Finite Group Databasebeta

← #11737561

Subgroups — #11737561

Order 896.

420 conjugacy classes of subgroups.

The subgroup lattice has 376 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90515C211yesmaximal, characteristic, complemented
2#90302C211yesmaximal, characteristic, complemented
3C7×#449C211yesmaximal, characteristic, complemented
4#90004C211yesmaximal, characteristic, complemented
5#89946C211yesmaximal, characteristic, complemented
6#89934C211yesmaximal, characteristic, complemented
7#89930C211yesmaximal, characteristic, complemented
8D8×Dic7C211yesmaximal, characteristic, complemented
9#89814C211yesmaximal, characteristic, complemented
10#89581C211yesmaximal, characteristic, complemented
11#89564C211yesmaximal, characteristic, complemented
12#89543C211yesmaximal, characteristic, complemented
13#89531C211yesmaximal, characteristic, complemented
14#89501C211yesmaximal, characteristic, complemented
15#89494C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C2×#1159—21no—
18C2×#1159C2211yescharacteristic
19C2×C7×SD16C2211yescharacteristic
20C2×C7×D8C2211yescharacteristic
21C7×#121C2211yescharacteristic
22#6509C2211yescharacteristic
23#6507C2211yescharacteristic
24#6502—21no—
25#6501C2211yescharacteristic
26#6498—21no—
27#6498—21no—
28D4×Dic7C2211yescharacteristic
29D4×Dic7C2211yescharacteristic
30C2×#1142C2211yescharacteristic
31#6492—21no—
32C4×C22⋊D7C2211yescharacteristic
33C2×C4⋊Dic7C2211yescharacteristic
34#6487C2211yescharacteristic
35C2×Dic7⋊C4—21no—
36C2×#1136C2211yescharacteristic
37C2×#1131C2211yescharacteristic
38D7×C4⋊C4C2211yescharacteristic
39#6451C2211yescharacteristic
40#6446—21no—
41#6441—21no—
42C7×#103C2211yescharacteristic
43C7×#102C2211yescharacteristic
44C7×C22⋊C8C2211yescharacteristic
45#6409C2211yescharacteristic
46#6409C2211yescharacteristic
47#6406C2211yescharacteristic
48#6406C2211yescharacteristic
49#6406C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic
51#6392C2211yescharacteristic
52#6392C2211yescharacteristic
53#6390C2211yescharacteristic
54#6390C2211yescharacteristic
55Dic7⋊C8C2211yescharacteristic
56C8×Dic7C2211yescharacteristic
57#6384C2211yescharacteristic
58#6381C2211yescharacteristic
59#3330—71nomaximal, Sylow
60C7×#41—22no—
61C7×#41—41no—
62C2×C7×Q8C2311yescharacteristic
63C2×C7×D4—21no—
64C2×C7×D4—21no—
65C2×C7×D4C2311yescharacteristic
66C2×C7×D4C2311yescharacteristic
67C22×C4×C7—21no—
68C22×C4×C7C2311yescharacteristic
69C2×C22⋊D7—21no—
70C2×C22⋊D7—21no—
71C2×C22⋊D7D411yescharacteristic
72C22×Dic7—21no—
73C22×Dic7—21no—
74C22×Dic7—21no—
75C22×Dic7—21no—
76#1159—22no—
77#1159—41no—
78#1159—44no—
79C2×C4×D7—21no—
80C2×C4×D7C2311yescharacteristic
81C2×Dic14—21no—
82C2×Dic14C2311yescharacteristic
83C7×SD16—41no—
84C7×D8—22no—
85C2×C7×C8C2311yescharacteristic
86C2×C7×C8C2311yescharacteristic
87C7×C4⋊C4C2311yescharacteristic
88C7×C22⋊C4—21no—
89C22⋊Dic7—21no—
90C22⋊Dic7—21no—
91C22⋊Dic7—21no—
92C22⋊Dic7—21no—
