Finite Group Databasebeta

← #34001534

Subgroups — #34001534

Order 18688.

462 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 108 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#22625859C211yesmaximal, characteristic
2#22625716C211yesmaximal, characteristic
3#22624749C214yesmaximal
4C4⋊Q8×Dic73C211yesmaximal, characteristic
5C2×#14556658—21no—
6C2×#14556646—21no—
7#17989453—21no—
8#17989453—21no—
9#17989453C2212yes—
10#17989453C2212yes—
11C2×Q8×Dic73—21no—
12#17988494—21no—
13#17988486—21no—
14#17988486C2212yes—
15#17988457C411yescharacteristic
16C4⋊C4×Dic73—21no—
17C4⋊C4×Dic73C412yes—
18C42×Dic73C2211yescharacteristic
19C2×C4⋊Q8×C73C411yescharacteristic
20C22×#10596833—22no—
21C22×#10596833—22no—
22#14556658—24no—
23#14556646—42no—
24Q8×Dic73—42no—
25Q8×Dic73—42no—
26C2×C4⋊Dic73—22no—
27C2×C4⋊Dic73—21no—
28C2×C4⋊Dic73—21no—
29C2×C4⋊Dic73C2×C412yes—
30C2×C4×Dic73—22no—
31C2×C4×Dic73C2×C411yescharacteristic
32C2×C4×Dic73D411yescharacteristic
33C2×C4×Dic73D411yescharacteristic
34C2×C4×Dic73D411yescharacteristic
35C2×C4×Dic73D411yescharacteristic
36C2×C4×Dic73C2311yescharacteristic
37#14556498—22no—
38#14556498—22no—
39C22×Q8×C73—21no—
40C4⋊Q8×C73—24no—
41C2×C4⋊C4×C73—21no—
42C2×C4⋊C4×C73C2×C412yes—
43C2×C42×C73C2×C411yescharacteristic
44C2×#10596833—24no—
45C2×#10596833—42no—
46C2×#10596833—42no—
47C2×#10596833—42no—
48C2×#10596833—42no—
49C2×#10596833—42no—
50C22×Dic73—22no—
51C22×Dic73C22⋊C411yescharacteristic
52C22×Dic73C22⋊C411yescharacteristic
53C22×Dic73C2×D411yescharacteristic
54C22×Dic73C2×D411yescharacteristic
55C4⋊Dic73—42no—
56C4⋊Dic73—42no—
57C4⋊Dic73—42no—
58C4⋊Dic73—42no—
59C4⋊Dic73—42no—
60C4×Dic73—22no—
61C4×Dic73—22no—
62C4×Dic73—22no—
63C4×Dic73—41no—
64C4×Dic73—41no—
65C4×Dic73—44no—
66C4×Dic73D812yes—
67C4×Dic73SD1612yes—
68C2×Q8×C73—24no—
69C2×Q8×C73—42no—
70C2×Q8×C73—42no—
71C22×C4×C73—22no—
72C22×C4×C73C22⋊C411yescharacteristic
73C22×C4×C73C22⋊C411yescharacteristic
74C22×C4×C73C22×C411yescharacteristic
75C4⋊C4×C73—24no—
76C4⋊C4×C73—22no—
77C4⋊C4×C73M1614yes—
78C42×C73—22no—
79C23×C73C2×C22⋊C411yescharacteristic
80Q8×C73—48no—
81C2×C4×C73—21no—
82C2×C4×C73—21no—
83C2×C4×C73—21no—
84C2×C4×C73—21no—
85C2×C4×C73—21no—
86C2×C4×C73—21no—
87C2×C4×C73—24no—
88C2×C4×C73—42no—
89C2×C4×C73—42no—
90C2×C4×C73#10212yes—
91C2×C4×C73#10412yes—
92C2×C4×C73C2×M1612yes—
93C2×Dic73—21no—
94C2×Dic73—21no—
95C2×Dic73—21no—
96C2×Dic73—21no—
97C2×Dic73—21no—
98C2×Dic73—21no—
99C2×Dic73—21no—
100C2×Dic73—21no—
101C2×Dic73—42no—
102C2×Dic73—42no—
103C2×Dic73—42no—
104C2×Dic73#10112yes—
105C2×Dic73#10212yes—
106C2×Dic73C2×D811yescharacteristic
107C2×Dic73C2×SD1611yescharacteristic
108C2×Dic73#13712yes—
109#10596833—44no—
110#10596833—82no—
111C22×C73#40711yescharacteristic
112C22×C73#41011yescharacteristic
113C22×C73C2×#10111yescharacteristic
114C22×C73C2×#10211yescharacteristic
115C22×C73#41911yescharacteristic
116C22×C73C2×#10411yescharacteristic
117C22×C73#42111yescharacteristic
118C4×C73—22no—
119C4×C73—22no—
120C4×C73—22no—
121C4×C73—44no—
122Dic73—22no—
123Dic73—21no—
124Dic73—21no—
125Dic73—22no—
126Dic73—22no—
127Dic73—82no—
128#7625—731nomaximal, Sylow
129C2×C73#146711yescharacteristic
130C2×C73#148611yescharacteristic
131C2×C73#149111yescharacteristic
