Finite Group Databasebeta

← #13403930

Subgroups — #13403930

Order 1792.

874 conjugacy classes of subgroups.

The subgroup lattice has 492 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11740705C211yesmaximal, characteristic
2C2×#89863C211yesmaximal, characteristic, complemented
3C2×C8×D28C211yesmaximal, characteristic
4#11731814C211yesmaximal, characteristic
5C7×#1990C211yesmaximal, characteristic, complemented
6C4×#6390C211yesmaximal, characteristic, complemented
7#11727321C211yesmaximal, characteristic, complemented
8#11726644C212yesmaximal, complemented
9#11726636C212yesmaximal, complemented
10#11726623C212yesmaximal, complemented
11#11726579C212yesmaximal, complemented
12C2×C7×C4⋊Q8C2211yescharacteristic
13C22×C8×D7C2212yes—
14C2×#6460C2212yes—
15C2×C4×D28C2211yescharacteristic
16C2×C4×Dic14C2211yescharacteristic
17C2×C4×C7×C8C2211yescharacteristic
18C7×#384C2211yescharacteristic
19#90003—22no—
20C2×(C2×D7)⋊C8C2212yes—
21C2×#6390C2211yescharacteristic
22C2×#6390C2211yescharacteristic
23#89863—24no—
24#89766—22no—
25#89760—21no—
26#89760—21no—
27#89759C2212yes—
28#89759C2212yes—
29#89759C2212yes—
30#89757—21no—
31#89749—21no—
32C2×#6384C2212yes—
33C2×#6381C2212yes—
34C2×#6381C2212yes—
35C4×C4⋊Dic7C2211yescharacteristic
36C2×C4⋊(C7⋊C8)C2211yescharacteristic
37C8×D28—24no—
38C7×#340C2212yes—
39C7×#340C2212yes—
40#89346C2211yescharacteristic
41#89346C2211yescharacteristic
42#89324C2212yes—
43#25340—71nomaximal, Sylow
44C22×C7×Q8—21no—
45C22×D28D411yescharacteristic
46C22×C4×D7C2312yes—
47C22×Dic14C2311yescharacteristic
48C22×C7×C8C2311yescharacteristic
49C22×C7×C8C2311yescharacteristic
50C7×C4⋊Q8—22no—
51C7×C4⋊Q8—22no—
52C2×C7×C4⋊C4—21no—
53C2×C7×C4⋊C4—21no—
54C2×C7×C4⋊C4—21no—
55C2×C7×C4⋊C4C2312yes—
56C2×C7×C4⋊C4C2312yes—
57C2×C42×C7C2311yescharacteristic
58C2×(C2×D7)⋊C4—22no—
59C2×(C2×D7)⋊C4—22no—
60C2×(C2×D7)⋊C4—22no—
61C2×(C2×D7)⋊C4—22no—
62C2×(C2×D7)⋊C4D412yes—
63C2×C4⋊Dic7—22no—
64C2×C4⋊Dic7D411yescharacteristic
65C2×C4⋊Dic7C2311yescharacteristic
66C2×C4⋊Dic7C2311yescharacteristic
67C2×C4⋊Dic7C2311yescharacteristic
68C2×Dic7⋊C4—21no—
69C2×C4×Dic7—21no—
70C2×C4×Dic7—21no—
71C22×C7⋊C8C2312yes—
72C2×C8×D7—24no—
73C2×C8×D7—24no—
74C2×C8×D7—42no—
75#6460—24no—
76#6460—42no—
77C4×D28—22no—
78C4×D28—22no—
79C4×Dic14—22no—
80C4×Dic14—22no—
81C4×C7×C8—22no—
82C7⋊#95—21no—
83C7⋊#95—22no—
84C7⋊#95—22no—
85C7⋊#95—22no—
86(C2×D7)⋊C8—24no—
87#6390—22no—
88#6390—22no—
89#6384—24no—
90#6381—24no—
91#6381—42no—
92C4⋊(C7⋊C8)D414yes—
93C2×#477—71no—
94C2×C8×D4—71no—
95#2563—71no—
96#2538—71no—
97#2076—72no—
98#2036—72no—
99#1990D711yescharacteristic, complemented
100#1947—71no—
101#1892—72no—
102#1808—72no—
103C4×#103—71no—
104C23×D7C2×D412yes—
105C2×C7×Q8—42no—
106C2×C7×Q8—81no—
