Finite Group Databasebeta

← #13252917

Subgroups — #13252917

Order 1792.

716 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739521C211yesmaximal, characteristic
2C7×#1797C211yesmaximal, characteristic
3C4×C7⋊#95C211yesmaximal, characteristic
4C2×#89262C214yesmaximal
5C22×C42×C7C2211yescharacteristic
6C2×#6487C2212yes—
7C22×C4×Dic7—21no—
8C2×#6432C411yescharacteristic, complemented
9C2×C7×C4⋊C8—21no—
10C2×C7×C22⋊C8C2212yes—
11#89984—21no—
12#89984—21no—
13C2×C7⋊#95—21no—
14C2×C7⋊#95C2212yes—
15#89707C411yescharacteristic, complemented
16#89701C412yescomplemented
17#89700—21no—
18#89700—21no—
19C7×#336—21no—
20#89262—28no—
21#9711—71nomaximal, Sylow
22C23×C4×C7D411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7C2311yescharacteristic
25C23×Dic7—22no—
26C22×Dic14C2×C412yes—
27C22×C7×C8—22no—
28C2×C42×C7—42no—
29C2×C42×C7C2×C411yescharacteristic
30C2×C42×C7C2×C411yescharacteristic
31C2×C42×C7D411yescharacteristic
32C2×C42×C7D411yescharacteristic
33C2×C22⋊Dic7—22no—
34C2×C4⋊Dic7—21no—
35C2×C4⋊Dic7—21no—
36C2×C4⋊Dic7—21no—
37C2×C4⋊Dic7—21no—
38C2×C4⋊Dic7C2×C412yes—
39#6487—24no—
40#6487—24no—
41C2×Dic7⋊C4—22no—
42C2×C4×Dic7—24no—
43C2×C4×Dic7—42no—
44C2×C4×Dic7—42no—
45#6432—42no—
46C7×C4⋊C8—42no—
47C7×C22⋊C8—24no—
48C7⋊#95—24no—
49C7⋊#95—22no—
50C7⋊#95—22no—
51C7⋊#95—22no—
52C7⋊#95D414yes—
53#2857—71no—
54#1797D711yescharacteristic
55C2×#328—74no—
56C4×#95—71no—
57C24×C7C2×D411yescharacteristic
58C22×C4×C7—22no—
59C22×C4×C7—22no—
60C22×C4×C7—22no—
61C22×C4×C7—22no—
62C22×C4×C7—41no—
63C22×C4×C7—41no—
64C22×C4×C7—41no—
65C22×C4×C7C22⋊C411yescharacteristic
66C22×C4×C7C22⋊C411yescharacteristic
67C22×C4×C7C22⋊C411yescharacteristic
68C22×C4×C7C22⋊C411yescharacteristic
69C22×C4×C7SD1612yes—
70C22×C4×C7Q1612yes—
71C22×C4×C7C22×C411yescharacteristic
72C22×C4×C7C2×D411yescharacteristic
73C22×C4×C7C2×D412yes—
74C22×C4×C7#4112yes—
75C22×Dic7—24no—
76C22×Dic7—24no—
77C22×Dic7—24no—
78C22×Dic7—22no—
79C22×Dic7—22no—
80C22×Dic7—42no—
81C22×Dic7—42no—
82C22×Dic7—42no—
83C22×Dic7—42no—
84C2×Dic14—24no—
85C2×Dic14—42no—
86C2×Dic14—42no—
87C2×C7×C8—42no—
88C2×C7×C8—42no—
89C2×C7×C8—42no—
90C42×C7—22no—
91C42×C7—22no—
92C42×C7—41no—
93C42×C7—41no—
94C22⋊Dic7—44no—
95C4⋊Dic7—24no—
96C4⋊Dic7—42no—
97C4⋊Dic7—42no—
98C4⋊Dic7—42no—
99C4⋊Dic7—42no—
100Dic7⋊C4—42no—
101Dic7⋊C4—42no—
102C4×Dic7—48no—
103C2×C4⋊Q8—71no—
104C2×C22⋊Q8—72no—
105C22×C42—141no—
106C22×C42C2×D711yescharacteristic
107C2×C4⋊C8—21no—
108C2×C22⋊C8C2×D712yes—
109#391—72no—
110#389—141no—
111#389—141no—
112#384—71no—
113#380—141no—
114#380—141no—
115C2×#95—72no—
116C2×#95—141no—
117#336—21no—
118#328—148no—
119C23×C7—21no—
120C23×C7—21no—
