Finite Group Databasebeta

← #13332412

Subgroups — #13332412

Order 1792.

878 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#557C211yesmaximal, characteristic
2#11740668C211yesmaximal, characteristic
3#11739892C211yesmaximal, characteristic
4#11739876C211yesmaximal, characteristic
5#11729692C214yesmaximal, complemented
6#11729690C214yesmaximal, complemented
7#11729527C211yesmaximal, characteristic
8#11729376C211yesmaximal, characteristic
9#11729375C211yesmaximal, characteristic
10C7×#557—22no—
11C7×#557—22no—
12C7×#557—22no—
13C7×#557C2214yes—
14C2×C7×C4⋊Q8—21no—
15C2×C7×C4⋊Q8C2211yescharacteristic
16C2×C7×#125—22no—
17C2×C7×#125—21no—
18C2×C4×C7×Q8—21no—
19C2×C4×C7×Q8—21no—
20C2×C4×C7×Q8C2211yescharacteristic
21C2×C4×C7×Q8C2211yescharacteristic
22#90161C2211yescharacteristic
23#90000—21no—
24#89997—21no—
25#89866—22no—
26#89865C2212yes—
27#89851C2212yes—
28#89803C2212yes—
29C4×#1144—22no—
30#89794—22no—
31#89793C2212yes—
32#89793C2212yes—
33#89792C2212yes—
34#89743—22no—
35#89742—21no—
36#89736—21no—
37#89697C2211yescharacteristic
38#89697C2211yescharacteristic
39#89696C2211yescharacteristic
40#89342C2214yes—
41Q8⋊(C7⋊C8)C2214yes—
42Q8⋊(C7⋊C8)C2214yes—
43#18761—71nomaximal, Sylow
44C22×C7×Q8—21no—
45C22×C7×Q8D411yescharacteristic
46C22×C7×Q8D411yescharacteristic
47C7×C4⋊Q8—24no—
48C7×C4⋊Q8—24no—
49C7×C4⋊Q8—21no—
50C7×C4⋊Q8—21no—
51C7×C4⋊Q8D412yes—
52C7×C4⋊Q8C2312yes—
53C7×#125—28no—
54C7×#125—24no—
55C7×#125—28no—
56C7×#125—24no—
57C4×C7×Q8—24no—
58C4×C7×Q8—24no—
59C4×C7×Q8—24no—
60C4×C7×Q8—24no—
61C4×C7×Q8—21no—
62C4×C7×Q8—21no—
63C4×C7×Q8—21no—
64C4×C7×Q8—21no—
65C4×C7×Q8D412yes—
66C4×C7×Q8D412yes—
67C4×C7×Q8C2312yes—
68C4×C7×Q8C2312yes—
69C2×C7×C4⋊C4—22no—
70C2×C7×C4⋊C4—22no—
71C2×C7×C4⋊C4—22no—
72C2×C7×C4⋊C4—21no—
73C2×C7×C4⋊C4—21no—
74C2×C7×C4⋊C4—21no—
75C2×C7×C4⋊C4D411yescharacteristic
76C2×C7×C4⋊C4D411yescharacteristic
77C2×C7×C4⋊C4D412yes—
78C2×C42×C7—22no—
79C2×C42×C7—21no—
80C2×C42×C7—21no—
81C2×C42×C7C2311yescharacteristic
82C2×#1144—22no—
83#6489—21no—
84#6487—21no—
85#6487—21no—
86C2×#1136—21no—
87#6433—21no—
88#6432C2311yescharacteristic
89#6432C2311yescharacteristic
90C4×Dic14—21no—
91C4×Dic14—21no—
92#6409—22no—
93#6409—22no—
94#6409—22no—
95C22⋊(C7⋊C8)—21no—
96C22⋊(C7⋊C8)—21no—
97#6384—24no—
98#6384—22no—
99#6382—24no—
100#6382—22no—
101#6381—22no—
102#6381—22no—
103C4⋊(C7⋊C8)—21no—
