Finite Group Databasebeta

← #13242269

Subgroups — #13242269

Order 1792.

562 conjugacy classes of subgroups.

The subgroup lattice has 322 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#11725444C214yesmaximal, complemented
5C2×C7×C4⋊Q8C411yescharacteristic
6C2×C7×C22⋊Q8—21no—
7C22×C42×C7C2211yescharacteristic
8C22×#1136—21no—
9C7×#391—21no—
10C7×#389—21no—
11C7×#389C2212yes—
12C7×#384C411yescharacteristic
13C7×#380—21no—
14C7×#380C412yes—
15C2×C22⋊(C7⋊C8)—21no—
16C2×C4×C7⋊C8—21no—
17#89319—21no—
18#89319C2212yes—
19#89319C2212yes—
20#7685—71nomaximal, Sylow
21C22×C7×Q8—21no—
22C23×C4×C7C2×C411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7D411yescharacteristic
25C7×C4⋊Q8—42no—
26C7×C22⋊Q8—42no—
27C7×C22⋊Q8—42no—
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×C4⋊C4C2×C412yes—
34C2×C7×C22⋊C4—22no—
35C2×C42×C7—42no—
36C2×C42×C7C2×C411yescharacteristic
37C2×C42×C7D411yescharacteristic
38C2×C42×C7D411yescharacteristic
39C2×C42×C7C2311yescharacteristic
40C2×#1136—24no—
41C2×#1136—42no—
42C2×#1136—42no—
43C22×C7⋊C8—22no—
44C22×C7⋊C8—22no—
45C7×#95—22no—
46C7×#95—22no—
47C22⋊(C7⋊C8)—24no—
48C4×C7⋊C8—42no—
49#2857D711yescharacteristic
50C4×C22⋊C8—71no—
51#1728—71no—
52#1271—74no—
53C24×C7C22⋊C411yescharacteristic
54C2×C7×Q8—42no—
55C2×C7×Q8—42no—
56C2×C7×Q8—42no—
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—22no—
65C22×C4×C7—41no—
66C22×C4×C7—41no—
67C22×C4×C7C22⋊C411yescharacteristic
68C22×C4×C7C22⋊C411yescharacteristic
69C22×C4×C7C22⋊C411yescharacteristic
70C22×C4×C7D812yes—
71C22×C4×C7SD1612yes—
72C22×C4×C7C22×C411yescharacteristic
73C22×C4×C7C2×D411yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C7×C4⋊C4—24no—
76C7×C4⋊C4—42no—
77C7×C4⋊C4—42no—
78C7×C4⋊C4—42no—
79C7×C4⋊C4—42no—
80C7×C4⋊C4—42no—
81C7×C4⋊C4—42no—
82C7×C22⋊C4—44no—
83C42×C7—22no—
84C42×C7—22no—
85C42×C7—41no—
86C42×C7—41no—
87#1136—48no—
88C2×C7⋊C8—24no—
89C2×C7⋊C8—24no—
90C2×C7⋊C8—42no—
91C2×C7⋊C8—42no—
92C2×C7⋊C8—42no—
93C2×C7⋊C8—42no—
94C22×M16—141no—
95C2×C4⋊Q8Dic711yescharacteristic
96C2×C22⋊Q8—21no—
97C22×C42C2×D711yescharacteristic
98C2×C22⋊C8—141no—
99C2×C4×C8—141no—
100#391—21no—
101#389—21no—
102#389C2×D712yes—
103#384Dic711yescharacteristic
104#380—21no—
105#380Dic712yes—
106#336—72no—
107#336—72no—
108#336—141no—
109C23×C7—21no—
110C23×C7—21no—
111C23×C7—21no—
112C23×C7—21no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7#10212yes—
116C23×C7C2×C22⋊C411yescharacteristic
117C7×Q8—84no—
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—21no—
127C2×C4×C7—21no—
128C2×C4×C7—24no—
129C2×C4×C7—24no—
130C2×C4×C7—44no—
131C2×C4×C7—44no—
132C2×C4×C7—42no—
133C2×C4×C7—42no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—42no—
137C2×C4×C7—42no—
138C2×C4×C7#9912yes—
139C2×C4×C7#10112yes—
140C2×C4×C7#10212yes—
141C2×C4×C7#10412yes—
142C2×C4×C7#10412yes—
143C2×C4×C7#13112yes—
144C2×C4×C7C2×D811yescharacteristic
145C2×C4×C7C2×SD1611yescharacteristic
146C2×C4×C7#13712yes—
147C7⋊C8—44no—
148C7⋊C8—44no—
149C22×Q8—21no—
150C23×C4C2×Dic711yescharacteristic
151C23×C4C22⋊D711yescharacteristic
152C23×C4C22⋊D711yescharacteristic
153C2×M16—144no—
154C2×M16—282no—
155C2×M16—282no—
156C22×C8—142no—
157C22×C8—142no—
158C4⋊Q8—42no—
159C22⋊Q8—42no—
160C22⋊Q8—42no—
161C2×C4⋊C4—21no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—21no—
164C2×C4⋊C4—22no—
165C2×C4⋊C4C2×Dic712yes—
