Finite Group Databasebeta

← #13242289

Subgroups — #13242289

Order 1792.

600 conjugacy classes of subgroups; only the 103 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 103 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11733919C211yesmaximal, characteristic
2D7×#336C211yesmaximal, characteristic, complemented
3#11726354C211yesmaximal, characteristic, complemented
4C7×#1270C211yesmaximal, characteristic, complemented
5#11725445C211yesmaximal, characteristic, complemented
6#11725286C211yesmaximal, characteristic, complemented
7#11725286C211yesmaximal, characteristic, complemented
8C2×#6501C411yescharacteristic
9C2×#6499C411yescharacteristic
10C22×C8×D7—21no—
11C2×#6460C411yescharacteristic
12C2×C42×D7C2211yescharacteristic
13C7×#390C2211yescharacteristic
14C2×(C2×D7)⋊C8—21no—
15C2×#6389—21no—
16#89749C411yescharacteristic
17#89728—21no—
18#89728—21no—
19#89723—21no—
20#89723—21no—
21#89701C2211yescharacteristic
22C7×#336C2211yescharacteristic
23C7×#336C2211yescharacteristic
24#89345—21no—
25#89319C2211yescharacteristic
26#89319C2211yescharacteristic
27#7690—71nomaximal, Sylow
28C22×C7×D4C2×C411yescharacteristic
29C22×C22⋊D7—21no—
30C22×C4×D7C2×C411yescharacteristic
31C22×C4×D7D411yescharacteristic
32C22×C4×D7D411yescharacteristic
33C22×Dic14C2×C411yescharacteristic
34C22×C7×C8—21no—
35C22×C7×C8—21no—
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4—21no—
39C2×C42×C7C2311yescharacteristic
40C2×C22⋊Dic7—21no—
41C2×C22⋊Dic7—21no—
42C2×C22⋊Dic7—21no—
43#6501—42no—
44#6499—42no—
45C2×(C2×D7)⋊C4—21no—
46C2×C4⋊Dic7—21no—
47C2×C4⋊Dic7C2×C411yescharacteristic
48C2×C4×Dic7C2×C411yescharacteristic
49C2×C4×Dic7D411yescharacteristic
50C2×C4×Dic7D411yescharacteristic
51C22×C7⋊C8—21no—
52C22×C7⋊C8—21no—
53C2×C8×D7—24no—
54C2×C8×D7—42no—
55C2×C8×D7—42no—
56#6460—42no—
57C42×D7—42no—
58C7⋊#95—21no—
59C7⋊#95—21no—
60C7⋊#95—21no—
61C7⋊#95—21no—
62(C2×D7)⋊C8—24no—
63#6389—42no—
64#2850—71no—
65#1749—71no—
66C2×#336—71no—
67#1272—71no—
68#1272—71no—
69#1270—71no—
70#1270D711yescharacteristic, complemented
71C24×C7—21no—
72C23×D7C22⋊C411yescharacteristic
73C2×C7×D4—21no—
74C2×C7×D4—21no—
75C2×C7×D4—41no—
76C2×C7×D4—41no—
77C22×C4×C7—21no—
78C22×C4×C7C22×C411yescharacteristic
79C22×C4×C7C2×D411yescharacteristic
80C22×C4×C7C2×D411yescharacteristic
81C2×C22⋊D7—42no—
82C2×C22⋊D7—42no—
83C2×C22⋊D7—42no—
84C22×Dic7—21no—
85C22×Dic7—21no—
86C22×Dic7C22⋊C411yescharacteristic
87C22×Dic7C22⋊C411yescharacteristic
88C22×Dic7C22⋊C411yescharacteristic
89C2×C4×D7—22no—
90C2×C4×D7—21no—
91C2×C4×D7—21no—
92C2×C4×D7—22no—
93C2×C4×D7—22no—
94C2×C4×D7—22no—
95C2×C4×D7—21no—
96C2×C4×D7—21no—
97C2×C4×D7—21no—
98C2×C4×D7—21no—
99C2×C4×D7—41no—
100C2×C4×D7—41no—
101C2×Dic14—21no—
102C2×Dic14—21no—
103C2×Dic14—41no—
104C2×Dic14—41no—
105C2×C7×C8—22no—
106C2×C7×C8—22no—
107C2×C7×C8—41no—
108C2×C7×C8—41no—
109C2×C7×C8—41no—
110C2×C7×C8—41no—
111C7×C4⋊C4—41no—
112C7×C22⋊C4—41no—
113C7×C22⋊C4—41no—
114C7×C22⋊C4—41no—
115C7×C22⋊C4—41no—
116C42×C7—41no—
117C22⋊Dic7—42no—
118C22⋊Dic7—42no—
119C22⋊Dic7—42no—
120(C2×D7)⋊C4—42no—
121C4⋊Dic7—42no—
122C4⋊Dic7—41no—
123C4×Dic7—22no—
124C4×Dic7—22no—
125C4×Dic7—41no—
126C2×C7⋊C8—22no—
127C2×C7⋊C8—22no—
128C2×C7⋊C8—41no—
129C2×C7⋊C8—41no—
130C2×C7⋊C8—41no—
131C2×C7⋊C8—41no—
132C8×D7—48no—
133C23×C8—141no—
134C2×#124—71no—
135C2×C22⋊Q8—71no—
