Finite Group Databasebeta

← #33241489

Subgroups — #33241489

Order 18688.

474 conjugacy classes of subgroups.

The subgroup lattice has 428 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620468C211yesmaximal, characteristic
2#22620464C211yesmaximal, characteristic
3#22612217C211yesmaximal, characteristic, complemented
4#22612216C211yesmaximal, characteristic, complemented
5#22612215C211yesmaximal, characteristic, complemented
6#22612209C211yesmaximal, characteristic, complemented
7#22612207C211yesmaximal, characteristic, complemented
8#22612205C211yesmaximal, characteristic, complemented
9#22612204C211yesmaximal, characteristic, complemented
10#22611718C211yesmaximal, characteristic
11#22611716C211yesmaximal, characteristic
12#22611714C211yesmaximal, characteristic
13#366×D73C211yesmaximal, characteristic
14#22611529C211yesmaximal, characteristic
15C73×#2255C211yesmaximal, characteristic, complemented
16#17989217—21no—
17#17989214—21no—
18C2×#14556530C2211yescharacteristic
19C2×C16×D73C2211yescharacteristic
20#17988997—21no—
21#17988995—21no—
22#17988994—21no—
23#17988993—21no—
24#17988992C2211yescharacteristic
25#17988991C2211yescharacteristic
26#17988991C2211yescharacteristic
27#17988989C2211yescharacteristic
28#17988988C2211yescharacteristic
29#17988988C2211yescharacteristic
30#17988628—21no—
31#17988626—21no—
32#17988624C2211yescharacteristic
33#17988624C2211yescharacteristic
34#107×D73C2211yescharacteristic
35#107×D73C2211yescharacteristic
36#17988541C2211yescharacteristic
37#17988415C2211yescharacteristic
38#17988415C2211yescharacteristic
39#17988414C2211yescharacteristic
40#17988414C2211yescharacteristic
41#17988413C2211yescharacteristic
42#17988413C2211yescharacteristic
43#17988412C2211yescharacteristic
44#17988412C2211yescharacteristic
45#17988412C2211yescharacteristic
46#17988412C2211yescharacteristic
47#17988412C2211yescharacteristic
48#17988412C2211yescharacteristic
49#17988362C2211yescharacteristic
50C4×#11924003C2211yescharacteristic
51C16×Dic73C2211yescharacteristic
52#500×C73C2211yescharacteristic
53#499×C73C2211yescharacteristic
54#367×C73C2211yescharacteristic
55#367×C73C2211yescharacteristic
56#366×C73C2211yescharacteristic
57#366×C73C2211yescharacteristic
58#346×C73C2211yescharacteristic
59C2×#11924011D411yescharacteristic
60C2×C8×D73C2311yescharacteristic
61#14556591—21no—
62#14556590—21no—
63#14556590—21no—
64#14556590—41no—
65#14556589—41no—
66#14556583—21no—
67#14556582—21no—
68#14556582—21no—
69#14556581—41no—
70#14556581—41no—
71#14556571—41no—
72#14556569—41no—
73#14556569—41no—
74#14556567—41no—
75#14556549—21no—
76#14556549—21no—
77#14556548D411yescharacteristic
78C4⋊C4×D73—21no—
79C4⋊C4×D73—21no—
80#14556530Q814yescomplemented
81C2×#11924003C2311yescharacteristic
82C2×#11924003C2311yescharacteristic
83C16×D73Q814yescomplemented
84#14556525—21no—
85#14556525C2311yescharacteristic
86#14556525C2311yescharacteristic
87#14556525C2311yescharacteristic
88#14556525C2311yescharacteristic
89#14556524—21no—
90#14556524—21no—
91#14556524—21no—
92#14556524—21no—
93#14556523—21no—
94#14556522—21no—
95#14556522—21no—
96#14556522—21no—
97#14556522—21no—
98#14556502D411yescharacteristic
99#14556501D411yescharacteristic
100C8×Dic73C2311yescharacteristic
101C4⋊Q8×C73—21no—
102#125×C73—21no—
103C2×C16×C73C2311yescharacteristic
104C2×C16×C73C2311yescharacteristic
105#107×C73—21no—
106#107×C73C2311yescharacteristic
107#107×C73C2311yescharacteristic
108#107×C73C2311yescharacteristic
109#107×C73C2311yescharacteristic
110#106×C73—21no—
111C4×C8×C73C2311yescharacteristic
