Finite Group Databasebeta

← #34007633

Subgroups — #34007633

Order 18688.

640 conjugacy classes of subgroups; only the 126 normal subgroups are shown.

normal subgroups only

This group has 640 conjugacy classes of subgroups — too many to draw. Only its 126 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#22625787C211yesmaximal, characteristic
2#22625657C211yesmaximal, characteristic
3C2×#17989432C214yesmaximal
4C2×#17988486C211yesmaximal, characteristic
5C2×#14556658—21no—
6C2×C22⋊#10596833C2212yes—
7C2×C22⋊#10596833C2212yes—
8C2×#14556646—21no—
9#17989453—21no—
10#17989453—21no—
11#17989432—28no—
12C22×C4⋊Dic73C411yescharacteristic
13C22×C4⋊Dic73C2212yes—
14C22×C4×Dic73C2211yescharacteristic
15#17988486—21no—
16#17988486—21no—
17#17988486—21no—
18#17988486—21no—
19C2×#14556498C412yes—
20C22×C4⋊C4×C73C411yescharacteristic
21C22×#10596833—22no—
22C22×#10596833—22no—
23#14556658—42no—
24C22⋊#10596833—24no—
25C22⋊#10596833—22no—
26C22⋊#10596833D414yes—
27#14556646—42no—
28C23×Dic73C2×C411yescharacteristic
29C23×Dic73C2×C412yes—
30C23×Dic73C2311yescharacteristic
31C2×C4⋊Dic73—24no—
32C2×C4⋊Dic73—22no—
33C2×C4⋊Dic73—22no—
34C2×C4⋊Dic73—24no—
35C2×C4⋊Dic73—21no—
36C2×C4⋊Dic73—21no—
37C2×C4⋊Dic73—42no—
38C2×C4×Dic73—42no—
39C2×C4×Dic73D411yescharacteristic
40C2×C4×Dic73D411yescharacteristic
41C2×C4×Dic73D411yescharacteristic
42C2×C4×Dic73D411yescharacteristic
43#14556498—22no—
44#14556498—22no—
45C23×C4×C73C2×C411yescharacteristic
46C23×C4×C73C2×C412yes—
47C2×C4⋊C4×C73—21no—
48C2×C4⋊C4×C73—21no—
49C2×C4⋊C4×C73—24no—
50C2×#10596833—24no—
51C2×#10596833—42no—
52C2×#10596833—42no—
53C2×#10596833—42no—
54C2×#10596833—42no—
55C2×#10596833—42no—
56C22×Dic73—22no—
57C22×Dic73—22no—
58C22×Dic73—22no—
59C22×Dic73—24no—
60C22×Dic73—41no—
61C22×Dic73—41no—
62C22×Dic73—41no—
63C22×Dic73—42no—
64C22×Dic73—42no—
65C22×Dic73C22⋊C411yescharacteristic
66C22×Dic73C22⋊C411yescharacteristic
67C22×Dic73C2×D411yescharacteristic
68C22×Dic73C2×D411yescharacteristic
69C22×Dic73C2×D412yes—
70C22×Dic73#4112yes—
71C4⋊Dic73—42no—
72C4⋊Dic73—42no—
73C4⋊Dic73—44no—
74C4⋊Dic73—42no—
75C4⋊Dic73—42no—
76C4×Dic73—22no—
77C4×Dic73—22no—
78C4×Dic73—41no—
79C4×Dic73—41no—
80C24×C73C22×C411yescharacteristic
81C22×C4×C73—22no—
82C22×C4×C73—22no—
83C22×C4×C73—22no—
84C22×C4×C73—22no—
85C22×C4×C73—24no—
86C22×C4×C73—22no—
87C22×C4×C73—24no—
88C22×C4×C73C22⋊C411yescharacteristic
89C22×C4×C73C22⋊C411yescharacteristic
90C22×C4×C73M1614yes—
91C4⋊C4×C73—24no—
92C4⋊C4×C73—24no—
93C23×C73—21no—
94C23×C73—21no—
95C23×C73—21no—
96C23×C73—21no—
97C23×C73C2×C22⋊C411yescharacteristic
