Finite Group Databasebeta

← #33002900

Subgroups — #33002900

Order 18688.

456 conjugacy classes of subgroups.

The subgroup lattice has 364 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22618832C211yesmaximal, characteristic
2#22611559C211yesmaximal, characteristic, complemented
3C2×#17988370C211yesmaximal, characteristic, complemented
4#22611551C211yesmaximal, characteristic, complemented
5#22608019C211yesmaximal, characteristic, complemented
6#22608018C211yesmaximal, characteristic, complemented
7C2×#504×C73C211yesmaximal, characteristic, complemented
8#17989385C411yescharacteristic
9C2×#14556597C2211yescharacteristic
10C2×#14556531C2211yescharacteristic
11C2×#14556531C2211yescharacteristic
12#17988707—22no—
13#17988702—22no—
14#17988635C411yescharacteristic
15#17988604C411yescharacteristic
16#17988603C411yescharacteristic
17#17988542C2211yescharacteristic
18#17988371—22no—
19#17988370—21no—
20#17988370—21no—
21#17988368—22no—
22C2×#131×C73C2211yescharacteristic
23#504×C73—22no—
24#504×C73—22no—
25C2×M32×C73—21no—
26C2×M32×C73C2211yescharacteristic
27C22×C16×C73—21no—
28C22×C16×C73C2211yescharacteristic
29C2×#11924028C2×C411yescharacteristic
30#14556622—21no—
31#14556621C812yes—
32#14556617—21no—
33#14556616C812yes—
34#14556611—22no—
35#14556597—21no—
36#14556597—21no—
37#14556597—21no—
38#14556597—21no—
39C2×#11924012C2×C411yescharacteristic
40C2×#11924012C2×C411yescharacteristic
41C2×#11924011—21no—
42C2×C8×D73—21no—
43#14556550C2×C411yescharacteristic
44#14556531—21no—
45#14556531—21no—
46#14556531—21no—
47#14556531—21no—
48C2×#11924003—21no—
49C2×#11924003—21no—
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—24no—
65M32×C73—21no—
66M32×C73D412yes—
67C2×C16×C73—24no—
68C2×C16×C73—21no—
69C2×C16×C73—21no—
70C2×C16×C73—21no—
71C2×C16×C73D411yescharacteristic
72C2×C16×C73D411yescharacteristic
73#11924028—21no—
74#11924028—21no—
75#11924028—21no—
76#11924028—21no—
77C2×Dic146C2×C811yescharacteristic
78C2×D292C2×C811yescharacteristic
79C2×C22⋊D73—21no—
80C2×C4×D73—21no—
81#11924018—22no—
82#11924016—22no—
83#11924015—22no—
84#11924013—22no—
85#11924012—21no—
86#11924012—21no—
87#11924012—21no—
88#11924012C2×C812yes—
89#11924011—41no—
90#11924011—41no—
91C2×#10596831—21no—
92C2×#10596831—21no—
93C8×D73—41no—
94C8×D73—41no—
95C4⋊Dic73—21no—
96C22⋊Dic73—21no—
97C4×Dic73—21no—
98#11924003—41no—
99#11924003—41no—
100#41×C73—21no—
101#41×C73—21no—
102#41×C73C22⋊C412yes—
103#41×C73C2×C812yes—
104C2×Q8×C73M1611yescharacteristic
105C2×D4×C73—21no—
106C2×D4×C73M1611yescharacteristic
107C22×C4×C73—21no—
108C22×C4×C73C22×C411yescharacteristic
109M16×C73—21no—
110M16×C73—22no—
111M16×C73—22no—
112M16×C73C22⋊C412yes—
113C2×C8×C73—21no—
114C2×C8×C73—22no—
115C2×C8×C73—22no—
116C2×C8×C73C2×D411yescharacteristic
117C2×C8×C73C2×D411yescharacteristic
118C2×C8×C73C2×D411yescharacteristic
119C2×C8×C73C2×D411yescharacteristic
120C2×C8×C73#4111yescharacteristic
