Finite Group Databasebeta

← #33242933

Subgroups — #33242933

Order 18688.

556 conjugacy classes of subgroups.

The subgroup lattice has 360 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22622362C211yesmaximal, characteristic, complemented
2#22612154C211yesmaximal, characteristic, complemented
3#22612119C212yesmaximal, complemented
4#22612073C211yesmaximal, characteristic, complemented
5#22612025C212yesmaximal, complemented
6#22611968C211yesmaximal, characteristic, complemented
7#22611871C212yesmaximal, complemented
8C4×#14556537C211yesmaximal, characteristic, complemented
9D4×#11924003C211yesmaximal, characteristic, complemented
10#22611638C212yesmaximal, complemented
11C73×#3287C211yesmaximal, characteristic
12#17989307—21no—
13C2×#14556537C2211yescharacteristic
14C2×#14556535C2212yes—
15C22×#11924003C2212yes—
16#17988992C2211yescharacteristic
17#17988929—22no—
18#17988873—21no—
19#17988828C2212yes—
20C4×Dic292C2211yescharacteristic
21#17988625C2212yes—
22#17988414C2212yes—
23#17988412C2211yescharacteristic
24C4⋊#11924003C2211yescharacteristic
25#17988391C2211yescharacteristic
26#17988391C2212yes—
27#17988390C2211yescharacteristic
28#17988388C2212yes—
29#17988388C2212yes—
30#17988388C2212yes—
31C22⋊#11924003C2212yes—
32C4×#11924003C2211yescharacteristic
33C2×#135×C73C2212yes—
34#550×C73—21no—
35#493×C73C2211yescharacteristic
36#466×C73C2212yes—
37#460×C73—21no—
38#449×C73—22no—
39C4×Q16×C73C2211yescharacteristic
40C8×D4×C73C2211yescharacteristic
41C2×Dic292C2311yescharacteristic
42#14556591—21no—
43#14556584—42no—
44#14556578—41no—
45#14556570—22no—
46C4×Dic146—21no—
47#14556556—41no—
48#14556550—22no—
49#14556537—22no—
50#14556535—42no—
51C2×#11924003—42no—
52C2×#11924003C2311yescharacteristic
53C2×#11924003C2311yescharacteristic
54C2×#11924003C2312yes—
55#14556525—21no—
56#14556525C2311yescharacteristic
57#14556525C2312yes—
58#14556523—22no—
59#14556516—22no—
60#14556516—21no—
61C2×#41×C73—22no—
62#135×C73—24no—
63#135×C73—42no—
64C2×Q16×C73C2311yescharacteristic
65C2×SD16×C73—22no—
66C2×D8×C73—21no—
67C2×D8×C73C2312yes—
68C22×C8×C73C2312yes—
69#127×C73—21no—
70#127×C73—41no—
71#121×C73—22no—
72#121×C73—42no—
73C4×Q8×C73—21no—
74C4×D4×C73—41no—
75C4×D4×C73D411yescharacteristic
76#107×C73C2311yescharacteristic
77C4⋊C8×C73D411yescharacteristic
78#103×C73—21no—
79#102×C73—22no—
80C22⋊C8×C73D412yes—
81C4×C8×C73C2311yescharacteristic
82C2×Dic146—21no—
83C2×Dic146—41no—
84Dic292—21no—
85Dic292—21no—
86C4⋊Dic73—21no—
87C4⋊Dic73—22no—
88C4⋊Dic73—41no—
89#11924007—42no—
90#11924007—41no—
91C22⋊Dic73—42no—
92C22⋊Dic73—42no—
93C4×Dic73—42no—
94C4×Dic73—41no—
95#11924003—21no—
96#11924003—21no—
97#11924003—22no—
98#41×C73—44no—
99#41×C73—82no—
100C2×Q8×C73—21no—
101C2×D4×C73—22no—
102C2×D4×C73—42no—
103C2×D4×C73—42no—
104C2×D4×C73—41no—
105C2×D4×C73—41no—
106C2×D4×C73D811yescharacteristic
107C22×C4×C73—42no—
