Finite Group Databasebeta

← #11737665

Subgroups — #11737665

Order 896.

394 conjugacy classes of subgroups.

The subgroup lattice has 372 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90523C211yesmaximal, characteristic
2#90313C211yesmaximal, characteristic, complemented
3C7×#452C211yesmaximal, characteristic, complemented
4#90011C211yesmaximal, characteristic, complemented
5#89957C211yesmaximal, characteristic, complemented
6#89953C211yesmaximal, characteristic, complemented
7#89947C211yesmaximal, characteristic, complemented
8#89938C211yesmaximal, characteristic, complemented
9#89819C211yesmaximal, characteristic, complemented
10#89570C211yesmaximal, characteristic, complemented
11#89564C211yesmaximal, characteristic, complemented
12#89539C211yesmaximal, characteristic, complemented
13#89532C211yesmaximal, characteristic, complemented
14#89505C211yesmaximal, characteristic, complemented
15#89490C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C2×#1157C2211yescharacteristic
18C2×C7×Q16C2211yescharacteristic
19C2×C7×SD16C2211yescharacteristic
20C7×C22⋊Q8C2211yescharacteristic
21#6510C2211yescharacteristic
22Q8×Dic7C2211yescharacteristic
23#6507C2211yescharacteristic
24#6507C2211yescharacteristic
25C2×#1144C2211yescharacteristic
26C2×#1144C2211yescharacteristic
27C2×#1143C2211yescharacteristic
28#6503C2211yescharacteristic
29#6502—21no—
30#6502—21no—
31#6499C2211yescharacteristic
32C2×#1142C2211yescharacteristic
33C4×C22⋊D7—21no—
34C4×C22⋊D7—21no—
35#6487C2211yescharacteristic
36C2×C4×Dic7C2211yescharacteristic
37C2×Dic28C2211yescharacteristic
38C2×#1132C2211yescharacteristic
39#6452—21no—
40#6451C2211yescharacteristic
41#6450—21no—
42#6450C2211yescharacteristic
43#6441—21no—
44#6440—21no—
45C7×#103C2211yescharacteristic
46C7×#102C2211yescharacteristic
47C7×C22⋊C8C2211yescharacteristic
48#6409C2211yescharacteristic
49#6406C2211yescharacteristic
50C22⋊(C7⋊C8)C2211yescharacteristic
51#6395C2211yescharacteristic
52#6390C2211yescharacteristic
53Dic7⋊C8C2211yescharacteristic
54Dic7⋊C8C2211yescharacteristic
55C8×Dic7C2211yescharacteristic
56C8×Dic7C2211yescharacteristic
57#6384C2211yescharacteristic
58#6383C2211yescharacteristic
59#3210—71nomaximal, Sylow
60C7×#41—41no—
61C7×#41—41no—
62C2×C7×Q8C2311yescharacteristic
63C2×C7×Q8C2311yescharacteristic
64C2×C7×D4—21no—
65C2×C7×D4C2311yescharacteristic
66C22×C4×C7—21no—
67C22×C4×C7C2311yescharacteristic
68C2×C22⋊D7—21no—
69C2×C22⋊D7—21no—
70C2×C22⋊D7—21no—
71C22×Dic7D411yescharacteristic
72C22×Dic7D411yescharacteristic
73#1157—41no—
74#1157—41no—
75C2×D28C2311yescharacteristic
76C2×C4×D7—21no—
77C2×Dic14—21no—
78C2×Dic14C2311yescharacteristic
79C2×Dic14C2311yescharacteristic
80C7×Q16—41no—
81C7×SD16—41no—
82C2×C7×C8C2311yescharacteristic
83C2×C7×C8C2311yescharacteristic
84C7×C4⋊C4—21no—
85C7×C4⋊C4C2311yescharacteristic
86C7×C22⋊C4—21no—
87C22⋊Dic7—21no—
88C22⋊Dic7—21no—
89C22⋊Dic7—21no—
