Finite Group Databasebeta

← #13251109

Subgroups — #13251109

Order 1792.

570 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 570 conjugacy classes of subgroups — too many to draw. Only its 127 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#11739830C211yesmaximal, characteristic
2#11739746C211yesmaximal, characteristic
3C7×#1799C211yesmaximal, characteristic
4#11727264C212yesmaximal, complemented
5#11725430C212yesmaximal, complemented
6C2×C4×C7×D4C2211yescharacteristic
7C2×#6487C2211yescharacteristic
8C2×C7×#99—21no—
9C7×#380C2212yes—
10#89992—21no—
11#89982—21no—
12C2×C7⋊#95C2211yescharacteristic
13C2×C22⋊(C7⋊C8)—21no—
14C2×C22⋊(C7⋊C8)C2212yes—
15D4×C7⋊C8—24no—
16#89758C411yescharacteristic, complemented
17#89751C411yescharacteristic, complemented
18#89721C411yescharacteristic, complemented
19#89719C411yescharacteristic, complemented
20#89714—21no—
21#89705—21no—
22#89697—21no—
23C7×#342—21no—
24#89247—24no—
25#9383—71nomaximal, Sylow
26C22×C7×D4D411yescharacteristic
27C23×C4×C7D411yescharacteristic
28C23×C4×C7C2311yescharacteristic
29C23×Dic7—21no—
30C22×Dic14C2×C411yescharacteristic
31C4×C7×D4—22no—
32C4×C7×D4—22no—
33C2×C7×C4⋊C4—22no—
34C2×C7×C4⋊C4C2×C411yescharacteristic
35C2×C7×C22⋊C4—22no—
36C2×C7×C22⋊C4C2×C411yescharacteristic
37C2×C7×C22⋊C4D411yescharacteristic
38C2×C42×C7D411yescharacteristic
39C2×C22⋊Dic7—21no—
40C2×C22⋊Dic7—21no—
41C2×C22⋊Dic7—21no—
42C2×C4⋊Dic7C2×C411yescharacteristic
43#6487—22no—
44#6487—22no—
45C2×Dic7⋊C4—21no—
46C2×Dic7⋊C4—21no—
47C2×Dic7⋊C4—21no—
48C22×C7⋊C8—22no—
49C22×C7⋊C8—22no—
50C7×#99—42no—
51C7×#95—22no—
52C7⋊#95—21no—
53C7⋊#95—21no—
54C7⋊#95—21no—
55C7⋊#95—21no—
56C7⋊#95—21no—
57C7⋊#95C2×C411yescharacteristic
58C7⋊#95C2×C411yescharacteristic
59C22⋊(C7⋊C8)—22no—
60C22⋊(C7⋊C8)—22no—
61C22⋊(C7⋊C8)D414yes—
62C4⋊(C7⋊C8)—22no—
63C4×C7⋊C8—22no—
64#2972—71no—
65#2564—71no—
66#1799D711yescharacteristic
67#1520—72no—
68#1294—72no—
69C24×C7C22⋊C411yescharacteristic
70C24×C7C2×D411yescharacteristic
71C2×C7×D4—22no—
72C2×C7×D4—22no—
73C2×C7×D4SD1612yes—
74C2×C7×D4Q1612yes—
75C22×C4×C7—21no—
76C22×C4×C7—21no—
77C22×C4×C7—42no—
78C22×C4×C7—81no—
79C22×C4×C7—81no—
80C22×C4×C7C22⋊C411yescharacteristic
81C22×C4×C7C22⋊C411yescharacteristic
82C22×C4×C7C22⋊C411yescharacteristic
83C22×C4×C7C22×C411yescharacteristic
84C22×C4×C7C2×D412yes—
85C22×C4×C7C2×D411yescharacteristic
86C22×C4×C7#4112yes—
87C22×Dic7—21no—
88C22×Dic7—21no—
89C22×Dic7—21no—
90C22×Dic7—21no—
91C22×Dic7—42no—
92C22×Dic7—42no—
93C22×Dic7—81no—
94C2×Dic14—22no—
95C2×Dic14—81no—
96C7×C4⋊C4—22no—
97C7×C4⋊C4—82no—
98C7×C22⋊C4—22no—
99C7×C22⋊C4—22no—
100C7×C22⋊C4—42no—
101C7×C22⋊C4—42no—
102C42×C7—21no—
103C42×C7—21no—
104C22⋊Dic7—42no—
105C22⋊Dic7—42no—
106C22⋊Dic7—81no—
107C4⋊Dic7—22no—
108Dic7⋊C4—42no—
109Dic7⋊C4—41no—
110Dic7⋊C4—41no—
111Dic7⋊C4—81no—
