Finite Group Databasebeta

← #13249557

Subgroups — #13249557

Order 1792.

630 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
1C7×#2855C211yesmaximal, characteristic
2C2×#89708C211yesmaximal, characteristic
3C2×#89333C212yesmaximal
4#11729285C211yesmaximal, characteristic
5#11725430C212yesmaximal
6C2×C7×C22⋊Q8C2211yescharacteristic
7C22×C42×C7C2211yescharacteristic
8C22×C4⋊Dic7—21no—
9C7×#391C412yescomplemented
10C7×#389C412yescomplemented
11C7×#387—22no—
12C7×#385—21no—
13C7×#381—21no—
14C7×#380C2211yescharacteristic
15C2×C7⋊#95—21no—
16C2×C7⋊#95C2212yes—
17C2×C22⋊(C7⋊C8)C2212yes—
18#89708—24no—
19#89333—24no—
20#89319—22no—
21#9152—71nomaximal, Sylow
22C22×C7×Q8C2×C411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C23×C4×C7C2311yescharacteristic
26C23×Dic7—22no—
27C7×C22⋊Q8—22no—
28C7×C22⋊Q8—22no—
29C2×C7×C4⋊C4—22no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4C2×C411yescharacteristic
32C2×C7×C4⋊C4C2×C412yes—
33C2×C7×C22⋊C4—21no—
34C2×C7×C22⋊C4—21no—
35C2×C42×C7—22no—
36C2×C42×C7—22no—
37C2×C42×C7C2×C412yes—
38C2×C42×C7D412yes—
39C2×C4⋊Dic7—24no—
40C2×C4⋊Dic7—42no—
41C2×C4⋊Dic7—42no—
42C22×C7⋊C8—22no—
43C7×#95—22no—
44C7×#95—22no—
45C7×#95—22no—
46C7×#95—21no—
47C7⋊#95—24no—
48C7⋊#95—22no—
49C7⋊#95D414yes—
50C22⋊(C7⋊C8)—22no—
51C22⋊(C7⋊C8)—22no—
52#2855D711yescharacteristic
53C2×#382—71no—
54#1817—71no—
55C2×#328—72no—
56#1294—72no—
57C24×C7C2×D411yescharacteristic
58C2×C7×Q8—22no—
59C2×C7×Q8—42no—
60C22×C4×C7—21no—
61C22×C4×C7—21no—
62C22×C4×C7—21no—
63C22×C4×C7—21no—
64C22×C4×C7—21no—
65C22×C4×C7—21no—
66C22×C4×C7—22no—
67C22×C4×C7—21no—
68C22×C4×C7—21no—
69C22×C4×C7—42no—
70C22×C4×C7—42no—
71C22×C4×C7C22⋊C412yes—
72C22×C4×C7C22⋊C412yes—
73C22×C4×C7SD1612yes—
74C22×C4×C7Q1612yes—
75C22×C4×C7C22×C411yescharacteristic
76C22×C4×C7C2×D411yescharacteristic
77C22×C4×C7C2×D412yes—
78C22×C4×C7#4112yes—
79C22×Dic7—24no—
80C22×Dic7—44no—
81C22×Dic7—44no—
82C22×Dic7—42no—
83C22×Dic7—42no—
84C7×C4⋊C4—22no—
85C7×C4⋊C4—42no—
86C7×C4⋊C4—44no—
87C7×C4⋊C4—42no—
88C7×C22⋊C4—41no—
89C7×C22⋊C4—41no—
90C7×C22⋊C4—41no—
91C7×C22⋊C4—41no—
92C42×C7—24no—
93C42×C7—24no—
94C4⋊Dic7—48no—
95C2×C7⋊C8—44no—
96C2×C7⋊C8—42no—
97C2×C22⋊Q8C2×D711yescharacteristic
98C22×C4⋊C4—141no—
99C22×C42C2×D711yescharacteristic
100C2×C22⋊C8—72no—
101#391Dic712yescomplemented
102#389Dic712yescomplemented
103#387—22no—
104#385—21no—
105#382—144no—
106#381—21no—
107#380C2×D711yescharacteristic
108C2×#95—72no—
109C2×#95—141no—
110#336—142no—
111#328—144no—
112C23×C7—22no—
113C23×C7—22no—
114C23×C7C2×C22⋊C411yescharacteristic
115C23×C7#12011yescharacteristic
116C23×C7#12311yescharacteristic
117C23×C7C2×SD1611yescharacteristic
118C23×C7C2×Q1611yescharacteristic
119C23×C7AGL(1,ℤ/8)11yescharacteristic
120C23×C7#13711yescharacteristic
121C7×Q8—82no—
122C2×C4×C7—22no—
123C2×C4×C7—22no—
124C2×C4×C7—22no—
125C2×C4×C7—22no—
126C2×C4×C7—22no—
127C2×C4×C7—22no—
128C2×C4×C7—22no—
129C2×C4×C7—22no—
130C2×C4×C7—22no—
131C2×C4×C7—22no—
132C2×C4×C7—22no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—41no—
138C2×C4×C7—41no—
139C2×C4×C7—42no—
140C2×C4×C7—42no—
141C2×C4×C7—42no—
142C2×C4×C7—42no—
143C2×C4×C7—42no—
144C2×C4×C7#10314yes—
145C2×C4×C7C4×D412yes—
146C2×C4×C7#12112yes—
147C2×Dic7—44no—
148C2×Dic7—44no—
149C2×Dic7—44no—
150C2×Dic7—84no—
151C2×Dic7—84no—
152C7⋊C8—82no—
153C22×Q8C2×Dic711yescharacteristic
154C23×C4—142no—
155C23×C4C22⋊D711yescharacteristic
156C23×C4C22⋊D711yescharacteristic
157C23×C4C22×D711yescharacteristic
158C22×C8—142no—
159C22⋊Q8—22no—
160C22⋊Q8—22no—
161C2×C4⋊C4—22no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—144no—
