Finite Group Databasebeta

← #32996618

Subgroups — #32996618

Order 18688.

572 conjugacy classes of subgroups; only the 95 normal subgroups are shown.

normal subgroups only

This group has 572 conjugacy classes of subgroups — too many to draw. Only its 95 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
1C4×#14556508C211yesmaximal, characteristic
2#22608269C211yesmaximal, characteristic
3#22607961C214yesmaximal, complemented
4C73×#2409C211yesmaximal, characteristic
5C2×#14556508—21no—
6#17988551—21no—
7C4×#11924012—21no—
8#17988356C2212yes—
9#17988313C2212yes—
10#17988311—21no—
11C23×Q8×C73C411yescharacteristic
12C2×C22⋊Q8×C73—21no—
13C2×C4×D4×C73C2211yescharacteristic
14#395×C73C2212yes—
15#393×C73C412yes—
16#391×C73—21no—
17#386×C73—21no—
18#385×C73—21no—
19#380×C73C411yescharacteristic
20C2×#11924012—22no—
21C2×#11924012—41no—
22#14556508—24no—
23C22⋊#10596831—22no—
24C22⋊#10596831—41no—
25#14556502—41no—
26C4×#10596831—41no—
27C22×Q8×C73—21no—
28C22×Q8×C73—41no—
29C22×Q8×C73—41no—
30C22×Q8×C73—44no—
31C22×Q8×C73C2×C412yes—
32C22×D4×C73D411yescharacteristic
33C23×C4×C73—21no—
34C23×C4×C73C2×C411yescharacteristic
35C23×C4×C73D411yescharacteristic
36C22⋊Q8×C73—44no—
37C4×D4×C73—41no—
38C4×D4×C73—41no—
39C2×C4⋊C4×C73—21no—
40C2×C4⋊C4×C73—22no—
41C2×C4⋊C4×C73C2311yescharacteristic
42C2×C22⋊C4×C73—21no—
43C2×C22⋊C4×C73—22no—
44C2×C22⋊C4×C73C2×C411yescharacteristic
45C2×C22⋊C4×C73D411yescharacteristic
46C2×C42×C73D411yescharacteristic
47#95×C73—22no—
48#95×C73—22no—
49#95×C73C2×C412yes—
50#11924012—24no—
51#11924012—42no—
52#11924012—82no—
53C2×#10596831—42no—
54C2×#10596831—42no—
55C24×C73C22⋊C411yescharacteristic
56C24×C73C22⋊C411yescharacteristic
57C2×Q8×C73—24no—
58C2×Q8×C73—44no—
59C2×Q8×C73—42no—
60C2×Q8×C73—42no—
61C2×Q8×C73—44no—
62C2×Q8×C73—48no—
63C2×Q8×C73—44no—
64C2×Q8×C73—41no—
65C2×Q8×C73—41no—
66C2×D4×C73—22no—
67C2×D4×C73—22no—
68C2×D4×C73—41no—
69C22×C4×C73—22no—
70C22×C4×C73—21no—
71C22×C4×C73—21no—
72C22×C4×C73—21no—
73C22×C4×C73—21no—
74C22×C4×C73—22no—
75C22×C4×C73—44no—
76C22×C4×C73—42no—
77C22×C4×C73—42no—
78C22×C4×C73—42no—
79C22×C4×C73C22⋊C411yescharacteristic
80C22×C4×C73C22⋊C411yescharacteristic
81C22×C4×C73C22×C411yescharacteristic
82C22×C4×C73C2×D411yescharacteristic
83C22×C4×C73C2×D411yescharacteristic
84C4⋊C4×C73—41no—
85C4⋊C4×C73—42no—
86C4⋊C4×C73—44no—
87C22⋊C4×C73—41no—
88C22⋊C4×C73—41no—
89C22⋊C4×C73—44no—
90C22⋊C4×C73—42no—
91C42×C73—21no—
92C42×C73—21no—
93C23×C73—21no—
94C23×C73—21no—
95C23×C73—21no—
96C23×C73—21no—
97C23×C73—21no—
98C23×C73—41no—