98C23×C73C4×D412yes—
99C23×C73C2×M1612yes—
100C23×C73#13112yes—
101C2×C4×C73—28no—
102C2×C4×C73—21no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—21no—
106C2×C4×C73—24no—
107C2×C4×C73—28no—
108C2×C4×C73—42no—
109C2×C4×C73—42no—
110C2×C4×C73—44no—
111C2×C4×C73#9912yes—
112C2×C4×C73#10112yes—
113C2×Dic73—21no—
114C2×Dic73—21no—
115C2×Dic73—21no—
116C2×Dic73—21no—
117C2×Dic73—21no—
118C2×Dic73—21no—
119C2×Dic73—21no—
120C2×Dic73—21no—
121C2×Dic73—21no—
122C2×Dic73—42no—
123C2×Dic73—42no—
124C2×Dic73—44no—
125C2×Dic73—42no—
126C2×Dic73—42no—
127C2×Dic73—42no—
128C2×Dic73—84no—
129C2×Dic73#10112yes—
130C2×Dic73#12112yes—
131C2×Dic73#12411yescharacteristic
132C2×Dic73#12711yescharacteristic
133#10596833—44no—
134#10596833—82no—
135C22×C73—22no—
136C22×C73—22no—
137C22×C73—22no—
138C22×C73—22no—
139C22×C73—22no—
140C22×C73—22no—
141C22×C73#39011yescharacteristic
142C22×C73#40711yescharacteristic
143C22×C73#40811yescharacteristic
144C22×C73C2×#9911yescharacteristic
145C22×C73#41011yescharacteristic
146C22×C73C2×#10111yescharacteristic
147C22×C73C2×#10111yescharacteristic
148C22×C73#43514yes—
149C4×C73—22no—
150C4×C73—22no—
151C4×C73—44no—
152Dic73—22no—
153Dic73—21no—
154Dic73—21no—
155Dic73—22no—
156Dic73—41no—
157Dic73—82no—
158#9654—731nomaximal, Sylow
159C2×C73—24no—
160C2×C73#146311yescharacteristic
161C2×C73#146711yescharacteristic
162C2×C73#147011yescharacteristic
163C2×C73#147211yescharacteristic
164C2×C73#195511yescharacteristic
165C2×C73#196411yescharacteristic
166C2×C73#196611yescharacteristic
167C2×#389—731no—
168#1797—731no—
169#1749—731no—
170C2×#324—734no—
171C73#965411yescharacteristic, complemented, Sylow
172C22×C4⋊C4—731no—
173C22×C4⋊C4—732no—
174C22×C4⋊C4#6440811yescharacteristic
175C22×C42—731no—
176C2×C4⋊C8—1461no—
177C2×C22⋊C8—732no—
178C2×C22⋊C8—732no—
179C2×#97—1461no—
180#389—1461no—
181#389—1461no—
182#389—1461no—
183#389—1461no—
184C2×#95—732no—
185#336—1461no—
186#336—1461no—
187#324—1468no—
188C23×C4—731no—
189C23×C4—732no—
190C23×C4—731no—
191C23×C4C2×#6440812yes—
192C23×C4C2×#6440811yescharacteristic
193C22×C8—1462no—
194C22×C8—1462no—
195C2×C4⋊C4—21no—
196C2×C4⋊C4—21no—
197C2×C4⋊C4—24no—
198C2×C4⋊C4—2922no—
199C2×C4⋊C4—1464no—
200C2×C4⋊C4—1464no—
201C2×C4⋊C4—1462no—
202C2×C4⋊C4—1462no—
203C2×C4⋊C4—1461no—
204C2×C4⋊C4—1461no—
205C2×C42—2922no—
206C2×C42—731no—
207C2×C42—731no—
208C2×C42—731no—
209C2×C42—731no—
210C4⋊C8—2922no—
211C22⋊C8—734no—
212C22⋊C8—1464no—
213C22⋊C8—1462no—
214#97—2922no—
215#95—1462no—
216#95—1462no—
217C24C22×#6440811yescharacteristic
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—24no—