112C2×C7⋊C8—24no—
113C2×C7⋊C8—82no—
114C2×C7⋊C8—82no—
115C2×C22⋊Q8—71no—
116C2×C4×D4C2×D711yescharacteristic
117C8×D4—144no—
118#431—141no—
119C2×#99—21no—
120C2×C22⋊C8—72no—
121C2×C22⋊C8—141no—
122#400—71no—
123#399—71no—
124#395—71no—
125#393—71no—
126#387—141no—
127#386—141no—
128#385—141no—
129#380—141no—
130#380C2×D712yes—
131C2×#95—71no—
132#342—21no—
133#328—144no—
134C23×C7—21no—
135C23×C7—21no—
136C23×C7—41no—
137C23×C7—41no—
138C23×C7—41no—
139C23×C7—41no—
140C23×C7#10312yes—
141C23×C7C2×C22⋊C411yescharacteristic
142C23×C7#12112yes—
143C23×C7C2×SD1611yescharacteristic
144C23×C7C2×Q1611yescharacteristic
145C23×C7#13512yes—
146C7×D4—24no—
147C7×D4—24no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—42no—
153C2×C4×C7—42no—
154C2×C4×C7—41no—
155C2×C4×C7—41no—
156C2×C4×C7—42no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—81no—
160C2×C4×C7—82no—
161C2×C4×C7—82no—
162C2×C4×C7—82no—
163C2×C4×C7#10312yes—
164C2×C4×C7C4×D412yes—
165C2×C4×C7#12011yescharacteristic
166C2×C4×C7#12311yescharacteristic
167C2×Dic7—22no—
168C2×Dic7—41no—
169C2×Dic7—41no—
170C2×Dic7—41no—
171C2×Dic7—41no—
172C2×Dic7—41no—
173C2×Dic7—41no—
174C2×Dic7—41no—
175C2×Dic7—41no—
176C2×Dic7—41no—
177C2×Dic7—41no—
178C2×Dic7—41no—
179C2×Dic7—84no—
180Dic14—82no—
181C7⋊C8—44no—
182C22×Q8—71no—
183C22×D4C22⋊D711yescharacteristic
184C23×C4—141no—
185C23×C4D2811yescharacteristic
186C23×C4C22×D711yescharacteristic
187C22×C8—142no—
188C22×C8—142no—
189C22⋊Q8—142no—
190C22⋊Q8—142no—
191C4×D4—22no—
192C4×D4—22no—
193C2×C4⋊C4—22no—
194C2×C4⋊C4—71no—
195C2×C4⋊C4—141no—
196C2×C4⋊C4—141no—
197C2×C4⋊C4—141no—
198C2×C4⋊C4C4×D711yescharacteristic
199C2×C22⋊C4—22no—
200C2×C22⋊C4—141no—
201C2×C22⋊C4—141no—
202C2×C22⋊C4—141no—
203C2×C22⋊C4C4×D711yescharacteristic
204C2×C22⋊C4C22⋊D711yescharacteristic
205C2×C42D2811yescharacteristic
206C4⋊C8—142no—
207#99—42no—
208C22⋊C8—74no—
209C22⋊C8—142no—
210C22⋊C8—142no—
211C4×C8—142no—
212#95—22no—
213#95—71no—
214#95—71no—
215#95—141no—
216#95—141no—
217#95—141no—
218#95—141no—
219#95—141no—
220C22×C7—21no—
221C22×C7—22no—
222C22×C7—21no—
223C22×C7—22no—
224C22×C7—22no—
225C22×C7—42no—
226C22×C7—41no—
227C22×C7—41no—
228C22×C7—42no—
229C22×C7—41no—
230C22×C7—41no—
231C22×C7—81no—
232C22×C7#38611yescharacteristic
233C22×C7C2×#10311yescharacteristic
234C22×C7#41611yescharacteristic
235C22×C7#46512yes—
236C22×C7#46712yes—
237C4×C7—22no—
238C4×C7—21no—
239C4×C7—22no—
240C4×C7—21no—
241C4×C7—41no—
242C4×C7—41no—
243C4×C7—162no—
244Dic7—42no—
245Dic7—81no—
246Dic7—81no—
247Dic7—81no—
248C24(C2×D7)⋊C411yescharacteristic
249C24C2×C22⋊D711yescharacteristic
250C2×Q8—142no—
251C2×Q8—561no—
252C2×D4—22no—
253C2×D4—22no—
254C2×D4#114212yes—
255C2×D4#114412yes—
256C22×C4—21no—
257C22×C4—21no—
258C22×C4—42no—
259C22×C4—81no—
260C22×C4—81no—
261C22×C4—141no—
