Finite Group Databasebeta

← #13242053

Subgroups — #13242053

Order 1792.

556 conjugacy classes of subgroups.

The subgroup lattice has 380 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C4×#6487C211yesmaximal, characteristic
2C7×#1728C211yesmaximal, characteristic
3#11729283C211yesmaximal, characteristic
4#11725286C212yesmaximal, complemented
5#11725284C212yesmaximal, complemented
6C22×C7×M16—21no—
7C22×C42×C7C2211yescharacteristic
8C2×#6487C411yescharacteristic
9C2×C4×Dic14C411yescharacteristic
10#89984C411yescharacteristic
11#89982—21no—
12C4×C22⋊Dic7—21no—
13C2×C22⋊(C7⋊C8)—21no—
14C4×C4⋊Dic7C411yescharacteristic
15#89705—21no—
16C4×Dic7⋊C4—21no—
17#89701C2211yescharacteristic
18#89700C2211yescharacteristic
19C2×#6377—21no—
20C7×#336—21no—
21C7×#336C2212yes—
22#89319C2212yes—
23#7634—71nomaximal, Sylow
24C23×C4×C7C2×C411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C23×C4×C7D411yescharacteristic
27C22×Dic14C2×C411yescharacteristic
28C2×C7×M16—24no—
29C2×C7×M16—42no—
30C2×C7×M16—42no—
31C22×C7×C8—22no—
32C2×C42×C7—42no—
33C2×C42×C7C2×C411yescharacteristic
34C2×C42×C7D411yescharacteristic
35C2×C42×C7D411yescharacteristic
36C2×C42×C7C2311yescharacteristic
37C2×C22⋊Dic7—21no—
38C2×C22⋊Dic7—21no—
39C2×C4⋊Dic7C2×C411yescharacteristic
40C2×C4⋊Dic7C2×C411yescharacteristic
41C2×C4⋊Dic7C2×C411yescharacteristic
42#6487—22no—
43#6487—22no—
44C2×Dic7⋊C4—21no—
45C2×Dic7⋊C4—21no—
46C2×Dic7⋊C4—21no—
47C2×C4×Dic7—21no—
48C2×C4×Dic7—21no—
49C22×C7⋊C8—22no—
50C4×Dic14—42no—
51C7⋊#95—21no—
52C7⋊#95—21no—
53C7⋊#95—21no—
54C7⋊#95—21no—
55C7⋊#95—21no—
56C22⋊(C7⋊C8)—24no—
57#6377—42no—
58C4×C22⋊Q8—71no—
59#1749—71no—
60#1728D711yescharacteristic
61#1272—72no—
62#1271—72no—
63C24×C7C22⋊C411yescharacteristic
64C22×C4×C7—22no—
65C22×C4×C7—22no—
66C22×C4×C7—21no—
67C22×C4×C7—21no—
68C22×C4×C7—22no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—22no—
72C22×C4×C7—41no—
73C22×C4×C7—41no—
74C22×C4×C7—41no—
75C22×C4×C7C22⋊C411yescharacteristic
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22⋊C411yescharacteristic
78C22×C4×C7C22×C411yescharacteristic
79C22×C4×C7C2×D411yescharacteristic
80C22×C4×C7C2×D411yescharacteristic
81C22×Dic7—21no—
82C22×Dic7—21no—
83C22×Dic7—21no—
84C22×Dic7—21no—
85C2×Dic14—21no—
86C2×Dic14—41no—
87C2×Dic14—41no—
88C2×Dic14M1612yescomplemented
89C7×M16—48no—
90C2×C7×C8—24no—
91C2×C7×C8—42no—
92C2×C7×C8—42no—
93C42×C7—22no—
94C42×C7—22no—
95C42×C7—41no—
96C42×C7—41no—
97C22⋊Dic7—24no—
98C22⋊Dic7—42no—
99C4⋊Dic7—21no—
100C4⋊Dic7—41no—
101C4⋊Dic7—41no—
102C4⋊Dic7M1612yescomplemented
103Dic7⋊C4—24no—
104Dic7⋊C4—42no—
105Dic7⋊C4—41no—
106Dic7⋊C4—41no—
107C4×Dic7—42no—
108C4×Dic7—42no—
109C2×C7⋊C8—24no—
110C2×C7⋊C8—42no—
111C2×C7⋊C8—42no—
112C22×M16—21no—
113C2×C22⋊Q8—71no—
114C2×C4×Q8—71no—
