Finite Group Databasebeta

← #11737641

Subgroups — #11737641

Order 896.

376 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×C22⋊D7C211yesmaximal, characteristic
2#90330C211yesmaximal, characteristic
3C7×#451C211yesmaximal, characteristic, complemented
4#89997C211yesmaximal, characteristic, complemented
5#89960C211yesmaximal, characteristic, complemented
6#89958C211yesmaximal, characteristic, complemented
7#89956C211yesmaximal, characteristic, complemented
8#89954C211yesmaximal, characteristic, complemented
9#89824C211yesmaximal, characteristic, complemented
10#89577C211yesmaximal, characteristic, complemented
11D7×#103C211yesmaximal, characteristic, complemented
12#89569C211yesmaximal, characteristic, complemented
13#89568C211yesmaximal, characteristic, complemented
14#89500C211yesmaximal, characteristic
15#89495C211yesmaximal, characteristic, complemented
16C22×C7×Q8C2211yescharacteristic
17C2×Q8×D7C2211yescharacteristic
18C2×C7×Q16C2211yescharacteristic
19C2×C7×Q16C2211yescharacteristic
20C7×C22⋊Q8C2211yescharacteristic
21#6509—21no—
22#6509C2211yescharacteristic
23#6509C2211yescharacteristic
24Q8×Dic7C2211yescharacteristic
25Q8×Dic7C2211yescharacteristic
26#6507—21no—
27#6507C2211yescharacteristic
28C2×#1144C2211yescharacteristic
29C4×C22⋊D7—21no—
30C4×C22⋊D7C2211yescharacteristic
31#6489C2211yescharacteristic
32#6487—21no—
33#6487C2211yescharacteristic
34C2×#1136C2211yescharacteristic
35C2×C8×D7C2211yescharacteristic
36#6455—21no—
37D7×C4⋊C4C2211yescharacteristic
38#6453C2211yescharacteristic
39#6452—21no—
40#6446—21no—
41#6442—21no—
42C7×#103C2211yescharacteristic
43C7×#103C2211yescharacteristic
44C7×C22⋊C8C2211yescharacteristic
45#6409C2211yescharacteristic
46#6409C2211yescharacteristic
47#6409C2211yescharacteristic
48#6409C2211yescharacteristic
49#6409C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic
51#6392C2211yescharacteristic
52#6391C2211yescharacteristic
53#6390C2211yescharacteristic
54#6390C2211yescharacteristic
55#6389C2211yescharacteristic
56Dic7⋊C8C2211yescharacteristic
57#6384C2211yescharacteristic
58#6382C2211yescharacteristic
59#3312—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8—22no—
62C2×C7×Q8—41no—
63C2×C7×Q8C2311yescharacteristic
64C2×C7×Q8C2311yescharacteristic
65C2×C7×Q8C2311yescharacteristic
66C22×C4×C7—21no—
67C22×C4×C7C2311yescharacteristic
68C2×C22⋊D7D411yescharacteristic
69Q8×D7—22no—
70Q8×D7—41no—
71C2×C4×D7—21no—
72C2×C4×D7—21no—
73C2×C4×D7C2311yescharacteristic
74C2×Dic14—21no—
75C2×Dic14C2311yescharacteristic
76C7×Q16—22no—
77C7×Q16—41no—
78C2×C7×C8C2311yescharacteristic
79C2×C7×C8C2311yescharacteristic
80C7×C4⋊C4—21no—
81C7×C4⋊C4C2311yescharacteristic
82C7×C22⋊C4—21no—
83C22⋊Dic7—21no—
84C22⋊Dic7—21no—
85C22⋊Dic7D411yescharacteristic
86#1144—41no—
87(C2×D7)⋊C4—21no—
88(C2×D7)⋊C4—21no—
89(C2×D7)⋊C4D411yescharacteristic
90C4⋊Dic7—21no—
91C4⋊Dic7—21no—
92C4⋊Dic7—21no—
93C4⋊Dic7C2311yescharacteristic
94C4⋊Dic7C2311yescharacteristic
95C4⋊Dic7C2311yescharacteristic
96Dic7⋊C4—21no—
97Dic7⋊C4—21no—
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
111C8×D7—41no—
112#552—71no—
113D4×Q8—71no—
114#491—71no—
115#485—71no—
116#483—71no—
117#479—71no—
118#470—71no—
119#467—71no—
120#462—71no—
121#451—71no—
122#451D711yescharacteristic, complemented
123#441—71no—
124#435—71no—
125#419—71no—
126C2×#103—71no—
127C23×C7C2×D411yescharacteristic
128C22×D7C2×D411yescharacteristic
129C7×Q8—21no—
130C7×Q8—21no—
131C7×Q8—21no—
132C7×Q8—21no—
133C7×Q8—21no—
134C7×Q8—42no—
135C7×Q8#4112yes—
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—42no—
143C2×C4×C7C2×D411yescharacteristic
144C2×C4×C7C2411yescharacteristic
145C22⋊D7—22no—
146C2×Dic7—21no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7C2×D411yescharacteristic
151C2×Dic7C2×D411yescharacteristic
152C2×Dic7C2×D411yescharacteristic
153C4×D7—21no—
154C4×D7—21no—
155C4×D7—42no—
156C4×D7—42no—
157Dic14—21no—
158Dic14—21no—
159Dic14—42no—
160C7×C8—21no—
161C7×C8—21no—
162C7⋊C8—21no—
163C7⋊C8—21no—
164C22×Q8—71no—
165C22×Q8C2×D711yescharacteristic
