Finite Group Databasebeta

← #11732397

Subgroups — #11732397

Order 896.

596 conjugacy classes of subgroups.

The subgroup lattice has 438 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6458C211yesmaximal, characteristic
2C7×#387C211yesmaximal, characteristic, complemented
3#89762C211yesmaximal, characteristic, complemented
4#89760C211yesmaximal, characteristic, complemented
5#89753C211yesmaximal, characteristic, complemented
6#89711C211yesmaximal, characteristic, complemented
7C4×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
8#89442C211yesmaximal, characteristic, complemented
9#89438C211yesmaximal, characteristic, complemented
10#89438C211yesmaximal, characteristic, complemented
11#89437C211yesmaximal, characteristic, complemented
12#89437C211yesmaximal, characteristic, complemented
13#89437C211yesmaximal, characteristic, complemented
14#89437C211yesmaximal, characteristic, complemented
15#89436C211yesmaximal, characteristic, complemented
16C22×D28C2211yescharacteristic
17C22×D28C2211yescharacteristic
18C22×D28C2211yescharacteristic
19C22×C4×D7C2211yescharacteristic
20C22×C4×D7C2211yescharacteristic
21C22×C4×D7C2211yescharacteristic
22C2×C7×C4⋊C4C2211yescharacteristic
23C2×C7×C4⋊C4C2211yescharacteristic
24C2×C42×C7C2211yescharacteristic
25C2×(C2×D7)⋊C4C2211yescharacteristic
26C2×(C2×D7)⋊C4C2211yescharacteristic
27C2×(C2×D7)⋊C4C2211yescharacteristic
28C2×(C2×D7)⋊C4C2211yescharacteristic
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×C4×Dic7C2211yescharacteristic
40C2×C4×Dic7C2211yescharacteristic
41#6458C418yescomplemented
42C7×#95C2211yescharacteristic
43C7×#95C2211yescharacteristic
44C7×#95C2211yescharacteristic
45C7×#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52#2333—71nomaximal, Sylow
53C23×D7C2311yescharacteristic
54C23×D7C2311yescharacteristic
55C23×D7C2311yescharacteristic
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×Dic7C2311yescharacteristic
64C22×Dic7C2311yescharacteristic
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C22×Dic7C2311yescharacteristic
68C2×D28—41no—
69C2×D28—41no—
70C2×D28—41no—
71C2×D28—41no—
72C2×D28—41no—
73C2×D28—41no—
74C2×D28C2×C414yes—
75C2×D28C2×C414yescomplemented
76C2×D28C2×C414yescomplemented
77C2×C4×D7—22no—
78C2×C4×D7—22no—
79C2×C4×D7—41no—
80C2×C4×D7—41no—
81C2×C4×D7—41no—
82C2×C4×D7—41no—
83C2×C4×D7—41no—
84C2×C4×D7—41no—
85C2×C4×D7C2×C414yescomplemented
86C7×C4⋊C4—41no—
87C7×C4⋊C4C2×C414yescomplemented
88C42×C7—41no—
89(C2×D7)⋊C4—22no—
90(C2×D7)⋊C4—22no—
91(C2×D7)⋊C4—22no—
92(C2×D7)⋊C4—22no—
93(C2×D7)⋊C4—41no—
94(C2×D7)⋊C4—41no—
95(C2×D7)⋊C4—41no—
96(C2×D7)⋊C4—41no—
97(C2×D7)⋊C4—41no—
98(C2×D7)⋊C4—41no—
99(C2×D7)⋊C4C2×C414yescomplemented
100(C2×D7)⋊C4C2×C414yescomplemented
101C4⋊Dic7—41no—
102C4×Dic7—41no—
103C4×Dic7—41no—
104C4×Dic7—41no—
105C2×#121—71no—
106#390—71no—
107#390—71no—
108#388—71no—
109#388—71no—
110#387—71no—
111#387D711yescharacteristic, complemented
112#386—71no—
113#386—71no—
114#386—71no—
115#386—71no—
