Finite Group Databasebeta

← #32983664

Subgroups — #32983664

Order 18688.

518 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 518 conjugacy classes of subgroups — too many to draw. Only its 111 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#22618960C211yesmaximal, characteristic
2#22608657C212yesmaximal, complemented
3#22608272C211yesmaximal, characteristic
4#22607787C212yesmaximal, complemented
5C73×#2547C211yesmaximal, characteristic
6#17988659—24no—
7#17988644C2212yes—
8#17988644C2212yes—
9#17988551—21no—
10C2×C22⋊#10596831—21no—
11#17988336—24no—
12#17988311—21no—
13#17988311—21no—
14C2×C4⋊Q8×C73C2211yescharacteristic
15C2×#124×C73C412yes—
16C2×C4×D4×C73C2211yescharacteristic
17#397×C73—21no—
18#397×C73—21no—
19#391×C73C412yes—
20#380×C73—21no—
21#380×C73—21no—
22C4×C4⋊C4×C73C2211yescharacteristic
23C2×#11924012—22no—
24C22×#10596831—22no—
25C4⋊#10596831—22no—
26C4⋊#10596831D414yes—
27C22⋊#10596831—22no—
28C22⋊#10596831—22no—
29C22⋊#10596831—22no—
30C22⋊#10596831—22no—
31#14556502—22no—
32#14556502—22no—
33C22×Q8×C73C2×C412yes—
34C22×D4×C73C2×C411yescharacteristic
35C23×C4×C73D411yescharacteristic
36C23×C4×C73D411yescharacteristic
37C4⋊Q8×C73—22no—
38C4⋊Q8×C73—22no—
39#124×C73—44no—
40C4×D4×C73—22no—
41C4×D4×C73—22no—
42C2×C4⋊C4×C73—21no—
43C2×C4⋊C4×C73—21no—
44C2×C4⋊C4×C73—21no—
45C2×C4⋊C4×C73C2×C411yescharacteristic
46C2×C22⋊C4×C73—22no—
47C2×C22⋊C4×C73—22no—
48C2×C22⋊C4×C73D411yescharacteristic
49C2×C22⋊C4×C73D411yescharacteristic
50C2×C42×C73C2×C412yes—
51C2×C42×C73C2311yescharacteristic
52#95×C73—22no—
53#95×C73—22no—
54#95×C73—21no—
55#11924012—82no—
56C2×#10596831—24no—
57C2×#10596831—42no—
58C2×#10596831—82no—
59C24×C73C22⋊C411yescharacteristic
60C24×C73C22⋊C411yescharacteristic
61C2×Q8×C73—24no—
62C2×Q8×C73—42no—
63C2×Q8×C73—42no—
64C2×D4×C73—22no—
65C2×D4×C73—22no—
66C2×D4×C73—41no—
67C22×C4×C73—21no—
68C22×C4×C73—21no—
69C22×C4×C73—21no—
70C22×C4×C73—21no—
71C22×C4×C73—22no—
72C22×C4×C73—22no—
73C22×C4×C73—42no—
74C22×C4×C73—42no—
75C22×C4×C73C22⋊C411yescharacteristic
76C22×C4×C73C22⋊C411yescharacteristic
77C22×C4×C73C22×C411yescharacteristic
78C22×C4×C73C2×D411yescharacteristic
79C22×C4×C73C2×D411yescharacteristic
80C4⋊C4×C73—22no—
81C4⋊C4×C73—42no—
82C4⋊C4×C73—41no—
83C4⋊C4×C73—42no—
84C4⋊C4×C73—41no—
85C22⋊C4×C73—22no—
86C22⋊C4×C73—22no—
87C22⋊C4×C73—44no—
88C22⋊C4×C73—44no—
89C42×C73—42no—
90C42×C73C2×D412yes—
91C42×C73#4112yes—
92C23×C73—21no—
93C23×C73—21no—
94C23×C73—21no—
95C23×C73—21no—
96C23×C73—21no—
97C23×C73—41no—
98C23×C73—41no—
99C23×C73—41no—
100C23×C73—41no—
101C23×C73#10012yes—
102C23×C73C2×C22⋊C411yescharacteristic
103Q8×C73—84no—
104D4×C73—44no—
105C2×C4×C73—21no—
106C2×C4×C73—21no—
107C2×C4×C73—22no—
108C2×C4×C73—21no—
109C2×C4×C73—21no—
110C2×C4×C73—41no—
111C2×C4×C73—42no—
112C2×C4×C73—42no—
113C2×C4×C73—41no—
114C2×C4×C73—41no—
115C2×C4×C73—42no—
116C2×C4×C73—42no—
117C2×C4×C73—42no—
118C2×C4×C73—42no—
119C2×C4×C73—42no—
120C2×C4×C73—42no—
121C2×C4×C73—81no—
122C2×C4×C73—81no—
123C2×C4×C73C4×D412yes—
124C2×C4×C73#12011yescharacteristic
125C2×C4×C73#12112yes—
126C2×C4×C73#12311yescharacteristic
127#10596831—44no—
128#10596831—82no—
129C22×C73—21no—
130C22×C73—21no—
131C22×C73—22no—
132C22×C73—42no—
133C22×C73—42no—
134C22×C73—41no—
135C22×C73—41no—
136C22×C73—41no—
137C22×C73—41no—
138C22×C73—41no—
139C22×C73—81no—
140C22×C73—81no—
141C22×C73#38611yescharacteristic
142C22×C73C2×#10011yescharacteristic
143C22×C73#41311yescharacteristic
144C4×C73—21no—
145C4×C73—22no—
146C4×C73—22no—
147C4×C73—41no—
148C4×C73—41no—
149C4×C73—82no—
150C4×C73—82no—
151C4×C73#45512yes—
152#9451—731nomaximal, Sylow
