Finite Group Databasebeta

← #11733744

Subgroups — #11733744

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×#6502C211yesmaximal, characteristic
2C7×#390C211yesmaximal, characteristic, complemented
3#89992C211yesmaximal, characteristic, complemented
4#89762C211yesmaximal, characteristic, complemented
5#89750C211yesmaximal, characteristic, complemented
6#89730C211yesmaximal, characteristic, complemented
7#89729C211yesmaximal, characteristic, complemented
8#89728C211yesmaximal, characteristic, complemented
9#89728C211yesmaximal, characteristic, complemented
10#89716C211yesmaximal, characteristic, complemented
11#89715C211yesmaximal, characteristic, complemented
12C22⋊C4×Dic7C211yesmaximal, characteristic, complemented
13C22⋊C4×Dic7C211yesmaximal, characteristic, complemented
14C4×(C2×D7)⋊C4C211yesmaximal, characteristic, complemented
15#89705C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C22×C22⋊D7C2211yescharacteristic
18C22×C22⋊D7C2211yescharacteristic
19C23×Dic7C2211yescharacteristic
20C23×Dic7C2211yescharacteristic
21C22×C4×D7C2211yescharacteristic
22C2×C7×C4⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C7×C22⋊C4C2211yescharacteristic
27C2×C42×C7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
31C2×C22⋊Dic7C2211yescharacteristic
32#6502C418yescomplemented
33C2×(C2×D7)⋊C4C2211yescharacteristic
34C2×(C2×D7)⋊C4C2211yescharacteristic
35C2×(C2×D7)⋊C4C2211yescharacteristic
36C2×(C2×D7)⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39C2×C4×Dic7C2211yescharacteristic
40C2×C4×Dic7C2211yescharacteristic
41C2×C4×Dic7C2211yescharacteristic
42C7⋊#95C2211yescharacteristic
43C7⋊#95C2211yescharacteristic
44C7⋊#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52#2333—71nomaximal, Sylow
53C24×C7C2311yescharacteristic
54C24×C7C2311yescharacteristic
55C23×D7C2311yescharacteristic
56C2×C7×D4—41no—
57C2×C7×D4—41no—
58C2×C7×D4C2×C414yescomplemented
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C2×C22⋊D7—41no—
65C2×C22⋊D7—41no—
66C2×C22⋊D7—41no—
67C2×C22⋊D7—41no—
68C2×C22⋊D7C2×C414yes—
69C2×C22⋊D7C2×C414yescomplemented
70C22×Dic7—22no—
71C22×Dic7—41no—
72C22×Dic7—41no—
73C22×Dic7—41no—
74C22×Dic7—41no—
75C22×Dic7C2×C414yescomplemented
76C22×Dic7C2311yescharacteristic
77C22×Dic7C2311yescharacteristic
78C22×Dic7C2311yescharacteristic
79C22×Dic7C2311yescharacteristic
80C22×Dic7C2311yescharacteristic
81C22×Dic7C2311yescharacteristic
82C22×Dic7C2311yescharacteristic
83C2×C4×D7—22no—
84C2×C4×D7—41no—
85C2×C4×D7—41no—
86C7×C4⋊C4—41no—
87C7×C22⋊C4—22no—
88C7×C22⋊C4—22no—
89C7×C22⋊C4—41no—
90C7×C22⋊C4—41no—
91C42×C7—41no—
92C22⋊Dic7—22no—
93C22⋊Dic7—41no—
94C22⋊Dic7—41no—
95C22⋊Dic7C2×C414yescomplemented
96(C2×D7)⋊C4—22no—
97(C2×D7)⋊C4—41no—
98(C2×D7)⋊C4—41no—
99(C2×D7)⋊C4C2×C414yescomplemented
100Dic7⋊C4—41no—
101Dic7⋊C4C2×C414yescomplemented
102C4×Dic7—41no—
103C4×Dic7—41no—
104C4×Dic7—41no—
105C2×#121—71no—
106#390—71no—
107#390D711yescharacteristic, complemented
