Finite Group Databasebeta

← #33001057

Subgroups — #33001057

Order 18688.

444 conjugacy classes of subgroups.

The subgroup lattice has 338 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618358C211yesmaximal, characteristic
2#22611560C211yesmaximal, characteristic
3#22611558C211yesmaximal, characteristic
4C2×#17988364C211yesmaximal, characteristic, complemented
5C2×#17988364C211yesmaximal, characteristic, complemented
6#22608017C211yesmaximal, characteristic, complemented
7C73×#1378C211yesmaximal, characteristic, complemented
8C2×#14556599C411yescharacteristic
9C2×#14556597C2211yescharacteristic
10#17989055C411yescharacteristic
11C2×#14556531C2211yescharacteristic
12C2×#14556531C2211yescharacteristic
13#17988544—21no—
14#17988543C411yescharacteristic
15#17988541—21no—
16M16×Dic73C411yescharacteristic
17C4×#11924011—21no—
18C2×#14556501C2211yescharacteristic
19#97×D73—21no—
20#17988372C414yescomplemented
21#17988370C414yescomplemented
22#17988364—21no—
23#17988364—21no—
24#17988364—21no—
25#17988364—21no—
26C2×M32×C73C2211yescharacteristic
27C2×M32×C73C2211yescharacteristic
28#405×C73C2211yescharacteristic
29C2×#11924028C2×C411yescharacteristic
30#14556599—22no—
31#14556599—22no—
32#14556597—21no—
33#14556597—21no—
34#14556597C2×C412yes—
35#14556597C2×C412yes—
36C2×#11924012C2×C411yescharacteristic
37C2×#11924011—21no—
38C2×#11924011—21no—
39C22×#10596831C2×C411yescharacteristic
40C2×C8×D73—21no—
41C2×C8×D73—21no—
42#14556556—21no—
43#14556553—21no—
44#14556551—22no—
45#14556548—21no—
46C2×C4×Dic73C2×C411yescharacteristic
47C42×D73—21no—
48#14556531—21no—
49#14556531—21no—
50#14556531—21no—
51#14556531C2×C412yescomplemented
52C2×#11924003—21no—
53C2×#11924003C2×C412yescomplemented
54#14556502—21no—
55#14556501—21no—
56#14556501D411yescharacteristic
57#14556501D411yescharacteristic
58#14556501D411yescharacteristic
59#14556501D411yescharacteristic
60C4×#10596831—21no—
61C8×Dic73—21no—
62C2×M16×C73C2×C411yescharacteristic
63C22×C8×C73C2311yescharacteristic
64#117×C73C2×C411yescharacteristic
65M32×C73—21no—
66M32×C73—21no—
67M32×C73—21no—
68M32×C73C2×C412yescomplemented
69C2×C16×C73—21no—
70C2×C16×C73C2×C412yescomplemented
71#97×C73—21no—
72C4×C8×C73—21no—
73#11924028—21no—
74#11924028—21no—
75#11924028C4212yes—
76#11924028M1612yes—
77C2×Dic146M1611yescharacteristic
78C2×D292M1611yescharacteristic
79C2×C22⋊D73—21no—
80C22×Dic73—21no—
81C2×C4×D73—21no—
82C2×C4×D73—22no—
83#11924012—21no—
84#11924012C4212yes—
85#11924011—24no—
86#11924011—22no—
87#11924011—22no—
88C2×#10596831—21no—
89C2×#10596831—21no—
90C2×#10596831—21no—
91C2×#10596831C22⋊C411yescharacteristic
92C2×#10596831C22⋊C411yescharacteristic
93C8×D73—24no—
94C8×D73—22no—
95C8×D73—22no—
96#11924007—22no—
97#11924005—22no—
98C4×Dic73—22no—
99C4×Dic73—21no—
100C4×Dic73C22⋊C411yescharacteristic
101C4×Dic73C22⋊C411yescharacteristic
102#11924003—22no—
103#11924003—22no—
104C22×C4×C73C22×C411yescharacteristic
