Finite Group Databasebeta

← #32980894

Subgroups — #32980894

Order 18688.

414 conjugacy classes of subgroups.

The subgroup lattice has 398 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22619101C211yesmaximal, characteristic
2#22610913C211yesmaximal, characteristic, complemented
3#22608417C211yesmaximal, characteristic
4#22608404C211yesmaximal, characteristic, complemented
5#22607825C211yesmaximal, characteristic, complemented
6#22607821C211yesmaximal, characteristic, complemented
7C73×#3172C211yesmaximal, characteristic
8#17988718—21no—
9#17988665—21no—
10#17988659—21no—
11#17988659—21no—
12#17988648—21no—
13#17988644C2211yescharacteristic
14#17988640C2211yescharacteristic
15#17988614—21no—
16#17988611C2211yescharacteristic
17#17988433C2211yescharacteristic
18#17988346—21no—
19#17988346—21no—
20#17988324—21no—
21#17988321—21no—
22Q8⋊#10596831—21no—
23Q8⋊#10596831—21no—
24#533×C73C2211yescharacteristic
25#529×C73C2211yescharacteristic
26#501×C73C411yescharacteristic, complemented
27#497×C73C411yescharacteristic, complemented
28#491×C73C411yescharacteristic, complemented
29#489×C73C411yescharacteristic, complemented
30#470×C73—21no—
31#468×C73—21no—
32C4×Q16×C73—21no—
33C4×SD16×C73—21no—
34C4×M16×C73C2211yescharacteristic
35C2×#11924012—21no—
36C2×#11924012D411yescharacteristic
37C2×#11924012D411yescharacteristic
38C22×#10596831—21no—
39C2×C4×Dic73—21no—
40C4⋊#10596831—21no—
41C4⋊#10596831—21no—
42C4⋊#10596831—21no—
43C4⋊#10596831—21no—
44C4⋊#10596831—21no—
45C4⋊#10596831D411yescharacteristic
46C4⋊#10596831D411yescharacteristic
47C22⋊#10596831—21no—
48C22⋊#10596831—21no—
49C22⋊#10596831—21no—
50C22⋊#10596831—21no—
51#14556502—21no—
52#14556502—21no—
53C4×#10596831—21no—
54C4×#10596831—21no—
55#14556498—21no—
56#14556498—21no—
57C2×Q16×C73—21no—
58C2×SD16×C73—21no—
59C2×M16×C73—21no—
60C4⋊Q8×C73C2×C411yescharacteristic
61C4⋊Q8×C73C2×C411yescharacteristic
62#126×C73—21no—
63#125×C73—21no—
64#125×C73C2×C411yescharacteristic
65#124×C73C2×C411yescharacteristic
66#123×C73—21no—
67C22⋊Q8×C73—21no—
68C22⋊Q8×C73—21no—
69C22⋊Q8×C73D411yescharacteristic
70#121×C73D411yescharacteristic
71C4×Q8×C73—21no—
72C4×Q8×C73—21no—
73C4×Q8×C73D411yescharacteristic
74C4×D4×C73—21no—
75C4×D4×C73D411yescharacteristic
76#117×C73—21no—
77#117×C73—21no—
78C2×C42×C73C2311yescharacteristic
79#107×C73—21no—
80#106×C73—21no—
81#103×C73—21no—
82#103×C73—21no—
83#103×C73—21no—
84#102×C73—21no—
85#97×C73C2×C411yescharacteristic
86#97×C73C2×C411yescharacteristic
87C4×C8×C73—21no—
88C22×Dic73—41no—
89#11924012—41no—
90#11924012—41no—
91#11924012—81no—
92C2×#10596831—21no—
93C2×#10596831—21no—
94C2×#10596831—21no—
95C2×#10596831—21no—
96C2×#10596831—41no—
97C2×#10596831—81no—
98C4×Dic73—22no—
99C4×Dic73—41no—
100C2×Q8×C73—21no—
101C2×Q8×C73—21no—
102C2×Q8×C73C22⋊C411yescharacteristic
103C2×D4×C73—21no—
104C2×D4×C73C22⋊C411yescharacteristic
105C22×C4×C73—21no—
106C22×C4×C73C2×D411yescharacteristic
107C22×C4×C73C2×D411yescharacteristic
108C22×C4×C73#4111yescharacteristic
109Q16×C73—42no—
110SD16×C73—42no—
111M16×C73—42no—
112C2×C8×C73—21no—
113C2×C8×C73—21no—
114C4⋊C4×C73—21no—
115C4⋊C4×C73—21no—
116C4⋊C4×C73—21no—
117C4⋊C4×C73—21no—
118C4⋊C4×C73—21no—
119C4⋊C4×C73—21no—
120C4⋊C4×C73—21no—
121C4⋊C4×C73—21no—