112C2×Dic146—41no—
113C2×Dic146—41no—
114C2×Dic146—41no—
115C2×C4×D73—41no—
116C2×C4×D73—41no—
117C2×C4×D73C2×D411yescharacteristic
118#11924011Q1614yes—
119C2×#10596831C2×D411yescharacteristic
120C2×#10596831C2×D411yescharacteristic
121C8×D73C2×Q812yes—
122C8×D73C2×Q812yes—
123C4⋊Dic73—21no—
124C4⋊Dic73—21no—
125C4⋊Dic73—21no—
126C4⋊Dic73—21no—
127C4⋊Dic73—41no—
128C4⋊Dic73—41no—
129C4⋊Dic73—41no—
130#11924007—41no—
131#11924007—41no—
132#11924007—41no—
133#11924007—41no—
134#11924007—41no—
135#11924007—41no—
136#11924007—41no—
137#11924007—41no—
138#11924007D811yescharacteristic
139#11924007D811yescharacteristic
140#11924005—41no—
141#11924005—41no—
142#11924005D811yescharacteristic
143#11924005D811yescharacteristic
144C4×Dic73—41no—
145C4×Dic73—41no—
146C4×Dic73C2×D411yescharacteristic
147#11924003C2×Q812yes—
148#11924003C2×Q812yes—
149C2×Q8×C73—41no—
150C2×C8×C73C2×D411yescharacteristic
151C2×C8×C73C2411yescharacteristic
152C4⋊C4×C73—21no—
153C4⋊C4×C73—21no—
154C4⋊C4×C73—21no—
155C4⋊C4×C73—21no—
156C4⋊C4×C73—41no—
157C4⋊C4×C73—41no—
158C4⋊C4×C73—41no—
159C42×C73C2×D411yescharacteristic
160C16×C73C2×Q812yes—
161C16×C73C2×Q812yes—
162C22×D73C2×D811yescharacteristic
163Q8×C73—161no—
164C2×C4×C73—41no—
165C2×C4×C73—41no—
166C2×C4×C73—41no—
167C2×C4×C73—41no—
168C2×C4×C73C2×D811yescharacteristic
169C2×C4×C73C2×D811yescharacteristic
170C2×C4×C73C22×D411yescharacteristic
171C2×Dic73—41no—
172C2×Dic73—41no—
173C2×Dic73—41no—
174C2×Dic73—41no—
175C2×Dic73C2×D811yescharacteristic
176C2×Dic73C2×D811yescharacteristic
177C2×Dic73C2×D811yescharacteristic
178C4×D73—161no—
179C4×D73—161no—
180C4×D73C4⋊Q812yes—
181C4×D73C2×Q1612yes—
182Dic146—161no—
183Dic146—161no—
184Dic146—161no—
185C8×C73C2×Q1612yes—
186C8×C73C22×Q811yescharacteristic
187C8×C73C22×Q811yescharacteristic
188#10596831C4⋊Q812yes—
189#10596831C2×Q1612yes—
190C22×C73C22×D811yescharacteristic
191C2×D73—21no—
192C2×D73—21no—
193C2×D73#50012yes—
194C4×C73—21no—
195C4×C73—21no—
196C4×C73—81no—
197C4×C73—81no—
198C4×C73—81no—
199C4×C73—81no—
200C4×C73C2×C4⋊Q811yescharacteristic
201C4×C73C22×Q1611yescharacteristic
202Dic73—21no—
203Dic73—21no—
204Dic73—81no—
205Dic73—81no—
206Dic73—81no—
207Dic73—81no—
208Dic73#50012yes—
209#33729—731nomaximal, Sylow
210C2×C73C2×#50011yescharacteristic
211C2×C73#341511yescharacteristic
212C2×C73#341711yescharacteristic
213D73—22no—
214#3164—731no—
215#3164—731no—
216#2255—731no—
217#2255—731no—
218#2255—731no—
219#2255D7311yescharacteristic, complemented
220#2253—731no—
221#2253—731no—
222#2252—731no—
223#2252—731no—
224#2159—731no—
225#2159—731no—
226#2158—731no—
227C2×#366—731no—
228#2108—731no—
229C73#3372911yescharacteristic, complemented, Sylow
230#544—1461no—
231#544—1461no—
232C2×M32—731no—
233C22×C16—731no—
234#501—1461no—
235#501—1461no—
236#501—1461no—
237#501—1461no—
238#500—731no—
239#500—731no—
240#500—731no—
241#500C2×D7311yescharacteristic
242#499—731no—
243#499—731no—
244#499—731no—
245#499C2×D7311yescharacteristic
246#428—1461no—
247#428—1461no—
248#427—731no—
249#427—731no—
250C2×#107—731no—
251C2×#107—731no—
252#405—731no—
253#367—731no—
254#367—731no—
255#367—731no—
256#367—731no—
257#367C2×D7311yescharacteristic
258#367C2×D7311yescharacteristic
259#366—731no—
260#366—731no—
261#366—731no—
262#366—731no—
263#366—731no—
264#366—731no—
265#366—731no—
266#366—731no—
267#366C2×D7311yescharacteristic
268#366C2×D7311yescharacteristic
269#346—731no—
270#346C2×D7311yescharacteristic
271C4×C16—731no—
272C4×C16—731no—
273C2×M16—731no—
274C22×C8—731no—
275C4⋊Q8—21no—
276C4⋊Q8—2921no—
277C4⋊Q8—2921no—
278C4⋊Q8—1461no—