153C2×C73—21no—
154C2×C73—21no—
155C2×C73—42no—
156C2×C73—81no—
157C2×C73#188511yescharacteristic
158C2×C73#190011yescharacteristic
159C2×C73#190911yescharacteristic
160#2973—731no—
161#2547D7311yescharacteristic
162#1558—732no—
163#1466—731no—
164#1285—732no—
165C73#945111yescharacteristic, complemented, Sylow
166C2×C4⋊Q8C2×D7311yescharacteristic
167C2×#124Dic7312yes—
168C2×C4×D4C2×D7311yescharacteristic
169#435—1464no—
170#432—732no—
171#432—732no—
172#407—1461no—
173C2×C22⋊C8—1461no—
174#397—21no—
175#397—21no—
176#391Dic7312yes—
177#380—21no—
178#380—21no—
179C4×C4⋊C4C2×D7311yescharacteristic
180#333—1464no—
181#323—1461no—
182#323—1461no—
183C22×Q8C2×Dic7312yes—
184C22×D4C2×Dic7311yescharacteristic
185C23×C4C22⋊D7311yescharacteristic
186C23×C4C22⋊D7311yescharacteristic
187C2×M16—1462no—
188C22×C8—1462no—
189C4⋊Q8—22no—
190C4⋊Q8—22no—
191#124—44no—
192C4×D4—22no—
193C4×D4—22no—
194C2×C4⋊C4—21no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4C2×Dic7311yescharacteristic
198C2×C22⋊C4—22no—
199C2×C22⋊C4—22no—
200C2×C22⋊C4C22⋊D7311yescharacteristic
201C2×C22⋊C4C22⋊D7311yescharacteristic
202C2×C42C2×Dic7312yes—
203C2×C42C22×D7311yescharacteristic
204C4⋊C8—734no—
205C4⋊C8—1462no—
206C22⋊C8—1462no—
207C22⋊C8—1462no—
208C22⋊C8—1462no—
209C22⋊C8—1462no—
210#97—1462no—
211#97—1462no—
212#95—22no—
213#95—22no—
214#95—21no—
215C24C22⋊Dic7311yescharacteristic
216C24C22⋊Dic7311yescharacteristic
217C2×Q8—24no—
218C2×Q8—42no—
219C2×Q8—42no—
220C2×D4—22no—
221C2×D4—22no—
222C2×D4—41no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—42no—
230C22×C4—42no—
231C22×C4C22⋊Dic7311yescharacteristic
232C22×C4C22⋊Dic7311yescharacteristic
233C22×C4C22×Dic7311yescharacteristic
234C22×C4C2×C22⋊D7311yescharacteristic
235C22×C4C2×C22⋊D7311yescharacteristic
236M16—5842no—
237C2×C8—2922no—
238C2×C8—5842no—
239C2×C8—1464no—
240C4⋊C4—22no—
241C4⋊C4—42no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—41no—
245C22⋊C4—22no—
246C22⋊C4—22no—
247C22⋊C4—44no—
248C22⋊C4—44no—
249C42—42no—
250C42D4×D7312yes—
251C42#1192402712yes—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—21no—
256C23—21no—
257C23—41no—
258C23—41no—
259C23—41no—
260C23—41no—
261C23#1455650812yes—
262C23C2×C22⋊Dic7311yescharacteristic
263Q8—84no—
264D4—44no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—22no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—81no—
282C2×C4—81no—
283C2×C4D4×Dic7312yes—
284C2×C4#1455655911yescharacteristic
285C2×C4#1455656312yes—
286C2×C4#1455657211yescharacteristic
287C8—2924no—
288C8—5842no—
289C22—21no—
290C22—21no—
291C22—22no—
292C22—42no—
293C22—42no—
294C22—41no—
295C22—41no—
296C22—41no—
297C22—41no—
298C22—41no—
299C22—81no—
300C22—81no—
301C22#1798846711yescharacteristic
302C22C2×#1455650811yescharacteristic
303C22#1798857111yescharacteristic
304C4—22no—
305C4—21no—
306C4—22no—
307C4—41no—
308C4—41no—
309C4—82no—
310C4—82no—
311C4#1798876412yes—
312C2—21no—
313C2—21no—
314C2—42no—
315C2—81no—
316C2#2261032011yescharacteristic
317C2#2261039511yescharacteristic
318C2#2261045411yescharacteristic
subgroupG/H
centerC22#17988467
commutatorC22×C4×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#2547C2
socleC22×C73#386
prime radicalC22×D4×C73C2×C4
cosocleC2×C42×C73C23
hypercenter#2547D73
hypocenterC73#9451
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2547

Complements

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

NH ≅ G/Ntype
2. #22608657C2semidirect
4. #22607787C2semidirect
165. C73#9451semidirect

Maximal subgroups

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

subgroupindex
#226189602
#226086572
#226082722
#226077872
C73×#25472
#945173