Finite Group Databasebeta

← #33104466

Subgroups — #33104466

Order 18688.

846 conjugacy classes of subgroups.

The subgroup lattice has 446 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17988703C211yesmaximal, characteristic
2#22618808C211yesmaximal, characteristic
3#22618805C211yesmaximal, characteristic
4#22618525C212yesmaximal, complemented
5#22610717C211yesmaximal, characteristic, complemented
6#22610167C212yesmaximal, complemented
7#22610161C212yesmaximal, complemented
8#22610094C211yesmaximal, characteristic, complemented
9#22609989C212yesmaximal, complemented
10#22609811C211yesmaximal, characteristic, complemented
11C73×#1952C211yesmaximal, characteristic, complemented
12C2×#14556613C2211yescharacteristic
13C2×#14556599C2212yes—
14#17989068C2212yes—
15#17989067C2212yes—
16#17988703C418yescomplemented
17#17988689—22no—
18#17988686—22no—
19C4×Dic292—21no—
20#17988681—21no—
21C4×D584—21no—
22#17988669—21no—
23C2×#14556526C2211yescharacteristic
24#17988629C2212yes—
25#17988628—22no—
26#17988626—22no—
27#17988624—22no—
28#17988623C2211yescharacteristic
29C2×#14556525C2211yescharacteristic
30C2×#14556522C2212yes—
31#106×D73—22no—
32C2×#14556517C2212yes—
33#17988589C2212yes—
34#17988586C2212yes—
35#17988542C2211yescharacteristic
36C2×C8×Dic73C2211yescharacteristic
37#17988452C2212yes—
38#17988355C2212yes—
39#17988348C2211yescharacteristic
40#17988346C2211yescharacteristic
41C2×#108×C73C2211yescharacteristic
42#428×C73C2212yes—
43C2×#107×C73C2211yescharacteristic
44#405×C73C2211yescharacteristic
45#343×C73C2212yes—
46C2×#11924028C2312yes—
47#14556613C2×C414yes—
48#14556613Q814yes—
49C2×Dic292D411yescharacteristic
50C2×#11924017D412yes—
51C2×D584D411yescharacteristic
52#14556599—24no—
53#14556599C2×C418yescomplemented
54C2×#11924012C2311yescharacteristic
55C2×#11924011—22no—
56C22×#10596831C2311yescharacteristic
57C2×C8×D73—22no—
58C4×Dic146—22no—
59#14556556—22no—
60C4×D292—22no—
61#14556553—22no—
62C4×C22⋊D73—42no—
63#14556551—42no—
64#14556550C2311yescharacteristic
65#14556549—22no—
66#14556549—22no—
67C2×C4⋊Dic73C2312yes—
68C4⋊C4×D73—22no—
69C4⋊C4×D73—22no—
70C2×C4×Dic73C2311yescharacteristic
71#14556526—22no—
72#14556525—22no—
73#14556525—21no—
74#14556525—21no—
75#14556524D411yescharacteristic
76#14556524D411yescharacteristic
77#14556523—22no—
78#14556522—22no—
79#14556522—22no—
80#14556522D411yescharacteristic
81#14556522D411yescharacteristic
82#14556517C2×C418yescomplemented
83#14556516—22no—
84#14556514—22no—
85#14556513—22no—
86#14556511—22no—
87#14556501—21no—
88C8×Dic73—21no—
89C8×Dic73C2×C414yescomplemented
90#14556498—42no—
91C2×M16×C73C2311yescharacteristic
92C2×M16×C73C2312yes—
93C22×C8×C73C2311yescharacteristic