99C23×C73—41no—
100C23×C73—41no—
101C23×C73—41no—
102C23×C73#10412yes—
103C23×C73C2×C22⋊C411yescharacteristic
104Q8×C73—44no—
105Q8×C73—44no—
106Q8×C73—44no—
107Q8×C73—44no—
108D4×C73—44no—
109C2×C4×C73—21no—
110C2×C4×C73—21no—
111C2×C4×C73—22no—
112C2×C4×C73—24no—
113C2×C4×C73—21no—
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—42no—
117C2×C4×C73—42no—
118C2×C4×C73—48no—
119C2×C4×C73—42no—
120C2×C4×C73—42no—
121C2×C4×C73—42no—
122C2×C4×C73—41no—
123C2×C4×C73—41no—
124C2×C4×C73—41no—
125C2×C4×C73—42no—
126C2×C4×C73—42no—
127C2×C4×C73—42no—
128C2×C4×C73—81no—
129C2×C4×C73#9911yescharacteristic
130C2×C4×C73#9911yescharacteristic
131C2×C4×C73#13712yes—
132#10596831—44no—
133#10596831—82no—
134C22×C73—21no—
135C22×C73—21no—
136C22×C73—22no—
137C22×C73—42no—
138C22×C73—42no—
139C22×C73—42no—
140C22×C73—41no—
141C22×C73—41no—
142C22×C73—41no—
143C22×C73—41no—
144C22×C73—41no—
145C22×C73—81no—
146C22×C73C2×#9911yescharacteristic
147C22×C73#41911yescharacteristic
148C22×C73C2×#10411yescharacteristic
149C4×C73—21no—
150C4×C73—22no—
151C4×C73—41no—
152C4×C73—44no—
153C4×C73—41no—
154C4×C73—41no—
155C4×C73—82no—
156C4×C73AΣL(1,𝔽16)12yes—
157#11791—731nomaximal, Sylow
158C2×C73—21no—
159C2×C73—21no—
160C2×C73—42no—
161C2×C73—81no—
162C2×C73#150811yescharacteristic
163C2×C73C2×AΣL(1,𝔽16)11yescharacteristic
164C2×C73#213511yescharacteristic
165#2409D7311yescharacteristic
166C4×#100—731no—
167#1465—731no—
168#1354—734no—
169C73#1179111yescharacteristic, complemented, Sylow
170C23×Q8Dic7311yescharacteristic
171C2×C22⋊Q8—21no—
172C2×C4×D4C2×D7311yescharacteristic
173C2×#100—1461no—
174#407—1461no—
175C4×M16—1461no—
176#395C2×D7312yes—
177#393Dic7312yes—
178#391—21no—
179#386—21no—
180#385—21no—
181#380Dic7311yescharacteristic
182#343—732no—
183#324—732no—
184#323—1461no—
185C22×Q8—21no—
186C22×Q8—41no—
187C22×Q8—41no—
188C22×Q8—44no—
189C22×Q8C2×Dic7312yes—
190C22×D4C22⋊D7311yescharacteristic
191C23×C4—21no—
192C23×C4C2×Dic7311yescharacteristic
193C23×C4C22⋊D7311yescharacteristic
194C2×M16—2921no—
195C2×M16—1462no—
196C22⋊Q8—44no—
197C4×D4—41no—
198C4×D4—41no—
199C2×C4⋊C4—21no—
200C2×C4⋊C4—22no—
201C2×C4⋊C4C22×D7311yescharacteristic
202C2×C22⋊C4—22no—
203C2×C22⋊C4—21no—
204C2×C22⋊C4C2×Dic7311yescharacteristic
205C2×C22⋊C4C22⋊D7311yescharacteristic
206C2×C42C22⋊D7311yescharacteristic
207#100—1464no—
208C22⋊C8—2921no—
209C22⋊C8—1462no—
210#97—2921no—
211C4×C8—2921no—
212#95—22no—
213#95—22no—
214#95C2×Dic7312yes—
215C24C22⋊Dic7311yescharacteristic
216C24C22⋊Dic7311yescharacteristic
217C2×Q8—24no—
