Finite Group Databasebeta

← #30743457

Subgroups — #30743457

Order 17152.

194 conjugacy classes of subgroups; only the 41 normal subgroups are shown.

normal subgroups only

This group has 194 conjugacy classes of subgroups — too many to draw. Only its 41 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#22561559C211yesmaximal, characteristic
2#22558153C211yesmaximal, characteristic
3C67×#1295C211yesmaximal, characteristic
4#16864027C411yescharacteristic
5#16863436—21no—
6#16863165C411yescharacteristic
7#16863156—21no—
8#16863122—21no—
9C2×#97×C67—21no—
10#382×C67C2211yescharacteristic
11C2×#11761612—21no—
12#14524042—41no—
13#14524034—41no—
14#14524022—41no—
15#14524002—21no—
16#14523976—41no—
17#14523974—41no—
18C22⋊(C67⋊C8)—41no—
19C22×C8×C67—41no—
20C2×C4⋊C4×C67C2×C411yescharacteristic
21C2×C4⋊C4×C67D411yescharacteristic
22C2×C42×C67D411yescharacteristic
23#97×C67—22no—
24#97×C67—41no—
25#95×C67—41no—
26C2×Dic134—41no—
27#11761612—22no—
28#11761612—41no—
29C2×C67⋊C8—41no—
30C2×C67⋊C8—81no—
31C4⋊Dic67—21no—
32C4⋊Dic67—21no—
33C4⋊Dic67—41no—
34Dic67⋊C4—42no—
35C22⋊Dic67—41no—
36C4×Dic67—41no—
37C22×C4×C67—21no—
38C22×C4×C67—21no—
39C22×C4×C67—21no—
40C22×C4×C67C22⋊C411yescharacteristic
41C2×C8×C67—42no—
42C2×C8×C67—42no—
43C2×C8×C67—42no—
44C4⋊C4×C67—41no—
45C4⋊C4×C67—41no—
46C42×C67—21no—
47C42×C67SD1611yescharacteristic
48C42×C67Q1611yescharacteristic
49C23×C67#9911yescharacteristic
50C2×C4×C67—21no—
51C2×C4×C67—21no—
52C2×C4×C67—21no—
53C2×C4×C67—21no—
54C2×C4×C67—41no—
55C2×C4×C67—41no—
56C2×C4×C67—41no—
57C2×C4×C67—41no—
58C2×C4×C67—81no—
59C2×C4×C67—81no—
60C2×C4×C67#10311yescharacteristic
61C2×C4×C67#10411yescharacteristic
62C2×Dic67—41no—
63C2×Dic67—41no—
64Dic134—161no—
65C8×C67—44no—
66C67⋊C8—42no—
67C67⋊C8—161no—
68C22×C67—21no—
69C22×C67—21no—
70C22×C67#32811yescharacteristic
71C22×C67AΣL(1,𝔽16)11yescharacteristic
72C22×C67#35611yescharacteristic
73C4×C67—21no—
74C4×C67—21no—
75C4×C67—41no—
76C4×C67—41no—
77C4×C67—81no—
78C4×C67—81no—
79Dic67—81no—
80Dic67—81no—
81#7442—671nomaximal, Sylow
82C2×C67—21no—
83C2×C67—21no—
84C2×C67#135411yescharacteristic
85C2×C67#140311yescharacteristic
86C2×C67#140411yescharacteristic
87#2061—671no—
88#1324—671no—
89#1295D6711yescharacteristic
90C67#744211yescharacteristic, complemented, Sylow
91#544—671no—
92#428—1341no—
93C2×#97—21no—
94#382C2×D6711yescharacteristic
95#343—671no—
96#339—1341no—
97#324—1341no—
98C2×M16—1341no—
99C22×C8—41no—
100C4⋊Q8—2681no—
101#125—2681no—
102C22⋊Q8—2681no—
103#117—1341no—
104C2×C4⋊C4C4×D6711yescharacteristic
105C2×C4⋊C4D26811yescharacteristic
106C2×C42C22⋊D6711yescharacteristic
107#107—2681no—
108#106—2681no—
109C22⋊C8—2681no—
110#97—22no—
111#97—41no—
112#95—41no—
113C2×Q8—2681no—
114C22×C4—21no—
115C22×C4—21no—
116C22×C4—21no—
117C22×C4(C2×D67)⋊C411yescharacteristic
118M16—2681no—
119M16—1342no—
120C2×C8—42no—
121C2×C8—42no—
122C2×C8—42no—
123C2×C8—2681no—
124C2×C8—5361no—
125C4⋊C4—41no—
126C4⋊C4—41no—
127C4⋊C4—2681no—
128C4⋊C4—2682no—
129C4⋊C4—1341no—
130C4⋊C4—1341no—
131C22⋊C4—2681no—
132C42—21no—
133C42—2681no—
134C42#1176161611yescharacteristic
135C42#1176161811yescharacteristic
136C23#1452395711yescharacteristic
137Q8—10721no—
138C2×C4—21no—
139C2×C4—21no—
140C2×C4—21no—
141C2×C4—21no—
142C2×C4—41no—
143C2×C4—41no—
144C2×C4—41no—
145C2×C4—41no—
146C2×C4—81no—
147C2×C4—81no—
148C2×C4—2681no—
149C2×C4—2681no—
150C2×C4#1452396611yescharacteristic
151C2×C4#1452397111yescharacteristic
152C8—44no—
153C8—2682no—
154C8—10721no—
155C22—21no—
156C22—21no—
157C22#1686313211yescharacteristic
158C22#1686318411yescharacteristic
159C22#1686319411yescharacteristic
160C4—21no—
161C4—21no—
162C4—41no—
163C4—41no—
164C4—81no—
165C4—81no—
166C4—5361no—
167C4—5361no—
168C2—21no—
169C2—21no—
170C2#2255823811yescharacteristic
171C2#2255835811yescharacteristic
172C2#2255836111yescharacteristic
subgroupG/H
centerC22#16863184
commutatorC2×C4⋊C4×C67C2×C4
Frattini#382C2×D67
FittingC67×#1295C2
socleC22×C67AΣL(1,𝔽16)
prime radicalC23×C67#99
cosocle#382×C67C22
hypercenter#1295D67
hypocenterC67#7442
LCSG ▷ C2×C4⋊C4×C67 ▷ C22×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C23 ◁ C22×C4 ◁ #382 ◁ #1295

Complements

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

NH ≅ G/Ntype
90. C67#7442semidirect

Maximal subgroups

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

subgroupindex
#225615592
#225581532
C67×#12952
#744267