Finite Group Databasebeta

← #33240689

Subgroups — #33240689

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#22618808C211yesmaximal, characteristic
2#22612092C212yesmaximal, complemented
3#22612085C212yesmaximal, complemented
4#22612080C212yesmaximal, complemented
5#22612065C212yesmaximal, complemented
6#22611720C211yesmaximal, characteristic, complemented
7C2×#17988412C211yesmaximal, characteristic, complemented
8#22611690C211yesmaximal, characteristic, complemented
9#22611641C211yesmaximal, characteristic
10#22611636C211yesmaximal, characteristic
11C73×#3164C211yesmaximal, characteristic, complemented
12C2×#14556613C2211yescharacteristic
13#17989067—21no—
14C2×#14556531C2211yescharacteristic
15C22×#11924003C2211yescharacteristic
16#17988689—22no—
17C4×Dic292C2212yes—
18C4×D584C2212yes—
19C2×#14556525C2211yescharacteristic
20#17988589—21no—
21#17988414C2212yes—
22#17988412C2212yes—
23#17988412C2212yes—
24#17988412C2212yes—
25C4⋊#11924003C2212yes—
26C4⋊#11924003C2212yes—
27#17988390C2211yescharacteristic
28#17988390C2212yes—
29#17988390C2211yescharacteristic
30#17988387C2211yescharacteristic
31#17988387C2212yes—
32#17988387C2211yescharacteristic
33#544×C73—21no—
34#501×C73—22no—
35#500×C73C2212yes—
36#499×C73C2212yes—
37#428×C73—21no—
38#427×C73C2211yescharacteristic
39C2×#107×C73C2211yescharacteristic
40#405×C73C2211yescharacteristic
41C2×#11924028—21no—
42#14556613—22no—
43#14556613Q814yescomplemented
44C2×Dic292C2311yescharacteristic
45C2×#11924017—21no—
46C2×D584C2311yescharacteristic
47C4×Dic146—22no—
48C4×D292—22no—
49#14556551—42no—
50#14556549—41no—
51C2×C4⋊Dic73—21no—
52C4⋊C4×D73—41no—
53#14556531—22no—
54C2×#11924003—22no—
55C2×#11924003C2311yescharacteristic
56C2×#11924003C2311yescharacteristic
57C2×#11924003C2312yes—
58#14556525—21no—
59#14556525C2312yes—
60#14556516—22no—
61#14556513—22no—
62C2×M16×C73D411yescharacteristic
63C22×C8×C73C2311yescharacteristic
64C4⋊Q8×C73—22no—
65C4⋊Q8×C73—41no—
66#125×C73—22no—
67#125×C73—41no—
68C22⋊Q8×C73—42no—
69#117×C73—21no—
70#117×C73D411yescharacteristic
71C2×C4⋊C4×C73—21no—
72#107×C73—21no—
73#107×C73—22no—
74#107×C73C2312yes—
75#107×C73C2311yescharacteristic
76#107×C73C2311yescharacteristic
77#106×C73—21no—
78#106×C73—22no—
79#97×C73D412yes—
80C4×C8×C73C2312yes—
81#11924028—24no—
82#11924028—42no—
83C2×Dic146—21no—
84C2×D292—21no—
85C2×C22⋊D73—41no—
86C22×Dic73—41no—
87C2×C4×D73—41no—
88C2×C4×D73—42no—
89Dic292—21no—
90Dic292C2×Q812yes—
91#11924017—24no—
92D584—21no—
93D584C2×Q812yes—
94C4⋊Dic73—22no—
95C4⋊Dic73—41no—
96#11924007—42no—
97#11924005—42no—
98C4×Dic73—42no—
99#11924003—21no—
100#11924003—21no—
101#11924003—22no—
102C2×Q8×C73—42no—
103C22×C4×C73—41no—
104C22×C4×C73C2×D411yescharacteristic
105M16×C73—22no—
106C2×C8×C73C2×D412yes—
107C2×C8×C73C2×D411yescharacteristic
