Finite Group Databasebeta

← #30767017

Subgroups — #30767017

Order 17152.

296 conjugacy classes of subgroups; only the 83 normal subgroups are shown.

normal subgroups only

This group has 296 conjugacy classes of subgroups — too many to draw. Only its 83 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#22562025C211yesmaximal, characteristic, complemented
2#22561814C211yesmaximal, characteristic, complemented
3#22558242C214yesmaximal, complemented
4C67×#3176C211yesmaximal, characteristic
5C2×#14523983—21no—
6#16863217—21no—
7#16863178—21no—
8#16863174—21no—
9#16863174C2212yes—
10#16863174C2212yes—
11#540×C67C412yes—
12#499×C67C411yescharacteristic
13#493×C67C411yescharacteristic
14#490×C67C2212yes—
15#469×C67—21no—
16#466×C67—21no—
17#442×C67—21no—
18C4×D8×C67—21no—
19#405×C67C2211yescharacteristic
20#14523983—22no—
21#14523983—22no—
22#14523983—22no—
23C2×C67⋊C16—22no—
24C2×D8×C67—21no—
25C2×D8×C67—21no—
26C2×M16×C67D411yescharacteristic
27C22×C8×C67D411yescharacteristic
28#127×C67C2×C412yes—
29#125×C67C2×C412yes—
30#123×C67—42no—
31#121×C67—22no—
32#121×C67—42no—
33C4×D4×C67—22no—
34#117×C67C2×C411yescharacteristic
35#107×C67—21no—
36#106×C67—21no—
37#102×C67—22no—
38#102×C67—22no—
39#97×C67D411yescharacteristic
40#97×C67C2311yescharacteristic
41C4×C8×C67C2×C411yescharacteristic
42C4×C8×C67D411yescharacteristic
43C67⋊C16—24no—
44C67⋊C16—42no—
45C2×D4×C67—22no—
46C2×D4×C67—42no—
47C2×D4×C67—42no—
48C22×C4×C67—42no—
49C22×C4×C67C22⋊C411yescharacteristic
50D8×C67—42no—
51D8×C67—41no—
52D8×C67—41no—
53M16×C67—41no—
54C2×C8×C67—21no—
55C2×C8×C67—21no—
56C2×C8×C67C22⋊C411yescharacteristic
57C2×C8×C67C22⋊C411yescharacteristic
58C2×C8×C67C2×D411yescharacteristic
59C2×C8×C67C2×D411yescharacteristic
60C4⋊C4×C67—21no—
61C4⋊C4×C67—22no—
62C4⋊C4×C67—42no—
63C22⋊C4×C67—21no—
64C22⋊C4×C67—42no—
65C22⋊C4×C67—42no—
66C42×C67C22⋊C411yescharacteristic
67C42×C67C22×C411yescharacteristic
68C23×C67—42no—
69C23×C67#10011yescharacteristic
70D4×C67—42no—
71D4×C67—42no—
72D4×C67—162no—
73D4×C67—162no—
74C2×C4×C67—21no—
75C2×C4×C67—21no—
76C2×C4×C67—21no—
77C2×C4×C67—21no—
78C2×C4×C67—42no—
79C2×C4×C67—162no—
80C2×C4×C67#10011yescharacteristic
81C2×C4×C67C2×C22⋊C411yescharacteristic
82C8×C67—21no—
83C8×C67—21no—
84C8×C67—21no—
85C8×C67#10411yescharacteristic
86C8×C67#10411yescharacteristic
87C22×C67—21no—
88C22×C67—41no—
89C22×C67—82no—
90C22×C67—82no—
91C22×C67—82no—
92C22×C67C2×#10011yescharacteristic
93C4×C67—21no—
94C4×C67—41no—
95C4×C67—41no—
96C4×C67—82no—
97C4×C67C2×#10411yescharacteristic
98C4×C67#42111yescharacteristic
99#11913—671nomaximal, Sylow
100C2×C67—21no—
101C2×C67—41no—
102C2×C67—162no—
103C2×C67#153011yescharacteristic
