Finite Group Databasebeta

← #30767348

Subgroups — #30767348

Order 17152.

292 conjugacy classes of subgroups; only the 87 normal subgroups are shown.

normal subgroups only

This group has 292 conjugacy classes of subgroups — too many to draw. Only its 87 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#22561975C211yesmaximal, characteristic, complemented
3#22558270C212yesmaximal, complemented
4#22558246C212yesmaximal, complemented
5C67×#3176C211yesmaximal, characteristic
6C2×#14523983—21no—
7#16863217—21no—
8#16863217C2212yes—
9#16863217C2212yes—
10#16863178—21no—
11#16863174—21no—
12#540×C67C412yes—
13#499×C67C2211yescharacteristic
14#493×C67C2211yescharacteristic
15#490×C67C412yes—
16#469×C67—21no—
17#466×C67—21no—
18#442×C67—21no—
19C4×D8×C67—21no—
20#405×C67C2211yescharacteristic
21#14523983—22no—
22#14523983—22no—
23#14523983—41no—
24C2×C67⋊C16—41no—
25C2×D8×C67—21no—
26C2×D8×C67—21no—
27C2×M16×C67D411yescharacteristic
28C22×C8×C67D411yescharacteristic
29#127×C67C2×C412yes—
30#125×C67C2×C412yes—
31#123×C67—42no—
32#121×C67—22no—
33#121×C67—42no—
34C4×D4×C67—22no—
35#117×C67C2×C411yescharacteristic
36#107×C67—21no—
37#106×C67—21no—
38#102×C67—22no—
39#102×C67—22no—
40#97×C67C2×C411yescharacteristic
41#97×C67D411yescharacteristic
42C4×C8×C67D411yescharacteristic
43C4×C8×C67C2311yescharacteristic
44C67⋊C16—42no—
45C67⋊C16—42no—
46C2×D4×C67—22no—
47C2×D4×C67—42no—
48C2×D4×C67—42no—
49C22×C4×C67—42no—
50C22×C4×C67C22⋊C411yescharacteristic
51D8×C67—42no—
52D8×C67—41no—
53D8×C67—41no—
54M16×C67—22no—
55C2×C8×C67—21no—
56C2×C8×C67—21no—
57C2×C8×C67C22⋊C411yescharacteristic
58C2×C8×C67C22⋊C411yescharacteristic
59C2×C8×C67C2×D411yescharacteristic
60C2×C8×C67C2×D411yescharacteristic
61C4⋊C4×C67—21no—
62C4⋊C4×C67—22no—
63C4⋊C4×C67—42no—
64C22⋊C4×C67—21no—
65C22⋊C4×C67—42no—
66C22⋊C4×C67—42no—
67C42×C67C22⋊C411yescharacteristic
68C42×C67C22×C411yescharacteristic
69C23×C67—42no—
70C23×C67#10011yescharacteristic
71D4×C67—42no—
72D4×C67—42no—
73D4×C67—162no—
74D4×C67—162no—
75C2×C4×C67—21no—
76C2×C4×C67—21no—
77C2×C4×C67—21no—
78C2×C4×C67—21no—
79C2×C4×C67—42no—
80C2×C4×C67—162no—
81C2×C4×C67#10011yescharacteristic
82C2×C4×C67C2×C22⋊C411yescharacteristic
83C8×C67—21no—
84C8×C67—21no—
85C8×C67#13512yes—
86C8×C67AGL(1,ℤ/8)11yescharacteristic
87C8×C67#13711yescharacteristic
88C22×C67—21no—
89C22×C67—41no—
90C22×C67—82no—
91C22×C67—82no—
92C22×C67—82no—
93C22×C67C2×#10011yescharacteristic
94C4×C67—21no—
95C4×C67—41no—
96C4×C67—41no—
97C4×C67—82no—
98C4×C67#41611yescharacteristic
99C4×C67#41711yescharacteristic
100#11989—671nomaximal, Sylow
101C2×C67—21no—
102C2×C67—41no—
103C2×C67—162no—
104C2×C67#155111yescharacteristic
105#3176D6711yescharacteristic
106#2166—671no—
107#2153—671no—
108#1366—672no—
109#1357—672no—
110C67#1198911yescharacteristic, complemented, Sylow
111#540Dic6712yes—
112C2×M32—1341no—
113#499C2×D6711yescharacteristic
114#493C2×D6711yescharacteristic
115#490Dic6712yes—
116#469—21no—
117#466—21no—
118#442—21no—
119C4×D8—21no—
120#405C2×D6711yescharacteristic
121#364—672no—
122#364—672no—
123#364—1341no—
124#349—1341no—
125#347—1341no—
126C2×D8—21no—
127C2×D8—21no—
128C2×M16C22⋊D6711yescharacteristic
129C22×C8C22⋊D6711yescharacteristic
130#127C2×Dic6712yes—
131#125C2×Dic6712yes—
132#123—42no—
133#121—22no—
134#121—42no—
135C4×D4—22no—
136#117C2×Dic6711yescharacteristic
137M32—2681no—
138M32—1342no—
139M32—1342no—
140C2×C16—2681no—
141#107—21no—
142#106—21no—
143#102—22no—
144#102—22no—
145#97C2×Dic6711yescharacteristic
146#97C22⋊D6711yescharacteristic
147C4×C8C22⋊D6711yescharacteristic
148C4×C8C22×D6711yescharacteristic
149C2×D4—22no—
150C2×D4—42no—
151C2×D4—42no—
152C22×C4—42no—
153C22×C4C22⋊Dic6711yescharacteristic
154D8—42no—
155D8—41no—
156D8—41no—
157M16—22no—
158C2×C8—21no—
159C2×C8—21no—
160C2×C8C22⋊Dic6711yescharacteristic
161C2×C8C22⋊Dic6711yescharacteristic
162C2×C8C2×C22⋊D6711yescharacteristic
163C2×C8C2×C22⋊D6711yescharacteristic
164C4⋊C4—21no—
165C4⋊C4—22no—
166C4⋊C4—42no—
167C22⋊C4—21no—
168C22⋊C4—42no—
169C22⋊C4—42no—
170C42C22⋊Dic6711yescharacteristic
171C42C22×Dic6711yescharacteristic
172C16—2682no—
173C16—2682no—
174C23—42no—
175C23#1452396011yescharacteristic
176D4—42no—
177D4—42no—
178D4—162no—
179D4—162no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—42no—
185C2×C4—162no—
186C2×C4#1452396011yescharacteristic
187C2×C4C2×C22⋊Dic6711yescharacteristic
188C8—21no—
189C8—21no—
190C8#1452406312yes—
191C8#1452406811yescharacteristic
192C8#1452407311yescharacteristic
193C22—21no—
194C22—41no—
195C22—82no—
196C22—82no—
197C22—82no—
198C22C2×#1452396011yescharacteristic
199C4—21no—
200C4—41no—
201C4—41no—
202C4—82no—
203C4#1686339211yescharacteristic
204C4#1686339711yescharacteristic
205C2—21no—
206C2—41no—
207C2—162no—
208C2#2255890411yescharacteristic
subgroupG/H
centerC2#22558904
commutatorC42×C67C22×C4
FrattiniC4×C8C22×D67
Fitting, prime radicalC67×#3176C2
socleC2×C67#1551
cosocleC4×C8×C67C23
hypercenter#3176D67
hypocenterC67#11989
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. #22561975C2semidirect
3. #22558270C2semidirect
4. #22558246C2semidirect
110. C67#11989semidirect

Maximal subgroups

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

subgroupindex
#225620252
#225619752
#225582702
#225582462
C67×#31762
#1198967