Finite Group Databasebeta

← #11731860

Subgroups — #11731860

Order 896.

528 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6441C211yesmaximal, characteristic
2C2×#6440C211yesmaximal, characteristic
3C7×#385C211yesmaximal, characteristic
4#89984C211yesmaximal, characteristic
5C2×C7⋊#95C211yesmaximal, characteristic
6#89757C211yesmaximal, characteristic, complemented
7#89755C211yesmaximal, characteristic, complemented
8#89755C211yesmaximal, characteristic, complemented
9#89746C211yesmaximal, characteristic, complemented
10#89722C211yesmaximal, characteristic
11#89715C211yesmaximal, characteristic
12#89431C211yesmaximal, characteristic, complemented
13#89428C211yesmaximal, characteristic, complemented
14#89425C211yesmaximal, characteristic, complemented
15#89425C211yesmaximal, characteristic, complemented
16C23×C4×C7C2211yescharacteristic
17C23×Dic7C2211yescharacteristic
18C23×Dic7C2211yescharacteristic
19C22×Dic14C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21C2×C7×C4⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25C2×C22⋊Dic7C2211yescharacteristic
26C2×C4⋊Dic7C2211yescharacteristic
27C2×C4⋊Dic7C2211yescharacteristic
28C2×C4⋊Dic7C2211yescharacteristic
29C2×C4⋊Dic7C2211yescharacteristic
30C2×C4⋊Dic7C2211yescharacteristic
31C2×Dic7⋊C4C2211yescharacteristic
32C2×Dic7⋊C4C2211yescharacteristic
33C2×Dic7⋊C4C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×C4×Dic7C2211yescharacteristic
36C2×C4×Dic7C2211yescharacteristic
37#6441—24no—
38#6440—24no—
39C7×#95C2211yescharacteristic
40C7×#95C2211yescharacteristic
41C7⋊#95C2211yescharacteristic
42C7⋊#95C2211yescharacteristic
43C7⋊#95C2211yescharacteristic
44C7⋊#95C2211yescharacteristic
45C7⋊#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53#2438—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C22×C4×C7—22no—
56C22×C4×C7—22no—
57C22×C4×C7—41no—
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C22×Dic7—22no—
65C22×Dic7—22no—
66C22×Dic7—22no—
67C22×Dic7—22no—
68C22×Dic7—41no—
69C22×Dic7Q814yescomplemented
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C22×Dic7C2311yescharacteristic
75C22×Dic7C2311yescharacteristic
76C22×Dic7C2311yescharacteristic
77C22×Dic7C2311yescharacteristic
78C2×Dic14—22no—
79C2×Dic14—22no—
80C2×Dic14—41no—
81C7×C4⋊C4—22no—
82C7×C4⋊C4—41no—
83C7×C22⋊C4—22no—
84C7×C22⋊C4D414yes—
85C22⋊Dic7—22no—
86C22⋊Dic7—22no—
87C4⋊Dic7—22no—
88C4⋊Dic7—22no—
89C4⋊Dic7—22no—
90C4⋊Dic7—41no—
91C4⋊Dic7—41no—
92Dic7⋊C4—22no—
93Dic7⋊C4—22no—
94Dic7⋊C4—22no—
95Dic7⋊C4—22no—
96C4×Dic7—22no—
97C4×Dic7—22no—
98C2×C22⋊Q8—71no—
99C2×#117—71no—
100#400—71no—
101#398—71no—
102#398—71no—
103#395—71no—
104#391—71no—
105#389—71no—
106#387—71no—
107#387—71no—
108#385—71no—
109#385D711yescharacteristic
110#381—71no—
111#380—71no—
112C2×#95—71no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7C2×D412yes—
118C23×C7C2×Q812yes—
119C23×C7#4112yes—
120C23×C7C2411yescharacteristic
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
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—44no—
137C2×C4×C7C2×D412yes—
138C2×C4×C7C2×D412yes—
139C2×C4×C7#4112yes—
140C2×Dic7—28no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
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—44no—
163C2×Dic7C2×Q812yes—
164C2×Dic7C2×Q812yes—
165C2×Dic7#4112yes—
166Dic14—44no—
167C22×Q8—71no—
168C23×C4—71no—
169C23×C4—71no—
170C23×C4C2×D711yescharacteristic
171C22⋊Q8—144no—
172#117—144no—
173C2×C4⋊C4—71no—
174C2×C4⋊C4—71no—
175C2×C4⋊C4—71no—
176C2×C4⋊C4—71no—
177C2×C4⋊C4—71no—
178C2×C4⋊C4—71no—
179C2×C4⋊C4—71no—
180C2×C4⋊C4—71no—
181C2×C4⋊C4—71no—
182C2×C4⋊C4C2×D711yescharacteristic
183C2×C4⋊C4C2×D711yescharacteristic
184C2×C22⋊C4—71no—
185C2×C22⋊C4—71no—
186C2×C22⋊C4C2×D711yescharacteristic
187C2×C22⋊C4C2×D711yescharacteristic
188C2×C42—71no—
189C2×C42—71no—
190#95—71no—
191#95—71no—
