Finite Group Databasebeta

← #13413021

Subgroups — #13413021

Order 1792.

806 conjugacy classes of subgroups.

The subgroup lattice has 458 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2552C211yesmaximal, characteristic, complemented
2C4⋊C4×D28C211yesmaximal, characteristic
3#11727379C212yesmaximal
4#11727376C212yesmaximal, complemented
5#11727338C212yesmaximal
6#11727338C211yesmaximal, characteristic
7#11727328C211yesmaximal, characteristic, complemented
8#11727325C212yesmaximal, complemented
9#11727238C211yesmaximal, characteristic, complemented
10C4×#6383C211yesmaximal, characteristic
11#11727220C211yesmaximal, characteristic, complemented
12C2×C7×#125—21no—
13C2×D7×C4⋊C4—21no—
14C2×C4×D28C2212yes—
15C2×C4×D28C2211yescharacteristic
16C7×#398—22no—
17C7×#389—21no—
18C7×#389C2212yes—
19C7×#384C2212yes—
20C7×#384C2211yescharacteristic
21C4×C7×C4⋊C4C2211yescharacteristic
22C4×C7×C4⋊C4C2211yescharacteristic
23#89759—21no—
24C2×#6383C2211yescharacteristic
25C2×#6383C2212yes—
26C2×#6383C2211yescharacteristic
27C2×#6381C2212yes—
28#89707C2211yescharacteristic
29#89700C2212yes—
30C2×C4⋊(C7⋊C8)C2211yescharacteristic
31C2×C4⋊(C7⋊C8)C2212yes—
32C2×C4×C7⋊C8C2211yescharacteristic
33#89440—22no—
34#89324C2211yescharacteristic
35#89324C2212yes—
36#89324C2212yes—
37#89324C2212yes—
38#89324C2212yes—
39#89324C2211yescharacteristic
40#89319C2212yes—
41#26106—71nomaximal, Sylow
42C22×D28C2311yescharacteristic
43C22×C4×D7—21no—
44C22×C4×D7—22no—
45C7×#125—44no—
46C2×C7×C4⋊C4—21no—
47C2×C7×C4⋊C4—22no—
48C2×C7×C4⋊C4—22no—
49C2×C7×C4⋊C4—22no—
50C2×C7×C4⋊C4—21no—
51C2×C7×C4⋊C4D411yescharacteristic
52C2×C7×C4⋊C4D411yescharacteristic
53C2×C7×C4⋊C4C2311yescharacteristic
54C2×C7×C4⋊C4C2311yescharacteristic
55C2×C7×C4⋊C4C2312yes—
56C2×C42×C7—21no—
57C2×C42×C7C2312yes—
58C2×C42×C7C2311yescharacteristic
59C2×(C2×D7)⋊C4—21no—
60C2×(C2×D7)⋊C4—22no—
61C2×C4⋊Dic7—21no—
62C2×C4⋊Dic7—21no—
63C2×C4⋊Dic7C2312yes—
64C2×C4⋊Dic7C2311yescharacteristic
65C2×Dic7⋊C4—22no—
66C22×C7⋊C8C2311yescharacteristic
67C22×C7⋊C8C2311yescharacteristic
68C22×C7⋊C8C2312yes—
69D7×C4⋊C4—44no—
70C4×D28—22no—
71C4×D28—24no—
72C4×D28—42no—
73C4×D28Q814yes—
74C7×#95—21no—
75C7×#95—22no—
76C7×#95D412yes—
77C7⋊#95—22no—
78#6383—22no—
79#6383—24no—
80#6383—22no—
81#6381—42no—
82C4⋊(C7⋊C8)—22no—
83C4⋊(C7⋊C8)—42no—
84C4×C7⋊C8—22no—
85#2552D711yescharacteristic, complemented
86D4×C4⋊C4—71no—
87#2099—72no—
88#2086—72no—
89#1947—71no—
90#1939—71no—
91#1862—72no—
92#1834—71no—
93#1802—71no—
94#1802—72no—
95C4×#102—71no—
96C23×D7—21no—
97C22×C4×C7—21no—
98C22×C4×C7—21no—
99C22×C4×C7—22no—
100C22×C4×C7C2×D412yes—
