Finite Group Databasebeta

← #13373113

Subgroups — #13373113

Order 1792.

842 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C4×#1132C211yesmaximal, characteristic
2#11731605C211yesmaximal, characteristic
3#11731602C211yesmaximal, characteristic, complemented
4C7×#1835C211yesmaximal, characteristic, complemented
5C8×C4⋊Dic7C211yesmaximal, characteristic, complemented
6#11727313C211yesmaximal, characteristic, complemented
7#11727310C211yesmaximal, characteristic
8#11726663C212yesmaximal
9#11726606C212yesmaximal
10#11726570C212yesmaximal, complemented
11#11726535C212yesmaximal, complemented
12C22×#1132C2211yescharacteristic
13C2×#6450C2212yes—
14C2×C4×D28C2211yescharacteristic
15C2×C4×Dic14C2211yescharacteristic
16C2×C4×C7×C8C2211yescharacteristic
17C7×#382C2211yescharacteristic
18C7×#382C2211yescharacteristic
19C2×#6395C2211yescharacteristic
20C2×#6391C2211yescharacteristic
21C2×#6390C2211yescharacteristic
22C2×C8×Dic7C2212yes—
23#89760C2212yes—
24C4⋊C4×Dic7C2212yes—
25#89745C2212yes—
26C2×#6384C2212yes—
27C2×#6383C2212yes—
28C4×C4⋊Dic7C2211yescharacteristic
29C2×C4⋊(C7⋊C8)C2211yescharacteristic
30C4×#1132C418yescomplemented
31#89442—22no—
32#89438—21no—
33#89436—21no—
34#89422—21no—
35#89421—21no—
36#89419—22no—
37C7×#340C2212yes—
38C7×#340C2212yes—
39#89345C2212yes—
40#89324C2212yes—
41#89324C2212yes—
42#22773—71nomaximal, Sylow
43C22×D28C2311yescharacteristic
44C22×C4×D7—21no—
45C22×Dic14C2311yescharacteristic
46C22×C7×C8C2311yescharacteristic
47C22×C7×C8C2311yescharacteristic
48C2×C7×C4⋊C4C2312yes—
49C2×C7×C4⋊C4C2312yes—
50C2×C42×C7C2311yescharacteristic
51C2×(C2×D7)⋊C4—21no—
52C2×(C2×D7)⋊C4—22no—
53C2×(C2×D7)⋊C4—22no—
54C2×C4⋊Dic7D412yes—
55C2×C4⋊Dic7C2311yescharacteristic
56C2×C4⋊Dic7C2311yescharacteristic
57C2×Dic7⋊C4—21no—
58C2×Dic7⋊C4—22no—
59C2×Dic7⋊C4—22no—
60C2×C4×Dic7—21no—
61C2×C4×Dic7—22no—
62C2×C4×Dic7—22no—
63C2×C4×Dic7C2312yes—
64C22×C7⋊C8C2312yes—
65C2×#1132—22no—
66C2×#1132—22no—
67C2×#1132C2×C414yes—
68#6450—24no—
69#6450—42no—
70C4×D28—22no—
71C4×D28C2×C414yes—
72C4×Dic14—22no—
73C4×Dic14C2×C414yes—
74C4×C7×C8C2×C414yes—
75C7×#95—21no—
76C7×#95—21no—
77C7×#95—21no—
78C7×#95—21no—
79C7⋊#95—22no—
80C7⋊#95—22no—
81C7⋊#95D411yescharacteristic
82C7⋊#95D411yescharacteristic
83#6395C2×C414yes—
84#6391C2×C414yes—
85#6390C2×C414yes—
86C8×Dic7—42no—
87#6384—24no—
88#6383—42no—
89C4⋊(C7⋊C8)—22no—
90C2×C4×SD16—71no—
91#2357—71no—
92#2351—71no—
93#1934—72no—
94#1919—72no—
95#1880—72no—
