Finite Group Databasebeta

← #29600681

Subgroups — #29600681

Order 15616.

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

normal subgroups only

This group has 288 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
1C2×#16840191C211yesmaximal, characteristic
2C2×Dic976C211yesmaximal, characteristic, complemented
3C2×#16839600C211yesmaximal, characteristic, complemented
4#21447911C211yesmaximal, characteristic, complemented
5#21443999C211yesmaximal, characteristic, complemented
6#21443956C211yesmaximal, characteristic, complemented
7C61×#1415C211yesmaximal, characteristic, complemented
8C2×#14519796—21no—
9#16840191—22no—
10#16840191—22no—
11C2×Dic488C2211yescharacteristic
12C2×Dic488C2211yescharacteristic
13C2×#14519721—21no—
14C2×C16×D61C2211yescharacteristic
15#16840004—21no—
16Dic976—22no—
17#16839600—22no—
18C2×#14519683C2211yescharacteristic
19#16839565—21no—
20C2×D16×C61C2211yescharacteristic
21C2×C32×C61C2211yescharacteristic
22C61×#366C2211yescharacteristic
23C2×#11758224—41no—
24#14519796—24no—
25#14519796—42no—
26C2×Dic244—21no—
27C2×Dic244—21no—
28C2×#11758200—41no—
29C2×C8×D61D411yescharacteristic
30#14519757—41no—
31Dic488—21no—
32Dic488—21no—
33Dic488—21no—
34Dic488D412yes—
35#14519721—24no—
36C2×#11758185D411yescharacteristic
37C16×D61—21no—
38C16×D61—21no—
39#14519700—41no—
40#14519683—21no—
41C2×D8×C61—21no—
42D16×C61—21no—
43D16×C61D412yes—
44C2×C16×C61C2311yescharacteristic
45#107×C61—21no—
46C32×C61—21no—
47C2×D4×C61—41no—
48C2×C22⋊D61—81no—
49C22×Dic61—81no—
50#11758224—44no—
51#11758224—82no—
52C2×C4×D61D811yescharacteristic
53C2×Dic122—41no—
54C2×Dic122—41no—
55D8×C61—22no—
56D8×C61—41no—
57C2×C8×C61C2×D411yescharacteristic
58C4⋊C4×C61—41no—
59#11758200—44no—
60#11758198—81no—
61C4⋊Dic61—81no—
62C2×#92418D811yescharacteristic
63Dic244—22no—
64Dic244—41no—
65Dic244—41no—
66Dic244—41no—
67C8×D61—21no—
68C8×D61—21no—
69C16×C61C2×D411yescharacteristic
70C16×C61C2×D411yescharacteristic
71#11758185—21no—
72C23×C61—81no—
73C22×D61D1611yescharacteristic
74D4×C61—42no—
75D4×C61—81no—
76C2×C4×C61—81no—
77C2×C4×C61C2×D811yescharacteristic
78C22⋊D61—84no—
79C2×Dic61—84no—
80C2×Dic61—81no—
81C2×Dic61—81no—
82C2×Dic61D1611yescharacteristic
83C4×D61—21no—
84C4×D61—21no—
85Dic122—42no—
86Dic122—81no—
87Dic122—81no—
88Dic122—81no—
89C8×C61#12011yescharacteristic
90C8×C61AGL(1,ℤ/8)11yescharacteristic
91#92418—21no—
92#13666—611nomaximal, Sylow
93C22×C61—82no—
94C22×C61—161no—
95C22×C61—161no—
96C22×C61C2×D1611yescharacteristic
97C2×D61—21no—
98C2×D61—21no—
99C2×D61—21no—
100C4×C61—161no—
101C4×C61#44711yescharacteristic
102C4×C61#50911yescharacteristic
103Dic61—21no—
104Dic61—82no—
105Dic61—161no—
106C2×#508—611no—
107C2×Q64—611no—
108C2×SD64—611no—
109#2216—611no—
110#1416—611no—
111#1415D6111yescharacteristic, complemented
112C22⋊C32—611no—
