Finite Group Databasebeta

← #29599405

Subgroups — #29599405

Order 15616.

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

normal subgroups only

This group has 436 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
1C2×#16840142C211yesmaximal, characteristic
2C2×#16839577C212yesmaximal
3#21447538C212yesmaximal
4#21443901C211yesmaximal, characteristic
5C61×#1945C211yesmaximal, characteristic
6C22×Dic244—21no—
7C22×Dic244C2212yes—
8C2×#14519777—21no—
9C2×#14519717C2212yes—
10#16840142—24no—
11#16839577—24no—
12#16839565—24no—
13C2×#108×C61C2211yescharacteristic
14C61×#427C2211yescharacteristic
15C61×#424—21no—
16C2×C4×C8×C61C2211yescharacteristic
17C61×#339—21no—
18C22×Dic122—22no—
19C2×Dic244—22no—
20C2×Dic244—22no—
21C2×Dic244—22no—
22C2×Dic244—22no—
23C2×Dic244—24no—
24C2×Dic244—22no—
25C2×Dic244—42no—
26C2×Dic244—42no—
27C2×Dic244D414yes—
28#14519777—24no—
29#14519777—42no—
30C2×C4⋊Dic61—42no—
31#14519717—22no—
32#14519717—22no—
33C2×#11758185—22no—
34C2×M16×C61—21no—
35C22×C8×C61D411yescharacteristic
36C22×C8×C61C2311yescharacteristic
37#117×C61—21no—
38C2×C42×C61D411yescharacteristic
39#108×C61—22no—
40#107×C61D412yes—
41#106×C61—21no—
42C4⋊C8×C61—42no—
43C4×C8×C61—22no—
44C22×C4×C61—21no—
45C22×C4×C61C2×D411yescharacteristic
46C22×Dic61—42no—
47C2×Dic122—24no—
48C2×Dic122—42no—
49C2×Dic122—42no—
50C2×Dic122—42no—
51C2×Dic122—42no—
52C2×Dic122—42no—
53M16×C61—41no—
54M16×C61—41no—
55C2×C8×C61—21no—
56C2×C8×C61—21no—
57C2×C8×C61—41no—
58C2×C8×C61D812yes—
59C2×C8×C61C2×D411yescharacteristic
60C2×C8×C61C2×D412yes—
61C2×C8×C61#4112yes—
62C2×C8×C61#4111yescharacteristic
63C4⋊C4×C61—22no—
64C22⋊C4×C61—41no—
65C42×C61—21no—
66C42×C61—41no—
67C42×C61SD1612yes—
68C4⋊Dic61—48no—
69Dic244—24no—
70Dic244—24no—
71Dic244—48no—
72Dic244—42no—
73Dic244—42no—
74#11758185—42no—
75C23×C61AGL(1,ℤ/8)11yescharacteristic
76C2×C4×C61—24no—
77C2×C4×C61—22no—
78C2×C4×C61—42no—
79C2×C4×C61#12011yescharacteristic
80C2×C4×C61#12111yescharacteristic
81C2×C4×C61#12411yescharacteristic
82C2×C4×C61C2×D811yescharacteristic
83C2×C4×C61C2×SD1611yescharacteristic
84C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
85C2×Dic61—44no—
86C2×Dic61—82no—
87C2×Dic61—82no—
88Dic122—44no—
89Dic122—44no—
90Dic122—82no—
91Dic122—82no—
92C8×C61—22no—
93C8×C61—81no—
94C8×C61#12112yes—
95C8×C61#12312yes—
96#13415—611nomaximal, Sylow
97C22×C61—22no—
98C22×C61#45011yescharacteristic
99C22×C61#45911yescharacteristic
100C22×C61#49011yescharacteristic
101C4×C61—24no—
102C4×C61—82no—
103C4×C61#39411yescharacteristic
104C4×C61#44711yescharacteristic
105C4×C61#46011yescharacteristic
106C4×C61#48611yescharacteristic
107Dic61—84no—
108C2×#494—611no—
109C2×#362—612no—
110#2141—612no—
111#1945D6111yescharacteristic
112#1376—611no—
113C2×C61—21no—
114C2×C61—21no—
115C2×C61#204111yescharacteristic
