Finite Group Databasebeta

← #22615669

Subgroups — #22615669

Order 9344.

824 conjugacy classes of subgroups; only the 157 normal subgroups are shown.

normal subgroups only

This group has 824 conjugacy classes of subgroups — too many to draw. Only its 157 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×#14556563C211yesmaximal, characteristic, complemented
2C2×#14556560C211yesmaximal, characteristic, complemented
3C22×C22⋊Dic73C211yesmaximal, characteristic, complemented
4#17988515C212yesmaximal, complemented
5#17988502C212yesmaximal, complemented
6#17988501C212yesmaximal, complemented
7#17988495C212yesmaximal, complemented
8#17988467C211yesmaximal, characteristic, complemented
9#17988464C211yesmaximal, characteristic, complemented
10#17988445C211yesmaximal, characteristic, complemented
11C2×#120×C73C211yesmaximal, characteristic, complemented
12C22×C22⋊D73C2211yescharacteristic
13C22×C22⋊D73C2212yes—
14C22×C22⋊D73C2211yescharacteristic, complemented
15C23×Dic73C2211yescharacteristic
16C23×Dic73C2211yescharacteristic
17#14556563—24no—
18#14556560—24no—
19C2×#11924007C2211yescharacteristic, complemented
20C2×#11924007C2212yescomplemented
21C2×C22⋊Dic73—24no—
22C2×C22⋊Dic73C2211yescharacteristic
23C2×C22⋊Dic73C2212yes—
24C2×C22⋊Dic73C2212yes—
25C2×C22⋊Dic73C2212yes—
26C2×C22⋊Dic73C2211yescharacteristic, complemented
27C2×C22⋊Dic73C2212yescomplemented
28C2×#11924005C2211yescharacteristic, complemented
29C2×#11924005C2212yescomplemented
30#14556498C2212yescomplemented
31#14556498C2212yescomplemented
32#14556498C2212yescomplemented
33C25×C73C2211yescharacteristic
34C22×D4×C73C2211yescharacteristic
35C22×D4×C73C2212yes—
36#120×C73—24no—
37C2×C22⋊C4×C73C2211yescharacteristic
38C2×C22⋊C4×C73C2212yes—
39C23×D73C2311yescharacteristic
40C2×C22⋊D73—24no—
41C2×C22⋊D73—22no—
42C2×C22⋊D73—22no—
43C2×C22⋊D73—22no—
44C2×C22⋊D73—22no—
45C2×C22⋊D73—42no—
46C2×C22⋊D73—42no—
47C2×C22⋊D73—42no—
48C22×Dic73—22no—
49C22×Dic73—42no—
50C22×Dic73—42no—
51C22×Dic73D414yescomplemented
52C22×Dic73C2312yes—
53C22×Dic73C2311yescharacteristic
54C22×Dic73C2312yes—
55C22×Dic73C2311yescharacteristic
56C22×Dic73C2311yescharacteristic
57#11924007—22no—
58#11924007—42no—
59C22⋊Dic73—24no—
60C22⋊Dic73—22no—
61C22⋊Dic73—24no—
62C22⋊Dic73—24no—
63C22⋊Dic73—22no—
64C22⋊Dic73—42no—
65#11924005—22no—
66#11924005—42no—
67C24×C73—22no—
68C24×C73—22no—
69C24×C73—42no—
70C24×C73—42no—
71C24×C73D414yescomplemented
72C24×C73C2311yescharacteristic
73C24×C73C2312yes—
74C24×C73C2311yescharacteristic
75C2×D4×C73—24no—
76C2×D4×C73—22no—
77C2×D4×C73—22no—
78C2×D4×C73—22no—
79C2×D4×C73—42no—
80C2×D4×C73—42no—
81C22×C4×C73C2311yescharacteristic
82C22×C4×C73C2312yes—
83C22⋊C4×C73—24no—
84C22⋊C4×C73—22no—
85C23×C73—22no—
86C23×C73—22no—
87C23×C73—28no—
88C23×C73—22no—
89C23×C73—22no—
90C23×C73—22no—
91C23×C73—22no—
92C23×C73—22no—
93C23×C73—22no—
94C23×C73—21no—
95C23×C73—21no—
96C23×C73—21no—
97C23×C73—22no—
98C23×C73—22no—
