Finite Group Databasebeta

← #32990382

Subgroups — #32990382

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#22620706C211yesmaximal, characteristic
2#22610916C212yesmaximal
3#22610354C212yesmaximal
4#22607799C211yesmaximal, characteristic
5C73×#1850C211yesmaximal, characteristic
6C2×#14556570C2212yes—
7#17988923—22no—
8#17988828—21no—
9#17988828—21no—
10#17988745—22no—
11C2×#14556516—22no—
12#17988447C2212yes—
13#17988353—22no—
14#17988353—22no—
15#17988350—22no—
16C23×C8×C73C2211yescharacteristic
17C2×C4⋊C8×C73—21no—
18C2×C22⋊C8×C73C2211yescharacteristic
19C4×C22⋊C4×C73C2211yescharacteristic
20#336×C73—21no—
21C22×Dic146D412yes—
22C2×Dic292—21no—
23C2×Dic292—21no—
24#14556570—22no—
25#14556570—22no—
26#14556570—22no—
27#14556570—22no—
28C2×C4⋊Dic73—22no—
29C2×C4⋊Dic73D412yes—
30C2×#11924007—22no—
31C2×C22⋊Dic73—22no—
32C2×C22⋊Dic73—22no—
33#14556525—21no—
34#14556525—21no—
35#14556516—22no—
36#14556516—22no—
37#14556516—44no—
38#14556498—22no—
39C23×C4×C73C2311yescharacteristic
40C22×C8×C73—21no—
41C22×C8×C73—22no—
42C22×C8×C73—42no—
43C22×C8×C73D411yescharacteristic
44C22×C8×C73D411yescharacteristic
45C2×C22⋊C4×C73D411yescharacteristic
46C2×C22⋊C4×C73D411yescharacteristic
47C2×C42×C73—21no—
48C4⋊C8×C73—42no—
49C22⋊C8×C73—22no—
50#95×C73—21no—
51C2×Dic146—22no—
52C2×Dic146—22no—
53C2×Dic146—42no—
54C2×Dic146—42no—
55C22×Dic73—22no—
56C22×Dic73—22no—
57C22×Dic73—42no—
58Dic292—81no—
59Dic292—81no—
60C4⋊Dic73—22no—
61C4⋊Dic73—22no—
62C4⋊Dic73—42no—
63C4⋊Dic73—42no—
64#11924007—42no—
65#11924007—42no—
66C22⋊Dic73—42no—
67C22⋊Dic73—42no—
68C22⋊Dic73—42no—
69C22⋊Dic73—42no—
70C24×C73C2×D411yescharacteristic
71C22×C4×C73—21no—
72C22×C4×C73—21no—
73C22×C4×C73—41no—
74C22×C4×C73—41no—
75C22×C4×C73C2×D412yes—
76C22×C4×C73C2×D411yescharacteristic
77C22×C4×C73#4111yescharacteristic
78C22×C4×C73#4112yes—
79C2×C8×C73—22no—
80C2×C8×C73—24no—
81C2×C8×C73—21no—
82C2×C8×C73—21no—
83C2×C8×C73—41no—
84C2×C8×C73—42no—
85C2×C8×C73—41no—
86C2×C8×C73—41no—
87C2×C8×C73—41no—
88C22⋊C4×C73—21no—
89C22⋊C4×C73—22no—
90C22⋊C4×C73Q1612yescomplemented
91C42×C73—41no—
92C42×C73—41no—
93C23×C73—41no—
94C23×C73—41no—
95C23×C73#12111yescharacteristic
96C23×C73#12112yes—
97C23×C73C2×Q1611yescharacteristic
98C23×C73#13511yescharacteristic
99C23×C73#13712yes—
100C2×C4×C73—21no—
101C2×C4×C73—22no—
102C2×C4×C73—21no—
103C2×C4×C73—22no—
104C2×C4×C73—22no—
105C2×C4×C73—41no—
106C2×C4×C73—42no—
107C2×C4×C73—41no—
108C2×C4×C73—41no—
109C2×C4×C73—41no—
110C2×C4×C73—41no—
111C2×C4×C73—41no—
112C2×C4×C73#12011yescharacteristic
113C2×C4×C73#12312yes—
114C2×C4×C73#12411yescharacteristic
115C2×Dic73—42no—
116C2×Dic73—42no—
117C2×Dic73—82no—
118C2×Dic73—82no—
119C2×Dic73—82no—
120C2×Dic73—82no—
121C2×Dic73—82no—
122Dic146—44no—
123Dic146—42no—
124Dic146—42no—
125C8×C73—42no—
126C8×C73—41no—
127C8×C73—41no—
128C22×C73—22no—
129C22×C73—21no—
130C22×C73—22no—
131C22×C73—21no—
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—42no—
143C4×C73—81no—
144C4×C73B(4,𝔽2)11yescharacteristic
145C4×C73#45811yescharacteristic
146Dic73—82no—
147Dic73—82no—
148Dic73—162no—
149#10566—731nomaximal, Sylow
150C2×C73—22no—
151C2×C73—21no—
152C2×C73—21no—
153C2×C73—22no—
154C2×C73#204411yescharacteristic
155C2×C73#205311yescharacteristic
156C2×C73#205611yescharacteristic
157#3189—731no—
158#1996—732no—
159#1892—732no—
160#1850D7311yescharacteristic
161#1289—731no—
162C73#1056611yescharacteristic, complemented, Sylow
163C23×C8C2×D7311yescharacteristic
