Finite Group Databasebeta

← #29592424

Subgroups — #29592424

Order 15616.

424 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 424 conjugacy classes of subgroups — too many to draw. Only its 59 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#21451192C211yesmaximal, characteristic, complemented
2#21444822C212yesmaximal
3#21443824C212yesmaximal, complemented
4#21443765C211yesmaximal, characteristic, complemented
5C61×#2039C211yesmaximal, characteristic
6C2×C4×Dic122—21no—
7#16840073—22no—
8#16839692—21no—
9#16839692C2212yes—
10#16839667—22no—
11#16839661—22no—
12#16839496—22no—
13#16839488C2212yes—
14C61×#537C2211yescharacteristic
15C2×#104×C61—21no—
16C61×#417—21no—
17C61×#384C2211yescharacteristic
18C61×#343C2211yescharacteristic
19C22×Dic122—21no—
20C4×Dic122—42no—
21C4×Dic122—42no—
22C2×C4⋊Dic61—21no—
23C2×C4⋊Dic61—21no—
24C2×C4⋊Dic61D412yes—
25C2×#11758196—22no—
26C2×C4×Dic61—22no—
27#14519710—42no—
28#14519698—42no—
29C22⋊#92418—22no—
30#14519684—22no—
31#14519684—21no—
32#14519684—22no—
33#14519684D412yes—
34C2×#41×C61—21no—
35C2×M16×C61—21no—
36C22⋊Q8×C61—41no—
37#121×C61—22no—
38#121×C61—41no—
39C2×C4⋊C4×C61—22no—
40C2×C4⋊C4×C61D411yescharacteristic
41C2×C4⋊C4×C61C2311yescharacteristic
42C2×C42×C61D411yescharacteristic
43#104×C61—42no—
44#103×C61—41no—
45#102×C61—41no—
46#41×C61—41no—
47#41×C61—41no—
48#41×C61—41no—
49#41×C61—41no—
50C2×Q8×C61—21no—
51C2×D4×C61—21no—
52C2×D4×C61—22no—
53C2×D4×C61—41no—
54C22×C4×C61—21no—
55C22×C4×C61—21no—
56C22×C4×C61—41no—
57C22×C4×C61C2×D411yescharacteristic
58C22×C4×C61C2×D412yes—
59C22×Dic61—22no—
60C22×Dic61—22no—
61C2×Dic122—24no—
62C2×Dic122—42no—
63C2×Dic122—42no—
64M16×C61—41no—
65M16×C61—41no—
66C2×C8×C61—41no—
67C4⋊C4×C61—22no—
68C4⋊C4×C61—41no—
69C4⋊C4×C61—44no—
70C4⋊C4×C61—41no—
71C22⋊C4×C61—42no—
72C42×C61—21no—
73C42×C61D812yes—
74C4⋊Dic61—42no—
75C4⋊Dic61—42no—
76C4⋊Dic61—81no—
77#11758196—42no—
78#11758196—42no—
79C4×Dic61—44no—
80C2×#92418—42no—
81C23×C61—41no—
82C23×C61#12011yescharacteristic
83Q8×C61—41no—
84Q8×C61—41no—
85D4×C61—41no—
86D4×C61—41no—
87D4×C61—82no—
88D4×C61—81no—
89D4×C61—81no—
90C2×C4×C61—22no—
91C2×C4×C61—22no—
92C2×C4×C61—22no—
93C2×C4×C61—21no—
94C2×C4×C61—21no—
95C2×C4×C61—22no—
96C2×C4×C61—42no—
97C2×C4×C61—42no—
98C2×C4×C61—41no—
99C2×C4×C61—41no—
100C2×C4×C61—81no—
101C2×C4×C61—81no—
102C2×C4×C61C2×D811yescharacteristic
103C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
104C2×Dic61—24no—
105C2×Dic61—42no—
106C2×Dic61—42no—
107C2×Dic61—42no—
108C2×Dic61—42no—
109C2×Dic61—42no—
110Dic122—48no—
111C8×C61—81no—
112#92418—82no—
113#12153—611nomaximal, Sylow
114C22×C61—21no—
115C22×C61—21no—
116C22×C61—81no—
117C22×C61—81no—
118C22×C61—81no—
119C22×C61#44711yescharacteristic
120C22×C61#45411yescharacteristic
121C22×C61B(4,𝔽2)11yescharacteristic
122C4×C61—24no—
123C4×C61—21no—
124C4×C61—21no—
125C4×C61—42no—
126C4×C61—81no—
127C4×C61—81no—
128Dic61—44no—
129Dic61—82no—
130#2582—611no—
131#2039D6111yescharacteristic
132#1610—612no—
133#1346—612no—
134#1325—611no—
135C2×C61—21no—
136C2×C61—21no—
137C2×C61—161no—
