Finite Group Databasebeta

← #29567780

Subgroups — #29567780

Order 15616.

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

normal subgroups only

This group has 270 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#21447484C211yesmaximal, characteristic
2#21446203C211yesmaximal, characteristic
3C61×#1536C211yesmaximal, characteristic
4#16840529C411yescharacteristic, complemented
5#16839779—21no—
6#16839736—21no—
7#16839736C411yescharacteristic, complemented
8#16839557—21no—
9#16839555C412yescomplemented
10#16839522—21no—
11C61×#532C2211yescharacteristic
12C61×#432—21no—
13C61×#329—21no—
14C61×#329C412yescomplemented
15#14519784—42no—
16C2×#11758194—21no—
17C2×#11758194—41no—
18#14519736—21no—
19#14519736—41no—
20#14519704—42no—
21#14519692—22no—
22#14519692—41no—
23#14519692—41no—
24C2×#41×C61C2×C411yescharacteristic
25C2×#41×C61D411yescharacteristic
26C4⋊Q8×C61Q811yescharacteristic
27#127×C61C2×C411yescharacteristic
28#124×C61C2×C411yescharacteristic
29#124×C61D411yescharacteristic
30#121×C61—41no—
31C4×D4×C61—41no—
32C2×C42×C61D411yescharacteristic
33C4⋊C8×C61—41no—
34C22⋊C8×C61—41no—
35#97×C61—22no—
36#41×C61—41no—
37#41×C61—41no—
38#41×C61—41no—
39#41×C61—41no—
40C2×Q8×C61C4⋊C411yescharacteristic
41C2×Q8×C61C4⋊C411yescharacteristic
42C2×D4×C61—21no—
43C2×D4×C61—21no—
44C2×D4×C61—41no—
45C2×D4×C61C4211yescharacteristic
46C2×D4×C61C22⋊C411yescharacteristic
47C22×C4×C61—21no—
48C22×C4×C61—21no—
49C22×C4×C61—21no—
50C22×C4×C61C22⋊C411yescharacteristic
51C2×C8×C61—42no—
52C4⋊C4×C61—41no—
53C22⋊C4×C61—41no—
54C22⋊C4×C61—41no—
55C42×C61—21no—
56C42×C61C22⋊C411yescharacteristic
57C42×C61C4⋊C411yescharacteristic
58C22⋊Dic61—42no—
59C22⋊Dic61—41no—
60C4⋊Dic61—41no—
61C4⋊Dic61—81no—
62C4×Dic61—41no—
63C4×Dic61—81no—
64#11758194—22no—
65#11758194—41no—
66#11758194—42no—
67C2×#92418—41no—
68C2×#92418—42no—
69C23×C61—21no—
70C23×C61—21no—
71C23×C61#9911yescharacteristic
72Q8×C61—41no—
73Q8×C61—41no—
74D4×C61—41no—
75D4×C61—41no—
76D4×C61—42no—
77D4×C61—81no—
78D4×C61—82no—
79C2×C4×C61—21no—
80C2×C4×C61—21no—
81C2×C4×C61—21no—
82C2×C4×C61—21no—
83C2×C4×C61—21no—
84C2×C4×C61—21no—
85C2×C4×C61—41no—
86C2×C4×C61—41no—
87C2×C4×C61—41no—
88C2×C4×C61—41no—
89C2×C4×C61—81no—
90C2×C4×C61#9511yescharacteristic
91C2×C4×C61#9911yescharacteristic
92C2×Dic61—81no—
93C2×Dic61—82no—
94C8×C61—82no—
95#92418—42no—
96#7494—611nomaximal, Sylow
97C22×C61—21no—
98C22×C61—41no—
99C22×C61—41no—
100C22×C61—41no—
101C22×C61—81no—
102C22×C61—81no—
103C22×C61#34211yescharacteristic
104C4×C61—21no—
105C4×C61—41no—
106C4×C61—41no—
107C4×C61—41no—
108C4×C61—41no—
109C4×C61AΣL(1,𝔽16)11yescharacteristic
110C4×C61#35211yescharacteristic
111Dic61—161no—
112Dic61—162no—
113#2126—611no—
114#1884—611no—
115#1536D6111yescharacteristic
116C2×C61—21no—
117C2×C61—41no—
118C2×C61—81no—
119C2×C61—81no—
