Finite Group Databasebeta

← #29600672

Subgroups — #29600672

Order 15616.

284 conjugacy classes of subgroups; only the 61 normal subgroups are shown.

normal subgroups only

This group has 284 conjugacy classes of subgroups — too many to draw. Only its 61 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×Q32×D61C211yesmaximal, characteristic
2C2×#16839602C211yesmaximal, characteristic, complemented
3#21447907C211yesmaximal, characteristic
4#21444001C211yesmaximal, characteristic, complemented
5#21444000C211yesmaximal, characteristic, complemented
6#21443956C211yesmaximal, characteristic
7C2×C61×Q64C211yesmaximal, characteristic, complemented
8C2×Q16×D61—21no—
9C2×Dic488C2211yescharacteristic
10C2×#14519723—21no—
11Q32×D61—22no—
12Q32×D61—22no—
13C2×C16×D61C2211yescharacteristic
14#16840000—21no—
15#16839602—22no—
16C2×#14519683C2211yescharacteristic
17#16839588C2211yescharacteristic
18#16839566—21no—
19C2×Q32×C61C2211yescharacteristic
20C2×Q32×C61C2211yescharacteristic
21C61×Q64—22no—
22C2×C32×C61C2211yescharacteristic
23C2×Q8×D61—41no—
24C2×Dic244—21no—
25C2×#11758202—41no—
26Q16×D61—24no—
27Q16×D61—42no—
28C2×C8×D61D411yescharacteristic
29#14519753—41no—
30Dic488—21no—
31Dic488D412yes—
32#14519723—24no—
33C2×#11758185D411yescharacteristic
34C16×D61—21no—
35C16×D61—21no—
36#14519711—21no—
37#14519701—41no—
38#14519683—21no—
39C2×Q16×C61—21no—
40C2×Q16×C61—21no—
41Q32×C61—21no—
42Q32×C61—21no—
43Q32×C61—21no—
44Q32×C61D412yes—
45C2×C16×C61C2311yescharacteristic
46C32×C61—21no—
47C2×Q8×C61—41no—
48C2×Q8×C61—41no—
49Q8×D61—44no—
50Q8×D61—82no—
51C2×C4×D61—81no—
52C2×C4×D61D811yescharacteristic
53C2×Dic122—41no—
54C2×Dic122—81no—
55Q16×C61—22no—
56Q16×C61—41no—
57Q16×C61—41no—
58Q16×C61—41no—
59C2×C8×C61C2×D411yescharacteristic
60#11758202—44no—
61#11758198—81no—
62C4⋊Dic61—41no—
63#11758196—81no—
64C2×#92418D811yescharacteristic
65Dic244—22no—
66Dic244—41no—
67C8×D61—21no—
68C8×D61—21no—
69C16×C61C2×D411yescharacteristic
70C16×C61C2×D411yescharacteristic
71#11758185—21no—
72C22×D61D1611yescharacteristic
73Q8×C61—42no—
74Q8×C61—81no—
75Q8×C61—81no—
76Q8×C61—81no—
77C2×C4×C61—81no—
78C2×C4×C61—81no—
79C2×C4×C61C2×D811yescharacteristic
80C2×Dic61—81no—
81C2×Dic61—81no—
82C2×Dic61D1611yescharacteristic
83C4×D61—21no—
84C4×D61—21no—
85C4×D61—84no—
86Dic122—42no—
87Dic122—84no—
88Dic122—81no—
89C8×C61#12011yescharacteristic
90C8×C61AGL(1,ℤ/8)11yescharacteristic
91#92418—21no—
92#13665—611nomaximal, Sylow
93C22×C61C2×D1611yescharacteristic
94C2×D61—21no—
95C2×D61—21no—
96C2×D61Q6412yescomplemented
97C4×C61—82no—
98C4×C61—161no—
99C4×C61#44711yescharacteristic
100C4×C61#50911yescharacteristic
101Dic61—21no—
102Dic61—82no—
103Dic61—161no—
104C22×Q32—611no—
105C2×Q64—611no—
106C2×Q64D6111yescharacteristic, complemented
107#2216—611no—
108#1416—611no—
