Finite Group Databasebeta

← #33240735

Subgroups — #33240735

Order 18688.

514 conjugacy classes of subgroups.

The subgroup lattice has 336 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#22620462C211yesmaximal, characteristic, complemented
2#22618849C211yesmaximal, characteristic
3#22612108C214yesmaximal, complemented
4#22612103C214yesmaximal, complemented
5#22611704C211yesmaximal, characteristic, complemented
6C2×#17988410C211yesmaximal, characteristic, complemented
7#22611668C211yesmaximal, characteristic
8C2×#17988398C211yesmaximal, characteristic
9C73×#2153C211yesmaximal, characteristic, complemented
10C2×#14556613C2211yescharacteristic
11#17989353—21no—
12#17989215—21no—
13#17988996—22no—
14#17988989C2212yescomplemented
15#17988985C2212yescomplemented
16#17988710—22no—
17#17988705C2214yes—
18C2×#14556527C2211yescharacteristic
19#17988629—21no—
20#17988625C2211yescharacteristic
21C2×#14556523C2211yescharacteristic
22#17988605—21no—
23#17988410C2212yescomplemented
24#17988410C2214yescomplemented
25#17988410C2212yescomplemented
26#17988398C2212yes—
27#17988398C2214yes—
28#17988398C2212yes—
29C2×M32×C73C2211yescharacteristic
30C2×M32×C73C2211yescharacteristic
31#405×C73C2211yescharacteristic
32#364×C73C2214yescomplemented
33C2×#11924028—21no—
34#14556613—21no—
35#14556613—21no—
36#14556613Q812yescomplemented
37#14556613C2312yes—
38C2×Dic292Q811yescharacteristic, complemented
39C2×#11924017—21no—
40C2×D584Q811yescharacteristic, complemented
41#14556599—42no—
42#14556597—24no—
43C2×#11924012—21no—
44M16×D73—42no—
45#14556591—22no—
46#14556590—41no—
47#14556583—22no—
48#14556582—41no—
49#14556568—42no—
50#14556550—21no—
51C2×C4⋊Dic73—21no—
52#14556527—21no—
53#14556527C2312yes—
54#14556525—21no—
55#14556525—22no—
56#14556523—21no—
57#14556523—22no—
58#14556523C2312yes—
59#14556523C2311yescharacteristic
60#14556523C2311yescharacteristic
61#14556519—24no—
62C2×M16×C73D411yescharacteristic
63C22×C8×C73C2311yescharacteristic
64#117×C73D411yescharacteristic
65M32×C73—21no—
66M32×C73—21no—
67M32×C73C2312yes—
68M32×C73C2312yes—
69C2×C16×C73—21no—
70C2×C16×C73—21no—
71#97×C73D412yes—
72C4×C8×C73C2312yes—
73#11924028—22no—
74#11924028—22no—
75#11924028—41no—
76#11924028—41no—
77C2×Dic146—21no—
78C2×Dic146—42no—
79C2×D292—21no—
80C2×C22⋊D73—41no—
81C22×Dic73—41no—
82C2×C4×D73—41no—
83Dic292—21no—
84Dic292C2×Q812yes—
85#11924017—22no—
86#11924017—22no—
87D584—21no—
88D584C2×Q812yes—
89#11924012—22no—
90#11924012—41no—
91#11924011—44no—
92C2×#10596831—41no—
93C8×D73—44no—
94C4⋊Dic73—22no—
95C4⋊Dic73—21no—
96C4⋊Dic73—21no—
97C4⋊Dic73—42no—
98C4⋊Dic73—41no—
99#11924007—42no—
100#11924007—42no—
101C22⋊Dic73—41no—
102C4×Dic73—41no—
103C22×C4×C73C2×D411yescharacteristic
104M16×C73SD1612yes—
105M16×C73SD1612yes—
106C2×C8×C73C2×D411yescharacteristic
107C2×C8×C73C2×D412yes—
108C2×C8×C73C2×Q811yescharacteristic
109C2×C8×C73C2×Q811yescharacteristic
110C2×C8×C73#4111yescharacteristic
