Finite Group Databasebeta

← #11730820

Subgroups — #11730820

Order 896.

560 conjugacy classes of subgroups.

The subgroup lattice has 404 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6492C211yesmaximal, characteristic
2C2×C4×C22⋊D7C211yesmaximal, characteristic
3C2×C7×#95C211yesmaximal, characteristic, complemented
4#89985C211yesmaximal, characteristic
5#89984C211yesmaximal, characteristic, complemented
6#89983C211yesmaximal, characteristic, complemented
7#89982C211yesmaximal, characteristic, complemented
8#89447C211yesmaximal, characteristic, complemented
9#89446C211yesmaximal, characteristic, complemented
10#89440C211yesmaximal, characteristic, complemented
11#89436C211yesmaximal, characteristic, complemented
12#89430C211yesmaximal, characteristic, complemented
13#89429C211yesmaximal, characteristic, complemented
14#89426C211yesmaximal, characteristic, complemented
15#89424C211yesmaximal, characteristic, complemented
16C23×C4×C7C2211yescharacteristic
17C23×C4×C7C2211yescharacteristic
18C23×C4×C7C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C22×C4×D7C2211yescharacteristic
21C2×C22⋊Dic7C2211yescharacteristic
22C2×C22⋊Dic7C2211yescharacteristic
23C2×C22⋊Dic7C2211yescharacteristic
24#6492—24no—
25C4×C22⋊D7—24no—
26C2×(C2×D7)⋊C4C2211yescharacteristic
27C2×(C2×D7)⋊C4C2211yescharacteristic
28C2×(C2×D7)⋊C4C2211yescharacteristic
29C2×(C2×D7)⋊C4C2211yescharacteristic
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×C4⋊Dic7C2211yescharacteristic
32C2×C4⋊Dic7C2211yescharacteristic
33C2×Dic7⋊C4C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×C4×Dic7C2211yescharacteristic
39C7×#95C2211yescharacteristic
40C7×#95C2211yescharacteristic
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#2419—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C23×D7C2311yescharacteristic
56C22×C4×C7—22no—
57C22×C4×C7—22no—
58C22×C4×C7—22no—
59C22×C4×C7—22no—
60C22×C4×C7—22no—
61C22×C4×C7—22no—
62C22×C4×C7—22no—
63C22×C4×C7—41no—
64C22×C4×C7—41no—
65C22×C4×C7C2311yescharacteristic
66C22×C4×C7C2311yescharacteristic
67C22×C4×C7C2311yescharacteristic
68C22×C4×C7C2311yescharacteristic
69C22×C4×C7C2311yescharacteristic
70C22×C4×C7C2311yescharacteristic
71C2×C22⋊D7—22no—
72C2×C22⋊D7—22no—
73C2×C22⋊D7D414yes—
74C22×Dic7C2311yescharacteristic
75C22×Dic7C2311yescharacteristic
76C22×Dic7C2311yescharacteristic
77C22×Dic7C2311yescharacteristic
78C22×Dic7C2311yescharacteristic
79C22×Dic7C2311yescharacteristic
80C22×Dic7C2311yescharacteristic
81C2×C4×D7—22no—
82C2×C4×D7—22no—
83C2×C4×D7—41no—
84C22⋊Dic7—22no—
85C22⋊Dic7—22no—
86C22⋊Dic7—22no—
87(C2×D7)⋊C4—22no—
88(C2×D7)⋊C4—22no—
89(C2×D7)⋊C4—22no—
90(C2×D7)⋊C4—22no—
91(C2×D7)⋊C4—41no—
92C4⋊Dic7—22no—
93C4⋊Dic7—41no—
94Dic7⋊C4—22no—
95Dic7⋊C4—22no—
96Dic7⋊C4—22no—
97Dic7⋊C4—22no—
98Dic7⋊C4—41no—
99C4×Dic7—22no—
100C2×#123—71no—
101C2×C4×D4—71no—
102#400—71no—
103#398—71no—
104#397—71no—
105#397—71no—
106#387—71no—
107#387—71no—
108#386—71no—
109#385—71no—
110#385—71no—
111#381—71no—
112#381—71no—
113#380—71no—
114C2×#95D711yescharacteristic, complemented
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7C2×D412yes—
120C23×C7#4112yes—
121C23×C7#4112yes—
122C23×C7C2411yescharacteristic
123C22×D7—21no—
124C22×D7—21no—
125C22×D7—21no—
126C22×D7—21no—
127C22×D7—21no—
128C22×D7—21no—
129C22×D7C2×D412yes—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
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—44no—
147C2×C4×C7—44no—
148C2×C4×C7—44no—
149C2×C4×C7#4112yes—
150C2×C4×C7#4112yes—
151C22⋊D7—28no—
152C2×Dic7—21no—
153C2×Dic7—21no—
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×Dic7C2×D412yes—
173C4×D7—44no—
174C22×D4—71no—
175C23×C4—71no—
176C23×C4C2×D711yescharacteristic
177C23×C4C2×D711yescharacteristic
178C23×C4C2×D711yescharacteristic
179#123—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×C22⋊C4—71no—
189C2×C22⋊C4—71no—
190C2×C22⋊C4—71no—
191C2×C22⋊C4—71no—
192C2×C22⋊C4—71no—
193C2×C22⋊C4—71no—
194C2×C22⋊C4—71no—
195C2×C22⋊C4—71no—
196C2×C42—71no—
197#95—71no—
198#95—71no—
199#95—71no—
200#95—71no—
201#95—71no—
202#95—71no—
203#95—71no—
204#95—71no—
205#95—71no—
206#95—71no—
207#95C2×D711yescharacteristic
