Finite Group Databasebeta

← #11737501

Subgroups — #11737501

Order 896.

386 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
1Q8×C22⋊D7C211yesmaximal, characteristic, complemented
2#90307C211yesmaximal, characteristic
3C7×#448C211yesmaximal, characteristic, complemented
4#89997C211yesmaximal, characteristic, complemented
5#89946C211yesmaximal, characteristic, complemented
6#89942C211yesmaximal, characteristic, complemented
7#89934C211yesmaximal, characteristic, complemented
8SD16×Dic7C211yesmaximal, characteristic, complemented
9#89814C211yesmaximal, characteristic, complemented
10#89587C211yesmaximal, characteristic, complemented
11#89574C211yesmaximal, characteristic, complemented
12#89566C211yesmaximal, characteristic, complemented
13#89564C211yesmaximal, characteristic, complemented
14#89501C211yesmaximal, characteristic, complemented
15#89495C211yesmaximal, characteristic, complemented
16C22×C7×Q8C2211yescharacteristic
17C2×Q8×D7C2211yescharacteristic
18C2×C7×SD16C2211yescharacteristic
19C2×C7×SD16C2211yescharacteristic
20C7×#121C2211yescharacteristic
21#6509—21no—
22#6509C2211yescharacteristic
23Q8×Dic7C2211yescharacteristic
24#6507—21no—
25#6507C2211yescharacteristic
26#6502—21no—
27#6501C2211yescharacteristic
28#6499—21no—
29D4×Dic7C2211yescharacteristic
30C2×#1142C2211yescharacteristic
31C4×C22⋊D7—21no—
32C4×C22⋊D7C2211yescharacteristic
33#6489C2211yescharacteristic
34#6487—21no—
35#6487C2211yescharacteristic
36C2×#1136C2211yescharacteristic
37C2×#1131C2211yescharacteristic
38#6455C2211yescharacteristic
39#6451C2211yescharacteristic
40#6448—21no—
41#6440—21no—
42C7×#103C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×C22⋊C8C2211yescharacteristic
45#6409C2211yescharacteristic
46#6409C2211yescharacteristic
47#6409C2211yescharacteristic
48#6409C2211yescharacteristic
49#6406C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic
51#6392C2211yescharacteristic
52#6391C2211yescharacteristic
53#6390C2211yescharacteristic
54#6390C2211yescharacteristic
55Dic7⋊C8C2211yescharacteristic
56C8×Dic7C2211yescharacteristic
57#6384C2211yescharacteristic
58#6381C2211yescharacteristic
59#3329—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8—22no—
62C2×C7×Q8—41no—
63C2×C7×Q8C2311yescharacteristic
64C2×C7×Q8C2311yescharacteristic
65C2×C7×D4—21no—
66C2×C7×D4C2311yescharacteristic
67C22×C4×C7—21no—
68C22×C4×C7C2311yescharacteristic
69C2×C22⋊D7—21no—
70C2×C22⋊D7D411yescharacteristic
71C22×Dic7—21no—
72Q8×D7—22no—
73Q8×D7—41no—
74C2×C4×D7—21no—
75C2×C4×D7C2311yescharacteristic
76C2×Dic14—21no—
77C2×Dic14—21no—
78C2×Dic14C2311yescharacteristic
79C7×SD16—22no—
80C7×SD16—41no—
81C2×C7×C8C2311yescharacteristic
82C2×C7×C8C2311yescharacteristic
83C7×C4⋊C4C2311yescharacteristic
84C7×C22⋊C4—21no—
85C22⋊Dic7—21no—
86C22⋊Dic7—21no—
87C22⋊Dic7—21no—
88C22⋊Dic7—21no—
89C22⋊Dic7D411yescharacteristic
90#1142—41no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4D411yescharacteristic
94C4⋊Dic7—21no—
95C4⋊Dic7C2311yescharacteristic
96C4⋊Dic7C2311yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4D411yescharacteristic
104C4×Dic7—21no—
105C4×Dic7—21no—
106C4×Dic7—21no—
107C4×Dic7C2311yescharacteristic
108#1136—41no—
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111#1131—41no—
112#555—71no—
113D4×Q8—71no—
114#487—71no—
115#485—71no—
116#485—71no—
117#477—71no—
118#468—71no—
119#468—71no—
120#461—71no—
121#448—71no—
122#448D711yescharacteristic, complemented
123C4×SD16—71no—
124#436—71no—
125#419—71no—
126#419—71no—
127C23×C7—21no—
128C23×C7C2×D411yescharacteristic
129C22×D7C2×D411yescharacteristic
130C7×Q8—21no—
131C7×Q8—21no—
132C7×Q8—21no—
133C7×Q8—42no—
134C7×Q8#4112yes—
135C7×D4—21no—
136C7×D4—21no—
137C7×D4—81no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—42no—
144C2×C4×C7C2×D411yescharacteristic
145C2×C4×C7C2411yescharacteristic
146C22⋊D7—22no—
147C22⋊D7—81no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7—42no—
153C2×Dic7C2×D411yescharacteristic
154C2×Dic7C2×D411yescharacteristic
155C2×Dic7C2×D411yescharacteristic
156C4×D7—21no—
157C4×D7—21no—
158C4×D7—42no—
159Dic14—21no—
160Dic14—21no—
161Dic14—42no—
162Dic14—81no—
163C7×C8—21no—
