Finite Group Databasebeta

← #13412770

Subgroups — #13412770

Order 1792.

714 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11731769C211yesmaximal, characteristic
2#11731170C211yesmaximal, characteristic, complemented
3C7×#1948C211yesmaximal, characteristic, complemented
4#11728340C212yesmaximal
5#11728318C212yesmaximal, complemented
6#11728233C211yesmaximal, characteristic
7#11728222C211yesmaximal, characteristic, complemented
8#11727331C211yesmaximal, characteristic, complemented
9C4×#6383C211yesmaximal, characteristic
10#11726655C212yesmaximal
11#11726544C212yesmaximal, complemented
12C2×#6455—21no—
13C2×C4×D28C2211yescharacteristic
14C2×C7×#106C2212yes—
15C2×C7×C4⋊C8C2211yescharacteristic
16C7×#384C2211yescharacteristic
17C4×C7×C4⋊C4C2211yescharacteristic
18C2×#6395C2212yes—
19C2×Dic7⋊C8C2212yes—
20#89762—22no—
21#89760C2212yes—
22#89759—21no—
23#89749C2212yes—
24C4⋊C4×Dic7C2212yes—
25#89745C2212yes—
26C2×#6383C2211yescharacteristic
27C2×#6383C2211yescharacteristic
28C2×#6381C2212yes—
29C2×#6381C2212yes—
30#89708—21no—
31#89707C2211yescharacteristic
32C4×C4⋊Dic7C2211yescharacteristic
33#89700—21no—
34C2×C4×C7⋊C8C2211yescharacteristic
35#89432—22no—
36C7×#340C2212yes—
37#89346C2212yes—
38#89345C2212yes—
39#89324C2211yescharacteristic
40#89324C2211yescharacteristic
41#26087—71nomaximal, Sylow
42C22×D28C2311yescharacteristic
43C22×C4×D7—21no—
44C22×C7×C8C2312yes—
45C2×C7×C4⋊C4—21no—
46C2×C7×C4⋊C4—21no—
47C2×C7×C4⋊C4C2311yescharacteristic
48C2×C7×C4⋊C4C2312yes—
49C2×C7×C4⋊C4C2311yescharacteristic
50C2×C42×C7—21no—
51C2×C42×C7C2311yescharacteristic
52C2×(C2×D7)⋊C4—21no—
53C2×(C2×D7)⋊C4—22no—
54C2×(C2×D7)⋊C4—22no—
55C2×C4⋊Dic7—21no—
56C2×C4⋊Dic7—21no—
57C2×C4⋊Dic7D411yescharacteristic
58C2×C4⋊Dic7D411yescharacteristic
59C2×C4⋊Dic7C2312yes—
60C2×C4⋊Dic7C2311yescharacteristic
61C2×Dic7⋊C4—22no—
62C2×C4×Dic7—22no—
63C2×C4×Dic7C2312yes—
64C22×C7⋊C8C2311yescharacteristic
65C22×C7⋊C8C2311yescharacteristic
66#6455—44no—
67C4×D28—22no—
68C4×D28Q814yes—
69C7×#106—42no—
70C7×C4⋊C8—22no—
71C7×#95—21no—
72C7⋊#95—22no—
73C7⋊#95—22no—
74C7⋊#95—22no—
75C7⋊#95—22no—
76C7⋊#95D412yes—
77#6395—42no—
78Dic7⋊C8—24no—
79#6383—22no—
80#6383—22no—
81#6381—24no—
82#6381—42no—
83C4×C7⋊C8—22no—
84#2571—71no—
85#2354—71no—
86#2092—72no—
87#2067—72no—
88#1949—72no—
89#1948D711yescharacteristic, complemented
90#1939—71no—
91#1921—72no—
92#1845—71no—
93#1802—71no—
94C4×#102—71no—
95C23×D7—21no—
96C22×C4×C7—21no—
97C22×C4×C7—21no—
98C22×C4×C7—22no—
99C22×C4×C7C2×D411yescharacteristic
100C22×C4×C7C2×D411yescharacteristic
101C22×C4×C7C2411yescharacteristic
102C22×Dic7—21no—
103C22×Dic7—22no—
104C22×Dic7C2×D412yes—
105C22×Dic7C2×D412yes—
106C2×D28—21no—
107C2×D28—21no—
108C2×D28—21no—
109C2×D28—21no—
110C2×D28—21no—
111C2×D28C2×Q812yes—
112C2×C4×D7—24no—
113C2×C4×D7—42no—
114C2×C4×D7—42no—
115C2×C7×C8—22no—
116C2×C7×C8—22no—
117C2×C7×C8—22no—
118C7×C4⋊C4—22no—
119C7×C4⋊C4—21no—
120C7×C4⋊C4—21no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—22no—
123C7×C4⋊C4—21no—
124C7×C4⋊C4—42no—
125C7×C4⋊C4—81no—
126C42×C7—42no—
127C42×C7C2×Q812yes—
128C42×C7#4112yes—
129(C2×D7)⋊C4—24no—
130(C2×D7)⋊C4—44no—
131(C2×D7)⋊C4—82no—
132C4⋊Dic7—22no—
133C4⋊Dic7—22no—
134C4⋊Dic7—21no—
135C4⋊Dic7—22no—
136C4⋊Dic7—22no—
137C4⋊Dic7—42no—
138C4⋊Dic7—81no—
139C4⋊Dic7C2×Q812yes—
140Dic7⋊C4—82no—
141C4×Dic7—22no—
142C4×Dic7—22no—
