Finite Group Databasebeta

← #34001672

Subgroups — #34001672

Order 18688.

568 conjugacy classes of subgroups; only the 124 normal subgroups are shown.

normal subgroups only

This group has 568 conjugacy classes of subgroups — too many to draw. Only its 124 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#22625787C211yesmaximal, characteristic
2C2×#17989453C211yesmaximal, characteristic
3#22624750C212yesmaximal, complemented
4#22624749C212yesmaximal, complemented
5#22618281C211yesmaximal, characteristic
6C23×#10596833—21no—
7C2×#14556658—21no—
8C2×C22⋊#10596833—21no—
9#17989453—21no—
10#17989453C2212yes—
11#17989453C2212yes—
12C2×#14556590C411yescharacteristic
13C2×#14556568—21no—
14C22×C4×Dic73C2211yescharacteristic
15#17988491—21no—
16#17988486C2211yescharacteristic
17#17988484—21no—
18#17988483C2211yescharacteristic
19#17988456C411yescharacteristic
20#17988447C411yescharacteristic
21#17988445—21no—
22#380×C73C411yescharacteristic
23C22×#10596833—24no—
24C22×#10596833—22no—
25C22×#10596833—22no—
26C22×#10596833—42no—
27C22×#10596833—42no—
28#14556658—42no—
29C22⋊#10596833—42no—
30C22×Dic146—21no—
31C23×Dic73D411yescharacteristic
32C23×Dic73D411yescharacteristic
33#14556590—42no—
34#14556568—42no—
35#14556568—42no—
36C2×C4⋊Dic73—21no—
37C2×C4⋊Dic73C2×C411yescharacteristic
38C2×C4⋊Dic73C2×C411yescharacteristic
39C2×#11924007—21no—
40C2×#11924007—21no—
41C2×#11924007—21no—
42C2×#11924007—21no—
43C2×C22⋊Dic73—21no—
44C2×C4×Dic73—42no—
45C2×C4×Dic73C2×C411yescharacteristic
46C2×C4×Dic73D411yescharacteristic
47C2×C4×Dic73D411yescharacteristic
48C2×C4×Dic73C2311yescharacteristic
49#14556498—21no—
50#14556498—21no—
51#14556498—21no—
52C23×C4×C73C2×C411yescharacteristic
53C2×C4⋊C4×C73C2×C411yescharacteristic
54C2×C4⋊C4×C73C2×C411yescharacteristic
55C2×C22⋊C4×C73—21no—
56#95×C73—21no—
57C2×#10596833—24no—
58C2×#10596833—42no—
59C2×#10596833—48no—
60C2×#10596833—42no—
61C2×#10596833—42no—
62C2×#10596833—42no—
63C2×#10596833—42no—
64C2×Dic146—42no—
65C2×Dic146—42no—
66C2×Dic146—42no—
67C22×Dic73—22no—
68C22×Dic73—22no—
69C22×Dic73—21no—
70C22×Dic73—21no—
71C22×Dic73—41no—
72C22×Dic73—41no—
73C22×Dic73C22⋊C411yescharacteristic
74C22×Dic73C22⋊C411yescharacteristic
75C22×Dic73D812yes—
76C22×Dic73SD1612yes—
77C22×Dic73SD1612yes—
78C22×Dic73Q1612yes—
79C22×Dic73C2×D411yescharacteristic
80C22×Dic73C2×D411yescharacteristic
81C4⋊Dic73—22no—
82C4⋊Dic73—41no—
83C4⋊Dic73—41no—
84C4⋊Dic73—41no—
85#11924007—42no—
86#11924007—42no—
87#11924007—42no—
88#11924007—41no—
89#11924007—41no—
90C22⋊Dic73—42no—
91C4×Dic73—22no—
92C4×Dic73—22no—
93C4×Dic73—41no—
94C4×Dic73—41no—
95C24×C73C22⋊C411yescharacteristic
96C22×C4×C73—22no—
97C22×C4×C73—22no—
98C22×C4×C73—21no—
99C22×C4×C73—21no—
100C22×C4×C73—41no—
101C22×C4×C73C22⋊C411yescharacteristic
102C22×C4×C73C22×C411yescharacteristic
103C4⋊C4×C73—22no—
104C4⋊C4×C73—41no—
105C22⋊C4×C73—42no—
