Finite Group Databasebeta

← #13508393

Subgroups — #13508393

Order 1792.

636 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
1C7×#3067C211yesmaximal, characteristic
2C2×#89967C211yesmaximal, characteristic, complemented
3C2×#89915C211yesmaximal, characteristic, complemented
4#11741225C211yesmaximal, characteristic, complemented
5#11729835C211yesmaximal, characteristic, complemented
6#11728850C214yesmaximal, complemented
7#11728847C214yesmaximal, complemented
8C2×#89291C211yesmaximal, characteristic, complemented
9#11728437C211yesmaximal, characteristic, complemented
10C7×#575—21no—
11C2×C7×#135C2211yescharacteristic
12C2×C7×#131C2211yescharacteristic
13C2×#6514—21no—
14C22×Dic28C2211yescharacteristic
15C7×#473C2212yes—
16C7×#472C2212yes—
17C7×#471—22no—
18C2×C7×#108C2211yescharacteristic
19C7×#413—21no—
20#89967—22no—
21#89967—22no—
22C2×#6403C2212yescomplemented
23C2×#6401C2212yescomplemented
24#89915—22no—
25#89915C2214yes—
26#89912—22no—
27#89912C2214yes—
28C2×#6398—21no—
29C2×#6393C2211yescharacteristic
30C2×#6386—22no—
31C2×#6386C2211yescharacteristic
32C2×#6386C2211yescharacteristic
33#89360C2212yescomplemented
34#89358C2212yescomplemented
35#89291C2214yescomplemented
36#89288C2214yescomplemented
37#33619—71nomaximal, Sylow
38C2×C7×#41—21no—
39C2×C7×#41D411yescharacteristic
40C22×Dic14—21no—
41C7×#137—22no—
42C7×#137—41no—
43C7×AGL(1,ℤ/8)—22no—
44C7×AGL(1,ℤ/8)—41no—
45C7×#135—21no—
46C7×#135—21no—
47C7×#135—21no—
48C7×#135—21no—
49C7×#135—42no—
50C2×C7×Q16C2311yescharacteristic
51C2×C7×SD16—21no—
52C2×C7×D8C2311yescharacteristic
53C7×#131—21no—
54C7×#131—21no—
55C7×#131D412yes—
56C7×#131C2312yes—
57C2×C7×M16—21no—
58C2×C7×M16—21no—
59C2×C7×M16D411yescharacteristic
60C22×C7×C8—21no—
61C22×C7×C8C2311yescharacteristic
62#6514—24no—
63#6514—42no—
64C2×#1144—41no—
65C2×#1142—41no—
66C2×#1136—21no—
67C2×Dic28—21no—
68C2×Dic28—21no—
69C2×Dic28—21no—
70C2×Dic28—21no—
71C2×Dic28—21no—
72C2×Dic28C2312yes—
73C7×#108—21no—
74C7×#108C2312yes—
75C7×#101—22no—
76C7×#100—22no—
77#6403—24no—
78#6401—24no—
79#6398—24no—
80#6393—21no—
81#6393C2312yes—
82#6386—21no—
83#6386—24no—
84#6386D412yes—
85#6386D412yes—
86#6386C2312yes—
87C2×C7⋊C16—21no—
88C2×C7⋊C16—24no—
89C2×C7⋊C16C2312yes—
90C2×#510—71no—
91#3407—71no—
92#3067D711yescharacteristic
93C2×#473—71no—
94#2223—74no—
95#2222—74no—
96#2161—71no—
97#2148—71no—
98C2×#362—71no—
99C7×#41—21no—
100C7×#41—21no—
101C7×#41—41no—
102C7×#41—41no—
103C7×#41—41no—
104C7×#41—41no—
105C7×#41D812yes—
106C7×#41C2×D412yes—
107C2×C7×Q8—21no—
108C2×C7×Q8D811yescharacteristic
109C2×C7×D4—21no—
110C2×C7×D4—21no—
111C2×C7×D4—41no—
112C2×C7×D4D811yescharacteristic
113C22×C4×C7—21no—
114C22×C4×C7—41no—
115C22×C4×C7C2×D411yescharacteristic
116C22×Dic7—41no—
117C2×Dic14—22no—
118C2×Dic14—41no—
119C2×Dic14—41no—
120C2×Dic14—41no—
121C2×Dic14—41no—
122C2×Dic14—41no—
123C7×Q16—21no—
124C7×Q16—21no—
125C7×Q16—42no—
126C7×SD16—42no—
127C7×SD16—42no—
128C7×SD16—41no—
129C7×SD16—41no—
130C7×D8—21no—
131C7×D8—21no—
132C7×D8—42no—
133C7×M16—21no—
134C7×M16—22no—
135C7×M16—22no—
136C7×M16—22no—
137C7×M16—41no—
138C7×M16C2×D412yes—
139C2×C7×C8—21no—
140C2×C7×C8—22no—
141C2×C7×C8—22no—
142C2×C7×C8—41no—
