Finite Group Databasebeta

← #13358424

Subgroups — #13358424

Order 1792.

1162 conjugacy classes of subgroups.

The subgroup lattice has 378 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#90161C211yesmaximal, characteristic
2C4×C7×#117C211yesmaximal, characteristic
3C2×#89696C211yesmaximal, characteristic
4#11729332C214yesmaximal
5#11727195C214yesmaximal, complemented
6#11727150C214yesmaximal, complemented
7C2×C7×#117C2212yes—
8C22×C42×C7C2211yescharacteristic
9C2×#6489C2212yes—
10C2×#6487—22no—
11C22×#1136C2212yes—
12#90161—22no—
13#90161—22no—
14#90161C412yescomplemented
15#90161C412yescomplemented
16#90161C2212yes—
17#90161C2212yes—
18C2×#6433C412yescomplemented
19C2×#6432C412yescomplemented
20C2×C4×Dic14—22no—
21C4×C7×C4⋊C4—22no—
22C4×C7×C22⋊C4—22no—
23C7×#376C2212yes—
24C43×C7C2212yes—
25#89696—22no—
26#89696C2212yes—
27#89696C2212yes—
28C2×C4⋊(C7⋊C8)—22no—
29#89318C22116yes—
30#21329—71nomaximal, Sylow
31C23×C4×C7D412yes—
32C23×C4×C7C2311yescharacteristic
33C22×Dic14—22no—
34C7×#117—24no—
35C7×#117Q818yescomplemented
36C2×C7×C4⋊C4—22no—
37C2×C7×C22⋊C4—22no—
38C2×C42×C7—24no—
39C2×C42×C7C2×C411yescharacteristic
40C2×C42×C7C2×C411yescharacteristic
41C2×C42×C7C2×C411yescharacteristic
42C2×C42×C7C2×C411yescharacteristic
43C2×C42×C7D412yes—
44C2×C42×C7D412yes—
45C2×C42×C7D412yes—
46C2×C42×C7C2314yes—
47C2×C42×C7C2311yescharacteristic
48C2×C42×C7C2311yescharacteristic
49C2×C22⋊Dic7—22no—
50#6489—22no—
51#6489—22no—
52#6489C2×C414yes—
53#6489C2314yes—
54C2×C4⋊Dic7—22no—
55C2×C4⋊Dic7C2×C414yes—
56#6487—24no—
57#6487—24no—
58#6487—24no—
59#6487—24no—
60C2×Dic7⋊C4—24no—
61C2×C4×Dic7—22no—
62C2×#1136—22no—
63C2×#1136—28no—
64C2×#1136C2314yes—
65C22×C7⋊C8—22no—
66#6433—24no—
67#6433C2×C414yes—
68#6433C2×C414yes—
69#6432—22no—
70#6432—22no—
71#6432C2×C414yes—
72#6432C2×C414yes—
73C4×Dic14—24no—
74C4×Dic14—24no—
75C4×Dic14—24no—
76C4×Dic14—24no—
77C7×#95—24no—
78C4⋊(C7⋊C8)—24no—
79C4⋊(C7⋊C8)—24no—
80C2×#533—71no—
81C2×#423—71no—
82C4×#117D711yescharacteristic
83#1995—74no—
84#1994—74no—
85#1733—74no—
86C24×C7C2×D411yescharacteristic
87C22×C4×C7—24no—
88C22×C4×C7—44no—
89C22×C4×C7C22⋊C412yes—
90C22×C4×C7C22⋊C412yes—
91C22×C4×C7C22⋊C412yes—
92C22×C4×C7C22⋊C412yes—
93C22×C4×C7C22×C411yescharacteristic
94C22×C4×C7C22×C411yescharacteristic
95C22×C4×C7C2×D412yes—
96C22×C4×C7C2×D412yes—
97C22×C4×C7C2×D411yescharacteristic
98C22×C4×C7C2×D411yescharacteristic
99C22×C4×C7C2×D411yescharacteristic
100C22×C4×C7C2×Q814yes—
101C22×C4×C7#4114yes—
102C22×C4×C7C2411yescharacteristic
103C22×Dic7—24no—
104C2×Dic14—24no—
105C2×Dic14—24no—
106C2×Dic14—48no—
107C7×C4⋊C4—28no—
108C7×C22⋊C4—28no—
109C42×C7—24no—
110C42×C7—48no—
111C42×C7C22⋊C412yes—
112C42×C7C22⋊C412yes—
113C42×C7C22⋊C412yes—
114C42×C7C22⋊C412yes—
115C42×C7C22×C411yescharacteristic
116C42×C7C22×C411yescharacteristic
117C42×C7C22×C411yescharacteristic
118C42×C7C22×C411yescharacteristic
119C42×C7C2×D412yes—
120C42×C7C2×D412yes—
121C42×C7C2×Q818yescomplemented
122C22⋊Dic7—24no—
123C22⋊Dic7—24no—
124C4⋊Dic7—24no—
125C4⋊Dic7—24no—