99C23×C73—48no—
100C23×C73—44no—
101C23×C73—44no—
102C23×C73—48no—
103C23×C73C2×D412yes—
104C23×C73C2×D414yes—
105C23×C73C2×D412yes—
106C23×C73#4112yes—
107C23×C73C2411yescharacteristic
108C22×D73—21no—
109C22×D73—21no—
110C22×D73—21no—
111C22×D73—22no—
112C22×D73—22no—
113D4×C73—48no—
114D4×C73—44no—
115C2×C4×C73—21no—
116C2×C4×C73—22no—
117C2×C4×C73—21no—
118C2×C4×C73—21no—
119C2×C4×C73—22no—
120C2×C4×C73—22no—
121C22⋊D73—44no—
122C22⋊D73—48no—
123C22⋊D73—44no—
124C2×Dic73—22no—
125C2×Dic73—22no—
126C2×Dic73—22no—
127C2×Dic73—21no—
128C2×Dic73—22no—
129C2×Dic73—21no—
130C2×Dic73—21no—
131C2×Dic73—22no—
132C2×Dic73—22no—
133C2×Dic73—22no—
134C2×Dic73—22no—
135C2×Dic73—44no—
136C2×Dic73—44no—
137C2×Dic73C2×D412yes—
138C2×Dic73C2×D412yes—
139C22×C73—22no—
140C22×C73—24no—
141C22×C73—22no—
142C22×C73—24no—
143C22×C73—24no—
144C22×C73—24no—
145C22×C73—24no—
146C22×C73—416no—
147C22×C73—42no—
148C22×C73—42no—
149C22×C73—42no—
150C22×C73—42no—
151C22×C73—41no—
152C22×C73—41no—
153C22×C73—42no—
154C22×C73—41no—
155C22×C73—42no—
156C22×C73#12114yes—
157C22×C73C22×D411yescharacteristic
158C22×C73C22×D411yescharacteristic
159C22×C73C2×#4111yescharacteristic
160C22×C73H(5,𝔽2)12yes—
161C22×C73H(5,𝔽2)12yes—
162C2×D73—41no—
163C2×D73—42no—
164C2×D73—41no—
165C2×D73—41no—
166C2×D73—42no—
167C4×C73—41no—
168C4×C73—42no—
169Dic73—22no—
170Dic73—22no—
171Dic73—41no—
172Dic73—42no—
173Dic73—42no—
174C2×C73—24no—
175C2×C73—44no—
176C2×C73—81no—
177C2×C73C2×#12111yescharacteristic
178C2×C73#53412yes—
179C2×C73#53411yescharacteristic
180C2×C73#53511yescharacteristic
181C2×C73#53812yes—
182D73—81no—
183#2617—731nomaximal, Sylow
184C73#261711yescharacteristic, complemented, Sylow
185C2×#121—731no—
186C2×#120—731no—
187C2×#120D7311yescharacteristic, complemented
188C22×C22⋊C4—731no—
189#397—732no—
190#394—732no—
191#394—732no—
192#392—732no—
193#386—731no—
194#386—731no—
195#380—731no—
196C25C2×D7311yescharacteristic
197C22×D4—732no—
198C22×D4—731no—
199C22×D4—731no—
200C22×D4C2×D7311yescharacteristic
201C22×D4C2×D7312yes—
202C23×C4—731no—
203C23×C4—731no—
204#121—1464no—
205#120—24no—
206#120—1464no—
207C2×C4⋊C4—732no—
208C2×C4⋊C4—731no—
209C2×C22⋊C4—732no—
210C2×C22⋊C4—732no—
211C2×C22⋊C4—732no—
212C2×C22⋊C4—732no—
213C2×C22⋊C4—731no—
214C2×C22⋊C4—732no—
215C2×C22⋊C4—731no—
216C2×C22⋊C4—731no—
217C2×C22⋊C4—1464no—
218C2×C22⋊C4C2×D7311yescharacteristic
219C2×C22⋊C4C2×D7312yes—
220#95—732no—
221#95—732no—
222#95—732no—
223C24—22no—
224C24—22no—
225C24—42no—
226C24—42no—
227C24—731no—
228C24C22⋊D7314yescomplemented
229C24C22×D7311yescharacteristic
230C24C22×D7312yes—
231C24C22×D7311yescharacteristic
232C2×D4—22no—
233C2×D4—24no—
234C2×D4—22no—
235C2×D4—22no—
236C2×D4—42no—
237C2×D4—42no—
238C2×D4—2922no—
239C2×D4—2922no—
240C2×D4—2922no—
241C2×D4—1462no—
242C2×D4—1462no—
243C2×D4—1464no—
244C2×D4—1462no—
245C2×D4—1462no—
246C22×C4—2922no—
247C22×C4—2922no—
248C22×C4—734no—
249C22×C4—731no—
250C22×C4—731no—
