Finite Group Databasebeta

← #11735564

Subgroups — #11735564

Order 896.

528 conjugacy classes of subgroups.

The subgroup lattice has 410 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#6460C211yesmaximal, characteristic, complemented
2C2×#6441C211yesmaximal, characteristic, complemented
3C7×#399C211yesmaximal, characteristic, complemented
4#89766C211yesmaximal, characteristic, complemented
5#89764C211yesmaximal, characteristic, complemented
6#89757C211yesmaximal, characteristic, complemented
7#89756C211yesmaximal, characteristic, complemented
8#89752C211yesmaximal, characteristic, complemented
9#89730C211yesmaximal, characteristic
10#89729C211yesmaximal, characteristic
11#89728C211yesmaximal, characteristic
12#89722C211yesmaximal, characteristic, complemented
13#89720C211yesmaximal, characteristic, complemented
14#89440C211yesmaximal, characteristic, complemented
15#89418C211yesmaximal, characteristic, complemented
16C22×C22⋊D7C2211yescharacteristic
17C23×Dic7C2211yescharacteristic
18C22×C4×D7C2211yescharacteristic
19C22×Dic14C2211yescharacteristic
20C2×C7×C4⋊C4C2211yescharacteristic
21C2×C7×C4⋊C4C2211yescharacteristic
22C2×C7×C4⋊C4C2211yescharacteristic
23C2×C7×C22⋊C4C2211yescharacteristic
24C2×C7×C22⋊C4C2211yescharacteristic
25C2×C7×C22⋊C4C2211yescharacteristic
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C22⋊Dic7C2211yescharacteristic
28C2×(C2×D7)⋊C4C2211yescharacteristic
29C2×(C2×D7)⋊C4C2211yescharacteristic
30C2×(C2×D7)⋊C4C2211yescharacteristic
31C2×(C2×D7)⋊C4C2211yescharacteristic
32C2×(C2×D7)⋊C4C2211yescharacteristic
33C2×C4⋊Dic7C2211yescharacteristic
34C2×C4⋊Dic7C2211yescharacteristic
35C2×C4⋊Dic7C2211yescharacteristic
36C2×C4⋊Dic7C2211yescharacteristic
37C2×Dic7⋊C4C2211yescharacteristic
38C2×Dic7⋊C4C2211yescharacteristic
39C2×Dic7⋊C4C2211yescharacteristic
40C2×Dic7⋊C4C2211yescharacteristic
41C2×Dic7⋊C4C2211yescharacteristic
42C2×C4×Dic7C2211yescharacteristic
43#6460—24no—
44#6441—24no—
45C7×#95C2211yescharacteristic
46C7⋊#95C2211yescharacteristic
47C7⋊#95C2211yescharacteristic
48C7⋊#95C2211yescharacteristic
49C7⋊#95C2211yescharacteristic
50C7⋊#95C2211yescharacteristic
51C7⋊#95C2211yescharacteristic
52C7⋊#95C2211yescharacteristic
53#2732—71nomaximal, Sylow
54C24×C7C2311yescharacteristic
55C23×D7C2311yescharacteristic
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×C4×C7C2311yescharacteristic
60C22×C4×C7C2311yescharacteristic
61C22×C4×C7C2311yescharacteristic
62C2×C22⋊D7—41no—
63C2×C22⋊D7—41no—
64C2×C22⋊D7Q814yescomplemented
65C22×Dic7—22no—
66C22×Dic7—41no—
67C22×Dic7—41no—
68C22×Dic7C2311yescharacteristic
69C22×Dic7C2311yescharacteristic
70C22×Dic7C2311yescharacteristic
71C22×Dic7C2311yescharacteristic
72C22×Dic7C2311yescharacteristic
73C22×Dic7C2311yescharacteristic
74C22×Dic7C2311yescharacteristic
75C2×C4×D7—22no—
76C2×C4×D7—41no—
77C2×C4×D7—41no—
78C2×Dic14—41no—
79C2×Dic14—41no—
80C2×Dic14D414yescomplemented
81C7×C4⋊C4—22no—
82C7×C4⋊C4—22no—
83C7×C4⋊C4—41no—
84C7×C22⋊C4—22no—
85C7×C22⋊C4—22no—
86C7×C22⋊C4—41no—
87C22⋊Dic7—22no—
88C22⋊Dic7—41no—
89(C2×D7)⋊C4—22no—
90(C2×D7)⋊C4—22no—
