Finite Group Databasebeta

← #22614820

Subgroups — #22614820

Order 9344.

800 conjugacy classes of subgroups; only the 161 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 161 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×#14556559C211yesmaximal, characteristic
2C2×D4×Dic73C211yesmaximal, characteristic, complemented
3C22×#11924005C211yesmaximal, characteristic, complemented
4#17988509C212yesmaximal, complemented
5#17988502C212yesmaximal, complemented
6#17988477C212yesmaximal, complemented
7#17988464C212yesmaximal, complemented
8#17988450C211yesmaximal, characteristic, complemented
9#17988445C211yesmaximal, characteristic, complemented
10#17988442C211yesmaximal, characteristic, complemented
11C2×#123×C73C211yesmaximal, characteristic, complemented
12C24×D73C2211yescharacteristic
13C22×C22⋊D73C2212yes—
14C23×Dic73C2211yescharacteristic
15C23×Dic73C2211yescharacteristic, complemented
16#14556559—24no—
17D4×Dic73—24no—
18C2×C4⋊Dic73C2211yescharacteristic, complemented
19C2×#11924007C2212yescomplemented
20C2×C22⋊Dic73C2211yescharacteristic
21C2×C22⋊Dic73C2211yescharacteristic
22C2×C22⋊Dic73C2211yescharacteristic, complemented
23C2×#11924005—24no—
24C2×#11924005C2212yes—
25C2×#11924005C2212yes—
26C2×#11924005C2212yes—
27C2×#11924005C2212yes—
28C2×#11924005C2212yes—
29C2×C4×Dic73C2211yescharacteristic, complemented
30#14556498C2212yescomplemented
31#14556498C2212yescomplemented
32#14556498C2212yescomplemented
33C22×D4×C73C2211yescharacteristic
34C23×C4×C73C2211yescharacteristic
35#123×C73—24no—
36C2×C4⋊C4×C73C2212yescomplemented
37C2×C22⋊C4×C73C2211yescharacteristic
38C2×C22⋊C4×C73C2212yescomplemented
39C23×D73—22no—
40C23×D73—24no—
41C23×D73—22no—
42C23×D73—22no—
43C23×D73—42no—
44C23×D73C2312yes—
45C2×C22⋊D73—24no—
46C2×C22⋊D73—24no—
47C2×C22⋊D73—42no—
48C22×Dic73—22no—
49C22×Dic73—42no—
50C22×Dic73—42no—
51C22×Dic73D414yescomplemented
52C22×Dic73C2312yes—
53C22×Dic73C2311yescharacteristic
54C22×Dic73C2311yescharacteristic
55C22×Dic73C2311yescharacteristic
56C22×Dic73C2311yescharacteristic
57C4⋊Dic73—22no—
58#11924007—42no—
59C22⋊Dic73—22no—
60C22⋊Dic73—22no—
61C22⋊Dic73—22no—
62#11924005—24no—
63#11924005—24no—
64#11924005—24no—
65#11924005—24no—
66#11924005—42no—
67C4×Dic73—22no—
68C24×C73C2311yescharacteristic
69C24×C73C2311yescharacteristic
70C2×D4×C73—22no—
71C2×D4×C73—22no—
72C2×D4×C73D414yes—
73C22×C4×C73—22no—
74C22×C4×C73—22no—
75C22×C4×C73—22no—
76C22×C4×C73C2312yes—
77C22×C4×C73C2312yes—
78C22×C4×C73C2311yescharacteristic
79C4⋊C4×C73—24no—
80C22⋊C4×C73—22no—
81C22⋊C4×C73—24no—
82C23×C73—22no—
83C23×C73—22no—
84C23×C73—21no—
85C23×C73—21no—
86C23×C73—22no—
87C23×C73—22no—
88C23×C73C2×D412yes—
89C23×C73C2×D412yes—
90C23×C73C2×D412yes—
91C23×C73#4112yes—
