Finite Group Databasebeta

← #29596283

Subgroups — #29596283

Order 15616.

566 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 566 conjugacy classes of subgroups — too many to draw. Only its 99 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#21448405C211yesmaximal, characteristic
2#21445743C211yesmaximal, characteristic
3#21443945C214yesmaximal, complemented
4C61×#1784C211yesmaximal, characteristic
5C2×#14519736C411yescharacteristic
6C2×#14519736C411yescharacteristic
7C2×#14519696—21no—
8C2×#14519696C2212yes—
9#16839726—21no—
10#16839669—21no—
11#16839669—21no—
12#16839669C412yes—
13#16839669C2212yes—
14#16839625—21no—
15#16839625—21no—
16C23×Q8×C61C2211yescharacteristic
17C2×#101×C61—21no—
18C2×#101×C61C2212yes—
19C61×#407—21no—
20C2×#11758194—22no—
21#14519736—42no—
22#14519736—42no—
23C2×C4⋊Dic61—21no—
24C2×C4⋊Dic61C2×C412yes—
25C2×C22⋊Dic61—21no—
26C2×C22⋊Dic61—21no—
27C2×C4×Dic61—21no—
28C2×C4×Dic61C2×C412yes—
29#14519696—24no—
30#14519696—42no—
31C22⋊#92418—42no—
32#14519684—22no—
33#14519684—22no—
34#14519684—22no—
35#14519684—22no—
36C22×Q8×C61—24no—
37C22×Q8×C61—42no—
38C22×Q8×C61—42no—
39C22×Q8×C61C2×C411yescharacteristic
40C22×Q8×C61D411yescharacteristic
41C22×Q8×C61D411yescharacteristic
42C22×Q8×C61C2311yescharacteristic
43C23×C4×C61C2×C411yescharacteristic
44C23×C4×C61D411yescharacteristic
45C23×C4×C61D411yescharacteristic
46C2×M16×C61—22no—
47#101×C61—24no—
48#101×C61—42no—
49C22⋊C8×C61—42no—
50C24×C61C22⋊C411yescharacteristic
51C2×Q8×C61—22no—
52C2×Q8×C61—22no—
53C2×Q8×C61—22no—
54C2×Q8×C61—22no—
55C2×Q8×C61—22no—
56C2×Q8×C61—41no—
57C2×Q8×C61—41no—
58C2×Q8×C61—41no—
59C2×Q8×C61—41no—
60C2×Q8×C61—41no—
61C2×Q8×C61—41no—
62C2×Q8×C61—416no—
63C2×Q8×C61#4114yes—
64C22×C4×C61—22no—
65C22×C4×C61—22no—
66C22×C4×C61—22no—
67C22×C4×C61—22no—
68C22×C4×C61—22no—
69C22×C4×C61—22no—
70C22×C4×C61—22no—
71C22×C4×C61—41no—
72C22×C4×C61—41no—
73C22×C4×C61C22⋊C411yescharacteristic
74C22×C4×C61C22⋊C411yescharacteristic
75C22×C4×C61C22⋊C411yescharacteristic
76C22×C4×C61C22×C411yescharacteristic
77C22×C4×C61C2×D411yescharacteristic
78C22×C4×C61C2×D411yescharacteristic
79C22×Dic61—22no—
80C22×Dic61—22no—
81M16×C61—42no—
82M16×C61—42no—
83C2×C8×C61—42no—
84C22⋊Dic61—42no—
85C22⋊Dic61—42no—
86C4⋊Dic61—42no—
87C4⋊Dic61—42no—
88C4×Dic61—42no—
89C4×Dic61—42no—
90#11758194—42no—
91#11758194—42no—
92C2×#92418—42no—
93C23×C61—21no—
94C23×C61—21no—
95C23×C61—42no—
96C23×C61#9912yes—
97C23×C61C2×C22⋊C411yescharacteristic
98Q8×C61—44no—
99Q8×C61—44no—
100Q8×C61—44no—
101Q8×C61—44no—
102C2×C4×C61—28no—
103C2×C4×C61—21no—
104C2×C4×C61—21no—
105C2×C4×C61—21no—
106C2×C4×C61—21no—
107C2×C4×C61—21no—
108C2×C4×C61—21no—
109C2×C4×C61—21no—
110C2×C4×C61—21no—
111C2×C4×C61—21no—
112C2×C4×C61—21no—
113C2×C4×C61—21no—
114C2×C4×C61—21no—
115C2×C4×C61—21no—
116C2×C4×C61—21no—
117C2×C4×C61—44no—
118C2×C4×C61—44no—
119C2×C4×C61#9912yes—
120C2×C4×C61#11712yes—
121C2×C4×C61#12312yes—
122C2×C4×C61#12312yes—
123C2×Dic61—44no—
124C2×Dic61—44no—
125C2×Dic61—42no—
126C2×Dic61—42no—
127C8×C61—82no—
128#92418—82no—
129#12862—611nomaximal, Sylow
130C22×C61—22no—
131C22×C61—22no—
132C22×C61—24no—
133C22×C61—42no—
134C22×C61—42no—
135C22×C61#38111yescharacteristic
136C22×C61C2×#9911yescharacteristic
137C22×C61C2×#9911yescharacteristic
138C4×C61—22no—
139C4×C61—22no—
140C4×C61—22no—
141C4×C61—22no—
142C4×C61—41no—
143C4×C61—41no—
144Dic61—82no—
145Dic61—82no—
146#2310—611no—
147#1784—611no—
