Finite Group Databasebeta

← #11734760

Subgroups — #11734760

Order 896.

576 conjugacy classes of subgroups.

The subgroup lattice has 412 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6451C211yesmaximal, characteristic, complemented
2C2×#6450C211yesmaximal, characteristic, complemented
3C2×#6448C211yesmaximal, characteristic
4C2×#6448C211yesmaximal, characteristic
5C7×#396C211yesmaximal, characteristic, complemented
6#89766C211yesmaximal, characteristic, complemented
7#89766C211yesmaximal, characteristic, complemented
8#89762C211yesmaximal, characteristic, complemented
9C4⋊C4×Dic7C211yesmaximal, characteristic, complemented
10#89729C211yesmaximal, characteristic
11#89723C211yesmaximal, characteristic, complemented
12#89720C211yesmaximal, characteristic, complemented
13#89717C211yesmaximal, characteristic, complemented
14#89441C211yesmaximal, characteristic, complemented
15#89418C211yesmaximal, characteristic, complemented
16C22×C22⋊D7C2211yescharacteristic
17C22×Dic14C2211yescharacteristic
18C22×Dic14C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21C2×C7×C4⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C22⋊Dic7C2211yescharacteristic
28C2×(C2×D7)⋊C4C2211yescharacteristic
29C2×(C2×D7)⋊C4C2211yescharacteristic
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×(C2×D7)⋊C4C2211yescharacteristic
33C2×(C2×D7)⋊C4C2211yescharacteristic
34C2×C4⋊Dic7C2211yescharacteristic
35C2×C4⋊Dic7C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39C2×Dic7⋊C4C2211yescharacteristic
40C2×C4×Dic7C2211yescharacteristic
41C2×C4×Dic7C2211yescharacteristic
42C2×C4×Dic7C2211yescharacteristic
43C2×C4×Dic7C2211yescharacteristic
44C2×C4×Dic7C2211yescharacteristic
45#6451—24no—
46#6450—24no—
47#6448—24no—
48#6448—24no—
49C7×#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53C7⋊#95C2211yescharacteristic
54C7⋊#95C2211yescharacteristic
55#2545—71nomaximal, Sylow
56C24×C7C2311yescharacteristic
57C23×D7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C2×C22⋊D7—22no—
65C2×C22⋊D7—22no—
66C2×C22⋊D7—22no—
67C22×Dic7C2311yescharacteristic
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C2×Dic14—22no—
75C2×Dic14—22no—
76C2×Dic14—22no—
77C2×Dic14—22no—
78C2×Dic14—22no—
79C2×Dic14D414yes—
80C7×C4⋊C4—22no—
81C7×C4⋊C4—22no—
82C7×C4⋊C4—22no—
83C7×C22⋊C4—22no—
84C7×C22⋊C4—22no—
85C7×C22⋊C4—22no—
86C22⋊Dic7—22no—
87C22⋊Dic7—22no—
88C22⋊Dic7—22no—
89(C2×D7)⋊C4—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—22no—
95C4⋊Dic7—22no—
96C4⋊Dic7—22no—
97Dic7⋊C4—22no—
98Dic7⋊C4—22no—
99Dic7⋊C4—22no—
100Dic7⋊C4—22no—
101C4×Dic7—22no—
102C4×Dic7—22no—
103C4×Dic7—22no—
104C4×Dic7—22no—
105C4×Dic7D414yes—
106C2×C4⋊Q8—71no—
107C2×#124—71no—
108C2×#124—71no—
109C2×C4×Q8—71no—
110#396—71no—
111#396—71no—
112#396—71no—
113#396D711yescharacteristic, complemented
114#391—71no—
115#390—71no—
116#388—71no—
117#388—71no—
118#388—71no—
119#388—71no—
120C4×C4⋊C4—71no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7C2411yescharacteristic
129C22×D7—21no—
130C22×D7—21no—
131C22×D7—21no—
132C22×D7—21no—
133C22×D7—21no—
134C22×D7—21no—
135C22×D7—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×C7C2×D412yes—
152C2×C4×C7C2×D412yes—
153C2×C4×C7#4112yes—
154C22⋊D7—44no—
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×Dic7C2×D412yes—
171C2×Dic7C2×D412yes—
172C2×Dic7C2×D412yes—
173C2×Dic7C2×D412yes—
174C2×Dic7#4112yes—
175C2×Dic7#4112yes—
176Dic14—28no—
177Dic14—44no—
178C22×Q8—71no—
179C22×Q8—71no—
180C22×D4—71no—
181C4⋊Q8—144no—
182#124—144no—
183#124—144no—
184C4×Q8—144no—
185C2×C4⋊C4—71no—
186C2×C4⋊C4—71no—
187C2×C4⋊C4—71no—
188C2×C4⋊C4—71no—
189C2×C4⋊C4—71no—
190C2×C4⋊C4—71no—
191C2×C4⋊C4C2×D711yescharacteristic
192C2×C4⋊C4C2×D711yescharacteristic
193C2×C4⋊C4C2×D711yescharacteristic
194C2×C22⋊C4—71no—
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×C22⋊C4C2×D711yescharacteristic
204C2×C22⋊C4C2×D711yescharacteristic
205C2×C22⋊C4C2×D711yescharacteristic
206C2×C42—71no—
207C2×C42—71no—
208C2×C42—71no—
209C2×C42—71no—
210C2×C42—71no—