104C4⋊(C7⋊C8)—21no—
105C4⋊(C7⋊C8)C2312yes—
106C4⋊(C7⋊C8)C2312yes—
107C4×C7⋊C8—21no—
108C4×C7⋊C8C2312yes—
109C2×#557D711yescharacteristic
110#3229—71no—
111#3104—71no—
112#3078—71no—
113#1653—74no—
114#1636—74no—
115#1546—71no—
116#1487—71no—
117#1479—71no—
118C2×C7×Q8—24no—
119C2×C7×Q8—21no—
120C2×C7×Q8—21no—
121C2×C7×Q8—22no—
122C2×C7×Q8—22no—
123C2×C7×Q8—22no—
124C2×C7×Q8—22no—
125C2×C7×Q8—42no—
126C2×C7×Q8—42no—
127C2×C7×Q8C2×D412yes—
128C2×C7×Q8C2×D412yes—
129C22×C4×C7—21no—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—21no—
133C22×C4×C7—22no—
134C22×C4×C7C2×D411yescharacteristic
135C22×C4×C7C2×D411yescharacteristic
136C22×C4×C7C2×D411yescharacteristic
137C2×Dic14—21no—
138C2×Dic14—21no—
139C7×C4⋊C4—28no—
140C7×C4⋊C4—28no—
141C7×C4⋊C4—28no—
142C7×C4⋊C4—22no—
143C7×C4⋊C4—22no—
144C7×C4⋊C4—22no—
145C7×C4⋊C4—21no—
146C7×C4⋊C4—21no—
147C7×C4⋊C4—22no—
148C7×C4⋊C4—22no—
149C7×C4⋊C4—22no—
150C7×C4⋊C4—22no—
151C7×C4⋊C4C2×D412yes—
152C7×C4⋊C4C2×D412yes—
153C7×C4⋊C4C2×D414yes—
154C42×C7—28no—
155C42×C7—22no—
156C42×C7—22no—
157C42×C7—22no—
158C42×C7—22no—
159C42×C7C2×D411yescharacteristic
160C42×C7C2×D411yescharacteristic
161C42×C7C2×D411yescharacteristic
162C42×C7C2411yescharacteristic
163C22⋊Dic7—41no—
164#1144—44no—
165C4⋊Dic7—21no—
166C4⋊Dic7—21no—
167C4⋊Dic7—21no—
168C4⋊Dic7—21no—
169Dic7⋊C4—41no—
170Dic7⋊C4—41no—
171C4×Dic7—41no—
172#1136—42no—
173C2×C7⋊C8—22no—
174C2×C7⋊C8—22no—
175C2×C7⋊C8—22no—
176#557—22no—
177#557—22no—
178#557—22no—
179#557C2×D714yes—
180#533—71no—
181C2×C4⋊Q8—21no—
182C2×C4⋊Q8C2×D711yescharacteristic
183C2×#125—22no—
184C2×#125—21no—
185C2×C4×Q8—21no—
186C2×C4×Q8—21no—
187C2×C4×Q8C2×D711yescharacteristic
188C2×C4×Q8C2×D711yescharacteristic
189#498—72no—
190#487—72no—
191#484—141no—
192#483—142no—
193#479—142no—
194#477—72no—
195#477—72no—
196#475—72no—
197#475—142no—
198#462—72no—
199#451—141no—
200C4×Q16—142no—
201#431—71no—
202#431—71no—
203#428—141no—
204#423—71no—
205#419—141no—
206#332—74no—
207Q8⋊C8—74no—
208Q8⋊C8—74no—
209C23×C7#12011yescharacteristic
210C7×Q8—24no—
211C7×Q8—22no—
212C7×Q8—22no—
213C7×Q8—22no—
214C7×Q8—42no—
215C7×Q8—42no—
216C7×Q8—42no—
217C7×Q8—44no—
218C7×Q8#13714yes—
219C2×C4×C7—24no—
220C2×C4×C7—22no—
221C2×C4×C7—22no—
222C2×C4×C7—22no—
223C2×C4×C7—22no—
224C2×C4×C7—22no—
225C2×C4×C7—24no—
226C2×C4×C7—41no—
227C2×C4×C7—42no—
