Finite Group Databasebeta

← #33004186

Subgroups — #33004186

Order 18688.

438 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618625C211yesmaximal, characteristic
2#22611696C211yesmaximal, characteristic, complemented
3#22611560C211yesmaximal, characteristic, complemented
4#22611558C211yesmaximal, characteristic, complemented
5#22611542C211yesmaximal, characteristic, complemented
6#22611535C211yesmaximal, characteristic, complemented
7C73×#2112C211yesmaximal, characteristic, complemented
8C2×#14556597C2211yescharacteristic
9#17989098C411yescharacteristic
10C2×#14556531C2211yescharacteristic
11C2×#14556531C2211yescharacteristic
12#17988658—22no—
13C8×C22⋊D73—22no—
14#17988613C411yescharacteristic
15#17988553C411yescharacteristic
16#17988551C411yescharacteristic
17#17988542C2211yescharacteristic
18#17988407—22no—
19#17988372—22no—
20#17988370—22no—
21C22⋊#11924003—22no—
22#17988364—22no—
23C23×C8×C73C2211yescharacteristic
24C2×M32×C73C2211yescharacteristic
25C2×M32×C73C2211yescharacteristic
26C22⋊C16×C73—22no—
27C2×#11924028C2×C411yescharacteristic
28#14556597—21no—
29#14556597—21no—
30#14556597—21no—
31#14556597—21no—
32C2×#11924012C2×C411yescharacteristic
33C2×#11924012C2×C411yescharacteristic
34C2×#11924011—21no—
35C2×C8×D73—21no—
36#14556574—21no—
37#14556570—21no—
38#14556565—21no—
39#14556560—21no—
40C4×C22⋊D73—22no—
41#14556550C2×C411yescharacteristic
42#14556531—21no—
43#14556531—21no—
44#14556531—21no—
45#14556531—21no—
46C2×#11924003—21no—
47C2×#11924003D412yescomplemented
48#14556520—22no—
49C22⋊#10596831—22no—
50#14556503—22no—
51#14556501D412yes—
52C8×Dic73D412yes—
53C23×C4×C73C2×C411yescharacteristic
54C22×C8×C73—21no—
55C22×C8×C73—21no—
56C22×C8×C73—21no—
57C22×C8×C73—21no—
58C22×C8×C73—21no—
59C22×C8×C73—21no—
60C22×C8×C73C2×C411yescharacteristic
61C22×C8×C73C2311yescharacteristic
62M32×C73—21no—
63M32×C73—21no—
64M32×C73—21no—
65M32×C73—21no—
66C2×C16×C73—21no—
67C2×C16×C73D412yescomplemented
68#11924028—21no—
69#11924028—21no—
70#11924028—21no—
71#11924028—21no—
72C2×Dic146M1611yescharacteristic
73C2×D292M1611yescharacteristic
74C2×C22⋊D73—22no—
75C2×C22⋊D73—21no—
76C2×C4×D73—21no—
77#11924012—21no—
78#11924012—21no—
79#11924012—21no—
80#11924012—21no—
81#11924011—41no—
82#11924011—41no—
83C2×#10596831—21no—
84C2×#10596831C22⋊C412yes—
85C8×D73—41no—
86C8×D73—41no—
87C4⋊Dic73—21no—
88#11924007—22no—
89C22⋊Dic73—22no—
90C22⋊Dic73—21no—
91#11924005—22no—
92C4×Dic73C22⋊C412yes—
93#11924003—22no—
94#11924003—41no—
95C24×C73M1611yescharacteristic
96C22×C4×C73—21no—
97C22×C4×C73—21no—
98C22×C4×C73—21no—
99C22×C4×C73—21no—
100C22×C4×C73—21no—
101C22×C4×C73—21no—
102C22×C4×C73M1611yescharacteristic
103C22×C4×C73C22×C411yescharacteristic
104C2×C8×C73—21no—
105C2×C8×C73—21no—
106C2×C8×C73—21no—
107C2×C8×C73—41no—
108C2×C8×C73—41no—
109C2×C8×C73—41no—
110C2×C8×C73—41no—
111C2×C8×C73C2×D411yescharacteristic
112C2×C8×C73C2×D411yescharacteristic
113C2×C8×C73C2×D411yescharacteristic
114C2×C8×C73C2×D411yescharacteristic