90#1144—41no—
91#1144—41no—
92#1143—41no—
93#1142—41no—
94(C2×D7)⋊C4—21no—
95(C2×D7)⋊C4—21no—
96C4⋊Dic7—21no—
97C4⋊Dic7C2311yescharacteristic
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104Dic7⋊C4—21no—
105C4×Dic7—21no—
106C4×Dic7—21no—
107C4×Dic7D411yescharacteristic
108C4×Dic7D411yescharacteristic
109C4×Dic7C2311yescharacteristic
110C4×Dic7C2311yescharacteristic
111C2×C7⋊C8C2311yescharacteristic
112C2×C7⋊C8C2311yescharacteristic
113Dic28—41no—
114#1132—41no—
115#533—71no—
116#532—71no—
117#494—71no—
118#492—71no—
119#487—71no—
120#486—71no—
121#464—71no—
122#464—71no—
123#464—71no—
124#464—71no—
125#452—71no—
126#452—71no—
127#452—71no—
128#452D711yescharacteristic, complemented
129#431—71no—
130C23×C7—21no—
131C23×C7C2×D411yescharacteristic
132C22×D7—21no—
133C7×Q8—21no—
134C7×Q8—21no—
135C7×Q8—21no—
136C7×Q8—21no—
137C7×D4—21no—
138C7×D4—21no—
139C7×D4—42no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—42no—
146C2×C4×C7C2×D411yescharacteristic
147C2×C4×C7C2411yescharacteristic
148C22⋊D7—42no—
149C22⋊D7—42no—
150C22⋊D7—42no—
151C2×Dic7—22no—
152C2×Dic7—22no—
153C2×Dic7—21no—
154C2×Dic7—21no—
155C2×Dic7—21no—
156C2×Dic7C2×D411yescharacteristic
157C2×Dic7C2×D411yescharacteristic
158C2×Dic7C2×D411yescharacteristic
159C2×Dic7C2×D411yescharacteristic
160D28—21no—
161D28—21no—
162C4×D7—42no—
163Dic14—21no—
164Dic14—21no—
165Dic14—21no—
166Dic14—21no—
167Dic14—42no—
168C7×C8—21no—
169C7×C8—21no—
170C7⋊C8—21no—
171C7⋊C8—21no—
172C2×#41—71no—
173C2×#41C2×D711yescharacteristic
174C2×Q16—71no—
175C2×Q16—71no—
176C2×Q16—71no—
177C2×Q16C2×D711yescharacteristic
178C2×SD16—71no—
179C2×SD16—71no—
180C2×SD16—71no—
181C2×SD16C2×D711yescharacteristic
182C4⋊Q8—71no—
183C4⋊Q8—71no—
184C4⋊Q8—71no—
185#127—71no—
186#125—141no—
187#124—71no—
188#124—71no—
189C22⋊Q8—71no—
190C22⋊Q8—141no—
191C22⋊Q8C2×D711yescharacteristic
192#121—141no—
193#121—141no—
194C4×Q8—71no—
195C4×Q8—71no—
196C4×Q8—141no—
197C4×D4—141no—
198C4×D4—141no—
199#117—141no—
200C2×C42—71no—
201C4⋊C8—71no—
202C4⋊C8—71no—
203#103—71no—
204#103—71no—
205#103—71no—
206#103C2×D711yescharacteristic
207#102—71no—
208#102—71no—
209#102—71no—
210#102C2×D711yescharacteristic
211C22⋊C8—71no—
212C22⋊C8C2×D711yescharacteristic
213C4×C8—71no—
214C4×C8—71no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7—21no—
218C22×C7—41no—
219C22×C7—41no—
220C22×C7—41no—
221C22×C7C22×D411yescharacteristic
222C2×D7—41no—
223C2×D7—41no—
224C2×D7—41no—
225C4×C7—21no—
226C4×C7—41no—
227C4×C7—41no—
228C4×C7—41no—
229C4×C7H(5,𝔽2)11yescharacteristic
230C4×C7H(5,𝔽2)11yescharacteristic
231Dic7—21no—
232Dic7—21no—
233Dic7—41no—
234Dic7—41no—
235Dic7—41no—
236Dic7#13512yes—
237Dic7#13712yes—
238#41—41no—
239#41—41no—
240#41—281no—
241#41—281no—
242C2×Q8—71no—
243C2×Q8—71no—
244C2×Q8—141no—
245C2×Q8C22×D711yescharacteristic
246C2×Q8C22×D711yescharacteristic
247C2×D4—21no—
248C2×D4—71no—
249C2×D4—141no—
250C2×D4—141no—
