Finite Group Databasebeta

← #13251516

Subgroups — #13251516

Order 1792.

904 conjugacy classes of subgroups; only the 115 normal subgroups are shown.

normal subgroups only

This group has 904 conjugacy classes of subgroups — too many to draw. Only its 115 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×#2435C211yesmaximal, characteristic
2#11741488C211yesmaximal, characteristic
3#11729883C211yesmaximal, characteristic
4#11729443C212yesmaximal, complemented
5#11725422C212yesmaximal, complemented
6C23×C7×D4C2211yescharacteristic
7C2×C7×#124C412yescomplemented
8C2×C7×C22⋊Q8C2211yescharacteristic
9C22×C22⋊Dic7—21no—
10C7×#394—22no—
11C7×#390C412yescomplemented
12C7×#379—21no—
13C7×#379—21no—
14C4×C7×C22⋊C4C2211yescharacteristic
15#90019—21no—
16#89992—24no—
17C2×#6407—22no—
18#89983C2212yes—
19#89979C2212yes—
20#89333—24no—
21#9459—71nomaximal, Sylow
22C25×C7D411yescharacteristic
23C25×C7D411yescharacteristic
24C22×C7×Q8C2×C411yescharacteristic, complemented
25C22×C7×D4—22no—
26C22×C7×D4—22no—
27C22×C7×D4—44no—
28C22×C7×D4C2×C412yes—
29C22×C7×D4D412yes—
30C23×C4×C7C2311yescharacteristic
31C23×Dic7—22no—
32C7×#124—44no—
33C7×C22⋊Q8—22no—
34C7×C22⋊Q8—22no—
35C2×C7×C4⋊C4—21no—
36C2×C7×C4⋊C4C2×C411yescharacteristic, complemented
37C2×C7×C22⋊C4—22no—
38C2×C7×C22⋊C4—22no—
39C2×C7×C22⋊C4—21no—
40C2×C7×C22⋊C4—21no—
41C2×C7×C22⋊C4—22no—
42C2×C7×C22⋊C4—22no—
43C2×C42×C7C2×C412yescomplemented
44C2×C22⋊Dic7—22no—
45C2×C22⋊Dic7—22no—
46C2×C22⋊Dic7—22no—
47C2×C22⋊Dic7—22no—
48C2×C22⋊Dic7—44no—
49C2×#1136—22no—
50C7×#95—21no—
51#6407—44no—
52C7⋊#95—22no—
53C7⋊#95D414yes—
54C22⋊(C7⋊C8)—22no—
55C22⋊(C7⋊C8)—22no—
56#2435D711yescharacteristic
57#2308—71no—
58#1783—71no—
59#1511—72no—
60#1284—72no—
61C24×C7—21no—
62C24×C7—24no—
63C24×C7—21no—
64C24×C7—21no—
65C24×C7—21no—
66C24×C7—42no—
67C24×C7—42no—
68C24×C7—42no—
69C24×C7—81no—
70C24×C7—81no—
71C24×C7C22⋊C412yes—
72C24×C7C22⋊C412yes—
73C24×C7C2×D411yescharacteristic
74C2×C7×Q8—22no—
75C2×C7×Q8—42no—
76C2×C7×D4—24no—
77C2×C7×D4—24no—
78C2×C7×D4—24no—
79C2×C7×D4—24no—
80C2×C7×D4—42no—
81C2×C7×D4—42no—
82C2×C7×D4—44no—
83C2×C7×D4—44no—
84C2×C7×D4—48no—
85C22×C4×C7—22no—
86C22×C4×C7—21no—
87C22×C4×C7—21no—
88C22×C4×C7—42no—
89C22×C4×C7C22×C411yescharacteristic
90C22×C4×C7C2×D411yescharacteristic
91C22×C4×C7C2×D412yes—
92C22×C4×C7#4112yes—
93C22×Dic7—24no—
94C22×Dic7—44no—
95C22×Dic7—82no—
96C22×Dic7—82no—
97C7×C4⋊C4—22no—
98C7×C4⋊C4—42no—
99C7×C22⋊C4—42no—
100C7×C22⋊C4—44no—
101C7×C22⋊C4—44no—
102C7×C22⋊C4—41no—
103C7×C22⋊C4—41no—
104C7×C22⋊C4—82no—
105C7×C22⋊C4—82no—
106C42×C7—42no—
107C22⋊Dic7—44no—
108C22⋊Dic7—44no—
109C22⋊Dic7—82no—
110C22⋊Dic7—82no—
111#1136—44no—
112C2×C7⋊C8—42no—
113C23×D4C2×D711yescharacteristic
114C2×#124Dic712yescomplemented
115C2×C22⋊Q8C2×D711yescharacteristic
116C22×C22⋊C4—141no—
117C2×#100—142no—
118#407—72no—
119#394—22no—
