Finite Group Databasebeta

← #30607271

Subgroups — #30607271

Order 15616.

342 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 342 conjugacy classes of subgroups — too many to draw. Only its 68 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#21461729C212yesmaximal
2#21460919C212yesmaximal
3#21460911C211yesmaximal, characteristic
4#21460908C211yesmaximal, characteristic
5#21454462C211yesmaximal, characteristic
6#16840748C2212yes—
7#16840741C2212yes—
8#16840633—22no—
9#16840629—22no—
10#16840230C2211yescharacteristic
11#16839822—22no—
12#16839814—22no—
13#16839789C411yescharacteristic, complemented
14#16839730C2211yescharacteristic
15#16839728C2211yescharacteristic
16M16×Dic61C411yescharacteristic, complemented
17C2×C8×Dic61C411yescharacteristic, complemented
18C61×#424C411yescharacteristic, complemented
19D61⋊M16—22no—
20#14519856—22no—
21#14519838—22no—
22#14519837—22no—
23#14519836—22no—
24#14519835—22no—
25C2×#11758222C2311yescharacteristic
26C2×#11758194C2×C411yescharacteristic
27C22×#92418C2×C411yescharacteristic
28#14519742—21no—
29#14519739—21no—
30#14519737—22no—
31#14519734D412yes—
32C2×C4×Dic61C2×C411yescharacteristic
33C42×D61D412yes—
34C4⋊#92418—22no—
35#14519689—22no—
36#14519689—22no—
37#14519687—21no—
38C8×Dic61—21no—
39C8×Dic61—21no—
40C8×Dic61—21no—
41C2×M16×C61C2×C411yescharacteristic
42C22×C8×C61C2×C411yescharacteristic
43#117×C61C2×C411yescharacteristic
44C4⋊C8×C61—22no—
45C22×C4×C61C22×C411yescharacteristic
46C2×C22⋊D61#4112yes—
47C22×Dic61—21no—
48#11758222—41no—
49#11758222—41no—
50C2×D244#4111yescharacteristic
51C2×C4×D61—22no—
52C2×C4×D61C2×D412yes—
53C2×Dic122#4111yescharacteristic
54C22⋊#6923—42no—
55#11758217—42no—
56C4⋊#6923—42no—
57C4×#6923—42no—
58#11758213—42no—
59D61⋊C8—42no—
60M16×C61—41no—
61C2×C8×C61—21no—
62C2×C8×C61—21no—
63C2×C8×C61—41no—
64C4⋊C4×C61—21no—
65C22⋊C4×C61—21no—
66C42×C61C22⋊C412yes—
67#11758198—22no—
68#11758196—22no—
69C4×Dic61—21no—
70C4×Dic61—22no—
71C4×Dic61C22⋊C411yescharacteristic
72C4×Dic61C22⋊C411yescharacteristic
73#11758194—41no—
74C2×#92418—21no—
75C2×#92418—21no—
76C2×#92418—41no—
77C23×C61#11711yescharacteristic
78C22×D61#12312yes—
79C2×#6923—82no—
80C2×C4×C61—21no—
81C2×C4×C61—21no—
82C2×C4×C61—22no—
83C2×C4×C61C2×C22⋊C411yescharacteristic
84C2×C4×C61#11711yescharacteristic
85C22⋊D61—42no—
86C2×Dic61—21no—
87C2×Dic61—21no—
88C2×Dic61—42no—
89C2×Dic61#12312yes—
90D244—41no—
91C4×D61—22no—
92C4×D61—22no—
93C4×D61—44no—
94Dic122—41no—
95#92420—82no—
96C8×C61—41no—
97C8×C61—41no—
98#92418—41no—
99#92418—41no—
100#12864—611nomaximal, Sylow
101C22×C61—21no—
102C22×C61—41no—
103C22×C61#38111yescharacteristic
104C2×D61—22no—
105C2×D61—42no—
106#6923—162no—
107C4×C61—21no—
108C4×C61—42no—
109C4×C61#41011yescharacteristic
110C4×C61#41011yescharacteristic
111Dic61—22no—
112Dic61—22no—
113Dic61—41no—
114#2922—611no—
115#1827—612no—
