Finite Group Databasebeta

← #33239436

Subgroups — #33239436

Order 18688.

652 conjugacy classes of subgroups.

The subgroup lattice has 400 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22623067C211yesmaximal, characteristic
2C2×#17988710C211yesmaximal, characteristic, complemented
3#22618733C211yesmaximal, characteristic, complemented
4#22611831C214yesmaximal, complemented
5#22611824C214yesmaximal, complemented
6#22611704C211yesmaximal, characteristic, complemented
7#22611681C211yesmaximal, characteristic, complemented
8#22611668C211yesmaximal, characteristic, complemented
9C2×#347×C73C211yesmaximal, characteristic, complemented
10C2×#14556613C2211yescharacteristic
11C2×#14556613C2211yescharacteristic
12C2×#14556597—21no—
13C4×#11924028—21no—
14#17989030C412yescomplemented
15#17989030C2212yes—
16#17989025C412yescomplemented
17#17989025C2212yes—
18#17989019C414yescomplemented
19#17988710—22no—
20#17988710C2214yes—
21#17988700C2212yescomplemented
22#17988695C2212yescomplemented
23#17988689—22no—
24C2×#14556527C2211yescharacteristic
25#17988625C2211yescharacteristic
26C2×#14556519—21no—
27#17988584—21no—
28#17988410C2214yescomplemented
29#17988404C2212yescomplemented
30#17988401C2212yescomplemented
31#17988398C2214yes—
32C2×M32×C73C2211yescharacteristic
33C2×M32×C73C2211yescharacteristic
34C2×#97×C73C2211yescharacteristic
35#347×C73C2214yescomplemented
36C2×#11924028—21no—
37C2×#11924028—21no—
38#14556613—21no—
39#14556613—21no—
40#14556613—21no—
41#14556613—21no—
42#14556613C2×C412yes—
43#14556613C2×C412yes—
44#14556613C2×C412yescomplemented
45#14556613C2312yes—
46C2×Dic292C2×C411yescharacteristic, complemented
47C2×Dic292C2311yescharacteristic
48C2×#11924017—21no—
49C2×#11924017—21no—
50C2×D584C2×C411yescharacteristic, complemented
51C2×D584C2311yescharacteristic
52#14556597—24no—
53#14556597—42no—
54C2×#11924012—21no—
55C2×#11924011—41no—
56C2×C8×D73—41no—
57C4×Dic146—22no—
58C4×D292—22no—
59C4×C22⋊D73—42no—
60#14556550—21no—
61#14556548—41no—
62C42×D73—41no—
63Dic584C2×C414yescomplemented
64#14556536C2×C414yescomplemented
65#14556536C2×C414yescomplemented
66D1168C2×C414yescomplemented
67#14556527—21no—
68#14556527C2312yes—
69#14556525—21no—
70#14556523C2312yes—
71#14556519—24no—
72#14556516—22no—
73#14556513—22no—
74C22×C8×C73D411yescharacteristic
75C22×C8×C73C2311yescharacteristic
76C2×C42×C73D411yescharacteristic
77M32×C73—21no—
78M32×C73C2×C412yescomplemented
79M32×C73C2312yes—
80M32×C73C2312yes—
81C2×C16×C73—21no—
82C2×C16×C73C2×C412yescomplemented
83#97×C73D412yes—
84#97×C73C2312yes—
85#11924028—22no—
86#11924028—22no—
87#11924028—22no—
88#11924028—22no—
89#11924028—41no—
90#11924028—41no—
91#11924028—41no—
92#11924028—41no—
93C2×Dic146—21no—
94C2×Dic146—21no—
95C2×D292—21no—
96C2×D292—21no—
97C2×C22⋊D73—41no—
98C2×C22⋊D73—41no—
99C2×C4×D73—41no—
100C2×C4×D73—41no—
101C2×C4×D73—42no—
102Dic292—21no—
103Dic292—21no—
104Dic292C22×C412yes—
105Dic292C22×C412yes—
106#11924017—22no—
107#11924017—22no—
108#11924017—22no—
109#11924017—22no—
110D584—21no—
111D584—21no—
112D584C22×C412yes—
113D584C22×C412yes—
114#11924012—22no—
115#11924012—41no—
116#11924011—44no—
117C2×#10596831—41no—
118C8×D73—44no—
119C4⋊Dic73—22no—
120#11924007—42no—
121C22⋊Dic73—41no—
122#11924005—42no—
123C4×Dic73—42no—
124C4×Dic73—41no—
125C22×C4×C73—21no—
126C22×C4×C73C2×D411yescharacteristic
127C2×C8×C73—21no—
128C2×C8×C73—21no—
129C2×C8×C73C22×C411yescharacteristic
130C2×C8×C73C22×C411yescharacteristic
131C2×C8×C73C2×D412yes—
132C2×C8×C73C2×D411yescharacteristic
133C2×C8×C73#4111yescharacteristic
134C2×C8×C73#4111yescharacteristic
135C2×C8×C73C2411yescharacteristic
136C42×C73—21no—
137C42×C73C2×D412yes—
138C16×C73—22no—
139C16×C73C22×C414yes—
140C23×C73AGL(1,ℤ/8)11yescharacteristic
141C22×D73—41no—
142C22×D73—41no—
143C2×C4×C73—22no—
144C2×C4×C73—81no—
145C2×C4×C73C4×D411yescharacteristic
146C2×C4×C73C4×D411yescharacteristic
147C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
148C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
149C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
150C2×C4×C73C22×D411yescharacteristic
151C22⋊D73—42no—
152C22⋊D73—42no—
153C22⋊D73—42no—
154C22⋊D73—42no—
155C2×Dic73—41no—
156C2×Dic73—41no—
157C2×Dic73—42no—
158D292—22no—
159D292—22no—
160D292—41no—
161D292—41no—
162C4×D73—42no—
163C4×D73—42no—
164C4×D73—42no—
165C4×D73—42no—
166C4×D73—84no—
167Dic146—22no—
168Dic146—22no—
169Dic146—41no—
170Dic146—41no—
171C8×C73—22no—
172C8×C73C4×D411yescharacteristic
173C8×C73C4×D411yescharacteristic
174C8×C73C23×C411yescharacteristic
175C8×C73C2×#4111yescharacteristic
176#10596831—42no—
177C22×C73—41no—
178C22×C73#44211yescharacteristic
179C22×C73#44211yescharacteristic