101C22×C4×C7C2×D411yescharacteristic
102C22×C4×C7C2×D412yes—
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7C2411yescharacteristic
105C22×Dic7—21no—
106C22×Dic7—22no—
107C2×D28—21no—
108C2×D28—21no—
109C2×D28—21no—
110C2×D28C2×Q812yes—
111C2×D28#4112yes—
112C2×D28#4112yes—
113C2×C4×D7—24no—
114C2×C4×D7—42no—
115C2×C4×D7—42no—
116C2×C4×D7—44no—
117C2×C4×D7—44no—
118C2×C4×D7—82no—
119C7×C4⋊C4—22no—
120C7×C4⋊C4—21no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—22no—
123C7×C4⋊C4—21no—
124C7×C4⋊C4—21no—
125C7×C4⋊C4—22no—
126C7×C4⋊C4—44no—
127C7×C4⋊C4—44no—
128C7×C4⋊C4—42no—
129C7×C4⋊C4—81no—
130C7×C4⋊C4—82no—
131C7×C4⋊C4D814yes—
132C42×C7—22no—
133C42×C7—22no—
134C42×C7—42no—
135C42×C7C2×Q812yes—
136C42×C7#4112yes—
137(C2×D7)⋊C4—24no—
138(C2×D7)⋊C4—44no—
139C4⋊Dic7—21no—
140C4⋊Dic7—22no—
141C4⋊Dic7—22no—
142C4⋊Dic7—42no—
143C4⋊Dic7—81no—
144C4⋊Dic7C2×Q812yes—
145Dic7⋊C4—44no—
146C2×C7⋊C8—21no—
147C2×C7⋊C8—21no—
148C2×C7⋊C8—22no—
149C2×C7⋊C8—21no—
150C2×C7⋊C8—21no—
151C2×C7⋊C8—22no—
152C2×C7⋊C8—21no—
153C2×C7⋊C8—22no—
154C2×C7⋊C8—21no—
155C2×#125—21no—
156C2×C4×D4—71no—
157C2×C4×D4—72no—
158C22×C4⋊C4—141no—
159C2×#107—72no—
160C2×C4⋊C8—71no—
161C2×C4⋊C8—72no—
162C2×#102—72no—
163C2×#102—71no—
164C2×#102—71no—
165C2×C4×C8—71no—
166#398—22no—
167#389—21no—
168#389—72no—
169#389C2×D712yes—
170#385—142no—
171#384—71no—
172#384C2×D711yescharacteristic
173#384C2×D712yes—
174#380—141no—
175C4×C4⋊C4C2×D711yescharacteristic
176C4×C4⋊C4C2×D711yescharacteristic
177#340—72no—
178#340—72no—
179#340—72no—
180#340—71no—
181#340—72no—
182#340—71no—
183#336—72no—
184C23×C7C22×D411yescharacteristic
185C22×D7—22no—
186C22×D7—22no—
187C22×D7—22no—
188C22×D7—41no—
189C22×D7—41no—
190C22×D7—41no—
191C22×D7—41no—
192C2×C4×C7—22no—
193C2×C4×C7—21no—
194C2×C4×C7—21no—
195C2×C4×C7—22no—
196C2×C4×C7—22no—
197C2×C4×C7—22no—
198C2×C4×C7—22no—
199C2×C4×C7—41no—
200C2×C4×C7—41no—
201C2×C4×C7—42no—
202C2×C4×C7—41no—
203C2×C4×C7—41no—
204C2×C4×C7—42no—
205C2×C4×C7—41no—
206C2×C4×C7—42no—
207C2×C4×C7—41no—
208C2×C4×C7C22⋊Q812yes—
209C2×C4×C7C22⋊Q812yes—
210C2×C4×C7C2×D814yes—
211C2×C4×C7C2×D812yes—
212C2×C4×C7#13712yes—
213C2×C4×C7C22×Q811yescharacteristic
214C2×C4×C7C2×#4111yescharacteristic
215C2×C4×C7C2×#4111yescharacteristic
216C2×C4×C7C2×#4111yescharacteristic
217C2×C4×C7H(5,𝔽2)11yescharacteristic
218C2×C4×C72−1+411yescharacteristic
219C2×Dic7—22no—
220C2×Dic7—41no—
221C2×Dic7—41no—
222C2×Dic7—42no—
223C2×Dic7—42no—
224C2×Dic7—42no—
225D28—22no—
226D28—22no—
227D28—22no—
228D28#12514yescomplemented
229C4×D7—48no—
230C4×D7—88no—
231C7⋊C8—41no—
232C7⋊C8—41no—
233C7⋊C8—42no—
234C22×D4—71no—
235C23×C4—141no—
