Finite Group Databasebeta

← #34019075

Subgroups — #34019075

Order 18688.

544 conjugacy classes of subgroups.

The subgroup lattice has 350 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22626835C212yesmaximal
2#22625556C212yesmaximal, complemented
3#22625555C212yesmaximal, complemented
4#22625545C212yesmaximal, complemented
5#22625542C212yesmaximal, complemented
6#22625493C211yesmaximal, characteristic
7#22625293C212yesmaximal
8#22625144C211yesmaximal, characteristic
9#22620512C211yesmaximal, characteristic
10#17989639—22no—
11#17989638C2212yes—
12#17989637—22no—
13#17989637C2212yes—
14#17989636C2212yes—
15Q8×#10596833C2212yes—
16#17989611—22no—
17#17989611—22no—
18#17989608C2212yes—
19#17989603C2211yescharacteristic
20#17989603C2211yescharacteristic
21#17989450C2211yescharacteristic
22#17989450C2211yescharacteristic
23#17989449C2212yes—
24#17989437C2212yes—
25#17989437C2212yes—
26Q8⋊#10596833C2212yes—
27Q8⋊#10596833C2212yes—
28#17989429C2212yes—
29#17989429C2212yes—
30#17989139C2212yes—
31#17988991C411yescharacteristic
32#17988965C411yescharacteristic
33#17988936C412yes—
34#17988828C411yescharacteristic
35#17988826C412yes—
36Q16×Dic73C2211yescharacteristic
37#17988681C2211yescharacteristic
38#17988679C2212yes—
39C2×C8×Dic73C2211yescharacteristic
40#467×C73C411yescharacteristic
41#14556658—22no—
42#14556658—22no—
43C4⋊#10596833—22no—
44C4⋊#10596833—22no—
45C4⋊#10596833—21no—
46C4⋊#10596833D411yescharacteristic
47C4⋊#10596833D411yescharacteristic
48C4⋊#10596833C2312yes—
49C4⋊#10596833C2312yes—
50C22⋊#10596833—22no—
51C22⋊#10596833—21no—
52C22⋊#10596833D412yes—
53#14556646—22no—
54#14556646—22no—
55#14556646—22no—
56C4×#10596833—22no—
57C4×#10596833—22no—
58C4×#10596833C2312yes—
59C4×#10596833C2312yes—
60C2×Dic292C2×C411yescharacteristic
61C2×#11924018C2×C412yes—
62C22×#10596831C2×C411yescharacteristic
63#14556590C2×C412yes—
64#14556589C2×C412yes—
65#14556581—22no—
66#14556570C2×C412yes—
67#14556568—22no—
68Q8×Dic73C2312yes—
69#14556556—22no—
70#14556556C2312yes—
71#14556547—22no—
72C2×C4×Dic73C2311yescharacteristic
73#14556525C2×C411yescharacteristic
74#14556524C2×C412yes—
75#14556516C2×C412yes—
76#14556515C2×C412yes—
77#14556514C2×C412yes—
78C8×Dic73C2×C411yescharacteristic
79C8×Dic73C2×C411yescharacteristic
80C8×Dic73C2311yescharacteristic
81C8×Dic73C2311yescharacteristic
82C2×Q16×C73C2×C411yescharacteristic
83C22×C8×C73C2×C411yescharacteristic
84C22⋊Q8×C73C2×C412yes—
85#107×C73C2×C411yescharacteristic
86#103×C73C2×C412yes—
87C2×#10596833—22no—
88C2×#10596833—22no—
89C2×#10596833—22no—
90C2×#10596833C2×D412yes—
91C2×#10596833C2×D412yes—
92C2×Dic146—22no—
93C2×Dic146C22×C412yes—
94C22×Dic73C2×D411yescharacteristic
95C22×Dic73#4111yescharacteristic
96Dic292—41no—
97#11924018—42no—
98C2×#10596831—41no—
99C2×#10596831C22×C411yescharacteristic