180C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
181C2×D73—42no—
182C2×D73—42no—
183C2×D73—82no—
184C2×D73—82no—
185C4×C73—42no—
186C4×C73#44211yescharacteristic
187C4×C73#44211yescharacteristic
188C4×C73C2×C4×D411yescharacteristic
189C4×C73C2×AGL(1,ℤ/8)11yescharacteristic
190Dic73—42no—
191Dic73—42no—
192Dic73—82no—
193#33468—731nomaximal, Sylow
194C2×C73—21no—
195C2×C73—21no—
196C2×C73—21no—
197C2×C73C2×#44211yescharacteristic
198D73—82no—
199D73—82no—
200#3414—731no—
201C2×#444—731no—
202#2943—731no—
203AGL(1,ℤ/16)—734no—
204#2180—734no—
205#2155—731no—
206#2148—731no—
207#2145—731no—
208C2×#347D7311yescharacteristic, complemented
209C73#3346811yescharacteristic, complemented, Sylow
210C2×#135—731no—
211C2×#135—731no—
212C2×#131—1461no—
213C4×#41—1461no—
214#510—732no—
215#510—732no—
216#509—732no—
217#509—732no—
218#508—734no—
219C2×M32C2×D7311yescharacteristic
220C2×M32C2×D7311yescharacteristic
221#444—734no—
222#444—1462no—
223#442—732no—
224#441—732no—
225#440—1462no—
226C2×#108—731no—
227#427—731no—
228C2×#104—1461no—
229#416—1461no—
230C2×#97C2×D7311yescharacteristic
231#365—734no—
232#362—732no—
233#361—732no—
234#360—734no—
235#347C2×D7314yescomplemented
236C2×#41—1461no—
237C2×#41—1461no—
238#135—732no—
239#135—732no—
240#135—732no—
241#135—732no—
242#135—1461no—
243#135—1461no—
244#135—1461no—
245#135—1461no—
246C2×Q16—731no—
247C2×Q16—731no—
248C2×SD16—1461no—
249C2×SD16—1461no—
250C2×D8—731no—
251C2×D8—731no—
252#131—2922no—
253#131—1464no—
254C2×M16—2921no—
255C2×M16—1461no—
256C22×C8—2921no—
257C22×C8D29211yescharacteristic
258C22×C8C22×D7311yescharacteristic
259C4×Q8—1462no—
260C4×D4—2922no—
261C4×D4—1462no—
262#117—2921no—
263#117—1461no—
264C2×C42—2921no—
265C2×C42D29211yescharacteristic
266Q32—734no—
267SD32—734no—
268SD32—734no—
269D16—734no—
270M32—21no—
271M32C4×D7312yescomplemented
272M32C22×D7312yes—
273M32C22×D7312yes—
274C2×C16—21no—
275C2×C16C4×D7312yescomplemented
276#108—732no—
277#108—1461no—
278#107—1461no—
279#106—732no—
280#104—1464no—
281#103—1462no—
282#102—1462no—
283#97D29212yes—
284#97C22×D7312yes—
285#41—2921no—
286#41—2921no—
287#41—2921no—
288#41—2921no—
289#41—1462no—
290#41—1462no—
291#41—1462no—
292#41—1462no—
293C2×Q8—1461no—
294C2×Q8—1461no—
295C2×D4—2921no—
296C2×D4—2921no—
297C2×D4—1461no—
298C2×D4—1461no—
299C22×C4—21no—
300C22×C4—2922no—
301C22×C4—2921no—
302C22×C4—2921no—
303C22×C4C2×D29211yescharacteristic
304Q16—732no—
305Q16—732no—
306Q16—1461no—
307Q16—1461no—
308SD16—1462no—
309SD16—1462no—
310SD16—1462no—
311SD16—1462no—
312D8—732no—
313D8—732no—
314D8—1461no—
315D8—1461no—
316M16—2924no—
317M16—2921no—
318M16—1462no—
319C2×C8—21no—
320C2×C8—21no—
321C2×C8—2924no—
322C2×C8—2921no—
323C2×C8C2×C4×D7311yescharacteristic
324C2×C8C2×C4×D7311yescharacteristic
325C2×C8C2×D29212yes—
326C2×C8C2×D29211yescharacteristic
327C2×C8#1192402811yescharacteristic
328C2×C8#1192402811yescharacteristic
329C2×C8C23×D7311yescharacteristic
330C4⋊C4—2922no—
331C4⋊C4—1462no—
332C22⋊C4—2922no—
333C22⋊C4—2921no—
334C42—21no—
335C42—2922no—
336C42—2921no—
337C42C2×D29212yes—
338C16—22no—
339C16C2×C4×D7314yes—
340C23—2921no—
341C23—2921no—
342C23#1455661911yescharacteristic
343Q8—2921no—
344Q8—2921no—
345Q8—1462no—
346Q8—1462no—
347D4—2922no—
348D4—2922no—
349D4—2922no—
350D4—2922no—
351D4—2921no—
352D4—2921no—
353D4—1462no—
354D4—1462no—
355C2×C4—22no—
356C2×C4—81no—
357C2×C4—2921no—
358C2×C4—2921no—
359C2×C4—2922no—
360C2×C4—2922no—
361C2×C4—2922no—
362C2×C4—2922no—
363C2×C4—2922no—
364C2×C4—5844no—
365C2×C4C4×D29211yescharacteristic
366C2×C4C4×D29211yescharacteristic
367C2×C4#1455661911yescharacteristic
368C2×C4#1455661911yescharacteristic
369C2×C4#1455661911yescharacteristic
370C2×C4C22×D29211yescharacteristic
371C8—22no—
372C8—2922no—
373C8C4×D29211yescharacteristic
374C8C4×D29211yescharacteristic
375C8C22×C4×D7311yescharacteristic
376C8C2×#1192402811yescharacteristic
377C22—41no—
378C22—2922no—
379C22—2922no—
380C22—5842no—
381C22—5842no—
382C22#1798870011yescharacteristic
383C22#1798870011yescharacteristic
384C22C2×#1455661911yescharacteristic
385C4—42no—
386C4—2922no—
387C4—2922no—
388C4—5842no—
389C4#1798870011yescharacteristic
390C4#1798870011yescharacteristic
391C4C2×C4×D29211yescharacteristic
392C4C2×#1455661911yescharacteristic
393C2—21no—
394C2—21no—
395C2—21no—
396C2—5842no—
397C2—5842no—
398C2C2×#1798870011yescharacteristic
subgroupG/H
centerC4#17988700
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC2×#347×C73C2
socleC2×C73C2×#442
prime radical#22623067C2
cosocleC2×C8×C73C24
hypercenterC2×#347D73
hypocenterC73#33468
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C4 ◁ C22×C4 ◁ C2×#97 ◁ C2×#347