107C2×C7×Q8—81no—
108C22×C4×C7—22no—
109C22×C4×C7—22no—
110C22×C4×C7C2×D411yescharacteristic
111C22×C4×C7C2×D411yescharacteristic
112C22×C4×C7C2411yescharacteristic
113C22×Dic7—21no—
114C22×Dic7—21no—
115C22×Dic7—21no—
116C22×Dic7—21no—
117C22×Dic7C2×D412yes—
118C2×D28—22no—
119C2×D28—22no—
120C2×D28Q1614yes—
121C2×C4×D7—22no—
122C2×C4×D7—22no—
123C2×C4×D7—22no—
124C2×C4×D7—22no—
125C2×C4×D7—22no—
126C2×C4×D7—22no—
127C2×Dic14—21no—
128C2×Dic14—21no—
129C2×Dic14—21no—
130C2×Dic14—21no—
131C2×Dic14—21no—
132C2×Dic14#4112yes—
133C2×C7×C8—21no—
134C2×C7×C8—21no—
135C2×C7×C8—21no—
136C2×C7×C8—21no—
137C2×C7×C8—21no—
138C2×C7×C8#4112yes—
139C7×C4⋊C4—22no—
140C7×C4⋊C4—22no—
141C7×C4⋊C4—22no—
142C7×C4⋊C4—22no—
143C7×C4⋊C4—42no—
144C7×C4⋊C4—42no—
145C7×C4⋊C4—81no—
146C42×C7C2×D412yes—
147C42×C7#4112yes—
148(C2×D7)⋊C4—24no—
149(C2×D7)⋊C4—44no—
150(C2×D7)⋊C4—82no—
151(C2×D7)⋊C4—82no—
152(C2×D7)⋊C4—82no—
153C4⋊Dic7—22no—
154C4⋊Dic7—21no—
155C4⋊Dic7—21no—
156C4⋊Dic7—21no—
157C4⋊Dic7—21no—
158C4⋊Dic7—21no—
159C4⋊Dic7—21no—
160C4⋊Dic7—44no—
161Dic7⋊C4—42no—
162C4×Dic7—42no—
163C4×Dic7—42no—
164C2×C7⋊C8—22no—
165C2×C7⋊C8—22no—
166C2×C7⋊C8C2×D414yes—
167C8×D7—48no—
168C23×C8—72no—
169C2×C4⋊Q8C2×D711yescharacteristic
170C2×C22⋊Q8—72no—
171C2×C4×Q8—71no—
172C2×C4×D4—71no—
173#477—144no—
174C8×D4—144no—
175C2×#107—72no—
176C2×#107—72no—
177C2×C4⋊C8—71no—
178C2×#103—71no—
179C2×#103—71no—
180C2×#103—72no—
181C2×C22⋊C8—72no—
182C2×C4×C8C2×D711yescharacteristic
183#396—142no—
184#393—142no—
185#390—141no—
186#390—141no—
187#389—141no—
188#384C2×D711yescharacteristic
189#382—141no—
190#380—72no—
191#380—72no—
192#380—72no—
193C4×C4⋊C4—71no—
194#340—71no—
195#340—71no—
196#340—72no—
197#340C2×D712yes—
198#340C2×D712yes—
199C23×C7C22×D411yescharacteristic
200C22×D7—22no—
201C22×D7—22no—
202C22×D7—22no—
203C22×D7—22no—
204C22×D7—22no—
205C22×D7—22no—
206C22×D7C2×Q1614yes—
207C7×Q8—84no—
208C2×C4×C7—21no—
209C2×C4×C7—21no—
210C2×C4×C7—42no—
211C2×C4×C7—42no—
212C2×C4×C7—42no—
213C2×C4×C7—42no—
214C2×C4×C7—42no—
215C2×C4×C7—42no—
216C2×C4×C7#12112yes—
217C2×C4×C7#12112yes—
218C2×C4×C7C2×Q1612yes—
219C2×C4×C7#13512yes—
220C2×C4×C7C22×D411yescharacteristic
221C2×C4×C7C2×#4111yescharacteristic
222C2×C4×C7C2×#4111yescharacteristic
223C2×C4×C7C2×#4111yescharacteristic
224C2×C4×C7H(5,𝔽2)11yescharacteristic
225C2×C4×C72−1+411yescharacteristic
226C2×Dic7—22no—
227C2×Dic7—22no—
228C2×Dic7—22no—
229C2×Dic7—22no—
230C2×Dic7—41no—
231C2×Dic7—41no—
232C2×Dic7—41no—
233C2×Dic7—41no—
234C2×Dic7—41no—
235C2×Dic7—41no—
236C2×Dic7—41no—
237C2×Dic7—41no—
238C2×Dic7—41no—
239C2×Dic7—41no—
240C2×Dic7—41no—
241D28—28no—
242C4×D7—42no—
243C4×D7—42no—
244C4×D7—42no—
245C4×D7—42no—
246Dic14—22no—
247Dic14—22no—
