Finite Group Databasebeta

← #11740028

Subgroups — #11740028

Order 896.

786 conjugacy classes of subgroups; only the 165 normal subgroups are shown.

normal subgroups only

This group has 786 conjugacy classes of subgroups — too many to draw. Only its 165 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×C7×#120C211yesmaximal, characteristic
2C22×C22⋊Dic7C211yesmaximal, characteristic, complemented
3C2×#6498C211yesmaximal, characteristic, complemented
4C2×#6440C211yesmaximal, characteristic, complemented
5#90019C211yesmaximal, characteristic, complemented
6#89994C212yesmaximal, complemented
7#89993C212yesmaximal, complemented
8#89992C211yesmaximal, characteristic, complemented
9#89722C211yesmaximal, characteristic, complemented
10#89720C212yesmaximal, complemented
11#89716C212yesmaximal, complemented
12C25×C7C2211yescharacteristic
13C22×C7×D4C2211yescharacteristic
14C22×C7×D4C2212yes—
15C23×Dic7C2211yescharacteristic
16C23×Dic7C2211yescharacteristic
17C7×#120—24no—
18C2×C7×C22⋊C4C2212yes—
19C2×C7×C22⋊C4C2211yescharacteristic
20C2×C22⋊Dic7—24no—
21C2×C22⋊Dic7C2211yescharacteristic
22C2×C22⋊Dic7C2212yes—
23C2×C22⋊Dic7C2212yes—
24C2×C22⋊Dic7C2211yescharacteristic
25C2×C22⋊Dic7C2212yes—
26C2×C22⋊Dic7C2212yes—
27C2×C22⋊Dic7C2211yescharacteristic, complemented
28C2×C22⋊Dic7C2211yescharacteristic, complemented
29C2×C22⋊Dic7C2212yescomplemented
30#6498—24no—
31C2×C4⋊Dic7C2212yescomplemented
32C2×Dic7⋊C4C2212yescomplemented
33C2×Dic7⋊C4C2211yescharacteristic, complemented
34C2×Dic7⋊C4C2211yescharacteristic, complemented
35C2×C4×Dic7C2212yescomplemented
36#6440—24no—
37C7⋊#95C2212yescomplemented
38C7⋊#95C2212yescomplemented
39#2411—71nomaximal, Sylow
40C24×C7—22no—
41C24×C7—24no—
42C24×C7—22no—
43C24×C7—22no—
44C24×C7—42no—
45C24×C7C2311yescharacteristic
46C24×C7C2312yes—
47C24×C7C2311yescharacteristic
48C2×C7×D4—22no—
49C2×C7×D4—22no—
50C2×C7×D4—24no—
51C2×C7×D4—22no—
52C2×C7×D4—24no—
53C2×C7×D4—42no—
54C22×C4×C7C2311yescharacteristic
55C22×C4×C7C2312yes—
56C22×Dic7—22no—
57C22×Dic7—22no—
58C22×Dic7—22no—
59C22×Dic7—42no—
60C22×Dic7D414yescomplemented
61C22×Dic7C2311yescharacteristic
62C22×Dic7C2312yes—
63C22×Dic7C2312yes—
64C22×Dic7C2311yescharacteristic
65C22×Dic7C2311yescharacteristic
66C22×Dic7C2311yescharacteristic
67C7×C22⋊C4—24no—
68C7×C22⋊C4D414yes—
69C22⋊Dic7—22no—
70C22⋊Dic7—22no—
71C22⋊Dic7—24no—
72C22⋊Dic7—22no—
73C22⋊Dic7—22no—
74C22⋊Dic7—24no—
75C22⋊Dic7—24no—
76C22⋊Dic7—42no—
77C22⋊Dic7—42no—
78C4⋊Dic7—42no—
79Dic7⋊C4—24no—
80Dic7⋊C4—22no—
81Dic7⋊C4—22no—
82C4×Dic7—24no—
83C2×#123—71no—
84C2×#120D711yescharacteristic
85C2×#117—71no—
86C22×C22⋊C4—71no—
87#396—72no—