100C2×#10596831C22×C411yescharacteristic
101C4⋊Dic73—22no—
102C4⋊Dic73C22×C412yes—
103#11924007—22no—
104#11924007—22no—
105#11924007—22no—
106#11924007—22no—
107#11924007—22no—
108C22⋊Dic73—22no—
109C4×Dic73—22no—
110C4×Dic73—22no—
111C4×Dic73C22×C411yescharacteristic
112C4×Dic73C2×D411yescharacteristic
113C4×Dic73#4111yescharacteristic
114C4×Dic73C2411yescharacteristic
115C2×Q8×C73C22×C412yes—
116C22×C4×C73C22×C411yescharacteristic
117Q16×C73—41no—
118C2×C8×C73—41no—
119C2×C8×C73C22×C411yescharacteristic
120C2×C8×C73C22×C411yescharacteristic
121C4⋊C4×C73—22no—
122C4⋊C4×C73C22×C412yes—
123C22⋊C4×C73—22no—
124C23×C73C4×D411yescharacteristic
125Q8×C73—22no—
126Q8×C73—22no—
127C2×C4×C73—21no—
128C2×C4×C73—21no—
129C2×C4×C73—22no—
130C2×C4×C73—22no—
131C2×C4×C73C4×D411yescharacteristic
132C2×C4×C73C23×C411yescharacteristic
133C2×Dic73—21no—
134C2×Dic73—21no—
135C2×Dic73—22no—
136C2×Dic73—22no—
137C2×Dic73—41no—
138C2×Dic73C4×D411yescharacteristic
139C2×Dic73C4×D411yescharacteristic
140C2×Dic73C22×D411yescharacteristic
141C2×Dic73C2×#4111yescharacteristic
142Dic146—22no—
143Dic146—22no—
144Dic146—82no—
145#10596833—22no—
146#10596833—22no—
147#10596833—42no—
148#10596833—42no—
149#10596833—42no—
150C8×C73—21no—
151C8×C73—21no—
152#10596831—21no—
153#10596831—21no—
154C22×C73—21no—
155C22×C73—21no—
156C22×C73—21no—
157C22×C73C2×C4×D411yescharacteristic
158C4×C73—21no—
159C4×C73—42no—
160C4×C73—42no—
161C4×C73#56811yescharacteristic
162C4×C73#56811yescharacteristic
163Dic73—21no—
164Dic73—21no—
165Dic73—21no—
166Dic73—42no—
167Dic73—42no—
168Dic73#57811yescharacteristic
169Dic73#57811yescharacteristic
170#17991—731nomaximal, Sylow
171C2×C73—41no—
172C2×C73#293211yescharacteristic
173C2×C73#295011yescharacteristic
174C2×C73#295711yescharacteristic
175#3170—731no—
176#2985—732no—
177#1587—732no—
178#1587—732no—
179#1584—732no—
180#1584—732no—
181#1568—731no—
182#1497—732no—
183#1452—731no—
184C73#1799111yescharacteristic, complemented, Sylow
185#533—732no—
186#500—731no—
187#494—731no—
188#487—732no—
189#467—731no—
190#467—732no—
191#467#6440811yescharacteristic
192#446—732no—
193#446—732no—
194#446—1462no—
195#446—1462no—
196C8×Q8—732no—
197C8×Q8—732no—
198C4×Q16—731no—
199C4×Q16—731no—
200C4×Q16—732no—
201#433—1462no—
202#433—1462no—
203#432—732no—
204#431—731no—
205#431—731no—
206C2×C4×C8—731no—
207#335—731no—
208#335—731no—
209#334—732no—
210Q8⋊C8—732no—
211Q8⋊C8—732no—
212Q8⋊C8—732no—
213Q8⋊C8—732no—
214#322—732no—
215#322—732no—
216C2×Q16—732no—
217C2×Q16—731no—
218C2×Q16C2×#6440811yescharacteristic
219C22×C8—731no—
220C22×C8C2×#6440811yescharacteristic
221C4⋊Q8—732no—
222C4⋊Q8—732no—
223#125—1462no—
224C22⋊Q8—732no—
225C22⋊Q8—1462no—
226C22⋊Q8C2×#6440812yes—
227C4×Q8—732no—
228C4×Q8—732no—
229C4×Q8—1462no—