104#3176D6711yescharacteristic
105#2166—671no—
106#2110—671no—
107#1355—674no—
108C67#1191311yescharacteristic, complemented, Sylow
109#540Dic6712yes—
110C2×M32—1341no—
111#499Dic6711yescharacteristic
112#493Dic6711yescharacteristic
113#490C2×D6712yes—
114#469—21no—
115#466—21no—
116#442—21no—
117C4×D8—21no—
118#405C2×D6711yescharacteristic
119#364—1341no—
120#349—1341no—
121#347—672no—
122#347—672no—
123#347—1341no—
124C2×D8—21no—
125C2×D8—21no—
126C2×M16C22⋊D6711yescharacteristic
127C22×C8C22⋊D6711yescharacteristic
128#127C2×Dic6712yes—
129#125C2×Dic6712yes—
130#123—42no—
131#121—22no—
132#121—42no—
133C4×D4—22no—
134#117C2×Dic6711yescharacteristic
135M32—1342no—
136M32—1342no—
137M32—1342no—
138C2×C16—1342no—
139#107—21no—
140#106—21no—
141#102—22no—
142#102—22no—
143#97C22⋊D6711yescharacteristic
144#97C22×D6711yescharacteristic
145C4×C8C2×Dic6711yescharacteristic
146C4×C8C22⋊D6711yescharacteristic
147C2×D4—22no—
148C2×D4—42no—
149C2×D4—42no—
150C22×C4—42no—
151C22×C4C22⋊Dic6711yescharacteristic
152D8—42no—
153D8—41no—
154D8—41no—
155M16—41no—
156C2×C8—21no—
157C2×C8—21no—
158C2×C8C22⋊Dic6711yescharacteristic
159C2×C8C22⋊Dic6711yescharacteristic
160C2×C8C2×C22⋊D6711yescharacteristic
161C2×C8C2×C22⋊D6711yescharacteristic
162C4⋊C4—21no—
163C4⋊C4—22no—
164C4⋊C4—42no—
165C22⋊C4—21no—
166C22⋊C4—42no—
167C22⋊C4—42no—
168C42C22⋊Dic6711yescharacteristic
169C42C22×Dic6711yescharacteristic
170C16—2682no—
171C16—1344no—
172C23—42no—
173C23#1452396011yescharacteristic
174D4—42no—
175D4—42no—
176D4—162no—
177D4—162no—
178C2×C4—21no—
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—42no—
183C2×C4—162no—
184C2×C4#1452396011yescharacteristic
185C2×C4C2×C22⋊Dic6711yescharacteristic
186C8—21no—
187C8—21no—
188C8—21no—
189C8#1452397011yescharacteristic
190C8#1452397011yescharacteristic
191C22—21no—
192C22—41no—
193C22—82no—
194C22—82no—
195C22—82no—
196C22C2×#1452396011yescharacteristic
197C4—21no—
198C4—41no—
199C4—41no—
200C4—82no—
201C4C2×#1452397011yescharacteristic
202C4#1686341311yescharacteristic
203C2—21no—
204C2—41no—
205C2—162no—
206C2#2255882011yescharacteristic
subgroupG/H
centerC2#22558820
commutatorC42×C67C22×C4
Frattini#97C22×D67
Fitting, prime radicalC67×#3176C2
socleC2×C67#1530
cosocle#97×C67C23
hypercenter#3176D67
hypocenterC67#11913
LCSG ▷ C42×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #3176

Complements

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

NH ≅ G/Ntype
1. #22562025C2semidirect
2. #22561814C2semidirect
3. #22558242C2semidirect
108. C67#11913semidirect

Maximal subgroups

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

subgroupindex
#225620252
#225618142
#225582422
C67×#31762
#1191367