88#394—72no—
89#390—72no—
90#388—72no—
91#386—71no—
92#385—71no—
93#379—71no—
94C23×C7—22no—
95C23×C7—22no—
96C23×C7—28no—
97C23×C7—22no—
98C23×C7—22no—
99C23×C7—22no—
100C23×C7—22no—
101C23×C7—22no—
102C23×C7—22no—
103C23×C7—21no—
104C23×C7—21no—
105C23×C7—21no—
106C23×C7—22no—
107C23×C7—22no—
108C23×C7—48no—
109C23×C7—48no—
110C23×C7—44no—
111C23×C7—44no—
112C23×C7C2×D412yes—
113C23×C7C2×D412yes—
114C23×C7#4114yes—
115C23×C7#4112yes—
116C23×C7C2411yescharacteristic
117C7×D4—48no—
118C7×D4—44no—
119C2×C4×C7—22no—
120C2×C4×C7—22no—
121C2×C4×C7—21no—
122C2×C4×C7—21no—
123C2×C4×C7—21no—
124C2×C4×C7C2×D414yes—
125C2×Dic7—28no—
126C2×Dic7—21no—
127C2×Dic7—21no—
128C2×Dic7—21no—
129C2×Dic7—21no—
130C2×Dic7—22no—
131C2×Dic7—21no—
132C2×Dic7—21no—
133C2×Dic7—22no—
134C2×Dic7—22no—
135C2×Dic7—22no—
136C2×Dic7—22no—
137C2×Dic7—22no—
138C2×Dic7—22no—
139C2×Dic7—22no—
140C2×Dic7—44no—
141C2×Dic7C2×D412yes—
142C2×Dic7C2×D412yes—
143C25C2×D711yescharacteristic
144C22×D4C2×D711yescharacteristic
145C22×D4C2×D712yes—
146C23×C4—71no—
147C23×C4—71no—
148#123—144no—
149#120—24no—
150#117—144no—
151C2×C4⋊C4—72no—
152C2×C4⋊C4—71no—
153C2×C4⋊C4—71no—
154C2×C4⋊C4—72no—
155C2×C22⋊C4—72no—
156C2×C22⋊C4—72no—
157C2×C22⋊C4—71no—
158C2×C22⋊C4—71no—
159C2×C22⋊C4—71no—
160C2×C22⋊C4—71no—
161C2×C22⋊C4—72no—
162C2×C22⋊C4—72no—
163C2×C22⋊C4—72no—
164C2×C22⋊C4—144no—
165C2×C22⋊C4C2×D712yes—
166C2×C22⋊C4C2×D711yescharacteristic
167C2×C42—72no—
168#95—72no—
169#95—72no—
170C22×C7—22no—
171C22×C7—22no—
172C22×C7—24no—
173C22×C7—24no—
174C22×C7—24no—
175C22×C7—24no—
176C22×C7—24no—
177C22×C7—416no—
178C22×C7—42no—
179C22×C7—42no—
180C22×C7—42no—
181C22×C7—42no—
182C22×C7—41no—
183C22×C7—41no—
184C22×C7—41no—
185C22×C7—42no—
186C22×C7—42no—
187C22×C7#12114yes—
188C22×C7C22×D411yescharacteristic
189C22×C7C22×D411yescharacteristic
190C22×C7C2×#4111yescharacteristic
191C22×C7C2×#4112yes—
192C22×C7H(5,𝔽2)12yes—
193C4×C7—24no—
194C4×C7—41no—
195Dic7—22no—
196Dic7—22no—
197Dic7—41no—
198Dic7—41no—
199Dic7—42no—
200Dic7—42no—
201C24—22no—
202C24—24no—
203C24—22no—
204C24—22no—
205C24—42no—
206C24C22×D711yescharacteristic
207C24C22×D712yes—
208C24C22×D711yescharacteristic
209C2×D4—22no—
210C2×D4—24no—
211C2×D4—22no—
212C2×D4—22no—
213C2×D4—24no—
214C2×D4—42no—
215C22×C4—74no—
216C22×C4—71no—
217C22×C4—71no—
218C22×C4—71no—
219C22×C4—72no—
220C22×C4—72no—
221C22×C4—71no—
222C22×C4—142no—
223C22×C4—142no—
224C22×C4—142no—
225C22×C4—282no—
226C22×C4C22×D711yescharacteristic
227C22×C4C22×D712yes—
228C4⋊C4—142no—
229C4⋊C4—142no—
230C4⋊C4—144no—
231C4⋊C4—282no—
232C22⋊C4—24no—
233C22⋊C4—144no—
234C22⋊C4—144no—
235C22⋊C4—142no—
