Finite Group Databasebeta

← #34013225

Subgroups — #34013225

Order 18688.

642 conjugacy classes of subgroups; only the 80 normal subgroups are shown.

normal subgroups only

This group has 642 conjugacy classes of subgroups — too many to draw. Only its 80 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#22625760C212yesmaximal
2#22625025C214yesmaximal, complemented
3#22612519C211yesmaximal, characteristic
4C2×#14556650C2214yes—
5C2×#14556649—22no—
6#17989530—22no—
7C22×#11924028C2211yescharacteristic
8C22×C4⋊Dic73C411yescharacteristic, complemented
9#17988493C412yescomplemented
10#17988489C2212yes—
11#17988464—22no—
12#17988444—22no—
13C2×#117×C73C411yescharacteristic, complemented
14C2×C22⋊#64408—24no—
15#14556650—28no—
16#14556649—44no—
17(C2×#64408)⋊C4—44no—
18C2×#11924028—24no—
19C2×#11924028—42no—
20C2×#11924028—42no—
21C22×Dic146C2×C411yescharacteristic
22C22×D292C2311yescharacteristic
23C22×C22⋊D73D412yes—
24C23×Dic73—21no—
25C22×C4×D73D412yes—
26C2×C4⋊Dic73—21no—
27C2×C4⋊Dic73—42no—
28C2×C4⋊Dic73C2×C412yes—
29C2×#11924005—24no—
30#14556498—24no—
31C23×C4×C73C2×C411yescharacteristic
32#117×C73—42no—
33C2×C4⋊C4×C73—21no—
34C2×C22⋊C4×C73—21no—
35C2×C42×C73C2×C412yes—
36C22×#64408—44no—
37C22⋊#64408—44no—
38C22⋊#64408—44no—
39C23×D73C2×D412yes—
40#11924028—416no—
41C2×Dic146—22no—
42C2×Dic146—41no—
43C2×Dic146—41no—
44C2×D292—41no—
45C2×D292—41no—
46C2×D292#4114yes—
47C2×C22⋊D73—24no—
48C2×C22⋊D73—24no—
49C2×C22⋊D73—42no—
50C22×Dic73—22no—
51C22×Dic73—42no—
52C22×Dic73—82no—
53C22×Dic73C22⋊C412yes—
54C2×C4×D73—24no—
55C2×C4×D73—24no—
56C2×C4×D73—42no—
57C4⋊Dic73—42no—
58C4⋊Dic73—42no—
59#11924005—84no—
60C24×C73C22⋊C411yescharacteristic
61C22×C4×C73—22no—
62C22×C4×C73—22no—
63C22×C4×C73—22no—
64C22×C4×C73—22no—
65C22×C4×C73C22⋊C411yescharacteristic
66C22×C4×C73C22×C411yescharacteristic
67C4⋊C4×C73—42no—
68C22⋊C4×C73—42no—
69C42×C73—42no—
70C23×C73—21no—
71C23×C73—21no—
72C23×C73—41no—
73C23×C73—41no—
74C23×C73#9912yes—
75C23×C73C2×C22⋊C411yescharacteristic
76C22×D73—22no—
77C22×D73—22no—
78C22×D73—42no—
79C22×D73—42no—
80C22×D73#12314yes—
81C2×#64408—84no—
82C2×#64408—84no—
83C2×#64408—84no—
84C2×C4×C73—28no—
85C2×C4×C73—21no—
86C2×C4×C73—21no—
87C2×C4×C73—21no—
88C2×C4×C73—21no—
89C2×C4×C73—42no—
90C2×C4×C73—42no—
91C2×C4×C73—42no—
92C2×C4×C73#9912yes—
93C2×C4×C73#11712yes—
94C22⋊D73—48no—
95C2×Dic73—22no—
96C2×Dic73—22no—
97C2×Dic73—22no—
98C2×Dic73—44no—
99C2×Dic73—42no—
100C2×Dic73—84no—
101D292—44no—
102C4×D73—48no—
103Dic146—44no—
104C22×C73—21no—
105C22×C73—21no—
106C22×C73—22no—
107C22×C73—42no—
108C22×C73—42no—
109C22×C73—41no—
110C22×C73—41no—
111C22×C73#38111yescharacteristic
112C22×C73C2×#9911yescharacteristic
113C22×C73C2×#9911yescharacteristic
114C2×D73—24no—
115C2×D73—42no—
116C2×D73—42no—
117C2×D73—44no—
118C2×D73—82no—
119#64408—164no—
120C4×C73—22no—
121C4×C73—22no—
122C4×C73—82no—
123Dic73—42no—
124Dic73—82no—
125#12859—731nomaximal, Sylow
126C2×C73—21no—
127C2×C73—21no—
128C2×C73—42no—
129C2×C73#181911yescharacteristic
130C2×C73#212811yescharacteristic
131C2×C73#212911yescharacteristic
