Finite Group Databasebeta

← #13413078

Subgroups — #13413078

Order 1792.

770 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
1C4⋊C4×D28C211yesmaximal, characteristic
2#11731170C211yesmaximal, characteristic, complemented
3C7×#1939C211yesmaximal, characteristic, complemented
4#11728234C212yesmaximal
5#11728233C211yesmaximal, characteristic
6#11728217C212yesmaximal, complemented
7#11728207C211yesmaximal, characteristic, complemented
8#11727516C211yesmaximal, characteristic, complemented
9C4×#6395C211yesmaximal, characteristic
10#11726424C212yesmaximal
11#11726398C212yesmaximal, complemented
12C2×D7×C4⋊C4—21no—
13C2×C4×D28C2212yes—
14C2×C4×D28C2211yescharacteristic
15C2×C7×C4⋊C8C2211yescharacteristic
16C2×C7×C4⋊C8C2212yes—
17C2×C4×C7×C8C2211yescharacteristic
18C4×C7×C4⋊C4C2211yescharacteristic
19C2×#6395C2211yescharacteristic
20C2×#6395C2212yes—
21C2×#6395C2211yescharacteristic
22C2×#6392C2212yes—
23#89759—21no—
24#89749—21no—
25#89745—21no—
26#89708C2212yes—
27#89707C2211yescharacteristic
28#89707C2211yescharacteristic
29C4×C4⋊Dic7C2211yescharacteristic
30#89700C2212yes—
31#89700C2212yes—
32#89440—22no—
33#89432—22no—
34C7×#336C2212yes—
35#89346C2211yescharacteristic
36#89346C2212yes—
37#89346C2212yes—
38#89346C2212yes—
39#89346C2211yescharacteristic
40#89346C2212yes—
41#26111—71nomaximal, Sylow
42C22×D28C2311yescharacteristic
43C22×C4×D7—21no—
44C22×C4×D7—22no—
45C22×C7×C8C2311yescharacteristic
46C22×C7×C8C2311yescharacteristic
47C22×C7×C8C2312yes—
48C2×C7×C4⋊C4D411yescharacteristic
49C2×C7×C4⋊C4D411yescharacteristic
50C2×C42×C7C2312yes—
51C2×C42×C7C2311yescharacteristic
52C2×(C2×D7)⋊C4—21no—
53C2×(C2×D7)⋊C4—22no—
54C2×C4⋊Dic7—21no—
55C2×C4⋊Dic7—21no—
56C2×C4⋊Dic7—21no—
57C2×C4⋊Dic7—21no—
58C2×C4⋊Dic7C2312yes—
59C2×C4⋊Dic7C2312yes—
60C2×C4⋊Dic7C2311yescharacteristic
61C2×C4⋊Dic7C2311yescharacteristic
62C2×C4⋊Dic7C2311yescharacteristic
63C2×Dic7⋊C4—22no—
64C2×Dic7⋊C4—22no—
65C2×C4×Dic7—21no—
66D7×C4⋊C4—44no—
67C4×D28—22no—
68C4×D28—24no—
69C4×D28—42no—
70C4×D28Q814yes—
71C7×C4⋊C8—22no—
72C7×C4⋊C8—42no—
73C4×C7×C8—22no—
74C7×#95D412yes—
75C7⋊#95—21no—
76C7⋊#95—22no—
77C7⋊#95—22no—
78C7⋊#95—22no—
79C7⋊#95—22no—
80#6395—22no—
81#6395—24no—
82#6395—22no—
83#6392—42no—
84#2571—71no—
85D4×C4⋊C4—71no—
86#2101—72no—
87#2087—72no—
88#1947—71no—
89#1939D711yescharacteristic, complemented
90#1862—72no—
91#1834—71no—
92#1804—72no—
93#1802—71no—
94C4×#102—71no—
95C23×D7—21no—
96C22×C4×C7C2×D412yes—
97C22×C4×C7C2×D412yes—
98C22×C4×C7C2×D411yescharacteristic
99C22×C4×C7C2×D411yescharacteristic
