Finite Group Databasebeta

← #13332496

Subgroups — #13332496

Order 1792.

922 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×Q82C211yesmaximal, characteristic
2#11740667C211yesmaximal, characteristic
3#11739892C211yesmaximal, characteristic
4#11739876C211yesmaximal, characteristic
5#11729695C214yesmaximal, complemented
6#11729694C214yesmaximal, complemented
7#11729528C211yesmaximal, characteristic
8#11729376C211yesmaximal, characteristic
9#11729375C211yesmaximal, characteristic
10C7×Q82—22no—
11C7×Q82—22no—
12C7×Q82—22no—
13C7×Q82C2214yes—
14C2×C7×C4⋊Q8—22no—
15C2×C7×C4⋊Q8—21no—
16C2×C7×C4⋊Q8—21no—
17C2×C7×C4⋊Q8C2211yescharacteristic
18C2×C4×C7×Q8—21no—
19C2×C4×C7×Q8—21no—
20C2×C4×C7×Q8C2211yescharacteristic
21C2×C4×C7×Q8C2211yescharacteristic
22#90161C2211yescharacteristic
23#90000—21no—
24#90000—22no—
25#89997—21no—
26#89864C2212yes—
27#89862—22no—
28#89853C2212yes—
29#89803C2212yes—
30C4×#1144—22no—
31#89794—22no—
32#89793C2212yes—
33#89793C2212yes—
34#89792C2212yes—
35#89742—21no—
36#89736—21no—
37#89697C2211yescharacteristic
38#89697C2211yescharacteristic
39#89696C2211yescharacteristic
40#89341C2214yes—
41Q8⋊(C7⋊C8)C2214yes—
42Q8⋊(C7⋊C8)C2214yes—
43#18769—71nomaximal, Sylow
44C22×C7×Q8—22no—
45C22×C7×Q8D411yescharacteristic
46C22×C7×Q8D411yescharacteristic
47C22×C7×Q8D412yes—
48C7×C4⋊Q8—28no—
49C7×C4⋊Q8—24no—
50C7×C4⋊Q8—24no—
51C7×C4⋊Q8—28no—
52C7×C4⋊Q8—24no—
53C7×C4⋊Q8—24no—
54C7×C4⋊Q8—21no—
55C7×C4⋊Q8—21no—
56C7×C4⋊Q8D412yes—
57C7×C4⋊Q8C2312yes—
58C4×C7×Q8—24no—
59C4×C7×Q8—24no—
60C4×C7×Q8—24no—
61C4×C7×Q8—24no—
62C4×C7×Q8—21no—
63C4×C7×Q8—21no—
64C4×C7×Q8—21no—
65C4×C7×Q8—21no—
66C4×C7×Q8D412yes—
67C4×C7×Q8D412yes—
68C4×C7×Q8C2312yes—
69C4×C7×Q8C2312yes—
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⋊C4—21no—
76C2×C7×C4⋊C4D411yescharacteristic
77C2×C7×C4⋊C4D411yescharacteristic
78C2×C42×C7—22no—
79C2×C42×C7—21no—
80C2×C42×C7—21no—
81C2×C42×C7C2311yescharacteristic
82C2×#1144—22no—
83C2×#1144—24no—
84#6489—21no—
85#6487—21no—
86#6487—21no—
87C2×#1136—21no—
88#6433—21no—
89#6432C2311yescharacteristic
90#6432C2311yescharacteristic
91C4×Dic14—21no—
92C4×Dic14—21no—
93#6409—24no—
94#6409—22no—
95#6409—22no—
96#6409—22no—
97C22⋊(C7⋊C8)—21no—
98C22⋊(C7⋊C8)—21no—
99#6384—22no—
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×Q82D711yescharacteristic
110#3209—71no—
111#3104—71no—
112#3078—71no—
113#1645—74no—
114#1633—74no—
115#1548—71no—
116#1487—71no—
117#1479—71no—
118C2×C7×Q8—24no—
119C2×C7×Q8—28no—
120C2×C7×Q8—21no—
121C2×C7×Q8—21no—
122C2×C7×Q8—22no—
123C2×C7×Q8—22no—
124C2×C7×Q8—82no—
125C2×C7×Q8—84no—
126C2×C7×Q8Q1614yescomplemented
127C2×C7×Q8C2×D412yes—
128C2×C7×Q8C2×D412yes—
129C2×C7×Q8C2×D414yes—
130C22×C4×C7—21no—
131C22×C4×C7—21no—
132C22×C4×C7—22no—
133C22×C4×C7—21no—
134C22×C4×C7—21no—
135C22×C4×C7C2×D411yescharacteristic
136C22×C4×C7C2×D411yescharacteristic
137C22×C4×C7C2×D411yescharacteristic
138C2×Dic14—21no—
139C2×Dic14—21no—
140C7×C4⋊C4—28no—
141C7×C4⋊C4—28no—
142C7×C4⋊C4—22no—
143C7×C4⋊C4—22no—
144C7×C4⋊C4—22no—
145C7×C4⋊C4—22no—
146C7×C4⋊C4—21no—
147C7×C4⋊C4—21no—
148C7×C4⋊C4—22no—
