Finite Group Databasebeta

← #32996683

Subgroups — #32996683

Order 18688.

488 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 488 conjugacy classes of subgroups — too many to draw. Only its 79 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#22608265C211yesmaximal, characteristic, complemented
3#22607933C214yesmaximal, complemented
4C73×#1822C211yesmaximal, characteristic, complemented
5C2×#14556588C411yescharacteristic
6C2×#14556565—21no—
7#17988644—21no—
8#17988504—21no—
9#17988504C2212yes—
10#17988496—21no—
11#17988496C412yes—
12#17988489—21no—
13#17988453C411yescharacteristic
14#17988313C2212yes—
15#17988311—21no—
16C2×#117×C73C2211yescharacteristic
17#407×C73—21no—
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—44no—
27C2×C4⋊Dic73—21no—
28C2×#11924005—22no—
29C2×#11924005—22no—
30C2×#11924005—22no—
31C4⋊#10596831—41no—
32C22⋊#10596831—22no—
33#14556502—41no—
34#14556498—21no—
35#14556498C2×C412yes—
36C23×C4×C73D411yescharacteristic
37C2×M16×C73—22no—
38#117×C73—41no—
39#117×C73—41no—
40C2×C4⋊C4×C73D411yescharacteristic
41C2×C4⋊C4×C73C2311yescharacteristic
42C2×C22⋊C4×C73C2×C411yescharacteristic
43C2×C22⋊C4×C73D411yescharacteristic
44C2×C42×C73C2×C411yescharacteristic
45C2×C42×C73D411yescharacteristic
46C22⋊C8×C73—41no—
47C22⋊C8×C73—41no—
48C23×D73—22no—
49C2×D292—42no—
50C2×D292—42no—
51C2×D292—42no—
52C2×D292—42no—
53C2×D292—41no—
54C2×D292—41no—
55C2×D292—82no—
56C2×D292—82no—
57C2×C22⋊D73—44no—
58C2×C22⋊D73—82no—
59C2×C22⋊D73—82no—
60C22×Dic73—22no—
61C2×#10596831—42no—
62C4⋊Dic73—42no—
63#11924005—44no—
64#11924005—82no—
65#11924005—82no—
66C24×C73C22⋊C411yescharacteristic
67C22×C4×C73—21no—
68C22×C4×C73—21no—
69C22×C4×C73—22no—
70C22×C4×C73—41no—
71C22×C4×C73C22⋊C411yescharacteristic
72C22×C4×C73C22⋊C411yescharacteristic
73C22×C4×C73C22⋊C411yescharacteristic
74C22×C4×C73C22×C411yescharacteristic
75C22×C4×C73C2×D411yescharacteristic
76C22×C4×C73C2×D411yescharacteristic
77M16×C73—42no—
78M16×C73—42no—
79C2×C8×C73—42no—
80C4⋊C4×C73—41no—
81C4⋊C4×C73—41no—
82C22⋊C4×C73—41no—
83C22⋊C4×C73—41no—
84C42×C73—21no—
85C42×C73—21no—
86C42×C73—21no—
87C42×C73—21no—
88C23×C73—21no—
89C23×C73—21no—
90C23×C73—21no—
91C23×C73—41no—
92C23×C73—41no—
93C23×C73C2×C22⋊C411yescharacteristic
94C22×D73—42no—
95C22×D73—42no—
96C22×D73—42no—
97C22×D73—42no—
98C22×D73—42no—
99C22×D73—42no—
100C22×D73—42no—
101C2×C4×C73—21no—
102C2×C4×C73—21no—
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—42no—
110C2×C4×C73—42no—
111C2×C4×C73—41no—
112C2×C4×C73—41no—
113C2×C4×C73—41no—
114C2×C4×C73—41no—
115C2×C4×C73#9911yescharacteristic
116C2×C4×C73#9911yescharacteristic
117C2×C4×C73AGL(1,ℤ/8)12yes—
118C22⋊D73—88no—
119C2×Dic73—42no—
120C2×Dic73—42no—
121C2×Dic73—42no—
122D292—82no—
123D292—82no—
124D292—84no—
125D292—84no—
126C8×C73—82no—
127#10596831—82no—
128C22×C73—21no—
129C22×C73—21no—
130C22×C73—42no—
131C22×C73—41no—
132C22×C73—41no—
133C22×C73—41no—
134C22×C73—81no—
135C22×C73C2×#9911yescharacteristic
136C22×C73#41811yescharacteristic
137C22×C73#42111yescharacteristic
138C2×D73—82no—
139C2×D73—82no—
140C2×D73—82no—
141C2×D73—82no—
142C2×D73—82no—
143C2×D73—82no—
144C2×D73—82no—
145C4×C73—22no—
146C4×C73—21no—
147C4×C73—21no—
148C4×C73—41no—
149C4×C73—41no—
150C4×C73—41no—
151C4×C73—41no—
152Dic73—82no—
153#11806—731nomaximal, Sylow
154C2×C73—21no—
155C2×C73—21no—
156C2×C73—81no—
157C2×C73#151011yescharacteristic
158C2×C73#212211yescharacteristic
159C2×C73#213411yescharacteristic
160D73—162no—
161#2619—731no—
