Finite Group Databasebeta

← #13263254

Subgroups — #13263254

Order 1792.

530 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 530 conjugacy classes of subgroups — too many to draw. Only its 95 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×#2321C211yesmaximal, characteristic
2#11729357C211yesmaximal, characteristic, complemented
3#11729356C211yesmaximal, characteristic, complemented
4#11725442C212yesmaximal, complemented
5#11725428C212yesmaximal, complemented
6C2×C7×#120C411yescharacteristic
7C2×C4×C7×D4C2211yescharacteristic
8C22×C7×C22⋊C4C411yescharacteristic
9C7×#390—21no—
10C7×#388—21no—
11C7×#388—21no—
12C7×#386—21no—
13C7×#386C411yescharacteristic
14C7×#385C2211yescharacteristic
15C7×#379C2211yescharacteristic
16C4×C7×C22⋊C4C411yescharacteristic
17#89979—21no—
18#89698—21no—
19#89323—21no—
20#89320—21no—
21#89320C2212yes—
22#89320C2212yes—
23#11527—71nomaximal, Sylow
24C25×C7C2×C411yescharacteristic
25C22×C7×D4—21no—
26C22×C7×D4C2×C411yescharacteristic
27C23×C4×C7C2×C411yescharacteristic
28C23×C4×C7C2×C411yescharacteristic
29C23×C4×C7D411yescharacteristic
30C7×#120—22no—
31C7×#120—22no—
32C4×C7×D4—41no—
33C4×C7×D4—41no—
34C2×C7×C4⋊C4—21no—
35C2×C7×C4⋊C4D411yescharacteristic
36C2×C7×C22⋊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—41no—
43C2×C7×C22⋊C4—41no—
44C2×C7×C22⋊C4C2×C411yescharacteristic
45C2×C7×C22⋊C4C2×C411yescharacteristic
46C2×C7×C22⋊C4D411yescharacteristic
47C2×C7×C22⋊C4C2311yescharacteristic
48C2×C42×C7—21no—
49C2×C42×C7D411yescharacteristic
50C2×#1136—41no—
51C7×#95—21no—
52C7×#95—21no—
53C7×#95—21no—
54C22⋊(C7⋊C8)—22no—
55C22⋊(C7⋊C8)—22no—
56C22⋊(C7⋊C8)—41no—
57C4⋊(C7⋊C8)—41no—
58#6377—41no—
59#2321D711yescharacteristic
60#1466—71no—
61#1465—71no—
62#1326—72no—
63#1316—72no—
64C24×C7—21no—
65C24×C7—21no—
66C24×C7—41no—
67C24×C7—41no—
68C24×C7—81no—
69C24×C7—81no—
70C24×C7C22⋊C411yescharacteristic
71C24×C7M1612yescomplemented
72C24×C7C22×C411yescharacteristic
73C2×C7×D4—21no—
74C2×C7×D4—21no—
75C2×C7×D4—24no—
76C2×C7×D4—41no—
77C2×C7×D4—41no—
78C2×C7×D4—81no—
79C2×C7×D4—81no—
80C22×C4×C7—21no—
81C22×C4×C7—21no—
82C22×C4×C7—21no—
83C22×C4×C7—21no—
84C22×C4×C7—21no—
85C22×C4×C7—21no—
86C22×C4×C7—21no—
87C22×C4×C7—41no—
88C22×C4×C7—81no—
89C22×C4×C7—81no—
90C22×C4×C7—81no—
91C22×C4×C7C22⋊C411yescharacteristic
92C22×C4×C7C22⋊C411yescharacteristic
93C22×C4×C7C22⋊C411yescharacteristic
94C22×C4×C7C2×D411yescharacteristic
95C22×C4×C7C2×D411yescharacteristic
96C7×C4⋊C4—41no—
97C7×C4⋊C4—81no—
98C7×C22⋊C4—24no—
99C7×C22⋊C4—21no—
100C7×C22⋊C4—41no—
101C7×C22⋊C4—41no—
102C7×C22⋊C4—42no—
103C7×C22⋊C4—41no—
104C7×C22⋊C4—41no—
105C7×C22⋊C4—41no—
106C7×C22⋊C4—81no—
107C7×C22⋊C4—81no—
108C7×C22⋊C4—81no—
109C7×C22⋊C4M1612yescomplemented
110C42×C7—21no—
111C42×C7—21no—
112C42×C7—81no—
113#1136—82no—
114C2×C7⋊C8—42no—
115C2×C7⋊C8—42no—
116C2×#120Dic711yescharacteristic
117C2×C4×D4C2×D711yescharacteristic
118C22×C22⋊C4Dic711yescharacteristic
119#432—141no—
120#407—141no—
121#390—21no—
122#388—21no—
123#388—21no—
124#386—21no—
125#386Dic711yescharacteristic
126#385C2×D711yescharacteristic
