Finite Group Databasebeta

← #11741065

Subgroups — #11741065

Order 896.

388 conjugacy classes of subgroups.

The subgroup lattice has 362 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#553C211yesmaximal, characteristic
2D4×Dic14C211yesmaximal, characteristic, complemented
3#89841C211yesmaximal, characteristic, complemented
4#89835C211yesmaximal, characteristic, complemented
5#89814C211yesmaximal, characteristic, complemented
6#89813C211yesmaximal, characteristic, complemented
7#89812C211yesmaximal, characteristic, complemented
8#89812C211yesmaximal, characteristic, complemented
9#89791C211yesmaximal, characteristic, complemented
10#89790C211yesmaximal, characteristic, complemented
11#89783C211yesmaximal, characteristic, complemented
12#89776C211yesmaximal, characteristic, complemented
13#89768C211yesmaximal, characteristic, complemented
14#89742C211yesmaximal, characteristic, complemented
15C2×#6384C211yesmaximal, characteristic, complemented
16C22×Dic14C2211yescharacteristic
17C22×Dic14C2211yescharacteristic
18C7×#126—21no—
19C7×#125C2211yescharacteristic
20C7×#123—21no—
21C7×#121—21no—
22C7×#121C2211yescharacteristic
23C7×#121C2211yescharacteristic
24C4×C7×D4—21no—
25C4×C7×D4C2211yescharacteristic
26C4×C7×D4C2211yescharacteristic
27C7×#117C2211yescharacteristic
28C2×C7×C4⋊C4C2211yescharacteristic
29D4×Dic7—21no—
30C2×#1142C2211yescharacteristic
31C2×#1142C2211yescharacteristic
32C2×#1142C2211yescharacteristic
33#6487C2211yescharacteristic
34#6487C2211yescharacteristic
35C2×#1136C2211yescharacteristic
36C22×C7⋊C8C2211yescharacteristic
37#6451—21no—
38#6441—21no—
39#6441—21no—
40#6432C2211yescharacteristic
41C4×Dic14C2211yescharacteristic
42#6406C2211yescharacteristic
43C22⋊(C7⋊C8)C2211yescharacteristic
44C22⋊(C7⋊C8)C2211yescharacteristic
45#6384C2211yescharacteristic
46#6384C2211yescharacteristic
47#6384C2211yescharacteristic
48#6384C2211yescharacteristic
49#6384C2211yescharacteristic
50#6384C2211yescharacteristic
51#6384C2211yescharacteristic
52#6384C2211yescharacteristic
53#6384C2211yescharacteristic
54#6382C2211yescharacteristic
55#6381C2211yescharacteristic
56#6381C2211yescharacteristic
57C4⋊(C7⋊C8)C2211yescharacteristic
58#6377C2211yescharacteristic
59#3320—71nomaximal, Sylow
60C2×C7×D4—21no—
61C2×C7×D4—21no—
62C2×C7×D4D411yescharacteristic
63C2×C7×D4C2311yescharacteristic
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7C2311yescharacteristic
67C22×C4×C7C2311yescharacteristic
68C22×Dic7—21no—
69C22×Dic7—21no—
70C2×Dic14—22no—
71C2×Dic14—22no—
72C2×Dic14—21no—
73C2×Dic14—21no—
74C2×Dic14—41no—
75C2×Dic14—41no—
76C2×Dic14C2311yescharacteristic
77C2×Dic14C2311yescharacteristic
78C2×Dic14C2311yescharacteristic
79C7×C4⋊C4—21no—
80C7×C4⋊C4—21no—
81C7×C4⋊C4—21no—
82C7×C4⋊C4D411yescharacteristic
83C7×C4⋊C4C2311yescharacteristic
84C7×C4⋊C4C2311yescharacteristic
85C7×C4⋊C4C2311yescharacteristic
86C7×C22⋊C4—21no—
87C7×C22⋊C4—21no—
88C7×C22⋊C4—21no—
