Finite Group Databasebeta

← #33239032

Subgroups — #33239032

Order 18688.

658 conjugacy classes of subgroups.

The subgroup lattice has 348 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1C2×#17989005C211yesmaximal, characteristic
2#22618528C211yesmaximal, characteristic
3#22618527C211yesmaximal, characteristic
4#22611719C211yesmaximal, characteristic, complemented
5#22611718C212yesmaximal, complemented
6#22611716C212yesmaximal, complemented
7#22611713C211yesmaximal, characteristic, complemented
8#22611712C212yesmaximal, complemented
9C2×#17988414C211yesmaximal, characteristic, complemented
10#367×D73C212yesmaximal, complemented
11C73×#2159C211yesmaximal, characteristic, complemented
12C2×#14556597C2211yescharacteristic
13#17989068—21no—
14#17989067—21no—
15#17989005C418yescomplemented
16C2×#14556530C2212yes—
17C22×#11924003C2211yescharacteristic
18C2×C16×D73C2212yes—
19#17988630—21no—
20#17988628—22no—
21#17988627—21no—
22#17988626—22no—
23#17988625C2211yescharacteristic
24#17988624C2212yes—
25#17988624C2212yes—
26C2×#14556525C2211yescharacteristic
27#107×D73C2212yes—
28#107×D73C2212yes—
29#17988415C2212yes—
30#17988414C2211yescharacteristic
31#17988414C2212yes—
32#17988414C2211yescharacteristic
33#17988414C2212yes—
34#17988413C2212yes—
35#17988412C2211yescharacteristic
36#17988412C2211yescharacteristic
37C2×M32×C73C2211yescharacteristic
38#427×C73C2211yescharacteristic
39C2×#107×C73C2211yescharacteristic
40#367×C73C2212yes—
41#366×C73C2212yes—
42C2×#11924028D411yescharacteristic
43#14556597C2×C414yes—
44#14556597Q814yes—
45C2×#11924012D411yescharacteristic
46C2×#11924011D412yes—
47C2×C8×D73C2312yes—
48C4×Dic146—41no—
49#14556556—41no—
50C4×D292—41no—
51#14556553—41no—
52C4×C22⋊D73—42no—
53#14556551—42no—
54#14556550—21no—
55#14556549—22no—
56#14556549—22no—
57C2×C4⋊Dic73—21no—
58C4⋊C4×D73—22no—
59C4⋊C4×D73—22no—
60#14556530C2×C418yescomplemented
61C2×#11924003C2×C414yescomplemented
62C2×#11924003C2311yescharacteristic
63C2×#11924003C2311yescharacteristic
64C16×D73C2×C418yescomplemented
65#14556525—21no—
66#14556525C2311yescharacteristic
67#14556525C2312yes—
68#14556525C2311yescharacteristic
69#14556524—22no—
70#14556524—22no—
71#14556523—21no—
72#14556522—22no—
73#14556522—22no—
74C22×C8×C73C2311yescharacteristic
75#117×C73—21no—
76C2×C4⋊C4×C73—21no—
77M32×C73C2×C414yescomplemented
78C2×C16×C73C2312yes—
79#107×C73—21no—
80#107×C73C2311yescharacteristic
81#107×C73C2312yes—
82#107×C73C2311yescharacteristic
83#106×C73—21no—
84#11924028C4⋊C414yes—
85#11924028Q1614yes—
86C2×Dic146D811yescharacteristic
87C2×D292D811yescharacteristic
88C2×C22⋊D73D812yes—
89C22×Dic73—41no—
90C2×C4×D73—42no—
91C2×C4×D73—42no—
92C2×C4×D73C2×D412yes—
93#11924012C4⋊C414yes—
94#11924011C4⋊C418yes—
95C2×#10596831C2×D412yes—
96C8×D73C22×C414yes—
97C8×D73C2×Q814yes—
98C4⋊Dic73—21no—
99C4⋊Dic73—22no—
100C4⋊Dic73—21no—
101C4⋊Dic73—41no—
102#11924007—42no—
103#11924007—42no—
104C22⋊Dic73—41no—
105#11924005—42no—
