Finite Group Databasebeta

← #13242278

Subgroups — #13242278

Order 1792.

562 conjugacy classes of subgroups.

The subgroup lattice has 318 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C7×#2857C211yesmaximal, characteristic
2C4×C22⋊(C7⋊C8)C211yesmaximal, characteristic
3#11729281C211yesmaximal, characteristic
4#11725448C214yesmaximal, complemented
5C2×C7×C4⋊Q8C411yescharacteristic
6C2×C7×C22⋊Q8C412yes—
7C22×C42×C7C2211yescharacteristic
8C22×#1136—21no—
9C7×#391C2212yes—
10C7×#389—21no—
11C7×#389—21no—
12C7×#384C411yescharacteristic
13C7×#380—21no—
14C7×#380—21no—
15C2×C22⋊(C7⋊C8)—21no—
16C2×C4×C7⋊C8—21no—
17#89319—21no—
18#89319C2212yes—
19#89319C2212yes—
20#7687—71nomaximal, Sylow
21C22×C7×Q8C2×C412yes—
22C23×C4×C7C2×C411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C7×C4⋊Q8—42no—
26C7×C22⋊Q8—44no—
27C2×C7×C4⋊C4—21no—
28C2×C7×C4⋊C4—21no—
29C2×C7×C4⋊C4—21no—
30C2×C7×C4⋊C4—21no—
31C2×C7×C4⋊C4—22no—
32C2×C7×C4⋊C4C2×C412yes—
33C2×C7×C22⋊C4—22no—
34C2×C42×C7—42no—
35C2×C42×C7C2×C411yescharacteristic
36C2×C42×C7D411yescharacteristic
37C2×C42×C7D411yescharacteristic
38C2×C42×C7C2311yescharacteristic
39C2×#1136—24no—
40C2×#1136—42no—
41C2×#1136—42no—
42C22×C7⋊C8—22no—
43C22×C7⋊C8—22no—
44C7×#95—22no—
45C7×#95—22no—
46C22⋊(C7⋊C8)—24no—
47C4×C7⋊C8—42no—
48#2857D711yescharacteristic
49C4×C22⋊C8—71no—
50#1728—71no—
51#1272—74no—
52C24×C7C22⋊C411yescharacteristic
53C2×C7×Q8—24no—
54C2×C7×Q8—42no—
55C2×C7×Q8—42no—
56C22×C4×C7—22no—
57C22×C4×C7—22no—
58C22×C4×C7—22no—
59C22×C4×C7—22no—
60C22×C4×C7—22no—
61C22×C4×C7—22no—
62C22×C4×C7—22no—
63C22×C4×C7—22no—
64C22×C4×C7—41no—
65C22×C4×C7—41no—
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×D411yescharacteristic
74C7×C4⋊C4—42no—
75C7×C4⋊C4—42no—
76C7×C4⋊C4—42no—
77C7×C4⋊C4—44no—
78C7×C4⋊C4—42no—
79C7×C4⋊C4—42no—
80C7×C22⋊C4—44no—
81C42×C7—22no—
82C42×C7—22no—
83C42×C7—41no—
84C42×C7—41no—
85#1136—48no—
86C2×C7⋊C8—24no—
87C2×C7⋊C8—24no—
88C2×C7⋊C8—42no—
89C2×C7⋊C8—42no—
90C2×C7⋊C8—42no—
91C2×C7⋊C8—42no—
92C22×M16—141no—
93C2×C4⋊Q8Dic711yescharacteristic
94C2×C22⋊Q8Dic712yes—
95C22×C42C2×D711yescharacteristic
96C2×C22⋊C8—141no—
97C2×C4×C8—141no—
98#391C2×D712yes—
99#389—21no—
100#389—21no—
101#384Dic711yescharacteristic
102#380—21no—
103#380—21no—
104#336—72no—
105#336—72no—
106#336—141no—
107C23×C7—21no—
108C23×C7—21no—
109C23×C7—21no—
110C23×C7—21no—
111C23×C7—21no—
112C23×C7—21no—
113C23×C7#10312yes—
114C23×C7C2×C22⋊C411yescharacteristic
115C7×Q8—84no—
116C2×C4×C7—21no—
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—24no—
127C2×C4×C7—24no—
128C2×C4×C7—44no—
129C2×C4×C7—44no—
130C2×C4×C7—42no—
131C2×C4×C7—42no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7#9912yes—
137C2×C4×C7#10012yes—
138C2×C4×C7#10312yes—
139C2×C4×C7#10412yes—
140C2×C4×C7#10412yes—
141C2×C4×C7#13112yes—
142C2×C4×C7C2×SD1611yescharacteristic
143C2×C4×C7C2×Q1611yescharacteristic
144C2×C4×C7#13712yes—
145C7⋊C8—44no—
146C7⋊C8—44no—
147C22×Q8C2×Dic712yes—
148C23×C4C2×Dic711yescharacteristic
149C23×C4C22⋊D711yescharacteristic
150C23×C4C22⋊D711yescharacteristic
151C2×M16—144no—
152C2×M16—282no—
153C2×M16—282no—
154C22×C8—142no—
155C22×C8—142no—
156C4⋊Q8—42no—
157C22⋊Q8—44no—
158C2×C4⋊C4—22no—
159C2×C4⋊C4—21no—
160C2×C4⋊C4—21no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4C2×Dic712yes—
164C2×C22⋊C4—22no—
165C2×C42—42no—
166C2×C42C2×Dic711yescharacteristic
167C2×C42C22⋊D711yescharacteristic