262C22×C4—141no—
263C22×C4—141no—
264C22×C4—141no—
265C22×C4—282no—
266C22×C4—282no—
267C22×C4—561no—
268C22×C4(C2×D7)⋊C411yescharacteristic
269C22×C4(C2×D7)⋊C411yescharacteristic
270C22×C4(C2×D7)⋊C411yescharacteristic
271C22×C4C2×C4×D711yescharacteristic
272C22×C4C2×D2811yescharacteristic
273C22×C4D4×D712yes—
274C22×C4#115912yes—
275C2×C8—144no—
276C2×C8—562no—
277C2×C8—562no—
278C4⋊C4—22no—
279C4⋊C4—82no—
280C4⋊C4—142no—
281C4⋊C4—282no—
282C4⋊C4—281no—
283C4⋊C4—281no—
284C4⋊C4—561no—
285C22⋊C4—22no—
286C22⋊C4—22no—
287C22⋊C4—42no—
288C22⋊C4—42no—
289C22⋊C4—282no—
290C22⋊C4—282no—
291C22⋊C4—561no—
292C42—21no—
293C42—21no—
294C2×C7—21no—
295C2×C7—22no—
296C2×C7—21no—
297C2×C7—22no—
298C2×C7—42no—
299C2×C7#188411yescharacteristic
300C2×C7#189211yescharacteristic
301C2×C7#191311yescharacteristic
302C23—21no—
303C23—21no—
304C23—41no—
305C23—41no—
306C23—41no—
307C23—41no—
308C23#638412yes—
309C23C2×(C2×D7)⋊C411yescharacteristic
310C23C2×#114211yescharacteristic
311C23#650212yes—
312C23C2×#114411yescharacteristic
313C23#651312yes—
314Q8—562no—
315D4—24no—
316D4—24no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—81no—
329C2×C4—82no—
330C2×C4—82no—
331C2×C4—82no—
332C2×C4—142no—
333C2×C4—281no—
334C2×C4—281no—
335C2×C4—281no—
336C2×C4—281no—
337C2×C4—281no—
338C2×C4—281no—
339C2×C4—281no—
340C2×C4—281no—
341C2×C4—281no—
342C2×C4—281no—
343C2×C4—281no—
344C2×C4—564no—
345C2×C4#638412yes—
346C2×C4(C22⋊D7)⋊C412yes—
347C2×C4#644511yescharacteristic
348C2×C4#644911yescharacteristic
349C8—284no—
350C7#938311yescharacteristic, complemented, Sylow
351C22—21no—
352C22—22no—
353C22—21no—
354C22—22no—
355C22—22no—
356C22—42no—
357C22—41no—
358C22—41no—
359C22—42no—
360C22—41no—
361C22—41no—
362C22—81no—
363C22#8973011yescharacteristic
364C22C2×#638411yescharacteristic
365C22#8977411yescharacteristic
366C22#8981312yes—
367C22#8982312yes—
368C4—21no—
369C4—21no—
370C4—22no—
371C4—22no—
372C4—41no—
373C4—41no—
374C4—162no—
375C4—282no—
376C4—561no—
377C4—561no—
378C4—561no—
379C2—22no—
380C2—21no—
381C2—21no—
382C2—22no—
383C2—42no—
384C2#1172726811yescharacteristic
385C2#1172777811yescharacteristic
386C2#1172779911yescharacteristic
subgroupG/H
centerC22#89730
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1799C2
socleC22×C7#386
prime radicalC22×C7×D4D4
cosocleC23×C4×C7C23
hypercenter#1799D7
hypocenterC7#9383
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1799

Complements

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

NH ≅ G/Ntype
4. #11727264C2semidirect
5. #11725430C2semidirect
16. #89758C4semidirect
17. #89751C4semidirect
18. #89721C4semidirect
19. #89719C4semidirect
350. C7#9383semidirect

Maximal subgroups

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

subgroupindex
#117398302
#117397462
C7×#17992
#117272642
#117254302
#93837