Finite Group Databasebeta

← #11741366

Subgroups — #11741366

Order 896.

500 conjugacy classes of subgroups.

The subgroup lattice has 418 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#135C211yesmaximal, characteristic, complemented
2#90520C211yesmaximal, characteristic, complemented
3#90515C211yesmaximal, characteristic, complemented
4C2×#6474C211yesmaximal, characteristic, complemented
5#90011C211yesmaximal, characteristic, complemented
6#90010C211yesmaximal, characteristic, complemented
7#89962C211yesmaximal, characteristic, complemented
8#89961C211yesmaximal, characteristic, complemented
9#89945C211yesmaximal, characteristic, complemented
10#89941C211yesmaximal, characteristic, complemented
11#89937C211yesmaximal, characteristic, complemented
12#89934C211yesmaximal, characteristic, complemented
13#89924C211yesmaximal, characteristic, complemented
14#89886C211yesmaximal, characteristic, complemented
15#89883C211yesmaximal, characteristic, complemented
16C2×C7×#41C2211yescharacteristic
17C2×C7×#41C2211yescharacteristic, complemented
18C2×#1159C2211yescharacteristic
19C2×D4×D7C2211yescharacteristic
20C7×#135—24no—
21C2×C7×Q16C2211yescharacteristic, complemented
22C2×C7×SD16C2211yescharacteristic
23C2×C7×SD16C2211yescharacteristic, complemented
24C2×C7×D8C2211yescharacteristic
25C22×C7×C8C2211yescharacteristic, complemented
26#6510C2211yescharacteristic
27#6509C2211yescharacteristic
28#6509C2211yescharacteristic
29#6507C2211yescharacteristic, complemented
30C2×#1144C2211yescharacteristic, complemented
31C2×#1143C2211yescharacteristic
32#6502—21no—
33#6502—21no—
34#6500—21no—
35#6498—21no—
36D4×Dic7C2211yescharacteristic
37D4×Dic7C2211yescharacteristic, complemented
38C2×#1142C2211yescharacteristic
39C2×#1141C2211yescharacteristic
40#6493C2211yescharacteristic
41#6492—21no—
42#6492—21no—
43C4×C22⋊D7C2211yescharacteristic
44C2×(C2×D7)⋊C4—21no—
45#6487C2211yescharacteristic, complemented
46C2×Dic7⋊C4—21no—
47#6474—24no—
48C2×#1132C2211yescharacteristic
49C2×#1131C2211yescharacteristic
50#6409C2211yescharacteristic, complemented
51#6409C2211yescharacteristic, complemented
52#6406C2211yescharacteristic
53#6406C2211yescharacteristic, complemented
54C22⋊(C7⋊C8)C2211yescharacteristic, complemented
55#6395C2211yescharacteristic
56(C2×D7)⋊C8C2211yescharacteristic
57#6391C2211yescharacteristic, complemented
58#6390C2211yescharacteristic, complemented
59#6389C2211yescharacteristic, complemented
60Dic7⋊C8C2211yescharacteristic, complemented
61#3267—71nomaximal, Sylow
62C23×D7—21no—
63C7×#41—22no—
64C7×#41—22no—
65C7×#41—41no—
66C7×#41—41no—
67C2×C7×Q8C2311yescharacteristic
68C2×C7×Q8C2311yescharacteristic
69C2×C7×D4—21no—
70C2×C7×D4—21no—
71C2×C7×D4C2311yescharacteristic
72C2×C7×D4C2311yescharacteristic
73C22×C4×C7—21no—
74C22×C4×C7—21no—
75C22×C4×C7C2311yescharacteristic
76C2×C22⋊D7—21no—
77C2×C22⋊D7—21no—
78C2×C22⋊D7—21no—
79C2×C22⋊D7D411yescharacteristic
80C22×Dic7—21no—
81C22×Dic7—21no—
82C22×Dic7—21no—
83#1159—22no—
84#1159—41no—
85D4×D7—22no—
86D4×D7—41no—
87C2×D28C2311yescharacteristic
88C2×C4×D7C2311yescharacteristic
89C2×Dic14C2311yescharacteristic
90C7×Q16—22no—
91C7×SD16—22no—
92C7×SD16—22no—
93C7×D8D414yescomplemented
94C2×C7×C8—22no—
95C2×C7×C8C2311yescharacteristic
96C2×C7×C8C2311yescharacteristic
97C22⋊Dic7—21no—
98C22⋊Dic7—21no—
99C22⋊Dic7—21no—
100C22⋊Dic7D411yescharacteristic
101#1144—41no—
102#1143—41no—
103#1142—22no—
104#1141—22no—
105(C2×D7)⋊C4—21no—
106(C2×D7)⋊C4—21no—
107(C2×D7)⋊C4—21no—
108(C2×D7)⋊C4—21no—
109(C2×D7)⋊C4—21no—
110(C2×D7)⋊C4D411yescharacteristic
111C4⋊Dic7C2311yescharacteristic
112C4⋊Dic7C2311yescharacteristic
113Dic7⋊C4—21no—
114Dic7⋊C4—21no—
115Dic7⋊C4—21no—
116Dic7⋊C4—21no—
117Dic7⋊C4—21no—
118Dic7⋊C4D411yescharacteristic
119C4×Dic7C2311yescharacteristic
120C2×C7⋊C8C2311yescharacteristic
121C2×C7⋊C8C2311yescharacteristic
122#1132—22no—
123#1131—22no—
124C2×AGL(1,ℤ/8)—71no—
125C2×#135D711yescharacteristic, complemented
126#547—71no—
127#546—71no—
128#497—71no—
129#470—71no—
130#465—71no—
131#463—71no—
132#461—71no—
133#452—71no—
134#452—71no—
135#450—71no—
136#449—71no—
137#442—71no—
138#435—71no—
139C23×C7—21no—
140C23×C7—21no—
141C23×C7C2×D411yescharacteristic
142C22×D7—21no—
143C22×D7—42no—
144C22×D7—42no—
145C22×D7—81no—
146C22×D7C2×D411yescharacteristic
147C7×Q8—21no—
148C7×Q8—21no—
149C7×Q8—21no—
150C7×Q8—21no—
151C7×D4—21no—
152C7×D4—21no—
153C7×D4—42no—
154C7×D4—42no—
155C7×D4C2×D412yes—
156C7×D4C2×D412yes—
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×C7C2411yescharacteristic
165C22⋊D7—41no—
