Finite Group Databasebeta

← #34001693

Subgroups — #34001693

Order 18688.

468 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 94 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#22625822C211yesmaximal, characteristic
2C4×C22⋊#10596833C211yesmaximal, characteristic
3#22624750C212yesmaximal, complemented
4#22624749C212yesmaximal, complemented
5#22618239C211yesmaximal, characteristic
6C2×C22⋊#10596833—21no—
7C2×C22⋊#10596833—21no—
8C2×C4×#10596833—21no—
9#17989453—21no—
10#17989453C2212yes—
11#17989453C2212yes—
12C2×#14556582C411yescharacteristic
13C22×C4×Dic73C2211yescharacteristic
14#17988486C2211yescharacteristic
15#17988483C2211yescharacteristic
16#17988474—21no—
17#17988470—21no—
18#17988462—21no—
19#17988459—21no—
20#17988452C411yescharacteristic
21#17988447C411yescharacteristic
22#380×C73C411yescharacteristic
23C22×#10596833—22no—
24C22×#10596833—22no—
25C22⋊#10596833—24no—
26C22⋊#10596833—42no—
27C4×#10596833—42no—
28C23×Dic73D411yescharacteristic
29C23×Dic73D411yescharacteristic
30#14556582—42no—
31C2×C4⋊Dic73—21no—
32C2×C4⋊Dic73—21no—
33C2×C4⋊Dic73C2×C411yescharacteristic
34C2×C4⋊Dic73C2×C411yescharacteristic
35C2×#11924007—21no—
36C2×#11924007—21no—
37C2×C22⋊Dic73—21no—
38C2×C4×Dic73—42no—
39C2×C4×Dic73C2×C411yescharacteristic
40C2×C4×Dic73D411yescharacteristic
41C2×C4×Dic73D411yescharacteristic
42C2×C4×Dic73C2311yescharacteristic
43#14556498—21no—
44#14556498—21no—
45#14556498—21no—
46#14556498—21no—
47#14556498—21no—
48C23×C4×C73C2×C411yescharacteristic
49C2×C4⋊C4×C73C2×C411yescharacteristic
50C2×C4⋊C4×C73C2×C411yescharacteristic
51C2×C22⋊C4×C73—21no—
52#95×C73—21no—
53C2×#10596833—24no—
54C2×#10596833—42no—
55C2×#10596833—42no—
56C2×#10596833—42no—
57C2×#10596833—42no—
58C2×#10596833—42no—
59C22×Dic73—22no—
60C22×Dic73—22no—
61C22×Dic73—22no—
62C22×Dic73—22no—
63C22×Dic73—21no—
64C22×Dic73—21no—
65C22×Dic73—41no—
66C22×Dic73—41no—
67C22×Dic73C22⋊C411yescharacteristic
68C22×Dic73C22⋊C411yescharacteristic
69C22×Dic73C2×D411yescharacteristic
70C22×Dic73C2×D411yescharacteristic
71C4⋊Dic73—22no—
72C4⋊Dic73—42no—
73C4⋊Dic73—42no—
74C4⋊Dic73—41no—
75#11924007—42no—
76#11924007—42no—
77C22⋊Dic73—42no—
78C4×Dic73—22no—
79C4×Dic73—22no—
80C4×Dic73—41no—
81C4×Dic73—41no—
82C24×C73C22⋊C411yescharacteristic
83C22×C4×C73—22no—
84C22×C4×C73—22no—
85C22×C4×C73—21no—
86C22×C4×C73—21no—
87C22×C4×C73—41no—
88C22×C4×C73C22⋊C411yescharacteristic
89C22×C4×C73C22×C411yescharacteristic
90C4⋊C4×C73—22no—
91C4⋊C4×C73—41no—
92C22⋊C4×C73—42no—
93C23×C73—21no—
94C23×C73—21no—
95C23×C73—21no—
96C23×C73—21no—
97C23×C73—21no—
98C23×C73—21no—
99C23×C73#10412yes—
100C23×C73C2×C22⋊C411yescharacteristic
