Finite Group Databasebeta

← #33241507

Subgroups — #33241507

Order 18688.

514 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#22620444C211yesmaximal, characteristic
2#500×D73C211yesmaximal, characteristic
3#22612218C211yesmaximal, characteristic, complemented
4#22612217C211yesmaximal, characteristic, complemented
5#22612214C211yesmaximal, characteristic, complemented
6#22612212C212yesmaximal, complemented
7#22612205C212yesmaximal, complemented
8#22611718C212yesmaximal
9#22611716C212yesmaximal
10#22611524C211yesmaximal, characteristic
11C73×#2255C211yesmaximal, characteristic, complemented
12#17989143—21no—
13C4⋊Q8×D73—21no—
14C2×#14556530C2212yes—
15#17988992C2211yescharacteristic
16#17988991—21no—
17#17988991C2212yes—
18#17988991C2211yescharacteristic
19#17988991C2211yescharacteristic
20#17988990—21no—
21#17988989C2211yescharacteristic
22#17988986—21no—
23#17988984—21no—
24#17988624C2211yescharacteristic
25#17988624C2211yescharacteristic
26#17988622—21no—
27#107×D73—21no—
28#107×D73C2212yes—
29C4×C8×D73C2211yescharacteristic
30#17988415C2212yes—
31#17988414C2212yes—
32#17988414C2212yes—
33#17988413C2212yes—
34#17988412C2212yes—
35#17988412C2212yes—
36#17988363C2211yescharacteristic
37C16×Dic73C2212yes—
38#500×C73C2211yescharacteristic
39#499×C73C2211yescharacteristic
40#367×C73C2212yes—
41#366×C73C2212yes—
42#346×C73C2211yescharacteristic
43C2×Q8×D73—41no—
44C2×C8×D73D411yescharacteristic
45C2×C8×D73C2311yescharacteristic
46#14556591—21no—
47#14556590—22no—
48#14556590—21no—
49#14556590—21no—
50#14556590—41no—
51#14556589—41no—
52#14556583—21no—
53#14556569—41no—
54#14556569—41no—
55#14556556—41no—
56#14556556—41no—
57#14556549—21no—
58#14556549—21no—
59C4⋊C4×D73—22no—
60C4⋊C4×D73—41no—
61C42×D73D411yescharacteristic
62#14556530Q818yescomplemented
63C2×#11924003C2312yes—
64#14556525—21no—
65#14556525C2312yes—
66#14556525C2311yescharacteristic
67#14556525C2311yescharacteristic
68#14556524—22no—
69#14556524—22no—
70#14556524—21no—
71#14556524—21no—
72#14556523—21no—
73#14556522—21no—
74#14556522—21no—
75C4×#10596831D411yescharacteristic
76C8×Dic73D411yescharacteristic
77C8×Dic73C2311yescharacteristic
78C4⋊Q8×C73—21no—
79#125×C73—21no—
80C2×C16×C73C2312yes—
81#107×C73—21no—
82#107×C73C2312yes—
83#107×C73C2311yescharacteristic
84#107×C73C2311yescharacteristic
85#106×C73—21no—
86C4×C8×C73C2311yescharacteristic
87C2×Dic146—41no—
88C2×Dic146—41no—
89C2×Dic146—41no—
90C2×Dic146—42no—
91Q8×D73—84no—
92C2×C4×D73—42no—
93C2×C4×D73D811yescharacteristic
94C2×C4×D73D811yescharacteristic
95C2×C4×D73C2×D411yescharacteristic
96C2×#10596831C2×D411yescharacteristic
97C2×#10596831C2×D411yescharacteristic
98C8×D73Q1614yes—
99C8×D73C2×Q814yes—
100C4⋊Dic73—21no—
101C4⋊Dic73—21no—
102C4⋊Dic73—22no—
103C4⋊Dic73—41no—
104#11924007—41no—
105#11924007—41no—
106#11924007—42no—