116#381—71no—
117C4×C22⋊C4—71no—
118C4×C22⋊C4—71no—
119C4×C22⋊C4—71no—
120C23×C7C2411yescharacteristic
121C22×D7—21no—
122C22×D7—21no—
123C22×D7—21no—
124C22×D7—21no—
125C22×D7—21no—
126C22×D7—21no—
127C22×D7—21no—
128C22×D7—21no—
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×D7C22×C412yes—
140C22×D7C22×C412yes—
141C22×D7C22×C412yes—
142C2×C4×C7—21no—
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×C7C22×C412yes—
160C2×C4×C7C22×C412yes—
161C2×C4×C7C22×C412yes—
162C2×C4×C7#4112yes—
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×Dic7C22×C412yes—
177C2×Dic7#4112yes—
178D28—44no—
179D28—44no—
180D28—44no—
181C4×D7—44no—
182C4×D7—44no—
183C4×D7—44no—
184C22×D4—71no—
185C22×D4—71no—
186C22×D4—71no—
187C23×C4—71no—
188C23×C4—71no—
189C23×C4—71no—
190#121—78no—
191C2×C4⋊C4—71no—
192C2×C4⋊C4C2×D711yescharacteristic
193C2×C4⋊C4C2×D711yescharacteristic
194C2×C22⋊C4—71no—
195C2×C22⋊C4—71no—
196C2×C22⋊C4—71no—
197C2×C22⋊C4—71no—
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×C42—71no—
207C2×C42—71no—
208C2×C42—71no—
209C2×C42C2×D711yescharacteristic
210#95—71no—
211#95—71no—
212#95—71no—
213#95—71no—
214#95—71no—
215#95—71no—
216#95C2×D711yescharacteristic
217#95C2×D711yescharacteristic
218#95C2×D711yescharacteristic
219#95C2×D711yescharacteristic
220C22×C7C23×C411yescharacteristic
221C22×C7C2×#4111yescharacteristic
222C22×C7C2×#4111yescharacteristic
223C22×C7H(5,𝔽2)11yescharacteristic
224C22×C7H(5,𝔽2)11yescharacteristic
225C22×C7H(5,𝔽2)11yescharacteristic
226C22×C7H(5,𝔽2)11yescharacteristic
227C2×D7—22no—
228C2×D7—22no—
229C2×D7—22no—
230C2×D7—22no—
231C2×D7—22no—
232C2×D7—22no—
233C2×D7—22no—
234C2×D7—22no—
235C2×D7—22no—
236C2×D7—41no—
237C2×D7—41no—
238C2×D7—41no—
239C2×D7—41no—
240C2×D7—41no—
241C2×D7—41no—
242C2×D7—41no—
243C2×D7—41no—
244C2×D7—41no—
245C2×D7—41no—
246C2×D7—41no—
247C2×D7—41no—
248C4×C7—22no—
249C4×C7—22no—
250C4×C7—22no—
251C4×C7—22no—
252C4×C7—41no—
253C4×C7—41no—
254C4×C7—41no—
255Dic7—22no—
256Dic7—22no—
257Dic7—41no—
258Dic7—41no—
259Dic7—41no—
260C24—71no—
261C24—71no—
262C24—71no—
263C2×D4—74no—
264C2×D4—74no—
265C2×D4—74no—
266C2×D4—281no—
267C2×D4—281no—
268C2×D4—281no—
269C2×D4—281no—
270C2×D4—281no—
271C2×D4—281no—
272C22×C4—71no—
273C22×C4—71no—
274C22×C4—74no—
275C22×C4—71no—
276C22×C4—71no—
277C22×C4—71no—
278C22×C4—142no—
279C22×C4—142no—
280C22×C4—281no—
281C22×C4—281no—
282C22×C4—281no—
283C22×C4—281no—
284C22×C4—281no—
285C22×C4—281no—
286C22×C4C22×D711yescharacteristic
287C22×C4C22×D711yescharacteristic
288C22×C4C22×D711yescharacteristic
289C22×C4C22×D711yescharacteristic
290C22×C4C22×D711yescharacteristic
291C22×C4C22×D711yescharacteristic
292C22×C4C22×D711yescharacteristic
293C4⋊C4—41no—
294C4⋊C4—281no—
295C4⋊C4C4×D714yescomplemented
296C22⋊C4—74no—
297C22⋊C4—74no—
298C22⋊C4—142no—
299C22⋊C4—142no—
300C22⋊C4—142no—
301C22⋊C4—142no—
302C22⋊C4—281no—
303C22⋊C4—281no—
304C22⋊C4—281no—
305C22⋊C4—281no—
306C22⋊C4—281no—
307C22⋊C4—281no—
308C42—41no—
