Finite Group Databasebeta

← #33241503

Subgroups — #33241503

Order 18688.

514 conjugacy classes of subgroups.

The subgroup lattice has 308 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620442C211yesmaximal, characteristic
2#500×D73C211yesmaximal, characteristic
3#22612215C214yesmaximal, complemented
4#22612213C211yesmaximal, characteristic, complemented
5#22612211C211yesmaximal, characteristic, complemented
6#22612204C211yesmaximal, characteristic, complemented
7#22611716C214yesmaximal
8C4×#14556530C211yesmaximal, characteristic
9C73×#2252C211yesmaximal, characteristic, complemented
10#17989142—21no—
11C4⋊Q8×D73—21no—
12C2×#14556530C2212yes—
13#17988992C2211yescharacteristic
14#17988992C2211yescharacteristic
15#17988991—21no—
16#17988991C2212yes—
17#17988990—21no—
18#17988988—21no—
19#17988988C2212yes—
20#17988983—21no—
21#17988624—21no—
22#17988624C2212yes—
23#107×D73—21no—
24#107×D73C2212yes—
25C4×C8×D73C2211yescharacteristic
26#17988415C2214yes—
27#17988414C2214yes—
28#17988412C2214yes—
29#17988362C2212yes—
30C4×#11924003C2211yescharacteristic
31#500×C73C2211yescharacteristic
32#500×C73C2211yescharacteristic
33#366×C73C2214yes—
34C4×C16×C73C2211yescharacteristic
35C2×Q8×D73—41no—
36C2×C8×D73D411yescharacteristic
37C2×C8×D73C2311yescharacteristic
38#14556591—21no—
39#14556591—21no—
40#14556590—22no—
41#14556590—41no—
42#14556589—41no—
43#14556582—22no—
44#14556571—41no—
45#14556567—41no—
46Q8×Dic73—41no—
47#14556556—41no—
48#14556549—22no—
49C4⋊C4×D73—22no—
50C4⋊C4×D73—41no—
51C42×D73D411yescharacteristic
52#14556530Q818yescomplemented
53C2×#11924003C2312yes—
54#14556525—21no—
55#14556525—21no—
56#14556525C2312yes—
57#14556525C2312yes—
58#14556524—22no—
59#14556524—22no—
60#14556522—22no—
61#14556522—22no—
62C4×#10596831D411yescharacteristic
63C8×Dic73D411yescharacteristic
64C8×Dic73C2311yescharacteristic
65C4⋊Q8×C73—21no—
66C4⋊Q8×C73—21no—
67C2×C16×C73C2312yes—
68#107×C73—21no—
69#107×C73—21no—
70#107×C73C2312yes—
71#107×C73C2312yes—
72C4×C8×C73C2311yescharacteristic
73C2×Dic146—41no—
74C2×Dic146—41no—
75C2×Dic146—42no—
76Q8×D73—84no—
77C2×C4×D73—42no—
78C2×C4×D73D811yescharacteristic
79C2×C4×D73D811yescharacteristic
80C2×C4×D73C2×D411yescharacteristic
81C2×#10596831C2×D411yescharacteristic
82C2×#10596831C2×D411yescharacteristic
83C8×D73Q1614yes—
84C8×D73C2×Q814yes—
85C4⋊Dic73—22no—
86C4⋊Dic73—22no—
87C4⋊Dic73—41no—
88C4⋊Dic73—41no—
89C4⋊Dic73—42no—
90#11924007—42no—
91#11924007—42no—
92#11924007—42no—
93#11924005—42no—
94C4×Dic73—42no—
95C4×Dic73D811yescharacteristic
96C4×Dic73D811yescharacteristic
97C4×Dic73C2×D411yescharacteristic
98#11924003C2×Q814yescomplemented
99C2×Q8×C73—41no—
100C2×Q8×C73—41no—
101C2×C8×C73C2×D411yescharacteristic
102C2×C8×C73C2411yescharacteristic
103C4⋊C4×C73—22no—
104C4⋊C4×C73—22no—
105C4⋊C4×C73—41no—
106C4⋊C4×C73—41no—
107C42×C73C2×D411yescharacteristic
108C16×C73C2×Q814yescomplemented
109C22×D73C2×D811yescharacteristic
110Q8×C73—82no—
111Q8×C73—82no—
112C2×C4×C73—42no—
113C2×C4×C73—42no—
114C2×C4×C73C2×D811yescharacteristic
115C2×C4×C73C2×D811yescharacteristic
116C2×C4×C73C22×D411yescharacteristic
117C2×Dic73—42no—
118C2×Dic73—42no—
119C2×Dic73C2×D811yescharacteristic
120C2×Dic73C2×D811yescharacteristic
121C2×Dic73C2×D811yescharacteristic
122C4×D73—22no—
123C4×D73—22no—
124C4×D73—84no—
125C4×D73C4⋊Q812yes—
126C4×D73C2×Q1612yes—
127Dic146—84no—
128Dic146—82no—
129Dic146—82no—
130C8×C73C2×Q1612yes—
131C8×C73C22×Q812yes—
132#10596831C4⋊Q812yes—
133#10596831C2×Q1612yes—
134C22×C73C22×D811yescharacteristic
135C2×D73—21no—
136C2×D73—21no—
137C2×D73#50012yes—
138C4×C73—82no—
139C4×C73—82no—
140C4×C73#50912yes—
141C4×C73#51012yes—
142C4×C73C2×C4⋊Q811yescharacteristic
143C4×C73C22×Q1611yescharacteristic
144Dic73—21no—
145Dic73—21no—
146Dic73—82no—
147Dic73—82no—