107#11924007—42no—
108#11924007—41no—
109#11924007—41no—
110#11924005—41no—
111#11924005—41no—
112C4×Dic73—41no—
113C4×Dic73—41no—
114C4×Dic73D811yescharacteristic
115C4×Dic73D811yescharacteristic
116C4×Dic73C2×D411yescharacteristic
117#11924003C2×Q814yescomplemented
118C2×Q8×C73—41no—
119C2×C8×C73C2×D411yescharacteristic
120C2×C8×C73C2411yescharacteristic
121C4⋊C4×C73—21no—
122C4⋊C4×C73—21no—
123C4⋊C4×C73—22no—
124C4⋊C4×C73—41no—
125C4⋊C4×C73—41no—
126C4⋊C4×C73—41no—
127C42×C73C2×D411yescharacteristic
128C16×C73C2×Q814yescomplemented
129C22×D73C2×D811yescharacteristic
130Q8×C73—82no—
131C2×C4×C73—41no—
132C2×C4×C73—41no—
133C2×C4×C73—42no—
134C2×C4×C73C2×D811yescharacteristic
135C2×C4×C73C2×D811yescharacteristic
136C2×C4×C73C22×D411yescharacteristic
137C2×Dic73—41no—
138C2×Dic73—41no—
139C2×Dic73—42no—
140C2×Dic73C2×D811yescharacteristic
141C2×Dic73C2×D811yescharacteristic
142C2×Dic73C2×D811yescharacteristic
143C4×D73—22no—
144C4×D73—22no—
145C4×D73—84no—
146C4×D73C4⋊Q812yes—
147C4×D73C2×Q1612yes—
148Dic146—82no—
149Dic146—84no—
150Dic146—82no—
151Dic146—82no—
152C8×C73C2×Q1612yes—
153C8×C73C22×Q812yes—
154#10596831C4⋊Q812yes—
155#10596831C2×Q1612yes—
156C22×C73C22×D811yescharacteristic
157C2×D73—21no—
158C2×D73—21no—
159C2×D73#50012yes—
160C4×C73—21no—
161C4×C73—21no—
162C4×C73—82no—
163C4×C73—81no—
164C4×C73—81no—
165C4×C73C2×C4⋊Q811yescharacteristic
166C4×C73C22×Q1611yescharacteristic
167Dic73—82no—
168Dic73—81no—
169Dic73—81no—
170Dic73#50012yes—
171Dic73#50912yes—
172Dic73#51012yes—
173#33731—731nomaximal, Sylow
174C2×C73C2×#50011yescharacteristic
175C2×C73C2×#50911yescharacteristic
176C2×C73C2×#51011yescharacteristic
177D73—22no—
178#3160—731no—
179C2×#500—731no—
180#2255—731no—
181#2255—731no—
182#2255—731no—
183#2255D7311yescharacteristic, complemented
184#2254—732no—
185#2252—732no—
186#2159—732no—
187#2159—732no—
188C4×M32—731no—
189C73#3373111yescharacteristic, complemented, Sylow
190#533—1461no—
191C2×C4⋊Q8—1461no—
192C2×M32—732no—
193#500—732no—
194#500—731no—
195#500—731no—
196#500—731no—
197#500—1461no—
198#500—1461no—
199#500C2×D7311yescharacteristic
200#499—731no—
201#499—1461no—
202#499C2×D7311yescharacteristic
203#498—1461no—
204#427—731no—
205#427—731no—
206#427—1461no—
207C2×#107—732no—
208C2×#107—1461no—
209C2×C4×C8—731no—
210#367—732no—
211#367—732no—
212#367—732no—
213#367C2×D7312yes—
214#366—732no—
215#366—732no—
216#366—732no—
217#366C2×D7312yes—
218#346—731no—
219#346C2×D7311yescharacteristic
220C4×C16—732no—
221C22×Q8—2921no—
222C22×C8—731no—
223C22×C8—731no—
224C4⋊Q8—21no—
225C4⋊Q8—2921no—
226C4⋊Q8—2921no—
227C4⋊Q8—1462no—
228C4⋊Q8—1461no—
229C4⋊Q8—1461no—
230C4⋊Q8—1461no—
231#125—21no—
232#125—1461no—
233C22⋊Q8—2921no—
234C22⋊Q8—2921no—
235C4×Q8—2921no—
236C4×Q8—2921no—
237#117—1461no—
238#117—1461no—
239C2×C4⋊C4—2921no—
