Finite Group Databasebeta

← #30770109

Subgroups — #30770109

Order 17152.

212 conjugacy classes of subgroups; only the 63 normal subgroups are shown.

normal subgroups only

This group has 212 conjugacy classes of subgroups — too many to draw. Only its 63 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#22562461C211yesmaximal, characteristic
2#22562114C211yesmaximal, characteristic
3#22559580C211yesmaximal, characteristic
4#22559576C211yesmaximal, characteristic
5#22558282C211yesmaximal, characteristic
6#22558259C211yesmaximal, characteristic
7C67×#1362C211yesmaximal, characteristic
8C2×#14523989—21no—
9C2×#14523989—21no—
10#16863776C2211yescharacteristic
11#16863682—21no—
12#16863678—21no—
13C8×Dic134C2211yescharacteristic
14Dic67⋊C16—21no—
15#16863171—21no—
16C4×C67⋊C16C2211yescharacteristic
17#16863149C2211yescharacteristic
18#16863142—21no—
19#16863141—21no—
20#494×C67C2211yescharacteristic
21C4⋊C16×C67C2211yescharacteristic
22#335×C67C2211yescharacteristic
23C2×Dic268—21no—
24C2×Dic268—21no—
25#14524043—21no—
26C4×Dic134D411yescharacteristic
27#14523989—42no—
28#14523989—42no—
29C2×C67⋊C16D411yescharacteristic
30C2×C67⋊C16D411yescharacteristic
31#14523975—41no—
32#14523974—41no—
33C4⋊(C67⋊C8)—21no—
34C4⋊(C67⋊C8)D411yescharacteristic
35Dic67⋊C8—21no—
36#14523968—41no—
37#14523966—41no—
38C8×Dic67—21no—
39C2×Q16×C67—21no—
40C2×Q16×C67—21no—
41C4⋊Q8×C67—21no—
42C2×C16×C67—21no—
43C4⋊C8×C67—21no—
44C4×C8×C67C2311yescharacteristic
45C2×Dic134—41no—
46C2×Dic134D811yescharacteristic
47Dic268—41no—
48Dic268—41no—
49Dic268—81no—
50C2×C67⋊C8—21no—
51C2×C67⋊C8—41no—
52C4⋊Dic67—41no—
53C4⋊Dic67—41no—
54C4⋊Dic67D811yescharacteristic
55Dic67⋊C4—21no—
56C4×Dic67—21no—
57C67⋊C16—21no—
58C67⋊C16SD1612yes—
59C2×Q8×C67—41no—
60Q16×C67—41no—
61Q16×C67—41no—
62Q16×C67—81no—
63C2×C8×C67—41no—
64C2×C8×C67C2×D411yescharacteristic
65C2×C8×C67#4111yescharacteristic
66C4⋊C4×C67—41no—
67C4⋊C4×C67—41no—
68C42×C67C2×D411yescharacteristic
69C16×C67—41no—
70Q8×C67—81no—
71Q8×C67—81no—
72C2×C4×C67—81no—
73C2×C4×C67#12111yescharacteristic
74C2×C4×C67C2×D811yescharacteristic
75C2×C4×C67#13511yescharacteristic
76C2×Dic67—21no—
77C2×Dic67—21no—
78C2×Dic67—81no—
79Dic134—21no—
80Dic134—21no—
81Dic134—81no—
82Dic134—81no—
83C8×C67—21no—
84C8×C67—81no—
85C8×C67C2×SD1611yescharacteristic
86C8×C67AGL(1,ℤ/8)11yescharacteristic
87C67⋊C8—41no—
88C67⋊C8—81no—
89C22×C67#46611yescharacteristic
90C4×C67—21no—
91C4×C67—161no—
92C4×C67#46011yescharacteristic
93C4×C67#47211yescharacteristic
94C4×C67#51011yescharacteristic
95C4×C67#51011yescharacteristic
96Dic67—41no—
97Dic67—41no—
98Dic67—161no—
99#12443—671nomaximal, Sylow
