Finite Group Databasebeta

← #13253878

Subgroups — #13253878

Order 1792.

598 conjugacy classes of subgroups.

The subgroup lattice has 346 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#11739521C211yesmaximal, characteristic
2C7×#1434C211yesmaximal, characteristic
3#11729283C211yesmaximal, characteristic
4#11725418C214yesmaximal
5C22×C42×C7C2211yescharacteristic
6C2×#6487—21no—
7C2×#6432C411yescharacteristic, complemented
8C2×C7×#95—21no—
9C2×C7×#95—21no—
10C2×C7×#95C2212yes—
11#89984—21no—
12#89984C2212yes—
13C2×C22⋊(C7⋊C8)C2212yes—
14#89707C411yescharacteristic, complemented
15#89701—21no—
16#89700—21no—
17#89700C412yescomplemented
18C2×#6377—21no—
19#89319—21no—
20#9888—71nomaximal, Sylow
21C23×C4×C7—22no—
22C23×C4×C7D411yescharacteristic
23C23×C4×C7D411yescharacteristic
24C23×C4×C7C2311yescharacteristic
25C22×Dic14—21no—
26C2×C42×C7—42no—
27C2×C42×C7C2×C411yescharacteristic
28C2×C42×C7C2×C411yescharacteristic
29C2×C42×C7D411yescharacteristic
30C2×C42×C7D411yescharacteristic
31C2×C22⋊Dic7—22no—
32C2×C4⋊Dic7—21no—
33C2×C4⋊Dic7—21no—
34C2×C4⋊Dic7—21no—
35C2×C4⋊Dic7C2×C412yes—
36C2×C4⋊Dic7C2×C412yes—
37#6487—44no—
38C2×Dic7⋊C4—22no—
39C22×C7⋊C8—22no—
40#6432—42no—
41C7×#95—24no—
42C7×#95—24no—
43C7×#95—42no—
44C7⋊#95—22no—
45C7⋊#95—22no—
46C22⋊(C7⋊C8)—22no—
47C22⋊(C7⋊C8)Q814yescomplemented
48#6377—42no—
49#2857—71no—
50#1749—71no—
51#1434D711yescharacteristic
52#1294—74no—
53C24×C7C2×D411yescharacteristic
54C22×C4×C7—22no—
55C22×C4×C7—22no—
56C22×C4×C7—22no—
57C22×C4×C7—22no—
58C22×C4×C7—24no—
59C22×C4×C7—41no—
60C22×C4×C7—41no—
61C22×C4×C7—41no—
62C22×C4×C7—41no—
63C22×C4×C7—42no—
64C22×C4×C7—42no—
65C22×C4×C7—42no—
66C22×C4×C7—42no—
67C22×C4×C7—42no—
68C22×C4×C7—42no—
69C22×C4×C7C22⋊C411yescharacteristic
70C22×C4×C7C22⋊C411yescharacteristic
71C22×C4×C7C22⋊C411yescharacteristic
72C22×C4×C7C22⋊C411yescharacteristic
73C22×C4×C7C22×C411yescharacteristic
74C22×C4×C7C2×D411yescharacteristic
75C22×C4×C7C2×Q812yes—
76C22×C4×C7#4112yes—
77C22×Dic7—22no—
78C22×Dic7—22no—
79C2×Dic14—42no—
80C2×Dic14—42no—
81C2×Dic14—42no—
82C42×C7—22no—
83C42×C7—22no—
84C42×C7—41no—
85C42×C7—41no—
86C22⋊Dic7—44no—
87C4⋊Dic7—42no—
88C4⋊Dic7—42no—
89C4⋊Dic7—42no—
90C4⋊Dic7—42no—
91C4⋊Dic7—42no—
92Dic7⋊C4—44no—
93C2×C7⋊C8—24no—
94C2×C7⋊C8—42no—
95C2×C7⋊C8—42no—
96C2×C4⋊Q8—71no—
97C2×C22⋊Q8—141no—
98C22×C42C2×D711yescharacteristic
99C2×C22⋊C8—72no—
100C2×#97—141no—
101#391—141no—
102#389—72no—
103#389—141no—
104#384—71no—
105#380—72no—
106#380—141no—
107C2×#95—21no—
108C2×#95—21no—
109C2×#95C2×D712yes—
