Finite Group Databasebeta

← #11740992

Subgroups — #11740992

Order 896.

488 conjugacy classes of subgroups.

The subgroup lattice has 416 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#548C211yesmaximal, characteristic, complemented
2C2×#6513C211yesmaximal, characteristic, complemented
3C2×#6505C211yesmaximal, characteristic, complemented
4#90244C211yesmaximal, characteristic, complemented
5#90010C211yesmaximal, characteristic, complemented
6#89999C211yesmaximal, characteristic, complemented
7#89830C211yesmaximal, characteristic, complemented
8#89826C211yesmaximal, characteristic, complemented
9#89820C211yesmaximal, characteristic, complemented
10#89819C211yesmaximal, characteristic, complemented
11#89811C211yesmaximal, characteristic, complemented
12#89809C211yesmaximal, characteristic, complemented
13#89802C211yesmaximal, characteristic, complemented
14#89799C211yesmaximal, characteristic, complemented
15#89783C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C22×C7×Q8C2211yescharacteristic
18C2×#1157C2211yescharacteristic
19C2×#1157C2211yescharacteristic
20C7×#124—21no—
21C7×#124C2211yescharacteristic
22C7×C22⋊Q8—21no—
23C7×C22⋊Q8C2211yescharacteristic
24C7×#121—21no—
25C7×#121C2211yescharacteristic
26C4×C7×Q8C2211yescharacteristic
27C4×C7×D4—21no—
28C4×C7×D4C2211yescharacteristic
29#6513—24no—
30C2×#1144C2211yescharacteristic
31C2×#1144C2211yescharacteristic
32#6505—24no—
33C2×#1143C2211yescharacteristic
34C2×#1143C2211yescharacteristic
35C2×#1143C2211yescharacteristic
36C2×#1143C2211yescharacteristic
37#6503—21no—
38C2×#1142C2211yescharacteristic
39C2×#1141C2211yescharacteristic
40C2×#1141C2211yescharacteristic
41#6493C2211yescharacteristic
42#6493C2211yescharacteristic
43C2×#1136C2211yescharacteristic
44C22×C7⋊C8C2211yescharacteristic
45#6456—21no—
46#6447—21no—
47#6447—21no—
48#6437C2211yescharacteristic
49C4×Dic14C2211yescharacteristic
50#6409C2211yescharacteristic
51#6409C2211yescharacteristic
52#6406C2211yescharacteristic
53C22⋊(C7⋊C8)C2211yescharacteristic
54C22⋊(C7⋊C8)C2211yescharacteristic
55#6384C2211yescharacteristic
56#6383C2211yescharacteristic
57#6383C2211yescharacteristic
58#6382C2211yescharacteristic
59C4⋊(C7⋊C8)C2211yescharacteristic
60#6377C2211yescharacteristic
61#3271—71nomaximal, Sylow
62C7×#41—22no—
63C7×#41—41no—
64C2×C7×Q8—21no—
65C2×C7×Q8—22no—
66C2×C7×Q8—41no—
67C2×C7×Q8C2311yescharacteristic
68C2×C7×Q8C2311yescharacteristic
69C2×C7×D4—21no—
70C2×C7×D4—21no—
71C2×C7×D4D411yescharacteristic
72C2×C7×D4C2311yescharacteristic
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7C2311yescharacteristic
76C22×C4×C7C2311yescharacteristic
77C2×C22⋊D7—21no—
78C2×C22⋊D7—21no—
79C2×C22⋊D7—21no—
80C2×C22⋊D7—21no—
81#1157—22no—
82#1157—22no—
83#1157—41no—
84#1157—41no—
85C2×D28—21no—
86C2×D28—21no—
87C2×D28C2311yescharacteristic
88C2×D28C2311yescharacteristic
89C2×C4×D7—21no—
90C2×C4×D7—21no—
91C2×Dic14C2311yescharacteristic
92C7×C4⋊C4—21no—
93C7×C4⋊C4—21no—
94C7×C4⋊C4D411yescharacteristic
95C7×C4⋊C4C2311yescharacteristic
96C7×C22⋊C4—21no—
97C7×C22⋊C4—21no—
98C7×C22⋊C4—21no—
99C7×C22⋊C4—21no—
100C7×C22⋊C4D411yescharacteristic
101C7×C22⋊C4D411yescharacteristic
102C42×C7—21no—
103C42×C7C2311yescharacteristic
104#1144—22no—
105#1144D414yescomplemented
