Finite Group Databasebeta

← #32983798

Subgroups — #32983798

Order 18688.

610 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 610 conjugacy classes of subgroups — too many to draw. Only its 111 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#22614218C211yesmaximal, characteristic
2#22610031C212yesmaximal
3#22608261C211yesmaximal, characteristic
4#22607784C212yesmaximal
5C73×#2274C211yesmaximal, characteristic
6C2×#14556550C2211yescharacteristic
7#17988551C2212yes—
8#17988526C412yescomplemented
9#17988517C412yescomplemented
10#17988468—22no—
11#17988460—21no—
12#17988450—21no—
13#17988450C2211yescharacteristic
14#17988347—24no—
15#17988313—22no—
16C2×#117×C73—21no—
17C22×C4⋊C4×C73C2211yescharacteristic
18C4×C4⋊C4×C73—24no—
19C4×C22⋊C4×C73—21no—
20C4×C22⋊C4×C73C2212yes—
21C2×#11924012—22no—
22#14556550—22no—
23#14556550—22no—
24C2×C4⋊Dic73C2×C411yescharacteristic
25C2×C4⋊Dic73C2×C411yescharacteristic
26C2×#11924007—22no—
27C2×C22⋊Dic73—21no—
28C2×C22⋊Dic73—21no—
29C2×C4×Dic73—21no—
30C22⋊#10596831—22no—
31C22⋊#10596831—22no—
32#14556498—22no—
33#14556498—22no—
34#14556498—22no—
35#14556498—21no—
36#14556498C2×C412yes—
37C23×C4×C73D411yescharacteristic
38C23×C4×C73D411yescharacteristic
39C23×C4×C73C2311yescharacteristic
40#117×C73—28no—
41C2×C4⋊C4×C73—22no—
42C2×C4⋊C4×C73—22no—
43C2×C4⋊C4×C73—22no—
44C2×C4⋊C4×C73C2×C412yes—
45C2×C4⋊C4×C73D412yes—
46C2×C22⋊C4×C73—22no—
47C2×C42×C73—22no—
48C2×C42×C73—22no—
49C2×C42×C73D414yes—
50#95×C73—22no—
51#95×C73—22no—
52C22×Dic73—22no—
53C22×Dic73—21no—
54C22×Dic73—21no—
55#11924012—44no—
56C2×#10596831—42no—
57C4⋊Dic73—22no—
58C4⋊Dic73—22no—
59#11924007—44no—
60C22⋊Dic73—42no—
61C22⋊Dic73—41no—
62C22⋊Dic73—41no—
63C4×Dic73—42no—
64C24×C73C2×D411yescharacteristic
65C22×C4×C73—21no—
66C22×C4×C73—21no—
67C22×C4×C73—24no—
68C22×C4×C73—24no—
69C22×C4×C73—21no—
70C22×C4×C73—21no—
71C22×C4×C73—42no—
72C22×C4×C73—42no—
73C22×C4×C73C22⋊C412yes—
74C22×C4×C73C22⋊C412yes—
75C22×C4×C73C22×C411yescharacteristic
76C22×C4×C73C2×D412yes—
77C22×C4×C73C2×D411yescharacteristic
78C22×C4×C73#4112yes—
79C4⋊C4×C73—28no—
80C4⋊C4×C73—24no—
81C4⋊C4×C73—24no—
82C22⋊C4×C73—28no—
83C42×C73—28no—
84C42×C73—44no—
85C42×C73—82no—
86C23×C73—22no—
87C23×C73—22no—
88C23×C73—41no—
89C23×C73C2×C22⋊C411yescharacteristic
90C23×C73#12011yescharacteristic
91C23×C73#12311yescharacteristic
92C2×C4×C73—22no—
93C2×C4×C73—22no—
94C2×C4×C73—24no—
95C2×C4×C73—22no—
96C2×C4×C73—22no—
97C2×C4×C73—24no—
98C2×C4×C73—28no—
99C2×C4×C73—22no—
100C2×C4×C73—41no—
101C2×C4×C73—41no—
102C2×C4×C73—42no—
103C2×C4×C73—44no—
104C2×C4×C73—44no—
105C2×C4×C73—42no—
106C2×C4×C73—42no—
107C2×C4×C73—81no—
108C2×C4×C73#10414yes—
109C2×C4×C73C4×D412yes—
110C2×C4×C73#12112yes—
111C2×Dic73—42no—
112C2×Dic73—42no—
113C2×Dic73—42no—
114C2×Dic73—41no—
115C2×Dic73—41no—
116C2×Dic73—42no—
117C2×Dic73—42no—
118#10596831—82no—
119C22×C73—21no—
120C22×C73—21no—
121C22×C73—21no—
122C22×C73—21no—
123C22×C73—21no—
124C22×C73—21no—
125C22×C73—21no—
126C22×C73—44no—
127C22×C73#38611yescharacteristic
128C22×C73C2×#10412yes—
129C22×C73#45812yes—
130C4×C73—22no—
131C4×C73—22no—
132C4×C73—24no—
133C4×C73—48no—
134C4×C73—42no—
135Dic73—82no—
136Dic73—81no—
137Dic73—81no—
138#9478—731nomaximal, Sylow
139C2×C73—22no—
140C2×C73—22no—
141C2×C73—41no—
142C2×C73#189011yescharacteristic