164C2×C4⋊C4—282no—
165C2×C4⋊C4—282no—
166C2×C4⋊C4C2×Dic711yescharacteristic
167C2×C4⋊C4C2×Dic712yes—
168C2×C22⋊C4—21no—
169C2×C22⋊C4—21no—
170C2×C42—22no—
171C2×C42—22no—
172C2×C42C2×Dic712yes—
173C2×C42C22⋊D712yes—
174C22⋊C8—142no—
175C22⋊C8—142no—
176#95—22no—
177#95—22no—
178#95—22no—
179#95—21no—
180#95—74no—
181#95—144no—
182#95—142no—
183C22×C7—21no—
184C22×C7—21no—
185C22×C7—21no—
186C22×C7—21no—
187C22×C7—22no—
188C22×C7—22no—
189C22×C7—22no—
190C22×C7—22no—
191C22×C7#38611yescharacteristic
192C22×C7#41012yes—
193C22×C7C2×#10311yescharacteristic
194C22×C7#41711yescharacteristic
195C22×C7#42112yes—
196C22×C7#45011yescharacteristic
197C22×C7#45111yescharacteristic
198C22×C7#48311yescharacteristic
199C22×C7#48411yescharacteristic
200C4×C7—24no—
201C4×C7—22no—
202C4×C7—22no—
203C4×C7—42no—
204C4×C7—81no—
205C4×C7—81no—
206C4×C7—82no—
207Dic7—84no—
208C24C2×C22⋊D711yescharacteristic
209C2×Q8—22no—
210C2×Q8—42no—
211C22×C4—21no—
212C22×C4—21no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—21no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—42no—
221C22×C4—42no—
222C22×C4—144no—
223C22×C4—284no—
224C22×C4—284no—
225C22×C4—282no—
226C22×C4—282no—
227C22×C4#114312yes—
228C22×C4#114412yes—
229C22×C4C22⋊Dic712yes—
230C22×C4C22⋊Dic712yes—
231C22×C4D4×D712yes—
232C22×C4#115912yes—
233C22×C4C22×Dic711yescharacteristic
234C22×C4C2×C22⋊D711yescharacteristic
235C2×C8—284no—
236C2×C8—282no—
237C4⋊C4—22no—
238C4⋊C4—42no—
239C4⋊C4—44no—
240C4⋊C4—42no—
241C4⋊C4—288no—
242C22⋊C4—41no—
243C22⋊C4—41no—
244C22⋊C4—41no—
245C22⋊C4—41no—
246C42—24no—
247C42—24no—
248C2×C7—21no—
249C2×C7—21no—
250C2×C7—21no—
251C2×C7—21no—
252C2×C7#151212yes—
253C2×C7#151812yes—
254C2×C7#187211yescharacteristic
255C2×C7#189711yescharacteristic
256C2×C7#189911yescharacteristic
257C23—22no—
258C23—22no—
259C23#649811yescharacteristic
260C23#650011yescharacteristic
261C23C2×#114311yescharacteristic
262C23C2×#114411yescharacteristic
263C23#651211yescharacteristic
264C23#651411yescharacteristic
265C23C2×C22⋊Dic711yescharacteristic
266Q8—82no—
267C2×C4—22no—
268C2×C4—22no—
269C2×C4—22no—
270C2×C4—22no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—284no—
290C2×C4—284no—
291C2×C4—284no—
292C2×C4—564no—
293C2×C4—564no—
294C2×C4#640914yes—
295C2×C4D4×Dic712yes—
296C2×C4#650212yes—
297C8—562no—
298C7#915211yescharacteristic, complemented, Sylow
299C22—21no—
300C22—21no—
301C22—21no—
302C22—21no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22#8981511yescharacteristic
308C22#8981611yescharacteristic
309C22#8981811yescharacteristic
310C22#8982211yescharacteristic
311C22#8999211yescharacteristic
312C22C2×#640911yescharacteristic
313C22#9000411yescharacteristic
314C22#9000812yes—
315C22#9001212yes—
316C4—24no—
317C4—22no—
318C4—22no—
319C4—42no—
320C4—81no—
321C4—81no—
322C4—82no—
323C4—564no—
324C2—21no—
325C2—21no—
326C2—21no—
327C2—21no—
328C2#1172944611yescharacteristic
329C2#1172945911yescharacteristic
330C2#1172946611yescharacteristic
331C2#1172951012yes—
332C2#1172959212yes—
subgroupG/H
centerC23C2×C22⋊Dic7
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#2855C2
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenter#2855D7
hypocenterC7#9152
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2855

Complements

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

NH ≅ G/Ntype
9. C7×#391C4semidirect
10. C7×#389C4semidirect
101. #391Dic7semidirect
102. #389Dic7semidirect
298. C7#9152semidirect

Maximal subgroups

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

subgroupindex
C7×#28552
C2×#897082
C2×#893332
#117292852
#117254302
#91527