Finite Group Databasebeta

← #34012809

Subgroups — #34012809

Order 18688.

1018 conjugacy classes of subgroups; only the 108 normal subgroups are shown.

normal subgroups only

This group has 1018 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
1C2×#17989498C214yesmaximal, complemented
2#22625765C212yesmaximal
3C22×#14556591C211yesmaximal, characteristic
4C2×#14556652—22no—
5C2×#14556652C2214yes—
6#17989557—22no—
7#17989498—28no—
8C23×Dic146C411yescharacteristic
9C23×Dic146C2211yescharacteristic
10C2×#14556591—22no—
11C2×#14556591—28no—
12C2×#14556591C412yes—
13C2×#14556591C2212yes—
14C22×C4⋊Dic73—22no—
15C22×C42×C73C411yescharacteristic
16C2×#11924037—24no—
17#14556652—28no—
18#14556652—44no—
19C22⋊#10596833—44no—
20C22×Dic146—24no—
21C22×Dic146—24no—
22C22×Dic146—21no—
23C22×Dic146—42no—
24C22×Dic146—42no—
25C22×Dic146—42no—
26C22×Dic146—42no—
27C22×Dic146C2×C412yes—
28C22×Dic146C2×C411yescharacteristic
29C22×Dic146D412yes—
30C22×Dic146C2311yescharacteristic
31C23×Dic73—21no—
32C23×Dic73D412yes—
33#14556591—216no—
34#14556591—24no—
35#14556591—24no—
36#14556591—24no—
37#14556591D418yes—
38C2×C4⋊Dic73—216no—
39C2×C4⋊Dic73—24no—
40C2×C4⋊Dic73—24no—
41C23×C4×C73—21no—
42C23×C4×C73C2×C411yescharacteristic
43C2×C42×C73—21no—
44C2×C42×C73—24no—
45C2×C42×C73C2×C412yes—
46#11924037—44no—
47#11924037—44no—
48C2×#10596833—44no—
49C2×Dic146—24no—
50C2×Dic146—24no—
51C2×Dic146—22no—
52C2×Dic146—22no—
53C2×Dic146—24no—
54C2×Dic146—41no—
55C2×Dic146—41no—
56C2×Dic146—42no—
57C2×Dic146—41no—
58C2×Dic146—41no—
59C2×Dic146—416no—
60C2×Dic146—42no—
61C2×Dic146—416no—
62C2×Dic146—42no—
63C2×Dic146—42no—
64C2×Dic146—42no—
65C2×Dic146C22⋊C414yes—
66C2×Dic146C2×D414yes—
67C22×Dic73—24no—
68C22×Dic73—24no—
69C22×Dic73—22no—
70C22×Dic73—24no—
71C22×Dic73—42no—
72C22×Dic73—42no—
73C22×Dic73—42no—
74C22×Dic73C22⋊C412yes—
75C22×Dic73C2×D412yes—
76C4⋊Dic73—216no—
77C4⋊Dic73—216no—
78C24×C73C22⋊C411yescharacteristic
79C22×C4×C73—24no—
80C22×C4×C73—28no—
81C22×C4×C73—22no—
82C22×C4×C73—22no—
83C22×C4×C73—22no—
84C22×C4×C73—22no—
85C22×C4×C73C22⋊C411yescharacteristic
86C22×C4×C73C22×C411yescharacteristic
87C42×C73—24no—
88C42×C73—22no—
89C42×C73C22⋊C414yes—
90C23×C73—24no—
91C23×C73—21no—
92C23×C73—21no—
93C23×C73#9912yes—
94C23×C73C2×C22⋊C411yescharacteristic
95C2×C4×C73—216no—
96C2×C4×C73—28no—
97C2×C4×C73—28no—
98C2×C4×C73—21no—
99C2×C4×C73—24no—
100C2×C4×C73—21no—
101C2×C4×C73—21no—
102C2×C4×C73—21no—
103C2×C4×C73#9912yes—
104C2×C4×C73C2×C22⋊C412yes—
105C2×Dic73—22no—
106C2×Dic73—22no—
107C2×Dic73—24no—
108C2×Dic73—22no—
109C2×Dic73—22no—
110C2×Dic73—22no—
111C2×Dic73—48no—
112C2×Dic73—48no—
113C2×Dic73—42no—
114C2×Dic73—42no—
115C2×Dic73#12014yes—
116Dic146—44no—
117Dic146—48no—
118Dic146—44no—
119Dic146—48no—
120Dic146—44no—
121Dic146—44no—
122#10596833—84no—
123C22×C73—24no—
124C22×C73—24no—
125C22×C73—22no—
126C22×C73—22no—
127C22×C73—22no—
128C22×C73#35614yescomplemented
129C22×C73#37911yescharacteristic
130C22×C73C2×#9911yescharacteristic
131C22×C73C2×#9911yescharacteristic
132C4×C73—28no—
133C4×C73—22no—
134C4×C73—22no—
135Dic73—24no—
136Dic73—42no—
137Dic73—44no—
138#12700—731nomaximal, Sylow
139C2×C73—21no—
140C2×C73—24no—
141C2×C73—21no—
142C2×C73#178111yescharacteristic
143C2×C73C2×#35612yes—
144C22×C4⋊Q8—731no—
