Finite Group Databasebeta

← #11738916

Subgroups — #11738916

Order 896.

398 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#90320C211yesmaximal, characteristic
2#90195C211yesmaximal, characteristic, complemented
3C7×#484C211yesmaximal, characteristic, complemented
4#89817C211yesmaximal, characteristic, complemented
5#89739C211yesmaximal, characteristic, complemented
6#89657C211yesmaximal, characteristic, complemented
7#89654C211yesmaximal, characteristic, complemented
8#89653C211yesmaximal, characteristic, complemented
9D7×#107C211yesmaximal, characteristic, complemented
10#89587C211yesmaximal, characteristic, complemented
11#89581C211yesmaximal, characteristic, complemented
12#89576C211yesmaximal, characteristic, complemented
13#89575C211yesmaximal, characteristic, complemented
14#89509C211yesmaximal, characteristic, complemented
15#89499C211yesmaximal, characteristic, complemented
16C2×#1161C2211yescharacteristic
17C2×#1157C2211yescharacteristic
18C7×C22⋊Q8C2211yescharacteristic
19C2×C7×C4⋊C4C2211yescharacteristic
20#6509C2211yescharacteristic
21#6493C2211yescharacteristic
22C4×C22⋊D7—21no—
23C4×C22⋊D7—21no—
24#6489C2211yescharacteristic
25C2×#1131C2211yescharacteristic
26C2×C8×D7C2211yescharacteristic
27#6459—21no—
28#6458—21no—
29#6458C2211yescharacteristic
30#6456—21no—
31#6455C2211yescharacteristic
32#6455C2211yescharacteristic
33D7×C4⋊C4C2211yescharacteristic
34D7×C4⋊C4C2211yescharacteristic
35D7×C4⋊C4C2211yescharacteristic
36#6450—21no—
37#6447—21no—
38#6447—21no—
39C7×#107C2211yescharacteristic
40C7×#107C2211yescharacteristic
41C7×#103C2211yescharacteristic
42C7×#103C2211yescharacteristic
43C7×C22⋊C8C2211yescharacteristic
44#6409C2211yescharacteristic
45#6409C2211yescharacteristic
46C22⋊(C7⋊C8)C2211yescharacteristic
47#6395C2211yescharacteristic
48#6395C2211yescharacteristic
49(C2×D7)⋊C8C2211yescharacteristic
50(C2×D7)⋊C8C2211yescharacteristic
51#6392C2211yescharacteristic
52#6391C2211yescharacteristic
53#6383C2211yescharacteristic
54#6383C2211yescharacteristic
55#6382C2211yescharacteristic
56#6381C2211yescharacteristic
57#6381C2211yescharacteristic
58#6381C2211yescharacteristic
59#3098—71nomaximal, Sylow
60C2×C7×Q8C2311yescharacteristic
61C22×C4×C7—21no—
62C22×C4×C7C2311yescharacteristic
63C2×C22⋊D7—21no—
64C2×C22⋊D7D411yescharacteristic
65C2×C22⋊D7D411yescharacteristic
66#1161—22no—
67#1161—22no—
68#1157—22no—
69#1157—22no—
70C2×D28—21no—
71C2×D28—21no—
72C2×D28D411yescharacteristic
73C2×D28C2311yescharacteristic
74C2×C4×D7—21no—
75C2×C4×D7—21no—
76C2×C4×D7—21no—
77C2×C4×D7—21no—
78C2×C4×D7C2311yescharacteristic
79C2×C4×D7C2311yescharacteristic
80C2×Dic14D411yescharacteristic
81C2×C7×C8C2311yescharacteristic
82C2×C7×C8C2311yescharacteristic
83C7×C4⋊C4—21no—
84C7×C4⋊C4—21no—
85C7×C4⋊C4C2311yescharacteristic
86C7×C4⋊C4C2311yescharacteristic
87C7×C4⋊C4C2311yescharacteristic
