Finite Group Databasebeta

← #11733112

Subgroups — #11733112

Order 896.

544 conjugacy classes of subgroups.

The subgroup lattice has 388 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6451C211yesmaximal, characteristic, complemented
2C2×#6441C211yesmaximal, characteristic
3C2×#6441C211yesmaximal, characteristic
4C2×#6432C211yesmaximal, characteristic, complemented
5C7×#388C211yesmaximal, characteristic
6#89751C211yesmaximal, characteristic, complemented
7#89723C211yesmaximal, characteristic
8#89722C211yesmaximal, characteristic
9#89721C211yesmaximal, characteristic
10#89719C211yesmaximal, characteristic
11#89705C211yesmaximal, characteristic, complemented
12#89433C211yesmaximal, characteristic, complemented
13#89429C211yesmaximal, characteristic, complemented
14#89428C211yesmaximal, characteristic, complemented
15#89426C211yesmaximal, characteristic, complemented
16C23×Dic7C2211yescharacteristic
17C22×Dic14C2211yescharacteristic
18C22×Dic14C2211yescharacteristic
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×C22⋊Dic7C2211yescharacteristic
28C2×C4⋊Dic7C2211yescharacteristic
29C2×C4⋊Dic7C2211yescharacteristic
30C2×C4⋊Dic7C2211yescharacteristic
31C2×C4⋊Dic7C2211yescharacteristic
32C2×C4⋊Dic7C2211yescharacteristic
33C2×C4⋊Dic7C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39C2×Dic7⋊C4C2211yescharacteristic
40C2×C4×Dic7C2211yescharacteristic
41#6451—24no—
42#6441—24no—
43#6441—24no—
44#6432—24no—
45C7×#95C2211yescharacteristic
46C7×#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53C7⋊#95C2211yescharacteristic
54C7⋊#95C2211yescharacteristic
55#2677—71nomaximal, Sylow
56C24×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×Dic7—22no—
64C22×Dic7—22no—
65C22×Dic7—41no—
66C22×Dic7C2311yescharacteristic
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×Dic14—22no—
80C2×Dic14—41no—
81C2×Dic14—41no—
82C2×Dic14—41no—
83C7×C4⋊C4—22no—
84C7×C22⋊C4—22no—
85C7×C22⋊C4—22no—
86C7×C22⋊C4Q814yescomplemented
87C42×C7—22no—
88C22⋊Dic7—22no—
89C22⋊Dic7—22no—
90C22⋊Dic7—41no—
91C4⋊Dic7—22no—
92C4⋊Dic7—22no—
93C4⋊Dic7—22no—
94C4⋊Dic7—22no—
95C4⋊Dic7—22no—
96C4⋊Dic7D414yes—
97Dic7⋊C4—22no—
98Dic7⋊C4—22no—
99Dic7⋊C4—22no—
100Dic7⋊C4—22no—
101Dic7⋊C4—41no—
102Dic7⋊C4D414yes—
103C4×Dic7—22no—
104C2×C4⋊Q8—71no—
105C2×C4⋊Q8—71no—
106C2×C22⋊Q8—71no—
107C2×C22⋊Q8—71no—
108#399—71no—
109#398—71no—
110#398—71no—
111#395—71no—
112#395—71no—
113#393—71no—
114#388—71no—
115#388D711yescharacteristic
116#387—71no—
117#387—71no—
118#385—71no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7—21no—
124C23×C7—21no—
125C23×C7C2×Q812yes—
126C23×C7C2411yescharacteristic
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7C2×D412yes—
142C2×C4×C7C2×D412yes—
143C2×C4×C7C2×Q812yes—
144C2×C4×C7C2×Q812yes—
145C2×Dic7—21no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7—21no—
153C2×Dic7—21no—
154C2×Dic7—21no—
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—44no—
166C2×Dic7C2×D412yes—
167C2×Dic7C2×D412yes—
168C2×Dic7C2×D412yes—
169C2×Dic7C2×D412yes—
170Dic14—44no—
171Dic14—44no—
172Dic14—44no—
173C22×Q8—71no—
174C22×Q8—71no—
175C22×Q8—71no—
176C23×C4—71no—
177C4⋊Q8—144no—
178C4⋊Q8—144no—
179C22⋊Q8—144no—
180C22⋊Q8—144no—
181C2×C4⋊C4—71no—
182C2×C4⋊C4—71no—
183C2×C4⋊C4—71no—
184C2×C4⋊C4—71no—
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⋊C4—71no—
192C2×C4⋊C4—71no—
193C2×C4⋊C4C2×D711yescharacteristic
194C2×C22⋊C4—71no—
195C2×C22⋊C4—71no—
196C2×C22⋊C4—71no—
197C2×C22⋊C4C2×D711yescharacteristic
198C2×C22⋊C4C2×D711yescharacteristic
199C2×C22⋊C4C2×D711yescharacteristic
200C2×C42—71no—
201C2×C42C2×D711yescharacteristic
202#95—71no—
203#95—71no—
204#95—71no—
205#95—71no—
206#95—71no—
207#95—71no—
208#95—71no—
209#95—71no—
210#95C2×D711yescharacteristic
211#95C2×D711yescharacteristic
212C22×C7—22no—
213C22×C7—22no—
