Finite Group Databasebeta

← #11740029

Subgroups — #11740029

Order 896.

814 conjugacy classes of subgroups.

The subgroup lattice has 352 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#120C211yesmaximal, characteristic
2C22×C22⋊Dic7C211yesmaximal, characteristic, complemented
3C22×C22⋊Dic7C211yesmaximal, characteristic, complemented
4C2×#6498C211yesmaximal, characteristic, complemented
5#89994C212yesmaximal, complemented
6#89992C211yesmaximal, characteristic, complemented
7#89992C212yesmaximal, complemented
8#89724C212yesmaximal, complemented
9#89718C211yesmaximal, characteristic, complemented
10#89715C211yesmaximal, characteristic, complemented
11#89714C212yesmaximal, complemented
12C25×C7C2211yescharacteristic
13C22×C7×D4C2211yescharacteristic
14C22×C7×D4C2211yescharacteristic
15C22×C7×D4C2211yescharacteristic
16C23×Dic7C2211yescharacteristic
17C23×Dic7C2212yes—
18C23×Dic7C2211yescharacteristic
19C7×#120—24no—
20C2×C7×C22⋊C4C2211yescharacteristic
21C2×C7×C22⋊C4C2211yescharacteristic
22C2×C7×C22⋊C4C2211yescharacteristic
23C2×C22⋊Dic7—24no—
24C2×C22⋊Dic7—24no—
25C2×C22⋊Dic7C2212yes—
26C2×C22⋊Dic7C2211yescharacteristic
27C2×C22⋊Dic7C2212yes—
28C2×C22⋊Dic7C2212yes—
29C2×C22⋊Dic7C2211yescharacteristic
30C2×C22⋊Dic7C2212yescomplemented
31C2×C22⋊Dic7C2212yescomplemented
32#6498—24no—
33C2×Dic7⋊C4C2212yescomplemented
34C2×Dic7⋊C4C2212yescomplemented
35C7⋊#95C2212yescomplemented
36C7⋊#95C2212yescomplemented
37C7⋊#95C2212yescomplemented
38C7⋊#95C2212yescomplemented
39#2412—71nomaximal, Sylow
40C24×C7—22no—
41C24×C7—22no—
42C24×C7—24no—
43C24×C7—42no—
44C24×C7D414yes—
45C24×C7C2311yescharacteristic
46C24×C7C2311yescharacteristic
47C24×C7C2311yescharacteristic
48C24×C7C2311yescharacteristic
49C2×C7×D4—22no—
50C2×C7×D4—22no—
51C2×C7×D4—22no—
52C2×C7×D4—22no—
53C2×C7×D4—22no—
54C2×C7×D4—24no—
55C2×C7×D4—42no—
56C22×C4×C7C2311yescharacteristic
57C22×C4×C7C2311yescharacteristic
58C22×C4×C7C2311yescharacteristic
59C22×Dic7—22no—
60C22×Dic7—22no—
61C22×Dic7—22no—
62C22×Dic7—24no—
63C22×Dic7—22no—
64C22×Dic7—24no—
65C22×Dic7—42no—
66C22×Dic7—42no—
67C22×Dic7C2312yes—
68C22×Dic7C2312yes—
69C22×Dic7C2312yes—
70C22×Dic7C2312yes—
71C7×C22⋊C4—22no—
72C7×C22⋊C4—22no—
73C7×C22⋊C4—22no—
74C22⋊Dic7—22no—
75C22⋊Dic7—24no—
76C22⋊Dic7—24no—
77C22⋊Dic7—24no—
78C22⋊Dic7—24no—
79C22⋊Dic7—42no—
80C22⋊Dic7D414yescomplemented
81Dic7⋊C4—24no—
82Dic7⋊C4—42no—
83C2×#123—71no—
84C2×#120D711yescharacteristic
85C22×C22⋊C4—71no—
86C22×C22⋊C4—71no—
87#397—72no—
88#394—72no—
89#386—71no—
90#386—72no—
91#385—71no—
92#381—71no—
93#380—72no—
94C23×C7—21no—
95C23×C7—21no—
96C23×C7—22no—
97C23×C7—22no—
98C23×C7—22no—
99C23×C7—28no—
100C23×C7—28no—
101C23×C7—21no—
102C23×C7—21no—
103C23×C7—22no—
