Finite Group Databasebeta

← #34008191

Subgroups — #34008191

Order 18688.

424 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 102 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#22625716C211yesmaximal, characteristic
2#22625648C211yesmaximal, characteristic
3#22624807C214yesmaximal
4#22618292C211yesmaximal, characteristic
5C2×#14556658—21no—
6C2×#14556646C2212yes—
7C2×#14556646C2212yes—
8#17989453—21no—
9#17989453—21no—
10#17989453—21no—
11C2×#14556582C412yes—
12#17988486—21no—
13#17988486—21no—
14#17988483—21no—
15#17988483—21no—
16#17988457C411yescharacteristic
17#17988452C2212yes—
18C42×Dic73C2211yescharacteristic
19#384×C73C411yescharacteristic
20C22×#10596833—22no—
21C22×#10596833—22no—
22#14556658—24no—
23#14556646—22no—
24#14556646—24no—
25#14556646Q814yes—
26#14556582—44no—
27C2×C4⋊Dic73—22no—
28C2×C4⋊Dic73—21no—
29C2×C4⋊Dic73—21no—
30C2×C4⋊Dic73C2×C412yes—
31C2×#11924007—22no—
32C2×C4×Dic73C2×C411yescharacteristic
33C2×C4×Dic73D411yescharacteristic
34C2×C4×Dic73D411yescharacteristic
35C2×C4×Dic73D411yescharacteristic
36C2×C4×Dic73D411yescharacteristic
37C2×C4×Dic73C2311yescharacteristic
38#14556498—22no—
39#14556498—22no—
40C2×C4⋊C4×C73—21no—
41C2×C4⋊C4×C73—21no—
42C2×C4⋊C4×C73C2×C412yes—
43C2×C42×C73C2×C411yescharacteristic
44C2×#10596833—24no—
45C2×#10596833—24no—
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—44no—
56C4⋊Dic73—42no—
57C4⋊Dic73—42no—
58C4⋊Dic73—42no—
59#11924007—44no—
60C4×Dic73—22no—
61C4×Dic73—22no—
62C4×Dic73—41no—
63C4×Dic73—41no—
64C4×Dic73—41no—
65C4×Dic73C2×Q812yes—
66C4×Dic73#4112yes—
67C22×C4×C73—22no—
68C22×C4×C73C22⋊C411yescharacteristic
69C22×C4×C73C22⋊C411yescharacteristic
70C22×C4×C73C22×C411yescharacteristic
71C4⋊C4×C73—42no—
72C4⋊C4×C73—42no—
73C4⋊C4×C73—42no—
74C42×C73—22no—
75C23×C73C2×C22⋊C411yescharacteristic
76C2×C4×C73—21no—
77C2×C4×C73—21no—
78C2×C4×C73—21no—
79C2×C4×C73—21no—
80C2×C4×C73—21no—
81C2×C4×C73—21no—
82C2×C4×C73—42no—
83C2×C4×C73—42no—
84C2×C4×C73—42no—
85C2×C4×C73#10112yes—
86C2×C4×C73#10412yes—
87C2×C4×C73C4×Q812yes—
88C2×Dic73—21no—
89C2×Dic73—21no—
90C2×Dic73—21no—
91C2×Dic73—21no—
92C2×Dic73—21no—
93C2×Dic73—21no—
94C2×Dic73—21no—
95C2×Dic73—21no—
96C2×Dic73—21no—
97C2×Dic73—42no—
98C2×Dic73—42no—
99C2×Dic73—42no—
100C2×Dic73#10012yes—
101C2×Dic73C22⋊Q812yes—
102C2×Dic73#12411yescharacteristic
103C2×Dic73C4⋊Q811yescharacteristic
104#10596833—44no—
105#10596833—44no—
106C22×C73#39111yescharacteristic
107C22×C73C2×#10011yescharacteristic
108C22×C73C2×#10111yescharacteristic
109C22×C73C2×#10411yescharacteristic
110C22×C73#42111yescharacteristic
111C22×C73#42111yescharacteristic
112C22×C73#42111yescharacteristic
113C4×C73—22no—
114C4×C73—22no—
115C4×C73—22no—
116C4×C73—82no—
117Dic73—21no—
118Dic73—21no—
119Dic73—22no—
120Dic73—22no—
121Dic73—41no—
122Dic73—82no—
123#9894—731nomaximal, Sylow
124C2×C73#152811yescharacteristic
125C2×C73#153011yescharacteristic
126C2×C73#153411yescharacteristic
127C2×C73#153611yescharacteristic
128C2×C73#198611yescharacteristic
129C2×C73#199511yescharacteristic