110#336—141no—
111C23×C7—21no—
112C23×C7—21no—
113C23×C7—21no—
114C23×C7—21no—
115C23×C7C2×C22⋊C411yescharacteristic
116C23×C7#12411yescharacteristic
117C23×C7C4⋊Q811yescharacteristic
118C23×C7AGL(1,ℤ/8)12yes—
119C23×C7#13712yes—
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—21no—
129C2×C4×C7—21no—
130C2×C4×C7—21no—
131C2×C4×C7—21no—
132C2×C4×C7—21no—
133C2×C4×C7—21no—
134C2×C4×C7—42no—
135C2×C4×C7—42no—
136C2×C4×C7—44no—
137C2×C4×C7—44no—
138C2×C4×C7—44no—
139C2×C4×C7—44no—
140C2×C4×C7—44no—
141C2×C4×C7—82no—
142C2×C4×C7—82no—
143C2×C4×C7—82no—
144C2×C4×C7—82no—
145C2×C4×C7#9912yes—
146C2×C4×C7#10412yes—
147C2×C4×C7C4×Q812yes—
148C2×C4×C7C22⋊Q812yes—
149C2×Dic7—42no—
150C2×Dic7—42no—
151C2×Dic7—42no—
152C2×Dic7—42no—
153C2×Dic7—42no—
154C2×Dic7—42no—
155Dic14—84no—
156C7⋊C8—44no—
157C22×Q8—141no—
158C23×C4—22no—
159C23×C4C22⋊D711yescharacteristic
160C23×C4C22⋊D711yescharacteristic
161C23×C4C22×D711yescharacteristic
162C22×C8—142no—
163C4⋊Q8—282no—
164C22⋊Q8—284no—
165C2×C4⋊C4—72no—
166C2×C4⋊C4—72no—
167C2×C4⋊C4—142no—
168C2×C4⋊C4—141no—
169C2×C4⋊C4—141no—
170C2×C4⋊C4—141no—
171C2×C22⋊C4—142no—
172C2×C42—42no—
173C2×C42C4×D711yescharacteristic
174C2×C42C4×D711yescharacteristic
175C2×C42D2811yescharacteristic
176C2×C42D2811yescharacteristic
177C22⋊C8—74no—
178C22⋊C8—142no—
179#97—282no—
180#95—24no—
181#95—24no—
182#95—42no—
183#95—142no—
184#95—142no—
185C22×C7—21no—
186C22×C7—21no—
187C22×C7—21no—
188C22×C7—21no—
189C22×C7—22no—
190C22×C7—22no—
191C22×C7—22no—
192C22×C7—22no—
193C22×C7#39111yescharacteristic
194C22×C7C2×#9911yescharacteristic
195C22×C7#41011yescharacteristic
196C22×C7#41811yescharacteristic
197C22×C7#41911yescharacteristic
198C22×C7C2×#10411yescharacteristic
199C22×C7#42111yescharacteristic
200C22×C7#48912yes—
201C22×C7#50112yes—
202C4×C7—22no—
203C4×C7—22no—
204C4×C7—22no—
205C4×C7—22no—
206C4×C7—41no—
207C4×C7—41no—
208C4×C7—84no—
209Dic7—82no—
210Dic7—82no—
211C24C2×C22⋊D711yescharacteristic
212C2×Q8—282no—
213C2×Q8—282no—
214C2×Q8—282no—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—24no—
220C22×C4—41no—
221C22×C4—41no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—42no—
225C22×C4—42no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—42no—
229C22×C4—42no—
230C22×C4—142no—
231C22×C4—142no—
232C22×C4(C2×D7)⋊C411yescharacteristic
233C22×C4(C2×D7)⋊C411yescharacteristic
234C22×C4(C2×D7)⋊C411yescharacteristic
235C22×C4(C2×D7)⋊C411yescharacteristic
236C22×C4C2×C4×D711yescharacteristic
237C22×C4C2×D2811yescharacteristic
238C22×C4#115912yes—
239C22×C4Q8×D712yes—