214C22×C7—22no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7C22×D411yescharacteristic
220C22×C7C22×D411yescharacteristic
221C22×C7C22×Q811yescharacteristic
222C22×C7H(5,𝔽2)11yescharacteristic
223C22×C72−1+411yescharacteristic
224C22×C72−1+411yescharacteristic
225C22×C72−1+411yescharacteristic
226C4×C7—22no—
227C4×C7—22no—
228C4×C7—22no—
229C4×C7—22no—
230C4×C7—41no—
231C4×C7—41no—
232Dic7—22no—
233Dic7—22no—
234Dic7—22no—
235Dic7—22no—
236Dic7—41no—
237Dic7—41no—
238Dic7—41no—
239Dic7—41no—
240C24C22×D711yescharacteristic
241C2×Q8—142no—
242C2×Q8—142no—
243C2×Q8—142no—
244C2×Q8—142no—
245C2×Q8—142no—
246C2×Q8—142no—
247C2×Q8—281no—
248C2×Q8—281no—
249C2×Q8—281no—
250C22×C4—71no—
251C22×C4—71no—
252C22×C4—71no—
253C22×C4—71no—
254C22×C4—71no—
255C22×C4—71no—
256C22×C4—71no—
257C22×C4—71no—
258C22×C4—142no—
259C22×C4—142no—
260C22×C4—281no—
261C22×C4C22×D711yescharacteristic
262C22×C4C22×D711yescharacteristic
263C22×C4C22×D711yescharacteristic
264C22×C4C22×D711yescharacteristic
265C22×C4C22×D711yescharacteristic
266C22×C4C22×D711yescharacteristic
267C4⋊C4—22no—
268C4⋊C4—74no—
269C4⋊C4—74no—
270C4⋊C4—142no—
271C4⋊C4—142no—
272C4⋊C4—142no—
273C4⋊C4—142no—
274C4⋊C4—142no—
275C4⋊C4—142no—
276C4⋊C4—142no—
277C4⋊C4—142no—
278C4⋊C4—142no—
279C4⋊C4—281no—
280C22⋊C4—22no—
281C22⋊C4—22no—
282C22⋊C4—142no—
283C22⋊C4—142no—
284C22⋊C4—281no—
285C22⋊C4Dic1414yescomplemented
286C42—22no—
287C42—142no—
288C2×C7—42no—
289C2×C7#53611yescharacteristic
290C2×C7#53611yescharacteristic
291C2×C7#54411yescharacteristic
292C2×C7D4211yescharacteristic
293C2×C7D4×Q811yescharacteristic
294C2×C7D4×Q811yescharacteristic
295C2×C7#56311yescharacteristic
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23—21no—
302C23C2×Dic1412yes—
303C23C23×D711yescharacteristic
304Q8—284no—
305Q8—284no—
306Q8—284no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—72no—
322C2×C4—72no—
323C2×C4—72no—
324C2×C4—72no—
325C2×C4—141no—
326C2×C4—141no—
327C2×C4—141no—
328C2×C4—141no—
329C2×C4—141no—
330C2×C4—141no—
331C2×C4—141no—
332C2×C4—141no—
333C2×C4—141no—
334C2×C4—141no—
335C2×C4—141no—
336C2×C4—141no—
337C2×C4—141no—
338C2×C4—141no—
339C2×C4—141no—
340C2×C4—141no—
341C2×C4—141no—
342C2×C4—141no—
343C2×C4—141no—
344C2×C4—141no—
345C2×C4—284no—
346C2×C4C2×Dic1412yes—
347C2×C4C2×Dic1412yes—
348C2×C4D4×D712yes—
349C2×C4D4×D712yes—
350C7#267711yescharacteristic, complemented, Sylow
351C22—22no—
352C22—22no—
353C22—22no—
354C22—41no—
355C22—41no—
356C22—41no—
357C22—41no—
358C22C22×Dic1411yescharacteristic
359C22C2×D4×D711yescharacteristic
360C22C2×D4×D711yescharacteristic
361C22#655311yescharacteristic
362C22#655411yescharacteristic
363C22#655411yescharacteristic
364C22#655411yescharacteristic
365C4—22no—
366C4—22no—
367C4—22no—
368C4—22no—
369C4—41no—
370C4—41no—
371C4—142no—
372C4—142no—
373C4—142no—
374C4—142no—
375C4—281no—
376C4—281no—
377C4—281no—
378C4—281no—
379C2—42no—
380C2#9021011yescharacteristic
381C2D4×Dic1411yescharacteristic
382C2D4×Dic1411yescharacteristic
383C2#9030011yescharacteristic
384C2#9033911yescharacteristic
385C2#9036611yescharacteristic
386C2#9040311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#388C2
prime radicalC24×C7C23
hypercenter#388D7
hypocenterC7#2677
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #388

Complements

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

NH ≅ G/Ntype
1. C2×#6451C2semidirect
4. C2×#6432C2semidirect
6. #89751C2semidirect
11. #89705C2semidirect
12. #89433C2semidirect
13. #89429C2semidirect
14. #89428C2semidirect
15. #89426C2semidirect
86. C7×C22⋊C4Q8semidirect
285. C22⋊C4Dic14semidirect
350. C7#2677semidirect

Maximal subgroups

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

subgroupindex
C2×#64512
C2×#64412
C2×#64322
C7×#3882
#897512
#897232
#897222
#897212
#897192
#897052
#894332
#894292
#894282
#894262
#26777