Finite Group Databasebeta

← #33240692

Subgroups — #33240692

Order 18688.

534 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
1#22620471C211yesmaximal, characteristic, complemented
2#22618805C211yesmaximal, characteristic
3#22612098C212yesmaximal, complemented
4#22612087C212yesmaximal, complemented
5#22612082C212yesmaximal, complemented
6#22612067C212yesmaximal, complemented
7C2×#17988412C211yesmaximal, characteristic, complemented
8#22611691C211yesmaximal, characteristic, complemented
9#22611644C211yesmaximal, characteristic
10#22611636C211yesmaximal, characteristic
11C73×#2159C211yesmaximal, characteristic, complemented
12C2×#14556613C2211yescharacteristic
13#17989215—21no—
14#17989068—21no—
15C22×#11924003C2211yescharacteristic
16#17988994—22no—
17#17988991C2212yes—
18#17988988C2212yes—
19#17988686—22no—
20#17988681C2212yes—
21#17988669C2212yes—
22#17988630—21no—
23C2×#14556525C2211yescharacteristic
24#17988586—21no—
25#17988542C2211yescharacteristic
26#17988412C2212yes—
27#17988412C2212yes—
28#17988405C2212yes—
29#17988405C2212yes—
30#17988392C2212yes—
31#17988390C2211yescharacteristic
32#17988390C2211yescharacteristic
33#17988389C2212yes—
34#17988387C2211yescharacteristic
35#17988387C2211yescharacteristic
36C2×M32×C73C2211yescharacteristic
37#427×C73C2211yescharacteristic
38C2×#107×C73C2211yescharacteristic
39#367×C73C2212yes—
40#366×C73C2212yes—
41C2×#11924028—21no—
42#14556613—22no—
43#14556613Q814yescomplemented
44C2×Dic292C2311yescharacteristic
45C2×#11924017—21no—
46C2×D584C2311yescharacteristic
47C2×#11924012D411yescharacteristic
48#14556591—41no—
49#14556590—22no—
50#14556583—41no—
51#14556582—22no—
52#14556568—42no—
53#14556556—22no—
54#14556553—22no—
55C4×C22⋊D73—42no—
56#14556550D411yescharacteristic
57#14556549—41no—
58C2×C4⋊Dic73—21no—
59C4⋊C4×D73—41no—
60C2×#11924003—22no—
61C2×#11924003C2311yescharacteristic
62C2×#11924003C2311yescharacteristic
63#14556525—21no—
64#14556525C2312yes—
65#14556524—22no—
66#14556522—22no—
67#14556514—22no—
68#14556511—22no—
69#14556501D412yes—
70C8×Dic73C2312yes—
71C22×C8×C73C2311yescharacteristic
72#117×C73—21no—
73C2×C4⋊C4×C73—21no—
74M32×C73—22no—
75C2×C16×C73C2312yes—
76#107×C73—21no—
77#107×C73C2311yescharacteristic
78#107×C73C2311yescharacteristic
79#107×C73C2312yes—
80#106×C73—21no—
81#11924028—24no—
82#11924028—42no—
83C2×Dic146—21no—
84C2×Dic146—42no—
85C2×D292—21no—
86C2×C22⋊D73—41no—
87C22×Dic73—41no—
88C2×C4×D73—41no—
89C2×C4×D73—42no—
90Dic292—21no—
91Dic292C2×Q812yes—
92#11924017—24no—
93D584—21no—
94D584C2×Q812yes—
95#11924012—22no—
96C2×#10596831C2×D412yes—
97C4⋊Dic73—22no—
98C4⋊Dic73—41no—
99C4⋊Dic73—42no—
100C4⋊Dic73D811yescharacteristic
101C4⋊Dic73D811yescharacteristic
102#11924007—42no—
103#11924007—42no—
104#11924007—42no—