164C7×C8—21no—
165C7⋊C8—21no—
166C7⋊C8—21no—
167C22×Q8—71no—
168C22×Q8C2×D711yescharacteristic
169C2×SD16—71no—
170C2×SD16C2×D711yescharacteristic
171C2×SD16C2×D711yescharacteristic
172C2×M16—71no—
173C2×M16—71no—
174C4⋊Q8—71no—
175C4⋊Q8—71no—
176C4⋊Q8—141no—
177#124—141no—
178#124—141no—
179C22⋊Q8—71no—
180C22⋊Q8—71no—
181C22⋊Q8—141no—
182C22⋊Q8—141no—
183#121—71no—
184#121—141no—
185#121C2×D711yescharacteristic
186C4×Q8—71no—
187C4×D4—71no—
188C4×D4—71no—
189C4×D4—141no—
190#117—71no—
191#117—71no—
192#117—141no—
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#103C2×D711yescharacteristic
205#103C2×D711yescharacteristic
206#102—71no—
207C22⋊C8—71no—
208C22⋊C8C2×D711yescharacteristic
209C4×C8—71no—
210C22×C7—21no—
211C22×C7—21no—
212C22×C7—21no—
213C22×C7—41no—
214C22×C7—41no—
215C22×C7—41no—
216C22×C7C22×D411yescharacteristic
217C2×D7—21no—
218C2×D7—21no—
219C2×D7—21no—
220C4×C7—22no—
221C4×C7—21no—
222C4×C7—41no—
223C4×C7—41no—
224C4×C7C2×#4111yescharacteristic
225C4×C7H(5,𝔽2)11yescharacteristic
226Dic7—21no—
227Dic7—21no—
228Dic7—41no—
229Dic7—41no—
230Dic7—41no—
231Dic7—41no—
232Dic7#13712yes—
233C2×Q8—21no—
234C2×Q8—22no—
235C2×Q8—41no—
236C2×Q8—71no—
237C2×Q8—141no—
238C2×Q8—141no—
239C2×Q8—142no—
240C2×Q8—281no—
241C2×Q8C22×D711yescharacteristic
242C2×Q8C22×D711yescharacteristic
243C2×D4—21no—
244C2×D4—71no—
245C2×D4—141no—
246C2×D4C22×D711yescharacteristic
247C22×C4—21no—
248C22×C4—71no—
249C22×C4—141no—
250C22×C4—141no—
251C22×C4C22×D711yescharacteristic
252SD16—22no—
253SD16—41no—
254SD16—281no—
255M16—281no—
256M16—281no—
257C2×C8—71no—
258C2×C8—71no—
259C2×C8C22×D711yescharacteristic
260C2×C8C22×D711yescharacteristic
261C4⋊C4—71no—
262C4⋊C4—71no—
263C4⋊C4—71no—
264C4⋊C4—71no—
265C4⋊C4—141no—
266C4⋊C4—141no—
267C4⋊C4—141no—
268C4⋊C4—141no—
269C4⋊C4—141no—
270C4⋊C4—141no—
271C4⋊C4C22×D711yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—71no—
274C22⋊C4—71no—
275C22⋊C4—141no—
276C22⋊C4—141no—
277C22⋊C4—141no—
278C22⋊C4—141no—
279C22⋊C4—141no—
280C22⋊C4—141no—
281C42—71no—
282C42—141no—
283C42—141no—
284C42—141no—
285C2×C7—41no—
286C2×C7—81no—
287C2×C7#54611yescharacteristic
288C2×C7C2×#13711yescharacteristic
289C2×C7#57711yescharacteristic
290D7—41no—
291C23—21no—
292C23—71no—
293C23D4×D711yescharacteristic
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—42no—
298Q8—141no—
299Q8—141no—
300Q8—282no—
301Q8—561no—
302Q8#115912yes—
303D4—21no—
304D4—21no—
305D4—81no—
306D4—142no—
307D4—561no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—42no—
314C2×C4—71no—
315C2×C4—71no—
316C2×C4—71no—
317C2×C4—141no—
318C2×C4—141no—
319C2×C4—141no—
320C2×C4—141no—
321C2×C4—141no—
322C2×C4—141no—
323C2×C4—282no—
324C2×C4—282no—
325C2×C4D4×D711yescharacteristic
326C2×C4C23×D711yescharacteristic
327C8—21no—
328C8—21no—
329C8—141no—
330C8—141no—
331C7#332911yescharacteristic, complemented, Sylow
332C22—21no—
333C22—21no—
334C22—21no—
335C22—41no—
336C22—41no—
337C22—41no—
338C22—141no—
339C22—141no—
340C22—141no—
341C22C2×D4×D711yescharacteristic
342C4—21no—
343C4—22no—
344C4—41no—
345C4—41no—
346C4—72no—
347C4—141no—
348C4—141no—
349C4—281no—
350C4—281no—
351C4—281no—
352C4—281no—
353C4C2×#115911yescharacteristic
354C4#654811yescharacteristic
355C2—41no—
356C2—81no—
357C2—281no—
358C2#9028211yescharacteristic
359C2#9045311yescharacteristic
360C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#448C2
socleC22×C7C22×D4
prime radical#90307C2
hypercenter#448D7
hypocenterC7#3329
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #448

Complements

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

NH ≅ G/Ntype
1. Q8×C22⋊D7C2semidirect
3. C7×#448C2semidirect
4. #89997C2semidirect
5. #89946C2semidirect
6. #89942C2semidirect
7. #89934C2semidirect
8. SD16×Dic7C2semidirect
9. #89814C2semidirect
10. #89587C2semidirect
11. #89574C2semidirect
12. #89566C2semidirect
13. #89564C2semidirect
14. #89501C2semidirect
15. #89495C2semidirect
122. #448D7semidirect
331. C7#3329semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊D72
#903072
C7×#4482
#899972
#899462
#899422
#899342
SD16×Dic72
#898142
#895872
#895742
#895662
#895642
#895012
#894952
#33297