Finite Group Databasebeta

← #11731862

Subgroups — #11731862

Order 896.

520 conjugacy classes of subgroups.

The subgroup lattice has 374 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C22×Dic7⋊C4C211yesmaximal, characteristic
2C2×#6452C211yesmaximal, characteristic, complemented
3C7×#385C211yesmaximal, characteristic
4#89984C211yesmaximal, characteristic
5#89756C211yesmaximal, characteristic, complemented
6#89752C211yesmaximal, characteristic, complemented
7#89744C211yesmaximal, characteristic, complemented
8#89718C211yesmaximal, characteristic
9#89714C211yesmaximal, characteristic
10#89714C211yesmaximal, characteristic
11C22⋊C4×Dic7C211yesmaximal, characteristic
12#89431C211yesmaximal, characteristic, complemented
13#89430C211yesmaximal, characteristic, complemented
14#89422C211yesmaximal, characteristic, complemented
15#89419C211yesmaximal, characteristic, complemented
16C23×C4×C7C2211yescharacteristic
17C23×Dic7C2211yescharacteristic
18C23×Dic7C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C22⋊Dic7C2211yescharacteristic
24C2×C22⋊Dic7C2211yescharacteristic
25C2×C4⋊Dic7C2211yescharacteristic
26C2×C4⋊Dic7C2211yescharacteristic
27C2×Dic7⋊C4—24no—
28C2×Dic7⋊C4C2211yescharacteristic
29C2×Dic7⋊C4C2211yescharacteristic
30C2×Dic7⋊C4C2211yescharacteristic
31C2×Dic7⋊C4C2211yescharacteristic
32C2×Dic7⋊C4C2211yescharacteristic
33C2×Dic7⋊C4C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×Dic7⋊C4C2211yescharacteristic
36C2×Dic7⋊C4C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×C4×Dic7C2211yescharacteristic
39C2×C4×Dic7C2211yescharacteristic
40#6452—24no—
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#2442—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×Dic7Q814yes—
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C22×Dic7C2311yescharacteristic
75C22×Dic7C2311yescharacteristic
76C22×Dic7C2311yescharacteristic
77C22×Dic7C2311yescharacteristic
78C7×C4⋊C4—22no—
79C7×C4⋊C4—41no—
80C7×C22⋊C4—22no—
81C7×C22⋊C4—22no—
82C22⋊Dic7—22no—
83C22⋊Dic7—22no—
84C4⋊Dic7—22no—
85C4⋊Dic7—41no—
86Dic7⋊C4—22no—
87Dic7⋊C4—22no—
88Dic7⋊C4—22no—
89Dic7⋊C4—22no—
90Dic7⋊C4—22no—
91Dic7⋊C4—22no—
92Dic7⋊C4—22no—
93Dic7⋊C4—41no—
94Dic7⋊C4—41no—
95Dic7⋊C4D414yes—
96C4×Dic7—22no—
97C4×Dic7—22no—
98C2×#125—71no—
99C22×C4⋊C4—71no—
100#400—71no—
101#400—71no—
102#398—71no—
103#398—71no—
104#389—71no—
105#389—71no—
106#385—71no—
107#385D711yescharacteristic
108#382—71no—
109#380—71no—
110#380—71no—
111#380—71no—
112C4×C22⋊C4—71no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7C2×Q812yes—
118C23×C7#4112yes—
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—21no—
137C2×C4×C7—21no—
138C2×C4×C7—44no—
139C2×C4×C7C2×D412yes—
140C2×Dic7—21no—
141C2×Dic7—21no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—28no—
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—44no—
161C2×Dic7C2×D412yes—
162C2×Dic7C2×D412yes—
163C2×Dic7C2×Q812yes—
164C2×Dic7C2×Q812yes—
165C2×Dic7#4112yes—
166C23×C4—71no—
167C23×C4—71no—
168C23×C4C2×D711yescharacteristic
169#125—144no—
170C2×C4⋊C4—71no—
171C2×C4⋊C4—71no—
172C2×C4⋊C4—71no—
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⋊C4—144no—
183C2×C4⋊C4C2×D711yescharacteristic
184C2×C4⋊C4C2×D711yescharacteristic
185C2×C22⋊C4—71no—
186C2×C22⋊C4—71no—
187C2×C22⋊C4C2×D711yescharacteristic
188C2×C22⋊C4C2×D711yescharacteristic
189C2×C42—71no—
190C2×C42—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#95C2×D711yescharacteristic
202#95C2×D711yescharacteristic
203C22×C7—22no—