96#1835D711yescharacteristic, complemented
97#1805—71no—
98#1804—71no—
99C8×C4⋊C4—71no—
100C4×#103—72no—
101C23×D7—21no—
102C22×C4×C7—22no—
103C22×C4×C7—22no—
104C22×C4×C7C2×D411yescharacteristic
105C22×C4×C7C2×D411yescharacteristic
106C22×C4×C7C2411yescharacteristic
107C22×Dic7—21no—
108C22×Dic7—21no—
109C22×Dic7—21no—
110C22×Dic7C2×D412yes—
111C22×Dic7C2×D412yes—
112C2×D28—21no—
113C2×D28—21no—
114C2×D28—21no—
115C2×D28—21no—
116C2×D28—21no—
117C2×D28C22×C412yes—
118C2×C4×D7—24no—
119C2×C4×D7—82no—
120C2×Dic14—21no—
121C2×Dic14—21no—
122C2×Dic14—21no—
123C2×Dic14C22×C412yes—
124C2×Dic14#4112yes—
125C2×Dic14#4112yes—
126C2×C7×C8—21no—
127C2×C7×C8—21no—
128C2×C7×C8—21no—
129C2×C7×C8—21no—
130C2×C7×C8C22×C412yes—
131C2×C7×C8C22×C412yes—
132C7×C4⋊C4—22no—
133C7×C4⋊C4—22no—
134C7×C4⋊C4—22no—
135C7×C4⋊C4—22no—
136C42×C7C22×C412yes—
137C42×C7#4112yes—
138(C2×D7)⋊C4—24no—
139(C2×D7)⋊C4—44no—
140(C2×D7)⋊C4—82no—
141C4⋊Dic7—21no—
142C4⋊Dic7—21no—
143C4⋊Dic7—24no—
144C4⋊Dic7C22×C412yes—
145C4⋊Dic7C22×C412yes—
146Dic7⋊C4—24no—
147Dic7⋊C4—44no—
148Dic7⋊C4—44no—
149C4×Dic7—24no—
150C4×Dic7—22no—
151C4×Dic7—22no—
152C4×Dic7—44no—
153C4×Dic7—82no—
154C2×C7⋊C8—22no—
155C2×C7⋊C8—22no—
156C2×C7⋊C8—22no—
157#1132—28no—
158C22×SD16—71no—
159C2×C4×Q8—72no—
160C2×C4×Q8—71no—
161C2×C4×D4—71no—
162C4×SD16—78no—
163C2×#106—71no—
164C2×C4⋊C8—71no—
165C2×#103—71no—
166C2×#103—72no—
167C2×#102—71no—
168C2×#102—72no—
169C2×C4×C8—72no—
170C2×C4×C8C2×D711yescharacteristic
171#390—72no—
172#389—72no—
173#389—142no—
174#388—142no—
175#382—141no—
176#382C2×D711yescharacteristic
177#382C2×D711yescharacteristic
178#381—141no—
179C4×C4⋊C4—71no—
180C4×C4⋊C4—72no—
181C4×C4⋊C4—141no—
182C4×C22⋊C4—141no—
183#340—72no—
184#340—72no—
185#340C2×D712yes—
186#340C2×D712yes—
187#336—72no—
188C23×C7C22×D411yescharacteristic
189C22×D7—22no—
190C22×D7—41no—
191C22×D7—41no—
192C22×D7—42no—
193C22×D7—42no—
194C2×C4×C7—22no—
195C2×C4×C7—22no—
196C2×C4×C7—42no—
197C2×C4×C7—42no—
198C2×C4×C7—42no—
199C2×C4×C7—42no—
200C2×C4×C7—42no—
201C2×C4×C7—42no—
202C2×C4×C7C4×D412yes—
203C2×C4×C7C4×D412yes—
204C2×C4×C7C23×C411yescharacteristic
205C2×C4×C7C2×#4111yescharacteristic
206C2×C4×C7C2×#4111yescharacteristic
207C2×C4×C7C2×#4111yescharacteristic
208C2×C4×C7H(5,𝔽2)11yescharacteristic
209C2×C4×C72−1+411yescharacteristic
210C2×Dic7—22no—
211C2×Dic7—22no—
212C2×Dic7—22no—
213C2×Dic7—22no—
214C2×Dic7—22no—
215C2×Dic7—22no—
216C2×Dic7—22no—
217C2×Dic7—22no—
218C2×Dic7—22no—
219C2×Dic7—22no—
