Finite Group Databasebeta

← #34004766

Subgroups — #34004766

Order 18688.

1026 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 1026 conjugacy classes of subgroups — too many to draw. Only its 116 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#22625765C212yesmaximal
2#22625181C212yesmaximal
3#22625179C212yesmaximal
4C22×#14556590C211yesmaximal, characteristic
5C2×#14556652—24no—
6#17989557C2214yes—
7#17989433—24no—
8#17989432—24no—
9C23×Dic146C411yescharacteristic
10C23×Dic146C2211yescharacteristic
11C2×#14556590—24no—
12C2×#14556590—24no—
13C2×#14556590—24no—
14C22×C4⋊Dic73C2211yescharacteristic
15C22×#11924007C411yescharacteristic
16C22×#11924007C2211yescharacteristic
17C22×C4×Dic73C411yescharacteristic
18C22×C4⋊C4×C73C411yescharacteristic
19C2×#11924037—24no—
20#14556652—48no—
21C22⋊#10596833—24no—
22C22⋊#10596833—24no—
23C22×Dic146—22no—
24C22×Dic146—22no—
25C22×Dic146—22no—
26C22×Dic146—22no—
27C22×Dic146—22no—
28C22×Dic146—44no—
29C22×Dic146—44no—
30C22×Dic146D414yes—
31C23×Dic73C2×C411yescharacteristic
32C23×Dic73C2×C411yescharacteristic
33C23×Dic73C2×C411yescharacteristic
34C23×Dic73C2311yescharacteristic
35#14556590—232no—
36C2×C4⋊Dic73—22no—
37C2×C4⋊Dic73—22no—
38C2×C4⋊Dic73D414yes—
39C2×#11924007—22no—
40C2×#11924007—22no—
41C2×#11924007—22no—
42C2×#11924007—22no—
43C2×#11924007—22no—
44C2×#11924007D414yes—
45C2×C4×Dic73—22no—
46C2×C4×Dic73—22no—
47C2×C4×Dic73—22no—
48C23×C4×C73C2×C411yescharacteristic
49C23×C4×C73C2×C411yescharacteristic
50C23×C4×C73C2×C411yescharacteristic
51C2×C4⋊C4×C73—22no—
52C2×C4⋊C4×C73—22no—
53C2×C4⋊C4×C73—22no—
54#11924037—48no—
55C2×#10596833—44no—
56C2×Dic146—28no—
57C2×Dic146—28no—
58C2×Dic146—28no—
59C2×Dic146—44no—
60C2×Dic146—48no—
61C2×Dic146—48no—
62C2×Dic146—48no—
63C2×Dic146—48no—
64C2×Dic146—44no—
65C2×Dic146—44no—
66C22×Dic73—21no—
67C22×Dic73—21no—
68C22×Dic73—21no—
69C22×Dic73—21no—
70C22×Dic73—24no—
71C22×Dic73—24no—
72C22×Dic73—21no—
73C22×Dic73—21no—
74C22×Dic73—42no—
75C22×Dic73—42no—
76C22×Dic73C22⋊C412yes—
77C22×Dic73C22⋊C412yes—
78C22×Dic73C22⋊C412yes—
79C22×Dic73C2×D412yes—
80C22×Dic73C2×D412yes—
81C22×Dic73C2×D412yes—
82C4⋊Dic73—28no—
83#11924007—28no—
84#11924007—28no—
85C4×Dic73—28no—
86C24×C73C22×C411yescharacteristic
87C22×C4×C73—21no—
88C22×C4×C73—21no—
89C22×C4×C73—24no—
90C22×C4×C73—21no—
91C22×C4×C73—21no—
92C22×C4×C73—21no—
93C22×C4×C73—21no—
94C22×C4×C73—42no—
95C22×C4×C73—42no—
96C22×C4×C73C22⋊C412yes—
97C22×C4×C73C22⋊C412yes—
98C22×C4×C73C22⋊C412yes—
99C4⋊C4×C73—28no—