138C2×C61#159811yescharacteristic
139C2×C61#219811yescharacteristic
140C2×C61#220511yescharacteristic
141#537C2×D6111yescharacteristic
142C2×C4×Q8—1221no—
143#480—1222no—
144C2×#104—21no—
145#417—21no—
146#398—612no—
147#398—1221no—
148#391—1222no—
149#389—1222no—
150#384C2×D6111yescharacteristic
151#343C2×D6111yescharacteristic
152#327—1222no—
153#324—612no—
154C61#1215311yescharacteristic, complemented, Sylow
155C2×#41—21no—
156C22×Q8—1221no—
157C2×M16—21no—
158C22⋊Q8—41no—
159#121—22no—
160#121—41no—
161C4×Q8—2442no—
162C4×Q8—2442no—
163C2×C4⋊C4—22no—
164C2×C4⋊C4—612no—
165C2×C4⋊C4—1221no—
166C2×C4⋊C4—1221no—
167C2×C4⋊C4—1222no—
168C2×C4⋊C4C22⋊D6111yescharacteristic
169C2×C4⋊C4C22×D6111yescharacteristic
170C2×C42—1222no—
171C2×C42C22⋊D6111yescharacteristic
172#107—2442no—
173#104—42no—
174#103—41no—
175#102—41no—
176#102—2442no—
177C22⋊C8—1222no—
178#95—612no—
179#95—1222no—
180#95—1221no—
181#95—1222no—
182#41—41no—
183#41—41no—
184#41—41no—
185#41—41no—
186C2×Q8—21no—
187C2×Q8—1224no—
188C2×Q8—2442no—
189C2×Q8—2442no—
190C2×D4—21no—
191C2×D4—22no—
192C2×D4—41no—
193C22×C4—21no—
194C22×C4—21no—
195C22×C4—41no—
196C22×C4—1222no—
197C22×C4—1222no—
198C22×C4D4×D6112yes—
199C22×C4C2×C22⋊D6111yescharacteristic
200M16—41no—
201M16—41no—
202C2×C8—41no—
203C2×C8—2442no—
204C4⋊C4—22no—
205C4⋊C4—41no—
206C4⋊C4—41no—
207C4⋊C4—44no—
208C4⋊C4—4881no—
209C4⋊C4—2442no—
210C4⋊C4—2442no—
211C4⋊C4—2442no—
212C4⋊C4—2442no—
213C22⋊C4—42no—
214C42—21no—
215C42—2444no—
216C42#1175819912yes—
217C23—41no—
218C23#1451974511yescharacteristic
219Q8—41no—
220Q8—41no—
221Q8—2448no—
222D4—41no—
223D4—41no—
224D4—82no—
225D4—81no—
226D4—81no—
227C2×C4—22no—
228C2×C4—22no—
229C2×C4—21no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—42no—
234C2×C4—42no—
235C2×C4—41no—
236C2×C4—41no—
237C2×C4—81no—
238C2×C4—81no—
239C2×C4—1224no—
240C2×C4—2442no—
241C2×C4—2442no—
242C2×C4—2442no—
243C2×C4—2442no—
244C2×C4—2442no—
245C2×C4C2×#1175819911yescharacteristic
246C2×C4#1451980211yescharacteristic
247C8—81no—
248C8—4882no—
249C22—21no—
250C22—21no—
251C22—81no—
252C22—81no—
253C22—81no—
254C22#1683989511yescharacteristic
255C22#1683993411yescharacteristic
256C22#1683994811yescharacteristic
257C4—24no—
258C4—21no—
259C4—21no—
260C4—42no—
261C4—81no—
262C4—81no—
263C4—4882no—
264C4—2444no—
265C2—21no—
266C2—21no—
267C2—161no—
268C2#2144475311yescharacteristic
269C2#2144781411yescharacteristic
270C2#2144784811yescharacteristic
subgroupG/H
centerC22#16839948
commutator, cosocleC2×C4⋊C4×C61C23
FrattiniC2×C4⋊C4C22×D61
FittingC61×#2039C2
socleC22×C61B(4,𝔽2)
prime radicalC61×#537C22
hypercenter#2039D61
hypocenterC61#12153
LCSG ▷ C2×C4⋊C4×C61 ▷ C23×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C23 ◁ C2×C4⋊C4 ◁ #2039

Complements

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

NH ≅ G/Ntype
1. #21451192C2semidirect
3. #21443824C2semidirect
4. #21443765C2semidirect
154. C61#12153semidirect

Maximal subgroups

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

subgroupindex
#214511922
#214448222
#214438242
#214437652
C61×#20392
#1215361