Finite Group Databasebeta

← #11740209

Subgroups — #11740209

Order 896.

648 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×C22⋊Q8C211yesmaximal, characteristic, complemented
2C2×#6509C211yesmaximal, characteristic, complemented
3C22×Dic7⋊C4C211yesmaximal, characteristic, complemented
4C2×(C22⋊D7)⋊C4C211yesmaximal, characteristic
5#90001C211yesmaximal, characteristic, complemented
6#89985C211yesmaximal, characteristic
7#89766C211yesmaximal, characteristic, complemented
8#89764C211yesmaximal, characteristic, complemented
9#89761C211yesmaximal, characteristic, complemented
10#89756C211yesmaximal, characteristic, complemented
11#89750C211yesmaximal, characteristic, complemented
12#89745C211yesmaximal, characteristic, complemented
13#89730C211yesmaximal, characteristic
14#89722C211yesmaximal, characteristic, complemented
15#89718C211yesmaximal, characteristic, complemented
16C22×C7×Q8C2211yescharacteristic
17C23×C4×C7C2211yescharacteristic
18C22×C22⋊D7C2211yescharacteristic
19C23×Dic7C2211yescharacteristic
20C23×Dic7C2211yescharacteristic
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
29#6509—24no—
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×(C2×D7)⋊C4C2211yescharacteristic
33C2×(C2×D7)⋊C4C2211yescharacteristic
34C2×(C2×D7)⋊C4C2211yescharacteristic
35C2×C4⋊Dic7C2211yescharacteristic
36C2×C4⋊Dic7C2211yescharacteristic
37C2×Dic7⋊C4—24no—
38C2×Dic7⋊C4C2211yescharacteristic
39C2×Dic7⋊C4C2211yescharacteristic
40C2×Dic7⋊C4C2211yescharacteristic
41C2×Dic7⋊C4C2211yescharacteristic
42C2×Dic7⋊C4C2211yescharacteristic
43C2×Dic7⋊C4C2211yescharacteristic
44C2×Dic7⋊C4C2211yescharacteristic
45C2×C4×Dic7C2211yescharacteristic
46(C22⋊D7)⋊C4—24no—
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53C7⋊#95C2211yescharacteristic
54C7⋊#95C2211yescharacteristic
55#2449—71nomaximal, Sylow
56C24×C7C2311yescharacteristic
57C23×D7C2311yescharacteristic
58C2×C7×Q8—22no—
59C2×C7×Q8—22no—
60C2×C7×Q8—41no—
61C22×C4×C7—22no—
62C22×C4×C7—22no—
63C22×C4×C7—41no—
64C22×C4×C7C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C22×C4×C7C2311yescharacteristic
67C22×C4×C7C2311yescharacteristic
68C22×C4×C7C2311yescharacteristic
69C22×C4×C7C2311yescharacteristic
70C2×C22⋊D7—22no—
71C2×C22⋊D7—22no—
72C2×C22⋊D7Q814yescomplemented
73C22×Dic7—22no—
74C22×Dic7—22no—
75C22×Dic7—22no—
76C22×Dic7—22no—
77C22×Dic7—22no—
78C22×Dic7—41no—
79C22×Dic7C2311yescharacteristic
80C22×Dic7C2311yescharacteristic
81C22×Dic7C2311yescharacteristic
82C22×Dic7C2311yescharacteristic
83C22×Dic7C2311yescharacteristic
84C22×Dic7C2311yescharacteristic
85C22×Dic7C2311yescharacteristic
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—
95(C2×D7)⋊C4—22no—
96(C2×D7)⋊C4—22no—
97(C2×D7)⋊C4—22no—
98(C2×D7)⋊C4—22no—
99(C2×D7)⋊C4—41no—
100C4⋊Dic7—22no—
101C4⋊Dic7—22no—
102Dic7⋊C4—22no—
103Dic7⋊C4—22no—
104Dic7⋊C4—22no—
105Dic7⋊C4—22no—
106Dic7⋊C4—22no—
107Dic7⋊C4—41no—
108Dic7⋊C4D414yescomplemented
109C4×Dic7—22no—
110C2×C22⋊Q8—71no—
111C2×C22⋊Q8D711yescharacteristic, complemented
112C2×C4×D4—71no—
113C22×C4⋊C4—71no—
114#399—71no—
115#398—71no—
116#396—71no—
117#395—71no—
