Finite Group Databasebeta

← #30766683

Subgroups — #30766683

Order 17152.

280 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 280 conjugacy classes of subgroups — too many to draw. Only its 79 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#22562021C211yesmaximal, characteristic
2#22560358C211yesmaximal, characteristic
3#22558203C212yesmaximal
4#22558197C212yesmaximal
5C67×#3185C211yesmaximal, characteristic
6#16863217—21no—
7#16863178C2212yes—
8#16863174—21no—
9#16863167—21no—
10#16863156—21no—
11C67⋊#338C2211yescharacteristic
12#16863153C2211yescharacteristic
13#533×C67—21no—
14#498×C67C411yescharacteristic, complemented
15#494×C67C411yescharacteristic, complemented
16#487×C67—21no—
17#470×C67—21no—
18#470×C67C2212yes—
19#441×C67—21no—
20#441×C67C412yescomplemented
21C4×M16×C67C2211yescharacteristic
22C2×#11761612—21no—
23#14524002—21no—
24#14523983—22no—
25C2×Q16×C67—21no—
26C2×Q16×C67C2×C412yes—
27C2×M16×C67D411yescharacteristic
28C2×M16×C67C2311yescharacteristic
29C4⋊Q8×C67—21no—
30C4⋊Q8×C67—21no—
31#125×C67—41no—
32C22⋊Q8×C67—22no—
33C22⋊Q8×C67—41no—
34C4×Q8×C67—22no—
35C4×Q8×C67—41no—
36#117×C67—41no—
37C2×C42×C67D411yescharacteristic
38#106×C67—21no—
39#106×C67C2×C412yes—
40#103×C67—22no—
41#103×C67—22no—
42#97×C67C2×C411yescharacteristic
43#97×C67D411yescharacteristic
44C4×C8×C67C2×C411yescharacteristic
45C4×C8×C67D411yescharacteristic
46#11761612—41no—
47#11761612—41no—
48C2×C67⋊C8—41no—
49C4⋊Dic67—41no—
50C22⋊Dic67—41no—
51C4×Dic67—41no—
52C67⋊C16—42no—
53C2×Q8×C67—22no—
54C2×Q8×C67—41no—
55C22×C4×C67—21no—
56C22×C4×C67C2×D411yescharacteristic
57Q16×C67—42no—
58Q16×C67—42no—
59M16×C67—41no—
60M16×C67—41no—
61C2×C8×C67C22⋊C411yescharacteristic
62C2×C8×C67C22⋊C411yescharacteristic
63C2×C8×C67C22×C411yescharacteristic
64C2×C8×C67C2×D411yescharacteristic
65C4⋊C4×C67—22no—
66C4⋊C4×C67—41no—
67C4⋊C4×C67—41no—
68C4⋊C4×C67—42no—
69C4⋊C4×C67—41no—
70C4⋊C4×C67—42no—
71C22⋊C4×C67—42no—
72C42×C67—21no—
73C42×C67—42no—
74C42×C67C22⋊C411yescharacteristic
75C42×C67C22⋊C411yescharacteristic
76C23×C67AGL(1,ℤ/8)11yescharacteristic
77Q8×C67—42no—
78Q8×C67—42no—
79Q8×C67—82no—
80C2×C4×C67—21no—
81C2×C4×C67—21no—
82C2×C4×C67—21no—
83C2×C4×C67—21no—
84C2×C4×C67—42no—
85C2×C4×C67—42no—
86C2×C4×C67—42no—
87C2×C4×C67C2×C22⋊C411yescharacteristic
88C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
89C2×Dic67—81no—
90C8×C67—21no—
91C8×C67—21no—
92C8×C67—21no—
93C8×C67—21no—
94C67⋊C8—81no—
95C22×C67—21no—
96C22×C67—41no—
97C22×C67#41811yescharacteristic
98C4×C67—22no—
99C4×C67—21no—
100C4×C67—41no—
101C4×C67—82no—
102C4×C67—82no—
103C4×C67#41011yescharacteristic
104C4×C67#42111yescharacteristic
105Dic67—161no—
106#11838—671nomaximal, Sylow
107C2×C67—21no—
108C2×C67—41no—
109C2×C67#151411yescharacteristic
110#3185D6711yescharacteristic
111#2165—671no—
112#1841—671no—