122C4⋊C4×C73—41no—
123C4⋊C4×C73—41no—
124C4⋊C4×C73C22⋊C411yescharacteristic
125C4⋊C4×C73C22⋊C411yescharacteristic
126C22⋊C4×C73—21no—
127C22⋊C4×C73—21no—
128C22⋊C4×C73—21no—
129C22⋊C4×C73—21no—
130C22⋊C4×C73—21no—
131C22⋊C4×C73—41no—
132C42×C73—21no—
133C42×C73—41no—
134C42×C73C22×C411yescharacteristic
135C42×C73C2×D411yescharacteristic
136C42×C73C2×D411yescharacteristic
137C42×C73#4111yescharacteristic
138C23×C73—21no—
139C23×C73#12111yescharacteristic
140Q8×C73—41no—
141Q8×C73—41no—
142Q8×C73—41no—
143Q8×C73—42no—
144D4×C73—41no—
145D4×C73—81no—
146C2×C4×C73—21no—
147C2×C4×C73—21no—
148C2×C4×C73—21no—
149C2×C4×C73—21no—
150C2×C4×C73—21no—
151C2×C4×C73—41no—
152C2×C4×C73—41no—
153C2×C4×C73—41no—
154C2×C4×C73—41no—
155C2×C4×C73—41no—
156C2×C4×C73—41no—
157C2×C4×C73C2×C22⋊C411yescharacteristic
158C2×C4×C73C4×D411yescharacteristic
159C2×C4×C73C4×D411yescharacteristic
160C2×C4×C73#12011yescharacteristic
161C2×C4×C73#12111yescharacteristic
162C2×C4×C73#12311yescharacteristic
163C2×Dic73—42no—
164C2×Dic73—81no—
165C2×Dic73—81no—
166C8×C73—22no—
167C8×C73—41no—
168#10596831—41no—
169#10596831—41no—
170#10596831—41no—
171#10596831—41no—
172#10596831—81no—
173C22×C73—21no—
174C22×C73—21no—
175C22×C73—21no—
176C22×C73—41no—
177C22×C73—41no—
178C22×C73—41no—
179C22×C73#38611yescharacteristic
180C4×C73—21no—
181C4×C73—21no—
182C4×C73—21no—
183C4×C73—21no—
184C4×C73—21no—
185C4×C73—41no—
186C4×C73—41no—
187C4×C73—41no—
188C4×C73—81no—
189C4×C73—81no—
190C4×C73#45511yescharacteristic
191C4×C73#45611yescharacteristic
192Dic73—82no—
193#8998—731nomaximal, Sylow
194C2×C73—41no—
195C2×C73—81no—
196C2×C73#188111yescharacteristic
197C2×C73#188211yescharacteristic
198C2×C73#188411yescharacteristic
199#3172D7311yescharacteristic
200#2995—731no—
201#1995—731no—
202#1500—731no—
203#1497—731no—
204#1300—731no—
205#1299—731no—
206C73#899811yescharacteristic, complemented, Sylow
207#533C2×D7311yescharacteristic
208#529C2×D7311yescharacteristic
209#501Dic7311yescharacteristic, complemented
210#497Dic7311yescharacteristic, complemented
211#491Dic7311yescharacteristic, complemented
212#489Dic7311yescharacteristic, complemented
213#470—21no—
214#468—21no—
215#446—1461no—
216C4×Q16—21no—
217C4×SD16—21no—
218#436—1461no—
219#435—1461no—
220#435—1461no—
221#433—1461no—
222#432—731no—
223#431—731no—
224#424—1461no—
225#423—731no—
226C4×M16C2×D7311yescharacteristic
227#376—731no—
228#339—1461no—
229#339—1461no—
230#328—1461no—
231#327—1461no—
232Q8⋊C8—1461no—
233Q8⋊C8—1461no—
234C2×Q16—21no—
235C2×SD16—21no—
236C2×M16—21no—
237C2×M16—731no—
238C2×M16—731no—
239C2×M16—1461no—
240C22×C8—1461no—
241C4⋊Q8C2×Dic7311yescharacteristic
242C4⋊Q8C2×Dic7311yescharacteristic
243#126—21no—
244#125—21no—
245#125C2×Dic7311yescharacteristic
246#124C2×Dic7311yescharacteristic
247#123—21no—
248C22⋊Q8—21no—
249C22⋊Q8—21no—
250C22⋊Q8C22⋊D7311yescharacteristic
251#121C22⋊D7311yescharacteristic
252C4×Q8—21no—
253C4×Q8—21no—
254C4×Q8C22⋊D7311yescharacteristic
255C4×D4—21no—
256C4×D4C22⋊D7311yescharacteristic
257#117—21no—
258#117—21no—
259C2×C42—1461no—
260C2×C42C22×D7311yescharacteristic
261#107—21no—
262#106—21no—
263C4⋊C8—731no—
264C4⋊C8—731no—
265C4⋊C8—1461no—
266C4⋊C8—1461no—
267C4⋊C8—1461no—
268C4⋊C8—1461no—
269C4⋊C8—1461no—
