Finite Group Databasebeta

← #13412759

Subgroups — #13412759

Order 1792.

750 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
1#11731769C211yesmaximal, characteristic
2#11731161C211yesmaximal, characteristic, complemented
3C7×#1947C211yesmaximal, characteristic, complemented
4#11728395C212yesmaximal
5#11728369C212yesmaximal, complemented
6#11728233C211yesmaximal, characteristic
7#11728222C211yesmaximal, characteristic, complemented
8#11727331C211yesmaximal, characteristic, complemented
9C4×#6383C211yesmaximal, characteristic
10#11726655C212yesmaximal
11#11726541C212yesmaximal, complemented
12C2×#6455—21no—
13C2×C4×D28C2211yescharacteristic
14C2×#6433—21no—
15C2×C7×#107C2212yes—
16C2×C7×C4⋊C8C2211yescharacteristic
17C7×#384C2211yescharacteristic
18C4×C7×C4⋊C4C2211yescharacteristic
19C2×#6395C2212yes—
20C2×Dic7⋊C8C2212yes—
21#89762—22no—
22#89760C2212yes—
23#89759—21no—
24#89748C2212yes—
25C4⋊C4×Dic7C2212yes—
26#89745C2212yes—
27C2×#6383C2211yescharacteristic
28C2×#6383C2211yescharacteristic
29C2×#6381C2212yes—
30C2×#6381C2212yes—
31#89707C2211yescharacteristic
32C4×C4⋊Dic7C2211yescharacteristic
33#89700—21no—
34C2×C4×C7⋊C8C2211yescharacteristic
35#89429—22no—
36C7×#340C2212yes—
37#89346C2212yes—
38#89345C2212yes—
39#89324C2211yescharacteristic
40#89324C2211yescharacteristic
41#26086—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×Dic7⋊C4—22no—
63C2×Dic7⋊C4—22no—
64C2×C4×Dic7—22no—
65C2×C4×Dic7C2312yes—
66C22×C7⋊C8C2311yescharacteristic
67C22×C7⋊C8C2311yescharacteristic
68#6455—44no—
69C4×D28—22no—
70C4×D28Q814yes—
71#6433—44no—
72C7×#107—42no—
73C7×C4⋊C8—22no—
74C7×#95—21no—
75C7⋊#95—22no—
76C7⋊#95—22no—
77C7⋊#95D412yes—
78#6395—42no—
79Dic7⋊C8—24no—
80#6383—22no—
81#6383—22no—
82#6381—24no—
83#6381—42no—
84C4×C7⋊C8—22no—
85#2552—71no—
86#2354—71no—
87#2093—72no—
88#2067—72no—
89#1947—72no—
90#1947D711yescharacteristic, complemented
91#1939—71no—
92#1918—72no—
93#1845—71no—
94#1802—71no—
95C4×#102—71no—
96C23×D7—21no—
97C22×C4×C7—21no—
98C22×C4×C7—22no—
99C22×C4×C7—21no—
100C22×C4×C7C2×D411yescharacteristic
101C22×C4×C7C2×D411yescharacteristic
102C22×C4×C7C2411yescharacteristic
103C22×Dic7—21no—
104C22×Dic7—22no—
105C22×Dic7C2×D412yes—
106C22×Dic7C2×D412yes—
107C2×D28—21no—
108C2×D28—21no—
109C2×D28—21no—
110C2×D28—21no—
111C2×D28—21no—
112C2×D28C2×Q812yes—
113C2×C4×D7—24no—
114C2×C4×D7—42no—
115C2×C4×D7—42no—
116C2×C7×C8—22no—
117C2×C7×C8—22no—
118C2×C7×C8—22no—
119C7×C4⋊C4—22no—
120C7×C4⋊C4—21no—
121C7×C4⋊C4—21no—
122C7×C4⋊C4—21no—
123C7×C4⋊C4—22no—
124C7×C4⋊C4—21no—
125C7×C4⋊C4—42no—
126C7×C4⋊C4—81no—
127C42×C7—42no—
128C42×C7C2×Q812yes—
129C42×C7#4112yes—
130(C2×D7)⋊C4—24no—
131(C2×D7)⋊C4—44no—
132(C2×D7)⋊C4—82no—
133C4⋊Dic7—22no—
134C4⋊Dic7—21no—
135C4⋊Dic7—22no—
136C4⋊Dic7—22no—
137C4⋊Dic7—42no—
138C4⋊Dic7—81no—
139C4⋊Dic7D814yes—
140C4⋊Dic7C2×Q812yes—
141Dic7⋊C4—44no—
142Dic7⋊C4—44no—
143Dic7⋊C4—82no—
144C4×Dic7—22no—
145C4×Dic7—22no—
146C4×Dic7—44no—
147C2×C7⋊C8—21no—
148C2×C7⋊C8—21no—
149C2×C7⋊C8—21no—
150C2×C7⋊C8—21no—
151C2×C7⋊C8#4112yes—
152C2×C7⋊C8#4112yes—
153C2×#125—141no—
154C2×C4×D4—71no—
155C2×#117—141no—
156C2×#107—72no—
157C2×#107—72no—
158C2×#107C2×D712yes—
159C2×C4⋊C8—72no—
160C2×C4⋊C8C2×D711yescharacteristic
161C2×#102—71no—