88C7×C22⋊C4—21no—
89C22⋊Dic7—21no—
90(C2×D7)⋊C4—21no—
91(C2×D7)⋊C4—21no—
92(C2×D7)⋊C4—21no—
93(C2×D7)⋊C4—21no—
94(C2×D7)⋊C4—21no—
95C4⋊Dic7C2311yescharacteristic
96C4⋊Dic7C2311yescharacteristic
97C4⋊Dic7C2311yescharacteristic
98Dic7⋊C4—21no—
99Dic7⋊C4—21no—
100Dic7⋊C4—21no—
101Dic7⋊C4—21no—
102C4×Dic7—21no—
103C4×Dic7—21no—
104C4×Dic7—21no—
105C2×C7⋊C8C2311yescharacteristic
106C2×C7⋊C8C2311yescharacteristic
107#1131—22no—
108C8×D7—22no—
109#537—71no—
110#520—71no—
111#485—71no—
112#485—71no—
113#484—71no—
114#484D711yescharacteristic, complemented
115#482—71no—
116#482—71no—
117#480—71no—
118#480—71no—
119#428—71no—
120C2×#107—71no—
121#417—71no—
122#416—71no—
123#408—71no—
124C23×C7C2×D411yescharacteristic
125C22×D7—21no—
126C22×D7C2×D411yescharacteristic
127C22×D7C2×D411yescharacteristic
128C7×Q8—21no—
129C7×Q8—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—21no—
135C2×C4×C7—21no—
136C2×C4×C7—81no—
137C2×C4×C7C2×D411yescharacteristic
138C2×C4×C7C2411yescharacteristic
139C22⋊D7—22no—
140C22⋊D7—22no—
141C22⋊D7—81no—
142C2×Dic7—21no—
143C2×Dic7—21no—
144C2×Dic7—21no—
145C2×Dic7C2×D411yescharacteristic
146C2×Dic7C2×D411yescharacteristic
147D28—22no—
148D28—21no—
149D28—21no—
150D28—42no—
151D28—81no—
152C4×D7—21no—
153C4×D7—21no—
154C4×D7—42no—
155C4×D7—81no—
156C4×D7—81no—
157C4×D7—81no—
158C4×D7#4112yes—
159C4×D7#4112yes—
160Dic14—22no—
161C7×C8—21no—
162C7×C8—21no—
163C7⋊C8—21no—
164C7⋊C8—21no—
165C2×#41—71no—
166C2×#41—71no—
167C2×M16—71no—
168C22×C8—71no—
169C22⋊Q8—71no—
170C22⋊Q8—141no—
171C22⋊Q8C2×D711yescharacteristic
172#121—71no—
173#121—71no—
174#121—141no—
175#121—141no—
176#121—141no—
177C4×Q8—141no—
178C4×D4—141no—
179C4×D4—141no—
180C4×D4—141no—
181#117—71no—
182#117—71no—
183#117—71no—
184C2×C4⋊C4—71no—
185C2×C4⋊C4—71no—
186C2×C4⋊C4—71no—
187C2×C4⋊C4C2×D711yescharacteristic
188#107—71no—
189#107—71no—
190#107—71no—
191#107—71no—
192#107C2×D711yescharacteristic
193#107C2×D711yescharacteristic
194#106—71no—
195#106—71no—
196#103—71no—
197#103—71no—
198#103C2×D711yescharacteristic
199#103C2×D711yescharacteristic
200#102—71no—
201#102—71no—
202#102—71no—
203#102—71no—
204C22⋊C8—71no—
205C22⋊C8—71no—
206C22⋊C8—71no—
207C22⋊C8C2×D711yescharacteristic
208C22×C7—21no—
209C22×C7—21no—
210C22×C7—21no—
211C22×C7C22×D411yescharacteristic
212C2×D7—21no—
213C2×D7—21no—
214C2×D7—21no—
215C2×D7—21no—
216C2×D7—21no—
217C2×D7—41no—
218C2×D7—41no—
219C2×D7—41no—
220C2×D7#12312yes—
221C4×C7—21no—
222C4×C7—41no—
223C4×C7—41no—
224C4×C7—41no—
225C4×C7—41no—
226C4×C7C2×#4111yescharacteristic
227C4×C7C2×#4111yescharacteristic
228Dic7—21no—
229Dic7—41no—
230Dic7—41no—
231Dic7—41no—
232Dic7#12312yes—
233#41—142no—
234#41—142no—
235#41—142no—
236#41—142no—
237C2×Q8—71no—
238C2×Q8C22×D711yescharacteristic
239C2×D4—71no—
240C2×D4—71no—
241C2×D4—71no—
242C2×D4—71no—
243C2×D4—141no—
244C2×D4—141no—
245C2×D4—141no—
