Finite Group Databasebeta

← #32990379

Subgroups — #32990379

Order 18688.

490 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 490 conjugacy classes of subgroups — too many to draw. Only its 99 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#22610917C212yesmaximal
2#22610355C212yesmaximal
3#22610108C211yesmaximal, characteristic
4#22607799C211yesmaximal, characteristic
5C73×#3189C211yesmaximal, characteristic
6C23×#10596831C2211yescharacteristic
7C2×#14556520—21no—
8C2×#14556515—22no—
9#17988447C2212yes—
10C4×C22⋊Dic73C2211yescharacteristic
11#17988354—22no—
12#17988354—22no—
13#17988349—22no—
14#17988341—21no—
15C2×C22⋊Q8×C73C2212yes—
16#484×C73—22no—
17#467×C73—21no—
18#467×C73—21no—
19#451×C73—22no—
20C2×C22⋊C8×C73C2211yescharacteristic
21C22×#10596831—22no—
22C22×#10596831—21no—
23C22×#10596831—42no—
24C2×C4⋊Dic73—22no—
25C2×C22⋊Dic73—22no—
26C2×C22⋊Dic73D411yescharacteristic
27C2×C22⋊Dic73D411yescharacteristic
28C2×C4×Dic73—21no—
29#14556520—42no—
30#14556515—44no—
31#14556498—21no—
32#14556498—22no—
33C22×Q8×C73D412yes—
34C23×C4×C73C2311yescharacteristic
35C2×Q16×C73—21no—
36C2×Q16×C73—21no—
37C22×C8×C73D411yescharacteristic
38C22×C8×C73D411yescharacteristic
39C22⋊Q8×C73—22no—
40C22⋊Q8×C73—22no—
41C22⋊Q8×C73—22no—
42C22⋊Q8×C73—22no—
43C2×C4⋊C4×C73—22no—
44C2×C4⋊C4×C73D412yes—
45C2×C22⋊C4×C73—22no—
46#107×C73—21no—
47#107×C73—21no—
48#103×C73—22no—
49#103×C73—22no—
50C22⋊C8×C73—22no—
51C22×Dic73—21no—
52C22×Dic73—21no—
53C22×Dic73—42no—
54C2×#10596831—24no—
55C2×#10596831—22no—
56C2×#10596831—41no—
57C2×#10596831—42no—
58C2×#10596831—41no—
59C4⋊Dic73—42no—
60C4⋊Dic73—42no—
61C22⋊Dic73—21no—
62C22⋊Dic73—22no—
63C22⋊Dic73—42no—
64C22⋊Dic73—42no—
65C22⋊Dic73Q1612yescomplemented
66C4×Dic73—41no—
67C4×Dic73—41no—
68C24×C73C2×D411yescharacteristic
69C2×Q8×C73—22no—
70C2×Q8×C73—22no—
71C2×Q8×C73—42no—
72C2×Q8×C73—42no—
73C22×C4×C73—22no—
74C22×C4×C73—22no—
75C22×C4×C73—41no—
76C22×C4×C73—41no—
77C22×C4×C73C2×D412yes—
78C22×C4×C73C2×D411yescharacteristic
79C22×C4×C73#4111yescharacteristic
80C22×C4×C73#4112yes—
81Q16×C73—81no—
82Q16×C73—81no—
83C2×C8×C73—21no—
84C2×C8×C73—21no—
85C2×C8×C73—41no—
86C2×C8×C73—41no—
87C4⋊C4×C73—22no—
88C4⋊C4×C73—22no—
89C4⋊C4×C73—42no—
90C4⋊C4×C73—42no—
91C22⋊C4×C73—42no—
92C22⋊C4×C73—42no—
93C23×C73—41no—
94C23×C73—41no—
95C23×C73#12111yescharacteristic
96C23×C73#12112yes—
97C23×C73C2×Q1611yescharacteristic
98C23×C73#13511yescharacteristic
99C23×C73#13712yes—
100Q8×C73—44no—
101Q8×C73—42no—
102Q8×C73—42no—
103C2×C4×C73—21no—
104C2×C4×C73—22no—
105C2×C4×C73—21no—
106C2×C4×C73—22no—
107C2×C4×C73—41no—
108C2×C4×C73—42no—
109C2×C4×C73—42no—
110C2×C4×C73—42no—
111C2×C4×C73—82no—
112C2×C4×C73—82no—
113C2×C4×C73#12011yescharacteristic
114C2×C4×C73#12312yes—
115C2×C4×C73#12411yescharacteristic
116C2×Dic73—22no—
117C2×Dic73—41no—
118C2×Dic73—41no—
119C2×Dic73—41no—
120C2×Dic73—41no—
121C2×Dic73—41no—
122C2×Dic73—82no—
123C2×Dic73—82no—
124C2×Dic73—82no—
125C8×C73—41no—
126C8×C73—41no—
127#10596831—42no—
128C22×C73—21no—
129C22×C73—22no—
130C22×C73—21no—
131C22×C73—22no—
132C22×C73—41no—
133C22×C73—41no—
134C22×C73—42no—
135C22×C73#39411yescharacteristic
136C22×C73#46212yes—
137C22×C73#46412yes—
138C22×C73#46711yescharacteristic
139C22×C73#47011yescharacteristic
140C4×C73—22no—
141C4×C73—21no—
142C4×C73—82no—
143C4×C73—82no—
144C4×C73B(4,𝔽2)11yescharacteristic
145C4×C73#45811yescharacteristic
146Dic73—42no—
147Dic73—81no—
148Dic73—162no—
149#10566—731nomaximal, Sylow
150C2×C73—21no—
151C2×C73—21no—
152C2×C73—22no—
153C2×C73—22no—