148Dic73#50012yes—
149#33731—731nomaximal, Sylow
150C2×C73C2×#50011yescharacteristic
151C2×C73C2×#50911yescharacteristic
152C2×C73C2×#51011yescharacteristic
153D73—22no—
154#3160—731no—
155C2×#500—731no—
156#2255—734no—
157#2254—731no—
158#2254—731no—
159#2252—731no—
160#2252D7311yescharacteristic, complemented
161#2159—734no—
162C4×M32—731no—
163C73#3373111yescharacteristic, complemented, Sylow
164#533—1461no—
165C2×C4⋊Q8—1461no—
166C2×M32—732no—
167#500—732no—
168#500—731no—
169#500—731no—
170#500—1461no—
171#500—1461no—
172#500C2×D7311yescharacteristic
173#500C2×D7311yescharacteristic
174#499—732no—
175#499—1461no—
176#498—1461no—
177#427—732no—
178#427—1461no—
179C2×#107—732no—
180C2×#107—1461no—
181C2×C4×C8—731no—
182#367—734no—
183#367—734no—
184#366—734no—
185#366C2×D7314yes—
186#346—732no—
187C4×C16—731no—
188C4×C16C2×D7311yescharacteristic
189C22×Q8—2921no—
190C22×C8—731no—
191C22×C8—731no—
192C4⋊Q8—21no—
193C4⋊Q8—21no—
194C4⋊Q8—2921no—
195C4⋊Q8—2921no—
196C4⋊Q8—1462no—
197C4⋊Q8—1461no—
198C4⋊Q8—1461no—
199#125—1462no—
200C22⋊Q8—2921no—
201C22⋊Q8—2921no—
202C4×Q8—2921no—
203C4×Q8—2921no—
204#117—1462no—
205C2×C4⋊C4—2921no—
206C2×C4⋊C4—1462no—
207C2×C42—731no—
208M32—738no—
209C2×C16—732no—
210C2×C16C22×D7312yes—
211#107—21no—
212#107—21no—
213#107—732no—
214#107—732no—
215#107—1462no—
216#107—1462no—
217#107—1461no—
218#107—1461no—
219#107C22×D7312yes—
220#107C22×D7312yes—
221#106—1462no—
222#106—1462no—
223C4×C8—731no—
224C4×C8—731no—
225C4×C8—731no—
226C4×C8C22×D7311yescharacteristic
227C2×Q8—41no—
228C2×Q8—41no—
229C2×Q8—2922no—
230C2×Q8—2921no—
231C2×Q8—2921no—
232C2×Q8—5844no—
233C22×C4—2922no—
234C22×C4—731no—
235C22×C4—731no—
236C22×C4—731no—
237C2×C8—731no—
238C2×C8—731no—
239C2×C8—734no—
240C2×C8—734no—
241C2×C8D4×D7311yescharacteristic
242C2×C8C23×D7311yescharacteristic
243C4⋊C4—22no—
244C4⋊C4—22no—
245C4⋊C4—41no—
246C4⋊C4—41no—
247C4⋊C4—2922no—
248C4⋊C4—2922no—
249C4⋊C4—2922no—
250C4⋊C4—2922no—
251C4⋊C4—2921no—
252C4⋊C4—2921no—
253C4⋊C4—1462no—
254C4⋊C4—1462no—
255C22⋊C4—2922no—
256C42—2922no—
257C42—731no—
258C42—731no—
259C42—731no—
260C42D4×D7311yescharacteristic
261C16—734no—
262C16Q8×D7314yescomplemented
263C23—731no—
264Q8—82no—
265Q8—82no—
266Q8—5844no—
267Q8—5842no—
268Q8—5842no—
269C2×C4—42no—
270C2×C4—42no—
271C2×C4—2922no—
272C2×C4—2922no—
273C2×C4—5844no—
274C2×C4—731no—
275C2×C4—732no—
276C2×C4—732no—
277C2×C4—731no—
278C2×C4—731no—
279C2×C4—1462no—
280C2×C4—1462no—
281C2×C4D8×D7311yescharacteristic
282C2×C4D8×D7311yescharacteristic
283C2×C4C2×D4×D7311yescharacteristic
284C8—732no—
285C8—732no—
286C8Q16×D7312yes—
287C8C2×Q8×D7312yes—
288C22—732no—
289C22—1461no—
290C22—1461no—
291C22C2×D8×D7311yescharacteristic
292C4—82no—
293C4—82no—
294C4—5842no—
295C4—5842no—
296C4—732no—
297C4—1461no—
298C4—1461no—
299C4#1798902112yes—
300C4#1798902912yes—
301C4C4⋊Q8×D7311yescharacteristic
302C4C2×Q16×D7311yescharacteristic
303C2—1462no—
304C2#500×D7311yescharacteristic
305C2C2×#1798902111yescharacteristic
306C2C2×#1798902911yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2252C2
socleC22×C73C22×D8
prime radicalC22×D73C2×D8
hypercenter#2252D73
hypocenterC73#33731
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #2252

Complements

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

NH ≅ G/Ntype
3. #22612215C2semidirect
4. #22612213C2semidirect
5. #22612211C2semidirect
6. #22612204C2semidirect
9. C73×#2252C2semidirect
52. #14556530Q8semidirect
98. #11924003C2×Q8semidirect
108. C16×C73C2×Q8semidirect
160. #2252D73semidirect
163. C73#33731semidirect
262. C16Q8×D73semidirect

Maximal subgroups

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

subgroupindex
#226204422
#500×D732
#226122152
#226122132
#226122112
#226122042
#226117162
C4×#145565302
C73×#22522
#3373173