Finite Group Databasebeta

← C2×#11729274 #13508180

Subgroups — C2×#11729274 #13508180

Order 1792.

686 conjugacy classes of subgroups.

The subgroup lattice has 414 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#510C211yesmaximal, characteristic
2C2×#89968C211yesmaximal, characteristic
3C2×#89967C211yesmaximal, characteristic
4C2×#89953C211yesmaximal, characteristic, complemented
5C2×#89360C211yesmaximal, characteristic, complemented
6#11729274C218yesmaximal, direct product factor
7C2×#89312C211yesmaximal, characteristic, complemented
8C2×#89306C211yesmaximal, characteristic, complemented
9C2×C7×#135C2211yescharacteristic
10C22×C7×Q16C2211yescharacteristic
11C2×#6511—21no—
12C2×#6507—21no—
13C22×#1144—21no—
14C22×Dic28C2211yescharacteristic
15C7×#510—22no—
16C7×#510C2214yes—
17C2×C7×Q32—21no—
18C2×C7×SD32—21no—
19C2×C7×M32C2211yescharacteristic
20C2×#6411—21no—
21#89968—22no—
22#89968C2214yes—
23#89967—22no—
24#89967C2214yes—
25#89953—22no—
26#89953C2214yescomplemented
27C2×#6403—21no—
28C2×#6401—21no—
29C2×#6393C2211yescharacteristic
30C2×C8×Dic7C2211yescharacteristic
31C2×#6386C2211yescharacteristic
32#89360C2214yescomplemented
33#89312C2214yescomplemented
34#89306C2214yescomplemented
35C2×#2214—71nomaximal, Sylow
36C2×C7×#41—21no—
37C22×C7×Q8—21no—
38C22×Dic14—21no—
39C7×#135—21no—
40C7×#135—21no—
41C7×#135D412yes—
42C7×#135C2312yes—
43C2×C7×Q16—21no—
44C2×C7×Q16—21no—
45C2×C7×Q16—21no—
46C2×C7×Q16—21no—
47C2×C7×Q16—21no—
48C2×C7×Q16D411yescharacteristic
49C2×C7×Q16C2312yes—
50C2×C7×SD16—21no—
51C2×C7×D8D411yescharacteristic
52C22×C7×C8C2311yescharacteristic
53#6511—24no—
54#6511—42no—
55#6507—24no—
56#6507—42no—
57C2×#1144—24no—
58C2×#1144—41no—
59C2×#1144—41no—
60C2×#1144—42no—
61C2×Dic7⋊C4—42no—
62C2×C4×Dic7D411yescharacteristic
63C2×#1136—21no—
64C2×#1136—41no—
65C22×C7⋊C8—41no—
66C22×C7⋊C8D411yescharacteristic
67C2×Dic28—21no—
68C2×Dic28—21no—
69C2×Dic28—21no—
70C2×Dic28—21no—
71C2×Dic28—21no—
72C2×Dic28C2312yes—
73C7×Q32—24no—
74C7×SD32—24no—
75C7×M32—21no—
76C7×M32C2312yes—
77C2×C7×C16—21no—
78#6411—24no—
79#6403—24no—
80#6401—24no—
81#6393—21no—
82#6393C2312yes—
83C8×Dic7D412yes—
84C8×Dic7C2312yes—
85#6386—21no—
86#6386C2312yes—
87C2×C7⋊C16—21no—
88C2×#510—71no—
89C2×#510D711yescharacteristic
90C2×#494—71no—
91C2×#443—71no—
92#2214—78no—
93C2×#364—71no—
94C2×#362—71no—
95C2×#362—71no—
96C7×#41—22no—
97C7×#41—22no—
98C7×#41—41no—
99C7×#41—41no—
100C2×C7×Q8—21no—
101C2×C7×Q8—22no—
102C2×C7×Q8—41no—
103C2×C7×Q8—41no—
104C2×C7×Q8—41no—
105C2×C7×Q8—41no—
106C2×C7×Q8—41no—
107C2×C7×D4—21no—
108C2×C7×D4—41no—
109C22×C4×C7—41no—
110C22×C4×C7—41no—
111C22×C4×C7C2×D411yescharacteristic
112C22×Dic7—21no—
113C22×Dic7—41no—
114C2×Dic14—22no—
115C2×Dic14—41no—
116C2×Dic14—41no—
117C2×Dic14—41no—
118C2×Dic14—41no—
119C2×Dic14—41no—
120C7×Q16—21no—
121C7×Q16—22no—
122C7×Q16—22no—
123C7×Q16—41no—
124C7×Q16—41no—
125C7×Q16C2×D412yes—
126C7×SD16—22no—
127C7×SD16—22no—
128C7×D8—21no—
129C7×D8C2×D412yes—
130C2×C7×C8C2×D411yescharacteristic
131C2×C7×C8C2×D411yescharacteristic
132C2×C7×C8C2×D411yescharacteristic
133C2×C7×C8#4111yescharacteristic
