Finite Group Databasebeta

← #11733113

Subgroups — #11733113

Order 896.

568 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×#6460C211yesmaximal, characteristic, complemented
2C2×#6449C211yesmaximal, characteristic
3C2×#6446C211yesmaximal, characteristic
4C2×#6441C211yesmaximal, characteristic, complemented
5C7×#388C211yesmaximal, characteristic, complemented
6#89755C211yesmaximal, characteristic, complemented
7#89733C211yesmaximal, characteristic
8#89725C211yesmaximal, characteristic, complemented
9#89722C211yesmaximal, characteristic, complemented
10#89712C211yesmaximal, characteristic, complemented
11#89707C211yesmaximal, characteristic, complemented
12#89446C211yesmaximal, characteristic, complemented
13#89440C211yesmaximal, characteristic, complemented
14#89433C211yesmaximal, characteristic, complemented
15#89428C211yesmaximal, characteristic, complemented
16C22×C22⋊D7C2211yescharacteristic
17C23×Dic7C2211yescharacteristic
18C22×C4×D7C2211yescharacteristic
19C22×Dic14C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C42×C7C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×(C2×D7)⋊C4C2211yescharacteristic
28C2×(C2×D7)⋊C4C2211yescharacteristic
29C2×(C2×D7)⋊C4C2211yescharacteristic
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×C4⋊Dic7C2211yescharacteristic
33C2×C4⋊Dic7C2211yescharacteristic
34C2×C4⋊Dic7C2211yescharacteristic
35C2×C4⋊Dic7C2211yescharacteristic
36C2×C4⋊Dic7C2211yescharacteristic
37C2×C4⋊Dic7C2211yescharacteristic
38C2×C4⋊Dic7C2211yescharacteristic
39C2×C4⋊Dic7C2211yescharacteristic
40C2×Dic7⋊C4C2211yescharacteristic
41C2×Dic7⋊C4C2211yescharacteristic
42C2×Dic7⋊C4C2211yescharacteristic
43#6460—24no—
44#6449—24no—
45#6446—24no—
46#6441—24no—
47C7×#95C2211yescharacteristic
48C7×#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53C7⋊#95C2211yescharacteristic
54C7⋊#95C2211yescharacteristic
55#2680—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—41no—
67C22×Dic7—22no—
68C22×Dic7—22no—
69C22×Dic7—41no—
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C22×Dic7C2311yescharacteristic
75C22×Dic7C2311yescharacteristic
76C22×Dic7C2311yescharacteristic
77C2×C4×D7—22no—
78C2×C4×D7—22no—
79C2×C4×D7—41no—
80C2×Dic14—22no—
81C2×Dic14—22no—
82C2×Dic14—41no—
83C7×C4⋊C4—22no—
84C7×C22⋊C4—22no—
85C7×C22⋊C4—22no—
86C7×C22⋊C4D414yes—
87C42×C7—22no—
88C22⋊Dic7—22no—
89C22⋊Dic7—22no—
90(C2×D7)⋊C4—22no—
91(C2×D7)⋊C4—22no—
92(C2×D7)⋊C4—22no—
93(C2×D7)⋊C4—22no—
94(C2×D7)⋊C4D414yes—
95C4⋊Dic7—22no—
96C4⋊Dic7—22no—
97C4⋊Dic7—22no—
98C4⋊Dic7—22no—
99C4⋊Dic7—22no—
100C4⋊Dic7—22no—
101C4⋊Dic7—41no—
102C4⋊Dic7—41no—
103Dic7⋊C4—22no—
104Dic7⋊C4—22no—
105Dic7⋊C4—22no—
106C2×#123—71no—
107C2×#123—71no—
108C2×C22⋊Q8—71no—
109C2×C22⋊Q8—71no—
110#398—71no—
111#398—71no—
112#397—71no—
113#397—71no—
114#395—71no—
115#394—71no—
116#388—71no—
117#388D711yescharacteristic, complemented
118#385—71no—
119#385—71no—
120#384—71no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7—21no—
126C23×C7—21no—
127C23×C7C2×D412yes—
128C23×C7C2411yescharacteristic
129C22×D7—21no—
130C22×D7—21no—
131C22×D7—21no—
132C22×D7—21no—
133C22×D7—21no—
134C22×D7—21no—
135C22×D7C2×D412yes—
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×C7C2×D412yes—
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×Dic7—21no—
171C2×Dic7—21no—
172C2×Dic7—21no—
173C2×Dic7—21no—
174C2×Dic7—21no—
175C2×Dic7—44no—
176C2×Dic7C2×D412yes—
177C4×D7—44no—
178Dic14—44no—
179C22×Q8—71no—
180C22×D4—71no—
181C23×C4—71no—
182C23×C4—71no—
183#123—144no—
184#123—144no—
185C22⋊Q8—144no—
186C22⋊Q8—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⋊C4—71no—
194C2×C4⋊C4—71no—
195C2×C4⋊C4—71no—
196C2×C4⋊C4—71no—
197C2×C4⋊C4—71no—
198C2×C4⋊C4C2×D711yescharacteristic
199C2×C22⋊C4—71no—
200C2×C22⋊C4—71no—
201C2×C22⋊C4—71no—
202C2×C22⋊C4—71no—
203C2×C22⋊C4—71no—
204C2×C22⋊C4—71no—
205C2×C22⋊C4—71no—
206C2×C22⋊C4C2×D711yescharacteristic
207C2×C22⋊C4C2×D711yescharacteristic
208C2×C22⋊C4C2×D711yescharacteristic
209C2×C42C2×D711yescharacteristic
210#95—71no—
211#95—71no—
212#95—71no—