118#389—71no—
119#387—71no—
120#386—71no—
121#386—71no—
122#385—71no—
123#385—71no—
124#385—71no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7C2×D412yes—
130C23×C7C2×Q812yes—
131C23×C7#4112yes—
132C23×C7C2411yescharacteristic
133C22×D7—21no—
134C22×D7—21no—
135C22×D7—21no—
136C22×D7—21no—
137C22×D7—21no—
138C22×D7—21no—
139C22×D7C2×Q812yes—
140C7×Q8—44no—
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×C7—21no—
153C2×C4×C7—21no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—44no—
157C2×C4×C7C2×D412yes—
158C2×C4×C7C2×D412yes—
159C2×C4×C7C2×D412yes—
160C22⋊D7—28no—
161C2×Dic7—28no—
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×Dic7—21no—
174C2×Dic7—21no—
175C2×Dic7—21no—
176C2×Dic7—21no—
177C2×Dic7—21no—
178C2×Dic7—21no—
179C2×Dic7—44no—
180C2×Dic7C2×D412yes—
181C2×Dic7C2×D412yes—
182C2×Dic7C2×Q812yes—
183C2×Dic7#4112yes—
184C4×D7—44no—
185C22×Q8C2×D711yescharacteristic
186C22×D4—71no—
187C23×C4—71no—
188C23×C4—71no—
189C23×C4—71no—
190C23×C4C2×D711yescharacteristic
191C22⋊Q8—24no—
192C22⋊Q8—144no—
193C4×D4—144no—
194C2×C4⋊C4—71no—
195C2×C4⋊C4—71no—
196C2×C4⋊C4—71no—
197C2×C4⋊C4—71no—
198C2×C4⋊C4—71no—
199C2×C4⋊C4—71no—
200C2×C4⋊C4—71no—
201C2×C4⋊C4—71no—
202C2×C4⋊C4—71no—
203C2×C4⋊C4—144no—
204C2×C4⋊C4C2×D711yescharacteristic
205C2×C4⋊C4C2×D711yescharacteristic
206C2×C4⋊C4C2×D711yescharacteristic
207C2×C22⋊C4—71no—
208C2×C22⋊C4—71no—
209C2×C22⋊C4—71no—
210C2×C22⋊C4—71no—
211C2×C22⋊C4—71no—
212C2×C22⋊C4—71no—
213C2×C22⋊C4C2×D711yescharacteristic
214C2×C22⋊C4C2×D711yescharacteristic
215C2×C42—71no—
216#95—71no—
217#95—71no—
218#95—71no—
219#95—71no—
220#95—71no—
221#95—71no—
222#95—71no—
223#95—71no—
224C22×C7—22no—
225C22×C7—22no—
226C22×C7—22no—
227C22×C7—22no—
228C22×C7—22no—
229C22×C7—22no—
230C22×C7C22⋊Q814yescomplemented
231C22×C7C22×D411yescharacteristic
232C22×C7C22×D411yescharacteristic
233C22×C7C22×Q811yescharacteristic
234C22×C7C2×#4111yescharacteristic
235C22×C7C2×#4111yescharacteristic
236C22×C7H(5,𝔽2)11yescharacteristic
237C22×C72−1+411yescharacteristic
238C2×D7—22no—
239C2×D7—22no—
240C2×D7—22no—
241C2×D7—41no—
242C2×D7—41no—
243C2×D7—41no—
244C2×D7—41no—
245C4×C7—22no—
246C4×C7—22no—
247C4×C7—22no—
248C4×C7—41no—
249C4×C7—41no—
250C4×C7—41no—
251Dic7—22no—
252Dic7—22no—
253Dic7—22no—
254Dic7—22no—
255Dic7—41no—
256Dic7—41no—
257Dic7—41no—
258C24—71no—
259C24C22×D711yescharacteristic
260C2×Q8—22no—
261C2×Q8—22no—
262C2×Q8—41no—
263C2×D4—74no—
264C2×D4—142no—
265C2×D4—142no—
266C22×C4—22no—
267C22×C4—22no—
268C22×C4—41no—
269C22×C4—71no—
270C22×C4—71no—
271C22×C4—71no—
272C22×C4—71no—
273C22×C4—71no—
274C22×C4—71no—
275C22×C4—71no—
276C22×C4—142no—
277C22×C4—142no—
278C22×C4—142no—
279C22×C4—142no—
280C22×C4—142no—
281C22×C4—142no—
282C22×C4—142no—
283C22×C4—281no—
284C22×C4—281no—
285C22×C4C22×D711yescharacteristic
286C22×C4C22×D711yescharacteristic
287C22×C4C22×D711yescharacteristic
288C22×C4C22×D711yescharacteristic
289C22×C4C22×D711yescharacteristic
290C22×C4C22×D711yescharacteristic
291C4⋊C4—22no—
292C4⋊C4—22no—
293C4⋊C4—22no—
294C4⋊C4—74no—
295C4⋊C4—142no—
296C4⋊C4—142no—