126C4⋊Dic7—24no—
127C4⋊Dic7C22×C418yes—
128Dic7⋊C4—28no—
129Dic7⋊C4—28no—
130C4×Dic7—24no—
131C4×Dic7—24no—
132#1136—28no—
133#1136—44no—
134C2×C7⋊C8—24no—
135C2×C7⋊C8—44no—
136C22×M16—72no—
137#533—72no—
138#533—72no—
139#533—72no—
140#533—72no—
141#533—142no—
142#533—142no—
143C2×C4⋊Q8—72no—
144C2×#125—72no—
145C2×C22⋊Q8—142no—
146C2×C4×Q8—142no—
147C2×#117—72no—
148C2×#117C2×D712yes—
149C22×C42C2×D711yescharacteristic
150#423—72no—
151#423—72no—
152#423—142no—
153C2×C4⋊C8—142no—
154C4×C4⋊C4—22no—
155C4×C22⋊C4—22no—
156#376C2×D712yes—
157C43C2×D712yes—
158#339—716no—
159C23×C7—42no—
160C23×C7C2×C22⋊C411yescharacteristic
161C23×C7C2×C22⋊C411yescharacteristic
162C23×C7C22⋊Q814yes—
163C23×C7C22×D411yescharacteristic
164C2×C4×C7—24no—
165C2×C4×C7—24no—
166C2×C4×C7—24no—
167C2×C4×C7—28no—
168C2×C4×C7—42no—
169C2×C4×C7—42no—
170C2×C4×C7—42no—
171C2×C4×C7—48no—
172C2×C4×C7C2×C22⋊C412yes—
173C2×C4×C7C2×C22⋊C412yes—
174C2×C4×C7C2×C22⋊C412yes—
175C2×C4×C7C2×C22⋊C411yescharacteristic
176C2×C4×C7C2×C22⋊C411yescharacteristic
177C2×C4×C7C2×C22⋊C412yes—
178C2×C4×C7C4×Q814yes—
179C2×C4×C7C4×Q814yes—
180C2×C4×C7C22⋊Q814yes—
181C2×C4×C7#12412yes—
182C2×C4×C7#12412yes—
183C2×C4×C7C4⋊Q812yes—
184C2×C4×C7C4⋊Q812yes—
185C2×C4×C7C23×C411yescharacteristic
186C2×C4×C7C22×D411yescharacteristic
187C2×C4×C7C22×Q812yes—
188C2×C4×C7C2×#4112yes—
189C2×Dic7—28no—
190C2×Dic7—48no—
191Dic14—48no—
192Dic14—48no—
193C7⋊C8—44no—
194C22×Q8—142no—
195C23×C4C22⋊D712yes—
196C23×C4C22×D711yescharacteristic
197C2×M16—74no—
198C2×M16—142no—
199C2×M16—148no—
200C22×C8—142no—
201C4⋊Q8—74no—
202C4⋊Q8—74no—
203C4⋊Q8—142no—
204C4⋊Q8—142no—
205#125—74no—
206#125—74no—
207#125—144no—
208C22⋊Q8—144no—
209C22⋊Q8—144no—
210C22⋊Q8—144no—
211C22⋊Q8—144no—
212C4×Q8—144no—
213C4×Q8—144no—
214C4×Q8—144no—
215C4×Q8—144no—
216#117—24no—
217#117—74no—
218#117—74no—
219#117—142no—
220#117—142no—
221#117Dic1418yescomplemented
222C2×C4⋊C4—22no—
223C2×C4⋊C4—74no—
224C2×C4⋊C4—144no—
225C2×C4⋊C4—142no—
226C2×C22⋊C4—22no—
227C2×C22⋊C4—142no—
228C2×C42—24no—
229C2×C42—142no—
230C2×C42C4×D711yescharacteristic
231C2×C42C4×D711yescharacteristic
232C2×C42C4×D711yescharacteristic
233C2×C42C4×D711yescharacteristic
234C2×C42D2812yes—
235C2×C42D2812yes—
236C2×C42C22⋊D712yes—
237C2×C42C22×D714yes—
238C2×C42C22×D711yescharacteristic
239C2×C42C22×D711yescharacteristic
240C4⋊C8—144no—
241C4⋊C8—144no—
242#95—24no—
243C22×C7—22no—
244C22×C7—22no—
245C22×C7—22no—
246C22×C7—44no—
247C22×C7#39112yes—
248C22×C7#39112yes—
249C22×C7C22×C22⋊C411yescharacteristic
250C22×C7C2×C22⋊Q812yes—
251C4×C7—22no—
252C4×C7—22no—
253C4×C7—22no—
254C4×C7—48no—
255C4×C7#39112yes—
256C4×C7#39112yes—
257C4×C7#42114yes—
258C4×C7C2×C4×Q812yes—
259C4×C7C2×#12411yescharacteristic
260C4×C7C2×C4⋊Q811yescharacteristic
261Dic7—48no—
262C24C2×C22⋊D711yescharacteristic
263C2×Q8—144no—
264C2×Q8—144no—
265C2×Q8—288no—
266C22×C4—24no—
267C22×C4—44no—
268C22×C4—144no—
269C22×C4(C2×D7)⋊C412yes—
270C22×C4(C2×D7)⋊C412yes—
271C22×C4(C2×D7)⋊C412yes—
272C22×C4(C2×D7)⋊C412yes—
273C22×C4C2×Dic1414yes—