108C22×C4×C73C2×D412yes—
109Q16×C73—21no—
110Q16×C73#4112yes—
111SD16×C73—44no—
112D8×C73—22no—
113D8×C73—22no—
114D8×C73—81no—
115C2×C8×C73—22no—
116C2×C8×C73—22no—
117C2×C8×C73C2×D411yescharacteristic
118C2×C8×C73C2×D412yes—
119C2×C8×C73C2411yescharacteristic
120C4⋊C4×C73—21no—
121C4⋊C4×C73—41no—
122C4⋊C4×C73D811yescharacteristic
123C22⋊C4×C73—42no—
124C22⋊C4×C73D812yes—
125C42×C73—41no—
126C42×C73C2×D411yescharacteristic
127C23×C73—42no—
128C23×C73C2×D812yes—
129Q8×C73—22no—
130Q8×C73—41no—
131D4×C73—22no—
132D4×C73—42no—
133D4×C73—42no—
134D4×C73—84no—
135D4×C73—82no—
136D4×C73—161no—
137D4×C73—162no—
138C2×C4×C73—22no—
139C2×C4×C73—22no—
140C2×C4×C73—41no—
141C2×C4×C73—41no—
142C2×C4×C73—84no—
143C2×C4×C73C2×D811yescharacteristic
144C2×C4×C73C2×D812yes—
145C2×C4×C73C2×D811yescharacteristic
146C2×C4×C73C22×D411yescharacteristic
147C2×Dic73—41no—
148C2×Dic73—41no—
149C2×Dic73—42no—
150Dic146—41no—
151Dic146—41no—
152Dic146—82no—
153C8×C73—22no—
154C8×C73—21no—
155C8×C73C2×#4111yescharacteristic
156C8×C73H(5,𝔽2)11yescharacteristic
157C22×C73—22no—
158C22×C73—22no—
159C22×C73—22no—
160C22×C73—82no—
161C22×C73—82no—
162C22×C73—82no—
163C22×C73C22×D811yescharacteristic
164C4×C73—21no—
165C4×C73—22no—
166C4×C73—42no—
167C4×C73—81no—
168C4×C73#50812yes—
169C4×C73#54611yescharacteristic
170C4×C73#57711yescharacteristic
171Dic73—82no—
172Dic73—81no—
173Dic73—81no—
174#33871—731nomaximal, Sylow
175C2×C73—42no—
176C2×C73—162no—
177C2×C73#329411yescharacteristic
178C2×C73C2×#50811yescharacteristic
179C2×C73#341611yescharacteristic
180#3342—731no—
181#3287D7311yescharacteristic
182#2240—731no—
183#2234—732no—
184#2225—731no—
185#2217—732no—
186#2207—731no—
187#2189—732no—
188C4×Q32—731no—
189D4×C16—731no—
190#2138—732no—
191C73#3387111yescharacteristic, complemented, Sylow
192C2×#135C2×D7312yes—
193#556—1461no—
194#550—21no—
195C2×Q32—731no—
196C2×SD32—732no—
197C22×C16—732no—
198#500—731no—
199#493C2×D7311yescharacteristic
200#485—1462no—
201#475—1461no—
202#467—732no—
203#466C2×D7312yes—
204#460—21no—
205#449—22no—
206C4×Q16—731no—
207C4×Q16C2×D7311yescharacteristic
208C8×D4C2×D7311yescharacteristic
209#427—732no—
210#367—732no—
211#366—731no—
212C4⋊C16—731no—
213#358—731no—
214#358—732no—
215#358—731no—
216#357—732no—
217#357—732no—
218#357—732no—
219C22⋊C16—732no—
220C4×C16—731no—
221C2×#41—22no—
222#135—24no—
223#135—42no—
224C2×Q16—731no—
225C2×Q16C22×D7311yescharacteristic
226C2×SD16—22no—
227C2×D8—21no—
228C2×D8C22×D7312yes—
229C22×C8C22×D7312yes—
230C4⋊Q8—1461no—
231#127—21no—
232#127—41no—
233#126—2922no—
234#124—2921no—
235C22⋊Q8—1462no—
236#121—22no—
237#121—42no—
238C4×Q8—21no—
239C4×Q8—2921no—
240C4×Q8—1461no—
241C4×D4—41no—
242C4×D4C22⋊D7311yescharacteristic
243#117—1462no—
244Q32—1462no—
245SD32—2922no—
246C2×C16—2922no—
247C2×C16—731no—
248C2×C16—731no—
249C2×C16—732no—
250#107—731no—