148#1784D6111yescharacteristic
149#1394—614no—
150C2×C61—21no—
151C2×C61—21no—
152C2×C61—42no—
153C2×C61#181911yescharacteristic
154C2×C61#213312yes—
155C23×Q8C2×D6111yescharacteristic
156C2×#117—611no—
157C2×#117—611no—
158C2×#101—21no—
159C2×#101—612no—
160C2×#101—1221no—
161C2×#101C2×D6112yes—
162#407—21no—
163#407—1221no—
164#391—612no—
165#391—612no—
166#391—1221no—
167#391—1221no—
168#381—1221no—
169#381—1221no—
170C61#1286211yescharacteristic, complemented, Sylow
171C22×Q8—24no—
172C22×Q8—42no—
173C22×Q8—42no—
174C22×Q8C4×D6111yescharacteristic
175C22×Q8D24411yescharacteristic
176C22×Q8C22⋊D6111yescharacteristic
177C22×Q8C22×D6111yescharacteristic
178C23×C4C4×D6111yescharacteristic
179C23×C4D24411yescharacteristic
180C23×C4C22⋊D6111yescharacteristic
181C2×M16—22no—
182C2×M16—1222no—
183#117—2442no—
184#117—2442no—
185C2×C4⋊C4—612no—
186C2×C4⋊C4—1221no—
187C2×C22⋊C4—1221no—
188C2×C22⋊C4—1221no—
189C2×C42—612no—
190C2×C42—1221no—
191#101—24no—
192#101—42no—
193#101—1224no—
194#101—2442no—
195C22⋊C8—42no—
196C22⋊C8—2442no—
197#95—1222no—
198#95—1222no—
199#95—1222no—
200#95—1222no—
201C24#1175819811yescharacteristic
202C2×Q8—22no—
203C2×Q8—22no—
204C2×Q8—22no—
205C2×Q8—22no—
206C2×Q8—22no—
207C2×Q8—41no—
208C2×Q8—41no—
209C2×Q8—41no—
210C2×Q8—41no—
211C2×Q8—41no—
212C2×Q8—41no—
213C2×Q8—416no—
214C2×Q8#1175822414yes—
215C22×C4—22no—
216C22×C4—22no—
217C22×C4—22no—
218C22×C4—22no—
219C22×C4—22no—
220C22×C4—22no—
221C22×C4—22no—
222C22×C4—41no—
223C22×C4—41no—
224C22×C4—1222no—
225C22×C4—1222no—
226C22×C4#1175819811yescharacteristic
227C22×C4#1175819811yescharacteristic
228C22×C4#1175819811yescharacteristic
229C22×C4C2×C4×D6111yescharacteristic
230C22×C4C2×D24411yescharacteristic
231C22×C4C2×C22⋊D6111yescharacteristic
232M16—42no—
233M16—42no—
234M16—2442no—
235M16—2442no—
236C2×C8—42no—
237C2×C8—2442no—
238C4⋊C4—2442no—
239C4⋊C4—2442no—
240C22⋊C4—2442no—
241C22⋊C4—2442no—
242C42—2442no—
243C42—2442no—
244C23—21no—
245C23—21no—
246C23—42no—
247C23#1451969112yes—
248C23C2×#1175819811yescharacteristic
249Q8—44no—
250Q8—44no—
251Q8—44no—
252Q8—44no—
253C2×C4—28no—
254C2×C4—21no—
255C2×C4—21no—
256C2×C4—21no—
257C2×C4—21no—
258C2×C4—21no—
259C2×C4—21no—
260C2×C4—21no—
261C2×C4—21no—
262C2×C4—21no—
263C2×C4—21no—
264C2×C4—21no—
265C2×C4—21no—
266C2×C4—21no—
267C2×C4—21no—
268C2×C4—44no—
269C2×C4—44no—
270C2×C4—2444no—
271C2×C4—2444no—
272C2×C4—2442no—
273C2×C4—2442no—
274C2×C4#1451969112yes—
275C2×C4#1451973312yes—
276C2×C4#1451975812yes—
277C2×C4#1451976112yes—
278C8—82no—
279C8—4882no—
280C22—22no—
281C22—22no—
282C22—24no—
283C22—42no—
284C22—42no—
285C22#1683962311yescharacteristic
286C22C2×#1451969111yescharacteristic
287C22C2×#1451969111yescharacteristic
288C4—22no—
289C4—22no—
290C4—22no—
291C4—22no—
292C4—41no—
293C4—41no—
294C4—4882no—
295C4—4882no—
296C2—21no—
297C2—21no—
298C2—42no—
299C2#2144589111yescharacteristic
300C2#2144750612yes—
subgroupG/H
centerC22C2×#14519691
commutatorC22×C4×C61C22×C4
FrattiniC22×Q8C22×D61
FittingC61×#1784C2
socleC22×C61C2×#99
prime radicalC24×C61C22⋊C4
cosocleC22×Q8×C61C23
hypercenter#1784D61
hypocenterC61#12862
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C24 ◁ C23×Q8 ◁ #1784

Complements

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

NH ≅ G/Ntype
3. #21443945C2semidirect
170. C61#12862semidirect

Maximal subgroups

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

subgroupindex
#214484052
#214457432
#214439452
C61×#17842
#1286261