220C2×Dic7—42no—
221C2×Dic7—42no—
222C2×Dic7#13514yes—
223D28—22no—
224D28—22no—
225D28—41no—
226D28—41no—
227C4×D7—84no—
228Dic14—22no—
229Dic14—22no—
230Dic14—22no—
231Dic14#11714yes—
232C7×C8—22no—
233C7×C8—22no—
234C7⋊C8—42no—
235C22×Q8—71no—
236C22×D4—71no—
237C23×C4—141no—
238C2×SD16—74no—
239C2×SD16—142no—
240C2×SD16—142no—
241C22×C8—72no—
242C22×C8C22×D711yescharacteristic
243C22×C8C22×D711yescharacteristic
244C4×Q8—74no—
245C4×Q8—142no—
246C4×Q8—144no—
247C4×Q8—282no—
248C4×D4—74no—
249C4×D4—142no—
250C2×C4⋊C4—71no—
251C2×C4⋊C4—71no—
252C2×C4⋊C4—72no—
253C2×C4⋊C4—141no—
254C2×C4⋊C4—142no—
255C2×C4⋊C4—142no—
256C2×C4⋊C4C22×D712yes—
257C2×C4⋊C4C22×D712yes—
258C2×C22⋊C4—141no—
259C2×C22⋊C4—142no—
260C2×C22⋊C4—142no—
261C2×C42—72no—
262C2×C42—141no—
263C2×C42—142no—
264C2×C42—142no—
265C2×C42C22×D711yescharacteristic
266#106—74no—
267C4⋊C8—142no—
268#103—74no—
269#103—144no—
270#102—74no—
271#102—282no—
272C4×C8—282no—
273C4×C8C4×D714yes—
274#95—21no—
275#95—21no—
276#95—21no—
277#95—21no—
278#95—71no—
279#95—71no—
280#95—142no—
281#95—142no—
282C22×C7C2×C4×D411yescharacteristic
283C22×C7#54611yescharacteristic
284C22×C7#54811yescharacteristic
285C22×C7C2×#13512yes—
286C22×C7#57712yes—
287C2×D7—42no—
288C2×D7—42no—
289C2×D7—42no—
290C2×D7—82no—
291C2×D7—82no—
292C4×C7—22no—
293C4×C7—22no—
294C4×C7—82no—
295C4×C7—82no—
296C4×C7C2×#11711yescharacteristic
297C4×C7#52011yescharacteristic
298C4×C7#55211yescharacteristic
299C4×C7#55311yescharacteristic
300Dic7—24no—
301Dic7—24no—
302Dic7—42no—
303Dic7—42no—
304Dic7—42no—
305C24—141no—
306C2×Q8—72no—
307C2×Q8—72no—
308C2×Q8—72no—
309C2×Q8—141no—
310C2×Q8—141no—
311C2×Q8—141no—
312C2×D4—72no—
313C2×D4—141no—
314C2×D4—141no—
315C2×D4—141no—
316C2×D4—141no—
317C2×D4—141no—
318C22×C4—22no—
319C22×C4—22no—
320C22×C4—72no—
321C22×C4—72no—
322C22×C4—141no—
323C22×C4—141no—
324C22×C4—141no—
325C22×C4—144no—
326C22×C4—562no—
327C22×C4D4×D711yescharacteristic
328C22×C4D4×D711yescharacteristic
329C22×C4C23×D711yescharacteristic
330SD16—148no—
331C2×C8—21no—
332C2×C8—21no—
333C2×C8—21no—
334C2×C8—21no—
335C2×C8—142no—
336C2×C8—142no—
337C2×C8—142no—
338C2×C8C2×C4×D712yes—
339C2×C8C2×C4×D712yes—
340C4⋊C4—22no—
341C4⋊C4—22no—
342C4⋊C4—22no—
343C4⋊C4—22no—
344C4⋊C4—72no—
345C4⋊C4—72no—
346C4⋊C4—144no—
347C4⋊C4—144no—
348C4⋊C4—141no—
349C4⋊C4—141no—
350C4⋊C4—284no—
351C4⋊C4—284no—
352C22⋊C4—144no—
353C22⋊C4—284no—
354C22⋊C4—562no—
355C42—144no—
356C42—142no—
357C42—142no—
358C42—284no—
359C42—562no—
