Finite Group Databasebeta

← #13263332

Subgroups — #13263332

Order 1792.

700 conjugacy classes of subgroups; only the 91 normal subgroups are shown.

normal subgroups only

This group has 700 conjugacy classes of subgroups — too many to draw. Only its 91 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
1C7×#2405C211yesmaximal, characteristic
2#11729293C211yesmaximal, characteristic, complemented
3C2×#89320C211yesmaximal, characteristic, complemented
4#11725442C212yesmaximal, complemented
5#11725428C212yesmaximal, complemented
6C23×C7×D4C411yescharacteristic
7C2×C7×#123—21no—
8C22×C7×C22⋊C4C2211yescharacteristic
9C7×#394—21no—
10C7×#394—21no—
11C7×#386—21no—
12C7×#385C411yescharacteristic
13C7×#385C2211yescharacteristic
14C7×#379C411yescharacteristic
15C7×#379C411yescharacteristic
16C7×#379C2211yescharacteristic
17#89979—21no—
18C2×C22⋊(C7⋊C8)—21no—
19#89320—21no—
20#89320—21no—
21#89320C2212yes—
22#89320C2212yes—
23#11545—71nomaximal, Sylow
24C25×C7C2×C411yescharacteristic
25C25×C7C2×C411yescharacteristic
26C22×C7×D4—21no—
27C22×C7×D4—21no—
28C22×C7×D4—44no—
29C22×C7×D4—41no—
30C22×C7×D4—41no—
31C23×C4×C7C2×C411yescharacteristic
32C23×C4×C7D411yescharacteristic
33C23×C4×C7D411yescharacteristic
34C7×#123—44no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4—21no—
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4—21no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—21no—
42C2×C7×C22⋊C4—21no—
43C2×C7×C22⋊C4—41no—
44C2×C7×C22⋊C4—41no—
45C2×C7×C22⋊C4C2×C411yescharacteristic
46C2×C7×C22⋊C4C2×C411yescharacteristic
47C2×C7×C22⋊C4C2×C411yescharacteristic
48C2×C7×C22⋊C4D411yescharacteristic
49C2×C7×C22⋊C4D411yescharacteristic
50C2×C7×C22⋊C4C2311yescharacteristic
51C2×#1136—41no—
52C22×C7⋊C8—41no—
53C7×#95—21no—
54C7×#95—21no—
55C22⋊(C7⋊C8)—22no—
56C22⋊(C7⋊C8)—22no—
57C22⋊(C7⋊C8)—41no—
58C22⋊(C7⋊C8)—41no—
59#2405D711yescharacteristic
60#1462—71no—
61C2×#323—71no—
62#1326—72no—
63#1316—72no—
64C24×C7—21no—
65C24×C7—21no—
66C24×C7—21no—
67C24×C7—22no—
68C24×C7—22no—
69C24×C7—41no—
70C24×C7—42no—
71C24×C7—41no—
72C24×C7—42no—
73C24×C7—41no—
74C24×C7—81no—
75C24×C7—81no—
76C24×C7C22⋊C411yescharacteristic
77C24×C7C22⋊C411yescharacteristic
78C24×C7C22×C411yescharacteristic
79C2×C7×D4—24no—
80C2×C7×D4—42no—
81C2×C7×D4—42no—
82C2×C7×D4—42no—
83C2×C7×D4—42no—
84C2×C7×D4—44no—
85C2×C7×D4—44no—
86C2×C7×D4—44no—
87C2×C7×D4—41no—
88C2×C7×D4—41no—
89C2×C7×D4—44no—
90C22×C4×C7—21no—
91C22×C4×C7—21no—
92C22×C4×C7—21no—
