Finite Group Databasebeta

← #13257168

Subgroups — #13257168

Order 1792.

572 conjugacy classes of subgroups.

The subgroup lattice has 454 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739921C211yesmaximal, characteristic
2C7×#1803C211yesmaximal, characteristic, complemented
3C2×Q8⋊(C7⋊C8)C211yesmaximal, characteristic, complemented
4C2×#89280C211yesmaximal, characteristic
5#11727334C211yesmaximal, characteristic
6#11727322C211yesmaximal, characteristic, complemented
7#11725510C211yesmaximal, characteristic, complemented
8C2×C4×C7×Q8C2211yescharacteristic
9C2×#6510—21no—
10C2×#6437C2211yescharacteristic
11C2×C7×C4⋊C8C2211yescharacteristic
12C2×C7×#103—21no—
13C7×#384C2211yescharacteristic
14#89763—22no—
15#89760—21no—
16C2×#6383—22no—
17C2×#6382—21no—
18#89711—21no—
19#89711—21no—
20C4×C4⋊Dic7C2211yescharacteristic
21C2×C4⋊(C7⋊C8)C2211yescharacteristic
22C2×C4×C7⋊C8C2211yescharacteristic
23C7×#340—21no—
24Q8⋊(C7⋊C8)—22no—
25#89324—21no—
26#89280—22no—
27#10410—71nomaximal, Sylow
28C22×C7×Q8D411yescharacteristic
29C22×D28—21no—
30C22×D28—21no—
31C22×D28D412yes—
32C22×C7×C8—21no—
33C4×C7×Q8—21no—
34C4×C7×Q8—21no—
35C4×C7×Q8—21no—
36C4×C7×Q8—21no—
37C2×C7×C4⋊C4—21no—
38C2×C7×C4⋊C4—21no—
39C2×C7×C4⋊C4—21no—
40C2×C7×C4⋊C4—21no—
41C2×C7×C4⋊C4D411yescharacteristic
42C2×C42×C7—21no—
43C2×C42×C7C2311yescharacteristic
44#6510—42no—
45#6510—42no—
46C2×(C2×D7)⋊C4—22no—
47C2×(C2×D7)⋊C4—22no—
48C2×(C2×D7)⋊C4—22no—
49C2×(C2×D7)⋊C4—22no—
50C2×C4⋊Dic7D411yescharacteristic
51C2×C4⋊Dic7D411yescharacteristic
52C2×C4×Dic7—21no—
53C22×C7⋊C8—21no—
54C22×C7⋊C8—21no—
55#6437—21no—
56#6437—21no—
57#6437—21no—
58#6437—21no—
59C7×C4⋊C8—21no—
60C7×C4⋊C8—21no—
61C7×#103—42no—
62C7⋊#95—21no—
63#6383—44no—
64#6382—42no—
65C4⋊(C7⋊C8)—21no—
66C4⋊(C7⋊C8)D412yes—
67C4×C7⋊C8—21no—
68C4×C7⋊C8—21no—
69#2548—71no—
70#1957—71no—
71#1948—71no—
72#1803D711yescharacteristic, complemented
73C2×#331—71no—
74C2×Q8⋊C8—71no—
75#1279—71no—
76C23×D7—22no—
77C2×C7×Q8—21no—
78C2×C7×Q8—21no—
79C2×C7×Q8—21no—
80C2×C7×Q8—21no—
81C2×C7×Q8—21no—
82C2×C7×Q8SD1612yes—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7—41no—
86C22×C4×C7C2×D411yescharacteristic
87C22×C4×C7C2×D411yescharacteristic
88C22×C4×C7C2×D411yescharacteristic
89C22×Dic7—21no—
90C22×Dic7—21no—
91C2×D28—22no—
92C2×D28—22no—
93C2×D28—42no—
94C2×D28—42no—
95C2×D28—41no—
96C2×D28—41no—
97C2×D28—41no—
98C2×D28—41no—
99C2×D28—41no—
100C2×D28—41no—
101C2×D28—41no—
102C2×D28—41no—
103C2×D28—41no—
104C2×D28—41no—
105C2×D28—41no—
106C2×D28—41no—
107C2×C7×C8—41no—
108C2×C7×C8—41no—
109C2×C7×C8—41no—
110C7×C4⋊C4—21no—
