Finite Group Databasebeta

← #29599533

Subgroups — #29599533

Order 15616.

438 conjugacy classes of subgroups; only the 73 normal subgroups are shown.

normal subgroups only

This group has 438 conjugacy classes of subgroups — too many to draw. Only its 73 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#21447469C212yesmaximal, complemented
2#21446347C212yesmaximal, complemented
3#21446038C211yesmaximal, characteristic, complemented
4#21443940C211yesmaximal, characteristic, complemented
5C61×#3199C211yesmaximal, characteristic
6#16840529C2211yescharacteristic
7#16839810—21no—
8#16839767—22no—
9#16839746C2211yescharacteristic
10#16839736C2212yes—
11#16839549—24no—
12#16839527—21no—
13#16839518—22no—
14C61×#535C2212yes—
15C61×#482—22no—
16C61×#469—21no—
17C61×#466—21no—
18C61×#447—22no—
19C61×#408C2211yescharacteristic
20#14519784—42no—
21C2×#11758194—22no—
22C2×#11758194—21no—
23C2×#11758194—21no—
24#14519736—22no—
25#14519712—42no—
26#14519698—44no—
27#14519696—21no—
28#14519694D412yes—
29#14519692—22no—
30#14519692—22no—
31C2×#41×C61C2311yescharacteristic
32C22×D4×C61D412yes—
33C2×D8×C61—21no—
34C2×D8×C61—21no—
35C2×M16×C61D411yescharacteristic
36C22×C8×C61D411yescharacteristic
37#127×C61—22no—
38#124×C61—22no—
39#121×C61—22no—
40#121×C61—22no—
41#120×C61—22no—
42#120×C61—22no—
43#117×C61D412yes—
44#107×C61—21no—
45#106×C61—21no—
46#102×C61—22no—
47#102×C61—22no—
48C22⋊C8×C61—22no—
49C24×C61—22no—
50#41×C61—41no—
51#41×C61—41no—
52C2×Q8×C61#4111yescharacteristic
53C2×D4×C61—22no—
54C2×D4×C61—22no—
55C2×D4×C61—42no—
56C2×D4×C61—42no—
57C2×D4×C61—42no—
58C2×D4×C61—42no—
59C2×D4×C61C2×D411yescharacteristic
60C2×D4×C61#4112yes—
61C22×C4×C61C2×D411yescharacteristic
62C22×C4×C61C2×D412yes—
63D8×C61—81no—
64D8×C61—81no—
65M16×C61—41no—
66C2×C8×C61—21no—
67C2×C8×C61—21no—
68C2×C8×C61—41no—
69C4⋊C4×C61—22no—
70C22⋊C4×C61—22no—
71C22⋊C4×C61—42no—
72C22⋊C4×C61—42no—
73C42×C61—22no—
74C22⋊Dic61—42no—
75C4⋊Dic61—42no—
76C4×Dic61—42no—
77#11758194—22no—
78#11758194—22no—
79#11758194—41no—
80#11758194—42no—
81#11758194—41no—
82C2×#92418—24no—
83C2×#92418—41no—
84C2×#92418—41no—
85C23×C61—42no—
86C23×C61—82no—
87C23×C61—162no—
88C23×C61#12112yes—
89C23×C61#12411yescharacteristic
90Q8×C61—41no—
91D4×C61—41no—
92D4×C61—42no—
93D4×C61—44no—
94D4×C61—42no—
95D4×C61—42no—
96D4×C61—162no—
97D4×C61—162no—
98C2×C4×C61—22no—
99C2×C4×C61—21no—
100C2×C4×C61—21no—
101C2×C4×C61—22no—
102C2×C4×C61—42no—
103C2×C4×C61#12011yescharacteristic
104C2×C4×C61#12111yescharacteristic
105C2×C4×C61#12312yes—
106C2×Dic61—82no—
107C8×C61—41no—
108C8×C61—41no—
109#92418—42no—
110#92418—42no—
111#13436—611nomaximal, Sylow
112C22×C61—22no—
113C22×C61—21no—
114C22×C61—41no—
115C22×C61—42no—
116C22×C61—82no—
117C22×C61—82no—
118C22×C61—82no—
119C22×C61—164no—
120C22×C61#39411yescharacteristic
121C4×C61—21no—
122C4×C61—22no—
123C4×C61—82no—
124C4×C61B(4,𝔽2)11yescharacteristic
125C4×C61#45811yescharacteristic
126Dic61—162no—
127#3199D6111yescharacteristic
128#2122—612no—
