Finite Group Databasebeta

← #30767032

Subgroups — #30767032

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#22562021C212yesmaximal, complemented
2#22558244C212yesmaximal, complemented
3#22558242C212yesmaximal, complemented
4C67×#3142C211yesmaximal, characteristic
5#16863217—22no—
6#16863178—22no—
7#16863174C2214yes—
8#532×C67C411yescharacteristic
9C2×#125×C67C411yescharacteristic
10#491×C67C411yescharacteristic
11#491×C67C2211yescharacteristic
12#490×C67C411yescharacteristic
13#490×C67C2211yescharacteristic
14#488×C67—22no—
15#417×C67—22no—
16C4×M16×C67C2211yescharacteristic
17#14523983—24no—
18C2×#41×C67—21no—
19C2×M16×C67D412yes—
20C4⋊Q8×C67C2×C411yescharacteristic
21#127×C67C2×C411yescharacteristic
22#125×C67—21no—
23#125×C67—41no—
24#125×C67C2×C411yescharacteristic
25#125×C67C2×C411yescharacteristic
26#124×C67—21no—
27#121×C67—41no—
28C4×D4×C67—41no—
29C2×C4⋊C4×C67—21no—
30C2×C4⋊C4×C67—41no—
31C2×C42×C67C2×C411yescharacteristic
32#103×C67—22no—
33#103×C67—22no—
34#102×C67—22no—
35#102×C67—22no—
36#97×C67C2×C411yescharacteristic
37#97×C67C2311yescharacteristic
38C4×C8×C67D412yes—
39C67⋊C16—44no—
40#41×C67—44no—
41C2×Q8×C67—21no—
42C2×D4×C67—21no—
43C2×D4×C67—41no—
44C2×D4×C67—41no—
45C22×C4×C67—21no—
46C22×C4×C67—41no—
47C22×C4×C67—41no—
48C22×C4×C67C22⋊C411yescharacteristic
49M16×C67—42no—
50C2×C8×C67C22⋊C412yes—
51C2×C8×C67C2×D412yes—
52C4⋊C4×C67—21no—
53C4⋊C4×C67—21no—
54C4⋊C4×C67—41no—
55C4⋊C4×C67—41no—
56C4⋊C4×C67—41no—
57C4⋊C4×C67—42no—
58C4⋊C4×C67—41no—
59C22⋊C4×C67—42no—
60C42×C67—21no—
61C42×C67C22⋊C411yescharacteristic
62C42×C67C22×C411yescharacteristic
63C23×C67—41no—
64C23×C67#10011yescharacteristic
65Q8×C67—42no—
66D4×C67—42no—
67D4×C67—82no—
68D4×C67—82no—
69C2×C4×C67—22no—
70C2×C4×C67—21no—
71C2×C4×C67—21no—
72C2×C4×C67—41no—
73C2×C4×C67—41no—
74C2×C4×C67—41no—
75C2×C4×C67—82no—
76C2×C4×C67—81no—
77C2×C4×C67—82no—
78C2×C4×C67#10011yescharacteristic
79C2×C4×C67C2×C22⋊C411yescharacteristic
80C8×C67—22no—
81C8×C67—22no—
82C22×C67—21no—
83C22×C67—41no—
84C22×C67—82no—
85C22×C67—81no—
86C22×C67C2×#10011yescharacteristic
87C4×C67—21no—
88C4×C67—41no—
89C4×C67—41no—
90C4×C67—81no—
91C4×C67—81no—
92C4×C67—81no—
93C4×C67#42112yes—
94#11917—671nomaximal, Sylow
95C2×C67—21no—
96C2×C67—41no—
97C2×C67—161no—
98C2×C67#153311yescharacteristic
99#3142D6711yescharacteristic
100#2165—672no—
101#1356—672no—
102#1355—672no—
103C67#1191711yescharacteristic, complemented, Sylow
104#532Dic6711yescharacteristic
105C2×#125Dic6711yescharacteristic
106#491Dic6711yescharacteristic
107#491C2×D6711yescharacteristic
108#490Dic6711yescharacteristic
109#490C2×D6711yescharacteristic
110#488—22no—
111#417—22no—
112C4×M16C2×D6711yescharacteristic
113#364—1342no—
114#349—1342no—
115#347—674no—
116C2×#41—21no—
117C2×M16C22⋊D6712yes—
118C4⋊Q8C2×Dic6711yescharacteristic
119#127C2×Dic6711yescharacteristic
120#125—21no—
121#125—41no—
122#125C2×Dic6711yescharacteristic
123#125C2×Dic6711yescharacteristic
124#124—21no—
125#121—41no—
126C4×D4—41no—
127C2×C4⋊C4—21no—
128C2×C4⋊C4—41no—
129C2×C42C2×Dic6711yescharacteristic
130M32—1344no—
131#103—22no—
132#103—22no—
133#102—22no—
134#102—22no—
135#97C2×Dic6711yescharacteristic
136#97C22×D6711yescharacteristic
137C4×C8C22⋊D6712yes—
138#41—44no—
139C2×Q8—21no—
140C2×D4—21no—
141C2×D4—41no—
142C2×D4—41no—
143C22×C4—21no—
144C22×C4—41no—
145C22×C4—41no—
146C22×C4C22⋊Dic6711yescharacteristic
147M16—42no—
148C2×C8C22⋊Dic6712yes—
149C2×C8C2×C22⋊D6712yes—
150C4⋊C4—21no—
151C4⋊C4—21no—
152C4⋊C4—41no—
153C4⋊C4—41no—
154C4⋊C4—41no—
155C4⋊C4—42no—
156C4⋊C4—41no—
157C22⋊C4—42no—
158C42—21no—
159C42C22⋊Dic6711yescharacteristic
160C42C22×Dic6711yescharacteristic
161C16—2684no—
162C23—41no—
163C23#1452396011yescharacteristic
164Q8—42no—
165D4—42no—
166D4—82no—
167D4—82no—
168C2×C4—22no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—41no—
172C2×C4—41no—
173C2×C4—41no—
174C2×C4—82no—
175C2×C4—81no—
176C2×C4—82no—
177C2×C4#1452396011yescharacteristic
178C2×C4C2×C22⋊Dic6711yescharacteristic
179C8—22no—
180C8—22no—
181C22—21no—
182C22—41no—
183C22—82no—
184C22—81no—
185C22C2×#1452396011yescharacteristic
186C4—21no—
187C4—41no—
188C4—41no—
189C4—81no—
190C4—81no—
191C4—81no—
192C4#1686341312yes—
193C2—21no—
194C2—41no—
195C2—161no—
196C2#2255883311yescharacteristic
subgroupG/H
centerC2#22558833
commutatorC42×C67C22×C4
Frattini#97C22×D67
FittingC67×#3142C2
socleC2×C67#1533
prime radical#532×C67C4
cosocle#97×C67C23
hypercenter#3142D67
hypocenterC67#11917
LCSG ▷ C42×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ C2×C4 ◁ C4×M16 ◁ #3142

Complements

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

NH ≅ G/Ntype
1. #22562021C2semidirect
2. #22558244C2semidirect
3. #22558242C2semidirect
103. C67#11917semidirect

Maximal subgroups

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

subgroupindex
#225620212
#225582442
#225582422
C67×#31422
#1191767