Finite Group Databasebeta

← #13247192

Subgroups — #13247192

Order 1792.

588 conjugacy classes of subgroups.

The subgroup lattice has 334 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739507C211yesmaximal, characteristic
2C7×#1817C211yesmaximal, characteristic
3#11729288C211yesmaximal, characteristic
4#11725481C214yesmaximal
5C22×C42×C7C2211yescharacteristic
6C2×#6489—21no—
7C2×C7×C22⋊C8C2212yes—
8#89984—21no—
9#89984C2212yes—
10C2×C7⋊#95—21no—
11C2×C7⋊#95—21no—
12C2×C7⋊#95C2212yes—
13#89708—21no—
14#89707C411yescharacteristic, complemented
15#89707C411yescharacteristic, complemented
16C4×C4⋊Dic7—21no—
17C4×C4⋊Dic7C412yescomplemented
18C7×#336—21no—
19C7×#336—21no—
20#8717—71nomaximal, Sylow
21C23×C4×C7D411yescharacteristic
22C23×C4×C7D411yescharacteristic
23C23×C4×C7C2311yescharacteristic
24C23×Dic7—22no—
25C22×C7×C8—22no—
26C2×C42×C7—24no—
27C2×C42×C7C2×C411yescharacteristic
28C2×C42×C7C2×C411yescharacteristic
29C2×C42×C7D411yescharacteristic
30C2×C42×C7D411yescharacteristic
31C2×C22⋊Dic7—22no—
32#6489—44no—
33C2×C4⋊Dic7—21no—
34C2×C4⋊Dic7—21no—
35C2×C4⋊Dic7—21no—
36C2×C4⋊Dic7—21no—
37C2×C4⋊Dic7C2×C412yes—
38C2×C4⋊Dic7C2×C412yes—
39C2×C4×Dic7—22no—
40C7×C22⋊C8—24no—
41C7⋊#95—24no—
42C7⋊#95—22no—
43C7⋊#95—22no—
44C7⋊#95—24no—
45C7⋊#95—42no—
46#2285—71no—
47#1817D711yescharacteristic
48#1434—71no—
49#1294—74no—
50C24×C7C2×D411yescharacteristic
51C22×C4×C7—22no—
52C22×C4×C7—22no—
53C22×C4×C7—22no—
54C22×C4×C7—22no—
55C22×C4×C7—41no—
56C22×C4×C7—41no—
57C22×C4×C7—41no—
58C22×C4×C7C22⋊C411yescharacteristic
59C22×C4×C7C22⋊C411yescharacteristic
60C22×C4×C7C22⋊C411yescharacteristic
61C22×C4×C7C22⋊C411yescharacteristic
62C22×C4×C7SD1612yes—
63C22×C4×C7Q1612yes—
64C22×C4×C7C22×C411yescharacteristic
65C22×C4×C7C2×D411yescharacteristic
66C22×C4×C7#4114yes—
67C22×Dic7—24no—
68C22×Dic7—22no—
69C22×Dic7—22no—
70C22×Dic7—42no—
71C22×Dic7—42no—
72C22×Dic7—42no—
73C22×Dic7—42no—
74C22×Dic7—42no—
75C22×Dic7—42no—
76C2×C7×C8—42no—
77C2×C7×C8—42no—
78C2×C7×C8—42no—
79C42×C7—22no—
80C42×C7—22no—
81C42×C7—22no—
82C42×C7—22no—
83C22⋊Dic7—44no—
84C4⋊Dic7—42no—
85C4⋊Dic7—42no—
86C4⋊Dic7—42no—
87C4⋊Dic7—42no—
88C4⋊Dic7—42no—
89C4⋊Dic7—42no—
90C4×Dic7—44no—
91C2×#117—141no—
92C22×C42C2×D711yescharacteristic
93C2×C22⋊C8C2×D712yes—
94#384—71no—
95#384—71no—
96#382—141no—
97#380—72no—
98#380—141no—
99C4×C4⋊C4—72no—
100C4×C4⋊C4—141no—
101C2×#95—72no—
102C2×#95—141no—
103C2×#95—141no—
104#336—21no—
105#336—21no—
106C23×C7—21no—
107C23×C7—21no—
108C23×C7—21no—
109C23×C7—21no—
110C23×C7C2×C22⋊C411yescharacteristic
111C23×C7#12312yes—
112C23×C7C2×SD1611yescharacteristic
113C23×C7C2×Q1611yescharacteristic
114C23×C7AGL(1,ℤ/8)12yes—
115C2×C4×C7—21no—
116C2×C4×C7—21no—
117C2×C4×C7—21no—
118C2×C4×C7—21no—
119C2×C4×C7—21no—
120C2×C4×C7—21no—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—24no—
126C2×C4×C7—24no—
127C2×C4×C7—21no—
128C2×C4×C7—21no—
129C2×C4×C7—21no—
130C2×C4×C7—44no—
131C2×C4×C7—44no—
132C2×C4×C7#10312yes—
133C2×C4×C7#10312yes—
134C2×C4×C7#10412yes—
135C2×C4×C7#11712yes—
136C2×C4×C7#12312yes—
137C2×Dic7—44no—
138C2×Dic7—44no—
139C2×Dic7—44no—
140C2×Dic7—42no—
141C2×Dic7—42no—
142C2×Dic7—42no—
143C2×Dic7—42no—
144C2×Dic7—42no—
145C2×Dic7—42no—
146C2×Dic7—82no—
147C2×Dic7—82no—
148C2×Dic7—82no—
149C2×Dic7—82no—
150C7×C8—82no—
151C23×C4—142no—
152C23×C4D2811yescharacteristic
153C23×C4D2811yescharacteristic
154C23×C4C22×D711yescharacteristic
155C22×C8—22no—
156#117—284no—
157C2×C4⋊C4—72no—
158C2×C4⋊C4—72no—
159C2×C4⋊C4—141no—
160C2×C4⋊C4—141no—
161C2×C4⋊C4—141no—
162C2×C4⋊C4—141no—
163C2×C22⋊C4—142no—
164C2×C42—24no—
165C2×C42—142no—
166C2×C42C4×D711yescharacteristic
167C2×C42C4×D711yescharacteristic
168C2×C42C22⋊D711yescharacteristic