166C2×C4⋊C4C2×Dic712yes—
167C2×C22⋊C4—22no—
168C2×C42—42no—
169C2×C42C2×Dic711yescharacteristic
170C2×C42C22⋊D711yescharacteristic
171C2×C42C22⋊D711yescharacteristic
172C2×C42C22×D711yescharacteristic
173C22⋊C8—144no—
174C4×C8—282no—
175#95—22no—
176#95—22no—
177C22×C7—22no—
178C22×C7—22no—
179C22×C7—22no—
180C22×C7—41no—
181C22×C7—41no—
182C22×C7—41no—
183C22×C7—41no—
184C22×C7#40811yescharacteristic
185C22×C7C2×#9911yescharacteristic
186C22×C7C2×#10111yescharacteristic
187C22×C7C2×#10211yescharacteristic
188C22×C7#41911yescharacteristic
189C22×C7C2×#10411yescharacteristic
190C22×C7C2×#10411yescharacteristic
191C4×C7—22no—
192C4×C7—22no—
193C4×C7—21no—
194C4×C7—21no—
195C4×C7—22no—
196C4×C7—22no—
197C4×C7—82no—
198C4×C7—82no—
199C4×C7#32714yes—
200C24C22⋊Dic711yescharacteristic
201C2×Q8—42no—
202C2×Q8—42no—
203C2×Q8—42no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4—22no—
207C22×C4—22no—
208C22×C4—22no—
209C22×C4—22no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—41no—
213C22×C4—41no—
214C22×C4#114112yes—
215C22×C4#114212yes—
216C22×C4C22⋊Dic711yescharacteristic
217C22×C4C22⋊Dic711yescharacteristic
218C22×C4C22⋊Dic711yescharacteristic
219C22×C4C22×Dic711yescharacteristic
220C22×C4C2×C22⋊D711yescharacteristic
221C22×C4C2×C22⋊D711yescharacteristic
222M16—288no—
223C2×C8—144no—
224C2×C8—144no—
225C2×C8—282no—
226C2×C8—282no—
227C2×C8—282no—
228C2×C8—282no—
229C4⋊C4—24no—
230C4⋊C4—42no—
231C4⋊C4—42no—
232C4⋊C4—42no—
233C4⋊C4—42no—
234C4⋊C4—42no—
235C4⋊C4—42no—
236C22⋊C4—44no—
237C42—22no—
238C42—22no—
239C42—41no—
240C42—41no—
241C2×C7—42no—
242C2×C7#147211yescharacteristic
243C2×C7#148511yescharacteristic
244C2×C7#149211yescharacteristic
245C2×C7C2×#32711yescharacteristic
246C2×C7#150811yescharacteristic
247C2×C7#152611yescharacteristic
248C2×C7#154611yescharacteristic
249C23—21no—
250C23—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23#640612yes—
256C23C2×C22⋊Dic711yescharacteristic
257Q8—84no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—24no—
269C2×C4—24no—
270C2×C4—44no—
271C2×C4—44no—
272C2×C4—42no—
273C2×C4—42no—
274C2×C4—42no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4#640612yes—
279C2×C4#640812yes—
280C2×C4#641012yes—
281C2×C4#641112yes—
282C2×C4#641112yes—
283C2×C4C2×#114111yescharacteristic
284C2×C4C2×#114211yescharacteristic
285C2×C4#650512yes—
286C2×C4#651112yes—
287C8—284no—
288C8—284no—
289C7#768511yescharacteristic, complemented, Sylow
290C22—22no—
291C22—22no—
292C22—22no—
293C22—41no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22C2×#640611yescharacteristic
298C22C2×#640811yescharacteristic
299C22#8999711yescharacteristic
300C22C2×#641011yescharacteristic
301C22#9000611yescharacteristic
302C22C2×#641111yescharacteristic
303C22C2×#641111yescharacteristic
304C4—21no—
305C4—21no—
306C4—22no—
307C4—22no—
308C4—22no—
309C4—22no—
310C4—82no—
311C4—82no—
312C4#8933214yes—
313C2—42no—
314C2C2×#8933211yescharacteristic
315C2#1172944211yescharacteristic
316C2#1172944511yescharacteristic
317C2#1172952711yescharacteristic
318C2#1172955611yescharacteristic
319C2#1172957511yescharacteristic
320C2#1172957911yescharacteristic
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#7685
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. #11725444C2semidirect
289. C7#7685semidirect

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
#117254442
#76857