Finite Group Databasebeta

← #32982060

Subgroups — #32982060

Order 18688.

578 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 578 conjugacy classes of subgroups — too many to draw. Only its 111 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#22615673C211yesmaximal, characteristic
2#22612534C211yesmaximal, characteristic, complemented
3#22608500C212yesmaximal
4#22607799C212yesmaximal, complemented
5C2×#324×C73C211yesmaximal, characteristic, complemented
6C2×#14556574—21no—
7C2×#14556565C2211yescharacteristic
8#17988517C412yescomplemented
9#17988509—22no—
10#17988504C412yescomplemented
11#17988473—24no—
12#17988466—21no—
13#17988450—21no—
14#17988447C2211yescharacteristic
15#17988447C2212yes—
16C4×C22⋊Dic73—21no—
17#17988322—24no—
18C22×C4⋊C4×C73C2211yescharacteristic
19C2×C22⋊C8×C73C2212yes—
20#324×C73—22no—
21C22×D292C2×C411yescharacteristic
22C22×C22⋊D73—21no—
23#14556574—44no—
24#14556565—22no—
25#14556565—22no—
26C2×C4⋊Dic73—22no—
27C2×C4⋊Dic73—21no—
28C2×C4⋊Dic73C2×C411yescharacteristic
29C2×#11924007—22no—
30C2×C22⋊Dic73—22no—
31C2×C22⋊Dic73—21no—
32C2×#11924005—22no—
33C2×#11924005—22no—
34C2×#11924005—21no—
35C2×C4×Dic73—22no—
36#14556498—22no—
37#14556498—24no—
38#14556498—22no—
39#14556498C2×C412yes—
40#14556498D414yes—
41C23×C4×C73D411yescharacteristic
42C23×C4×C73D411yescharacteristic
43C23×C4×C73C2311yescharacteristic
44C22×C8×C73—22no—
45C2×C4⋊C4×C73—22no—
46C2×C4⋊C4×C73—22no—
47C2×C4⋊C4×C73C2×C412yes—
48C2×C4⋊C4×C73D412yes—
49C22⋊C8×C73—22no—
50C22⋊C8×C73—22no—
51C23×D73—21no—
52C2×D292—22no—
53C2×D292—81no—
54C2×C22⋊D73—41no—
55C2×C22⋊D73—41no—
56C2×C22⋊D73—42no—
57C2×C22⋊D73—42no—
58C22×Dic73—21no—
59C22×Dic73—22no—
60C22×Dic73—24no—
61C4⋊Dic73—22no—
62C4⋊Dic73—82no—
63C4⋊Dic73—81no—
64#11924007—44no—
65C22⋊Dic73—42no—
66C22⋊Dic73—42no—
67C22⋊Dic73—42no—
68#11924005—44no—
69#11924005—42no—
70#11924005—82no—
71C4×Dic73—82no—
72C24×C73C2×D411yescharacteristic
73C22×C4×C73—21no—
74C22×C4×C73—21no—
75C22×C4×C73—21no—
76C22×C4×C73—22no—
77C22×C4×C73—22no—
78C22×C4×C73—21no—
79C22×C4×C73—81no—
80C22×C4×C73—81no—
81C22×C4×C73C22⋊C412yes—
82C22×C4×C73C22⋊C412yes—
83C22×C4×C73C22×C411yescharacteristic
84C22×C4×C73C2×D412yes—
85C22×C4×C73C2×D411yescharacteristic
86C22×C4×C73#4112yes—
87C2×C8×C73—42no—
88C2×C8×C73—82no—
89C4⋊C4×C73—24no—
90C4⋊C4×C73—24no—
91C23×C73—41no—
92C23×C73—41no—
93C23×C73C2×C22⋊C411yescharacteristic
94C23×C73#12011yescharacteristic
95C23×C73#12311yescharacteristic
96C23×C73#13512yes—
97C23×C73AGL(1,ℤ/8)12yes—
98C22×D73—42no—
99C22×D73—42no—
100C22×D73—41no—
101C22×D73—41no—
102C22×D73—41no—
103C2×C4×C73—22no—
104C2×C4×C73—22no—
105C2×C4×C73—22no—
106C2×C4×C73—22no—
107C2×C4×C73—42no—
108C2×C4×C73—41no—
109C2×C4×C73—42no—
110C2×C4×C73—41no—
111C2×C4×C73—42no—
112C2×C4×C73—42no—
113C2×C4×C73—42no—
114C2×C4×C73—42no—
115C2×C4×C73—81no—
116C2×C4×C73C4×D412yes—
117C2×C4×C73#12112yes—
118C22⋊D73—82no—
119C22⋊D73—82no—
120C2×Dic73—44no—
121C2×Dic73—42no—
122C2×Dic73—44no—
123C2×Dic73—42no—
124C2×Dic73—41no—
125C2×Dic73—41no—
126C2×Dic73—42no—
127C2×Dic73—41no—
128C2×Dic73—44no—
129D292—82no—
130C8×C73—82no—
131C22×C73—21no—
132C22×C73—21no—
133C22×C73—21no—
134C22×C73—21no—
135C22×C73—21no—
136C22×C73—21no—
137C22×C73—42no—
138C22×C73—42no—
139C22×C73#38611yescharacteristic
140C22×C73#41611yescharacteristic
141C22×C73#41811yescharacteristic
142C22×C73#44912yes—
143C22×C73#48212yes—
144C2×D73—82no—
145C2×D73—82no—
146C2×D73—81no—
147C2×D73—81no—
148C2×D73—81no—
149C4×C73—22no—
150C4×C73—22no—
151C4×C73—42no—
152C4×C73—42no—
153Dic73—82no—
154Dic73—84no—
155Dic73—81no—
156#9206—731nomaximal, Sylow
157C2×C73—21no—
158C2×C73—22no—
159C2×C73—21no—
160C2×C73—22no—
161C2×C73#187511yescharacteristic
162C2×C73#191211yescharacteristic