120#390Dic712yescomplemented
121#386—144no—
122#381—72no—
123#379—21no—
124#379—21no—
125#379—141no—
126C4×C22⋊C4C2×D711yescharacteristic
127#328—144no—
128C23×C7—22no—
129C23×C7—22no—
130C23×C7—24no—
131C23×C7—22no—
132C23×C7—22no—
133C23×C7—22no—
134C23×C7—22no—
135C23×C7—22no—
136C23×C7—22no—
137C23×C7—22no—
138C23×C7—41no—
139C23×C7—42no—
140C23×C7—42no—
141C23×C7—42no—
142C23×C7—42no—
143C23×C7—42no—
144C23×C7—42no—
145C23×C7—42no—
146C23×C7—42no—
147C23×C7—42no—
148C23×C7—42no—
149C23×C7—81no—
150C23×C7—81no—
151C23×C7—81no—
152C23×C7—81no—
153C23×C7—88no—
154C23×C7—88no—
155C23×C7#9914yes—
156C23×C7C2×C22⋊C411yescharacteristic
157C23×C7#12011yescharacteristic
158C23×C7#12311yescharacteristic
159C7×Q8—82no—
160C7×D4—48no—
161C7×D4—48no—
162C2×C4×C7—22no—
163C2×C4×C7—22no—
164C2×C4×C7—42no—
165C2×C4×C7—42no—
166C2×C4×C7—42no—
167C2×C4×C7—42no—
168C2×C4×C7—42no—
169C2×C4×C7—41no—
170C2×C4×C7—41no—
171C2×C4×C7—42no—
172C2×C4×C7—42no—
173C2×C4×C7C4×D412yes—
174C2×C4×C7#12112yes—
175C2×Dic7—44no—
176C2×Dic7—44no—
177C2×Dic7—44no—
178C2×Dic7—88no—
179C7⋊C8—82no—
180C25C22⋊D711yescharacteristic
181C25C22⋊D711yescharacteristic
182C22×Q8C2×Dic711yescharacteristic, complemented
183C22×D4—22no—
184C22×D4—22no—
185C22×D4—44no—
186C22×D4C2×Dic712yes—
187C22×D4C22⋊D712yes—
188C23×C4—142no—
189C23×C4C22×D711yescharacteristic
190C2×M16—142no—
191#124—44no—
192C22⋊Q8—22no—
193C22⋊Q8—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4C2×Dic711yescharacteristic, complemented
196C2×C22⋊C4—22no—
197C2×C22⋊C4—22no—
198C2×C22⋊C4—21no—
199C2×C22⋊C4—21no—
200C2×C22⋊C4—22no—
201C2×C22⋊C4—22no—
202C2×C22⋊C4—142no—
203C2×C22⋊C4—142no—
204C2×C22⋊C4—142no—
205C2×C22⋊C4—142no—
206C2×C22⋊C4—284no—
207C2×C42C2×Dic712yescomplemented
208#100—284no—
209C22⋊C8—142no—
210C22⋊C8—142no—
211#95—21no—
212#95—74no—
213#95—142no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7—21no—
218C22×C7—21no—
219C22×C7—21no—
220C22×C7—24no—
221C22×C7—21no—
222C22×C7—44no—
223C22×C7—44no—
224C22×C7—44no—
225C22×C7—44no—
226C22×C7—44no—
227C22×C7—42no—
228C22×C7—42no—
229C22×C7—42no—
230C22×C7—42no—
231C22×C7—42no—
232C22×C7—82no—
233C22×C7—82no—
234C22×C7—84no—
235C22×C7—84no—
236C22×C7—84no—
237C22×C7#38611yescharacteristic
238C22×C7C2×#9912yes—
239C22×C7#45512yes—
240C4×C7—22no—
241C4×C7—22no—
242C4×C7—82no—
243C4×C7—81no—
244C4×C7—81no—
245Dic7—84no—
246C24—21no—
247C24—24no—
248C24—21no—
249C24—21no—
250C24—21no—
251C24—42no—
252C24—42no—
253C24—42no—
254C24—81no—
255C24—81no—
256C24C22⋊Dic712yes—
257C24C22⋊Dic712yes—
258C24C2×C22⋊D711yescharacteristic
259C2×Q8—22no—
260C2×Q8—42no—
261C2×D4—24no—
262C2×D4—24no—
263C2×D4—24no—
264C2×D4—24no—
265C2×D4—42no—
266C2×D4—42no—
267C2×D4—44no—
268C2×D4—44no—
269C2×D4—48no—
270C22×C4—22no—
271C22×C4—21no—
272C22×C4—21no—
273C22×C4—42no—
274C22×C4—144no—
275C22×C4—284no—
276C22×C4—562no—