127#379C2×D711yescharacteristic
128C4×C22⋊C4Dic711yescharacteristic
129#324—141no—
130#323—72no—
131#323—72no—
132#323—141no—
133C23×C7—21no—
134C23×C7—21no—
135C23×C7—22no—
136C23×C7—21no—
137C23×C7—21no—
138C23×C7—21no—
139C23×C7—41no—
140C23×C7—41no—
141C23×C7—41no—
142C23×C7—41no—
143C23×C7—41no—
144C23×C7—41no—
145C23×C7—41no—
146C23×C7—41no—
147C23×C7—42no—
148C23×C7—42no—
149C23×C7—41no—
150C23×C7—41no—
151C23×C7—82no—
152C23×C7—82no—
153C23×C7—82no—
154C23×C7—82no—
155C23×C7—84no—
156C23×C7C2×C22⋊C411yescharacteristic
157C23×C7C2×M1611yescharacteristic
158C23×C7C2×M1611yescharacteristic
159C7×D4—42no—
160C7×D4—42no—
161C7×D4—84no—
162C2×C4×C7—21no—
163C2×C4×C7—22no—
164C2×C4×C7—21no—
165C2×C4×C7—21no—
166C2×C4×C7—21no—
167C2×C4×C7—21no—
168C2×C4×C7—21no—
169C2×C4×C7—41no—
170C2×C4×C7—42no—
171C2×C4×C7—41no—
172C2×C4×C7—41no—
173C2×C4×C7—41no—
174C2×C4×C7—41no—
175C2×C4×C7—41no—
176C2×C4×C7—41no—
177C2×C4×C7—41no—
178C2×C4×C7—41no—
179C2×C4×C7—41no—
180C2×C4×C7—41no—
181C2×C4×C7—41no—
182C2×C4×C7—81no—
183C2×C4×C7—81no—
184C2×C4×C7—81no—
185C2×C4×C7—82no—
186C2×C4×C7#9911yescharacteristic
187C2×C4×C7#9911yescharacteristic
188C7⋊C8—82no—
189C7⋊C8—82no—
190C25C2×Dic711yescharacteristic
191C22×D4—21no—
192C22×D4C2×Dic711yescharacteristic
193C23×C4C2×Dic711yescharacteristic
194C23×C4C2×Dic711yescharacteristic
195C23×C4C22⋊D711yescharacteristic
196C2×M16—281no—
197#120—22no—
198#120—22no—
199C4×D4—41no—
200C4×D4—41no—
201C2×C4⋊C4—21no—
202C2×C4⋊C4C22⋊D711yescharacteristic
203C2×C22⋊C4—21no—
204C2×C22⋊C4—21no—
205C2×C22⋊C4—21no—
206C2×C22⋊C4—21no—
207C2×C22⋊C4—21no—
208C2×C22⋊C4—21no—
209C2×C22⋊C4—41no—
210C2×C22⋊C4—41no—
211C2×C22⋊C4C2×Dic711yescharacteristic
212C2×C22⋊C4C2×Dic711yescharacteristic
213C2×C22⋊C4C22⋊D711yescharacteristic
214C2×C22⋊C4C22×D711yescharacteristic
215C2×C42—21no—
216C2×C42C22⋊D711yescharacteristic
217C4⋊C8—281no—
218C22⋊C8—142no—
219C22⋊C8—142no—
220C22⋊C8—281no—
221#97—281no—
222#95—21no—
223#95—21no—
224#95—21no—
225C22×C7—21no—
226C22×C7—21no—
227C22×C7—21no—
228C22×C7—21no—
229C22×C7—41no—
230C22×C7—41no—
231C22×C7—42no—
232C22×C7—42no—
233C22×C7—42no—
234C22×C7—42no—
235C22×C7—41no—
236C22×C7—41no—
237C22×C7—41no—
238C22×C7—41no—
239C22×C7—41no—
240C22×C7—42no—
241C22×C7—81no—
242C22×C7—81no—
243C22×C7—81no—
244C22×C7—84no—
245C22×C7—84no—
246C22×C7—81no—
247C22×C7—81no—
248C22×C7#40711yescharacteristic
249C22×C7C2×#9911yescharacteristic
250C22×C7#41311yescharacteristic
251C4×C7—21no—
252C4×C7—21no—
253C4×C7—42no—
254C4×C7—41no—
255C4×C7—41no—
256C4×C7—41no—
257C4×C7—41no—
258C4×C7—81no—
259C4×C7—81no—
260C24—21no—
261C24—21no—
262C24—41no—
263C24—41no—
264C24—81no—
265C24—81no—
266C24#113612yescomplemented
267C24C22⋊Dic711yescharacteristic
268C24C22×Dic711yescharacteristic
269C2×D4—21no—
270C2×D4—21no—
271C2×D4—24no—
272C2×D4—41no—
273C2×D4—41no—
274C2×D4—81no—
275C2×D4—81no—
276C22×C4—21no—
277C22×C4—21no—
278C22×C4—21no—
279C22×C4—21no—
280C22×C4—21no—
281C22×C4—21no—
282C22×C4—21no—
283C22×C4—41no—
284C22×C4—81no—
285C22×C4—81no—
286C22×C4—81no—