166C22⋊D7—42no—
167C22⋊D7—42no—
168C22⋊D7—81no—
169C2×Dic7—21no—
170C2×Dic7—21no—
171C2×Dic7—21no—
172C2×Dic7—42no—
173C2×Dic7—42no—
174C2×Dic7—81no—
175C2×Dic7C2×D411yescharacteristic
176C2×Dic7C2×D411yescharacteristic
177C2×Dic7C2×D411yescharacteristic
178D28—21no—
179D28—21no—
180C4×D7—21no—
181C4×D7—21no—
182Dic14—21no—
183Dic14—21no—
184C7×C8—21no—
185C7×C8C2×D412yes—
186C7⋊C8—21no—
187C7⋊C8—21no—
188C2×#41—71no—
189C2×#41C2×D711yescharacteristic
190C2×#41C2×D711yescharacteristic, complemented
191C22×D4—71no—
192AGL(1,ℤ/8)—144no—
193#135—24no—
194C2×Q16—71no—
195C2×Q16C2×D711yescharacteristic, complemented
196C2×SD16—71no—
197C2×SD16—71no—
198C2×SD16—71no—
199C2×SD16C2×D711yescharacteristic
200C2×SD16C2×D711yescharacteristic, complemented
201C2×D8—71no—
202C2×D8C2×D711yescharacteristic
203C2×M16—71no—
204C22×C8C2×D711yescharacteristic, complemented
205C4⋊Q8—71no—
206#124—71no—
207#123—141no—
208#123—141no—
209#123—141no—
210C22⋊Q8—71no—
211C22⋊Q8—71no—
212C22⋊Q8—71no—
213#121—71no—
214#121—141no—
215#121—141no—
216#120—141no—
217C4×D4—71no—
218C4×D4—71no—
219C4×D4—71no—
220C2×C4⋊C4—141no—
221C2×C22⋊C4—141no—
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—22no—
234C22×C7—22no—
235C22×C7—21no—
236C22×C7—21no—
237C22×C7—21no—
238C22×C7—41no—
239C22×C7—41no—
240C22×C7—41no—
241C22×C7—41no—
242C22×C7C22×D411yescharacteristic
243C2×D7—21no—
244C2×D7—21no—
245C2×D7—21no—
246C2×D7—41no—
247C2×D7—41no—
248C2×D7—41no—
249C2×D7—84no—
250C4×C7—21no—
251C4×C7—41no—
252C4×C7—41no—
253C4×C7C22×D411yescharacteristic
254C4×C7H(5,𝔽2)11yescharacteristic
255Dic7—21no—
256Dic7—21no—
257Dic7—21no—
258Dic7—41no—
259Dic7—41no—
260Dic7—41no—
261C24—141no—
262#41—22no—
263#41—22no—
264#41—41no—
265#41—41no—
266#41—142no—
267#41—281no—
268C2×Q8—71no—
269C2×Q8C22×D711yescharacteristic
270C2×Q8C22×D711yescharacteristic
271C2×D4—21no—
272C2×D4—21no—
273C2×D4—71no—
274C2×D4—71no—
275C2×D4—141no—
276C2×D4—142no—
277C2×D4—141no—
278C2×D4—141no—
279C2×D4—281no—
280C2×D4C22×D711yescharacteristic
281C2×D4C22×D711yescharacteristic
282C22×C4—21no—
283C22×C4—21no—
284C22×C4—71no—
285C22×C4—141no—
286C22×C4—141no—
287C22×C4—141no—
288C22×C4C22×D711yescharacteristic
289Q16—22no—
290Q16—281no—
291SD16—22no—
292SD16—22no—
293SD16—142no—
294SD16—142no—
295SD16—281no—
296D8—142no—
297D8C22⋊D714yescomplemented
298M16—142no—
299C2×C8—22no—
300C2×C8—71no—
301C2×C8—71no—
302C2×C8C22×D711yescharacteristic
303C2×C8C22×D711yescharacteristic
304C4⋊C4—71no—
305C4⋊C4—71no—
306C4⋊C4—71no—
307C4⋊C4—141no—
308C4⋊C4—141no—
309C4⋊C4—141no—
310C4⋊C4—141no—
311C4⋊C4—141no—
312C22⋊C4—71no—
313C22⋊C4—71no—
314C22⋊C4—141no—
315C22⋊C4—141no—
316C22⋊C4—141no—
317C22⋊C4—141no—
318C22⋊C4—141no—
319C22⋊C4—141no—
320C22⋊C4—141no—
321C22⋊C4—141no—
322C42—71no—
323C2×C7—42no—
324C2×C7—41no—
325C2×C7—42no—
326C2×C7D4211yescharacteristic
327C2×C7#57511yescharacteristic
328C2×C7#57711yescharacteristic
329D7—41no—
330D7—81no—
331C23—21no—
332C23—21no—
333C23—71no—
334C23—141no—
335C23—282no—
336C23—282no—
337C23—561no—
338C23D4×D711yescharacteristic
339Q8—21no—
340Q8—21no—
341Q8—21no—
342Q8—21no—
343Q8—141no—
344Q8—141no—
345D4—21no—
346D4—21no—
347D4—42no—
348D4—42no—
349D4—141no—
350D4—141no—
351D4—281no—
352D4—282no—
353D4—282no—
354D4—561no—
355D4C2×C22⋊D712yes—
356D4C2×C22⋊D712yes—
357C2×C4—21no—
358C2×C4—21no—
359C2×C4—21no—
360C2×C4—21no—
361C2×C4—42no—
362C2×C4—42no—
363C2×C4—71no—
364C2×C4—71no—
365C2×C4—71no—
366C2×C4—141no—
367C2×C4—141no—
368C2×C4—141no—
369C2×C4—141no—
370C2×C4—141no—
371C2×C4—282no—
372C2×C4—282no—
373C2×C4—561no—
374C2×C4D4×D711yescharacteristic
375C2×C4C23×D711yescharacteristic
376C8—21no—
377C8—141no—
378C8—141no—
379C8C2×C22⋊D712yes—
380C7#326711yescharacteristic, complemented, Sylow
381C22—21no—
382C22—21no—
383C22—21no—
384C22—22no—
385C22—22no—
386C22—41no—
387C22—41no—
388C22—41no—
389C22—41no—
390C22—141no—
391C22—141no—
392C22—141no—
393C22—281no—
394C22—281no—
395C22—281no—
396C22—564no—
397C22C2×D4×D711yescharacteristic
398C4—21no—
399C4—41no—
400C4—41no—
401C4—141no—
402C4—141no—
403C4—141no—
404C4—281no—
405C4—281no—
406C4—281no—
407C4#654811yescharacteristic
408C4C22×C22⋊D711yescharacteristic
409C2—41no—
410C2—42no—
411C2—42no—
412C2—281no—
413C2—561no—
414C2#9045911yescharacteristic
415C2#9046111yescharacteristic
416C2D4×C22⋊D711yescharacteristic
subgroupG/H
centerC22C2×D4×D7
commutator, cosocleC2×C4×C7C24
FrattiniC2×C4C23×D7
FittingC2×C7×#135C2
socleC22×C7C22×D4
hypercenterC2×#135D7
hypocenterC7#3267
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#135

