Finite Group Databasebeta

← C2×#11728851 #13508367

Subgroups — C2×#11728851 #13508367

Order 1792.

658 conjugacy classes of subgroups.

The subgroup lattice has 408 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#473C211yesmaximal, characteristic, complemented
2C22×#6402C211yesmaximal, characteristic, complemented
3C2×#89914C211yesmaximal, characteristic
4C2×#89912C211yesmaximal, characteristic, complemented
5C2×#89360C211yesmaximal, characteristic, complemented
6#11728851C218yesmaximal, direct product factor
7C2×#89290C211yesmaximal, characteristic, complemented
8C2×#89288C211yesmaximal, characteristic, complemented
9C2×C7×#137—21no—
10C22×C7×Q16C2211yescharacteristic
11C2×C7×#131C2211yescharacteristic
12C2×#6512—21no—
13C22×D56C2211yescharacteristic
14C7×#473—22no—
15C7×#473C2214yescomplemented
16C2×C7×#108C2211yescharacteristic
17C2×C7×#101—21no—
18C2×#6402—21no—
19C2×#6402—21no—
20C2×#6402—22no—
21C2×#6402C2214yescomplemented
22#89914—22no—
23#89914C2214yes—
24#89912—22no—
25#89912C2214yescomplemented
26C2×#6397—21no—
27C2×#6393C2211yescharacteristic
28C2×#6386—21no—
29C2×#6386C2211yescharacteristic
30C22×C7⋊C16—21no—
31C22×C7⋊C16C2211yescharacteristic
32#89360C2214yescomplemented
33#89290C2214yescomplemented
34#89288C2214yescomplemented
35C2×#2223—71nomaximal, Sylow
36C2×C7×#41D411yescharacteristic
37C22×C7×Q8—21no—
38C22×D28—21no—
39C7×#137—24no—
40C7×#137—42no—
41C2×C7×Q16—21no—
42C2×C7×Q16—21no—
43C2×C7×Q16—21no—
44C2×C7×Q16—21no—
45C2×C7×Q16—21no—
46C2×C7×Q16—41no—
47C2×C7×Q16C2312yes—
48C2×C7×SD16—41no—
49C7×#131—21no—
50C7×#131—21no—
51C7×#131D412yes—
52C7×#131C2312yes—
53C2×C7×M16—21no—
54C2×C7×M16—21no—
55C2×C7×M16D411yescharacteristic
56C22×C7×C8—21no—
57C22×C7×C8C2311yescharacteristic
58#6512—24no—
59#6512—42no—
60C2×#1143—41no—
61C2×#1141—41no—
62C2×#1136—21no—
63C2×D56—21no—
64C2×D56—21no—
65C2×D56—21no—
66C2×D56—21no—
67C2×D56—21no—
68C2×D56C2312yes—
69C7×#108—21no—
70C7×#108C2312yes—
71C7×#101—24no—
72#6402—24no—
73#6402—24no—
74#6397—24no—
75#6393—21no—
76#6393C2312yes—
77#6386—21no—
78#6386—24no—
79#6386C2312yes—
80C2×C7⋊C16—21no—
81C2×C7⋊C16—24no—
82C2×C7⋊C16D411yescharacteristic
83C2×C7⋊C16D411yescharacteristic
84C2×C7⋊C16D412yes—
85C2×C7⋊C16C2312yes—
86C22×SD32—71no—
87C2×#504—71no—
88C2×#473D711yescharacteristic, complemented
89C2×#472—71no—
90#2223—78no—
91C2×#368—71no—
92C2×#362—71no—
93C2×#361—71no—
94C23×D7—41no—
95C7×#41—21no—
96C7×#41—21no—
97C7×#41D812yes—
98C7×#41C2×D412yes—
99C2×C7×Q8—22no—
100C2×C7×Q8—41no—
101C2×C7×Q8—41no—
102C2×C7×Q8—41no—
103C2×C7×Q8—41no—
104C2×C7×Q8—41no—
105C2×C7×Q8D811yescharacteristic
106C2×C7×D4—21no—
107C2×C7×D4D811yescharacteristic
108C22×C4×C7—21no—
109C22×C4×C7—41no—
110C22×C4×C7C2×D411yescharacteristic
111C2×D28—22no—
112C2×D28—41no—
113C2×D28—41no—
114C2×D28—41no—
115C2×D28—41no—
116C2×D28—41no—
117C7×Q16—22no—
118C7×Q16—22no—
119C7×Q16—44no—
120C7×Q16—41no—
121C7×Q16—41no—
122C7×SD16—44no—
123C7×M16—21no—
124C7×M16—22no—
125C7×M16—22no—
126C7×M16—22no—
127C7×M16—41no—
128C7×M16C2×D412yes—
129C2×C7×C8—21no—
130C2×C7×C8—22no—
131C2×C7×C8—22no—
132C2×C7×C8—41no—
133C2×C7×C8C2×D411yescharacteristic