228C2×C4×C7—42no—
229C2×C4×C7—42no—
230C2×C4×C7—44no—
231C2×C4×C7#12011yescharacteristic
232C2×C4×C7#12011yescharacteristic
233C2×C4×C7#12011yescharacteristic
234C2×C4×C7#13512yes—
235C2×C4×C7#13712yes—
236C2×C4×C7C22×D411yescharacteristic
237C2×C4×C7C22×D411yescharacteristic
238C2×C4×C7C22×D411yescharacteristic
239C2×Dic7—41no—
240C2×Dic7—41no—
241Dic14—42no—
242Dic14—42no—
243C7⋊C8—42no—
244C7⋊C8—42no—
245C7⋊C8—42no—
246C22×Q8—21no—
247C22×Q8C22⋊D711yescharacteristic
248C22×Q8C22⋊D711yescharacteristic
249C2×Q16—142no—
250C2×M16—141no—
251C4⋊Q8—24no—
252C4⋊Q8—24no—
253C4⋊Q8—21no—
254C4⋊Q8—21no—
255C4⋊Q8—71no—
256C4⋊Q8—71no—
257C4⋊Q8C22⋊D712yes—
258C4⋊Q8C22×D712yes—
259#125—28no—
260#125—24no—
261#125—28no—
262#125—24no—
263#125—141no—
264C22⋊Q8—141no—
265C22⋊Q8—141no—
266C4×Q8—24no—
267C4×Q8—24no—
268C4×Q8—24no—
269C4×Q8—24no—
270C4×Q8—21no—
271C4×Q8—21no—
272C4×Q8—21no—
273C4×Q8—21no—
274C4×Q8—141no—
275C4×Q8—141no—
276C4×Q8C22⋊D712yes—
277C4×Q8C22⋊D712yes—
278C4×Q8C22×D712yes—
279C4×Q8C22×D712yes—
280#117—141no—
281C2×C4⋊C4—22no—
282C2×C4⋊C4—22no—
283C2×C4⋊C4—22no—
284C2×C4⋊C4—21no—
285C2×C4⋊C4—21no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4C22⋊D711yescharacteristic
288C2×C4⋊C4C22⋊D711yescharacteristic
289C2×C4⋊C4C22⋊D712yes—
290C2×C42—22no—
291C2×C42—21no—
292C2×C42—21no—
293C2×C42C22×D711yescharacteristic
294#107—142no—
295#107—142no—
296#106—142no—
297#106—144no—
298C4⋊C8—72no—
299C4⋊C8—72no—
300C4⋊C8—141no—
301C4⋊C8—141no—
302#103—142no—
303#103—142no—
304#103—142no—
305#103—142no—
306#103—144no—
307C22⋊C8—141no—
308C22⋊C8—141no—
309C4×C8—72no—
310C4×C8—141no—
311C22×C7—21no—
312C22×C7—21no—
313C22×C7#45212yes—
314C22×C7C2×#12011yescharacteristic
315C4×C7—24no—
316C4×C7—21no—
317C4×C7—21no—
318C4×C7—21no—
319C4×C7—42no—
320C4×C7—44no—
321C4×C7—42no—
322C4×C7—42no—
323C4×C7#45212yes—
324C4×C7C2×#13511yescharacteristic
325C4×C7C2×#13712yes—
326C4×C7C2×#13711yescharacteristic
327Dic7—81no—
328Dic7—81no—
329C2×Q8—24no—
330C2×Q8—21no—
331C2×Q8—21no—
332C2×Q8—22no—
333C2×Q8—22no—
334C2×Q8—22no—
335C2×Q8—22no—
336C2×Q8—42no—
337C2×Q8—42no—
338C2×Q8—141no—
339C2×Q8—141no—
340C2×Q8C2×C22⋊D712yes—
341C2×Q8C2×C22⋊D712yes—
342C22×C4—21no—
343C22×C4—21no—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4C2×C22⋊D711yescharacteristic
348C22×C4C2×C22⋊D711yescharacteristic
349C22×C4C2×C22⋊D711yescharacteristic