204C22×C7—22no—
205C22×C7—22no—
206C22×C7—22no—
207C22×C7—22no—
208C22×C7—22no—
209C22×C7#12514yescomplemented
210C22×C7C22×D411yescharacteristic
211C22×C7C22×Q811yescharacteristic
212C22×C7C2×#4111yescharacteristic
213C22×C7C2×#4111yescharacteristic
214C22×C7C2×#4111yescharacteristic
215C22×C7H(5,𝔽2)11yescharacteristic
216C22×C72−1+411yescharacteristic
217C4×C7—22no—
218C4×C7—41no—
219C4×C7—41no—
220C4×C7—41no—
221C4×C7—41no—
222C4×C7—41no—
223Dic7—22no—
224Dic7—22no—
225Dic7—41no—
226Dic7—41no—
227Dic7—41no—
228Dic7—41no—
229Dic7—41no—
230Dic7C22⋊Q814yes—
231C24C22×D711yescharacteristic
232C22×C4—22no—
233C22×C4—22no—
234C22×C4—41no—
235C22×C4—71no—
236C22×C4—74no—
237C22×C4—71no—
238C22×C4—71no—
239C22×C4—71no—
240C22×C4—71no—
241C22×C4—71no—
242C22×C4—71no—
243C22×C4—71no—
244C22×C4—142no—
245C22×C4—142no—
246C22×C4—142no—
247C22×C4—142no—
248C22×C4—281no—
249C22×C4C22×D711yescharacteristic
250C22×C4C22×D711yescharacteristic
251C22×C4C22×D711yescharacteristic
252C22×C4C22×D711yescharacteristic
253C22×C4C22×D711yescharacteristic
254C22×C4C22×D711yescharacteristic
255C4⋊C4—22no—
256C4⋊C4—41no—
257C4⋊C4—74no—
258C4⋊C4—142no—
259C4⋊C4—142no—
260C4⋊C4—142no—
261C4⋊C4—142no—
262C4⋊C4—142no—
263C4⋊C4—142no—
264C4⋊C4—142no—
265C4⋊C4—142no—
266C4⋊C4—281no—
267C4⋊C4—281no—
268C4⋊C4—281no—
269C22⋊C4—22no—
270C22⋊C4—22no—
271C22⋊C4—142no—
272C22⋊C4—142no—
273C42—142no—
274C42—142no—
275C2×C7—24no—
276C2×C7C2×C22⋊Q811yescharacteristic
277C2×C7C2×#12511yescharacteristic
278C2×C7#54411yescharacteristic
279C2×C7#54611yescharacteristic
280C2×C7#54811yescharacteristic
281C2×C7#55211yescharacteristic
282C2×C7#55311yescharacteristic
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23#115712yes—
288C23#115912yes—
289C23Q8×D712yes—
290C23C23×D711yescharacteristic
291C2×C4—21no—
292C2×C4—21no—
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—72no—
313C2×C4—72no—
314C2×C4—148no—
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×C4D4×D712yes—
336C7#244211yescharacteristic, complemented, Sylow
337C22—22no—
338C22—22no—
339C22—22no—
340C22—22no—
341C22—22no—
342C22—22no—
343C22#645214yescomplemented
344C22C2×#115711yescharacteristic
345C22C2×D4×D711yescharacteristic
346C22C2×#115911yescharacteristic
347C22#654811yescharacteristic
348C22C2×Q8×D711yescharacteristic
349C22D7×#4111yescharacteristic
350C22#655411yescharacteristic
351C4—22no—
352C4—41no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—74no—
358C4—142no—
359C4—142no—
360C4—281no—
361C4—281no—
362C4—281no—
363C4—281no—
364C4—281no—
365C2—24no—
366C2C2×#645211yescharacteristic
367C2#9023111yescharacteristic
368C2#9024311yescharacteristic
369C2#9028211yescharacteristic
370C2#9031411yescharacteristic
371C2D7×C22⋊Q811yescharacteristic
372C2#9034111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#385C2
prime radicalC24×C7C23
hypercenter#385D7
hypocenterC7#2442
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #385

Complements

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

NH ≅ G/Ntype
2. C2×#6452C2semidirect
5. #89756C2semidirect
6. #89752C2semidirect
7. #89744C2semidirect
12. #89431C2semidirect
13. #89430C2semidirect
14. #89422C2semidirect
15. #89419C2semidirect
209. C22×C7#125semidirect
336. C7#2442semidirect
343. C22#6452semidirect

Maximal subgroups

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

subgroupindex
C22×Dic7⋊C42
C2×#64522
C7×#3852
#899842
#897562
#897522
#897442
#897182
#897142
C22⋊C4×Dic72
#894312
#894302
#894222
#894192
#24427