Finite Group Databasebeta

← #33004672

Subgroups — #33004672

Order 18688.

420 conjugacy classes of subgroups; only the 111 normal subgroups are shown.

normal subgroups only

This group has 420 conjugacy classes of subgroups — too many to draw. Only its 111 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#22608058C214yesmaximal
4C73×#2567C211yesmaximal, characteristic
5#17988611—21no—
6C2×#14556510C2212yes—
7C2×#14556510C2212yes—
8C4×#11924012—21no—
9#17988356—21no—
10#17988356—21no—
11C2×C4⋊Q8×C73C411yescharacteristic
12C2×C4×Q8×C73C2211yescharacteristic
13#395×C73—21no—
14#395×C73—21no—
15#391×C73C2212yes—
16#389×C73—21no—
17#389×C73—21no—
18#384×C73C412yes—
19C4×C4⋊C4×C73C411yescharacteristic
20C2×#11924012—22no—
21C2×#11924012—22no—
22C4⋊#10596831—41no—
23C4⋊#10596831—41no—
24#14556510—22no—
25#14556510—24no—
26#14556510Q814yescomplemented
27#14556502—41no—
28C4×#10596831—41no—
29C22×Q8×C73—21no—
30C22×Q8×C73C2311yescharacteristic
31C4⋊Q8×C73—41no—
32C4⋊Q8×C73—41no—
33C4×Q8×C73—41no—
34C4×Q8×C73—41no—
35C2×C4⋊C4×C73—21no—
36C2×C4⋊C4×C73—21no—
37C2×C4⋊C4×C73—22no—
38C2×C4⋊C4×C73—22no—
39C2×C4⋊C4×C73C2×C412yes—
40C2×C4⋊C4×C73C2×C411yescharacteristic
41C2×C4⋊C4×C73D411yescharacteristic
42C2×C4⋊C4×C73D411yescharacteristic
43C2×C42×C73C2×C411yescharacteristic
44C2×C42×C73C2×C412yes—
45C2×C42×C73D411yescharacteristic
46C2×C42×C73D411yescharacteristic
47#95×C73—21no—
48#95×C73—22no—
49#95×C73—22no—
50#11924012—24no—
51#11924012—42no—
52#11924012—42no—
53#11924012—42no—
54C2×#10596831—42no—
55C2×#10596831—42no—
56C2×Q8×C73—41no—
57C2×Q8×C73—41no—
58C2×Q8×C73—42no—
59C2×Q8×C73—82no—
60C2×Q8×C73C2×Q812yes—
61C2×Q8×C73#4112yes—
62C22×C4×C73—22no—
63C22×C4×C73—22no—
64C22×C4×C73C22⋊C411yescharacteristic
65C22×C4×C73C22⋊C411yescharacteristic
66C22×C4×C73C22⋊C411yescharacteristic
67C22×C4×C73C22⋊C411yescharacteristic
68C22×C4×C73C22×C411yescharacteristic
69C22×C4×C73C2×D411yescharacteristic
70C22×C4×C73C2×D411yescharacteristic
71C4⋊C4×C73—24no—
72C4⋊C4×C73—41no—
73C4⋊C4×C73—41no—
74C4⋊C4×C73—41no—
75C4⋊C4×C73—42no—
76C4⋊C4×C73—41no—
77C4⋊C4×C73—41no—
78C4⋊C4×C73—82no—
79C4⋊C4×C73—82no—
80C42×C73—21no—
81C42×C73—21no—
82C42×C73—21no—
83C42×C73—21no—
84C42×C73—21no—
85C42×C73—21no—
86C42×C73—42no—
87C23×C73C2×C22⋊C411yescharacteristic
88Q8×C73—42no—
89Q8×C73—42no—
90Q8×C73—84no—
91C2×C4×C73—21no—
92C2×C4×C73—21no—
93C2×C4×C73—21no—
94C2×C4×C73—21no—
95C2×C4×C73—21no—
96C2×C4×C73—21no—
97C2×C4×C73—22no—
98C2×C4×C73—24no—
99C2×C4×C73—21no—
100C2×C4×C73—21no—
101C2×C4×C73—21no—
102C2×C4×C73—42no—
103C2×C4×C73—42no—
104C2×C4×C73—41no—
105C2×C4×C73—41no—
106C2×C4×C73—41no—
107C2×C4×C73—42no—
108C2×C4×C73—42no—
109C2×C4×C73—42no—
110C2×C4×C73#9911yescharacteristic
111C2×C4×C73#9911yescharacteristic
112C2×C4×C73C4×Q812yes—
113C2×C4×C73C22⋊Q812yes—
114C2×C4×C73#12411yescharacteristic
115C2×C4×C73C4⋊Q811yescharacteristic
116#10596831—44no—
117#10596831—82no—
118C22×C73—21no—
119C22×C73—21no—
120C22×C73#39111yescharacteristic
121C22×C73C2×#9911yescharacteristic
122C22×C73#41011yescharacteristic
123C4×C73—21no—
124C4×C73—22no—
125C4×C73—21no—
126C4×C73—22no—
127C4×C73—21no—
128C4×C73—44no—
129C4×C73—41no—
130C4×C73—41no—
131C4×C73—41no—
132C4×C73—82no—
133C4×C73#35612yes—
134#13302—731nomaximal, Sylow
135C2×C73—21no—
136C2×C73—21no—
137C2×C73#199611yescharacteristic