120C2×C61#139011yescharacteristic
121#568—611no—
122#532C2×D6111yescharacteristic
123#432—21no—
124#421—1221no—
125#410—611no—
126#410—1221no—
127#354—1221no—
128#353—612no—
129#339—1221no—
130#329—21no—
131#329Dic6112yescomplemented
132C61#749411yescharacteristic, complemented, Sylow
133C2×#41C4×D6111yescharacteristic
134C2×#41D24411yescharacteristic
135#131—2442no—
136C2×M16—1221no—
137C2×M16—2441no—
138C4⋊Q8Dic12211yescharacteristic
139#127C4×D6111yescharacteristic
140#124C2×Dic6111yescharacteristic
141#124C22⋊D6111yescharacteristic
142#121—41no—
143C4×D4—41no—
144#117—1221no—
145#117—2441no—
146C2×C42C22⋊D6111yescharacteristic
147C4⋊C8—41no—
148#104—2442no—
149#99—1222no—
150#99—2441no—
151#99—2441no—
152C22⋊C8—41no—
153#97—22no—
154#41—41no—
155#41—41no—
156#41—41no—
157#41—41no—
158C2×Q8#1175819611yescharacteristic
159C2×Q8C4⋊Dic6111yescharacteristic
160C2×D4—21no—
161C2×D4—21no—
162C2×D4—41no—
163C2×D4C4×Dic6111yescharacteristic
164C2×D4#1175819811yescharacteristic
165C22×C4—21no—
166C22×C4—21no—
167C22×C4—21no—
168C22×C4#1175819811yescharacteristic
169M16—1222no—
170M16—2441no—
171M16—2442no—
172C2×C8—42no—
173C2×C8—2441no—
174C2×C8—2442no—
175C4⋊C4—41no—
176C4⋊C4—4881no—
177C4⋊C4—2441no—
178C22⋊C4—41no—
179C22⋊C4—41no—
180C22⋊C4—2442no—
181C22⋊C4—2441no—
182C42—21no—
183C42—4881no—
184C42—2441no—
185C42#1175819611yescharacteristic
186C42C22⋊Dic6111yescharacteristic
187C23—21no—
188C23—21no—
189C23#1451969111yescharacteristic
190Q8—41no—
191Q8—41no—
192D4—41no—
193D4—41no—
194D4—42no—
195D4—81no—
196D4—82no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—21no—
202C2×C4—21no—
203C2×C4—41no—
204C2×C4—41no—
205C2×C4—41no—
206C2×C4—41no—
207C2×C4—81no—
208C2×C4—4881no—
209C2×C4—4882no—
210C2×C4#1451968411yescharacteristic
211C2×C4#1451969111yescharacteristic
212C8—82no—
213C8—2442no—
214C22—21no—
215C22—41no—
216C22—41no—
217C22—41no—
218C22—81no—
219C22—81no—
220C22#1683952811yescharacteristic
221C4—21no—
222C4—41no—
223C4—41no—
224C4—41no—
225C4—41no—
226C4—9761no—
227C4—9762no—
228C4#1683955011yescharacteristic
229C4#1683955311yescharacteristic
230C2—21no—
231C2—41no—
232C2—81no—
233C2—81no—
234C2#2144393811yescharacteristic
subgroupG/H
centerC2#21443938
commutatorC2×D4×C61C42
Frattini#532C2×D61
FittingC61×#1536C2
socleC2×C61#1390
prime radical, cosocleC61×#532C22
hypercenter#1536D61
hypocenterC61#7494
LCSG ▷ C2×D4×C61 ▷ C2×C4×C61 ▷ C22×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #1536

Complements

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

NH ≅ G/Ntype
4. #16840529C4semidirect
7. #16839736C4semidirect
9. #16839555C4semidirect
14. C61×#329C4semidirect
131. #329Dic61semidirect
132. C61#7494semidirect

Maximal subgroups

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

subgroupindex
#214474842
#214462032
C61×#15362
#749461