Finite Group Databasebeta

← #32996194

Subgroups — #32996194

Order 18688.

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

normal subgroups only

This group has 552 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#22615681C211yesmaximal, characteristic
2#22608245C211yesmaximal, characteristic, complemented
3#22607933C214yesmaximal, complemented
4C73×#1742C211yesmaximal, characteristic, complemented
5C2×#14556588C411yescharacteristic
6C2×#14556565—21no—
7#17988640—21no—
8#17988504—21no—
9#17988504C2212yes—
10#17988496—21no—
11#17988496C412yes—
12#17988489—21no—
13#17988453C411yescharacteristic
14#17988313C2212yes—
15#17988311—21no—
16C22×M16×C73—21no—
17C2×#117×C73C2211yescharacteristic
18#343×C73—21no—
19#343×C73C2212yes—
20C22×D292—21no—
21C22×D292—21no—
22C22×D292C2×C412yes—
23C22×C22⋊D73—22no—
24#14556588—41no—
25#14556588—41no—
26#14556565—42no—
27#14556565—42no—
28C2×C4⋊Dic73—21no—
29C2×#11924005—22no—
30C2×#11924005—22no—
31C2×#11924005—22no—
32C4⋊#10596831—41no—
33C22⋊#10596831—22no—
34C4×#10596831—41no—
35#14556498—21no—
36#14556498C2×C412yes—
37C23×C4×C73D411yescharacteristic
38C2×M16×C73—24no—
39C2×M16×C73—22no—
40C2×M16×C73—41no—
41C2×M16×C73—42no—
42C22×C8×C73—41no—
43#117×C73—41no—
44#117×C73—41no—
45C2×C4⋊C4×C73D411yescharacteristic
46C2×C4⋊C4×C73C2311yescharacteristic
47C2×C22⋊C4×C73C2×C411yescharacteristic
48C2×C22⋊C4×C73D411yescharacteristic
49C2×C42×C73C2×C411yescharacteristic
50C2×C42×C73D411yescharacteristic
51C23×D73—22no—
52C2×D292—42no—
53C2×D292—42no—
54C2×D292—41no—
55C2×D292—41no—
56C2×D292—42no—
57C2×D292—42no—
58C2×D292—82no—
59C2×D292—82no—
60C2×C22⋊D73—44no—
61C2×C22⋊D73—82no—
62C2×C22⋊D73—82no—
63C22×Dic73—22no—
64C2×#10596831—42no—
65C4⋊Dic73—42no—
66#11924005—42no—
67#11924005—42no—
68#11924005—82no—
69#11924005—82no—
70C24×C73C22⋊C411yescharacteristic
71C22×C4×C73—21no—
72C22×C4×C73—21no—
73C22×C4×C73—22no—
74C22×C4×C73C22⋊C411yescharacteristic
75C22×C4×C73C22⋊C411yescharacteristic
76C22×C4×C73C22⋊C411yescharacteristic
77C22×C4×C73D812yes—
78C22×C4×C73SD1612yes—
79C22×C4×C73C22×C411yescharacteristic
80C22×C4×C73C2×D411yescharacteristic
81C22×C4×C73C2×D411yescharacteristic
82M16×C73—24no—
83M16×C73—44no—
84M16×C73—42no—
85C2×C8×C73—42no—
86C2×C8×C73—44no—
87C4⋊C4×C73—41no—
88C4⋊C4×C73—41no—
89C22⋊C4×C73—41no—
90C22⋊C4×C73—41no—
91C42×C73—21no—
92C42×C73—21no—
93C42×C73—21no—
94C42×C73—21no—
95C23×C73—21no—
96C23×C73—21no—
97C23×C73—41no—
98C23×C73—41no—
99C23×C73#10212yes—
100C23×C73C2×C22⋊C411yescharacteristic
101C22×D73—42no—
102C22×D73—42no—
103C22×D73—42no—
104C22×D73—42no—
105C22×D73—42no—