248Dic14—41no—
249Dic14—41no—
250C7×C8—22no—
251C7×C8—41no—
252C7⋊C8—24no—
253C22×Q8—21no—
254C22×Q8—71no—
255C22×D4—71no—
256C23×C4—72no—
257C22×C8—72no—
258C22×C8—144no—
259C22×C8—144no—
260C22×C8—282no—
261C22×C8C22×D711yescharacteristic
262C22×C8C22×D711yescharacteristic
263C4⋊Q8—22no—
264C4⋊Q8—22no—
265C22⋊Q8—144no—
266C22⋊Q8—282no—
267C4×Q8—142no—
268C4×Q8—142no—
269C4×D4—142no—
270C4×D4—142no—
271C2×C4⋊C4—21no—
272C2×C4⋊C4—21no—
273C2×C4⋊C4—21no—
274C2×C4⋊C4—71no—
275C2×C4⋊C4—71no—
276C2×C4⋊C4—71no—
277C2×C4⋊C4—71no—
278C2×C4⋊C4—142no—
279C2×C4⋊C4—141no—
280C2×C4⋊C4C22×D712yes—
281C2×C4⋊C4C22×D712yes—
282C2×C22⋊C4—72no—
283C2×C22⋊C4—142no—
284C2×C22⋊C4—142no—
285C2×C22⋊C4—142no—
286C2×C22⋊C4—142no—
287C2×C42—141no—
288C2×C42—141no—
289C2×C42C22×D711yescharacteristic
290#107—144no—
291#107—282no—
292C4⋊C8—74no—
293#103—144no—
294#103—142no—
295#103—142no—
296C22⋊C8—144no—
297C4×C8—22no—
298#95—142no—
299#95—142no—
300#95—142no—
301#95—141no—
302C22×C7C2×#12111yescharacteristic
303C22×C7#54611yescharacteristic
304C22×C7#54711yescharacteristic
305C22×C7C22×Q1611yescharacteristic
306C22×C7C2×#13511yescharacteristic
307C22×C7#57611yescharacteristic
308C22×C7#57711yescharacteristic
309C2×D7—24no—
310C2×D7—24no—
311C2×D7—24no—
312C2×D7—42no—
313C2×D7—42no—
314C2×D7—42no—
315C2×D7—42no—
316C4×C7—22no—
317C4×C7—82no—
318C4×C7—82no—
319C4×C7#46714yes—
320C4×C7#52911yescharacteristic
321C4×C7#54211yescharacteristic
322C4×C7#54611yescharacteristic
323C4×C7#54811yescharacteristic
324Dic7—42no—
325Dic7—42no—
326Dic7—81no—
327Dic7—81no—
328Dic7—81no—
329C24—72no—
330C2×Q8—42no—
331C2×Q8—72no—
332C2×Q8—81no—
333C2×Q8—81no—
334C2×Q8—141no—
335C2×Q8—141no—
336C2×Q8—141no—
337C2×Q8—141no—
338C2×Q8—141no—
339C2×D4—74no—
340C2×D4—142no—
341C2×D4—142no—
342C22×C4—22no—
343C22×C4—22no—
344C22×C4—72no—
345C22×C4—141no—
346C22×C4—142no—
347C22×C4—142no—
348C22×C4—142no—
349C22×C4—142no—
350C22×C4—142no—
351C22×C4—142no—
352C22×C4—141no—
353C22×C4—141no—
354C22×C4—141no—
355C22×C4D4×D711yescharacteristic
356C22×C4D4×D711yescharacteristic
357C22×C4C23×D711yescharacteristic
358C2×C8—21no—
359C2×C8—21no—
360C2×C8—21no—
361C2×C8—21no—
362C2×C8—21no—
363C2×C8—74no—
364C2×C8—142no—
365C2×C8—142no—
366C2×C8—288no—
367C2×C8#116112yes—
368C4⋊C4—22no—
369C4⋊C4—22no—
370C4⋊C4—22no—
371C4⋊C4—22no—
372C4⋊C4—42no—
373C4⋊C4—42no—
374C4⋊C4—81no—
375C4⋊C4—142no—
376C4⋊C4—141no—
377C4⋊C4—141no—
378C4⋊C4—141no—
379C4⋊C4—141no—
380C4⋊C4—141no—
381C4⋊C4—141no—
382C4⋊C4—284no—
383C4⋊C4—282no—
384C22⋊C4—144no—
385C22⋊C4—284no—
386C22⋊C4—562no—
387C22⋊C4—562no—
388C22⋊C4—562no—
389C42—282no—
390C42—282no—
391C42#115712yes—
392C42D4×D712yes—
393C2×C7#242711yescharacteristic
394C2×C7C2×#46711yescharacteristic