121C23×C7—21no—
122C23×C7—21no—
123C23×C7C2×C22⋊C411yescharacteristic
124C23×C7#12112yes—
125C23×C7C2×SD1611yescharacteristic
126C23×C7C2×Q1611yescharacteristic
127C23×C7#13712yes—
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—21no—
135C2×C4×C7—21no—
136C2×C4×C7—24no—
137C2×C4×C7—24no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—42no—
142C2×C4×C7—42no—
143C2×C4×C7—44no—
144C2×C4×C7#9912yes—
145C2×C4×C7#10312yes—
146C2×C4×C7#10312yes—
147C2×C4×C7#10412yes—
148C2×C4×C7#10412yes—
149C2×C4×C7C4×D412yes—
150C2×C4×C7#12411yescharacteristic
151C2×C4×C7#12711yescharacteristic
152C2×Dic7—28no—
153C2×Dic7—44no—
154C2×Dic7—44no—
155C2×Dic7—44no—
156C2×Dic7—44no—
157C2×Dic7—44no—
158C2×Dic7—44no—
159C2×Dic7—42no—
160C2×Dic7—42no—
161C2×Dic7—42no—
162C2×Dic7—42no—
163C2×Dic7—42no—
164C2×Dic7—42no—
165Dic14—84no—
166C7×C8—82no—
167C22×Q8—72no—
168C23×C4—142no—
169C23×C4D2811yescharacteristic
170C23×C4D2811yescharacteristic
171C23×C4C22×D711yescharacteristic
172C22×C8—22no—
173C4⋊Q8—282no—
174C22⋊Q8—144no—
175C22⋊Q8—144no—
176C2×C4⋊C4—72no—
177C2×C4⋊C4—141no—
178C2×C4⋊C4—141no—
179C2×C4⋊C4—141no—
180C2×C4⋊C4—141no—
181C2×C4⋊C4—142no—
182C2×C22⋊C4—142no—
183C2×C42—42no—
184C2×C42—144no—
185C2×C42—282no—
186C2×C42—282no—
187C2×C42C4×D711yescharacteristic
188C2×C42C4×D711yescharacteristic
189C2×C42C22⋊D711yescharacteristic
190C2×C42C22⋊D711yescharacteristic
191C4⋊C8—42no—
192C22⋊C8—24no—
193#95—74no—
194#95—144no—
195#95—142no—
196#95—142no—
197#95—142no—
198C22×C7—22no—
199C22×C7—22no—
200C22×C7—21no—
201C22×C7—21no—
202C22×C7—21no—
203C22×C7—21no—
204C22×C7—22no—
205C22×C7—22no—
206C22×C7#39011yescharacteristic
207C22×C7C2×#9911yescharacteristic
208C22×C7#41011yescharacteristic
209C22×C7C2×#10311yescharacteristic
210C22×C7#41911yescharacteristic
211C22×C7C2×#10411yescharacteristic
212C22×C7C2×#10411yescharacteristic
213C22×C7#46112yes—
214C22×C7#46212yes—
215C4×C7—21no—
216C4×C7—22no—
217C4×C7—21no—
218C4×C7—22no—
219C4×C7—22no—
220C4×C7—41no—
221C4×C7#32814yescomplemented
222Dic7—48no—
223Dic7—82no—
224Dic7—82no—
225C24C2×D2811yescharacteristic
226C2×Q8—144no—
227C2×Q8—282no—
228C2×Q8—282no—
229C22×C4—22no—
230C22×C4—22no—
231C22×C4—22no—
232C22×C4—22no—
233C22×C4—41no—
234C22×C4—41no—
235C22×C4—41no—
236C22×C4—144no—
237C22×C4—144no—
238C22×C4—144no—
239C22×C4—142no—
240C22×C4—142no—
241C22×C4—282no—
242C22×C4—282no—
243C22×C4—282no—
244C22×C4—282no—
245C22×C4#113212yes—
246C22×C4Dic2812yes—
247C22×C4(C2×D7)⋊C411yescharacteristic
248C22×C4(C2×D7)⋊C411yescharacteristic
249C22×C4(C2×D7)⋊C411yescharacteristic
250C22×C4(C2×D7)⋊C411yescharacteristic
251C22×C4C2×C4×D711yescharacteristic