94#117×C73C2311yescharacteristic
95C2×C4⋊C4×C73C2312yes—
96#108×C73C2×C414yescomplemented
97#107×C73—21no—
98#107×C73—21no—
99#107×C73—22no—
100#106×C73—22no—
101#97×C73—21no—
102C4×C8×C73—21no—
103#11924028—22no—
104#11924028—22no—
105#11924028C22×C414yes—
106#11924028C2×Q814yes—
107C2×Dic146C2×D412yes—
108C2×D292C2×D412yes—
109C2×C22⋊D73—22no—
110C22×Dic73—21no—
111C22×Dic73—22no—
112C2×C4×D73—22no—
113C2×C4×D73—42no—
114C2×C4×D73—42no—
115Dic292C4⋊C414yes—
116#11924017C4⋊C418yes—
117D584C4⋊C414yes—
118#11924012—21no—
119#11924012—21no—
120#11924011—28no—
121C2×#10596831—21no—
122C2×#10596831C22×C412yes—
123C2×#10596831C2×D411yescharacteristic
124C2×#10596831C2×D411yescharacteristic
125C2×#10596831#4112yes—
126C8×D73—28no—
127C4⋊Dic73—22no—
128C4⋊Dic73—22no—
129C4⋊Dic73C2×D411yescharacteristic
130C4⋊Dic73C2×D411yescharacteristic
131#11924007—42no—
132#11924007—42no—
133C22⋊Dic73—21no—
134#11924005—42no—
135#11924005—42no—
136C4×Dic73—21no—
137C4×Dic73—21no—
138C4×Dic73—42no—
139C4×Dic73—42no—
140C4×Dic73C22×C412yes—
141C22×C4×C73—22no—
142C22×C4×C73C2411yescharacteristic
143M16×C73—22no—
144M16×C73—21no—
145M16×C73—21no—
146M16×C73C22×C414yes—
147C2×C8×C73—22no—
148C2×C8×C73—21no—
149C2×C8×C73C22×C412yes—
150C2×C8×C73C2×D411yescharacteristic
151C2×C8×C73C2×D411yescharacteristic
152C2×C8×C73C2×Q812yes—
153C4⋊C4×C73—22no—
154C4⋊C4×C73—22no—
155C4⋊C4×C73C2×D411yescharacteristic
156C4⋊C4×C73C2×D411yescharacteristic
157C22⋊C4×C73—21no—
158C42×C73—21no—
159C23×C73H(5,𝔽2)11yescharacteristic
160C22×D73—22no—
161C2×C4×C73—21no—
162C2×C4×C73—21no—
163C2×C4×C73—42no—
164C2×C4×C73—82no—
165C2×C4×C73C23×C411yescharacteristic
166C2×C4×C73C22×D411yescharacteristic
167C2×C4×C73C22×D411yescharacteristic
168C2×C4×C73C22×Q811yescharacteristic
169C2×C4×C73C2×#4111yescharacteristic
170C2×C4×C732−1+411yescharacteristic
171C22⋊D73—24no—
172C22⋊D73—24no—
173C2×Dic73—22no—
174C2×Dic73—22no—
175C2×Dic73—21no—
176C2×Dic73—21no—
177C2×Dic73—41no—
178C2×Dic73—41no—
179C2×Dic73—42no—
180C2×Dic73—82no—
181D292—22no—
182D292C2×C4⋊C414yes—
183C4×D73—24no—
184C4×D73—24no—
185C4×D73—162no—
186C4×D73—162no—
187Dic146—22no—
188Dic146C2×C4⋊C414yes—
189C8×C73—24no—
190C8×C73—41no—
191C8×C73C2×C4⋊C412yes—
192C8×C73C2×C4⋊C412yes—
193#10596831—41no—
194#10596831C4×D412yes—
195#10596831C4×D412yes—
196C22×C73—21no—
197C22×C73—21no—
198C22×C73#52011yescharacteristic
199C22×C73D4211yescharacteristic
200C22×C73#55411yescharacteristic
201C2×D73—24no—
202C2×D73—42no—
203C2×D73—42no—
204C4×C73—41no—
205C4×C73—41no—
206C4×C73—82no—
207C4×C73C22×C4⋊C411yescharacteristic
208C4×C73C2×C4×D411yescharacteristic
209C4×C73#54711yescharacteristic
210C4×C73D4×Q811yescharacteristic
