Finite Group Databasebeta

← #33060890

Subgroups — #33060890

Order 18688.

870 conjugacy classes of subgroups; only the 207 normal subgroups are shown.

normal subgroups only

This group has 870 conjugacy classes of subgroups — too many to draw. Only its 207 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#22621079C212yesmaximal
2#22615143C211yesmaximal, characteristic
3#22608973C218yesmaximal
4#22608430C212yesmaximal
5#22608255C211yesmaximal, characteristic
6C73×#3502C211yesmaximal, characteristic
7C2×#14556591C2211yescharacteristic
8#17988990—22no—
9#17988937—22no—
10#17988920C2218yes—
11#17988882—24no—
12#17988640C2212yes—
13#17988526C2214yes—
14#17988494—21no—
15#17988493—22no—
16#17988452—21no—
17C4×C4⋊Dic73C2211yescharacteristic
18#17988324C2218yes—
19#17988313C2214yes—
20Q82×C73—22no—
21#557×C73—22no—
22#557×C73—24no—
23#544×C73—21no—
24#544×C73C2214yes—
25#533×C73—22no—
26#533×C73C2212yes—
27#519×C73—21no—
28C2×C4×Q8×C73C2211yescharacteristic
29C22×Dic146—21no—
30#14556591—22no—
31#14556591—22no—
32C2×C4⋊Dic73—21no—
33C2×C4⋊Dic73—21no—
34C2×C4⋊Dic73C2312yes—
35C2×C4×Dic73—21no—
36#14556524—28no—
37C4⋊#10596831—22no—
38#14556515—28no—
39C22⋊#10596831C2314yes—
40C4×#10596831—22no—
41#14556498—24no—
42#14556498—24no—
43#14556498C2312yes—
44C22×Q8×C73D411yescharacteristic
45C4⋊Q8×C73—22no—
46C4⋊Q8×C73—22no—
47C4⋊Q8×C73—24no—
48C4⋊Q8×C73—28no—
49C4⋊Q8×C73D412yes—
50C4⋊Q8×C73D412yes—
51#125×C73—24no—
52#125×C73—24no—
53#125×C73—28no—
54#125×C73—22no—
55C22⋊Q8×C73—22no—
56C22⋊Q8×C73—24no—
57C22⋊Q8×C73C2314yes—
58C4×Q8×C73—24no—
59C4×Q8×C73—24no—
60C4×Q8×C73—24no—
61C4×Q8×C73—24no—
62C4×Q8×C73—22no—
63C4×Q8×C73—22no—
64C4×Q8×C73—22no—
65C4×Q8×C73D414yes—
66#117×C73—24no—
67#117×C73—22no—
68C2×C4⋊C4×C73D411yescharacteristic
69C2×C4⋊C4×C73C2312yes—
70C2×C42×C73D412yes—
71C2×C42×C73C2311yescharacteristic
72C2×Dic146—44no—
73C2×Dic146—41no—
74C2×Dic146—41no—
75C22×Dic73—22no—
76C22×Dic73—22no—
77C2×#10596831—24no—
78C4⋊Dic73—24no—
79C4⋊Dic73—42no—
80C4⋊Dic73—42no—
81C4×Dic73—42no—
82C2×Q8×C73—24no—
83C2×Q8×C73—22no—
84C2×Q8×C73—22no—
85C2×Q8×C73—22no—
86C2×Q8×C73Q1614yes—
87C2×Q8×C73C2×D414yes—
88C22×C4×C73C2×D411yescharacteristic
89C22×C4×C73C2×D412yes—
90C22×C4×C73C2×D412yes—
91C22×C4×C73C2×D411yescharacteristic
92C22×C4×C73C2411yescharacteristic
93C4⋊C4×C73—24no—
94C4⋊C4×C73—24no—
95C4⋊C4×C73—24no—
96C4⋊C4×C73—28no—
97C4⋊C4×C73—22no—
98C4⋊C4×C73—22no—
99C4⋊C4×C73—22no—