93C22×C4×C7—21no—
94C22×C4×C7—21no—
95C22×C4×C7—21no—
96C22×C4×C7—21no—
97C22×C4×C7—42no—
98C22×C4×C7—41no—
99C22×C4×C7—42no—
100C22×C4×C7—81no—
101C22×C4×C7C22⋊C411yescharacteristic
102C22×C4×C7C22⋊C411yescharacteristic
103C22×C4×C7C2×D411yescharacteristic
104C22×C4×C7C2×D411yescharacteristic
105C7×C4⋊C4—42no—
106C7×C4⋊C4—42no—
107C7×C22⋊C4—22no—
108C7×C22⋊C4—41no—
109C7×C22⋊C4—41no—
110C7×C22⋊C4—41no—
111C7×C22⋊C4—41no—
112C7×C22⋊C4—42no—
113C7×C22⋊C4—42no—
114C7×C22⋊C4—42no—
115C7×C22⋊C4—41no—
116C7×C22⋊C4—81no—
117C7×C22⋊C4—81no—
118C7×C22⋊C4—81no—
119#1136—82no—
120C2×C7⋊C8—42no—
121C2×C7⋊C8—42no—
122C2×C7⋊C8—82no—
123C23×D4Dic711yescharacteristic
124C2×#123—21no—
125C22×C22⋊C4C2×D711yescharacteristic
126#407—141no—
127C2×C22⋊C8—141no—
128#394—21no—
129#394—21no—
130#386—21no—
131#385Dic711yescharacteristic
132#385C2×D711yescharacteristic
133#379Dic711yescharacteristic
134#379Dic711yescharacteristic
135#379C2×D711yescharacteristic
136#323—72no—
137#323—72no—
138#323—141no—
139#323—141no—
140C23×C7—22no—
141C23×C7—21no—
142C23×C7—21no—
143C23×C7—21no—
144C23×C7—21no—
145C23×C7—22no—
146C23×C7—21no—
147C23×C7—21no—
148C23×C7—21no—
149C23×C7—41no—
150C23×C7—42no—
151C23×C7—42no—
152C23×C7—41no—
153C23×C7—41no—
154C23×C7—41no—
155C23×C7—41no—
156C23×C7—41no—
157C23×C7—42no—
158C23×C7—41no—
159C23×C7—41no—
160C23×C7—42no—
161C23×C7—42no—
162C23×C7—42no—
163C23×C7—41no—
164C23×C7—41no—
165C23×C7—41no—
166C23×C7—81no—
167C23×C7—81no—
168C23×C7—82no—
169C23×C7—84no—
170C23×C7—84no—
171C23×C7—84no—
172C23×C7—84no—
173C23×C7—82no—
174C23×C7#9911yescharacteristic
175C23×C7#9911yescharacteristic
176C23×C7C2×C22⋊C411yescharacteristic
177C23×C7#13112yes—
178C7×D4—44no—
179C7×D4—48no—
180C7×D4—44no—
181C2×C4×C7—22no—
182C2×C4×C7—21no—
183C2×C4×C7—21no—
184C2×C4×C7—21no—
185C2×C4×C7—21no—
186C2×C4×C7—42no—
187C2×C4×C7—41no—
188C2×C4×C7—42no—
189C2×C4×C7—42no—
190C2×C4×C7—41no—
191C2×C4×C7—41no—
192C2×C4×C7—41no—
193C2×C4×C7—41no—
194C2×C4×C7—41no—
195C2×C4×C7—42no—
196C2×C4×C7—41no—
197C2×C4×C7—41no—
198C2×C4×C7—41no—
199C2×C4×C7—41no—
200C2×C4×C7—41no—
201C2×C4×C7—41no—
202C2×C4×C7—81no—
203C2×C4×C7—81no—
204C7⋊C8—82no—
205C7⋊C8—82no—
206C25C2×Dic711yescharacteristic
207C25C2×Dic711yescharacteristic
208C22×D4—21no—
209C22×D4—21no—
210C22×D4—44no—
211C22×D4—41no—
212C22×D4—41no—
213C23×C4C2×Dic711yescharacteristic
214C23×C4C22⋊D711yescharacteristic
215C23×C4C22⋊D711yescharacteristic
216C2×M16—281no—
217C22×C8—281no—
218#123—44no—
