Finite Group Databasebeta

← #11734698

Subgroups — #11734698

Order 896.

544 conjugacy classes of subgroups.

The subgroup lattice has 386 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6509C211yesmaximal, characteristic
2C2×#6460C211yesmaximal, characteristic
3C7×#395C211yesmaximal, characteristic, complemented
4#90002C211yesmaximal, characteristic, complemented
5#89761C211yesmaximal, characteristic
6#89759C211yesmaximal, characteristic
7#89757C211yesmaximal, characteristic, complemented
8#89756C211yesmaximal, characteristic, complemented
9#89744C211yesmaximal, characteristic, complemented
10#89440C211yesmaximal, characteristic
11#89438C211yesmaximal, characteristic
12D7×#95C211yesmaximal, characteristic
13#89433C211yesmaximal, characteristic, complemented
14#89428C211yesmaximal, characteristic, complemented
15#89418C211yesmaximal, characteristic, complemented
16C22×C7×Q8C2211yescharacteristic
17C22×C4×D7C2211yescharacteristic
18C22×C4×D7C2211yescharacteristic
19C22×C4×D7C2211yescharacteristic
20C22×Dic14C2211yescharacteristic
21C2×C7×C4⋊C4C2211yescharacteristic
22C2×C7×C4⋊C4C2211yescharacteristic
23C2×C7×C4⋊C4C2211yescharacteristic
24#6509—24no—
25C2×(C2×D7)⋊C4C2211yescharacteristic
26C2×(C2×D7)⋊C4C2211yescharacteristic
27C2×(C2×D7)⋊C4C2211yescharacteristic
28C2×(C2×D7)⋊C4C2211yescharacteristic
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×C4×Dic7C2211yescharacteristic
39C2×C4×Dic7C2211yescharacteristic
40#6460—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#2549—71nomaximal, Sylow
54C23×D7C2311yescharacteristic
55C2×C7×Q8—22no—
56C2×C7×Q8—22no—
57C2×C7×Q8—41no—
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C22×C4×C7C2311yescharacteristic
63C22×C4×C7C2311yescharacteristic
64C22×C4×C7C2311yescharacteristic
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C22×Dic7C2311yescharacteristic
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C2×C4×D7—22no—
73C2×C4×D7—22no—
74C2×C4×D7—22no—
75C2×C4×D7—22no—
76C2×C4×D7—41no—
77C2×C4×D7—41no—
78C2×C4×D7—41no—
79C2×C4×D7—41no—
80C2×C4×D7Q814yescomplemented
81C2×Dic14—22no—
82C2×Dic14—22no—
83C2×Dic14—41no—
84C7×C4⋊C4—22no—
85C7×C4⋊C4—22no—
86C7×C4⋊C4—41no—
87(C2×D7)⋊C4—22no—
88(C2×D7)⋊C4—22no—
89(C2×D7)⋊C4D414yes—
90(C2×D7)⋊C4Q814yescomplemented
91C4⋊Dic7—22no—
92C4⋊Dic7—22no—
93C4⋊Dic7—22no—
94C4⋊Dic7—41no—
95C4⋊Dic7—41no—
96Dic7⋊C4—22no—
97Dic7⋊C4—22no—
98Dic7⋊C4—22no—
99Dic7⋊C4—41no—
100C4×Dic7—41no—
101C4×Dic7—41no—
102C2×C22⋊Q8—71no—
103C2×C22⋊Q8—71no—
104#398—71no—
105#398—71no—
106#395—71no—
107#395D711yescharacteristic, complemented
108#391—71no—
109#391—71no—
110#389—71no—
111#389—71no—
112#385—71no—
113#385—71no—
114#380—71no—
115C4×C22⋊C4—71no—
116C2×#95—71no—
117C23×C7C2411yescharacteristic
118C22×D7—21no—
119C22×D7—21no—
120C22×D7—21no—
121C22×D7—21no—
122C22×D7C2×D412yes—
123C22×D7C2×Q812yes—
124C22×D7C2×Q812yes—
125C7×Q8—44no—
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—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7C2×D412yes—
143C2×C4×C7C2×Q812yes—
144C2×C4×C7C2×Q812yes—
145C2×C4×C7#4112yes—
146C2×C4×C7#4112yes—
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×Dic7C2×D412yes—
166C2×Dic7C2×Q812yes—
167C2×Dic7C2×Q812yes—
168C4×D7—44no—
169C4×D7—44no—
170C4×D7—44no—
171Dic14—44no—
172C22×Q8—71no—
173C22×Q8C2×D711yescharacteristic
174C23×C4—71no—
175C23×C4—71no—
176C23×C4—71no—
177C22⋊Q8—144no—
178C22⋊Q8—144no—
179C2×C4⋊C4—71no—
180C2×C4⋊C4—71no—
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⋊C4C2×D711yescharacteristic
189C2×C4⋊C4C2×D711yescharacteristic
190C2×C4⋊C4C2×D711yescharacteristic
191C2×C22⋊C4—71no—
192C2×C22⋊C4—71no—
193C2×C22⋊C4—71no—
194C2×C22⋊C4—71no—
195C2×C42—71no—
196C2×C42—71no—
197#95—71no—
198#95—71no—
199#95—71no—
200#95—71no—
201#95—71no—
202#95—71no—
203#95—71no—
204#95—71no—
205#95—71no—
206#95C2×D711yescharacteristic