100C23×C73—22no—
101C23×C73—22no—
102C23×C73—22no—
103C23×C73C2×C22⋊C411yescharacteristic
104C23×C73C2×C22⋊C411yescharacteristic
105C23×C73C2×C22⋊C411yescharacteristic
106C2×C4×C73—24no—
107C2×C4×C73—24no—
108C2×C4×C73—22no—
109C2×C4×C73—22no—
110C2×C4×C73—22no—
111C2×C4×C73—22no—
112C2×C4×C73—24no—
113C2×C4×C73—42no—
114C2×C4×C73—42no—
115C2×C4×C73—42no—
116C2×C4×C73—42no—
117C2×C4×C73—42no—
118C2×C4×C73#10114yescomplemented
119C2×Dic73—24no—
120C2×Dic73—24no—
121C2×Dic73—24no—
122C2×Dic73—24no—
123C2×Dic73—22no—
124C2×Dic73—22no—
125C2×Dic73—22no—
126C2×Dic73—24no—
127C2×Dic73—22no—
128C2×Dic73—22no—
129C2×Dic73—42no—
130C2×Dic73—42no—
131C2×Dic73—42no—
132C2×Dic73—42no—
133C2×Dic73—42no—
134C2×Dic73#9914yes—
135C2×Dic73#10114yescomplemented
136C2×Dic73#12014yes—
137Dic146—416no—
138Dic146—416no—
139#10596833—84no—
140C22×C73—28no—
141C22×C73—21no—
142C22×C73—21no—
143C22×C73—21no—
144C22×C73—21no—
145C22×C73—21no—
146C22×C73—21no—
147C22×C73#37911yescharacteristic
148C22×C73C2×#9912yes—
149C22×C73C2×#10112yes—
150C22×C73C2×#10112yes—
151C4×C73—24no—
152C4×C73—42no—
153C4×C73—42no—
154Dic73—24no—
155Dic73—24no—
156Dic73—24no—
157Dic73—42no—
158#8551—731nomaximal, Sylow
159C2×C73—22no—
160C2×C73—22no—
161C2×C73—22no—
162C2×C73#178111yescharacteristic
163C2×C73#178411yescharacteristic
164C2×C73#178411yescharacteristic
165C22×C4⋊Q8—731no—
166#1784—732no—
167#1463—732no—
168#1463—732no—
169C73#855111yescharacteristic, complemented, Sylow
170C23×Q8—731no—
171C23×Q8—731no—
172C2×C4⋊Q8—1464no—
173C2×C4⋊Q8—1464no—
174C2×C4⋊Q8—1464no—
175C22×C4⋊C4—731no—
176C22×C4⋊C4—731no—
177C22×C4⋊C4—731no—
178C22×C4⋊C4#6440811yescharacteristic
179C22×C42—731no—
180C2×#101—1464no—
181#407—734no—
182#324—1464no—
183#324—1464no—
184C22×Q8—2924no—
185C22×Q8—2924no—
186C22×Q8—734no—
187C22×Q8—1462no—
188C22×Q8—1462no—
189C22×Q8—1462no—
190C22×Q8—1462no—
191C22×Q8—1462no—
192C23×C4—731no—
193C23×C4—731no—
194C23×C4—731no—
195C23×C4—731no—
196C23×C4C2×#6440811yescharacteristic
197C23×C4C2×#6440811yescharacteristic
198C23×C4C2×#6440811yescharacteristic
199C2×M16—1464no—
200C4⋊Q8—14632no—
201C2×C4⋊C4—22no—
202C2×C4⋊C4—22no—
203C2×C4⋊C4—22no—
204C2×C4⋊C4—734no—
205C2×C4⋊C4—734no—
206C2×C4⋊C4—1462no—
207C2×C4⋊C4—1462no—
208C2×C4⋊C4—1462no—
209C2×C4⋊C4—1462no—
210C2×C4⋊C4—1462no—
211C2×C4⋊C4—1462no—
212C2×C4⋊C4—1462no—
213C2×C42—1462no—
214C2×C42—1462no—
215C2×C42—1462no—
216#101—2928no—
217C22⋊C8—1464no—
218C22⋊C8—1464no—
219C24C22×#6440811yescharacteristic
220C2×Q8—2924no—
221C2×Q8—2924no—
