Finite Group Databasebeta

← #32997033

Subgroups — #32997033

Order 18688.

460 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 460 conjugacy classes of subgroups — too many to draw. Only its 99 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#22614239C211yesmaximal, characteristic
2#22608249C211yesmaximal, characteristic
3#22607936C214yesmaximal, complemented
4C73×#1800C211yesmaximal, characteristic
5C2×#14556572—21no—
6C22×C4⋊Dic73C411yescharacteristic
7C2×C22⋊#10596831—21no—
8#17988517C2212yes—
9#17988512C412yes—
10#17988485—21no—
11#17988448—21no—
12#17988447—21no—
13#17988447C411yescharacteristic
14#17988313C2212yes—
15#17988311—21no—
16C2×C4×D4×C73C2211yescharacteristic
17#423×C73—21no—
18C2×#100×C73—21no—
19#343×C73C2212yes—
20C23×Dic73—21no—
21C22×#10596831—41no—
22#14556572—42no—
23#14556572—42no—
24C2×C4⋊Dic73—21no—
25C2×C4⋊Dic73—21no—
26C2×C4⋊Dic73—41no—
27C2×C4⋊Dic73—41no—
28C2×C4⋊Dic73C2×C412yes—
29C2×#11924007—22no—
30C2×C22⋊Dic73—22no—
31C2×C22⋊Dic73—21no—
32C22⋊#10596831—22no—
33C22⋊#10596831—41no—
34#14556498—22no—
35#14556498—22no—
36#14556498C2×C412yes—
37C22×D4×C73D411yescharacteristic
38C23×C4×C73C2×C411yescharacteristic
39C23×C4×C73D411yescharacteristic
40C2×M16×C73—22no—
41C4×D4×C73—41no—
42C4×D4×C73—41no—
43C2×C4⋊C4×C73C2311yescharacteristic
44C2×C22⋊C4×C73C2×C411yescharacteristic
45C2×C22⋊C4×C73D411yescharacteristic
46C2×C42×C73D411yescharacteristic
47C4⋊C8×C73—41no—
48C4⋊C8×C73—41no—
49#100×C73—42no—
50C22×Dic73—22no—
51C22×Dic73—22no—
52C22×Dic73—42no—
53C22×Dic73—42no—
54C22×Dic73—42no—
55C2×#10596831—42no—
56C2×#10596831—82no—
57C4⋊Dic73—42no—
58C4⋊Dic73—41no—
59C4⋊Dic73—41no—
60C4⋊Dic73—42no—
61#11924007—42no—
62#11924007—42no—
63C22⋊Dic73—42no—
64C22⋊Dic73—44no—
65C24×C73C22⋊C411yescharacteristic
66C24×C73C22⋊C411yescharacteristic
67C2×D4×C73—41no—
68C2×D4×C73—41no—
69C2×D4×C73D812yes—
70C2×D4×C73SD1612yes—
71C22×C4×C73—21no—
72C22×C4×C73—21no—
73C22×C4×C73—21no—
74C22×C4×C73—21no—
75C22×C4×C73—42no—
76C22×C4×C73—42no—
77C22×C4×C73C22⋊C411yescharacteristic
78C22×C4×C73C22⋊C411yescharacteristic
79C22×C4×C73C22×C411yescharacteristic
80C22×C4×C73C2×D411yescharacteristic
81C22×C4×C73C2×D411yescharacteristic
82M16×C73—42no—
83M16×C73—42no—
84C2×C8×C73—42no—
85C4⋊C4×C73—41no—
86C22⋊C4×C73—41no—
87C22⋊C4×C73—41no—
88C42×C73—21no—
89C42×C73—21no—
90C23×C73—21no—
91C23×C73—21no—
92C23×C73—21no—
93C23×C73—21no—
94C23×C73—41no—
95C23×C73—41no—
96C23×C73—41no—
97C23×C73—41no—
98C23×C73#10212yes—
99C23×C73#10212yes—
100C23×C73C2×C22⋊C411yescharacteristic
101D4×C73—42no—
102D4×C73—42no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—22no—
106C2×C4×C73—21no—
107C2×C4×C73—21no—
108C2×C4×C73—21no—
109C2×C4×C73—42no—
110C2×C4×C73—42no—
111C2×C4×C73—42no—
112C2×C4×C73—41no—
113C2×C4×C73—41no—
114C2×C4×C73—41no—
115C2×C4×C73—81no—
116C2×C4×C73#9911yescharacteristic
117C2×C4×C73#9911yescharacteristic
118C2×C4×C73C2×D811yescharacteristic
119C2×C4×C73C2×SD1611yescharacteristic
120C2×Dic73—42no—
121C2×Dic73—42no—
122C2×Dic73—42no—
123C2×Dic73—42no—
124C2×Dic73—42no—
125C2×Dic73—42no—
126C2×Dic73—84no—
127C8×C73—82no—
128#10596831—82no—
129C22×C73—21no—
130C22×C73—21no—
131C22×C73—22no—
132C22×C73—22no—
133C22×C73—42no—
134C22×C73—42no—
135C22×C73—42no—
136C22×C73—41no—
137C22×C73—41no—
138C22×C73—41no—
139C22×C73—41no—
140C22×C73—81no—
141C22×C73C2×#9911yescharacteristic
142C22×C73C2×#10211yescharacteristic
143C22×C73#42111yescharacteristic
144C4×C73—21no—
145C4×C73—21no—
146C4×C73—21no—
147C4×C73—41no—
148C4×C73—41no—
149C4×C73—41no—
150C4×C73#35212yes—
151Dic73—82no—
152Dic73—82no—
153#11881—731nomaximal, Sylow
