Finite Group Databasebeta

← #29689181

Subgroups — #29689181

Order 15616.

1270 conjugacy classes of subgroups; only the 231 normal subgroups are shown.

normal subgroups only

This group has 1270 conjugacy classes of subgroups — too many to draw. Only its 231 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#21459463C211yesmaximal, characteristic
2#21454295C212yesmaximal, complemented
3#21446368C214yesmaximal, complemented
4#21446234C214yesmaximal, complemented
5#21445928C212yesmaximal, complemented
6#21445747C211yesmaximal, characteristic, complemented
7C61×#1785C211yesmaximal, characteristic, complemented
8C2×#14519822C2211yescharacteristic
9#16840528C2214yes—
10#16840411—24no—
11#16840400—24no—
12#16840336C2212yes—
13#16840321C414yes—
14#16840273C2212yes—
15C2×#14519760C414yes—
16#16840237C2211yescharacteristic
17#16839744C2214yes—
18#16839743—24no—
19C2×#14519696C2212yescomplemented
20C2×#14519694C2212yes—
21#100×D61—24no—
22#16839726C2212yescomplemented
23#16839534C2214yescomplemented
24#16839533C2214yescomplemented
25C22×#41×C61C2211yescharacteristic
26C2×#101×C61C2212yescomplemented
27C2×#100×C61C2212yes—
28C61×#407C2212yescomplemented
29#14519822—42no—
30#14519822—42no—
31C2×#11758222C2312yes—
32C2×#11758224D414yes—
33C22×C22⋊D61C2×C414yes—
34C2×D4×D61D414yes—
35#14519785—44no—
36#14519783—44no—
37C2×#11758194C2314yes—
38M16×D61—28no—
39#14519775—22no—
40#14519766—22no—
41#14519760—28no—
42#14519760C2×C414yes—
43#14519760C2×C418yes—
44#14519758—24no—
45#14519756—22no—
46#14519753—24no—
47#14519751—22no—
48#14519749—28no—
49#14519746C2×C414yes—
50#14519745—24no—
51D4×Dic61—24no—
52C4×C22⋊D61—24no—
53#14519736C2312yes—
54C2×#11758196—24no—
55C2×C22⋊Dic61C2×C414yes—
56C2×#11758198—24no—
57#14519696—42no—
58#14519695—24no—
59#14519694—24no—
60#14519693—24no—
61C22⋊#92418—42no—
62C2×#41×C61—24no—
63C2×#41×C61—42no—
64C2×#41×C61—42no—
65C22×Q8×C61C2311yescharacteristic
66C22×D4×C61C2×C412yes—
67C22×D4×C61C2311yescharacteristic
68C23×C4×C61C2×C412yes—
69C23×C4×C61C2311yescharacteristic
70C2×M16×C61C2314yes—
71#101×C61—42no—
72#100×C61—24no—
73C22⋊C8×C61—42no—
74C24×C61C22×C411yescharacteristic
75C24×C61C22×C412yes—
76C23×D61C22⋊C414yes—
77#41×C61—416no—
78C2×Q8×C61—22no—
79C2×Q8×C61—22no—
80C2×Q8×C61—22no—
81C2×D4×C61—24no—
82C2×D4×C61—22no—
83C2×D4×C61—22no—
84C2×D4×C61—42no—
85C2×D4×C61—42no—
86C2×D4×C61C2×D414yes—
87C22×C4×C61—24no—
88C22×C4×C61—22no—
89C22×C4×C61—22no—
90C22×C4×C61—22no—
91C22×C4×C61—42no—
92C22×C4×C61—42no—
93C22×C4×C61C22×C411yescharacteristic
94C22×C4×C61C22×C412yes—
95C22×C4×C61C2411yescharacteristic
96C2×C22⋊D61—24no—
97C2×C22⋊D61—44no—
98C2×C22⋊D61—44no—
99C2×C22⋊D61C22⋊C418yes—
100C2×C22⋊D61C22×C414yes—
101C22×Dic61—24no—
102C22×Dic61C22⋊C414yes—
103#11758224—48no—
104D4×D61—48no—
105#11758222—24no—
106#11758222—24no—
107C2×D244C2×D412yes—
108C2×C4×D61C2×D414yes—
109C2×Dic122C2×D412yes—
110M16×C61—24no—
111M16×C61—24no—
112C2×C8×C61—24no—
113C22⋊Dic61—24no—
