Finite Group Databasebeta

← #11733366

Subgroups — #11733366

Order 896.

562 conjugacy classes of subgroups.

The subgroup lattice has 440 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6447C211yesmaximal, characteristic
2C2×#6438C211yesmaximal, characteristic, complemented
3C7×#388C211yesmaximal, characteristic, complemented
4#89761C211yesmaximal, characteristic, complemented
5#89759C211yesmaximal, characteristic, complemented
6#89733C211yesmaximal, characteristic, complemented
7#89733C211yesmaximal, characteristic, complemented
8#89731C211yesmaximal, characteristic, complemented
9#89728C211yesmaximal, characteristic, complemented
10#89719C211yesmaximal, characteristic, complemented
11#89705C211yesmaximal, characteristic, complemented
12#89446C211yesmaximal, characteristic, complemented
13#89437C211yesmaximal, characteristic, complemented
14#89436C211yesmaximal, characteristic, complemented
15#89428C211yesmaximal, characteristic, complemented
16C22×C22⋊D7C2211yescharacteristic
17C22×C22⋊D7C2211yescharacteristic
18C22×D28C2211yescharacteristic
19C22×C4×D7C2211yescharacteristic
20C22×C4×D7C2211yescharacteristic
21C22×Dic14C2211yescharacteristic
22C2×C7×C4⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C42×C7C2211yescharacteristic
27C2×C22⋊Dic7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×(C2×D7)⋊C4C2211yescharacteristic
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×(C2×D7)⋊C4C2211yescharacteristic
33C2×(C2×D7)⋊C4C2211yescharacteristic
34C2×(C2×D7)⋊C4C2211yescharacteristic
35C2×(C2×D7)⋊C4C2211yescharacteristic
36C2×(C2×D7)⋊C4C2211yescharacteristic
37C2×C4⋊Dic7C2211yescharacteristic
38C2×C4⋊Dic7C2211yescharacteristic
39C2×Dic7⋊C4C2211yescharacteristic
40C2×Dic7⋊C4C2211yescharacteristic
41#6447—24no—
42#6438—24no—
43C7×#95C2211yescharacteristic
44C7×#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53#2686—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C23×D7C2311yescharacteristic
56C23×D7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C2×C22⋊D7—22no—
64C2×C22⋊D7—22no—
65C2×C22⋊D7—22no—
66C2×C22⋊D7—41no—
67C2×C22⋊D7—41no—
68C2×C22⋊D7—41no—
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C22×Dic7C2311yescharacteristic
75C2×D28—41no—
76C2×D28—41no—
77C2×D28D414yes—
78C2×C4×D7—22no—
79C2×C4×D7—22no—
80C2×C4×D7—22no—
81C2×C4×D7—41no—
82C2×C4×D7—41no—
83C2×C4×D7—41no—
84C2×Dic14—22no—
85C2×Dic14—41no—
86C2×Dic14—41no—
87C7×C4⋊C4—41no—
88C7×C22⋊C4—22no—
89C7×C22⋊C4—22no—
90C7×C22⋊C4—41no—
91C42×C7—22no—
92C22⋊Dic7—41no—
93C22⋊Dic7—41no—
94(C2×D7)⋊C4—22no—
95(C2×D7)⋊C4—22no—
96(C2×D7)⋊C4—22no—
97(C2×D7)⋊C4—22no—
98(C2×D7)⋊C4—22no—
99(C2×D7)⋊C4—22no—
100(C2×D7)⋊C4—41no—
101(C2×D7)⋊C4—41no—
102C4⋊Dic7—41no—
103C4⋊Dic7—41no—
104Dic7⋊C4—22no—
105Dic7⋊C4—22no—
106C2×#124—71no—
107C2×#121—71no—
108#397—71no—
109#395—71no—
110#394—71no—
111#394—71no—
112#394—71no—
113#393—71no—
114#388D711yescharacteristic, complemented
115#387—71no—
116#386—71no—
117#386—71no—
118#385—71no—