223C22×C4—22no—
224C22×C4—24no—
225C22×C4—2921no—
226C22×C4—2921no—
227C22×C4—2921no—
228C22×C4—2922no—
229C22×C4—2922no—
230C22×C4—732no—
231C22×C4—732no—
232C22×C4—731no—
233C22×C4—731no—
234C22×C4—731no—
235C22×C4—731no—
236C22×C4—1462no—
237C22×C4—1462no—
238C22×C4—1462no—
239C22×C4—1464no—
240C22×C4C22⋊#6440811yescharacteristic
241C22×C4C22⋊#6440811yescharacteristic
242C22×C4#1192403714yes—
243C2×C8—2922no—
244C2×C8—2922no—
245C2×C8—2922no—
246C2×C8—2922no—
247C2×C8—2922no—
248C2×C8—1464no—
249C4⋊C4—24no—
250C4⋊C4—24no—
251C4⋊C4—2922no—
252C4⋊C4—2922no—
253C4⋊C4—2924no—
254C4⋊C4—2922no—
255C4⋊C4—2922no—
256C42—2921no—
257C42—2921no—
258C42—1462no—
259C42—1462no—
260C23—21no—
261C23—21no—
262C23—21no—
263C23—21no—
264C23C2×C22⋊#6440811yescharacteristic
265C23D4×#6440812yes—
266C23C2×#1192403712yes—
267C23#1455667812yes—
268C2×C4—28no—
269C2×C4—21no—
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—24no—
274C2×C4—28no—
275C2×C4—42no—
276C2×C4—42no—
277C2×C4—44no—
278C2×C4—2924no—
279C2×C4—2922no—
280C2×C4—2922no—
281C2×C4—2922no—
282C2×C4—2922no—
283C2×C4—2922no—
284C2×C4—5844no—
285C2×C4—731no—
286C2×C4—732no—
287C2×C4—731no—
288C2×C4—732no—
289C2×C4—1461no—
290C2×C4—1461no—
291C2×C4—1461no—
292C2×C4—1461no—
293C2×C4—1461no—
294C2×C4—1461no—
295C2×C4—1461no—
296C2×C4—1461no—
297C2×C4—1461no—
298C2×C4#1455665012yes—
299C2×C4#1455665212yes—
300C8—2924no—
301C8—5842no—
302C22—22no—
303C22—22no—
304C22—22no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22#1798953911yescharacteristic
309C22#1798955711yescharacteristic
310C22#1798956111yescharacteristic
311C22C2×#1455665011yescharacteristic
312C22#1798956811yescharacteristic
313C22C2×#1455665211yescharacteristic
314C22D73⋊#10111yescharacteristic
315C22#1798961814yes—
316C4—22no—
317C4—22no—
318C4—44no—
319C4—2921no—
320C4—5842no—
321C4—1462no—
322C4—1461no—
323C4—1461no—
324C4—1462no—
325C2—24no—
326C2#2262517911yescharacteristic
327C2#2262519411yescharacteristic
328C2#2262520411yescharacteristic
329C2#2262520911yescharacteristic
330C2#2262605511yescharacteristic
331C2#2262607111yescharacteristic
332C2#2262607411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutator, prime radicalC24×C73C22×C4
FrattiniC24C22×#64408
FittingC22×C4⋊C4×C73C4
socleC23×C73C2×C22⋊C4
cosocleC23×Dic73C23
hypercenterC22×C4⋊C4#64408
hypocenterC73#9654
LCSG ▷ C24×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
171. C73#9654semidirect

Maximal subgroups

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

subgroupindex
#226257872
#226256572
C2×#179894322
C2×#179884862
#965473