219C2×C4⋊C4—21no—
220C2×C4⋊C4—21no—
221C2×C22⋊C4—21no—
222C2×C22⋊C4—21no—
223C2×C22⋊C4—21no—
224C2×C22⋊C4—21no—
225C2×C22⋊C4—21no—
226C2×C22⋊C4—21no—
227C2×C22⋊C4—41no—
228C2×C22⋊C4—41no—
229C2×C22⋊C4C2×Dic711yescharacteristic
230C2×C22⋊C4C2×Dic711yescharacteristic
231C2×C22⋊C4C2×Dic711yescharacteristic
232C2×C22⋊C4C22⋊D711yescharacteristic
233C2×C22⋊C4C22⋊D711yescharacteristic
234C2×C22⋊C4C22×D711yescharacteristic
235C22⋊C8—142no—
236C22⋊C8—142no—
237C22⋊C8—281no—
238C22⋊C8—281no—
239#95—21no—
240#95—21no—
241C22×C7—21no—
242C22×C7—21no—
243C22×C7—21no—
244C22×C7—21no—
245C22×C7—21no—
246C22×C7—21no—
247C22×C7—22no—
248C22×C7—21no—
249C22×C7—42no—
250C22×C7—42no—
251C22×C7—42no—
252C22×C7—42no—
253C22×C7—42no—
254C22×C7—42no—
255C22×C7—41no—
256C22×C7—41no—
257C22×C7—42no—
258C22×C7—41no—
259C22×C7—41no—
260C22×C7—41no—
261C22×C7—84no—
262C22×C7—81no—
263C22×C7—81no—
264C22×C7—81no—
265C22×C7—81no—
266C22×C7—84no—
267C22×C7—84no—
268C22×C7—84no—
269C22×C7—81no—
270C22×C7—81no—
271C22×C7AΣL(1,𝔽16)12yes—
272C22×C7#40811yescharacteristic
273C22×C7C2×#9911yescharacteristic
274C22×C7#41311yescharacteristic
275C4×C7—41no—
276C4×C7—42no—
277C4×C7—41no—
278C4×C7—41no—
279C4×C7—41no—
280C4×C7—81no—
281C4×C7—81no—
282C24—21no—
283C24—21no—
284C24—21no—
285C24—22no—
286C24—22no—
287C24—41no—
288C24—41no—
289C24—42no—
290C24—42no—
291C24—41no—
292C24—81no—
293C24—81no—
294C24C22⋊Dic711yescharacteristic
295C24C22⋊Dic711yescharacteristic
296C24C22×Dic711yescharacteristic
297C2×D4—24no—
298C2×D4—42no—
299C2×D4—42no—
300C2×D4—42no—
301C2×D4—42no—
302C2×D4—44no—
303C2×D4—44no—
304C2×D4—44no—
305C2×D4—41no—
306C2×D4—41no—
307C2×D4—44no—
308C22×C4—21no—
309C22×C4—21no—
310C22×C4—21no—
311C22×C4—21no—
312C22×C4—21no—
313C22×C4—21no—
314C22×C4—21no—
315C22×C4—42no—
316C22×C4—41no—
317C22×C4—42no—
318C22×C4—81no—
319C22×C4C22⋊Dic711yescharacteristic
320C22×C4C22⋊Dic711yescharacteristic
321C22×C4C2×C22⋊D711yescharacteristic
322C22×C4C2×C22⋊D711yescharacteristic
323M16—562no—
324C2×C8—282no—
325C2×C8—282no—
326C2×C8—562no—
327C4⋊C4—42no—
328C4⋊C4—42no—
329C22⋊C4—22no—
330C22⋊C4—41no—
331C22⋊C4—41no—
332C22⋊C4—41no—
333C22⋊C4—41no—
334C22⋊C4—42no—
335C22⋊C4—42no—
336C22⋊C4—42no—
337C22⋊C4—41no—
338C22⋊C4—81no—
339C22⋊C4—81no—
340C22⋊C4—81no—
341C2×C7—21no—
342C2×C7—21no—
343C2×C7—22no—
344C2×C7—41no—
345C2×C7—42no—
346C2×C7—82no—
347C2×C7—81no—
348C2×C7#146811yescharacteristic
349C2×C7C2×AΣL(1,𝔽16)11yescharacteristic
350C2×C7#212111yescharacteristic