119#381—71no—
120#380—71no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7C2411yescharacteristic
129C22×D7—21no—
130C22×D7—21no—
131C22×D7—21no—
132C22×D7—21no—
133C22×D7—21no—
134C22×D7—21no—
135C22×D7—21no—
136C22×D7—21no—
137C22×D7—21no—
138C22×D7—21no—
139C22×D7—21no—
140C22×D7—21no—
141C22×D7C2×D412yes—
142C22×D7C2×D412yes—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7C2×D412yes—
160C2×C4×C7#4112yes—
161C22⋊D7—44no—
162C22⋊D7—44no—
163C2×Dic7—21no—
164C2×Dic7—21no—
165C2×Dic7—21no—
166C2×Dic7—21no—
167C2×Dic7—21no—
168C2×Dic7—21no—
169C2×Dic7—21no—
170C2×Dic7—21no—
171C2×Dic7—21no—
172C2×Dic7—21no—
173C2×Dic7—21no—
174C2×Dic7—21no—
175C2×Dic7—21no—
176C2×Dic7—21no—
177C2×Dic7—21no—
178C2×Dic7—21no—
179C2×Dic7—21no—
180C2×Dic7#4112yes—
181D28—44no—
182C4×D7—44no—
183C4×D7—44no—
184Dic14—44no—
185C22×Q8—71no—
186C22×D4—71no—
187C22×D4—71no—
188C22×D4—71no—
189C23×C4—71no—
190C23×C4—71no—
191#124—144no—
192#121—144no—
193C2×C4⋊C4—71no—
194C2×C4⋊C4—71no—
195C2×C4⋊C4—71no—
196C2×C4⋊C4—71no—
197C2×C4⋊C4C2×D711yescharacteristic
198C2×C22⋊C4—71no—
199C2×C22⋊C4—71no—
200C2×C22⋊C4—71no—
201C2×C22⋊C4—71no—
202C2×C22⋊C4—71no—
203C2×C22⋊C4—71no—
204C2×C22⋊C4—71no—
205C2×C22⋊C4—71no—
206C2×C22⋊C4—71no—
207C2×C22⋊C4—71no—
208C2×C22⋊C4C2×D711yescharacteristic
209C2×C22⋊C4C2×D711yescharacteristic
210C2×C22⋊C4C2×D711yescharacteristic
211C2×C42C2×D711yescharacteristic
212#95—71no—
213#95—71no—
214#95—71no—
215#95—71no—
216#95—71no—
217#95—71no—
218#95—71no—
219#95—71no—
220#95C2×D711yescharacteristic
221#95C2×D711yescharacteristic
222C22×C7—41no—
223C22×C7—41no—
224C22×C7—41no—
225C22×C7—41no—
226C22×C7—41no—
227C22×C7—41no—
228C22×C7—41no—
229C22×C7C22×D411yescharacteristic
230C22×C7C2×#4111yescharacteristic
231C22×C7C2×#4111yescharacteristic
232C22×C7H(5,𝔽2)11yescharacteristic
233C22×C7H(5,𝔽2)11yescharacteristic
234C22×C7H(5,𝔽2)11yescharacteristic
235C22×C72−1+411yescharacteristic
236C2×D7—22no—
237C2×D7—22no—
238C2×D7—22no—
239C2×D7—22no—
240C2×D7—22no—
241C2×D7—22no—
242C2×D7—41no—
243C2×D7—41no—
244C2×D7—41no—
245C2×D7—41no—
246C2×D7—41no—
247C2×D7—41no—
248C2×D7—41no—
249C2×D7—41no—
250C4×C7—22no—
251C4×C7—22no—
252C4×C7—41no—
253C4×C7—41no—
254C4×C7—41no—
255C4×C7—41no—
256Dic7—22no—
257Dic7—41no—
258Dic7—41no—
259Dic7—41no—
260Dic7—41no—
261Dic7—41no—
262C24—71no—
263C24—71no—
264C24C22×D711yescharacteristic
265C2×Q8—142no—
266C2×Q8—281no—
267C2×Q8—281no—
268C2×D4—74no—
269C2×D4—142no—
270C2×D4—142no—
271C2×D4—142no—
272C2×D4—281no—
273C2×D4—281no—
274C2×D4—281no—
275C2×D4—281no—
276C2×D4—281no—
277C22×C4—71no—
278C22×C4—71no—
279C22×C4—71no—
280C22×C4—71no—
281C22×C4—71no—
282C22×C4—71no—
283C22×C4—142no—
284C22×C4—142no—
285C22×C4—142no—
286C22×C4—281no—
287C22×C4—281no—
288C22×C4—281no—
289C22×C4C22×D711yescharacteristic
290C22×C4C22×D711yescharacteristic
291C22×C4C22×D711yescharacteristic
292C22×C4C22×D711yescharacteristic
293C22×C4C22×D711yescharacteristic
294C22×C4C22×D711yescharacteristic