162C2×#102—71no—
163C2×#102—72no—
164C2×C4×C8—71no—
165#398—142no—
166#390—72no—
167#389—72no—
168#389—141no—
169#388—142no—
170#384—72no—
171#384—71no—
172#384C2×D711yescharacteristic
173#380—141no—
174C4×C4⋊C4—71no—
175C4×C4⋊C4—72no—
176C4×C4⋊C4C2×D711yescharacteristic
177#340—72no—
178#340—71no—
179#340—71no—
180#340C2×D712yes—
181#336—72no—
182C23×C7C22×D411yescharacteristic
183C22×D7—22no—
184C22×D7—41no—
185C22×D7—41no—
186C22×D7—41no—
187C22×D7—41no—
188C22×D7—41no—
189C22×D7—41no—
190C2×C4×C7—21no—
191C2×C4×C7—21no—
192C2×C4×C7—41no—
193C2×C4×C7—42no—
194C2×C4×C7—41no—
195C2×C4×C7—42no—
196C2×C4×C7—41no—
197C2×C4×C7—41no—
198C2×C4×C7—41no—
199C2×C4×C7—42no—
200C2×C4×C7—41no—
201C2×C4×C7C22⋊Q812yes—
202C2×C4×C7C22⋊Q812yes—
203C2×C4×C7C2×D812yes—
204C2×C4×C7#13712yes—
205C2×C4×C7C22×Q811yescharacteristic
206C2×C4×C7C2×#4111yescharacteristic
207C2×C4×C7C2×#4111yescharacteristic
208C2×C4×C7C2×#4111yescharacteristic
209C2×C4×C7H(5,𝔽2)11yescharacteristic
210C2×C4×C72−1+411yescharacteristic
211C2×Dic7—22no—
212C2×Dic7—22no—
213C2×Dic7—22no—
214C2×Dic7—22no—
215C2×Dic7—22no—
216C2×Dic7—22no—
217C2×Dic7—41no—
218C2×Dic7—41no—
219C2×Dic7—42no—
220C2×Dic7—42no—
221C2×Dic7—42no—
222C2×Dic7C2×D814yes—
223D28—22no—
224D28—22no—
225D28—41no—
226D28—41no—
227C4×D7—48no—
228C7×C8—42no—
229C7⋊C8—22no—
230C7⋊C8—22no—
231C22×D4—71no—
232C23×C4—141no—
233C22×C8—71no—
234C22×C8—71no—
235C22×C8C22×D712yes—
236#125—284no—
237C4×D4—74no—
238C4×D4—142no—
239#117—284no—
240C2×C4⋊C4—21no—
241C2×C4⋊C4—21no—
242C2×C4⋊C4—71no—
243C2×C4⋊C4—71no—
244C2×C4⋊C4—71no—
245C2×C4⋊C4—72no—
246C2×C4⋊C4—142no—
247C2×C4⋊C4—142no—
248C2×C4⋊C4—141no—
249C2×C4⋊C4—142no—
250C2×C4⋊C4—141no—
251C2×C4⋊C4C22×D711yescharacteristic
252C2×C4⋊C4C22×D711yescharacteristic
253C2×C4⋊C4C22×D712yes—
254C2×C22⋊C4—141no—
255C2×C22⋊C4—142no—
256C2×C22⋊C4—142no—
257C2×C42—21no—
258C2×C42—72no—
259C2×C42—142no—
260C2×C42C22×D711yescharacteristic
261#107—42no—
262#107—144no—
263#107—282no—
264C4⋊C8—22no—
265C4⋊C8—144no—
266#102—142no—
267#102—142no—
268#102—282no—
269C4×C8—142no—
270#95—21no—
271#95—72no—
272#95—142no—
273#95—142no—
274C22×C7C2×C22⋊Q811yescharacteristic
275C22×C7#54711yescharacteristic
276C22×C7#55011yescharacteristic
277C22×C7C22×D811yescharacteristic
278C22×C7C2×#13711yescharacteristic
279C22×C7#57611yescharacteristic
280C22×C7#57811yescharacteristic
281C2×D7—42no—
282C2×D7—42no—
283C2×D7—42no—
284C2×D7—81no—
285C2×D7—81no—
286C2×D7—81no—
287C2×D7—81no—
288C4×C7—22no—
289C4×C7—81no—
290C4×C7—82no—
291C4×C7—81no—
292C4×C7#47414yes—
293C4×C7#53311yescharacteristic
294C4×C7#54411yescharacteristic
295C4×C7#55211yescharacteristic
296C4×C7#55311yescharacteristic
297Dic7—24no—
298Dic7—42no—
299Dic7—42no—
300Dic7—82no—
301C24—141no—
302C2×D4—72no—
303C2×D4—141no—
304C2×D4—141no—
305C2×D4—141no—
306C2×D4—141no—
307C2×D4—141no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—22no—
311C22×C4—72no—
312C22×C4—72no—
313C22×C4—141no—
314C22×C4—144no—
315C22×C4—142no—
316C22×C4—282no—
317C22×C4—282no—
318C22×C4D4×D711yescharacteristic
319C22×C4D4×D711yescharacteristic
320C22×C4C23×D711yescharacteristic
321C2×C8—22no—
322C2×C8—22no—
323C2×C8—22no—
324C2×C8—72no—
325C2×C8—72no—
326C2×C8—141no—
327C2×C8—141no—