108C2×C8×C73C2×Q812yes—
109C2×C8×C73#4112yes—
110C2×C8×C73C2411yescharacteristic
111C4⋊C4×C73—21no—
112C4⋊C4×C73—22no—
113C4⋊C4×C73—21no—
114C4⋊C4×C73—41no—
115C4⋊C4×C73—42no—
116C4⋊C4×C73—42no—
117C4⋊C4×C73—42no—
118C4⋊C4×C73D811yescharacteristic
119C4⋊C4×C73D811yescharacteristic
120C22⋊C4×C73—41no—
121C22⋊C4×C73D812yes—
122C42×C73—41no—
123C42×C73C2×D412yes—
124C23×C73C2×D811yescharacteristic
125C22×D73—41no—
126Q8×C73—162no—
127C2×C4×C73—41no—
128C2×C4×C73—41no—
129C2×C4×C73—42no—
130C2×C4×C73—161no—
131C2×C4×C73C22⋊Q812yes—
132C2×C4×C73C2×D812yes—
133C2×C4×C73C2×D812yes—
134C2×C4×C73C2×D811yescharacteristic
135C2×C4×C73#13712yes—
136C2×C4×C73C22×D411yescharacteristic
137C22⋊D73—44no—
138C2×Dic73—41no—
139C2×Dic73—42no—
140C2×Dic73—161no—
141D292—22no—
142D292—41no—
143C4×D73—44no—
144C4×D73—162no—
145Dic146—22no—
146Dic146—41no—
147C8×C73—22no—
148C8×C73C22⋊Q812yes—
149C8×C73C22×Q811yescharacteristic
150C8×C73C2×#4111yescharacteristic
151C22×C73—21no—
152C22×C73—21no—
153C22×C73#47412yes—
154C22×C73C22×D811yescharacteristic
155C2×D73—42no—
156C2×D73—81no—
157C2×D73—81no—
158C4×C73—22no—
159C4×C73—22no—
160C4×C73—82no—
161C4×C73—81no—
162C4×C73—81no—
163C4×C73#47412yes—
164C4×C73C2×C22⋊Q811yescharacteristic
165C4×C73C2×#13711yescharacteristic
166Dic73—42no—
167Dic73—82no—
168#33633—731nomaximal, Sylow
169C2×C73—22no—
170C2×C73C2×#47411yescharacteristic
171C2×C73#341511yescharacteristic
172C2×C73#341711yescharacteristic
173D73—82no—
174#3164D7311yescharacteristic, complemented
175#2951—731no—
176#2229—732no—
177#2228—732no—
178#2227—732no—
179#2224—732no—
180#2159—731no—
181C2×#366—731no—
182#2151—731no—
183#2139—731no—
184#2138—731no—
185C73#3363311yescharacteristic, complemented, Sylow
186C2×#135—731no—
187#544—21no—
188#520—1461no—
189C2×M32—731no—
190C22×C16—731no—
191#501—22no—
192#500C2×D7312yes—
193#499C2×D7312yes—
194#440—1462no—
195C4×Q16—732no—
196C4×D8—732no—
197#428—21no—
198#427C2×D7311yescharacteristic
199C2×#107—731no—
200C2×#107C2×D7311yescharacteristic
201#417—1461no—
202#405C2×D7311yescharacteristic
203#367—732no—
204#366—732no—
205#366—732no—
206#366—732no—
207C4⋊C16—732no—
208C4⋊C16—732no—
209#358—731no—
210#358—731no—
211#358—732no—
212#357—731no—
213#357—731no—
214#357—732no—
215C2×#41—1461no—
216#135—734no—
217#135—1462no—
218C2×Q16—731no—
219C2×SD16—1461no—
220C2×D8—731no—
221C2×M16C22⋊D7311yescharacteristic
222C22×C8C22×D7311yescharacteristic
223C4⋊Q8—22no—
224C4⋊Q8—41no—
225#125—22no—
226#125—41no—
227C22⋊Q8—42no—
228C4×Q8—1462no—
229C4×D4—2922no—
230C4×D4—1462no—
231#117—21no—
232#117—2921no—
233#117C22⋊D7311yescharacteristic
234C2×C4⋊C4—21no—
235C2×C4⋊C4—2921no—
236C2×C4⋊C4—1461no—
237M32—1462no—
238C2×C16—731no—
239C2×C16—731no—
240C2×C16—732no—
241C2×C16—1462no—
242#107—21no—
243#107—22no—
