Finite Group Databasebeta

← #11740998

Subgroups — #11740998

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×D4×Q8C211yesmaximal, characteristic, complemented
2#90256C211yesmaximal, characteristic
3#89999C211yesmaximal, characteristic, complemented
4#89999C211yesmaximal, characteristic, complemented
5#89997C211yesmaximal, characteristic, complemented
6C2×#6409C211yesmaximal, characteristic, complemented
7#89855C211yesmaximal, characteristic, complemented
8#89854C211yesmaximal, characteristic, complemented
9#89821C211yesmaximal, characteristic, complemented
10#89820C211yesmaximal, characteristic, complemented
11#89817C211yesmaximal, characteristic, complemented
12#89816C211yesmaximal, characteristic, complemented
13#89798C211yesmaximal, characteristic, complemented
14#89794C211yesmaximal, characteristic, complemented
15#89783C211yesmaximal, characteristic, complemented
16C22×C7×Q8C2211yescharacteristic
17C22×C7×Q8C2211yescharacteristic
18C7×C4⋊Q8—21no—
19C7×C4⋊Q8C2211yescharacteristic
20C7×C22⋊Q8—21no—
21C7×C22⋊Q8—21no—
22C7×C22⋊Q8C2211yescharacteristic
23C7×C22⋊Q8C2211yescharacteristic
24C4×C7×Q8C2211yescharacteristic
25C4×C7×D4—21no—
26C4×C7×D4C2211yescharacteristic
27C2×#1143C2211yescharacteristic
28C2×#1143C2211yescharacteristic
29C2×#1143C2211yescharacteristic
30#6502—21no—
31#6493C2211yescharacteristic
32#6493C2211yescharacteristic
33#6489C2211yescharacteristic
34C2×C4⋊Dic7C2211yescharacteristic
35C2×#1136C2211yescharacteristic
36C22×C7⋊C8C2211yescharacteristic
37#6461—21no—
38#6446—21no—
39(C22⋊D7)⋊C4—21no—
40C4×D28C2211yescharacteristic
41#6433C2211yescharacteristic
42#6409C2211yescharacteristic
43#6409C2211yescharacteristic
44#6409C2211yescharacteristic
45#6409C2211yescharacteristic
46#6409C2211yescharacteristic
47#6409C2211yescharacteristic
48#6409C2211yescharacteristic
49#6409C2211yescharacteristic
50#6409C2211yescharacteristic
51C22⋊(C7⋊C8)C2211yescharacteristic
52C22⋊(C7⋊C8)C2211yescharacteristic
53#6383C2211yescharacteristic
54#6382C2211yescharacteristic
55#6381C2211yescharacteristic
56#6381C2211yescharacteristic
57C4⋊(C7⋊C8)C2211yescharacteristic
58#6377C2211yescharacteristic
59#3320—71nomaximal, Sylow
60C2×C7×Q8—21no—
61C2×C7×Q8—21no—
62C2×C7×Q8—22no—
63C2×C7×Q8—22no—
64C2×C7×Q8—41no—
65C2×C7×Q8—41no—
66C2×C7×Q8C2311yescharacteristic
67C2×C7×Q8C2311yescharacteristic
68C2×C7×Q8C2311yescharacteristic
69C2×C7×D4D411yescharacteristic
70C22×C4×C7—21no—
71C22×C4×C7—21no—
72C22×C4×C7C2311yescharacteristic
73C22×C4×C7C2311yescharacteristic
74C2×C22⋊D7—21no—
75C2×C22⋊D7—21no—
76C22×Dic7—21no—
77C2×D28C2311yescharacteristic
78C2×C4×D7—21no—
79C7×C4⋊C4—21no—
80C7×C4⋊C4—21no—
81C7×C4⋊C4—21no—
82C7×C4⋊C4—21no—
83C7×C4⋊C4—21no—
84C7×C4⋊C4D411yescharacteristic
85C7×C4⋊C4C2311yescharacteristic
86C7×C22⋊C4—21no—
87C7×C22⋊C4—21no—
88C7×C22⋊C4D411yescharacteristic
89C7×C22⋊C4D411yescharacteristic
90C42×C7—21no—
91C42×C7C2311yescharacteristic
92C22⋊Dic7—21no—
93#1143—22no—
94#1143—41no—
95#1143—41no—
96(C2×D7)⋊C4—21no—
97(C2×D7)⋊C4—21no—
98(C2×D7)⋊C4—21no—
99C4⋊Dic7—21no—
100C4⋊Dic7C2311yescharacteristic
101C4⋊Dic7C2311yescharacteristic
102C4⋊Dic7C2311yescharacteristic
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#553—71no—
113D4×Q8D711yescharacteristic, complemented
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×C7C2×D411yescharacteristic
128C23×C7C2×D411yescharacteristic
129C22×D7—21no—
130C7×Q8—21no—
131C7×Q8—21no—
132C7×Q8—21no—
133C7×Q8—21no—
134C7×Q8—21no—
135C7×Q8—42no—
136C7×Q8—42no—
137C7×Q8#4112yes—
138C7×D4—22no—
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—42no—
148C2×C4×C7—42no—
149C2×C4×C7C2×D411yescharacteristic
150C2×C4×C7C2×D411yescharacteristic
