Finite Group Databasebeta

← #32983863

Subgroups — #32983863

Order 18688.

544 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 544 conjugacy classes of subgroups — too many to draw. Only its 107 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×#2873C211yesmaximal, 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—
20C22×M16×C73—21no—
21C2×#117×C73C2211yescharacteristic
22#405×C73—24no—
23C4×M16×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—
48C2×M16×C73—24no—
49C2×M16×C73—24no—
50C2×M16×C73—22no—
51C22×C8×C73—22no—
52#117×C73—22no—
53#117×C73—22no—
54C2×C4⋊C4×C73C2×C411yescharacteristic
55C2×C4⋊C4×C73D411yescharacteristic
56C2×C22⋊C4×C73C2×C411yescharacteristic
57C2×C22⋊C4×C73D411yescharacteristic
58C2×C42×C73D411yescharacteristic
59C2×C42×C73C2311yescharacteristic
60#97×C73—22no—
61#97×C73—22no—
62C4×C8×C73—22no—
63C4×C8×C73D414yes—
64C2×Dic146—22no—
65C2×Dic146—42no—
66C22×Dic73—21no—
67C22×Dic73—21no—
68C22×Dic73—21no—
69C22×Dic73—21no—
70C2×#10596831—42no—
71C4⋊Dic73—22no—
72#11924007—42no—
73#11924007—42no—
74#11924007—41no—
75#11924007—41no—
76C22⋊Dic73—42no—
77C22⋊Dic73—42no—
78C4×Dic73—42no—
79C24×C73C22⋊C411yescharacteristic
80C22×C4×C73—22no—
81C22×C4×C73—22no—
82C22×C4×C73—21no—
83C22×C4×C73—21no—
84C22×C4×C73C22⋊C411yescharacteristic
85C22×C4×C73C22⋊C411yescharacteristic
86C22×C4×C73C22⋊C411yescharacteristic
87C22×C4×C73C22×C411yescharacteristic
88C22×C4×C73C2×D411yescharacteristic
89C22×C4×C73C2×D411yescharacteristic
90M16×C73—28no—
91M16×C73—24no—
92M16×C73—24no—
93C2×C8×C73—28no—
94C2×C8×C73—24no—
95C4⋊C4×C73—22no—
96C4⋊C4×C73—22no—
97C22⋊C4×C73—22no—
98C22⋊C4×C73—22no—
99C42×C73—21no—
100C42×C73—21no—
101C42×C73C2×D412yes—
102C42×C73#4112yes—
103C23×C73—21no—
104C23×C73—21no—
105C23×C73—41no—
106C23×C73—41no—
107C23×C73#10412yes—
108C23×C73C2×C22⋊C411yescharacteristic
109C2×C4×C73—22no—
110C2×C4×C73—22no—
111C2×C4×C73—24no—
112C2×C4×C73—21no—
113C2×C4×C73—21no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—21no—
117C2×C4×C73—21no—
118C2×C4×C73—41no—
119C2×C4×C73—41no—
120C2×C4×C73—41no—
121C2×C4×C73—41no—
122C2×C4×C73#10412yes—
123C2×C4×C73C4×D412yes—
124C2×C4×C73#12011yescharacteristic
125C2×C4×C73#12112yes—
126C2×C4×C73#12311yescharacteristic
127C2×Dic73—22no—
128C2×Dic73—41no—
129C2×Dic73—41no—
130C2×Dic73—41no—
131C2×Dic73—41no—
132C2×Dic73—41no—
133C2×Dic73—41no—
134C2×Dic73—41no—
135C2×Dic73—41no—
136C2×Dic73—41no—
137C2×Dic73—41no—
138C2×Dic73—41no—
139Dic146—24no—
140Dic146—42no—
141C8×C73—28no—
142#10596831—82no—
143C22×C73—21no—
144C22×C73—21no—
145C22×C73—22no—
146C22×C73—42no—
147C22×C73—41no—
148C22×C73—41no—
149C22×C73—81no—
150C22×C73#38611yescharacteristic
151C22×C73C2×#10411yescharacteristic
152C22×C73C2×#10411yescharacteristic
153C4×C73—22no—
154C4×C73—21no—
155C4×C73—22no—
156C4×C73—21no—
157C4×C73—22no—
158C4×C73—41no—
159C4×C73—41no—
160Dic73—42no—
161Dic73—81no—
162Dic73—81no—
163Dic73—81no—
164#9491—731nomaximal, Sylow
165C2×C73—21no—
166C2×C73—21no—
167C2×C73—42no—
168C2×C73#189111yescharacteristic
169C2×C73#189611yescharacteristic
170C2×C73#190911yescharacteristic
171#2873D7311yescharacteristic
172#2599—731no—
173#1491—732no—
174#1464—731no—
175#1288—732no—
176C73#949111yescharacteristic, complemented, Sylow
177C22×M16—21no—
178C2×C4×Q8—731no—
179C2×#117C2×D7311yescharacteristic
180#432—732no—
181#405—24no—
182C4×M16—21no—
183C4×M16C2×D7312yes—
184#398—731no—