134C2×C7×C8#4111yescharacteristic
135C2×C7×C8C2411yescharacteristic
136#1144—44no—
137#1144—44no—
138Dic7⋊C4—48no—
139C4×Dic7—21no—
140C4×Dic7C2×D412yes—
141#1136—22no—
142#1136—41no—
143#1136—44no—
144C2×C7⋊C8—41no—
145C2×C7⋊C8—44no—
146C2×C7⋊C8D811yescharacteristic
147C2×C7⋊C8D811yescharacteristic
148C2×C7⋊C8D812yes—
149C2×C7⋊C8C2×D412yes—
150Dic28—22no—
151Dic28—22no—
152Dic28—41no—
153Dic28—41no—
154C7×C16—22no—
155C7⋊C16—22no—
156C2×#135C2×D711yescharacteristic
157C22×Q16—71no—
158C22×Q16—141no—
159C22×Q16C2×D711yescharacteristic
160C2×#131—141no—
161C2×C4⋊Q8—141no—
162#510—22no—
163#510—74no—
164#510—142no—
165#510C2×D714yes—
166C2×Q32—21no—
167C2×Q32—141no—
168C2×SD32—21no—
169C2×SD32—141no—
170C2×M32—71no—
171C2×M32C2×D711yescharacteristic
172#494—74no—
173#494—142no—
174#443—74no—
175#443—142no—
176C2×#108—71no—
177C2×#104—141no—
178C2×C4×C8—71no—
179#364—74no—
180#362—74no—
181#362—74no—
182C23×C7—41no—
183C23×C7AGL(1,ℤ/8)11yescharacteristic
184C7×Q8—22no—
185C7×Q8—41no—
186C7×Q8—42no—
187C7×Q8—42no—
188C7×Q8—81no—
189C7×Q8—81no—
190C7×D4—22no—
191C7×D4—42no—
192C7×D4—42no—
193C7×D4—41no—
194C2×C4×C7—41no—
195C2×C4×C7—42no—
196C2×C4×C7—42no—
197C2×C4×C7—42no—
198C2×C4×C7—81no—
199C2×C4×C7—81no—
200C2×C4×C7#12111yescharacteristic
201C2×C4×C7#12111yescharacteristic
202C2×C4×C7C2×D811yescharacteristic
203C2×C4×C7C2×D811yescharacteristic
204C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
205C2×C4×C7C22×D411yescharacteristic
206C2×Dic7—24no—
207C2×Dic7—22no—
208C2×Dic7—42no—
209C2×Dic7—81no—
210C2×Dic7—81no—
211Dic14—42no—
212Dic14—42no—
213Dic14—81no—
214Dic14—81no—
215C7×C8#12111yescharacteristic
216C7×C8#12111yescharacteristic
217C7×C8C22×D411yescharacteristic
218C7×C8C2×#4111yescharacteristic
219C7⋊C8—42no—
220C7⋊C8C2×D812yes—
221C7⋊C8C2×D812yes—
222C2×#41—21no—
223C22×Q8—21no—
224C22×Q8—141no—
225#135—21no—
226#135—21no—
227#135C22⋊D712yes—
228#135C22×D712yes—
229C2×Q16—21no—
230C2×Q16—21no—
231C2×Q16—21no—
232C2×Q16—21no—
233C2×Q16—21no—
234C2×Q16—72no—
235C2×Q16—144no—
236C2×Q16—141no—
237C2×Q16—141no—
238C2×Q16—141no—
239C2×Q16—141no—
240C2×Q16—141no—
241C2×Q16—281no—
242C2×Q16—281no—
243C2×Q16—282no—
244C2×Q16C22⋊D711yescharacteristic
245C2×Q16C22×D712yes—
246C2×SD16—21no—
247C2×D8C22⋊D711yescharacteristic
248#131—144no—
249#131—282no—
250C2×M16—141no—
251C2×M16—281no—
252C22×C8—71no—
253C22×C8—281no—
254C22×C8C22×D711yescharacteristic
255C4⋊Q8—144no—
256C4⋊Q8—282no—
257C2×C4⋊C4—282no—
258C2×C42—71no—
259Q32—24no—
260Q32—144no—
261SD32—24no—
262SD32—144no—
263M32—21no—
264M32—72no—
265M32—141no—
266M32C22×D712yes—
267C2×C16—21no—
268C2×C16—141no—
269#108—72no—
270#108—141no—
271#104—144no—
272C4×C8—72no—
273C4×C8—72no—
274C22×C7—22no—
275C22×C7—42no—
276C22×C7—82no—
277C22×C7#45911yescharacteristic
278C22×C7#45911yescharacteristic
279C22×C7C2×AGL(1,ℤ/8)11yescharacteristic
280C4×C7—42no—
281C4×C7—82no—
282C4×C7#45911yescharacteristic
283C4×C7#45911yescharacteristic
284C4×C7C2×#12111yescharacteristic
285C4×C7C22×D811yescharacteristic
286Dic7—24no—
287Dic7—82no—
288#41—22no—
289#41—22no—
290#41—41no—
291#41—41no—
292C2×Q8—21no—
293C2×Q8—22no—
