Finite Group Databasebeta

← #13253977

Subgroups — #13253977

Order 1792.

592 conjugacy classes of subgroups.

The subgroup lattice has 354 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739521C211yesmaximal, characteristic
2C7×#1797C211yesmaximal, characteristic
3#11729288C211yesmaximal, characteristic
4#11725481C214yesmaximal
5C22×C42×C7C2211yescharacteristic
6C2×#6487—21no—
7C2×#6432C411yescharacteristic, complemented
8C2×C7×C4⋊C8—21no—
9C2×C7×C22⋊C8C2212yes—
10#89984—21no—
11#89984C2212yes—
12C2×C7⋊#95—21no—
13C2×C7⋊#95—21no—
14C2×C7⋊#95C2212yes—
15#89707C411yescharacteristic, complemented
16#89701—21no—
17#89700—21no—
18#89700C412yescomplemented
19C7×#336—21no—
20#9911—71nomaximal, Sylow
21C23×C4×C7D411yescharacteristic
22C23×C4×C7D411yescharacteristic
23C23×C4×C7C2311yescharacteristic
24C23×Dic7—22no—
25C22×Dic14—21no—
26C22×C7×C8—22no—
27C2×C42×C7—42no—
28C2×C42×C7C2×C411yescharacteristic
29C2×C42×C7C2×C411yescharacteristic
30C2×C42×C7D411yescharacteristic
31C2×C42×C7D411yescharacteristic
32C2×C22⋊Dic7—22no—
33C2×C4⋊Dic7—21no—
34C2×C4⋊Dic7—21no—
35C2×C4⋊Dic7—21no—
36C2×C4⋊Dic7C2×C412yes—
37C2×C4⋊Dic7C2×C412yes—
38#6487—44no—
39C2×Dic7⋊C4—22no—
40#6432—42no—
41C7×C4⋊C8—42no—
42C7×C22⋊C8—24no—
43C7⋊#95—22no—
44C7⋊#95—22no—
45C7⋊#95—22no—
46C7⋊#95—42no—
47C7⋊#95—42no—
48C7⋊#95Q814yescomplemented
49#2857—71no—
50#1797D711yescharacteristic
51#1434—71no—
52#1294—74no—
53C24×C7C2×D411yescharacteristic
54C22×C4×C7—22no—
55C22×C4×C7—22no—
56C22×C4×C7—41no—
57C22×C4×C7—41no—
58C22×C4×C7—41no—
59C22×C4×C7—41no—
60C22×C4×C7—41no—
61C22×C4×C7C22⋊C411yescharacteristic
62C22×C4×C7C22⋊C411yescharacteristic
63C22×C4×C7C22⋊C411yescharacteristic
64C22×C4×C7C22⋊C411yescharacteristic
65C22×C4×C7SD1612yes—
66C22×C4×C7Q1612yes—
67C22×C4×C7C22×C411yescharacteristic
68C22×C4×C7C2×D411yescharacteristic
69C22×C4×C7C2×Q812yes—
70C22×C4×C7#4112yes—
71C22×Dic7—22no—
72C22×Dic7—22no—
73C22×Dic7—24no—
74C22×Dic7—42no—
75C22×Dic7—42no—
76C22×Dic7—42no—
77C22×Dic7—42no—
78C22×Dic7—42no—
79C22×Dic7—42no—
80C2×Dic14—42no—
81C2×Dic14—42no—
82C2×Dic14—42no—
83C2×C7×C8—42no—
84C2×C7×C8—42no—
85C2×C7×C8—42no—
86C42×C7—22no—
87C42×C7—22no—
88C42×C7—41no—
89C42×C7—41no—
90C22⋊Dic7—44no—
91C4⋊Dic7—42no—
92C4⋊Dic7—42no—
93C4⋊Dic7—42no—
94C4⋊Dic7—42no—
95C4⋊Dic7—42no—
96Dic7⋊C4—44no—
97C2×C4⋊Q8—71no—
98C2×C22⋊Q8—141no—
99C22×C42C2×D711yescharacteristic
100C2×C4⋊C8—21no—
101C2×C22⋊C8C2×D712yes—
102#391—141no—
103#389—72no—
104#389—141no—
105#384—71no—
106#380—72no—
107#380—141no—
108C2×#95—72no—
109C2×#95—141no—
110C2×#95—141no—
111#336—21no—
112C23×C7—21no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7—21no—
116C23×C7C2×C22⋊C411yescharacteristic
117C23×C7C22⋊Q812yes—
118C23×C7C2×SD1611yescharacteristic
119C23×C7C2×Q1611yescharacteristic
120C23×C7AGL(1,ℤ/8)12yes—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7—44no—
139C2×C4×C7#9912yes—
140C2×C4×C7#10312yes—
141C2×C4×C7#10312yes—
142C2×C4×C7C4×Q812yes—
143C2×C4×C7#12411yescharacteristic
144C2×C4×C7C4⋊Q811yescharacteristic
145C2×Dic7—44no—
146C2×Dic7—44no—
147C2×Dic7—42no—
148C2×Dic7—42no—
149C2×Dic7—42no—
150C2×Dic7—42no—
151C2×Dic7—42no—
152C2×Dic7—42no—
153C2×Dic7—44no—
154C2×Dic7—82no—
155C2×Dic7—82no—
156C2×Dic7—82no—
157C2×Dic7—82no—
158Dic14—84no—
159C7×C8—82no—
160C22×Q8—141no—
161C23×C4—142no—
162C23×C4D2811yescharacteristic
163C23×C4D2811yescharacteristic
164C23×C4C22×D711yescharacteristic
165C22×C8—22no—
166C4⋊Q8—282no—
167C22⋊Q8—284no—
168C2×C4⋊C4—72no—
169C2×C4⋊C4—72no—
170C2×C4⋊C4—142no—
171C2×C4⋊C4—141no—
172C2×C4⋊C4—141no—
173C2×C4⋊C4—141no—
174C2×C22⋊C4—142no—
175C2×C42—42no—
176C2×C42C4×D711yescharacteristic
177C2×C42C4×D711yescharacteristic
178C2×C42C22⋊D711yescharacteristic
179C2×C42C22⋊D711yescharacteristic
180C4⋊C8—42no—
181C22⋊C8—24no—