113#1342—672no—
114#1340—672no—
115C67#1183811yescharacteristic, complemented, Sylow
116#533—21no—
117#498Dic6711yescharacteristic, complemented
118#494Dic6711yescharacteristic, complemented
119#487—21no—
120#470—21no—
121#470C2×D6712yes—
122#441—21no—
123#441Dic6712yescomplemented
124C4×M16C2×D6711yescharacteristic
125#364—1341no—
126#349—672no—
127#347—1341no—
128#344—1341no—
129#339—1341no—
130#338—671no—
131H(3,ℤ/4)—671no—
132C2×Q16—21no—
133C2×Q16C2×Dic6712yes—
134C2×M16—1341no—
135C2×M16C22⋊D6711yescharacteristic
136C2×M16C22×D6711yescharacteristic
137C4⋊Q8—21no—
138C4⋊Q8—21no—
139#125—41no—
140C22⋊Q8—22no—
141C22⋊Q8—41no—
142C4×Q8—22no—
143C4×Q8—41no—
144#117—41no—
145#117—1341no—
146C2×C42C22⋊D6711yescharacteristic
147M32—1342no—
148#106—21no—
149#106C2×Dic6712yes—
150#103—22no—
151#103—22no—
152#97C2×Dic6711yescharacteristic
153#97C22⋊D6711yescharacteristic
154C4×C8C2×Dic6711yescharacteristic
155C4×C8C22⋊D6711yescharacteristic
156C2×Q8—22no—
157C2×Q8—41no—
158C22×C4—21no—
159C22×C4C2×C22⋊D6711yescharacteristic
160Q16—42no—
161Q16—42no—
162M16—41no—
163M16—41no—
164M16—2681no—
165M16—2681no—
166C2×C8—2681no—
167C2×C8C22⋊Dic6711yescharacteristic
168C2×C8C22⋊Dic6711yescharacteristic
169C2×C8C22×Dic6711yescharacteristic
170C2×C8C2×C22⋊D6711yescharacteristic
171C4⋊C4—22no—
172C4⋊C4—41no—
173C4⋊C4—41no—
174C4⋊C4—42no—
175C4⋊C4—41no—
176C4⋊C4—42no—
177C4⋊C4—2681no—
178C22⋊C4—42no—
179C22⋊C4—2681no—
180C42—21no—
181C42—42no—
182C42—2681no—
183C42C22⋊Dic6711yescharacteristic
184C42C22⋊Dic6711yescharacteristic
185C16—2682no—
186C23#1452406911yescharacteristic
187Q8—42no—
188Q8—42no—
189Q8—82no—
190C2×C4—21no—
191C2×C4—21no—
192C2×C4—21no—
193C2×C4—21no—
194C2×C4—42no—
195C2×C4—42no—
196C2×C4—42no—
197C2×C4—5361no—
198C2×C4C2×C22⋊Dic6711yescharacteristic
199C2×C4#1452406911yescharacteristic
200C8—21no—
201C8—21no—
202C8—21no—
203C8—21no—
204C8—5361no—
205C22—21no—
206C22—41no—
207C22#1686340111yescharacteristic
208C4—22no—
209C4—21no—
210C4—41no—
211C4—82no—
212C4—82no—
213C4—10721no—
214C4#1686337011yescharacteristic
215C4#1686341311yescharacteristic
216C2—21no—
217C2—41no—
218C2#2255875611yescharacteristic
subgroupG/H
centerC2#22558756
commutatorC2×C8×C67C22×C4
FrattiniC2×M16C22×D67
FittingC67×#3185C2
socleC2×C67#1514
prime radicalC23×C67AGL(1,ℤ/8)
cosocleC2×M16×C67C23
hypercenter#3185D67
hypocenterC67#11838
LCSG ▷ C2×C8×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3185

Complements

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

NH ≅ G/Ntype
14. #498×C67C4semidirect
15. #494×C67C4semidirect
20. #441×C67C4semidirect
115. C67#11838semidirect
117. #498Dic67semidirect
118. #494Dic67semidirect
123. #441Dic67semidirect

Maximal subgroups

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

subgroupindex
#225620212
#225603582
#225582032
#225581972
C67×#31852
#1183867