Complements

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

NH ≅ G/Ntype
2. C2×#17988710C2semidirect
3. #22618733C2semidirect
4. #22611831C2semidirect
5. #22611824C2semidirect
6. #22611704C2semidirect
7. #22611681C2semidirect
8. #22611668C2semidirect
9. C2×#347×C73C2semidirect
14. #17989030C4semidirect
16. #17989025C4semidirect
18. #17989019C4semidirect
21. #17988700C22semidirect
22. #17988695C22semidirect
28. #17988410C22semidirect
29. #17988404C22semidirect
30. #17988401C22semidirect
35. #347×C73C22semidirect
44. #14556613C2×C4semidirect
46. C2×Dic292C2×C4semidirect
50. C2×D584C2×C4semidirect
63. Dic584C2×C4semidirect
64. #14556536C2×C4semidirect
65. #14556536C2×C4semidirect
66. D1168C2×C4semidirect
78. M32×C73C2×C4semidirect
82. C2×C16×C73C2×C4semidirect
208. C2×#347D73semidirect
209. C73#33468semidirect
235. #347C2×D73semidirect
271. M32C4×D73semidirect
275. C2×C16C4×D73semidirect

Maximal subgroups

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

subgroupindex
#226230672
C2×#179887102
#226187332
#226118312
#226118242
#226117042
#226116812
#226116682
C2×#347×C732
#3346873