Finite Group Databasebeta

← #11732055

Subgroups — #11732055

Order 896.

584 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊D7C211yesmaximal, characteristic
2C2×#6460C211yesmaximal, characteristic, complemented
3C7×#385C211yesmaximal, characteristic, complemented
4#89984C211yesmaximal, characteristic, complemented
5#89761C211yesmaximal, characteristic, complemented
6#89756C211yesmaximal, characteristic, complemented
7#89753C211yesmaximal, characteristic, complemented
8#89728C211yesmaximal, characteristic
9#89728C211yesmaximal, characteristic
10#89720C211yesmaximal, characteristic, complemented
11#89719C211yesmaximal, characteristic, complemented
12#89440C211yesmaximal, characteristic, complemented
13D7×#95C211yesmaximal, characteristic, complemented
14#89431C211yesmaximal, characteristic, complemented
15#89418C211yesmaximal, characteristic, complemented
16C23×C4×C7C2211yescharacteristic
17C22×C22⋊D7C2211yescharacteristic
18C22×C4×D7C2211yescharacteristic
19C22×C4×D7C2211yescharacteristic
20C22×C4×D7C2211yescharacteristic
21C22×Dic14C2211yescharacteristic
22C2×C7×C4⋊C4C2211yescharacteristic
23C2×C7×C4⋊C4C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C22⋊Dic7C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C4×C22⋊D7—24no—
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×(C2×D7)⋊C4C2211yescharacteristic
33C2×C4⋊Dic7C2211yescharacteristic
34C2×C4⋊Dic7C2211yescharacteristic
35C2×C4⋊Dic7C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39C2×C4×Dic7C2211yescharacteristic
40#6460—24no—
41C7×#95C2211yescharacteristic
42C7×#95C2211yescharacteristic
43C7⋊#95C2211yescharacteristic
44C7⋊#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53#2450—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C23×D7C2311yescharacteristic
56C22×C4×C7—22no—
57C22×C4×C7—22no—
58C22×C4×C7—41no—
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C2×C22⋊D7—22no—
66C2×C22⋊D7—22no—
67C2×C22⋊D7Q814yescomplemented
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C22×Dic7C2311yescharacteristic
75C2×C4×D7—22no—
76C2×C4×D7—22no—
77C2×C4×D7—22no—
78C2×C4×D7—22no—
79C2×C4×D7—22no—
80C2×C4×D7—41no—
81C2×C4×D7—41no—
82C2×C4×D7—41no—
83C2×C4×D7—41no—
84C2×Dic14—22no—
85C2×Dic14—22no—
86C2×Dic14—41no—
87C7×C4⋊C4—22no—
88C7×C4⋊C4—22no—
89C7×C22⋊C4—22no—
90C7×C22⋊C4—41no—
91C22⋊Dic7—22no—
92C22⋊Dic7—41no—
93C22⋊Dic7—41no—
94(C2×D7)⋊C4—22no—
95(C2×D7)⋊C4—22no—
96(C2×D7)⋊C4D414yes—
97C4⋊Dic7—22no—
98C4⋊Dic7—22no—
99C4⋊Dic7—41no—
100Dic7⋊C4—22no—
101Dic7⋊C4—41no—
102Dic7⋊C4—41no—
103C4×Dic7—22no—
104C2×C22⋊Q8—71no—
105C2×C4×D4—71no—
106#400—71no—
107#398—71no—
108#396—71no—
109#393—71no—
110#391—71no—
111#387—71no—
112#386—71no—
113#386—71no—
114#385—71no—
115#385—71no—
116#385D711yescharacteristic, complemented
117#380—71no—
118C2×#95—71no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7C2×Q812yes—
126C23×C7C2411yescharacteristic
127C22×D7—21no—
128C22×D7—21no—
129C22×D7—21no—
130C22×D7—21no—
131C22×D7C2×D412yes—
132C22×D7C2×Q812yes—
133C22×D7#4112yes—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
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—44no—
150C2×C4×C7C2×D412yes—
151C2×C4×C7#4112yes—
152C2×C4×C7#4112yes—
153C22⋊D7—28no—
154C2×Dic7—21no—
155C2×Dic7—21no—
156C2×Dic7—21no—
157C2×Dic7—21no—
158C2×Dic7—21no—
159C2×Dic7—21no—
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×Dic7C2×D412yes—
174C2×Dic7C2×Q812yes—
175C4×D7—44no—
176C4×D7—44no—
177C4×D7—44no—
178Dic14—44no—
179C22×Q8—71no—
180C22×D4—71no—
181C23×C4—71no—
182C23×C4—71no—
183C23×C4—71no—
184C23×C4C2×D711yescharacteristic
185C22⋊Q8—144no—
186C4×D4—144no—
187C2×C4⋊C4—71no—
188C2×C4⋊C4—71no—
189C2×C4⋊C4—71no—
190C2×C4⋊C4—71no—
191C2×C4⋊C4—71no—
192C2×C4⋊C4—71no—
193C2×C4⋊C4C2×D711yescharacteristic
194C2×C4⋊C4C2×D711yescharacteristic
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⋊C4C2×D711yescharacteristic
202C2×C22⋊C4C2×D711yescharacteristic
203C2×C42—71no—
204#95—71no—
205#95—71no—
206#95—71no—
207#95—71no—
208#95—71no—
209#95—71no—
210#95—71no—
211#95—71no—
212#95—71no—
213#95—71no—
214#95C2×D711yescharacteristic
215#95C2×D711yescharacteristic
