Finite Group Databasebeta

← #28605270

Subgroups — #28605270

Order 15104.

938 conjugacy classes of subgroups.

The subgroup lattice has 366 nodes — too large to display as a diagram.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1Q8×#11745664C212yesmaximal
2#21425090C214yesmaximal
3#21425086C212yesmaximal
4#21424092C211yesmaximal, characteristic
5#21424092C212yesmaximal
6C4×#14273646C212yesmaximal
7C4×#14273646C211yesmaximal, characteristic
8C2×C59×#557C211yesmaximal, characteristic
9#16837533—24no—
10#16837533—22no—
11Q8×C59⋊C8—22no—
12#16837426C2212yes—
13#16837426C2214yes—
14C2×#14273646C2212yes—
15C2×#14273646C2211yescharacteristic
16C2×#14273646C2211yescharacteristic
17C4×#11745664C2212yes—
18C4×#11745664C2214yes—
19#16837171C2214yes—
20#16837171C2212yes—
21#16837171C2214yes—
22#16837171C2212yes—
23C59×#557—22no—
24C59×#557—24no—
25C59×#557—22no—
26C2×C4⋊Q8×C59C412yes—
27C2×C4⋊Q8×C59C2211yescharacteristic
28C2×#125×C59C414yes—
29C2×#125×C59C2212yes—
30C2×C4×Q8×C59C412yes—
31C2×C4×Q8×C59C2212yes—
32C2×C4×Q8×C59C2211yescharacteristic
33C2×C4×Q8×C59C2211yescharacteristic
34C2×#11745664C2314yes—
35C2×#11745664C2312yes—
36C2×#11745664C2312yes—
37C4⋊(C59⋊C8)—24no—
38C4⋊(C59⋊C8)—22no—
39C4⋊(C59⋊C8)—24no—
40C4⋊(C59⋊C8)—22no—
41#14273646—24no—
42#14273646—22no—
43#14273646Q814yescomplemented
44#14273638—24no—
45#14273638—22no—
46C4×C59⋊C8—24no—
47C4×C59⋊C8—22no—
48C22×Q8×C59C2×C412yes—
49C22×Q8×C59C2×C411yescharacteristic
50C22×Q8×C59C2311yescharacteristic
51C4⋊Q8×C59—24no—
52C4⋊Q8×C59—24no—
53C4⋊Q8×C59—22no—
54C4⋊Q8×C59—22no—
55#125×C59—28no—
56#125×C59—28no—
57#125×C59—24no—
58#125×C59—24no—
59C4×Q8×C59—24no—
60C4×Q8×C59—22no—
61C4×Q8×C59—22no—
62C4×Q8×C59—24no—
63C4×Q8×C59—24no—
64C4×Q8×C59—22no—
65C4×Q8×C59—22no—
66C4×Q8×C59D412yes—
67C4×Q8×C59D414yes—
68C4×Q8×C59D412yes—
69C2×C4⋊C4×C59C2×C414yes—
70C2×C4⋊C4×C59C2×C412yes—
71C2×C4⋊C4×C59C2×C414yes—
72C2×C4⋊C4×C59C2×C412yes—
73C2×C4⋊C4×C59C2×C414yes—
74C2×C4⋊C4×C59C2×C411yescharacteristic
75C2×C4⋊C4×C59C2×C412yes—
76C2×C4⋊C4×C59C2311yescharacteristic
77C2×C4⋊C4×C59C2312yes—
78C2×C42×C59C2×C414yes—
79C2×C42×C59C2×C412yes—
80C2×C42×C59C2312yes—
81C2×C42×C59C2311yescharacteristic
82C2×Q8×C59—22no—
83C2×Q8×C59—22no—
84C2×Q8×C59—22no—
85C2×Q8×C59—22no—
86C2×Q8×C59—42no—
87C2×Q8×C59—44no—
88C2×Q8×C59C22⋊C412yes—
89C2×Q8×C59C22⋊C412yes—
90C2×Q8×C59C2×Q812yes—
91C2×Q8×C59#4112yes—
92C22×C4×C59C22×C411yescharacteristic
93C22×C4×C59C22×C412yes—
94C22×C4×C59C22×C412yes—
95C22×C4×C59C22×C414yes—
96C22×C4×C59C22×C412yes—
97C22×C4×C59C22×C411yescharacteristic
98C22×C4×C59C22×C412yes—
99C22×C4×C59C2411yescharacteristic
100C4⋊C4×C59—24no—
101C4⋊C4×C59—24no—
102C4⋊C4×C59—24no—
103C4⋊C4×C59—28no—
104C4⋊C4×C59—28no—
105C4⋊C4×C59—24no—
106C4⋊C4×C59—22no—
107C4⋊C4×C59—24no—
108C4⋊C4×C59C22⋊C412yes—
109C4⋊C4×C59C22⋊C411yescharacteristic
110C4⋊C4×C59C22⋊C411yescharacteristic
111C4⋊C4×C59C22⋊C414yes—
112C4⋊C4×C59C22⋊C414yes—