105M16×C73—21no—
106M16×C73M1612yes—
107C2×C8×C73—21no—
108C2×C8×C73C22×C411yescharacteristic
109C2×C8×C73C22×C411yescharacteristic
110C2×C8×C73C2×D411yescharacteristic
111C2×C8×C73C2×D411yescharacteristic
112C2×C8×C73#4111yescharacteristic
113C2×C8×C73#4111yescharacteristic
114C4⋊C4×C73—21no—
115C22⋊C4×C73M1612yes—
116C42×C73—21no—
117C16×C73—22no—
118C16×C73—22no—
119C23×C73C2×M1611yescharacteristic
120C22×D73—21no—
121C2×C4×C73—22no—
122C2×C4×C73C2×C4211yescharacteristic
123C2×C4×C73C2×C22⋊C411yescharacteristic
124C2×C4×C73#11711yescharacteristic
125C2×C4×C73C2×M1611yescharacteristic
126C2×C4×C73C2×M1611yescharacteristic
127C2×C4×C73C2×M1611yescharacteristic
128C22⋊D73—22no—
129C22⋊D73—22no—
130C2×Dic73—21no—
131C2×Dic73—21no—
132C2×Dic73—21no—
133C2×Dic73—81no—
134D292—21no—
135D292#9712yes—
136C4×D73—22no—
137C4×D73—22no—
138C4×D73—82no—
139Dic146—21no—
140Dic146#9712yes—
141C8×C73—22no—
142C8×C73C4×D411yescharacteristic
143C8×C73C4×D411yescharacteristic
144C8×C73C4×D411yescharacteristic
145C8×C73C4×D411yescharacteristic
146#10596831—22no—
147#10596831—21no—
148#10596831—21no—
149C22×C73—41no—
150C22×C73C4×M1611yescharacteristic
151C22×C73#40711yescharacteristic
152C22×C73#43211yescharacteristic
153C2×D73—22no—
154C2×D73—42no—
155C4×C73—42no—
156C4×C73C4×C22⋊C411yescharacteristic
157C4×C73C2×#9711yescharacteristic
158C4×C73#40711yescharacteristic
159C4×C73#43211yescharacteristic
160Dic73—22no—
161Dic73—41no—
162Dic73—41no—
163#12609—731nomaximal, Sylow
164C2×C73—21no—
165C2×C73—21no—
166C2×C73—21no—
167C2×C73#174911yescharacteristic
168D73—42no—
169#2875—731no—
170#2117—731no—
171#2117—731no—
172C2×#347—731no—
173C2×#347—731no—
174#1378—731no—
175#1378D7311yescharacteristic, complemented
176C73#1260911yescharacteristic, complemented, Sylow
177C2×#131—731no—
178C2×#131—731no—
179C4×#41—731no—
180C2×M32—731no—
181C2×M32—731no—
182C2×M32C2×D7311yescharacteristic
183C2×M32C2×D7311yescharacteristic
184#405—731no—
185#405—1461no—
186#405—1461no—
187#405C2×D7311yescharacteristic
188C4×M16—731no—
189C4×M16—1461no—
190C2×#97—731no—
191C2×#97—1461no—
192#350—734no—
193#350—734no—
194#347—1461no—
195#347—1461no—
196#347—1461no—
197#347—1461no—
198C2×#41—731no—
199#131—732no—
200#131—732no—
201#131—1462no—
202#131—1462no—
203#131—1461no—
204#131—1461no—
205C2×M16—731no—
206C2×M16—1461no—
207C2×M16—1461no—
208C2×M16C4×D7311yescharacteristic
209C22×C8—731no—
210C22×C8—1461no—
211C22×C8—1461no—
212C22×C8C22×D7311yescharacteristic
213C4×Q8—1461no—
214C4×D4—1461no—
215C4×D4—1462no—
216#117—1461no—
217#117C4×D7311yescharacteristic
218C2×C42—731no—
219C2×C42—1461no—
220M32—21no—
221M32—21no—
222M32—21no—
223M32—732no—
224M32—1461no—
225M32—1461no—
226M32—1461no—
227M32C4×D7312yescomplemented
228C2×C16—21no—
229C2×C16—732no—
230C2×C16—1461no—
231C2×C16C4×D7312yescomplemented
232#97—21no—
233#97—731no—
234#97—731no—
235#97—731no—
236#97—731no—
237#97—1461no—
238#97—1461no—
239C4×C8—21no—