208#95C2×D711yescharacteristic
209#95C2×D711yescharacteristic
210#95C2×D711yescharacteristic
211C22×C7—22no—
212C22×C7—22no—
213C22×C7—22no—
214C22×C7—22no—
215C22×C7—22no—
216C22×C7—22no—
217C22×C7#12314yes—
218C22×C7C22×D411yescharacteristic
219C22×C7C2×#4111yescharacteristic
220C22×C7C2×#4111yescharacteristic
221C22×C7C2×#4111yescharacteristic
222C22×C7C2×#4111yescharacteristic
223C22×C7H(5,𝔽2)11yescharacteristic
224C22×C72−1+411yescharacteristic
225C2×D7—22no—
226C2×D7—22no—
227C2×D7—22no—
228C2×D7—41no—
229C2×D7—41no—
230C2×D7—41no—
231C2×D7—41no—
232C4×C7—22no—
233C4×C7—22no—
234C4×C7—41no—
235C4×C7—41no—
236C4×C7—41no—
237C4×C7—41no—
238Dic7—22no—
239Dic7—41no—
240Dic7—41no—
241Dic7—41no—
242Dic7—41no—
243Dic7—41no—
244Dic7—41no—
245C24—71no—
246C24C22×D711yescharacteristic
247C2×D4—74no—
248C2×D4—142no—
249C2×D4—142no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—22no—
253C22×C4—22no—
254C22×C4—22no—
255C22×C4—22no—
256C22×C4—22no—
257C22×C4—41no—
258C22×C4—41no—
259C22×C4—71no—
260C22×C4—71no—
261C22×C4—71no—
262C22×C4—71no—
263C22×C4—71no—
264C22×C4—71no—
265C22×C4—71no—
266C22×C4—142no—
267C22×C4—142no—
268C22×C4—281no—
269C22×C4C22×D711yescharacteristic
270C22×C4C22×D711yescharacteristic
271C22×C4C22×D711yescharacteristic
272C22×C4C22×D711yescharacteristic
273C22×C4C22×D711yescharacteristic
274C22×C4C22×D711yescharacteristic
275C4⋊C4—142no—
276C4⋊C4—142no—
277C4⋊C4—142no—
278C4⋊C4—142no—
279C4⋊C4—142no—
280C4⋊C4—281no—
281C4⋊C4—281no—
282C22⋊C4—142no—
283C22⋊C4—142no—
284C22⋊C4—142no—
285C22⋊C4—142no—
286C22⋊C4—142no—
287C22⋊C4—142no—
288C22⋊C4—142no—
289C22⋊C4—281no—
290C42—142no—
291C2×C7—24no—
292C2×C7C2×#12311yescharacteristic
293C2×C7#52911yescharacteristic
294C2×C7#53911yescharacteristic
295C2×C7#54611yescharacteristic
296C2×C7#54711yescharacteristic
297C2×C7#55111yescharacteristic
298C2×C7#55211yescharacteristic
299D7—42no—
300C23—21no—
301C23—21no—
302C23—21no—
303C23—21no—
304C23—72no—
305C23—141no—
306C23—141no—
307C23—141no—
308C23—141no—
309C23—141no—
310C23—141no—
311C23#115712yes—
312C23D4×D712yes—
313C23#115912yes—
314C23C23×D711yescharacteristic
315D4—148no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
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—44no—
333C2×C4—44no—
334C2×C4—44no—
335C2×C4—72no—
336C2×C4—141no—
337C2×C4—141no—
338C2×C4—141no—
339C2×C4—141no—
340C2×C4—141no—
341C2×C4—141no—
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—284no—
357C2×C4#115712yes—
358C2×C4#115712yes—
359C7#241911yescharacteristic, complemented, Sylow
360C22—22no—
361C22—22no—
362C22—22no—
363C22—22no—
364C22—22no—
365C22—22no—
366C22—142no—
367C22—142no—
368C22—142no—
369C22—281no—
370C22—281no—
371C22—281no—
372C22—281no—
373C22#644614yes—
374C22C2×#115711yescharacteristic
375C22C2×#115711yescharacteristic
376C22C2×#115711yescharacteristic
377C22C2×D4×D711yescharacteristic
378C22C2×#115911yescharacteristic
379C22#654811yescharacteristic
380C22#655111yescharacteristic
381C4—22no—
382C4—22no—
383C4—41no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—142no—
388C4—281no—
389C4—281no—
390C4—281no—
391C4—281no—
392C4—281no—
393C4—281no—
394C2—24no—
395C2—282no—
396C2#9017011yescharacteristic
397C2C2×#644611yescharacteristic
398C2#9018111yescharacteristic
399C2#9018611yescharacteristic
400C2#9019811yescharacteristic
401C2#9020411yescharacteristic
402C2#9020611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×#95C2
prime radicalC22×C22⋊D7C22
hypercenterC2×#95D7
hypocenterC7#2419
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#95

Complements

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

NH ≅ G/Ntype
3. C2×C7×#95C2semidirect
5. #89984C2semidirect
6. #89983C2semidirect
7. #89982C2semidirect
8. #89447C2semidirect
9. #89446C2semidirect
10. #89440C2semidirect
11. #89436C2semidirect
12. #89430C2semidirect
13. #89429C2semidirect
14. #89426C2semidirect
15. #89424C2semidirect
114. C2×#95D7semidirect
359. C7#2419semidirect

Maximal subgroups

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

subgroupindex
C2×#64922
C2×C4×C22⋊D72
C2×C7×#952
#899852
#899842
#899832
#899822
#894472
#894462
#894402
#894362
#894302
#894292
#894262
#894242
#24197