Finite Group Databasebeta

← #11734352

Subgroups — #11734352

Order 896.

696 conjugacy classes of subgroups.

The subgroup lattice has 490 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6503C211yesmaximal, characteristic
2C2×#6498C211yesmaximal, characteristic, complemented
3C2×#6457C211yesmaximal, characteristic, complemented
4C2×(C22⋊D7)⋊C4C211yesmaximal, characteristic, complemented
5C7×#394C211yesmaximal, characteristic, complemented
6#89744C211yesmaximal, characteristic, complemented
7#89731C211yesmaximal, characteristic, complemented
8#89731C211yesmaximal, characteristic, complemented
9#89730C211yesmaximal, characteristic, complemented
10#89729C211yesmaximal, characteristic, complemented
11#89729C211yesmaximal, characteristic, complemented
12#89729C211yesmaximal, characteristic, complemented
13#89728C211yesmaximal, characteristic, complemented
14#89444C211yesmaximal, characteristic, complemented
15#89421C211yesmaximal, characteristic, complemented
16C22×C7×D4C2211yescharacteristic
17C22×C22⋊D7C2211yescharacteristic
18C22×C22⋊D7C2211yescharacteristic
19C22×C22⋊D7C2211yescharacteristic
20C22×C22⋊D7C2211yescharacteristic
21C23×Dic7C2211yescharacteristic
22C22×D28C2211yescharacteristic
23C22×C4×D7C2211yescharacteristic
24C2×C7×C4⋊C4C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C7×C22⋊C4C2211yescharacteristic
27C2×C7×C22⋊C4C2211yescharacteristic
28C2×C7×C22⋊C4C2211yescharacteristic
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2211yescharacteristic
31C2×C22⋊Dic7C2211yescharacteristic
32#6503—24no—
33#6498—24no—
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×Dic7⋊C4C2211yescharacteristic
42C2×Dic7⋊C4C2211yescharacteristic
43C2×Dic7⋊C4C2211yescharacteristic
44C2×Dic7⋊C4C2211yescharacteristic
45C2×Dic7⋊C4C2211yescharacteristic
46C2×C4×Dic7C2211yescharacteristic
47C2×C4×Dic7C2211yescharacteristic
48C2×C4×Dic7C2211yescharacteristic
49#6457—24no—
50(C22⋊D7)⋊C4—24no—
51C7×#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53C7⋊#95C2211yescharacteristic
54C7⋊#95C2211yescharacteristic
55#2445—71nomaximal, Sylow
56C24×C7C2311yescharacteristic
57C24×C7C2311yescharacteristic
58C23×D7C2311yescharacteristic
59C23×D7C2311yescharacteristic
60C2×C7×D4—22no—
61C2×C7×D4—22no—
62C2×C7×D4—41no—
63C22×C4×C7C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C22×C4×C7C2311yescharacteristic
66C22×C4×C7C2311yescharacteristic
67C22×C4×C7C2311yescharacteristic
68C2×C22⋊D7—22no—
69C2×C22⋊D7—22no—
70C2×C22⋊D7—22no—
71C2×C22⋊D7—22no—
72C2×C22⋊D7—22no—
73C2×C22⋊D7—22no—
74C2×C22⋊D7—22no—
75C2×C22⋊D7—22no—
76C2×C22⋊D7—22no—
77C2×C22⋊D7—41no—
78C2×C22⋊D7—41no—
79C2×C22⋊D7D414yes—
80C22×Dic7—22no—
81C22×Dic7—22no—
82C22×Dic7—41no—
83C22×Dic7C2311yescharacteristic