115C2×C8×C73#4111yescharacteristic
116C2×C8×C73#4111yescharacteristic
117C16×C73—22no—
118C16×C73—41no—
119C23×C73—21no—
120C23×C73—21no—
121C23×C73—21no—
122C23×C73—42no—
123C23×C73C2×M1611yescharacteristic
124C22×D73—21no—
125C2×C4×C73—21no—
126C2×C4×C73—21no—
127C2×C4×C73—21no—
128C2×C4×C73—41no—
129C2×C4×C73—41no—
130C2×C4×C73—41no—
131C2×C4×C73—41no—
132C2×C4×C73C2×C22⋊C411yescharacteristic
133C2×C4×C73C4×D411yescharacteristic
134C2×C4×C73C4×D411yescharacteristic
135C2×C4×C73C2×M1611yescharacteristic
136C2×C4×C73#13111yescharacteristic
137C2×C4×C73#13111yescharacteristic
138C22⋊D73—44no—
139C22⋊D73—41no—
140C22⋊D73—41no—
141C2×Dic73—22no—
142C2×Dic73—21no—
143D292—21no—
144D292—21no—
145C4×D73—41no—
146C4×D73—41no—
147Dic146—21no—
148Dic146—21no—
149C8×C73—41no—
150C8×C73#12111yescharacteristic
151C8×C73#12111yescharacteristic
152C8×C73#12411yescharacteristic
153C8×C73#12711yescharacteristic
154#10596831—22no—
155#10596831—41no—
156C22×C73—22no—
157C22×C73—22no—
158C22×C73—22no—
159C22×C73—41no—
160C22×C73—41no—
161C22×C73—41no—
162C22×C73—41no—
163C22×C73—41no—
164C22×C73#40711yescharacteristic
165C22×C73#43511yescharacteristic
166C22×C73#43511yescharacteristic
167C2×D73—42no—
168C2×D73—41no—
169C4×C73—41no—
170C4×C73#39011yescharacteristic
171C4×C73#40811yescharacteristic
172C4×C73#43511yescharacteristic
173C4×C73#43511yescharacteristic
174Dic73—42no—
175Dic73—41no—
176#13200—731nomaximal, Sylow
177C2×C73—21no—
178C2×C73—21no—
179C2×C73—21no—
180C2×C73—42no—
181C2×C73#195511yescharacteristic
182D73—81no—
183#2929—731no—
184#2153—731no—
185#2117—731no—
186#2117—731no—
187#2112—731no—
188#2112D7311yescharacteristic, complemented
189#2110—731no—
190C73#1320011yescharacteristic, complemented, Sylow
191C2×#131—731no—
192C23×C8C2×D7311yescharacteristic
193#525—731no—
194C2×M32—731no—
195C2×M32—731no—
196C2×M32C2×D7311yescharacteristic
197C2×M32C2×D7311yescharacteristic
198#435—1462no—
199C8×D4—1462no—
200#424—731no—
201#408—731no—
202#407—731no—
203#405—731no—
204#364—1462no—
205#350—1462no—
206#350—1462no—
207C22⋊C16—22no—
208C22⋊C16—1462no—
209#347—1462no—
210C2×#41—731no—
211C23×C4C4×D7311yescharacteristic
212#131—1461no—
213#131—1461no—
214#131—1461no—
215#131—1461no—
216C2×M16—731no—
217C2×M16—731no—
218C2×M16—1461no—
219C22×C8—21no—
220C22×C8—21no—
221C22×C8—21no—
222C22×C8—21no—
223C22×C8—21no—
224C22×C8—21no—
225C22×C8—1461no—
226C22×C8C4×D7311yescharacteristic
227C22×C8C22×D7311yescharacteristic
228#123—1461no—
229C22⋊Q8—1461no—
230#121—1461no—
231#120—1461no—
232C4×D4—1462no—
233#117—731no—
234M32—21no—
235M32—21no—
236M32—21no—
237M32—21no—
238M32—1461no—
239M32—1461no—
240M32—1461no—
241M32—1461no—
242C2×C16—21no—
243C2×C16—732no—
244C2×C16—1461no—
245C2×C16C22⋊D7312yescomplemented
246C4⋊C8—1462no—
247C22⋊C8—1462no—
248C22⋊C8—1462no—
249#97—732no—
250C4×C8—732no—
251C24#1192401111yescharacteristic
252#41—1461no—
253#41—1461no—
254#41—1461no—
255#41—1461no—
256C2×Q8—731no—
257C2×D4—731no—
