Finite Group Databasebeta

← #34007587

Subgroups — #34007587

Order 18688.

582 conjugacy classes of subgroups; only the 134 normal subgroups are shown.

normal subgroups only

This group has 582 conjugacy classes of subgroups — too many to draw. Only its 134 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#22625864C211yesmaximal, characteristic, complemented
2#22625645C211yesmaximal, characteristic, complemented
3C2×Q8⋊#10596833C212yesmaximal, complemented
4C2×#17989435C212yesmaximal
5#22612474C211yesmaximal, characteristic
6C2×#14556658—22no—
7C2×C4⋊#10596833C2212yes—
8C2×C4×#10596833C2212yes—
9#17989453—22no—
10Q8⋊#10596833—24no—
11#17989435—24no—
12C2×#14556580C411yescharacteristic
13C2×Q8×Dic73C2211yescharacteristic
14C2×#14556553C2211yescharacteristic
15#17988489C411yescharacteristic
16#17988479—22no—
17#17988452C411yescharacteristic
18#17988434—22no—
19C42×Dic73C2211yescharacteristic
20#391×C73C411yescharacteristic
21C22×#10596833—22no—
22C22×#10596833—22no—
23#14556658—44no—
24C4⋊#10596833—22no—
25C4⋊#10596833—22no—
26C4×#10596833—22no—
27C4×#10596833D414yes—
28C22×D292C2×C411yescharacteristic
29C22×C4×D73—21no—
30#14556580—22no—
31#14556580C814yescomplemented
32Q8×Dic73—22no—
33Q8×Dic73—22no—
34#14556553—22no—
35#14556553—22no—
36C2×C4⋊Dic73—21no—
37C2×C4⋊Dic73C2×C411yescharacteristic
38C2×#11924005—22no—
39C2×#11924005—21no—
40C2×#11924005—22no—
41C2×C4×Dic73—21no—
42C2×C4×Dic73C2×C411yescharacteristic
43C2×C4×Dic73D412yes—
44C2×C4×Dic73D412yes—
45C2×C4×Dic73C2311yescharacteristic
46#14556498—22no—
47C22×Q8×C73C2×C411yescharacteristic
48C2×C4⋊C4×C73C2×C411yescharacteristic
49C2×C42×C73C2×C411yescharacteristic
50#95×C73—22no—
51C2×#10596833—24no—
52C2×#10596833—44no—
53C2×#10596833—42no—
54C2×#10596833—44no—
55C23×D73—21no—
56C2×D292—22no—
57C2×D292—22no—
58C2×D292—21no—
59C2×D292C2×C812yes—
60C22×Dic73—21no—
61C22×Dic73C22⋊C412yes—
62C22×Dic73C2×D411yescharacteristic
63C22×Dic73C2×D411yescharacteristic
64C2×C4×D73—41no—
65C2×C4×D73—41no—
66C2×C4×D73—42no—
67C2×C4×D73—42no—
68C4⋊Dic73—22no—
69C4⋊Dic73—42no—
70#11924005—28no—
71#11924005—44no—
72#11924005—42no—
73C4×Dic73—22no—
74C4×Dic73—22no—
75C4×Dic73—42no—
76C4×Dic73—41no—
77C4×Dic73—41no—
78C4×Dic73C2×C812yes—
79C4×Dic73M1612yes—
80C4×Dic73C2×D412yes—
81C4×Dic73#4112yes—
82C2×Q8×C73—21no—
83C2×Q8×C73—22no—
84C2×Q8×C73—22no—
85C2×Q8×C73C2×C812yes—
86C22×C4×C73—21no—
87C22×C4×C73—21no—
88C22×C4×C73C22⋊C412yes—
89C22×C4×C73C22×C411yescharacteristic
90C4⋊C4×C73—22no—
91C42×C73—22no—
92C23×C73C2×C22⋊C411yescharacteristic
93C22×D73—22no—
94C22×D73—41no—
95C22×D73—41no—
96C22×D73—42no—
97C22×D73—42no—
98Q8×C73—24no—
99Q8×C73—42no—
100C2×C4×C73—22no—
101C2×C4×C73—22no—
102C2×C4×C73—22no—
103C2×C4×C73—22no—
104C2×C4×C73—41no—
105C2×C4×C73—41no—
106C2×C4×C73—41no—
107C2×C4×C73—42no—
108C2×C4×C73C4×D412yes—
109C2×C4×C73C22×C811yescharacteristic
110C2×C4×C73C2×M1611yescharacteristic
111C2×C4×C73#13112yes—
112C2×Dic73—22no—
113C2×Dic73—22no—
114C2×Dic73—22no—
115C2×Dic73—41no—
116C2×Dic73—42no—
117C2×Dic73C22⋊C814yes—
118C2×Dic73#12112yes—
119C2×Dic73#12411yescharacteristic
120C2×Dic73#12711yescharacteristic
121C2×Dic73AGL(1,ℤ/8)12yes—
122C2×Dic73#13712yes—
123D292—24no—
124D292—42no—
125C4×D73—82no—
126C4×D73—81no—
127C4×D73—81no—
128#10596833—44no—
129#10596833—82no—
130C22×C73#39011yescharacteristic
131C22×C73C2×C22⋊C811yescharacteristic
132C22×C73#40811yescharacteristic
133C22×C73#41712yes—
134C22×C73#42112yes—
135C2×D73—42no—
136C2×D73—44no—
137C2×D73—81no—
138C2×D73—81no—
139C2×D73—82no—
140C4×C73—42no—
141C4×C73—42no—
142C4×C73—81no—
143C4×C73C8×D412yes—
144C4×C73#43612yes—
145Dic73—22no—
146Dic73—24no—
147Dic73—81no—
148Dic73#48912yes—
149Dic73#49611yescharacteristic
150Dic73#49711yescharacteristic