185#397—731no—
186#395—731no—
187#393—731no—
188#388—1461no—
189#387—1461no—
190#387—1461no—
191#385—1461no—
192#383—731no—
193Q8⋊C8—1464no—
194#324—1461no—
195#323—1461no—
196C22×Q8—731no—
197C23×C4C22⋊D7311yescharacteristic
198C2×M16—24no—
199C2×M16—24no—
200C2×M16—22no—
201C2×M16—22no—
202C22×C8—22no—
203C4×Q8—1462no—
204C4×Q8—1462no—
205#117—22no—
206#117—22no—
207C2×C4⋊C4—731no—
208C2×C4⋊C4—1461no—
209C2×C4⋊C4—1461no—
210C2×C4⋊C4—1461no—
211C2×C4⋊C4C4×D7311yescharacteristic
212C2×C4⋊C4D29211yescharacteristic
213C2×C22⋊C4—1461no—
214C2×C22⋊C4—1461no—
215C2×C22⋊C4C4×D7311yescharacteristic
216C2×C22⋊C4D29211yescharacteristic
217C2×C42—1461no—
218C2×C42C22⋊D7311yescharacteristic
219C2×C42C22×D7311yescharacteristic
220C4⋊C8—1462no—
221C22⋊C8—1462no—
222#97—22no—
223#97—22no—
224#97—1462no—
225C4×C8—22no—
226C4×C8C22⋊D7314yes—
227#95—731no—
228#95—731no—
229#95—1461no—
230#95—1461no—
231#95—1461no—
232#95—1461no—
233#95—1461no—
234#95—1461no—
235C24#1192400511yescharacteristic
236C2×Q8—2922no—
237C2×Q8—1462no—
238C22×C4—22no—
239C22×C4—22no—
240C22×C4—21no—
241C22×C4—21no—
242C22×C4—1461no—
243C22×C4—1461no—
244C22×C4—1461no—
245C22×C4—1461no—
246C22×C4#1192400511yescharacteristic
247C22×C4#1192400511yescharacteristic
248C22×C4#1192400511yescharacteristic
249C22×C4C2×C4×D7311yescharacteristic
250C22×C4C2×C22⋊D7311yescharacteristic
251C22×C4C2×D29211yescharacteristic
252M16—28no—
253M16—24no—
254M16—24no—
255C2×C8—28no—
256C2×C8—24no—
257C2×C8—2922no—
258C4⋊C4—22no—
259C4⋊C4—22no—
260C4⋊C4—2922no—
261C4⋊C4—2922no—
262C4⋊C4—2921no—
263C4⋊C4—2921no—
264C4⋊C4—1462no—
265C22⋊C4—22no—
266C22⋊C4—22no—
267C22⋊C4—2922no—
268C22⋊C4—2922no—
269C42—21no—
270C42—21no—
271C42—2922no—
272C42C2×C22⋊D7312yes—
273C42#1192402812yes—
274C23—21no—
275C23—21no—
276C23—41no—
277C23—41no—
278C23#1455651912yes—
279C23C2×#1192400511yescharacteristic
280Q8—2922no—
281Q8—1464no—
282C2×C4—22no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—21no—
288C2×C4—21no—
289C2×C4—21no—
290C2×C4—21no—
291C2×C4—41no—
292C2×C4—41no—
293C2×C4—41no—
294C2×C4—41no—
295C2×C4—2921no—
296C2×C4—2921no—
297C2×C4—2921no—
298C2×C4—2921no—
299C2×C4—2921no—
300C2×C4—2921no—
301C2×C4—2921no—
302C2×C4—2921no—
303C2×C4—2921no—
304C2×C4—2921no—
305C2×C4—2921no—
306C2×C4—1462no—
307C2×C4#1455651912yes—
308C2×C4C4×C22⋊D7312yes—
309C2×C4#1455656011yescharacteristic
310C2×C4#1455656512yes—
311C2×C4#1455657411yescharacteristic
312C8—28no—
313C8—5842no—
314C22—21no—
315C22—21no—
316C22—22no—
317C22—42no—
318C22—41no—
319C22—41no—
320C22—81no—
321C22#1798846611yescharacteristic
322C22C2×#1455651911yescharacteristic
323C22C2×#1455651911yescharacteristic
324C4—22no—
325C4—22no—
326C4—21no—
327C4—22no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—2922no—
332C4—5841no—
333C4—5841no—
334C4—5841no—
335C2—21no—
336C2—21no—
337C2—42no—
338C2#2261034911yescharacteristic
339C2#2261037411yescharacteristic
340C2#2261045311yescharacteristic
subgroupG/H
centerC22#17988466
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#2873C2
socleC22×C73#386
prime radicalC24×C73C22⋊C4
cosocleC2×C42×C73C23
hypercenter#2873D73
hypocenterC73#9491
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2873

Complements

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

NH ≅ G/Ntype
2. #22608379C2semidirect
4. #22607796C2semidirect
176. C73#9491semidirect

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×#28732
#949173