113C42×C59—24no—
114C42×C59—24no—
115C42×C59—22no—
116C42×C59—22no—
117C42×C59C2×D414yes—
118C42×C59C2×D414yes—
119C42×C59C2×D412yes—
120C42×C59C2×D412yes—
121#11745664—22no—
122#11745664—24no—
123#11745664—22no—
124#11745664—24no—
125#11745664—22no—
126#11745664C2×Q814yes—
127C2×C59⋊C8—22no—
128C2×C59⋊C8—24no—
129C2×C59⋊C8—22no—
130C23×C59C23×C411yescharacteristic
131Q8×C59—24no—
132Q8×C59—24no—
133Q8×C59—24no—
134Q8×C59—42no—
135Q8×C59—44no—
136Q8×C59—44no—
137C2×C4×C59—22no—
138C2×C4×C59—22no—
139C2×C4×C59—22no—
140C2×C4×C59—22no—
141C2×C4×C59—24no—
142C2×C4×C59—42no—
143C2×C4×C59—41no—
144C2×C4×C59—42no—
145C2×C4×C59—42no—
146C2×C4×C59—44no—
147C2×C4×C59C2×C22⋊C412yes—
148C2×C4×C59C2×C22⋊C412yes—
149C2×C4×C59C2×C22⋊C411yescharacteristic
150C2×C4×C59C2×C22⋊C412yes—
151C2×C4×C59C2×C22⋊C414yes—
152C2×C4×C59C2×C22⋊C411yescharacteristic
153C2×C4×C59C4×Q814yes—
154C2×C4×C59C22×D412yes—
155C2×C4×C59C22×Q811yescharacteristic
156C2×C4×C59C2×#4111yescharacteristic
157C2×C4×C592−1+412yes—
158C59⋊C8—24no—
159C59⋊C8—44no—
160C59⋊C8—42no—
161#22517—591nomaximal, Sylow
162C22×C59—21no—
163C22×C59—21no—
164C22×C59C22×C22⋊C411yescharacteristic
165C22×C59C2×C4×Q811yescharacteristic
166C22×C59#51911yescharacteristic
167C4×C59—24no—
168C4×C59—21no—
169C4×C59—24no—
170C4×C59—22no—
171C4×C59—22no—
172C4×C59—22no—
173C4×C59—44no—
174C4×C59—42no—
175C4×C59#41312yes—
176C4×C59#54812yes—
177C4×C59D4×Q812yes—
178C2×#557D5911yescharacteristic
179Q8×M16—592no—
180#1998—594no—
181#1997—592no—
182#1801—592no—
183#1801—591no—
184C4×#101—592no—
185C4×#101—591no—
186C2×C59—21no—
187C2×C59—21no—
188C2×C59Q8×C22⋊C411yescharacteristic
189C2×C59C2×#41311yescharacteristic
190C2×C59#288911yescharacteristic
191#557—22no—
192#557—24no—
193#557—22no—
194C2×C4⋊Q8Dic5912yes—
195C2×C4⋊Q8C2×D5911yescharacteristic
196C2×#125Dic5914yes—
197C2×#125C2×D5912yes—
198C2×C4×Q8Dic5912yes—
199C2×C4×Q8C2×D5912yes—
200C2×C4×Q8C2×D5911yescharacteristic
201C2×C4×Q8C2×D5911yescharacteristic
202#446—1184no—
203#446—1182no—
204C8×Q8—1182no—
205#423—592no—
206#423—594no—
207C2×#101—591no—
208C2×#101—591no—
209C2×#101—592no—
210C4×M16—592no—
211C4×M16—594no—
212#343—594no—
213#343—592no—
214#343—594no—
215#343—592no—
216C59#2251711yescharacteristic, complemented, Sylow
217C22×Q8C2×Dic5912yes—
218C22×Q8C2×Dic5911yescharacteristic
219C22×Q8C22×D5911yescharacteristic
220C2×M16—592no—
221C2×M16—592no—
222C2×M16—594no—
223C4⋊Q8—24no—
224C4⋊Q8—24no—
225C4⋊Q8—22no—
226C4⋊Q8—22no—
227#125—28no—
228#125—28no—
229#125—24no—
230#125—24no—
231C4×Q8—24no—
232C4×Q8—22no—
233C4×Q8—22no—
234C4×Q8—24no—
235C4×Q8—24no—
236C4×Q8—22no—
237C4×Q8—22no—
238C4×Q8C22⋊D5912yes—
239C4×Q8C22⋊D5914yes—
240C4×Q8C22⋊D5912yes—
241C2×C4⋊C4C2×Dic5914yes—
242C2×C4⋊C4C2×Dic5912yes—
243C2×C4⋊C4C2×Dic5914yes—
244C2×C4⋊C4C2×Dic5912yes—
245C2×C4⋊C4C2×Dic5914yes—
246C2×C4⋊C4C2×Dic5911yescharacteristic
247C2×C4⋊C4C2×Dic5912yes—
248C2×C4⋊C4C22×D5911yescharacteristic