240C2×C8—144no—
241C2×C8—282no—
242C2×C8—282no—
243C4⋊C4—282no—
244C4⋊C4—282no—
245C4⋊C4—284no—
246C4⋊C4—282no—
247C4⋊C4—282no—
248C4⋊C4—282no—
249C22⋊C4—284no—
250C42—22no—
251C42—22no—
252C42—41no—
253C42—41no—
254C2×C7—22no—
255C2×C7—22no—
256C2×C7#150611yescharacteristic
257C2×C7#150711yescharacteristic
258C2×C7#151111yescharacteristic
259C2×C7#151411yescharacteristic
260C2×C7#198611yescharacteristic
261C2×C7#199311yescharacteristic
262C2×C7#199611yescharacteristic
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23C2×(C2×D7)⋊C411yescharacteristic
268C23#649911yescharacteristic
269C23#650711yescharacteristic
270C23#651212yes—
271C23#651412yes—
272Q8—564no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—21no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—42no—
288C2×C4—42no—
289C2×C4—44no—
290C2×C4—44no—
291C2×C4—44no—
292C2×C4—44no—
293C2×C4—44no—
294C2×C4—82no—
295C2×C4—82no—
296C2×C4—82no—
297C2×C4—82no—
298C2×C4—282no—
299C2×C4—282no—
300C2×C4—282no—
301C2×C4—282no—
302C2×C4—282no—
303C2×C4—282no—
304C2×C4#637912yes—
305C2×C4#638012yes—
306C2×C4#645012yes—
307C2×C4#646012yes—
308C8—284no—
309C7#988811yescharacteristic, complemented, Sylow
310C22—22no—
311C22—22no—
312C22—21no—
313C22—21no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—22no—
318C22C2×#637911yescharacteristic
319C22C2×#638011yescharacteristic
320C22#8974611yescharacteristic
321C22#8977311yescharacteristic
322C22#8977611yescharacteristic
323C22#8977711yescharacteristic
324C22#8977811yescharacteristic
325C22#8982712yes—
326C22#8983712yes—
327C4—22no—
328C4—22no—
329C4—22no—
330C4—22no—
331C4—41no—
332C4—41no—
333C4—84no—
334C4—562no—
335C4—562no—
336C2—22no—
337C2—22no—
338C2#1172718711yescharacteristic
339C2#1172719811yescharacteristic
340C2#1172724711yescharacteristic
341C2#1172728311yescharacteristic
342C2#1172728711yescharacteristic
343C2#1172730911yescharacteristic
344C2#1172731811yescharacteristic
subgroupG/H
centerC23C2×(C2×D7)⋊C4
commutatorC22×C4×C7C22×C4
FrattiniC23×C4C22×D7
FittingC7×#1434C2
socleC23×C7C2×C22⋊C4
prime radicalC24×C7C2×D4
cosocleC23×C4×C7C23
hypercenter#1434D7
hypocenterC7#9888
LCSG ▷ C22×C4×C7 ▷ C22×C7 ▷ C7
UCSC1 ◁ C23 ◁ C22×C42 ◁ #1434

Complements

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

NH ≅ G/Ntype
7. C2×#6432C4semidirect
14. #89707C4semidirect
17. #89700C4semidirect
47. C22⋊(C7⋊C8)Q8semidirect
309. C7#9888semidirect

Maximal subgroups

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

subgroupindex
#117395212
C7×#14342
#117292832
#117254182
#98887