100C4⋊C4×C73—22no—
101C4⋊C4×C73—24no—
102C4⋊C4×C73—24no—
103C4⋊C4×C73—24no—
104C4⋊C4×C73C2×D414yes—
105C22⋊C4×C73—24no—
106C42×C73—28no—
107C42×C73—22no—
108C42×C73—22no—
109C42×C73—24no—
110C42×C73C2×D412yes—
111C42×C73C2×D412yes—
112C23×C73C22×D411yescharacteristic
113Q8×C73—28no—
114Q8×C73—44no—
115Q8×C73—44no—
116Q8×C73—84no—
117C2×C4×C73—22no—
118C2×C4×C73—22no—
119C2×C4×C73—22no—
120C2×C4×C73—24no—
121C2×C4×C73—24no—
122C2×C4×C73—22no—
123C2×C4×C73—41no—
124C2×C4×C73—42no—
125C2×C4×C73#12011yescharacteristic
126C2×C4×C73#12012yes—
127C2×C4×C73#12011yescharacteristic
128C2×C4×C73C2×Q1614yes—
129C2×C4×C73C2×Q1612yes—
130C2×C4×C73#13712yes—
131C2×C4×C73C22×D412yes—
132C2×Dic73—42no—
133C2×Dic73—42no—
134C2×Dic73—44no—
135C2×Dic73—42no—
136C2×Dic73—42no—
137Dic146—82no—
138Dic146—82no—
139#10596831—44no—
140C22×C73—21no—
141C22×C73—21no—
142C22×C73C2×#12011yescharacteristic
143C22×C73C22×Q1611yescharacteristic
144C22×C73C2×#13711yescharacteristic
145C4×C73—24no—
146C4×C73—21no—
147C4×C73—22no—
148C4×C73—44no—
149C4×C73—42no—
150C4×C73—44no—
151C4×C73#45114yes—
152C4×C73#45412yes—
153Dic73—82no—
154Dic73—82no—
155#18411—731nomaximal, Sylow
156C2×C73—21no—
157C2×C73—21no—
158C2×C73C2×#45111yescharacteristic
159C2×C73C2×#45411yescharacteristic
160C2×C73#302811yescharacteristic
161#3502D7311yescharacteristic
162#3230—732no—
163#2568—731no—
164#1617—738no—
165#1503—732no—
166#1461—731no—
167C73#1841111yescharacteristic, complemented, Sylow
168Q82—22no—
169#557—22no—
170#557—24no—
171#544—21no—
172#544C2×D7314yes—
173#533—22no—
174#533C2×D7312yes—
175C2×C4⋊Q8—731no—
176#519—21no—
177C2×C4×Q8C2×D7311yescharacteristic
178#500—1462no—
179#487—1462no—
180#484—738no—
181#477—1464no—
182#431—732no—
183#400—734no—
184#391—1461no—
185#391—1462no—
186#382—1461no—
187C4×C4⋊C4—731no—
188#328—738no—
189#324—734no—
190C22×Q8—1461no—
191C22×Q8C22⋊D7311yescharacteristic
192C4⋊Q8—22no—
193C4⋊Q8—22no—
194C4⋊Q8—24no—
195C4⋊Q8—28no—
196C4⋊Q8—1462no—
197C4⋊Q8—1462no—
198C4⋊Q8C22⋊D7312yes—
199C4⋊Q8C22⋊D7312yes—
200#125—24no—
201#125—24no—
202#125—28no—
203#125—22no—
204C22⋊Q8—22no—
205C22⋊Q8—24no—
206C22⋊Q8C22×D7314yes—
207C4×Q8—24no—
208C4×Q8—24no—
209C4×Q8—24no—
210C4×Q8—24no—
211C4×Q8—22no—
212C4×Q8—22no—
213C4×Q8—22no—
214C4×Q8C22⋊D7314yes—
215#117—24no—
216#117—22no—
217C2×C4⋊C4—732no—
218C2×C4⋊C4—1461no—
219C2×C4⋊C4—1461no—
220C2×C4⋊C4C22⋊D7311yescharacteristic
221C2×C4⋊C4C22×D7312yes—
222C2×C42—1461no—
