Finite Group Databasebeta

← #11731871

Subgroups — #11731871

Order 896.

548 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×C22⋊D7C211yesmaximal, characteristic
2C2×#6452C211yesmaximal, characteristic, complemented
3C7×#385C211yesmaximal, characteristic, complemented
4#89984C211yesmaximal, characteristic, complemented
5#89766C211yesmaximal, characteristic, complemented
6#89761C211yesmaximal, characteristic, complemented
7C4⋊C4×Dic7C211yesmaximal, characteristic, complemented
8#89729C211yesmaximal, characteristic
9#89729C211yesmaximal, characteristic
10#89721C211yesmaximal, characteristic, complemented
11#89720C211yesmaximal, characteristic, complemented
12#89449C211yesmaximal, characteristic, complemented
13#89439C211yesmaximal, characteristic, complemented
14#89419C211yesmaximal, characteristic, complemented
15#89419C211yesmaximal, characteristic, complemented
16C23×C4×C7C2211yescharacteristic
17C22×C22⋊D7C2211yescharacteristic
18C22×C4×D7C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C22⋊Dic7C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C4×C22⋊D7—24no—
27C2×(C2×D7)⋊C4C2211yescharacteristic
28C2×(C2×D7)⋊C4C2211yescharacteristic
29C2×(C2×D7)⋊C4C2211yescharacteristic
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×C4⋊Dic7C2211yescharacteristic
33C2×C4⋊Dic7C2211yescharacteristic
34C2×C4⋊Dic7C2211yescharacteristic
35C2×C4⋊Dic7C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39C2×Dic7⋊C4C2211yescharacteristic
40C2×Dic7⋊C4C2211yescharacteristic
41C2×Dic7⋊C4C2211yescharacteristic
42C2×Dic7⋊C4C2211yescharacteristic
43C2×C4×Dic7C2211yescharacteristic
44C2×C4×Dic7C2211yescharacteristic
45C2×C4×Dic7C2211yescharacteristic
46#6452—24no—
47C7×#95C2211yescharacteristic
48C7×#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53#2454—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⋊D7Q814yes—
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—41no—
78C7×C4⋊C4—22no—
79C7×C4⋊C4—22no—
80C7×C22⋊C4—22no—
81C7×C22⋊C4—41no—
82C22⋊Dic7—22no—
83C22⋊Dic7—41no—
84C22⋊Dic7—41no—
85(C2×D7)⋊C4—22no—
86(C2×D7)⋊C4—22no—
87(C2×D7)⋊C4—41no—
88(C2×D7)⋊C4—41no—
89(C2×D7)⋊C4—41no—
90C4⋊Dic7—22no—
91C4⋊Dic7—22no—
92C4⋊Dic7—41no—
93C4⋊Dic7—41no—
94Dic7⋊C4—22no—
95Dic7⋊C4—22no—
96Dic7⋊C4—22no—
97Dic7⋊C4—22no—
98Dic7⋊C4—41no—
99Dic7⋊C4—41no—
100Dic7⋊C4D414yes—
101C4×Dic7—22no—
102C4×Dic7—22no—
103C4×Dic7—22no—
104C2×#125—71no—
105C2×C4×D4—71no—
106#399—71no—
107#399—71no—
108#396—71no—
109#396—71no—
110#390—71no—
111#390—71no—
112#389—71no—
113#389—71no—
114#385—71no—
115#385D711yescharacteristic, complemented
116#380—71no—
117#380—71no—
118C4×C4⋊C4—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×D7—21no—
132C22×D7—21no—
133C22×D7C2×Q812yes—
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×Dic7C2×D412yes—
172C2×Dic7C2×D412yes—
173C2×Dic7C2×Q812yes—
174C2×Dic7#4112yes—
175C4×D7—44no—
176C22×D4—71no—
177C23×C4—71no—
178C23×C4C2×D711yescharacteristic
179#125—144no—
180C4×D4—144no—
181C2×C4⋊C4—71no—
182C2×C4⋊C4—71no—
183C2×C4⋊C4—71no—
184C2×C4⋊C4—71no—
185C2×C4⋊C4—71no—
186C2×C4⋊C4—71no—
187C2×C4⋊C4—71no—
188C2×C4⋊C4—71no—
189C2×C4⋊C4—71no—
190C2×C4⋊C4—71no—
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⋊C4C2×D711yescharacteristic
203C2×C22⋊C4C2×D711yescharacteristic
204C2×C42—71no—
205C2×C42—71no—
206C2×C42—71no—
207#95—71no—
208#95—71no—
209#95—71no—
210#95—71no—
211#95C2×D711yescharacteristic
212#95C2×D711yescharacteristic
213C22×C7—22no—
214C22×C7—22no—
215C22×C7—22no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—41no—
220C22×C7C22×D411yescharacteristic
221C22×C7C22×Q811yescharacteristic
222C22×C7C2×#4111yescharacteristic
223C22×C7C2×#4111yescharacteristic