143C4×Dic7—44no—
144C2×C7⋊C8—21no—
145C2×C7⋊C8—21no—
146C2×C7⋊C8—21no—
147C2×C7⋊C8—21no—
148C2×C7⋊C8#4112yes—
149C2×C7⋊C8#4112yes—
150C2×C4×D4—71no—
151C2×#117—141no—
152C2×#107—72no—
153C2×#107—72no—
154C2×#106C2×D712yes—
155C2×C4⋊C8—72no—
156C2×C4⋊C8C2×D711yescharacteristic
157C2×#102—71no—
158C2×#102—71no—
159C2×#102—72no—
160C2×C4×C8—71no—
161#400—142no—
162#390—72no—
163#389—72no—
164#389—141no—
165#388—142no—
166#384—71no—
167#384C2×D711yescharacteristic
168#382—72no—
169#382—141no—
170#380—141no—
171C4×C4⋊C4—71no—
172C4×C4⋊C4—72no—
173C4×C4⋊C4C2×D711yescharacteristic
174#340—72no—
175#340—71no—
176#340—71no—
177#340C2×D712yes—
178#336—72no—
179C23×C7C22×D411yescharacteristic
180C22×D7—22no—
181C22×D7—41no—
182C22×D7—41no—
183C22×D7—41no—
184C22×D7—41no—
185C22×D7—41no—
186C22×D7—41no—
187C2×C4×C7—21no—
188C2×C4×C7—21no—
189C2×C4×C7—41no—
190C2×C4×C7—41no—
191C2×C4×C7—42no—
192C2×C4×C7—41no—
193C2×C4×C7—41no—
194C2×C4×C7—42no—
195C2×C4×C7—41no—
196C2×C4×C7—42no—
197C2×C4×C7—41no—
198C2×C4×C7C22⋊Q812yes—
199C2×C4×C7C22⋊Q812yes—
200C2×C4×C7#13512yes—
201C2×C4×C7AGL(1,ℤ/8)12yes—
202C2×C4×C7C22×Q811yescharacteristic
203C2×C4×C7C2×#4111yescharacteristic
204C2×C4×C7C2×#4111yescharacteristic
205C2×C4×C7C2×#4111yescharacteristic
206C2×C4×C7H(5,𝔽2)11yescharacteristic
207C2×C4×C72−1+411yescharacteristic
208C2×Dic7—22no—
209C2×Dic7—22no—
210C2×Dic7—22no—
211C2×Dic7—22no—
212C2×Dic7—22no—
213C2×Dic7—22no—
214C2×Dic7—22no—
215C2×Dic7—41no—
216C2×Dic7—41no—
217C2×Dic7—42no—
218C2×Dic7—42no—
219C2×Dic7—42no—
220D28—22no—
221D28—22no—
222D28—41no—
223D28—41no—
224C4×D7—48no—
225C7×C8—42no—
226C7⋊C8—22no—
227C7⋊C8—22no—
228C22×D4—71no—
229C23×C4—141no—
230C22×C8—71no—
231C22×C8—71no—
232C22×C8C22×D712yes—
233C4×D4—74no—
234C4×D4—142no—
235#117—284no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—21no—
238C2×C4⋊C4—71no—
239C2×C4⋊C4—71no—
240C2×C4⋊C4—71no—
241C2×C4⋊C4—72no—
242C2×C4⋊C4—141no—
243C2×C4⋊C4—142no—
244C2×C4⋊C4—141no—
245C2×C4⋊C4C22×D711yescharacteristic
246C2×C4⋊C4C22×D711yescharacteristic
247C2×C4⋊C4C22×D712yes—
248C2×C22⋊C4—141no—
249C2×C22⋊C4—142no—
250C2×C22⋊C4—142no—
251C2×C42—21no—
252C2×C42—72no—
253C2×C42—142no—
254C2×C42C22×D711yescharacteristic
255#107—144no—
256#107—282no—
257#106—42no—
258C4⋊C8—22no—
259C4⋊C8—144no—
260#102—142no—
261#102—142no—
262#102—282no—
263C4×C8—142no—
264#95—21no—
265#95—72no—
266#95—142no—
267#95—142no—
268#95—142no—
269#95—142no—
270C22×C7C2×C22⋊Q811yescharacteristic
271C22×C7#54711yescharacteristic
272C22×C7#55011yescharacteristic
273C22×C7C2×#13511yescharacteristic
274C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
275C22×C7#57611yescharacteristic
276C22×C7#57811yescharacteristic
277C2×D7—42no—
278C2×D7—42no—
279C2×D7—42no—
280C2×D7—81no—
281C2×D7—81no—
282C2×D7—81no—
283C2×D7—81no—
284C4×C7—22no—
285C4×C7—81no—
286C4×C7—81no—
287C4×C7—82no—
288C4×C7#47814yes—
289C4×C7#53311yescharacteristic
290C4×C7#54411yescharacteristic
291C4×C7#55211yescharacteristic
292C4×C7#55311yescharacteristic
293Dic7—42no—
294Dic7—42no—
295Dic7—42no—
296Dic7—82no—
297C24—141no—
298C2×D4—72no—
299C2×D4—141no—
300C2×D4—141no—
301C2×D4—141no—
302C2×D4—141no—
303C2×D4—141no—
304C22×C4—21no—
305C22×C4—21no—
306C22×C4—22no—
307C22×C4—72no—
308C22×C4—72no—
309C22×C4—141no—
310C22×C4—144no—
311C22×C4—142no—
312C22×C4—282no—
313C22×C4—282no—