251C2×D4—141no—
252C2×D4C22×D711yescharacteristic
253C22×C4—21no—
254C22×C4—71no—
255C22×C4—71no—
256C22×C4—141no—
257C22×C4C22×D711yescharacteristic
258Q16—41no—
259Q16—281no—
260Q16—281no—
261Q16—281no—
262SD16—41no—
263SD16—281no—
264SD16—281no—
265SD16—281no—
266C2×C8—71no—
267C2×C8—71no—
268C2×C8C22×D711yescharacteristic
269C2×C8C22×D711yescharacteristic
270C4⋊C4—21no—
271C4⋊C4—71no—
272C4⋊C4—141no—
273C4⋊C4—141no—
274C4⋊C4—141no—
275C4⋊C4—141no—
276C4⋊C4—141no—
277C4⋊C4—141no—
278C4⋊C4—141no—
279C4⋊C4—141no—
280C4⋊C4C22×D711yescharacteristic
281C22⋊C4—21no—
282C22⋊C4—141no—
283C22⋊C4—141no—
284C22⋊C4—141no—
285C22⋊C4—141no—
286C22⋊C4—141no—
287C42—71no—
288C42—71no—
289C42—71no—
290C42—71no—
291C42—141no—
292C42—141no—
293C2×C7—41no—
294C2×C7—81no—
295C2×C7#53511yescharacteristic
296C2×C7C2×#13511yescharacteristic
297C2×C7C2×#13711yescharacteristic
298D7—81no—
299C23—21no—
300C23—141no—
301C23D4×D711yescharacteristic
302Q8—21no—
303Q8—21no—
304Q8—21no—
305Q8—21no—
306Q8—141no—
307Q8—141no—
308Q8—141no—
309Q8—141no—
310Q8—282no—
311D4—21no—
312D4—21no—
313D4—42no—
314D4—141no—
315D4—141no—
316D4—282no—
317D4—282no—
318D4—282no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—21no—
322C2×C4—21no—
323C2×C4—21no—
324C2×C4—42no—
325C2×C4—71no—
326C2×C4—71no—
327C2×C4—71no—
328C2×C4—71no—
329C2×C4—142no—
330C2×C4—142no—
331C2×C4—141no—
332C2×C4—141no—
333C2×C4—141no—
334C2×C4—282no—
335C2×C4D4×D711yescharacteristic
336C2×C4C23×D711yescharacteristic
337C8—21no—
338C8—21no—
339C8—141no—
340C8—141no—
341C7#321011yescharacteristic, complemented, Sylow
342C22—21no—
343C22—21no—
344C22—21no—
345C22—41no—
346C22—41no—
347C22—41no—
348C22—281no—
349C22—281no—
350C22—281no—
351C22C2×D4×D711yescharacteristic
352C4—21no—
353C4—41no—
354C4—41no—
355C4—41no—
356C4—72no—
357C4—72no—
358C4—141no—
359C4—141no—
360C4—281no—
361C4—281no—
362C4—281no—
363C4#654811yescharacteristic
364C4#654811yescharacteristic
365C2—41no—
366C2—81no—
367C2—561no—
368C2#9028311yescharacteristic
369C2D7×#13511yescharacteristic
370C2D7×#13711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#452C2
socleC22×C7C22×D4
prime radical#90523C2
hypercenter#452D7
hypocenterC7#3210
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #452

Complements

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

NH ≅ G/Ntype
2. #90313C2semidirect
3. C7×#452C2semidirect
4. #90011C2semidirect
5. #89957C2semidirect
6. #89953C2semidirect
7. #89947C2semidirect
8. #89938C2semidirect
9. #89819C2semidirect
10. #89570C2semidirect
11. #89564C2semidirect
12. #89539C2semidirect
13. #89532C2semidirect
14. #89505C2semidirect
15. #89490C2semidirect
128. #452D7semidirect
341. C7#3210semidirect

Maximal subgroups

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

subgroupindex
#905232
#903132
C7×#4522
#900112
#899572
#899532
#899472
#899382
#898192
#895702
#895642
#895392
#895322
#895052
#894902
#32107