Finite Group Databasebeta

← #33062246

Subgroups — #33062246

Order 18688.

1004 conjugacy classes of subgroups; only the 195 normal subgroups are shown.

normal subgroups only

This group has 1004 conjugacy classes of subgroups — too many to draw. Only its 195 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#22620388C211yesmaximal, characteristic
2#22620053C212yesmaximal
3#22609071C214yesmaximal, complemented
4#22609009C214yesmaximal, complemented
5#22608657C211yesmaximal, characteristic, complemented
6#22608378C212yesmaximal, complemented
7C2×Q82×C73C211yesmaximal, characteristic, complemented
8C2×#14556622—22no—
9#17989137C2211yescharacteristic
10#17988979—22no—
11#17988966C2212yes—
12#17988961C2212yes—
13#17988900—22no—
14#17988882C2214yes—
15#17988787C2214yes—
16#17988726—22no—
17#17988685—24no—
18#17988644C2212yes—
19C4×#11924012C2211yescharacteristic
20#17988336C2214yes—
21Q8⋊#10596831C2218yes—
22Q82×C73—22no—
23Q82×C73—24no—
24Q82×C73C2214yes—
25C2×C4⋊Q8×C73—22no—
26C2×C4⋊Q8×C73—22no—
27C2×C4⋊Q8×C73C2211yescharacteristic
28C2×C4×Q8×C73—22no—
29C2×C4×Q8×C73C2212yes—
30C2×#11924028—21no—
31#14556622—48no—
32C2×#11924018—24no—
33C2×#11924015—24no—
34C2×#11924015—24no—
35C2×#11924012—21no—
36#14556591C2311yescharacteristic
37#14556588C2311yescharacteristic
38#14556579—21no—
39#14556565—22no—
40C4×D292—22no—
41#14556524—24no—
42C4⋊#10596831C2314yes—
43#14556515—24no—
44#14556511—24no—
45C22⋊#10596831—22no—
46#14556502—22no—
47#14556502—21no—
48C4×#10596831C2312yes—
49C22×Q8×C73—22no—
50C22×Q8×C73D412yes—
51C22×Q8×C73D412yes—
52C4⋊Q8×C73—28no—
53C4⋊Q8×C73—28no—
54C4⋊Q8×C73—28no—
55C4⋊Q8×C73—28no—
56C4⋊Q8×C73—22no—
57C4⋊Q8×C73D412yes—
58C4⋊Q8×C73C2312yes—
59C4×Q8×C73—28no—
60C4×Q8×C73—28no—
61C4×Q8×C73—22no—
62C4×Q8×C73—22no—
63C4×Q8×C73D414yes—
64C4×Q8×C73C2314yes—
65C2×C4⋊C4×C73—24no—
66C2×C4⋊C4×C73—22no—
67C2×C4⋊C4×C73—22no—
68C2×C4⋊C4×C73D412yes—
69C2×C42×C73—22no—
70C2×C42×C73—22no—
71C2×C42×C73C2311yescharacteristic
72#11924028—44no—
73C2×Dic146—21no—
74C2×D292—22no—
75C2×D292—21no—
76C2×C22⋊D73—41no—
77C2×C4×D73—41no—
78#11924018—48no—
79#11924015—48no—
80#11924015—48no—
81#11924012—24no—
82C2×#10596831—22no—
83C2×#10596831—24no—
84C4⋊Dic73—22no—
85#11924005—42no—
86C2×Q8×C73—28no—
87C2×Q8×C73—28no—
88C2×Q8×C73—22no—
89C2×Q8×C73—22no—
90C2×Q8×C73—44no—
91C2×Q8×C73—84no—
92C2×Q8×C73C2×D414yes—
93C2×Q8×C73C2×D414yes—
94C22×C4×C73—22no—
95C22×C4×C73—22no—
96C22×C4×C73—22no—
