Finite Group Databasebeta

← #29577433

Subgroups — #29577433

Order 15616.

450 conjugacy classes of subgroups; only the 119 normal subgroups are shown.

normal subgroups only

This group has 450 conjugacy classes of subgroups — too many to draw. Only its 119 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#21451031C211yesmaximal, characteristic
2#21445815C211yesmaximal, characteristic
3#21444503C212yesmaximal
4#21443721C212yesmaximal
5C61×#2983C211yesmaximal, characteristic
6C2×#14519777C2211yescharacteristic
7#16839786C2212yes—
8C2×#14519696—21no—
9#16839701C414yes—
10#16839669—21no—
11#16839668—22no—
12#16839627—21no—
13C4×C4⋊Dic61C2211yescharacteristic
14#16839531—21no—
15#16839509—24no—
16C2×C4×Q8×C61C2211yescharacteristic
17C61×#446—24no—
18C61×#432—22no—
19C61×#431C2212yes—
20C22×Dic122—21no—
21C2×#11758194—22no—
22#14519777—22no—
23#14519777—22no—
24C2×C4⋊Dic61—21no—
25C2×C4⋊Dic61—21no—
26C2×C4⋊Dic61C2×C412yes—
27C2×C4×Dic61—21no—
28C4⋊#92418—22no—
29C4⋊#92418—22no—
30#14519696—42no—
31#14519684—24no—
32#14519684—24no—
33#14519684C2×C412yes—
34C22×Q8×C61D411yescharacteristic
35C4×Q8×C61—22no—
36C4×Q8×C61—22no—
37C2×C4⋊C4×C61C2×C412yes—
38C2×C4⋊C4×C61D411yescharacteristic
39C2×C42×C61D412yes—
40C2×C42×C61C2311yescharacteristic
41C4⋊C8×C61—24no—
42C4⋊C8×C61D414yes—
43C22⋊C8×C61—22no—
44#97×C61—24no—
45C4×C8×C61—22no—
46C2×Q8×C61—22no—
47C2×Q8×C61—21no—
48C2×Q8×C61—21no—
49C2×Q8×C61—22no—
50C22×C4×C61C22⋊C412yes—
51C22×C4×C61C22⋊C412yes—
52C22×C4×C61C22×C411yescharacteristic
53C22×C4×C61C2×D411yescharacteristic
54C22×C4×C61C2×D411yescharacteristic
55C22×Dic61—22no—
56C22×Dic61—22no—
57C2×Dic122—44no—
58C2×Dic122—41no—
59C2×Dic122—41no—
60C2×C8×C61—24no—
61C4⋊C4×C61—24no—
62C4⋊C4×C61—22no—
63C42×C61—22no—
64C42×C61—22no—
65C42×C61C2×D412yes—
66C42×C61#4112yes—
67C4⋊Dic61—24no—
68C4⋊Dic61—42no—
69C4⋊Dic61—42no—
70C4×Dic61—41no—
71C4×Dic61—41no—
72#11758194—42no—
73#11758194—42no—
74C2×#92418—42no—
75C23×C61C2×C22⋊C411yescharacteristic
76Q8×C61—42no—
77Q8×C61—42no—
78C2×C4×C61—22no—
79C2×C4×C61—22no—
80C2×C4×C61—22no—
81C2×C4×C61—22no—
82C2×C4×C61—22no—
83C2×C4×C61—41no—
84C2×C4×C61—42no—
85C2×C4×C61C4×D412yes—
86C2×C4×C61#12011yescharacteristic
87C2×C4×C61#12112yes—
88C2×C4×C61#12311yescharacteristic
89C2×C4×C61AGL(1,ℤ/8)11yescharacteristic
90C2×C4×C61#13712yes—
91C2×C4×C61#13711yescharacteristic
92C2×Dic61—42no—
93C2×Dic61—44no—
94C2×Dic61—42no—
95C2×Dic61—42no—
96C2×Dic61—42no—
97Dic122—82no—
98Dic122—82no—
99C8×C61—44no—
100#92418—82no—
101#9404—611nomaximal, Sylow
102C22×C61—21no—
103C22×C61—21no—
104C22×C61#38611yescharacteristic
105C22×C61#41711yescharacteristic
106C22×C61#41911yescharacteristic
107C4×C61—21no—
108C4×C61—22no—
109C4×C61—42no—
110C4×C61—42no—
111C4×C61#45412yes—
112C4×C61#46812yes—
113C4×C61#47012yes—
114Dic61—82no—
115Dic61—82no—
116#2983D6111yescharacteristic
117#2568—611no—
118#1801—611no—
119#1549—612no—
120#1303—612no—
121C2×C61—21no—
