Finite Group Databasebeta

← #33004677

Subgroups — #33004677

Order 18688.

396 conjugacy classes of subgroups; only the 107 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 107 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#22609927C211yesmaximal, characteristic
2C4×#14556510C211yesmaximal, characteristic
3#22608058C212yesmaximal
4#22608058C212yesmaximal
5C73×#2565C211yesmaximal, characteristic
6#17988611—21no—
7C2×#14556510C2212yes—
8C2×#14556510C2212yes—
9C4×#11924012—21no—
10#17988356—21no—
11#17988356—21no—
12C2×#125×C73C411yescharacteristic
13C2×C4×Q8×C73C2211yescharacteristic
14#398×C73—21no—
15#398×C73—21no—
16#391×C73C2211yescharacteristic
17#391×C73C2211yescharacteristic
18#387×C73—21no—
19#387×C73—21no—
20#384×C73C411yescharacteristic
21#382×C73C411yescharacteristic
22C4×C4⋊C4×C73C411yescharacteristic
23C2×#11924012—22no—
24C2×#11924012—22no—
25C4⋊#10596831—41no—
26C4⋊#10596831—41no—
27#14556510—22no—
28#14556510—24no—
29#14556510Q814yes—
30#14556502—41no—
31C4×#10596831—41no—
32C22×Q8×C73C2311yescharacteristic
33#125×C73—41no—
34#125×C73—41no—
35C4×Q8×C73—41no—
36C4×Q8×C73—41no—
37C2×C4⋊C4×C73—21no—
38C2×C4⋊C4×C73—21no—
39C2×C4⋊C4×C73—21no—
40C2×C4⋊C4×C73—21no—
41C2×C4⋊C4×C73—21no—
42C2×C4⋊C4×C73C2×C411yescharacteristic
43C2×C4⋊C4×C73C2×C411yescharacteristic
44C2×C4⋊C4×C73C2×C411yescharacteristic
45C2×C4⋊C4×C73D411yescharacteristic
46C2×C4⋊C4×C73D411yescharacteristic
47C2×C42×C73C2×C411yescharacteristic
48C2×C42×C73C2×C411yescharacteristic
49C2×C42×C73C2×C411yescharacteristic
50C2×C42×C73D411yescharacteristic
51C2×C42×C73D411yescharacteristic
52#95×C73—21no—
53#95×C73—21no—
54#95×C73—21no—
55#95×C73—21no—
56#95×C73—21no—
57#95×C73—21no—
58#95×C73—21no—
59#11924012—24no—
60#11924012—42no—
61#11924012—42no—
62#11924012—42no—
63C2×#10596831—42no—
64C2×#10596831—42no—
65C2×Q8×C73—41no—
66C2×Q8×C73—41no—
67C2×Q8×C73C2×Q812yes—
68C2×Q8×C73#4112yes—
69C22×C4×C73—21no—
70C22×C4×C73—21no—
71C22×C4×C73—21no—
72C22×C4×C73—21no—
73C22×C4×C73C22⋊C411yescharacteristic
74C22×C4×C73C22⋊C411yescharacteristic
75C22×C4×C73C22⋊C411yescharacteristic
76C22×C4×C73C22⋊C411yescharacteristic
77C22×C4×C73C22×C411yescharacteristic
78C22×C4×C73C2×D411yescharacteristic
79C22×C4×C73C2×D411yescharacteristic
80C4⋊C4×C73—41no—
81C4⋊C4×C73—41no—
82C4⋊C4×C73—41no—
83C4⋊C4×C73—41no—
84C4⋊C4×C73—41no—
85C4⋊C4×C73—42no—
86C4⋊C4×C73—42no—
87C4⋊C4×C73—42no—
88C4⋊C4×C73—42no—
89C4⋊C4×C73—41no—
90C4⋊C4×C73—41no—
91C42×C73—21no—
92C42×C73—21no—
93C42×C73—21no—
94C42×C73—21no—
95C42×C73—21no—
96C42×C73—21no—
97C42×C73—41no—
98C42×C73—41no—
99C23×C73C2×C22⋊C411yescharacteristic
100Q8×C73—42no—
101Q8×C73—42no—
102C2×C4×C73—21no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—21no—
106C2×C4×C73—21no—
107C2×C4×C73—21no—
108C2×C4×C73—22no—
109C2×C4×C73—21no—
110C2×C4×C73—21no—
111C2×C4×C73—21no—
112C2×C4×C73—41no—
113C2×C4×C73—41no—
114C2×C4×C73—41no—
115C2×C4×C73—41no—
116C2×C4×C73—41no—
117C2×C4×C73—41no—
118C2×C4×C73—41no—
119C2×C4×C73—41no—
120C2×C4×C73—41no—
121C2×C4×C73—41no—
122C2×C4×C73—42no—
123C2×C4×C73—41no—
124C2×C4×C73—41no—
125C2×C4×C73—41no—
126C2×C4×C73#9911yescharacteristic
127C2×C4×C73#9911yescharacteristic
128C2×C4×C73C4×Q812yes—
129C2×C4×C73C22⋊Q812yes—
130C2×C4×C73#12411yescharacteristic
131C2×C4×C73C4⋊Q811yescharacteristic
132#10596831—44no—
133#10596831—82no—
134C22×C73—21no—
135C22×C73—21no—
136C22×C73#39111yescharacteristic
137C22×C73C2×#9911yescharacteristic
138C22×C73#41011yescharacteristic
139C4×C73—21no—
140C4×C73—21no—
141C4×C73—22no—
142C4×C73—21no—
143C4×C73—22no—
144C4×C73—21no—
145C4×C73—42no—
146C4×C73—41no—
147C4×C73—41no—
148C4×C73—41no—
149C4×C73—81no—
150C4×C73—81no—
151C4×C73—81no—
152#13303—731nomaximal, Sylow
153C2×C73—21no—
154C2×C73—21no—