100C22×C4×C7C2411yescharacteristic
101C22×Dic7—21no—
102C22×Dic7—21no—
103C22×Dic7—21no—
104C22×Dic7—22no—
105C22×Dic7—22no—
106C2×D28—21no—
107C2×D28—21no—
108C2×D28—21no—
109C2×D28C2×Q812yes—
110C2×D28#4112yes—
111C2×D28#4112yes—
112C2×C4×D7—24no—
113C2×C4×D7—42no—
114C2×C4×D7—42no—
115C2×C4×D7—44no—
116C2×C4×D7—44no—
117C2×C4×D7—82no—
118C2×C7×C8—21no—
119C2×C7×C8—21no—
120C2×C7×C8—22no—
121C2×C7×C8—21no—
122C2×C7×C8—21no—
123C2×C7×C8—22no—
124C2×C7×C8—21no—
125C2×C7×C8—22no—
126C2×C7×C8—21no—
127C7×C4⋊C4—22no—
128C7×C4⋊C4—22no—
129C42×C7—22no—
130C42×C7—22no—
131C42×C7C2×Q812yes—
132C42×C7#4112yes—
133(C2×D7)⋊C4—24no—
134(C2×D7)⋊C4—44no—
135C4⋊Dic7—21no—
136C4⋊Dic7—22no—
137C4⋊Dic7—21no—
138C4⋊Dic7—22no—
139C4⋊Dic7—21no—
140C4⋊Dic7—22no—
141C4⋊Dic7—21no—
142C4⋊Dic7—22no—
143C4⋊Dic7—21no—
144C4⋊Dic7—42no—
145C4⋊Dic7—42no—
146C4⋊Dic7—81no—
147C4⋊Dic7—81no—
148C4⋊Dic7C2×Q812yes—
149Dic7⋊C4—44no—
150Dic7⋊C4—82no—
151C4×Dic7—42no—
152C2×C4×D4—71no—
153C2×C4×D4—72no—
154C22×C4⋊C4—141no—
155C2×#107—72no—
156C2×C4⋊C8C2×D711yescharacteristic
157C2×C4⋊C8C2×D712yes—
158C2×#102—72no—
159C2×#102—71no—
160C2×#102—71no—
161C2×C4×C8C2×D711yescharacteristic
162#400—142no—
163#389—72no—
164#389—72no—
165#389—141no—
166#385—142no—
167#384—71no—
168#384—71no—
169#382—72no—
170#382—141no—
171#380—141no—
172C4×C4⋊C4—71no—
173C4×C4⋊C4C2×D711yescharacteristic
174#340—72no—
175#340—72no—
176#340—72no—
177#340—71no—
178#340—72no—
179#340—71no—
180#336C2×D712yes—
181C23×C7C22×D411yescharacteristic
182C22×D7—22no—
183C22×D7—22no—
184C22×D7—22no—
185C22×D7—41no—
186C22×D7—41no—
187C22×D7—41no—
188C22×D7—41no—
189C2×C4×C7—21no—
190C2×C4×C7—21no—
191C2×C4×C7—22no—
192C2×C4×C7—22no—
193C2×C4×C7—22no—
194C2×C4×C7—22no—
195C2×C4×C7—22no—
196C2×C4×C7—22no—
197C2×C4×C7C22⋊Q812yes—
198C2×C4×C7C22⋊Q812yes—
199C2×C4×C7#13512yes—
200C2×C4×C7AGL(1,ℤ/8)12yes—
201C2×C4×C7C22×Q811yescharacteristic
202C2×C4×C7C2×#4111yescharacteristic
203C2×C4×C7C2×#4111yescharacteristic
204C2×C4×C7C2×#4111yescharacteristic
205C2×C4×C7H(5,𝔽2)11yescharacteristic
206C2×C4×C72−1+411yescharacteristic
207C2×Dic7—22no—
208C2×Dic7—41no—
209C2×Dic7—41no—
210C2×Dic7—42no—
211C2×Dic7—41no—
212C2×Dic7—42no—
213C2×Dic7—41no—
214C2×Dic7—42no—
215C2×Dic7—41no—
216C2×Dic7—41no—
217C2×Dic7—42no—
218C2×Dic7—41no—
219C2×Dic7—42no—
220C2×Dic7—42no—
221C2×Dic7—41no—
222D28—22no—
223D28—22no—
224D28—22no—
225D28#12514yes—
226C4×D7—48no—
227C4×D7—88no—
228C7×C8—41no—
229C7×C8—41no—
230C7×C8—42no—
231C22×D4—71no—
232C23×C4—141no—
233C23×C4—142no—
234C22×C8C22×D711yescharacteristic