246C22×C4—21no—
247C22×C4—71no—
248C22×C4—71no—
249C22×C4—141no—
250C22×C4—141no—
251C22×C4—141no—
252C22×C4—141no—
253C22×C4C22×D711yescharacteristic
254M16—142no—
255C2×C8—71no—
256C2×C8—71no—
257C2×C8—142no—
258C2×C8C22×D711yescharacteristic
259C2×C8C22×D711yescharacteristic
260C4⋊C4—21no—
261C4⋊C4—21no—
262C4⋊C4—71no—
263C4⋊C4—71no—
264C4⋊C4—71no—
265C4⋊C4—141no—
266C4⋊C4—141no—
267C4⋊C4—141no—
268C4⋊C4—141no—
269C4⋊C4C22×D711yescharacteristic
270C4⋊C4C22×D711yescharacteristic
271C4⋊C4C22×D711yescharacteristic
272C22⋊C4—21no—
273C22⋊C4—141no—
274C22⋊C4—141no—
275C22⋊C4—141no—
276C22⋊C4—141no—
277C22⋊C4—141no—
278C22⋊C4—141no—
279C42—141no—
280C42—141no—
281C42—141no—
282C2×C7—41no—
283C2×C7C2×#12311yescharacteristic
284C2×C7#57611yescharacteristic
285C2×C7#57811yescharacteristic
286D7—22no—
287D7—41no—
288D7—81no—
289C23—71no—
290C23—71no—
291C23—141no—
292C23D4×D711yescharacteristic
293Q8—21no—
294Q8—21no—
295Q8—142no—
296D4—142no—
297D4—142no—
298D4—142no—
299D4—141no—
300D4—141no—
301D4—282no—
302D4—561no—
303D4—561no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—71no—
311C2×C4—72no—
312C2×C4—72no—
313C2×C4—71no—
314C2×C4—81no—
315C2×C4—141no—
316C2×C4—141no—
317C2×C4—141no—
318C2×C4—141no—
319C2×C4—141no—
320C2×C4—282no—
321C2×C4—561no—
322C2×C4—561no—
323C2×C4—561no—
324C2×C4D4×D711yescharacteristic
325C2×C4C23×D711yescharacteristic
326C8—21no—
327C8—21no—
328C8—141no—
329C8—141no—
330C7#309811yescharacteristic, complemented, Sylow
331C22—21no—
332C22—21no—
333C22—21no—
334C22—72no—
335C22—141no—
336C22—141no—
337C22—141no—
338C22—141no—
339C22—141no—
340C22—281no—
341C22—281no—
342C22—281no—
343C22C2×D4×D711yescharacteristic
344C4—21no—
345C4—41no—
346C4—41no—
347C4—41no—
348C4—41no—
349C4—72no—
350C4—141no—
351C4—281no—
352C4—281no—
353C4—281no—
354C4D7×#4111yescharacteristic
355C4D7×#4111yescharacteristic
356C2—41no—
357C2—142no—
358C2—281no—
359C2—561no—
360C2D7×#12311yescharacteristic
361C2#9045711yescharacteristic
362C2#9046511yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#484C2
socleC22×C7C22×D4
prime radical#90320C2
hypercenter#484D7
hypocenterC7#3098
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ #484

Complements

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

NH ≅ G/Ntype
2. #90195C2semidirect
3. C7×#484C2semidirect
4. #89817C2semidirect
5. #89739C2semidirect
6. #89657C2semidirect
7. #89654C2semidirect
8. #89653C2semidirect
9. D7×#107C2semidirect
10. #89587C2semidirect
11. #89581C2semidirect
12. #89576C2semidirect
13. #89575C2semidirect
14. #89509C2semidirect
15. #89499C2semidirect
114. #484D7semidirect
330. C7#3098semidirect

Maximal subgroups

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

subgroupindex
#903202
#901952
C7×#4842
#898172
#897392
#896572
#896542
#896532
D7×#1072
#895872
#895812
#895762
#895752
#895092
#894992
#30987