Finite Group Databasebeta

← #11740271

Subgroups — #11740271

Order 896.

618 conjugacy classes of subgroups.

The subgroup lattice has 452 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#6510C211yesmaximal, characteristic, complemented
3C2×#6492C211yesmaximal, characteristic, complemented
4C2×#6447C211yesmaximal, characteristic
5#90001C211yesmaximal, characteristic, complemented
6#89985C211yesmaximal, characteristic, complemented
7#89766C211yesmaximal, characteristic, complemented
8#89764C211yesmaximal, characteristic, complemented
9#89763C211yesmaximal, characteristic, complemented
10#89761C211yesmaximal, characteristic, complemented
11#89759C211yesmaximal, characteristic, complemented
12#89744C211yesmaximal, characteristic, complemented
13#89731C211yesmaximal, characteristic, complemented
14#89731C211yesmaximal, characteristic, complemented
15#89723C211yesmaximal, characteristic, complemented
16C22×C7×Q8C2211yescharacteristic
17C23×C4×C7C2211yescharacteristic
18C22×C22⋊D7C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C22×D28C2211yescharacteristic
21C22×C4×D7C2211yescharacteristic
22C7×C22⋊Q8—24no—
23C2×C7×C4⋊C4C2211yescharacteristic
24C2×C7×C4⋊C4C2211yescharacteristic
25C2×C7×C4⋊C4C2211yescharacteristic
26C2×C7×C22⋊C4C2211yescharacteristic
27C2×C7×C22⋊C4C2211yescharacteristic
28C2×C22⋊Dic7C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30#6510—24no—
31#6492—24no—
32C2×(C2×D7)⋊C4C2211yescharacteristic
33C2×(C2×D7)⋊C4C2211yescharacteristic
34C2×(C2×D7)⋊C4C2211yescharacteristic
35C2×(C2×D7)⋊C4C2211yescharacteristic
36C2×(C2×D7)⋊C4C2211yescharacteristic
37C2×(C2×D7)⋊C4C2211yescharacteristic
38C2×(C2×D7)⋊C4C2211yescharacteristic
39C2×(C2×D7)⋊C4C2211yescharacteristic
40C2×(C2×D7)⋊C4C2211yescharacteristic
41C2×C4⋊Dic7C2211yescharacteristic
42C2×Dic7⋊C4C2211yescharacteristic
43C2×Dic7⋊C4C2211yescharacteristic
44C2×Dic7⋊C4C2211yescharacteristic
45C2×Dic7⋊C4C2211yescharacteristic
46C2×Dic7⋊C4C2211yescharacteristic
47C2×C4×Dic7C2211yescharacteristic
48#6447—24no—
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53C7⋊#95C2211yescharacteristic
54C7⋊#95C2211yescharacteristic
55#2672—71nomaximal, Sylow
56C24×C7C2311yescharacteristic
57C23×D7C2311yescharacteristic
58C23×D7C2311yescharacteristic
59C2×C7×Q8—22no—
60C2×C7×Q8—22no—
61C2×C7×Q8—41no—
62C22×C4×C7—22no—
63C22×C4×C7—22no—
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⋊D7—22no—
74C2×C22⋊D7—22no—
75C2×C22⋊D7—41no—
76C2×C22⋊D7—41no—
77C22×Dic7C2311yescharacteristic
78C22×Dic7C2311yescharacteristic
79C22×Dic7C2311yescharacteristic
80C22×Dic7C2311yescharacteristic
81C22×Dic7C2311yescharacteristic
82C22×Dic7C2311yescharacteristic
83C2×D28—22no—
84C2×D28—22no—
85C2×D28—41no—
86C2×C4×D7—22no—
87C2×C4×D7—22no—
88C2×C4×D7—41no—
89C7×C4⋊C4—22no—
90C7×C4⋊C4—22no—
91C7×C4⋊C4—22no—
92C7×C22⋊C4—22no—
93C7×C22⋊C4D414yes—
94C22⋊Dic7—22no—
95C22⋊Dic7—41no—
96(C2×D7)⋊C4—22no—
97(C2×D7)⋊C4—22no—
98(C2×D7)⋊C4—22no—
99(C2×D7)⋊C4—22no—
100(C2×D7)⋊C4—22no—
101(C2×D7)⋊C4—41no—
102(C2×D7)⋊C4—41no—
103(C2×D7)⋊C4—41no—
104(C2×D7)⋊C4D414yescomplemented
105C4⋊Dic7—41no—
106Dic7⋊C4—22no—
107Dic7⋊C4—22no—
108Dic7⋊C4—22no—
109Dic7⋊C4—22no—
110Dic7⋊C4—41no—
111C4×Dic7—22no—
112C2×#124—71no—
113C2×#123—71no—
114C2×C22⋊Q8D711yescharacteristic, complemented
115C2×#121—71no—
116#399—71no—
117#396—71no—
118#395—71no—
119#394—71no—
120#394—71no—
121#394—71no—
122#389—71no—
