Finite Group Databasebeta

← #13249694

Subgroups — #13249694

Order 1792.

516 conjugacy classes of subgroups.

The subgroup lattice has 426 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739971C211yesmaximal, characteristic
2#11739960C211yesmaximal, characteristic, complemented
3C7×#1477C211yesmaximal, characteristic, complemented
4#11728528C211yesmaximal, characteristic, complemented
5C2×#89280C211yesmaximal, characteristic, complemented
6#11725551C211yesmaximal, characteristic, complemented
7#11725511C211yesmaximal, characteristic, complemented
8C4×C7×#41C2211yescharacteristic
9C2×#6512—21no—
10C2×#6437C2211yescharacteristic
11C7×#431C2211yescharacteristic
12C2×C7×C4⋊C8C2211yescharacteristic
13#90006—21no—
14#89799—21no—
15#89799C412yes—
16#89787—21no—
17#89787C412yes—
18#89773—21no—
19#89772—22no—
20#89769—21no—
21#89698—21no—
22#89697—21no—
23#89696C2211yescharacteristic
24#89696C2211yescharacteristic
25C2×C4⋊(C7⋊C8)C2211yescharacteristic
26C7×Q8⋊C8—21no—
27C7×D4⋊C8—21no—
28#89280—21no—
29#89280—21no—
30#89280—22no—
31#9174—71nomaximal, Sylow
32C2×C7×#41D411yescharacteristic
33C22×D28—21no—
34C22×D28—21no—
35C22×D28—21no—
36C22×C7×C8—21no—
37C4×C7×Q8C2×C411yescharacteristic
38C4×C7×Q8D411yescharacteristic
39C4×C7×D4—21no—
40C4×C7×D4—21no—
41C4×C7×D4C2×C411yescharacteristic
42C4×C7×D4D411yescharacteristic
43C7×#117—21no—
44C7×#117D411yescharacteristic
45C2×C42×C7—21no—
46C2×C42×C7C2311yescharacteristic
47#6512—42no—
48#6512—42no—
49C2×#1143—22no—
50C2×#1141—22no—
51C2×#1136—21no—
52C2×#1136—21no—
53C22×C7⋊C8—21no—
54#6437—21no—
55#6437—21no—
56#6437—21no—
57#6437C2×C412yes—
58C7×C4⋊C8—21no—
59C7×C4⋊C8—21no—
60C7×C4⋊C8—21no—
61C7×C22⋊C8—21no—
62C4×C7×C8—21no—
63C22⋊(C7⋊C8)—22no—
64C22⋊(C7⋊C8)—22no—
65#6383—22no—
66#6383—22no—
67C4⋊(C7⋊C8)—21no—
68C4⋊(C7⋊C8)—21no—
69C4⋊(C7⋊C8)—21no—
70C4⋊(C7⋊C8)C2×C412yes—
71C4⋊(C7⋊C8)D411yescharacteristic
72C4⋊(C7⋊C8)D411yescharacteristic
73C4⋊(C7⋊C8)D412yes—
74#6377—22no—
75C4×C7⋊C8—22no—
76#3038—71no—
77#2925—71no—
78#1545—71no—
79C2×#331—71no—
80#1477D711yescharacteristic, complemented
81#1301—71no—
82#1292—71no—
83C23×D7—41no—
84C7×#41—21no—
85C7×#41—21no—
86C7×#41—21no—
87C7×#41—21no—
88C2×C7×Q8C22⋊C411yescharacteristic
89C2×C7×D4—21no—
90C2×C7×D4C22⋊C411yescharacteristic
91C22×C4×C7—21no—
92C22×C4×C7—21no—
93C22×C4×C7—21no—
94C22×C4×C7C2×D411yescharacteristic
95C22×C4×C7#4111yescharacteristic
96C22×C4×C7#4111yescharacteristic
97C2×D28—22no—
98C2×D28—22no—
99C2×D28—22no—
100C2×D28—41no—
101C2×D28—41no—
102C2×D28—41no—
103C2×D28—41no—
104C2×D28—41no—
105C2×D28—41no—
106C2×D28—41no—
107C2×D28—82no—
108C2×D28—81no—
109C2×D28—81no—
110C2×C7×C8—41no—
111C2×C7×C8—41no—
112C2×C7×C8—81no—
113C7×C4⋊C4—21no—
114C7×C4⋊C4—21no—
115C7×C4⋊C4C22⋊C411yescharacteristic
116C7×C4⋊C4C22⋊C411yescharacteristic
117C7×C22⋊C4—21no—
118C7×C22⋊C4—21no—
119C7×C22⋊C4—21no—
120C42×C7—21no—
121C42×C7—21no—
122C42×C7—21no—
123C42×C7C22×C411yescharacteristic
124C42×C7C2×D411yescharacteristic
125C42×C7C2×D411yescharacteristic
126C42×C7C2×D411yescharacteristic
127#1143—44no—
128#1141—44no—
129#1136—41no—
130#1136—41no—
131#1136—41no—
132#1136—41no—
133C2×C7⋊C8—22no—
134C2×C7⋊C8—21no—
135C2×C7⋊C8—21no—
136C2×C7⋊C8—22no—