132C2×C73#150211yescharacteristic
133C2×C73#151511yescharacteristic
134C2×C73#153111yescharacteristic
135C2×C73#154611yescharacteristic
136C4×C4⋊Q8—731no—
137#1839—731no—
138#1767—731no—
139#1271—734no—
140C73#762511yescharacteristic, complemented, Sylow
141C2×C4⋊Q8#6440811yescharacteristic
142C2×C4×Q8—1461no—
143C2×C4⋊C8—1461no—
144C2×#97—1461no—
145#391—1461no—
146#389—732no—
147#389—1461no—
148#384—731no—
149C4×C4⋊C4—732no—
150C4×C4⋊C4—1461no—
151C43—731no—
152#336—732no—
153#336—732no—
154#336—1461no—
155#336—1461no—
156C22×Q8—21no—
157C22×C8—1462no—
158C22×C8—1462no—
159C4⋊Q8—24no—
160C4×Q8—2922no—
161C4×Q8—2922no—
162C2×C4⋊C4—21no—
163C2×C4⋊C4—732no—
164C2×C4⋊C4—1462no—
165C2×C4⋊C4—1461no—
166C2×C4⋊C4—1461no—
167C2×C4⋊C4C2×#6440812yes—
168C2×C42—731no—
169C2×C42—731no—
170C2×C42—731no—
171C2×C42—731no—
172C2×C42—731no—
173C2×C42—731no—
174C2×C42—1462no—
175C2×C42C2×#6440811yescharacteristic
176C4⋊C8—1464no—
177#97—2922no—
178#95—1462no—
179#95—1462no—
180C2×Q8—24no—
181C2×Q8—42no—
182C2×Q8—42no—
183C22×C4—22no—
184C22×C4—731no—
185C22×C4—731no—
186C22×C4—731no—
187C22×C4—731no—
188C22×C4—1462no—
189C22×C4C22⋊#6440811yescharacteristic
190C22×C4C22⋊#6440811yescharacteristic
191C22×C4C22×#6440811yescharacteristic
192C2×C8—2922no—
193C2×C8—2922no—
194C2×C8—2922no—
195C2×C8—2922no—
196C2×C8—2922no—
197C2×C8—1464no—
198C4⋊C4—24no—
199C4⋊C4—22no—
200C4⋊C4—2922no—
201C4⋊C4—2922no—
202C4⋊C4—2922no—
203C4⋊C4—2922no—
204C4⋊C4—2922no—
205C4⋊C4#1192403714yes—
206C42—22no—
207C42—2921no—
208C42—2921no—
209C42—2924no—
210C42—732no—
211C42—732no—
212C42—1462no—
213C42—1462no—
214C42—1462no—
215C23C2×C22⋊#6440811yescharacteristic
216Q8—48no—
217C2×C4—21no—
218C2×C4—21no—
219C2×C4—21no—
220C2×C4—21no—
221C2×C4—21no—
222C2×C4—21no—
223C2×C4—24no—
224C2×C4—42no—
225C2×C4—42no—
226C2×C4—2922no—
227C2×C4—2922no—
228C2×C4—2922no—
229C2×C4—732no—
230C2×C4—732no—
231C2×C4—731no—
232C2×C4—731no—
233C2×C4—732no—
234C2×C4—1461no—
235C2×C4—1461no—
236C2×C4—1461no—
237C2×C4—1461no—
238C2×C4—1461no—
239C2×C4—1461no—
240C2×C4—1461no—
241C2×C4—1461no—
242C2×C4#1455665312yes—
243C2×C4#1455665612yes—
244C2×C4C2×#1192403712yes—
245C8—2924no—
246C8—5842no—
247C22#1798955711yescharacteristic
248C22#1798956811yescharacteristic
249C22D73⋊#10111yescharacteristic
250C22C2×#1455665311yescharacteristic
251C22#1798958011yescharacteristic
252C22C2×#1455665611yescharacteristic
253C22#1798958511yescharacteristic
254C4—22no—
255C4—22no—
256C4—22no—
257C4—44no—
258C4—5842no—
259C4—1461no—
260C4—1461no—
261C4—1462no—
262C4—1462no—
263C4—1462no—
264C2#2262519411yescharacteristic
265C2#2262525611yescharacteristic
266C2#2262527311yescharacteristic
267C2#2262531211yescharacteristic
268C2#2262535411yescharacteristic
269C2#2262540511yescharacteristic
270C2#2262544111yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC2×C4⋊Q8×C73C4
socle, prime radicalC23×C73C2×C22⋊C4
cosocleC2×C4×Dic73C23
hypercenterC2×C4⋊Q8#64408
hypocenterC73#7625
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ C2×C4⋊Q8

Complements

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

NH ≅ G/Ntype
140. C73#7625semidirect

Maximal subgroups

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

subgroupindex
#226258592
#226257162
#226247492
C4⋊Q8×Dic732
#762573