106#1143—22no—
107#1143—22no—
108#1143—22no—
109#1143—41no—
110#1142—22no—
111#1141—22no—
112#1141—41no—
113(C2×D7)⋊C4—21no—
114(C2×D7)⋊C4—21no—
115C4⋊Dic7C2311yescharacteristic
116Dic7⋊C4—21no—
117C4×Dic7—21no—
118#1136—22no—
119C2×C7⋊C8—22no—
120C2×C7⋊C8C2311yescharacteristic
121C2×C7⋊C8C2311yescharacteristic
122C2×C7⋊C8C2311yescharacteristic
123C2×C7⋊C8C2311yescharacteristic
124C2×#137—71no—
125C2×#135—71no—
126#550—71no—
127#548D711yescharacteristic, complemented
128#496—71no—
129#469—71no—
130#465—71no—
131#464—71no—
132#460—71no—
133#452—71no—
134#449—71no—
135#449—71no—
136#448—71no—
137#441—71no—
138#435—71no—
139C23×C7—21no—
140C23×C7C2×D411yescharacteristic
141C23×C7C2×D411yescharacteristic
142C22×D7—21no—
143C22×D7—21no—
144C7×Q8—21no—
145C7×Q8—21no—
146C7×Q8—21no—
147C7×Q8—42no—
148C7×Q8C2×D412yes—
149C7×D4—21no—
150C7×D4—21no—
151C7×D4—41no—
152C7×D4—42no—
153C7×D4—42no—
154C2×C4×C7—21no—
155C2×C4×C7—21no—
156C2×C4×C7—21no—
157C2×C4×C7—21no—
158C2×C4×C7—21no—
159C2×C4×C7—21no—
160C2×C4×C7—21no—
161C2×C4×C7—42no—
162C2×C4×C7—42no—
163C2×C4×C7C2×D411yescharacteristic
164C2×C4×C7C2×D411yescharacteristic
165C2×C4×C7C2×D411yescharacteristic
166C2×C4×C7C2×D411yescharacteristic
167C2×C4×C7C2411yescharacteristic
168C22⋊D7—42no—
169C22⋊D7—42no—
170C22⋊D7—81no—
171C22⋊D7—81no—
172C2×Dic7—21no—
173C2×Dic7—21no—
174D28—21no—
175D28—21no—
176D28—21no—
177D28—21no—
178D28—42no—
179D28—81no—
180C4×D7—42no—
181C4×D7—42no—
182Dic14—21no—
183Dic14C2×D412yes—
184C7⋊C8—21no—
185C7⋊C8—21no—
186C7⋊C8—21no—
187C7⋊C8C2×D412yes—
188C2×#41—71no—
189C2×#41—71no—
190C2×#41C2×D711yescharacteristic
191C22×Q8C2×D711yescharacteristic
192#137—144no—
193#135—144no—
194C2×Q16—71no—
195C2×Q16—71no—
196C2×SD16—71no—
197C2×SD16—71no—
198C2×SD16—71no—
199C2×SD16—71no—
200C2×SD16—71no—
201C2×D8—71no—
202C2×D8—71no—
203C2×M16—71no—
204C22×C8—71no—
205#127—71no—
206#127—141no—
207#124—21no—
208#124C2×D711yescharacteristic
209C22⋊Q8—21no—
210C22⋊Q8C2×D711yescharacteristic
211#121—21no—
212#121—71no—
213#121—71no—
214#121—141no—
215#121—141no—
216#121C2×D711yescharacteristic
217C4×Q8—71no—
218C4×Q8C2×D711yescharacteristic
219C4×D4—21no—
220C4×D4—141no—
221C4×D4C2×D711yescharacteristic
222#106—71no—
223C4⋊C8—71no—
224#103—71no—
225#103—71no—
226#103—71no—
227#102—71no—
228#102—71no—
229#102—71no—
230C22⋊C8—71no—
231C22⋊C8—71no—
232#97—71no—
233C22×C7—21no—
234C22×C7—21no—
235C22×C7—21no—
236C22×C7—21no—
237C22×C7—21no—
238C22×C7—21no—
239C22×C7—41no—
240C22×C7—41no—
241C22×C7—41no—
242C22×C7C22×D411yescharacteristic
243C2×D7—41no—
244C2×D7—41no—
245C2×D7—41no—
246C2×D7—41no—
247C2×D7—41no—
248C2×D7—41no—
249C4×C7—21no—
250C4×C7—21no—
251C4×C7—22no—
252C4×C7—21no—
253C4×C7—21no—
254C4×C7—41no—
255C4×C7—41no—
256C4×C7C22×D411yescharacteristic
257C4×C7H(5,𝔽2)11yescharacteristic
258Dic7—22no—
259Dic7—41no—
260#41—22no—
261#41—41no—
262#41—142no—
263#41—142no—
264#41—281no—
265#41—281no—
266C2×Q8—21no—
267C2×Q8—22no—
268C2×Q8—41no—
269C2×Q8—71no—
270C2×Q8C22×D711yescharacteristic
271C2×Q8C22×D711yescharacteristic
272C2×D4—21no—
273C2×D4—21no—
274C2×D4—71no—
275C2×D4—71no—
276C2×D4—141no—
277C2×D4—141no—
278C2×D4—141no—
279C2×D4—141no—
280C2×D4—141no—
281C2×D4—141no—
282C2×D4C22⋊D711yescharacteristic
283C2×D4C22×D711yescharacteristic