108#388—71no—
109#388—71no—
110#387—71no—
111#387—71no—
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×C7—21no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—21no—
130C23×C7—21no—
131C23×C7—21no—
132C23×C7C22×C412yes—
133C23×C7C22×C412yes—
134C23×C7C2411yescharacteristic
135C22×D7—21no—
136C22×D7—21no—
137C22×D7—21no—
138C22×D7—21no—
139C22×D7—21no—
140C22×D7—21no—
141C22×D7C22×C412yes—
142C7×D4—44no—
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×C7C22×C412yes—
157C2×C4×C7#4112yes—
158C22⋊D7—44no—
159C22⋊D7—44no—
160C2×Dic7—21no—
161C2×Dic7—21no—
162C2×Dic7—21no—
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—44no—
178C2×Dic7—44no—
179C2×Dic7C22×C412yes—
180C2×Dic7C22×C412yes—
181C2×Dic7C22×C412yes—
182C2×Dic7#4112yes—
183C4×D7—44no—
184C22×D4—71no—
185C22×D4—71no—
186C22×D4C2×D711yescharacteristic
187C23×C4—71no—
188C23×C4—71no—
189C23×C4—71no—
190#121—78no—
191C2×C4⋊C4—71no—
192C2×C4⋊C4—71no—
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⋊C4C2×D711yescharacteristic
203C2×C22⋊C4C2×D711yescharacteristic
204C2×C22⋊C4C2×D711yescharacteristic
205C2×C22⋊C4C2×D711yescharacteristic
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#95—71no—
217#95—71no—
218#95—71no—
219#95—71no—
220C22×C7—22no—
221C22×C7—22no—
222C22×C7—22no—
223C22×C7—22no—
224C22×C7—22no—
225C22×C7—22no—
226C22×C7—41no—
227C22×C7—41no—
228C22×C7—41no—
229C22×C7—41no—
230C22×C7—41no—
231C22×C7—41no—
232C22×C7—41no—
233C22×C7—41no—
234C22×C7C23×C411yescharacteristic
235C22×C7C2×#4111yescharacteristic
236C22×C7C2×#4111yescharacteristic
237C22×C7H(5,𝔽2)11yescharacteristic
238C22×C7H(5,𝔽2)11yescharacteristic
239C22×C7H(5,𝔽2)11yescharacteristic
240C22×C7H(5,𝔽2)11yescharacteristic
241C2×D7—22no—
242C2×D7—22no—
243C2×D7—22no—
244C2×D7—41no—
245C2×D7—41no—
246C2×D7—41no—
247C2×D7—41no—
248C4×C7—22no—
249C4×C7—22no—
250C4×C7—41no—
251C4×C7—41no—
252C4×C7—41no—
253Dic7—22no—
254Dic7—22no—
255Dic7—22no—
256Dic7—22no—
257Dic7—41no—
258Dic7—41no—
259Dic7—41no—
260C24—71no—
261C24C22×D711yescharacteristic
262C24C22×D711yescharacteristic
263C2×D4—41no—
264C2×D4—41no—
265C2×D4—74no—
266C2×D4—74no—
267C2×D4—281no—
268C2×D4—281no—
269C2×D4—281no—
270C2×D4—281no—
271C2×D4C4×D714yescomplemented
272C22×C4—71no—
273C22×C4—71no—
274C22×C4—71no—
275C22×C4—71no—
276C22×C4—74no—
277C22×C4—71no—
278C22×C4—71no—
279C22×C4—71no—
280C22×C4—142no—
281C22×C4—142no—
282C22×C4—281no—
283C22×C4—281no—
284C22×C4—281no—
285C22×C4—281no—
286C22×C4—281no—
287C22×C4—281no—
288C22×C4C22×D711yescharacteristic
289C22×C4C22×D711yescharacteristic
290C22×C4C22×D711yescharacteristic
291C22×C4C22×D711yescharacteristic
292C22×C4C22×D711yescharacteristic
293C4⋊C4—41no—
294C4⋊C4—74no—
295C4⋊C4—281no—
296C22⋊C4—22no—
297C22⋊C4—22no—
298C22⋊C4—41no—
299C22⋊C4—41no—
300C22⋊C4—74no—
301C22⋊C4—74no—
302C22⋊C4—142no—
303C22⋊C4—142no—
304C22⋊C4—281no—
305C22⋊C4—281no—
306C22⋊C4—281no—
307C22⋊C4—281no—
308C42—41no—
309C42—281no—