223C2×C42C22⋊D7312yes—
224C2×C42C22×D7311yescharacteristic
225#107—1468no—
226C4⋊C8—1462no—
227#103—1468no—
228C22⋊C8—734no—
229C4×C8—1462no—
230#95—732no—
231#95—1464no—
232#95—1464no—
233C2×Q8—24no—
234C2×Q8—22no—
235C2×Q8—22no—
236C2×Q8—22no—
237C2×Q8—2924no—
238C2×Q8—2921no—
239C2×Q8—2921no—
240C2×Q8#1192401814yes—
241C2×Q8C2×C22⋊D7314yes—
242C22×C4—1462no—
243C22×C4—1462no—
244C22×C4C2×C22⋊D7311yescharacteristic
245C22×C4C2×C22⋊D7312yes—
246C22×C4C2×C22⋊D7312yes—
247C22×C4C2×C22⋊D7311yescharacteristic
248C22×C4C23×D7311yescharacteristic
249C2×C8—1464no—
250C4⋊C4—24no—
251C4⋊C4—24no—
252C4⋊C4—24no—
253C4⋊C4—28no—
254C4⋊C4—22no—
255C4⋊C4—22no—
256C4⋊C4—22no—
257C4⋊C4—22no—
258C4⋊C4—24no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—2922no—
262C4⋊C4—2922no—
263C4⋊C4—1464no—
264C4⋊C4C2×C22⋊D7314yes—
265C22⋊C4—24no—
266C42—28no—
267C42—22no—
268C42—22no—
269C42—24no—
270C42—2922no—
271C42C2×C22⋊D7312yes—
272C42C2×C22⋊D7312yes—
273C23C22×C22⋊D7311yescharacteristic
274Q8—28no—
275Q8—44no—
276Q8—44no—
277Q8—84no—
278Q8—5842no—
279Q8—5842no—
280C2×C4—22no—
281C2×C4—22no—
282C2×C4—22no—
283C2×C4—24no—
284C2×C4—24no—
285C2×C4—22no—
286C2×C4—41no—
287C2×C4—42no—
288C2×C4—2922no—
289C2×C4—2924no—
290C2×C4—2922no—
291C2×C4—2922no—
292C2×C4—2922no—
293C2×C4#1455656011yescharacteristic
294C2×C4#1455656012yes—
295C2×C4#1455656011yescharacteristic
296C2×C4C2×#1192401814yes—
297C2×C4C2×#1192401812yes—
298C2×C4#1455662212yes—
299C2×C4C22×C22⋊D7312yes—
300C8—2924no—
301C22—21no—
302C22—21no—
303C22C2×#1455656011yescharacteristic
304C22C22×#1192401811yescharacteristic
305C22C2×#1455662211yescharacteristic
306C4—21no—
307C4—22no—
308C4—24no—
309C4—44no—
310C4—42no—
311C4—44no—
312C4—5842no—
313C4—5842no—
314C4#1798874314yes—
315C4#1798876012yes—
316C2—21no—
317C2—21no—
318C2C2×#1798874311yescharacteristic
319C2C2×#1798876011yescharacteristic
320C2#2261933711yescharacteristic
subgroupG/H
centerC22C2×#14556560
commutator, cosocleC22×C4×C73C24
FrattiniC22×C4C23×D73
FittingC73×#3502C2
socleC22×C73C2×#120
prime radicalC23×C73C22×D4
hypercenter#3502D73
hypocenterC73#18411
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ #3502

Complements

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

NH ≅ G/Ntype
167. C73#18411semidirect

Maximal subgroups

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

subgroupindex
#226210792
#226151432
#226089732
#226084302
#226082552
C73×#35022
#1841173