114C22⋊Dic61—24no—
115C22⋊Dic61C22×C414yes—
116#11758198—24no—
117#11758198—28no—
118C4⋊Dic61—22no—
119#11758196—24no—
120#11758196—28no—
121C4×Dic61—22no—
122#11758194—24no—
123#11758194—24no—
124C2×#92418—24no—
125#11758189—48no—
126C8×D61—48no—
127C23×C61—22no—
128C23×C61—22no—
129C23×C61—21no—
130C23×C61—21no—
131C23×C61—21no—
132C23×C61—81no—
133C23×C61—82no—
134C23×C61C2×C22⋊C414yes—
135C23×C61C23×C411yescharacteristic
136C22×D61—44no—
137C22×D61—84no—
138C22×D61C2×C22⋊C414yes—
139Q8×C61—44no—
140D4×C61—28no—
141D4×C61—48no—
142C2×C4×C61—21no—
143C2×C4×C61—22no—
144C2×C4×C61—21no—
145C2×C4×C61—21no—
146C2×C4×C61—22no—
147C2×C4×C61—22no—
148C2×C4×C61—48no—
149C2×C4×C61—44no—
150C2×C4×C61C22×D412yes—
151C2×C4×C61H(5,𝔽2)12yes—
152C2×C4×C61H(5,𝔽2)12yes—
153C22⋊D61—24no—
154C22⋊D61—24no—
155C22⋊D61—48no—
156C22⋊D61—84no—
157C2×Dic61—24no—
158C2×Dic61—44no—
159C2×Dic61—84no—
160C2×Dic61C2×C22⋊C414yes—
161D244—24no—
162C4×D61—28no—
163Dic122—24no—
164C8×C61—44no—
165#92418—44no—
166#21779—611nomaximal, Sylow
167C22×C61—24no—
168C22×C61—41no—
169C22×C61—44no—
170C22×C61—42no—
171C22×C61—42no—
172C22×C61—42no—
173C22×C61—41no—
174C22×C61—82no—
175C22×C61—82no—
176C22×C61C22×C22⋊C411yescharacteristic
177C22×C61#51811yescharacteristic
178C22×C61#51811yescharacteristic
179C2×D61—24no—
180C2×D61—44no—
181C2×D61—164no—
182C4×C61—41no—
183C4×C61—42no—
184C4×C61#53414yes—
185Dic61—24no—
186Dic61—44no—
187#3485—611no—
188#2888—612no—
189#1915—614no—
190#1891—614no—
191#1829—612no—
192#1785—611no—
193#1785D6111yescharacteristic, complemented
194C2×C61—21no—
195C2×C61—21no—
196C2×C61—21no—
197C2×C61—44no—
198C2×C61—81no—
199C2×C61#230811yescharacteristic
200D61—44no—
201C2×H(5,𝔽2)—611no—
202C22×#41C2×D6111yescharacteristic
203#568—614no—
204#547—1224no—
205#546—1224no—
206#536—612no—
207#534—614no—
208#525—612no—
209C2×#123—614no—
210#518—611no—
211#413—614no—
212#413—1224no—
213C2×#101—612no—
214C2×#101C2×D6112yescomplemented
215C2×#100—612no—
216C2×#100—1224no—
217C2×#100C2×D6112yes—
218#407—612no—
219#407C2×D6112yescomplemented
220#344—614no—
221#344—614no—
222C61#2177911yescharacteristic, complemented, Sylow
223H(5,𝔽2)—2442no—
224H(5,𝔽2)—2442no—
225C2×#41—24no—
226C2×#41—42no—
227C2×#41—42no—
228C2×#41—614no—
229C2×#41—612no—
230C22×Q8C22×D6111yescharacteristic
231C22×D4—614no—
232C22×D4—614no—
233C22×D4C4×D6112yes—
234C22×D4C22×D6111yescharacteristic
235C23×C4C4×D6112yes—
236C23×C4C22×D6111yescharacteristic
237#131—2444no—
238#131—2444no—
239C2×M16—614no—
240C2×M16—1228no—
241C2×M16C22×D6114yes—
242C4⋊Q8—1222no—
243#124—1222no—
244#123—614no—
245#123—618no—
246#123—1228no—
247#123—1224no—
248C22⋊Q8—1222no—
249C22⋊Q8—1224no—
250#121—1222no—
251#121—1228no—
252#120—614no—
253#120—1224no—
254C4×D4—1224no—
255C4×D4—1224no—
256#117—612no—
257C2×C4⋊C4—1224no—
258C2×C22⋊C4—614no—
259C2×C22⋊C4—1224no—
260#101—42no—
261#101—1224no—
262#101—2442no—
263#100—24no—
264#100—1224no—
265#100—1224no—
266C22⋊C8—42no—
