Finite Group Databasebeta

← #22615677

Subgroups — #22615677

Order 9344.

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

normal subgroups only

This group has 632 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×#14556574C211yesmaximal, characteristic, complemented
2C2×#14556563C211yesmaximal, characteristic
3C22×C4⋊Dic73C211yesmaximal, characteristic, complemented
4#17988520C212yesmaximal, complemented
5#17988510C212yesmaximal, complemented
6#17988509C212yesmaximal, complemented
7#17988503C212yesmaximal, complemented
8#17988467C211yesmaximal, characteristic, complemented
9#17988464C211yesmaximal, characteristic, complemented
10#17988445C211yesmaximal, characteristic, complemented
11C2×#123×C73C211yesmaximal, characteristic, complemented
12C22×C22⋊D73C2211yescharacteristic
13C22×C22⋊D73C2211yescharacteristic
14C23×Dic73C2211yescharacteristic
15C23×Dic73C2211yescharacteristic
16#14556574—24no—
17#14556563—24no—
18C2×C4⋊Dic73—24no—
19C2×C4⋊Dic73C2212yes—
20C2×C4⋊Dic73C2212yes—
21C2×#11924007C2212yes—
22C2×#11924007C2211yescharacteristic
23C2×#11924007C2212yes—
24C2×C22⋊Dic73C2211yescharacteristic
25C2×C22⋊Dic73C2211yescharacteristic
26C2×C22⋊Dic73C2212yes—
27C2×#11924005C2212yes—
28C2×#11924005C2211yescharacteristic
29C2×#11924005C2212yes—
30#14556498C2212yes—
31#14556498C2212yes—
32#14556498C2212yes—
33C22×D4×C73C2211yescharacteristic
34C23×C4×C73C2211yescharacteristic
35#123×C73—24no—
36C2×C4⋊C4×C73C2212yes—
37C2×C22⋊C4×C73C2211yescharacteristic
38C2×C22⋊C4×C73C2212yes—
39C23×D73C2311yescharacteristic
40C2×C22⋊D73—22no—
41C2×C22⋊D73—22no—
42C2×C22⋊D73—22no—
43C2×C22⋊D73—22no—
44C2×C22⋊D73—42no—
45C22×Dic73—22no—
46C22×Dic73—42no—
47C22×Dic73—42no—
48C22×Dic73D414yes—
49C22×Dic73C2312yes—
50C22×Dic73C2312yes—
51C22×Dic73C2311yescharacteristic
52C22×Dic73C2311yescharacteristic
53C22×Dic73C2311yescharacteristic
54C4⋊Dic73—24no—
55C4⋊Dic73—24no—
56#11924007—24no—
57#11924007—22no—
58#11924007—42no—
59C22⋊Dic73—22no—
60C22⋊Dic73—22no—
61C22⋊Dic73—42no—
62#11924005—24no—
63#11924005—22no—
64#11924005—42no—
65C24×C73C2311yescharacteristic
66C24×C73C2311yescharacteristic
67C2×D4×C73—22no—
68C2×D4×C73—22no—
69C2×D4×C73—22no—
70C22×C4×C73—22no—
71C22×C4×C73—22no—
72C22×C4×C73D414yescomplemented
73C22×C4×C73C2312yes—
74C22×C4×C73C2311yescharacteristic
75C22×C4×C73C2312yes—
76C4⋊C4×C73—24no—
77C22⋊C4×C73—22no—
78C22⋊C4×C73—24no—
79C23×C73—22no—
80C23×C73—22no—
81C23×C73—21no—
82C23×C73—21no—
83C23×C73—21no—
84C23×C73—22no—
85C23×C73—22no—
86C23×C73C2×D412yes—
87C23×C73C2×D412yes—
88C23×C73#4112yes—
89C23×C73C2411yescharacteristic
90C22×D73—21no—
91C22×D73—21no—
92C22×D73—21no—
93C22×D73—22no—
94C22×D73—22no—
95D4×C73—44no—
96C2×C4×C73—28no—
97C2×C4×C73—22no—
98C2×C4×C73—22no—
99C2×C4×C73—22no—
100C2×C4×C73—21no—
101C2×C4×C73—21no—
102C2×C4×C73—21no—
103C2×C4×C73—22no—
104C2×C4×C73—22no—
105C2×C4×C73C2×D414yes—
106C22⋊D73—44no—
107C22⋊D73—44no—
108C2×Dic73—22no—
109C2×Dic73—22no—
110C2×Dic73—22no—
111C2×Dic73—21no—
112C2×Dic73—21no—
113C2×Dic73—22no—
114C2×Dic73—21no—
115C2×Dic73—22no—
116C2×Dic73—22no—
117C2×Dic73—22no—
118C2×Dic73—22no—
119C2×Dic73—44no—
120C2×Dic73—44no—
121C2×Dic73C2×D412yes—
122C2×Dic73C2×D412yes—
123C22×C73—22no—
124C22×C73—22no—
125C22×C73—24no—
126C22×C73—24no—
127C22×C73—41no—
128C22×C73—41no—
129C22×C73—41no—
130C22×C73—42no—
131C22×C73—42no—
132C22×C73#12114yes—
133C22×C73C22×D411yescharacteristic
134C22×C73C22×D411yescharacteristic
135C22×C73C2×#4111yescharacteristic
136C22×C73H(5,𝔽2)12yes—
137C22×C732−1+412yes—
138C2×D73—41no—
139C2×D73—41no—
140C2×D73—41no—
141C2×D73—42no—
142C2×D73—42no—
143C4×C73—24no—
144C4×C73—41no—
145C4×C73—42no—
146Dic73—22no—
147Dic73—22no—
148Dic73—41no—
149Dic73—42no—
150Dic73—42no—
151C2×C73—24no—
152C2×C73—81no—
153C2×C73C2×#12111yescharacteristic