143C2×C7×C8C2×D411yescharacteristic
144C2×C7×C8C2×D411yescharacteristic
145C2×C7×C8C2×D411yescharacteristic
146C2×C7×C8#4111yescharacteristic
147C2×C7×C8#4111yescharacteristic
148C2×C7×C8C2411yescharacteristic
149#1144—44no—
150#1142—44no—
151#1136—22no—
152#1136—41no—
153C2×C7⋊C8—41no—
154Dic28—22no—
155Dic28—22no—
156Dic28—41no—
157Dic28—41no—
158C7⋊C16—22no—
159C7⋊C16C2×D414yes—
160#575—21no—
161C2×#137—141no—
162C2×#135C2×D711yescharacteristic
163C22×Q16—71no—
164C2×#131C2×D711yescharacteristic
165#510—142no—
166#510—142no—
167C2×Q32—72no—
168C2×SD32—72no—
169#504—74no—
170#504—142no—
171C2×M32—71no—
172C2×M32—71no—
173C2×M32—142no—
174#473—74no—
175#473—142no—
176#473C2×D712yes—
177#472C2×D712yes—
178#471—22no—
179C2×#108—71no—
180C2×#108C2×D711yescharacteristic
181#413—21no—
182C2×#101—141no—
183#368—74no—
184#362—72no—
185#362—74no—
186#361—72no—
187C23×C7—21no—
188C23×C7—41no—
189C23×C7#13511yescharacteristic
190C7×Q8—21no—
191C7×Q8—41no—
192C7×Q8—41no—
193C7×Q8C2×D812yes—
194C7×D4—22no—
195C7×D4—22no—
196C7×D4—21no—
197C7×D4—41no—
198C7×D4—41no—
199C7×D4—81no—
200C7×D4—81no—
201C7×D4C2×D812yes—
202C2×C4×C7—21no—
203C2×C4×C7—22no—
204C2×C4×C7—22no—
205C2×C4×C7—41no—
206C2×C4×C7—81no—
207C2×C4×C7—81no—
208C2×C4×C7#12111yescharacteristic
209C2×C4×C7#12111yescharacteristic
210C2×C4×C7C2×D811yescharacteristic
211C2×C4×C7C2×D811yescharacteristic
212C2×C4×C7#13511yescharacteristic
213C2×C4×C7C22×D411yescharacteristic
214C2×Dic7—42no—
215C2×Dic7—81no—
216C2×Dic7—81no—
217Dic14—42no—
218Dic14—42no—
219Dic14—81no—
220Dic14—81no—
221C7×C8—22no—
222C7×C8—42no—
223C7×C8#12111yescharacteristic
224C7×C8#12111yescharacteristic
225C7×C8C22×D411yescharacteristic
226C7×C8C2×#4111yescharacteristic
227C7⋊C8—42no—
228C2×#41—21no—
229C2×#41C22⋊D711yescharacteristic
230C22×Q8—141no—
231#137—22no—
232#137—41no—
233#137—144no—
234#137—282no—
235AGL(1,ℤ/8)—22no—
236AGL(1,ℤ/8)—41no—
237#135—21no—
238#135—21no—
239#135—21no—
240#135—21no—
241#135—42no—
242C2×Q16—72no—
243C2×Q16—141no—
244C2×Q16—141no—
245C2×Q16—141no—
246C2×Q16—141no—
247C2×Q16—141no—
248C2×Q16—281no—
249C2×Q16C22×D711yescharacteristic
250C2×SD16—21no—
251C2×SD16—281no—
252C2×D8C22×D711yescharacteristic
253#131—21no—
254#131—21no—
255#131C22⋊D712yes—
256#131C22×D712yes—
257C2×M16—21no—
258C2×M16—21no—
259C2×M16—141no—
260C2×M16C22⋊D711yescharacteristic
261C22×C8—21no—
262C22×C8C22×D711yescharacteristic
263Q32—144no—
264SD32—144no—
265M32—72no—
266M32—72no—
267M32—72no—
268M32—144no—
269M32—141no—
270C2×C16—72no—
271C2×C16—144no—
272C2×C16—141no—
273#108—21no—
274#108—72no—
275#108—141no—
276#108C22×D712yes—
277#101—22no—
278#101—144no—
279#100—22no—
280C22×C7—22no—
281C22×C7—41no—
282C22×C7—81no—
283C22×C7—81no—
284C22×C7—161no—
285C22×C7#46611yescharacteristic
286C22×C7#46611yescharacteristic
287C22×C7C2×#13511yescharacteristic
288C4×C7—22no—
289C4×C7—81no—
290C4×C7#46611yescharacteristic
291C4×C7#46611yescharacteristic
292C4×C7C2×#12111yescharacteristic
293C4×C7C22×D811yescharacteristic
294Dic7—82no—
295#41—21no—
296#41—21no—
297#41—41no—
298#41—41no—
299#41—41no—
300#41—41no—
301#41#114112yes—
302#41C2×C22⋊D712yes—
303C2×Q8—21no—
304C2×Q8—142no—
305C2×Q8—281no—