163C2×C73#191311yescharacteristic
164D73—161no—
165#2618—731no—
166#2313—731no—
167#1519—732no—
168C2×#324D7311yescharacteristic, complemented
169#1289—732no—
170C73#920611yescharacteristic, complemented, Sylow
171C2×#123—1461no—
172C2×#121—731no—
173C22×C4⋊C4C2×D7311yescharacteristic
174C2×C22⋊C8C2×D7312yes—
175#398—732no—
176#396—1462no—
177#394—732no—
178#387—1464no—
179#386—1461no—
180#381—1461no—
181#380—731no—
182#380—732no—
183C4×C22⋊C4—1461no—
184#327—1464no—
185#324—22no—
186C22×D4—731no—
187C22×D4—1461no—
188C23×C4D29211yescharacteristic
189C23×C4D29211yescharacteristic
190C23×C4C22×D7311yescharacteristic
191C22×C8—22no—
192#123—2924no—
193#121—1462no—
194#121—1462no—
195C2×C4⋊C4—22no—
196C2×C4⋊C4—22no—
197C2×C4⋊C4—731no—
198C2×C4⋊C4—1462no—
199C2×C4⋊C4—1462no—
200C2×C4⋊C4—1461no—
201C2×C4⋊C4C4×D7312yes—
202C2×C4⋊C4C22⋊D7312yes—
203C2×C22⋊C4—1461no—
204C2×C22⋊C4—1462no—
205C2×C22⋊C4—1462no—
206C2×C22⋊C4—1461no—
207C2×C22⋊C4—1462no—
208C2×C42—1462no—
209C22⋊C8—22no—
210C22⋊C8—22no—
211#95—732no—
212#95—734no—
213#95—1462no—
214#95—1464no—
215#95—1462no—
216C24—1461no—
217C24C2×D29211yescharacteristic
218C2×D4—2921no—
219C2×D4—2921no—
220C2×D4—2922no—
221C2×D4—2922no—
222C2×D4—5841no—
223C2×D4—1462no—
224C22×C4—21no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—22no—
228C22×C4—22no—
229C22×C4—21no—
230C22×C4—81no—
231C22×C4—81no—
232C22×C4—1462no—
233C22×C4—1461no—
234C22×C4—1464no—
235C22×C4#1192400512yes—
236C22×C4#1192400512yes—
237C22×C4C2×C4×D7311yescharacteristic
238C22×C4D4×D7312yes—
239C22×C4C2×C22⋊D7311yescharacteristic
240C22×C4#1192402712yes—
241C2×C8—42no—
242C2×C8—82no—
243C4⋊C4—24no—
244C4⋊C4—24no—
245C4⋊C4—2924no—
246C4⋊C4—5842no—
247C4⋊C4—5841no—
248C4⋊C4—1462no—
249C22⋊C4—2924no—
250C22⋊C4—2922no—
251C22⋊C4—2922no—
252C22⋊C4—2922no—
253C22⋊C4—2922no—
254C22⋊C4—5842no—
255C42—5842no—
256C23—41no—
257C23—41no—
258C23—2922no—
259C23—2922no—
260C23—2921no—
261C23—2921no—
262C23—2921no—
263C23C2×#1192400511yescharacteristic
264C23#1455656111yescharacteristic
265C23#1455657511yescharacteristic
266C23#1455661312yes—
267C23#1455661912yes—
268D4—5842no—
269D4—5842no—
270D4—5842no—
271C2×C4—22no—
272C2×C4—22no—
273C2×C4—22no—
274C2×C4—22no—
275C2×C4—42no—
276C2×C4—41no—
277C2×C4—42no—
278C2×C4—41no—
279C2×C4—42no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—81no—
284C2×C4—2924no—
285C2×C4—2922no—
286C2×C4—2924no—
287C2×C4—2922no—
288C2×C4—2921no—
289C2×C4—2924no—
290C2×C4—2921no—
291C2×C4—2922no—
292C2×C4—2921no—
293C2×C4#1455655112yes—
294C2×C4#1455656312yes—
295C8—82no—
296C22—21no—
297C22—21no—
298C22—21no—
299C22—21no—
300C22—21no—
301C22—21no—
302C22—42no—
303C22—42no—
304C22—5842no—
305C22—5842no—
306C22—5841no—
307C22—5841no—
308C22—5841no—
309C22#1798846411yescharacteristic
310C22#1798858411yescharacteristic
311C22#1798859311yescharacteristic
312C22#1798873412yes—
313C22#1798891212yes—
314C4—22no—
315C4—22no—
316C4—42no—
317C4—42no—
318C4—5842no—
319C4—5844no—
320C4—5841no—
321C2—21no—
322C2—22no—
323C2—21no—
324C2—22no—
325C2—11681no—
326C2#2261025111yescharacteristic
327C2#2261046611yescharacteristic
328C2#2261047111yescharacteristic
subgroupG/H
centerC22#17988464
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC2×#324×C73C2
socleC22×C73#386
prime radicalC2×#14556565C22
cosocleC23×C4×C73C23
hypercenterC2×#324D73
hypocenterC73#9206
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
2. #22612534C2semidirect
4. #22607799C2semidirect
5. C2×#324×C73C2semidirect
8. #17988517C4semidirect
10. #17988504C4semidirect
168. C2×#324D73semidirect
170. C73#9206semidirect

Maximal subgroups

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

subgroupindex
#226156732
#226125342
#226085002
#226077992
C2×#324×C732
#920673