137C2×C7⋊C8—41no—
138C2×C7⋊C8—41no—
139C2×AGL(1,ℤ/8)—141no—
140C2×#127—71no—
141C4×#41C2×D711yescharacteristic
142#460—72no—
143#460—141no—
144#459—72no—
145#459—141no—
146#433—142no—
147#432—141no—
148#431—141no—
149#431C2×D711yescharacteristic
150#423—71no—
151#423—71no—
152C2×C4⋊C8—71no—
153C2×C4⋊C8C2×D711yescharacteristic
154#418—141no—
155#408—141no—
156#405—141no—
157#331—141no—
158#331—141no—
159#331—142no—
160Q8⋊C8—21no—
161D4⋊C8—21no—
162C23×C7—21no—
163C23×C7#12311yescharacteristic
164C22×D7—42no—
165C22×D7—82no—
166C22×D7—82no—
167C22×D7—81no—
168C22×D7—81no—
169C7×Q8—21no—
170C7×Q8—21no—
171C7×D4—21no—
172C7×D4—21no—
173C7×D4—41no—
174C7×D4—41no—
175C2×C4×C7—21no—
176C2×C4×C7—21no—
177C2×C4×C7—21no—
178C2×C4×C7—21no—
179C2×C4×C7—21no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—41no—
183C2×C4×C7—41no—
184C2×C4×C7—41no—
185C2×C4×C7—81no—
186C2×C4×C7C2×C22⋊C411yescharacteristic
187C2×C4×C7C4×D411yescharacteristic
188C2×C4×C7C4×D411yescharacteristic
189C2×C4×C7#12011yescharacteristic
190C2×C4×C7#12111yescharacteristic
191C2×C4×C7#12111yescharacteristic
192C2×C4×C7#13511yescharacteristic
193C2×C4×C7#13511yescharacteristic
194C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
195C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
196D28—42no—
197D28—42no—
198D28—82no—
199D28—82no—
200D28—82no—
201D28—81no—
202D28—81no—
203D28—81no—
204D28—81no—
205C7×C8—81no—
206C7×C8—81no—
207C7⋊C8—42no—
208C7⋊C8—41no—
209C7⋊C8—41no—
210C7⋊C8—42no—
211C2×#41D2811yescharacteristic
212C22×D4—141no—
213C22×D4—141no—
214C22×D4—141no—
215AGL(1,ℤ/8)—282no—
216AGL(1,ℤ/8)—282no—
217C2×SD16—142no—
218C2×D8—142no—
219C2×M16—141no—
220C2×M16—141no—
221C22×C8—21no—
222C22×C8—141no—
223#127—72no—
224#127—141no—
225#127—141no—
226#127—141no—
227C4×Q8C4×D711yescharacteristic
228C4×Q8C22⋊D711yescharacteristic
229C4×D4—21no—
230C4×D4—21no—
231C4×D4C4×D711yescharacteristic
232C4×D4C22⋊D711yescharacteristic
233#117—21no—
234#117D2811yescharacteristic
235C2×C42—21no—
236C2×C42C22×D711yescharacteristic
237C4⋊C8—21no—
238C4⋊C8—21no—
239C4⋊C8—21no—
240C4⋊C8—72no—
241C4⋊C8—71no—
242C4⋊C8—71no—
243C4⋊C8—72no—
244C4⋊C8—141no—
245C4⋊C8—141no—
246C4⋊C8—141no—
247#102—142no—
248#102—142no—
249C22⋊C8—21no—
250C22⋊C8—142no—
251C22⋊C8—142no—
252#97—142no—
253C4×C8—21no—
254C4×C8—142no—
255C22×C7—21no—
256C22×C7—21no—
257C22×C7—41no—
258C22×C7—41no—
259C22×C7—41no—
260C22×C7#38611yescharacteristic
261C22×C7#48211yescharacteristic
262C22×C7#48211yescharacteristic
263C2×D7—82no—
264C2×D7—82no—
265C2×D7—82no—
266C2×D7—162no—
267C2×D7—162no—
268C4×C7—21no—
269C4×C7—21no—
270C4×C7—21no—
271C4×C7—41no—
272C4×C7—41no—
273C4×C7—41no—
274C4×C7#41611yescharacteristic
275C4×C7#41811yescharacteristic
276C4×C7#44911yescharacteristic
277C4×C7#44911yescharacteristic
278C4×C7#47111yescharacteristic
279C4×C7#47211yescharacteristic
280C24—281no—
281#41—21no—
282#41—21no—
283#41—21no—
284#41—21no—
285C2×Q8(C2×D7)⋊C411yescharacteristic
286C2×D4—21no—
287C2×D4—142no—
288C2×D4—142no—
289C2×D4—142no—
290C2×D4—281no—
291C2×D4—281no—
292C2×D4—281no—
293C2×D4—281no—
294C2×D4—281no—
295C2×D4—281no—
296C2×D4—281no—
297C2×D4—561no—
298C2×D4—561no—
299C2×D4—562no—
300C2×D4(C2×D7)⋊C411yescharacteristic