111C7×C4⋊C4—21no—
112C7×C4⋊C4—42no—
113C7×C4⋊C4—41no—
114C7×C4⋊C4—41no—
115C7×C4⋊C4—41no—
116C7×C4⋊C4—41no—
117C7×C4⋊C4—41no—
118C7×C4⋊C4—41no—
119C42×C7—41no—
120C42×C7—41no—
121C42×C7C2×D411yescharacteristic
122C42×C7#4111yescharacteristic
123C42×C7#4111yescharacteristic
124C42×C7#4111yescharacteristic
125(C2×D7)⋊C4—44no—
126(C2×D7)⋊C4—44no—
127(C2×D7)⋊C4—82no—
128(C2×D7)⋊C4—82no—
129C4⋊Dic7—21no—
130C4⋊Dic7—21no—
131C4⋊Dic7—21no—
132C4⋊Dic7—21no—
133C4×Dic7—41no—
134C4×Dic7—41no—
135C2×C7⋊C8—22no—
136C2×C7⋊C8—22no—
137C2×C7⋊C8—41no—
138C2×C7⋊C8—41no—
139C2×C7⋊C8—41no—
140C2×C7⋊C8—41no—
141C2×#127—71no—
142C2×#124—141no—
143C2×C4×Q8C2×D711yescharacteristic
144C2×#106—141no—
145C2×C4⋊C8—71no—
146C2×C4⋊C8C2×D711yescharacteristic
147C2×#103—21no—
148C2×#102—142no—
149C2×C4×C8—71no—
150#394—142no—
151#390—141no—
152#390—141no—
153#390—141no—
154#384C2×D711yescharacteristic
155C4×C4⋊C4—71no—
156#340—21no—
157#340—141no—
158#331—142no—
159Q8⋊C8—142no—
160C23×C7#12011yescharacteristic
161C22×D7—42no—
162C22×D7—42no—
163C22×D7—42no—
164C22×D7—42no—
165C22×D7—42no—
166C22×D7—42no—
167C22×D7—42no—
168C7×Q8—22no—
169C7×Q8—22no—
170C7×Q8—41no—
171C7×Q8—41no—
172C2×C4×C7—21no—
173C2×C4×C7—21no—
174C2×C4×C7—22no—
175C2×C4×C7—41no—
176C2×C4×C7—41no—
177C2×C4×C7—41no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—81no—
181C2×C4×C7—81no—
182C2×C4×C7—81no—
183C2×C4×C7#12111yescharacteristic
184C2×C4×C7#12111yescharacteristic
185C2×C4×C7#12111yescharacteristic
186C2×C4×C7#12311yescharacteristic
187C2×C4×C7#12311yescharacteristic
188C2×C4×C7#12411yescharacteristic
189C2×C4×C7C2×SD1611yescharacteristic
190C2×C4×C7#13511yescharacteristic
191C2×C4×C7#13511yescharacteristic
192C2×C4×C7#13511yescharacteristic
193C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
194C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
195C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
196C2×C4×C7#13711yescharacteristic
197C2×Dic7—41no—
198C2×Dic7—41no—
199C2×Dic7—41no—
200C2×Dic7—41no—
201C2×Dic7—41no—
202C2×Dic7—41no—
203D28—44no—
204D28—82no—
205D28—81no—
206D28—81no—
207D28—81no—
208D28—81no—
209D28—81no—
210D28—81no—
211D28—81no—
212D28—81no—
213C7×C8—81no—
214C7⋊C8—42no—
215C7⋊C8—42no—
216C22×Q8C22⋊D711yescharacteristic
217C22×D4—72no—
218C22×D4—141no—
219C22×D4—141no—
220C22×C8—21no—
221C22×C8—141no—
222C22×C8—141no—
223#127—141no—
224#127—141no—
225#127—141no—
226#127—141no—
227#124—282no—
228#124—282no—
229C4×Q8—21no—
230C4×Q8—21no—
231C4×Q8—21no—
232C4×Q8—21no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—21no—
236C2×C4⋊C4—21no—