182#95—74no—
183#95—142no—
184#95—142no—
185#95—142no—
186#95—282no—
187#95—282no—
188C22×C7—22no—
189C22×C7—22no—
190C22×C7—22no—
191C22×C7—22no—
192C22×C7—21no—
193C22×C7—21no—
194C22×C7—21no—
195C22×C7—21no—
196C22×C7#39111yescharacteristic
197C22×C7C2×#9911yescharacteristic
198C22×C7#41011yescharacteristic
199C22×C7C2×#10311yescharacteristic
200C22×C7#41811yescharacteristic
201C22×C7#42111yescharacteristic
202C22×C7#42111yescharacteristic
203C22×C7#47612yes—
204C22×C7#47712yes—
205C4×C7—21no—
206C4×C7—22no—
207C4×C7—21no—
208C4×C7—22no—
209C4×C7—22no—
210C4×C7—22no—
211C4×C7—41no—
212Dic7—84no—
213Dic7—82no—
214Dic7—82no—
215C24C2×D2811yescharacteristic
216C2×Q8—282no—
217C2×Q8—282no—
218C2×Q8—282no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—41no—
225C22×C4—41no—
226C22×C4—144no—
227C22×C4—142no—
228C22×C4—142no—
229C22×C4—282no—
230C22×C4—282no—
231C22×C4—282no—
232C22×C4—282no—
233C22×C4—282no—
234C22×C4—282no—
235C22×C4#113212yes—
236C22×C4Dic2812yes—
237C22×C4(C2×D7)⋊C411yescharacteristic
238C22×C4(C2×D7)⋊C411yescharacteristic
239C22×C4(C2×D7)⋊C411yescharacteristic
240C22×C4(C2×D7)⋊C411yescharacteristic
241C22×C4C2×C4×D711yescharacteristic
242C22×C4#115912yes—
243C22×C4Q8×D712yes—
244C22×C4C2×C22⋊D711yescharacteristic
245C2×C8—42no—
246C2×C8—42no—
247C2×C8—42no—
248C4⋊C4—282no—
249C4⋊C4—282no—
250C4⋊C4—284no—
251C4⋊C4—282no—
252C4⋊C4—282no—
253C4⋊C4—282no—
254C22⋊C4—284no—
255C42—22no—
256C42—22no—
257C42—41no—
258C42—41no—
259C2×C7—22no—
260C2×C7—22no—
261C2×C7#151011yescharacteristic
262C2×C7#151411yescharacteristic
263C2×C7#151811yescharacteristic
264C2×C7#152011yescharacteristic
265C2×C7#198711yescharacteristic
266C2×C7#199511yescharacteristic
267C2×C7#199611yescharacteristic
268C23—21no—
269C23—21no—
270C23—21no—
271C23—21no—
272C23#646012yes—
273C23C2×#113211yescharacteristic
274C23C2×Dic2811yescharacteristic
275C23#647112yes—
276C23C2×(C2×D7)⋊C411yescharacteristic
277Q8—564no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—42no—
294C2×C4—42no—
295C2×C4—44no—
296C2×C4—284no—
297C2×C4—284no—
298C2×C4—284no—
299C2×C4—282no—
300C2×C4—282no—
301C2×C4—282no—
302C2×C4—282no—
303C2×C4—282no—
304C2×C4—282no—
305C2×C4—562no—
306C2×C4—562no—
307C2×C4—562no—
308C2×C4—562no—
309C2×C4#638012yes—
310C2×C4#639012yes—
311C2×C4#639012yes—
312C2×C4#645012yes—
313C2×C4#649911yescharacteristic
314C2×C4#650711yescharacteristic
315C8—82no—
316C7#991111yescharacteristic, complemented, Sylow
317C22—21no—
318C22—21no—
319C22—22no—
320C22—22no—
321C22—21no—
322C22—21no—
323C22—22no—
324C22—22no—
325C22#8961412yes—
326C22#8962212yes—
327C22C2×#638011yescharacteristic
328C22#8974611yescharacteristic
329C22#8977811yescharacteristic
330C22C2×#639011yescharacteristic
331C22#8990311yescharacteristic
332C22#8991111yescharacteristic
333C22#8991111yescharacteristic
334C4—22no—
335C4—22no—
336C4—22no—
337C4—21no—
338C4—22no—
339C4—21no—
340C4—41no—
341C4—564no—
342C4—562no—
343C4—562no—
344C2—22no—
345C2—22no—
346C2#1172728311yescharacteristic
347C2#1172770011yescharacteristic
348C2#1172773211yescharacteristic
349C2#1172821311yescharacteristic
350C2#1172826911yescharacteristic
351C2#1172854711yescharacteristic
352C2#1172858911yescharacteristic
subgroupG/H
centerC23C2×(C2×D7)⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1797C2
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenter#1797D7
hypocenterC7#9911
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1797

Complements

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

NH ≅ G/Ntype
7. C2×#6432C4semidirect
15. #89707C4semidirect
18. #89700C4semidirect
48. C7⋊#95Q8semidirect
316. C7#9911semidirect

Maximal subgroups

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

subgroupindex
#117395212
C7×#17972
#117292882
#117254812
#99117