216C22×C7—22no—
217C22×C7—22no—
218C22×C7—22no—
219C22×C7—41no—
220C22×C7—41no—
221C22×C7—41no—
222C22×C7—41no—
223C22×C7C22×D411yescharacteristic
224C22×C7C22×Q811yescharacteristic
225C22×C7C2×#4111yescharacteristic
226C22×C7C2×#4111yescharacteristic
227C22×C7C2×#4111yescharacteristic
228C22×C7H(5,𝔽2)11yescharacteristic
229C22×C7H(5,𝔽2)11yescharacteristic
230C2×D7—22no—
231C2×D7—22no—
232C2×D7—22no—
233C2×D7—22no—
234C2×D7—22no—
235C2×D7—22no—
236C2×D7C22⋊Q814yes—
237C4×C7—22no—
238C4×C7—22no—
239C4×C7—22no—
240C4×C7—41no—
241C4×C7—41no—
242C4×C7—41no—
243Dic7—22no—
244Dic7—22no—
245Dic7—41no—
246Dic7—41no—
247Dic7—41no—
248Dic7—41no—
249Dic7—41no—
250C24—71no—
251C24C22×D711yescharacteristic
252C2×Q8—142no—
253C2×Q8—142no—
254C2×Q8—281no—
255C2×D4—74no—
256C2×D4—142no—
257C2×D4—142no—
258C22×C4—22no—
259C22×C4—22no—
260C22×C4—41no—
261C22×C4—71no—
262C22×C4—71no—
263C22×C4—71no—
264C22×C4—71no—
265C22×C4—71no—
266C22×C4—71no—
267C22×C4—71no—
268C22×C4—142no—
269C22×C4—142no—
270C22×C4—142no—
271C22×C4—142no—
272C22×C4—142no—
273C22×C4—281no—
274C22×C4—281no—
275C22×C4—281no—
276C22×C4—281no—
277C22×C4C22×D711yescharacteristic
278C22×C4C22×D711yescharacteristic
279C22×C4C22×D711yescharacteristic
280C22×C4C22×D711yescharacteristic
281C22×C4C22×D711yescharacteristic
282C22×C4C22×D711yescharacteristic
283C4⋊C4—22no—
284C4⋊C4—22no—
285C4⋊C4—142no—
286C4⋊C4—142no—
287C4⋊C4—142no—
288C4⋊C4—281no—
289C4⋊C4—281no—
290C4⋊C4—281no—
291C22⋊C4—22no—
292C22⋊C4—41no—
293C22⋊C4—74no—
294C22⋊C4—142no—
295C22⋊C4—142no—
296C22⋊C4—142no—
297C22⋊C4—281no—
298C22⋊C4—281no—
299C42—142no—
300C2×C7—42no—
301C2×C7C2×C22⋊Q811yescharacteristic
302C2×C7#52911yescharacteristic
303C2×C7#53811yescharacteristic
304C2×C7#54611yescharacteristic
305C2×C7#54611yescharacteristic
306C2×C7#55411yescharacteristic
307C2×C7#55411yescharacteristic
308D7—24no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—72no—
316C23—72no—
317C23—72no—
318C23—141no—
319C23—141no—
320C23—141no—
321C23—141no—
322C23Q8×D712yes—
323C23C23×D711yescharacteristic
324Q8—284no—
325D4—148no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—44no—
342C2×C4—72no—
343C2×C4—72no—
344C2×C4—141no—
345C2×C4—141no—
346C2×C4—141no—
347C2×C4—141no—
348C2×C4—141no—
349C2×C4—141no—
350C2×C4—141no—
351C2×C4—141no—
352C2×C4—141no—
353C2×C4—141no—
354C2×C4—141no—
355C2×C4—141no—
356C2×C4—141no—
357C2×C4—141no—
358C2×C4—141no—
359C2×C4—141no—
360C2×C4—141no—
361C2×C4—141no—
362C2×C4—141no—
363C2×C4—284no—
364C2×C4—284no—
365C2×C4—284no—
366C2×C4#115712yes—
367C2×C4D4×D712yes—
368C2×C4#115912yes—
369C7#245011yescharacteristic, complemented, Sylow
370C22—22no—
371C22—22no—
372C22—22no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—74no—
378C22—142no—
379C22—142no—
380C22—142no—
381C22—142no—
382C22—142no—
383C22—142no—
384C22C2×#115711yescharacteristic
385C22C2×D4×D711yescharacteristic
386C22C2×#115911yescharacteristic
387C22#654811yescharacteristic
388C22C2×Q8×D711yescharacteristic
389C22D7×#4111yescharacteristic
390C22#655311yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—142no—
398C4—142no—
399C4—281no—
400C4—281no—
401C4—281no—
402C4—281no—
403C4—281no—
404C2—42no—
405C2—144no—
406C2#9020211yescharacteristic
407C2#9024111yescharacteristic
408C2#9024811yescharacteristic
409C2#9028211yescharacteristic
410C2D7×C22⋊Q811yescharacteristic
411C2#9032611yescharacteristic
412C2#9034811yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#385C2
prime radicalC22×C22⋊D7C22
hypercenter#385D7
hypocenterC7#2450
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#6460C2semidirect
3. C7×#385C2semidirect
4. #89984C2semidirect
5. #89761C2semidirect
6. #89756C2semidirect
7. #89753C2semidirect
10. #89720C2semidirect
11. #89719C2semidirect
12. #89440C2semidirect
13. D7×#95C2semidirect
14. #89431C2semidirect
15. #89418C2semidirect
67. C2×C22⋊D7Q8semidirect
116. #385D7semidirect
369. C7#2450semidirect

Maximal subgroups

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

subgroupindex
C2×C4×C22⋊D72
C2×#64602
C7×#3852
#899842
#897612
#897562
#897532
#897282
#897202
#897192
#894402
D7×#952
#894312
#894182
#24507