121C2×C8×C73#4111yescharacteristic
122C16×C73—21no—
123C16×C73—22no—
124C16×C73—21no—
125C23×C73—21no—
126C23×C73#13111yescharacteristic
127C22×D73—21no—
128Q8×C73—21no—
129Q8×C73C22⋊C812yes—
130D4×C73—22no—
131D4×C73—22no—
132D4×C73—21no—
133D4×C73C22⋊C812yes—
134C2×C4×C73—21no—
135C2×C4×C73—22no—
136C2×C4×C73—22no—
137C2×C4×C73C2×C22⋊C411yescharacteristic
138C2×C4×C73C4×D411yescharacteristic
139C2×C4×C73C4×D411yescharacteristic
140C2×C4×C73C22×C811yescharacteristic
141C2×C4×C73C2×M1611yescharacteristic
142C2×C4×C73#13111yescharacteristic
143C22⋊D73—41no—
144C22⋊D73—41no—
145C2×Dic73—21no—
146C2×Dic73—41no—
147D292—21no—
148D292—21no—
149C4×D73—41no—
150C4×D73—41no—
151Dic146—21no—
152Dic146—21no—
153C8×C73—22no—
154C8×C73#12011yescharacteristic
155C8×C73#12111yescharacteristic
156C8×C73#12111yescharacteristic
157C8×C73#12311yescharacteristic
158#10596831—22no—
159#10596831—41no—
160C22×C73—22no—
161C22×C73—41no—
162C22×C73—42no—
163C22×C73#40811yescharacteristic
164C22×C73C8×D411yescharacteristic
165C22×C73#43511yescharacteristic
166C2×D73—41no—
167C2×D73—81no—
168C4×C73—22no—
169C4×C73#38611yescharacteristic
170C4×C73C2×C22⋊C811yescharacteristic
171C4×C73C8×D411yescharacteristic
172C4×C73#43511yescharacteristic
173Dic73—41no—
174Dic73—81no—
175#12987—731nomaximal, Sylow
176C2×C73—21no—
177C2×C73—21no—
178C2×C73—21no—
179C2×C73—42no—
180C2×C73#186911yescharacteristic
181D73—81no—
182C2×#504D7311yescharacteristic, complemented
183#2955—731no—
184#2117—731no—
185C2×#350—731no—
186#2115—731no—
187#1379—731no—
188#1378—731no—
189C73#1298711yescharacteristic, complemented, Sylow
190#575—731no—
191C2×#131—731no—
192C2×#131C2×D7311yescharacteristic
193#504—22no—
194#504—22no—
195C2×M32—21no—
196C2×M32—731no—
197C2×M32—731no—
198C2×M32C2×D7311yescharacteristic
199C22×C16—21no—
200C22×C16C2×D7311yescharacteristic
201#444—1462no—
202#443—1462no—
203#430—731no—
204#421—731no—
205#421—731no—
206#405—731no—
207#350—1462no—
208#350—1461no—
209#350—1461no—
210#349—1462no—
211C2×#41—731no—
212C2×#41C4×D7311yescharacteristic
213#137—732no—
214#137—1461no—
215AGL(1,ℤ/8)—732no—
216AGL(1,ℤ/8)—1461no—
217#135—1462no—
218#131—21no—
219#131—21no—
220#131—1461no—
221#131—1461no—
222#131—1461no—
223#131—1461no—
224#131D29212yes—
225#131C22⋊D7312yes—
226C2×M16—21no—
227C2×M16—731no—
228C2×M16—731no—
229C2×M16—1461no—
230C2×M16C4×D7311yescharacteristic
231C22×C8—21no—
232C22×C8—1461no—
233C22×C8C22×D7311yescharacteristic
234#117—731no—
235M32—24no—
236M32—21no—
237M32—1461no—
238M32—1461no—
239M32—1461no—
240M32—1461no—
241M32C22⋊D7312yes—
242C2×C16—24no—
243C2×C16—21no—
244C2×C16—21no—
245C2×C16—21no—
246C2×C16—1461no—
247C2×C16—1461no—
248C2×C16C22⋊D7311yescharacteristic
249C2×C16C22⋊D7311yescharacteristic
250#108—1462no—
251#104—1462no—
252#104—1462no—
253#97—1461no—
254C4×C8—1461no—
255#41—21no—
256#41—21no—
257#41—1461no—
258#41—1461no—
259#41—1461no—