218C2×Q8—44no—
219C2×Q8—42no—
220C2×Q8—42no—
221C2×Q8—44no—
222C2×Q8—48no—
223C2×Q8—44no—
224C2×Q8—41no—
225C2×Q8—41no—
226C2×D4—22no—
227C2×D4—22no—
228C2×D4—41no—
229C22×C4—22no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—21no—
233C22×C4—21no—
234C22×C4—22no—
235C22×C4—44no—
236C22×C4—42no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4C22⋊Dic7311yescharacteristic
240C22×C4C22⋊Dic7311yescharacteristic
241C22×C4C22×Dic7311yescharacteristic
242C22×C4C2×C22⋊D7311yescharacteristic
243C22×C4C2×C22⋊D7311yescharacteristic
244M16—2922no—
245M16—5842no—
246M16—1464no—
247C2×C8—2922no—
248C2×C8—2922no—
249C4⋊C4—41no—
250C4⋊C4—42no—
251C4⋊C4—44no—
252C22⋊C4—41no—
253C22⋊C4—41no—
254C22⋊C4—44no—
255C22⋊C4—42no—
256C42—21no—
257C42—21no—
258C23—21no—
259C23—21no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—41no—
264C23—41no—
265C23—41no—
266C23—41no—
267C23#1455651812yes—
268C23C2×C22⋊Dic7311yescharacteristic
269Q8—44no—
270Q8—44no—
271Q8—44no—
272Q8—44no—
273D4—44no—
274C2×C4—21no—
275C2×C4—21no—
276C2×C4—22no—
277C2×C4—24no—
278C2×C4—21no—
279C2×C4—21no—
280C2×C4—21no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—48no—
284C2×C4—42no—
285C2×C4—42no—
286C2×C4—42no—
287C2×C4—41no—
288C2×C4—41no—
289C2×C4—41no—
290C2×C4—42no—
291C2×C4—42no—
292C2×C4—42no—
293C2×C4—81no—
294C2×C4#1455650611yescharacteristic
295C2×C4#1455650611yescharacteristic
296C2×C4#1455662212yes—
297C8—2924no—
298C8—5842no—
299C22—21no—
300C22—21no—
301C22—22no—
302C22—42no—
303C22—42no—
304C22—42no—
305C22—41no—
306C22—41no—
307C22—41no—
308C22—41no—
309C22—41no—
310C22—81no—
311C22C2×#1455650611yescharacteristic
312C22#1798859611yescharacteristic
313C22C2×#1455651811yescharacteristic
314C4—22no—
315C4—21no—
316C4—41no—
317C4—44no—
318C4—41no—
319C4—41no—
320C4—82no—
321C4#1798837412yes—
322C2—21no—
323C2—21no—
324C2—42no—
325C2—81no—
326C2#2260845111yescharacteristic
327C2C2×#1798837411yescharacteristic
328C2#2261162611yescharacteristic
subgroupG/H
centerC22C2×#14556506
commutatorC22×C4×C73C22×C4
FrattiniC2×C4⋊C4C22×D73
FittingC73×#2409C2
socleC22×C73C2×#99
prime radicalC22×D4×C73D4
cosocleC2×C4⋊C4×C73C23
hypercenter#2409D73
hypocenterC73#11791
LCSG ▷ C22×C4×C73 ▷ C23×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×D4 ◁ #2409

Complements

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

NH ≅ G/Ntype
3. #22607961C2semidirect
169. C73#11791semidirect

Maximal subgroups

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

subgroupindex
C4×#145565082
#226082692
#226079612
C73×#24092
#1179173