149C7×C4⋊C4—22no—
150C7×C4⋊C4—22no—
151C7×C4⋊C4—22no—
152C7×C4⋊C4C2×D412yes—
153C7×C4⋊C4C2×D412yes—
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—
165#1144—84no—
166C4⋊Dic7—21no—
167C4⋊Dic7—21no—
168C4⋊Dic7—21no—
169C4⋊Dic7—21no—
170Dic7⋊C4—41no—
171Dic7⋊C4—41no—
172C4×Dic7—41no—
173#1136—42no—
174C2×C7⋊C8—22no—
175C2×C7⋊C8—22no—
176C2×C7⋊C8—22no—
177Q82—22no—
178Q82—22no—
179Q82—22no—
180Q82C2×D714yes—
181#533—71no—
182C2×C4⋊Q8—22no—
183C2×C4⋊Q8—21no—
184C2×C4⋊Q8—21no—
185C2×C4⋊Q8C2×D711yescharacteristic
186C2×C4×Q8—21no—
187C2×C4×Q8—21no—
188C2×C4×Q8C2×D711yescharacteristic
189C2×C4×Q8C2×D711yescharacteristic
190#494—72no—
191#487—72no—
192#484—141no—
193#479—142no—
194#477—72no—
195#477—72no—
196#475—72no—
197#462—72no—
198#462—142no—
199#451—141no—
200#451—142no—
201C4×Q16—142no—
202#431—71no—
203#431—71no—
204#428—141no—
205#423—71no—
206#419—141no—
207#333—74no—
208Q8⋊C8—74no—
209Q8⋊C8—74no—
210C23×C7#12011yescharacteristic
211C7×Q8—24no—
212C7×Q8—22no—
213C7×Q8—22no—
214C7×Q8—24no—
215C7×Q8—44no—
216C7×Q8—82no—
217C7×Q8—88no—
218C7×Q8C2×Q1614yes—
219C7×Q8#13714yes—
220C2×C4×C7—24no—
221C2×C4×C7—22no—
222C2×C4×C7—22no—
223C2×C4×C7—24no—
224C2×C4×C7—22no—
225C2×C4×C7—24no—
226C2×C4×C7—22no—
227C2×C4×C7—22no—
228C2×C4×C7—41no—
229C2×C4×C7—44no—
230C2×C4×C7—42no—
231C2×C4×C7—42no—
232C2×C4×C7#12011yescharacteristic
233C2×C4×C7#12011yescharacteristic
234C2×C4×C7#12011yescharacteristic
235C2×C4×C7C2×Q1612yes—
236C2×C4×C7#13712yes—
237C2×C4×C7C22×D411yescharacteristic
238C2×C4×C7C22×D411yescharacteristic
239C2×C4×C7C22×D411yescharacteristic
240C2×Dic7—41no—
241C2×Dic7—41no—
242Dic14—42no—
243Dic14—42no—
244C7⋊C8—42no—
245C7⋊C8—42no—
246C7⋊C8—42no—
247C22×Q8—22no—
248C22×Q8C22⋊D711yescharacteristic
249C22×Q8C22⋊D711yescharacteristic
250C22×Q8C22⋊D712yes—
251C2×Q16—142no—
252C2×Q16—144no—
253C2×M16—141no—
254C4⋊Q8—28no—
255C4⋊Q8—24no—
256C4⋊Q8—24no—
257C4⋊Q8—28no—
258C4⋊Q8—24no—
259C4⋊Q8—24no—
260C4⋊Q8—21no—
261C4⋊Q8—21no—
262C4⋊Q8—71no—
263C4⋊Q8—71no—
264C4⋊Q8C22⋊D712yes—
265C4⋊Q8C22×D712yes—
266#125—141no—
267C22⋊Q8—141no—
268C22⋊Q8—141no—
269C4×Q8—24no—
270C4×Q8—24no—
271C4×Q8—24no—
272C4×Q8—24no—
273C4×Q8—21no—
274C4×Q8—21no—
275C4×Q8—21no—
276C4×Q8—21no—
277C4×Q8—141no—
278C4×Q8—141no—
279C4×Q8C22⋊D712yes—
280C4×Q8C22⋊D712yes—
281C4×Q8C22×D712yes—
282C4×Q8C22×D712yes—
283#117—141no—
284C2×C4⋊C4—22no—
285C2×C4⋊C4—22no—
286C2×C4⋊C4—21no—
287C2×C4⋊C4—21no—
288C2×C4⋊C4—21no—
289C2×C4⋊C4—21no—
290C2×C4⋊C4C22⋊D711yescharacteristic
291C2×C4⋊C4C22⋊D711yescharacteristic
292C2×C42—22no—
293C2×C42—21no—
294C2×C42—21no—
295C2×C42C22×D711yescharacteristic
296#107—142no—
297#107—142no—
298#106—142no—
299C4⋊C8—72no—
300C4⋊C8—72no—
301C4⋊C8—141no—
302C4⋊C8—141no—
303#103—142no—
304#103—142no—
305#103—142no—
306#103—144no—
307#103—142no—
308C22⋊C8—141no—
309C22⋊C8—141no—
310C4×C8—72no—
311C4×C8—141no—
312C22×C7—21no—
313C22×C7—21no—
314C22×C7#45112yes—
315C22×C7C2×#12011yescharacteristic
316C4×C7—24no—
317C4×C7—24no—
318C4×C7—21no—
319C4×C7—21no—
320C4×C7—21no—
321C4×C7—44no—
322C4×C7—42no—
323C4×C7—42no—
324C4×C7#45112yes—