350Q16—284no—
351M16—282no—
352C2×C8—142no—
353C2×C8—142no—
354C2×C8—142no—
355C4⋊C4—28no—
356C4⋊C4—28no—
357C4⋊C4—28no—
358C4⋊C4—22no—
359C4⋊C4—22no—
360C4⋊C4—22no—
361C4⋊C4—21no—
362C4⋊C4—21no—
363C4⋊C4—22no—
364C4⋊C4—22no—
365C4⋊C4—22no—
366C4⋊C4—22no—
367C4⋊C4—141no—
368C4⋊C4—141no—
369C4⋊C4—141no—
370C4⋊C4—141no—
371C4⋊C4—281no—
372C4⋊C4—281no—
373C4⋊C4C2×C22⋊D714yes—
374C4⋊C4C2×C22⋊D712yes—
375C4⋊C4C2×C22⋊D712yes—
376C22⋊C4—281no—
377C42—28no—
378C42—22no—
379C42—22no—
380C42—22no—
381C42—22no—
382C42—281no—
383C42C2×C22⋊D711yescharacteristic
384C42C2×C22⋊D711yescharacteristic
385C42C2×C22⋊D711yescharacteristic
386C42C23×D711yescharacteristic
387C2×C7—22no—
388C2×C7C2×#45211yescharacteristic
389C2×C7#300411yescharacteristic
390C2×C7#302011yescharacteristic
391C23#651611yescharacteristic
392Q8—24no—
393Q8—22no—
394Q8—22no—
395Q8—22no—
396Q8—42no—
397Q8—42no—
398Q8—42no—
399Q8—44no—
400Q8—282no—
401Q8—282no—
402Q8#650514yes—
403C2×C4—24no—
404C2×C4—22no—
405C2×C4—22no—
406C2×C4—22no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—24no—
410C2×C4—41no—
411C2×C4—42no—
412C2×C4—42no—
413C2×C4—42no—
414C2×C4—44no—
415C2×C4—281no—
416C2×C4—281no—
417C2×C4#651312yes—
418C2×C4#651412yes—
419C2×C4#651611yescharacteristic
420C2×C4#651611yescharacteristic
421C2×C4#651611yescharacteristic
422C2×C4C22×C22⋊D711yescharacteristic
423C2×C4C22×C22⋊D711yescharacteristic
424C2×C4C22×C22⋊D711yescharacteristic
425C8—282no—
426C8—282no—
427C8—282no—
428C7#1876111yescharacteristic, complemented, Sylow
429C22—21no—
430C22—21no—
431C22#9001112yes—
432C22C2×#651611yescharacteristic
433C4—24no—
434C4—21no—
435C4—21no—
436C4—21no—
437C4—42no—
438C4—44no—
439C4—42no—
440C4—42no—
441C4—561no—
442C4—561no—
443C4#9001112yes—
444C4C2×#650512yes—
445C4C2×#651311yescharacteristic
446C4C2×#651411yescharacteristic
447C2—22no—
448C2#1174150411yescharacteristic
449C2C2×#9001111yescharacteristic
450C2#1174156111yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×#557C2
socleC22×C7C2×#120
prime radicalC23×C7#120
hypercenterC2×#557D7
hypocenterC7#18761
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
5. #11729692C2semidirect
6. #11729690C2semidirect
428. C7#18761semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5572
#117406682
#117398922
#117398762
#117296922
#117296902
#117295272
#117293762
#117293752
#187617