207#95C2×D711yescharacteristic
208#95C2×D711yescharacteristic
209C22×C7C22×D411yescharacteristic
210C22×C7C22×Q811yescharacteristic
211C22×C7C22×Q811yescharacteristic
212C22×C7C2×#4111yescharacteristic
213C22×C7C2×#4111yescharacteristic
214C22×C7H(5,𝔽2)11yescharacteristic
215C22×C7H(5,𝔽2)11yescharacteristic
216C2×D7—22no—
217C2×D7—22no—
218C2×D7—22no—
219C2×D7—22no—
220C2×D7—22no—
221C2×D7—22no—
222C2×D7C4⋊Q814yes—
223C4×C7—22no—
224C4×C7—22no—
225C4×C7—22no—
226C4×C7—22no—
227C4×C7—22no—
228C4×C7—41no—
229C4×C7—41no—
230Dic7—22no—
231Dic7—22no—
232Dic7—22no—
233Dic7—41no—
234Dic7—41no—
235Dic7—41no—
236Dic7—41no—
237C24—71no—
238C2×Q8—22no—
239C2×Q8—22no—
240C2×Q8—41no—
241C2×Q8—142no—
242C2×Q8—142no—
243C2×Q8—281no—
244C22×C4—71no—
245C22×C4—71no—
246C22×C4—71no—
247C22×C4—74no—
248C22×C4—71no—
249C22×C4—71no—
250C22×C4—71no—
251C22×C4—71no—
252C22×C4—142no—
253C22×C4—142no—
254C22×C4—142no—
255C22×C4—142no—
256C22×C4—281no—
257C22×C4—281no—
258C22×C4—281no—
259C22×C4—281no—
260C22×C4C22×D711yescharacteristic
261C22×C4C22×D711yescharacteristic
262C22×C4C22×D711yescharacteristic
263C22×C4C22×D711yescharacteristic
264C22×C4C22×D711yescharacteristic
265C22×C4C22×D711yescharacteristic
266C22×C4C22×D711yescharacteristic
267C4⋊C4—22no—
268C4⋊C4—22no—
269C4⋊C4—41no—
270C4⋊C4—142no—
271C4⋊C4—142no—
272C4⋊C4—142no—
273C4⋊C4—142no—
274C4⋊C4—142no—
275C4⋊C4—142no—
276C4⋊C4—281no—
277C4⋊C4—281no—
278C4⋊C4—281no—
279C22⋊C4—74no—
280C22⋊C4—74no—
281C22⋊C4—142no—
282C22⋊C4—142no—
283C42—281no—
284C42—281no—
285C2×C7C2×C4⋊Q811yescharacteristic
286C2×C7#53311yescharacteristic
287C2×C7#54311yescharacteristic
288C2×C7#54611yescharacteristic
289C2×C7#54611yescharacteristic
290C2×C7#55411yescharacteristic
291C2×C7#55411yescharacteristic
292D7—24no—
293C23—72no—
294C23—72no—
295C23—72no—
296C23—141no—
297C23—141no—
298C23—141no—
299C23—141no—
300C23C23×D711yescharacteristic
301Q8—44no—
302Q8—284no—
303C2×C4—21no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
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—72no—
320C2×C4—72no—
321C2×C4—72no—
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—141no—
335C2×C4—141no—
336C2×C4—141no—
337C2×C4—141no—
338C2×C4—141no—
339C2×C4—141no—
340C2×C4—284no—
341C2×C4—284no—
342C2×C4—284no—
343C2×C4D4×D712yes—
344C2×C4#115912yes—
345C2×C4Q8×D712yes—
346C2×C4Q8×D712yes—
347C2×C4#116112yes—
348C7#254911yescharacteristic, complemented, Sylow
349C22—74no—
350C22—142no—
351C22—142no—
352C22—142no—
353C22—142no—
354C22—142no—
355C22—142no—
356C22C2×D4×D711yescharacteristic
357C22C2×#115911yescharacteristic
358C22#654811yescharacteristic
359C22C2×Q8×D711yescharacteristic
360C22C2×Q8×D711yescharacteristic
361C22C2×#116111yescharacteristic
362C22#655311yescharacteristic
363C4—22no—
364C4—22no—
365C4—22no—
366C4—22no—
367C4—22no—
368C4—41no—
369C4—41no—
370C4—142no—
371C4—142no—
372C4—142no—
373C4—281no—
374C4—281no—
375C4—281no—
376C4—281no—
377C2—144no—
378C2#9028211yescharacteristic
379C2#9032111yescharacteristic
380C2#9032511yescharacteristic
381C2#9032611yescharacteristic
382C2D7×C4⋊Q811yescharacteristic
383C2#9041911yescharacteristic
384C2#9042111yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#395C2
prime radicalC23×D7C23
hypercenter#395D7
hypocenterC7#2549
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
3. C7×#395C2semidirect
4. #90002C2semidirect
7. #89757C2semidirect
8. #89756C2semidirect
9. #89744C2semidirect
13. #89433C2semidirect
14. #89428C2semidirect
15. #89418C2semidirect
80. C2×C4×D7Q8semidirect
90. (C2×D7)⋊C4Q8semidirect
107. #395D7semidirect
348. C7#2549semidirect

Maximal subgroups

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

subgroupindex
C2×#65092
C2×#64602
C7×#3952
#900022
#897612
#897592
#897572
#897562
#897442
#894402
#894382
D7×#952
#894332
#894282
#894182
#25497