123#388—71no—
124#386—71no—
125#385—71no—
126#380—71no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—21no—
130C23×C7—21no—
131C23×C7—21no—
132C23×C7—21no—
133C23×C7C2×D412yes—
134C23×C7C2411yescharacteristic
135C22×D7—21no—
136C22×D7—21no—
137C22×D7—21no—
138C22×D7—21no—
139C22×D7—21no—
140C22×D7—21no—
141C22×D7—21no—
142C22×D7—21no—
143C22×D7—21no—
144C22×D7—21no—
145C22×D7—21no—
146C22×D7—21no—
147C22×D7—21no—
148C22×D7C2×D412yes—
149C7×Q8—44no—
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×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—21no—
160C2×C4×C7—21no—
161C2×C4×C7—21no—
162C2×C4×C7—21no—
163C2×C4×C7—21no—
164C2×C4×C7—21no—
165C2×C4×C7—44no—
166C2×C4×C7C2×D412yes—
167C2×C4×C7C2×D412yes—
168C2×C4×C7C2×D412yes—
169C22⋊D7—44no—
170C22⋊D7—44no—
171C2×Dic7—21no—
172C2×Dic7—21no—
173C2×Dic7—21no—
174C2×Dic7—21no—
175C2×Dic7—21no—
176C2×Dic7—21no—
177C2×Dic7—21no—
178C2×Dic7—21no—
179C2×Dic7—21no—
180C2×Dic7—21no—
181C2×Dic7—21no—
182C2×Dic7—21no—
183C2×Dic7—21no—
184C2×Dic7—21no—
185C2×Dic7—21no—
186C2×Dic7—21no—
187C2×Dic7C2×D412yes—
188C2×Dic7#4112yes—
189D28—44no—
190C4×D7—44no—
191C22×Q8C2×D711yescharacteristic
192C22×D4—71no—
193C22×D4—71no—
194C22×D4—71no—
195C23×C4—71no—
196C23×C4C2×D711yescharacteristic
197#124—144no—
198#123—144no—
199C22⋊Q8—24no—
200#121—144no—
201C2×C4⋊C4—71no—
202C2×C4⋊C4—71no—
203C2×C4⋊C4—71no—
204C2×C4⋊C4—71no—
205C2×C4⋊C4—71no—
206C2×C4⋊C4—71no—
207C2×C4⋊C4C2×D711yescharacteristic
208C2×C4⋊C4C2×D711yescharacteristic
209C2×C4⋊C4C2×D711yescharacteristic
210C2×C22⋊C4—71no—
211C2×C22⋊C4—71no—
212C2×C22⋊C4—71no—
213C2×C22⋊C4—71no—
214C2×C22⋊C4—71no—
215C2×C22⋊C4—71no—
216C2×C22⋊C4—71no—
217C2×C22⋊C4—71no—
218C2×C22⋊C4—71no—
219C2×C22⋊C4—71no—
220C2×C22⋊C4—71no—
221C2×C22⋊C4C2×D711yescharacteristic
222C2×C22⋊C4C2×D711yescharacteristic
223C2×C42—71no—
224#95—71no—
225#95—71no—
226#95—71no—
227#95—71no—
228#95—71no—
229#95—71no—
230C22×C7—22no—
231C22×C7—22no—
232C22×C7—22no—
233C22×C7—41no—
234C22×C7—41no—
235C22×C7—41no—
236C22×C7—41no—
237C22×C7C22×D411yescharacteristic
238C22×C7C22×D411yescharacteristic
239C22×C7C2×#4111yescharacteristic
240C22×C7H(5,𝔽2)11yescharacteristic
241C22×C7H(5,𝔽2)11yescharacteristic
242C22×C72−1+411yescharacteristic
243C22×C72−1+411yescharacteristic
244C2×D7—22no—
245C2×D7—22no—
246C2×D7—22no—
247C2×D7—41no—
248C2×D7—41no—
249C2×D7—41no—
250C2×D7—41no—
251C2×D7—41no—
252C2×D7—41no—
253C2×D7—41no—
254C2×D7—41no—
255C2×D7—41no—
256C2×D7—41no—
257C2×D7—41no—
258C4×C7—22no—
259C4×C7—22no—
260C4×C7—22no—
261C4×C7—41no—
262C4×C7—41no—
263C4×C7—41no—
264Dic7—22no—
265Dic7—22no—
266Dic7—41no—
267Dic7—41no—
268Dic7—41no—
269Dic7—41no—
270C24—71no—
271C24—71no—
272C24C22×D711yescharacteristic
273C2×Q8—22no—
274C2×Q8—22no—
275C2×Q8—41no—
276C2×D4—142no—
277C2×D4—142no—
278C2×D4—142no—
279C2×D4—142no—
280C2×D4—142no—
281C2×D4—142no—
282C2×D4—281no—
283C2×D4—281no—
284C2×D4—281no—
285C22×C4—22no—
286C22×C4—22no—
287C22×C4—41no—
288C22×C4—71no—
289C22×C4—71no—
290C22×C4—71no—
291C22×C4—71no—
292C22×C4—71no—
293C22×C4—71no—
294C22×C4—142no—
295C22×C4—142no—
296C22×C4—281no—
297C22×C4C22×D711yescharacteristic
298C22×C4C22×D711yescharacteristic
299C22×C4C22×D711yescharacteristic
300C22×C4C22×D711yescharacteristic
301C22×C4C22×D711yescharacteristic
302C22×C4C22×D711yescharacteristic
303C4⋊C4—22no—
304C4⋊C4—22no—
305C4⋊C4—22no—
306C4⋊C4—142no—
