Finite Group Databasebeta

← #32997004

Subgroups — #32997004

Order 18688.

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

normal subgroups only

This group has 472 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#22609919C211yesmaximal, characteristic
3#22607932C214yesmaximal, complemented
4C73×#1460C211yesmaximal, characteristic
5C2×#14556572—21no—
6C22×C4⋊Dic73C411yescharacteristic, complemented
7C2×#14556506—21no—
8#17988517C412yescomplemented
9#17988512C2212yes—
10#17988485—21no—
11#17988448—21no—
12#17988447—21no—
13#17988447—21no—
14#17988447C411yescharacteristic, complemented
15#17988354C2212yes—
16C2×C4×D4×C73C2211yescharacteristic
17#431×C73—21no—
18#323×C73—21no—
19#323×C73C2212yes—
20C23×Dic73—21no—
21#14556572—42no—
22#14556572—42no—
23C2×C4⋊Dic73—21no—
24C2×C4⋊Dic73—21no—
25C2×C4⋊Dic73—41no—
26C2×C4⋊Dic73—41no—
27C2×C4⋊Dic73—41no—
28C2×C4⋊Dic73C2×C412yes—
29C2×#11924007—22no—
30C2×C22⋊Dic73—21no—
31C2×C22⋊Dic73—22no—
32C2×C22⋊Dic73—22no—
33#14556506—42no—
34#14556498—22no—
35#14556498—22no—
36#14556498—41no—
37#14556498C2×C412yes—
38C22×D4×C73D411yescharacteristic
39C23×C4×C73C2×C411yescharacteristic
40C23×C4×C73D411yescharacteristic
41C4×D4×C73—41no—
42C4×D4×C73—41no—
43C2×C4⋊C4×C73C2×C411yescharacteristic
44C2×C22⋊C4×C73D411yescharacteristic
45C2×C22⋊C4×C73C2311yescharacteristic
46C2×C42×C73D411yescharacteristic
47C4⋊C8×C73—41no—
48C22⋊C8×C73—22no—
49C4×C8×C73—41no—
50C22×Dic73—22no—
51C22×Dic73—22no—
52C22×Dic73—42no—
53C22×Dic73—42no—
54C22×Dic73—42no—
55C22×Dic73—42no—
56C4⋊Dic73—42no—
57C4⋊Dic73—41no—
58C4⋊Dic73—41no—
59C4⋊Dic73—42no—
60C4⋊Dic73—82no—
61#11924007—44no—
62C22⋊Dic73—42no—
63C22⋊Dic73—42no—
64C22⋊Dic73—42no—
65C22⋊Dic73—42no—
66C22⋊Dic73—42no—
67C24×C73C22⋊C411yescharacteristic
68C24×C73C2×D411yescharacteristic
69C2×D4×C73—41no—
70C2×D4×C73—41no—
71C2×D4×C73D812yes—
72C2×D4×C73SD1612yes—
73C22×C4×C73—21no—
74C22×C4×C73—21no—
75C22×C4×C73—21no—
76C22×C4×C73—21no—
77C22×C4×C73—42no—
78C22×C4×C73—42no—
79C22×C4×C73C22⋊C411yescharacteristic
80C22×C4×C73C22⋊C411yescharacteristic
81C22×C4×C73C22⋊C411yescharacteristic
82C22×C4×C73C22×C411yescharacteristic
83C22×C4×C73C2×D411yescharacteristic
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×C73C2×C22⋊C411yescharacteristic
100C23×C73C2×D811yescharacteristic
101C23×C73C2×SD1611yescharacteristic
102D4×C73—42no—
103D4×C73—42no—
104C2×C4×C73—21no—
105C2×C4×C73—21no—
106C2×C4×C73—22no—
107C2×C4×C73—21no—
108C2×C4×C73—21no—
109C2×C4×C73—21no—
110C2×C4×C73—42no—
111C2×C4×C73—42no—
112C2×C4×C73—42no—
113C2×C4×C73—41no—
114C2×C4×C73—41no—
115C2×C4×C73—41no—
116C2×C4×C73—81no—
117C2×C4×C73#9911yescharacteristic
118C2×C4×C73#9911yescharacteristic
119C2×C4×C73#10212yes—
120C2×Dic73—42no—
121C2×Dic73—42no—
122C2×Dic73—42no—
123C2×Dic73—42no—
124C2×Dic73—42no—
125C2×Dic73—42no—
126C2×Dic73—84no—
127C2×Dic73—82no—
128C2×Dic73—82no—
129C2×Dic73—82no—
130C8×C73—82no—
131C22×C73—21no—
132C22×C73—21no—
133C22×C73—22no—
134C22×C73—21no—
135C22×C73—21no—
136C22×C73—42no—
137C22×C73—41no—
138C22×C73—41no—
139C22×C73—42no—
140C22×C73—41no—
141C22×C73—41no—
142C22×C73—41no—
143C22×C73—41no—
144C22×C73—81no—
145C22×C73C2×#9911yescharacteristic
146C22×C73C2×#10211yescharacteristic
147C22×C73#42111yescharacteristic
148C4×C73—21no—
149C4×C73—22no—
150C4×C73—41no—
151C4×C73—41no—
152C4×C73—41no—
153C4×C73#35212yes—
154Dic73—82no—
155Dic73—82no—
156Dic73—162no—
157#11874—731nomaximal, Sylow
158C2×C73—21no—
159C2×C73—21no—
160C2×C73—41no—
161C2×C73—42no—
162C2×C73—41no—
163C2×C73#151911yescharacteristic
164C2×C73C2×#35211yescharacteristic
165C2×C73#212911yescharacteristic
166#2486—731no—
167#1799—731no—
168#1460D7311yescharacteristic
169#1344—734no—
170C73#1187411yescharacteristic, complemented, Sylow
