Finite Group Databasebeta

← #13253988

Subgroups — #13253988

Order 1792.

548 conjugacy classes of subgroups.

The subgroup lattice has 390 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2851C211yesmaximal, characteristic
2#11729288C211yesmaximal, characteristic
3#11729284C211yesmaximal, characteristic
4#11725430C212yesmaximal
5#11725430C212yesmaximal
6C2×C7×#125C411yescharacteristic, complemented
7C22×C42×C7C2211yescharacteristic
8C7×#389C411yescharacteristic, complemented
9C7×#389C411yescharacteristic, complemented
10C7×#387—21no—
11C7×#387—21no—
12C7×#385—21no—
13C7×#385—21no—
14C7×#382C411yescharacteristic, complemented
15C7×#380C2211yescharacteristic
16C7×#380C2211yescharacteristic
17C2×C7⋊#95—21no—
18C2×C7⋊#95—21no—
19C2×C7⋊#95C2212yes—
20C2×C22⋊(C7⋊C8)C2212yes—
21C2×C4⋊(C7⋊C8)—21no—
22#89319—21no—
23#9913—71nomaximal, Sylow
24C23×C4×C7D411yescharacteristic
25C23×C4×C7D411yescharacteristic
26C23×C4×C7C2311yescharacteristic
27C23×Dic7—22no—
28C7×#125—42no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—21no—
32C2×C7×C4⋊C4—21no—
33C2×C7×C4⋊C4C2×C411yescharacteristic
34C2×C7×C4⋊C4C2×C411yescharacteristic
35C2×C7×C4⋊C4C2×C411yescharacteristic
36C2×C7×C4⋊C4C2×C411yescharacteristic
37C2×C7×C22⋊C4—21no—
38C2×C7×C22⋊C4—21no—
39C2×C42×C7—42no—
40C2×C42×C7C2×C411yescharacteristic
41C2×C42×C7C2×C411yescharacteristic
42C2×C42×C7D411yescharacteristic
43C2×C42×C7D411yescharacteristic
44C22×C7⋊C8—22no—
45C7×#95—21no—
46C7×#95—21no—
47C7×#95—21no—
48C7×#95—21no—
49C7×#95—21no—
50C7×#95—21no—
51C7⋊#95—22no—
52C7⋊#95—42no—
53C7⋊#95—42no—
54C7⋊#95Q814yes—
55C22⋊(C7⋊C8)—24no—
56C4⋊(C7⋊C8)—42no—
57#2851D711yescharacteristic
58#1797—71no—
59#1434—71no—
60#1294—72no—
61#1294—72no—
62C24×C7C2×D411yescharacteristic
63C22×C4×C7—22no—
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7—22no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—21no—
70C22×C4×C7—21no—
71C22×C4×C7—41no—
72C22×C4×C7—41no—
73C22×C4×C7—41no—
74C22×C4×C7—41no—
75C22×C4×C7—41no—
76C22×C4×C7C22⋊C411yescharacteristic
77C22×C4×C7C22⋊C411yescharacteristic
78C22×C4×C7C22⋊C411yescharacteristic
79C22×C4×C7C22⋊C411yescharacteristic
80C22×C4×C7C22×C411yescharacteristic
81C22×C4×C7C2×D411yescharacteristic
82C22×C4×C7C2×Q812yes—
83C22×C4×C7#4112yes—
84C22×Dic7—24no—
85C22×Dic7—42no—
86C22×Dic7—42no—
87C22×Dic7—42no—
88C22×Dic7—42no—
89C22×Dic7—42no—
90C22×Dic7—42no—
91C7×C4⋊C4—41no—
92C7×C4⋊C4—41no—
93C7×C4⋊C4—41no—
94C7×C4⋊C4—41no—
95C7×C4⋊C4—42no—
96C7×C4⋊C4—42no—
97C7×C4⋊C4—42no—
98C7×C4⋊C4—42no—
99C7×C22⋊C4—42no—
100C7×C22⋊C4—41no—
101C7×C22⋊C4—41no—
102C42×C7—22no—
103C42×C7—22no—
104C42×C7—41no—
105C42×C7—41no—
106C2×C7⋊C8—42no—
107C2×C7⋊C8—42no—
108C2×C7⋊C8—42no—
109C2×#125Dic711yescharacteristic, complemented
110C22×C42C2×D711yescharacteristic
111C2×C4⋊C8—141no—
112C2×C22⋊C8—72no—
113#389Dic711yescharacteristic, complemented
114#389Dic711yescharacteristic, complemented
115#387—21no—
116#387—21no—
117#385—21no—
118#385—21no—
119#382Dic711yescharacteristic, complemented
120#380C2×D711yescharacteristic