168C2×C42C22⋊D711yescharacteristic
169C2×C42C22×D711yescharacteristic
170C22⋊C8—144no—
171C4×C8—282no—
172#95—22no—
173#95—22no—
174C22×C7—22no—
175C22×C7—22no—
176C22×C7—22no—
177C22×C7—41no—
178C22×C7—41no—
179C22×C7—41no—
180C22×C7—41no—
181C22×C7#40811yescharacteristic
182C22×C7C2×#9911yescharacteristic
183C22×C7C2×#10011yescharacteristic
184C22×C7C2×#10311yescharacteristic
185C22×C7#41911yescharacteristic
186C22×C7C2×#10411yescharacteristic
187C22×C7C2×#10411yescharacteristic
188C4×C7—22no—
189C4×C7—22no—
190C4×C7—21no—
191C4×C7—21no—
192C4×C7—22no—
193C4×C7—22no—
194C4×C7—82no—
195C4×C7—82no—
196C4×C7#32814yes—
197C24C22⋊Dic711yescharacteristic
198C2×Q8—24no—
199C2×Q8—42no—
200C2×Q8—42no—
201C22×C4—22no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—41no—
210C22×C4—41no—
211C22×C4#114312yes—
212C22×C4#114412yes—
213C22×C4C22⋊Dic711yescharacteristic
214C22×C4C22⋊Dic711yescharacteristic
215C22×C4C22⋊Dic711yescharacteristic
216C22×C4C22×Dic711yescharacteristic
217C22×C4C2×C22⋊D711yescharacteristic
218C22×C4C2×C22⋊D711yescharacteristic
219M16—288no—
220C2×C8—144no—
221C2×C8—144no—
222C2×C8—282no—
223C2×C8—282no—
224C2×C8—282no—
225C2×C8—282no—
226C4⋊C4—42no—
227C4⋊C4—42no—
228C4⋊C4—44no—
229C4⋊C4—42no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C22⋊C4—44no—
233C42—22no—
234C42—22no—
235C42—41no—
236C42—41no—
237C2×C7—42no—
238C2×C7#147111yescharacteristic
239C2×C7#148511yescharacteristic
240C2×C7#149311yescharacteristic
241C2×C7C2×#32811yescharacteristic
242C2×C7#150811yescharacteristic
243C2×C7#152511yescharacteristic
244C2×C7#154811yescharacteristic
245C23—21no—
246C23—21no—
247C23—21no—
248C23—21no—
249C23—21no—
250C23—21no—
251C23#640912yes—
252C23C2×C22⋊Dic711yescharacteristic
253Q8—84no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—24no—
265C2×C4—24no—
266C2×C4—44no—
267C2×C4—44no—
268C2×C4—42no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4#640712yes—
275C2×C4#640812yes—
276C2×C4#640912yes—
277C2×C4#641112yes—
278C2×C4#641112yes—
279C2×C4C2×#114311yescharacteristic
280C2×C4#650512yes—
281C2×C4C2×#114411yescharacteristic
282C2×C4#651112yes—
283C8—284no—
284C8—284no—
285C7#768711yescharacteristic, complemented, Sylow
286C22—22no—
287C22—22no—
288C22—22no—
289C22—41no—
290C22—41no—
291C22—41no—
292C22—41no—
293C22C2×#640711yescharacteristic
294C22C2×#640811yescharacteristic
295C22C2×#640911yescharacteristic
296C22#8999711yescharacteristic
297C22#9000611yescharacteristic
298C22C2×#641111yescharacteristic
299C22C2×#641111yescharacteristic
300C4—22no—
301C4—22no—
302C4—21no—
303C4—21no—
304C4—22no—
305C4—22no—
306C4—82no—
307C4—82no—
308C4#8933314yes—
309C2—42no—
310C2#1172943011yescharacteristic
311C2C2×#8933311yescharacteristic
312C2#1172944211yescharacteristic
313C2#1172952811yescharacteristic
314C2#1172955111yescharacteristic
315C2#1172957911yescharacteristic
316C2#1172958011yescharacteristic
subgroupG/H
centerC23C2×C22⋊Dic7
commutatorC22×C4×C7C22×C4
FrattiniC2×C42C22×D7
FittingC7×#2857C2
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C22⋊C4
cosocleC2×C42×C7C23
hypercenter#2857D7
hypocenterC7#7687
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #2857

Complements

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

NH ≅ G/Ntype
4. #11725448C2semidirect
285. C7#7687semidirect

Maximal subgroups

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

subgroupindex
C7×#28572
C4×C22⋊(C7⋊C8)2
#117292812
#117254482
#76877