211#95—71no—
212#95—71no—
213#95—71no—
214#95—71no—
215#95—71no—
216#95C2×D711yescharacteristic
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—41no—
220C22×C7—41no—
221C22×C7—41no—
222C22×C7—41no—
223C22×C7—41no—
224C22×C7C22×D411yescharacteristic
225C22×C7C22×D411yescharacteristic
226C22×C7C2×#4111yescharacteristic
227C22×C7C2×#4111yescharacteristic
228C22×C7C2×#4111yescharacteristic
229C22×C72−1+411yescharacteristic
230C22×C72−1+411yescharacteristic
231C2×D7—41no—
232C2×D7—41no—
233C2×D7—41no—
234C2×D7—41no—
235C2×D7—41no—
236C2×D7—41no—
237C2×D7—41no—
238C4×C7—22no—
239C4×C7—22no—
240C4×C7—22no—
241C4×C7—41no—
242C4×C7—41no—
243C4×C7—41no—
244Dic7—22no—
245Dic7—22no—
246Dic7—22no—
247Dic7—41no—
248Dic7—41no—
249Dic7—41no—
250Dic7#12114yes—
251C24—71no—
252C24C22×D711yescharacteristic
253C2×Q8—74no—
254C2×Q8—142no—
255C2×Q8—142no—
256C2×Q8—142no—
257C2×Q8—142no—
258C2×Q8—142no—
259C2×D4—142no—
260C2×D4—142no—
261C2×D4—142no—
262C22×C4—71no—
263C22×C4—71no—
264C22×C4—71no—
265C22×C4—71no—
266C22×C4—71no—
267C22×C4—71no—
268C22×C4—71no—
269C22×C4C22×D711yescharacteristic
270C22×C4C22×D711yescharacteristic
271C22×C4C22×D711yescharacteristic
272C22×C4C22×D711yescharacteristic
273C22×C4C22×D711yescharacteristic
274C22×C4C22×D711yescharacteristic
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—22no—
278C4⋊C4—142no—
279C4⋊C4—142no—
280C4⋊C4—142no—
281C4⋊C4—142no—
282C4⋊C4—142no—
283C4⋊C4—142no—
284C22⋊C4—22no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—142no—
288C22⋊C4—142no—
289C22⋊C4—142no—
290C22⋊C4—142no—
291C22⋊C4—142no—
292C22⋊C4—142no—
293C22⋊C4—142no—
294C22⋊C4—142no—
295C22⋊C4—142no—
296C42—74no—
297C42—142no—
298C42—142no—
299C42—142no—
300C42—142no—
301C2×C7—81no—
302C2×C7C2×#12111yescharacteristic
303C2×C7#53211yescharacteristic
304C2×C7#53611yescharacteristic
305C2×C7#54811yescharacteristic
306C2×C7#54811yescharacteristic
307C2×C7#55611yescharacteristic
308C2×C7#55611yescharacteristic
309D7—81no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—141no—
318C23—141no—
319C23—141no—
320C23—141no—
321C23—141no—
322C23—141no—
323C23—141no—
324C23C23×D711yescharacteristic
325Q8—148no—
326Q8—284no—
327D4—284no—
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—72no—
344C2×C4—72no—
345C2×C4—72no—
346C2×C4—72no—
347C2×C4—72no—
348C2×C4—72no—
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—141no—
363C2×C4—141no—
364C2×C4D4×D712yes—
365C2×C4D4×D712yes—
366C2×C4#115912yes—
367C7#254511yescharacteristic, complemented, Sylow
368C22—41no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—41no—
374C22—41no—
375C22—281no—
376C22—281no—
377C22—281no—
378C22—281no—
379C22—281no—
380C22—281no—
381C22—281no—
382C22C2×D4×D711yescharacteristic
383C22C2×D4×D711yescharacteristic
384C22C2×#115911yescharacteristic
385C22#655111yescharacteristic
386C22D7×#4111yescharacteristic
387C22D7×#4111yescharacteristic
388C22#655411yescharacteristic
389C4—22no—
390C4—22no—
391C4—22no—
392C4—41no—
393C4—41no—
394C4—41no—
395C4—74no—
396C4—142no—
397C4—142no—
398C4—142no—
399C4—281no—
400C4—281no—
401C4—281no—
402C2—81no—
403C2—561no—
404C2#9028711yescharacteristic
405C2#9028911yescharacteristic
406C2D7×#12111yescharacteristic
407C2#9031911yescharacteristic
408C2#9032411yescharacteristic
409C2#9033211yescharacteristic
410C2#9039311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#396C2
prime radicalC22×C22⋊D7C22
hypercenter#396D7
hypocenterC7#2545
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#6451C2semidirect
2. C2×#6450C2semidirect
5. C7×#396C2semidirect
6. #89766C2semidirect
7. #89766C2semidirect
8. #89762C2semidirect
9. C4⋊C4×Dic7C2semidirect
11. #89723C2semidirect
12. #89720C2semidirect
13. #89717C2semidirect
14. #89441C2semidirect
15. #89418C2semidirect
113. #396D7semidirect
367. C7#2545semidirect

Maximal subgroups

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

subgroupindex
C2×#64512
C2×#64502
C2×#64482
C7×#3962
#897662
#897622
C4⋊C4×Dic72
#897292
#897232
#897202
#897172
#894412
#894182
#25457