—
251#102—24no—
252#102—142no—
253C22⋊C8—72no—
254C22⋊C8—144no—
255C22⋊C8—141no—
256C4×C8—72no—
257C4×C8—141no—
258C22×C7—21no—
259C22×C7—21no—
260C22×C7—24no—
261C22×C7—42no—
262C22×C7—42no—
263C22×C7—84no—
264C22×C7#46512yes—
265C22×C7C2×#12111yescharacteristic
266C2×D7—81no—
267C2×D7—81no—
268C2×D7—81no—
269C4×C7—21no—
270C4×C7—21no—
271C4×C7—21no—
272C4×C7—42no—
273C4×C7—82no—
274C4×C7#46512yes—
275C4×C7C22×SD1611yescharacteristic
276C4×C7C2×#13511yescharacteristic
277C4×C7#57612yes—
278Dic7—81no—
279C24—21no—
280#41—282no—
281#41—282no—
282C2×Q8—22no—
283C2×Q8—41no—
284C2×Q8—82no—
285C2×Q8—141no—
286C2×D4—21no—
287C2×D4—22no—
288C2×D4—141no—
289C2×D4—141no—
290C2×D4—141no—
291C2×D4—281no—
292C2×D4#114314yescomplemented
293C2×D4C2×C22⋊D712yes—
294C22×C4—21no—
295C22×C4—22no—
296C22×C4—22no—
297C22×C4—41no—
298C22×C4—82no—
299C22×C4—281no—
300C22×C4D4×D711yescharacteristic
301C22×C4#115911yescharacteristic
302C22×C4C2×C22⋊D711yescharacteristic
303SD16—284no—
304SD16—564no—
305D8—284no—
306M16—282no—
307C2×C8—22no—
308C2×C8—42no—
309C2×C8—74no—
310C2×C8—142no—
311C2×C8—142no—
312C2×C8—564no—
313C4⋊C4—21no—
314C4⋊C4—22no—
315C4⋊C4—22no—
316C4⋊C4—41no—
317C4⋊C4—41no—
318C4⋊C4—141no—
319C4⋊C4—141no—
320C4⋊C4C2×C22⋊D712yes—
321C22⋊C4—22no—
322C22⋊C4—22no—
323C22⋊C4—281no—
324C22⋊C4—281no—
325C42D4×D711yescharacteristic
326C42#115911yescharacteristic
327C42C2×C22⋊D711yescharacteristic
328C42C23×D711yescharacteristic
329C2×C7—22no—
330C2×C7—44no—
331C2×C7#304111yescharacteristic
332C2×C7C2×#46511yescharacteristic
333C2×C7#306911yescharacteristic
334D7—161no—
335C23—22no—
336C23—24no—
337C23—81no—
338C23—81no—
339C23—281no—
340C23#650211yescharacteristic
341Q8—44no—
342Q8—82no—
343Q8—281no—
344Q8—281no—
345D4—22no—
346D4—24no—
347D4—282no—
348D4—281no—
349D4—281no—
350D4—561no—
351D4—562no—
352D4C2×#114314yes—
353C2×C4—22no—
354C2×C4—22no—
355C2×C4—41no—
356C2×C4—44no—
357C2×C4—42no—
358C2×C4—82no—
359C2×C4—81no—
360C2×C4—82no—
361C2×C4—281no—
362C2×C4—562no—
363C2×C4#650211yescharacteristic
364C2×C4#650211yescharacteristic
365C2×C4#650211yescharacteristic
366C2×C4C2×#114312yes—
367C2×C4#651312yes—
368C2×C4C2×D4×D711yescharacteristic
369C2×C4C2×#115911yescharacteristic
370C2×C4C22×C22⋊D711yescharacteristic
371C8—42no—
372C8—144no—
373C8—282no—
374C8—282no—
375C7#1938511yescharacteristic, complemented, Sylow
376C22—21no—
377C22—21no—
378C22—24no—
379C22—42no—
380C22—42no—
381C22—84no—
382C22—561no—
383C22—561no—
384C22—561no—
385C22#8982012yes—
386C22C2×#650211yescharacteristic
387C4—21no—
388C4—21no—
389C4—21no—
390C4—42no—
391C4—82no—
392C4—561no—
393C4#8982012yes—
394C4#9044812yes—
395C4C22×#114311yescharacteristic
396C4C2×#651311yescharacteristic
397C2—22no—
398C2—44no—
399C2—1121no—
400C2C2×#8982011yescharacteristic
401C2#1174127111yescharacteristic
402C2#1174142311yescharacteristic
subgroupG/H
centerC22C2×#6502
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×#474C2
socleC22×C7C2×#121
prime radical#11739752C2
hypercenterC2×#474D7
hypocenterC7#19385
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#474

Complements

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

NH ≅ G/Ntype
1. C2×C7×#474C2semidirect
2. #11740658C2semidirect
4. #11739746C2semidirect
5. #11729529C2semidirect
6. #11728856C2semidirect
7. #11728854C2semidirect
8. #11728185C2semidirect
9. #11728175C2semidirect
21. #89857C22semidirect
23. #89853C22semidirect
33. #89342C22semidirect
34. #89280C22semidirect
35. #89277C22semidirect
36. #89275C22semidirect
86. C2×#474D7semidirect
100. C2×C7×D4SD16semidirect
292. C2×D4#1143semidirect
375. C7#19385semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#4742
#117406582
#117397522
#117397462
#117295292
#117288562
#117288542
#117281852
#117281752
#193857