129#1910—612no—
130#1858—611no—
131#1391—611no—
132C2×C61—21no—
133C2×C61—42no—
134C2×C61—41no—
135C2×C61—162no—
136C2×C61#204411yescharacteristic
137#568—611no—
138#535C2×D6112yes—
139#482—22no—
140#469—21no—
141#466—21no—
142#447—22no—
143#430—1221no—
144#418—1222no—
145#413—611no—
146#410—612no—
147#408C2×D6111yescharacteristic
148AΣL(1,𝔽16)—1224no—
149#341—1221no—
150H(3,ℤ/4)—1222no—
151C61#1343611yescharacteristic, complemented, Sylow
152C2×#41C22×D6111yescharacteristic
153C22×D4C22⋊D6112yes—
154C2×D8—21no—
155C2×D8—21no—
156#131—2442no—
157C2×M16—1222no—
158C2×M16—1221no—
159C2×M16—1221no—
160C2×M16C22⋊D6111yescharacteristic
161C22×C8C22⋊D6111yescharacteristic
162#127—22no—
163#124—22no—
164#121—22no—
165#121—22no—
166#120—22no—
167#120—22no—
168#117—1222no—
169#117C22⋊D6112yes—
170#108—2442no—
171#107—21no—
172#106—21no—
173#102—22no—
174#102—22no—
175#102—2444no—
176#101—1221no—
177#100—612no—
178#99—1222no—
179#99—1222no—
180C22⋊C8—22no—
181C24—22no—
182#41—41no—
183#41—41no—
184C2×Q8#1175822411yescharacteristic
185C2×D4—22no—
186C2×D4—22no—
187C2×D4—42no—
188C2×D4—42no—
189C2×D4—42no—
190C2×D4—42no—
191C2×D4D4×D6111yescharacteristic
192C2×D4#1175822412yes—
193C22×C4C2×C22⋊D6111yescharacteristic
194C22×C4C2×C22⋊D6112yes—
195D8—81no—
196D8—81no—
197M16—41no—
198M16—1222no—
199M16—1222no—
200M16—2441no—
201M16—2442no—
202M16—2441no—
203C2×C8—21no—
204C2×C8—21no—
205C2×C8—41no—
206C2×C8—1224no—
207C2×C8—2441no—
208C2×C8—2441no—
209C4⋊C4—22no—
210C4⋊C4—2442no—
211C22⋊C4—22no—
212C22⋊C4—42no—
213C22⋊C4—42no—
214C22⋊C4—2442no—
215C42—22no—
216C42—2442no—
217C23—42no—
218C23—82no—
219C23—162no—
220C23#1451974912yes—
221C23#1451976411yescharacteristic
222Q8—41no—
223D4—41no—
224D4—42no—
225D4—44no—
226D4—42no—
227D4—42no—
228D4—162no—
229D4—162no—
230C2×C4—22no—
231C2×C4—21no—
232C2×C4—22no—
233C2×C4—21no—
234C2×C4—42no—
235C2×C4—4882no—
236C2×C4#1451974611yescharacteristic
237C2×C4#1451974811yescharacteristic
238C2×C4#1451975812yes—
239C8—41no—
240C8—41no—
241C8—2442no—
242C8—2442no—
243C22—22no—
244C22—21no—
245C22—41no—
246C22—42no—
247C22—82no—
248C22—82no—
249C22—82no—
250C22—164no—
251C22#1683967611yescharacteristic
252C4—21no—
253C4—22no—
254C4—82no—
255C4—9762no—
256C4#1683994911yescharacteristic
257C4#1683995211yescharacteristic
258C2—21no—
259C2—42no—
260C2—41no—
261C2—162no—
262C2#2144706711yescharacteristic
subgroupG/H
centerC2#21447067
commutator, cosocleC2×#41×C61C23
FrattiniC2×#41C22×D61
Fitting, prime radicalC61×#3199C2
socleC2×C61#2044
hypercenter#3199D61
hypocenterC61#13436
LCSG ▷ C2×#41×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×#41 ◁ #3199

Complements

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

NH ≅ G/Ntype
1. #21447469C2semidirect
2. #21446347C2semidirect
3. #21446038C2semidirect
4. #21443940C2semidirect
151. C61#13436semidirect

Maximal subgroups

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

subgroupindex
#214474692
#214463472
#214460382
#214439402
C61×#31992
#1343661