309C42—281no—
310C42—281no—
311C42—281no—
312C2×C7C4×#4111yescharacteristic
313C2×C7#51811yescharacteristic
314C2×C7#51811yescharacteristic
315C2×C7#53811yescharacteristic
316C2×C7#53811yescharacteristic
317C2×C7#54011yescharacteristic
318C2×C7#54011yescharacteristic
319D7—42no—
320D7—42no—
321D7—42no—
322C23—72no—
323C23—72no—
324C23—72no—
325C23—141no—
326C23—141no—
327C23—141no—
328C23—141no—
329C23—141no—
330C23—141no—
331C23—141no—
332C23—141no—
333C23—141no—
334C23—141no—
335C23—141no—
336C23—141no—
337C23—141no—
338C23—141no—
339C23—141no—
340C23—141no—
341C23—141no—
342C23—141no—
343C23C23×D711yescharacteristic
344D4—284no—
345D4—284no—
346D4—284no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
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—72no—
365C2×C4—72no—
366C2×C4—141no—
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—284no—
380C2×C4—284no—
381C2×C4—284no—
382C2×C4C2×C4×D712yes—
383C2×C4C2×C4×D712yes—
384C2×C4C2×C4×D712yes—
385C2×C4#116112yes—
386C7#233311yescharacteristic, complemented, Sylow
387C22—142no—
388C22—142no—
389C22—142no—
390C22—142no—
391C22—142no—
392C22—142no—
393C22—142no—
394C22—142no—
395C22—142no—
396C22—281no—
397C22—281no—
398C22—281no—
399C22—281no—
400C22—281no—
401C22—281no—
402C22—281no—
403C22—281no—
404C22—281no—
405C22—281no—
406C22—281no—
407C22—281no—
408C22C22×C4×D711yescharacteristic
409C22#654811yescharacteristic
410C22C2×#116111yescharacteristic
411C22D7×#4111yescharacteristic
412C22#655311yescharacteristic
413C22#655311yescharacteristic
414C22#655311yescharacteristic
415C4—22no—
416C4—22no—
417C4—22no—
418C4—22no—
419C4—41no—
420C4—41no—
421C4—41no—
422C4—142no—
423C4—142no—
424C4—281no—
425C4—281no—
426C4—281no—
427C2—282no—
428C2—282no—
429C2—282no—
430C2#9021211yescharacteristic
431C2#9023611yescharacteristic
432C2C4×#116111yescharacteristic
433C2#9033511yescharacteristic
434C2#9035411yescharacteristic
435C2#9038911yescharacteristic
436C2#9039511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutatorC22×C7C23×C4
FittingC7×#387C2
socle, cosocleC23×C7C24
prime radicalC2×#6458C2
hypercenter#387D7
hypocenterC7#2333
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
2. C7×#387C2semidirect
3. #89762C2semidirect
4. #89760C2semidirect
5. #89753C2semidirect
6. #89711C2semidirect
7. C4×(C2×D7)⋊C4C2semidirect
8. #89442C2semidirect
9. #89438C2semidirect
10. #89438C2semidirect
11. #89437C2semidirect
12. #89437C2semidirect
13. #89437C2semidirect
14. #89437C2semidirect
15. #89436C2semidirect
41. #6458C4semidirect
75. C2×D28C2×C4semidirect
76. C2×D28C2×C4semidirect
85. C2×C4×D7C2×C4semidirect
87. C7×C4⋊C4C2×C4semidirect
99. (C2×D7)⋊C4C2×C4semidirect
100. (C2×D7)⋊C4C2×C4semidirect
111. #387D7semidirect
295. C4⋊C4C4×D7semidirect
386. C7#2333semidirect

Maximal subgroups

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

subgroupindex
C2×#64582
C7×#3872
#897622
#897602
#897532
#897112
C4×(C2×D7)⋊C42
#894422
#894382
#894372
#894362
#23337