287C22×C4C22⋊Dic711yescharacteristic
288C22×C4C22⋊Dic711yescharacteristic
289C22×C4C22⋊Dic711yescharacteristic
290C22×C4C2×C22⋊D711yescharacteristic
291C22×C4C2×C22⋊D711yescharacteristic
292M16—562no—
293C2×C8—282no—
294C2×C8—282no—
295C4⋊C4—41no—
296C4⋊C4—81no—
297C22⋊C4—24no—
298C22⋊C4—21no—
299C22⋊C4—41no—
300C22⋊C4—41no—
301C22⋊C4—42no—
302C22⋊C4—41no—
303C22⋊C4—41no—
304C22⋊C4—41no—
305C22⋊C4—81no—
306C22⋊C4—81no—
307C22⋊C4—81no—
308C22⋊C4#113612yescomplemented
309C42—21no—
310C42—21no—
311C42—81no—
312C2×C7—21no—
313C2×C7—21no—
314C2×C7—41no—
315C2×C7—41no—
316C2×C7—81no—
317C2×C7—82no—
318C2×C7#146511yescharacteristic
319C2×C7#212111yescharacteristic
320C2×C7#212211yescharacteristic
321C23—21no—
322C23—21no—
323C23—22no—
324C23—21no—
325C23—21no—
326C23—21no—
327C23—41no—
328C23—41no—
329C23—41no—
330C23—41no—
331C23—41no—
332C23—41no—
333C23—41no—
334C23—41no—
335C23—42no—
336C23—42no—
337C23—41no—
338C23—41no—
339C23—82no—
340C23—82no—
341C23—82no—
342C23—82no—
343C23—84no—
344C23C2×#113611yescharacteristic
345C23C2×#113611yescharacteristic
346C23C2×C22⋊Dic711yescharacteristic
347D4—42no—
348D4—42no—
349D4—84no—
350C2×C4—21no—
351C2×C4—22no—
352C2×C4—21no—
353C2×C4—21no—
354C2×C4—21no—
355C2×C4—21no—
356C2×C4—21no—
357C2×C4—41no—
358C2×C4—42no—
359C2×C4—41no—
360C2×C4—41no—
361C2×C4—41no—
362C2×C4—41no—
363C2×C4—41no—
364C2×C4—41no—
365C2×C4—41no—
366C2×C4—41no—
367C2×C4—41no—
368C2×C4—41no—
369C2×C4—41no—
370C2×C4—81no—
371C2×C4—81no—
372C2×C4—81no—
373C2×C4—82no—
374C2×C4#640811yescharacteristic
375C2×C4#640811yescharacteristic
376C8—562no—
377C8—562no—
378C7#1152711yescharacteristic, complemented, Sylow
379C22—21no—
380C22—21no—
381C22—21no—
382C22—21no—
383C22—41no—
384C22—41no—
385C22—42no—
386C22—42no—
387C22—42no—
388C22—42no—
389C22—41no—
390C22—41no—
391C22—41no—
392C22—41no—
393C22—42no—
394C22—41no—
395C22—81no—
396C22—81no—
397C22—81no—
398C22—84no—
399C22—84no—
400C22—81no—
401C22—81no—
402C22#8997911yescharacteristic
403C22C2×#640811yescharacteristic
404C22#9000911yescharacteristic
405C4—21no—
406C4—21no—
407C4—42no—
408C4—41no—
409C4—41no—
410C4—41no—
411C4—41no—
412C4—81no—
413C4—81no—
414C2—21no—
415C2—21no—
416C2—41no—
417C2—41no—
418C2—81no—
419C2—82no—
420C2#1172935611yescharacteristic
421C2#1172960111yescharacteristic
422C2#1172961311yescharacteristic
subgroupG/H
centerC22C2×#6408
commutatorC24×C7C22×C4
FrattiniC2×C22⋊C4C22×D7
FittingC7×#2321C2
socleC22×C7C2×#99
prime radicalC2×C7×#120C4
cosocleC2×C7×C22⋊C4C23
hypercenter#2321D7
hypocenterC7#11527
LCSG ▷ C24×C7 ▷ C22×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2321

Complements

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

NH ≅ G/Ntype
2. #11729357C2semidirect
3. #11729356C2semidirect
4. #11725442C2semidirect
5. #11725428C2semidirect
71. C24×C7M16semidirect
109. C7×C22⋊C4M16semidirect
266. C24#1136semidirect
308. C22⋊C4#1136semidirect
378. C7#11527semidirect

Maximal subgroups

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

subgroupindex
C7×#23212
#117293572
#117293562
#117254422
#117254282
#115277