Finite Group Databasebeta

← #22615732

Subgroups — #22615732

Order 9344.

464 conjugacy classes of subgroups; only the 133 normal subgroups are shown.

normal subgroups only

This group has 464 conjugacy classes of subgroups — too many to draw. Only its 133 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×#14556569C211yesmaximal, characteristic
2C2×#14556549C211yesmaximal, characteristic
3#17988525C211yesmaximal, characteristic, complemented
4#17988519C211yesmaximal, characteristic, complemented
5#17988517C211yesmaximal, characteristic, complemented
6#17988516C212yesmaximal
7#17988514C212yesmaximal
8#17988491C212yesmaximal, complemented
9#17988484C212yesmaximal, complemented
10#17988446C211yesmaximal, characteristic
11#400×C73C211yesmaximal, characteristic, complemented
12C22×Dic146C2211yescharacteristic
13C22×C4×D73C2211yescharacteristic
14#14556569—24no—
15#14556549—24no—
16C2×C4⋊Dic73C2211yescharacteristic
17C2×C4⋊Dic73C2212yes—
18C2×#11924007C2212yes—
19C2×#11924007C2211yescharacteristic
20C2×#11924007C2211yescharacteristic
21C2×#11924007C2212yes—
22C2×#11924005C2211yescharacteristic
23C2×#11924005C2212yes—
24C2×#11924005C2211yescharacteristic
25C2×#11924005C2212yes—
26C2×C4×Dic73C2212yes—
27#14556498C2211yescharacteristic
28#14556498C2212yes—
29#14556498C2212yes—
30#14556498C2212yes—
31#14556498C2212yes—
32C2×C4⋊C4×C73C2211yescharacteristic
33C2×C4⋊C4×C73C2211yescharacteristic
34#95×C73C2211yescharacteristic
35#95×C73C2212yes—
36#95×C73C2212yes—
37C23×D73C2311yescharacteristic
38C2×Dic146—22no—
39C2×Dic146—24no—
40C22×Dic73C2312yes—
41C22×Dic73C2311yescharacteristic
42C22×Dic73C2312yes—
43C22×Dic73C2311yescharacteristic
44C22×Dic73C2311yescharacteristic
45C2×C4×D73—24no—
46C2×C4×D73D414yes—
47C4⋊Dic73—22no—
48C4⋊Dic73—42no—
49#11924007—22no—
50#11924007—22no—
51#11924007—24no—
52#11924007—42no—
53#11924005—24no—
54#11924005—22no—
55#11924005—22no—
56#11924005—42no—
57C4×Dic73—24no—
58C22×C4×C73C2312yes—
59C22×C4×C73C2312yes—
60C22×C4×C73C2311yescharacteristic
61C22×C4×C73C2311yescharacteristic
62C22×C4×C73C2311yescharacteristic
63C4⋊C4×C73—22no—
64C4⋊C4×C73—22no—
65C23×C73C2411yescharacteristic
66C22×D73—22no—
67C22×D73—21no—
68C22×D73—22no—
69C22×D73—21no—
70C22×D73C2×D412yes—
71C2×C4×C73—22no—
72C2×C4×C73—22no—
73C2×C4×C73—21no—
74C2×C4×C73—21no—
75C2×C4×C73—22no—
76C2×C4×C73—22no—
77C2×C4×C73—22no—
78C2×C4×C73—21no—
79C2×C4×C73—24no—
80C2×C4×C73—22no—
81C2×C4×C73—21no—
82C2×C4×C73C2×D412yes—
83C2×Dic73—22no—
84C2×Dic73—21no—
85C2×Dic73—21no—
86C2×Dic73—22no—
87C2×Dic73—22no—
88C2×Dic73—22no—
89C2×Dic73—22no—
90C2×Dic73—21no—
91C2×Dic73—22no—
92C2×Dic73—22no—
93C2×Dic73—21no—
94C2×Dic73C2×D412yes—
95C2×Dic73#4114yes—
96C4×D73—28no—
97Dic146—44no—
98C22×C73C22×D411yescharacteristic
99C22×C73C2×#4112yes—
100C22×C73H(5,𝔽2)11yescharacteristic
101C22×C732−1+412yes—
102C22×C732−1+411yescharacteristic
103C2×D73—22no—
104C2×D73—22no—
105C2×D73—22no—
106C2×D73—42no—
107C2×D73—42no—
108C4×C73—22no—
109C4×C73—41no—
110C4×C73—42no—
111C4×C73—41no—
112C4×C73—42no—