267C22⋊C8—2442no—
268C24—614no—
269C24C2×C4×D6111yescharacteristic
270C24C2×C4×D6112yes—
271#41—416no—
272#41—1224no—
273#41—1224no—
274#41—2448no—
275C2×Q8—22no—
276C2×Q8—22no—
277C2×Q8—22no—
278C2×Q8—612no—
279C2×D4—24no—
280C2×D4—22no—
281C2×D4—22no—
282C2×D4—42no—
283C2×D4—42no—
284C2×D4—614no—
285C2×D4—618no—
286C2×D4—612no—
287C2×D4—1224no—
288C2×D4—2448no—
289C2×D4—2444no—
290C2×D4—2444no—
291C2×D4D4×D6114yes—
292C22×C4—24no—
293C22×C4—22no—
294C22×C4—22no—
295C22×C4—22no—
296C22×C4—42no—
297C22×C4—42no—
298C22×C4—614no—
299C22×C4—614no—
300C22×C4—1224no—
301C22×C4C2×C4×D6111yescharacteristic
302C22×C4C2×C4×D6112yes—
303C22×C4C23×D6111yescharacteristic
304M16—24no—
305M16—24no—
306M16—1224no—
307M16—1224no—
308M16—2448no—
309C2×C8—24no—
310C2×C8—1224no—
311C2×C8—2448no—
312C4⋊C4—1224no—
313C4⋊C4—1228no—
314C4⋊C4—1222no—
315C22⋊C4—614no—
316C22⋊C4—1224no—
317C22⋊C4—1228no—
318C22⋊C4—1224no—
319C22⋊C4—1224no—
320C42—1222no—
321C23—22no—
322C23—22no—
323C23—21no—
324C23—21no—
325C23—21no—
326C23—81no—
327C23—82no—
328C23—614no—
329C23—4884no—
330C23—2444no—
331C23C22⋊C4×D6114yes—
332C23C22×C4×D6111yescharacteristic
333Q8—44no—
334Q8—1224no—
335D4—28no—
336D4—48no—
337D4—1224no—
338D4—1224no—
339D4—1224no—
340D4—4884no—
341D4—2448no—
342C2×C4—21no—
343C2×C4—22no—
344C2×C4—21no—
345C2×C4—21no—
346C2×C4—22no—
347C2×C4—22no—
348C2×C4—48no—
349C2×C4—44no—
350C2×C4—614no—
351C2×C4—1228no—
352C2×C4—1224no—
353C2×C4—4884no—
354C2×C4—2444no—
355C2×C4C2×D4×D6112yes—
356C2×C4#1451982212yes—
357C2×C4#1451982212yes—
358C8—44no—
359C8—2444no—
360C22—24no—
361C22—41no—
362C22—44no—
363C22—42no—
364C22—42no—
365C22—42no—
366C22—41no—
367C22—82no—
368C22—82no—
369C22—1224no—
370C22—9764no—
371C22—2444no—
372C22C2×C22⋊C4×D6111yescharacteristic
373C22#1684023511yescharacteristic
374C22#1684023511yescharacteristic
375C4—41no—
376C4—42no—
377C4—1224no—
378C4—2444no—
379C4#1684032214yes—
380C2—21no—
381C2—21no—
382C2—21no—
383C2—44no—
384C2—81no—
385C2—2444no—
386C2#2144839511yescharacteristic
subgroupG/H
centerC2#21448395
commutatorC23×C61C23×C4
FrattiniC22×C4C23×D61
FittingC61×#1785C2
socleC2×C61#2308
prime radical#21459463C2
cosocleC22×C4×C61C24
hypercenter#1785D61
hypocenterC61#21779
LCSG ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22×C4 ◁ #1785

Complements

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

NH ≅ G/Ntype
2. #21454295C2semidirect
3. #21446368C2semidirect
4. #21446234C2semidirect
5. #21445928C2semidirect
6. #21445747C2semidirect
7. C61×#1785C2semidirect
19. C2×#14519696C22semidirect
22. #16839726C22semidirect
23. #16839534C22semidirect
24. #16839533C22semidirect
26. C2×#101×C61C22semidirect
28. C61×#407C22semidirect
193. #1785D61semidirect
214. C2×#101C2×D61semidirect
219. #407C2×D61semidirect
222. C61#21779semidirect

Maximal subgroups

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

subgroupindex
#214594632
#214542952
#214463682
#214462342
#214459282
#214457472
C61×#17852
#2177961