116#1395—612no—
117#1392—611no—
118#1391—611no—
119C2×C61—21no—
120C2×C61—41no—
121C2×C61#182011yescharacteristic
122D61—42no—
123C4×#41—611no—
124#433—1222no—
125#431—1222no—
126#424—611no—
127#424#692311yescharacteristic, complemented
128#413—612no—
129#410—612no—
130#408—611no—
131#408—611no—
132C4×M16—611no—
133C2×C4×C8—611no—
134#339—1222no—
135H(3,ℤ/4)—1222no—
136C61#1286411yescharacteristic, complemented, Sylow
137C2×#41—611no—
138C2×M16—611no—
139C2×M16—1222no—
140C2×M16C2×#692311yescharacteristic
141C22×C8—611no—
142C22×C8C2×#692311yescharacteristic
143C4×Q8—1221no—
144C4×D4—1222no—
145C4×D4—1221no—
146#117—612no—
147#117—1222no—
148#117C2×#692311yescharacteristic
149C2×C42—611no—
150C2×C42—612no—
151C4⋊C8—22no—
152C4⋊C8—1222no—
153#101—1222no—
154#100—1222no—
155#99—1222no—
156#99—1222no—
157C22⋊C8—1222no—
158C22⋊C8—1222no—
159#97—1221no—
160C4×C8—1221no—
161C4×C8—1221no—
162C4×C8—1221no—
163#41—2441no—
164#41—2441no—
165C2×Q8—611no—
166C2×D4—612no—
167C2×D4—611no—
168C22×C4—612no—
169C22×C4—1222no—
170C22×C4—1221no—
171C22×C4C22×#692311yescharacteristic
172M16—41no—
173M16—2442no—
174M16—2442no—
175M16—2441no—
176C2×C8—21no—
177C2×C8—21no—
178C2×C8—41no—
179C2×C8—1221no—
180C2×C8—1221no—
181C2×C8—2442no—
182C2×C8—2441no—
183C4⋊C4—21no—
184C4⋊C4—1222no—
185C4⋊C4—2442no—
186C22⋊C4—21no—
187C22⋊C4—1222no—
188C22⋊C4—2442no—
189C42—611no—
190C42—611no—
191C42—1222no—
192C42—1221no—
193C42—2442no—
194C42C22⋊#692312yes—
195C23—612no—
196C23#1451985611yescharacteristic
197Q8—2441no—
198D4—2441no—
199D4—2442no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—22no—
203C2×C4—612no—
204C2×C4—1221no—
205C2×C4—1222no—
206C2×C4—1222no—
207C2×C4—1221no—
208C2×C4—4882no—
209C2×C4—2444no—
210C2×C4—2442no—
211C2×C4C2×C22⋊#692311yescharacteristic
212C2×C4#1451985611yescharacteristic
213C8—41no—
214C8—41no—
215C8—4882no—
216C8—2441no—
217C8—2441no—
218C22—21no—
219C22—41no—
220C22—1222no—
221C22—2442no—
222C22#1684070511yescharacteristic
223C4—21no—
224C4—42no—
225C4—1222no—
226C4—1222no—
227C4—9762no—
228C4—2441no—
229C4#1684074211yescharacteristic
230C4#1684074311yescharacteristic
231C2—21no—
232C2—41no—
233C2—2442no—
234C2#2146171711yescharacteristic
subgroupG/H
centerC4#16840743
commutatorC22×C4×C61C22×C4
FrattiniC22×C4C22×#6923
FittingC61×#424C4
socleC2×C61#1820
prime radical, cosocleC2×#11758222C23
hypercenter#424#6923
hypocenterC61#12864
LCSG ▷ C22×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C4 ◁ C2×C4 ◁ #117 ◁ #424

Complements

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

NH ≅ G/Ntype
13. #16839789C4semidirect
16. M16×Dic61C4semidirect
17. C2×C8×Dic61C4semidirect
18. C61×#424C4semidirect
127. #424#6923semidirect
136. C61#12864semidirect

Maximal subgroups

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

subgroupindex
#214617292
#214609192
#214609112
#214609082
#214544622
#1286461