100C2×C67#166811yescharacteristic
101C2×C67#221911yescharacteristic
102C2×C67#222211yescharacteristic
103#2247—671no—
104#2183—671no—
105#1670—671no—
106#1670—671no—
107#1372—671no—
108#1362—671no—
109#1362D6711yescharacteristic
110C67#1244311yescharacteristic, complemented, Sylow
111C2×Q32—1341no—
112C2×Q32—1341no—
113#494—671no—
114#494C2×D6711yescharacteristic
115#475—1341no—
116#475—1341no—
117C8×Q8—671no—
118C4⋊C16—1341no—
119C4⋊C16C2×D6711yescharacteristic
120#346—1341no—
121C4×C16—671no—
122#335—671no—
123#335C2×D6711yescharacteristic
124#332—1341no—
125#332—1341no—
126C2×Q16—21no—
127C2×Q16—21no—
128C2×Q16—1341no—
129C2×Q16—1341no—
130C4⋊Q8—21no—
131C4⋊Q8—1341no—
132C4×Q8—671no—
133Q32—2682no—
134Q32—2682no—
135C2×C16—21no—
136C2×C16—671no—
137C2×C16—671no—
138#106—2681no—
139#106—2681no—
140C4⋊C8—21no—
141C4⋊C8—671no—
142C4⋊C8—1341no—
143C4⋊C8—1341no—
144#103—2681no—
145#103—2681no—
146C4×C8—1341no—
147C4×C8C22×D6711yescharacteristic
148C2×Q8—41no—
149C2×Q8—2681no—
150C2×Q8—671no—
151Q16—41no—
152Q16—41no—
153Q16—81no—
154Q16—2681no—
155Q16—2681no—
156Q16—5361no—
157C2×C8—41no—
158C2×C8—2681no—
159C2×C8—1341no—
160C2×C8D4×D6711yescharacteristic
161C2×C8#1176162811yescharacteristic
162C4⋊C4—41no—
163C4⋊C4—41no—
164C4⋊C4—2681no—
165C4⋊C4—2681no—
166C4⋊C4—671no—
167C4⋊C4—1341no—
168C42—1341no—
169C42D4×D6711yescharacteristic
170C16—41no—
171C16—672no—
172C16—1341no—
173Q8—81no—
174Q8—81no—
175Q8—5361no—
176Q8—5361no—
177Q8—1341no—
178Q8—1341no—
179C2×C4—81no—
180C2×C4—5361no—
181C2×C4—1341no—
182C2×C4—1341no—
183C2×C4#1452401611yescharacteristic
184C2×C4D8×D6711yescharacteristic
185C2×C4#1452406411yescharacteristic
186C8—21no—
187C8—81no—
188C8—2681no—
189C8—5361no—
190C8SD16×D6711yescharacteristic
191C8#1452406711yescharacteristic
192C22#1686363111yescharacteristic
193C4—21no—
194C4—2681no—
195C4—2681no—
196C4—161no—
197C4—10721no—
198C4#1686359511yescharacteristic
199C4#1686366511yescharacteristic
200C4#1686383511yescharacteristic
201C4#1686383811yescharacteristic
202C2#2255956811yescharacteristic
203C2#2256231611yescharacteristic
204C2#2256233311yescharacteristic
subgroupG/H
centerC22#16863631
commutator, cosocleC4×C8×C67C23
FrattiniC4×C8C22×D67
FittingC67×#1362C2
socle, prime radicalC22×C67#466
hypercenter#1362D67
hypocenterC67#12443
LCSG ▷ C4×C8×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1362

Complements

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

NH ≅ G/Ntype
110. C67#12443semidirect

Maximal subgroups

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

subgroupindex
#225624612
#225621142
#225595802
#225595762
#225582822
#225582592
C67×#13622
#1244367