224C22×C7C2×#4111yescharacteristic
225C22×C7H(5,𝔽2)11yescharacteristic
226C22×C7H(5,𝔽2)11yescharacteristic
227C2×D7—22no—
228C2×D7—22no—
229C2×D7—22no—
230C2×D7—41no—
231C2×D7—41no—
232C2×D7—41no—
233C2×D7—41no—
234C4×C7—22no—
235C4×C7—22no—
236C4×C7—22no—
237C4×C7—41no—
238C4×C7—41no—
239C4×C7—41no—
240Dic7—22no—
241Dic7—41no—
242Dic7—41no—
243Dic7—41no—
244Dic7—41no—
245Dic7—41no—
246Dic7C22⋊Q814yes—
247C24—71no—
248C24C22×D711yescharacteristic
249C2×D4—74no—
250C2×D4—142no—
251C2×D4—142no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—41no—
255C22×C4—71no—
256C22×C4—71no—
257C22×C4—71no—
258C22×C4—71no—
259C22×C4—71no—
260C22×C4—71no—
261C22×C4—71no—
262C22×C4—142no—
263C22×C4—142no—
264C22×C4—281no—
265C22×C4C22×D711yescharacteristic
266C22×C4C22×D711yescharacteristic
267C22×C4C22×D711yescharacteristic
268C22×C4C22×D711yescharacteristic
269C22×C4C22×D711yescharacteristic
270C22×C4C22×D711yescharacteristic
271C4⋊C4—22no—
272C4⋊C4—22no—
273C4⋊C4—74no—
274C4⋊C4—142no—
275C4⋊C4—142no—
276C4⋊C4—142no—
277C4⋊C4—142no—
278C4⋊C4—142no—
279C4⋊C4—142no—
280C4⋊C4—281no—
281C4⋊C4—281no—
282C4⋊C4—281no—
283C4⋊C4—281no—
284C22⋊C4—22no—
285C22⋊C4—41no—
286C22⋊C4—142no—
287C22⋊C4—142no—
288C22⋊C4—142no—
289C22⋊C4—281no—
290C22⋊C4—281no—
291C22⋊C4—281no—
292C22⋊C4—281no—
293C22⋊C4—281no—
294C42—142no—
295C42—142no—
296C42—142no—
297C2×C7—42no—
298C2×C7C2×C22⋊Q811yescharacteristic
299C2×C7#52911yescharacteristic
300C2×C7#54011yescharacteristic
301C2×C7#54611yescharacteristic
302C2×C7#55011yescharacteristic
303C2×C7#55411yescharacteristic
304C2×C7#55411yescharacteristic
305D7—42no—
306C23—21no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—72no—
313C23—141no—
314C23—141no—
315C23—141no—
316C23—141no—
317C23—141no—
318C23—141no—
319C23Q8×D712yes—
320C23C23×D711yescharacteristic
321D4—148no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—21no—
325C2×C4—21no—
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—44no—
338C2×C4—72no—
339C2×C4—72no—
340C2×C4—72no—
341C2×C4—72no—
342C2×C4—141no—
343C2×C4—141no—
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—284no—
360C2×C4#115712yes—
361C2×C4D4×D712yes—
362C2×C4#115912yes—
363C7#245411yescharacteristic, complemented, Sylow
364C22—22no—
365C22—22no—
366C22—22no—
367C22—41no—
368C22—41no—
369C22—41no—
370C22—41no—
371C22—142no—
372C22—142no—
373C22—142no—
374C22—281no—
375C22—281no—
376C22—281no—
377C22—281no—
378C22C2×#115711yescharacteristic
379C22C2×D4×D711yescharacteristic
380C22C2×#115911yescharacteristic
381C22#654811yescharacteristic
382C22C2×Q8×D711yescharacteristic
383C22D7×#4111yescharacteristic
384C22#655311yescharacteristic
385C4—22no—
386C4—22no—
387C4—22no—
388C4—41no—
389C4—41no—
390C4—41no—
391C4—74no—
392C4—142no—
393C4—281no—
394C4—281no—
395C4—281no—
396C4—281no—
397C4—281no—
398C2—42no—
399C2—282no—
400C2#9020211yescharacteristic
401C2#9022911yescharacteristic
402C2#9024411yescharacteristic
403C2#9028211yescharacteristic
404C2D7×C22⋊Q811yescharacteristic
405C2#9032611yescharacteristic
406C2#9035411yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#385C2
prime radicalC22×C22⋊D7C22
hypercenter#385D7
hypocenterC7#2454
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#6452C2semidirect
3. C7×#385C2semidirect
4. #89984C2semidirect
5. #89766C2semidirect
6. #89761C2semidirect
7. C4⋊C4×Dic7C2semidirect
10. #89721C2semidirect
11. #89720C2semidirect
12. #89449C2semidirect
13. #89439C2semidirect
14. #89419C2semidirect
15. #89419C2semidirect
115. #385D7semidirect
363. C7#2454semidirect

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×#64522
C7×#3852
#899842
#897662
#897612
C4⋊C4×Dic72
#897292
#897212
#897202
#894492
#894392
#894192
#24547