235C22×C8C22×D712yes—
236C22×C8C22×D711yescharacteristic
237C4×D4—74no—
238C4×D4—144no—
239C4×D4—142no—
240C4×D4—282no—
241C2×C4⋊C4—71no—
242C2×C4⋊C4—72no—
243C2×C4⋊C4—71no—
244C2×C4⋊C4—72no—
245C2×C4⋊C4—71no—
246C2×C4⋊C4—142no—
247C2×C4⋊C4—141no—
248C2×C4⋊C4—141no—
249C2×C4⋊C4—142no—
250C2×C4⋊C4—141no—
251C2×C4⋊C4—141no—
252C2×C4⋊C4—284no—
253C2×C4⋊C4D2811yescharacteristic
254C2×C4⋊C4D2811yescharacteristic
255C2×C22⋊C4—141no—
256C2×C22⋊C4—142no—
257C2×C42—141no—
258C2×C42C22×D712yes—
259C2×C42C22×D711yescharacteristic
260#107—282no—
261C4⋊C8—22no—
262C4⋊C8—42no—
263#102—144no—
264#102—142no—
265#102—142no—
266C4×C8—22no—
267#95—141no—
268#95—142no—
269#95—142no—
270#95—142no—
271#95—142no—
272#95D2812yes—
273C22×C7C2×C22⋊Q811yescharacteristic
274C22×C7#54611yescharacteristic
275C22×C7#54811yescharacteristic
276C22×C7C2×#13511yescharacteristic
277C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
278C22×C7#57611yescharacteristic
279C22×C7#57811yescharacteristic
280C2×D7—24no—
281C2×D7—42no—
282C2×D7—42no—
283C2×D7—42no—
284C2×D7—42no—
285C2×D7—42no—
286C2×D7—42no—
287C4×C7—22no—
288C4×C7—42no—
289C4×C7—42no—
290C4×C7#47814yes—
291C4×C7C2×#12511yescharacteristic
292C4×C7#54411yescharacteristic
293C4×C7#55211yescharacteristic
294C4×C7#55311yescharacteristic
295Dic7—42no—
296Dic7—82no—
297Dic7—81no—
298Dic7—82no—
299Dic7—81no—
300C24—141no—
301C2×D4—72no—
302C2×D4—72no—
303C2×D4—72no—
304C2×D4—141no—
305C2×D4—141no—
306C2×D4—141no—
307C22×C4—141no—
308C22×C4—144no—
309C22×C4—141no—
310C22×C4—141no—
311C22×C4—142no—
312C22×C4—142no—
313C22×C4—284no—
314C22×C4—284no—
315C22×C4—282no—
316C22×C4—282no—
317C22×C4—562no—
318C22×C4C2×D2812yes—
319C22×C4C2×D2812yes—
320C22×C4C2×D2811yescharacteristic
321C22×C4C2×D2811yescharacteristic
322C22×C4C23×D711yescharacteristic
323C2×C8—21no—
324C2×C8—21no—
325C2×C8—22no—
326C2×C8—21no—
327C2×C8—22no—
328C2×C8—21no—
329C2×C8—22no—
330C2×C8—21no—
331C2×C8—21no—
332C4⋊C4—22no—
333C4⋊C4—22no—
334C4⋊C4—72no—
335C4⋊C4—141no—
336C4⋊C4—141no—
337C4⋊C4—142no—
338C4⋊C4—141no—
339C4⋊C4—142no—
340C4⋊C4—141no—
341C4⋊C4—141no—
342C4⋊C4—142no—
343C4⋊C4—142no—
344C4⋊C4—284no—
345C4⋊C4—282no—
346C4⋊C4—282no—
347C4⋊C4—562no—
348C4⋊C4—561no—
349C4⋊C4—561no—
350C22⋊C4—144no—
351C22⋊C4—284no—
352C42—22no—
353C42—22no—
354C42—282no—
355C42#115912yes—
356C42Q8×D712yes—
357C2×C7#244211yescharacteristic
358C2×C7C2×#47811yescharacteristic
359C2×C7#308311yescharacteristic
360C2×C7#330211yescharacteristic
361C2×C7#332511yescharacteristic
362C2×C7#336111yescharacteristic
363C2×C7#336711yescharacteristic
