Finite Group Databasebeta

← #30599754

Subgroups — #30599754

Order 15616.

396 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 396 conjugacy classes of subgroups — too many to draw. Only its 82 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#21462365C211yesmaximal, characteristic
2#21462002C211yesmaximal, characteristic
3#21461742C212yesmaximal
4#21460696C212yesmaximal
5#330×D61C211yesmaximal, characteristic
6#16840721C2211yescharacteristic
7#16840713—22no—
8#16840711—22no—
9#16840711—22no—
10#16840708C2211yescharacteristic
11#16840706C2212yes—
12#16840636—22no—
13#16840636—22no—
14#16840635—22no—
15#125×D61C2211yescharacteristic
16#97×D61C2212yes—
17#16839504C411yescharacteristic, complemented
18#16839503C412yescomplemented
19C61×#330C411yescharacteristic, complemented
20C2×D61⋊C8—21no—
21C2×C4⋊#6923—21no—
22C2×C4⋊#6923—22no—
23C2×C4⋊#6923—22no—
24C2×C4⋊#6923D412yes—
25C2×C4⋊#6923D412yes—
26C2×C4×#6923—22no—
27#14519832—22no—
28(C2×#6923)⋊C4—22no—
29(C2×#6923)⋊C4—22no—
30(C2×#6923)⋊C4—22no—
31(C2×#6923)⋊C4—22no—
32C2×C8×D61—22no—
33#14519769C2×C411yescharacteristic
34#14519768Q812yes—
35#14519767—22no—
36C4⋊C4×D61—22no—
37C4⋊C4×D61D412yes—
38C42×D61C2311yescharacteristic
39#14519688C2×C412yes—
40#14519687—22no—
41#125×C61C2×C411yescharacteristic
42#97×C61C2×C412yes—
43C22×#6923—22no—
44C22×#6923—22no—
45C22×#6923—41no—
46C2×C4×D61—22no—
47C2×C4×D61C2×D412yes—
48C2×C4×D61C2×D411yescharacteristic
49C2×#92420—41no—
50C4⋊#6923—42no—
51C4⋊#6923—42no—
52C4⋊#6923—44no—
53C4⋊#6923—42no—
54C4⋊#6923—82no—
55C4×#6923—82no—
56D61⋊C8—22no—
57D61⋊C8—41no—
58C2×C8×C61—22no—
59C4⋊C4×C61—22no—
60C4⋊C4×C61C4⋊C412yes—
61C42×C61C22×C411yescharacteristic
62C4⋊Dic61—22no—
63C4⋊Dic61C4⋊C412yes—
64#11758196—22no—
65#11758196—22no—
66#11758196—22no—
67C4×Dic61C2×Q811yescharacteristic
68C4×Dic61#4112yes—
69C2×#92418—22no—
70C8×D61—42no—
71C8×D61—42no—
72C22×D61#12011yescharacteristic
73C2×#6923—42no—
74C2×#6923—42no—
75C2×#6923—42no—
76C2×#6923—42no—
77C2×#6923—42no—
78C2×#6923—42no—
79C2×#6923—81no—
80C2×#6923—82no—
81C2×C4×C61—22no—
82C2×C4×C61C2×C4⋊C411yescharacteristic
83C2×C4×C61C4×D412yes—
84C2×Dic61—22no—
85C2×Dic61C22⋊Q812yes—
86C2×Dic61#12311yescharacteristic
87C4×D61—22no—
88C4×D61—21no—
89C4×D61—22no—
90C4×D61—21no—
91C4×D61—42no—
92C4×D61—42no—
93#92420—42no—
94C8×C61—42no—
95#92418—42no—
96#8915—611nomaximal, Sylow
97C22×C61#38511yescharacteristic
98C2×D61#45411yescharacteristic
99C2×D61#45412yes—
100C2×D61#45512yes—
101C2×D61#45611yescharacteristic
102#6923—82no—
103#6923—82no—
104#6923—161no—
105C4×C61—21no—
106C4×C61—22no—
107C4×C61—42no—
108Dic61—21no—
109Dic61—22no—
110Dic61—42no—
111#2382—611no—
112#1992—611no—
113#1836—612no—
114C2×#330—611no—