236C22⋊C4—142no—
237C22⋊C4—142no—
238C22⋊C4—144no—
239C22⋊C4—142no—
240C22⋊C4—282no—
241C22⋊C4—282no—
242C22⋊C4C22⋊D714yes—
243C42—144no—
244C2×C7—24no—
245C2×C7—44no—
246C2×C7—81no—
247C2×C7C2×#12111yescharacteristic
248C2×C7#52511yescharacteristic
249C2×C7#53511yescharacteristic
250C2×C7#54612yes—
251C2×C7#55112yes—
252C23—22no—
253C23—22no—
254C23—28no—
255C23—22no—
256C23—22no—
257C23—22no—
258C23—22no—
259C23—22no—
260C23—22no—
261C23—21no—
262C23—21no—
263C23—22no—
264C23—22no—
265C23—21no—
266C23—48no—
267C23—48no—
268C23—44no—
269C23—44no—
270C23D4×D712yes—
271C23#115914yes—
272C23#115912yes—
273C23C2×C22⋊D712yes—
274C23C23×D711yescharacteristic
275D4—48no—
276D4—44no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—21no—
282C2×C4—72no—
283C2×C4—72no—
284C2×C4—148no—
285C2×C4—141no—
286C2×C4—141no—
287C2×C4—141no—
288C2×C4—141no—
289C2×C4—141no—
290C2×C4—141no—
291C2×C4—142no—
292C2×C4—142no—
293C2×C4—142no—
294C2×C4—142no—
295C2×C4—142no—
296C2×C4—142no—
297C2×C4—142no—
298C2×C4—142no—
299C2×C4—284no—
300C2×C4C2×C22⋊D714yes—
301C7#241111yescharacteristic, complemented, Sylow
302C22—24no—
303C22—22no—
304C22—24no—
305C22—22no—
306C22—24no—
307C22—24no—
308C22—24no—
309C22—416no—
310C22—42no—
311C22—42no—
312C22—42no—
313C22—42no—
314C22—41no—
315C22—42no—
316C22—41no—
317C22—42no—
318C22—41no—
319C22#650114yes—
320C22C2×D4×D711yescharacteristic
321C22C2×#115911yescharacteristic
322C22C2×#115912yes—
323C22#654812yes—
324C22C22×C22⋊D711yescharacteristic
325C4—24no—
326C4—41no—
327C4—142no—
328C4—142no—
329C4—282no—
330C4—281no—
331C4—282no—
332C4—281no—
333C2—24no—
334C2—44no—
335C2—81no—
336C2#9027711yescharacteristic
337C2#9027812yes—
338C2#9028311yescharacteristic
339C2C2×#650111yescharacteristic
340C2#9049712yes—
subgroupG/H
center, FrattiniC23C23×D7
commutator, socle, cosocleC23×C7C24
Fitting, prime radicalC2×C7×#120C2
hypercenterC2×#120D7
hypocenterC7#2411
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. C2×#6498C2semidirect
4. C2×#6440C2semidirect
5. #90019C2semidirect
6. #89994C2semidirect
7. #89993C2semidirect
8. #89992C2semidirect
9. #89722C2semidirect
10. #89720C2semidirect
11. #89716C2semidirect
27. C2×C22⋊Dic7C22semidirect
28. C2×C22⋊Dic7C22semidirect
29. C2×C22⋊Dic7C22semidirect
31. C2×C4⋊Dic7C22semidirect
32. C2×Dic7⋊C4C22semidirect
33. C2×Dic7⋊C4C22semidirect
34. C2×Dic7⋊C4C22semidirect
35. C2×C4×Dic7C22semidirect
37. C7⋊#95C22semidirect
38. C7⋊#95C22semidirect
60. C22×Dic7D4semidirect
301. C7#2411semidirect

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
C2×#64402
#900192
#899942
#899932
#899922
#897222
#897202
#897162
#24117