97C22×C4×C73C2×D412yes—
98C22×C4×C73C2×D411yescharacteristic
99C4⋊C4×C73—216no—
100C4⋊C4×C73—24no—
101C4⋊C4×C73—24no—
102C4⋊C4×C73—22no—
103C4⋊C4×C73—24no—
104C4⋊C4×C73—24no—
105C4⋊C4×C73C2×D414yes—
106C42×C73—28no—
107C42×C73—24no—
108C42×C73—24no—
109C42×C73C2×D411yescharacteristic
110C42×C73C2×D412yes—
111C42×C73C2411yescharacteristic
112C23×C73#12011yescharacteristic
113C22×D73—41no—
114Q8×C73—28no—
115Q8×C73—24no—
116Q8×C73—44no—
117Q8×C73—82no—
118Q8×C73—88no—
119Q8×C73#13718yescomplemented
120C2×C4×C73—28no—
121C2×C4×C73—24no—
122C2×C4×C73—22no—
123C2×C4×C73—24no—
124C2×C4×C73—24no—
125C2×C4×C73—24no—
126C2×C4×C73—44no—
127C2×C4×C73—44no—
128C2×C4×C73#12012yes—
129C2×C4×C73#12011yescharacteristic
130C2×C4×C73C22×D411yescharacteristic
131C2×C4×C73C22×D412yes—
132C22⋊D73—82no—
133C2×Dic73—41no—
134D292—42no—
135D292—44no—
136C4×D73—82no—
137Dic146—42no—
138#10596831—24no—
139#10596831—44no—
140C22×C73—22no—
141C22×C73#45312yes—
142C22×C73C2×#12011yescharacteristic
143C2×D73—82no—
144C2×D73—81no—
145C4×C73—28no—
146C4×C73—22no—
147C4×C73—21no—
148C4×C73—21no—
149C4×C73—44no—
150C4×C73—44no—
151C4×C73C2×#13714yes—
152Dic73—81no—
153#18546—731nomaximal, Sylow
154C2×C73—22no—
155C2×C73#300412yes—
156C2×C73C2×#45311yescharacteristic
157D73—161no—
158C2×Q82D7311yescharacteristic, complemented
159#3152—731no—
160#3112—732no—
161#1633—734no—
162#1623—734no—
163#1558—731no—
164#1491—732no—
165C73#1854611yescharacteristic, complemented, Sylow
166C2×#137—1462no—
167Q82—22no—
168Q82—24no—
169Q82C2×D7314yes—
170#532—731no—
171C2×C4⋊Q8—22no—
172C2×C4⋊Q8—22no—
173C2×C4⋊Q8C2×D7311yescharacteristic
174C2×C4×Q8—22no—
175C2×C4×Q8C2×D7312yes—
176#497—1462no—
177#494—732no—
178#492—732no—
179#480—1462no—
180#477—734no—
181#460—734no—
182#448—1462no—
183#440—1464no—
184#432—732no—
185C4×M16—731no—
186#333—734no—
187Q8⋊C8—738no—
188C2×#41—1461no—
189C22×Q8—22no—
190C22×Q8C22⋊D7312yes—
191C22×Q8C22⋊D7312yes—
192#137—2928no—
193C2×Q16—1464no—
194C2×SD16—1464no—
195C2×SD16—1464no—
196C2×M16—1461no—
197C4⋊Q8—28no—
198C4⋊Q8—28no—
199C4⋊Q8—28no—
200C4⋊Q8—28no—
201C4⋊Q8—22no—
202C4⋊Q8—731no—
203C4⋊Q8C22⋊D7312yes—
204C4⋊Q8C22×D7312yes—
205#127—731no—
206#124—1461no—
207#121—1462no—
208C4×Q8—28no—
209C4×Q8—28no—
210C4×Q8—22no—
211C4×Q8—22no—
212C4×Q8C22⋊D7314yes—
213C4×Q8C22×D7314yes—
214C4×D4—1462no—
215C2×C4⋊C4—24no—
216C2×C4⋊C4—22no—
217C2×C4⋊C4—22no—
218C2×C4⋊C4C22⋊D7312yes—
219C2×C42—22no—
220C2×C42—22no—