93C22⋊Dic7D411yescharacteristic
94#1142—41no—
95(C2×D7)⋊C4—21no—
96(C2×D7)⋊C4D411yescharacteristic
97C4⋊Dic7—21no—
98C4⋊Dic7C2311yescharacteristic
99C4⋊Dic7C2311yescharacteristic
100C4⋊Dic7C2311yescharacteristic
101Dic7⋊C4—21no—
102Dic7⋊C4—21no—
103Dic7⋊C4—21no—
104Dic7⋊C4—21no—
105Dic7⋊C4—21no—
106Dic7⋊C4—21no—
107Dic7⋊C4D411yescharacteristic
108C4×Dic7C2311yescharacteristic
109#1136—41no—
110C2×C7⋊C8C2311yescharacteristic
111C2×C7⋊C8C2311yescharacteristic
112#1131—41no—
113#547—71no—
114#547—71no—
115#487—71no—
116#484—71no—
117#484—71no—
118#474—71no—
119#468—71no—
120#468—71no—
121#461—71no—
122#449—71no—
123#449D711yescharacteristic, complemented
124C4×D8—71no—
125#436—71no—
126#417—71no—
127#417—71no—
128C23×C7—21no—
129C23×C7—21no—
130C23×C7C2×D411yescharacteristic
131C22×D7C2×D411yescharacteristic
132C7×Q8—21no—
133C7×Q8—21no—
134C7×D4—21no—
135C7×D4—21no—
136C7×D4—21no—
137C7×D4—42no—
138C7×D4—42no—
139C7×D4#4112yescomplemented
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—42no—
145C2×C4×C7C2×D411yescharacteristic
146C2×C4×C7C2411yescharacteristic
147C22⋊D7—22no—
148C22⋊D7—42no—
149C22⋊D7—42no—
150C2×Dic7—21no—
151C2×Dic7—21no—
152C2×Dic7—21no—
153C2×Dic7—21no—
154C2×Dic7—42no—
155C2×Dic7—42no—
156C2×Dic7—42no—
157C2×Dic7—42no—
158C2×Dic7C2×D411yescharacteristic
159C2×Dic7C2×D411yescharacteristic
160C2×Dic7C2×D411yescharacteristic
161C4×D7—21no—
162C4×D7—21no—
163C4×D7—42no—
164Dic14—21no—
165Dic14—21no—
166Dic14—42no—
167C7×C8—21no—
168C7×C8—21no—
169C7⋊C8—21no—
170C7⋊C8—21no—
171C2×#41—71no—
172C2×#41—141no—
173C2×#41C2×D711yescharacteristic
174C2×SD16—71no—
175C2×SD16C2×D711yescharacteristic
176C2×D8C2×D711yescharacteristic
177C2×M16—71no—
178C2×M16—71no—
179C4⋊Q8—71no—
180C4⋊Q8—71no—
181#123—141no—
182#123—141no—
183#123—141no—
184#123—141no—
185C22⋊Q8—71no—
186C22⋊Q8—71no—
187C22⋊Q8—141no—
188#121—71no—
189#121—141no—
190#121C2×D711yescharacteristic
191C4×D4—71no—
192C4×D4—71no—
193C4×D4—71no—
194C2×C4⋊C4—71no—
195C2×C4⋊C4—71no—
196C2×C4⋊C4—141no—
197#107—71no—
198#107—71no—
199#107—71no—
200C4⋊C8—71no—
201#103—71no—
202#103—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—
214C22×C7—22no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7—21no—
218C22×C7—41no—
219C22×C7—41no—
220C22×C7—41no—
221C22×C7—41no—
222C22×C7—41no—
223C22×C7C22×D411yescharacteristic
224C2×D7—21no—
225C2×D7—21no—
226C2×D7—21no—
227C4×C7—21no—
228C4×C7—41no—
229C4×C7—41no—
230C4×C7C2×#4111yescharacteristic
231C4×C7H(5,𝔽2)11yescharacteristic
232Dic7—21no—
233Dic7—21no—
234Dic7—41no—
235Dic7—41no—
236Dic7—41no—
237Dic7—41no—
238Dic7AGL(1,ℤ/8)12yes—
239#41—22no—
240#41—41no—
241#41—142no—
242#41—281no—
243#41—284no—
244C2×Q8—71no—
245C2×Q8—141no—
246C2×Q8C22×D711yescharacteristic
247C2×D4—21no—
248C2×D4—21no—
249C2×D4—71no—
250C2×D4—141no—
251C2×D4—141no—
252C2×D4C22×D711yescharacteristic
253C2×D4C22×D711yescharacteristic
254C22×C4—21no—
255C22×C4—71no—