154C2×C73#204411yescharacteristic
155C2×C73#205311yescharacteristic
156C2×C73#205611yescharacteristic
157#3189D7311yescharacteristic
158#1996—732no—
159#1892—732no—
160#1850—731no—
161#1289—731no—
162C73#1056611yescharacteristic, complemented, Sylow
163C23×C8—731no—
164C2×C22⋊Q8C2×D7312yes—
165#484—22no—
166#467—21no—
167#467—21no—
168#451—22no—
169C2×C4⋊C8—1461no—
170C2×#103—1462no—
171C2×C22⋊C8C2×D7311yescharacteristic
172#380—732no—
173C4×C22⋊C4—731no—
174#342—1462no—
175#342—1462no—
176#340—1462no—
177#336—1461no—
178C22×Q8C22⋊D7312yes—
179C23×C4C22×D7311yescharacteristic
180C2×Q16—21no—
181C2×Q16—21no—
182C22×C8—2922no—
183C22×C8—1461no—
184C22×C8—1462no—
185C22×C8C22⋊D7311yescharacteristic
186C22×C8C22⋊D7311yescharacteristic
187C22⋊Q8—22no—
188C22⋊Q8—22no—
189C22⋊Q8—22no—
190C22⋊Q8—22no—
191C2×C4⋊C4—22no—
192C2×C4⋊C4—1462no—
193C2×C4⋊C4C22⋊D7312yes—
194C2×C22⋊C4—22no—
195C2×C22⋊C4—731no—
196C2×C22⋊C4—731no—
197C2×C22⋊C4—1462no—
198C2×C42—1461no—
199#107—21no—
200#107—21no—
201C4⋊C8—2922no—
202#103—22no—
203#103—22no—
204#103—2924no—
205C22⋊C8—22no—
206#95—1462no—
207#95—1461no—
208C24D4×D7311yescharacteristic
209C2×Q8—22no—
210C2×Q8—22no—
211C2×Q8—42no—
212C2×Q8—42no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—2922no—
218C22×C4—1461no—
219C22×C4—1461no—
220C22×C4C2×C22⋊D7312yes—
221C22×C4C2×C22⋊D7311yescharacteristic
222C22×C4#1192402711yescharacteristic
223C22×C4#1192402712yes—
224Q16—81no—
225Q16—81no—
226C2×C8—21no—
227C2×C8—21no—
228C2×C8—41no—
229C2×C8—41no—
230C2×C8—2921no—
231C2×C8—2922no—
232C2×C8—2921no—
233C2×C8—1464no—
234C2×C8—1462no—
235C4⋊C4—22no—
236C4⋊C4—22no—
237C4⋊C4—42no—
238C4⋊C4—42no—
239C4⋊C4—2922no—
240C4⋊C4—2922no—
241C22⋊C4—42no—
242C22⋊C4—42no—
243C22⋊C4—2922no—
244C22⋊C4—2922no—
245C22⋊C4—732no—
246C22⋊C4—1462no—
247C22⋊C4—1461no—
248C42—2921no—
249C42—2921no—
250C23—41no—
251C23—41no—
252C23#1455656211yescharacteristic
253C23#1455656312yes—
254C23Q16×D7311yescharacteristic
255C23#1455661411yescharacteristic
256C23#1455662312yes—
257Q8—44no—
258Q8—42no—
259Q8—42no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—41no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—82no—
269C2×C4—82no—
270C2×C4—2921no—
271C2×C4—2921no—
272C2×C4—2921no—
273C2×C4—2921no—
274C2×C4—2921no—
275C2×C4—5842no—
276C2×C4—5842no—
277C2×C4—5842no—
278C2×C4—1462no—
279C2×C4#1455656011yescharacteristic
280C2×C4#1455657212yes—
281C2×C4#1455657811yescharacteristic
282C8—41no—
283C8—41no—
284C8—2922no—
285C22—22no—
286C22—21no—
287C22—22no—
288C22—21no—
289C22—41no—
290C22—41no—
291C22—42no—
292C22#1798850111yescharacteristic
293C22#1798879512yes—
294C22#1798881112yes—
295C22#1798882511yescharacteristic
296C22#1798884211yescharacteristic
297C4—22no—
298C4—21no—
299C4—82no—
300C4—82no—
301C4—2922no—
302C4—5841no—
303C4—11682no—
304C4#1798877411yescharacteristic
305C4#1798877711yescharacteristic
306C2—21no—
307C2—21no—
308C2—22no—
309C2—22no—
310C2#2261117911yescharacteristic
311C2#2261123411yescharacteristic
312C2#2261125111yescharacteristic
subgroupG/H
centerC22#17988501
commutator, cosocleC23×C4×C73C23
FrattiniC23×C4C22×D73
FittingC73×#3189C2
socleC22×C73#394
prime radicalC24×C73C2×D4
hypercenter#3189D73
hypocenterC73#10566
LCSG ▷ C23×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #3189

Complements

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

NH ≅ G/Ntype
65. C22⋊Dic73Q16semidirect
162. C73#10566semidirect

Maximal subgroups

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

subgroupindex
#226109172
#226103552
#226101082
#226077992
C73×#31892
#1056673