143C2×C73#189611yescharacteristic
144C2×C73#191211yescharacteristic
145#2484—731no—
146#2274D7311yescharacteristic
147#1826—732no—
148#1463—731no—
149#1284—732no—
150C73#947811yescharacteristic, complemented, Sylow
151C2×#117—21no—
152C2×#117—731no—
153C22×C4⋊C4C2×D7311yescharacteristic
154#407—732no—
155#400—732no—
156#398—732no—
157#387—1462no—
158#385—1461no—
159#381—731no—
160#381—1461no—
161C4×C4⋊C4—24no—
162C4×C22⋊C4—21no—
163C4×C22⋊C4C2×D7312yes—
164#339—1464no—
165#324—1462no—
166C23×C4C22⋊D7311yescharacteristic
167C23×C4C22⋊D7311yescharacteristic
168C23×C4C22×D7311yescharacteristic
169C2×M16—1462no—
170#117—28no—
171#117—1462no—
172#117—1462no—
173C2×C4⋊C4—22no—
174C2×C4⋊C4—22no—
175C2×C4⋊C4—22no—
176C2×C4⋊C4—731no—
177C2×C4⋊C4—731no—
178C2×C4⋊C4—1462no—
179C2×C4⋊C4C4×D7312yes—
180C2×C4⋊C4D29212yes—
181C2×C22⋊C4—22no—
182C2×C22⋊C4—1461no—
183C2×C22⋊C4—1461no—
184C2×C42—22no—
185C2×C42—22no—
186C2×C42—1461no—
187C2×C42C22⋊D7314yes—
188C22⋊C8—1462no—
189C22⋊C8—1462no—
190#95—22no—
191#95—22no—
192#95—732no—
193#95—1462no—
194#95—1462no—
195#95—1462no—
196#95—1461no—
197C24C2×C22⋊D7311yescharacteristic
198C22×C4—21no—
199C22×C4—21no—
200C22×C4—24no—
201C22×C4—24no—
202C22×C4—21no—
203C22×C4—21no—
204C22×C4—42no—
205C22×C4—42no—
206C22×C4—1462no—
207C22×C4—1461no—
208C22×C4—1461no—
209C22×C4#1192400512yes—
210C22×C4#1192400512yes—
211C22×C4C2×C4×D7311yescharacteristic
212C22×C4C2×C22⋊D7312yes—
213C22×C4C2×D29211yescharacteristic
214C22×C4#1192402812yes—
215M16—2924no—
216C2×C8—2922no—
217C4⋊C4—28no—
218C4⋊C4—24no—
219C4⋊C4—24no—
220C4⋊C4—2924no—
221C4⋊C4—1462no—
222C4⋊C4—1462no—
223C22⋊C4—28no—
224C22⋊C4—2922no—
225C22⋊C4—2921no—
226C22⋊C4—2921no—
227C42—28no—
228C42—44no—
229C42—82no—
230C42—2922no—
231C23—22no—
232C23—22no—
233C23—41no—
234C23C2×#1192400511yescharacteristic
235C23#1455656011yescharacteristic
236C23#1455657411yescharacteristic
237C2×C4—22no—
238C2×C4—22no—
239C2×C4—24no—
240C2×C4—22no—
241C2×C4—24no—
242C2×C4—22no—
243C2×C4—28no—
244C2×C4—22no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—44no—
249C2×C4—44no—
250C2×C4—42no—
251C2×C4—42no—
252C2×C4—81no—
253C2×C4—2922no—
254C2×C4—2922no—
255C2×C4—2922no—
256C2×C4—2921no—
257C2×C4—2921no—
258C2×C4—2922no—
259C2×C4—2922no—
260C2×C4#1455651914yes—
261C2×C4C4×C22⋊D7312yes—
262C2×C4#1455656512yes—
263C8—5842no—
264C22—21no—
265C22—21no—
266C22—21no—
267C22—21no—
268C22—21no—
269C22—21no—
270C22—21no—
271C22—44no—
272C22#1798846611yescharacteristic
273C22C2×#1455651912yes—
274C22#1798877712yes—
275C4—22no—
276C4—22no—
277C4—24no—
278C4—48no—
279C4—42no—
280C4—5842no—
281C4—5841no—
282C4—5841no—
283C2—22no—
284C2—22no—
285C2—41no—
286C2#2261034411yescharacteristic
287C2#2261037411yescharacteristic
288C2#2261046811yescharacteristic
subgroupG/H
centerC22#17988466
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC73×#2274C2
socleC22×C73#386
prime radicalC24×C73C2×D4
cosocleC23×C4×C73C23
hypercenter#2274D73
hypocenterC73#9478
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #2274

Complements

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

NH ≅ G/Ntype
8. #17988526C4semidirect
9. #17988517C4semidirect
150. C73#9478semidirect

Maximal subgroups

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

subgroupindex
#226142182
#226100312
#226082612
#226077842
C73×#22742
#947873