256C22×C4—141no—
257C22×C4—141no—
258C22×C4—141no—
259C22×C4—141no—
260C22×C4—141no—
261C22×C4C22×D711yescharacteristic
262SD16—41no—
263SD16—281no—
264D8—22no—
265M16—281no—
266M16—281no—
267C2×C8—71no—
268C2×C8—71no—
269C2×C8C22×D711yescharacteristic
270C2×C8C22×D711yescharacteristic
271C4⋊C4—71no—
272C4⋊C4—71no—
273C4⋊C4—71no—
274C4⋊C4—71no—
275C4⋊C4—141no—
276C4⋊C4—141no—
277C4⋊C4—141no—
278C4⋊C4—141no—
279C4⋊C4—141no—
280C4⋊C4—141no—
281C4⋊C4—141no—
282C4⋊C4C22×D711yescharacteristic
283C22⋊C4—21no—
284C22⋊C4—71no—
285C22⋊C4—71no—
286C22⋊C4—141no—
287C22⋊C4—141no—
288C22⋊C4—141no—
289C22⋊C4—141no—
290C22⋊C4—141no—
291C42—71no—
292C2×C7—41no—
293C2×C7—42no—
294C2×C7—81no—
295C2×C7#54611yescharacteristic
296C2×C7C2×AGL(1,ℤ/8)11yescharacteristic
297C2×C7#57811yescharacteristic
298D7—41no—
299C23—21no—
300C23—21no—
301C23—71no—
302C23D4×D711yescharacteristic
303Q8—21no—
304Q8—21no—
305Q8—141no—
306Q8—141no—
307Q8—282no—
308D4—21no—
309D4—21no—
310D4—21no—
311D4—42no—
312D4—42no—
313D4—142no—
314D4—282no—
315D4—282no—
316D4#115912yescomplemented
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—71no—
323C2×C4—71no—
324C2×C4—71no—
325C2×C4—141no—
326C2×C4—141no—
327C2×C4—141no—
328C2×C4—141no—
329C2×C4—141no—
330C2×C4—141no—
331C2×C4—282no—
332C2×C4—282no—
333C2×C4—282no—
334C2×C4—282no—
335C2×C4—282no—
336C2×C4D4×D711yescharacteristic
337C2×C4C23×D711yescharacteristic
338C8—21no—
339C8—21no—
340C8—141no—
341C8—141no—
342C7#333011yescharacteristic, complemented, Sylow
343C22—21no—
344C22—21no—
345C22—21no—
346C22—22no—
347C22—41no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22—141no—
353C22—141no—
354C22—141no—
355C22C2×D4×D711yescharacteristic
356C4—21no—
357C4—41no—
358C4—41no—
359C4—72no—
360C4—141no—
361C4—141no—
362C4—281no—
363C4—281no—
364C4—281no—
365C4—281no—
366C4C2×#115911yescharacteristic
367C4#654811yescharacteristic
368C2—41no—
369C2—42no—
370C2—81no—
371C2—281no—
372C2#9028211yescharacteristic
373C2#9046211yescharacteristic
374C2D7×AGL(1,ℤ/8)11yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#449C2
socleC22×C7C22×D4
hypercenter#449D7
hypocenterC7#3330
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #449

Complements

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

NH ≅ G/Ntype
1. #90515C2semidirect
2. #90302C2semidirect
3. C7×#449C2semidirect
4. #90004C2semidirect
5. #89946C2semidirect
6. #89934C2semidirect
7. #89930C2semidirect
8. D8×Dic7C2semidirect
9. #89814C2semidirect
10. #89581C2semidirect
11. #89564C2semidirect
12. #89543C2semidirect
13. #89531C2semidirect
14. #89501C2semidirect
15. #89494C2semidirect
123. #449D7semidirect
139. C7×D4#41semidirect
316. D4#1159semidirect
342. C7#3330semidirect

Maximal subgroups

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

subgroupindex
#905152
#903022
C7×#4492
#900042
#899462
#899342
#899302
D8×Dic72
#898142
#895812
#895642
#895432
#895312
#895012
#894942
#33307