Finite Group Databasebeta

← #29600359

Subgroups — #29600359

Order 15616.

326 conjugacy classes of subgroups; only the 59 normal subgroups are shown.

normal subgroups only

This group has 326 conjugacy classes of subgroups — too many to draw. Only its 59 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#21456163C211yesmaximal, characteristic
2#21447481C212yesmaximal
3#21443990C211yesmaximal, characteristic, complemented
4#21443989C212yesmaximal, complemented
5C61×#1412C211yesmaximal, characteristic, complemented
6#16840318C411yescharacteristic, complemented
7#16840310C2211yescharacteristic
8#16840132C411yescharacteristic, complemented
9#16840131C411yescharacteristic, complemented
10#16840126C2211yescharacteristic
11#16840120—22no—
12#16839755—22no—
13#16839734—22no—
14#97×D61C411yescharacteristic, complemented
15#16839556—22no—
16#16839555C2212yes—
17C61×#490C2211yescharacteristic
18C61×#353C2212yes—
19C2×#11758224D412yes—
20C2×C8×D61—21no—
21#14519777C2×C411yescharacteristic
22#14519776—21no—
23#14519769C2×C411yescharacteristic
24#14519764D412yes—
25#14519755—41no—
26#14519748—41no—
27#14519742—41no—
28D4×Dic61—41no—
29#14519735—21no—
30#14519733—42no—
31C42×D61C2×C411yescharacteristic
32#14519702—22no—
33#14519700—22no—
34#14519698—22no—
35#14519692—22no—
36#14519691—42no—
37#14519688C2×C411yescharacteristic
38#14519687—21no—
39#127×C61C2311yescharacteristic
40#125×C61C2×C411yescharacteristic
41#102×C61—22no—
42#99×C61—22no—
43#97×C61C2×C411yescharacteristic
44C2×D4×C61—41no—
45C2×D4×C61C2×D412yes—
46C2×C22⋊D61—22no—
47C22×Dic61—22no—
48#11758224—42no—
49#11758224—42no—
50C2×C4×D61—21no—
51C2×C4×D61C22⋊C411yescharacteristic
52C2×Dic122—41no—
53C2×Dic122C22⋊C412yes—
54C2×C8×C61—21no—
55C4⋊C4×C61—21no—
56C4⋊C4×C61—41no—
57C22⋊C4×C61—42no—
58C42×C61C22×C411yescharacteristic
59C22⋊Dic61—42no—
60C22⋊Dic61—42no—
61#11758198—41no—
62C4⋊Dic61—21no—
63C4⋊Dic61—41no—
64#11758196—41no—
65#11758196—41no—
66#11758196—82no—
67C4×Dic61—21no—
68C4×Dic61—41no—
69C4×Dic61—82no—
70C4×Dic61C22⋊C411yescharacteristic
71C2×#92418—21no—
72C8×D61—41no—
73C8×D61—41no—
74C23×C61—22no—
75C22×D61#9911yescharacteristic
76D4×C61—42no—
77D4×C61—82no—
78C2×C4×C61—21no—
79C2×C4×C61—41no—
80C2×C4×C61—82no—
81C2×C4×C61C2×C22⋊C411yescharacteristic
82C22⋊D61—82no—
83C2×Dic61—21no—
84C2×Dic61—22no—
85C2×Dic61—42no—
86C2×Dic61—42no—
87C2×Dic61—41no—
88C2×Dic61—82no—
89C2×Dic61#9911yescharacteristic
90C4×D61—21no—
91C4×D61—21no—
92C4×D61—81no—
93Dic122—42no—
94Dic122—82no—
95C8×C61—41no—
96#92418—41no—
97#13594—611nomaximal, Sylow
98C22×C61—42no—
99C22×C61—82no—
100C22×C61C2×#9911yescharacteristic
101C2×D61—21no—
102C2×D61—41no—
103C4×C61—21no—
104C4×C61—41no—
105C4×C61—81no—
106C4×C61—162no—
107Dic61—41no—
108Dic61—42no—
109Dic61—81no—
110Dic61—162no—
111Dic61#35411yescharacteristic
112Dic61#35411yescharacteristic