364D7—44no—
365C23—142no—
366C23—142no—
367C23—142no—
368C23—281no—
369C23—281no—
370C23—281no—
371C23—281no—
372C23C22×D2811yescharacteristic
373D4—74no—
374D4—142no—
375D4—142no—
376D4—142no—
377C2×C4—21no—
378C2×C4—21no—
379C2×C4—22no—
380C2×C4—22no—
381C2×C4—22no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—22no—
385C2×C4—142no—
386C2×C4—281no—
387C2×C4—281no—
388C2×C4—288no—
389C2×C4—282no—
390C2×C4—281no—
391C2×C4—281no—
392C2×C4—282no—
393C2×C4—282no—
394C2×C4—281no—
395C2×C4—282no—
396C2×C4—281no—
397C2×C4—282no—
398C2×C4—281no—
399C2×C4—282no—
400C2×C4—281no—
401C2×C4—568no—
402C2×C4#646012yes—
403C2×C4#646012yes—
404C2×C4#646712yes—
405C2×C4#647112yes—
406C2×C4C2×#115911yescharacteristic
407C2×C4#654811yescharacteristic
408C2×C4C2×Q8×D711yescharacteristic
409C2×C4#655111yescharacteristic
410C2×C4D7×#4111yescharacteristic
411C2×C4D7×#4111yescharacteristic
412C8—41no—
413C8—42no—
414C8—41no—
415C7#2611111yescharacteristic, complemented, Sylow
416C22—144no—
417C22—282no—
418C22—282no—
419C22—282no—
420C22—282no—
421C22—282no—
422C22—282no—
423C22C2×#646011yescharacteristic
424C22#9024511yescharacteristic
425C22#9026711yescharacteristic
426C22C2×#646711yescharacteristic
427C22C2×#647111yescharacteristic
428C22#9044311yescharacteristic
429C22#9044411yescharacteristic
430C4—22no—
431C4—42no—
432C4—42no—
433C4—282no—
434C4—562no—
435C4—561no—
436C4—562no—
437C4—561no—
438C4#8961914yes—
439C4#9030511yescharacteristic
440C4#9033011yescharacteristic
441C4D7×#12511yescharacteristic
442C4#9042011yescharacteristic
443C2—284no—
444C2#1173344811yescharacteristic
445C2C2×#8961911yescharacteristic
446C2#1173653611yescharacteristic
447C2#1173688711yescharacteristic
448C2#1173700111yescharacteristic
449C2#1173737211yescharacteristic
450C2#1173741511yescharacteristic
subgroupG/H
centerC23C22×D28
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1939C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#1939D7
hypocenterC7#26111
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C4×C4⋊C4 ◁ #1939

Complements

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

NH ≅ G/Ntype
2. #11731170C2semidirect
3. C7×#1939C2semidirect
6. #11728217C2semidirect
7. #11728207C2semidirect
8. #11727516C2semidirect
11. #11726398C2semidirect
89. #1939D7semidirect
415. C7#26111semidirect

Maximal subgroups

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

subgroupindex
C4⋊C4×D282
#117311702
C7×#19392
#117282342
#117282332
#117282172
#117282072
#117275162
C4×#63952
#117264242
#117263982
#261117