258C2×D4—1462no—
259C2×D4—1461no—
260C22×C4—21no—
261C22×C4—21no—
262C22×C4—21no—
263C22×C4—21no—
264C22×C4—21no—
265C22×C4—21no—
266C22×C4—1461no—
267C22×C4#1192401111yescharacteristic
268C22×C4C2×C4×D7311yescharacteristic
269M16—2921no—
270M16—2921no—
271M16—1461no—
272M16—1461no—
273M16—1461no—
274M16—1461no—
275C2×C8—21no—
276C2×C8—21no—
277C2×C8—21no—
278C2×C8—41no—
279C2×C8—41no—
280C2×C8—41no—
281C2×C8—41no—
282C2×C8—2921no—
283C2×C8—2921no—
284C2×C8—732no—
285C2×C8—1461no—
286C2×C8D4×D7311yescharacteristic
287C2×C8D4×D7311yescharacteristic
288C2×C8D4×D7311yescharacteristic
289C2×C8C2×C22⋊D7311yescharacteristic
290C2×C8#1192402711yescharacteristic
291C2×C8#1192402811yescharacteristic
292C4⋊C4—1462no—
293C4⋊C4—1461no—
294C22⋊C4—1462no—
295C22⋊C4—1462no—
296C22⋊C4—1461no—
297C42—732no—
298C16—22no—
299C16—41no—
300C16—2921no—
301C16—1462no—
302C23—21no—
303C23—21no—
304C23—21no—
305C23—42no—
306C23—1461no—
307C23C2×#1192401111yescharacteristic
308Q8—1461no—
309Q8—1461no—
310D4—2924no—
311D4—2921no—
312D4—2921no—
313D4—1461no—
314D4—1461no—
315C2×C4—21no—
316C2×C4—21no—
317C2×C4—21no—
318C2×C4—41no—
319C2×C4—41no—
320C2×C4—41no—
321C2×C4—41no—
322C2×C4—2921no—
323C2×C4—2921no—
324C2×C4—1462no—
325C2×C4—1461no—
326C2×C4C22⋊C4×D7311yescharacteristic
327C2×C4#1455655111yescharacteristic
328C2×C4C4×C22⋊D7311yescharacteristic
329C2×C4M16×D7311yescharacteristic
330C2×C4#1455659711yescharacteristic
331C2×C4#1455659911yescharacteristic
332C8—41no—
333C8—2921no—
334C8—1462no—
335C8#1455656311yescharacteristic
336C8#1455656411yescharacteristic
337C8#1455657711yescharacteristic
338C8#1455658711yescharacteristic
339C22—22no—
340C22—22no—
341C22—22no—
342C22—41no—
343C22—41no—
344C22—41no—
345C22—41no—
346C22—41no—
347C22—2922no—
348C22—2921no—
349C22#1798854911yescharacteristic
350C22#1798865511yescharacteristic
351C22#1798865811yescharacteristic
352C4—41no—
353C4—2922no—
354C4—2921no—
355C4#1798848711yescharacteristic
356C4#1798855211yescharacteristic
357C4#1798865611yescharacteristic
358C4#1798866111yescharacteristic
359C2—21no—
360C2—21no—
361C2—21no—
362C2—42no—
363C2—5841no—
364C2#2261072711yescharacteristic
subgroupG/H
centerC8#14556587
commutatorC22×C4×C73C22×C4
FrattiniC22×C8C22×D73
FittingC73×#2112C2
socleC2×C73#1955
prime radical#17989098C4
cosocleC22×C8×C73C23
hypercenter#2112D73
hypocenterC73#13200
LCSG ▷ C22×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C8 ◁ C22×C8 ◁ #2112

Complements

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

NH ≅ G/Ntype
2. #22611696C2semidirect
3. #22611560C2semidirect
4. #22611558C2semidirect
5. #22611542C2semidirect
6. #22611535C2semidirect
7. C73×#2112C2semidirect
47. C2×#11924003D4semidirect
67. C2×C16×C73D4semidirect
188. #2112D73semidirect
190. C73#13200semidirect
245. C2×C16C22⋊D73semidirect

Maximal subgroups

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

subgroupindex
#226186252
#226116962
#226115602
#226115582
#226115422
#226115352
C73×#21122
#1320073