Finite Group Databasebeta

← #11737693

Subgroups — #11737693

Order 896.

386 conjugacy classes of subgroups.

The subgroup lattice has 358 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90515C211yesmaximal, characteristic
2#90332C211yesmaximal, characteristic, complemented
3C7×#452C211yesmaximal, characteristic, complemented
4#90005C211yesmaximal, characteristic, complemented
5#89960C211yesmaximal, characteristic, complemented
6#89954C211yesmaximal, characteristic, complemented
7#89945C211yesmaximal, characteristic, complemented
8SD16×Dic7C211yesmaximal, characteristic, complemented
9#89823C211yesmaximal, characteristic, complemented
10#89587C211yesmaximal, characteristic, complemented
11#89564C211yesmaximal, characteristic, complemented
12#89544C211yesmaximal, characteristic, complemented
13#89531C211yesmaximal, characteristic, complemented
14(C22⋊D7)⋊C8C211yesmaximal, characteristic, complemented
15#89495C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C2×#1159C2211yescharacteristic
18C2×C7×Q16C2211yescharacteristic
19C2×C7×SD16C2211yescharacteristic
20C7×C22⋊Q8C2211yescharacteristic
21#6509C2211yescharacteristic
22#6509C2211yescharacteristic
23Q8×Dic7C2211yescharacteristic
24#6507C2211yescharacteristic
25C2×#1144C2211yescharacteristic
26#6502—21no—
27#6498—21no—
28D4×Dic7C2211yescharacteristic
29#6492—21no—
30C4×C22⋊D7C2211yescharacteristic
31#6489C2211yescharacteristic
32#6487C2211yescharacteristic
33C2×Dic7⋊C4—21no—
34C22×C7⋊C8C2211yescharacteristic
35C2×C8×D7C2211yescharacteristic
36#6461—21no—
37#6455C2211yescharacteristic
38#6451C2211yescharacteristic
39#6450—21no—
40#6448—21no—
41#6442—21no—
42C7×#103C2211yescharacteristic
43C7×#102C2211yescharacteristic
44C7×C22⋊C8C2211yescharacteristic
45#6409C2211yescharacteristic
46#6409C2211yescharacteristic
47#6409C2211yescharacteristic
48#6406C2211yescharacteristic
49#6406C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic
51#6392C2211yescharacteristic
52#6391C2211yescharacteristic
53#6390C2211yescharacteristic
54#6390C2211yescharacteristic
55Dic7⋊C8C2211yescharacteristic
56C8×Dic7C2211yescharacteristic
57#6384C2211yescharacteristic
58#6381C2211yescharacteristic
59#3302—71nomaximal, Sylow
60C7×#41—22no—
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⋊D7D411yescharacteristic
70C22×Dic7—21no—
71C22×Dic7—21no—
72#1159—22no—
73#1159—41no—
74C2×C4×D7C2311yescharacteristic
75C2×Dic14—21no—
76C2×Dic14C2311yescharacteristic
77C7×Q16—41no—
78C7×SD16—22no—
79C2×C7×C8C2311yescharacteristic
80C2×C7×C8C2311yescharacteristic
81C7×C4⋊C4—21no—
82C7×C4⋊C4C2311yescharacteristic
83C7×C22⋊C4—21no—
84C22⋊Dic7—21no—
85C22⋊Dic7—21no—
86C22⋊Dic7—21no—
87C22⋊Dic7D411yescharacteristic
88#1144—41no—
89(C2×D7)⋊C4—21no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4D411yescharacteristic
93C4⋊Dic7—21no—
94C4⋊Dic7C2311yescharacteristic
95C4⋊Dic7C2311yescharacteristic
96C4⋊Dic7C2311yescharacteristic
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
108C2×C7⋊C8—41no—
109C2×C7⋊C8C2311yescharacteristic
110C2×C7⋊C8C2311yescharacteristic
111C8×D7—41no—
112#556—71no—
113#547—71no—
114#487—71no—
115#485—71no—
116#485—71no—
117#474—71no—
118#467—71no—
119#467—71no—
120#462—71no—
121#452—71no—
122#452D711yescharacteristic, complemented
123C4×SD16—71no—
124C8×D4—71no—
125#416—71no—
126#416—71no—
127C23×C7—21no—
128C23×C7C2×D411yescharacteristic
129C22×D7C2×D411yescharacteristic
130C7×Q8—21no—
131C7×Q8—21no—
132C7×Q8—21no—
133C7×Q8—21no—
134C7×D4—21no—
135C7×D4—42no—
136C7×D4#4112yes—
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—
146C22⋊D7—42no—
147C2×Dic7—21no—
148C2×Dic7—21no—
149C2×Dic7—21no—
150C2×Dic7—21no—
151C2×Dic7—42no—
152C2×Dic7—42no—
153C2×Dic7C2×D411yescharacteristic
154C2×Dic7C2×D411yescharacteristic
155C2×Dic7C2×D411yescharacteristic
156C4×D7—21no—
157C4×D7—21no—
158Dic14—21no—
159Dic14—21no—
160Dic14—42no—
161C7×C8—21no—
162C7×C8—21no—
163C7⋊C8—21no—
164C7⋊C8—21no—
165C2×#41—71no—
166C2×#41C2×D711yescharacteristic
167C2×Q16—71no—
168C2×Q16C2×D711yescharacteristic
