Finite Group Databasebeta

← #34009982

Subgroups — #34009982

Order 18688.

440 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 440 conjugacy classes of subgroups — too many to draw. Only its 90 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#22625174C212yesmaximal, complemented
2#22624874C212yesmaximal, complemented
3#22624860C212yesmaximal
4#22612645C211yesmaximal, characteristic
5#17989603—22no—
6#17989432—22no—
7#17989432C2214yes—
8C2×#14556580C411yescharacteristic
9C2×Q8×Dic73C2211yescharacteristic
10C2×#14556553C411yescharacteristic
11#17988489C2211yescharacteristic
12#17988476—22no—
13#17988461—22no—
14#17988456C2211yescharacteristic
15#17988433C411yescharacteristic
16#391×C73C411yescharacteristic
17C4⋊#10596833—42no—
18C22⋊#10596833—24no—
19C4×#10596833—42no—
20C22×D292C2×C411yescharacteristic
21C22×C4×D73—21no—
22#14556580—22no—
23#14556580C814yescomplemented
24Q8×Dic73—42no—
25#14556553—42no—
26C2×C4⋊Dic73D412yes—
27C2×C4⋊Dic73C2311yescharacteristic
28C2×#11924007—22no—
29C2×#11924005—21no—
30C2×#11924005—22no—
31C2×#11924005—22no—
32C2×C4×Dic73C2×C411yescharacteristic
33C2×C4×Dic73C2×C411yescharacteristic
34C2×C4×Dic73D412yes—
35#14556498—21no—
36#14556498—21no—
37C22×Q8×C73C2×C411yescharacteristic
38C2×C4⋊C4×C73C2×C411yescharacteristic
39C2×C42×C73C2×C411yescharacteristic
40#95×C73—22no—
41C2×#10596833—44no—
42C23×D73—21no—
43C2×D292—21no—
44C2×D292—41no—
45C2×D292—41no—
46C2×D292C2×C812yes—
47C22×Dic73—21no—
48C22×Dic73C22⋊C411yescharacteristic
49C22×Dic73C22⋊C411yescharacteristic
50C22×Dic73C2×D412yes—
51C2×C4×D73—42no—
52C2×C4×D73—82no—
53C4⋊Dic73—42no—
54C4⋊Dic73—41no—
55#11924007—82no—
56#11924005—28no—
57#11924005—42no—
58#11924005—82no—
59C4×Dic73—21no—
60C4×Dic73—22no—
61C4×Dic73—22no—
62C4×Dic73—22no—
63C4×Dic73C2×C812yes—
64C2×Q8×C73—41no—
65C2×Q8×C73—41no—
66C2×Q8×C73C2×C812yes—
67C2×Q8×C73M1612yes—
68C22×C4×C73—21no—
69C22×C4×C73—21no—
70C22×C4×C73C22⋊C412yes—
71C22×C4×C73C22×C411yescharacteristic
72C4⋊C4×C73—41no—
73C42×C73—41no—
74C23×C73C2×C22⋊C411yescharacteristic
75C22×D73—22no—
76C22×D73—42no—
77C22×D73—42no—
78C22×D73—41no—
79C22×D73—41no—
80Q8×C73—42no—
81Q8×C73—42no—
82C2×C4×C73—21no—
83C2×C4×C73—22no—
84C2×C4×C73—21no—
85C2×C4×C73—22no—
86C2×C4×C73—41no—
87C2×C4×C73—41no—
88C2×C4×C73—42no—
89C2×C4×C73—41no—
90C2×C4×C73—41no—
91C2×C4×C73C22⋊C814yes—
92C2×C4×C73C22×C811yescharacteristic
93C2×C4×C73C2×M1611yescharacteristic
94C2×Dic73—22no—
95C2×Dic73—21no—
96C2×Dic73—22no—
97C2×Dic73—41no—
98C2×Dic73—41no—
99C2×Dic73—41no—
100C2×Dic73—42no—
101C2×Dic73—41no—
102C2×Dic73#9912yes—
103D292—42no—
104D292—42no—
105C4×D73—84no—
106#10596833—84no—
107C22×C73—21no—
108C22×C73—21no—
109C22×C73C2×C22⋊C811yescharacteristic
110C22×C73C2×#9911yescharacteristic
111C22×C73#41311yescharacteristic
112C2×D73—44no—
113C2×D73—42no—
114C2×D73—82no—
115C2×D73—81no—
116C2×D73—81no—
117C4×C73—21no—
118C4×C73—21no—
119C4×C73—24no—
120C4×C73—81no—
121C4×C73—81no—
122Dic73—22no—
123Dic73—42no—
124Dic73—42no—
125Dic73—41no—
126#11480—731nomaximal, Sylow
127C2×C73—21no—
128C2×C73—21no—
129C2×C73#145911yescharacteristic
130C2×C73#212811yescharacteristic
