Finite Group Databasebeta

← #11734741

Subgroups — #11734741

Order 896.

508 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
1C2×#6450C211yesmaximal, characteristic, complemented
2C2×#6441C211yesmaximal, characteristic
3C7×#396C211yesmaximal, characteristic
4#89753C211yesmaximal, characteristic, complemented
5#89751C211yesmaximal, characteristic, complemented
6#89751C211yesmaximal, characteristic, complemented
7#89750C211yesmaximal, characteristic, complemented
8#89746C211yesmaximal, characteristic, complemented
9#89719C211yesmaximal, characteristic
10#89718C211yesmaximal, characteristic
11#89717C211yesmaximal, characteristic
12#89716C211yesmaximal, characteristic
13C22⋊C4×Dic7C211yesmaximal, characteristic
14#89424C211yesmaximal, characteristic, complemented
15#89419C211yesmaximal, characteristic, complemented
16C23×Dic7C2211yescharacteristic
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×C4⋊Dic7C2211yescharacteristic
29C2×Dic7⋊C4C2211yescharacteristic
30C2×Dic7⋊C4C2211yescharacteristic
31C2×Dic7⋊C4C2211yescharacteristic
32C2×Dic7⋊C4C2211yescharacteristic
33C2×Dic7⋊C4C2211yescharacteristic
34C2×Dic7⋊C4C2211yescharacteristic
35C2×C4×Dic7C2211yescharacteristic
36C2×C4×Dic7C2211yescharacteristic
37C2×C4×Dic7C2211yescharacteristic
38C2×C4×Dic7C2211yescharacteristic
39#6450—24no—
40#6441—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#2465—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C22×C4×C7C2311yescharacteristic
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×Dic7—22no—
62C22×Dic7—22no—
63C22×Dic7—41no—
64C22×Dic7C2311yescharacteristic
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C22×Dic7C2311yescharacteristic
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C2×Dic14—22no—
73C2×Dic14—22no—
74C2×Dic14—22no—
75C2×Dic14—22no—
76C2×Dic14—22no—
77C2×Dic14—41no—
78C7×C4⋊C4—22no—
79C7×C4⋊C4—22no—
80C7×C4⋊C4—41no—
81C7×C22⋊C4—22no—
82C7×C22⋊C4—22no—
83C7×C22⋊C4—41no—
84C22⋊Dic7—22no—
85C22⋊Dic7—22no—
86C22⋊Dic7Q814yescomplemented
87C4⋊Dic7—22no—
88Dic7⋊C4—22no—
89Dic7⋊C4—22no—
90Dic7⋊C4—22no—
91Dic7⋊C4—22no—
92Dic7⋊C4—41no—
93Dic7⋊C4D414yes—
94C4×Dic7—22no—
95C4×Dic7—22no—
96C4×Dic7—22no—
97C4×Dic7—41no—
98C2×C22⋊Q8—71no—
99C2×C4×Q8—71no—
100#396D711yescharacteristic
101#395—71no—
102#395—71no—
103#393—71no—
104#391—71no—
105#389—71no—
106#388—71no—
107#388—71no—
108#387—71no—
109#387—71no—
110#387—71no—
111#385—71no—
112C4×C22⋊C4—71no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7—21no—
116C23×C7—21no—
117C23×C7—21no—
118C23×C7—21no—
119C23×C7C2×Q812yes—
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×C7C2×D412yes—
138C2×C4×C7#4112yes—
139C2×Dic7—21no—
140C2×Dic7—21no—
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—44no—
158C2×Dic7C2×D412yes—
159C2×Dic7C2×D412yes—
160C2×Dic7C2×Q812yes—
161C2×Dic7C2×Q812yes—
162C2×Dic7#4112yes—
163C2×Dic7#4112yes—
164Dic14—28no—
165Dic14—44no—
166C22×Q8—71no—
167C22×Q8—71no—
168C23×C4—71no—
169C22⋊Q8—144no—
170C4×Q8—144no—
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⋊C4C2×D711yescharacteristic
179C2×C4⋊C4C2×D711yescharacteristic
180C2×C4⋊C4C2×D711yescharacteristic
181C2×C22⋊C4—71no—
182C2×C22⋊C4—71no—
183C2×C22⋊C4—71no—
184C2×C22⋊C4C2×D711yescharacteristic
185C2×C22⋊C4C2×D711yescharacteristic