91(C2×D7)⋊C4—22no—
92(C2×D7)⋊C4—41no—
93(C2×D7)⋊C4—41no—
94C4⋊Dic7—22no—
95C4⋊Dic7—22no—
96C4⋊Dic7—22no—
97C4⋊Dic7—41no—
98Dic7⋊C4—22no—
99Dic7⋊C4—22no—
100Dic7⋊C4—41no—
101Dic7⋊C4—41no—
102Dic7⋊C4—41no—
103C4×Dic7—41no—
104C2×C22⋊Q8—71no—
105C2×C22⋊Q8—71no—
106#399—71no—
107#399D711yescharacteristic, complemented
108#398—71no—
109#398—71no—
110#396—71no—
111#396—71no—
112#391—71no—
113#390—71no—
114#389—71no—
115#386—71no—
116#386—71no—
117#385—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
127C22×D7—21no—
128C22×D7—21no—
129C22×D7—21no—
130C22×D7—21no—
131C22×D7—21no—
132C22×D7—21no—
133C22×D7C2×Q812yes—
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×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7C2×D412yes—
151C2×C4×C7#4112yes—
152C22⋊D7—44no—
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—21no—
166C2×Dic7—21no—
167C2×Dic7—21no—
168C2×Dic7—21no—
169C2×Dic7—21no—
170C2×Dic7—21no—
171C2×Dic7—44no—
172C2×Dic7C2×D412yes—
173C2×Dic7C2×D412yes—
174C2×Dic7C2×Q812yes—
175C4×D7—44no—
176Dic14—44no—
177C22×Q8—71no—
178C22×D4—71no—
179C23×C4—71no—
180C23×C4—71no—
181C22⋊Q8—144no—
182C22⋊Q8—144no—
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⋊C4C2×D711yescharacteristic
193C2×C4⋊C4C2×D711yescharacteristic
194C2×C4⋊C4C2×D711yescharacteristic
195C2×C22⋊C4—71no—
196C2×C22⋊C4—71no—
197C2×C22⋊C4—71no—
198C2×C22⋊C4—71no—
199C2×C22⋊C4—71no—
200C2×C22⋊C4—71no—
201C2×C22⋊C4—71no—
202C2×C22⋊C4C2×D711yescharacteristic
203C2×C22⋊C4C2×D711yescharacteristic
204C2×C22⋊C4C2×D711yescharacteristic
205C2×C42—71no—
206#95—71no—
207#95—71no—
208#95—71no—
209#95—71no—
210#95—71no—
211#95—71no—
212#95—71no—
213#95C2×D711yescharacteristic
214C22×C7—22no—
215C22×C7—22no—
216C22×C7—22no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7—41no—
220C22×C7—41no—
221C22×C7C22×D411yescharacteristic
222C22×C7C22×Q811yescharacteristic
223C22×C7C2×#4111yescharacteristic
224C22×C7H(5,𝔽2)11yescharacteristic
225C22×C7H(5,𝔽2)11yescharacteristic
226C22×C7H(5,𝔽2)11yescharacteristic
227C22×C72−1+411yescharacteristic
228C2×D7—22no—
229C2×D7—22no—
230C2×D7—22no—
231C2×D7—41no—
232C2×D7—41no—
233C2×D7—41no—
234C2×D7—41no—
235C4×C7—22no—
236C4×C7—22no—
237C4×C7—41no—
238C4×C7—41no—
239C4×C7—41no—
240C4×C7—41no—
241Dic7—22no—
242Dic7—22no—
243Dic7—22no—
244Dic7—41no—
245Dic7—41no—
246Dic7—41no—
247Dic7—41no—
248C24—71no—
249C24C22×D711yescharacteristic
250C2×Q8—74no—
251C2×Q8—281no—
252C2×Q8—281no—
253C2×D4—74no—
254C2×D4—281no—
255C2×D4—281no—
256C22×C4—71no—
257C22×C4—71no—
258C22×C4—71no—
259C22×C4—71no—
260C22×C4—71no—
261C22×C4—71no—
262C22×C4—71no—
263C22×C4—142no—
264C22×C4—142no—
265C22×C4—281no—
266C22×C4—281no—
267C22×C4—281no—
268C22×C4—281no—
269C22×C4C22×D711yescharacteristic
270C22×C4C22×D711yescharacteristic
271C22×C4C22×D711yescharacteristic
272C22×C4C22×D711yescharacteristic
273C22×C4C22×D711yescharacteristic
274C22×C4C22×D711yescharacteristic
275C4⋊C4—22no—
276C4⋊C4—22no—
277C4⋊C4—41no—
278C4⋊C4—142no—
279C4⋊C4—142no—
280C4⋊C4—142no—
281C4⋊C4—142no—
282C4⋊C4—142no—
283C4⋊C4—281no—
284C4⋊C4—281no—
285C4⋊C4—281no—