131C2×C73#213311yescharacteristic
132D73—82no—
133#2326—731no—
134#1461—732no—
135#1325—732no—
136#1318—732no—
137C73#1148011yescharacteristic, complemented, Sylow
138C2×#124—731no—
139C2×C4×Q8—731no—
140C2×C4×D4—731no—
141#431—1462no—
142#391#6440811yescharacteristic
143#390—731no—
144#388—1462no—
145#385—1462no—
146#384—731no—
147#376—731no—
148#324—734no—
149#324—1462no—
150C22×Q8C2×#6440811yescharacteristic
151C22×D4—731no—
152C23×C4—1461no—
153#124—734no—
154#124—1462no—
155C4×Q8—2922no—
156C4×D4—2922no—
157C2×C4⋊C4—732no—
158C2×C4⋊C4—731no—
159C2×C4⋊C4—1462no—
160C2×C4⋊C4C2×#6440811yescharacteristic
161C2×C22⋊C4—1461no—
162C2×C22⋊C4—1462no—
163C2×C22⋊C4—1462no—
164C2×C42—731no—
165C2×C42—732no—
166C2×C42—731no—
167C2×C42C2×#6440811yescharacteristic
168C4⋊C8—2922no—
169C22⋊C8—1464no—
170C4×C8—2922no—
171#95—22no—
172#95—1461no—
173#95—1461no—
174C24—1461no—
175C2×Q8—41no—
176C2×Q8—41no—
177C2×Q8#1192403612yes—
178C2×Q8#1192403812yes—
179C2×D4—2921no—
180C2×D4—2921no—
181C2×D4—732no—
182C2×D4—1461no—
183C22×C4—21no—
184C22×C4—21no—
185C22×C4—2922no—
186C22×C4—5842no—
187C22×C4—731no—
188C22×C4—731no—
189C22×C4—732no—
190C22×C4—1461no—
191C22×C4C22⋊#6440812yes—
192C22×C4C22×#6440811yescharacteristic
193C2×C8—2924no—
194C4⋊C4—41no—
195C4⋊C4—2921no—
196C4⋊C4—2922no—
197C4⋊C4—5842no—
198C22⋊C4—2922no—
199C22⋊C4—5842no—
200C22⋊C4—1468no—
201C42—41no—
202C42—732no—
203C42—1462no—
204C42—1462no—
205C42—1462no—
206C42—1461no—
207C23—2922no—
208C23—2922no—
209C23—2921no—
210C23—2921no—
211C23—1462no—
212C23C2×C22⋊#6440811yescharacteristic
213Q8—42no—
214Q8—42no—
215D4—2922no—
216D4—2922no—
217C2×C4—21no—
218C2×C4—22no—
219C2×C4—21no—
220C2×C4—22no—
221C2×C4—41no—
222C2×C4—42no—
223C2×C4—41no—
224C2×C4—41no—
225C2×C4—41no—
226C2×C4—2921no—
227C2×C4—2921no—
228C2×C4—2922no—
229C2×C4—2921no—
230C2×C4—2921no—
231C2×C4—5844no—
232C2×C4—732no—
233C2×C4—1462no—
234C2×C4—1462no—
235C2×C4—1461no—
236C2×C4#1455664814yes—
237C2×C4C2×#1192403611yescharacteristic
238C2×C4C2×#1192403811yescharacteristic
239C8—5844no—
240C22—21no—
241C22—21no—
242C22—2924no—
243C22—2922no—
244C22—5842no—
245C22—5841no—
246C22—5841no—
247C22C2×#1455664811yescharacteristic
248C22D73⋊#9911yescharacteristic
249C22#1798957411yescharacteristic
250C4—21no—
251C4—21no—
252C4—24no—
253C4—81no—
254C4—81no—
255C4—2922no—
256C4—2922no—
257C4—2921no—
258C4—1462no—
259C2—21no—
260C2—21no—
261C2—5842no—
262C2#2262516911yescharacteristic
263C2#2262621111yescharacteristic
264C2#2262622911yescharacteristic
subgroupG/H
centerC22D73⋊#99
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
Fitting#391×C73C4
socleC22×C73C2×#99
prime radicalC22×D292C2×C4
cosocleC2×C4⋊Dic73C23
hypercenter#391#64408
hypocenterC73#11480
LCSG ▷ C2×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C22×Q8 ◁ #391

Complements

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

NH ≅ G/Ntype
1. #22625174C2semidirect
2. #22624874C2semidirect
23. #14556580C8semidirect
137. C73#11480semidirect

Maximal subgroups

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

subgroupindex
#226251742
#226248742
#226248602
#226126452
#1148073