169C2×SD16C2×D711yescharacteristic
170C22×C8—71no—
171C22×C8—71no—
172C4⋊Q8—71no—
173C4⋊Q8—71no—
174#126—141no—
175#126—141no—
176#124—141no—
177#123—141no—
178#123—141no—
179C22⋊Q8—71no—
180C22⋊Q8—71no—
181C22⋊Q8—71no—
182C22⋊Q8C2×D711yescharacteristic
183#121—141no—
184C4×Q8—71no—
185C4×Q8—141no—
186C4×D4—71no—
187C4×D4—71no—
188#117—71no—
189#117—71no—
190C2×C4⋊C4—141no—
191#107—71no—
192#107—71no—
193#106—71no—
194C4⋊C8—71no—
195#103—71no—
196#103—71no—
197#103—71no—
198#103—71no—
199#103—71no—
200#103—71no—
201#103C2×D711yescharacteristic
202#102—71no—
203#102—71no—
204#102C2×D711yescharacteristic
205C22⋊C8—71no—
206C22⋊C8C2×D711yescharacteristic
207C4×C8—71no—
208C22×C7—22no—
209C22×C7—21no—
210C22×C7—21no—
211C22×C7—21no—
212C22×C7—41no—
213C22×C7—41no—
214C22×C7C22×D411yescharacteristic
215C2×D7—21no—
216C2×D7—21no—
217C2×D7—21no—
218C4×C7—21no—
219C4×C7—41no—
220C4×C7—41no—
221C4×C7—41no—
222C4×C7C2×#4111yescharacteristic
223C4×C7H(5,𝔽2)11yescharacteristic
224Dic7—21no—
225Dic7—21no—
226Dic7—41no—
227Dic7—41no—
228Dic7—41no—
229Dic7—41no—
230Dic7#13512yes—
231#41—22no—
232#41—41no—
233#41—142no—
234#41—281no—
235C2×Q8—71no—
236C2×Q8—141no—
237C2×Q8C22×D711yescharacteristic
238C2×Q8C22×D711yescharacteristic
239C2×D4—21no—
240C2×D4—71no—
241C2×D4—141no—
242C2×D4C22×D711yescharacteristic
243C22×C4—21no—
244C22×C4—71no—
245C22×C4—141no—
246C22×C4—141no—
247C22×C4C22×D711yescharacteristic
248Q16—41no—
249Q16—281no—
250SD16—22no—
251C2×C8—71no—
252C2×C8—71no—
253C2×C8—281no—
254C2×C8—281no—
255C2×C8C22×D711yescharacteristic
256C2×C8C22×D711yescharacteristic
257C4⋊C4—21no—
258C4⋊C4—71no—
259C4⋊C4—71no—
260C4⋊C4—71no—
261C4⋊C4—71no—
262C4⋊C4—141no—
263C4⋊C4—141no—
264C4⋊C4—141no—
265C4⋊C4—141no—
266C4⋊C4—141no—
267C4⋊C4—141no—
268C4⋊C4—141no—
269C4⋊C4C22×D711yescharacteristic
270C22⋊C4—21no—
271C22⋊C4—71no—
272C22⋊C4—71no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C22⋊C4—141no—
277C22⋊C4—141no—
278C22⋊C4—141no—
279C42—71no—
280C42—141no—
281C42—141no—
282C42—141no—
283C2×C7—41no—
284C2×C7—42no—
285C2×C7#54611yescharacteristic
286C2×C7C2×#13511yescharacteristic
287C2×C7#57811yescharacteristic
288D7—41no—
289C23—21no—
290C23—71no—
291C23D4×D711yescharacteristic
292Q8—21no—
293Q8—21no—
294Q8—21no—
295Q8—21no—
296Q8—141no—
297Q8—141no—
298Q8—282no—
299D4—21no—
300D4—42no—
301D4—142no—
302D4—282no—
303D4#115912yes—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—42no—
310C2×C4—71no—
311C2×C4—71no—
312C2×C4—71no—
313C2×C4—141no—
314C2×C4—141no—
315C2×C4—141no—
316C2×C4—141no—
317C2×C4—141no—
318C2×C4—141no—
319C2×C4—282no—
320C2×C4—282no—
321C2×C4D4×D711yescharacteristic
322C2×C4C23×D711yescharacteristic
323C8—21no—
324C8—21no—
325C8—141no—
326C8—141no—
327C7#330211yescharacteristic, complemented, Sylow
328C22—21no—
329C22—21no—
330C22—21no—
331C22—22no—
332C22—41no—
333C22—41no—
334C22—141no—
335C22—141no—
336C22—141no—
337C22C2×D4×D711yescharacteristic
338C4—21no—
339C4—41no—
340C4—41no—
341C4—41no—
342C4—72no—
343C4—141no—
344C4—141no—
345C4—281no—
346C4—281no—
347C4—281no—
348C4—281no—
349C4C2×#115911yescharacteristic
350C4#654811yescharacteristic
351C2—41no—
352C2—42no—
353C2—281no—
354C2#9028211yescharacteristic
355C2D7×#13511yescharacteristic
356C2#9047011yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#452C2
socleC22×C7C22×D4
prime radical#90515C2
hypercenter#452D7
hypocenterC7#3302
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. #90332C2semidirect
3. C7×#452C2semidirect
4. #90005C2semidirect
5. #89960C2semidirect
6. #89954C2semidirect
7. #89945C2semidirect
8. SD16×Dic7C2semidirect
9. #89823C2semidirect
10. #89587C2semidirect
11. #89564C2semidirect
12. #89544C2semidirect
13. #89531C2semidirect
14. (C22⋊D7)⋊C8C2semidirect
15. #89495C2semidirect
122. #452D7semidirect
327. C7#3302semidirect

Maximal subgroups

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

subgroupindex
#905152
#903322
C7×#4522
#900052
#899602
#899542
#899452
SD16×Dic72
#898232
#895872
#895642
#895442
#895312
(C22⋊D7)⋊C82
#894952
#33027