Finite Group Databasebeta

← #32983873

Subgroups — #32983873

Order 18688.

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

normal subgroups only

This group has 458 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#22615508C211yesmaximal, characteristic
2#22608379C212yesmaximal, complemented
3#22608265C211yesmaximal, characteristic
4#22607796C212yesmaximal, complemented
5C73×#2921C211yesmaximal, characteristic
6C2×C4×Dic146C2211yescharacteristic
7#17988644C2212yes—
8#17988517C411yescharacteristic
9#17988512C411yescharacteristic
10#17988507C411yescharacteristic
11#17988499C411yescharacteristic
12#17988475—21no—
13#17988471—21no—
14#17988468—21no—
15#17988460—21no—
16#17988455C2211yescharacteristic
17#17988319—24no—
18#17988313—21no—
19#17988311—21no—
20C2×#117×C73C2211yescharacteristic
21#433×C73—24no—
22#423×C73—21no—
23#407×C73—21no—
24C4×M16×C73C2212yes—
25C22×Dic146C2×C411yescharacteristic
26C4×Dic146—22no—
27C4×Dic146—22no—
28C2×C4⋊Dic73C2×C411yescharacteristic
29C2×#11924007—21no—
30C2×#11924007—21no—
31C2×#11924007—21no—
32C2×C22⋊Dic73—21no—
33C2×C22⋊Dic73—21no—
34C2×C4×Dic73—21no—
35C4⋊#10596831—22no—
36C22⋊#10596831—22no—
37#14556502—22no—
38#14556498—21no—
39#14556498—21no—
40#14556498—21no—
41#14556498—21no—
42#14556498—21no—
43#14556498—21no—
44#14556498C2×C411yescharacteristic
45#14556498C2×C411yescharacteristic
46C23×C4×C73D411yescharacteristic
47C2×M16×C73—22no—
48#117×C73—22no—
49#117×C73—22no—
50C2×C4⋊C4×C73C2×C411yescharacteristic
51C2×C4⋊C4×C73D411yescharacteristic
52C2×C22⋊C4×C73C2×C411yescharacteristic
53C2×C22⋊C4×C73D411yescharacteristic
54C2×C42×C73D411yescharacteristic
55C2×C42×C73C2311yescharacteristic
56C4⋊C8×C73—24no—
57C22⋊C8×C73—24no—
58#97×C73—22no—
59C4×C8×C73D414yes—
60C2×Dic146—22no—
61C2×Dic146—42no—
62C22×Dic73—21no—
63C22×Dic73—21no—
64C22×Dic73—21no—
65C22×Dic73—21no—
66C2×#10596831—42no—
67C4⋊Dic73—22no—
68#11924007—42no—
69#11924007—42no—
70#11924007—41no—
71#11924007—41no—
72C22⋊Dic73—42no—
73C22⋊Dic73—42no—
74C4×Dic73—42no—
75C24×C73C22⋊C411yescharacteristic
76C22×C4×C73—21no—
77C22×C4×C73—21no—
78C22×C4×C73—41no—
79C22×C4×C73—41no—
80C22×C4×C73C22⋊C411yescharacteristic
81C22×C4×C73C22⋊C411yescharacteristic
82C22×C4×C73C22⋊C411yescharacteristic
83C22×C4×C73C22×C411yescharacteristic
84C22×C4×C73C2×D411yescharacteristic
85C22×C4×C73C2×D411yescharacteristic
86M16×C73—42no—
87M16×C73—42no—
88C2×C8×C73—24no—
89C4⋊C4×C73—22no—
90C4⋊C4×C73—22no—
91C22⋊C4×C73—22no—
92C22⋊C4×C73—22no—
93C42×C73—21no—
94C42×C73—21no—
95C42×C73C2×D412yes—
96C42×C73#4112yes—
97C23×C73—21no—
98C23×C73—21no—
99C23×C73—21no—
100C23×C73—41no—
101C23×C73—41no—
102C23×C73C2×C22⋊C411yescharacteristic
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—21no—
106C2×C4×C73—21no—
107C2×C4×C73—21no—
108C2×C4×C73—21no—
109C2×C4×C73—21no—
110C2×C4×C73—41no—
111C2×C4×C73—41no—
112C2×C4×C73—42no—
113C2×C4×C73—42no—
114C2×C4×C73—41no—
115C2×C4×C73—41no—
116C2×C4×C73C4×D412yes—
117C2×C4×C73#12011yescharacteristic
118C2×C4×C73#12112yes—
119C2×C4×C73#12311yescharacteristic
120C2×Dic73—22no—
121C2×Dic73—41no—
122C2×Dic73—41no—
123C2×Dic73—41no—
124C2×Dic73—41no—
125C2×Dic73—41no—
126C2×Dic73—41no—
127C2×Dic73—41no—
128C2×Dic73—41no—
129C2×Dic73—41no—
130C2×Dic73—41no—
131C2×Dic73—41no—
132Dic146—24no—
133Dic146—42no—
134C8×C73—44no—
135#10596831—82no—
136C22×C73—21no—
137C22×C73—21no—
138C22×C73—42no—
139C22×C73—41no—
140C22×C73—41no—
141C22×C73—41no—
142C22×C73—81no—
143C22×C73#38611yescharacteristic
144C22×C73#42111yescharacteristic
145C22×C73#42111yescharacteristic
146C4×C73—22no—
147C4×C73—21no—
148C4×C73—21no—
149C4×C73—22no—
150C4×C73—41no—
151C4×C73—41no—
152C4×C73—41no—
153Dic73—42no—
154Dic73—81no—
155Dic73—81no—
