Finite Group Databasebeta

← #33002852

Subgroups — #33002852

Order 18688.

456 conjugacy classes of subgroups.

The subgroup lattice has 356 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22623034C211yesmaximal, characteristic, complemented
2#22618832C211yesmaximal, characteristic
3#22611558C211yesmaximal, characteristic, complemented
4#22611544C211yesmaximal, characteristic, complemented
5#22608019C211yesmaximal, characteristic, complemented
6#22608018C211yesmaximal, characteristic, complemented
7C2×#350×C73C211yesmaximal, characteristic, complemented
8#17989385C411yescharacteristic
9C2×#14556597C2211yescharacteristic
10#17989004—22no—
11#17989004—22no—
12C2×#14556531—22no—
13C2×#14556531C2211yescharacteristic
14C2×#14556531C2211yescharacteristic
15#17988707—22no—
16#17988702—22no—
17#17988635C411yescharacteristic
18#17988604C411yescharacteristic
19#17988603C411yescharacteristic
20#17988542C2211yescharacteristic
21#17988370—22no—
22#17988366—22no—
23C2×#131×C73C2211yescharacteristic
24C2×M32×C73C2211yescharacteristic
25C22×C16×C73C2211yescharacteristic
26#350×C73—22no—
27C2×#11924028C2×C411yescharacteristic
28#14556622—21no—
29#14556621C812yes—
30#14556617—21no—
31#14556616C812yes—
32#14556611—22no—
33#14556597—21no—
34#14556597—21no—
35#14556597—21no—
36#14556597—21no—
37C2×#11924012C2×C411yescharacteristic
38C2×#11924012C2×C411yescharacteristic
39C2×#11924011—21no—
40C2×C8×D73—21no—
41#14556550C2×C411yescharacteristic
42#14556531—24no—
43#14556531—21no—
44#14556531—21no—
45#14556531—21no—
46#14556531D412yes—
47C2×#11924003—24no—
48C2×#11924003—21no—
49C2×#11924003D412yes—
50#14556526—22no—
51#14556518—22no—
52#14556517—22no—
53#14556501—21no—
54C8×Dic73—21no—
55C2×#41×C73C2×C411yescharacteristic
56#131×C73—21no—
57#131×C73—21no—
58#131×C73D412yes—
59#131×C73D412yes—
60C2×M16×C73—21no—
61C2×M16×C73C2×C411yescharacteristic
62C22×C8×C73—21no—
63C22×C8×C73C2311yescharacteristic
64M32×C73—21no—
65M32×C73—21no—
66C2×C16×C73—21no—
67C2×C16×C73—21no—
68C2×C16×C73—21no—
69C2×C16×C73—21no—
70#11924028—21no—
71#11924028—21no—
72#11924028—21no—
73#11924028—21no—
74C2×Dic146C2×C811yescharacteristic
75C2×D292C2×C811yescharacteristic
76C2×C22⋊D73—21no—
77C2×C4×D73—21no—
78#11924018—22no—
79#11924016—22no—
80#11924015—22no—
81#11924013—22no—
82#11924012—21no—
83#11924012—21no—
84#11924012—21no—
85#11924012C2×C812yes—
86#11924011—41no—
87#11924011—41no—
88C2×#10596831—21no—
89C2×#10596831—21no—
90C8×D73—41no—
91C8×D73—41no—
92C4⋊Dic73—21no—
93C22⋊Dic73—21no—
94C4×Dic73—21no—
95#11924003—22no—
96#11924003—22no—
97#41×C73—21no—
98#41×C73—21no—
99#41×C73C22⋊C412yes—
100#41×C73C2×C812yes—
101C2×Q8×C73M1611yescharacteristic
102C2×D4×C73—21no—
103C2×D4×C73M1611yescharacteristic
104C22×C4×C73—21no—
105C22×C4×C73C22×C411yescharacteristic
106M16×C73—21no—
107M16×C73—22no—
108M16×C73—22no—
109M16×C73C22⋊C412yes—
110C2×C8×C73—21no—
111C2×C8×C73—22no—
112C2×C8×C73—22no—