186C2×C22⋊C4C2×D711yescharacteristic
187C2×C42—71no—
188C2×C42—71no—
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#95—71no—
202#95C2×D711yescharacteristic
203C22×C7—22no—
204C22×C7—22no—
205C22×C7—22no—
206C22×C7—41no—
207C22×C7—41no—
208C22×C7—41no—
209C22×C7—41no—
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—22no—
219C4×C7—41no—
220C4×C7—41no—
221C4×C7—41no—
222C4×C7—41no—
223Dic7—22no—
224Dic7—22no—
225Dic7—22no—
226Dic7—22no—
227Dic7—22no—
228Dic7—22no—
229Dic7—41no—
230Dic7—41no—
231C24C22×D711yescharacteristic
232C2×Q8—142no—
233C2×Q8—142no—
234C2×Q8—142no—
235C2×Q8—142no—
236C2×Q8—142no—
237C2×Q8—281no—
238C22×C4—71no—
239C22×C4—71no—
240C22×C4—71no—
241C22×C4—71no—
242C22×C4—71no—
243C22×C4—71no—
244C22×C4—71no—
245C22×C4—71no—
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—22no—
257C4⋊C4—41no—
258C4⋊C4—74no—
259C4⋊C4—142no—
260C4⋊C4—142no—
261C4⋊C4—142no—
262C4⋊C4—142no—
263C4⋊C4—142no—
264C4⋊C4—281no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—41no—
268C22⋊C4—74no—
269C22⋊C4—142no—
270C22⋊C4—142no—
271C42—142no—
272C42—142no—
273C42—142no—
274C42—281no—
275C2×C7—42no—
276C2×C7#52511yescharacteristic
277C2×C7#53311yescharacteristic
278C2×C7#54211yescharacteristic
279C2×C7#54611yescharacteristic
280C2×C7D4×Q811yescharacteristic
281C2×C7#55411yescharacteristic
282C2×C7#55611yescharacteristic
283C23—21no—
284C23—21no—
285C23—21no—
286C23—21no—
287C23—21no—
288C23—21no—
289C23Q8×D712yes—
290C23C23×D711yescharacteristic
291Q8—148no—
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—21no—
309C2×C4—72no—
310C2×C4—72no—
311C2×C4—72no—
312C2×C4—72no—
313C2×C4—72no—
314C2×C4—72no—
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—284no—
334C2×C4D4×D712yes—
335C2×C4#115912yes—
336C7#246511yescharacteristic, complemented, Sylow
337C22—22no—
338C22—22no—
339C22—22no—
340C22—41no—
341C22—41no—
342C22—41no—
343C22—41no—
344C22C2×D4×D711yescharacteristic
345C22C2×#115911yescharacteristic
346C22#654811yescharacteristic
347C22C2×Q8×D711yescharacteristic
348C22D7×#4111yescharacteristic
349C22D7×#4111yescharacteristic
350C22#655411yescharacteristic
351C4—22no—
352C4—22no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—41no—
357C4—142no—
358C4—142no—
359C4—142no—
360C4—142no—
361C4—142no—
362C4—142no—
363C4—281no—
364C4—281no—
365C2—42no—
366C2#9029411yescharacteristic
367C2#9029711yescharacteristic
368C2#9030111yescharacteristic
369C2#9031311yescharacteristic
370C2#9032311yescharacteristic
371C2#9032711yescharacteristic
372C2#9039311yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#396C2
prime radicalC24×C7C23
hypercenter#396D7
hypocenterC7#2465
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #396

Complements

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

NH ≅ G/Ntype
1. C2×#6450C2semidirect
4. #89753C2semidirect
5. #89751C2semidirect
6. #89751C2semidirect
7. #89750C2semidirect
8. #89746C2semidirect
14. #89424C2semidirect
15. #89419C2semidirect
86. C22⋊Dic7Q8semidirect
336. C7#2465semidirect

Maximal subgroups

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

subgroupindex
C2×#64502
C2×#64412
C7×#3962
#897532
#897512
#897502
#897462
#897192
#897182
#897172
#897162
C22⋊C4×Dic72
#894242
#894192
#24657