236C23×C4—142no—
237C22×C8—71no—
238C22×C8—72no—
239C22×C8—71no—
240#125—44no—
241C4×D4—74no—
242C4×D4—144no—
243C4×D4—142no—
244C4×D4—282no—
245C2×C4⋊C4—21no—
246C2×C4⋊C4—22no—
247C2×C4⋊C4—22no—
248C2×C4⋊C4—22no—
249C2×C4⋊C4—21no—
250C2×C4⋊C4—71no—
251C2×C4⋊C4—72no—
252C2×C4⋊C4—142no—
253C2×C4⋊C4—141no—
254C2×C4⋊C4—141no—
255C2×C4⋊C4—284no—
256C2×C4⋊C4C22⋊D711yescharacteristic
257C2×C4⋊C4C22⋊D711yescharacteristic
258C2×C4⋊C4C22×D711yescharacteristic
259C2×C4⋊C4C22×D712yes—
260C2×C4⋊C4C22×D711yescharacteristic
261C2×C22⋊C4—141no—
262C2×C22⋊C4—142no—
263C2×C42—21no—
264C2×C42C22×D712yes—
265C2×C42C22×D711yescharacteristic
266#107—282no—
267C4⋊C8—142no—
268C4⋊C8—282no—
269#102—144no—
270#102—142no—
271#102—142no—
272C4×C8—142no—
273#95—21no—
274#95—22no—
275#95—142no—
276#95C22⋊D712yes—
277C22×C7C2×C22⋊Q811yescharacteristic
278C22×C7#54611yescharacteristic
279C22×C7#54811yescharacteristic
280C22×C7C22×D811yescharacteristic
281C22×C7C2×#13711yescharacteristic
282C22×C7#57711yescharacteristic
283C22×C7#57711yescharacteristic
284C2×D7—24no—
285C2×D7—42no—
286C2×D7—42no—
287C2×D7—42no—
288C2×D7—42no—
289C2×D7—42no—
290C2×D7—42no—
291C4×C7—22no—
292C4×C7—24no—
293C4×C7—42no—
294C4×C7—81no—
295C4×C7—82no—
296C4×C7—81no—
297C4×C7#47414yes—
298C4×C7C2×#12511yescharacteristic
299C4×C7#54411yescharacteristic
300C4×C7#55211yescharacteristic
301C4×C7#55311yescharacteristic
302Dic7—42no—
303Dic7—82no—
304C24—141no—
305C2×D4—72no—
306C2×D4—72no—
307C2×D4—72no—
308C2×D4—141no—
309C2×D4—141no—
310C2×D4—141no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—22no—
314C22×C4—141no—
315C22×C4—144no—
316C22×C4—142no—
317C22×C4—284no—
318C22×C4—282no—
319C22×C4—282no—
320C22×C4—284no—
321C22×C4—562no—
322C22×C4C2×C22⋊D712yes—
323C22×C4C2×C22⋊D712yes—
324C22×C4C2×C22⋊D711yescharacteristic
325C22×C4C2×C22⋊D711yescharacteristic
326C22×C4C23×D711yescharacteristic
327C2×C8—141no—
328C2×C8—141no—
329C2×C8—142no—
330C2×C8—141no—
331C2×C8—142no—
332C2×C8—141no—
333C2×C8—141no—
334C2×C8—142no—
335C2×C8—141no—
336C4⋊C4—22no—
337C4⋊C4—21no—
338C4⋊C4—21no—
339C4⋊C4—22no—
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—22no—
343C4⋊C4—44no—
344C4⋊C4—44no—
345C4⋊C4—42no—
346C4⋊C4—72no—
347C4⋊C4—81no—
348C4⋊C4—82no—
349C4⋊C4—141no—
350C4⋊C4—142no—
351C4⋊C4—142no—
352C4⋊C4—284no—
353C4⋊C4—282no—
354C4⋊C4—561no—
355C4⋊C4#114114yes—
356C22⋊C4—144no—
357C22⋊C4—284no—
358C42—22no—
359C42—22no—
360C42—42no—
361C42Q8×D712yes—
362C42#116112yes—
363C2×C7#244211yescharacteristic
364C2×C7C2×#47411yescharacteristic
365C2×C7#308411yescharacteristic
366C2×C7#329411yescharacteristic
367C2×C7#333111yescharacteristic