105C22⋊Dic73D812yes—
106#11924005—42no—
107C4×Dic73—42no—
108C4×Dic73C2×D412yes—
109#11924003—21no—
110#11924003—21no—
111C22×C4×C73—41no—
112C22×C4×C73C2×D411yescharacteristic
113C2×C8×C73C2×D411yescharacteristic
114C2×C8×C73C2×Q812yes—
115C2×C8×C73#4112yes—
116C2×C8×C73C2411yescharacteristic
117C4⋊C4×C73—21no—
118C4⋊C4×C73—21no—
119C4⋊C4×C73—22no—
120C4⋊C4×C73—41no—
121C22⋊C4×C73—41no—
122C42×C73—41no—
123C16×C73—22no—
124C23×C73C2×D811yescharacteristic
125C22×D73—41no—
126C2×C4×C73—41no—
127C2×C4×C73—41no—
128C2×C4×C73—42no—
129C2×C4×C73—161no—
130C2×C4×C73C22⋊Q812yes—
131C2×C4×C73C2×D811yescharacteristic
132C2×C4×C73#13712yes—
133C2×C4×C73C22×D411yescharacteristic
134C22⋊D73—44no—
135C2×Dic73—41no—
136C2×Dic73—42no—
137C2×Dic73—161no—
138C2×Dic73C2×D812yes—
139C2×Dic73C2×D812yes—
140D292—22no—
141D292—41no—
142C4×D73—44no—
143C4×D73—162no—
144Dic146—22no—
145Dic146—41no—
146Dic146—162no—
147C8×C73C22⋊Q812yes—
148C8×C73C22×Q811yescharacteristic
149C8×C73C2×#4111yescharacteristic
150#10596831—22no—
151C22×C73—21no—
152C22×C73—21no—
153C22×C73#47412yes—
154C22×C73C22×D811yescharacteristic
155C2×D73—42no—
156C2×D73—81no—
157C2×D73—81no—
158C4×C73—81no—
159C4×C73—82no—
160C4×C73—81no—
161C4×C73#47412yes—
162C4×C73C2×C22⋊Q811yescharacteristic
163C4×C73C2×#13711yescharacteristic
164Dic73—22no—
165Dic73—22no—
166Dic73—42no—
167Dic73—82no—
168#33633—731nomaximal, Sylow
169C2×C73—22no—
170C2×C73C2×#47411yescharacteristic
171C2×C73#341511yescharacteristic
172C2×C73#341711yescharacteristic
173D73—82no—
174#3164—731no—
175#2951—731no—
176#2229—732no—
177#2228—732no—
178#2227—732no—
179#2224—732no—
180#2159D7311yescharacteristic, complemented
181C2×#366—731no—
182#2151—731no—
183#2139—731no—
184#2138—731no—
185C73#3363311yescharacteristic, complemented, Sylow
186C2×#135—731no—
187#544—1461no—
188#520—1461no—
189C2×M32C2×D7311yescharacteristic
190C22×C16—731no—
191#501—1462no—
192#500—732no—
193#499—732no—
194#440—1462no—
195C4×Q16—732no—
196C4×D8—732no—
197#428—1461no—
198#427C2×D7311yescharacteristic
199C2×#107—731no—
200C2×#107C2×D7311yescharacteristic
201#417—1461no—
202#405—731no—
203#367C2×D7312yes—
204#366—732no—
205#366—732no—
206#366C2×D7312yes—
207C4⋊C16—732no—
208C4⋊C16—732no—
209#358—732no—
210#358—731no—
211#358—731no—
212#357—732no—
213#357—731no—
214#357—731no—
215C2×#41—1461no—
216#135—734no—
217#135—1462no—
218C2×Q16—731no—
219C2×SD16—1461no—
220C2×D8—731no—
221C2×M16—731no—
222C22×C8C22×D7311yescharacteristic
223C4⋊Q8—2921no—
224C4⋊Q8—1462no—
225#125—2921no—
226#125—1462no—
227C22⋊Q8—2922no—
228C4×Q8—1462no—
229C4×D4—2922no—
230C4×D4—1462no—
231#117—21no—
232#117—2921no—
233#117—731no—
234C2×C4⋊C4—21no—
235C2×C4⋊C4—2921no—
236C2×C4⋊C4—1461no—
237M32—22no—
238C2×C16—731no—
239C2×C16—731no—
