Finite Group Databasebeta

← #30607919

Subgroups — #30607919

Order 15616.

274 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 274 conjugacy classes of subgroups — too many to draw. Only its 86 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×#16840688C211yesmaximal, characteristic
2#21462027C211yesmaximal, characteristic
3#21461937C212yesmaximal
4#21460895C212yesmaximal
5#368×D61C211yesmaximal, characteristic
6D61⋊M32C2211yescharacteristic
7#16840773—22no—
8C2×#14519846C2212yes—
9#16840725C2211yescharacteristic
10#16840688—21no—
11#16840688—21no—
12#16840649—21no—
13C16×#6923—21no—
14#16840641—22no—
15C2×C16×D61C2211yescharacteristic
16#108×D61C2212yes—
17#16839592C411yescharacteristic, complemented
18#16839591C412yescomplemented
19C61×#368C411yescharacteristic, complemented
20D61⋊M16D411yescharacteristic
21#14519856D411yescharacteristic
22C2×C4⋊#6923—22no—
23#14519852Q812yes—
24#14519851—21no—
25D61⋊C16—21no—
26#14519846—22no—
27#14519846—22no—
28#14519846—42no—
29#14519845—42no—
30#14519831—21no—
31C8×#6923—21no—
32M16×D61—22no—
33C2×C8×D61C2311yescharacteristic
34C2×#11758185C2×C411yescharacteristic
35C16×D61D411yescharacteristic
36C16×D61D411yescharacteristic
37C16×D61Q811yescharacteristic
38C16×D61Q811yescharacteristic
39#14519713C2×C412yes—
40#14519712—22no—
41C2×C16×C61C2×C411yescharacteristic
42#108×C61C2×C412yes—
43C22×#6923—42no—
44C2×C4×D61C2×D411yescharacteristic
45C22⋊#6923—21no—
46#11758217Q1612yes—
47C4⋊#6923—42no—
48C4⋊#6923—42no—
49C4⋊#6923D811yescharacteristic
50C4⋊#6923D811yescharacteristic
51C4×#6923—21no—
52#11758213—21no—
53D61⋊C8—21no—
54M16×C61—22no—
55C2×C8×C61C22×C411yescharacteristic
56#11758194—22no—
57C2×#92418#4111yescharacteristic
58#11758189—42no—
59C8×D61—42no—
60C8×D61C2×D411yescharacteristic
61C8×D61C2×Q811yescharacteristic
62C8×D61C2×Q811yescharacteristic
63C8×D61#4111yescharacteristic
64#11758187—22no—
65C16×C61C4⋊C411yescharacteristic
66C16×C61C4⋊C411yescharacteristic
67#11758185C4⋊C411yescharacteristic
68#11758185C4⋊C411yescharacteristic
69C22×D61#13511yescharacteristic
70C2×#6923—21no—
71C2×#6923—21no—
72C2×#6923—82no—
73C2×#6923—162no—
74C2×C4×C61C4×D411yescharacteristic
75C2×Dic61C2×Q1611yescharacteristic
76C4×D61#12111yescharacteristic
77C4×D61C4⋊Q811yescharacteristic
78C4×D61C2×D811yescharacteristic
79C4×D61#13511yescharacteristic
80#92420—22no—
81C8×C61—42no—
82C8×C61C2×C4⋊C411yescharacteristic
83C8×C61C4×Q811yescharacteristic
84#92418—42no—
85#92418C22⋊Q811yescharacteristic
86#92418#12511yescharacteristic
87#13171—611nomaximal, Sylow
88C22×C61C4×Q1611yescharacteristic
89C2×D61—21no—
90C2×D61—21no—
91C2×D61#46611yescharacteristic
92C2×D61#49911yescharacteristic
93#6923—41no—
94#6923—41no—
95#6923—162no—
96C4×C61#38911yescharacteristic