132D73—44no—
133#2311—731no—
134#1782—732no—
135#1393—734no—
136C73#1285911yescharacteristic, complemented, Sylow
137C22×#41—731no—
138C2×#117#6440811yescharacteristic, complemented
139C22×C4⋊C4—731no—
140C2×#99—734no—
141C2×#99—1462no—
142#391—732no—
143#390—732no—
144#386—1462no—
145#381—1462no—
146#380—1462no—
147C2×#41—2922no—
148C2×#41—2922no—
149C2×#41—1464no—
150C22×Q8—731no—
151C22×D4—732no—
152C22×D4—731no—
153C23×C4—732no—
154C23×C4—1461no—
155C23×C4C2×#6440811yescharacteristic
156#117—42no—
157C2×C4⋊C4—21no—
158C2×C4⋊C4—2922no—
159C2×C4⋊C4—732no—
160C2×C4⋊C4—1461no—
161C2×C22⋊C4—21no—
162C2×C22⋊C4—1464no—
163C2×C22⋊C4—1464no—
164C2×C42C2×#6440812yes—
165#99—2924no—
166#99—1468no—
167#95—2924no—
168#95—1464no—
169C24—732no—
170C24C22⋊#6440811yescharacteristic
171#41—29216no—
172C2×Q8—2921no—
173C2×Q8—2921no—
174C2×Q8—1462no—
175C2×D4—2921no—
176C2×D4—2921no—
177C2×D4—2922no—
178C2×D4—734no—
179C2×D4—1464no—
180C2×D4—1464no—
181C22×C4—22no—
182C22×C4—22no—
183C22×C4—22no—
184C22×C4—22no—
185C22×C4—2922no—
186C22×C4—2922no—
187C22×C4—2924no—
188C22×C4—5842no—
189C22×C4—732no—
190C22×C4—1464no—
191C22×C4—1462no—
192C22×C4—1464no—
193C22×C4C22⋊#6440811yescharacteristic
194C22×C4C22×#6440811yescharacteristic
195C4⋊C4—42no—
196C4⋊C4—2922no—
197C4⋊C4—2922no—
198C22⋊C4—42no—
199C22⋊C4—2924no—
200C22⋊C4—2924no—
201C22⋊C4—5844no—
202C42—42no—
203C23—21no—
204C23—21no—
205C23—41no—
206C23—41no—
207C23—2922no—
208C23—2922no—
209C23—734no—
210C23—1462no—
211C23—1462no—
212C23#1455664912yes—
213C23C2×C22⋊#6440811yescharacteristic
214Q8—2924no—
215D4—2924no—
216D4—2928no—
217C2×C4—28no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—42no—
223C2×C4—42no—
224C2×C4—42no—
225C2×C4—2928no—
226C2×C4—2924no—
227C2×C4—2922no—
228C2×C4—5844no—
229C2×C4—5844no—
230C2×C4—5844no—
231C2×C4—5844no—
232C2×C4—1462no—
233C2×C4—1462no—
234C2×C4—1462no—
235C2×C4#1455665012yes—
236C2×C4#1455667012yes—
237C22—21no—
238C22—21no—
239C22—22no—
240C22—42no—
241C22—42no—
242C22—41no—
243C22—41no—
244C22—2924no—
245C22—2922no—
246C22—2922no—
247C22—5842no—
248C22—1464no—
249C22#1798953011yescharacteristic
250C22C2×#1455664911yescharacteristic
251C22C2×#1455665011yescharacteristic
252C4—22no—
253C4—22no—
254C4—82no—
255C4—2922no—
256C4—5842no—
257C4—11684no—
258C2—21no—
259C2—21no—
260C2—42no—
261C2—2924no—
262C2#2262582611yescharacteristic
263C2#2262621211yescharacteristic
264C2#2262621511yescharacteristic
subgroupG/H
centerC22C2×#14556650
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC2×#117×C73C4
socleC22×C73C2×#99
prime radicalC22×#11924028C22
cosocleC22×D292C23
hypercenterC2×#117#64408
hypocenterC73#12859
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C23×C4 ◁ C2×#117

Complements

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

NH ≅ G/Ntype
2. #22625025C2semidirect
8. C22×C4⋊Dic73C4semidirect
9. #17988493C4semidirect
13. C2×#117×C73C4semidirect
136. C73#12859semidirect
138. C2×#117#64408semidirect

Maximal subgroups

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

subgroupindex
#226257602
#226250252
#226125192
#1285973