294C2×Q8—41no—
295C2×Q8—41no—
296C2×Q8—41no—
297C2×Q8—41no—
298C2×Q8—41no—
299C2×Q8—142no—
300C2×Q8—281no—
301C2×Q8—281no—
302C2×Q8—281no—
303C2×Q8—281no—
304C2×Q8—281no—
305C2×D4—21no—
306C2×D4—41no—
307C22×C4—41no—
308C22×C4—41no—
309C22×C4—141no—
310C22×C4—281no—
311C22×C4D4×D711yescharacteristic
312Q16—21no—
313Q16—22no—
314Q16—22no—
315Q16—41no—
316Q16—41no—
317Q16—142no—
318Q16—142no—
319Q16—284no—
320Q16—284no—
321Q16—281no—
322Q16—281no—
323Q16C2×C22⋊D712yes—
324SD16—22no—
325SD16—22no—
326D8—21no—
327D8C2×C22⋊D712yes—
328M16—142no—
329M16—284no—
330M16—281no—
331C2×C8—71no—
332C2×C8—71no—
333C2×C8—72no—
334C2×C8—72no—
335C2×C8—284no—
336C2×C8—281no—
337C2×C8D4×D711yescharacteristic
338C2×C8#115911yescharacteristic
339C2×C8#115911yescharacteristic
340C2×C8C2×C22⋊D711yescharacteristic
341C2×C8C2×C22⋊D711yescharacteristic
342C2×C8C23×D711yescharacteristic
343C4⋊C4—288no—
344C42—72no—
345C42—141no—
346C16—22no—
347C16—142no—
348C2×C7—21no—
349C2×C7—21no—
350C2×C7—82no—
351C2×C7#221412yescomplemented
352C2×C7C2×#45911yescharacteristic
353C23—41no—
354C23#647411yescharacteristic
355Q8—22no—
356Q8—41no—
357Q8—42no—
358Q8—42no—
359Q8—81no—
360Q8—81no—
361Q8—282no—
362Q8—282no—
363Q8—561no—
364Q8—561no—
365D4—22no—
366D4—42no—
367D4—42no—
368D4—41no—
369C2×C4—41no—
370C2×C4—42no—
371C2×C4—42no—
372C2×C4—42no—
373C2×C4—81no—
374C2×C4—81no—
375C2×C4—144no—
376C2×C4—142no—
377C2×C4—282no—
378C2×C4—561no—
379C2×C4—561no—
380C2×C4D7×D811yescharacteristic
381C2×C4D7×D811yescharacteristic
382C2×C4#647411yescharacteristic
383C2×C4#650211yescharacteristic
384C2×C4#650211yescharacteristic
385C2×C4C2×D4×D711yescharacteristic
386C8—72no—
387C8—72no—
388C8—282no—
389C8#650211yescharacteristic
390C8#650211yescharacteristic
391C8C2×#115911yescharacteristic
392C8C22×C22⋊D711yescharacteristic
393C7C2×#221411yescharacteristic, complemented, Sylow
394C22—22no—
395C22—42no—
396C22—82no—
397C22#8992211yescharacteristic
398C22#8992211yescharacteristic
399C22C2×#647411yescharacteristic
400C4—42no—
401C4—82no—
402C4—144no—
403C4—562no—
404C4#8992211yescharacteristic
405C4#8992211yescharacteristic
406C4C2×D7×D811yescharacteristic
407C4C2×#650211yescharacteristic
408C2—21no—
409C2—21no—
410C2—82no—
411C2#1172927412yesdirect product factor
412C2C2×#8992211yescharacteristic
subgroupG/H
centerC22#89922
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
FittingC2×C7×#510C2
socleC22×C7#459
prime radicalC2×C7×#135C22
hypercenterC2×#510D7
hypocenterC7C2×#2214
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ C2×#510

Complements

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

NH ≅ G/Ntype
4. C2×#89953C2semidirect
5. C2×#89360C2semidirect
6. #11729274C2direct (N × H)
7. C2×#89312C2semidirect
8. C2×#89306C2semidirect
26. #89953C22semidirect
32. #89360C22semidirect
33. #89312C22semidirect
34. #89306C22semidirect
351. C2×C7#2214semidirect
393. C7C2×#2214semidirect
411. C2#11729274direct (N × H)

Maximal subgroups

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

subgroupindex
C2×C7×#5102
C2×#899682
C2×#899672
C2×#899532
C2×#893602
#117292742
C2×#893122
C2×#893062
C2×#22147