113Dic73—41no—
114Dic73—42no—
115Dic73—41no—
116Dic73—42no—
117Dic73#12314yes—
118C2×C73C2×#12311yescharacteristic
119C2×C73#53611yescharacteristic
120C2×C73#53711yescharacteristic
121C2×C73#54112yes—
122C2×C73#54212yes—
123D73—42no—
124#2625—731nomaximal, Sylow
125C73#262511yescharacteristic, complemented, Sylow
126C2×C22⋊Q8—731no—
127C2×#117—731no—
128#400—731no—
129#400D7311yescharacteristic, complemented
130#398—731no—
131#398—731no—
132#397—732no—
133#397—732no—
134#391—732no—
135#389—732no—
136#380—731no—
137C22×Q8—731no—
138C23×C4—731no—
139C22⋊Q8—1464no—
140#117—1464no—
141C2×C4⋊C4—732no—
142C2×C4⋊C4—732no—
143C2×C4⋊C4—731no—
144C2×C4⋊C4—731no—
145C2×C4⋊C4—731no—
146C2×C4⋊C4—732no—
147C2×C4⋊C4C2×D7311yescharacteristic
148C2×C4⋊C4C2×D7311yescharacteristic
149C2×C22⋊C4—732no—
150C2×C22⋊C4—732no—
151C2×C22⋊C4—731no—
152C2×C22⋊C4—731no—
153C2×C42—732no—
154#95—731no—
155#95—732no—
156#95—732no—
157#95—732no—
158#95—732no—
159#95C2×D7311yescharacteristic
160#95C2×D7312yes—
161#95C2×D7312yes—
162C24—731no—
163C2×Q8—1462no—
164C2×Q8—1464no—
165C22×C4—731no—
166C22×C4—734no—
167C22×C4—732no—
168C22×C4—731no—
169C22×C4—732no—
170C22×C4—731no—
171C22×C4—1464no—
172C22×C4C22×D7312yes—
173C22×C4C22×D7311yescharacteristic
174C22×C4C22×D7312yes—
175C22×C4C22×D7311yescharacteristic
176C22×C4C22×D7311yescharacteristic
177C4⋊C4—22no—
178C4⋊C4—22no—
179C4⋊C4—2922no—
180C4⋊C4—2922no—
181C4⋊C4—1464no—
182C4⋊C4—1462no—
183C4⋊C4—1462no—
184C4⋊C4—1462no—
185C22⋊C4—2922no—
186C22⋊C4—1464no—
187C22⋊C4—1462no—
188C22⋊C4—1462no—
189C42—1464no—
190C23—732no—
191C23—1462no—
192C23—1461no—
193C23—1462no—
194C23—1461no—
195C23C23×D7311yescharacteristic
196Q8—2924no—
197C2×C4—22no—
198C2×C4—22no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—21no—
205C2×C4—24no—
206C2×C4—22no—
207C2×C4—21no—
208C2×C4—732no—
209C2×C4—734no—
210C2×C4—1468no—
211C2×C4—1462no—
212C2×C4—1461no—
213C2×C4—1461no—
214C2×C4—1462no—
215C2×C4—1462no—
216C2×C4—1462no—
217C2×C4—1462no—
218C2×C4—1461no—
219C2×C4—1462no—
220C2×C4—1462no—
221C2×C4—1461no—
222C2×C4D4×D7312yes—
223C22—2922no—
224C22—2922no—
225C22—1462no—
226C22—1462no—
227C22—1462no—
228C22C2×D4×D7311yescharacteristic
229C22#41×D7312yes—
230C22#1455663711yescharacteristic
231C22#1455663812yes—
232C22#1455663911yescharacteristic
233C4—22no—
234C4—42no—
235C4—41no—
236C4—42no—
237C4—41no—
238C4—2922no—
239C4—2921no—
240C4—2922no—
241C4—2921no—
242C4—734no—
243C2—2922no—
244C2#123×D7311yescharacteristic
245C2#1798915711yescharacteristic
246C2#1798916711yescharacteristic
247C2#1798919612yes—
248C2#1798920312yes—
subgroupG/H
center, FrattiniC23C23×D73
commutator, socleC23×C73C24
Fitting#400×C73C2
hypercenter#400D73
hypocenterC73#2625
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ #400

Complements

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

NH ≅ G/Ntype
3. #17988525C2semidirect
4. #17988519C2semidirect
5. #17988517C2semidirect
8. #17988491C2semidirect
9. #17988484C2semidirect
11. #400×C73C2semidirect
125. C73#2625semidirect
129. #400D73semidirect