211Dic73—24no—
212Dic73—42no—
213Dic73—41no—
214Dic73—41no—
215Dic73—82no—
216#22690—731nomaximal, Sylow
217C2×C73—21no—
218C2×C73—21no—
219C2×C73D4×C4⋊C411yescharacteristic
220C2×C73#329011yescharacteristic
221C2×C73#329011yescharacteristic
222D73—44no—
223C2×#443—731no—
224#2951—731no—
225#2951—731no—
226#2912—732no—
227#1952—731no—
228#1952D7311yescharacteristic, complemented
229#1860—732no—
230#1860—732no—
231#1846—731no—
232#1815—732no—
233#1772—731no—
234C73#2269011yescharacteristic, complemented, Sylow
235C2×#135—731no—
236C2×#131—732no—
237#520—732no—
238#520—732no—
239#443—738no—
240#440—1462no—
241#440—1462no—
242C4×Q16—1461no—
243C4×Q16—1461no—
244C4×D8—1461no—
245C4×D8—1461no—
246C2×#108—731no—
247C2×#108C2×D7311yescharacteristic
248#428—732no—
249#428—1462no—
250#428—1462no—
251#428C2×D7312yes—
252#427—731no—
253#427—1462no—
254C2×#107—731no—
255C2×#107C2×D7311yescharacteristic
256C2×#106—732no—
257C2×#106—1462no—
258C2×#104—732no—
259#417—732no—
260#417—732no—
261#405—731no—
262#405C2×D7311yescharacteristic
263C2×C4×C8—731no—
264#382—732no—
265#343—732no—
266#343C2×D7312yes—
267#339—731no—
268#339—731no—
269C2×#41—732no—
270#135—734no—
271#135—734no—
272C2×Q16—731no—
273C2×SD16—732no—
274C2×D8—731no—
275#131—738no—
276#131—1464no—
277C2×M16—731no—
278C2×M16—1462no—
279C2×M16C22×D7311yescharacteristic
280C2×M16C22×D7312yes—
281C22×C8—731no—
282C22×C8—1462no—
283C22×C8C22×D7311yescharacteristic
284C4×Q8—1462no—
285C4×Q8—1462no—
286C4×D4—2922no—
287C4×D4—2922no—
288C4×D4—1462no—
289C4×D4—1462no—
290#117—731no—
291#117—1462no—
292#117—1462no—
293#117C22×D7311yescharacteristic
294C2×C4⋊C4—732no—
295C2×C4⋊C4—1462no—
296C2×C4⋊C4—1462no—
297C2×C4⋊C4C22×D7312yes—
298C2×C42—731no—
299#108—1462no—
300#108C4×D7314yescomplemented
301#107—21no—
302#107—21no—
303#107—22no—
304#107—731no—
305#107—731no—
306#107—1462no—
307#107—1461no—
308#107—1461no—
309#106—22no—
310#106—731no—
311#106—731no—
312#106—1462no—
313#106—1462no—
314#106—1462no—
315#104—738no—
316#103—1462no—
317#103—1462no—
318#102—1462no—
319#102—1462no—
320#97—21no—
321#97—1461no—
322C4×C8—21no—
323C4×C8—734no—
324C4×C8—1461no—
325#95—2922no—
326#41—734no—
327#41—734no—
328#41—1462no—
329#41—1462no—
330C2×Q8—732no—
331C2×D4—732no—
332C2×D4—1462no—
333C22×C4—22no—
334C22×C4—2922no—
335C22×C4—2922no—
336C22×C4—1461no—
337C22×C4—1462no—
338C22×C4—1462no—
339C22×C4C23×D7311yescharacteristic
340Q16—734no—
341SD16—738no—
342D8—734no—
343M16—22no—
344M16—21no—
345M16—21no—
346M16—1468no—
347M16—1461no—
348M16—1461no—
349M16C2×C4×D7314yes—
350C2×C8—22no—
351C2×C8—21no—
352C2×C8—732no—
353C2×C8—731no—
354C2×C8—731no—
355C2×C8—732no—
356C2×C8—1468no—
357C2×C8—1461no—
358C2×C8C2×C4×D7312yes—