237C2×C4⋊C4—71no—
238C2×C4⋊C4—71no—
239C2×C4⋊C4C22⋊D711yescharacteristic
240C2×C22⋊C4—142no—
241C2×C22⋊C4—142no—
242C2×C22⋊C4—142no—
243C2×C22⋊C4—142no—
244C2×C42—21no—
245C2×C42—141no—
246C2×C42C22×D711yescharacteristic
247#106—282no—
248C4⋊C8—21no—
249C4⋊C8—21no—
250C4⋊C8—72no—
251C4⋊C8—141no—
252#103—42no—
253#102—284no—
254C4×C8—141no—
255C4×C8—141no—
256#95—141no—
257C22×C7#39411yescharacteristic
258C22×C7#44911yescharacteristic
259C22×C7#44911yescharacteristic
260C22×C7#45011yescharacteristic
261C22×C7#45211yescharacteristic
262C22×C7#45311yescharacteristic
263C22×C7#45511yescharacteristic
264C2×D7—82no—
265C2×D7—82no—
266C2×D7—82no—
267C2×D7—82no—
268C2×D7—82no—
269C2×D7—82no—
270C2×D7—82no—
271C4×C7—21no—
272C4×C7—21no—
273C4×C7—42no—
274C4×C7—81no—
275C4×C7—161no—
276C4×C7#46311yescharacteristic
277C4×C7#46411yescharacteristic
278C4×C7#46511yescharacteristic
279C4×C7#46911yescharacteristic
280C4×C7#48111yescharacteristic
281C4×C7#48211yescharacteristic
282C4×C7#48811yescharacteristic
283C4×C7#48911yescharacteristic
284Dic7—81no—
285Dic7—81no—
286C24—142no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8—21no—
290C2×Q8—21no—
291C2×Q8—21no—
292C2×Q8#114312yes—
293C2×D4—142no—
294C2×D4—142no—
295C2×D4—282no—
296C2×D4—282no—
297C2×D4—281no—
298C2×D4—281no—
299C2×D4—281no—
300C2×D4—281no—
301C2×D4—281no—
302C2×D4—281no—
303C2×D4—281no—
304C2×D4—281no—
305C2×D4—281no—
306C2×D4—281no—
307C2×D4—281no—
308C2×D4—281no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—41no—
312C22×C4—141no—
313C22×C4—141no—
314C22×C4D4×D711yescharacteristic
315C22×C4D4×D711yescharacteristic
316C22×C4C2×C22⋊D711yescharacteristic
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—142no—
321C2×C8—142no—
322C2×C8—281no—
323C2×C8—281no—
324C2×C8—281no—
325C2×C8—281no—
326C4⋊C4—21no—
327C4⋊C4—21no—
328C4⋊C4—42no—
329C4⋊C4—41no—
330C4⋊C4—41no—
331C4⋊C4—41no—
332C4⋊C4—41no—
333C4⋊C4—41no—
334C4⋊C4—41no—
335C4⋊C4—141no—
336C4⋊C4—141no—
337C4⋊C4—141no—
338C4⋊C4—141no—
339C22⋊C4—284no—
340C22⋊C4—284no—
341C22⋊C4—562no—
342C22⋊C4—562no—
343C42—41no—
344C42—41no—
345C42—281no—
346C42—281no—
347C42#115711yescharacteristic
348C42#115711yescharacteristic
349C42D4×D711yescharacteristic
350C42#115911yescharacteristic
351C2×C7#162011yescharacteristic
352C2×C7#163111yescharacteristic
353C2×C7#164011yescharacteristic
354C2×C7#164211yescharacteristic
355C2×C7#203411yescharacteristic
356C2×C7#203711yescharacteristic
357C2×C7#204211yescharacteristic
358D7—162no—
359C23—282no—
360C23—282no—
361C23—282no—
362C23—282no—
363C23—282no—
364C23—282no—
365C23—282no—