113C2×C61—162no—
114C2×C61#218411yescharacteristic
115C2×C61#226211yescharacteristic
116C2×C61#226411yescharacteristic
117D61—41no—
118C2×#135—1221no—
119#508—1222no—
120#508—1222no—
121C2×Q32—611no—
122C2×Q32—611no—
123C2×SD32—1221no—
124C2×D16C2×D6111yescharacteristic
125C22×C16—611no—
126#467—1221no—
127Q64—1222no—
128SD64—1222no—
129C2×C32—611no—
130C2×C32C2×D6111yescharacteristic
131#366C2×D6111yescharacteristic
132#358—1221no—
133C61#1366611yescharacteristic, complemented, Sylow
134C2×#41—2441no—
135#135—1224no—
136#135—2442no—
137C2×Q16—1221no—
138C2×Q16—1221no—
139C2×SD16—2441no—
140C2×D8—21no—
141C22×C8—611no—
142C22⋊Q8—2441no—
143Q32—612no—
144Q32—1221no—
145Q32—1221no—
146Q32—1221no—
147SD32—1224no—
148D16—21no—
149D16D24412yes—
150C2×C16—611no—
151C2×C16—1221no—
152C2×C16—1221no—
153C2×C16C22×D6111yescharacteristic
154#107—21no—
155#103—2441no—
156C32—21no—
157C32—1221no—
158#41—4882no—
159#41—2444no—
160C2×Q8—2441no—
161C2×Q8—2441no—
162C2×D4—41no—
163C2×D4—4881no—
164C22×C4—611no—
165C22×C4—4881no—
166Q16—1222no—
167Q16—2441no—
168Q16—2441no—
169Q16—2441no—
170SD16—2444no—
171D8—22no—
172D8—41no—
173C2×C8—611no—
174C2×C8—1221no—
175C2×C8—1221no—
176C2×C8D4×D6111yescharacteristic
177C4⋊C4—41no—
178C4⋊C4—4881no—
179C22⋊C4—4881no—
180C16—1221no—
181C16C2×D24411yescharacteristic
182C16D4×D6111yescharacteristic
183C23—81no—
184C23—611no—
185Q8—4881no—
186Q8—4881no—
187Q8—4881no—
188Q8—2442no—
189D4—42no—
190D4—81no—
191D4—4884no—
192C2×C4—81no—
193C2×C4—611no—
194C2×C4—1221no—
195C2×C4—1221no—
196C2×C4—4884no—
197C2×C4—4881no—
198C2×C4—4881no—
199C2×C4D8×D6111yescharacteristic
200C8—1221no—
201C8#1451974711yescharacteristic
202C8#1451980411yescharacteristic
203C22—82no—
204C22—161no—
205C22—161no—
206C22—1221no—
207C22—1221no—
208C22—1221no—
209C22D16×D6111yescharacteristic
210C4—161no—
211C4—1221no—
212C4—9761no—
213C4—4882no—
214C4#1683989711yescharacteristic
215C4#1684019911yescharacteristic
216C2—162no—
217C2—2441no—
218C2#2144773211yescharacteristic
219C2#2144812511yescharacteristic
220C2#2144813511yescharacteristic
subgroupG/H
centerC22D16×D61
commutator, cosocleC2×C16×C61C23
FrattiniC2×C16C22×D61
FittingC61×#1415C2
socleC22×C61C2×D16
prime radicalC2×#16840191C2
hypercenter#1415D61
hypocenterC61#13666
LCSG ▷ C2×C16×C61 ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ #1415

Complements

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

NH ≅ G/Ntype
2. C2×Dic976C2semidirect
3. C2×#16839600C2semidirect
4. #21447911C2semidirect
5. #21443999C2semidirect
6. #21443956C2semidirect
7. C61×#1415C2semidirect
111. #1415D61semidirect
133. C61#13666semidirect

Maximal subgroups

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

subgroupindex
C2×#168401912
C2×Dic9762
C2×#168396002
#214479112
#214439992
#214439562
C61×#14152
#1366661