251#107—732no—
252#107—1461no—
253#107C22×D7311yescharacteristic
254#106—1462no—
255C4⋊C8C22⋊D7311yescharacteristic
256#103—21no—
257#103—1461no—
258#103—1462no—
259#102—22no—
260C22⋊C8C22⋊D7312yes—
261C4×C8C22×D7311yescharacteristic
262#41—44no—
263#41—82no—
264C2×Q8—21no—
265C2×Q8—2921no—
266C2×Q8—1461no—
267C2×D4—22no—
268C2×D4—42no—
269C2×D4—42no—
270C2×D4—41no—
271C2×D4—41no—
272C2×D4#1192401311yescharacteristic
273C22×C4—42no—
274C22×C4C2×C22⋊D7312yes—
275Q16—21no—
276Q16—1461no—
277Q16—1461no—
278Q16#1192402712yes—
279SD16—44no—
280D8—22no—
281D8—22no—
282D8—81no—
283C2×C8—22no—
284C2×C8—22no—
285C2×C8C2×C22⋊D7311yescharacteristic
286C2×C8C2×C22⋊D7312yes—
287C2×C8C23×D7311yescharacteristic
288C4⋊C4—21no—
289C4⋊C4—41no—
290C4⋊C4—2922no—
291C4⋊C4—2921no—
292C4⋊C4—2921no—
293C4⋊C4—1461no—
294C4⋊C4—1462no—
295C4⋊C4#1192401311yescharacteristic
296C22⋊C4—42no—
297C22⋊C4—2922no—
298C22⋊C4—2922no—
299C22⋊C4#1192401312yes—
300C42—41no—
301C42—2922no—
302C42—2921no—
303C42C2×C22⋊D7311yescharacteristic
304C16—1461no—
305C16—1461no—
306C16—1462no—
307C23—42no—
308C23C2×#1192401312yes—
309Q8—22no—
310Q8—41no—
311Q8—2921no—
312Q8—2921no—
313Q8—5842no—
314D4—22no—
315D4—42no—
316D4—42no—
317D4—84no—
318D4—82no—
319D4—161no—
320D4—162no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—41no—
324C2×C4—41no—
325C2×C4—84no—
326C2×C4—2921no—
327C2×C4—2921no—
328C2×C4—2922no—
329C2×C4C2×#1192401312yes—
330C2×C4C2×#1192401311yescharacteristic
331C2×C4C2×#1192401311yescharacteristic
332C2×C4C22×C22⋊D7311yescharacteristic
333C8—21no—
334C8—22no—
335C8C2×#1192402711yescharacteristic
336C8#1455663611yescharacteristic
337C22—22no—
338C22—22no—
339C22—22no—
340C22—82no—
341C22—82no—
342C22—82no—
343C22C22×#1192401311yescharacteristic
344C4—22no—
345C4—21no—
346C4—42no—
347C4—81no—
348C4—5841no—
349C4—5842no—
350C4—5841no—
351C4#1798901712yes—
352C4#1798923111yescharacteristic
353C4#1798939811yescharacteristic
354C2—42no—
355C2—162no—
356C2#2262174411yescharacteristic
357C2C2×#1798901711yescharacteristic
358C2#2262307811yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
Fitting, prime radicalC73×#3287C2
socleC22×C73C22×D8
hypercenter#3287D73
hypocenterC73#33871
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ C4×D4 ◁ C8×D4 ◁ #3287

Complements

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

NH ≅ G/Ntype
1. #22622362C2semidirect
2. #22612154C2semidirect
3. #22612119C2semidirect
4. #22612073C2semidirect
5. #22612025C2semidirect
6. #22611968C2semidirect
7. #22611871C2semidirect
8. C4×#14556537C2semidirect
9. D4×#11924003C2semidirect
10. #22611638C2semidirect
191. C73#33871semidirect

Maximal subgroups

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

subgroupindex
#226223622
#226121542
#226121192
#226120732
#226120252
#226119682
#226118712
C4×#145565372
D4×#119240032
#226116382
C73×#32872
#3387173