122C2×C61—21no—
123C2×C61#188811yescharacteristic
124C2×C61#189411yescharacteristic
125C2×C61#191511yescharacteristic
126C2×C4⋊Q8—611no—
127C2×C4×Q8C2×D6111yescharacteristic
128#446—24no—
129#432—22no—
130#431C2×D6112yes—
131#423—612no—
132C2×#101—1221no—
133#400—614no—
134#391—1221no—
135#391—1222no—
136#382—1221no—
137C4×C4⋊C4—611no—
138#343—1221no—
139#332—1224no—
140C61#940411yescharacteristic, complemented, Sylow
141C22×Q8—1221no—
142C22×Q8D24411yescharacteristic
143C2×M16—1222no—
144C4⋊Q8—1222no—
145C4⋊Q8—1222no—
146C4×Q8—22no—
147C4×Q8—22no—
148C2×C4⋊C4—612no—
149C2×C4⋊C4—1221no—
150C2×C4⋊C4—1221no—
151C2×C4⋊C4C4×D6112yes—
152C2×C4⋊C4D24411yescharacteristic
153C2×C42—1221no—
154C2×C42C22⋊D6112yes—
155C2×C42C22×D6111yescharacteristic
156C4⋊C8—24no—
157C4⋊C8—1222no—
158C4⋊C8—1222no—
159C4⋊C8C22⋊D6114yes—
160#101—2442no—
161C22⋊C8—22no—
162#97—24no—
163C4×C8—22no—
164#95—612no—
165#95—1224no—
166#95—1224no—
167C2×Q8—21no—
168C2×Q8—22no—
169C2×Q8—22no—
170C2×Q8—21no—
171C2×Q8—2444no—
172C2×Q8—2441no—
173C2×Q8—2441no—
174C22×C4—1222no—
175C22×C4—1222no—
176C22×C4#1175819812yes—
177C22×C4#1175819812yes—
178C22×C4C2×C4×D6111yescharacteristic
179C22×C4C2×D24411yescharacteristic
180C22×C4C2×C22⋊D6111yescharacteristic
181M16—2442no—
182M16—2442no—
183C2×C8—24no—
184C2×C8—2442no—
185C4⋊C4—24no—
186C4⋊C4—22no—
187C4⋊C4—1224no—
188C4⋊C4—2442no—
189C4⋊C4—2442no—
190C42—22no—
191C42—22no—
192C42—2441no—
193C42—2441no—
194C42#1175822212yes—
195C42C2×C22⋊D6112yes—
196C23C2×#1175819811yescharacteristic
197Q8—42no—
198Q8—42no—
199Q8—4882no—
200Q8—4882no—
201C2×C4—22no—
202C2×C4—22no—
203C2×C4—22no—
204C2×C4—22no—
205C2×C4—22no—
206C2×C4—41no—
207C2×C4—42no—
208C2×C4—2442no—
209C2×C4—2442no—
210C2×C4—2444no—
211C2×C4—2442no—
212C2×C4—2442no—
213C2×C4C4×C22⋊D6112yes—
214C2×C4#1451974611yescharacteristic
215C2×C4#1451975112yes—
216C2×C4#1451976011yescharacteristic
217C2×C4#1451980511yescharacteristic
218C2×C4#1451981012yes—
219C2×C4#1451981011yescharacteristic
220C8—44no—
221C8—4882no—
222C22—21no—
223C22—21no—
224C22#1683964111yescharacteristic
225C22#1683976411yescharacteristic
226C22#1683977211yescharacteristic
227C4—21no—
228C4—22no—
229C4—42no—
230C4—42no—
231C4—4882no—
232C4—4882no—
233C4#1683993512yes—
234C4#1684000812yes—
235C4#1684002012yes—
236C2—21no—
237C2—21no—
238C2#2144622211yescharacteristic
239C2#2144625211yescharacteristic
240C2#2144637111yescharacteristic
subgroupG/H
centerC22#16839641
commutatorC22×C4×C61C22×C4
FrattiniC2×C42C22×D61
FittingC61×#2983C2
socleC22×C61#386
prime radicalC23×C61C2×C22⋊C4
cosocleC2×C42×C61C23
hypercenter#2983D61
hypocenterC61#9404
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C42 ◁ #2983

Complements

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

NH ≅ G/Ntype
140. C61#9404semidirect

Maximal subgroups

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

subgroupindex
#214510312
#214458152
#214445032
#214437212
C61×#29832
#940461