274C22×C4C2×C4×D711yescharacteristic
275C22×C4C2×C4×D711yescharacteristic
276C22×C4C2×D2811yescharacteristic
277C22×C4C2×D2811yescharacteristic
278C22×C4#115714yes—
279C22×C4C2×C22⋊D712yes—
280C22×C4C2×C22⋊D712yes—
281C22×C4C2×C22⋊D711yescharacteristic
282C22×C4C23×D711yescharacteristic
283M16—148no—
284M16—284no—
285C2×C8—144no—
286C2×C8—284no—
287C4⋊C4—28no—
288C4⋊C4—78no—
289C4⋊C4—148no—
290C4⋊C4—148no—
291C4⋊C4—144no—
292C4⋊C4—144no—
293C4⋊C4—144no—
294C22⋊C4—28no—
295C22⋊C4—144no—
296C22⋊C4—144no—
297C42—24no—
298C42—48no—
299C42—144no—
300C42—144no—
301C42(C2×D7)⋊C412yes—
302C42(C2×D7)⋊C412yes—
303C42(C2×D7)⋊C412yes—
304C42(C2×D7)⋊C412yes—
305C42C2×Dic1418yescomplemented
306C42C2×C4×D711yescharacteristic
307C42C2×C4×D711yescharacteristic
308C42C2×C4×D711yescharacteristic
309C42C2×C4×D711yescharacteristic
310C42C2×D2812yes—
311C42C2×D2812yes—
312C2×C7—22no—
313C2×C7—22no—
314C2×C7—22no—
315C2×C7C2×#39111yescharacteristic
316C2×C7C2×#42112yes—
317C23—42no—
318C23#648714yes—
319C23C2×(C2×D7)⋊C411yescharacteristic
320C23C2×(C2×D7)⋊C411yescharacteristic
321C23C22×C22⋊D711yescharacteristic
322Q8—288no—
323Q8—288no—
324C2×C4—24no—
325C2×C4—24no—
326C2×C4—24no—
327C2×C4—28no—
328C2×C4—42no—
329C2×C4—42no—
330C2×C4—42no—
331C2×C4—48no—
332C2×C4—148no—
333C2×C4—288no—
334C2×C4C4×Dic1414yes—
335C2×C4C4×Dic1414yes—
336C2×C4#643212yes—
337C2×C4#643212yes—
338C2×C4#643812yes—
339C2×C4#643812yes—
340C2×C4#648714yes—
341C2×C4C2×(C2×D7)⋊C412yes—
342C2×C4C2×(C2×D7)⋊C412yes—
343C2×C4C2×(C2×D7)⋊C412yes—
344C2×C4C2×(C2×D7)⋊C411yescharacteristic
345C2×C4C2×(C2×D7)⋊C411yescharacteristic
346C2×C4C2×(C2×D7)⋊C412yes—
347C2×C4C22×Dic1412yes—
348C2×C4C22×C4×D711yescharacteristic
349C2×C4C22×D2811yescharacteristic
350C2×C4C2×#115712yes—
351C8—284no—
352C7#2132911yescharacteristic, complemented, Sylow
353C22—22no—
354C22—22no—
355C22—22no—
356C22—44no—
357C22#8970112yes—
358C22#8970112yes—
359C22C2×#648712yes—
360C22C22×(C2×D7)⋊C411yescharacteristic
361C4—22no—
362C4—22no—
363C4—22no—
364C4—48no—
365C4—288no—
366C4#8970112yes—
367C4#8970112yes—
368C4#8977714yes—
369C4C2×C4×Dic1412yes—
370C4C2×#643211yescharacteristic
371C4C2×#643811yescharacteristic
372C2—22no—
373C2—22no—
374C2—22no—
375C2C2×#8970111yescharacteristic
376C2C2×#8977712yes—
subgroupG/H
centerC2×C4C22×D28
commutatorC2×C4×C7C23×C4
FrattiniC22×C4C23×D7
FittingC4×C7×#117C2
socleC22×C7C22×C22⋊C4
prime radicalC24×C7C2×D4
cosocleC22×C4×C7C24
hypercenterC4×#117D7
hypocenterC7#21329
LCSG ▷ C2×C4×C7 ▷ C2×C7 ▷ C7
UCSC1 ◁ C2×C4 ◁ C22×C42 ◁ C4×#117

Complements

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

NH ≅ G/Ntype
5. #11727195C2semidirect
6. #11727150C2semidirect
14. #90161C4semidirect
15. #90161C4semidirect
18. C2×#6433C4semidirect
19. C2×#6432C4semidirect
35. C7×#117Q8semidirect
121. C42×C7C2×Q8semidirect
221. #117Dic14semidirect
305. C42C2×Dic14semidirect
352. C7#21329semidirect

Maximal subgroups

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

subgroupindex
C2×#901612
C4×C7×#1172
C2×#896962
#117293322
#117271952
#117271502
#213297