106C23×C73—21no—
107C23×C73—21no—
108C23×C73—21no—
109C23×C73—21no—
110C23×C73#10112yes—
111C23×C73#10212yes—
112C23×C73#10312yes—
113C23×C73C2×C22⋊C411yescharacteristic
114C2×C4×C73—21no—
115C2×C4×C73—21no—
116C2×C4×C73—21no—
117C2×C4×C73—21no—
118C2×C4×C73—21no—
119C2×C4×C73—44no—
120C2×C4×C73—41no—
121C2×C4×C73—41no—
122C2×C4×C73—41no—
123C2×C4×C73—41no—
124C2×C4×C73—41no—
125C2×C4×C73—41no—
126C2×C4×C73#13112yes—
127C2×Dic73—21no—
128C2×Dic73—21no—
129C2×Dic73—21no—
130C2×Dic73—21no—
131C2×Dic73—21no—
132C2×Dic73—21no—
133C2×Dic73—24no—
134C2×Dic73—24no—
135C2×Dic73—42no—
136C2×Dic73—42no—
137C2×Dic73—41no—
138C2×Dic73—41no—
139C2×Dic73—41no—
140C2×Dic73—41no—
141C2×Dic73—41no—
142C2×Dic73—41no—
143C2×Dic73#9912yes—
144C2×Dic73#10212yes—
145C2×Dic73#10312yes—
146C2×Dic73#10412yes—
147C2×Dic73C2×D811yescharacteristic
148C2×Dic73C2×SD1611yescharacteristic
149C2×Dic73C2×SD1611yescharacteristic
150C2×Dic73C2×Q1611yescharacteristic
151Dic146—84no—
152#10596833—44no—
153#10596833—82no—
154C22×C73—22no—
155C22×C73—22no—
156C22×C73—22no—
157C22×C73—22no—
158C22×C73—22no—
159C22×C73—22no—
160C22×C73#33214yes—
161C22×C73#40811yescharacteristic
162C22×C73C2×#9911yescharacteristic
163C22×C73C2×#10111yescharacteristic
164C22×C73C2×#10211yescharacteristic
165C22×C73C2×#10311yescharacteristic
166C22×C73C2×#10411yescharacteristic
167C22×C73#42111yescharacteristic
168C4×C73—22no—
169C4×C73—41no—
170C4×C73—81no—
171C4×C73—81no—
172Dic73—21no—
173Dic73—21no—
174Dic73—21no—
175Dic73—22no—
176Dic73—21no—
177Dic73—81no—
178Dic73—81no—
179Dic73#32814yes—
180#7665—731nomaximal, Sylow
181C2×C73—24no—
182C2×C73#147211yescharacteristic
183C2×C73#148111yescharacteristic
184C2×C73#149211yescharacteristic
185C2×C73C2×#32811yescharacteristic
186C2×C73#151911yescharacteristic
187C2×C73#153111yescharacteristic
188C2×C73C2×#33211yescharacteristic
189#2857—731no—
190#1797—731no—
191C2×#336—731no—
192#1271—732no—
193#1271—732no—
194C73#766511yescharacteristic, complemented, Sylow
195C23×C8—1461no—
196C2×C4⋊Q8—731no—
197C2×C22⋊Q8—1461no—
198C22×C42—731no—
199C2×C4⋊C8—1461no—
200C2×C22⋊C8—1461no—
201#391—1461no—
202#389—731no—
203#389—731no—
204#389—1461no—
205#384—731no—
206#380—731no—
207#380—1461no—
208#380#6440811yescharacteristic
209#336—732no—
210#336—732no—
211#336—1461no—
212C22×Q8—1461no—
213C23×C4—731no—
214C23×C4—731no—
215C23×C4C2×#6440811yescharacteristic
216C22×C8—2922no—
217C22×C8—2922no—
218C22×C8—1462no—
219C22×C8—1462no—
220C22×C8—1464no—
221C4⋊Q8—2922no—
222C22⋊Q8—2922no—
223C22⋊Q8—2922no—
224C2×C4⋊C4—731no—
225C2×C4⋊C4—731no—
226C2×C4⋊C4—1461no—
227C2×C4⋊C4—1461no—
228C2×C4⋊C4—1461no—
229C2×C4⋊C4—1461no—
230C2×C4⋊C4—1461no—
231C2×C4⋊C4C2×#6440811yescharacteristic
232C2×C4⋊C4C2×#6440811yescharacteristic
233C2×C22⋊C4—21no—
234C2×C22⋊C4—1461no—
235C2×C42—2922no—
236C2×C42—731no—
237C2×C42—731no—
238C2×C42—731no—
239C2×C42—731no—
