Finite Group Databasebeta

← #30767786

Subgroups — #30767786

Order 17152.

234 conjugacy classes of subgroups; only the 69 normal subgroups are shown.

normal subgroups only

This group has 234 conjugacy classes of subgroups — too many to draw. Only its 69 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#22562083C211yesmaximal, characteristic
2#22559122C211yesmaximal, characteristic, complemented
3#22559033C211yesmaximal, characteristic, complemented
4#22558274C211yesmaximal, characteristic, complemented
5#22558177C211yesmaximal, characteristic, complemented
6#22558129C211yesmaximal, characteristic, complemented
7C67×#1333C211yesmaximal, characteristic, complemented
8C2×#14523984C411yescharacteristic
9C4×D536C2211yescharacteristic
10#16863473—21no—
11D4×C67⋊C8—21no—
12#16863223C411yescharacteristic
13#16863197C411yescharacteristic
14#16863196C411yescharacteristic
15#16863172C2211yescharacteristic
16#16863147C2211yescharacteristic
17D4⋊(C67⋊C8)—21no—
18#16863123—21no—
19#16863117C2211yescharacteristic
20C4×D8×C67C2211yescharacteristic
21#346×C67C2211yescharacteristic
22#334×C67C2211yescharacteristic
23C2×D536C2×C411yescharacteristic
24C2×#11761611—41no—
25C22×C67⋊C8—41no—
26C4×D268—21no—
27#14523984—22no—
28C2×C67⋊C16C2×C411yescharacteristic
29C2×C67⋊C16C2×C411yescharacteristic
30#14523977C2×C411yescharacteristic
31C4⋊(C67⋊C8)—21no—
32#14523965—21no—
33C22⋊(C67⋊C8)—41no—
34(C2×D67)⋊C8—41no—
35C4×C67⋊C8D411yescharacteristic
36C4×C67⋊C8D411yescharacteristic
37C2×D8×C67C2×C411yescharacteristic
38C4×D4×C67—21no—
39C2×C16×C67—21no—
40#107×C67C2×C411yescharacteristic
41C4⋊C8×C67—21no—
42#102×C67—21no—
43C4×C8×C67C2311yescharacteristic
44C2×D268—21no—
45C2×C4×D67—41no—
46D536—21no—
47D536—21no—
48#11761611—82no—
49C2×C67⋊C8—21no—
50C2×C67⋊C8—21no—
51C2×C67⋊C8—41no—
52C2×C67⋊C8—82no—
53C4⋊Dic67—21no—
54(C2×D67)⋊C4—41no—
55C67⋊C16—21no—
56C67⋊C16—21no—
57C2×D4×C67—21no—
58C22×C4×C67—41no—
59D8×C67—21no—
60D8×C67M1612yescomplemented
61C2×C8×C67—41no—
62C2×C8×C67C22×C411yescharacteristic
63C2×C8×C67#4111yescharacteristic
64C4⋊C4×C67—21no—
65C22⋊C4×C67—41no—
66C42×C67C2×D411yescharacteristic
67C16×C67—22no—
68C23×C67—41no—
69C22×D67—41no—
70D4×C67—22no—
71D4×C67—41no—
72C2×C4×C67—41no—
73C2×C4×C67—81no—
74C2×C4×C67—81no—
75C2×C4×C67C4×D411yescharacteristic
76C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
77C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
78C2×Dic67—41no—
79D268—41no—
80D268—41no—
81C4×D67—81no—
82C4×D67—81no—
83C8×C67—21no—
84C8×C67—81no—
85C8×C67C2×M1611yescharacteristic
86C8×C67#13111yescharacteristic
87C67⋊C8—41no—
88C67⋊C8—41no—
89C67⋊C8—81no—
90C22×C67—42no—
91C22×C67—81no—
92C22×C67—81no—
93C22×C67#44211yescharacteristic
94C2×D67—81no—
95C2×D67—81no—
96C2×D67—81no—
97C4×C67—21no—
98C4×C67—21no—
99C4×C67—81no—
100C4×C67#43511yescharacteristic
101C4×C67#44311yescharacteristic
102Dic67—81no—
103#12078—671nomaximal, Sylow
104C2×C67—82no—
105C2×C67#158611yescharacteristic
106C2×C67AGL(1,ℤ/16)11yescharacteristic