106#11924005—42no—
107C4×Dic73—41no—
108C4×Dic73—42no—
109C4×Dic73—42no—
110#11924003C22×C412yes—
111#11924003C22×C412yes—
112C22×C4×C73—41no—
113C22×C4×C73C2×D411yescharacteristic
114C2×C8×C73C22×C412yes—
115C2×C8×C73C2×D411yescharacteristic
116C2×C8×C73C2×Q812yes—
117C2×C8×C73C2411yescharacteristic
118C4⋊C4×C73—21no—
119C4⋊C4×C73—21no—
120C4⋊C4×C73—22no—
121C4⋊C4×C73—41no—
122C22⋊C4×C73—41no—
123C42×C73—41no—
124C16×C73C22×C414yes—
125C23×C73C2×D811yescharacteristic
126C22×D73C2×D812yes—
127C2×C4×C73—41no—
128C2×C4×C73—41no—
129C2×C4×C73—42no—
130C2×C4×C73—161no—
131C2×C4×C73C2×C4⋊C412yes—
132C2×C4×C73C2×D811yescharacteristic
133C2×C4×C73C2×Q1612yes—
134C2×C4×C73C22×D411yescharacteristic
135C22⋊D73#10718yes—
136C2×Dic73—41no—
137C2×Dic73—42no—
138C2×Dic73—41no—
139C2×Dic73—161no—
140C2×Dic73C2×D812yes—
141D292#10714yes—
142C4×D73—162no—
143C4×D73—162no—
144C4×D73C2×C4⋊C414yes—
145C4×D73C2×Q1614yes—
146Dic146#10714yes—
147C8×C73C2×C4⋊C412yes—
148C8×C73C23×C411yescharacteristic
149C8×C73C22×Q811yescharacteristic
150#10596831C2×C4⋊C414yes—
151C22×C73—21no—
152C22×C73—21no—
153C22×C73C2×#10712yes—
154C22×C73C22×D811yescharacteristic
155C2×D73—22no—
156C2×D73—22no—
157C2×D73C2×#10714yes—
158C4×C73—81no—
159C4×C73—82no—
160C4×C73—81no—
161C4×C73C2×#10712yes—
162C4×C73C22×C4⋊C411yescharacteristic
163C4×C73C22×Q1611yescharacteristic
164Dic73—81no—
165Dic73—82no—
166Dic73—81no—
167Dic73C2×#10714yes—
168#33411—731nomaximal, Sylow
169C2×C73—22no—
170C2×C73C22×#10711yescharacteristic
171C2×C73#341611yescharacteristic
172C2×C73#341611yescharacteristic
173D73—24no—
174C2×#504—731no—
175#2913—731no—
176#2913—731no—
177#2159—731no—
178#2159—732no—
179#2159—732no—
180#2159D7311yescharacteristic, complemented
181#2158—731no—
182#2158—732no—
183C2×#367—731no—
184C2×#367—732no—
185C73#3341111yescharacteristic, complemented, Sylow
186C2×#131—731no—
187#520—1461no—
188#520—1461no—
189#504—738no—
190C2×M32—732no—
191C2×M32C2×D7311yescharacteristic
192C22×C16—731no—
193C22×C16—732no—
194#428—1462no—
195#428—1462no—
196#428—1461no—
197#428—1461no—
198#427—731no—
199#427—732no—
200#427—732no—
201#427C2×D7311yescharacteristic
202C2×#107—731no—
203C2×#107—732no—
204C2×#107—732no—
205C2×#107C2×D7311yescharacteristic
206#367—732no—
207#367—731no—
208#367—732no—
209#367—732no—
210#367—731no—
211#367C2×D7312yes—
212#366—732no—
213#366—731no—
214#366—731no—
215#366C2×D7312yes—
216C2×#41—731no—
217#131—734no—
218#131—734no—
219C2×M16—731no—
220C2×M16—732no—
221C22×C8—732no—
222C22×C8C22×D7311yescharacteristic
223C4×Q8—2921no—
224C4×Q8—2921no—
225C4×D4—2921no—
226C4×D4—2922no—
227C4×D4—2921no—
228C4×D4—2922no—
229#117—21no—
230#117—1462no—
231#117—1462no—
232#117—1461no—
233C2×C4⋊C4—21no—
234C2×C4⋊C4—1462no—
235C2×C4⋊C4—1462no—
236C2×C4⋊C4—1461no—
237M32—738no—
238M32C4×D7314yescomplemented
239C2×C16—734no—
240C2×C16—731no—
241C2×C16—731no—
242C2×C16—738no—
243C2×C16C22×D7312yes—
244#107—21no—
245#107—731no—