240C4⋊C8—2922no—
241C22⋊C8—2922no—
242#95—21no—
243#95—1461no—
244#95—1461no—
245#95—1461no—
246C24C22⋊#6440811yescharacteristic
247C2×Q8—2922no—
248C2×Q8—2922no—
249C2×Q8—2922no—
250C22×C4—22no—
251C22×C4—22no—
252C22×C4—21no—
253C22×C4—21no—
254C22×C4—41no—
255C22×C4—2921no—
256C22×C4—2921no—
257C22×C4—732no—
258C22×C4—732no—
259C22×C4—731no—
260C22×C4—732no—
261C22×C4—732no—
262C22×C4—731no—
263C22×C4—731no—
264C22×C4—731no—
265C22×C4—1461no—
266C22×C4—1461no—
267C22×C4—1462no—
268C22×C4—1462no—
269C22×C4C22⋊#6440811yescharacteristic
270C22×C4C22×#6440811yescharacteristic
271C2×C8—2922no—
272C2×C8—2928no—
273C2×C8—2922no—
274C2×C8—2922no—
275C2×C8—2922no—
276C2×C8—2922no—
277C2×C8—1464no—
278C4⋊C4—22no—
279C4⋊C4—41no—
280C4⋊C4—2921no—
281C4⋊C4—2922no—
282C4⋊C4—2922no—
283C4⋊C4—2921no—
284C4⋊C4—2921no—
285C4⋊C4—2922no—
286C4⋊C4—2921no—
287C4⋊C4—2921no—
288C4⋊C4—1462no—
289C22⋊C4—42no—
290C22⋊C4—2922no—
291C42—2921no—
292C42—2921no—
293C42—1462no—
294C42—1462no—
295C23—21no—
296C23—21no—
297C23—21no—
298C23—21no—
299C23#1455665212yes—
300C23#1455665312yes—
301C23#1455665412yes—
302C23C2×C22⋊#6440811yescharacteristic
303Q8—5844no—
304C2×C4—21no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—21no—
308C2×C4—21no—
309C2×C4—44no—
310C2×C4—41no—
311C2×C4—41no—
312C2×C4—41no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—41no—
316C2×C4—2921no—
317C2×C4—2921no—
318C2×C4—2921no—
319C2×C4—2921no—
320C2×C4—2921no—
321C2×C4—2921no—
322C2×C4—2922no—
323C2×C4—2922no—
324C2×C4—732no—
325C2×C4—732no—
326C2×C4—732no—
327C2×C4—731no—
328C2×C4—731no—
329C2×C4—732no—
330C2×C4—731no—
331C2×C4—731no—
332C2×C4—1464no—
333C2×C4—1464no—
334C2×C4—1461no—
335C2×C4—1461no—
336C2×C4—1461no—
337C2×C4—1461no—
338C2×C4—1461no—
339C2×C4—1461no—
340C2×C4#1455667812yes—
341C8—2924no—
342C8—5842no—
343C22—22no—
344C22—22no—
345C22—22no—
346C22—22no—
347C22—22no—
348C22—22no—
349C22#1798944614yes—
350C22#1798956011yescharacteristic
351C22D73⋊#9911yescharacteristic
352C22C2×#1455665211yescharacteristic
353C22C2×#1455665311yescharacteristic
354C22C2×#1455665411yescharacteristic
355C22D73⋊#10411yescharacteristic
356C22#1798958411yescharacteristic
357C4—22no—
358C4—41no—
359C4—81no—
360C4—81no—
361C4—5841no—
362C4—5841no—
363C4—734no—
364C4—1461no—
365C4—1461no—
366C4—1461no—
367C4—1462no—
368C4—1461no—
369C2—24no—
370C2#2262521011yescharacteristic
371C2#2262523911yescharacteristic
372C2#2262527611yescharacteristic
373C2D73⋊#32811yescharacteristic
374C2#2262536611yescharacteristic
375C2#2262540311yescharacteristic
376C2C2×#1798944611yescharacteristic
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#7665
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
194. C73#7665semidirect

Maximal subgroups

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

subgroupindex
#226257872
C2×#179894532
#226247502
#226247492
#226182812
#766573