130C2×C73#199711yescharacteristic
131#2859—731no—
132#1767—731no—
133#1745—731no—
134#1295—734no—
135C73#989411yescharacteristic, complemented, Sylow
136C2×#125—732no—
137C2×C4⋊C8—1461no—
138C2×#97—732no—
139C2×#97—732no—
140#389—1461no—
141#389—1461no—
142#389—1461no—
143#389—1461no—
144#384—731no—
145#384#6440811yescharacteristic
146#382—732no—
147C43—731no—
148#336—1461no—
149#336—1461no—
150#336—1461no—
151C22×C8—1462no—
152C22×C8—1462no—
153#125—2924no—
154C2×C4⋊C4—21no—
155C2×C4⋊C4—21no—
156C2×C4⋊C4—732no—
157C2×C4⋊C4—1462no—
158C2×C4⋊C4—1462no—
159C2×C4⋊C4—1461no—
160C2×C4⋊C4—1461no—
161C2×C4⋊C4C2×#6440812yes—
162C2×C42—731no—
163C2×C42—731no—
164C2×C42—731no—
165C2×C42—731no—
166C2×C42—731no—
167C2×C42—731no—
168C2×C42C2×#6440811yescharacteristic
169C4⋊C8—1464no—
170#97—734no—
171#97—1464no—
172#97—1462no—
173#95—1462no—
174#95—1462no—
175C22×C4—22no—
176C22×C4—731no—
177C22×C4—731no—
178C22×C4—731no—
179C22×C4—731no—
180C22×C4—1462no—
181C22×C4C22⋊#6440811yescharacteristic
182C22×C4C22⋊#6440811yescharacteristic
183C22×C4C22×#6440811yescharacteristic
184C2×C8—2922no—
185C2×C8—2922no—
186C2×C8—2922no—
187C2×C8—2922no—
188C2×C8—1464no—
189C2×C8—1464no—
190C4⋊C4—42no—
191C4⋊C4—42no—
192C4⋊C4—42no—
193C4⋊C4—2922no—
194C4⋊C4—2922no—
195C4⋊C4—2922no—
196C4⋊C4—2924no—
197C4⋊C4—2924no—
198C42—22no—
199C42—2921no—
200C42—2921no—
201C42—2921no—
202C42—732no—
203C42—732no—
204C42—1462no—
205C42—1462no—
206C23C2×C22⋊#6440811yescharacteristic
207C2×C4—21no—
208C2×C4—21no—
209C2×C4—21no—
210C2×C4—21no—
211C2×C4—21no—
212C2×C4—21no—
213C2×C4—42no—
214C2×C4—42no—
215C2×C4—42no—
216C2×C4—2922no—
217C2×C4—2922no—
218C2×C4—2922no—
219C2×C4—731no—
220C2×C4—732no—
221C2×C4—731no—
222C2×C4—732no—
223C2×C4—1461no—
224C2×C4—1461no—
225C2×C4—1461no—
226C2×C4—1461no—
227C2×C4—1461no—
228C2×C4—1461no—
229C2×C4—1461no—
230C2×C4—1461no—
231C2×C4—1461no—
232C2×C4#1455665212yes—
233C2×C4#1455665512yes—
234C2×C4Q8×#6440812yes—
235C8—2924no—
236C8—2924no—
237C22(C2×#64408)⋊Q811yescharacteristic
238C22D73⋊#10011yescharacteristic
239C22C2×#1455665211yescharacteristic
240C22C2×#1455665511yescharacteristic
241C22#1798958511yescharacteristic
242C22#1798958611yescharacteristic
243C22#1798958611yescharacteristic
244C4—22no—
245C4—22no—
246C4—22no—
247C4—82no—
248C4—2921no—
249C4—5842no—
250C4—1461no—
251C4—1461no—
252C4—1462no—
253C4—1462no—
254C2#2262539311yescharacteristic
255C2#2262540111yescharacteristic
256C2#2262541311yescharacteristic
257C2#2262542111yescharacteristic
258C2#2262610711yescharacteristic
259C2#2262611911yescharacteristic
260C2#2262612411yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
Fitting#384×C73C4
socle, prime radicalC23×C73C2×C22⋊C4
cosocleC2×C4×Dic73C23
hypercenter#384#64408
hypocenterC73#9894
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C2×C42 ◁ #384

Complements

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

NH ≅ G/Ntype
135. C73#9894semidirect

Maximal subgroups

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

subgroupindex
#226257162
#226256482
#226248072
#226182922
#989473