286C4⋊C4—281no—
287C22⋊C4—22no—
288C22⋊C4—22no—
289C22⋊C4—41no—
290C22⋊C4—142no—
291C22⋊C4—142no—
292C22⋊C4—142no—
293C22⋊C4—142no—
294C22⋊C4—281no—
295C22⋊C4—281no—
296C22⋊C4—281no—
297C42—281no—
298C2×C7—42no—
299C2×C7#53411yescharacteristic
300C2×C7#53911yescharacteristic
301C2×C7#54311yescharacteristic
302C2×C7D4×Q811yescharacteristic
303C2×C7#55011yescharacteristic
304C2×C7#55411yescharacteristic
305C2×C7#56211yescharacteristic
306D7—42no—
307C23—21no—
308C23—21no—
309C23—21no—
310C23—21no—
311C23—21no—
312C23—21no—
313C23—72no—
314C23—141no—
315C23—141no—
316C23—141no—
317C23—141no—
318C23—141no—
319C23—141no—
320C23Q8×D712yes—
321C23C23×D711yescharacteristic
322Q8—284no—
323D4—284no—
324C2×C4—21no—
325C2×C4—21no—
326C2×C4—21no—
327C2×C4—21no—
328C2×C4—21no—
329C2×C4—21no—
330C2×C4—21no—
331C2×C4—21no—
332C2×C4—21no—
333C2×C4—21no—
334C2×C4—21no—
335C2×C4—21no—
336C2×C4—21no—
337C2×C4—21no—
338C2×C4—21no—
339C2×C4—21no—
340C2×C4—72no—
341C2×C4—72no—
342C2×C4—72no—
343C2×C4—141no—
344C2×C4—141no—
345C2×C4—141no—
346C2×C4—141no—
347C2×C4—141no—
348C2×C4—141no—
349C2×C4—141no—
350C2×C4—141no—
351C2×C4—141no—
352C2×C4—141no—
353C2×C4—141no—
354C2×C4—141no—
355C2×C4—141no—
356C2×C4—141no—
357C2×C4—141no—
358C2×C4—141no—
359C2×C4—141no—
360C2×C4—141no—
361C2×C4—284no—
362C2×C4—284no—
363C2×C4D4×D712yes—
364C2×C4#115912yes—
365C7#273211yescharacteristic, complemented, Sylow
366C22—22no—
367C22—22no—
368C22—22no—
369C22—41no—
370C22—41no—
371C22—41no—
372C22—41no—
373C22—142no—
374C22—142no—
375C22—142no—
376C22—281no—
377C22—281no—
378C22—281no—
379C22—281no—
380C22C2×D4×D711yescharacteristic
381C22C2×#115911yescharacteristic
382C22#654811yescharacteristic
383C22C2×Q8×D711yescharacteristic
384C22#655311yescharacteristic
385C22#655311yescharacteristic
386C22#655411yescharacteristic
387C4—22no—
388C4—22no—
389C4—41no—
390C4—41no—
391C4—41no—
392C4—41no—
393C4—142no—
394C4—142no—
395C4—142no—
396C4—281no—
397C4—281no—
398C4—281no—
399C4—281no—
400C2—42no—
401C2—282no—
402C2#9032311yescharacteristic
403C2#9032511yescharacteristic
404C2#9032611yescharacteristic
405C2#9034411yescharacteristic
406C2#9035611yescharacteristic
407C2#9035811yescharacteristic
408C2#9040911yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
FittingC7×#399C2
prime radicalC22×C22⋊D7C22
hypercenter#399D7
hypocenterC7#2732
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ #399

Complements

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

NH ≅ G/Ntype
1. C2×#6460C2semidirect
2. C2×#6441C2semidirect
3. C7×#399C2semidirect
4. #89766C2semidirect
5. #89764C2semidirect
6. #89757C2semidirect
7. #89756C2semidirect
8. #89752C2semidirect
12. #89722C2semidirect
13. #89720C2semidirect
14. #89440C2semidirect
15. #89418C2semidirect
64. C2×C22⋊D7Q8semidirect
80. C2×Dic14D4semidirect
107. #399D7semidirect
365. C7#2732semidirect

Maximal subgroups

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

subgroupindex
C2×#64602
C2×#64412
C7×#3992
#897662
#897642
#897572
#897562
#897522
#897302
#897292
#897282
#897222
#897202
#894402
#894182
#27327