368C2×C7#335811yescharacteristic
369C2×C7#337211yescharacteristic
370D7—44no—
371C23—142no—
372C23—142no—
373C23—142no—
374C23—281no—
375C23—281no—
376C23—281no—
377C23—281no—
378C23C22×C22⋊D711yescharacteristic
379D4—74no—
380D4—142no—
381D4—142no—
382D4—142no—
383C2×C4—22no—
384C2×C4—21no—
385C2×C4—21no—
386C2×C4—22no—
387C2×C4—22no—
388C2×C4—22no—
389C2×C4—22no—
390C2×C4—41no—
391C2×C4—41no—
392C2×C4—42no—
393C2×C4—41no—
394C2×C4—41no—
395C2×C4—42no—
396C2×C4—41no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—142no—
400C2×C4—281no—
401C2×C4—281no—
402C2×C4—288no—
403C2×C4—282no—
404C2×C4—282no—
405C2×C4—282no—
406C2×C4—568no—
407C2×C4C2×#114114yes—
408C2×C4C2×#114112yes—
409C2×C4#650512yes—
410C2×C4#650912yes—
411C2×C4#650912yes—
412C2×C4C2×Q8×D711yescharacteristic
413C2×C4C2×#116111yescharacteristic
414C2×C4D7×#4111yescharacteristic
415C2×C4D7×#4111yescharacteristic
416C2×C4#655311yescharacteristic
417C2×C4#655411yescharacteristic
418C8—281no—
419C8—281no—
420C8—282no—
421C7#2610611yescharacteristic, complemented, Sylow
422C22—144no—
423C22—282no—
424C22—282no—
425C22—282no—
426C22—282no—
427C22—282no—
428C22—282no—
429C22C22×#114111yescharacteristic
430C22C2×#650511yescharacteristic
431C22C2×#650911yescharacteristic
432C22#9052011yescharacteristic
433C22#9052211yescharacteristic
434C22#9052711yescharacteristic
435C22#9052811yescharacteristic
436C4—24no—
437C4—22no—
438C4—42no—
439C4—81no—
440C4—81no—
441C4—82no—
442C4—282no—
443C4—562no—
444C4#8985914yes—
445C4D7×#12511yescharacteristic
446C4#9038011yescharacteristic
447C4#9038511yescharacteristic
448C4#9038611yescharacteristic
449C2—284no—
450C2#1174047311yescharacteristic
451C2C2×#8985911yescharacteristic
452C2#1174092111yescharacteristic
453C2#1174097011yescharacteristic
454C2#1174102511yescharacteristic
455C2#1174108111yescharacteristic
456C2#1174113711yescharacteristic
subgroupG/H
centerC23C22×C22⋊D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#2552C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#2552D7
hypocenterC7#26106
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #2552

Complements

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

NH ≅ G/Ntype
1. C7×#2552C2semidirect
4. #11727376C2semidirect
7. #11727328C2semidirect
8. #11727325C2semidirect
9. #11727238C2semidirect
11. #11727220C2semidirect
85. #2552D7semidirect
228. D28#125semidirect
421. C7#26106semidirect

Maximal subgroups

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

subgroupindex
C7×#25522
C4⋊C4×D282
#117273792
#117273762
#117273382
#117273282
#117273252
#117272382
C4×#63832
#117272202
#261067