307C4⋊C4—142no—
308C4⋊C4—142no—
309C4⋊C4—142no—
310C4⋊C4—281no—
311C4⋊C4—281no—
312C22⋊C4—22no—
313C22⋊C4—74no—
314C22⋊C4—142no—
315C22⋊C4—142no—
316C22⋊C4—142no—
317C22⋊C4—142no—
318C22⋊C4—142no—
319C22⋊C4—142no—
320C22⋊C4—281no—
321C22⋊C4—281no—
322C22⋊C4—281no—
323C22⋊C4—281no—
324C22⋊C4C22⋊D714yes—
325C42—142no—
326C2×C7—42no—
327C2×C7#53611yescharacteristic
328C2×C7#53711yescharacteristic
329C2×C7#54211yescharacteristic
330C2×C7D4211yescharacteristic
331C2×C7#54611yescharacteristic
332C2×C7#54711yescharacteristic
333C2×C7#56211yescharacteristic
334D7—42no—
335D7—81no—
336C23—21no—
337C23—21no—
338C23—21no—
339C23—21no—
340C23—21no—
341C23—21no—
342C23—72no—
343C23—141no—
344C23—141no—
345C23—141no—
346C23—141no—
347C23—141no—
348C23—141no—
349C23—141no—
350C23—141no—
351C23—141no—
352C23—141no—
353C23—141no—
354C23—141no—
355C23—141no—
356C23C2×C22⋊D712yes—
357C23C23×D711yescharacteristic
358Q8—44no—
359D4—284no—
360D4—284no—
361D4—284no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—21no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—21no—
371C2×C4—21no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—44no—
378C2×C4—72no—
379C2×C4—72no—
380C2×C4—141no—
381C2×C4—141no—
382C2×C4—141no—
383C2×C4—141no—
384C2×C4—141no—
385C2×C4—141no—
386C2×C4—141no—
387C2×C4—141no—
388C2×C4—141no—
389C2×C4—141no—
390C2×C4—141no—
391C2×C4—141no—
392C2×C4—141no—
393C2×C4—141no—
394C2×C4—141no—
395C2×C4—141no—
396C2×C4—284no—
397C2×C4D4×D712yes—
398C2×C4C2×C22⋊D712yes—
399C2×C4C2×C22⋊D712yes—
400C7#267211yescharacteristic, complemented, Sylow
401C22—22no—
402C22—22no—
403C22—22no—
404C22—41no—
405C22—41no—
406C22—41no—
407C22—41no—
408C22—142no—
409C22—142no—
410C22—142no—
411C22—281no—
412C22—281no—
413C22—281no—
414C22—281no—
415C22—281no—
416C22—281no—
417C22—281no—
418C22—281no—
419C22—281no—
420C22—281no—
421C22—281no—
422C22C2×D4×D711yescharacteristic
423C22#654811yescharacteristic
424C22#655111yescharacteristic
425C22#655111yescharacteristic
426C22D7×#4111yescharacteristic
427C22#655311yescharacteristic
428C22C22×C22⋊D711yescharacteristic
429C4—22no—
430C4—22no—
431C4—22no—
432C4—41no—
433C4—41no—
434C4—41no—
435C4—142no—
436C4—142no—
437C4—281no—
438C4—281no—
439C4—281no—
440C4—281no—
441C2—42no—
442C2—282no—
443C2—561no—
444C2#9032011yescharacteristic
445C2#9032211yescharacteristic
446C2#9033111yescharacteristic
447C2#9033811yescharacteristic
448C2D4×C22⋊D711yescharacteristic
449C2#9050711yescharacteristic
450C2#9052011yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×C22⋊Q8C2
prime radicalC2×#6447C2
hypercenterC2×C22⋊Q8D7
hypocenterC7#2672
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C22⋊Q8

Complements

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

NH ≅ G/Ntype
1. C2×C7×C22⋊Q8C2semidirect
2. C2×#6510C2semidirect
3. C2×#6492C2semidirect
5. #90001C2semidirect
6. #89985C2semidirect
7. #89766C2semidirect
8. #89764C2semidirect
9. #89763C2semidirect
10. #89761C2semidirect
11. #89759C2semidirect
12. #89744C2semidirect
13. #89731C2semidirect
14. #89731C2semidirect
15. #89723C2semidirect
104. (C2×D7)⋊C4D4semidirect
114. C2×C22⋊Q8D7semidirect
400. C7#2672semidirect

Maximal subgroups

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

subgroupindex
C2×C7×C22⋊Q82
C2×#65102
C2×#64922
C2×#64472
#900012
#899852
#897662
#897642
#897632
#897612
#897592
#897442
#897312
#897232
#26727