240C2×C16—1462no—
241C2×C16C22×D7312yes—
242#107—21no—
243#107—732no—
244#107—1462no—
245#107—1461no—
246#107C22×D7311yescharacteristic
247#107C22×D7311yescharacteristic
248#107C22×D7312yes—
249#106—21no—
250#106—1462no—
251#103—1462no—
252#102—1462no—
253#97—732no—
254C4×C8—732no—
255#41—2922no—
256#41—1464no—
257C2×Q8—2922no—
258C2×Q8—1461no—
259C2×D4—2921no—
260C2×D4—1461no—
261C22×C4—41no—
262C22×C4—2922no—
263C22×C4—2921no—
264C22×C4—2921no—
265C22×C4D4×D7311yescharacteristic
266Q16—732no—
267Q16—1461no—
268SD16—1464no—
269D8—732no—
270D8—1461no—
271M16—1462no—
272C2×C8—732no—
273C2×C8D4×D7311yescharacteristic
274C2×C8Q8×D7312yes—
275C2×C8#1192402812yes—
276C2×C8C23×D7311yescharacteristic
277C4⋊C4—22no—
278C4⋊C4—21no—
279C4⋊C4—21no—
280C4⋊C4—41no—
281C4⋊C4—2922no—
282C4⋊C4—2922no—
283C4⋊C4—2922no—
284C4⋊C4—2922no—
285C4⋊C4—2921no—
286C4⋊C4—731no—
287C4⋊C4—731no—
288C4⋊C4—1462no—
289C22⋊C4—41no—
290C22⋊C4—2922no—
291C22⋊C4—732no—
292C42—41no—
293C42—2922no—
294C42—732no—
295C16—22no—
296C16—1461no—
297C16—1461no—
298C23—2921no—
299C23D8×D7311yescharacteristic
300Q8—2921no—
301Q8—11682no—
302Q8—1462no—
303D4—2924no—
304D4—2921no—
305D4—1462no—
306C2×C4—41no—
307C2×C4—41no—
308C2×C4—42no—
309C2×C4—161no—
310C2×C4—2921no—
311C2×C4—2924no—
312C2×C4—2922no—
313C2×C4—732no—
314C2×C4—732no—
315C2×C4—11682no—
316C2×C4—11681no—
317C2×C4#1455656912yes—
318C2×C4D8×D7311yescharacteristic
319C2×C4#1455662312yes—
320C2×C4C2×D4×D7311yescharacteristic
321C8—1462no—
322C8#1455656912yes—
323C8C2×Q8×D7311yescharacteristic
324C8C2×#1192402811yescharacteristic
325C22—21no—
326C22—21no—
327C22—2922no—
328C22—5841no—
329C22—5841no—
330C22#1798886612yes—
331C22C2×D8×D7311yescharacteristic
332C4—82no—
333C4—81no—
334C4—81no—
335C4—2922no—
336C4—5842no—
337C4—1462no—
338C4—1462no—
339C4#1798886612yes—
340C4C2×#1455656911yescharacteristic
341C4C2×#1455662311yescharacteristic
342C2—22no—
343C2—5842no—
344C2C2×#1798886611yescharacteristic
345C2#2262307111yescharacteristic
346C2#2262308311yescharacteristic
subgroupG/H
centerC22C2×D8×D73
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2159C2
socleC22×C73C22×D8
prime radicalC2×#14556613C22
hypercenter#2159D73
hypocenterC73#33633
LCSG ▷ C2×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
1. #22620471C2semidirect
3. #22612098C2semidirect
4. #22612087C2semidirect
5. #22612082C2semidirect
6. #22612067C2semidirect
7. C2×#17988412C2semidirect
8. #22611691C2semidirect
11. C73×#2159C2semidirect
43. #14556613Q8semidirect
180. #2159D73semidirect
185. C73#33633semidirect

Maximal subgroups

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

subgroupindex
#226204712
#226188052
#226120982
#226120872
#226120822
#226120672
C2×#179884122
#226116912
#226116442
#226116362
C73×#21592
#3363373