Finite Group Databasebeta

← #32982003

Subgroups — #32982003

Order 18688.

458 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 458 conjugacy classes of subgroups — too many to draw. Only its 119 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#22618569C211yesmaximal, characteristic
2#22608616C212yesmaximal, complemented
3#22608245C211yesmaximal, characteristic
4#22607789C212yesmaximal, complemented
5C73×#2628C211yesmaximal, characteristic
6#17988648—24no—
7#17988640C2212yes—
8#17988611C2212yes—
9#17988551—21no—
10C4×#11924012—21no—
11#17988333—24no—
12#17988313—21no—
13#17988311—21no—
14C2×C4⋊Q8×C73C2211yescharacteristic
15C2×#117×C73C2211yescharacteristic
16#400×C73C412yes—
17#399×C73C412yes—
18#395×C73—21no—
19#393×C73—21no—
20#389×C73—21no—
21#382×C73C2211yescharacteristic
22#380×C73—21no—
23C2×#11924012—22no—
24C4⋊#10596831—22no—
25C4⋊#10596831—22no—
26C4⋊#10596831D414yes—
27C22⋊#10596831—24no—
28C22⋊#10596831—22no—
29#14556502—22no—
30C4×#10596831—22no—
31C4×#10596831—22no—
32C22×Q8×C73—21no—
33C23×C4×C73D411yescharacteristic
34C4⋊Q8×C73—22no—
35C4⋊Q8×C73—22no—
36#117×C73—22no—
37#117×C73—22no—
38C2×C4⋊C4×C73—21no—
39C2×C4⋊C4×C73—21no—
40C2×C4⋊C4×C73—22no—
41C2×C4⋊C4×C73C2×C412yes—
42C2×C4⋊C4×C73C2×C411yescharacteristic
43C2×C4⋊C4×C73D411yescharacteristic
44C2×C22⋊C4×C73—22no—
45C2×C22⋊C4×C73C2×C411yescharacteristic
46C2×C22⋊C4×C73D411yescharacteristic
47C2×C42×C73D411yescharacteristic
48C2×C42×C73C2311yescharacteristic
49#95×C73—22no—
50#95×C73—22no—
51#95×C73—21no—
52#95×C73C2×C412yes—
53#11924012—42no—
54#11924012—42no—
55C2×#10596831—24no—
56C2×#10596831—42no—
57C24×C73C22⋊C411yescharacteristic
58C2×Q8×C73—42no—
59C2×Q8×C73—82no—
60C22×C4×C73—21no—
61C22×C4×C73—21no—
62C22×C4×C73—22no—
63C22×C4×C73—22no—
64C22×C4×C73—22no—
65C22×C4×C73—22no—
66C22×C4×C73C22⋊C411yescharacteristic
67C22×C4×C73C22⋊C411yescharacteristic
68C22×C4×C73C22⋊C411yescharacteristic
69C22×C4×C73C22×C411yescharacteristic
70C22×C4×C73C2×D411yescharacteristic
71C22×C4×C73C2×D411yescharacteristic
72C4⋊C4×C73—22no—
73C4⋊C4×C73—22no—
74C4⋊C4×C73—24no—
75C4⋊C4×C73—41no—
76C4⋊C4×C73—41no—
77C4⋊C4×C73—42no—
78C4⋊C4×C73—82no—
79C22⋊C4×C73—22no—
80C22⋊C4×C73—22no—
81C22⋊C4×C73—82no—
82C42×C73—21no—
83C42×C73—21no—
84C42×C73C2×D412yes—
85C42×C73#4112yes—
86C23×C73—21no—
87C23×C73—21no—
88C23×C73—21no—
89C23×C73—41no—
90C23×C73—41no—
91C23×C73C2×C22⋊C411yescharacteristic
92Q8×C73—84no—
93C2×C4×C73—21no—
94C2×C4×C73—21no—
95C2×C4×C73—21no—
96C2×C4×C73—21no—
97C2×C4×C73—21no—
98C2×C4×C73—41no—
99C2×C4×C73—41no—
100C2×C4×C73—42no—
101C2×C4×C73—42no—
102C2×C4×C73—41no—
103C2×C4×C73—41no—
104C2×C4×C73—42no—
105C2×C4×C73—42no—
106C2×C4×C73—42no—
107C2×C4×C73—42no—
108C2×C4×C73—42no—
109C2×C4×C73—42no—
110C2×C4×C73C4×D412yes—
111C2×C4×C73#12011yescharacteristic
112C2×C4×C73#12112yes—
113C2×C4×C73#12311yescharacteristic
114C2×C4×C73#13512yes—
115C2×C4×C73#13712yes—
116#10596831—44no—
117#10596831—82no—
118C22×C73—21no—
119C22×C73—21no—
120C22×C73—42no—
121C22×C73—41no—
122C22×C73—41no—
123C22×C73—41no—
124C22×C73—81no—
125C22×C73#38611yescharacteristic
126C22×C73#41611yescharacteristic
127C22×C73#41911yescharacteristic
128C4×C73—22no—
129C4×C73—22no—
130C4×C73—41no—
131C4×C73—41no—
132C4×C73—41no—
133C4×C73—82no—
134C4×C73—82no—
135C4×C73#45212yes—
136C4×C73#48512yes—
137#9194—731nomaximal, Sylow
138C2×C73—21no—
139C2×C73—21no—
140C2×C73—81no—
141C2×C73#187311yescharacteristic