156Dic73—81no—
157#9493—731nomaximal, Sylow
158C2×C73—21no—
159C2×C73—21no—
160C2×C73—81no—
161C2×C73#189111yescharacteristic
162C2×C73#189711yescharacteristic
163C2×C73#190911yescharacteristic
164#2921D7311yescharacteristic
165#2599—731no—
166#1491—732no—
167#1464—731no—
168#1288—732no—
169C73#949311yescharacteristic, complemented, Sylow
170C2×C4×Q8—731no—
171C2×#117C2×D7311yescharacteristic
172#433—24no—
173#432—732no—
174#423—21no—
175#407—21no—
176C4×M16C2×D7312yes—
177#398—731no—
178#397—731no—
179#395—731no—
180#393—731no—
181#388—1461no—
182#387—1461no—
183#387—1461no—
184#385—1461no—
185#383—731no—
186Q8⋊C8—1464no—
187#324—1461no—
188#323—1461no—
189C22×Q8—731no—
190C23×C4C22⋊D7311yescharacteristic
191C2×M16—22no—
192C4×Q8—1462no—
193C4×Q8—1462no—
194#117—22no—
195#117—22no—
196C2×C4⋊C4—731no—
197C2×C4⋊C4—1461no—
198C2×C4⋊C4—1461no—
199C2×C4⋊C4—1461no—
200C2×C4⋊C4C4×D7311yescharacteristic
201C2×C4⋊C4D29211yescharacteristic
202C2×C22⋊C4—1461no—
203C2×C22⋊C4—1461no—
204C2×C22⋊C4C4×D7311yescharacteristic
205C2×C22⋊C4D29211yescharacteristic
206C2×C42—1461no—
207C2×C42C22⋊D7311yescharacteristic
208C2×C42C22×D7311yescharacteristic
209C4⋊C8—24no—
210C4⋊C8—1462no—
211C22⋊C8—24no—
212C22⋊C8—1462no—
213#97—22no—
214#97—1462no—
215C4×C8C22⋊D7314yes—
216#95—731no—
217#95—731no—
218#95—1461no—
219#95—1461no—
220#95—1461no—
221#95—1461no—
222#95—1461no—
223#95—1461no—
224C24#1192400511yescharacteristic
225C2×Q8—2922no—
226C2×Q8—1462no—
227C22×C4—21no—
228C22×C4—21no—
229C22×C4—41no—
230C22×C4—41no—
231C22×C4—1461no—
232C22×C4—1461no—
233C22×C4—1461no—
234C22×C4—1461no—
235C22×C4#1192400511yescharacteristic
236C22×C4#1192400511yescharacteristic
237C22×C4#1192400511yescharacteristic
238C22×C4C2×C4×D7311yescharacteristic
239C22×C4C2×C22⋊D7311yescharacteristic
240C22×C4C2×D29211yescharacteristic
241M16—42no—
242M16—42no—
243C2×C8—24no—
244C2×C8—2922no—
245C4⋊C4—22no—
246C4⋊C4—22no—
247C4⋊C4—2922no—
248C4⋊C4—2922no—
249C4⋊C4—2921no—
250C4⋊C4—2921no—
251C4⋊C4—1462no—
252C22⋊C4—22no—
253C22⋊C4—22no—
254C22⋊C4—2922no—
255C22⋊C4—2922no—
256C42—21no—
257C42—21no—
258C42—2922no—
259C42C2×C22⋊D7312yes—
260C42#1192402812yes—
261C23—21no—
262C23—21no—
263C23—21no—
264C23—41no—
265C23—41no—
266C23C2×#1192400511yescharacteristic
267Q8—2922no—
268Q8—1464no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—2921no—
283C2×C4—2921no—
284C2×C4—2921no—
285C2×C4—2921no—
286C2×C4—2921no—
287C2×C4—2921no—
288C2×C4—2921no—
289C2×C4—2921no—
290C2×C4—2921no—
291C2×C4—2921no—
292C2×C4—2921no—
293C2×C4—1462no—
294C2×C4C4×C22⋊D7312yes—
295C2×C4#1455656011yescharacteristic
296C2×C4#1455656512yes—
297C2×C4#1455657411yescharacteristic
298C8—44no—
299C8—5842no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—41no—
306C22—81no—
307C22#1798846611yescharacteristic
308C22#1798860511yescharacteristic
309C22#1798860511yescharacteristic
310C4—22no—
311C4—22no—
312C4—21no—
313C4—21no—
314C4—41no—
315C4—41no—
316C4—41no—
317C4—2922no—
318C4—5841no—
319C4—5841no—
320C4—5841no—
321C2—21no—
322C2—21no—
323C2—81no—
324C2#2261034911yescharacteristic
325C2#2261037911yescharacteristic
326C2#2261045311yescharacteristic
subgroupG/H
centerC22#17988466
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#2921C2
socleC22×C73#386
prime radicalC24×C73C22⋊C4
cosocleC2×C42×C73C23
hypercenter#2921D73
hypocenterC73#9493
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2921

Complements

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

NH ≅ G/Ntype
2. #22608379C2semidirect
4. #22607796C2semidirect
169. C73#9493semidirect

Maximal subgroups

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

subgroupindex
#226155082
#226083792
#226082652
#226077962
C73×#29212
#949373