279C4⋊Q8—1461no—
280C4⋊Q8—1461no—
281#125—21no—
282#125—2921no—
283#125—2921no—
284#125—1461no—
285#125—1461no—
286#125—1461no—
287C22⋊Q8—2921no—
288C22⋊Q8—2921no—
289C22⋊Q8—2921no—
290C22⋊Q8—2921no—
291#117—731no—
292#117—1461no—
293#117—1461no—
294C2×C4⋊C4—1461no—
295C2×C4⋊C4—1461no—
296M32—734no—
297C2×C16—731no—
298C2×C16—731no—
299C2×C16—734no—
300C2×C16C22×D7311yescharacteristic
301C2×C16C22×D7311yescharacteristic
302#107—21no—
303#107—731no—
304#107—731no—
305#107—731no—
306#107—731no—
307#107—1461no—
308#107—1461no—
309#107—1461no—
310#107—1461no—
311#107—1461no—
312#107C22×D7311yescharacteristic
313#107C22×D7311yescharacteristic
314#107C22×D7311yescharacteristic
315#107C22×D7311yescharacteristic
316#106—21no—
317#106—1461no—
318#106—1461no—
319#106—1461no—
320#106—1461no—
321#106—1461no—
322#97—731no—
323#97—731no—
324C4×C8—731no—
325C4×C8C22×D7311yescharacteristic
326C2×Q8—41no—
327C2×Q8—2921no—
328C2×Q8—2921no—
329C2×Q8—2921no—
330C22×C4—2921no—
331C22×C4—2921no—
332C22×C4—731no—
333M16—734no—
334C2×C8—731no—
335C2×C8—731no—
336C2×C8—732no—
337C2×C8—732no—
338C2×C8D4×D7311yescharacteristic
339C2×C8C23×D7311yescharacteristic
340C4⋊C4—21no—
341C4⋊C4—21no—
342C4⋊C4—21no—
343C4⋊C4—21no—
344C4⋊C4—41no—
345C4⋊C4—41no—
346C4⋊C4—41no—
347C4⋊C4—2921no—
348C4⋊C4—2921no—
349C4⋊C4—2921no—
350C4⋊C4—2921no—
351C4⋊C4—2921no—
352C4⋊C4—2921no—
353C4⋊C4—2921no—
354C4⋊C4—2921no—
355C4⋊C4—2921no—
356C4⋊C4—2921no—
357C4⋊C4—2921no—
358C4⋊C4—731no—
359C4⋊C4—731no—
360C4⋊C4—1461no—
361C4⋊C4—1461no—
362C4⋊C4—1461no—
363C4⋊C4—1461no—
364C22⋊C4—2921no—
365C22⋊C4—2921no—
366C22⋊C4—731no—
367C22⋊C4—731no—
368C42—2921no—
369C42—2921no—
370C42—731no—
371C42D4×D7311yescharacteristic
372C16—732no—
373C16—732no—
374C16Q8×D7312yes—
375C16Q8×D7312yes—
376C23—731no—
377Q8—161no—
378Q8—11681no—
379Q8—11681no—
380Q8—11681no—
381C2×C4—41no—
382C2×C4—41no—
383C2×C4—41no—
384C2×C4—41no—
385C2×C4—2921no—
386C2×C4—2921no—
387C2×C4—2921no—
388C2×C4—2921no—
389C2×C4—731no—
390C2×C4—732no—
391C2×C4—732no—
392C2×C4—731no—
393C2×C4—731no—
394C2×C4—11681no—
395C2×C4—11681no—
396C2×C4D8×D7311yescharacteristic
397C2×C4D8×D7311yescharacteristic
398C2×C4C2×D4×D7311yescharacteristic
399C8—732no—
400C8—732no—
401C8Q16×D7312yes—
402C8C2×Q8×D7311yescharacteristic
403C8C2×Q8×D7311yescharacteristic
404C22—732no—
405C22—1461no—
406C22—1461no—
407C22C2×D8×D7311yescharacteristic
408C4—21no—
409C4—21no—
410C4—81no—
411C4—81no—
412C4—81no—
413C4—81no—
414C4—5841no—
415C4—5841no—
416C4—5841no—
417C4—5841no—
418C4—732no—
419C4—1461no—
420C4—1461no—
421C4C4⋊Q8×D7311yescharacteristic
422C4C2×Q16×D7311yescharacteristic
423C2—1462no—
424C2#500×D7311yescharacteristic
425C2#2262307111yescharacteristic
426C2#2262308311yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2255C2
socleC22×C73C22×D8
prime radicalC22×D73C2×D8
hypercenter#2255D73
hypocenterC73#33729
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2255

Complements

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

NH ≅ G/Ntype
3. #22612217C2semidirect
4. #22612216C2semidirect
5. #22612215C2semidirect
6. #22612209C2semidirect
7. #22612207C2semidirect
8. #22612205C2semidirect
9. #22612204C2semidirect
15. C73×#2255C2semidirect
80. #14556530Q8semidirect
83. C16×D73Q8semidirect
219. #2255D73semidirect
229. C73#33729semidirect

Maximal subgroups

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

subgroupindex
#226204682
#226204642
#226122172
#226122162
#226122152
#226122092
#226122072
#226122052
#226122042
#226117182
#226117162
#226117142
#366×D732
#226115292
C73×#22552
#3372973