Finite Group Databasebeta

← #11732767

Subgroups — #11732767

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×#6459C211yesmaximal, characteristic, complemented
2C2×C4×D28C211yesmaximal, characteristic
3C7×#387C211yesmaximal, characteristic, complemented
4#89766C211yesmaximal, characteristic, complemented
5#89761C211yesmaximal, characteristic, complemented
6#89761C211yesmaximal, characteristic, complemented
7#89701C211yesmaximal, characteristic, complemented
8#89445C211yesmaximal, characteristic, complemented
9#89440C211yesmaximal, characteristic, complemented
10#89439C211yesmaximal, characteristic, complemented
11#89437C211yesmaximal, characteristic
12#89437C211yesmaximal, characteristic
13D7×#95C211yesmaximal, characteristic, complemented
14#89432C211yesmaximal, characteristic, complemented
15#89429C211yesmaximal, characteristic, complemented
16C22×D28C2211yescharacteristic
17C22×C4×D7C2211yescharacteristic
18C22×C4×D7C2211yescharacteristic
19C22×C4×D7C2211yescharacteristic
20C22×C4×D7C2211yescharacteristic
21C22×Dic14C2211yescharacteristic
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×C4⋊Dic7C2211yescharacteristic
34C2×C4⋊Dic7C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39#6459—24no—
40C4×D28—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
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
68C22×Dic7C2311yescharacteristic
69C2×D28—22no—
70C2×D28—22no—
71C2×D28Q814yescomplemented
72C2×C4×D7—22no—
73C2×C4×D7—22no—
74C2×C4×D7—22no—
75C2×C4×D7—22no—
76C2×C4×D7—22no—
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×Dic14—22no—
85C2×Dic14—22no—
86C2×Dic14—41no—
87C7×C4⋊C4—22no—
88C7×C4⋊C4—22no—
89C42×C7—22no—
90(C2×D7)⋊C4—22no—
91(C2×D7)⋊C4—22no—
92(C2×D7)⋊C4—22no—
93(C2×D7)⋊C4—22no—
94(C2×D7)⋊C4—41no—
95(C2×D7)⋊C4—41no—
96(C2×D7)⋊C4—41no—
97(C2×D7)⋊C4D414yes—
98C4⋊Dic7—22no—
99C4⋊Dic7—41no—
100Dic7⋊C4—22no—
101Dic7⋊C4—22no—
102Dic7⋊C4—41no—
103Dic7⋊C4—41no—
104C2×C22⋊Q8—71no—
105C2×C4×D4—71no—
106#400—71no—
107#398—71no—
108#396—71no—
109#393—71no—
110#391—71no—
111#387D711yescharacteristic, complemented
112#386—71no—
113#386—71no—
114#385—71no—
115#385—71no—
116#385—71no—
117#380—71no—
118C2×#95—71no—
119C23×C7C2411yescharacteristic
120C22×D7—21no—
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×D7C2×D412yes—
131C22×D7C2×Q812yes—
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—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7C2×D412yes—
153C2×C4×C7C2×Q812yes—
154C2×C4×C7#4112yes—
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×Dic7#4112yes—
173D28—28no—
174C4×D7—44no—
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×C4—71no—
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⋊C4—71no—
202C2×C22⋊C4—71no—
203C2×C42C2×D711yescharacteristic
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#95C2×D711yescharacteristic
213#95C2×D711yescharacteristic
214#95C2×D711yescharacteristic
215#95C2×D711yescharacteristic
216C22×C7C22×D411yescharacteristic
217C22×C7C22×Q811yescharacteristic
218C22×C7C2×#4111yescharacteristic
219C22×C7C2×#4111yescharacteristic
220C22×C7C2×#4111yescharacteristic
221C22×C7H(5,𝔽2)11yescharacteristic
222C22×C7H(5,𝔽2)11yescharacteristic
223C2×D7—22no—
224C2×D7—22no—
225C2×D7—22no—
226C2×D7—22no—
227C2×D7—22no—
228C2×D7—22no—
229C2×D7—22no—
230C2×D7—22no—
231C2×D7—22no—
232C2×D7—41no—