109#1416—611no—
110C22⋊C32—611no—
111C2×C61#218411yescharacteristic
112C2×C61C2×Q6411yescharacteristic
113C2×C61#226411yescharacteristic
114D61—22no—
115C22×Q16—1221no—
116C2×Q32—611no—
117C2×Q32—1221no—
118C2×Q32—1222no—
119C2×Q32—1222no—
120C2×Q32C2×D6111yescharacteristic
121C2×Q32C2×D6111yescharacteristic
122C22×C16—611no—
123#467—1221no—
124Q64—22no—
125Q64—1222no—
126C2×C32—611no—
127C2×C32C2×D6111yescharacteristic
128#366—611no—
129#358—1221no—
130C61#1366511yescharacteristic, complemented, Sylow
131C22×Q8—2441no—
132C2×Q16—21no—
133C2×Q16—21no—
134C2×Q16—1224no—
135C2×Q16—1221no—
136C2×Q16—2441no—
137C2×Q16—2442no—
138C22×C8—611no—
139C22⋊Q8—2441no—
140Q32—21no—
141Q32—21no—
142Q32—21no—
143Q32—612no—
144Q32—1224no—
145Q32—1221no—
146Q32C22⋊D6112yes—
147C2×C16—611no—
148C2×C16—1221no—
149C2×C16—1221no—
150C2×C16C22×D6111yescharacteristic
151#107—1221no—
152#103—2441no—
153C32—21no—
154C32—1221no—
155C2×Q8—41no—
156C2×Q8—41no—
157C2×Q8—4881no—
158C2×Q8—4882no—
159C2×Q8—2444no—
160C2×Q8—2441no—
161C22×C4—611no—
162C22×C4—4881no—
163Q16—22no—
164Q16—41no—
165Q16—41no—
166Q16—41no—
167Q16—1222no—
168Q16—2444no—
169Q16—2441no—
170C2×C8—611no—
171C2×C8—1221no—
172C2×C8—1221no—
173C2×C8D4×D6111yescharacteristic
174C4⋊C4—4881no—
175C4⋊C4—2441no—
176C22⋊C4—4881no—
177C16—1221no—
178C16D4×D6111yescharacteristic
179C16C2×C22⋊D6111yescharacteristic
180C23—611no—
181Q8—42no—
182Q8—81no—
183Q8—81no—
184Q8—81no—
185Q8—4884no—
186Q8—4881no—
187Q8—2442no—
188C2×C4—81no—
189C2×C4—81no—
190C2×C4—611no—
191C2×C4—1221no—
192C2×C4—1221no—
193C2×C4—4884no—
194C2×C4—4881no—
195C2×C4—4881no—
196C2×C4D8×D6111yescharacteristic
197C8—1221no—
198C8#1451974511yescharacteristic
199C8#1451980111yescharacteristic
200C22—612no—
201C22—1221no—
202C22—1221no—
203C22D16×D6111yescharacteristic
204C4—82no—
205C4—161no—
206C4—1221no—
207C4—9761no—
208C4—4882no—
209C4#1683989411yescharacteristic
210C4#1684019611yescharacteristic
211C2—1222no—
212C2#2144772911yescharacteristic
213C2Q64×D6111yescharacteristic
214C2#2144813811yescharacteristic
subgroupG/H
centerC22D16×D61
commutator, cosocleC2×C16×C61C23
FrattiniC2×C16C22×D61
FittingC2×C61×Q64C2
socleC22×C61C2×D16
prime radicalC22×D61D16
hypercenterC2×Q64D61
hypocenterC61#13665
LCSG ▷ C2×C16×C61 ▷ C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C22 ◁ C2×C4 ◁ C2×C8 ◁ C2×C16 ◁ C2×Q64

Complements

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

NH ≅ G/Ntype
2. C2×#16839602C2semidirect
4. #21444001C2semidirect
5. #21444000C2semidirect
7. C2×C61×Q64C2semidirect
96. C2×D61Q64semidirect
106. C2×Q64D61semidirect
130. C61#13665semidirect

Maximal subgroups

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

subgroupindex
C2×Q32×D612
C2×#168396022
#214479072
#214440012
#214440002
#214439562
C2×C61×Q642
#1366561