395C2×C7#306211yescharacteristic
396C2×C7#329911yescharacteristic
397C2×C7#330511yescharacteristic
398C2×C7#333811yescharacteristic
399C2×C7#334011yescharacteristic
400D7—44no—
401C23—74no—
402C23—142no—
403C23—142no—
404C23—142no—
405C23—142no—
406C23—142no—
407C23—142no—
408C23C2×D4×D711yescharacteristic
409Q8—84no—
410Q8—142no—
411Q8—142no—
412Q8—281no—
413Q8—281no—
414D4—148no—
415C2×C4—21no—
416C2×C4—21no—
417C2×C4—42no—
418C2×C4—42no—
419C2×C4—42no—
420C2×C4—42no—
421C2×C4—42no—
422C2×C4—42no—
423C2×C4—142no—
424C2×C4—142no—
425C2×C4—142no—
426C2×C4—142no—
427C2×C4—281no—
428C2×C4—281no—
429C2×C4—282no—
430C2×C4—282no—
431C2×C4—282no—
432C2×C4—282no—
433C2×C4—281no—
434C2×C4—281no—
435C2×C4—281no—
436C2×C4—281no—
437C2×C4—281no—
438C2×C4—281no—
439C2×C4—281no—
440C2×C4—281no—
441C2×C4—281no—
442C2×C4#644712yes—
443C2×C4#644712yes—
444C2×C4#647912yes—
445C2×C4D7×Q1612yes—
446C2×C4C2×#115711yescharacteristic
447C2×C4C2×D4×D711yescharacteristic
448C2×C4C2×#116111yescharacteristic
449C2×C4#655111yescharacteristic
450C2×C4D7×#4111yescharacteristic
451C2×C4#655311yescharacteristic
452C8—22no—
453C8—41no—
454C8—144no—
455C7#2534011yescharacteristic, complemented, Sylow
456C22—144no—
457C22—144no—
458C22—144no—
459C22—282no—
460C22—282no—
461C22—282no—
462C22—282no—
463C22C2×#644711yescharacteristic
464C22#9036711yescharacteristic
465C22#9041711yescharacteristic
466C22#9044811yescharacteristic
467C22C2×#647911yescharacteristic
468C22#9045311yescharacteristic
469C22C2×D7×Q1611yescharacteristic
470C4—22no—
471C4—82no—
472C4—82no—
473C4—282no—
474C4—282no—
475C4—561no—
476C4—561no—
477C4—561no—
478C4#8958514yes—
479C4#9027111yescharacteristic
480C4#9027311yescharacteristic
481C4#9032111yescharacteristic
482C4#9032411yescharacteristic
483C2—284no—
484C2#1173406911yescharacteristic
485C2#1173619711yescharacteristic
486C2C2×#8958511yescharacteristic
487C2#1173902411yescharacteristic
488C2#1173905811yescharacteristic
489C2#1173934511yescharacteristic
490C2#1173939211yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1990C2
socleC23×C7C22×D4
prime radicalC22×D28D4
hypercenter#1990D7
hypocenterC7#25340
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1990

Complements

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

NH ≅ G/Ntype
2. C2×#89863C2semidirect
5. C7×#1990C2semidirect
6. C4×#6390C2semidirect
7. #11727321C2semidirect
8. #11726644C2semidirect
9. #11726636C2semidirect
10. #11726623C2semidirect
11. #11726579C2semidirect
99. #1990D7semidirect
455. C7#25340semidirect

Maximal subgroups

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

subgroupindex
#117407052
C2×#898632
C2×C8×D282
#117318142
C7×#19902
C4×#63902
#117273212
#117266442
#117266362
#117266232
#117265792
#253407