Finite Group Databasebeta

← #22616900

Subgroups — #22616900

Order 9344.

512 conjugacy classes of subgroups; only the 137 normal subgroups are shown.

normal subgroups only

This group has 512 conjugacy classes of subgroups — too many to draw. Only its 137 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×#14556567C212yesmaximal, complemented
2#17988527C212yesmaximal, complemented
3#17988517C211yesmaximal, characteristic, complemented
4#17988514C212yesmaximal, complemented
5#17988504C211yesmaximal, characteristic
6#17988478C212yesmaximal, complemented
7#17988465C212yesmaximal
8#17988461C212yesmaximal, complemented
9#391×C73C211yesmaximal, characteristic, complemented
10C22×D292C2211yescharacteristic
11C22×C4×D73C2212yes—
12#14556567—28no—
13C2×C4⋊Dic73C2212yes—
14C2×C4⋊Dic73C2211yescharacteristic
15C2×#11924007C2212yes—
16C2×#11924007C2212yes—
17C2×#11924005C2212yes—
18C2×#11924005C2212yes—
19C2×#11924005C2212yes—
20C2×#11924005C2212yes—
21C2×#11924005C2212yes—
22#14556498C2212yes—
23#14556498C2212yes—
24#14556498C2212yes—
25#14556498C2211yescharacteristic
26#14556498C2211yescharacteristic
27C22×Q8×C73C2211yescharacteristic
28C2×C4⋊C4×C73C2211yescharacteristic
29C2×C42×C73C2211yescharacteristic
30#95×C73C2212yes—
31#95×C73C2212yes—
32C23×D73C2312yes—
33C2×D292—24no—
34C2×D292—41no—
35C22×Dic73C2312yes—
36C22×Dic73C2312yes—
37C22×Dic73C2312yes—
38C2×C4×D73—24no—
39C2×C4×D73—42no—
40C2×C4×D73—42no—
41C4⋊Dic73—24no—
42C4⋊Dic73—41no—
43#11924007—24no—
44#11924007—24no—
45#11924005—24no—
46#11924005—24no—
47#11924005—24no—
48#11924005—24no—
49#11924005—42no—
50C2×Q8×C73—41no—
51C2×Q8×C73—41no—
52C2×Q8×C73D414yescomplemented
53C22×C4×C73C2312yes—
54C22×C4×C73C2311yescharacteristic
55C22×C4×C73C2311yescharacteristic
56C22×C4×C73C2312yes—
57C22×C4×C73C2311yescharacteristic
58C4⋊C4×C73—41no—
59C42×C73—41no—
60C23×C73C2411yescharacteristic
61C22×D73—22no—
62C22×D73—22no—
63C22×D73—22no—
64C22×D73—22no—
65C22×D73—22no—
66C22×D73—22no—
67C22×D73#4114yes—
68Q8×C73—44no—
69C2×C4×C73—21no—
70C2×C4×C73—21no—
71C2×C4×C73—22no—
72C2×C4×C73—22no—
73C2×C4×C73—22no—
74C2×C4×C73—22no—
75C2×C4×C73—22no—
76C2×C4×C73—22no—
77C2×C4×C73—21no—
78C2×C4×C73—21no—
79C2×C4×C73—22no—
80C2×C4×C73C2×D412yes—
81C2×C4×C73C2×D414yes—
82C2×Dic73—22no—
83C2×Dic73—22no—
84C2×Dic73—22no—
85C2×Dic73—22no—
86C2×Dic73—22no—
87C2×Dic73—22no—
88C2×Dic73—22no—
89C2×Dic73—22no—
90C2×Dic73—22no—
91D292—44no—
92C4×D73—48no—
93C22×C73C22×D411yescharacteristic
94C22×C73C2×#4112yes—
95C22×C73H(5,𝔽2)11yescharacteristic
96C22×C73H(5,𝔽2)11yescharacteristic
97C22×C732−1+412yes—
98C2×D73—24no—
99C2×D73—24no—
100C2×D73—24no—
101C2×D73—42no—
102C2×D73—42no—
103C2×D73—42no—
104C2×D73—42no—
105C4×C73—24no—
106C4×C73—22no—
107C4×C73—42no—
108C4×C73—41no—
109C4×C73—41no—
110Dic73—42no—
111Dic73—42no—
112Dic73—42no—
113C2×C73#53411yescharacteristic
114C2×C73#53912yes—
115C2×C73#54812yes—
116C2×C73#55111yescharacteristic