101C2×C4×C73—21no—
102C2×C4×C73—21no—
103C2×C4×C73—21no—
104C2×C4×C73—21no—
105C2×C4×C73—44no—
106C2×C4×C73—41no—
107C2×C4×C73—41no—
108C2×C4×C73—41no—
109C2×C4×C73—41no—
110C2×C4×C73—41no—
111C2×C4×C73—41no—
112C2×C4×C73#10112yes—
113C2×C4×C73#13112yes—
114C2×Dic73—21no—
115C2×Dic73—21no—
116C2×Dic73—21no—
117C2×Dic73—21no—
118C2×Dic73—21no—
119C2×Dic73—21no—
120C2×Dic73—21no—
121C2×Dic73—21no—
122C2×Dic73—21no—
123C2×Dic73—44no—
124C2×Dic73—44no—
125C2×Dic73—41no—
126C2×Dic73—41no—
127C2×Dic73—41no—
128C2×Dic73—41no—
129C2×Dic73—41no—
130C2×Dic73—41no—
131C2×Dic73#9912yes—
132C2×Dic73#13512yes—
133C2×Dic73#13512yes—
134#10596833—44no—
135#10596833—82no—
136C22×C73—22no—
137C22×C73—22no—
138C22×C73—22no—
139C22×C73—41no—
140C22×C73—41no—
141C22×C73—41no—
142C22×C73—41no—
143C22×C73#40811yescharacteristic
144C22×C73C2×#9911yescharacteristic
145C22×C73C2×#10111yescharacteristic
146C22×C73#41611yescharacteristic
147C22×C73#41611yescharacteristic
148C22×C73C2×#10411yescharacteristic
149C22×C73#42111yescharacteristic
150C4×C73—22no—
151C4×C73—22no—
152C4×C73—81no—
153C4×C73—81no—
154Dic73—22no—
155Dic73—22no—
156Dic73—22no—
157Dic73—41no—
158Dic73—81no—
159Dic73—81no—
160#7671—731nomaximal, Sylow
161C2×C73—42no—
162C2×C73#147211yescharacteristic
163C2×C73#148211yescharacteristic
164C2×C73#149411yescharacteristic
165C2×C73#150111yescharacteristic
166C2×C73#151611yescharacteristic
167C2×C73#152811yescharacteristic
168C2×C73#154411yescharacteristic
169#2851—731no—
170#1817—731no—
171C4×C22⋊C8—731no—
172#1271—732no—
173#1271—732no—
174C73#767111yescharacteristic, complemented, Sylow
175C2×#125—731no—
176C22×C42—731no—
177C2×C22⋊C8—1461no—
178C2×C22⋊C8—1461no—
179C2×C4×C8—1461no—
180#389—731no—
181#389—731no—
182#387—1461no—
183#387—1461no—
184#385—1461no—
185#385—1461no—
186#382—731no—
187#380—731no—
188#380#6440811yescharacteristic
189#336—732no—
190#336—732no—
191#336—1461no—
192C23×C4—731no—
193C23×C4—731no—
194C23×C4C2×#6440811yescharacteristic
195C22×C8—1462no—
196C22×C8—1462no—
197#125—2922no—
198C2×C4⋊C4—731no—
199C2×C4⋊C4—731no—
200C2×C4⋊C4—1461no—
201C2×C4⋊C4—1461no—
202C2×C4⋊C4—1461no—
203C2×C4⋊C4—1461no—
204C2×C4⋊C4C2×#6440811yescharacteristic
205C2×C4⋊C4C2×#6440811yescharacteristic
206C2×C22⋊C4—21no—
207C2×C22⋊C4—1461no—
208C2×C42—2922no—
209C2×C42—731no—
210C2×C42—731no—
211C2×C42—731no—
212C2×C42—731no—
213C22⋊C8—2922no—
214C22⋊C8—1464no—
215C4×C8—2922no—
216#95—21no—
217#95—1461no—
218#95—1461no—
219#95—1461no—
220#95—1461no—
221#95—1461no—
222C24C22⋊#6440811yescharacteristic
223C22×C4—22no—
224C22×C4—22no—
225C22×C4—21no—
226C22×C4—21no—
227C22×C4—41no—
228C22×C4—2921no—
229C22×C4—2921no—