107C2×C67AGL(1,ℤ/16)11yescharacteristic
108D67—161no—
109#2177—671no—
110#1594—671no—
111#1580—671no—
112#1368—671no—
113#1333—671no—
114#1333D6711yescharacteristic, complemented
115#1313—671no—
116C67#1207811yescharacteristic, complemented, Sylow
117C2×D16—671no—
118C4×D8—671no—
119C4×D8C2×D6711yescharacteristic
120#436—1341no—
121C8×D4—1341no—
122#367—671no—
123#357—671no—
124#357—671no—
125#346—671no—
126#346C2×D6711yescharacteristic
127#334—671no—
128#334C2×D6711yescharacteristic
129D4⋊C8—1341no—
130D4⋊C8—1341no—
131#322—671no—
132C2×D8—671no—
133C2×D8C4×D6711yescharacteristic
134C2×M16—2681no—
135C22×C8—2681no—
136C4×D4—21no—
137C4×D4—1341no—
138D16—1342no—
139C2×C16—21no—
140C2×C16—671no—
141C2×C16—671no—
142#107—671no—
143#107C4×D6711yescharacteristic
144C4⋊C8—21no—
145C4⋊C8—1341no—
146#102—21no—
147#102—1341no—
148C22⋊C8—2681no—
149C22⋊C8—2681no—
150C4×C8—671no—
151C4×C8—671no—
152C4×C8C22×D6711yescharacteristic
153C2×D4—21no—
154C2×D4—1341no—
155C22×C4—41no—
156C22×C4—2681no—
157D8—21no—
158D8—1341no—
159D8—1341no—
160D8#1176161112yescomplemented
161M16—5362no—
162C2×C8—41no—
163C2×C8—2681no—
164C2×C8—5362no—
165C2×C8—1341no—
166C2×C8—1341no—
167C2×C8C2×C4×D6711yescharacteristic
168C2×C8#1176162711yescharacteristic
169C4⋊C4—21no—
170C4⋊C4—1341no—
171C22⋊C4—41no—
172C22⋊C4—2681no—
173C42D4×D6711yescharacteristic
174C16—22no—
175C16—1341no—
176C16—1341no—
177C23—41no—
178C23—2681no—
179D4—22no—
180D4—41no—
181D4—2681no—
182D4—2681no—
183C2×C4—41no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—2681no—
187C2×C4—5361no—
188C2×C4—5361no—
189C2×C4(C22⋊D67)⋊C411yescharacteristic
190C2×C4#1452406711yescharacteristic
191C2×C4#1452407011yescharacteristic
192C8—21no—
193C8—81no—
194C8—2681no—
195C8—2681no—
196C8—5361no—
197C8C2×#1176161111yescharacteristic
198C8#1452405111yescharacteristic
199C22—42no—
200C22—81no—
201C22—81no—
202C22—5361no—
203C22—5361no—
204C22—5361no—
205C22#1686350511yescharacteristic
206C4—21no—
207C4—21no—
208C4—81no—
209C4—5361no—
210C4#1686346411yescharacteristic
211C4#1686351011yescharacteristic
212C2—82no—
213C2—10721no—
214C2#2255907011yescharacteristic
215C2#2256210411yescharacteristic
216C2#2256211011yescharacteristic
subgroupG/H
centerC22#16863505
commutatorC2×C8×C67C22×C4
FrattiniC4×C8C22×D67
FittingC67×#1333C2
socleC22×C67#442
prime radicalC2×#14523984C4
cosocleC4×C8×C67C23
hypercenter#1333D67
hypocenterC67#12078
LCSG ▷ C2×C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ C4×C8 ◁ #1333

Complements

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

NH ≅ G/Ntype
2. #22559122C2semidirect
3. #22559033C2semidirect
4. #22558274C2semidirect
5. #22558177C2semidirect
6. #22558129C2semidirect
7. C67×#1333C2semidirect
60. D8×C67M16semidirect
114. #1333D67semidirect
116. C67#12078semidirect
160. D8#11761611semidirect

Maximal subgroups

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

subgroupindex
#225620832
#225591222
#225590332
#225582742
#225581772
#225581292
C67×#13332
#1207867