89C7×C22⋊C4—21no—
90C7×C22⋊C4D411yescharacteristic
91C7×C22⋊C4D411yescharacteristic
92C42×C7—21no—
93C42×C7C2311yescharacteristic
94C22⋊Dic7—21no—
95C22⋊Dic7—21no—
96#1142—22no—
97#1142—41no—
98#1142—41no—
99C4⋊Dic7—21no—
100C4⋊Dic7—21no—
101C4⋊Dic7C2311yescharacteristic
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104Dic7⋊C4—21no—
105C4×Dic7—21no—
106#1136—41no—
107C2×C7⋊C8—41no—
108C2×C7⋊C8C2311yescharacteristic
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111C2×C7⋊C8C2311yescharacteristic
112#553D711yescharacteristic
113D4×Q8—71no—
114#491—71no—
115#485—71no—
116#484—71no—
117#479—71no—
118#468—71no—
119#465—71no—
120#461—71no—
121#448—71no—
122#448—71no—
123#440—71no—
124#435—71no—
125#419—71no—
126C2×#103—71no—
127C23×C7—21no—
128C23×C7C2×D411yescharacteristic
129C23×C7C2×D411yescharacteristic
130C7×D4—22no—
131C7×D4—21no—
132C7×D4—21no—
133C7×D4—42no—
134C7×D4—81no—
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—81no—
143C2×C4×C7—81no—
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2×D411yescharacteristic
146C2×C4×C7C2×D411yescharacteristic
147C2×C4×C7C2×D411yescharacteristic
148C2×C4×C7C2411yescharacteristic
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7—21no—
153C2×Dic7—42no—
154C2×Dic7—42no—
155Dic14—21no—
156Dic14—21no—
157Dic14—21no—
158Dic14—21no—
159Dic14—21no—
160Dic14—42no—
161Dic14—42no—
162Dic14#4112yes—
163C7⋊C8—21no—
164C7⋊C8—21no—
165C7⋊C8—21no—
166C7⋊C8—21no—
167C22×Q8—71no—
168C22×Q8—71no—
169C2×SD16—71no—
170C2×SD16—71no—
171C2×SD16—71no—
172C2×M16—71no—
173C22×C8—71no—
174C4⋊Q8—71no—
175C4⋊Q8—141no—
176#126—21no—
177#125C2×D711yescharacteristic
178#123—21no—
179C22⋊Q8—71no—
180C22⋊Q8—71no—
181C22⋊Q8—141no—
182C22⋊Q8—141no—
183#121—21no—
184#121C2×D711yescharacteristic
185#121C2×D711yescharacteristic
186C4×Q8—71no—
187C4×D4—21no—
188C4×D4—141no—
189C4×D4C2×D711yescharacteristic
190C4×D4C2×D711yescharacteristic
191#117C2×D711yescharacteristic
192C2×C4⋊C4C2×D711yescharacteristic
193#107—71no—
194#107—71no—
195#106—71no—
196C4⋊C8—71no—
197#103—71no—
198#103—71no—
199#103—71no—
200#103—71no—
201#103—71no—
202#103—71no—
203#103—71no—
204#103—71no—
205#103—71no—
206#102—71no—
207C22⋊C8—71no—
208C22⋊C8—71no—
209#97—71no—
210C22×C7—21no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—21no—
214C22×C7—21no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7#13712yes—
219C22×C7C22×D411yescharacteristic
220C4×C7—21no—
221C4×C7—21no—
222C4×C7—21no—
223C4×C7—21no—
224C4×C7—41no—
225C4×C7—41no—
226C4×C7—41no—
227C4×C7C2×#4111yescharacteristic
228C4×C7H(5,𝔽2)11yescharacteristic
229Dic7—22no—
230Dic7—41no—
231Dic7—41no—
232Dic7—41no—
233C2×Q8—71no—
234C2×Q8—71no—
235C2×Q8—71no—
236C2×Q8—142no—
237C2×Q8—142no—
238C2×Q8—141no—
239C2×Q8—141no—
240C2×Q8—281no—
241C2×Q8—281no—
242C2×D4—21no—
243C2×D4—21no—
244C2×D4C22⋊D711yescharacteristic
245C2×D4C22×D711yescharacteristic