121#380C2×D711yescharacteristic
122C2×#95—72no—
123C2×#95—141no—
124C2×#95—141no—
125#336—141no—
126C23×C7—21no—
127C23×C7—21no—
128C23×C7—21no—
129C23×C7—21no—
130C23×C7C2×C22⋊C411yescharacteristic
131C23×C7C22⋊Q812yes—
132C23×C7#13512yes—
133C23×C7AGL(1,ℤ/8)11yescharacteristic
134C23×C7#13711yescharacteristic
135C2×C4×C7—21no—
136C2×C4×C7—21no—
137C2×C4×C7—21no—
138C2×C4×C7—21no—
139C2×C4×C7—21no—
140C2×C4×C7—21no—
141C2×C4×C7—21no—
142C2×C4×C7—21no—
143C2×C4×C7—21no—
144C2×C4×C7—21no—
145C2×C4×C7—21no—
146C2×C4×C7—21no—
147C2×C4×C7—21no—
148C2×C4×C7—21no—
149C2×C4×C7—21no—
150C2×C4×C7—42no—
151C2×C4×C7—42no—
152C2×C4×C7—44no—
153C2×C4×C7—42no—
154C2×C4×C7—42no—
155C2×C4×C7—41no—
156C2×C4×C7—41no—
157C2×C4×C7—41no—
158C2×C4×C7—41no—
159C2×C4×C7—41no—
160C2×C4×C7—41no—
161C2×C4×C7—41no—
162C2×C4×C7—41no—
163C2×C4×C7—41no—
164C2×C4×C7—41no—
165C2×C4×C7—41no—
166C2×C4×C7—41no—
167C2×C4×C7#9912yes—
168C2×C4×C7C4×Q812yes—
169C2×C4×C7#12411yescharacteristic
170C2×C4×C7C4⋊Q811yescharacteristic
171C2×Dic7—44no—
172C2×Dic7—44no—
173C2×Dic7—44no—
174C2×Dic7—82no—
175C2×Dic7—82no—
176C2×Dic7—82no—
177C2×Dic7—82no—
178C7⋊C8—82no—
179C23×C4—142no—
180C23×C4C22⋊D711yescharacteristic
181C23×C4C22⋊D711yescharacteristic
182C23×C4C22×D711yescharacteristic
183C22×C8—142no—
184#125—42no—
185C2×C4⋊C4—21no—
186C2×C4⋊C4—21no—
187C2×C4⋊C4—21no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4C2×Dic711yescharacteristic
190C2×C4⋊C4C2×Dic711yescharacteristic
191C2×C4⋊C4C2×Dic711yescharacteristic
192C2×C4⋊C4C2×Dic711yescharacteristic
193C2×C22⋊C4—21no—
194C2×C22⋊C4—21no—
195C2×C42—42no—
196C2×C42C2×Dic711yescharacteristic
197C2×C42C2×Dic711yescharacteristic
198C2×C42C22⋊D711yescharacteristic
199C2×C42C22⋊D711yescharacteristic
200C4⋊C8—282no—
201C22⋊C8—144no—
202#95—21no—
203#95—21no—
204#95—21no—
205#95—21no—
206#95—21no—
207#95—21no—
208#95—74no—
209#95—142no—
210#95—282no—
211#95—282no—
212C22×C7—22no—
213C22×C7—22no—
214C22×C7—21no—
215C22×C7—21no—
216C22×C7—21no—
217C22×C7—21no—
218C22×C7—22no—
219C22×C7—22no—
220C22×C7#39111yescharacteristic
221C22×C7C2×#9911yescharacteristic
222C22×C7#41011yescharacteristic
223C22×C7#41611yescharacteristic
224C22×C7#41711yescharacteristic
225C22×C7#42111yescharacteristic
226C22×C7#42111yescharacteristic
227C22×C7#47812yes—
228C22×C7#47912yes—
229C4×C7—21no—
230C4×C7—22no—
231C4×C7—21no—
232C4×C7—22no—
233C4×C7—41no—
234C4×C7—41no—
235C4×C7—41no—
236C4×C7—81no—
237C4×C7—81no—
238C4×C7—81no—
239C4×C7—81no—
240Dic7—84no—
241C24C2×C22⋊D711yescharacteristic
242C22×C4—21no—
243C22×C4—22no—
244C22×C4—21no—
245C22×C4—22no—
246C22×C4—21no—
247C22×C4—21no—
248C22×C4—21no—
249C22×C4—21no—
250C22×C4—41no—
251C22×C4—41no—
252C22×C4—41no—
253C22×C4—41no—
254C22×C4—41no—
255C22×C4—144no—
256C22×C4—282no—
257C22×C4—282no—
258C22×C4—282no—
259C22×C4—282no—
260C22×C4—282no—
261C22×C4—282no—
262C22×C4C22⋊Dic711yescharacteristic
263C22×C4C22⋊Dic711yescharacteristic
264C22×C4C22⋊Dic711yescharacteristic
265C22×C4C22⋊Dic711yescharacteristic
266C22×C4Q8×D712yes—
267C22×C4#116112yes—