249C2×C4⋊C4C22×D5912yes—
250C2×C42C2×Dic5914yes—
251C2×C42C2×Dic5912yes—
252C2×C42C22×D5912yes—
253C2×C42C22×D5911yescharacteristic
254C4⋊C8—1184no—
255C4⋊C8—1182no—
256C4⋊C8—1184no—
257C4⋊C8—1182no—
258#101—594no—
259#101—1182no—
260#101—1184no—
261#97—1184no—
262#97—1182no—
263C4×C8—1184no—
264C4×C8—1182no—
265C2×Q8—22no—
266C2×Q8—22no—
267C2×Q8—22no—
268C2×Q8—22no—
269C2×Q8—42no—
270C2×Q8—44no—
271C2×Q8C22⋊Dic5912yes—
272C2×Q8C22⋊Dic5912yes—
273C2×Q8Q8×D5912yes—
274C2×Q8#1174568912yes—
275C22×C4C22×Dic5911yescharacteristic
276C22×C4C22×Dic5912yes—
277C22×C4C22×Dic5912yes—
278C22×C4C22×Dic5914yes—
279C22×C4C22×Dic5912yes—
280C22×C4C22×Dic5911yescharacteristic
281C22×C4C22×Dic5912yes—
282C22×C4C23×D5911yescharacteristic
283M16—594no—
284M16—1182no—
285M16—1182no—
286M16—1184no—
287M16—1182no—
288M16—1184no—
289C2×C8—1182no—
290C2×C8—1184no—
291C2×C8—1182no—
292C4⋊C4—24no—
293C4⋊C4—24no—
294C4⋊C4—24no—
295C4⋊C4—28no—
296C4⋊C4—28no—
297C4⋊C4—24no—
298C4⋊C4—22no—
299C4⋊C4—24no—
300C4⋊C4C22⋊Dic5912yes—
301C4⋊C4C22⋊Dic5911yescharacteristic
302C4⋊C4C22⋊Dic5911yescharacteristic
303C4⋊C4C22⋊Dic5914yes—
304C4⋊C4C22⋊Dic5914yes—
305C42—24no—
306C42—24no—
307C42—22no—
308C42—22no—
309C42C2×C22⋊D5914yes—
310C42C2×C22⋊D5912yes—
311C42C2×C22⋊D5914yes—
312C42C2×C22⋊D5912yes—
313C23C23×Dic5911yescharacteristic
314Q8—24no—
315Q8—24no—
316Q8—24no—
317Q8—42no—
318Q8—44no—
319Q8—44no—
320C2×C4—22no—
321C2×C4—22no—
322C2×C4—22no—
323C2×C4—22no—
324C2×C4—24no—
325C2×C4—42no—
326C2×C4—41no—
327C2×C4—42no—
328C2×C4—42no—
329C2×C4—44no—
330C2×C4C2×C22⋊Dic5912yes—
331C2×C4C2×C22⋊Dic5912yes—
332C2×C4C2×C22⋊Dic5911yescharacteristic
333C2×C4C2×C22⋊Dic5912yes—
334C2×C4C2×C22⋊Dic5914yes—
335C2×C4C2×C22⋊Dic5911yescharacteristic
336C2×C4Q8×Dic5914yes—
337C2×C4C22×C22⋊D5912yes—
338C2×C4C2×Q8×D5911yescharacteristic
339C2×C4C2×#1174568911yescharacteristic
340C2×C4#1427377512yes—
341C8—1184no—
342C8—2364no—
343C8—2362no—
344C22—21no—
345C22—21no—
346C22C22×C22⋊Dic5911yescharacteristic
347C22C2×Q8×Dic5911yescharacteristic
348C22#1683788011yescharacteristic
349C4—22no—
350C4—22no—
351C4—22no—
352C4—24no—
353C4—21no—
354C4—24no—
355C4—42no—
356C4—44no—
357C4#1683738612yes—
358C4#1683806312yes—
359C4Q8×C22⋊D5912yes—
360C2—21no—
361C2—21no—
362C2Q8×C22⋊Dic5911yescharacteristic
363C2C2×#1683738611yescharacteristic
364C2#2143257911yescharacteristic
subgroupG/H
centerC22C22×C22⋊Dic59
commutator, prime radicalC23×C59C23×C4
FrattiniC22×C4C23×D59
FittingC2×C59×#557C2
socleC22×C59C22×C22⋊C4
cosocleC22×C4×C59C24
hypercenterC2×#557D59
hypocenterC59#22517
LCSG ▷ C23×C59 ▷ C2×C59 ▷ C59
UCSC1 ◁ C22 ◁ C2×C4×Q8 ◁ C2×#557

Complements

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

NH ≅ G/Ntype
43. #14273646Q8semidirect
216. C59#22517semidirect

Maximal subgroups

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

subgroupindex
Q8×#117456642
#214250902
#214250862
#214240922
C4×#142736462
C2×C59×#5572
#2251759