Finite Group Databasebeta

← #13247185

Subgroups — #13247185

Order 1792.

588 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739507C211yesmaximal, characteristic
2C7×#1434C211yesmaximal, characteristic
3#11729285C211yesmaximal, characteristic
4#11725418C214yesmaximal
5C22×C42×C7C2211yescharacteristic
6C2×#6489—21no—
7C2×C7×#95—21no—
8C2×C7×#95—21no—
9C2×C7×#95C2212yes—
10#89984—21no—
11#89984C2212yes—
12C2×C22⋊(C7⋊C8)C2212yes—
13#89708—21no—
14#89707C411yescharacteristic, complemented
15#89707C411yescharacteristic, complemented
16C4×C4⋊Dic7—21no—
17C4×C4⋊Dic7C412yescomplemented
18#89319—21no—
19#89319—21no—
20#8716—71nomaximal, Sylow
21C23×C4×C7—22no—
22C23×C4×C7D411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7C2311yescharacteristic
25C2×C42×C7—24no—
26C2×C42×C7C2×C411yescharacteristic
27C2×C42×C7C2×C411yescharacteristic
28C2×C42×C7D411yescharacteristic
29C2×C42×C7D411yescharacteristic
30C2×C22⋊Dic7—22no—
31#6489—42no—
32#6489—42no—
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—
40C22×C7⋊C8—22no—
41C7×#95—24no—
42C7×#95—24no—
43C7×#95—42no—
44C7⋊#95—22no—
45C7⋊#95—22no—
46C22⋊(C7⋊C8)—24no—
47#2285—71no—
48#1817—71no—
49#1434D711yescharacteristic
50#1294—74no—
51C24×C7C2×D411yescharacteristic
52C22×C4×C7—22no—
53C22×C4×C7—22no—
54C22×C4×C7—22no—
55C22×C4×C7—22no—
56C22×C4×C7—24no—
57C22×C4×C7—41no—
58C22×C4×C7—41no—
59C22×C4×C7—41no—
60C22×C4×C7—42no—
61C22×C4×C7—42no—
62C22×C4×C7—42no—
63C22×C4×C7—42no—
64C22×C4×C7—42no—
65C22×C4×C7—42no—
66C22×C4×C7C22⋊C411yescharacteristic
67C22×C4×C7C22⋊C411yescharacteristic
68C22×C4×C7C22⋊C411yescharacteristic
69C22×C4×C7C22⋊C411yescharacteristic
70C22×C4×C7D812yes—
71C22×C4×C7SD1612yes—
72C22×C4×C7C22×C411yescharacteristic
73C22×C4×C7C2×D411yescharacteristic
74C22×C4×C7#4114yes—
75C22×Dic7—22no—
76C22×Dic7—22no—
77C42×C7—22no—
78C42×C7—22no—
79C42×C7—22no—
80C42×C7—22no—
81C22⋊Dic7—42no—
82C22⋊Dic7—42no—
83C4⋊Dic7—42no—
84C4⋊Dic7—42no—
85C4⋊Dic7—42no—
86C4⋊Dic7—42no—
87C4⋊Dic7—42no—
88C4⋊Dic7—42no—
89C4×Dic7—44no—
90C2×C7⋊C8—42no—
91C2×C7⋊C8—42no—
92C2×C7⋊C8—42no—
93C2×#117—141no—
94C22×C42C2×D711yescharacteristic
95C2×C22⋊C8—72no—
96#384—71no—
97#384—71no—
98#382—141no—
99#380—72no—
100#380—141no—
101C4×C4⋊C4—72no—
102C4×C4⋊C4—141no—
103C2×#95—21no—
104C2×#95—21no—
105C2×#95C2×D712yes—
106#336—141no—
107#336—141no—
108C23×C7—21no—
109C23×C7—21no—
110C23×C7—21no—
111C23×C7—21no—
112C23×C7C2×C22⋊C411yescharacteristic
113C23×C7#12312yes—
114C23×C7C2×D811yescharacteristic
115C23×C7C2×SD1611yescharacteristic
116C23×C7#13712yes—
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—21no—
126C2×C4×C7—21no—
127C2×C4×C7—24no—
128C2×C4×C7—24no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—44no—
133C2×C4×C7—44no—
134C2×C4×C7—44no—
135C2×C4×C7—44no—
136C2×C4×C7—44no—
137C2×C4×C7—82no—
138C2×C4×C7—82no—
139C2×C4×C7—82no—
140C2×C4×C7—82no—
141C2×C4×C7#10212yes—
142C2×C4×C7#10212yes—
143C2×C4×C7#10412yes—
144C2×C4×C7#11712yes—
145C2×C4×C7#12312yes—
146C2×Dic7—42no—
147C2×Dic7—42no—
148C2×Dic7—42no—
149C2×Dic7—42no—
150C2×Dic7—42no—
151C2×Dic7—42no—
152C7⋊C8—82no—
153C23×C4—22no—
154C23×C4C22⋊D711yescharacteristic
155C23×C4C22⋊D711yescharacteristic
156C23×C4C22×D711yescharacteristic
157C22×C8—142no—
158#117—282no—
159#117—282no—
160C2×C4⋊C4—72no—
161C2×C4⋊C4—72no—
162C2×C4⋊C4—141no—
163C2×C4⋊C4—141no—
164C2×C4⋊C4—141no—
165C2×C4⋊C4—141no—
166C2×C22⋊C4—142no—
167C2×C42—24no—
168C2×C42—142no—
169C2×C42C4×D711yescharacteristic
170C2×C42C4×D711yescharacteristic
171C2×C42D2811yescharacteristic
172C2×C42D2811yescharacteristic
173C22⋊C8—144no—
174#95—24no—
175#95—24no—
176#95—42no—
177#95—142no—
178#95—142no—