295C4⋊C4—41no—
296C4⋊C4—142no—
297C4⋊C4—142no—
298C4⋊C4—281no—
299C4⋊C4—281no—
300C22⋊C4—22no—
301C22⋊C4—22no—
302C22⋊C4—41no—
303C22⋊C4—142no—
304C22⋊C4—142no—
305C22⋊C4—142no—
306C22⋊C4—142no—
307C22⋊C4—142no—
308C22⋊C4—142no—
309C22⋊C4—281no—
310C22⋊C4—281no—
311C22⋊C4—281no—
312C22⋊C4—281no—
313C42—22no—
314C2×C7—81no—
315C2×C7#53411yescharacteristic
316C2×C7#53811yescharacteristic
317C2×C7#54211yescharacteristic
318C2×C7#54611yescharacteristic
319C2×C7#54711yescharacteristic
320C2×C7#55511yescharacteristic
321C2×C7#56211yescharacteristic
322D7—42no—
323D7—42no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—72no—
332C23—72no—
333C23—141no—
334C23—141no—
335C23—141no—
336C23—141no—
337C23—141no—
338C23—141no—
339C23—141no—
340C23—141no—
341C23—141no—
342C23—141no—
343C23—141no—
344C23—141no—
345C23C23×D711yescharacteristic
346Q8—284no—
347D4—284no—
348D4—284no—
349D4—284no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—72no—
367C2×C4—141no—
368C2×C4—141no—
369C2×C4—141no—
370C2×C4—141no—
371C2×C4—141no—
372C2×C4—141no—
373C2×C4—141no—
374C2×C4—141no—
375C2×C4—141no—
376C2×C4—141no—
377C2×C4—141no—
378C2×C4—141no—
379C2×C4—141no—
380C2×C4—141no—
381C2×C4—141no—
382C2×C4—141no—
383C2×C4—141no—
384C2×C4—284no—
385C2×C4—284no—
386C2×C4#115712yes—
387C2×C4D4×D712yes—
388C7#268611yescharacteristic, complemented, Sylow
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—142no—
397C22—142no—
398C22—142no—
399C22—142no—
400C22—142no—
401C22—142no—
402C22—281no—
403C22—281no—
404C22—281no—
405C22—281no—
406C22—281no—
407C22—281no—
408C22—281no—
409C22—281no—
410C22C2×#115711yescharacteristic
411C22C2×D4×D711yescharacteristic
412C22#654811yescharacteristic
413C22D7×#4111yescharacteristic
414C22#655311yescharacteristic
415C22#655311yescharacteristic
416C22#655411yescharacteristic
417C4—22no—
418C4—22no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—41no—
423C4—142no—
424C4—281no—
425C4—281no—
426C4—281no—
427C4—281no—
428C4—281no—
429C2—81no—
430C2—282no—
431C2—282no—
432C2#9022311yescharacteristic
433C2#9024211yescharacteristic
434C2#9025111yescharacteristic
435C2#9030311yescharacteristic
436C2#9034411yescharacteristic
437C2#9037511yescharacteristic
438C2#9040111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#388C2
prime radicalC2×#6447C2
hypercenter#388D7
hypocenterC7#2686
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
2. C2×#6438C2semidirect
3. C7×#388C2semidirect
4. #89761C2semidirect
5. #89759C2semidirect
6. #89733C2semidirect
7. #89733C2semidirect
8. #89731C2semidirect
9. #89728C2semidirect
10. #89719C2semidirect
11. #89705C2semidirect
12. #89446C2semidirect
13. #89437C2semidirect
14. #89436C2semidirect
15. #89428C2semidirect
114. #388D7semidirect
388. C7#2686semidirect

Maximal subgroups

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

subgroupindex
C2×#64472
C2×#64382
C7×#3882
#897612
#897592
#897332
#897312
#897282
#897192
#897052
#894462
#894372
#894362
#894282
#26867