Finite Group Databasebeta

← #32995224

Subgroups — #32995224

Order 18688.

386 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 386 conjugacy classes of subgroups — too many to draw. Only its 87 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
1#22618624C211yesmaximal, characteristic
2#22611540C212yesmaximal, complemented
3#22607873C211yesmaximal, characteristic, complemented
4#22607856C212yesmaximal, complemented
5C73×#1323C211yesmaximal, characteristic, complemented
6C2×#14556599C411yescharacteristic
7C22×C8×D73C2211yescharacteristic
8#17989099C411yescharacteristic
9C2×#14556530—22no—
10#17988657—22no—
11#17988652—22no—
12#17988553C2211yescharacteristic
13#17988550C411yescharacteristic
14#17988543C411yescharacteristic
15C22⋊#11924003C2212yes—
16#17988366—22no—
17#424×C73C2211yescharacteristic
18C22⋊C16×C73C2212yes—
19C2×#11924028C2×C411yescharacteristic
20C22×C4×D73C2×C411yescharacteristic
21#14556599—42no—
22C2×#11924012C2×C411yescharacteristic
23C2×#11924011—21no—
24C22×#10596831C2×C411yescharacteristic
25C2×C8×D73—21no—
26C2×C8×D73—42no—
27C2×C8×D73D412yes—
28C2×C8×D73D412yes—
29#14556575—21no—
30#14556571C812yes—
31#14556566C812yes—
32#14556561—21no—
33C4×D292—22no—
34#14556530—44no—
35C2×#11924003—22no—
36#14556503—22no—
37#14556503—22no—
38#14556502—21no—
39C4×#10596831—21no—
40C2×M16×C73C2×C411yescharacteristic
41C22×C8×C73C2311yescharacteristic
42#117×C73C2×C411yescharacteristic
43C2×C16×C73—22no—
44C4⋊C8×C73—22no—
45C23×D73M1611yescharacteristic
46#11924028—41no—
47#11924028—41no—
48C2×Dic146C2×C811yescharacteristic
49C2×D292—22no—
50C2×D292C2×C811yescharacteristic
51C2×C22⋊D73—21no—
52C22×Dic73M1611yescharacteristic
53C2×C4×D73—21no—
54C2×C4×D73—41no—
55C2×C4×D73—41no—
56C2×C4×D73C22⋊C412yes—
57C2×C4×D73C2×C812yes—
58#11924012—41no—
59#11924011—42no—
60C2×#10596831—22no—
61C2×#10596831—21no—
62C2×#10596831C22⋊C411yescharacteristic
63C2×#10596831C22⋊C411yescharacteristic
64C8×D73—24no—
65C8×D73—42no—
66C8×D73—42no—
67C4⋊Dic73—22no—
68#11924005—22no—
69#11924005—22no—
70#11924003—42no—
71C22×C4×C73C22×C411yescharacteristic
72M16×C73—41no—
73C2×C8×C73—21no—
74C2×C8×C73—41no—
75C2×C8×C73C2×D412yes—
76C4⋊C4×C73C2×C812yes—
77C22⋊C4×C73—21no—
78C42×C73—21no—
79C16×C73—42no—
80C23×C73C2×M1611yescharacteristic
81C22×D73—21no—
82C22×D73—41no—
83C22×D73—42no—
84C22×D73C22⋊C812yes—
85C2×C4×C73—21no—
86C2×C4×C73—21no—
87C2×C4×C73—22no—
88C2×C4×C73C2×C22⋊C411yescharacteristic
89C2×C4×C73C22×C811yescharacteristic
90C22⋊D73—42no—
91C2×Dic73—21no—
92C2×Dic73—41no—
93C2×Dic73C22⋊C812yes—
94D292—41no—
95D292—44no—
96C4×D73—22no—
97C4×D73—22no—
98C4×D73—42no—
99C4×D73—41no—
100C4×D73—41no—
101Dic146—41no—
102C8×C73—22no—
103C8×C73—41no—
104#10596831—21no—
105#10596831—41no—
106#10596831#9912yes—
107C22×C73—22no—
108C22×C73—21no—
109C22×C73C2×C22⋊C811yescharacteristic
110C2×D73—22no—
111C2×D73—22no—
112C2×D73—22no—
113C2×D73—44no—
114C2×D73—42no—
115C2×D73—41no—
116C4×C73—21no—
117C4×C73—42no—
118C4×C73C2×#9911yescharacteristic
119C4×C73#41311yescharacteristic
120Dic73—22no—
121Dic73—41no—