154C2×C73—21no—
155C2×C73—21no—
156C2×C73—42no—
157C2×C73—42no—
158C2×C73#151911yescharacteristic
159C2×C73C2×#35211yescharacteristic
160C2×C73#213511yescharacteristic
161#2486—731no—
162#1800D7311yescharacteristic
163#1459—731no—
164#1346—734no—
165C73#1188111yescharacteristic, complemented, Sylow
166C2×#123—1461no—
167C2×C4×D4C2×D7311yescharacteristic
168C22×C4⋊C4—731no—
169#423—21no—
170C2×#100—21no—
171C2×C22⋊C8—1461no—
172#398—732no—
173#397—732no—
174#389—1461no—
175#381—1461no—
176#380—731no—
177#380—1461no—
178#343C2×D7312yes—
179#324—732no—
180#323—1461no—
181C22×D4C22⋊D7311yescharacteristic
182C23×C4—1461no—
183C23×C4C4×D7311yescharacteristic
184C23×C4D29211yescharacteristic
185C2×M16—22no—
186C22×C8—2921no—
187#123—2922no—
188#123—2922no—
189C4×D4—41no—
190C4×D4—41no—
191C2×C4⋊C4—2921no—
192C2×C4⋊C4—2921no—
193C2×C4⋊C4—732no—
194C2×C4⋊C4—1461no—
195C2×C4⋊C4—1461no—
196C2×C4⋊C4—1462no—
197C2×C4⋊C4C22×D7311yescharacteristic
198C2×C22⋊C4—1461no—
199C2×C22⋊C4—1462no—
200C2×C22⋊C4C4×D7311yescharacteristic
201C2×C22⋊C4D29211yescharacteristic
202C2×C42C22⋊D7311yescharacteristic
203C4⋊C8—41no—
204C4⋊C8—41no—
205#100—42no—
206C22⋊C8—2921no—
207C22⋊C8—1462no—
208#95—732no—
209#95—1462no—
210#95—1462no—
211C24#1192400511yescharacteristic
212C24#1192400511yescharacteristic
213C2×D4—41no—
214C2×D4—41no—
215C2×D4#1192401312yes—
216C2×D4#1192401512yes—
217C22×C4—21no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—21no—
221C22×C4—42no—
222C22×C4—42no—
223C22×C4—2922no—
224C22×C4—2922no—
225C22×C4—2922no—
226C22×C4—1462no—
227C22×C4—1462no—
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—
237C2×C8—5842no—
238C4⋊C4—41no—
239C4⋊C4—2922no—
240C4⋊C4—2922no—
241C4⋊C4—2921no—
242C4⋊C4—2921no—
243C4⋊C4—2922no—
244C4⋊C4—2922no—
245C22⋊C4—41no—
246C22⋊C4—41no—
247C22⋊C4—2922no—
248C22⋊C4—2924no—
249C42—21no—
250C42—21no—
251C23—21no—
252C23—21no—
253C23—21no—
254C23—21no—
255C23—41no—
256C23—41no—
257C23—41no—
258C23—41no—
259C23#1455651112yes—
260C23#1455651112yes—
261C23C2×#1192400511yescharacteristic
262D4—42no—
263D4—42no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—42no—
271C2×C4—42no—
272C2×C4—42no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—81no—
277C2×C4—2922no—
278C2×C4—2922no—
279C2×C4—2922no—
280C2×C4—2922no—
281C2×C4—2922no—
282C2×C4—2922no—
283C2×C4—5844no—
284C2×C4#1455650511yescharacteristic
285C2×C4#1455650511yescharacteristic
286C2×C4C2×#1192401311yescharacteristic
287C2×C4C2×#1192401511yescharacteristic
288C8—82no—
289C8—5842no—
290C22—21no—
291C22—21no—
292C22—22no—
293C22—22no—
294C22—42no—
295C22—42no—
296C22—42no—
297C22—41no—
298C22—41no—
299C22—41no—
300C22—41no—
301C22—81no—
302C22C2×#1455650511yescharacteristic
303C22C2×#1455651111yescharacteristic
304C22#1798860511yescharacteristic
305C4—21no—
306C4—21no—
307C4—21no—
308C4—41no—
309C4—41no—
310C4—41no—
311C4—5842no—
312C4—5842no—
313C4#1798837812yes—
314C2—21no—
315C2—21no—
316C2—42no—
317C2—42no—
318C2#2260849811yescharacteristic
319C2C2×#1798837811yescharacteristic
320C2#2261162511yescharacteristic
subgroupG/H
centerC22C2×#14556505
commutatorC22×C4×C73C22×C4
FrattiniC2×C4⋊C4C22×D73
FittingC73×#1800C2
socleC22×C73C2×#99
prime radicalC22×D4×C73D4
cosocleC2×C4⋊C4×C73C23
hypercenter#1800D73
hypocenterC73#11881
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1800

Complements

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

NH ≅ G/Ntype
3. #22607936C2semidirect
165. C73#11881semidirect

Maximal subgroups

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

subgroupindex
#226142392
#226082492
#226079362
C73×#18002
#1188173