360C42C2×C4×D712yes—
361C42#116112yes—
362C2×C7#235211yescharacteristic
363C2×C7C4×#13511yescharacteristic
364C2×C7#294911yescharacteristic
365C2×C7#330112yes—
366C2×C7#336512yes—
367D7—82no—
368C23—142no—
369C23—281no—
370C23—282no—
371C23—282no—
372C23—281no—
373C23C2×D4×D711yescharacteristic
374Q8—74no—
375Q8—142no—
376Q8—142no—
377Q8—142no—
378D4—142no—
379D4—142no—
380D4—281no—
381D4—281no—
382C2×C4—22no—
383C2×C4—22no—
384C2×C4—42no—
385C2×C4—42no—
386C2×C4—42no—
387C2×C4—42no—
388C2×C4—42no—
389C2×C4—42no—
390C2×C4—74no—
391C2×C4—142no—
392C2×C4—142no—
393C2×C4—142no—
394C2×C4—142no—
395C2×C4—142no—
396C2×C4—142no—
397C2×C4—142no—
398C2×C4—142no—
399C2×C4—142no—
400C2×C4—142no—
401C2×C4—282no—
402C2×C4—282no—
403C2×C4—564no—
404C2×C4#645612yes—
405C2×C4#645612yes—
406C2×C4C22×C4×D711yescharacteristic
407C2×C4C2×#116111yescharacteristic
408C2×C4D7×#4111yescharacteristic
409C2×C4D7×#4111yescharacteristic
410C2×C4#655311yescharacteristic
411C2×C4#655411yescharacteristic
412C8—22no—
413C8—22no—
414C8—282no—
415C7#2277311yescharacteristic, complemented, Sylow
416C22—282no—
417C22—282no—
418C22—282no—
419C22—562no—
420C22—562no—
421C22C2×#645611yescharacteristic
422C22#9036711yescharacteristic
423C22#9036811yescharacteristic
424C22D7×#13512yes—
425C22#9046112yes—
426C4—22no—
427C4—22no—
428C4—82no—
429C4—82no—
430C4—144no—
431C4—144no—
432C4—282no—
433C4—282no—
434C4—282no—
435C4D7×#11711yescharacteristic
436C4#9021411yescharacteristic
437C4#9038511yescharacteristic
438C4#9038611yescharacteristic
439C2—562no—
440C2#1173167411yescharacteristic
441C2#1173661011yescharacteristic
442C2#1173661311yescharacteristic
443C2#1173908012yes—
444C2#1173910012yes—
subgroupG/H
centerC23C2×D4×D7
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC7×#1835C2
socleC23×C7C22×D4
prime radicalC22×D28C23
cosocleC22×C4×C7C24
hypercenter#1835D7
hypocenterC7#22773
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1835

Complements

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

NH ≅ G/Ntype
3. #11731602C2semidirect
4. C7×#1835C2semidirect
5. C8×C4⋊Dic7C2semidirect
6. #11727313C2semidirect
10. #11726570C2semidirect
11. #11726535C2semidirect
30. C4×#1132C4semidirect
96. #1835D7semidirect
415. C7#22773semidirect

Maximal subgroups

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

subgroupindex
C2×C4×#11322
#117316052
#117316022
C7×#18352
C8×C4⋊Dic72
#117273132
#117273102
#117266632
#117266062
#117265702
#117265352
#227737