115C22×C42C2×D711yescharacteristic
116C2×C22⋊C8—141no—
117C2×#97—141no—
118#391—71no—
119#389—71no—
120#387—141no—
121#385—141no—
122#380—71no—
123C4×C4⋊C4—71no—
124C4×C4⋊C4—141no—
125C4×C22⋊C4—141no—
126#336—21no—
127#336—72no—
128#336C2×D712yes—
129C23×C7—21no—
130C23×C7—21no—
131C23×C7—21no—
132C23×C7—21no—
133C23×C7—21no—
134C23×C7—21no—
135C23×C7#10412yes—
136C23×C7C2×C22⋊C411yescharacteristic
137C2×C4×C7—24no—
138C2×C4×C7—24no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—21no—
151C2×C4×C7—21no—
152C2×C4×C7—44no—
153C2×C4×C7—42no—
154C2×C4×C7—42no—
155C2×C4×C7#9912yes—
156C2×C4×C7#10412yes—
157C2×C4×C7C2×M1611yescharacteristic
158C2×C4×C7C2×M1611yescharacteristic
159C2×C4×C7AGL(1,ℤ/8)11yescharacteristic
160C2×C4×C7#13712yes—
161C2×C4×C7#13711yescharacteristic
162C2×Dic7—22no—
163C2×Dic7—22no—
164C2×Dic7—41no—
165C2×Dic7—41no—
166C2×Dic7—41no—
167C2×Dic7—41no—
168C2×Dic7—41no—
169C2×Dic7—41no—
170C2×Dic7—41no—
171C2×Dic7—41no—
172C2×Dic7—41no—
173C2×Dic7—41no—
174Dic14—42no—
175Dic14—42no—
176C7×C8—44no—
177C7⋊C8—44no—
178C22×Q8—71no—
179C23×C4C4×D711yescharacteristic
180C23×C4D2811yescharacteristic
181C23×C4C22⋊D711yescharacteristic
182C2×M16—24no—
183C2×M16—42no—
184C2×M16—42no—
185C22×C8—22no—
186C22×C8—142no—
187C22⋊Q8—142no—
188C22⋊Q8—142no—
189C4×Q8—282no—
190C2×C4⋊C4—71no—
191C2×C4⋊C4—71no—
192C2×C4⋊C4—71no—
193C2×C4⋊C4—141no—
194C2×C4⋊C4—141no—
195C2×C4⋊C4—141no—
196C2×C22⋊C4—141no—
197C2×C22⋊C4—141no—
198C2×C42—42no—
199C2×C42—141no—
200C2×C42—141no—
201C2×C42C4×D711yescharacteristic
202C2×C42D2811yescharacteristic
203C2×C42C22⋊D711yescharacteristic
204C2×C42C22×D711yescharacteristic
205C22⋊C8—144no—
206#97—282no—
207#95—141no—
208#95—141no—
209#95—141no—
210#95—141no—
211#95—141no—
212C22×C7—22no—
213C22×C7—22no—
214C22×C7—22no—
215C22×C7—41no—
216C22×C7—41no—
217C22×C7—41no—
218C22×C7—41no—
219C22×C7#40711yescharacteristic
220C22×C7C2×#9911yescharacteristic
221C22×C7#41311yescharacteristic
222C22×C7#41711yescharacteristic
223C22×C7#41911yescharacteristic
224C22×C7C2×#10411yescharacteristic
225C22×C7C2×#10411yescharacteristic
226C4×C7—21no—
227C4×C7—21no—
228C4×C7—21no—
229C4×C7—22no—
230C4×C7—21no—
231C4×C7—22no—
232C4×C7—22no—
233C4×C7—41no—
234Dic7—42no—
235Dic7—42no—
236Dic7—81no—
237Dic7—81no—
238C24(C2×D7)⋊C411yescharacteristic
239C2×Q8—72no—
240C2×Q8—141no—
241C2×Q8—281no—
242C2×Q8—281no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—21no—
246C22×C4—22no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—22no—
250C22×C4—22no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—141no—
255C22×C4—141no—
256C22×C4—141no—
257C22×C4—141no—
258C22×C4(C2×D7)⋊C411yescharacteristic
259C22×C4(C2×D7)⋊C411yescharacteristic
260C22×C4(C2×D7)⋊C411yescharacteristic