138C2×C73C2×#35611yescharacteristic
139C2×C73#213411yescharacteristic
140#2567D7311yescharacteristic
141#1801—731no—
142C4×#101—731no—
143#1394—734no—
144C73#1330211yescharacteristic, complemented, Sylow
145C2×C4⋊Q8Dic7311yescharacteristic
146C2×C4×Q8C2×D7311yescharacteristic
147#423—1461no—
148C2×#101—732no—
149C2×#101—732no—
150C4×M16—1461no—
151#395—21no—
152#395—21no—
153#391C2×D7312yes—
154#389—21no—
155#389—21no—
156#384Dic7312yes—
157C4×C4⋊C4Dic7311yescharacteristic
158#343—1461no—
159#343—1461no—
160C22×Q8—21no—
161C22×Q8C22×D7311yescharacteristic
162C2×M16—1462no—
163C2×M16—1462no—
164C4⋊Q8—41no—
165C4⋊Q8—41no—
166C4×Q8—41no—
167C4×Q8—41no—
168C2×C4⋊C4—21no—
169C2×C4⋊C4—21no—
170C2×C4⋊C4—22no—
171C2×C4⋊C4—22no—
172C2×C4⋊C4C2×Dic7312yes—
173C2×C4⋊C4C2×Dic7311yescharacteristic
174C2×C4⋊C4C22⋊D7311yescharacteristic
175C2×C4⋊C4C22⋊D7311yescharacteristic
176C2×C42C2×Dic7311yescharacteristic
177C2×C42C2×Dic7312yes—
178C2×C42C22⋊D7311yescharacteristic
179C2×C42C22⋊D7311yescharacteristic
180C4⋊C8—2921no—
181C4⋊C8—2921no—
182#101—734no—
183#101—1464no—
184#101—1462no—
185#97—2921no—
186C4×C8—2921no—
187#95—21no—
188#95—22no—
189#95—22no—
190C2×Q8—41no—
191C2×Q8—41no—
192C2×Q8—42no—
193C2×Q8—82no—
194C2×Q8Q8×D7312yes—
195C2×Q8#1192402912yes—
196C22×C4—22no—
197C22×C4—22no—
198C22×C4C22⋊Dic7311yescharacteristic
199C22×C4C22⋊Dic7311yescharacteristic
200C22×C4C22⋊Dic7311yescharacteristic
201C22×C4C22⋊Dic7311yescharacteristic
202C22×C4C22×Dic7311yescharacteristic
203C22×C4C2×C22⋊D7311yescharacteristic
204C22×C4C2×C22⋊D7311yescharacteristic
205M16—2922no—
206M16—2922no—
207M16—2922no—
208M16—1464no—
209C2×C8—2922no—
210C2×C8—2922no—
211C4⋊C4—24no—
212C4⋊C4—41no—
213C4⋊C4—41no—
214C4⋊C4—41no—
215C4⋊C4—42no—
216C4⋊C4—41no—
217C4⋊C4—41no—
218C4⋊C4—82no—
219C4⋊C4—82no—
220C42—21no—
221C42—21no—
222C42—21no—
223C42—21no—
224C42—21no—
225C42—21no—
226C42—42no—
227C23C2×C22⋊Dic7311yescharacteristic
228Q8—42no—
229Q8—42no—
230Q8—84no—
231C2×C4—21no—
232C2×C4—21no—
233C2×C4—21no—
234C2×C4—21no—
235C2×C4—21no—
236C2×C4—21no—
237C2×C4—22no—
238C2×C4—24no—
239C2×C4—21no—
240C2×C4—21no—
241C2×C4—21no—
242C2×C4—42no—
243C2×C4—42no—
244C2×C4—41no—
245C2×C4—41no—
246C2×C4—41no—
247C2×C4—42no—
248C2×C4—42no—
249C2×C4—42no—
250C2×C4#1455650611yescharacteristic
251C2×C4#1455650611yescharacteristic
252C2×C4Q8×Dic7312yes—
253C2×C4#1455656712yes—
254C2×C4#1455658011yescharacteristic
255C2×C4#1455658911yescharacteristic
256C8—2924no—
257C8—5842no—
258C22—21no—
259C22—21no—
260C22#1798849411yescharacteristic
261C22C2×#1455650611yescharacteristic
262C22#1798856111yescharacteristic
263C4—21no—
264C4—22no—
265C4—21no—
266C4—22no—
267C4—21no—
268C4—44no—
269C4—41no—
270C4—41no—
271C4—41no—
272C4—82no—
273C4#1798838612yes—
274C2—21no—
275C2—21no—
276C2#2261091711yescharacteristic
277C2C2×#1798838611yescharacteristic
278C2#2261162311yescharacteristic
subgroupG/H
centerC22C2×#14556506
commutatorC22×C4×C73C22×C4
FrattiniC22×Q8C22×D73
FittingC73×#2567C2
socleC22×C73C2×#99
prime radicalC23×C73C2×C22⋊C4
cosocleC22×Q8×C73C23
hypercenter#2567D73
hypocenterC73#13302
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×Q8 ◁ #2567

Complements

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

NH ≅ G/Ntype
26. #14556510Q8semidirect
144. C73#13302semidirect

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×#25672
#1330273