113#3135—611no—
114#2126—612no—
115#1412—611no—
116#1412—612no—
117#1412D6111yescharacteristic, complemented
118C2×C61—21no—
119C2×C61—82no—
120C2×C61C2×#35411yescharacteristic
121D61—41no—
122#533—611no—
123#532—611no—
124#491—611no—
125#491—611no—
126#490—611no—
127#490C2×D6111yescharacteristic
128#489—1222no—
129#416—1222no—
130#410—1222no—
131C2×#97—611no—
132#354—1222no—
133#353—612no—
134#353C2×D6112yes—
135C61#1359411yescharacteristic, complemented, Sylow
136C2×#41—612no—
137C22×C8—1221no—
138C4⋊Q8—611no—
139C4⋊Q8—1221no—
140#127C22×D6111yescharacteristic
141#125—611no—
142#125C4×D6111yescharacteristic
143#124—612no—
144C22⋊Q8—2441no—
145#121—2441no—
146C4×Q8—2441no—
147C4×D4—2441no—
148#117—1221no—
149#117—2442no—
150C2×C42—611no—
151#103—1222no—
152#103—1222no—
153#102—22no—
154#102—1222no—
155#99—22no—
156#99—1222no—
157#99—2442no—
158#97—611no—
159#97—1221no—
160#97C4×D6111yescharacteristic
161#41—2442no—
162#41—2442no—
163C2×Q8—612no—
164C2×Q8—2441no—
165C2×D4—41no—
166C2×D4—1222no—
167C2×D4D4×D6112yes—
168C22×C4—611no—
169C22×C4—1221no—
170C22×C4—1222no—
171C2×C8—21no—
172C2×C8—1221no—
173C2×C8—2441no—
174C2×C8—2441no—
175C4⋊C4—21no—
176C4⋊C4—41no—
177C4⋊C4—1221no—
178C4⋊C4—4882no—
179C4⋊C4—2441no—
180C4⋊C4—2441no—
181C4⋊C4—2441no—
182C22⋊C4—42no—
183C22⋊C4—2441no—
184C22⋊C4—2442no—
185C22⋊C4—2442no—
186C42—611no—
187C42—1221no—
188C42—4882no—
189C42—2441no—
190C42C2×C4×D6111yescharacteristic
191C23—22no—
192C23—611no—
193Q8—4882no—
194Q8—2442no—
195D4—42no—
196D4—82no—
197D4—4882no—
198C2×C4—21no—
199C2×C4—41no—
200C2×C4—82no—
201C2×C4—611no—
202C2×C4—1221no—
203C2×C4—1221no—
204C2×C4—1221no—
205C2×C4—1222no—
206C2×C4—4881no—
207C2×C4—4882no—
208C2×C4—2442no—
209C2×C4—2442no—
210C2×C4—2441no—
211C2×C4C22⋊C4×D6111yescharacteristic
212C8—41no—
213C8—2441no—
214C22—42no—
215C22—82no—
216C22—1221no—
217C22—2441no—
218C22#99×D6111yescharacteristic
219C4—21no—
220C4—41no—
221C4—81no—
222C4—162no—
223C4—611no—
224C4—611no—
225C4—9762no—
226C4—4881no—
227C4—2441no—
228C4—2442no—
229C2—21no—
230C2—82no—
231C2—2441no—
232C2#354×D6111yescharacteristic
subgroupG/H
centerC2#354×D61
commutatorC42×C61C22×C4
Frattini#127C22×D61
FittingC61×#1412C2
socleC2×C61C2×#354
prime radical#16840310C22
cosocle#127×C61C23
hypercenter#1412D61
hypocenterC61#13594
LCSG ▷ C42×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C22 ◁ C2×C4 ◁ #127 ◁ #1412

Complements

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

NH ≅ G/Ntype
3. #21443990C2semidirect
4. #21443989C2semidirect
5. C61×#1412C2semidirect
6. #16840318C4semidirect
8. #16840132C4semidirect
9. #16840131C4semidirect
14. #97×D61C4semidirect
117. #1412D61semidirect
135. C61#13594semidirect

Maximal subgroups

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

subgroupindex
#214561632
#214474812
#214439902
#214439892
C61×#14122
#1359461