92C23×C73C2411yescharacteristic
93C22×D73—28no—
94C22×D73—22no—
95C22×D73—22no—
96C22×D73—22no—
97C22×D73—22no—
98C22×D73—22no—
99C22×D73—22no—
100C22×D73—48no—
101C22×D73—48no—
102C22×D73—44no—
103C22×D73—44no—
104C22×D73#4114yes—
105D4×C73—28no—
106C2×C4×C73—21no—
107C2×C4×C73—21no—
108C2×C4×C73—22no—
109C2×C4×C73—22no—
110C2×C4×C73—22no—
111C2×C4×C73—22no—
112C2×C4×C73—22no—
113C2×C4×C73—22no—
114C2×C4×C73—44no—
115C2×C4×C73C2×D412yes—
116C22⋊D73—48no—
117C2×Dic73—22no—
118C2×Dic73—22no—
119C2×Dic73—21no—
120C2×Dic73—22no—
121C2×Dic73—21no—
122C2×Dic73—22no—
123C2×Dic73—22no—
124C2×Dic73—22no—
125C2×Dic73—22no—
126C2×Dic73—44no—
127C2×Dic73—44no—
128C2×Dic73C2×D412yes—
129C2×Dic73C2×D412yes—
130C22×C73—22no—
131C22×C73—22no—
132C22×C73—24no—
133C22×C73—24no—
134C22×C73—22no—
135C22×C73—24no—
136C22×C73—41no—
137C22×C73—41no—
138C22×C73—42no—
139C22×C73#12114yes—
140C22×C73C22×D411yescharacteristic
141C22×C73C22×D411yescharacteristic
142C22×C73C2×#4112yes—
143C22×C73C2×#4111yescharacteristic
144C22×C73H(5,𝔽2)12yes—
145C2×D73—24no—
146C2×D73—24no—
147C2×D73—24no—
148C2×D73—416no—
149C2×D73—42no—
150C2×D73—42no—
151C2×D73—42no—
152C2×D73—42no—
153C4×C73—22no—
154C4×C73—42no—
155C4×C73—42no—
156Dic73—22no—
157Dic73—22no—
158Dic73—41no—
159Dic73—41no—
160Dic73—42no—
161C2×C73—24no—
162C2×C73—42no—
163C2×C73C2×#12111yescharacteristic
164C2×C73#52511yescharacteristic
165C2×C73#53411yescharacteristic
166C2×C73#54612yes—
167C2×C73#55112yes—
168D73—44no—
169#2534—731nomaximal, Sylow
170C73#253411yescharacteristic, complemented, Sylow
171C2×#123D7311yescharacteristic, complemented
172C2×#120—731no—
173C2×C4×D4—731no—
174C22×C22⋊C4—731no—
175#396—732no—
176#394—732no—
177#388—732no—
178#386—732no—
179#381—731no—
180#380—731no—
181#379—731no—
182C25—731no—
183C22×D4—732no—
184C22×D4C2×D7311yescharacteristic
185C23×C4—731no—
186C23×C4—731no—
187C23×C4C2×D7311yescharacteristic
188#123—24no—
189#120—1464no—
190C4×D4—1464no—
191C2×C4⋊C4—732no—
192C2×C4⋊C4—731no—
193C2×C4⋊C4C2×D7312yescomplemented
194C2×C22⋊C4—732no—
195C2×C22⋊C4—731no—
196C2×C22⋊C4—731no—
197C2×C22⋊C4—731no—
198C2×C22⋊C4—732no—
199C2×C22⋊C4—732no—
200C2×C22⋊C4—732no—
201C2×C22⋊C4—732no—
202C2×C22⋊C4—1464no—
203C2×C22⋊C4C2×D7311yescharacteristic
204C2×C22⋊C4C2×D7312yescomplemented
205C2×C42—731no—
206#95—732no—
207#95—732no—
208#95—732no—
209C24—2922no—
210C24—732no—
211C24—1464no—
212C24—1462no—
213C24—1462no—
214C24—1462no—
215C24C22×D7311yescharacteristic
216C24C22×D7311yescharacteristic
217C2×D4—22no—
218C2×D4—22no—
219C2×D4—2922no—
220C2×D4—1464no—
221C2×D4—1464no—
222C2×D4C22⋊D7314yes—
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—22no—
226C22×C4—2922no—
227C22×C4—2922no—
228C22×C4—734no—
229C22×C4—731no—
230C22×C4—731no—