134C2×C7×C8C2×D411yescharacteristic
135C2×C7×C8C2×D411yescharacteristic
136C2×C7×C8#4111yescharacteristic
137C2×C7×C8#4111yescharacteristic
138C2×C7×C8C2411yescharacteristic
139#1143—44no—
140#1141—44no—
141#1136—22no—
142#1136—41no—
143C2×C7⋊C8—41no—
144D56—22no—
145D56—22no—
146D56—41no—
147D56—41no—
148C7⋊C16—22no—
149C7⋊C16C2×D412yes—
150C7⋊C16C2×D412yes—
151C2×#137—21no—
152C2×AGL(1,ℤ/8)—141no—
153C22×Q16C2×D711yescharacteristic
154C22×D8—71no—
155C2×#131C2×D711yescharacteristic
156C2×SD32—74no—
157C2×SD32—141no—
158C2×SD32—141no—
159C2×SD32—142no—
160#504—74no—
161#504—142no—
162C2×M32—71no—
163C2×M32—141no—
164C22×C16—71no—
165C22×C16—141no—
166#473—22no—
167#473C2×D714yescomplemented
168#472—74no—
169#472—142no—
170C2×#108—71no—
171C2×#108C2×D711yescharacteristic
172C2×#101—21no—
173C2×#100—141no—
174#368—74no—
175#362—74no—
176#361—74no—
177C23×C7—21no—
178C23×C7#13511yescharacteristic
179C22×D7—42no—
180C22×D7—81no—
181C22×D7—81no—
182C22×D7—162no—
183C7×Q8—21no—
184C7×Q8—42no—
185C7×Q8—42no—
186C7×Q8—81no—
187C7×Q8—81no—
188C7×Q8C2×D812yes—
189C7×D4—22no—
190C7×D4—22no—
191C7×D4—21no—
192C7×D4C2×D812yes—
193C2×C4×C7—21no—
194C2×C4×C7—22no—
195C2×C4×C7—22no—
196C2×C4×C7—42no—
197C2×C4×C7—81no—
198C2×C4×C7—81no—
199C2×C4×C7#12111yescharacteristic
200C2×C4×C7#12111yescharacteristic
201C2×C4×C7C2×D811yescharacteristic
202C2×C4×C7C2×D811yescharacteristic
203C2×C4×C7#13511yescharacteristic
204C2×C4×C7C22×D411yescharacteristic
205D28—42no—
206D28—42no—
207D28—81no—
208D28—81no—
209C7×C8—22no—
210C7×C8—42no—
211C7×C8#12111yescharacteristic
212C7×C8#12111yescharacteristic
213C7×C8C22×D411yescharacteristic
214C7×C8C2×#4111yescharacteristic
215C7⋊C8—42no—
216C2×#41C22⋊D711yescharacteristic
217C22×Q8—21no—
218C22×D4—141no—
219#137—24no—
220#137—42no—
221AGL(1,ℤ/8)—144no—
222AGL(1,ℤ/8)—282no—
223C2×Q16—21no—
224C2×Q16—21no—
225C2×Q16—21no—
226C2×Q16—21no—
227C2×Q16—21no—
228C2×Q16—41no—
229C2×Q16C22×D712yes—
230C2×SD16—41no—
231C2×SD16—281no—
232C2×D8—72no—
233C2×D8—141no—
234C2×D8—141no—
235C2×D8—141no—
236C2×D8—141no—
237C2×D8—141no—
238C2×D8—281no—
239#131—21no—
240#131—21no—
241#131C22⋊D712yes—
242#131C22×D712yes—
243C2×M16—21no—
244C2×M16—21no—
245C2×M16—141no—
246C2×M16C22⋊D711yescharacteristic
247C22×C8—21no—
248C22×C8C22×D711yescharacteristic
249SD32—144no—
250SD32—144no—
251M32—72no—
252M32—144no—
253M32—141no—
254C2×C16—71no—
255C2×C16—71no—
256C2×C16—72no—
257C2×C16—72no—
258C2×C16—144no—
259C2×C16—141no—
260#108—21no—
261#108—72no—
262#108—141no—
263#108C22×D712yes—
264#101—24no—
265#100—144no—
266C22×C7—22no—
267C22×C7—22no—
268C22×C7—42no—
269C22×C7#46611yescharacteristic
270C22×C7#46611yescharacteristic
271C22×C7C2×#13511yescharacteristic
272C2×D7—82no—
273C2×D7—162no—
274C2×D7—162no—
275C2×D7—162no—
276C4×C7—22no—
277C4×C7—82no—
278C4×C7#46611yescharacteristic
279C4×C7#46611yescharacteristic
280C4×C7C2×#12111yescharacteristic
281C4×C7C22×D811yescharacteristic
282C24—281no—
283#41—21no—
284#41—21no—
285#41#114112yes—
286#41C2×C22⋊D712yes—
287C2×Q8—22no—
288C2×Q8—41no—
289C2×Q8—41no—
290C2×Q8—41no—
291C2×Q8—41no—
292C2×Q8—41no—
293C2×Q8#114111yescharacteristic