328C2×C8—141no—
329C2×C8—141no—
330C4⋊C4—21no—
331C4⋊C4—22no—
332C4⋊C4—21no—
333C4⋊C4—21no—
334C4⋊C4—21no—
335C4⋊C4—22no—
336C4⋊C4—42no—
337C4⋊C4—74no—
338C4⋊C4—72no—
339C4⋊C4—81no—
340C4⋊C4—142no—
341C4⋊C4—141no—
342C4⋊C4—142no—
343C4⋊C4—142no—
344C4⋊C4—284no—
345C4⋊C4—284no—
346C4⋊C4—282no—
347C4⋊C4—561no—
348C4⋊C4—562no—
349C22⋊C4—144no—
350C22⋊C4—284no—
351C22⋊C4—562no—
352C42—42no—
353C42—142no—
354C42—142no—
355C42—284no—
356C42#115712yes—
357C42Q8×D712yes—
358C2×C7#244111yescharacteristic
359C2×C7C2×#47411yescharacteristic
360C2×C7#308311yescharacteristic
361C2×C7#333411yescharacteristic
362C2×C7#335711yescharacteristic
363C2×C7#339111yescharacteristic
364C2×C7#339811yescharacteristic
365D7—82no—
366C23—142no—
367C23—281no—
368C23—281no—
369C23—281no—
370C23—281no—
371C23—281no—
372C23—281no—
373C23C2×D4×D711yescharacteristic
374D4—142no—
375D4—142no—
376D4—281no—
377D4—281no—
378C2×C4—21no—
379C2×C4—21no—
380C2×C4—41no—
381C2×C4—41no—
382C2×C4—42no—
383C2×C4—42no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—42no—
388C2×C4—41no—
389C2×C4—74no—
390C2×C4—142no—
391C2×C4—142no—
392C2×C4—142no—
393C2×C4—142no—
394C2×C4—142no—
395C2×C4—142no—
396C2×C4—281no—
397C2×C4—281no—
398C2×C4—288no—
399C2×C4—282no—
400C2×C4—282no—
401C2×C4—282no—
402C2×C4#645912yes—
403C2×C4#645912yes—
404C2×C4D7×D812yes—
405C2×C4#648112yes—
406C2×C4C2×#115711yescharacteristic
407C2×C4#654811yescharacteristic
408C2×C4C2×Q8×D711yescharacteristic
409C2×C4#655111yescharacteristic
410C2×C4D7×#4111yescharacteristic
411C2×C4D7×#4111yescharacteristic
412C8—42no—
413C8—142no—
414C8—142no—
415C7#2608611yescharacteristic, complemented, Sylow
416C22—282no—
417C22—282no—
418C22—282no—
419C22—561no—
420C22—561no—
421C22—561no—
422C22—561no—
423C22C2×#645911yescharacteristic
424C22#9029011yescharacteristic
425C22#9032211yescharacteristic
426C22C2×D7×D811yescharacteristic
427C22C2×#648111yescharacteristic
428C22#9046511yescharacteristic
429C22#9047011yescharacteristic
430C4—22no—
431C4—81no—
432C4—81no—
433C4—82no—
434C4—144no—
435C4—282no—
436C4—282no—
437C4—562no—
438C4#8966214yes—
439C4#9024911yescharacteristic
440C4#9027211yescharacteristic
441C4#9037911yescharacteristic
442C4#9042011yescharacteristic
443C2—562no—
444C2#1173366911yescharacteristic
445C2C2×#8966211yescharacteristic
446C2#1173668811yescharacteristic
447C2#1173776911yescharacteristic
448C2#1173791811yescharacteristic
449C2#1173843111yescharacteristic
450C2#1173857511yescharacteristic
subgroupG/H
centerC23C2×D4×D7
commutator, cosocleC22×C4×C7C24
FrattiniC22×C4C23×D7
FittingC7×#1947C2
socleC23×C7C22×D4
prime radicalC22×D28C23
hypercenter#1947D7
hypocenterC7#26086
LCSG ▷ C22×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1947

Complements

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

NH ≅ G/Ntype
2. #11731161C2semidirect
3. C7×#1947C2semidirect
5. #11728369C2semidirect
7. #11728222C2semidirect
8. #11727331C2semidirect
11. #11726541C2semidirect
90. #1947D7semidirect
415. C7#26086semidirect

Maximal subgroups

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

subgroupindex
#117317692
#117311612
C7×#19472
#117283952
#117283692
#117282332
#117282222
#117273312
C4×#63832
#117266552
#117265412
#260867