113C2×C8×C73C2×D411yescharacteristic
114C2×C8×C73C2×D411yescharacteristic
115C2×C8×C73C2×D411yescharacteristic
116C2×C8×C73C2×D411yescharacteristic
117C2×C8×C73#4111yescharacteristic
118C2×C8×C73#4111yescharacteristic
119C16×C73—41no—
120C16×C73—41no—
121C23×C73—21no—
122C23×C73#13111yescharacteristic
123C22×D73—21no—
124Q8×C73—21no—
125Q8×C73C22⋊C812yes—
126D4×C73—22no—
127D4×C73—22no—
128D4×C73—21no—
129D4×C73C22⋊C812yes—
130C2×C4×C73—21no—
131C2×C4×C73—22no—
132C2×C4×C73—22no—
133C2×C4×C73C2×C22⋊C411yescharacteristic
134C2×C4×C73C4×D411yescharacteristic
135C2×C4×C73C4×D411yescharacteristic
136C2×C4×C73C22×C811yescharacteristic
137C2×C4×C73C2×M1611yescharacteristic
138C2×C4×C73#13111yescharacteristic
139C22⋊D73—41no—
140C22⋊D73—41no—
141C2×Dic73—21no—
142C2×Dic73—41no—
143D292—21no—
144D292—21no—
145C4×D73—41no—
146C4×D73—41no—
147Dic146—21no—
148Dic146—21no—
149C8×C73—22no—
150C8×C73#12011yescharacteristic
151C8×C73#12111yescharacteristic
152C8×C73#12111yescharacteristic
153C8×C73#12311yescharacteristic
154#10596831—22no—
155#10596831—41no—
156C22×C73—22no—
157C22×C73—41no—
158C22×C73—81no—
159C22×C73#40811yescharacteristic
160C22×C73C8×D411yescharacteristic
161C22×C73#43511yescharacteristic
162C2×D73—42no—
163C2×D73—41no—
164C4×C73—22no—
165C4×C73#38611yescharacteristic
166C4×C73C2×C22⋊C811yescharacteristic
167C4×C73C8×D411yescharacteristic
168C4×C73#43511yescharacteristic
169Dic73—41no—
170Dic73—81no—
171#12979—731nomaximal, Sylow
172C2×C73—21no—
173C2×C73—21no—
174C2×C73—21no—
175C2×C73—42no—
176C2×C73#186911yescharacteristic
177D73—81no—
178#3407—731no—
179#2955—731no—
180#2117—731no—
181C2×#350D7311yescharacteristic, complemented
182#2113—731no—
183#1379—731no—
184#1378—731no—
185C73#1297911yescharacteristic, complemented, Sylow
186#575—731no—
187C2×#131—731no—
188C2×#131C2×D7311yescharacteristic
189#504—1462no—
190#504—1462no—
191C2×M32—731no—
192C2×M32—731no—
193C2×M32—1462no—
194C2×M32C2×D7311yescharacteristic
195C22×C16C2×D7311yescharacteristic
196#444—1462no—
197#443—1462no—
198#430—731no—
199#421—731no—
200#421—731no—
201#405—731no—
202#350—22no—
203#350—1462no—
204C22⋊C16—1462no—
205C2×#41—731no—
206C2×#41C4×D7311yescharacteristic
207#137—732no—
208#137—1461no—
209AGL(1,ℤ/8)—732no—
210AGL(1,ℤ/8)—1461no—
211#135—1462no—
212#131—21no—
213#131—21no—
214#131—1461no—
215#131—1461no—
216#131—1461no—
217#131—1461no—
218#131D29212yes—
219#131C22⋊D7312yes—
220C2×M16—21no—
221C2×M16—731no—
222C2×M16—731no—
223C2×M16—1461no—
224C2×M16C4×D7311yescharacteristic
225C22×C8—21no—
226C22×C8—1461no—
227C22×C8C22×D7311yescharacteristic
228#117—731no—
229M32—21no—
230M32—21no—
231M32—732no—
232M32—1464no—
233M32—1461no—
234M32—1461no—
235M32—1461no—
236C2×C16—21no—
237C2×C16—21no—
238C2×C16—21no—
239C2×C16—21no—
240C2×C16—732no—
241C2×C16—1464no—
242C2×C16—1461no—
243#108—1462no—
244#104—1462no—
245#104—1462no—
246#97—1461no—
247C4×C8—1461no—
248#41—21no—