230C22×C4—731no—
231C22×C4—731no—
232C22×C4—731no—
233C22×C4—731no—
234C22×C4—1461no—
235C22×C4—1461no—
236C22×C4—1462no—
237C22×C4—1462no—
238C22×C4—1462no—
239C22×C4—1462no—
240C22×C4C22⋊#6440811yescharacteristic
241C22×C4C22×#6440811yescharacteristic
242C2×C8—2922no—
243C2×C8—2922no—
244C2×C8—2922no—
245C2×C8—2922no—
246C2×C8—2922no—
247C2×C8—1464no—
248C4⋊C4—22no—
249C4⋊C4—41no—
250C4⋊C4—2921no—
251C4⋊C4—2922no—
252C4⋊C4—2922no—
253C4⋊C4—2922no—
254C4⋊C4—2922no—
255C4⋊C4—1462no—
256C22⋊C4—42no—
257C22⋊C4—2922no—
258C42—2921no—
259C42—2921no—
260C42—1462no—
261C42—1462no—
262C23—21no—
263C23—21no—
264C23—21no—
265C23—21no—
266C23—21no—
267C23—21no—
268C23#1455665512yes—
269C23C2×C22⋊#6440811yescharacteristic
270C2×C4—21no—
271C2×C4—21no—
272C2×C4—21no—
273C2×C4—21no—
274C2×C4—44no—
275C2×C4—41no—
276C2×C4—41no—
277C2×C4—41no—
278C2×C4—41no—
279C2×C4—41no—
280C2×C4—41no—
281C2×C4—2924no—
282C2×C4—2921no—
283C2×C4—2921no—
284C2×C4—2921no—
285C2×C4—2921no—
286C2×C4—2921no—
287C2×C4—2921no—
288C2×C4—2924no—
289C2×C4—732no—
290C2×C4—732no—
291C2×C4—732no—
292C2×C4—1461no—
293C2×C4—1461no—
294C2×C4—1461no—
295C2×C4—1461no—
296C2×C4—1461no—
297C2×C4—1461no—
298C2×C4—1461no—
299C2×C4—1461no—
300C2×C4—1461no—
301C2×C4#1455665212yes—
302C2×C4#1455667812yes—
303C8—2924no—
304C8—5842no—
305C22—22no—
306C22—22no—
307C22—22no—
308C22—41no—
309C22—41no—
310C22—41no—
311C22—41no—
312C22#1798956011yescharacteristic
313C22D73⋊#9911yescharacteristic
314C22C2×#1455665211yescharacteristic
315C22#1798957711yescharacteristic
316C22#1798957711yescharacteristic
317C22C2×#1455665511yescharacteristic
318C22#1798958611yescharacteristic
319C4—22no—
320C4—22no—
321C4—81no—
322C4—81no—
323C4—2921no—
324C4—5841no—
325C4—5841no—
326C4—1462no—
327C4—1462no—
328C4—1462no—
329C2—42no—
330C2#2262521011yescharacteristic
331C2#2262524411yescharacteristic
332C2#2262528211yescharacteristic
333C2#2262530711yescharacteristic
334C2#2262535811yescharacteristic
335C2#2262539311yescharacteristic
336C2#2262543611yescharacteristic
subgroupG/H
centerC23C2×C22⋊#64408
commutatorC22×C4×C73C22×C4
FrattiniC22×C4C22×#64408
Fitting#380×C73C4
socleC23×C73C2×C22⋊C4
prime radicalC24×C73C22⋊C4
cosocleC2×C4×Dic73C23
hypercenter#380#64408
hypocenterC73#7671
LCSG ▷ C22×C4×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C23 ◁ C23×C4 ◁ #380

Complements

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

NH ≅ G/Ntype
3. #22624750C2semidirect
4. #22624749C2semidirect
174. C73#7671semidirect

Maximal subgroups

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

subgroupindex
#226258222
C4×C22⋊#105968332
#226247502
#226247492
#226182392
#767173