111C2×C8×C73#4111yescharacteristic
112C2×C8×C73C2411yescharacteristic
113C4⋊C4×C73—21no—
114C22⋊C4×C73—21no—
115C42×C73C2×D412yes—
116C16×C73—22no—
117C16×C73—22no—
118C23×C73AGL(1,ℤ/8)11yescharacteristic
119C22×D73—41no—
120C2×C4×C73—22no—
121C2×C4×C73C22⋊Q811yescharacteristic
122C2×C4×C73C22⋊Q811yescharacteristic
123C2×C4×C73C2×SD1611yescharacteristic
124C2×C4×C73C2×SD1611yescharacteristic
125C2×C4×C73AGL(1,ℤ/8)11yescharacteristic
126C2×C4×C73C22×D411yescharacteristic
127C22⋊D73—42no—
128C22⋊D73—42no—
129C2×Dic73—41no—
130C2×Dic73—41no—
131C2×Dic73—41no—
132C2×Dic73—42no—
133C2×Dic73—161no—
134D292—22no—
135D292—41no—
136C4×D73—42no—
137C4×D73—42no—
138Dic146—22no—
139Dic146—41no—
140Dic146—162no—
141C8×C73C22⋊Q811yescharacteristic
142C8×C73C22⋊Q811yescharacteristic
143C8×C73C2×SD1614yes—
144C8×C73C22×Q811yescharacteristic
145C8×C73C2×#4111yescharacteristic
146#10596831—42no—
147C22×C73—41no—
148C22×C73#47611yescharacteristic
149C22×C73#47611yescharacteristic
150C22×C73C2×AGL(1,ℤ/8)11yescharacteristic
151C2×D73—42no—
152C2×D73—161no—
153C4×C73—42no—
154C4×C73#47611yescharacteristic
155C4×C73#47611yescharacteristic
156C4×C73C2×C22⋊Q811yescharacteristic
157C4×C73C22×SD1611yescharacteristic
158Dic73—42no—
159Dic73—81no—
160Dic73—82no—
161Dic73—81no—
162#33638—731nomaximal, Sylow
163C2×C73—21no—
164C2×C73—21no—
165C2×C73—21no—
166C2×C73C2×#47611yescharacteristic
167D73—82no—
168#3163—731no—
169#2957—731no—
170#2231—734no—
171#2230—734no—
172#2155—731no—
173C2×#365—731no—
174#2153D7311yescharacteristic, complemented
175#2145—731no—
176C2×#360—731no—
177C73#3363811yescharacteristic, complemented, Sylow
178C2×#135—731no—
179#568—1461no—
180#544—1461no—
181C2×M32C2×D7311yescharacteristic
182C2×M32C2×D7311yescharacteristic
183#501—1462no—
184#499—732no—
185#498—732no—
186#444—1462no—
187#443—734no—
188C2×#108—731no—
189#428—1461no—
190#427—731no—
191C2×#106—731no—
192#421—1461no—
193#405C2×D7311yescharacteristic
194#365—734no—
195#365—732no—
196#365—732no—
197#364C2×D7314yescomplemented
198#360—734no—
199#360—732no—
200#360—732no—
201C2×#41—1461no—
202#135—732no—
203#135—732no—
204#135—1461no—
205#135—1461no—
206C2×Q16—731no—
207C2×SD16—1461no—
208C2×D8—731no—
209#131—2922no—
210#131—1464no—
211C2×M16—2922no—
212C2×M16—1461no—
213C2×M16D29211yescharacteristic
214C22×C8C22×D7311yescharacteristic
215C4⋊Q8—2921no—
216C4⋊Q8—1462no—
217#125—2921no—
218#125—1462no—
219C22⋊Q8—2922no—
220#117—1461no—
221#117D29211yescharacteristic
222C2×C4⋊C4—1461no—
223M32—21no—
224M32—21no—
225M32C22×D7312yes—
226M32C22×D7312yes—
227C2×C16—21no—
228C2×C16—21no—
229#108—732no—
230#108—1461no—
231#107—1461no—
232#107—1462no—
233#106—732no—
234#106—731no—
235#106—731no—
236#106—1461no—
237#106—1462no—
238#104—1464no—
239#97D29212yes—
240C4×C8C22×D7312yes—
241#41—2921no—
242#41—2921no—
243#41—1462no—
244#41—1462no—
245C2×Q8—2922no—
246C2×Q8—1461no—
247C2×D4—2921no—
248C2×D4—1461no—