104C23×C7—22no—
105C23×C7—21no—
106C23×C7—21no—
107C23×C7—21no—
108C23×C7—22no—
109C23×C7—22no—
110C23×C7—44no—
111C23×C7—48no—
112C23×C7—48no—
113C23×C7C2×D412yes—
114C23×C7C2×D412yes—
115C23×C7C2×D412yes—
116C23×C7C2×D412yes—
117C23×C7#4112yes—
118C23×C7#4114yes—
119C23×C7C2411yescharacteristic
120C7×D4—44no—
121C7×D4—44no—
122C7×D4—44no—
123C2×C4×C7—21no—
124C2×C4×C7—21no—
125C2×C4×C7—21no—
126C2×C4×C7—21no—
127C2×C4×C7—21no—
128C2×C4×C7—22no—
129C2×C4×C7—22no—
130C2×Dic7—22no—
131C2×Dic7—22no—
132C2×Dic7—22no—
133C2×Dic7—22no—
134C2×Dic7—22no—
135C2×Dic7—22no—
136C2×Dic7—22no—
137C2×Dic7—22no—
138C2×Dic7—22no—
139C2×Dic7—22no—
140C2×Dic7—22no—
141C2×Dic7—48no—
142C2×Dic7—44no—
143C2×Dic7—44no—
144C2×Dic7C2×D414yes—
145C25C2×D711yescharacteristic
146C22×D4C2×D711yescharacteristic
147C22×D4C2×D711yescharacteristic
148C22×D4C2×D711yescharacteristic
149C23×C4—71no—
150C23×C4—72no—
151C23×C4—71no—
152#123—144no—
153#120—24no—
154C2×C4⋊C4—72no—
155C2×C4⋊C4—72no—
156C2×C22⋊C4—71no—
157C2×C22⋊C4—72no—
158C2×C22⋊C4—71no—
159C2×C22⋊C4—72no—
160C2×C22⋊C4—72no—
161C2×C22⋊C4—72no—
162C2×C22⋊C4—72no—
163C2×C22⋊C4—144no—
164C2×C22⋊C4—144no—
165C2×C22⋊C4C2×D711yescharacteristic
166C2×C22⋊C4C2×D711yescharacteristic
167C2×C22⋊C4C2×D711yescharacteristic
168#95—72no—
169#95—72no—
170#95—72no—
171#95—72no—
172C22×C7—22no—
173C22×C7—22no—
174C22×C7—22no—
175C22×C7—22no—
176C22×C7—24no—
177C22×C7—24no—
178C22×C7—24no—
179C22×C7—24no—
180C22×C7—22no—
181C22×C7—22no—
182C22×C7—22no—
183C22×C7—416no—
184C22×C7—42no—
185C22×C7—42no—
186C22×C7—41no—
187C22×C7—41no—
188C22×C7—41no—
189C22×C7—42no—
190C22×C7—42no—
191C22×C7#12114yes—
192C22×C7#12314yescomplemented
193C22×C7C22×D411yescharacteristic
194C22×C7C22×D411yescharacteristic
195C22×C7C2×#4111yescharacteristic
196C22×C7C2×#4112yes—
197C22×C7H(5,𝔽2)12yes—
198C4×C7—41no—
199C4×C7—41no—
200C4×C7—41no—
201Dic7—24no—
202Dic7—42no—
203Dic7—42no—
204Dic7—42no—
205C24—22no—
206C24—24no—
207C24—22no—
208C24—42no—
209C24C22⋊D714yes—
210C24C22×D711yescharacteristic
211C24C22×D711yescharacteristic
212C24C22×D711yescharacteristic
213C24C22×D711yescharacteristic
214C2×D4—22no—
215C2×D4—24no—
216C2×D4—22no—
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—22no—
220C2×D4—42no—
221C22×C4—72no—
222C22×C4—72no—
223C22×C4—72no—
224C22×C4—72no—
225C22×C4—144no—
226C22×C4—142no—
227C22×C4—144no—
228C22×C4—142no—
229C22×C4—142no—
230C22×C4—142no—
231C22×C4—282no—
232C22×C4—282no—
233C22×C4C22×D711yescharacteristic
234C22×C4C22×D711yescharacteristic
235C22×C4C22×D711yescharacteristic
236C4⋊C4—144no—
237C4⋊C4—282no—
238C22⋊C4—22no—
239C22⋊C4—22no—
240C22⋊C4—22no—
241C22⋊C4—74no—
242C22⋊C4—144no—
243C22⋊C4—144no—
244C22⋊C4—142no—
245C22⋊C4—144no—
246C22⋊C4—144no—