284C22×C4—21no—
285C22×C4—21no—
286C22×C4—141no—
287C22×C4—141no—
288C22×C4C22×D711yescharacteristic
289C22×C4C22×D711yescharacteristic
290Q16—74no—
291Q16—142no—
292SD16—142no—
293SD16—142no—
294SD16—142no—
295SD16—142no—
296SD16—281no—
297D8—142no—
298D8—281no—
299M16—142no—
300C2×C8—71no—
301C2×C8—71no—
302C2×C8—71no—
303C2×C8—71no—
304C2×C8—142no—
305C4⋊C4—21no—
306C4⋊C4—21no—
307C4⋊C4—71no—
308C4⋊C4—141no—
309C4⋊C4C22⋊D711yescharacteristic
310C4⋊C4C22×D711yescharacteristic
311C22⋊C4—21no—
312C22⋊C4—21no—
313C22⋊C4—21no—
314C22⋊C4—21no—
315C22⋊C4—141no—
316C22⋊C4—141no—
317C22⋊C4C22⋊D711yescharacteristic
318C22⋊C4C22⋊D711yescharacteristic
319C42—21no—
320C42—141no—
321C42C22×D711yescharacteristic
322C2×C7—41no—
323C2×C7—41no—
324C2×C7—81no—
325C2×C7D4211yescharacteristic
326C2×C7#57511yescharacteristic
327C2×C7#57711yescharacteristic
328D7—81no—
329D7—81no—
330C23—21no—
331C23—141no—
332C23—141no—
333C23C2×C22⋊D711yescharacteristic
334C23C2×C22⋊D711yescharacteristic
335Q8—21no—
336Q8—21no—
337Q8—21no—
338Q8—42no—
339Q8—72no—
340Q8—141no—
341Q8D4×D712yes—
342D4—21no—
343D4—21no—
344D4—41no—
345D4—42no—
346D4—42no—
347D4—141no—
348D4—141no—
349D4—141no—
350D4—141no—
351D4—282no—
352D4—282no—
353D4—282no—
354D4—561no—
355D4—561no—
356D4—561no—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—21no—
362C2×C4—21no—
363C2×C4—21no—
364C2×C4—42no—
365C2×C4—42no—
366C2×C4—141no—
367C2×C4—141no—
368C2×C4—282no—
369C2×C4—282no—
370C2×C4C2×C22⋊D711yescharacteristic
371C2×C4C2×C22⋊D711yescharacteristic
372C2×C4C2×C22⋊D711yescharacteristic
373C2×C4C2×C22⋊D711yescharacteristic
374C2×C4C23×D711yescharacteristic
375C8—72no—
376C8—141no—
377C8—141no—
378C8—141no—
379C7#327111yescharacteristic, complemented, Sylow
380C22—21no—
381C22—21no—
382C22—21no—
383C22—21no—
384C22—21no—
385C22—21no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—281no—
390C22—281no—
391C22—281no—
392C22—281no—
393C22—281no—
394C22—281no—
395C22C22×C22⋊D711yescharacteristic
396C4—21no—
397C4—21no—
398C4—22no—
399C4—21no—
400C4—21no—
401C4—41no—
402C4—41no—
403C4—142no—
404C4—281no—
405C4C2×D4×D711yescharacteristic
406C4#654811yescharacteristic
407C2—41no—
408C2—41no—
409C2—81no—
410C2—561no—
411C2—561no—
412C2D4×C22⋊D711yescharacteristic
413C2#9051311yescharacteristic
414C2#9052811yescharacteristic
subgroupG/H
centerC22C22×C22⋊D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC7×#548C2
socleC22×C7C22×D4
hypercenter#548D7
hypocenterC7#3271
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C4×D4 ◁ #548

Complements

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

NH ≅ G/Ntype
1. C7×#548C2semidirect
2. C2×#6513C2semidirect
3. C2×#6505C2semidirect
4. #90244C2semidirect
5. #90010C2semidirect
6. #89999C2semidirect
7. #89830C2semidirect
8. #89826C2semidirect
9. #89820C2semidirect
10. #89819C2semidirect
11. #89811C2semidirect
12. #89809C2semidirect
13. #89802C2semidirect
14. #89799C2semidirect
15. #89783C2semidirect
105. #1144D4semidirect
127. #548D7semidirect
379. C7#3271semidirect

Maximal subgroups

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

subgroupindex
C7×#5482
C2×#65132
C2×#65052
#902442
#900102
#899992
#898302
#898262
#898202
#898192
#898112
#898092
#898022
#897992
#897832
#32717