366C23#650011yescharacteristic
367Q8—22no—
368Q8—22no—
369Q8—41no—
370Q8—41no—
371D4—284no—
372D4—562no—
373D4—561no—
374D4—561no—
375D4—561no—
376D4—561no—
377D4—561no—
378D4—561no—
379D4—561no—
380D4—561no—
381C2×C4—21no—
382C2×C4—21no—
383C2×C4—22no—
384C2×C4—41no—
385C2×C4—41no—
386C2×C4—41no—
387C2×C4—41no—
388C2×C4—41no—
389C2×C4—81no—
390C2×C4—81no—
391C2×C4—81no—
392C2×C4—281no—
393C2×C4—281no—
394C2×C4—281no—
395C2×C4—281no—
396C2×C4—281no—
397C2×C4—281no—
398C2×C4#644611yescharacteristic
399C2×C4#644711yescharacteristic
400C2×C4#644711yescharacteristic
401C2×C4#644811yescharacteristic
402C2×C4#647711yescharacteristic
403C2×C4#647911yescharacteristic
404C2×C4#648111yescharacteristic
405C2×C4#648211yescharacteristic
406C2×C4#649211yescharacteristic
407C2×C4#649511yescharacteristic
408C2×C4#650211yescharacteristic
409C2×C4C2×#114311yescharacteristic
410C2×C4#651211yescharacteristic
411C2×C4#651311yescharacteristic
412C8—81no—
413C8—282no—
414C8—282no—
415C7#1041011yescharacteristic, complemented, Sylow
416C22—562no—
417C22—562no—
418C22—562no—
419C22—562no—
420C22—562no—
421C22—562no—
422C22—562no—
423C22#8973111yescharacteristic
424C22#8980911yescharacteristic
425C22#8981811yescharacteristic
426C22#8994411yescharacteristic
427C22#8995911yescharacteristic
428C22#8997011yescharacteristic
429C22#8997411yescharacteristic
430C4—21no—
431C4—21no—
432C4—42no—
433C4—81no—
434C4—161no—
435C4—561no—
436C4—561no—
437C4#8956711yescharacteristic
438C4#8957011yescharacteristic
439C4#8957211yescharacteristic
440C4#8959111yescharacteristic
441C4#8974011yescharacteristic
442C4#8977511yescharacteristic
443C4#8981111yescharacteristic
444C4#8982011yescharacteristic
445C2—1122no—
446C2#1172727811yescharacteristic
447C2#1172803311yescharacteristic
448C2#1172806011yescharacteristic
449C2#1172878511yescharacteristic
450C2#1172881411yescharacteristic
451C2#1172887811yescharacteristic
452C2#1172891411yescharacteristic
subgroupG/H
centerC23#6500
commutator, cosocleC2×C42×C7C23
FrattiniC2×C42C22×D7
FittingC7×#1803C2
socleC23×C7#120
prime radicalC2×#6437C22
hypercenter#1803D7
hypocenterC7#10410
LCSG ▷ C2×C42×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1803

Complements

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

NH ≅ G/Ntype
2. C7×#1803C2semidirect
3. C2×Q8⋊(C7⋊C8)C2semidirect
6. #11727322C2semidirect
7. #11725510C2semidirect
72. #1803D7semidirect
415. C7#10410semidirect

Maximal subgroups

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

subgroupindex
#117399212
C7×#18032
C2×Q8⋊(C7⋊C8)2
C2×#892802
#117273342
#117273222
#117255102
#104107