Finite Group Databasebeta

← #22613570

Subgroups — #22613570

Order 9344.

800 conjugacy classes of subgroups; only the 181 normal subgroups are shown.

normal subgroups only

This group has 800 conjugacy classes of subgroups — too many to draw. Only its 181 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
1C22×Q8×D73C211yesmaximal, characteristic
2C2×#14556590C212yesmaximal, complemented
3C2×#14556569C211yesmaximal, characteristic
4#17988508C211yesmaximal, characteristic, complemented
5#17988507C211yesmaximal, characteristic, complemented
6#17988506C211yesmaximal, characteristic, complemented
7#17988491C212yesmaximal, complemented
8#17988461C212yesmaximal
9#17988458C212yesmaximal
10#17988434C211yesmaximal, characteristic
11#395×C73C211yesmaximal, characteristic, complemented
12C22×Dic146C2211yescharacteristic
13C22×Dic146C2211yescharacteristic
14C22×Dic146C2212yes—
15C2×Q8×D73—28no—
16C2×Q8×D73—24no—
17C22×C4×D73C2212yes—
18C22×C4×D73C2211yescharacteristic
19#14556590—28no—
20#14556569—24no—
21C2×C4⋊Dic73C2212yes—
22C2×#11924007C2212yes—
23C2×#11924007C2212yes—
24C2×#11924007C2211yescharacteristic
25C2×#11924007C2211yescharacteristic
26C2×#11924005C2211yescharacteristic
27C2×#11924005C2211yescharacteristic
28C2×#11924005C2212yes—
29C2×C4×Dic73C2212yes—
30#14556498C2211yescharacteristic
31#14556498C2212yes—
32#14556498C2212yes—
33#14556498C2212yes—
34C22×Q8×C73C2211yescharacteristic
35C2×C4⋊C4×C73C2212yes—
36C2×C4⋊C4×C73C2211yescharacteristic
37#95×C73C2211yescharacteristic
38#95×C73C2212yes—
39C23×D73C2311yescharacteristic
40C2×Dic146—22no—
41C2×Dic146—24no—
42C2×Dic146—24no—
43C2×Dic146—24no—
44C2×Dic146—22no—
45C2×Dic146—24no—
46C2×Dic146D418yes—
47Q8×D73—232no—
48C22×Dic73C2311yescharacteristic
49C22×Dic73C2311yescharacteristic
50C22×Dic73C2312yes—
51C22×Dic73C2312yes—
52C22×Dic73C2311yescharacteristic
53C2×C4×D73—24no—
54C2×C4×D73—24no—
55C2×C4×D73—24no—
56C2×C4×D73—24no—
57C2×C4×D73D414yes—
58C4⋊Dic73—24no—
59#11924007—22no—
60#11924007—24no—
61#11924007—22no—
62#11924007—24no—
63#11924005—22no—
64#11924005—22no—
65#11924005Q818yescomplemented
66C4×Dic73—24no—
67C2×Q8×C73—24no—
68C2×Q8×C73—22no—
69C22×C4×C73C2311yescharacteristic
70C22×C4×C73C2311yescharacteristic
71C22×C4×C73C2312yes—
72C22×C4×C73C2311yescharacteristic
73C22×C4×C73C2312yes—
74C4⋊C4×C73—22no—
75C4⋊C4×C73—24no—
76C23×C73C2411yescharacteristic
77C22×D73—22no—
78C22×D73—21no—
79C22×D73—21no—
80C22×D73C2×D412yes—
81C22×D73C2×Q814yes—
82Q8×C73—28no—
83C2×C4×C73—22no—
84C2×C4×C73—22no—
85C2×C4×C73—22no—
86C2×C4×C73—22no—
87C2×C4×C73—22no—
88C2×C4×C73—21no—
89C2×C4×C73—21no—
90C2×C4×C73—22no—
91C2×C4×C73—21no—
92C2×C4×C73—21no—
93C2×C4×C73C2×D414yes—
94C2×C4×C73C2×D412yes—
95C2×C4×C73C2×Q814yescomplemented
96C2×Dic73—22no—
97C2×Dic73—22no—
98C2×Dic73—22no—
99C2×Dic73—22no—
100C2×Dic73—21no—
101C2×Dic73—21no—
102C2×Dic73—22no—
103C2×Dic73—21no—
104C2×Dic73—21no—
105C2×Dic73C2×D414yes—
106C2×Dic73C2×D414yes—
107C2×Dic73C2×D412yes—
108C2×Dic73C2×Q814yescomplemented
109C4×D73—216no—
110C4×D73—28no—
111Dic146—28no—
112Dic146—216no—
113Dic146—44no—
114C22×C73C22×D411yescharacteristic
115C22×C73C22×D412yes—
116C22×C73C22×Q812yes—
117C22×C732−1+411yescharacteristic
118C22×C732−1+411yescharacteristic
119C2×D73—22no—
120C2×D73—22no—
121C2×D73—24no—
122C2×D73—24no—
123C2×D73C4⋊Q814yescomplemented
124C4×C73—22no—
125C4×C73—24no—
126C4×C73—24no—
127C4×C73—41no—
128C4×C73—41no—
129Dic73—24no—
130Dic73—24no—
131Dic73—41no—
132Dic73—41no—
133Dic73#12014yes—