117C2×C73#56311yescharacteristic
118D73—44no—
119#2728—731nomaximal, Sylow
120C73#272811yescharacteristic, complemented, Sylow
121C2×C22⋊Q8—732no—
122#400—732no—
123#398—731no—
124#397—732no—
125#394—731no—
126#391D7311yescharacteristic, complemented
127#388—732no—
128#386—732no—
129#385—732no—
130C22×Q8C2×D7311yescharacteristic
131C22×D4—731no—
132C23×C4—732no—
133C22⋊Q8—1468no—
134C2×C4⋊C4—731no—
135C2×C4⋊C4—732no—
136C2×C4⋊C4—732no—
137C2×C4⋊C4—732no—
138C2×C4⋊C4C2×D7311yescharacteristic
139C2×C22⋊C4—732no—
140C2×C22⋊C4—732no—
141C2×C22⋊C4—732no—
142C2×C22⋊C4—732no—
143C2×C22⋊C4—732no—
144C2×C42C2×D7311yescharacteristic
145#95—732no—
146#95—732no—
147#95—732no—
148#95—731no—
149#95—731no—
150#95C2×D7312yes—
151#95C2×D7312yes—
152C24—732no—
153C2×Q8—41no—
154C2×Q8—41no—
155C2×Q8D29214yescomplemented
156C2×D4—2921no—
157C2×D4—1464no—
158C22×C4—2922no—
159C22×C4—2922no—
160C22×C4—732no—
161C22×C4—732no—
162C22×C4—732no—
163C22×C4—1464no—
164C22×C4C22×D7311yescharacteristic
165C22×C4C22×D7312yes—
166C22×C4C22×D7311yescharacteristic
167C22×C4C22×D7312yes—
168C22×C4C22×D7311yescharacteristic
169C4⋊C4—41no—
170C4⋊C4—2921no—
171C4⋊C4—1464no—
172C4⋊C4—1464no—
173C4⋊C4—1464no—
174C22⋊C4—2922no—
175C22⋊C4—1464no—
176C22⋊C4—1464no—
177C22⋊C4—1464no—
178C22⋊C4—1464no—
179C42—41no—
180C23—734no—
181C23—1462no—
182C23—1462no—
183C23—1462no—
184C23—1462no—
185C23—1462no—
186C23—1462no—
187C23C23×D7311yescharacteristic
188Q8—44no—
189D4—2924no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—22no—
193C2×C4—22no—
194C2×C4—22no—
195C2×C4—22no—
196C2×C4—22no—
197C2×C4—22no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—22no—
201C2×C4—2928no—
202C2×C4—1462no—
203C2×C4—1462no—
204C2×C4—1462no—
205C2×C4—1462no—
206C2×C4—1462no—
207C2×C4—1462no—
208C2×C4—1462no—
209C2×C4—1462no—
210C2×C4—1462no—
211C2×C4C2×D29212yes—
212C2×C4C2×D29214yes—
213C22—2922no—
214C22—2922no—
215C22—2922no—
216C22—2922no—
217C22—1464no—
218C22—1464no—
219C22—1464no—
220C22C22×D29211yescharacteristic
221C22#41×D7312yes—
222C22#1455663611yescharacteristic
223C22#1455663611yescharacteristic
224C22#1455663912yes—
225C4—24no—
226C4—22no—
227C4—42no—
228C4—41no—
229C4—41no—
230C4—2922no—
231C4—2922no—
232C4—2922no—
233C2—2924no—
234C2#1798914911yescharacteristic
235C2#1798918312yes—
236C2#1798924912yes—
237C2#1798926711yescharacteristic
238C2#1798933611yescharacteristic
subgroupG/H
center, FrattiniC23C23×D73
commutator, socleC23×C73C24
Fitting#391×C73C2
hypercenter#391D73
hypocenterC73#2728
LCSG ▷ C23×C73 ▷ C73
UCSC1 ◁ C23 ◁ #391

Complements

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

NH ≅ G/Ntype
1. C2×#14556567C2semidirect
2. #17988527C2semidirect
3. #17988517C2semidirect
4. #17988514C2semidirect
6. #17988478C2semidirect
8. #17988461C2semidirect
9. #391×C73C2semidirect
52. C2×Q8×C73D4semidirect
120. C73#2728semidirect
126. #391D73semidirect
155. C2×Q8D292semidirect