231C22×C4—732no—
232C22×C4—731no—
233C22×C4—731no—
234C22×C4—1462no—
235C22×C4C22×D7312yes—
236C22×C4C22×D7312yes—
237C22×C4C22×D7311yescharacteristic
238C4⋊C4—24no—
239C4⋊C4—2922no—
240C4⋊C4—1462no—
241C22⋊C4—22no—
242C22⋊C4—24no—
243C22⋊C4—2922no—
244C22⋊C4—1464no—
245C22⋊C4—1464no—
246C22⋊C4—1464no—
247C22⋊C4—1462no—
248C22⋊C4—1462no—
249C22⋊C4—1462no—
250C22⋊C4—1464no—
251C42—1462no—
252C23—22no—
253C23—22no—
254C23—21no—
255C23—22no—
256C23—21no—
257C23—22no—
258C23—2924no—
259C23—2928no—
260C23—2928no—
261C23—2924no—
262C23—734no—
263C23—1468no—
264C23—1462no—
265C23—1462no—
266C23—1462no—
267C23—1462no—
268C23—1462no—
269C23—1462no—
270C23D4×D7312yes—
271C23C2×C22⋊D7312yes—
272C23C2×C22⋊D7312yes—
273C23#1192402712yes—
274C23C23×D7311yescharacteristic
275D4—28no—
276D4—2928no—
277C2×C4—21no—
278C2×C4—21no—
279C2×C4—22no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—22no—
285C2×C4—44no—
286C2×C4—2924no—
287C2×C4—2924no—
288C2×C4—732no—
289C2×C4—732no—
290C2×C4—1461no—
291C2×C4—1461no—
292C2×C4—1462no—
293C2×C4—1462no—
294C2×C4—1462no—
295C2×C4—1462no—
296C2×C4—1462no—
297C2×C4—1462no—
298C2×C4—1462no—
299C2×C4C2×C22⋊D7312yes—
300C22—22no—
301C22—24no—
302C22—24no—
303C22—22no—
304C22—24no—
305C22—22no—
306C22—41no—
307C22—42no—
308C22—41no—
309C22—29216no—
310C22—2922no—
311C22—2922no—
312C22—2922no—
313C22—2922no—
314C22—1464no—
315C22—1464no—
316C22—1464no—
317C22#1455656314yes—
318C22C2×D4×D7311yescharacteristic
319C22C22×C22⋊D7311yescharacteristic
320C22#41×D7312yes—
321C22C2×#1192402711yescharacteristic
322C22#1455663712yes—
323C4—22no—
324C4—42no—
325C4—42no—
326C4—2921no—
327C4—2922no—
328C4—2921no—
329C4—1462no—
330C4—1462no—
331C2—24no—
332C2—42no—
333C2—2924no—
334C2C2×#1455656311yescharacteristic
335C2#1798909511yescharacteristic
336C2#1798914811yescharacteristic
337C2#1798922512yes—
338C2#1798926012yes—
subgroupG/H
center, FrattiniC23C23×D73
commutator, socleC23×C73C24
FittingC2×#123×C73C2
hypercenterC2×#123D73
hypocenterC73#2534
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
2. C2×D4×Dic73C2semidirect
3. C22×#11924005C2semidirect
4. #17988509C2semidirect
5. #17988502C2semidirect
6. #17988477C2semidirect
7. #17988464C2semidirect
8. #17988450C2semidirect
9. #17988445C2semidirect
10. #17988442C2semidirect
11. C2×#123×C73C2semidirect
15. C23×Dic73C22semidirect
18. C2×C4⋊Dic73C22semidirect
19. C2×#11924007C22semidirect
22. C2×C22⋊Dic73C22semidirect
29. C2×C4×Dic73C22semidirect
30. #14556498C22semidirect
31. #14556498C22semidirect
32. #14556498C22semidirect
36. C2×C4⋊C4×C73C22semidirect
38. C2×C22⋊C4×C73C22semidirect
51. C22×Dic73D4semidirect
170. C73#2534semidirect
171. C2×#123D73semidirect
193. C2×C4⋊C4C2×D73semidirect
204. C2×C22⋊C4C2×D73semidirect