134C2×C73C2×#12011yescharacteristic
135C2×C73C2×C4⋊Q811yescharacteristic
136C2×C73#53611yescharacteristic
137C2×C73D4×Q812yes—
138C2×C73D4×Q812yes—
139D73—24no—
140#2434—731nomaximal, Sylow
141C73#243411yescharacteristic, complemented, Sylow
142C23×Q8—731no—
143C2×C4⋊Q8—732no—
144C2×C22⋊Q8—731no—
145#395—731no—
146#395—731no—
147#395—731no—
148#395D7311yescharacteristic, complemented
149#391—732no—
150#385—732no—
151#385—732no—
152C4×C22⋊C4—731no—
153C22×Q8—732no—
154C22×Q8—731no—
155C22×Q8—731no—
156C22×Q8—1464no—
157C22×Q8—1468no—
158C22×Q8C2×D7311yescharacteristic
159C23×C4—731no—
160C23×C4—732no—
161C4⋊Q8—1468no—
162C22⋊Q8—1464no—
163C2×C4⋊C4—731no—
164C2×C4⋊C4—731no—
165C2×C4⋊C4—732no—
166C2×C4⋊C4—732no—
167C2×C4⋊C4—732no—
168C2×C4⋊C4C2×D7311yescharacteristic
169C2×C4⋊C4C2×D7312yes—
170C2×C22⋊C4—732no—
171C2×C22⋊C4—731no—
172C2×C22⋊C4—731no—
173C2×C42—732no—
174#95—731no—
175#95—732no—
176#95—732no—
177#95—732no—
178#95C2×D7311yescharacteristic
179#95C2×D7312yes—
180C24—731no—
181C2×Q8—22no—
182C2×Q8—24no—
183C2×Q8—738no—
184C2×Q8—1462no—
185C2×Q8—1464no—
186C2×Q8—1464no—
187C2×Q8—1464no—
188C2×Q8—1464no—
189C2×Q8—14632no—
190C2×Q8—1462no—
191C22×C4—731no—
192C22×C4—732no—
193C22×C4—732no—
194C22×C4—734no—
195C22×C4—731no—
196C22×C4—731no—
197C22×C4—1464no—
198C22×C4—1464no—
199C22×C4—1464no—
200C22×C4—1464no—
201C22×C4C22×D7311yescharacteristic
202C22×C4C22×D7311yescharacteristic
203C22×C4C22×D7311yescharacteristic
204C22×C4C22×D7312yes—
205C22×C4C22×D7312yes—
206C4⋊C4—22no—
207C4⋊C4—24no—
208C4⋊C4—1462no—
209C4⋊C4—1464no—
210C4⋊C4—1464no—
211C4⋊C4—1462no—
212C4⋊C4—1464no—
213C22⋊C4—738no—
214C22⋊C4—1462no—
215C22⋊C4—1462no—
216C42—1464no—
217C23—732no—
218C23—734no—
219C23—1462no—
220C23—1461no—
221C23—1461no—
222C23C23×D7311yescharacteristic
223Q8—28no—
224Q8—2924no—
225Q8—14616no—
226Q8—1468no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—22no—
230C2×C4—22no—
231C2×C4—22no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—22no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—734no—
238C2×C4—732no—
239C2×C4—734no—
240C2×C4—734no—
241C2×C4—1468no—
242C2×C4—14616no—
243C2×C4—1461no—
244C2×C4—1461no—
245C2×C4—1462no—
246C2×C4—1461no—
247C2×C4—1461no—
248C2×C4—1462no—
249C2×C4—1462no—
250C2×C4—1462no—
251C2×C4—1462no—
252C2×C4D4×D7312yes—
253C2×C4D4×D7314yes—
254C2×C4Q8×D7314yescomplemented
255C22—734no—
256C22—1462no—
257C22—1464no—
258C22—1464no—
259C22—1462no—
260C22C2×D4×D7311yescharacteristic
261C22C2×D4×D7312yes—
262C22C2×Q8×D7312yes—
263C22#1455663811yescharacteristic
264C22#1455663911yescharacteristic
265C4—24no—
266C4—22no—
267C4—24no—
268C4—41no—
269C4—41no—
270C4—2921no—
271C4—2921no—
272C4—734no—
273C4—1464no—
274C4—1464no—
275C2—1464no—
276C2#120×D7311yescharacteristic
277C2C4⋊Q8×D7311yescharacteristic
278C2#1798916011yescharacteristic
279C2#1798925012yes—
280C2#1798925212yes—
subgroupG/H
center, FrattiniC23C23×D73
commutator, socleC23×C73C24
Fitting#395×C73C2
hypercenter#395D73
hypocenterC73#2434
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ #395

Complements

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

NH ≅ G/Ntype
2. C2×#14556590C2semidirect
4. #17988508C2semidirect
5. #17988507C2semidirect
6. #17988506C2semidirect
7. #17988491C2semidirect
11. #395×C73C2semidirect
65. #11924005Q8semidirect
95. C2×C4×C73C2×Q8semidirect
108. C2×Dic73C2×Q8semidirect
123. C2×D73C4⋊Q8semidirect
141. C73#2434semidirect
148. #395D73semidirect
254. C2×C4Q8×D73semidirect