Complements

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

NH ≅ G/Ntype
1. C2×C7×#135C2semidirect
2. #90520C2semidirect
3. #90515C2semidirect
4. C2×#6474C2semidirect
5. #90011C2semidirect
6. #90010C2semidirect
7. #89962C2semidirect
8. #89961C2semidirect
9. #89945C2semidirect
10. #89941C2semidirect
11. #89937C2semidirect
12. #89934C2semidirect
13. #89924C2semidirect
14. #89886C2semidirect
15. #89883C2semidirect
17. C2×C7×#41C22semidirect
21. C2×C7×Q16C22semidirect
23. C2×C7×SD16C22semidirect
25. C22×C7×C8C22semidirect
29. #6507C22semidirect
30. C2×#1144C22semidirect
37. D4×Dic7C22semidirect
45. #6487C22semidirect
50. #6409C22semidirect
51. #6409C22semidirect
53. #6406C22semidirect
54. C22⋊(C7⋊C8)C22semidirect
57. #6391C22semidirect
58. #6390C22semidirect
59. #6389C22semidirect
60. Dic7⋊C8C22semidirect
93. C7×D8D4semidirect
125. C2×#135D7semidirect
190. C2×#41C2×D7semidirect
195. C2×Q16C2×D7semidirect
200. C2×SD16C2×D7semidirect
204. C22×C8C2×D7semidirect
297. D8C22⋊D7semidirect
380. C7#3267semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1352
#905202
#905152
C2×#64742
#900112
#900102
#899622
#899612
#899452
#899412
#899372
#899342
#899242
#898862
#898832
#32677