301C22×C4—21no—
302C22×C4—21no—
303C22×C4—21no—
304C22×C4C2×D2811yescharacteristic
305C22×C4#115911yescharacteristic
306C22×C4#115911yescharacteristic
307SD16—284no—
308D8—284no—
309M16—281no—
310M16—281no—
311M16—281no—
312M16—281no—
313C2×C8—41no—
314C2×C8—41no—
315C2×C8—81no—
316C2×C8—141no—
317C2×C8—142no—
318C2×C8—141no—
319C2×C8—142no—
320C2×C8—281no—
321C2×C8—281no—
322C4⋊C4—21no—
323C4⋊C4—21no—
324C4⋊C4(C2×D7)⋊C411yescharacteristic
325C4⋊C4(C2×D7)⋊C411yescharacteristic
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—21no—
329C42—21no—
330C42—21no—
331C42—21no—
332C42C2×C4×D711yescharacteristic
333C42D4×D711yescharacteristic
334C42D4×D711yescharacteristic
335C42C2×C22⋊D711yescharacteristic
336C2×C7—21no—
337C2×C7—21no—
338C2×C7—81no—
339C2×C7#187511yescharacteristic
340C2×C7#190311yescharacteristic
341C2×C7#190711yescharacteristic
342D7—162no—
343C23—21no—
344C23—282no—
345C23—562no—
346C23—562no—
347C23—561no—
348C23—561no—
349C23#644911yescharacteristic
350Q8—21no—
351Q8—21no—
352D4—21no—
353D4—21no—
354D4—41no—
355D4—41no—
356D4—282no—
357D4—282no—
358D4—562no—
359D4—562no—
360D4—562no—
361D4—561no—
362D4—561no—
363D4—561no—
364D4—561no—
365C2×C4—21no—
366C2×C4—21no—
367C2×C4—21no—
368C2×C4—21no—
369C2×C4—21no—
370C2×C4—41no—
371C2×C4—41no—
372C2×C4—41no—
373C2×C4—41no—
374C2×C4—41no—
375C2×C4—81no—
376C2×C4(C22⋊D7)⋊C411yescharacteristic
377C2×C4(C22⋊D7)⋊C411yescharacteristic
378C2×C4#644511yescharacteristic
379C2×C4#646711yescharacteristic
380C2×C4#646711yescharacteristic
381C2×C4#647111yescharacteristic
382C2×C4#647111yescharacteristic
383C2×C4C2×(C2×D7)⋊C411yescharacteristic
384C2×C4#650211yescharacteristic
385C2×C4#650211yescharacteristic
386C8—81no—
387C8—81no—
388C8—282no—
389C8—281no—
390C8—281no—
391C8—282no—
392C7#917411yescharacteristic, complemented, Sylow
393C22—21no—
394C22—21no—
395C22—41no—
396C22—41no—
397C22—41no—
398C22—562no—
399C22—562no—
400C22—562no—
401C22—1122no—
402C22—1122no—
403C22#8950911yescharacteristic
404C22#8950911yescharacteristic
405C22#8973011yescharacteristic
406C4—21no—
407C4—21no—
408C4—21no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4#8950611yescharacteristic
413C4#8950611yescharacteristic
414C4#8988511yescharacteristic
415C4#8990311yescharacteristic
416C4#8997111yescharacteristic
417C4#8997511yescharacteristic
418C2—21no—
419C2—21no—
420C2—81no—
421C2—1122no—
422C2#1172768711yescharacteristic
423C2#1172805711yescharacteristic
424C2#1172815411yescharacteristic
subgroupG/H
centerC22#89730
commutatorC42×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1477C2
socleC22×C7#386
prime radical#11739971C2
cosocleC2×C42×C7C23
hypercenter#1477D7
hypocenterC7#9174
LCSG ▷ C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×C42 ◁ #1477

Complements

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

NH ≅ G/Ntype
2. #11739960C2semidirect
3. C7×#1477C2semidirect
4. #11728528C2semidirect
5. C2×#89280C2semidirect
6. #11725551C2semidirect
7. #11725511C2semidirect
80. #1477D7semidirect
392. C7#9174semidirect

Maximal subgroups

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

subgroupindex
#117399712
#117399602
C7×#14772
#117285282
C2×#892802
#117255512
#117255112
#91747