Finite Group Databasebeta

← #30776197

Subgroups — #30776197

Order 17152.

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

normal subgroups only

This group has 318 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#22561894C212yesmaximal
2#22558382C214yesmaximal, complemented
3C67×#3142C211yesmaximal, characteristic
4#16863380—22no—
5#16863195C2214yes—
6#16863193—22no—
7#532×C67C2211yescharacteristic
8C2×#125×C67C411yescharacteristic
9#491×C67C2212yes—
10#490×C67C412yes—
11#488×C67—22no—
12#417×C67—22no—
13C4×M16×C67C411yescharacteristic
14C2×#11761612—42no—
15#14523962—24no—
16#14523960—42no—
17C2×#41×C67D412yes—
18C2×M16×C67—21no—
19C4⋊Q8×C67C2311yescharacteristic
20#127×C67C2×C411yescharacteristic
21#125×C67—21no—
22#125×C67—41no—
23#125×C67C2×C412yes—
24#124×C67D412yes—
25#121×C67—41no—
26C4×D4×C67—41no—
27C2×C4⋊C4×C67—21no—
28C2×C4⋊C4×C67—41no—
29C2×C42×C67C2×C411yescharacteristic
30#103×C67—24no—
31#102×C67—24no—
32#97×C67C2×C412yes—
33C4×C8×C67—21no—
34#11761612—44no—
35#11761612—82no—
36C2×C67⋊C8—82no—
37#41×C67—42no—
38#41×C67—42no—
39C2×Q8×C67C2×D412yes—
40C2×D4×C67—22no—
41C2×D4×C67—41no—
42C2×D4×C67C22⋊C412yes—
43C22×C4×C67—21no—
44C22×C4×C67—22no—
45C22×C4×C67—41no—
46C22×C4×C67C22⋊C411yescharacteristic
47M16×C67—42no—
48C2×C8×C67—22no—
49C4⋊C4×C67—22no—
50C4⋊C4×C67—41no—
51C4⋊C4×C67—42no—
52C4⋊C4×C67—42no—
53C4⋊C4×C67—41no—
54C22⋊C4×C67—42no—
55C42×C67—21no—
56C42×C67C22⋊C411yescharacteristic
57C42×C67C22×C411yescharacteristic
58C23×C67—22no—
59C23×C67#9911yescharacteristic
60Q8×C67—42no—
61D4×C67—42no—
62D4×C67—82no—
63D4×C67—82no—
64C2×C4×C67—22no—
65C2×C4×C67—21no—
66C2×C4×C67—21no—
67C2×C4×C67—22no—
68C2×C4×C67—42no—
69C2×C4×C67—42no—
70C2×C4×C67—42no—
71C2×C4×C67—81no—
72C2×C4×C67—82no—
73C2×C4×C67#9911yescharacteristic
74C2×C4×C67C2×C22⋊C411yescharacteristic
75C8×C67—42no—
76C67⋊C8—84no—
77C22×C67—21no—
78C22×C67—41no—
79C22×C67—42no—
80C22×C67—82no—
81C22×C67C2×#9911yescharacteristic
82C4×C67—21no—
83C4×C67—41no—
84C4×C67—41no—
85C4×C67—42no—
86C4×C67—82no—
87C4×C67#35412yes—
88#13600—671nomaximal, Sylow
89C2×C67—21no—
90C2×C67—41no—
91C2×C67—82no—
92C2×C67C2×#35411yescharacteristic
93#3142D6711yescharacteristic
94#2132—672no—
95#1413—674no—
96C67#1360011yescharacteristic, complemented, Sylow
97#532C2×D6711yescharacteristic
98C2×#125Dic6711yescharacteristic
99#491C2×D6712yes—
100#490Dic6712yes—
101#488—22no—
102#417—22no—
103#413—1342no—
104C4×M16Dic6711yescharacteristic
105#356—674no—
106#355—1342no—
107C2×#41C22⋊D6712yes—
108C2×M16—21no—
109C2×M16—2682no—
110C4⋊Q8C22×D6711yescharacteristic
111#127C2×Dic6711yescharacteristic
112#125—21no—
113#125—41no—
114#125C2×Dic6712yes—
115#124C22⋊D6712yes—
116#121—41no—
117C4×D4—41no—
118C2×C4⋊C4—21no—
119C2×C4⋊C4—41no—
120C2×C42C2×Dic6711yescharacteristic
121#103—24no—
122#102—24no—
123#101—1344no—
124#100—2682no—
125#97C2×Dic6712yes—
126C4×C8—21no—
127#41—42no—
128#41—42no—
129C2×Q8C2×C22⋊D6712yes—
130C2×D4—22no—
131C2×D4—41no—
132C2×D4C22⋊Dic6712yes—
133C22×C4—21no—
134C22×C4—22no—
135C22×C4—41no—
136C22×C4C22⋊Dic6711yescharacteristic
137M16—42no—
138M16—2684no—
139M16—5362no—
140C2×C8—22no—
141C2×C8—5362no—
142C4⋊C4—22no—
143C4⋊C4—41no—
144C4⋊C4—42no—
145C4⋊C4—42no—
146C4⋊C4—41no—
147C22⋊C4—42no—
148C42—21no—
149C42C22⋊Dic6711yescharacteristic
150C42C22×Dic6711yescharacteristic
151C23—22no—
152C23#1452395811yescharacteristic
153Q8—42no—
154D4—42no—
155D4—82no—
156D4—82no—
157C2×C4—22no—
158C2×C4—21no—
159C2×C4—21no—
160C2×C4—22no—
161C2×C4—42no—
162C2×C4—42no—
163C2×C4—42no—
164C2×C4—81no—
165C2×C4—82no—
166C2×C4#1452395811yescharacteristic
167C2×C4C2×C22⋊Dic6711yescharacteristic
168C8—42no—
169C8—5364no—
170C22—21no—
171C22—41no—
172C22—42no—
173C22—82no—
174C22C2×#1452395811yescharacteristic
175C4—21no—
176C4—41no—
177C4—41no—
178C4—42no—
179C4—82no—
180C4#1686319112yes—
181C2—21no—
182C2—41no—
183C2—82no—
184C2C2×#1686319111yescharacteristic
subgroupG/H
centerC2C2×#16863191
commutatorC42×C67C22×C4
FrattiniC4⋊Q8C22×D67
FittingC67×#3142C2
socleC2×C67C2×#354
prime radical#532×C67C22
cosocleC4⋊Q8×C67C23
hypercenter#3142D67
hypocenterC67#13600
LCSG ▷ C42×C67 ▷ C2×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ C2×C4 ◁ C22×C4 ◁ #532 ◁ #3142

Complements

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

NH ≅ G/Ntype
2. #22558382C2semidirect
96. C67#13600semidirect

Maximal subgroups

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

subgroupindex
#225618942
#225583822
C67×#31422
#1360067