261C22×C4C2×C4×D711yescharacteristic
262C22×C4C2×D2811yescharacteristic
263C22×C4C2×C22⋊D711yescharacteristic
264M16—48no—
265C2×C8—24no—
266C2×C8—42no—
267C2×C8—42no—
268C2×C8—144no—
269C2×C8—282no—
270C2×C8—282no—
271C4⋊C4—72no—
272C4⋊C4—144no—
273C4⋊C4—141no—
274C4⋊C4—281no—
275C4⋊C4—281no—
276C4⋊C4—281no—
277C4⋊C4—281no—
278C4⋊C4—282no—
279C22⋊C4—144no—
280C22⋊C4—282no—
281C42—22no—
282C42—22no—
283C42—41no—
284C42—41no—
285C42—282no—
286C42—282no—
287C2×C7—42no—
288C2×C7#146511yescharacteristic
289C2×C7#148911yescharacteristic
290C2×C7#149111yescharacteristic
291C2×C7#150411yescharacteristic
292C2×C7#150811yescharacteristic
293C2×C7#152411yescharacteristic
294C2×C7#155911yescharacteristic
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23#637912yes—
302C23C2×(C2×D7)⋊C411yescharacteristic
303Q8—282no—
304Q8—282no—
305C2×C4—24no—
306C2×C4—24no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—44no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—142no—
324C2×C4—142no—
325C2×C4—281no—
326C2×C4—281no—
327C2×C4—281no—
328C2×C4—281no—
329C2×C4—281no—
330C2×C4—281no—
331C2×C4—281no—
332C2×C4—281no—
333C2×C4—281no—
334C2×C4—281no—
335C2×C4#638012yes—
336C2×C4#639912yes—
337C2×C4D7×M1611yescharacteristic
338C2×C4D7×M1611yescharacteristic
339C2×C4#647212yes—
340C2×C4#649511yescharacteristic
341C2×C4#650511yescharacteristic
342C8—44no—
343C8—284no—
344C7#763411yescharacteristic, complemented, Sylow
345C22—22no—
346C22—22no—
347C22—22no—
348C22—41no—
349C22—41no—
350C22—41no—
351C22—41no—
352C22C2×#637911yescharacteristic
353C22C2×#638011yescharacteristic
354C22#8973811yescharacteristic
355C22#8989211yescharacteristic
356C22#8989811yescharacteristic
357C22#8990411yescharacteristic
358C22C2×#639911yescharacteristic
359C4—21no—
360C4—21no—
361C4—22no—
362C4—21no—
363C4—21no—
364C4—22no—
365C4—22no—
366C4—41no—
367C4—282no—
368C4—282no—
369C4—561no—
370C4—561no—
371C2—42no—
372C2#1172715211yescharacteristic
373C2#1172761111yescharacteristic
374C2#1172761211yescharacteristic
375C2#1172772211yescharacteristic
376C2#1172773411yescharacteristic
377C2#1172825111yescharacteristic
378C2#1172826411yescharacteristic
subgroupG/H
centerC23C2×(C2×D7)⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#1728C2
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C22⋊C4
cosocleC2×C42×C7C23
hypercenter#1728D7
hypocenterC7#7634
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1728

Complements

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

NH ≅ G/Ntype
4. #11725286C2semidirect
5. #11725284C2semidirect
88. C2×Dic14M16semidirect
102. C4⋊Dic7M16semidirect
344. C7#7634semidirect

Maximal subgroups

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

subgroupindex
C4×#64872
C7×#17282
#117292832
#117252862
#117252842
#76347