136C2×#121—71no—
137C22×C42—71no—
138C2×C22⋊C8—141no—
139C2×#97—141no—
140#391—71no—
141#390C2×D711yescharacteristic
142#388—141no—
143#388—141no—
144#386—141no—
145#386—141no—
146#382—71no—
147#336—71no—
148#336—71no—
149#336—141no—
150#336C2×D711yescharacteristic
151#336C2×D711yescharacteristic
152C23×C7—41no—
153C23×C7—41no—
154C23×C7—41no—
155C23×C7—41no—
156C23×C7—41no—
157C23×C7—41no—
158C23×C7—41no—
159C23×C7C2×C22⋊C411yescharacteristic
160C22×D7—21no—
161C22×D7—21no—
162C22×D7—21no—
163C22×D7—21no—
164C22×D7#10012yes—
165C22×D7#10412yes—
166C22×D7#10412yes—
167C7×D4—81no—
168C7×D4—81no—
169C2×C4×C7—21no—
170C2×C4×C7—21no—
171C2×C4×C7—21no—
172C2×C4×C7—21no—
173C2×C4×C7—21no—
174C2×C4×C7—21no—
175C2×C4×C7—41no—
176C2×C4×C7—41no—
177C2×C4×C7—41no—
178C2×C4×C7#13111yescharacteristic
179C2×C4×C7#13111yescharacteristic
180C2×C4×C7#13511yescharacteristic
181C2×C4×C7#13511yescharacteristic
182C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
183C2×C4×C7#13711yescharacteristic
184C22⋊D7—84no—
185C2×Dic7—21no—
186C2×Dic7—21no—
187C2×Dic7—21no—
188C2×Dic7—21no—
189C2×Dic7—21no—
190C2×Dic7—21no—
191C2×Dic7—21no—
192C2×Dic7—21no—
193C2×Dic7—41no—
194C2×Dic7—41no—
195C2×Dic7—41no—
196C2×Dic7—41no—
197C2×Dic7—41no—
198C2×Dic7—41no—
199C2×Dic7#9912yes—
200C4×D7—24no—
201C4×D7—24no—
202C4×D7—42no—
203C4×D7—42no—
204C4×D7—42no—
205C4×D7—42no—
206Dic14—81no—
207Dic14—81no—
208C7×C8—42no—
209C7×C8—42no—
210C7⋊C8—42no—
211C7⋊C8—42no—
212C22×Q8—71no—
213C22×D4—141no—
214C22×D4C4×D711yescharacteristic
215C23×C4—71no—
216C23×C4—71no—
217C23×C4—71no—
218C22×C8—21no—
219C22×C8—21no—
220C22×C8—144no—
221C22×C8—141no—
222C22×C8—141no—
223C22×C8—282no—
224C22×C8—282no—
225#124—282no—
226C22⋊Q8—282no—
227#121—282no—
228C2×C4⋊C4—71no—
229C2×C4⋊C4—141no—
230C2×C4⋊C4C4×D711yescharacteristic
231C2×C22⋊C4—21no—
232C2×C22⋊C4—21no—
233C2×C22⋊C4—141no—
234C2×C22⋊C4—141no—
235C2×C22⋊C4—141no—
236C2×C22⋊C4—141no—
237C2×C42—71no—
238C2×C42—71no—
239C2×C42—71no—
240C2×C42—282no—
241C2×C42C22×D711yescharacteristic
242C22⋊C8—144no—
243#97—282no—
244#95—141no—
245#95—141no—
246#95—141no—
247#95—141no—
248C22×C7—81no—
249C22×C7—81no—
250C22×C7—81no—
251C22×C7—81no—
252C22×C7—81no—
253C22×C7—81no—
254C22×C7—81no—
255C22×C7#40811yescharacteristic
256C22×C7C2×#9911yescharacteristic
257C22×C7C2×#10011yescharacteristic
258C22×C7#41611yescharacteristic
259C22×C7#41711yescharacteristic
260C22×C7C2×#10411yescharacteristic
261C22×C7C2×#10411yescharacteristic
262C2×D7—22no—
263C2×D7—22no—
264C2×D7—22no—
265C2×D7—22no—
266C2×D7—22no—
267C2×D7—22no—
268C2×D7#32914yes—
269C4×C7—21no—
270C4×C7—21no—
271C4×C7—21no—
272C4×C7—21no—
273C4×C7—21no—
274C4×C7—21no—
275C4×C7—81no—
276Dic7—22no—
277Dic7—41no—
278Dic7—41no—
279Dic7—81no—
280Dic7—81no—
281C24—21no—
282C24—71no—
283C2×Q8—141no—
284C2×Q8—141no—
285C2×Q8—281no—
286C2×Q8—281no—
287C2×D4—21no—
288C2×D4—21no—
289C2×D4—41no—
290C2×D4—41no—
291C2×D4—282no—
292C2×D4—282no—
293C2×D4—282no—
294C22×C4—21no—
295C22×C4—71no—
296C22×C4—71no—
297C22×C4—71no—
298C22×C4—142no—
299C22×C4—142no—
300C22×C4—142no—
301C22×C4—142no—
302C22×C4—141no—
303C22×C4—141no—
304C22×C4—141no—
305C22×C4—141no—
306C22×C4—141no—
307C22×C4—141no—
308C22×C4—141no—
309C22×C4—141no—
310C22×C4—281no—
311C22×C4—281no—
312C22×C4C2×C4×D711yescharacteristic
313C22×C4D4×D711yescharacteristic