106C22×D73—42no—
107C22×D73—42no—
108C2×C4×C73—21no—
109C2×C4×C73—21no—
110C2×C4×C73—22no—
111C2×C4×C73—22no—
112C2×C4×C73—22no—
113C2×C4×C73—22no—
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#9911yescharacteristic
123C2×C4×C73#9911yescharacteristic
124C2×C4×C73#10212yes—
125C2×C4×C73#10412yes—
126C2×C4×C73C2×D811yescharacteristic
127C2×C4×C73C2×SD1611yescharacteristic
128C22⋊D73—88no—
129C2×Dic73—42no—
130C2×Dic73—42no—
131C2×Dic73—42no—
132D292—84no—
133D292—84no—
134D292—82no—
135D292—82no—
136C8×C73—44no—
137#10596831—82no—
138C22×C73—21no—
139C22×C73—21no—
140C22×C73—22no—
141C22×C73—42no—
142C22×C73—41no—
143C22×C73—41no—
144C22×C73—81no—
145C22×C73C2×#9911yescharacteristic
146C22×C73C2×#10211yescharacteristic
147C22×C73C2×#10411yescharacteristic
148C2×D73—82no—
149C2×D73—82no—
150C2×D73—82no—
151C2×D73—82no—
152C2×D73—82no—
153C2×D73—82no—
154C2×D73—82no—
155C4×C73—21no—
156C4×C73—21no—
157C4×C73—41no—
158C4×C73—41no—
159C4×C73—41no—
160C4×C73—41no—
161C4×C73#32712yes—
162C4×C73#32712yes—
163Dic73—82no—
164#11699—731nomaximal, Sylow
165C2×C73—21no—
166C2×C73—21no—
167C2×C73—42no—
168C2×C73C2×#32711yescharacteristic
169C2×C73#212211yescharacteristic
170C2×C73#213411yescharacteristic
171D73—162no—
172#2619—731no—
173#1742D7311yescharacteristic, complemented
174#1458—731no—
175#1345—734no—
176C73#1169911yescharacteristic, complemented, Sylow
177C22×M16—21no—
178C2×#127—731no—
179C2×#121—1461no—
180C2×#117C2×D7311yescharacteristic
181#431—1461no—
182#394—732no—
183#394—1461no—
184#392—732no—
185#392—1461no—
186#390—1461no—
187#382—731no—
188#343—21no—
189#343C2×D7312yes—
190#324—732no—
191#323—1461no—
192C22×D4—732no—
193C22×D4—1462no—
194C22×D4—1461no—
195C22×D4—1461no—
196C23×C4C22⋊D7311yescharacteristic
197C2×M16—24no—
198C2×M16—22no—
199C2×M16—41no—
200C2×M16—42no—
201C22×C8—41no—
202#127—2921no—
203#127—2921no—
204#121—2922no—
205#121—2922no—
206#117—41no—
207#117—41no—
208C2×C4⋊C4—1461no—
209C2×C4⋊C4C22⋊D7311yescharacteristic
210C2×C4⋊C4C22×D7311yescharacteristic
211C2×C22⋊C4—1462no—
212C2×C22⋊C4—1462no—
213C2×C22⋊C4—1462no—
214C2×C22⋊C4C4×D7311yescharacteristic
215C2×C22⋊C4D29211yescharacteristic
216C2×C42C4×D7311yescharacteristic
217C2×C42D29211yescharacteristic
218C4⋊C8—2921no—
219C22⋊C8—1462no—
220C4×C8—2921no—
221#95—732no—
222#95—1461no—
223C24—1462no—
224C24#1192400511yescharacteristic
225C2×D4—2922no—
226C2×D4—2922no—
227C2×D4—2924no—
228C2×D4—2921no—
229C2×D4—2921no—
230C2×D4—2922no—
231C2×D4—2922no—
232C2×D4—5842no—
233C2×D4—5842no—
234C2×D4—5842no—
235C2×D4—5842no—
236C22×C4—21no—
237C22×C4—21no—