155C2×C73#199611yescharacteristic
156C2×C73#213211yescharacteristic
157C2×C73#213311yescharacteristic
158#2565D7311yescharacteristic
159#1801—731no—
160C4×#101—731no—
161#1394—732no—
162#1394—732no—
163C73#1330311yescharacteristic, complemented, Sylow
164C2×#125Dic7311yescharacteristic
165C2×C4×Q8C2×D7311yescharacteristic
166#423—1461no—
167C2×#101—732no—
168C2×#101—732no—
169C4×M16—1461no—
170#398—21no—
171#398—21no—
172#391C2×D7311yescharacteristic
173#391C2×D7311yescharacteristic
174#387—21no—
175#387—21no—
176#384Dic7311yescharacteristic
177#382Dic7311yescharacteristic
178C4×C4⋊C4Dic7311yescharacteristic
179#343—1461no—
180#343—1461no—
181C22×Q8C22×D7311yescharacteristic
182C2×M16—1462no—
183C2×M16—1462no—
184#125—41no—
185#125—41no—
186C4×Q8—41no—
187C4×Q8—41no—
188C2×C4⋊C4—21no—
189C2×C4⋊C4—21no—
190C2×C4⋊C4—21no—
191C2×C4⋊C4—21no—
192C2×C4⋊C4—21no—
193C2×C4⋊C4C2×Dic7311yescharacteristic
194C2×C4⋊C4C2×Dic7311yescharacteristic
195C2×C4⋊C4C2×Dic7311yescharacteristic
196C2×C4⋊C4C22⋊D7311yescharacteristic
197C2×C4⋊C4C22⋊D7311yescharacteristic
198C2×C42C2×Dic7311yescharacteristic
199C2×C42C2×Dic7311yescharacteristic
200C2×C42C2×Dic7311yescharacteristic
201C2×C42C22⋊D7311yescharacteristic
202C2×C42C22⋊D7311yescharacteristic
203C4⋊C8—2921no—
204C4⋊C8—2921no—
205#101—734no—
206#101—1464no—
207#101—1462no—
208#97—2921no—
209C4×C8—2921no—
210#95—21no—
211#95—21no—
212#95—21no—
213#95—21no—
214#95—21no—
215#95—21no—
216#95—21no—
217C2×Q8—41no—
218C2×Q8—41no—
219C2×Q8Q8×D7312yes—
220C2×Q8#1192402912yes—
221C22×C4—21no—
222C22×C4—21no—
223C22×C4—21no—
224C22×C4—21no—
225C22×C4C22⋊Dic7311yescharacteristic
226C22×C4C22⋊Dic7311yescharacteristic
227C22×C4C22⋊Dic7311yescharacteristic
228C22×C4C22⋊Dic7311yescharacteristic
229C22×C4C22×Dic7311yescharacteristic
230C22×C4C2×C22⋊D7311yescharacteristic
231C22×C4C2×C22⋊D7311yescharacteristic
232M16—2922no—
233M16—2922no—
234M16—2922no—
235M16—1464no—
236C2×C8—2922no—
237C2×C8—2922no—
238C4⋊C4—41no—
239C4⋊C4—41no—
240C4⋊C4—41no—
241C4⋊C4—41no—
242C4⋊C4—41no—
243C4⋊C4—42no—
244C4⋊C4—42no—
245C4⋊C4—42no—
246C4⋊C4—42no—
247C4⋊C4—41no—
248C4⋊C4—41no—
249C42—21no—
250C42—21no—
251C42—21no—
252C42—21no—
253C42—21no—
254C42—21no—
255C42—41no—
256C42—41no—
257C23C2×C22⋊Dic7311yescharacteristic
258Q8—42no—
259Q8—42no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—22no—
267C2×C4—21no—
268C2×C4—21no—
269C2×C4—21no—
270C2×C4—41no—
271C2×C4—41no—
272C2×C4—41no—
273C2×C4—41no—
274C2×C4—41no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—42no—
281C2×C4—41no—
282C2×C4—41no—
283C2×C4—41no—
284C2×C4#1455650611yescharacteristic
285C2×C4#1455650611yescharacteristic
286C2×C4Q8×Dic7312yes—
287C2×C4#1455656712yes—
288C2×C4#1455658011yescharacteristic
289C2×C4#1455658911yescharacteristic
290C8—2924no—
291C8—5842no—
292C22—21no—
293C22—21no—
294C22#1798849411yescharacteristic
295C22C2×#1455650611yescharacteristic
296C22#1798856111yescharacteristic
297C4—21no—
298C4—21no—
299C4—22no—
300C4—21no—
301C4—22no—
302C4—21no—
303C4—42no—
304C4—41no—
305C4—41no—
306C4—41no—
307C4—81no—
308C4—81no—
309C4—81no—
310C2—21no—
311C2—21no—
312C2#2261091711yescharacteristic
313C2#2261161611yescharacteristic
314C2#2261162011yescharacteristic
subgroupG/H
centerC22C2×#14556506
commutatorC22×C4×C73C22×C4
FrattiniC22×Q8C22×D73
FittingC73×#2565C2
socleC22×C73C2×#99
prime radicalC23×C73C2×C22⋊C4
cosocleC22×Q8×C73C23
hypercenter#2565D73
hypocenterC73#13303
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2565

Complements

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

NH ≅ G/Ntype
163. C73#13303semidirect

Maximal subgroups

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

subgroupindex
#226099272
C4×#145565102
#226080582
C73×#25652
#1330373