249C22×C4—2921no—
250C22×C4—2921no—
251C22×C4C2×D29211yescharacteristic
252Q16—732no—
253Q16—1461no—
254SD16—1462no—
255SD16—1462no—
256D8—732no—
257D8—1461no—
258M16—2921no—
259M16—2924no—
260M16—1462no—
261M16#1192401712yes—
262M16#1192401712yes—
263C2×C8—2921no—
264C2×C8—2924no—
265C2×C8C2×D29211yescharacteristic
266C2×C8C2×D29212yes—
267C2×C8Q8×D7311yescharacteristic
268C2×C8Q8×D7311yescharacteristic
269C2×C8#1192402711yescharacteristic
270C2×C8#1192402711yescharacteristic
271C2×C8C23×D7311yescharacteristic
272C4⋊C4—21no—
273C4⋊C4—2922no—
274C4⋊C4—2922no—
275C4⋊C4—2922no—
276C4⋊C4—2921no—
277C4⋊C4—1462no—
278C4⋊C4—1461no—
279C4⋊C4—1461no—
280C22⋊C4—21no—
281C22⋊C4—2921no—
282C42—2921no—
283C42C2×D29212yes—
284C16—22no—
285C16—22no—
286C23—2921no—
287C23#1455661911yescharacteristic
288Q8—2921no—
289Q8—11682no—
290Q8—1462no—
291D4—2922no—
292D4—2922no—
293D4—2921no—
294D4—1462no—
295C2×C4—22no—
296C2×C4—2921no—
297C2×C4—2922no—
298C2×C4—2922no—
299C2×C4—2921no—
300C2×C4—2921no—
301C2×C4—2922no—
302C2×C4—11681no—
303C2×C4#1455657111yescharacteristic
304C2×C4#1455657111yescharacteristic
305C2×C4C2×#1192401711yescharacteristic
306C2×C4C2×#1192401711yescharacteristic
307C2×C4#1455661911yescharacteristic
308C2×C4C22×D29211yescharacteristic
309C8—2922no—
310C8#1455657111yescharacteristic
311C8#1455657111yescharacteristic
312C8C2×#1192401714yes—
313C8C2×Q8×D7311yescharacteristic
314C8C2×#1192402711yescharacteristic
315C22—41no—
316C22—2922no—
317C22—11681no—
318C22#1798887811yescharacteristic
319C22#1798887811yescharacteristic
320C22C2×#1455661911yescharacteristic
321C4—42no—
322C4—2922no—
323C4—5842no—
324C4—5841no—
325C4—5841no—
326C4#1798887811yescharacteristic
327C4#1798887811yescharacteristic
328C4C2×#1455657111yescharacteristic
329C4C22×#1192401711yescharacteristic
330C2—21no—
331C2—21no—
332C2—21no—
333C2—5842no—
334C2C2×#1798887811yescharacteristic
subgroupG/H
centerC2C2×#17988878
commutator, cosocleC2×C8×C73C24
FrattiniC2×C8C23×D73
FittingC73×#2153C2
socleC2×C73C2×#476
prime radicalC2×#14556613C22
hypercenter#2153D73
hypocenterC73#33638
LCSG ▷ C2×C8×C73 ▷ C4×C73 ▷ C2×C73 ▷ C73
UCSC1 ◁ C2 ◁ C22×C4 ◁ #405 ◁ #2153

Complements

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

NH ≅ G/Ntype
1. #22620462C2semidirect
3. #22612108C2semidirect
4. #22612103C2semidirect
5. #22611704C2semidirect
6. C2×#17988410C2semidirect
9. C73×#2153C2semidirect
14. #17988989C22semidirect
15. #17988985C22semidirect
23. #17988410C22semidirect
24. #17988410C22semidirect
25. #17988410C22semidirect
32. #364×C73C22semidirect
36. #14556613Q8semidirect
38. C2×Dic292Q8semidirect
40. C2×D584Q8semidirect
174. #2153D73semidirect
177. C73#33638semidirect
197. #364C2×D73semidirect

Maximal subgroups

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

subgroupindex
#226204622
#226188492
#226121082
#226121032
#226117042
C2×#179884102
#226116682
C2×#179883982
C73×#21532
#3363873