249#41—21no—
250#41—1461no—
251#41—1461no—
252#41—1461no—
253#41—1461no—
254#41#1192400512yes—
255#41C8×D7312yes—
256C2×Q8—731no—
257C2×Q8#1192401111yescharacteristic
258C2×D4—21no—
259C2×D4—731no—
260C2×D4—1461no—
261C2×D4#1192401111yescharacteristic
262C22×C4—21no—
263C22×C4—1461no—
264C22×C4C2×C4×D7311yescharacteristic
265Q16—1462no—
266SD16—1462no—
267SD16—1462no—
268D8—1462no—
269M16—21no—
270M16—22no—
271M16—22no—
272M16—2921no—
273M16—2921no—
274M16—732no—
275M16—1461no—
276M16—1461no—
277M16—1461no—
278M16#1192400512yes—
279C2×C8—21no—
280C2×C8—22no—
281C2×C8—22no—
282C2×C8—2921no—
283C2×C8—2921no—
284C2×C8—1461no—
285C2×C8—1461no—
286C2×C8D4×D7311yescharacteristic
287C2×C8D4×D7311yescharacteristic
288C2×C8C2×C22⋊D7311yescharacteristic
289C2×C8C2×D29211yescharacteristic
290C2×C8#1192402711yescharacteristic
291C2×C8#1192402711yescharacteristic
292C4⋊C4—1461no—
293C22⋊C4—1461no—
294C42—1461no—
295C16—41no—
296C16—41no—
297C16—1462no—
298C16—1462no—
299C23—21no—
300C23—1461no—
301C23#1455659911yescharacteristic
302Q8—21no—
303Q8—1461no—
304Q8—1461no—
305Q8#1455650312yes—
306D4—22no—
307D4—22no—
308D4—21no—
309D4—2921no—
310D4—2921no—
311D4—1461no—
312D4—1461no—
313D4#1455650312yes—
314C2×C4—21no—
315C2×C4—22no—
316C2×C4—22no—
317C2×C4—2921no—
318C2×C4—2921no—
319C2×C4—2921no—
320C2×C4—1461no—
321C2×C4C2×#1192400511yescharacteristic
322C2×C4#1455655111yescharacteristic
323C2×C4#1455655111yescharacteristic
324C2×C4C2×C8×D7311yescharacteristic
325C2×C4C2×#1192401111yescharacteristic
326C2×C4#1455659911yescharacteristic
327C8—22no—
328C8—2921no—
329C8—1462no—
330C8#1455656111yescharacteristic
331C8#1455656311yescharacteristic
332C8#1455656311yescharacteristic
333C8#1455657511yescharacteristic
334C22—22no—
335C22—41no—
336C22—81no—
337C22—2922no—
338C22—2921no—
339C22#1798855511yescharacteristic
340C22#1798865011yescharacteristic
341C22#1798865511yescharacteristic
342C4—22no—
343C4—2921no—
344C4—5841no—
345C4#1798846411yescharacteristic
346C4C2×#1455650311yescharacteristic
347C4#1798865011yescharacteristic
348C4#1798865511yescharacteristic
349C2—21no—
350C2—21no—
351C2—21no—
352C2—42no—
353C2—5841no—
354C2#2261022111yescharacteristic
subgroupG/H
centerC8#14556561
commutatorC2×C4×C73C22×C8
FrattiniC22×C8C22×D73
FittingC2×#350×C73C2
socleC2×C73#1869
prime radical#17989385C4
cosocleC22×C8×C73C23
hypercenterC2×#350D73
hypocenterC73#12979
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C8 ◁ C22×C8 ◁ C2×#350

Complements

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

NH ≅ G/Ntype
1. #22623034C2semidirect
3. #22611558C2semidirect
4. #22611544C2semidirect
5. #22608019C2semidirect
6. #22608018C2semidirect
7. C2×#350×C73C2semidirect
181. C2×#350D73semidirect
185. C73#12979semidirect

Maximal subgroups

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

subgroupindex
#226230342
#226188322
#226115582
#226115442
#226080192
#226080182
C2×#350×C732
#1297973