306C2×Q8—281no—
307C2×Q8—281no—
308C2×Q8—281no—
309C2×Q8—281no—
310C2×Q8#114111yescharacteristic
311C2×D4—21no—
312C2×D4—21no—
313C2×D4—41no—
314C2×D4#114111yescharacteristic
315C22×C4—21no—
316C22×C4—41no—
317C22×C4—281no—
318C22×C4C2×C22⋊D711yescharacteristic
319Q16—21no—
320Q16—21no—
321Q16—42no—
322Q16—142no—
323Q16—142no—
324Q16—284no—
325Q16—281no—
326Q16—281no—
327SD16—42no—
328SD16—42no—
329SD16—41no—
330SD16—41no—
331SD16—284no—
332D8—21no—
333D8—21no—
334D8—42no—
335M16—21no—
336M16—22no—
337M16—22no—
338M16—22no—
339M16—41no—
340M16—142no—
341M16—281no—
342M16C2×C22⋊D712yes—
343C2×C8—21no—
344C2×C8—22no—
345C2×C8—22no—
346C2×C8—41no—
347C2×C8—281no—
348C2×C8D4×D711yescharacteristic
349C2×C8D4×D711yescharacteristic
350C2×C8#115911yescharacteristic
351C2×C8#115911yescharacteristic
352C2×C8C2×C22⋊D711yescharacteristic
353C2×C8C23×D711yescharacteristic
354C16—74no—
355C16—142no—
356C2×C7—21no—
357C2×C7—21no—
358C2×C7—21no—
359C2×C7—42no—
360C2×C7—161no—
361C2×C7C2×#46611yescharacteristic
362C23—21no—
363C23—41no—
364C23#651311yescharacteristic
365Q8—21no—
366Q8—41no—
367Q8—41no—
368Q8—282no—
369Q8—282no—
370Q8—561no—
371Q8—561no—
372Q8C2×#114112yes—
373D4—22no—
374D4—22no—
375D4—21no—
376D4—41no—
377D4—41no—
378D4—81no—
379D4—81no—
380D4C2×#114112yes—
381C2×C4—21no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—81no—
386C2×C4—81no—
387C2×C4—282no—
388C2×C4—561no—
389C2×C4—561no—
390C2×C4C2×#114111yescharacteristic
391C2×C4C2×#114111yescharacteristic
392C2×C4#650211yescharacteristic
393C2×C4#650211yescharacteristic
394C2×C4#651311yescharacteristic
395C2×C4C22×C22⋊D711yescharacteristic
396C8—22no—
397C8—42no—
398C8—282no—
399C8#650211yescharacteristic
400C8#650211yescharacteristic
401C8C2×D4×D711yescharacteristic
402C8C2×#115911yescharacteristic
403C7#3361911yescharacteristic, complemented, Sylow
404C22—22no—
405C22—41no—
406C22—81no—
407C22—81no—
408C22—161no—
409C22#8981011yescharacteristic
410C22#8981011yescharacteristic
411C22C2×#651311yescharacteristic
412C4—22no—
413C4—81no—
414C4—562no—
415C4#8981011yescharacteristic
416C4#8981011yescharacteristic
417C4C22×#114111yescharacteristic
418C4C2×#650211yescharacteristic
419C2—21no—
420C2—21no—
421C2—21no—
422C2—42no—
423C2—161no—
424C2C2×#8981011yescharacteristic
subgroupG/H
centerC2C2×#89810
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
Fitting, prime radicalC7×#3067C2
socleC2×C7C2×#466
hypercenter#3067D7
hypocenterC7#33619
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2 ◁ C22×C4 ◁ C2×#131 ◁ #3067

Complements

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

NH ≅ G/Ntype
2. C2×#89967C2semidirect
3. C2×#89915C2semidirect
4. #11741225C2semidirect
5. #11729835C2semidirect
6. #11728850C2semidirect
7. #11728847C2semidirect
8. C2×#89291C2semidirect
9. #11728437C2semidirect
22. C2×#6403C22semidirect
23. C2×#6401C22semidirect
33. #89360C22semidirect
34. #89358C22semidirect
35. #89291C22semidirect
36. #89288C22semidirect
403. C7#33619semidirect

Maximal subgroups

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

subgroupindex
C7×#30672
C2×#899672
C2×#899152
#117412252
#117298352
#117288502
#117288472
C2×#892912
#117284372
#336197