Finite Group Databasebeta

← #13507651

Subgroups — #13507651

Order 1792.

1086 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
1C2×C7×#578C211yesmaximal, characteristic
2C2×#89967C211yesmaximal, characteristic, complemented
3C22×#6403C211yesmaximal, characteristic, complemented
4#11741252C211yesmaximal, characteristic, complemented
5#11729837C214yesmaximal, complemented
6#11729831C211yesmaximal, characteristic, complemented
7#11729818C214yesmaximal, complemented
8#11729815C211yesmaximal, characteristic, complemented
9#11729758C211yesmaximal, characteristic, complemented
10C2×C7×2−1+4—21no—
11C7×#578—24no—
12C7×#578—24no—
13C2×C7×#137—21no—
14C2×C7×#137—22no—
15C2×C7×#135C2212yes—
16C2×C7×#135C2211yescharacteristic
17C22×C7×Q16C2212yes—
18C22×C7×Q16C2211yescharacteristic
19C2×C7×#131C2211yescharacteristic
20#90516—21no—
21C22×Dic28C2211yescharacteristic
22#89967—24no—
23C2×#6403—24no—
24C2×#6403C2214yes—
25C2×#6403C2212yes—
26C2×#6401C2212yes—
27#89909—22no—
28#89892—21no—
29#89888C2214yes—
30#89878C2211yescharacteristic
31C2×#6386C2211yescharacteristic
32C22×C7⋊C16C2211yescharacteristic
33#89359C2214yes—
34#89359C2212yes—
35#89357C2212yes—
36C22⋊(C7⋊C16)C2214yes—
37#33525—71nomaximal, Sylow
38C7×2−1+4—28no—
39C7×2−1+4—44no—
40C2×C7×#41—22no—
41C2×C7×#41—21no—
42C2×C7×#41—42no—
43C2×C7×#41—41no—
44C2×C7×#41D411yescharacteristic
45C22×C7×Q8—22no—
46C22×C7×Q8—21no—
47C22×C7×Q8—41no—
48C22×Dic14—21no—
49C7×#137—28no—
50C7×#137—216no—
51C7×#135—24no—
52C7×#135—24no—
53C7×#135—22no—
54C7×#135D414yes—
55C2×C7×Q16—24no—
56C2×C7×Q16—24no—
57C2×C7×Q16—22no—
58C2×C7×Q16—21no—
59C2×C7×Q16—22no—
60C2×C7×Q16D414yes—
61C2×C7×Q16C2311yescharacteristic
62C2×C7×Q16C2312yes—
63C2×C7×SD16—22no—
64C2×C7×SD16—21no—
65C2×C7×D8C2311yescharacteristic
66C7×#131—22no—
67C7×#131—22no—
68C2×C7×M16D411yescharacteristic
69C2×C7×M16D412yes—
70C22×C7×C8C2312yes—
71C22×C7×C8C2311yescharacteristic
72#6507—41no—
73#6499—41no—
74#6498—42no—
75#6489—21no—
76#6487—24no—
77C2×Dic28—22no—
78C2×Dic28—22no—
79C2×Dic28—21no—
80C2×Dic28C2312yes—
81#6403—28no—
82#6403—24no—
83#6401—24no—
84#6392C2312yes—
85#6391—21no—
86#6390—24no—
87#6386—22no—
88C2×C7⋊C16—22no—
89C2×C7⋊C16C2312yes—
90C2×C7⋊C16C2312yes—
91C2×#578D711yescharacteristic
92C2×#510—71no—
93C22×Q32—71no—
94#3059—71no—
95#2189—74no—
96#2188—74no—
97#2141—71no—
98#2138—71no—
99#2113—71no—
100C7×#41—22no—
101C7×#41—22no—
102C7×#41—28no—
103C7×#41—24no—
104C7×#41—44no—
105C7×#41—416no—
106C7×#41—48no—
107C7×#41—42no—
108C2×C7×Q8—28no—
109C2×C7×Q8—24no—
110C2×C7×Q8—22no—
111C2×C7×Q8—21no—
112C2×C7×Q8—44no—
113C2×C7×Q8—48no—
114C2×C7×Q8—42no—
115C2×C7×Q8—41no—
116C2×C7×Q8—42no—
117C2×C7×Q8D811yescharacteristic
118C2×C7×D4—21no—
119C2×C7×D4—42no—
120C2×C7×D4—41no—
121C2×C7×D4D812yes—
122C2×C7×D4D811yescharacteristic
123C22×C4×C7—42no—
124C22×C4×C7—41no—
125C22×C4×C7—42no—
126C22×C4×C7—41no—
127C22×C4×C7C2×D412yes—
128C22×C4×C7C2×D411yescharacteristic
129C22×Dic7—41no—
130C2×Dic14—22no—
131C2×Dic14—42no—
132C2×Dic14—42no—
133C2×Dic14—41no—