240C4×C8—1461no—
241C4×C8—1461no—
242#41—732no—
243#41—732no—
244#41—1461no—
245#41—1461no—
246C2×Q8—731no—
247C2×D4—731no—
248C2×D4—1461no—
249C22×C4—1462no—
250C22×C4—1461no—
251C22×C4—1461no—
252C22×C4C2×C4×D7311yescharacteristic
253M16—21no—
254M16—732no—
255M16—1464no—
256M16—1461no—
257M16—1462no—
258M16—1462no—
259M16#1192401112yes—
260C2×C8—21no—
261C2×C8—731no—
262C2×C8—731no—
263C2×C8—1464no—
264C2×C8—1461no—
265C2×C8—1461no—
266C2×C8—1461no—
267C2×C8—1462no—
268C2×C8—1462no—
269C2×C8C2×C4×D7311yescharacteristic
270C2×C8C2×C4×D7311yescharacteristic
271C2×C8D4×D7311yescharacteristic
272C2×C8D4×D7311yescharacteristic
273C2×C8#1192402711yescharacteristic
274C2×C8#1192402911yescharacteristic
275C4⋊C4—21no—
276C4⋊C4—1462no—
277C22⋊C4—1462no—
278C22⋊C4#1192401112yes—
279C42—21no—
280C42—731no—
281C42—731no—
282C42—1462no—
283C42—1461no—
284C16—22no—
285C16—22no—
286C16—1462no—
287C16—1462no—
288C23—1461no—
289C23C2×#1192401111yescharacteristic
290Q8—732no—
291Q8—1461no—
292D4—732no—
293D4—1462no—
294D4—1462no—
295D4—1461no—
296C2×C4—22no—
297C2×C4—5842no—
298C2×C4—5841no—
299C2×C4—1461no—
300C2×C4—1461no—
301C2×C4—1461no—
302C2×C4—1462no—
303C2×C4—1462no—
304C2×C4C42×D7311yescharacteristic
305C2×C4C22⋊C4×D7311yescharacteristic
306C2×C4#1455654911yescharacteristic
307C2×C4M16×D7311yescharacteristic
308C2×C4M16×D7311yescharacteristic
309C2×C4C2×#1192401111yescharacteristic
310C8—22no—
311C8—1462no—
312C8—1461no—
313C8—1461no—
314C8#1455655111yescharacteristic
315C8#1455655111yescharacteristic
316C8#1455655311yescharacteristic
317C8#1455655311yescharacteristic
318C22—41no—
319C22—2922no—
320C22—1462no—
321C22C4×#1192401111yescharacteristic
322C22#1798854911yescharacteristic
323C22#1798864311yescharacteristic
324C4—42no—
325C4—2921no—
326C4—2921no—
327C4—1462no—
328C4#1798843411yescharacteristic
329C4#97×D7311yescharacteristic
330C4#1798854911yescharacteristic
331C4#1798864311yescharacteristic
332C2—21no—
333C2—21no—
334C2—21no—
335C2—2922no—
336C2#2260969211yescharacteristic
subgroupG/H
centerC4#17988434
commutatorC2×C4×C73C2×C42
FrattiniC22×C8C22×D73
FittingC73×#1378C2
socleC2×C73#1749
prime radicalC2×#11924028C2×C4
cosocleC22×C8×C73C23
hypercenter#1378D73
hypocenterC73#12609
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C22×C8 ◁ #1378

Complements

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

NH ≅ G/Ntype
4. C2×#17988364C2semidirect
5. C2×#17988364C2semidirect
6. #22608017C2semidirect
7. C73×#1378C2semidirect
20. #17988372C4semidirect
21. #17988370C4semidirect
51. #14556531C2×C4semidirect
53. C2×#11924003C2×C4semidirect
68. M32×C73C2×C4semidirect
70. C2×C16×C73C2×C4semidirect
175. #1378D73semidirect
176. C73#12609semidirect
227. M32C4×D73semidirect
231. C2×C16C4×D73semidirect

Maximal subgroups

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

subgroupindex
#226183582
#226115602
#226115582
C2×#179883642
#226080172
C73×#13782
#1260973