213#95—71no—
214#95—71no—
215#95—71no—
216#95C2×D711yescharacteristic
217#95C2×D711yescharacteristic
218C22×C7—22no—
219C22×C7—22no—
220C22×C7—22no—
221C22×C7—41no—
222C22×C7—41no—
223C22×C7—41no—
224C22×C7—41no—
225C22×C7C22×D411yescharacteristic
226C22×C7C22×D411yescharacteristic
227C22×C7C2×#4111yescharacteristic
228C22×C7H(5,𝔽2)11yescharacteristic
229C22×C72−1+411yescharacteristic
230C22×C72−1+411yescharacteristic
231C22×C72−1+411yescharacteristic
232C2×D7—22no—
233C2×D7—22no—
234C2×D7—22no—
235C2×D7—41no—
236C2×D7—41no—
237C2×D7—41no—
238C2×D7—41no—
239C4×C7—22no—
240C4×C7—22no—
241C4×C7—22no—
242C4×C7—22no—
243C4×C7—41no—
244C4×C7—41no—
245Dic7—22no—
246Dic7—41no—
247Dic7—41no—
248Dic7—41no—
249Dic7—41no—
250Dic7—41no—
251Dic7—41no—
252C24—71no—
253C24C22×D711yescharacteristic
254C2×Q8—142no—
255C2×Q8—142no—
256C2×Q8—281no—
257C2×D4—142no—
258C2×D4—142no—
259C2×D4—281no—
260C22×C4—71no—
261C22×C4—71no—
262C22×C4—71no—
263C22×C4—71no—
264C22×C4—71no—
265C22×C4—71no—
266C22×C4—71no—
267C22×C4—142no—
268C22×C4—142no—
269C22×C4—142no—
270C22×C4—142no—
271C22×C4—281no—
272C22×C4—281no—
273C22×C4C22×D711yescharacteristic
274C22×C4C22×D711yescharacteristic
275C22×C4C22×D711yescharacteristic
276C22×C4C22×D711yescharacteristic
277C22×C4C22×D711yescharacteristic
278C22×C4C22×D711yescharacteristic
279C4⋊C4—22no—
280C4⋊C4—142no—
281C4⋊C4—142no—
282C4⋊C4—142no—
283C4⋊C4—142no—
284C4⋊C4—142no—
285C4⋊C4—142no—
286C4⋊C4—142no—
287C4⋊C4—142no—
288C4⋊C4—142no—
289C4⋊C4—281no—
290C4⋊C4—281no—
291C22⋊C4—22no—
292C22⋊C4—22no—
293C22⋊C4—74no—
294C22⋊C4—142no—
295C22⋊C4—142no—
296C22⋊C4—142no—
297C22⋊C4—142no—
298C22⋊C4—142no—
299C22⋊C4—142no—
300C22⋊C4D2814yes—
301C42—22no—
302C2×C7—42no—
303C2×C7#53611yescharacteristic
304C2×C7#53611yescharacteristic
305C2×C7#53911yescharacteristic
306C2×C7D4211yescharacteristic
307C2×C7#54711yescharacteristic
308C2×C7#54711yescharacteristic
309C2×C7#56311yescharacteristic
310D7—42no—
311C23—21no—
312C23—21no—
313C23—21no—
314C23—21no—
315C23—21no—
316C23—21no—
317C23—72no—
318C23—141no—
319C23—141no—
320C23—141no—
321C23—141no—
322C23—141no—
323C23—141no—
324C23C2×D2812yes—
325C23C23×D711yescharacteristic
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—72no—
343C2×C4—141no—
344C2×C4—141no—
345C2×C4—141no—
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—141no—
363C2×C4—284no—
364C2×C4—284no—
365C2×C4C2×D2812yes—
366C2×C4C2×D2812yes—
367C2×C4D4×D712yes—
368C2×C4#115912yes—
369C7#268011yescharacteristic, complemented, Sylow
370C22—22no—
371C22—22no—
372C22—22no—
373C22—41no—
374C22—41no—
375C22—41no—
376C22—41no—
377C22—142no—
378C22—142no—
379C22—142no—
380C22—281no—
381C22—281no—
382C22—281no—
383C22—281no—
384C22C22×D2811yescharacteristic
385C22C2×D4×D711yescharacteristic
386C22C2×#115911yescharacteristic
387C22#655311yescharacteristic
388C22#655411yescharacteristic
389C22#655411yescharacteristic
390C22#655411yescharacteristic
391C4—22no—
392C4—22no—
393C4—22no—
394C4—22no—
395C4—41no—
396C4—41no—
397C4—142no—
398C4—281no—
399C4—281no—
400C4—281no—
401C4—281no—
402C4—281no—
403C4—281no—
404C2—42no—
405C2—282no—
406C2#9021711yescharacteristic
407C2D4×D2811yescharacteristic
408C2#9024611yescharacteristic
409C2#9030211yescharacteristic
410C2#9035611yescharacteristic
411C2#9036611yescharacteristic
412C2#9040311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#388C2
prime radicalC22×C22⋊D7C22
hypercenter#388D7
hypocenterC7#2680
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#6460C2semidirect
4. C2×#6441C2semidirect
5. C7×#388C2semidirect
6. #89755C2semidirect
8. #89725C2semidirect
9. #89722C2semidirect
10. #89712C2semidirect
11. #89707C2semidirect
12. #89446C2semidirect
13. #89440C2semidirect
14. #89433C2semidirect
15. #89428C2semidirect
117. #388D7semidirect
369. C7#2680semidirect

Maximal subgroups

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

subgroupindex
C2×#64602
C2×#64492
C2×#64462
C2×#64412
C7×#3882
#897552
#897332
#897252
#897222
#897122
#897072
#894462
#894402
#894332
#894282
#26807