314C22×C4D4×D711yescharacteristic
315C22×C4D4×D711yescharacteristic
316C22×C4C23×D711yescharacteristic
317C2×C8—22no—
318C2×C8—22no—
319C2×C8—22no—
320C2×C8—72no—
321C2×C8—72no—
322C2×C8—141no—
323C2×C8—141no—
324C2×C8—141no—
325C2×C8—141no—
326C4⋊C4—21no—
327C4⋊C4—22no—
328C4⋊C4—21no—
329C4⋊C4—21no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—42no—
333C4⋊C4—72no—
334C4⋊C4—81no—
335C4⋊C4—142no—
336C4⋊C4—142no—
337C4⋊C4—141no—
338C4⋊C4—142no—
339C4⋊C4—142no—
340C4⋊C4—282no—
341C4⋊C4—561no—
342C4⋊C4—562no—
343C22⋊C4—144no—
344C22⋊C4—284no—
345C22⋊C4—562no—
346C42—42no—
347C42—142no—
348C42—142no—
349C42—284no—
350C42#115712yes—
351C42Q8×D712yes—
352C2×C7#244111yescharacteristic
353C2×C7C2×#47811yescharacteristic
354C2×C7#308311yescharacteristic
355C2×C7#334111yescharacteristic
356C2×C7#335311yescharacteristic
357C2×C7#339311yescharacteristic
358C2×C7#339911yescharacteristic
359D7—82no—
360C23—142no—
361C23—281no—
362C23—281no—
363C23—281no—
364C23—281no—
365C23—281no—
366C23—281no—
367C23C2×D4×D711yescharacteristic
368D4—142no—
369D4—142no—
370D4—281no—
371D4—281no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—42no—
375C2×C4—41no—
376C2×C4—41no—
377C2×C4—42no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—42no—
382C2×C4—41no—
383C2×C4—142no—
384C2×C4—142no—
385C2×C4—142no—
386C2×C4—142no—
387C2×C4—142no—
388C2×C4—142no—
389C2×C4—142no—
390C2×C4—281no—
391C2×C4—281no—
392C2×C4—288no—
393C2×C4—282no—
394C2×C4—282no—
395C2×C4—282no—
396C2×C4#645912yes—
397C2×C4#645912yes—
398C2×C4#647712yes—
399C2×C4#647912yes—
400C2×C4C2×#115711yescharacteristic
401C2×C4#654811yescharacteristic
402C2×C4C2×Q8×D711yescharacteristic
403C2×C4#655111yescharacteristic
404C2×C4D7×#4111yescharacteristic
405C2×C4D7×#4111yescharacteristic
406C8—42no—
407C8—142no—
408C8—142no—
409C7#2608711yescharacteristic, complemented, Sylow
410C22—282no—
411C22—282no—
412C22—282no—
413C22—561no—
414C22—561no—
415C22—561no—
416C22—561no—
417C22C2×#645911yescharacteristic
418C22#9029011yescharacteristic
419C22#9032211yescharacteristic
420C22C2×#647711yescharacteristic
421C22C2×#647911yescharacteristic
422C22#9046511yescharacteristic
423C22#9047011yescharacteristic
424C4—22no—
425C4—81no—
426C4—81no—
427C4—82no—
428C4—282no—
429C4—282no—
430C4—282no—
431C4—562no—
432C4#8964314yes—
433C4#9024911yescharacteristic
434C4#9027211yescharacteristic
435C4#9037911yescharacteristic
436C4#9042011yescharacteristic
437C2—562no—
438C2#1173366911yescharacteristic
439C2C2×#8964311yescharacteristic
440C2#1173668811yescharacteristic
441C2#1173782411yescharacteristic
442C2#1173786611yescharacteristic
443C2#1173847711yescharacteristic
444C2#1173852711yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1948C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#1948D7
hypocenterC7#26087
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1948

Complements

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

NH ≅ G/Ntype
2. #11731170C2semidirect
3. C7×#1948C2semidirect
5. #11728318C2semidirect
7. #11728222C2semidirect
8. #11727331C2semidirect
11. #11726544C2semidirect
89. #1948D7semidirect
409. C7#26087semidirect

Maximal subgroups

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

subgroupindex
#117317692
#117311702
C7×#19482
#117283402
#117283182
#117282332
#117282222
#117273312
C4×#63832
#117266552
#117265442
#260877