145C2×#356—734no—
146#1784—732no—
147C73#1270011yescharacteristic, complemented, Sylow
148C23×Q8—731no—
149C23×Q8—731no—
150C2×C4⋊Q8—732no—
151C2×C4⋊Q8—732no—
152C2×C4⋊Q8—1468no—
153C2×C4⋊Q8—1462no—
154C22×C4⋊C4—1462no—
155C22×C42#6440811yescharacteristic
156C2×#101—734no—
157C2×#101—1462no—
158#407—1462no—
159#356—1468no—
160C22×Q8—2922no—
161C22×Q8—2922no—
162C22×Q8—2922no—
163C22×Q8—2922no—
164C22×Q8—732no—
165C22×Q8—731no—
166C22×Q8—731no—
167C22×Q8—732no—
168C22×Q8—1464no—
169C22×Q8—1464no—
170C22×Q8—1461no—
171C23×C4—21no—
172C23×C4—732no—
173C23×C4—1461no—
174C23×C4C2×#6440811yescharacteristic
175C2×M16—1464no—
176C4⋊Q8—738no—
177C4⋊Q8—1464no—
178C4⋊Q8—1464no—
179C4⋊Q8—14616no—
180C4⋊Q8—1464no—
181C2×C4⋊C4—1464no—
182C2×C4⋊C4—1464no—
183C2×C4⋊C4—14616no—
184C2×C42—21no—
185C2×C42—24no—
186C2×C42C2×#6440812yes—
187#101—2924no—
188#101—1468no—
189C22⋊C8—2924no—
190C24C22⋊#6440811yescharacteristic
191C2×Q8—29216no—
192C2×Q8—29216no—
193C2×Q8—2922no—
194C2×Q8—2922no—
195C2×Q8—2922no—
196C2×Q8—2921no—
197C2×Q8—2921no—
198C2×Q8—2922no—
199C2×Q8—2921no—
200C2×Q8—2921no—
201C2×Q8—2922no—
202C2×Q8—734no—
203C2×Q8—734no—
204C2×Q8—1464no—
205C2×Q8—1464no—
206C2×Q8—1464no—
207C2×Q8—1462no—
208C2×Q8—1462no—
209C22×C4—24no—
210C22×C4—28no—
211C22×C4—22no—
212C22×C4—22no—
213C22×C4—22no—
214C22×C4—22no—
215C22×C4—2922no—
216C22×C4—2922no—
217C22×C4—2922no—
218C22×C4—732no—
219C22×C4—732no—
220C22×C4—1464no—
221C22×C4—1464no—
222C22×C4—1464no—
223C22×C4—1462no—
224C22×C4C22⋊#6440811yescharacteristic
225C22×C4C22×#6440811yescharacteristic
226M16—2924no—
227M16—2924no—
228C2×C8—2924no—
229C4⋊C4—14616no—
230C4⋊C4—14616no—
231C42—24no—
232C42—22no—
233C42C22⋊#6440814yes—
234C23—24no—
235C23—21no—
236C23—21no—
237C23#1455665012yes—
238C23C2×C22⋊#6440811yescharacteristic
239Q8—2924no—
240Q8—2928no—
241Q8—2924no—
242Q8—2924no—
243Q8—2928no—
244Q8—2924no—
245C2×C4—28no—
246C2×C4—216no—
247C2×C4—28no—
248C2×C4—21no—
249C2×C4—24no—
250C2×C4—21no—
251C2×C4—21no—
252C2×C4—21no—
253C2×C4—2928no—
254C2×C4—2922no—
255C2×C4—2922no—
256C2×C4—2928no—
257C2×C4—734no—
258C2×C4—1462no—
259C2×C4—1464no—
260C2×C4—1462no—
261C2×C4—1462no—
262C2×C4—1462no—
263C2×C4—1462no—
264C2×C4#1455665012yes—
265C2×C4C2×C22⋊#6440812yes—
266C8—5844no—
267C22—24no—
268C22—24no—
269C22—22no—
270C22—22no—
271C22—22no—
272C22#1798949814yescomplemented
273C22#1798952811yescharacteristic
274C22C2×#1455665011yescharacteristic
275C22C2×#1455665011yescharacteristic
276C4—28no—
277C4—22no—
278C4—22no—
279C4—2924no—
280C4—2922no—
281C4—1464no—
282C2—21no—
283C2—24no—
284C2—21no—
285C2#2262575911yescharacteristic
286C2C2×#1798949812yes—
subgroupG/H
centerC22C2×#14556650
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
FittingC22×C42×C73C4
socleC22×C73C2×#99
prime radicalC24×C73C22⋊C4
cosocleC22×Dic146C23
hypercenterC22×C42#64408
hypocenterC73#12700
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24 ◁ C23×C4 ◁ C22×C42

Complements

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

NH ≅ G/Ntype
1. C2×#17989498C2semidirect
128. C22×C73#356semidirect
147. C73#12700semidirect
272. C22#17989498semidirect

Maximal subgroups

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

subgroupindex
C2×#179894982
#226257652
C22×#145565912
#1270073