151C2×C4×C7C2×D411yescharacteristic
152C2×C4×C7C2×D411yescharacteristic
153C2×C4×C7C2411yescharacteristic
154C22⋊D7—42no—
155C22⋊D7—81no—
156C2×Dic7—21no—
157C2×Dic7—21no—
158C2×Dic7—21no—
159C2×Dic7—81no—
160D28—21no—
161D28—21no—
162C4×D7—81no—
163C7⋊C8—21no—
164C7⋊C8—21no—
165C7⋊C8—21no—
166C7⋊C8—21no—
167C22×Q8C2×D711yescharacteristic
168C22×Q8C2×D711yescharacteristic
169C2×SD16—71no—
170C2×SD16—71no—
171C2×SD16—71no—
172C2×M16—71no—
173C22×C8—71no—
174C4⋊Q8—21no—
175C4⋊Q8C2×D711yescharacteristic
176#126—141no—
177#125—71no—
178#123—141no—
179C22⋊Q8—21no—
180C22⋊Q8—21no—
181C22⋊Q8C2×D711yescharacteristic
182C22⋊Q8C2×D711yescharacteristic
183#121—71no—
184#121—71no—
185#121—141no—
186C4×Q8C2×D711yescharacteristic
187C4×D4—21no—
188C4×D4—71no—
189C4×D4—141no—
190C4×D4C2×D711yescharacteristic
191#117—71no—
192C2×C4⋊C4—71no—
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#13712yes—
216C22×C7C22×D411yescharacteristic
217C2×D7—41no—
218C2×D7—41no—
219C2×D7—41no—
220C4×C7—21no—
221C4×C7—21no—
222C4×C7—22no—
223C4×C7—21no—
224C4×C7—21no—
225C4×C7—41no—
226C4×C7—41no—
227C4×C7—41no—
228C4×C7C2×#4111yescharacteristic
229C4×C7H(5,𝔽2)11yescharacteristic
230Dic7—41no—
231Dic7—41no—
232Dic7—41no—
233C2×Q8—21no—
234C2×Q8—21no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—41no—
238C2×Q8—41no—
239C2×Q8C22×D711yescharacteristic
240C2×Q8C22×D711yescharacteristic
241C2×Q8C22×D711yescharacteristic
242C2×D4—71no—
243C2×D4—141no—
244C2×D4—141no—
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—21no—
265C4⋊C4—21no—
266C4⋊C4—71no—
267C4⋊C4—71no—
268C4⋊C4—71no—
269C4⋊C4—141no—
270C4⋊C4—141no—
271C4⋊C4—141no—
272C4⋊C4C22⋊D711yescharacteristic
273C4⋊C4C22×D711yescharacteristic
274C22⋊C4—21no—
275C22⋊C4—21no—
276C22⋊C4—141no—
277C22⋊C4—141no—
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#54611yescharacteristic
288C2×C7C2×#13711yescharacteristic
289C2×C7#57711yescharacteristic
290D7—81no—
291C23—141no—
292C23C2×C22⋊D711yescharacteristic
293C23C2×C22⋊D711yescharacteristic
294Q8—21no—
295Q8—21no—
296Q8—21no—
297Q8—21no—
298Q8—21no—
299Q8—42no—
300Q8—42no—
301Q8#115912yes—
302D4—22no—
303D4—141no—
304D4—141no—
305D4—282no—
306D4—561no—
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—42no—
316C2×C4—42no—
317C2×C4—141no—
318C2×C4—141no—
319C2×C4—141no—
320C2×C4—561no—
321C2×C4—561no—
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—281no—
338C22—281no—
339C22—281no—
340C22#650512yes—
341C22C22×C22⋊D711yescharacteristic
342C4—21no—
343C4—21no—
344C4—22no—
345C4—21no—
346C4—21no—
347C4—41no—
348C4—41no—
349C4—41no—
350C4—281no—
351C4—281no—
352C4—281no—
353C4C2×#115911yescharacteristic
354C4#654811yescharacteristic
355C2—22no—
356C2—41no—
357C2—561no—
358C2#9049711yescharacteristic
359C2C2×#650511yescharacteristic
360C2#9052811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×D4×Q8C2
socleC22×C7C22×D4
prime radical#90256C2
hypercenterD4×Q8D7
hypocenterC7#3320
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ D4×Q8

Complements

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

NH ≅ G/Ntype
1. C7×D4×Q8C2semidirect
3. #89999C2semidirect
4. #89999C2semidirect
5. #89997C2semidirect
6. C2×#6409C2semidirect
7. #89855C2semidirect
8. #89854C2semidirect
9. #89821C2semidirect
10. #89820C2semidirect
11. #89817C2semidirect
12. #89816C2semidirect
13. #89798C2semidirect
14. #89794C2semidirect
15. #89783C2semidirect
113. D4×Q8D7semidirect
331. C7#3320semidirect

Maximal subgroups

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

subgroupindex
C7×D4×Q82
#902562
#899992
#899972
C2×#64092
#898552
#898542
#898212
#898202
#898172
#898162
#897982
#897942
#897832
#33207