84C22×Dic7C2311yescharacteristic
85C22×Dic7C2311yescharacteristic
86C22×Dic7C2311yescharacteristic
87C22×Dic7C2311yescharacteristic
88C22×Dic7C2311yescharacteristic
89C2×D28—22no—
90C2×D28—22no—
91C2×D28—41no—
92C2×C4×D7—22no—
93C2×C4×D7—22no—
94C2×C4×D7—41no—
95C7×C4⋊C4—22no—
96C7×C22⋊C4—22no—
97C7×C22⋊C4—41no—
98C7×C22⋊C4—41no—
99C7×C22⋊C4—41no—
100C22⋊Dic7—22no—
101C22⋊Dic7—22no—
102C22⋊Dic7—22no—
103(C2×D7)⋊C4—22no—
104(C2×D7)⋊C4—22no—
105(C2×D7)⋊C4—22no—
106(C2×D7)⋊C4—22no—
107(C2×D7)⋊C4—41no—
108(C2×D7)⋊C4—41no—
109(C2×D7)⋊C4—41no—
110Dic7⋊C4—22no—
111Dic7⋊C4—22no—
112Dic7⋊C4—22no—
113Dic7⋊C4—22no—
114Dic7⋊C4D414yescomplemented
115C4×Dic7—22no—
116C4×Dic7—22no—
117C4×Dic7—22no—
118C2×#127—71no—
119C2×#123—71no—
120C2×#123—71no—
121C2×C4×D4—71no—
122#394—71no—
123#394—71no—
124#394—71no—
125#394D711yescharacteristic, complemented
126#390—71no—
127#390—71no—
128#390—71no—
129#389—71no—
130#386—71no—
131#386—71no—
132C4×C4⋊C4—71no—
133C23×C7—21no—
134C23×C7—21no—
135C23×C7—21no—
136C23×C7—21no—
137C23×C7—21no—
138C23×C7—21no—
139C23×C7—21no—
140C23×C7—21no—
141C23×C7—21no—
142C23×C7—21no—
143C23×C7—21no—
144C23×C7—21no—
145C23×C7—21no—
146C23×C7C2×D412yes—
147C23×C7C2411yescharacteristic
148C22×D7—21no—
149C22×D7—21no—
150C22×D7—21no—
151C22×D7—21no—
152C22×D7—21no—
153C22×D7—21no—
154C22×D7—21no—
155C22×D7—21no—
156C22×D7—21no—
157C22×D7—21no—
158C22×D7—21no—
159C22×D7—21no—
160C22×D7—21no—
161C22×D7C2×D412yes—
162C7×D4—44no—
163C2×C4×C7—21no—
164C2×C4×C7—21no—
165C2×C4×C7—21no—
166C2×C4×C7—21no—
167C2×C4×C7—21no—
168C2×C4×C7—21no—
169C2×C4×C7—21no—
170C2×C4×C7—21no—
171C2×C4×C7—21no—
172C2×C4×C7—21no—
173C2×C4×C7—21no—
174C2×C4×C7—21no—
175C2×C4×C7—21no—
176C2×C4×C7—21no—
177C2×C4×C7C2×D412yes—
178C22⋊D7—28no—
179C22⋊D7—44no—
180C22⋊D7—44no—
181C22⋊D7—44no—
182C2×Dic7—21no—
183C2×Dic7—21no—
184C2×Dic7—21no—
185C2×Dic7—21no—
186C2×Dic7—21no—
187C2×Dic7—21no—
188C2×Dic7—21no—
189C2×Dic7—21no—
190C2×Dic7—21no—
191C2×Dic7—21no—
192C2×Dic7—21no—
193C2×Dic7—21no—
194C2×Dic7—44no—
195C2×Dic7C2×D412yes—
196C2×Dic7C2×D412yes—
197C2×Dic7C2×D412yes—
198C2×Dic7#4112yes—
199C2×Dic7#4112yes—
200C2×Dic7#4112yes—
201D28—44no—
202C4×D7—44no—
203C22×D4—71no—
204C22×D4—71no—
205C22×D4—71no—
206C22×D4—71no—
207C22×D4—71no—
208C22×D4C2×D711yescharacteristic
209C23×C4—71no—
210C23×C4—71no—
211#127—144no—
212#123—144no—
213#123—144no—
214C4×D4—144no—
215C2×C4⋊C4—71no—
216C2×C4⋊C4—71no—
217C2×C4⋊C4—71no—
218C2×C4⋊C4—71no—
219C2×C4⋊C4—71no—
220C2×C4⋊C4C2×D711yescharacteristic
221C2×C22⋊C4—71no—
222C2×C22⋊C4—71no—
223C2×C22⋊C4—71no—
224C2×C22⋊C4—71no—
225C2×C22⋊C4—71no—
226C2×C22⋊C4—71no—
227C2×C22⋊C4—71no—
228C2×C22⋊C4—71no—
229C2×C22⋊C4—71no—
230C2×C22⋊C4—71no—
231C2×C22⋊C4C2×D711yescharacteristic