97C4×C61C4×D811yescharacteristic
98Dic61#46711yescharacteristic
99Dic61#50011yescharacteristic
100C2×#368—611no—
101C2×#366—611no—
102#2108—611no—
103#1952—612no—
104#1387—612no—
105C2×C61—21no—
106C2×C61#194211yescharacteristic
107D61—21no—
108D61#222111yescharacteristic
109D61#222211yescharacteristic
110C2×M32—611no—
111C22×C16—611no—
112C2×#108—612no—
113#428—1222no—
114C2×#107—612no—
115#405—611no—
116#368—611no—
117#368—612no—
118#368#692311yescharacteristic, complemented
119#366—1221no—
120#366—1221no—
121#346—1221no—
122C4×C16—1221no—
123#343—1222no—
124C61#1317111yescharacteristic, complemented, Sylow
125C2×M16—611no—
126C2×M16—1222no—
127C22×C8—611no—
128#117—611no—
129C2×C4⋊C4—1222no—
130M32—612no—
131M32—1221no—
132C2×C16—611no—
133C2×C16—611no—
134C2×C16—611no—
135C2×C16—611no—
136C2×C16—611no—
137C2×C16—1221no—
138C2×C16C2×#692311yescharacteristic
139#108—612no—
140#108—1222no—
141#108C2×#692312yes—
142#107—1222no—
143#107—1222no—
144#107—2442no—
145#106—2442no—
146#97—1221no—
147C4×C8—1221no—
148C22×C4—611no—
149C22×C4—2442no—
150M16—22no—
151M16—612no—
152M16—1221no—
153M16—1222no—
154M16—2442no—
155C2×C8—611no—
156C2×C8—611no—
157C2×C8—611no—
158C2×C8—611no—
159C2×C8—611no—
160C2×C8—1221no—
161C2×C8—2442no—
162C2×C8C22×#692311yescharacteristic
163C4⋊C4—611no—
164C4⋊C4—611no—
165C4⋊C4—2442no—
166C4⋊C4—2442no—
167C22⋊C4—1221no—
168C42—1221no—
169C16—611no—
170C16—611no—
171C16—1222no—
172C16C4⋊#692311yescharacteristic
173C16C4⋊#692311yescharacteristic
174C23—611no—
175C2×C4—611no—
176C2×C4—611no—
177C2×C4—611no—
178C2×C4—611no—
179C2×C4—611no—
180C2×C4—1221no—
181C2×C4—1221no—
182C2×C4—9762no—
183C2×C4—4882no—
184C2×C4D4×#692311yescharacteristic
185C8—42no—
186C8—611no—
187C8—611no—
188C8—1222no—
189C8—2442no—
190C8C2×C4⋊#692311yescharacteristic
191C8Q8×#692311yescharacteristic
192C22—611no—
193C22—611no—
194C22—1221no—
195C22—1221no—
196C22Q16×#692311yescharacteristic
197C4—611no—
198C4—611no—
199C4—9762no—
200C4—2441no—
201C4—2441no—
202C4#1684071311yescharacteristic
203C4D8×#692311yescharacteristic
204C2—21no—
205C2—611no—
206C2—611no—
207C2—1221no—
208C2#2146192311yescharacteristic
subgroupG/H
centerC2#21461923
commutatorC2×C8×C61C22×C4
FrattiniC2×C8C22×#6923
FittingC61×#368C4
socleC2×C61#1942
prime radicalC22×D61#135
cosocleC2×C8×D61C23
hypercenter#368#6923
hypocenterC61#13171
LCSG ▷ C2×C8×C61 ▷ C4×C61 ▷ C2×C61 ▷ C61
UCSC1 ◁ C2 ◁ C2×C4 ◁ C2×C8 ◁ #368

Complements

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

NH ≅ G/Ntype
17. #16839592C4semidirect
18. #16839591C4semidirect
19. C61×#368C4semidirect
118. #368#6923semidirect
124. C61#13171semidirect

Maximal subgroups

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

subgroupindex
C2×#168406882
#214620272
#214619372
#214608952
#368×D612
#1317161