Finite Group Databasebeta

← #30767787

Subgroups — #30767787

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#22559034C211yesmaximal, characteristic, complemented
3#22558274C211yesmaximal, characteristic, complemented
4#22558176C211yesmaximal, characteristic, complemented
5#22558175C211yesmaximal, characteristic, complemented
6#22558130C211yesmaximal, characteristic, complemented
7C67×#1594C211yesmaximal, characteristic, complemented
8C2×#14523984C411yescharacteristic
9C4×D536C2211yescharacteristic
10C8×D268—21no—
11#16863223C411yescharacteristic
12#16863197C411yescharacteristic
13#16863196C411yescharacteristic
14#16863172C2211yescharacteristic
15#16863172C2211yescharacteristic
16#16863147C2211yescharacteristic
17#16863125—21no—
18C4×D8×C67C2211yescharacteristic
19#436×C67—21no—
20#334×C67C2211yescharacteristic
21D4⋊C8×C67—21no—
22#322×C67C2211yescharacteristic
23C2×D536C2×C411yescharacteristic
24C2×C8×D67—41no—
25C4×D268—21no—
26#14523984—22no—
27C2×C67⋊C16—21no—
28C2×C67⋊C16C2×C411yescharacteristic
29C2×C67⋊C16C2×C411yescharacteristic
30#14523977C2×C411yescharacteristic
31C4⋊(C67⋊C8)—21no—
32#14523965—21no—
33(C2×D67)⋊C8—41no—
34C2×D8×C67C2×C411yescharacteristic
35C2×M16×C67—41no—
36C4×D4×C67—21no—
37#107×C67C2×C411yescharacteristic
38C4⋊C8×C67—21no—
39#102×C67—21no—
40C22⋊C8×C67—41no—
41C4×C8×C67D411yescharacteristic
42C4×C8×C67D411yescharacteristic
43C4×C8×C67C2311yescharacteristic
44C2×D268—21no—
45C2×C4×D67—41no—
46D536—21no—
47D536M1612yescomplemented
48C2×C67⋊C8—41no—
49C8×D67—82no—
50C4⋊Dic67—21no—
51(C2×D67)⋊C4—41no—
52C67⋊C16—22no—
53C67⋊C16—21no—
54C67⋊C16—21no—
55C2×D4×C67—21no—
56C22×C4×C67—41no—
57D8×C67—21no—
58D8×C67—21no—
59M16×C67—82no—
60C2×C8×C67—21no—
61C2×C8×C67—21no—
62C2×C8×C67—41no—
63C2×C8×C67C22×C411yescharacteristic
64C2×C8×C67#4111yescharacteristic
65C4⋊C4×C67—21no—
66C22⋊C4×C67—41no—
67C42×C67C2×D411yescharacteristic
68C23×C67—41no—
69C22×D67—41no—
70D4×C67—41no—
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—22no—
80D268—41no—
81C4×D67—81no—
82C4×D67—81no—
83C8×C67—21no—
84C8×C67—41no—
85C8×C67—41no—
86C8×C67—81no—
87C8×C67C2×M1611yescharacteristic
88C8×C67#13111yescharacteristic
89C67⋊C8—81no—
90C22×C67—81no—
91C22×C67—81no—
92C22×C67—81no—
93C22×C67#44211yescharacteristic
94C2×D67—42no—
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—161no—
105C2×C67#158611yescharacteristic
106C2×C67AGL(1,ℤ/16)11yescharacteristic
107C2×C67AGL(1,ℤ/16)11yescharacteristic
108D67—82no—
109#2177—671no—
110#1594D6711yescharacteristic, complemented
111#1580—671no—
112#1368—671no—
113#1333—671no—
114#1333—671no—
115#1313—671no—
116C67#1207811yescharacteristic, complemented, Sylow
117C2×D16—671no—
118C4×D8—671no—
119C4×D8C2×D6711yescharacteristic
120#436—21no—
121C8×D4—1341no—
122#367—671no—
123#357—671no—
124#357—671no—
125#346—671no—
126#346—671no—
127#334—671no—
128#334C2×D6711yescharacteristic
129D4⋊C8—21no—
130D4⋊C8—1341no—
131#322C2×D6711yescharacteristic
132C2×D8—671no—
133C2×D8C4×D6711yescharacteristic
134C2×M16—41no—
135C22×C8—2681no—
136C4×D4—21no—
137C4×D4—1341no—
138D16—1342no—
139C2×C16—671no—
140C2×C16—671no—
141C2×C16—1341no—
142#107—671no—
143#107C4×D6711yescharacteristic
144C4⋊C8—21no—
145C4⋊C8—1341no—
146#102—21no—
147#102—1341no—
148C22⋊C8—41no—
149C22⋊C8—2681no—
150C4×C8C22⋊D6711yescharacteristic
151C4×C8C22⋊D6711yescharacteristic
152C4×C8C22×D6711yescharacteristic
153C2×D4—21no—
154C2×D4—1341no—
155C22×C4—41no—
156C22×C4—2681no—
157D8—21no—
158D8—21no—
159D8—672no—
160D8—1341no—
161M16—82no—
162C2×C8—21no—
163C2×C8—21no—
164C2×C8—41no—
165C2×C8—2681no—
166C2×C8—5362no—
167C2×C8C2×C4×D6711yescharacteristic
168C2×C8#1176162811yescharacteristic
169C4⋊C4—21no—
170C4⋊C4—1341no—
171C22⋊C4—41no—
172C22⋊C4—2681no—
173C42C2×C22⋊D6711yescharacteristic
174C16—1342no—
175C16—1341no—
176C16—1341no—
177C23—41no—
178C23—2681no—
179D4—41no—
180D4—41no—
181D4—2681no—
182D4—1342no—
183C2×C4—41no—
184C2×C4—81no—
185C2×C4—81no—
186C2×C4—2681no—
187C2×C4—5361no—
188C2×C4—5361no—
189C2×C4C4×C22⋊D6711yescharacteristic
190C2×C4#1452406811yescharacteristic
191C2×C4#1452406911yescharacteristic
192C8—21no—
193C8—41no—
194C8—41no—
195C8—81no—
196C8—5361no—
197C8M16×D6711yescharacteristic
198C8#1452405111yescharacteristic
199C22—81no—
200C22—81no—
201C22—81no—
202C22—2682no—
203C22—5361no—
204C22—5361no—
205C22#1686350611yescharacteristic
206C4—21no—
207C4—21no—
208C4—81no—
209C4—5361no—
210C4#1686346511yescharacteristic
211C4#1686351111yescharacteristic
212C2—161no—
213C2—5362no—
214C2#2255907111yescharacteristic
215C2#2256210511yescharacteristic
216C2#2256210711yescharacteristic
subgroupG/H
centerC22#16863506
commutatorC2×C8×C67C22×C4
FrattiniC4×C8C22×D67
FittingC67×#1594C2
socleC22×C67#442
prime radicalC2×#14523984C4
cosocleC4×C8×C67C23
hypercenter#1594D67
hypocenterC67#12078
LCSG ▷ C2×C8×C67 ▷ C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C42 ◁ #322 ◁ #1594

Complements

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

NH ≅ G/Ntype
2. #22559034C2semidirect
3. #22558274C2semidirect
4. #22558176C2semidirect
5. #22558175C2semidirect
6. #22558130C2semidirect
7. C67×#1594C2semidirect
47. D536M16semidirect
110. #1594D67semidirect
116. C67#12078semidirect

Maximal subgroups

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

subgroupindex
#225620832
#225590342
#225582742
#225581762
#225581752
#225581302
C67×#15942
#1207867