260#41—1461no—
261#41#1192400512yes—
262#41C8×D7312yes—
263C2×Q8—731no—
264C2×Q8#1192401111yescharacteristic
265C2×D4—21no—
266C2×D4—731no—
267C2×D4—1461no—
268C2×D4#1192401111yescharacteristic
269C22×C4—21no—
270C22×C4—1461no—
271C22×C4C2×C4×D7311yescharacteristic
272Q16—1462no—
273SD16—1462no—
274SD16—1462no—
275D8—1462no—
276M16—21no—
277M16—22no—
278M16—22no—
279M16—2921no—
280M16—2921no—
281M16—732no—
282M16—1461no—
283M16—1461no—
284M16—1461no—
285M16#1192400512yes—
286C2×C8—21no—
287C2×C8—22no—
288C2×C8—22no—
289C2×C8—2921no—
290C2×C8—2921no—
291C2×C8—1461no—
292C2×C8—1461no—
293C2×C8C2×C22⋊D7311yescharacteristic
294C2×C8C2×C22⋊D7311yescharacteristic
295C2×C8C2×C22⋊D7311yescharacteristic
296C2×C8C2×D29211yescharacteristic
297C2×C8#1192402811yescharacteristic
298C2×C8#1192402811yescharacteristic
299C4⋊C4—1461no—
300C22⋊C4—1461no—
301C42—1461no—
302C16—21no—
303C16—22no—
304C16—21no—
305C16—2921no—
306C16—2921no—
307C23—21no—
308C23—1461no—
309C23#1455659711yescharacteristic
310Q8—21no—
311Q8—1461no—
312Q8—1461no—
313Q8#1455650312yes—
314D4—22no—
315D4—22no—
316D4—21no—
317D4—2921no—
318D4—2921no—
319D4—1461no—
320D4—1461no—
321D4#1455650312yes—
322C2×C4—21no—
323C2×C4—22no—
324C2×C4—22no—
325C2×C4—2921no—
326C2×C4—2921no—
327C2×C4—2921no—
328C2×C4—1461no—
329C2×C4C2×#1192400511yescharacteristic
330C2×C4C4×C22⋊D7311yescharacteristic
331C2×C4C4×C22⋊D7311yescharacteristic
332C2×C4C2×C8×D7311yescharacteristic
333C2×C4C2×#1192401111yescharacteristic
334C2×C4#1455659711yescharacteristic
335C8—22no—
336C8—2921no—
337C8—1462no—
338C8#1455656011yescharacteristic
339C8#1455656511yescharacteristic
340C8#1455656511yescharacteristic
341C8#1455657411yescharacteristic
342C22—22no—
343C22—41no—
344C22—42no—
345C22—2921no—
346C22—5841no—
347C22#1798855311yescharacteristic
348C22C8×C22⋊D7311yescharacteristic
349C22#1798865811yescharacteristic
350C4—22no—
351C4—2921no—
352C4—5841no—
353C4#1798846611yescharacteristic
354C4C2×#1455650311yescharacteristic
355C4C8×C22⋊D7311yescharacteristic
356C4#1798865811yescharacteristic
357C2—21no—
358C2—21no—
359C2—21no—
360C2—42no—
361C2—5841no—
362C2#2261022311yescharacteristic
subgroupG/H
centerC8#14556565
commutatorC2×C4×C73C22×C8
FrattiniC22×C8C22×D73
FittingC2×#504×C73C2
socleC2×C73#1869
prime radical#17989385C4
cosocleC22×C8×C73C23
hypercenterC2×#504D73
hypocenterC73#12987
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C8 ◁ C22×C8 ◁ C2×#504

Complements

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

NH ≅ G/Ntype
2. #22611559C2semidirect
3. C2×#17988370C2semidirect
4. #22611551C2semidirect
5. #22608019C2semidirect
6. #22608018C2semidirect
7. C2×#504×C73C2semidirect
182. C2×#504D73semidirect
189. C73#12987semidirect

Maximal subgroups

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

subgroupindex
#226188322
#226115592
C2×#179883702
#226115512
#226080192
#226080182
C2×#504×C732
#1298773