115#1295—612no—
116C2×C61#185711yescharacteristic
117C2×C61#186012yes—
118D61#165811yescharacteristic
119D61#166211yescharacteristic
120D61#166412yes—
121C2×#125—611no—
122C2×#97—612no—
123C2×#97—611no—
124#389—1222no—
125#387—1222no—
126#387—1222no—
127#384—611no—
128#382—612no—
129#340—1222no—
130#340—1222no—
131#340—1222no—
132#330—611no—
133#330—612no—
134#330#692311yescharacteristic, complemented
135C61#891511yescharacteristic, complemented, Sylow
136C22×C8—1222no—
137C22×C8—1221no—
138#125—612no—
139#125—611no—
140#125—1222no—
141#125C2×#692311yescharacteristic
142C2×C4⋊C4—612no—
143C2×C4⋊C4—612no—
144C2×C4⋊C4—612no—
145C2×C4⋊C4—1222no—
146C2×C4⋊C4—1222no—
147C2×C4⋊C4—1221no—
148C2×C4⋊C4—1222no—
149C2×C42—611no—
150C2×C42—1222no—
151#97—612no—
152#97—1222no—
153#97—1222no—
154#97C2×#692312yes—
155#95—1222no—
156#95—1222no—
157#95—1222no—
158#95—1222no—
159C22×C4—611no—
160C22×C4—612no—
161C22×C4—1222no—
162C22×C4—1222no—
163C22×C4—1222no—
164C22×C4—2441no—
165C2×C8—22no—
166C2×C8—1222no—
167C2×C8—1222no—
168C2×C8—2442no—
169C2×C8—2442no—
170C2×C8—2441no—
171C2×C8—2441no—
172C4⋊C4—22no—
173C4⋊C4—612no—
174C4⋊C4—1222no—
175C4⋊C4—1222no—
176C4⋊C4—1222no—
177C4⋊C4—1222no—
178C4⋊C4—4882no—
179C4⋊C4—2442no—
180C4⋊C4—2444no—
181C4⋊C4—2442no—
182C4⋊C4—2442no—
183C4⋊C4C4⋊#692312yes—
184C42—611no—
185C42—612no—
186C42—4882no—
187C42C22×#692311yescharacteristic
188C23—611no—
189C2×C4—22no—
190C2×C4—612no—
191C2×C4—611no—
192C2×C4—1221no—
193C2×C4—1222no—
194C2×C4—1222no—
195C2×C4—1221no—
196C2×C4—1222no—
197C2×C4—4881no—
198C2×C4—4882no—
199C2×C4—2442no—
200C2×C4—2442no—
201C2×C4—2442no—
202C2×C4—2442no—
203C2×C4—2442no—
204C2×C4—2442no—
205C2×C4—2442no—
206C2×C4—2442no—
207C2×C4C2×C4⋊#692311yescharacteristic
208C2×C4D4×#692312yes—
209C8—42no—
210C8—2442no—
211C8—2442no—
212C22—611no—
213C22—611no—
214C22—612no—
215C22—612no—
216C22#1684070911yescharacteristic
217C4—21no—
218C4—22no—
219C4—42no—
220C4—1221no—
221C4—1222no—
222C4—9761no—
223C4—4882no—
224C4—4882no—
225C4—2442no—
226C2—611no—
227C2—611no—
228C2—612no—
229C2#2146177511yescharacteristic
230C2#2146177912yes—
subgroupG/H
centerC22#16840709
commutatorC42×C61C22×C4
FrattiniC42C22×#6923
FittingC61×#330C4
socleC22×C61#385
prime radicalC22×D61#120
cosocleC42×D61C23
hypercenter#330#6923
hypocenterC61#8915
LCSG ▷ C42×C61 ▷ C22×C61 ▷ C61
UCSC1 ◁ C22 ◁ C42 ◁ #330

Complements

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

NH ≅ G/Ntype
17. #16839504C4semidirect
18. #16839503C4semidirect
19. C61×#330C4semidirect
134. #330#6923semidirect
135. C61#8915semidirect

Maximal subgroups

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

subgroupindex
#214623652
#214620022
#214617422
#214606962
#330×D612
#891561