270#103—21no—
271#103—21no—
272#103—21no—
273#102—21no—
274C22⋊C8—1461no—
275C22⋊C8—1461no—
276C22⋊C8—1461no—
277C22⋊C8—1461no—
278#97—1461no—
279#97—1461no—
280#97C2×Dic7311yescharacteristic
281#97C2×Dic7311yescharacteristic
282C4×C8—21no—
283C4×C8—1461no—
284C4×C8—1461no—
285#95—1461no—
286#95—1461no—
287C2×Q8—21no—
288C2×Q8—21no—
289C2×Q8C22⋊Dic7311yescharacteristic
290C2×D4—21no—
291C2×D4C22⋊Dic7311yescharacteristic
292C22×C4—21no—
293C22×C4—2921no—
294C22×C4D4×D7311yescharacteristic
295C22×C4C2×C22⋊D7311yescharacteristic
296C22×C4#1192402711yescharacteristic
297Q16—42no—
298SD16—42no—
299M16—42no—
300M16—2921no—
301M16—2921no—
302M16—5841no—
303C2×C8—21no—
304C2×C8—21no—
305C2×C8—2921no—
306C2×C8—5841no—
307C2×C8—1461no—
308C2×C8—1461no—
309C2×C8—1461no—
310C2×C8—1461no—
311C4⋊C4—21no—
312C4⋊C4—21no—
313C4⋊C4—21no—
314C4⋊C4—21no—
315C4⋊C4—21no—
316C4⋊C4—21no—
317C4⋊C4—21no—
318C4⋊C4—21no—
319C4⋊C4—41no—
320C4⋊C4—41no—
321C4⋊C4C22⋊Dic7311yescharacteristic
322C4⋊C4C22⋊Dic7311yescharacteristic
323C22⋊C4—21no—
324C22⋊C4—21no—
325C22⋊C4—21no—
326C22⋊C4—21no—
327C22⋊C4—21no—
328C22⋊C4—41no—
329C42—21no—
330C42—41no—
331C42—2921no—
332C42—1462no—
333C42C22×Dic7311yescharacteristic
334C42D4×D7311yescharacteristic
335C42C2×C22⋊D7311yescharacteristic
336C42#1192402711yescharacteristic
337C23—21no—
338C23#1455656311yescharacteristic
339Q8—41no—
340Q8—41no—
341Q8—41no—
342Q8—42no—
343D4—41no—
344D4—81no—
345C2×C4—21no—
346C2×C4—21no—
347C2×C4—21no—
348C2×C4—21no—
349C2×C4—21no—
350C2×C4—41no—
351C2×C4—41no—
352C2×C4—41no—
353C2×C4—41no—
354C2×C4—41no—
355C2×C4—41no—
356C2×C4—2922no—
357C2×C4—5841no—
358C2×C4—5841no—
359C2×C4C2×C22⋊Dic7311yescharacteristic
360C2×C4D4×Dic7311yescharacteristic
361C2×C4D4×Dic7311yescharacteristic
362C2×C4#1455655911yescharacteristic
363C2×C4#1455656311yescharacteristic
364C2×C4#1455657211yescharacteristic
365C8—22no—
366C8—41no—
367C8—2921no—
368C8—2921no—
369C8—2921no—
370C8—2921no—
371C8—5841no—
372C22—21no—
373C22—21no—
374C22—21no—
375C22—41no—
376C22—41no—
377C22—41no—
378C22#1798846711yescharacteristic
379C4—21no—
380C4—21no—
381C4—21no—
382C4—21no—
383C4—21no—
384C4—41no—
385C4—41no—
386C4—41no—
387C4—81no—
388C4—81no—
389C4—5842no—
390C4#1798876311yescharacteristic
391C4#1798876811yescharacteristic
392C2—41no—
393C2—81no—
394C2#2261029511yescharacteristic
395C2#2261030211yescharacteristic
396C2#2261031511yescharacteristic
subgroupG/H
centerC2×C4#14556559
commutatorC42×C73C22×C4
FrattiniC2×C42C22×D73
FittingC73×#3172C2
socleC22×C73#386
prime radical#121×C73D4
cosocleC2×C42×C73C23
hypercenter#3172D73
hypocenterC73#8998
LCSG ▷ C42×C73 ▷ C22×C73 ▷ C73
UCSC1 ◁ C2×C4 ◁ C2×C42 ◁ #3172

Complements

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

NH ≅ G/Ntype
2. #22610913C2semidirect
4. #22608404C2semidirect
5. #22607825C2semidirect
6. #22607821C2semidirect
26. #501×C73C4semidirect
27. #497×C73C4semidirect
28. #491×C73C4semidirect
29. #489×C73C4semidirect
206. C73#8998semidirect
209. #501Dic73semidirect
210. #497Dic73semidirect
211. #491Dic73semidirect
212. #489Dic73semidirect

Maximal subgroups

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

subgroupindex
#226191012
#226109132
#226084172
#226084042
#226078252
#226078212
C73×#31722
#899873