246#107—731no—
247#107—732no—
248#107—1462no—
249#107—1462no—
250#107—1461no—
251#107C22×D7311yescharacteristic
252#107C22×D7311yescharacteristic
253#107C22×D7312yes—
254#106—21no—
255#106—1462no—
256#106—1462no—
257#106—1461no—
258#41—734no—
259#41—734no—
260C2×Q8—731no—
261C2×D4—732no—
262C2×D4—731no—
263C22×C4—41no—
264C22×C4—2922no—
265C22×C4—2922no—
266C22×C4—2921no—
267C22×C4—732no—
268C22×C4D4×D7311yescharacteristic
269M16—734no—
270M16—738no—
271C2×C8—732no—
272C2×C8—734no—
273C2×C8—734no—
274C2×C8C2×C4×D7312yes—
275C2×C8D4×D7311yescharacteristic
276C2×C8Q8×D7312yes—
277C2×C8C23×D7311yescharacteristic
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—22no—
281C4⋊C4—41no—
282C4⋊C4—2922no—
283C4⋊C4—2922no—
284C4⋊C4—2921no—
285C4⋊C4—1461no—
286C4⋊C4—1462no—
287C4⋊C4—1461no—
288C22⋊C4—41no—
289C22⋊C4—2922no—
290C22⋊C4—2922no—
291C22⋊C4—2921no—
292C42—41no—
293C42—2922no—
294C42—2922no—
295C42—2921no—
296C16—732no—
297C16—732no—
298C16C2×C4×D7314yes—
299C23—732no—
300C23D8×D7311yescharacteristic
301Q8—734no—
302D4—738no—
303D4—734no—
304C2×C4—41no—
305C2×C4—41no—
306C2×C4—42no—
307C2×C4—161no—
308C2×C4—2921no—
309C2×C4—2922no—
310C2×C4—2921no—
311C2×C4—732no—
312C2×C4—734no—
313C2×C4—734no—
314C2×C4—11682no—
315C2×C4—11682no—
316C2×C4—11681no—
317C2×C4C4⋊C4×D7312yes—
318C2×C4D8×D7311yescharacteristic
319C2×C4Q16×D7312yes—
320C2×C4C2×D4×D7311yescharacteristic
321C8—734no—
322C8C4⋊C4×D7312yes—
323C8C22×C4×D7311yescharacteristic
324C8C2×Q8×D7311yescharacteristic
325C22—21no—
326C22—21no—
327C22—734no—
328C22—1462no—
329C22—1462no—
330C22#107×D7312yes—
331C22C2×D8×D7311yescharacteristic
332C4—81no—
333C4—82no—
334C4—81no—
335C4—5842no—
336C4—5841no—
337C4—5841no—
338C4—734no—
339C4#107×D7312yes—
340C4C2×C4⋊C4×D7311yescharacteristic
341C4C2×Q16×D7311yescharacteristic
342C2—22no—
343C2—1464no—
344C2C2×#107×D7311yescharacteristic
345C2#2262307611yescharacteristic
346C2#2262307711yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutatorC8×C73C23×C4
FrattiniC2×C8C23×D73
FittingC73×#2159C2
socleC22×C73C22×D8
prime radicalC2×#11924028D4
cosocleC2×C8×C73C24
hypercenter#2159D73
hypocenterC73#33411
LCSG ▷ C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C22×C4 ◁ C22×C8 ◁ #2159

Complements

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

NH ≅ G/Ntype
4. #22611719C2semidirect
5. #22611718C2semidirect
6. #22611716C2semidirect
7. #22611713C2semidirect
8. #22611712C2semidirect
9. C2×#17988414C2semidirect
10. #367×D73C2semidirect
11. C73×#2159C2semidirect
15. #17989005C4semidirect
60. #14556530C2×C4semidirect
61. C2×#11924003C2×C4semidirect
64. C16×D73C2×C4semidirect
77. M32×C73C2×C4semidirect
180. #2159D73semidirect
185. C73#33411semidirect
238. M32C4×D73semidirect

Maximal subgroups

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

subgroupindex
C2×#179890052
#226185282
#226185272
#226117192
#226117182
#226117162
#226117132
#226117122
C2×#179884142
#367×D732
C73×#21592
#3341173