Finite Group Databasebeta

← #34009799

Subgroups — #34009799

Order 18688.

526 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 526 conjugacy classes of subgroups — too many to draw. Only its 90 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#22625163C212yesmaximal
2#22624858C212yesmaximal
3#22624857C212yesmaximal
4#22612516C211yesmaximal, characteristic
5#17989604—22no—
6#17989433—22no—
7#17989430C2214yes—
8C23×Dic146C411yescharacteristic
9C2×#14556547C411yescharacteristic
10C2×#14556547C2211yescharacteristic
11#17988491—22no—
12#17988491—22no—
13#17988450C2211yescharacteristic
14#17988448C411yescharacteristic
15#17988448C2211yescharacteristic
16#381×C73C411yescharacteristic
17#14556658—42no—
18C22⋊#10596833—24no—
19C4×#10596833—42no—
20C22×Dic146—21no—
21C22×Dic146—21no—
22C22×Dic146—22no—
23C22×Dic146—42no—
24C22×Dic146—42no—
25C22×Dic146C814yescomplemented
26C23×Dic73C2×C411yescharacteristic
27C23×Dic73C2×C411yescharacteristic
28#14556547—42no—
29#14556547—42no—
30C2×#11924007—21no—
31C2×#11924007D412yes—
32C2×C22⋊Dic73C2×C411yescharacteristic
33C2×C22⋊Dic73C2311yescharacteristic
34C2×C4×Dic73—21no—
35C2×C4×Dic73D412yes—
36#14556498—22no—
37#14556498—22no—
38#14556498—22no—
39C23×C4×C73C2×C411yescharacteristic
40C2×C22⋊C4×C73C2×C411yescharacteristic
41C2×C22⋊C4×C73C2×C411yescharacteristic
42#95×C73—22no—
43C2×#10596833—44no—
44C2×Dic146—24no—
45C2×Dic146—24no—
46C2×Dic146—42no—
47C2×Dic146—42no—
48C2×Dic146—48no—
49C2×Dic146—42no—
50C2×Dic146—42no—
51C2×Dic146—48no—
52C22×Dic73—22no—
53C22×Dic73—22no—
54C22×Dic73—21no—
55C22×Dic73—21no—
56C22×Dic73—21no—
57C22×Dic73—41no—
58C22×Dic73—41no—
59C22×Dic73C22⋊C411yescharacteristic
60C22×Dic73C22⋊C411yescharacteristic
61C22×Dic73C2×C812yes—
62C22×Dic73C2×C812yes—
63C22×Dic73M1612yes—
64C22×Dic73C2×D412yes—
65#11924007—42no—
66#11924007—42no—
67C22⋊Dic73—41no—
68C22⋊Dic73—41no—
69C4×Dic73—22no—
70C4×Dic73—22no—
71C4×Dic73—42no—
72C24×C73C22×C411yescharacteristic
73C22×C4×C73—21no—
74C22×C4×C73—21no—
75C22×C4×C73—21no—
76C22×C4×C73—41no—
77C22×C4×C73—41no—
78C22×C4×C73C22⋊C412yes—
79C22×C4×C73C2×C812yes—
80C22⋊C4×C73—41no—
81C22⋊C4×C73—41no—
82C23×C73—21no—
83C23×C73—21no—
84C23×C73—42no—
85C23×C73C2×C22⋊C411yescharacteristic
86C23×C73C22×C811yescharacteristic
87C23×C73C2×M1611yescharacteristic
88C2×C4×C73—22no—
89C2×C4×C73—22no—
90C2×C4×C73—42no—
91C2×C4×C73—41no—
92C2×C4×C73—41no—
93C2×C4×C73—42no—
94C2×C4×C73—42no—
95C2×C4×C73—42no—
96C2×C4×C73—41no—
97C2×Dic73—21no—
98C2×Dic73—21no—
99C2×Dic73—24no—
100C2×Dic73—24no—
101C2×Dic73—22no—
102C2×Dic73—22no—
103C2×Dic73—42no—
104C2×Dic73—41no—
105C2×Dic73—41no—
106C2×Dic73—42no—
107C2×Dic73—42no—
108C2×Dic73—42no—
109C2×Dic73—41no—
110C2×Dic73C22⋊C812yes—
111C2×Dic73C22⋊C812yes—
112C2×Dic73#9912yes—
113C2×Dic73#10012yes—
114Dic146—48no—
115Dic146—48no—
116#10596833—84no—
117C22×C73—22no—
118C22×C73—21no—
119C22×C73—21no—
120C22×C73—41no—
121C22×C73—41no—
122C22×C73—42no—
123C22×C73—42no—
124C22×C73C2×C22⋊C811yescharacteristic
125C22×C73C2×#9911yescharacteristic
126C22×C73C2×#10011yescharacteristic
127C4×C73—42no—
128C4×C73—42no—
129C4×C73—81no—
130Dic73—22no—
131Dic73—42no—
132Dic73—42no—
133Dic73—81no—
134Dic73#32314yes—
135#11428—731nomaximal, Sylow
136C2×C73—21no—