154C2×C73#53411yescharacteristic
155C2×C73#53611yescharacteristic
156C2×C73#53712yes—
157C2×C73#53912yes—
158D73—81no—
159#2618—731nomaximal, Sylow
160C73#261811yescharacteristic, complemented, Sylow
161C2×#123—731no—
162C2×#123D7311yescharacteristic, complemented
163C2×#121—731no—
164C22×C4⋊C4—731no—
165#398—732no—
166#396—732no—
167#396—732no—
168#394—732no—
169#386—731no—
170#386—731no—
171#380—731no—
172C22×D4—731no—
173C22×D4—731no—
174C22×D4C2×D7311yescharacteristic
175C23×C4—731no—
176C23×C4—731no—
177C23×C4C2×D7311yescharacteristic
178#123—24no—
179#123—1464no—
180#121—1464no—
181C2×C4⋊C4—732no—
182C2×C4⋊C4—732no—
183C2×C4⋊C4—732no—
184C2×C4⋊C4—731no—
185C2×C4⋊C4—732no—
186C2×C4⋊C4—1464no—
187C2×C4⋊C4C2×D7312yes—
188C2×C22⋊C4—732no—
189C2×C22⋊C4—732no—
190C2×C22⋊C4—731no—
191C2×C22⋊C4—731no—
192C2×C22⋊C4—731no—
193C2×C22⋊C4—732no—
194C2×C22⋊C4C2×D7311yescharacteristic
195C2×C22⋊C4C2×D7312yes—
196#95—732no—
197#95—732no—
198#95—732no—
199C24—731no—
200C24C22×D7311yescharacteristic
201C24C22×D7311yescharacteristic
202C2×D4—22no—
203C2×D4—22no—
204C2×D4—22no—
205C2×D4—2922no—
206C2×D4—1462no—
207C2×D4—1462no—
208C2×D4—1462no—
209C2×D4—1462no—
210C22×C4—22no—
211C22×C4—22no—
212C22×C4—2922no—
213C22×C4—2922no—
214C22×C4—734no—
215C22×C4—731no—
216C22×C4—731no—
217C22×C4—732no—
218C22×C4—732no—
219C22×C4—731no—
220C22×C4—1462no—
221C22×C4C22⋊D7314yescomplemented
222C22×C4C22×D7312yes—
223C22×C4C22×D7311yescharacteristic
224C22×C4C22×D7312yes—
225C4⋊C4—24no—
226C4⋊C4—2922no—
227C4⋊C4—1462no—
228C4⋊C4—1464no—
229C4⋊C4—1464no—
230C4⋊C4—1464no—
231C22⋊C4—22no—
232C22⋊C4—24no—
233C22⋊C4—2922no—
234C22⋊C4—2922no—
235C22⋊C4—1462no—
236C22⋊C4—1462no—
237C22⋊C4—1462no—
238C22⋊C4—1464no—
239C23—22no—
240C23—22no—
241C23—21no—
242C23—21no—
243C23—21no—
244C23—22no—
245C23—22no—
246C23—1461no—
247C23—1461no—
248C23—1461no—
249C23—1462no—
250C23—1462no—
251C23D4×D7312yes—
252C23C2×C22⋊D7312yes—
253C23#1192402712yes—
254C23C23×D7311yescharacteristic
255D4—44no—
256D4—2924no—
257D4—2924no—
258C2×C4—28no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—21no—
262C2×C4—22no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—22no—
266C2×C4—22no—
267C2×C4—2924no—
268C2×C4—2924no—
269C2×C4—732no—
270C2×C4—732no—
271C2×C4—1461no—
272C2×C4—1462no—
273C2×C4—1462no—
274C2×C4—1461no—
275C2×C4—1461no—
276C2×C4—1462no—
277C2×C4—1462no—
278C2×C4—1462no—
279C2×C4—1462no—
280C2×C4—1462no—
281C2×C4—1462no—
282C2×C4C2×C22⋊D7314yes—
283C22—22no—
284C22—22no—
285C22—24no—
286C22—24no—
287C22—41no—
288C22—41no—
289C22—42no—
290C22—42no—
291C22—41no—
292C22—2921no—
293C22—2921no—
294C22—2922no—
295C22—2922no—
296C22—2921no—
297C22#1455656314yes—
298C22C2×D4×D7311yescharacteristic
299C22C22×C22⋊D7311yescharacteristic
300C22C2×#1192402711yescharacteristic
301C22#1455663712yes—
302C22#1455663812yes—
303C4—24no—
304C4—42no—
305C4—41no—
306C4—2922no—
307C4—2922no—
308C4—2921no—
309C4—1462no—
310C4—1462no—
311C2—24no—
312C2—81no—
313C2—5841no—
314C2C2×#1455656311yescharacteristic
315C2#1798914811yescharacteristic
316C2#1798915911yescharacteristic
317C2#1798916512yes—
318C2#1798918112yes—
subgroupG/H
center, FrattiniC23C23×D73
commutator, socleC23×C73C24
FittingC2×#123×C73C2
hypercenterC2×#123D73
hypocenterC73#2618
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×#123

Complements

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

NH ≅ G/Ntype
1. C2×#14556574C2semidirect
3. C22×C4⋊Dic73C2semidirect
4. #17988520C2semidirect
5. #17988510C2semidirect
6. #17988509C2semidirect
7. #17988503C2semidirect
8. #17988467C2semidirect
9. #17988464C2semidirect
10. #17988445C2semidirect
11. C2×#123×C73C2semidirect
72. C22×C4×C73D4semidirect
160. C73#2618semidirect
162. C2×#123D73semidirect
221. C22×C4C22⋊D73semidirect