351C23—21no—
352C23—21no—
353C23—21no—
354C23—21no—
355C23—22no—
356C23—22no—
357C23—21no—
358C23—21no—
359C23—21no—
360C23—41no—
361C23—42no—
362C23—42no—
363C23—41no—
364C23—41no—
365C23—41no—
366C23—41no—
367C23—41no—
368C23—42no—
369C23—41no—
370C23—41no—
371C23—42no—
372C23—42no—
373C23—41no—
374C23—41no—
375C23—41no—
376C23—42no—
377C23—81no—
378C23—81no—
379C23—82no—
380C23—84no—
381C23—84no—
382C23—84no—
383C23—84no—
384C23—82no—
385C23#640811yescharacteristic
386C23#640811yescharacteristic
387C23#651112yes—
388C23C2×C22⋊Dic711yescharacteristic
389D4—44no—
390D4—48no—
391D4—44no—
392C2×C4—22no—
393C2×C4—21no—
394C2×C4—21no—
395C2×C4—21no—
396C2×C4—21no—
397C2×C4—42no—
398C2×C4—41no—
399C2×C4—42no—
400C2×C4—41no—
401C2×C4—42no—
402C2×C4—41no—
403C2×C4—41no—
404C2×C4—41no—
405C2×C4—41no—
406C2×C4—42no—
407C2×C4—41no—
408C2×C4—41no—
409C2×C4—41no—
410C2×C4—41no—
411C2×C4—41no—
412C2×C4—41no—
413C2×C4—81no—
414C2×C4—81no—
415C8—562no—
416C8—562no—
417C7#1154511yescharacteristic, complemented, Sylow
418C22—21no—
419C22—21no—
420C22—21no—
421C22—21no—
422C22—21no—
423C22—22no—
424C22—21no—
425C22—21no—
426C22—42no—
427C22—42no—
428C22—42no—
429C22—42no—
430C22—42no—
431C22—42no—
432C22—41no—
433C22—41no—
434C22—42no—
435C22—41no—
436C22—41no—
437C22—41no—
438C22—84no—
439C22—81no—
440C22—81no—
441C22—81no—
442C22—81no—
443C22—84no—
444C22—84no—
445C22—84no—
446C22—81no—
447C22—81no—
448C22#8933112yes—
449C22C2×#640811yescharacteristic
450C22#9000611yescharacteristic
451C22#9000911yescharacteristic
452C4—41no—
453C4—42no—
454C4—41no—
455C4—41no—
456C4—41no—
457C4—81no—
458C4—81no—
459C2—21no—
460C2—21no—
461C2—22no—
462C2—41no—
463C2—42no—
464C2—82no—
465C2—81no—
466C2C2×#8933111yescharacteristic
467C2#1172945611yescharacteristic
468C2#1172960111yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC24×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
FittingC7×#2405C2
socleC22×C7C2×#99
prime radicalC23×C7×D4C4
cosocleC2×C7×C22⋊C4C23
hypercenter#2405D7
hypocenterC7#11545
LCSG ▷ C24×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C24 ◁ C22×C22⋊C4 ◁ #2405

Complements

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

NH ≅ G/Ntype
2. #11729293C2semidirect
3. C2×#89320C2semidirect
4. #11725442C2semidirect
5. #11725428C2semidirect
417. C7#11545semidirect

Maximal subgroups

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

subgroupindex
C7×#24052
#117292932
C2×#893202
#117254422
#117254282
#115457