314C22×C4D4×D711yescharacteristic
315C2×C8—22no—
316C2×C8—22no—
317C2×C8—41no—
318C2×C8—41no—
319C2×C8—41no—
320C2×C8—41no—
321C2×C8—142no—
322C2×C8—142no—
323C2×C8—288no—
324C2×C8—281no—
325C2×C8—281no—
326C2×C8—281no—
327C2×C8—281no—
328C4⋊C4—41no—
329C4⋊C4—281no—
330C4⋊C4—282no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—41no—
335C22⋊C4—282no—
336C22⋊C4—282no—
337C22⋊C4—282no—
338C22⋊C4—282no—
339C42—41no—
340C42—142no—
341C42—142no—
342C42—281no—
343C2×C7—161no—
344C2×C7#147111yescharacteristic
345C2×C7#148311yescharacteristic
346C2×C7#149411yescharacteristic
347C2×C7#150111yescharacteristic
348C2×C7#150411yescharacteristic
349C2×C7C2×#32911yescharacteristic
350C2×C7#155111yescharacteristic
351D7—24no—
352C23—41no—
353C23—41no—
354C23—41no—
355C23—41no—
356C23—41no—
357C23—41no—
358C23—41no—
359C23—72no—
360C23—72no—
361C23—72no—
362C23—141no—
363C23—141no—
364C23—141no—
365C23—141no—
366C23D7×C22⋊C411yescharacteristic
367Q8—561no—
368Q8—561no—
369D4—81no—
370D4—81no—
371D4—564no—
372C2×C4—21no—
373C2×C4—21no—
374C2×C4—21no—
375C2×C4—21no—
376C2×C4—21no—
377C2×C4—21no—
378C2×C4—41no—
379C2×C4—41no—
380C2×C4—41no—
381C2×C4—72no—
382C2×C4—144no—
383C2×C4—144no—
384C2×C4—141no—
385C2×C4—141no—
386C2×C4—141no—
387C2×C4—141no—
388C2×C4—141no—
389C2×C4—141no—
390C2×C4—141no—
391C2×C4—141no—
392C2×C4—282no—
393C2×C4—282no—
394C2×C4—282no—
395C2×C4—282no—
396C2×C4—281no—
397C2×C4—281no—
398C2×C4—281no—
399C2×C4—281no—
400C2×C4—281no—
401C2×C4—281no—
402C2×C4#646411yescharacteristic
403C2×C4#647011yescharacteristic
404C2×C4#647411yescharacteristic
405C2×C4#647511yescharacteristic
406C2×C4#647811yescharacteristic
407C2×C4#647911yescharacteristic
408C8—42no—
409C8—42no—
410C8—282no—
411C8—282no—
412C7#769011yescharacteristic, complemented, Sylow
413C22—74no—
414C22—81no—
415C22—81no—
416C22—81no—
417C22—81no—
418C22—81no—
419C22—81no—
420C22—81no—
421C22—142no—
422C22—142no—
423C22—142no—
424C22—142no—
425C22—142no—
426C22—142no—
427C22#8949911yescharacteristic
428C22D7×#9911yescharacteristic
429C22D7×#10011yescharacteristic
430C22#8954311yescharacteristic
431C22#8954411yescharacteristic
432C22D7×#10411yescharacteristic
433C22D7×#10411yescharacteristic
434C4—21no—
435C4—21no—
436C4—21no—
437C4—21no—
438C4—21no—
439C4—21no—
440C4—81no—
441C4—142no—
442C4—281no—
443C4—281no—
444C4—561no—
445C4—561no—
446C2—144no—
447C2—161no—
448C2#1172587711yescharacteristic
449C2#1172592411yescharacteristic
450C2#1172593111yescharacteristic
451C2#1172605511yescharacteristic
452C2#1172605611yescharacteristic
453C2D7×#32911yescharacteristic
454C2#1172613911yescharacteristic
subgroupG/H
centerC23D7×C22⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1270C2
socleC23×C7C2×C22⋊C4
prime radicalC2×#6501C4
cosocleC2×C42×C7C23
hypercenter#1270D7
hypocenterC7#7690
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×C42 ◁ #1270

Complements

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

NH ≅ G/Ntype
2. D7×#336C2semidirect
3. #11726354C2semidirect
4. C7×#1270C2semidirect
5. #11725445C2semidirect
6. #11725286C2semidirect
7. #11725286C2semidirect
70. #1270D7semidirect
412. C7#7690semidirect

Maximal subgroups

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

subgroupindex
#117339192
D7×#3362
#117263542
C7×#12702
#117254452
#117252862
#76907