Finite Group Databasebeta

← #30770387

Subgroups — #30770387

Order 17152.

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

normal subgroups only

This group has 226 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#22562357C211yesmaximal, characteristic
2#22562251C211yesmaximal, characteristic
3#22559803C211yesmaximal, characteristic, complemented
4#22559093C211yesmaximal, characteristic
5#22558278C211yesmaximal, characteristic, complemented
6#22558268C211yesmaximal, characteristic, complemented
7C67×#1362C211yesmaximal, characteristic, complemented
8C2×#14523986—21no—
9#16863801C2211yescharacteristic
10#16863792—21no—
11#16863748—21no—
12C4×D536C2211yescharacteristic
13#16863473—21no—
14#16863224—21no—
15C4⋊(C67⋊C16)C2211yescharacteristic
16#16863200—21no—
17#16863198—21no—
18#16863123—21no—
19C8×C67⋊C8C2211yescharacteristic
20#494×C67C2211yescharacteristic
21C4⋊C16×C67C2211yescharacteristic
22#335×C67C2211yescharacteristic
23C2×D536D411yescharacteristic
24C2×#11761611—41no—
25#14524043—21no—
26C4×D268—21no—
27#14523986—42no—
28C2×C67⋊C16—21no—
29#14523977—21no—
30#14523977—21no—
31#14523977D411yescharacteristic
32#14523974—41no—
33#14523974—41no—
34#14523966—42no—
35#14523965—21no—
36(C2×D67)⋊C8—41no—
37C4×C67⋊C8D411yescharacteristic
38C4×C67⋊C8D411yescharacteristic
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×D268—21no—
47C2×C4×D67—41no—
48D536—21no—
49D536—21no—
50#11761611—82no—
51C2×C67⋊C8—21no—
52C2×C67⋊C8D811yescharacteristic
53C2×C67⋊C8D811yescharacteristic
54C4⋊Dic67—21no—
55C4⋊Dic67—41no—
56C4⋊Dic67—41no—
57(C2×D67)⋊C4—41no—
58C67⋊C16—41no—
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—
70C22×D67—41no—
71Q8×C67—81no—
72Q8×C67—81no—
73C2×C4×C67—81no—
74C2×C4×C67#12111yescharacteristic
75C2×C4×C67C2×D811yescharacteristic
76C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
77C2×Dic67—41no—
78C2×Dic67—81no—
79D268—41no—
80D268—41no—
81C4×D67—81no—
82C4×D67—81no—
83Dic134—82no—
84C8×C67—21no—
85C8×C67—81no—
86C8×C67AGL(1,ℤ/8)11yescharacteristic
87C8×C67#13711yescharacteristic
88C67⋊C8—21no—
89C67⋊C8—41no—
90C67⋊C8SD3212yescomplemented
91C22×C67#45911yescharacteristic
92C2×D67—81no—
93C2×D67—81no—
94C2×D67—81no—
95C4×C67—21no—
96C4×C67—161no—
97C4×C67#46811yescharacteristic
98C4×C67#47111yescharacteristic
99C4×C67C2×SD3211yescharacteristic
100C4×C67#51011yescharacteristic
101Dic67—81no—
102Dic67—161no—
103#12487—671nomaximal, Sylow
104C2×C67#168511yescharacteristic
105C2×C67#220811yescharacteristic
106C2×C67#221211yescharacteristic
107D67—161no—
108#2226—671no—
109#2208—671no—
110#1707—671no—
111#1589—671no—
112#1370—671no—
113#1365—671no—
114#1362D6711yescharacteristic, complemented
115C67#1248711yescharacteristic, complemented, Sylow
116C2×SD32—1341no—
117#500—671no—
118#498—1341no—
119#494C2×D6711yescharacteristic
120#487—1341no—
121C4×D8—671no—
122#436—1341no—
123#367—1341no—
124C4⋊C16—671no—
125C4⋊C16C2×D6711yescharacteristic
126#358—1341no—
127#357—1341no—
128#335C2×D6711yescharacteristic
129D4⋊C8—1341no—
130C82—671no—
131C2×Q16—21no—
132C2×Q16—21no—
133C2×D8—671no—
134C2×M16—2681no—
135C4⋊Q8—21no—
136C4⋊Q8—1341no—
137C4×D4—1341no—
138SD32—2682no—
139C2×C16—21no—
140C2×C16—1341no—
141#107—671no—
142#107—1341no—
143#107—1341no—
144#106—2681no—
145#106—2681no—
146C4⋊C8—21no—
147#103—2682no—
148#102—1341no—
149C22⋊C8—2681no—
150C4×C8—671no—
151C4×C8—671no—
152C4×C8C22×D6711yescharacteristic
153C2×Q8—41no—
154C2×Q8—2681no—
155C2×D4—1341no—
156C22×C4—2681no—
157Q16—41no—
158Q16—41no—
159Q16—81no—
160D8—1341no—
161D8—1341no—
162M16—5362no—
163C2×C8—41no—
164C2×C8—671no—
165C2×C8—671no—
166C2×C8—1341no—
167C2×C8D4×D6711yescharacteristic
168C2×C8#1176162811yescharacteristic
169C4⋊C4—41no—
170C4⋊C4—41no—
171C4⋊C4—2681no—
172C4⋊C4—2681no—
173C4⋊C4—1341no—
174C22⋊C4—2681no—
175C42D4×D6711yescharacteristic
176C16—41no—
177C16—2681no—
178C23—2681no—
179Q8—81no—
180Q8—81no—
181Q8—5362no—
182D4—2681no—
183D4—2681no—
184C2×C4—81no—
185C2×C4—2681no—
186C2×C4—5361no—
187C2×C4—5361no—
188C2×C4—5361no—
189C2×C4#1452401611yescharacteristic
190C2×C4D8×D6711yescharacteristic
191C2×C4#1452407011yescharacteristic
192C8—21no—
193C8—81no—
194C8—2681no—
195C8—672no—
196C8—1341no—
197C8#1452406711yescharacteristic
198C8#1452407211yescharacteristic
199C22—5361no—
200C22—5361no—
201C22—5361no—
202C22#1686359011yescharacteristic
203C4—21no—
204C4—161no—
205C4—5361no—
206C4—10721no—
207C4#1686364111yescharacteristic
208C4#1686365811yescharacteristic
209C4SD32×D6711yescharacteristic
210C4#1686383811yescharacteristic
211C2—10721no—
212C2#2255967311yescharacteristic
213C2#2256225311yescharacteristic
214C2#2256228111yescharacteristic
subgroupG/H
centerC22#16863590
commutator, cosocleC4×C8×C67C23
FrattiniC4×C8C22×D67
FittingC67×#1362C2
socleC22×C67#459
prime radicalC2×D536D4
hypercenter#1362D67
hypocenterC67#12487
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
3. #22559803C2semidirect
5. #22558278C2semidirect
6. #22558268C2semidirect
7. C67×#1362C2semidirect
90. C67⋊C8SD32semidirect
114. #1362D67semidirect
115. C67#12487semidirect

Maximal subgroups

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

subgroupindex
#225623572
#225622512
#225598032
#225590932
#225582782
#225582682
C67×#13622
#1248767