122#11469—731nomaximal, Sylow
123C2×C73—41no—
124C2×C73#145911yescharacteristic
125C2×C73#211712yes—
126D73—42no—
127D73—81no—
128#2929—731no—
129#2112—732no—
130#1323—731no—
131#1323D7311yescharacteristic, complemented
132#1315—732no—
133C73#1146911yescharacteristic, complemented, Sylow
134C2×#131—731no—
135C23×C8—731no—
136#525—731no—
137C2×M32—1462no—
138#435—1462no—
139C8×D4—1462no—
140#424C2×D7311yescharacteristic
141#408—731no—
142#407—731no—
143#405—731no—
144C22⋊C16—732no—
145C22⋊C16—1462no—
146C22⋊C16C2×D7312yes—
147C2×#41—731no—
148C23×C4—731no—
149#131—2922no—
150C2×M16—731no—
151C2×M16—1461no—
152C2×M16C4×D7311yescharacteristic
153C22×C8—2922no—
154C22×C8—732no—
155C22×C8—731no—
156C22×C8—732no—
157C22×C8—1461no—
158C22×C8C22×D7311yescharacteristic
159#123—1461no—
160C22⋊Q8—732no—
161#121—732no—
162#120—1461no—
163C4×D4—1462no—
164#117C4×D7311yescharacteristic
165M32—2924no—
166C2×C16—22no—
167C2×C16—1462no—
168C4⋊C8—22no—
169C22⋊C8—1462no—
170C22⋊C8—1462no—
171#97—1461no—
172C4×C8—1461no—
173C24—731no—
174#41—2921no—
175#41—2921no—
176C2×Q8—731no—
177C2×D4—731no—
178C2×D4—1462no—
179C2×D4—1461no—
180C22×C4—2921no—
181C22×C4—2921no—
182C22×C4—731no—
183C22×C4—732no—
184C22×C4—732no—
185C22×C4—1461no—
186C22×C4C2×C4×D7311yescharacteristic
187M16—41no—
188M16—2922no—
189M16—2921no—
190C2×C8—21no—
191C2×C8—41no—
192C2×C8—2922no—
193C2×C8—2922no—
194C2×C8—731no—
195C2×C8—731no—
196C2×C8—1464no—
197C2×C8—1461no—
198C2×C8—1462no—
199C2×C8D4×D7312yes—
200C4⋊C4—1462no—
201C4⋊C4C8×D7312yes—
202C22⋊C4—21no—
203C22⋊C4—1462no—
204C22⋊C4—1462no—
205C42—21no—
206C16—42no—
207C16—2922no—
208C23—2921no—
209C23—2922no—
210C23—732no—
211C23—1461no—
212C23M16×D7311yescharacteristic
213Q8—2921no—
214D4—2921no—
215D4—2924no—
216D4—2922no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—22no—
220C2×C4—2922no—
221C2×C4—2921no—
222C2×C4—2921no—
223C2×C4—2921no—
224C2×C4—732no—
225C2×C4—1461no—
226C2×C4—1462no—
227C2×C4—1462no—
228C2×C4C22⋊C4×D7311yescharacteristic
229C2×C4C2×C8×D7311yescharacteristic
230C8—22no—
231C8—41no—
232C8—2921no—
233C8—732no—
234C8—1461no—
235C22—22no—
236C22—21no—
237C22—2924no—
238C22—2922no—
239C22—2921no—
240C22—1462no—
241C22—1462no—
242C22—1462no—
243C22C22⋊C8×D7311yescharacteristic
244C4—21no—
245C4—42no—
246C4—2921no—
247C4—1462no—
248C4#99×D7311yescharacteristic
249C4#1798857011yescharacteristic
250C2—41no—
251C2—2922no—
252C2—5841no—
253C2#2260824811yescharacteristic
254C2#2261155712yes—
subgroupG/H
centerC2×C4C22⋊C4×D73
commutatorC2×C4×C73C22×C8
FrattiniC22×C8C22×D73
FittingC73×#1323C2
socleC22×C73C2×C22⋊C8
prime radical#17989099C4
cosocleC22×C8×C73C23
hypercenter#1323D73
hypocenterC73#11469
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C22×C8 ◁ #1323

Complements

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

NH ≅ G/Ntype
2. #22611540C2semidirect
3. #22607873C2semidirect
4. #22607856C2semidirect
5. C73×#1323C2semidirect
131. #1323D73semidirect
133. C73#11469semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#226186242
#226115402
#226078732
#226078562
C73×#13232
#1146973