222C2×Q8—2928no—
223C2×Q8—2928no—
224C2×Q8—2928no—
225C2×Q8—2924no—
226C2×Q8—2928no—
227C2×Q8—1468no—
228C2×Q8—1468no—
229C2×Q8—1468no—
230C22×C4—21no—
231C22×C4—21no—
232C22×C4—24no—
233C22×C4—21no—
234C22×C4—21no—
235C22×C4—21no—
236C22×C4—21no—
237C22×C4—42no—
238C22×C4—42no—
239C22×C4—2922no—
240C22×C4—2922no—
241C22×C4—732no—
242C22×C4—732no—
243C22×C4—732no—
244C22×C4—732no—
245C22×C4—732no—
246C22×C4—732no—
247C22×C4—1461no—
248C22×C4—1461no—
249C22×C4—1464no—
250C22×C4—1464no—
251C22×C4—1461no—
252C22×C4—1461no—
253C22×C4—1461no—
254C22×C4—1461no—
255C22×C4C22⋊#6440812yes—
256C22×C4C22⋊#6440812yes—
257C22×C4C22⋊#6440812yes—
258M16—2928no—
259C2×C8—2924no—
260C4⋊C4—28no—
261C4⋊C4—1468no—
262C4⋊C4—1468no—
263C4⋊C4—1468no—
264C42—1468no—
265C23—22no—
266C23—22no—
267C23—22no—
268C23C2×C22⋊#6440811yescharacteristic
269C23C2×C22⋊#6440811yescharacteristic
270C23C2×C22⋊#6440811yescharacteristic
271Q8—29216no—
272Q8—29216no—
273C2×C4—24no—
274C2×C4—24no—
275C2×C4—22no—
276C2×C4—22no—
277C2×C4—22no—
278C2×C4—22no—
279C2×C4—24no—
280C2×C4—42no—
281C2×C4—42no—
282C2×C4—42no—
283C2×C4—42no—
284C2×C4—42no—
285C2×C4—2922no—
286C2×C4—2922no—
287C2×C4—2922no—
288C2×C4—2922no—
289C2×C4—2922no—
290C2×C4—734no—
291C2×C4—734no—
292C2×C4—734no—
293C2×C4—1462no—
294C2×C4—1464no—
295C2×C4—1464no—
296C2×C4—1464no—
297C2×C4—1464no—
298C2×C4—1462no—
299C2×C4—1462no—
300C2×C4—1464no—
301C2×C4—1462no—
302C2×C4—1462no—
303C2×C4#1455665214yescomplemented
304C8—5844no—
305C22—28no—
306C22—21no—
307C22—21no—
308C22—21no—
309C22—21no—
310C22—21no—
311C22—21no—
312C22#1798952811yescharacteristic
313C22D73⋊#9912yes—
314C22C2×#1455665212yes—
315C22D73⋊#10112yes—
316C4—24no—
317C4—42no—
318C4—42no—
319C4—2922no—
320C4—1464no—
321C4—1464no—
322C4—1464no—
323C2—22no—
324C2—22no—
325C2—22no—
326C2#2262575811yescharacteristic
327C2#2262576511yescharacteristic
328C2#2262576611yescharacteristic
subgroupG/H
centerC22#17989528
commutator, prime radicalC24×C73C22×C4
FrattiniC24C22×#64408
FittingC22×C4⋊C4×C73C4
socleC22×C73#379
cosocleC23×Dic73C23
hypercenterC22×C4⋊C4#64408
hypocenterC73#8551
LCSG ▷ C24×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C22 ◁ C24 ◁ C22×C4⋊C4

Complements

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

NH ≅ G/Ntype
118. C2×C4×C73#101semidirect
135. C2×Dic73#101semidirect
169. C73#8551semidirect
303. C2×C4#14556652semidirect

Maximal subgroups

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

subgroupindex
#226257652
#226251812
#226251792
C22×#145565902
#855173