169C2×C42C22⋊D711yescharacteristic
170C22⋊C8—24no—
171#95—144no—
172#95—142no—
173#95—142no—
174#95—144no—
175#95—282no—
176C22×C7—22no—
177C22×C7—22no—
178C22×C7—21no—
179C22×C7—21no—
180C22×C7—22no—
181C22×C7—22no—
182C22×C7—21no—
183C22×C7—21no—
184C22×C7#38111yescharacteristic
185C22×C7#41011yescharacteristic
186C22×C7#41011yescharacteristic
187C22×C7C2×#10311yescharacteristic
188C22×C7#41811yescharacteristic
189C22×C7C2×#10411yescharacteristic
190C22×C7#42111yescharacteristic
191C22×C7#48112yes—
192C22×C7#48412yes—
193C4×C7—22no—
194C4×C7—22no—
195C4×C7—22no—
196C4×C7—22no—
197C4×C7—22no—
198C4×C7—41no—
199Dic7—84no—
200Dic7—82no—
201Dic7—82no—
202C24C2×D2811yescharacteristic
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—41no—
208C22×C4—41no—
209C22×C4—41no—
210C22×C4—144no—
211C22×C4—142no—
212C22×C4—142no—
213C22×C4—282no—
214C22×C4—282no—
215C22×C4—282no—
216C22×C4—282no—
217C22×C4—282no—
218C22×C4—282no—
219C22×C4#113212yes—
220C22×C4Dic2812yes—
221C22×C4(C2×D7)⋊C411yescharacteristic
222C22×C4(C2×D7)⋊C411yescharacteristic
223C22×C4(C2×D7)⋊C411yescharacteristic
224C22×C4(C2×D7)⋊C411yescharacteristic
225C22×C4C2×C4×D711yescharacteristic
226C22×C4#115914yes—
227C22×C4C2×C22⋊D711yescharacteristic
228C2×C8—42no—
229C2×C8—42no—
230C2×C8—42no—
231C4⋊C4—282no—
232C4⋊C4—282no—
233C4⋊C4—282no—
234C4⋊C4—282no—
235C4⋊C4—282no—
236C4⋊C4—282no—
237C22⋊C4—284no—
238C42—22no—
239C42—22no—
240C42—22no—
241C42—22no—
242C42—284no—
243C2×C7—22no—
244C2×C7—22no—
245C2×C7#150311yescharacteristic
246C2×C7#150511yescharacteristic
247C2×C7#151411yescharacteristic
248C2×C7#151811yescharacteristic
249C2×C7#182011yescharacteristic
250C2×C7#182411yescharacteristic
251C2×C7#182611yescharacteristic
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23#644912yes—
257C23C2×#113211yescharacteristic
258C23C2×Dic2811yescharacteristic
259C23#647112yes—
260C23C2×(C2×D7)⋊C411yescharacteristic
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—24no—
268C2×C4—24no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—44no—
277C2×C4—44no—
278C2×C4—284no—
279C2×C4—284no—
280C2×C4—284no—
281C2×C4—282no—
282C2×C4—282no—
283C2×C4—282no—
284C2×C4—282no—
285C2×C4—282no—
286C2×C4—282no—
287C2×C4—562no—
288C2×C4—562no—
289C2×C4—562no—
290C2×C4—562no—
291C2×C4#639012yes—
292C2×C4#639012yes—
293C2×C4#639912yes—
294C2×C4#644012yes—
295C2×C4#649812yes—
296C8—82no—
297C7#871711yescharacteristic, complemented, Sylow
298C22—22no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22#8949412yes—
307C22#8950712yes—
308C22#8971511yescharacteristic
309C22#8977811yescharacteristic
310C22#8977811yescharacteristic
311C22C2×#639011yescharacteristic
312C22#8990311yescharacteristic
313C22C2×#639911yescharacteristic
314C22#8991111yescharacteristic
315C4—22no—
316C4—22no—
317C4—22no—
318C4—22no—
319C4—22no—
320C4—41no—
321C4—564no—
322C4—562no—
323C4—562no—
324C2—22no—
325C2—22no—
326C2#1172728811yescharacteristic
327C2#1172767911yescharacteristic
328C2#1172772911yescharacteristic
329C2#1172854611yescharacteristic
330C2#1172854711yescharacteristic
331C2#1172858611yescharacteristic
332C2#1172858911yescharacteristic
subgroupG/H
centerC23C2×(C2×D7)⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1817C2
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenter#1817D7
hypocenterC7#8717
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1817

Complements

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

NH ≅ G/Ntype
14. #89707C4semidirect
15. #89707C4semidirect
17. C4×C4⋊Dic7C4semidirect
297. C7#8717semidirect

Maximal subgroups

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

subgroupindex
#117395072
C7×#18172
#117292882
#117254812
#87177