277C22×C4—562no—
278C22×C4D4×D712yes—
279C22×C4#115912yes—
280C22×C4C22×Dic711yescharacteristic
281C22×C4C2×C22⋊D711yescharacteristic
282M16—284no—
283C2×C8—282no—
284C4⋊C4—22no—
285C4⋊C4—42no—
286C22⋊C4—42no—
287C22⋊C4—44no—
288C22⋊C4—44no—
289C22⋊C4—41no—
290C22⋊C4—41no—
291C22⋊C4—82no—
292C22⋊C4—82no—
293C22⋊C4—284no—
294C22⋊C4—284no—
295C22⋊C4—562no—
296C22⋊C4—562no—
297C42—42no—
298C2×C7—22no—
299C2×C7—22no—
300C2×C7—41no—
301C2×C7—44no—
302C2×C7—82no—
303C2×C7#188511yescharacteristic
304C2×C7#189811yescharacteristic
305C2×C7#191111yescharacteristic
306C23—22no—
307C23—22no—
308C23—24no—
309C23—22no—
310C23—22no—
311C23—22no—
312C23—22no—
313C23—22no—
314C23—22no—
315C23—22no—
316C23—41no—
317C23—42no—
318C23—42no—
319C23—42no—
320C23—42no—
321C23—42no—
322C23—42no—
323C23—42no—
324C23—42no—
325C23—42no—
326C23—42no—
327C23—81no—
328C23—81no—
329C23—81no—
330C23—81no—
331C23—88no—
332C23—88no—
333C23#640814yes—
334C23#649811yescharacteristic
335C23#650011yescharacteristic
336C23C2×C22⋊Dic711yescharacteristic
337Q8—82no—
338D4—48no—
339D4—48no—
340C2×C4—22no—
341C2×C4—22no—
342C2×C4—42no—
343C2×C4—42no—
344C2×C4—42no—
345C2×C4—42no—
346C2×C4—42no—
347C2×C4—42no—
348C2×C4—42no—
349C2×C4—41no—
350C2×C4—41no—
351C2×C4—284no—
352C2×C4—284no—
353C2×C4—284no—
354C2×C4—568no—
355C2×C4D4×Dic712yes—
356C2×C4#650212yes—
357C8—562no—
358C7#945911yescharacteristic, complemented, Sylow
359C22—21no—
360C22—21no—
361C22—21no—
362C22—21no—
363C22—21no—
364C22—24no—
365C22—21no—
366C22—21no—
367C22—44no—
368C22—44no—
369C22—44no—
370C22—44no—
371C22—44no—
372C22—42no—
373C22—42no—
374C22—42no—
375C22—42no—
376C22—42no—
377C22—82no—
378C22—82no—
379C22—84no—
380C22—84no—
381C22—84no—
382C22#8983312yes—
383C22C2×#640812yes—
384C22#8999211yescharacteristic
385C4—22no—
386C4—22no—
387C4—82no—
388C4—81no—
389C4—81no—
390C4—564no—
391C2—22no—
392C2—22no—
393C2—41no—
394C2—44no—
395C2—82no—
396C2#1172942411yescharacteristic
397C2#1172947711yescharacteristic
398C2#1172960711yescharacteristic
subgroupG/H
centerC22#89992
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2435C2
socleC22×C7#386
prime radicalC23×C7×D4C22
cosocleC23×C4×C7C23
hypercenter#2435D7
hypocenterC7#9459
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2435

Complements

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

NH ≅ G/Ntype
4. #11729443C2semidirect
5. #11725422C2semidirect
7. C2×C7×#124C4semidirect
11. C7×#390C4semidirect
24. C22×C7×Q8C2×C4semidirect
36. C2×C7×C4⋊C4C2×C4semidirect
43. C2×C42×C7C2×C4semidirect
114. C2×#124Dic7semidirect
120. #390Dic7semidirect
182. C22×Q8C2×Dic7semidirect
195. C2×C4⋊C4C2×Dic7semidirect
207. C2×C42C2×Dic7semidirect
358. C7#9459semidirect

Maximal subgroups

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

subgroupindex
C7×#24352
#117414882
#117298832
#117294432
#117254222
#94597