359C2×C8D4×D7311yescharacteristic
360C2×C8D4×D7311yescharacteristic
361C2×C8Q8×D7312yes—
362C4⋊C4—22no—
363C4⋊C4—22no—
364C4⋊C4—2922no—
365C4⋊C4—2922no—
366C4⋊C4—731no—
367C4⋊C4—731no—
368C4⋊C4—1462no—
369C4⋊C4—1462no—
370C4⋊C4D4×D7311yescharacteristic
371C4⋊C4D4×D7311yescharacteristic
372C22⋊C4—21no—
373C22⋊C4—2922no—
374C22⋊C4—2922no—
375C22⋊C4—1461no—
376C42—21no—
377C42—2922no—
378C42—2922no—
379C42—732no—
380C42—1461no—
381C42—1461no—
382C23—1462no—
383C23#1455663711yescharacteristic
384Q8—734no—
385Q8—1462no—
386D4—734no—
387D4—1464no—
388D4—1464no—
389D4—1462no—
390C2×C4—21no—
391C2×C4—21no—
392C2×C4—42no—
393C2×C4—82no—
394C2×C4—2921no—
395C2×C4—2921no—
396C2×C4—2922no—
397C2×C4—5842no—
398C2×C4—11682no—
399C2×C4—11682no—
400C2×C4—1461no—
401C2×C4—1461no—
402C2×C4—1462no—
403C2×C4—1462no—
404C2×C4—1464no—
405C2×C4—1464no—
406C2×C4C22×C4×D7311yescharacteristic
407C2×C4C2×D4×D7311yescharacteristic
408C2×C4C2×D4×D7311yescharacteristic
409C2×C4C2×Q8×D7311yescharacteristic
410C2×C4#41×D7311yescharacteristic
411C2×C4#1455663911yescharacteristic
412C8—24no—
413C8—41no—
414C8—2921no—
415C8—732no—
416C8—732no—
417C8C4⋊C4×D7312yes—
418C8C4⋊C4×D7312yes—
419C22—21no—
420C22—21no—
421C22—2922no—
422C22—2922no—
423C22—1464no—
424C22#1798906911yescharacteristic
425C22#1798922111yescharacteristic
426C22#1798929311yescharacteristic
427C4—41no—
428C4—41no—
429C4—82no—
430C4—2922no—
431C4—2921no—
432C4—2921no—
433C4—5842no—
434C4—1464no—
435C4C2×C4⋊C4×D7311yescharacteristic
436C4C4×D4×D7311yescharacteristic
437C4#1798923911yescharacteristic
438C4#1798925211yescharacteristic
439C2—21no—
440C2—21no—
441C2—2924no—
442C2#2261283211yescharacteristic
443C2#2262171211yescharacteristic
444C2#2262172011yescharacteristic
subgroupG/H
centerC22#17989221
commutatorC2×C4×C73C23×C4
FrattiniC22×C4C23×D73
FittingC73×#1952C2
socleC22×C73D42
prime radicalC2×#14556613C22
cosocleC22×C4×C73C24
hypercenter#1952D73
hypocenterC73#22690
LCSG ▷ C2×C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ #1952

Complements

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

NH ≅ G/Ntype
4. #22618525C2semidirect
5. #22610717C2semidirect
6. #22610167C2semidirect
7. #22610161C2semidirect
8. #22610094C2semidirect
9. #22609989C2semidirect
10. #22609811C2semidirect
11. C73×#1952C2semidirect
16. #17988703C4semidirect
53. #14556599C2×C4semidirect
82. #14556517C2×C4semidirect
89. C8×Dic73C2×C4semidirect
96. #108×C73C2×C4semidirect
228. #1952D73semidirect
234. C73#22690semidirect
300. #108C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179887032
#226188082
#226188052
#226185252
#226107172
#226101672
#226101612
#226100942
#226099892
#226098112
C73×#19522
#2269073