297C4⋊C4—142no—
298C4⋊C4—142no—
299C4⋊C4—142no—
300C4⋊C4—142no—
301C4⋊C4—142no—
302C4⋊C4—281no—
303C22⋊C4—22no—
304C22⋊C4—142no—
305C22⋊C4—142no—
306C22⋊C4—142no—
307C22⋊C4—142no—
308C22⋊C4—142no—
309C22⋊C4—281no—
310C22⋊C4C22⋊D714yes—
311C42—142no—
312C2×C7—24no—
313C2×C7C2×C22⋊Q811yescharacteristic
314C2×C7#52511yescharacteristic
315C2×C7#53911yescharacteristic
316C2×C7D4211yescharacteristic
317C2×C7#54711yescharacteristic
318C2×C7D4×Q811yescharacteristic
319C2×C7#55411yescharacteristic
320D7—42no—
321C23—21no—
322C23—21no—
323C23—21no—
324C23—21no—
325C23—72no—
326C23—141no—
327C23—141no—
328C23—141no—
329C23—141no—
330C23—141no—
331C23—141no—
332C23Q8×D712yes—
333C23#116112yes—
334C23C2×C22⋊D712yes—
335C23C23×D711yescharacteristic
336Q8—44no—
337D4—148no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—21no—
341C2×C4—21no—
342C2×C4—21no—
343C2×C4—21no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—21no—
351C2×C4—21no—
352C2×C4—21no—
353C2×C4—44no—
354C2×C4—72no—
355C2×C4—72no—
356C2×C4—72no—
357C2×C4—72no—
358C2×C4—148no—
359C2×C4—141no—
360C2×C4—141no—
361C2×C4—141no—
362C2×C4—141no—
363C2×C4—141no—
364C2×C4—141no—
365C2×C4—141no—
366C2×C4—141no—
367C2×C4—141no—
368C2×C4—141no—
369C2×C4—141no—
370C2×C4—141no—
371C2×C4—141no—
372C2×C4—141no—
373C2×C4—141no—
374C2×C4—141no—
375C2×C4—141no—
376C2×C4—284no—
377C2×C4—284no—
378C2×C4D4×D712yes—
379C2×C4C2×C22⋊D712yes—
380C2×C4C2×C22⋊D712yes—
381C7#244911yescharacteristic, complemented, Sylow
382C22—22no—
383C22—22no—
384C22—22no—
385C22—22no—
386C22—22no—
387C22—22no—
388C22—142no—
389C22—142no—
390C22—142no—
391C22—281no—
392C22—281no—
393C22—281no—
394C22—281no—
395C22#650914yescomplemented
396C22C2×D4×D711yescharacteristic
397C22#654811yescharacteristic
398C22C2×Q8×D711yescharacteristic
399C22C2×#116111yescharacteristic
400C22D7×#4111yescharacteristic
401C22#655411yescharacteristic
402C22C22×C22⋊D711yescharacteristic
403C4—22no—
404C4—22no—
405C4—22no—
406C4—41no—
407C4—41no—
408C4—41no—
409C4—142no—
410C4—142no—
411C4—142no—
412C4—142no—
413C4—281no—
414C4—281no—
415C4—281no—
416C2—24no—
417C2—282no—
418C2#9031811yescharacteristic
419C2#9032311yescharacteristic
420C2#9032711yescharacteristic
421C2#9033611yescharacteristic
422C2D4×C22⋊D711yescharacteristic
423C2C2×#650911yescharacteristic
424C2#9051511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC2×C7×C22⋊Q8C2
prime radicalC22×C22⋊D7C22
hypercenterC2×C22⋊Q8D7
hypocenterC7#2449
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×#6509C2semidirect
3. C22×Dic7⋊C4C2semidirect
5. #90001C2semidirect
7. #89766C2semidirect
8. #89764C2semidirect
9. #89761C2semidirect
10. #89756C2semidirect
11. #89750C2semidirect
12. #89745C2semidirect
14. #89722C2semidirect
15. #89718C2semidirect
72. C2×C22⋊D7Q8semidirect
108. Dic7⋊C4D4semidirect
111. C2×C22⋊Q8D7semidirect
230. C22×C7C22⋊Q8semidirect
381. C7#2449semidirect
395. C22#6509semidirect

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×#65092
C22×Dic7⋊C42
C2×(C22⋊D7)⋊C42
#900012
#899852
#897662
#897642
#897612
#897562
#897502
#897452
#897302
#897222
#897182
#24497