166C2×Q16—71no—
167C2×Q16C2×D711yescharacteristic
168C2×Q16C2×D711yescharacteristic
169C2×M16—71no—
170C22×C8—71no—
171C4⋊Q8—71no—
172C4⋊Q8—141no—
173#126—141no—
174#125—71no—
175#125—141no—
176#123—141no—
177C22⋊Q8—71no—
178C22⋊Q8—71no—
179C22⋊Q8—71no—
180C22⋊Q8—141no—
181C22⋊Q8—141no—
182C22⋊Q8C2×D711yescharacteristic
183C4×Q8—71no—
184C4×Q8—71no—
185C4×D4—71no—
186C4×D4—141no—
187#117—71no—
188#117—141no—
189C2×C4⋊C4—71no—
190#107—71no—
191#106—71no—
192#106—71no—
193C4⋊C8—71no—
194#103—71no—
195#103—71no—
196#103—71no—
197#103—71no—
198#103—71no—
199#103—71no—
200#103—71no—
201#103—71no—
202#103C2×D711yescharacteristic
203#103C2×D711yescharacteristic
204C22⋊C8—71no—
205C22⋊C8C2×D711yescharacteristic
206#97—71no—
207C22×C7—21no—
208C22×C7—21no—
209C22×C7—21no—
210C22×C7C22×D411yescharacteristic
211C2×D7—21no—
212C2×D7—21no—
213C2×D7#13712yes—
214C4×C7—22no—
215C4×C7—21no—
216C4×C7—41no—
217C4×C7—41no—
218C4×C7—41no—
219C4×C7C2×#4111yescharacteristic
220C4×C7H(5,𝔽2)11yescharacteristic
221Dic7—21no—
222Dic7—21no—
223Dic7—21no—
224Dic7—41no—
225Dic7—41no—
226Dic7—41no—
227Dic7—41no—
228C2×Q8—21no—
229C2×Q8—22no—
230C2×Q8—41no—
231C2×Q8—71no—
232C2×Q8—141no—
233C2×Q8—142no—
234C2×Q8—281no—
235C2×Q8C22×D711yescharacteristic
236C2×Q8C22×D711yescharacteristic
237C2×Q8C22×D711yescharacteristic
238C2×D4—71no—
239C22×C4—21no—
240C22×C4—71no—
241C22×C4—141no—
242C22×C4—141no—
243C22×C4C22×D711yescharacteristic
244Q16—22no—
245Q16—41no—
246Q16—281no—
247M16—281no—
248C2×C8—71no—
249C2×C8—71no—
250C2×C8—281no—
251C2×C8C22×D711yescharacteristic
252C2×C8C22×D711yescharacteristic
253C4⋊C4—21no—
254C4⋊C4—71no—
255C4⋊C4—71no—
256C4⋊C4—71no—
257C4⋊C4—71no—
258C4⋊C4—141no—
259C4⋊C4—141no—
260C4⋊C4—141no—
261C4⋊C4—141no—
262C4⋊C4—141no—
263C4⋊C4—141no—
264C4⋊C4—141no—
265C4⋊C4—141no—
266C4⋊C4—141no—
267C4⋊C4—141no—
268C4⋊C4C22×D711yescharacteristic
269C22⋊C4—21no—
270C22⋊C4—71no—
271C22⋊C4—71no—
272C22⋊C4—141no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C42—71no—
277C42—141no—
278C42—141no—
279C42—141no—
280C2×C7—41no—
281C2×C7#54611yescharacteristic
282C2×C7C2×#13711yescharacteristic
283C2×C7#57811yescharacteristic
284D7—22no—
285C23—71no—
286C23D4×D711yescharacteristic
287Q8—21no—
288Q8—21no—
289Q8—21no—
290Q8—21no—
291Q8—21no—
292Q8—42no—
293Q8—141no—
294Q8—141no—
295Q8—282no—
296Q8#115912yes—
297D4—142no—
298C2×C4—21no—
299C2×C4—21no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—21no—
304C2×C4—42no—
305C2×C4—71no—
306C2×C4—71no—
307C2×C4—71no—
308C2×C4—141no—
309C2×C4—141no—
310C2×C4—141no—
311C2×C4—141no—
312C2×C4—141no—
313C2×C4—141no—
314C2×C4—282no—
315C2×C4—282no—
316C2×C4D4×D711yescharacteristic
317C2×C4C23×D711yescharacteristic
318C8—21no—
319C8—21no—
320C8—141no—
321C8—141no—
322C7#331211yescharacteristic, complemented, Sylow
323C22—21no—
324C22—21no—
325C22—21no—
326C22—72no—
327C22—141no—
328C22—141no—
329C22C2×D4×D711yescharacteristic
330C4—21no—
331C4—22no—
332C4—41no—
333C4—41no—
334C4—41no—
335C4—141no—
336C4—141no—
337C4—141no—
338C4—281no—
339C4—281no—
340C4—281no—
341C4—281no—
342C4C2×#115911yescharacteristic
343C4#654811yescharacteristic
344C2—41no—
345C2—142no—
346C2#9028211yescharacteristic
347C2#9045711yescharacteristic
348C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#451C2
socleC22×C7C22×D4
prime radicalC2×C22⋊D7D4
hypercenter#451D7
hypocenterC7#3312
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #451

Complements

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

NH ≅ G/Ntype
3. C7×#451C2semidirect
4. #89997C2semidirect
5. #89960C2semidirect
6. #89958C2semidirect
7. #89956C2semidirect
8. #89954C2semidirect
9. #89824C2semidirect
10. #89577C2semidirect
11. D7×#103C2semidirect
12. #89569C2semidirect
13. #89568C2semidirect
15. #89495C2semidirect
122. #451D7semidirect
322. C7#3312semidirect

Maximal subgroups

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

subgroupindex
Q8×C22⋊D72
#903302
C7×#4512
#899972
#899602
#899582
#899562
#899542
#898242
#895772
D7×#1032
#895692
#895682
#895002
#894952
#33127