251C22×C4—732no—
252C22×C4—731no—
253C22×C4—732no—
254C22×C4—1462no—
255C22×C4C22×D7311yescharacteristic
256C22×C4C22×D7312yes—
257C4⋊C4—2922no—
258C4⋊C4—1462no—
259C22⋊C4—24no—
260C22⋊C4—22no—
261C22⋊C4—2922no—
262C22⋊C4—2922no—
263C22⋊C4—1462no—
264C22⋊C4—1464no—
265C22⋊C4—1464no—
266C22⋊C4—1462no—
267C22⋊C4—1464no—
268C22⋊C4—1462no—
269C23—22no—
270C23—22no—
271C23—28no—
272C23—22no—
273C23—22no—
274C23—22no—
275C23—22no—
276C23—22no—
277C23—22no—
278C23—21no—
279C23—21no—
280C23—21no—
281C23—22no—
282C23—22no—
283C23—48no—
284C23—44no—
285C23—44no—
286C23—48no—
287C23—1461no—
288C23—1461no—
289C23—1461no—
290C23—1462no—
291C23—1462no—
292C23D4×D7312yes—
293C23C2×C22⋊D7314yes—
294C23C2×C22⋊D7312yes—
295C23#1192402712yes—
296C23C23×D7311yescharacteristic
297D4—48no—
298D4—44no—
299D4—2928no—
300D4—2924no—
301D4—2924no—
302C2×C4—21no—
303C2×C4—22no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—22no—
307C2×C4—22no—
308C2×C4—2924no—
309C2×C4—2924no—
310C2×C4—732no—
311C2×C4—732no—
312C2×C4—1462no—
313C2×C4—1461no—
314C2×C4—1462no—
315C2×C4—1461no—
316C2×C4—1461no—
317C2×C4—1462no—
318C2×C4—1462no—
319C2×C4—1462no—
320C2×C4—1462no—
321C2×C4—1462no—
322C2×C4—1462no—
323C22—24no—
324C22—22no—
325C22—22no—
326C22—24no—
327C22—24no—
328C22—24no—
329C22—24no—
330C22—416no—
331C22—42no—
332C22—42no—
333C22—42no—
334C22—42no—
335C22—42no—
336C22—41no—
337C22—41no—
338C22—42no—
339C22—41no—
340C22—2922no—
341C22—2921no—
342C22—2921no—
343C22—2922no—
344C22—2921no—
345C22#1455656314yes—
346C22C2×D4×D7311yescharacteristic
347C22C22×C22⋊D7311yescharacteristic
348C22C2×#1192402711yescharacteristic
349C22#1455663612yes—
350C22#1455663612yes—
351C4—42no—
352C4—41no—
353C4—2922no—
354C4—2922no—
355C4—2921no—
356C4—1462no—
357C4—1462no—
358C2—24no—
359C2—44no—
360C2—81no—
361C2—5841no—
362C2C2×#1455656311yescharacteristic
363C2#1798914612yes—
364C2#1798914711yescharacteristic
365C2#1798915211yescharacteristic
366C2#1798917212yes—
subgroupG/H
center, FrattiniC23C23×D73
commutator, socleC23×C73C24
FittingC2×#120×C73C2
hypercenterC2×#120D73
hypocenterC73#2617
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#120

Complements

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

NH ≅ G/Ntype
1. C2×#14556563C2semidirect
2. C2×#14556560C2semidirect
3. C22×C22⋊Dic73C2semidirect
4. #17988515C2semidirect
5. #17988502C2semidirect
6. #17988501C2semidirect
7. #17988495C2semidirect
8. #17988467C2semidirect
9. #17988464C2semidirect
10. #17988445C2semidirect
11. C2×#120×C73C2semidirect
14. C22×C22⋊D73C22semidirect
19. C2×#11924007C22semidirect
20. C2×#11924007C22semidirect
26. C2×C22⋊Dic73C22semidirect
27. C2×C22⋊Dic73C22semidirect
28. C2×#11924005C22semidirect
29. C2×#11924005C22semidirect
30. #14556498C22semidirect
31. #14556498C22semidirect
32. #14556498C22semidirect
51. C22×Dic73D4semidirect
71. C24×C73D4semidirect
184. C73#2617semidirect
187. C2×#120D73semidirect
228. C24C22⋊D73semidirect