325C4×C7C22×Q1611yescharacteristic
326C4×C7C2×#13711yescharacteristic
327C4×C7C2×#13712yes—
328Dic7—81no—
329Dic7—81no—
330C2×Q8—24no—
331C2×Q8—28no—
332C2×Q8—22no—
333C2×Q8—21no—
334C2×Q8—21no—
335C2×Q8—22no—
336C2×Q8—82no—
337C2×Q8—84no—
338C2×Q8—141no—
339C2×Q8—141no—
340C2×Q8#114414yescomplemented
341C2×Q8C2×C22⋊D712yes—
342C2×Q8C2×C22⋊D712yes—
343C2×Q8C2×C22⋊D714yes—
344C22×C4—21no—
345C22×C4—21no—
346C22×C4—22no—
347C22×C4—21no—
348C22×C4—21no—
349C22×C4C2×C22⋊D711yescharacteristic
350C22×C4C2×C22⋊D711yescharacteristic
351C22×C4C2×C22⋊D711yescharacteristic
352Q16—284no—
353Q16—564no—
354M16—282no—
355C2×C8—142no—
356C2×C8—142no—
357C2×C8—142no—
358C4⋊C4—28no—
359C4⋊C4—28no—
360C4⋊C4—22no—
361C4⋊C4—22no—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—21no—
365C4⋊C4—21no—
366C4⋊C4—22no—
367C4⋊C4—22no—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—141no—
371C4⋊C4—141no—
372C4⋊C4—141no—
373C4⋊C4—141no—
374C4⋊C4—281no—
375C4⋊C4—281no—
376C4⋊C4C2×C22⋊D712yes—
377C4⋊C4C2×C22⋊D712yes—
378C22⋊C4—281no—
379C42—28no—
380C42—22no—
381C42—22no—
382C42—22no—
383C42—22no—
384C42—281no—
385C42C2×C22⋊D711yescharacteristic
386C42C2×C22⋊D711yescharacteristic
387C42C2×C22⋊D711yescharacteristic
388C42C23×D711yescharacteristic
389C2×C7—22no—
390C2×C7C2×#45111yescharacteristic
391C2×C7#300411yescharacteristic
392C2×C7#301911yescharacteristic
393C23#651611yescharacteristic
394Q8—24no—
395Q8—22no—
396Q8—22no—
397Q8—24no—
398Q8—44no—
399Q8—82no—
400Q8—88no—
401Q8—282no—
402Q8—282no—
403Q8#650514yes—
404Q8C2×#114414yes—
405C2×C4—24no—
406C2×C4—24no—
407C2×C4—22no—
408C2×C4—22no—
409C2×C4—22no—
410C2×C4—24no—
411C2×C4—22no—
412C2×C4—22no—
413C2×C4—41no—
414C2×C4—44no—
415C2×C4—42no—
416C2×C4—42no—
417C2×C4—281no—
418C2×C4—281no—
419C2×C4#650512yes—
420C2×C4C2×#114412yes—
421C2×C4#651611yescharacteristic
422C2×C4#651611yescharacteristic
423C2×C4#651611yescharacteristic
424C2×C4C22×C22⋊D711yescharacteristic
425C2×C4C22×C22⋊D711yescharacteristic
426C2×C4C22×C22⋊D711yescharacteristic
427C8—282no—
428C8—282no—
429C8—282no—
430C7#1876911yescharacteristic, complemented, Sylow
431C22—21no—
432C22—21no—
433C22#9000012yes—
434C22C2×#651611yescharacteristic
435C4—24no—
436C4—21no—
437C4—24no—
438C4—21no—
439C4—21no—
440C4—44no—
441C4—42no—
442C4—42no—
443C4—561no—
444C4—561no—
445C4#9000012yes—
446C4C2×#650512yes—
447C4C2×#650511yescharacteristic
448C4C22×#114411yescharacteristic
449C2—22no—
450C2#1174150411yescharacteristic
451C2C2×#9000011yescharacteristic
452C2#1174154911yescharacteristic
subgroupG/H
centerC22C2×#6516
commutator, cosocleC42×C7C24
FrattiniC42C23×D7
FittingC2×C7×Q82C2
socleC22×C7C2×#120
prime radicalC23×C7#120
hypercenterC2×Q82D7
hypocenterC7#18769
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
5. #11729695C2semidirect
6. #11729694C2semidirect
126. C2×C7×Q8Q16semidirect
340. C2×Q8#1144semidirect
430. C7#18769semidirect

Maximal subgroups

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

subgroupindex
C2×C7×Q822
#117406672
#117398922
#117398762
#117296952
#117296942
#117295282
#117293762
#117293752
#187697