238C22×C4—22no—
239C22×C4—1462no—
240C22×C4#1192400511yescharacteristic
241C22×C4#1192400511yescharacteristic
242C22×C4#1192400511yescharacteristic
243C22×C4#1192401312yes—
244C22×C4#1192401512yes—
245C22×C4C2×C4×D7311yescharacteristic
246C22×C4C2×C22⋊D7311yescharacteristic
247C22×C4C2×D29211yescharacteristic
248M16—24no—
249M16—44no—
250M16—42no—
251C2×C8—42no—
252C2×C8—44no—
253C2×C8—2922no—
254C4⋊C4—41no—
255C4⋊C4—41no—
256C4⋊C4—2922no—
257C22⋊C4—41no—
258C22⋊C4—41no—
259C22⋊C4—2922no—
260C22⋊C4—2922no—
261C22⋊C4—5842no—
262C22⋊C4—5842no—
263C42—21no—
264C42—21no—
265C42—21no—
266C42—21no—
267C23—21no—
268C23—21no—
269C23—41no—
270C23—41no—
271C23—2922no—
272C23—2922no—
273C23—2922no—
274C23—2922no—
275C23—2922no—
276C23—2922no—
277C23—2922no—
278C23#1455651112yes—
279C23C2×#1192400511yescharacteristic
280D4—5844no—
281D4—5844no—
282D4—5848no—
283D4—5842no—
284D4—5842no—
285C2×C4—21no—
286C2×C4—21no—
287C2×C4—22no—
288C2×C4—22no—
289C2×C4—22no—
290C2×C4—22no—
291C2×C4—21no—
292C2×C4—21no—
293C2×C4—21no—
294C2×C4—21no—
295C2×C4—41no—
296C2×C4—41no—
297C2×C4—41no—
298C2×C4—41no—
299C2×C4—2922no—
300C2×C4—2922no—
301C2×C4—2922no—
302C2×C4#1455650511yescharacteristic
303C2×C4#1455650511yescharacteristic
304C2×C4#1455651112yes—
305C2×C4#1455651912yes—
306C2×C4C2×#1192401311yescharacteristic
307C2×C4C2×#1192401511yescharacteristic
308C8—44no—
309C8—5842no—
310C22—21no—
311C22—21no—
312C22—22no—
313C22—42no—
314C22—41no—
315C22—41no—
316C22—81no—
317C22—5842no—
318C22—5842no—
319C22—5842no—
320C22—5842no—
321C22—5842no—
322C22—5842no—
323C22—5842no—
324C22C2×#1455650511yescharacteristic
325C22C2×#1455651111yescharacteristic
326C22C2×#1455651911yescharacteristic
327C4—21no—
328C4—21no—
329C4—41no—
330C4—41no—
331C4—41no—
332C4—41no—
333C4—5842no—
334C4#1798832012yes—
335C4#1798832012yes—
336C2—21no—
337C2—21no—
338C2—42no—
339C2—11682no—
340C2C2×#1798832011yescharacteristic
341C2#2261158211yescharacteristic
342C2#2261162211yescharacteristic
subgroupG/H
centerC22C2×#14556505
commutatorC22×C4×C73C22×C4
FrattiniC2×C4⋊C4C22×D73
FittingC73×#1742C2
socleC22×C73C2×#99
prime radical#22615681C2
cosocleC2×C4⋊C4×C73C23
hypercenter#1742D73
hypocenterC73#11699
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1742

Complements

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

NH ≅ G/Ntype
2. #22608245C2semidirect
3. #22607933C2semidirect
4. C73×#1742C2semidirect
173. #1742D73semidirect
176. C73#11699semidirect

Maximal subgroups

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

subgroupindex
#226156812
#226082452
#226079332
C73×#17422
#1169973