233C2×D7—41no—
234C2×D7—41no—
235C2×D7—41no—
236C2×D7C22⋊Q814yes—
237C4×C7—22no—
238C4×C7—22no—
239C4×C7—22no—
240C4×C7—41no—
241C4×C7—41no—
242C4×C7—41no—
243C4×C7—41no—
244Dic7—22no—
245Dic7—22no—
246Dic7—41no—
247Dic7—41no—
248Dic7—41no—
249Dic7—41no—
250C24—71no—
251C24—71no—
252C2×Q8—142no—
253C2×Q8—142no—
254C2×Q8—281no—
255C2×D4—74no—
256C2×D4—142no—
257C2×D4—142no—
258C22×C4—71no—
259C22×C4—71no—
260C22×C4—71no—
261C22×C4—71no—
262C22×C4—71no—
263C22×C4—71no—
264C22×C4—142no—
265C22×C4—142no—
266C22×C4—142no—
267C22×C4—142no—
268C22×C4—142no—
269C22×C4—142no—
270C22×C4—142no—
271C22×C4—281no—
272C22×C4—281no—
273C22×C4—281no—
274C22×C4—281no—
275C22×C4—281no—
276C22×C4C22×D711yescharacteristic
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—74no—
292C22⋊C4—142no—
293C22⋊C4—142no—
294C22⋊C4—142no—
295C22⋊C4—142no—
296C22⋊C4—281no—
297C22⋊C4—281no—
298C22⋊C4—281no—
299C42—22no—
300C2×C7C2×C22⋊Q811yescharacteristic
301C2×C7#52911yescharacteristic
302C2×C7#53811yescharacteristic
303C2×C7#54611yescharacteristic
304C2×C7#54611yescharacteristic
305C2×C7#55411yescharacteristic
306C2×C7#55411yescharacteristic
307D7—24no—
308D7—42no—
309C23—72no—
310C23—72no—
311C23—72no—
312C23—72no—
313C23—141no—
314C23—141no—
315C23—141no—
316C23—141no—
317C23—141no—
318C23—141no—
319C23—141no—
320C23—141no—
321C23—141no—
322C23—141no—
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—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—72no—
345C2×C4—72no—
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—284no—
363C2×C4—284no—
364C2×C4—284no—
365C2×C4—284no—
366C2×C4#115712yes—
367C2×C4D4×D712yes—
368C2×C4Q8×D712yes—
369C7#245011yescharacteristic, complemented, Sylow
370C22—74no—
371C22—142no—
372C22—142no—
373C22—142no—
374C22—142no—
375C22—142no—
376C22—142no—
377C22—142no—
378C22—142no—
379C22—142no—
380C22—281no—
381C22—281no—
382C22—281no—
383C22—281no—
384C22C2×#115711yescharacteristic
385C22C2×D4×D711yescharacteristic
386C22C2×Q8×D711yescharacteristic
387C22D7×#4111yescharacteristic
388C22D7×#4111yescharacteristic
389C22#655311yescharacteristic
390C22#655311yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—41no—
395C4—41no—
396C4—41no—
397C4—41no—
398C4—142no—
399C4—142no—
400C4—281no—
401C4—281no—
402C4—281no—
403C4—281no—
404C2—144no—
405C2—282no—
406C2#9021911yescharacteristic
407C2#9024111yescharacteristic
408C2#9027011yescharacteristic
409C2D7×C22⋊Q811yescharacteristic
410C2#9034511yescharacteristic
411C2#9038111yescharacteristic
412C2#9039511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#387C2
prime radicalC22×D28C22
hypercenter#387D7
hypocenterC7#2450
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #387

Complements

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

NH ≅ G/Ntype
1. C2×#6459C2semidirect
3. C7×#387C2semidirect
4. #89766C2semidirect
5. #89761C2semidirect
6. #89761C2semidirect
7. #89701C2semidirect
8. #89445C2semidirect
9. #89440C2semidirect
10. #89439C2semidirect
13. D7×#95C2semidirect
14. #89432C2semidirect
15. #89429C2semidirect
71. C2×D28Q8semidirect
111. #387D7semidirect
369. C7#2450semidirect

Maximal subgroups

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

subgroupindex
C2×#64592
C2×C4×D282
C7×#3872
#897662
#897612
#897012
#894452
#894402
#894392
#894372
D7×#952
#894322
#894292
#24507