221C2×C42C22×D7311yescharacteristic
222#107—1464no—
223C4⋊C8—734no—
224#103—1464no—
225#102—1464no—
226C22⋊C8—1462no—
227#97—1461no—
228#97—1462no—
229C4×C8—732no—
230#41—2924no—
231C2×Q8—28no—
232C2×Q8—28no—
233C2×Q8—22no—
234C2×Q8—22no—
235C2×Q8—44no—
236C2×Q8—84no—
237C2×Q8—1461no—
238C2×Q8C2×C22⋊D7314yes—
239C2×Q8C2×C22⋊D7314yes—
240C2×D4—2921no—
241C2×D4—1462no—
242C2×D4—1461no—
243C22×C4—22no—
244C22×C4—22no—
245C22×C4—22no—
246C22×C4—2921no—
247C22×C4C2×C22⋊D7312yes—
248C22×C4C2×C22⋊D7311yescharacteristic
249Q16—2928no—
250SD16—2928no—
251SD16—2928no—
252M16—1464no—
253C2×C8—1464no—
254C2×C8—1462no—
255C4⋊C4—216no—
256C4⋊C4—24no—
257C4⋊C4—24no—
258C4⋊C4—22no—
259C4⋊C4—24no—
260C4⋊C4—24no—
261C4⋊C4—1462no—
262C4⋊C4C2×C22⋊D7314yes—
263C22⋊C4—2922no—
264C42—28no—
265C42—24no—
266C42—24no—
267C42C2×C22⋊D7312yes—
268C42C2×C22⋊D7311yescharacteristic
269C42C23×D7311yescharacteristic
270C23—2921no—
271C23#1455656011yescharacteristic
272Q8—28no—
273Q8—24no—
274Q8—44no—
275Q8—82no—
276Q8—88no—
277Q8—2922no—
278Q8#1455662218yescomplemented
279D4—2922no—
280D4—2924no—
281D4—5842no—
282C2×C4—28no—
283C2×C4—22no—
284C2×C4—24no—
285C2×C4—24no—
286C2×C4—24no—
287C2×C4—24no—
288C2×C4—44no—
289C2×C4—44no—
290C2×C4—2921no—
291C2×C4—5842no—
292C2×C4#1455656012yes—
293C2×C4#1455656011yescharacteristic
294C2×C4C22×C22⋊D7311yescharacteristic
295C2×C4C22×C22⋊D7312yes—
296C8—2924no—
297C8—1464no—
298C22—22no—
299C22—5842no—
300C22—5841no—
301C22#1798875512yes—
302C22C2×#1455656011yescharacteristic
303C4—28no—
304C4—22no—
305C4—21no—
306C4—21no—
307C4—44no—
308C4—44no—
309C4—5841no—
310C4C2×#1455662214yes—
311C2—22no—
312C2—11681no—
313C2#2261916512yes—
314C2C2×#1798875511yescharacteristic
subgroupG/H
centerC22C2×#14556560
commutator, cosocleC42×C73C24
FrattiniC42C23×D73
FittingC2×Q82×C73C2
socleC22×C73C2×#120
prime radical#17989137C22
hypercenterC2×Q82D73
hypocenterC73#18546
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C2×C42 ◁ C2×Q82

Complements

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

NH ≅ G/Ntype
3. #22609071C2semidirect
4. #22609009C2semidirect
5. #22608657C2semidirect
6. #22608378C2semidirect
7. C2×Q82×C73C2semidirect
119. Q8×C73#137semidirect
158. C2×Q82D73semidirect
165. C73#18546semidirect
278. Q8#14556622semidirect

Maximal subgroups

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

subgroupindex
#226203882
#226200532
#226090712
#226090092
#226086572
#226083782
C2×Q82×C732
#1854673