179C22×C7—22no—
180C22×C7—22no—
181C22×C7—21no—
182C22×C7—21no—
183C22×C7—22no—
184C22×C7—22no—
185C22×C7—21no—
186C22×C7—21no—
187C22×C7#38111yescharacteristic
188C22×C7#41011yescharacteristic
189C22×C7#41011yescharacteristic
190C22×C7C2×#10211yescharacteristic
191C22×C7#41911yescharacteristic
192C22×C7C2×#10411yescharacteristic
193C22×C7#42111yescharacteristic
194C22×C7#48012yes—
195C22×C7#48312yes—
196C4×C7—22no—
197C4×C7—22no—
198C4×C7—22no—
199C4×C7—22no—
200C4×C7—22no—
201C4×C7—41no—
202C4×C7—84no—
203Dic7—82no—
204Dic7—82no—
205C24C2×C22⋊D711yescharacteristic
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—24no—
211C22×C4—41no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4—42no—
215C22×C4—42no—
216C22×C4—42no—
217C22×C4—42no—
218C22×C4—42no—
219C22×C4—42no—
220C22×C4—142no—
221C22×C4—142no—
222C22×C4(C2×D7)⋊C411yescharacteristic
223C22×C4(C2×D7)⋊C411yescharacteristic
224C22×C4(C2×D7)⋊C411yescharacteristic
225C22×C4(C2×D7)⋊C411yescharacteristic
226C22×C4#114112yes—
227C22×C4#114312yes—
228C22×C4C2×C4×D711yescharacteristic
229C22×C4C2×D2811yescharacteristic
230C22×C4#115914yes—
231C2×C8—282no—
232C2×C8—282no—
233C2×C8—282no—
234C4⋊C4—282no—
235C4⋊C4—282no—
236C4⋊C4—282no—
237C4⋊C4—282no—
238C4⋊C4—282no—
239C4⋊C4—282no—
240C22⋊C4—282no—
241C22⋊C4—282no—
242C42—22no—
243C42—22no—
244C42—22no—
245C42—22no—
246C42—284no—
247C2×C7—22no—
248C2×C7—22no—
249C2×C7#150211yescharacteristic
250C2×C7#150611yescharacteristic
251C2×C7#151511yescharacteristic
252C2×C7#151711yescharacteristic
253C2×C7#182011yescharacteristic
254C2×C7#182311yescharacteristic
255C2×C7#182611yescharacteristic
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23C2×(C2×D7)⋊C411yescharacteristic
261C23C2×#114111yescharacteristic
262C23#649812yes—
263C23C2×#114311yescharacteristic
264C23#651412yes—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—24no—
276C2×C4—24no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—44no—
281C2×C4—44no—
282C2×C4—44no—
283C2×C4—44no—
284C2×C4—44no—
285C2×C4—82no—
286C2×C4—82no—
287C2×C4—82no—
288C2×C4—82no—
289C2×C4—282no—
290C2×C4—282no—
291C2×C4—282no—
292C2×C4—282no—
293C2×C4—282no—
294C2×C4—282no—
295C2×C4#637912yes—
296C2×C4#638312yes—
297C2×C4#638312yes—
298C2×C4#644012yes—
299C2×C4#644912yes—
300C8—562no—
301C7#871611yescharacteristic, complemented, Sylow
302C22—22no—
303C22—22no—
304C22—21no—
305C22—21no—
306C22—22no—
307C22—21no—
308C22—21no—
309C22—22no—
310C22C2×#637911yescharacteristic
311C22#8971511yescharacteristic
312C22C2×#638311yescharacteristic
313C22#8977611yescharacteristic
314C22#8977711yescharacteristic
315C22#8977811yescharacteristic
316C22#8977811yescharacteristic
317C22#8980512yes—
318C22#8981512yes—
319C4—22no—
320C4—22no—
321C4—22no—
322C4—22no—
323C4—22no—
324C4—41no—
325C4—84no—
326C4—562no—
327C4—562no—
328C2—22no—
329C2—22no—
330C2#1172718711yescharacteristic
331C2#1172719611yescharacteristic
332C2#1172722311yescharacteristic
333C2#1172725511yescharacteristic
334C2#1172728811yescharacteristic
335C2#1172729911yescharacteristic
336C2#1172732011yescharacteristic
subgroupG/H
centerC23C2×(C2×D7)⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1434C2
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenter#1434D7
hypocenterC7#8716
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1434

Complements

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

NH ≅ G/Ntype
14. #89707C4semidirect
15. #89707C4semidirect
17. C4×C4⋊Dic7C4semidirect
301. C7#8716semidirect

Maximal subgroups

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

subgroupindex
#117395072
C7×#14342
#117292852
#117254182
#87167