232C2×C22⋊C4C2×D711yescharacteristic
233C2×C22⋊C4C2×D711yescharacteristic
234C2×C22⋊C4C2×D711yescharacteristic
235C2×C42—71no—
236C2×C42—71no—
237C2×C42—71no—
238#95—71no—
239#95—71no—
240#95—71no—
241#95C2×D711yescharacteristic
242C22×C7—22no—
243C22×C7—22no—
244C22×C7—22no—
245C22×C7—41no—
246C22×C7—41no—
247C22×C7—41no—
248C22×C7—41no—
249C22×C7—41no—
250C22×C7—41no—
251C22×C7—41no—
252C22×C7—41no—
253C22×C7—41no—
254C22×C7—41no—
255C22×C7—41no—
256C22×C7C22×D411yescharacteristic
257C22×C7C22×D411yescharacteristic
258C22×C7C2×#4111yescharacteristic
259C22×C7C2×#4111yescharacteristic
260C22×C7C2×#4111yescharacteristic
261C22×C7H(5,𝔽2)11yescharacteristic
262C22×C7H(5,𝔽2)11yescharacteristic
263C2×D7—22no—
264C2×D7—22no—
265C2×D7—22no—
266C2×D7—41no—
267C2×D7—41no—
268C2×D7—41no—
269C2×D7—41no—
270C2×D7—41no—
271C2×D7—41no—
272C2×D7—41no—
273C2×D7—41no—
274C2×D7—41no—
275C2×D7—41no—
276C2×D7—41no—
277C4×C7—22no—
278C4×C7—41no—
279C4×C7—41no—
280C4×C7—41no—
281C4×C7—41no—
282Dic7—22no—
283Dic7—22no—
284Dic7—22no—
285Dic7—41no—
286Dic7—41no—
287Dic7#12314yescomplemented
288C24—71no—
289C24—71no—
290C24C22×D711yescharacteristic
291C24C22×D711yescharacteristic
292C2×D4—22no—
293C2×D4—22no—
294C2×D4—41no—
295C2×D4—74no—
296C2×D4—142no—
297C2×D4—142no—
298C2×D4—142no—
299C2×D4—142no—
300C2×D4—142no—
301C2×D4—142no—
302C2×D4—142no—
303C2×D4—142no—
304C2×D4—142no—
305C2×D4—142no—
306C2×D4—142no—
307C2×D4—281no—
308C2×D4—281no—
309C2×D4—281no—
310C22×C4—71no—
311C22×C4—71no—
312C22×C4—71no—
313C22×C4—71no—
314C22×C4—71no—
315C22×C4—71no—
316C22×C4—142no—
317C22×C4—142no—
318C22×C4—142no—
319C22×C4—142no—
320C22×C4—281no—
321C22×C4—281no—
322C22×C4C22×D711yescharacteristic
323C22×C4C22×D711yescharacteristic
324C22×C4C22×D711yescharacteristic
325C22×C4C22×D711yescharacteristic
326C22×C4C22×D711yescharacteristic
327C4⋊C4—22no—
328C4⋊C4—74no—
329C4⋊C4—142no—
330C4⋊C4—142no—
331C4⋊C4—142no—
332C4⋊C4—142no—
333C22⋊C4—22no—
334C22⋊C4—41no—
335C22⋊C4—41no—
336C22⋊C4—41no—
337C22⋊C4—142no—
338C22⋊C4—142no—
339C22⋊C4—142no—
340C22⋊C4—142no—
341C22⋊C4—142no—
342C22⋊C4—142no—
343C22⋊C4—142no—
344C22⋊C4—281no—
345C22⋊C4—281no—
346C22⋊C4—281no—
347C42—142no—
348C42—142no—
349C42—142no—
350C2×C7—42no—
351C2×C7—81no—
352C2×C7C2×#12311yescharacteristic
353C2×C7#53211yescharacteristic
354C2×C7#54011yescharacteristic
355C2×C7D4211yescharacteristic
356C2×C7#54611yescharacteristic
357C2×C7#55011yescharacteristic
358C2×C7#55911yescharacteristic
359D7—42no—
360D7—81no—
361C23—21no—
362C23—21no—
363C23—21no—
364C23—21no—
365C23—21no—
366C23—21no—
367C23—21no—
368C23—21no—
369C23—21no—
370C23—21no—
371C23—21no—
372C23—21no—
373C23—21no—
374C23—72no—
375C23—141no—
376C23—141no—
377C23—141no—
378C23—141no—
379C23—141no—