164C2×C22⋊Q8—732no—
165#484—1462no—
166#467—1461no—
167#467—1461no—
168#451—1462no—
169C2×C4⋊C8—21no—
170C2×#103—1462no—
171C2×C22⋊C8C2×D7311yescharacteristic
172#380—732no—
173C4×C22⋊C4C2×D7311yescharacteristic
174#342—1462no—
175#342—1462no—
176#340—1462no—
177#336—21no—
178C22×Q8—732no—
179C23×C4C22×D7311yescharacteristic
180C2×Q16—1461no—
181C2×Q16—1461no—
182C22×C8—22no—
183C22×C8—21no—
184C22×C8—42no—
185C22×C8C22⋊D7311yescharacteristic
186C22×C8C22⋊D7311yescharacteristic
187C22⋊Q8—1462no—
188C22⋊Q8—1462no—
189C22⋊Q8—1462no—
190C22⋊Q8—1462no—
191C2×C4⋊C4—732no—
192C2×C4⋊C4—1462no—
193C2×C4⋊C4—1462no—
194C2×C22⋊C4—1462no—
195C2×C22⋊C4—1462no—
196C2×C22⋊C4D29211yescharacteristic
197C2×C22⋊C4D29211yescharacteristic
198C2×C42—21no—
199#107—1461no—
200#107—1461no—
201C4⋊C8—42no—
202#103—2924no—
203#103—1462no—
204#103—1462no—
205C22⋊C8—22no—
206#95—21no—
207#95—1462no—
208C24C2×D29211yescharacteristic
209C2×Q8—2922no—
210C2×Q8—2922no—
211C2×Q8—1462no—
212C2×Q8—1462no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—41no—
216C22×C4—41no—
217C22×C4—2922no—
218C22×C4—1462no—
219C22×C4—1462no—
220C22×C4D4×D7312yes—
221C22×C4C2×C22⋊D7311yescharacteristic
222C22×C4#1192402811yescharacteristic
223C22×C4#1192402912yes—
224Q16—5841no—
225Q16—5841no—
226C2×C8—24no—
227C2×C8—22no—
228C2×C8—21no—
229C2×C8—21no—
230C2×C8—41no—
231C2×C8—42no—
232C2×C8—41no—
233C2×C8—41no—
234C2×C8—41no—
235C4⋊C4—2922no—
236C4⋊C4—2922no—
237C4⋊C4—2922no—
238C4⋊C4—2922no—
239C4⋊C4—1462no—
240C4⋊C4—1462no—
241C22⋊C4—21no—
242C22⋊C4—22no—
243C22⋊C4—2922no—
244C22⋊C4—2922no—
245C22⋊C4—2922no—
246C22⋊C4—2922no—
247C22⋊C4Dic29212yescomplemented
248C42—41no—
249C42—41no—
250C23—41no—
251C23—41no—
252C23#1455656511yescharacteristic
253C23#1455656612yes—
254C23C2×Dic29211yescharacteristic
255C23#1455661311yescharacteristic
256C23#1455662412yes—
257Q8—2924no—
258Q8—2922no—
259Q8—2922no—
260C2×C4—21no—
261C2×C4—22no—
262C2×C4—21no—
263C2×C4—22no—
264C2×C4—22no—
265C2×C4—41no—
266C2×C4—42no—
267C2×C4—41no—
268C2×C4—41no—
269C2×C4—41no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—2922no—
273C2×C4—2922no—
274C2×C4—5842no—
275C2×C4—5842no—
276C2×C4—5842no—
277C2×C4—5842no—
278C2×C4—5842no—
279C2×C4#1455655911yescharacteristic
280C2×C4#1455657612yes—
281C2×C4#1455658011yescharacteristic
282C8—42no—
283C8—41no—
284C8—41no—
285C22—21no—
286C22—22no—
287C22—21no—
288C22—22no—
289C22—41no—
290C22—41no—
291C22—42no—
292C22#1798850411yescharacteristic
293C22#1798879812yes—
294C22#1798881012yes—
295C22#1798882811yescharacteristic
296C22#1798884511yescharacteristic
297C4—22no—
298C4—21no—
299C4—42no—
300C4—81no—
301C4—5842no—
302C4—5842no—
303C4—11682no—
304C4#1798877311yescharacteristic
305C4#1798877611yescharacteristic
306C2—21no—
307C2—21no—
308C2—22no—
309C2—22no—
310C2#2261118211yescharacteristic
311C2#2261123711yescharacteristic
312C2#2261125411yescharacteristic
subgroupG/H
centerC22#17988504
commutator, cosocleC23×C4×C73C23
FrattiniC23×C4C22×D73
FittingC73×#1850C2
socleC22×C73#394
prime radicalC24×C73C2×D4
hypercenter#1850D73
hypocenterC73#10566
LCSG ▷ C23×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ #1850

Complements

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

NH ≅ G/Ntype
90. C22⋊C4×C73Q16semidirect
162. C73#10566semidirect
247. C22⋊C4Dic292semidirect

Maximal subgroups

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

subgroupindex
#226207062
#226109162
#226103542
#226077992
C73×#18502
#1056673