252C22×C4D4×D712yes—
253C22×C4#116112yes—
254C22×C4C2×C22⋊D711yescharacteristic
255C2×C8—42no—
256C2×C8—42no—
257C2×C8—42no—
258C4⋊C4—144no—
259C4⋊C4—282no—
260C4⋊C4—282no—
261C4⋊C4—282no—
262C4⋊C4—282no—
263C4⋊C4—282no—
264C4⋊C4—282no—
265C22⋊C4—284no—
266C42—22no—
267C42—22no—
268C42—41no—
269C42—41no—
270C42—288no—
271C2×C7—22no—
272C2×C7—22no—
273C2×C7C2×#32811yescharacteristic
274C2×C7#150311yescharacteristic
275C2×C7#150611yescharacteristic
276C2×C7#150811yescharacteristic
277C2×C7#196211yescharacteristic
278C2×C7#196411yescharacteristic
279C2×C7#198111yescharacteristic
280C23—21no—
281C23—21no—
282C23—21no—
283C23—21no—
284C23#645812yes—
285C23C2×#113211yescharacteristic
286C23C2×Dic2811yescharacteristic
287C23#647212yes—
288C23C2×(C2×D7)⋊C411yescharacteristic
289Q8—564no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—21no—
296C2×C4—21no—
297C2×C4—21no—
298C2×C4—24no—
299C2×C4—24no—
300C2×C4—21no—
301C2×C4—21no—
302C2×C4—21no—
303C2×C4—42no—
304C2×C4—42no—
305C2×C4—44no—
306C2×C4—148no—
307C2×C4—284no—
308C2×C4—284no—
309C2×C4—284no—
310C2×C4—284no—
311C2×C4—284no—
312C2×C4—284no—
313C2×C4—282no—
314C2×C4—282no—
315C2×C4—282no—
316C2×C4—282no—
317C2×C4—282no—
318C2×C4—282no—
319C2×C4#638012yes—
320C2×C4#639012yes—
321C2×C4#639012yes—
322C2×C4#639912yes—
323C2×C4#639912yes—
324C2×C4#645612yes—
325C2×C4#650311yescharacteristic
326C2×C4#651011yescharacteristic
327C8—82no—
328C7#971111yescharacteristic, complemented, Sylow
329C22—21no—
330C22—21no—
331C22—22no—
332C22—22no—
333C22—21no—
334C22—21no—
335C22—22no—
336C22—22no—
337C22#8961312yes—
338C22#8962112yes—
339C22C2×#638011yescharacteristic
340C22#8976011yescharacteristic
341C22#8977811yescharacteristic
342C22C2×#639011yescharacteristic
343C22#8989211yescharacteristic
344C22C2×#639911yescharacteristic
345C22C2×#639911yescharacteristic
346C4—22no—
347C4—22no—
348C4—21no—
349C4—22no—
350C4—21no—
351C4—41no—
352C4—288no—
353C4—562no—
354C4—562no—
355C4#8926214yescomplemented
356C2—22no—
357C2—22no—
358C2#1172728211yescharacteristic
359C2C2×#8926211yescharacteristic
360C2#1172773411yescharacteristic
361C2#1172821011yescharacteristic
362C2#1172826611yescharacteristic
363C2#1172854611yescharacteristic
364C2#1172859011yescharacteristic
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#9711
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
8. C2×#6432C4semidirect
15. #89707C4semidirect
16. #89701C4semidirect
221. C4×C7#328semidirect
328. C7#9711semidirect
355. C4#89262semidirect

Maximal subgroups

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

subgroupindex
#117395212
C7×#17972
C4×C7⋊#952
C2×#892622
#97117