192#95—71no—
193#95—71no—
194#95—71no—
195#95—71no—
196#95—71no—
197#95—71no—
198#95—71no—
199#95—71no—
200#95—71no—
201#95—71no—
202#95C2×D711yescharacteristic
203#95C2×D711yescharacteristic
204C22×C7—22no—
205C22×C7—22no—
206C22×C7—22no—
207C22×C7—22no—
208C22×C7—22no—
209C22×C7—22no—
210C22×C7C22⋊Q814yes—
211C22×C7C22×D411yescharacteristic
212C22×C7C22×Q811yescharacteristic
213C22×C7C2×#4111yescharacteristic
214C22×C7C2×#4111yescharacteristic
215C22×C7C2×#4111yescharacteristic
216C22×C7H(5,𝔽2)11yescharacteristic
217C22×C72−1+411yescharacteristic
218C4×C7—22no—
219C4×C7—22no—
220C4×C7—22no—
221C4×C7—41no—
222C4×C7—41no—
223C4×C7—41no—
224Dic7—22no—
225Dic7—22no—
226Dic7—22no—
227Dic7—41no—
228Dic7—41no—
229Dic7—41no—
230Dic7—41no—
231Dic7—41no—
232C24C22×D711yescharacteristic
233C2×Q8—142no—
234C2×Q8—142no—
235C2×Q8—281no—
236C22×C4—22no—
237C22×C4—22no—
238C22×C4—41no—
239C22×C4—74no—
240C22×C4—71no—
241C22×C4—71no—
242C22×C4—71no—
243C22×C4—71no—
244C22×C4—71no—
245C22×C4—71no—
246C22×C4—71no—
247C22×C4—71no—
248C22×C4—142no—
249C22×C4—142no—
250C22×C4—142no—
251C22×C4—142no—
252C22×C4—281no—
253C22×C4C22×D711yescharacteristic
254C22×C4C22×D711yescharacteristic
255C22×C4C22×D711yescharacteristic
256C22×C4C22×D711yescharacteristic
257C22×C4C22×D711yescharacteristic
258C22×C4C22×D711yescharacteristic
259C4⋊C4—22no—
260C4⋊C4—41no—
261C4⋊C4—142no—
262C4⋊C4—142no—
263C4⋊C4—142no—
264C4⋊C4—142no—
265C4⋊C4—142no—
266C4⋊C4—142no—
267C4⋊C4—142no—
268C4⋊C4—281no—
269C4⋊C4—281no—
270C22⋊C4—22no—
271C22⋊C4—142no—
272C22⋊C4—142no—
273C22⋊C4D2814yes—
274C42—142no—
275C42—142no—
276C2×C7—24no—
277C2×C7C2×C22⋊Q811yescharacteristic
278C2×C7#53311yescharacteristic
279C2×C7#54411yescharacteristic
280C2×C7#54611yescharacteristic
281C2×C7#54711yescharacteristic
282C2×C7#55111yescharacteristic
283C2×C7#55211yescharacteristic
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23C2×D2812yes—
289C23#115912yes—
290C23Q8×D712yes—
291C23C23×D711yescharacteristic
292Q8—284no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—44no—
309C2×C4—72no—
310C2×C4—72no—
311C2×C4—72no—
312C2×C4—148no—
313C2×C4—141no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—141no—
317C2×C4—141no—
318C2×C4—141no—
319C2×C4—141no—
320C2×C4—141no—
321C2×C4—141no—
322C2×C4—141no—
323C2×C4—141no—
324C2×C4—141no—
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—284no—
335C2×C4C2×D2812yes—
336C2×C4C2×D2812yes—
337C2×C4#115912yes—
338C7#243811yescharacteristic, complemented, Sylow
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22—22no—
344C22—22no—
345C22#646014yes—
346C22C22×D2811yescharacteristic
347C22C2×#115911yescharacteristic
348C22C2×#115911yescharacteristic
349C22#654811yescharacteristic
350C22C2×Q8×D711yescharacteristic
351C22D7×#4111yescharacteristic
352C22#655411yescharacteristic
353C4—22no—
354C4—22no—
355C4—22no—
356C4—41no—
357C4—41no—
358C4—41no—
359C4—142no—
360C4—142no—
361C4—142no—
362C4—281no—
363C4—281no—
364C4—281no—
365C4—281no—
366C4—281no—
367C2—24no—
368C2C2×#646011yescharacteristic
369C2#9024511yescharacteristic
370C2#9024611yescharacteristic
371C2#9027811yescharacteristic
372C2#9031311yescharacteristic
373C2#9031411yescharacteristic
374C2#9034111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#385C2
prime radicalC24×C7C23
hypercenter#385D7
hypocenterC7#2438
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
6. #89757C2semidirect
7. #89755C2semidirect
8. #89755C2semidirect
9. #89746C2semidirect
12. #89431C2semidirect
13. #89428C2semidirect
14. #89425C2semidirect
15. #89425C2semidirect
69. C22×Dic7Q8semidirect
338. C7#2438semidirect

Maximal subgroups

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

subgroupindex
C2×#64412
C2×#64402
C7×#3852
#899842
C2×C7⋊#952
#897572
#897552
#897462
#897222
#897152
#894312
#894282
#894252
#24387