Finite Group Databasebeta

← #32980976

Subgroups — #32980976

Order 18688.

756 conjugacy classes of subgroups; only the 127 normal subgroups are shown.

normal subgroups only

This group has 756 conjugacy classes of subgroups — too many to draw. Only its 127 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
1C2×#17988507C211yesmaximal, characteristic
2#22612534C211yesmaximal, characteristic
3#22609867C212yesmaximal
4#22607811C212yesmaximal
5C2×#324×C73C211yesmaximal, characteristic
6C23×Dic146C2211yescharacteristic
7C22×#11924007—21no—
8C2×#14556516—24no—
9#17988507—22no—
10#17988507C414yescomplemented
11#17988473—24no—
12#17988450—21no—
13#17988447C2212yes—
14C4×C22⋊Dic73—21no—
15C2×#14556498—21no—
16C2×#14556498C2211yescharacteristic
17C22×C4⋊C4×C73C2211yescharacteristic
18C2×C22⋊C8×C73C2212yes—
19#324×C73—22no—
20C22×Dic146—21no—
21C22×Dic146—22no—
22C22×Dic146—22no—
23C22×Dic146—44no—
24C22×Dic146C2×C412yes—
25C23×Dic73—21no—
26C23×Dic73—21no—
27C2×C4⋊Dic73D414yes—
28C2×#11924007—24no—
29C2×#11924007—44no—
30C2×C22⋊Dic73—22no—
31C2×C4×Dic73—22no—
32#14556516—48no—
33#14556498—22no—
34#14556498—24no—
35#14556498—24no—
36#14556498—21no—
37#14556498—22no—
38#14556498C2×C412yes—
39C23×C4×C73D411yescharacteristic
40C23×C4×C73D411yescharacteristic
41C23×C4×C73C2311yescharacteristic
42C22×C8×C73—22no—
43C2×C4⋊C4×C73—22no—
44C2×C4⋊C4×C73—22no—
45C2×C4⋊C4×C73C2×C412yes—
46C2×C4⋊C4×C73D412yes—
47C22⋊C8×C73—22no—
48C22⋊C8×C73—22no—
49C2×Dic146—28no—
50C2×Dic146—24no—
51C2×Dic146—44no—
52C2×Dic146—48no—
53C2×Dic146—44no—
54C2×Dic146—44no—
55C2×Dic146—42no—
56C22×Dic73—24no—
57C22×Dic73—24no—
58C22×Dic73—22no—
59C22×Dic73—22no—
60C22×Dic73—24no—
61C22×Dic73—42no—
62C22×Dic73—42no—
63C22×Dic73—41no—
64C22×Dic73—41no—
65C22×Dic73—42no—
66C4⋊Dic73—44no—
67#11924007—48no—
68C22⋊Dic73—42no—
69C22⋊Dic73—42no—
70C4×Dic73—82no—
71C24×C73C2×D411yescharacteristic
72C22×C4×C73—21no—
73C22×C4×C73—21no—
74C22×C4×C73—21no—
75C22×C4×C73—22no—
76C22×C4×C73—22no—
77C22×C4×C73—21no—
78C22×C4×C73—42no—
79C22×C4×C73—42no—
80C22×C4×C73C22⋊C412yes—
81C22×C4×C73C22⋊C412yes—
82C22×C4×C73C22×C411yescharacteristic
83C22×C4×C73C2×D412yes—
84C22×C4×C73C2×D411yescharacteristic
85C22×C4×C73#4112yes—
86C2×C8×C73—44no—
87C2×C8×C73—42no—
88C4⋊C4×C73—24no—
89C4⋊C4×C73—24no—
90C23×C73—22no—
91C23×C73—22no—
92C23×C73C2×C22⋊C411yescharacteristic
93C23×C73#12011yescharacteristic
94C23×C73#12311yescharacteristic
95C23×C73#13512yes—
96C23×C73#13712yes—
97C2×C4×C73—24no—
98C2×C4×C73—22no—
99C2×C4×C73—22no—
100C2×C4×C73—22no—
101C2×C4×C73—22no—
102C2×C4×C73—41no—
103C2×C4×C73—41no—
104C2×C4×C73—42no—
105C2×C4×C73—42no—
106C2×C4×C73—42no—
107C2×C4×C73—42no—
108C2×C4×C73—42no—
109C2×C4×C73—81no—
110C2×C4×C73C4×D412yes—
111C2×C4×C73#12112yes—
112C2×Dic73—24no—
113C2×Dic73—24no—
114C2×Dic73—42no—
115C2×Dic73—48no—
116C2×Dic73—42no—
117C2×Dic73—44no—
118C2×Dic73—42no—
119C2×Dic73—42no—
120C2×Dic73—44no—
121C2×Dic73—44no—
122C2×Dic73—44no—
123C2×Dic73—44no—
124Dic146—28no—
125Dic146—48no—
126Dic146—44no—
127C8×C73—82no—
128C22×C73—24no—
129C22×C73—24no—
130C22×C73—21no—
131C22×C73—21no—
132C22×C73—21no—
133C22×C73—21no—
134C22×C73#38611yescharacteristic
135C22×C73#41611yescharacteristic
136C22×C73#41911yescharacteristic
137C22×C73#45212yes—
138C22×C73#45612yes—
139C22×C73B(4,𝔽2)12yes—
140C22×C73#48512yes—
141C4×C73—22no—
142C4×C73—22no—
143C4×C73—42no—
144C4×C73—42no—
145Dic73—44no—
146Dic73—44no—
147Dic73—84no—
148#9014—731nomaximal, Sylow
149C2×C73—21no—
150C2×C73—21no—
151C2×C73—22no—
152C2×C73#160314yes—
153C2×C73#187311yescharacteristic
154C2×C73#188611yescharacteristic