116C2×C61#221412yes—
117C22×Q16—612no—
118C22×Q16—1221no—
119C2×C4⋊Q8—1221no—
120C2×M32—612no—
121#494—1224no—
122C2×#108C2×D6111yescharacteristic
123#427C2×D6111yescharacteristic
124#424—21no—
125C2×C4×C8C2×D6111yescharacteristic
126#362—1224no—
127#358—1224no—
128#339—21no—
129C61#1341511yescharacteristic, complemented, Sylow
130C22×Q8—1222no—
131C2×Q16—614no—
132C2×Q16—1224no—
133C2×Q16—1222no—
134C2×Q16—1222no—
135C2×Q16—1222no—
136C2×Q16—1222no—
137C2×Q16—1222no—
138C2×Q16—2442no—
139C2×Q16—2442no—
140C2×M16—21no—
141C22×C8C22⋊D6111yescharacteristic
142C22×C8C22×D6111yescharacteristic
143C4⋊Q8—1224no—
144C4⋊Q8—2442no—
145#117—21no—
146C2×C4⋊C4—2442no—
147C2×C42C22⋊D6111yescharacteristic
148M32—1222no—
149M32—1222no—
150C2×C16—1222no—
151#108—22no—
152#107D24412yes—
153#106—21no—
154C4⋊C8—42no—
155C4×C8—22no—
156C2×Q8—1224no—
157C2×Q8—2442no—
158C2×Q8—2442no—
159C2×Q8—2442no—
160C2×Q8—2442no—
161C2×Q8—2442no—
162C22×C4—21no—
163C22×C4—2442no—
164C22×C4C2×C22⋊D6111yescharacteristic
165Q16—1224no—
166Q16—1224no—
167Q16—2448no—
168Q16—2442no—
169Q16—2442no—
170M16—41no—
171M16—41no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—41no—
175C2×C8#1175819912yes—
176C2×C8C2×D24411yescharacteristic
177C2×C8#1175822211yescharacteristic
178C2×C8D4×D6112yes—
179C2×C8#1175822612yes—
180C4⋊C4—22no—
181C4⋊C4—2448no—
182C22⋊C4—41no—
183C42—21no—
184C42—41no—
185C42#1175820112yes—
186C16—2442no—
187C23#1451980211yescharacteristic
188Q8—4882no—
189Q8—4882no—
190Q8—2444no—
191Q8—2444no—
192C2×C4—24no—
193C2×C4—22no—
194C2×C4—42no—
195C2×C4—4882no—
196C2×C4—4882no—
197C2×C4—2444no—
198C2×C4#1451974511yescharacteristic
199C2×C4#1451975111yescharacteristic
200C2×C4#1451976611yescharacteristic
201C2×C4C2×#1175819911yescharacteristic
202C2×C4C2×#1175820111yescharacteristic
203C2×C4#1451980211yescharacteristic
204C8—22no—
205C8—81no—
206C8#1451975212yes—
207C8#1451976212yes—
208C22—22no—
209C22#1683991211yescharacteristic
210C22#1683995711yescharacteristic
211C22#1684012811yescharacteristic
212C4—24no—
213C4—82no—
214C4—4884no—
215C4#1683967911yescharacteristic
216C4#1683989511yescharacteristic
217C4#1683996211yescharacteristic
218C4#1684011011yescharacteristic
219C2—21no—
220C2—21no—
221C2#2144705311yescharacteristic
222C2#2144790012yes—
subgroupG/H
centerC22#16839957
commutator, cosocleC22×C8×C61C23
FrattiniC22×C8C22×D61
FittingC61×#1945C2
socleC22×C61#459
prime radicalC23×C61AGL(1,ℤ/8)
hypercenter#1945D61
hypocenterC61#13415
LCSG ▷ C22×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×C4×C8 ◁ #1945

Complements

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

NH ≅ G/Ntype
129. C61#13415semidirect

Maximal subgroups

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

subgroupindex
C2×#168401422
C2×#168395772
#214475382
#214439012
C61×#19452
#1341561