134C7×Q16—28no—
135C7×Q16—24no—
136C7×Q16—21no—
137C7×Q16—22no—
138C7×Q16C2×D412yes—
139C7×Q16C2×D414yes—
140C7×SD16—28no—
141C7×SD16—24no—
142C7×D8—21no—
143C7×D8C2×D412yes—
144C7×M16—22no—
145C7×M16—24no—
146C2×C7×C8—22no—
147C2×C7×C8—22no—
148C2×C7×C8C2×D412yes—
149C2×C7×C8C2×D411yescharacteristic
150C2×C7×C8C2×D412yes—
151C2×C7×C8C2×D412yes—
152C2×C7×C8C2411yescharacteristic
153C22⋊Dic7—44no—
154C22⋊Dic7—41no—
155C4⋊Dic7—22no—
156Dic7⋊C4—44no—
157C4×Dic7—41no—
158Dic28—22no—
159Dic28—22no—
160Dic28—42no—
161C7⋊C16—22no—
162C7⋊C16—22no—
163C2×2−1+4—21no—
164#578—24no—
165#578—24no—
166C2×#137—21no—
167C2×#137—22no—
168C2×#135C2×D712yes—
169C2×#135C2×D711yescharacteristic
170C22×Q16—71no—
171C22×Q16C2×D712yes—
172C22×Q16C2×D711yescharacteristic
173C2×#131C2×D711yescharacteristic
174#536—141no—
175#510—144no—
176C2×Q32—74no—
177C2×Q32—72no—
178C2×Q32—144no—
179C2×SD32—72no—
180C2×M32—71no—
181C22×C16—71no—
182#470—142no—
183#467—74no—
184#427—71no—
185#419—141no—
186#358—72no—
187#358—74no—
188#357—72no—
189C22⋊C16—74no—
190C23×C7—41no—
191C23×C7C2×D812yes—
192C23×C7C2×D811yescharacteristic
193C7×Q8—22no—
194C7×Q8—24no—
195C7×Q8—22no—
196C7×Q8—48no—
197C7×Q8—41no—
198C7×Q8—42no—
199C7×Q8—44no—
200C7×D4—24no—
201C7×D4—22no—
202C7×D4—22no—
203C7×D4—48no—
204C7×D4—44no—
205C7×D4—41no—
206C2×C4×C7—22no—
207C2×C4×C7—22no—
208C2×C4×C7—48no—
209C2×C4×C7—48no—
210C2×C4×C7—42no—
211C2×C4×C7—41no—
212C2×C4×C7—44no—
213C2×C4×C7—44no—
214C2×C4×C7#12012yes—
215C2×C4×C7C2×D812yes—
216C2×C4×C7C2×D811yescharacteristic
217C2×C4×C7AGL(1,ℤ/8)12yes—
218C2×C4×C7C22×D411yescharacteristic
219C2×Dic7—42no—
220C2×Dic7—42no—
221C2×Dic7—82no—
222Dic14—42no—
223Dic14—42no—
224Dic14—82no—
225C7×C8—22no—
226C7×C8#12012yes—
227C7×C8C22×D411yescharacteristic
228C7×C8C22×D411yescharacteristic
2292−1+4—28no—
2302−1+4—44no—
231C2×#41—22no—
232C2×#41—21no—
233C2×#41—42no—
234C2×#41—41no—
235C2×#41C22⋊D711yescharacteristic
236C22×Q8—22no—
237C22×Q8—21no—
238C22×Q8—41no—
239C22×Q8—141no—
240#137—28no—
241#137—216no—
242#135—24no—
243#135—24no—
244#135—22no—
245#135C22⋊D714yes—
246C2×Q16—24no—
247C2×Q16—24no—
248C2×Q16—22no—
249C2×Q16—21no—
250C2×Q16—22no—
251C2×Q16—72no—
252C2×Q16—142no—
253C2×Q16—142no—
254C2×Q16—141no—
255C2×Q16C22⋊D714yes—
256C2×Q16C22×D711yescharacteristic
257C2×Q16C22×D712yes—
258C2×SD16—22no—
259C2×SD16—21no—
260C2×D8C22×D711yescharacteristic
261#131—22no—
262#131—22no—
263C2×M16C22⋊D711yescharacteristic
264C2×M16C22⋊D712yes—
265C22×C8C22×D712yes—
266C22×C8C22×D711yescharacteristic
267C4⋊Q8—281no—
268#124—281no—
269#123—282no—
270C22⋊Q8—144no—
271#117—141no—
272Q32—148no—
273Q32—144no—
274SD32—144no—
275M32—142no—
276C2×C16—72no—
277C2×C16—72no—
278C2×C16—142no—
279#107—72no—
280#106—141no—
281#103—144no—
282C22×C7—22no—
283C22×C7—24no—
284C22×C7—22no—
285C22×C7—42no—
286C22×C7—82no—
287C22×C7#44712yes—
288C22×C7C22×D811yescharacteristic
289C4×C7—22no—
290C4×C7—44no—
291C4×C7—42no—
292C4×C7#44712yes—
293C4×C7C2×#12011yescharacteristic
294C4×C7C2×AGL(1,ℤ/8)11yescharacteristic