294C2×D4—21no—
295C2×D4—142no—
296C2×D4—281no—
297C2×D4—281no—
298C2×D4—281no—
299C2×D4—281no—
300C2×D4—281no—
301C2×D4#114111yescharacteristic
302C22×C4—21no—
303C22×C4—41no—
304C22×C4C2×C22⋊D711yescharacteristic
305Q16—22no—
306Q16—22no—
307Q16—44no—
308Q16—41no—
309Q16—41no—
310SD16—44no—
311SD16—284no—
312D8—142no—
313D8—142no—
314D8—284no—
315D8—281no—
316D8—281no—
317M16—21no—
318M16—22no—
319M16—22no—
320M16—22no—
321M16—41no—
322M16—142no—
323M16—281no—
324M16C2×C22⋊D712yes—
325C2×C8—21no—
326C2×C8—22no—
327C2×C8—22no—
328C2×C8—41no—
329C2×C8—281no—
330C2×C8D4×D711yescharacteristic
331C2×C8D4×D711yescharacteristic
332C2×C8#115911yescharacteristic
333C2×C8#115911yescharacteristic
334C2×C8C2×C22⋊D711yescharacteristic
335C2×C8C23×D711yescharacteristic
336C16—72no—
337C16—72no—
338C16—142no—
339C2×C7—21no—
340C2×C7—21no—
341C2×C7—42no—
342C2×C7#222312yescomplemented
343C2×C7C2×#46611yescharacteristic
344D7—162no—
345C23—21no—
346C23—282no—
347C23—561no—
348C23—561no—
349C23—1122no—
350C23#651311yescharacteristic
351Q8—21no—
352Q8—42no—
353Q8—42no—
354Q8—81no—
355Q8—81no—
356Q8C2×#114112yes—
357D4—22no—
358D4—22no—
359D4—21no—
360D4—282no—
361D4—282no—
362D4—561no—
363D4—561no—
364D4C2×#114112yes—
365C2×C4—21no—
366C2×C4—22no—
367C2×C4—22no—
368C2×C4—42no—
369C2×C4—81no—
370C2×C4—81no—
371C2×C4C2×#114111yescharacteristic
372C2×C4C2×#114111yescharacteristic
373C2×C4#650211yescharacteristic
374C2×C4#650211yescharacteristic
375C2×C4#651311yescharacteristic
376C2×C4C22×C22⋊D711yescharacteristic
377C8—22no—
378C8—42no—
379C8—282no—
380C8#650211yescharacteristic
381C8#650211yescharacteristic
382C8C2×D4×D711yescharacteristic
383C8C2×#115911yescharacteristic
384C7C2×#222311yescharacteristic, complemented, Sylow
385C22—22no—
386C22—22no—
387C22—42no—
388C22—562no—
389C22—1122no—
390C22—1122no—
391C22—1122no—
392C22#8981011yescharacteristic
393C22#8981011yescharacteristic
394C22C2×#651311yescharacteristic
395C4—22no—
396C4—82no—
397C4#8981011yescharacteristic
398C4#8981011yescharacteristic
399C4C22×#114111yescharacteristic
400C4C2×#650211yescharacteristic
401C2—21no—
402C2—21no—
403C2—42no—
404C2—1122no—
405C2#1172885112yesdirect product factor
406C2C2×#8981011yescharacteristic
subgroupG/H
centerC22#89810
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
FittingC2×C7×#473C2
socleC22×C7#466
prime radicalC2×#89914C2
hypercenterC2×#473D7
hypocenterC7C2×#2223
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#131 ◁ C2×#473

Complements

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

NH ≅ G/Ntype
1. C2×C7×#473C2semidirect
2. C22×#6402C2semidirect
4. C2×#89912C2semidirect
5. C2×#89360C2semidirect
6. #11728851C2direct (N × H)
7. C2×#89290C2semidirect
8. C2×#89288C2semidirect
15. C7×#473C22semidirect
21. C2×#6402C22semidirect
25. #89912C22semidirect
32. #89360C22semidirect
33. #89290C22semidirect
34. #89288C22semidirect
88. C2×#473D7semidirect
167. #473C2×D7semidirect
342. C2×C7#2223semidirect
384. C7C2×#2223semidirect
405. C2#11728851direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#4732
C22×#64022
C2×#899142
C2×#899122
C2×#893602
#117288512
C2×#892902
C2×#892882
C2×#22237