151#9640—731nomaximal, Sylow
152C2×C73#147712yes—
153C2×C73#149512yes—
154C2×C73#195411yescharacteristic
155C2×C73#195911yescharacteristic
156C2×C73#196311yescharacteristic
157D73—82no—
158C4×#124—731no—
159#1842—731no—
160#1744—731no—
161C2×Q8⋊C8—732no—
162C2×D4⋊C8—732no—
163C73#964011yescharacteristic, complemented, Sylow
164C2×#124—731no—
165C2×C4×Q8—731no—
166C2×C4×D4—731no—
167C2×C4⋊C8—732no—
168C2×C4⋊C8—1462no—
169C2×C4×C8—732no—
170#391#6440811yescharacteristic
171#390—731no—
172#388—1462no—
173#382—731no—
174C4×C22⋊C4—1462no—
175C43—731no—
176#336—1462no—
177Q8⋊C8—1464no—
178D4⋊C8—1464no—
179C22×Q8C2×#6440811yescharacteristic
180C22×D4—731no—
181C23×C4—1461no—
182C22×C8—1462no—
183C22×C8—1462no—
184#124—734no—
185#124—1462no—
186C4×Q8—1462no—
187C4×Q8—1462no—
188C4×D4—1462no—
189C4×D4—1462no—
190C2×C4⋊C4—731no—
191C2×C4⋊C4—1461no—
192C2×C4⋊C4C2×#6440811yescharacteristic
193C2×C22⋊C4—1461no—
194C2×C22⋊C4—1462no—
195C2×C22⋊C4—1462no—
196C2×C42—732no—
197C2×C42—731no—
198C2×C42—731no—
199C2×C42—732no—
200C2×C42—1461no—
201C2×C42C2×#6440811yescharacteristic
202C4⋊C8—2924no—
203C4⋊C8—1462no—
204C4⋊C8—1462no—
205C4×C8—734no—
206C4×C8—1462no—
207#95—22no—
208#95—1462no—
209C24—1461no—
210C2×Q8—21no—
211C2×Q8—22no—
212C2×Q8—22no—
213C2×Q8#1192403612yes—
214C2×D4—732no—
215C2×D4—1461no—
216C2×D4—1462no—
217C2×D4—1462no—
218C22×C4—21no—
219C22×C4—21no—
220C22×C4—2921no—
221C22×C4—2921no—
222C22×C4—2922no—
223C22×C4—2922no—
224C22×C4—731no—
225C22×C4—732no—
226C22×C4—731no—
227C22×C4—1461no—
228C22×C4C22⋊#6440812yes—
229C22×C4C22×#6440811yescharacteristic
230C2×C8—2924no—
231C2×C8—2924no—
232C2×C8—2922no—
233C2×C8—1464no—
234C4⋊C4—22no—
235C4⋊C4—2922no—
236C4⋊C4—1462no—
237C22⋊C4—2924no—
238C22⋊C4—2922no—
239C22⋊C4—1468no—
240C42—22no—
241C42—2922no—
242C42—2921no—
243C42—2921no—
244C42—732no—
245C42—732no—
246C42—732no—
247C42—732no—
248C42—1462no—
249C42—1462no—
250C23—2921no—
251C23—2921no—
252C23—2922no—
253C23—2922no—
254C23—1462no—
255C23C2×C22⋊#6440811yescharacteristic
256Q8—24no—
257Q8—42no—
258D4—2922no—
259D4—1464no—
260C2×C4—22no—
261C2×C4—22no—
262C2×C4—22no—
263C2×C4—22no—
264C2×C4—41no—
265C2×C4—41no—
266C2×C4—41no—
267C2×C4—42no—
268C2×C4—2921no—
269C2×C4—2922no—
270C2×C4—5841no—
271C2×C4—5842no—
272C2×C4—5841no—
273C2×C4—731no—
274C2×C4—731no—
275C2×C4—732no—
276C2×C4—732no—
277C2×C4—732no—
278C2×C4—734no—
279C2×C4—1462no—
280C2×C4—1462no—
281C2×C4—1462no—
282C2×C4D4×#6440812yes—
283C2×C4C2×#1192403611yescharacteristic
284C2×C4D73⋊M1611yescharacteristic
285C2×C4#1455667812yes—
286C8—2924no—
287C8—5842no—
288C22—2922no—
289C22—2924no—
290C22—5842no—
291C22—5841no—
292C22—5841no—
293C22#1798953911yescharacteristic
294C22#1798955511yescharacteristic
295C22#1798956011yescharacteristic
296C22#1798957812yes—
297C22#1798958612yes—
298C4—42no—
299C4—42no—
300C4—81no—
301C4—5841no—
302C4—731no—
303C4—732no—
304C4—731no—
305C4—1462no—
306C4—1464no—
307C4#1798961412yes—
308C4#1798962012yes—
309C2—5842no—
310C2#2262522912yes—
311C2#2262528812yes—
312C2#2262605411yescharacteristic
313C2#2262606111yescharacteristic
314C2#2262606711yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC2×C4×C73C22×C8
FrattiniC22×C4C22×#64408
Fitting#391×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC22×D292C2×C4
cosocleC2×C4×Dic73C23
hypercenter#391#64408
hypocenterC73#9640
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #391

Complements

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

NH ≅ G/Ntype
1. #22625864C2semidirect
2. #22625645C2semidirect
3. C2×Q8⋊#10596833C2semidirect
31. #14556580C8semidirect
163. C73#9640semidirect

Maximal subgroups

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

subgroupindex
#226258642
#226256452
C2×Q8⋊#105968332
C2×#179894352
#226124742
#964073