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—141no—
249C22×C4—141no—
250C22×C4C22×D711yescharacteristic
251C22×C4C22×D711yescharacteristic
252SD16—142no—
253SD16—281no—
254SD16—281no—
255M16—281no—
256C2×C8—71no—
257C2×C8—71no—
258C2×C8—71no—
259C2×C8—71no—
260C2×C8—281no—
261C4⋊C4—21no—
262C4⋊C4—21no—
263C4⋊C4—21no—
264C4⋊C4—71no—
265C4⋊C4—141no—
266C4⋊C4—141no—
267C4⋊C4—141no—
268C4⋊C4—141no—
269C4⋊C4—141no—
270C4⋊C4C22⋊D711yescharacteristic
271C4⋊C4C22×D711yescharacteristic
272C4⋊C4C22×D711yescharacteristic
273C4⋊C4C22×D711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—21no—
277C22⋊C4—21no—
278C22⋊C4—141no—
279C22⋊C4—141no—
280C22⋊C4C22⋊D711yescharacteristic
281C22⋊C4C22⋊D711yescharacteristic
282C42—21no—
283C42—141no—
284C42C22×D711yescharacteristic
285C2×C7—22no—
286C2×C7—41no—
287C2×C7—81no—
288C2×C7#54611yescharacteristic
289C2×C7C2×#13711yescharacteristic
290C2×C7#57711yescharacteristic
291C23—21no—
292C23C2×C22⋊D711yescharacteristic
293C23C2×C22⋊D711yescharacteristic
294Q8—72no—
295Q8—141no—
296Q8—141no—
297Q8—141no—
298Q8—141no—
299Q8—141no—
300Q8—282no—
301Q8—282no—
302D4—22no—
303D4—21no—
304D4—21no—
305D4—42no—
306D4—81no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—81no—
315C2×C4—81no—
316C2×C4—141no—
317C2×C4—141no—
318C2×C4—141no—
319C2×C4—141no—
320C2×C4—282no—
321C2×C4—282no—
322C2×C4C2×C22⋊D711yescharacteristic
323C2×C4C2×C22⋊D711yescharacteristic
324C2×C4C2×C22⋊D711yescharacteristic
325C2×C4C2×C22⋊D711yescharacteristic
326C2×C4C23×D711yescharacteristic
327C8—141no—
328C8—141no—
329C8—141no—
330C8—141no—
331C7#332011yescharacteristic, complemented, Sylow
332C22—21no—
333C22—21no—
334C22—21no—
335C22—21no—
336C22—21no—
337C22—41no—
338C22—41no—
339C22—41no—
340C22#651412yes—
341C22C22×C22⋊D711yescharacteristic
342C4—21no—
343C4—21no—
344C4—21no—
345C4—21no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—142no—
350C4—281no—
351C4—281no—
352C4—281no—
353C4D7×#4111yescharacteristic
354C4#655311yescharacteristic
355C2—22no—
356C2—41no—
357C2—81no—
358C2C2×#651411yescharacteristic
359C2#9052011yescharacteristic
360C2#9052711yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
Fitting, prime radicalC7×#553C2
socleC22×C7C22×D4
hypercenter#553D7
hypocenterC7#3320
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #553

Complements

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

NH ≅ G/Ntype
2. D4×Dic14C2semidirect
3. #89841C2semidirect
4. #89835C2semidirect
5. #89814C2semidirect
6. #89813C2semidirect
7. #89812C2semidirect
8. #89812C2semidirect
9. #89791C2semidirect
10. #89790C2semidirect
11. #89783C2semidirect
12. #89776C2semidirect
13. #89768C2semidirect
14. #89742C2semidirect
15. C2×#6384C2semidirect
331. C7#3320semidirect

Maximal subgroups

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

subgroupindex
C7×#5532
D4×Dic142
#898412
#898352
#898142
#898132
#898122
#897912
#897902
#897832
#897762
#897682
#897422
C2×#63842
#33207