Finite Group Databasebeta

← #32983324

Subgroups — #32983324

Order 18688.

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

normal subgroups only

This group has 532 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#22615629C211yesmaximal, characteristic
2#22608532C212yesmaximal, complemented
3#22608265C211yesmaximal, characteristic
4#22607825C212yesmaximal, complemented
5C73×#2873C211yesmaximal, characteristic
6C2×#14556583C2211yescharacteristic
7#17988644C2212yes—
8#17988526C411yescharacteristic
9#17988523C411yescharacteristic
10#17988517C411yescharacteristic
11#17988512C411yescharacteristic
12#17988482—21no—
13#17988481—21no—
14#17988471—21no—
15#17988460—21no—
16#17988433C2211yescharacteristic
17#17988328—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—
25#14556583—22no—
26#14556583—22no—
27C2×C4⋊Dic73C2×C411yescharacteristic
28C2×C4⋊Dic73C2×C411yescharacteristic
29C2×#11924007—21no—
30C2×#11924007—21no—
31C2×#11924007—21no—
32C2×#11924007—21no—
33C2×C22⋊Dic73—21no—
34C2×C22⋊Dic73—21no—
35C2×C4×Dic73—21no—
36C4⋊#10596831—22no—
37C22⋊#10596831—22no—
38#14556502—22no—
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×C73D414yes—
62C4×C8×C73—22no—
63C4×C8×C73—22no—
64C22×Dic73—21no—
65C22×Dic73—21no—
66C22×Dic73—21no—
67C22×Dic73—21no—
68C2×#10596831—42no—
69C4⋊Dic73—22no—
70C4⋊Dic73—22no—
71#11924007—42no—
72#11924007—42no—
73#11924007—41no—
74#11924007—42no—
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—
139C8×C73—28no—
140#10596831—82no—
141C22×C73—21no—
142C22×C73—21no—
143C22×C73—22no—
144C22×C73—42no—
145C22×C73—41no—
146C22×C73—41no—
147C22×C73—81no—
148C22×C73#38611yescharacteristic
149C22×C73C2×#10411yescharacteristic
150C22×C73C2×#10411yescharacteristic
151C4×C73—22no—
152C4×C73—21no—
153C4×C73—22no—
154C4×C73—21no—
155C4×C73—22no—
156C4×C73—41no—
157C4×C73—41no—
158Dic73—42no—
159Dic73—81no—
160Dic73—81no—
161Dic73—81no—
162#9388—731nomaximal, Sylow
163C2×C73—21no—
164C2×C73—21no—
165C2×C73—42no—
166C2×C73#188411yescharacteristic
167C2×C73#189611yescharacteristic
168C2×C73#191511yescharacteristic
169#2873D7311yescharacteristic
170#2611—731no—
171#1528—732no—
172#1464—731no—
173#1300—732no—
174C73#938811yescharacteristic, complemented, Sylow
175C22×M16—21no—
176C2×#125—731no—
177C2×#117C2×D7311yescharacteristic
178#432—732no—
179#405—24no—
180C4×M16—21no—
181C4×M16C2×D7312yes—
182#400—731no—
183#399—731no—
184#398—731no—
185#397—731no—
186#389—1461no—
187#388—1461no—
188#387—1461no—
189#385—1461no—
190#376—731no—
191#330—1464no—
192#324—1461no—
193#323—1461no—
194C23×C4C22⋊D7311yescharacteristic
195C2×M16—24no—
196C2×M16—24no—
197C2×M16—22no—
198C2×M16—22no—
199C22×C8—22no—
200#125—1462no—
201#125—1462no—
202#117—22no—
203#117—22no—
204C2×C4⋊C4—731no—
205C2×C4⋊C4—731no—
206C2×C4⋊C4—1461no—
207C2×C4⋊C4—1461no—
208C2×C4⋊C4—1461no—
209C2×C4⋊C4—1461no—
210C2×C4⋊C4C4×D7311yescharacteristic
211C2×C4⋊C4D29211yescharacteristic
212C2×C22⋊C4—1461no—
213C2×C22⋊C4—1461no—
214C2×C22⋊C4C4×D7311yescharacteristic
215C2×C22⋊C4D29211yescharacteristic
216C2×C42—1461no—
217C2×C42C22⋊D7311yescharacteristic
218C2×C42C22×D7311yescharacteristic
219C4⋊C8—1462no—
220C22⋊C8—1462no—
221#97—22no—
222#97—1462no—
223#97C22⋊D7314yes—
224C4×C8—22no—
225C4×C8—22no—
226#95—731no—
227#95—731no—
228#95—1461no—
229#95—1461no—
230#95—1461no—
231#95—1461no—
232#95—1461no—
233C24#1192400511yescharacteristic
234C22×C4—22no—
235C22×C4—22no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—1461no—
239C22×C4—1461no—
240C22×C4—1461no—
241C22×C4—1461no—
242C22×C4#1192400511yescharacteristic
243C22×C4#1192400511yescharacteristic
244C22×C4#1192400511yescharacteristic
245C22×C4C2×C4×D7311yescharacteristic
246C22×C4C2×C22⋊D7311yescharacteristic
247C22×C4C2×D29211yescharacteristic
248M16—28no—
249M16—24no—
250M16—24no—
251C2×C8—28no—
252C2×C8—24no—
253C2×C8—2922no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—2922no—
257C4⋊C4—2922no—
258C4⋊C4—2921no—
259C4⋊C4—2922no—
260C4⋊C4—2921no—
261C4⋊C4—1462no—
262C4⋊C4—1462no—
263C22⋊C4—22no—
264C22⋊C4—22no—
265C22⋊C4—2922no—
266C22⋊C4—2922no—
267C42—21no—
268C42—21no—
269C42—2922no—
270C42C2×C22⋊D7312yes—
271C42#1192402812yes—
272C23—21no—
273C23—21no—
274C23—41no—
275C23—41no—
276C23#1455651912yes—
277C23C2×#1192400511yescharacteristic
278C2×C4—22no—
279C2×C4—22no—
280C2×C4—24no—
281C2×C4—21no—
282C2×C4—21no—
283C2×C4—21no—
284C2×C4—21no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—41no—
291C2×C4—2921no—
292C2×C4—2921no—
293C2×C4—2921no—
294C2×C4—2921no—
295C2×C4—2921no—
296C2×C4—2921no—
297C2×C4—2921no—
298C2×C4—2921no—
299C2×C4—2921no—
300C2×C4—2921no—
301C2×C4—2921no—
302C2×C4—1462no—
303C2×C4#1455651912yes—
304C2×C4C4×C22⋊D7312yes—
305C2×C4#1455656011yescharacteristic
306C2×C4#1455656512yes—
307C2×C4#1455657411yescharacteristic
308C8—28no—
309C8—5842no—
310C22—21no—
311C22—21no—
312C22—22no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22#1798846611yescharacteristic
318C22C2×#1455651911yescharacteristic
319C22C2×#1455651911yescharacteristic
320C4—22no—
321C4—22no—
322C4—21no—
323C4—22no—
324C4—21no—
325C4—41no—
326C4—41no—
327C4—2922no—
328C4—5841no—
329C4—5841no—
330C4—5841no—
331C2—21no—
332C2—21no—
333C2—42no—
334C2#2261031411yescharacteristic
335C2#2261037411yescharacteristic
336C2#2261048311yescharacteristic
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#9388
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. #22608532C2semidirect
4. #22607825C2semidirect
174. C73#9388semidirect

Maximal subgroups

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

subgroupindex
#226156292
#226085322
#226082652
#226078252
C73×#28732
#938873