137C2×C73—21no—
138C2×C73—41no—
139C2×C73—41no—
140C2×C73C2×#32311yescharacteristic
141C2×C73#212312yes—
142#2310—731no—
143#1458—732no—
144#1317—732no—
145#1317—732no—
146C73#1142811yescharacteristic, complemented, Sylow
147C23×Q8—731no—
148C2×#117—731no—
149C2×#117—731no—
150#431—1462no—
151#391—1462no—
152#391—1462no—
153#381—731no—
154#381—731no—
155#381—731no—
156#381#6440811yescharacteristic
157#324—1462no—
158#323—734no—
159C22×Q8—2922no—
160C22×Q8—2922no—
161C22×Q8—734no—
162C22×Q8—1461no—
163C22×Q8—1461no—
164C22×Q8—1462no—
165C23×C4—731no—
166C23×C4—731no—
167C23×C4C2×#6440811yescharacteristic
168#117—2922no—
169#117—2922no—
170C2×C4⋊C4—732no—
171C2×C4⋊C4—1461no—
172C2×C22⋊C4—731no—
173C2×C22⋊C4—731no—
174C2×C22⋊C4C2×#6440811yescharacteristic
175C2×C22⋊C4C2×#6440811yescharacteristic
176C2×C42—732no—
177C2×C42—1461no—
178C4⋊C8—2922no—
179C22⋊C8—1464no—
180C4×C8—2922no—
181#95—22no—
182#95—1462no—
183#95—1462no—
184#95—1462no—
185C24C22×#6440811yescharacteristic
186C2×Q8—2922no—
187C2×Q8—2922no—
188C2×Q8—2928no—
189C2×Q8—2928no—
190C2×Q8—2922no—
191C2×Q8—2922no—
192C2×Q8—1464no—
193C2×Q8—1464no—
194C22×C4—21no—
195C22×C4—21no—
196C22×C4—21no—
197C22×C4—41no—
198C22×C4—41no—
199C22×C4—2921no—
200C22×C4—2921no—
201C22×C4—732no—
202C22×C4—732no—
203C22×C4—731no—
204C22×C4—732no—
205C22×C4—732no—
206C22×C4—731no—
207C22×C4—1461no—
208C22×C4—1461no—
209C22×C4—1461no—
210C22×C4—1462no—
211C22×C4—1462no—
212C22×C4C22⋊#6440812yes—
213C22×C4#1192403612yes—
214C2×C8—2924no—
215C4⋊C4—2922no—
216C4⋊C4—2922no—
217C22⋊C4—41no—
218C22⋊C4—41no—
219C22⋊C4—2921no—
220C22⋊C4—2921no—
221C42—2922no—
222C42—1462no—
223C42—1462no—
224C23—21no—
225C23—21no—
226C23—42no—
227C23C2×C22⋊#6440811yescharacteristic
228C23C2×#1192403611yescharacteristic
229C23D73⋊M1611yescharacteristic
230Q8—2928no—
231Q8—2928no—
232C2×C4—22no—
233C2×C4—22no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—42no—
238C2×C4—42no—
239C2×C4—42no—
240C2×C4—41no—
241C2×C4—2922no—
242C2×C4—2921no—
243C2×C4—2921no—
244C2×C4—2922no—
245C2×C4—2922no—
246C2×C4—2922no—
247C2×C4—2921no—
248C2×C4—732no—
249C2×C4—732no—
250C2×C4—732no—
251C2×C4—732no—
252C2×C4—1462no—
253C2×C4—1462no—
254C2×C4—1464no—
255C2×C4—1464no—
256C2×C4—1461no—
257C2×C4—1461no—
258C8—5844no—
259C22—22no—
260C22—21no—
261C22—21no—
262C22—41no—
263C22—41no—
264C22—42no—
265C22—42no—
266C22#1798955511yescharacteristic
267C22D73⋊#9911yescharacteristic
268C22D73⋊#10011yescharacteristic
269C4—42no—
270C4—42no—
271C4—81no—
272C4—2922no—
273C4—2922no—
274C4—5841no—
275C4—734no—
276C4—1462no—
277C2—21no—
278C2—21no—
279C2—41no—
280C2—41no—
281C2#2262515711yescharacteristic
282C2#2262619412yes—
subgroupG/H
centerC22D73⋊#99
commutatorC23×C73C22×C8
FrattiniC24C22×#64408
Fitting#381×C73C4
socleC22×C73C2×#99
prime radicalC24×C73C22×C4
cosocleC2×C22⋊Dic73C23
hypercenter#381#64408
hypocenterC73#11428
LCSG ▷ C23×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C23 ◁ C2×C22⋊C4 ◁ #381

Complements

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

NH ≅ G/Ntype
25. C22×Dic146C8semidirect
146. C73#11428semidirect

Maximal subgroups

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

subgroupindex
#226251632
#226248582
#226248572
#226125162
#1142873