155C2×C73#188911yescharacteristic
156C2×#395—731no—
157#2313—731no—
158#1787—732no—
159C2×#324D7311yescharacteristic
160#1294—732no—
161C73#901411yescharacteristic, complemented, Sylow
162C23×Q8—731no—
163C22×C4⋊C4—1461no—
164C22×C4⋊C4C2×D7311yescharacteristic
165C2×#103—1464no—
166C2×C22⋊C8C2×D7312yes—
167#395—734no—
168#395—1462no—
169#387—1464no—
170#381—1461no—
171#380—732no—
172C4×C22⋊C4—1461no—
173C2×#95—731no—
174C2×#95—1461no—
175#324—22no—
176C22×Q8—2924no—
177C22×Q8—732no—
178C22×Q8—1462no—
179C22×Q8—1462no—
180C22×Q8—1461no—
181C23×C4—1461no—
182C23×C4—1461no—
183C23×C4D29211yescharacteristic
184C23×C4D29211yescharacteristic
185C23×C4C22×D7311yescharacteristic
186C22×C8—22no—
187C2×C4⋊C4—22no—
188C2×C4⋊C4—22no—
189C2×C4⋊C4—2924no—
190C2×C4⋊C4—734no—
191C2×C4⋊C4—1464no—
192C2×C4⋊C4C4×D7312yes—
193C2×C4⋊C4C22⋊D7312yes—
194C2×C22⋊C4—1462no—
195C2×C42—1462no—
196#103—2928no—
197C22⋊C8—22no—
198C22⋊C8—22no—
199#95—732no—
200#95—1462no—
201#95—1464no—
202#95—1464no—
203#95—1462no—
204#95—1461no—
205C24C2×D29211yescharacteristic
206C2×Q8—2924no—
207C2×Q8—2928no—
208C2×Q8—2924no—
209C2×Q8—2924no—
210C2×Q8—2922no—
211C2×Q8—1468no—
212C2×Q8—1464no—
213C22×C4—21no—
214C22×C4—21no—
215C22×C4—21no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—21no—
219C22×C4—42no—
220C22×C4—42no—
221C22×C4—2922no—
222C22×C4—2922no—
223C22×C4—2921no—
224C22×C4—2921no—
225C22×C4—2922no—
226C22×C4—1464no—
227C22×C4—1464no—
228C22×C4—1464no—
229C22×C4—1462no—
230C22×C4—1462no—
231C22×C4#1192400512yes—
232C22×C4#1192400512yes—
233C22×C4C2×C4×D7311yescharacteristic
234C22×C4D4×D7312yes—
235C22×C4C2×C22⋊D7311yescharacteristic
236C22×C4#1192402712yes—
237C2×C8—44no—
238C2×C8—42no—
239C4⋊C4—24no—
240C4⋊C4—24no—
241C4⋊C4—2924no—
242C4⋊C4—2928no—
243C22⋊C4—2922no—
244C22⋊C4—2922no—
245C42—5842no—
246C23—22no—
247C23—22no—
248C23C2×#1192400511yescharacteristic
249C23#1455656111yescharacteristic
250C23#1455657511yescharacteristic
251C23#1455661312yes—
252C23#1455662412yes—
253Q8—2928no—
254Q8—2924no—
255Q8—1468no—
256C2×C4—24no—
257C2×C4—22no—
258C2×C4—22no—
259C2×C4—22no—
260C2×C4—22no—
261C2×C4—41no—
262C2×C4—41no—
263C2×C4—42no—
264C2×C4—42no—
265C2×C4—42no—
266C2×C4—42no—
267C2×C4—42no—
268C2×C4—81no—
269C2×C4—2922no—
270C2×C4—2922no—
271C2×C4—2924no—
272C2×C4—2928no—
273C2×C4—2922no—
274C2×C4—2924no—
275C2×C4—2922no—
276C2×C4—2924no—
277C2×C4—2924no—
278C2×C4—2924no—
279C2×C4—1464no—
280C2×C4—1464no—
281C2×C4#1455655112yes—
282C2×C4#1455656312yes—
283C8—82no—
284C22—24no—
285C22—24no—
286C22—21no—
287C22—21no—
288C22—21no—
289C22—21no—
290C22#1798846411yescharacteristic
291C22#1798858411yescharacteristic
292C22#1798859711yescharacteristic
293C22#1798875112yes—
294C22#1798877212yes—
295C22#1798877512yes—
296C22#1798892912yes—
297C4—22no—
298C4—22no—
299C4—42no—
300C4—42no—
301C4—2924no—
302C4—2924no—
303C4—5844no—
304C2—21no—
305C2—22no—
306C2—21no—
307C2#2260889614yes—
308C2#2261024111yescharacteristic
309C2#2261032211yescharacteristic
310C2#2261033711yescharacteristic
subgroupG/H
centerC23#14556561
commutatorC22×C4×C73C22×C4
FrattiniC23×C4C22×D73
FittingC2×#324×C73C2
socleC23×C73#120
prime radicalC24×C73C2×D4
cosocleC23×C4×C73C23
hypercenterC2×#324D73
hypocenterC73#9014
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C2×#324

Complements

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

NH ≅ G/Ntype
10. #17988507C4semidirect
161. C73#9014semidirect

Maximal subgroups

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

subgroupindex
C2×#179885072
#226125342
#226098672
#226078112
C2×#324×C732
#901473