171C2×#123—1461no—
172C2×C4×D4C2×D7311yescharacteristic
173C22×C4⋊C4—731no—
174#431—21no—
175C2×#99—1461no—
176#398—732no—
177#397—732no—
178#389—1461no—
179#381—1461no—
180#380—731no—
181#380—1461no—
182#380—1461no—
183#342—732no—
184#323—21no—
185#323C2×D7312yes—
186C22×D4D29211yescharacteristic
187C23×C4—1461no—
188C23×C4C4×D7311yescharacteristic
189C23×C4D29211yescharacteristic
190#123—2922no—
191#123—2922no—
192C4×D4—41no—
193C4×D4—41no—
194C2×C4⋊C4—2921no—
195C2×C4⋊C4—2921no—
196C2×C4⋊C4—2921no—
197C2×C4⋊C4—732no—
198C2×C4⋊C4—1462no—
199C2×C4⋊C4—1461no—
200C2×C4⋊C4—1461no—
201C2×C4⋊C4C4×D7311yescharacteristic
202C2×C22⋊C4—1462no—
203C2×C22⋊C4—1461no—
204C2×C22⋊C4—1462no—
205C2×C22⋊C4C22⋊D7311yescharacteristic
206C2×C22⋊C4C22×D7311yescharacteristic
207C2×C42C22⋊D7311yescharacteristic
208C4⋊C8—41no—
209#99—2922no—
210C22⋊C8—22no—
211C4×C8—41no—
212#95—2921no—
213#95—732no—
214#95—1462no—
215#95—1462no—
216C24#1192400511yescharacteristic
217C24C2×D29211yescharacteristic
218C2×D4—41no—
219C2×D4—41no—
220C2×D4D58412yes—
221C2×D4#1192401712yes—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—42no—
227C22×C4—42no—
228C22×C4—2922no—
229C22×C4—2922no—
230C22×C4—2922no—
231C22×C4—2922no—
232C22×C4—1462no—
233C22×C4—1462no—
234C22×C4#1192400511yescharacteristic
235C22×C4#1192400511yescharacteristic
236C22×C4#1192400511yescharacteristic
237C22×C4C2×C4×D7311yescharacteristic
238C22×C4C2×C22⋊D7311yescharacteristic
239C2×C8—42no—
240C4⋊C4—41no—
241C4⋊C4—2922no—
242C4⋊C4—2922no—
243C4⋊C4—2921no—
244C4⋊C4—2921no—
245C4⋊C4—2924no—
246C4⋊C4—5842no—
247C22⋊C4—41no—
248C22⋊C4—41no—
249C22⋊C4—2922no—
250C22⋊C4—2922no—
251C22⋊C4—2922no—
252C22⋊C4—2922no—
253C22⋊C4—2922no—
254C42—21no—
255C42—21no—
256C23—21no—
257C23—21no—
258C23—21no—
259C23—21no—
260C23—41no—
261C23—41no—
262C23—41no—
263C23—41no—
264C23#1455651312yes—
265C23C2×#1192400511yescharacteristic
266C23C2×D58411yescharacteristic
267C23C2×#1192401711yescharacteristic
268D4—42no—
269D4—42no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—22no—
273C2×C4—21no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—42no—
277C2×C4—42no—
278C2×C4—42no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—41no—
282C2×C4—81no—
283C2×C4—2922no—
284C2×C4—2922no—
285C2×C4—2922no—
286C2×C4—2922no—
287C2×C4—2922no—
288C2×C4—2922no—
289C2×C4—5844no—
290C2×C4—5842no—
291C2×C4—5842no—
292C2×C4—5842no—
293C2×C4#1455650511yescharacteristic
294C2×C4#1455650511yescharacteristic
295C2×C4#1455651312yes—
296C8—82no—
297C22—21no—
298C22—21no—
299C22—22no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—41no—
304C22—41no—
305C22—42no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22C2×#1455650511yescharacteristic
312C22C2×#1455651311yescharacteristic
313C22#1798860311yescharacteristic
314C4—21no—
315C4—22no—
316C4—41no—
317C4—41no—
318C4—41no—
319C4—5842no—
320C4—5842no—
321C4—11682no—
322C4#1798837812yes—
323C2—21no—
324C2—21no—
325C2—41no—
326C2—42no—
327C2—41no—
328C2#2260850011yescharacteristic
329C2C2×#1798837811yescharacteristic
330C2#2261160511yescharacteristic
subgroupG/H
centerC22C2×#14556505
commutatorC22×C4×C73C22×C4
FrattiniC2×C22⋊C4C22×D73
FittingC73×#1460C2
socleC22×C73C2×#99
prime radicalC22×D4×C73D4
cosocleC2×C22⋊C4×C73C23
hypercenter#1460D73
hypocenterC73#11874
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #1460

Complements

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

NH ≅ G/Ntype
3. #22607932C2semidirect
6. C22×C4⋊Dic73C4semidirect
8. #17988517C4semidirect
14. #17988447C4semidirect
170. C73#11874semidirect

Maximal subgroups

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

subgroupindex
#226142392
#226099192
#226079322
C73×#14602
#1187473