247C22⋊C4—282no—
248C2×C7—24no—
249C2×C7—24no—
250C2×C7—42no—
251C2×C7—81no—
252C2×C7C2×#12111yescharacteristic
253C2×C7C2×#12311yescharacteristic
254C2×C7#53411yescharacteristic
255C2×C7#54612yes—
256C2×C7#55112yes—
257C23—21no—
258C23—22no—
259C23—21no—
260C23—22no—
261C23—22no—
262C23—28no—
263C23—28no—
264C23—21no—
265C23—22no—
266C23—21no—
267C23—22no—
268C23—21no—
269C23—22no—
270C23—21no—
271C23—22no—
272C23—21no—
273C23—44no—
274C23—48no—
275C23—48no—
276C23D4×D712yes—
277C23#115914yes—
278C23#115912yes—
279C23C2×C22⋊D712yes—
280C23C2×C22⋊D712yes—
281C23C2×C22⋊D712yes—
282C23C23×D711yescharacteristic
283D4—44no—
284D4—44no—
285D4—44no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—21no—
289C2×C4—22no—
290C2×C4—21no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—74no—
294C2×C4—142no—
295C2×C4—142no—
296C2×C4—142no—
297C2×C4—142no—
298C2×C4—142no—
299C2×C4—142no—
300C2×C4—142no—
301C2×C4—142no—
302C2×C4—142no—
303C2×C4—142no—
304C2×C4—142no—
305C2×C4—288no—
306C2×C4—284no—
307C2×C4—284no—
308C7#241211yescharacteristic, complemented, Sylow
309C22—22no—
310C22—22no—
311C22—22no—
312C22—22no—
313C22—24no—
314C22—24no—
315C22—24no—
316C22—24no—
317C22—22no—
318C22—22no—
319C22—22no—
320C22—416no—
321C22—42no—
322C22—42no—
323C22—41no—
324C22—41no—
325C22—41no—
326C22—42no—
327C22—42no—
328C22#649814yescomplemented
329C22#650214yes—
330C22C2×D4×D711yescharacteristic
331C22C2×#115912yes—
332C22C2×#115911yescharacteristic
333C22#654812yes—
334C22C22×C22⋊D711yescharacteristic
335C4—41no—
336C4—41no—
337C4—41no—
338C4—144no—
339C4—282no—
340C4—282no—
341C4—282no—
342C2—24no—
343C2—24no—
344C2—42no—
345C2—81no—
346C2#9027812yes—
347C2#9028212yes—
348C2C2×#649811yescharacteristic
349C2C2×#650211yescharacteristic
350C2#9049511yescharacteristic
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
Fitting, prime radicalC2×C7×#120C2
hypercenterC2×#120D7
hypocenterC7#2412
LCSG ▷ C23×C7 ▷ C7
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
2. C22×C22⋊Dic7C2semidirect
3. C22×C22⋊Dic7C2semidirect
4. C2×#6498C2semidirect
5. #89994C2semidirect
6. #89992C2semidirect
7. #89992C2semidirect
8. #89724C2semidirect
9. #89718C2semidirect
10. #89715C2semidirect
11. #89714C2semidirect
30. C2×C22⋊Dic7C22semidirect
31. C2×C22⋊Dic7C22semidirect
33. C2×Dic7⋊C4C22semidirect
34. C2×Dic7⋊C4C22semidirect
35. C7⋊#95C22semidirect
36. C7⋊#95C22semidirect
37. C7⋊#95C22semidirect
38. C7⋊#95C22semidirect
80. C22⋊Dic7D4semidirect
192. C22×C7#123semidirect
308. C7#2412semidirect
328. C22#6498semidirect

Maximal subgroups

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

subgroupindex
C2×C7×#1202
C22×C22⋊Dic72
C2×#64982
#899942
#899922
#897242
#897182
#897152
#897142
#24127