295Dic7—82no—
296Dic7—82no—
297#41—22no—
298#41—22no—
299#41—28no—
300#41—24no—
301#41—44no—
302#41—416no—
303#41—48no—
304#41—42no—
305C2×Q8—28no—
306C2×Q8—24no—
307C2×Q8—22no—
308C2×Q8—21no—
309C2×Q8—44no—
310C2×Q8—48no—
311C2×Q8—42no—
312C2×Q8—41no—
313C2×Q8—42no—
314C2×Q8—142no—
315C2×Q8—281no—
316C2×Q8—282no—
317C2×Q8—282no—
318C2×Q8#114111yescharacteristic
319C2×D4—21no—
320C2×D4—42no—
321C2×D4—41no—
322C2×D4#114112yes—
323C2×D4#114111yescharacteristic
324C22×C4—42no—
325C22×C4—41no—
326C22×C4—42no—
327C22×C4—41no—
328C22×C4—281no—
329C22×C4C2×C22⋊D712yes—
330C22×C4C2×C22⋊D711yescharacteristic
331Q16—28no—
332Q16—24no—
333Q16—21no—
334Q16—22no—
335Q16—142no—
336Q16—142no—
337Q16—282no—
338Q16C2×C22⋊D712yes—
339Q16C2×C22⋊D714yes—
340SD16—28no—
341SD16—24no—
342D8—21no—
343D8C2×C22⋊D712yes—
344M16—22no—
345M16—24no—
346C2×C8—22no—
347C2×C8—22no—
348C2×C8C2×C22⋊D712yes—
349C2×C8C2×C22⋊D711yescharacteristic
350C2×C8C2×C22⋊D712yes—
351C2×C8C2×C22⋊D712yes—
352C2×C8C23×D711yescharacteristic
353C4⋊C4—142no—
354C4⋊C4—284no—
355C22⋊C4—281no—
356C22⋊C4—284no—
357C42—281no—
358C16—142no—
359C16—142no—
360C2×C7—22no—
361C2×C7—42no—
362C2×C7—82no—
363C2×C7C2×#44711yescharacteristic
364C2×C7#341712yes—
365C23—41no—
366C23C2×#114112yes—
367C23C2×#114111yescharacteristic
368Q8—22no—
369Q8—24no—
370Q8—22no—
371Q8—48no—
372Q8—41no—
373Q8—42no—
374Q8—44no—
375Q8—282no—
376Q8—282no—
377Q8—562no—
378D4—24no—
379D4—22no—
380D4—22no—
381D4—48no—
382D4—44no—
383D4—41no—
384C2×C4—22no—
385C2×C4—22no—
386C2×C4—48no—
387C2×C4—48no—
388C2×C4—42no—
389C2×C4—41no—
390C2×C4—44no—
391C2×C4—44no—
392C2×C4—282no—
393C2×C4—282no—
394C2×C4—562no—
395C2×C4C2×#114112yes—
396C2×C4C2×#114111yescharacteristic
397C2×C4#649512yes—
398C2×C4#651612yes—
399C2×C4C22×C22⋊D711yescharacteristic
400C8—22no—
401C8#651612yes—
402C8C22×C22⋊D711yescharacteristic
403C8C22×C22⋊D711yescharacteristic
404C7#3352511yescharacteristic, complemented, Sylow
405C22—22no—
406C22—24no—
407C22—22no—
408C22—42no—
409C22—82no—
410C22#8998912yes—
411C22C22×#114111yescharacteristic
412C4—22no—
413C4—44no—
414C4—42no—
415C4—562no—
416C4—562no—
417C4#8998912yes—
418C4C2×#649511yescharacteristic
419C4C2×#651611yescharacteristic
420C2—22no—
421C2—42no—
422C2—82no—
423C2#1174146212yes—
424C2C2×#8998911yescharacteristic
subgroupG/H
centerC22C22×#1141
commutator, cosocleC2×C7×C8C24
FrattiniC2×C8C23×D7
Fitting, prime radicalC2×C7×#578C2
socleC22×C7C22×D8
hypercenterC2×#578D7
hypocenterC7#33525
LCSG ▷ C2×C7×C8 ▷ C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C2×#41 ◁ C2×#131 ◁ C2×#578

Complements

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

NH ≅ G/Ntype
2. C2×#89967C2semidirect
3. C22×#6403C2semidirect
4. #11741252C2semidirect
5. #11729837C2semidirect
6. #11729831C2semidirect
7. #11729818C2semidirect
8. #11729815C2semidirect
9. #11729758C2semidirect
404. C7#33525semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#5782
C2×#899672
C22×#64032
#117412522
#117298372
#117298312
#117298182
#117298152
#117297582
#335257