230#117—1462no—
231C2×C42—731no—
232#107—731no—
233#107—732no—
234#107C2×#6440811yescharacteristic
235C4⋊C8—731no—
236C4⋊C8—731no—
237C4⋊C8—732no—
238C4⋊C8—732no—
239C4⋊C8—1461no—
240C4⋊C8—1462no—
241C4⋊C8—1462no—
242C4⋊C8—1462no—
243C4⋊C8—1462no—
244#103—732no—
245#103—732no—
246#103—732no—
247#103C2×#6440812yes—
248C22⋊C8—732no—
249C22⋊C8—1461no—
250C22⋊C8—1462no—
251#97—1462no—
252#97—1462no—
253#97—1462no—
254C4×C8—732no—
255C4×C8—731no—
256C4×C8—731no—
257C4×C8—731no—
258C4×C8—731no—
259C4×C8—732no—
260C4×C8—1462no—
261C4×C8—1462no—
262C2×Q8—732no—
263C2×Q8—1462no—
264C2×Q8C22×#6440812yes—
265C22×C4—731no—
266C22×C4—731no—
267C22×C4C22×#6440811yescharacteristic
268Q16—41no—
269Q16—2921no—
270Q16—2922no—
271C2×C8—41no—
272C2×C8—2921no—
273C2×C8—732no—
274C2×C8—732no—
275C2×C8—731no—
276C2×C8—731no—
277C2×C8—1462no—
278C2×C8—1462no—
279C2×C8—1462no—
280C2×C8C22×#6440811yescharacteristic
281C2×C8C22×#6440811yescharacteristic
282C4⋊C4—22no—
283C4⋊C4—732no—
284C4⋊C4—1462no—
285C4⋊C4—1462no—
286C4⋊C4—1462no—
287C4⋊C4—1462no—
288C4⋊C4—1462no—
289C4⋊C4—1462no—
290C4⋊C4C22×#6440812yes—
291C22⋊C4—22no—
292C22⋊C4—1462no—
293C42—731no—
294C42—731no—
295C42—731no—
296C42—731no—
297C42—1462no—
298C42—1462no—
299C23D4×#6440811yescharacteristic
300Q8—22no—
301Q8—22no—
302Q8—5842no—
303Q8—1462no—
304Q8—1462no—
305C2×C4—21no—
306C2×C4—21no—
307C2×C4—22no—
308C2×C4—22no—
309C2×C4—2921no—
310C2×C4—731no—
311C2×C4—731no—
312C2×C4—731no—
313C2×C4—731no—
314C2×C4—1462no—
315C2×C4—1462no—
316C2×C4—1461no—
317C2×C4—1461no—
318C2×C4D4×#6440811yescharacteristic
319C2×C4C23×#6440811yescharacteristic
320C8—21no—
321C8—21no—
322C8—2922no—
323C8—2922no—
324C8—2922no—
325C8—1462no—
326C8—1462no—
327C8—1461no—
328C8—1461no—
329C22—21no—
330C22—21no—
331C22—21no—
332C22C2×D4×#6440811yescharacteristic
333C4—21no—
334C4—42no—
335C4—42no—
336C4—2922no—
337C4—2922no—
338C4—731no—
339C4—731no—
340C4—1461no—
341C4—1461no—
342C4—1461no—
343C4#1798966511yescharacteristic
344C4#1798966611yescharacteristic
345C2—41no—
346C2#2262665911yescharacteristic
347C2#2262669211yescharacteristic
348C2#2262670811yescharacteristic
subgroupG/H
centerC22C2×D4×#64408
commutatorC2×C4×C73C23×C4
FrattiniC2×C4C23×#64408
Fitting#467×C73C4
socleC22×C73C2×C4×D4
prime radicalC23×C73C4×D4
cosocleC4×Dic73C24
hypercenter#467#64408
hypocenterC73#17991
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #467

Complements

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

NH ≅ G/Ntype
2. #22625556C2semidirect
3. #22625555C2semidirect
4. #22625545C2semidirect
5. #22625542C2semidirect
184. C73#17991semidirect

Maximal subgroups

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

subgroupindex
#226268352
#226255562
#226255552
#226255452
#226255422
#226254932
#226252932
#226251442
#226205122
#1799173