380C23—141no—
381C23—141no—
382C23—141no—
383C23—141no—
384C23—141no—
385C23—141no—
386C23—141no—
387C23—141no—
388C23D4×D712yes—
389C23C23×D711yescharacteristic
390D4—44no—
391D4—148no—
392D4—284no—
393D4—284no—
394D4—284no—
395D4—284no—
396C2×C4—21no—
397C2×C4—21no—
398C2×C4—21no—
399C2×C4—21no—
400C2×C4—21no—
401C2×C4—21no—
402C2×C4—21no—
403C2×C4—21no—
404C2×C4—21no—
405C2×C4—21no—
406C2×C4—21no—
407C2×C4—21no—
408C2×C4—21no—
409C2×C4—21no—
410C2×C4—72no—
411C2×C4—72no—
412C2×C4—72no—
413C2×C4—72no—
414C2×C4—72no—
415C2×C4—72no—
416C2×C4—141no—
417C2×C4—141no—
418C2×C4—141no—
419C2×C4—141no—
420C2×C4—141no—
421C2×C4—141no—
422C2×C4—141no—
423C2×C4—141no—
424C2×C4—141no—
425C2×C4—141no—
426C2×C4—141no—
427C2×C4—141no—
428C2×C4—284no—
429C2×C4—284no—
430C2×C4D4×D712yes—
431C7#244511yescharacteristic, complemented, Sylow
432C22—22no—
433C22—22no—
434C22—22no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—41no—
439C22—41no—
440C22—41no—
441C22—41no—
442C22—41no—
443C22—41no—
444C22—41no—
445C22—41no—
446C22—142no—
447C22—142no—
448C22—142no—
449C22—281no—
450C22—281no—
451C22—281no—
452C22—281no—
453C22—281no—
454C22—281no—
455C22—281no—
456C22—281no—
457C22—281no—
458C22—281no—
459C22—281no—
460C22C2×D4×D711yescharacteristic
461C22C2×D4×D711yescharacteristic
462C22#654811yescharacteristic
463C22#654811yescharacteristic
464C22D7×#4111yescharacteristic
465C22D7×#4111yescharacteristic
466C22D7×#4111yescharacteristic
467C4—22no—
468C4—41no—
469C4—41no—
470C4—41no—
471C4—41no—
472C4—74no—
473C4—142no—
474C4—142no—
475C4—142no—
476C4—281no—
477C4—281no—
478C2—42no—
479C2—81no—
480C2—282no—
481C2—561no—
482C2#9028111yescharacteristic
483C2#9029011yescharacteristic
484C2#9030111yescharacteristic
485C2#9030611yescharacteristic
486C2D7×#12311yescharacteristic
487C2#9035511yescharacteristic
488C2#9035911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#394C2
prime radicalC2×#6503C2
hypercenter#394D7
hypocenterC7#2445
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #394

Complements

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

NH ≅ G/Ntype
2. C2×#6498C2semidirect
3. C2×#6457C2semidirect
4. C2×(C22⋊D7)⋊C4C2semidirect
5. C7×#394C2semidirect
6. #89744C2semidirect
7. #89731C2semidirect
8. #89731C2semidirect
9. #89730C2semidirect
10. #89729C2semidirect
11. #89729C2semidirect
12. #89729C2semidirect
13. #89728C2semidirect
14. #89444C2semidirect
15. #89421C2semidirect
114. Dic7⋊C4D4semidirect
125. #394D7semidirect
287. Dic7#123semidirect
431. C7#2445semidirect

Maximal subgroups

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

subgroupindex
C2×#65032
C2×#64982
C2×#64572
C2×(C22⋊D7)⋊C42
C7×#3942
#897442
#897312
#897302
#897292
#897282
#894442
#894212
#24457