142C2×C73#191011yescharacteristic
143C2×C73#191511yescharacteristic
144#2921—731no—
145#2628D7311yescharacteristic
146#1549—732no—
147#1458—731no—
148#1286—732no—
149C73#919411yescharacteristic, complemented, Sylow
150C2×C4⋊Q8C2×D7311yescharacteristic
151C2×#117C2×D7311yescharacteristic
152#433—1464no—
153#431—732no—
154#423—732no—
155#407—1461no—
156C4×M16—1461no—
157#400Dic7312yes—
158#399Dic7312yes—
159#395—21no—
160#393—21no—
161#389—21no—
162#382C2×D7311yescharacteristic
163#380—21no—
164#332—1464no—
165#324—1461no—
166#323—1461no—
167C22×Q8—21no—
168C23×C4C22⋊D7311yescharacteristic
169C2×M16—1462no—
170C4⋊Q8—22no—
171C4⋊Q8—22no—
172#117—22no—
173#117—22no—
174C2×C4⋊C4—21no—
175C2×C4⋊C4—21no—
176C2×C4⋊C4—22no—
177C2×C4⋊C4C2×Dic7312yes—
178C2×C4⋊C4C2×Dic7311yescharacteristic
179C2×C4⋊C4C22⋊D7311yescharacteristic
180C2×C22⋊C4—22no—
181C2×C22⋊C4C2×Dic7311yescharacteristic
182C2×C22⋊C4C22⋊D7311yescharacteristic
183C2×C42C22⋊D7311yescharacteristic
184C2×C42C22×D7311yescharacteristic
185C4⋊C8—734no—
186C4⋊C8—1462no—
187C4⋊C8—1462no—
188C22⋊C8—1462no—
189C22⋊C8—1464no—
190#97—1462no—
191C4×C8—1462no—
192C4×C8—1462no—
193#95—22no—
194#95—22no—
195#95—21no—
196#95C2×Dic7312yes—
197C24C22⋊Dic7311yescharacteristic
198C2×Q8—42no—
199C2×Q8—82no—
200C22×C4—21no—
201C22×C4—21no—
202C22×C4—22no—
203C22×C4—22no—
204C22×C4—22no—
205C22×C4—22no—
206C22×C4C22⋊Dic7311yescharacteristic
207C22×C4C22⋊Dic7311yescharacteristic
208C22×C4C22⋊Dic7311yescharacteristic
209C22×C4C22×Dic7311yescharacteristic
210C22×C4C2×C22⋊D7311yescharacteristic
211C22×C4C2×C22⋊D7311yescharacteristic
212M16—2922no—
213M16—2922no—
214C2×C8—2922no—
215C2×C8—1464no—
216C4⋊C4—22no—
217C4⋊C4—22no—
218C4⋊C4—24no—
219C4⋊C4—41no—
220C4⋊C4—41no—
221C4⋊C4—42no—
222C4⋊C4—82no—
223C22⋊C4—22no—
224C22⋊C4—22no—
225C22⋊C4—82no—
226C42—21no—
227C42—21no—
228C42D4×D7312yes—
229C42#1192402712yes—
230C23—21no—
231C23—21no—
232C23—21no—
233C23—41no—
234C23—41no—
235C23C2×C22⋊Dic7311yescharacteristic
236Q8—84no—
237C2×C4—21no—
238C2×C4—21no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—41no—
243C2×C4—41no—
244C2×C4—42no—
245C2×C4—42no—
246C2×C4—41no—
247C2×C4—41no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—42no—
253C2×C4—42no—
254C2×C4D4×Dic7312yes—
255C2×C4#1455655911yescharacteristic
256C2×C4#1455656312yes—
257C2×C4#1455657211yescharacteristic
258C2×C4#1455661112yes—
259C2×C4#1455662212yes—
260C8—2924no—
261C8—5842no—
262C22—21no—
263C22—21no—
264C22—42no—
265C22—41no—
266C22—41no—
267C22—41no—
268C22—81no—
269C22#1798846711yescharacteristic
270C22#1798858311yescharacteristic
271C22#1798859611yescharacteristic
272C4—22no—
273C4—22no—
274C4—41no—
275C4—41no—
276C4—41no—
277C4—82no—
278C4—82no—
279C4#1798874712yes—
280C4#1798892512yes—
281C2—21no—
282C2—21no—
283C2—81no—
284C2#2261024411yescharacteristic
285C2#2261045911yescharacteristic
286C2#2261048411yescharacteristic
subgroupG/H
centerC22#17988467
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#2628C2
socleC22×C73#386
prime radicalC24×C73C22⋊C4
cosocleC2×C42×C73C23
hypercenter#2628D73
hypocenterC73#9194
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2628

Complements

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

NH ≅ G/Ntype
2. #22608616C2semidirect
4. #22607789C2semidirect
149. C73#9194semidirect

Maximal subgroups

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

subgroupindex
#226185692
#226086162
#226082452
#226077892
C73×#26282
#919473