162#1822D7311yescharacteristic, complemented
163#1464—731no—
164#1345—734no—
165C73#1180611yescharacteristic, complemented, Sylow
166C2×#127—731no—
167C2×#121—1461no—
168C2×#117C2×D7311yescharacteristic
169#432—1461no—
170#407—21no—
171#394—732no—
172#394—1461no—
173#392—732no—
174#392—1461no—
175#390—1461no—
176#382—731no—
177#343—21no—
178#343C2×D7312yes—
179#324—732no—
180#323—1461no—
181C22×D4—732no—
182C22×D4—1462no—
183C22×D4—1461no—
184C22×D4—1461no—
185C23×C4C22⋊D7311yescharacteristic
186C2×M16—22no—
187#127—2921no—
188#127—2921no—
189#121—2924no—
190#117—41no—
191#117—41no—
192C2×C4⋊C4—1461no—
193C2×C4⋊C4C22⋊D7311yescharacteristic
194C2×C4⋊C4C22×D7311yescharacteristic
195C2×C22⋊C4—1462no—
196C2×C22⋊C4—1462no—
197C2×C22⋊C4—1462no—
198C2×C22⋊C4C4×D7311yescharacteristic
199C2×C22⋊C4D29211yescharacteristic
200C2×C42C4×D7311yescharacteristic
201C2×C42D29211yescharacteristic
202C4⋊C8—2921no—
203C22⋊C8—41no—
204C22⋊C8—41no—
205C22⋊C8—1462no—
206#97—2921no—
207#95—732no—
208#95—1461no—
209C24—1462no—
210C24#1192400511yescharacteristic
211C2×D4—2922no—
212C2×D4—2922no—
213C2×D4—2922no—
214C2×D4—2922no—
215C2×D4—2924no—
216C2×D4—2921no—
217C2×D4—2921no—
218C2×D4—5842no—
219C2×D4—5842no—
220C2×D4—5842no—
221C2×D4—5842no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—22no—
225C22×C4—41no—
226C22×C4—1462no—
227C22×C4#1192400511yescharacteristic
228C22×C4#1192400511yescharacteristic
229C22×C4#1192400511yescharacteristic
230C22×C4C2×C4×D7311yescharacteristic
231C22×C4C2×C22⋊D7311yescharacteristic
232C22×C4C2×D29211yescharacteristic
233M16—42no—
234M16—42no—
235C2×C8—42no—
236C2×C8—2922no—
237C4⋊C4—41no—
238C4⋊C4—41no—
239C4⋊C4—2922no—
240C22⋊C4—41no—
241C22⋊C4—41no—
242C22⋊C4—2924no—
243C22⋊C4—5842no—
244C22⋊C4—5842no—
245C42—21no—
246C42—21no—
247C42—21no—
248C42—21no—
249C23—21no—
250C23—21no—
251C23—21no—
252C23—41no—
253C23—41no—
254C23—2922no—
255C23—2922no—
256C23—2922no—
257C23—2922no—
258C23—2922no—
259C23—2922no—
260C23—2922no—
261C23C2×#1192400511yescharacteristic
262D4—5842no—
263D4—5842no—
264D4—5844no—
265D4—5844no—
266D4—5848no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—2922no—
282C2×C4—2922no—
283C2×C4—2922no—
284C2×C4#1455650511yescharacteristic
285C2×C4#1455650511yescharacteristic
286C2×C4#1455661612yes—
287C8—82no—
288C8—5842no—
289C22—21no—
290C22—21no—
291C22—42no—
292C22—41no—
293C22—41no—
294C22—41no—
295C22—81no—
296C22—5842no—
297C22—5842no—
298C22—5842no—
299C22—5842no—
300C22—5842no—
301C22—5842no—
302C22—5842no—
303C22C2×#1455650511yescharacteristic
304C22#1798859111yescharacteristic
305C22#1798860511yescharacteristic
306C4—22no—
307C4—21no—
308C4—21no—
309C4—41no—
310C4—41no—
311C4—41no—
312C4—41no—
313C4—5842no—
314C2—21no—
315C2—21no—
316C2—81no—
317C2—11682no—
318C2#2260845911yescharacteristic
319C2#2261158211yescharacteristic
320C2#2261162211yescharacteristic
subgroupG/H
centerC22C2×#14556505
commutatorC22×C4×C73C22×C4
FrattiniC2×C4⋊C4C22×D73
FittingC73×#1822C2
socleC22×C73C2×#99
prime radical#22615681C2
cosocleC2×C4⋊C4×C73C23
hypercenter#1822D73
hypocenterC73#11806
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1822

Complements

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

NH ≅ G/Ntype
2. #22608265C2semidirect
3. #22607933C2semidirect
4. C73×#1822C2semidirect
162. #1822D73semidirect
165. C73#11806semidirect

Maximal subgroups

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

subgroupindex
#226156812
#226082652
#226079332
C73×#18222
#1180673