244#107—732no—
245#107—1461no—
246#107C22×D7312yes—
247#107C22×D7311yescharacteristic
248#107C22×D7311yescharacteristic
249#106—21no—
250#106—22no—
251#103—1462no—
252#102—1462no—
253#97C22⋊D7312yes—
254C4×C8C22×D7312yes—
255#41—2922no—
256#41—1464no—
257C2×Q8—42no—
258C2×Q8—1461no—
259C2×D4—2921no—
260C2×D4—1461no—
261C22×C4—41no—
262C22×C4—2921no—
263C22×C4—2921no—
264C22×C4—2922no—
265C22×C4C2×C22⋊D7311yescharacteristic
266Q16—732no—
267Q16—1461no—
268SD16—1464no—
269D8—732no—
270D8—1461no—
271M16—22no—
272C2×C8C2×C22⋊D7312yes—
273C2×C8C2×C22⋊D7311yescharacteristic
274C2×C8Q8×D7312yes—
275C2×C8#1192402912yes—
276C2×C8C23×D7311yescharacteristic
277C4⋊C4—21no—
278C4⋊C4—22no—
279C4⋊C4—21no—
280C4⋊C4—41no—
281C4⋊C4—42no—
282C4⋊C4—42no—
283C4⋊C4—42no—
284C4⋊C4—2922no—
285C4⋊C4—2921no—
286C4⋊C4—1462no—
287C4⋊C4#1192401311yescharacteristic
288C4⋊C4#1192401311yescharacteristic
289C22⋊C4—41no—
290C22⋊C4—2922no—
291C22⋊C4#1192401312yes—
292C42—41no—
293C42—2922no—
294C42C2×C22⋊D7312yes—
295C16—1461no—
296C16—1461no—
297C16—1462no—
298C23—2921no—
299C23C2×#1192401311yescharacteristic
300Q8—162no—
301Q8—2921no—
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—11682no—
314C2×C4—11681no—
315C2×C4#1455656712yes—
316C2×C4C2×#1192401312yes—
317C2×C4C2×#1192401312yes—
318C2×C4C2×#1192401311yescharacteristic
319C2×C4#1455662212yes—
320C2×C4C22×C22⋊D7311yescharacteristic
321C8—22no—
322C8#1455656712yes—
323C8C2×Q8×D7311yescharacteristic
324C8C2×#1192402911yescharacteristic
325C22—21no—
326C22—21no—
327C22—2922no—
328C22—5841no—
329C22—5841no—
330C22#1798886412yes—
331C22C22×#1192401311yescharacteristic
332C4—22no—
333C4—22no—
334C4—82no—
335C4—81no—
336C4—81no—
337C4—2922no—
338C4—5842no—
339C4#1798886412yes—
340C4C2×#1455656711yescharacteristic
341C4C2×#1455662211yescharacteristic
342C2—22no—
343C2—5842no—
344C2C2×#1798886411yescharacteristic
345C2#2262307011yescharacteristic
346C2#2262308211yescharacteristic
subgroupG/H
centerC22C22×#11924013
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#3164C2
socleC22×C73C22×D8
prime radicalC2×#14556613C22
hypercenter#3164D73
hypocenterC73#33633
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C22 ◁ #117 ◁ #405 ◁ #3164

Complements

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

NH ≅ G/Ntype
2. #22612092C2semidirect
3. #22612085C2semidirect
4. #22612080C2semidirect
5. #22612065C2semidirect
6. #22611720C2semidirect
7. C2×#17988412C2semidirect
8. #22611690C2semidirect
11. C73×#3164C2semidirect
43. #14556613Q8semidirect
174. #3164D73semidirect
185. C73#33633semidirect

Maximal subgroups

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

subgroupindex
#226188082
#226120922
#226120852
#226120802
#226120652
#226117202
C2×#179884122
#226116902
#226116412
#226116362
C73×#31642
#3363373