240C2×C4⋊C4—1462no—
241C2×C42—731no—
242M32—738no—
243C2×C16—732no—
244C2×C16C22×D7312yes—
245#107—21no—
246#107—731no—
247#107—732no—
248#107—731no—
249#107—1462no—
250#107—1462no—
251#107—1461no—
252#107—1461no—
253#107—1461no—
254#107C22×D7311yescharacteristic
255#107C22×D7312yes—
256#107C22×D7311yescharacteristic
257#106—21no—
258#106—1461no—
259#106—1461no—
260#106—1461no—
261C4×C8—731no—
262C4×C8—731no—
263C4×C8—731no—
264C4×C8C22×D7311yescharacteristic
265C2×Q8—41no—
266C2×Q8—2921no—
267C2×Q8—2921no—
268C2×Q8—2922no—
269C2×Q8—2921no—
270C2×Q8—5844no—
271C22×C4—2922no—
272C22×C4—731no—
273C22×C4—731no—
274C22×C4—731no—
275C2×C8—731no—
276C2×C8—731no—
277C2×C8—734no—
278C2×C8—734no—
279C2×C8D4×D7311yescharacteristic
280C2×C8C23×D7311yescharacteristic
281C4⋊C4—21no—
282C4⋊C4—21no—
283C4⋊C4—22no—
284C4⋊C4—41no—
285C4⋊C4—41no—
286C4⋊C4—41no—
287C4⋊C4—2922no—
288C4⋊C4—2922no—
289C4⋊C4—2921no—
290C4⋊C4—2921no—
291C4⋊C4—2921no—
292C4⋊C4—2921no—
293C4⋊C4—2921no—
294C4⋊C4—1461no—
295C4⋊C4—1461no—
296C4⋊C4—1462no—
297C22⋊C4—2921no—
298C22⋊C4—2921no—
299C42—2921no—
300C42—2921no—
301C42—731no—
302C42—731no—
303C42—731no—
304C42D4×D7311yescharacteristic
305C16—734no—
306C16Q8×D7314yescomplemented
307C23—731no—
308Q8—82no—
309Q8—5842no—
310Q8—5844no—
311Q8—5842no—
312Q8—5842no—
313C2×C4—41no—
314C2×C4—41no—
315C2×C4—42no—
316C2×C4—2921no—
317C2×C4—2921no—
318C2×C4—2922no—
319C2×C4—5844no—
320C2×C4—731no—
321C2×C4—731no—
322C2×C4—731no—
323C2×C4—732no—
324C2×C4—732no—
325C2×C4—1462no—
326C2×C4—1462no—
327C2×C4D8×D7311yescharacteristic
328C2×C4D8×D7311yescharacteristic
329C2×C4C2×D4×D7311yescharacteristic
330C8—732no—
331C8—732no—
332C8Q16×D7312yes—
333C8C2×Q8×D7312yes—
334C22—732no—
335C22—1461no—
336C22—1461no—
337C22C2×D8×D7311yescharacteristic
338C4—21no—
339C4—21no—
340C4—82no—
341C4—81no—
342C4—81no—
343C4—5842no—
344C4—5841no—
345C4—5841no—
346C4—732no—
347C4—732no—
348C4—732no—
349C4C4⋊Q8×D7311yescharacteristic
350C4C2×Q16×D7311yescharacteristic
351C2—1462no—
352C2#500×D7311yescharacteristic
353C2#509×D7311yescharacteristic
354C2#510×D7311yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2255C2
socleC22×C73C22×D8
prime radicalC22×D73C2×D8
hypercenter#2255D73
hypocenterC73#33731
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. #22612218C2semidirect
4. #22612217C2semidirect
5. #22612214C2semidirect
6. #22612212C2semidirect
7. #22612205C2semidirect
11. C73×#2255C2semidirect
62. #14556530Q8semidirect
117. #11924003C2×Q8semidirect
128. C16×C73C2×Q8semidirect
183. #2255D73semidirect
189. C73#33731semidirect
306. C16Q8×D73semidirect

Maximal subgroups

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

subgroupindex
#226204442
#500×D732
#226122182
#226122172
#226122142
#226122122
#226122052
#226117182
#226117162
#226115242
C73×#22552
#3373173