268C22×C4C22×Dic711yescharacteristic
269C22×C4C2×C22⋊D711yescharacteristic
270C2×C8—282no—
271C2×C8—282no—
272C2×C8—282no—
273C4⋊C4—41no—
274C4⋊C4—41no—
275C4⋊C4—41no—
276C4⋊C4—41no—
277C4⋊C4—42no—
278C4⋊C4—42no—
279C4⋊C4—42no—
280C4⋊C4—42no—
281C22⋊C4—41no—
282C22⋊C4—42no—
283C22⋊C4—41no—
284C42—22no—
285C42—22no—
286C42—41no—
287C42—41no—
288C2×C7—22no—
289C2×C7—22no—
290C2×C7#151211yescharacteristic
291C2×C7#151311yescharacteristic
292C2×C7#151611yescharacteristic
293C2×C7#152111yescharacteristic
294C2×C7#198911yescharacteristic
295C2×C7#199511yescharacteristic
296C2×C7#199611yescharacteristic
297C23—21no—
298C23—21no—
299C23—21no—
300C23—21no—
301C23#650912yes—
302C23#651211yescharacteristic
303C23#651312yes—
304C23#651411yescharacteristic
305C23C2×C22⋊Dic711yescharacteristic
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—21no—
310C2×C4—21no—
311C2×C4—21no—
312C2×C4—21no—
313C2×C4—21no—
314C2×C4—21no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—21no—
319C2×C4—21no—
320C2×C4—21no—
321C2×C4—42no—
322C2×C4—42no—
323C2×C4—44no—
324C2×C4—42no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—41no—
328C2×C4—41no—
329C2×C4—41no—
330C2×C4—41no—
331C2×C4—41no—
332C2×C4—41no—
333C2×C4—41no—
334C2×C4—41no—
335C2×C4—41no—
336C2×C4—41no—
337C2×C4—41no—
338C2×C4—284no—
339C2×C4—284no—
340C2×C4—284no—
341C2×C4—562no—
342C2×C4—562no—
343C2×C4—562no—
344C2×C4—562no—
345C2×C4#640812yes—
346C2×C4#650711yescharacteristic
347C2×C4Q8×Dic712yes—
348C2×C4#651011yescharacteristic
349C8—562no—
350C7#991311yescharacteristic, complemented, Sylow
351C22—22no—
352C22—21no—
353C22—21no—
354C22—22no—
355C22—21no—
356C22—21no—
357C22—22no—
358C22—22no—
359C22#8983512yes—
360C22#8983812yes—
361C22C2×#640811yescharacteristic
362C22#9000211yescharacteristic
363C22#9000411yescharacteristic
364C22#9000511yescharacteristic
365C22#9000811yescharacteristic
366C22#9000811yescharacteristic
367C22#9001211yescharacteristic
368C4—21no—
369C4—21no—
370C4—22no—
371C4—22no—
372C4—41no—
373C4—41no—
374C4—41no—
375C4—81no—
376C4—81no—
377C4—81no—
378C4—81no—
379C4—564no—
380C2—22no—
381C2—22no—
382C2#1172945511yescharacteristic
383C2#1172946311yescharacteristic
384C2#1172946411yescharacteristic
385C2#1172949211yescharacteristic
386C2#1172951011yescharacteristic
387C2#1172951111yescharacteristic
388C2#1172959411yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic7
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2851C2
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenter#2851D7
hypocenterC7#9913
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2851

Complements

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

NH ≅ G/Ntype
6. C2×C7×#125C4semidirect
8. C7×#389C4semidirect
9. C7×#389C4semidirect
14. C7×#382C4semidirect
109. C2×#125Dic7semidirect
113. #389Dic7semidirect
114. #389Dic7semidirect
119. #382Dic7semidirect
350. C7#9913semidirect

Maximal subgroups

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

subgroupindex
C7×#28512
#117292882
#117292842
#117254302
#99137