310C42—281no—
311C42—281no—
312C2×C7—42no—
313C2×C7—42no—
314C2×C7C4×#4111yescharacteristic
315C2×C7#51811yescharacteristic
316C2×C7#51811yescharacteristic
317C2×C7#53811yescharacteristic
318C2×C7#53811yescharacteristic
319C2×C7#54011yescharacteristic
320C2×C7#54011yescharacteristic
321D7—42no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—21no—
328C23—21no—
329C23—21no—
330C23—21no—
331C23—21no—
332C23—21no—
333C23—21no—
334C23—72no—
335C23—141no—
336C23—141no—
337C23—141no—
338C23—141no—
339C23—141no—
340C23—141no—
341C23C2×C4×D712yes—
342C23C2×C4×D712yes—
343C23C23×D711yescharacteristic
344D4—44no—
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—72no—
361C2×C4—72no—
362C2×C4—72no—
363C2×C4—72no—
364C2×C4—141no—
365C2×C4—141no—
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—141no—
380C2×C4—141no—
381C2×C4—284no—
382C2×C4—284no—
383C2×C4—284no—
384C2×C4C2×C4×D712yes—
385C2×C4#115912yes—
386C7#233311yescharacteristic, complemented, Sylow
387C22—22no—
388C22—22no—
389C22—22no—
390C22—22no—
391C22—22no—
392C22—22no—
393C22—41no—
394C22—41no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—41no—
399C22—41no—
400C22—41no—
401C22—142no—
402C22—142no—
403C22—142no—
404C22—281no—
405C22—281no—
406C22—281no—
407C22—281no—
408C22C22×C4×D711yescharacteristic
409C22C2×#115911yescharacteristic
410C22#654811yescharacteristic
411C22#654811yescharacteristic
412C22#654811yescharacteristic
413C22D7×#4111yescharacteristic
414C22#655311yescharacteristic
415C4—22no—
416C4—22no—
417C4—41no—
418C4—41no—
419C4—41no—
420C4—142no—
421C4—142no—
422C4—142no—
423C4—142no—
424C4—281no—
425C4—281no—
426C4—281no—
427C2—42no—
428C2—42no—
429C2—282no—
430C2#9017711yescharacteristic
431C2C4×#115911yescharacteristic
432C2#9023611yescharacteristic
433C2#9030611yescharacteristic
434C2#9030811yescharacteristic
435C2#9036911yescharacteristic
436C2#9041011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutatorC22×C7C23×C4
FittingC7×#390C2
socle, cosocleC23×C7C24
prime radicalC2×#6502C2
hypercenter#390D7
hypocenterC7#2333
LCSG ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ #390

Complements

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

NH ≅ G/Ntype
2. C7×#390C2semidirect
3. #89992C2semidirect
4. #89762C2semidirect
5. #89750C2semidirect
6. #89730C2semidirect
7. #89729C2semidirect
8. #89728C2semidirect
9. #89728C2semidirect
10. #89716C2semidirect
11. #89715C2semidirect
12. C22⋊C4×Dic7C2semidirect
13. C22⋊C4×Dic7C2semidirect
14. C4×(C2×D7)⋊C4C2semidirect
15. #89705C2semidirect
32. #6502C4semidirect
58. C2×C7×D4C2×C4semidirect
69. C2×C22⋊D7C2×C4semidirect
75. C22×Dic7C2×C4semidirect
95. C22⋊Dic7C2×C4semidirect
99. (C2×D7)⋊C4C2×C4semidirect
101. Dic7⋊C4C2×C4semidirect
107. #390D7semidirect
271. C2×D4C4×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×#65022
C7×#3902
#899922
#897622
#897502
#897302
#897292
#897282
#897162
#897152
C22⋊C4×Dic72
C4×(C2×D7)⋊C42
#897052
#23337