Finite Group Databasebeta

← #30766372

Subgroups — #30766372

Order 17152.

316 conjugacy classes of subgroups; only the 75 normal subgroups are shown.

normal subgroups only

This group has 316 conjugacy classes of subgroups — too many to draw. Only its 75 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#22570859C211yesmaximal, characteristic
2#22562025C211yesmaximal, characteristic, complemented
3#22558199C212yesmaximal, complemented
4#22558193C212yesmaximal, complemented
5C67×#1747C211yesmaximal, characteristic, complemented
6#16864001—21no—
7#16863798C411yescharacteristic, complemented
8#16863773C411yescharacteristic, complemented
9#16863761—21no—
10#16863649C2212yes—
11#16863632—21no—
12#16863509C412yescomplemented
13C4×D536—21no—
14#16863217—21no—
15#16863178C2212yes—
16#16863174—21no—
17#405×C67—21no—
18#405×C67C2211yescharacteristic
19#338×C67C2211yescharacteristic
20H(3,ℤ/4)×C67—21no—
21H(3,ℤ/4)×C67C2211yescharacteristic
22C2×D536—21no—
23C2×D536C2×C412yes—
24#14524040—21no—
25#14524035—21no—
26#14524028—42no—
27#14524017—22no—
28#14524016—42no—
29C4×D268—22no—
30#14523983—22no—
31#14523977—21no—
32#14523975C2×C412yes—
33#14523965—22no—
34#14523965—22no—
35C2×M16×C67—21no—
36C2×M16×C67C2311yescharacteristic
37C22×C8×C67D411yescharacteristic
38#117×C67—21no—
39#117×C67D411yescharacteristic
40#97×C67—22no—
41#97×C67C2×C411yescharacteristic
42#97×C67D411yescharacteristic
43C4×C8×C67—22no—
44C4×C8×C67C2×C411yescharacteristic
45C4×C8×C67D411yescharacteristic
46C2×D268—22no—
47C2×D268—42no—
48C2×C22⋊D67—42no—
49C2×C4×D67—42no—
50D536—42no—
51D536—41no—
52D536—41no—
53C4⋊Dic67—22no—
54Dic67⋊C4—42no—
55(C2×D67)⋊C4—42no—
56(C2×D67)⋊C4—42no—
57C67⋊C16—42no—
58C22×C4×C67C2×D411yescharacteristic
59M16×C67—41no—
60M16×C67—41no—
61M16×C67—41no—
62C2×C8×C67—21no—
63C2×C8×C67—21no—
64C2×C8×C67—22no—
65C2×C8×C67C22⋊C411yescharacteristic
66C2×C8×C67C22⋊C411yescharacteristic
67C2×C8×C67C22×C411yescharacteristic
68C2×C8×C67C2×D411yescharacteristic
69C4⋊C4×C67—21no—
70C4⋊C4×C67—41no—
71C22⋊C4×C67—21no—
72C22⋊C4×C67—41no—
73C42×C67—22no—
74C42×C67C22⋊C411yescharacteristic
75C42×C67C22⋊C411yescharacteristic
76C23×C67AGL(1,ℤ/8)11yescharacteristic
77C22×D67—42no—
78C2×C4×C67—21no—
79C2×C4×C67—21no—
80C2×C4×C67—21no—
81C2×C4×C67—21no—
82C2×C4×C67—42no—
83C2×C4×C67C2×C22⋊C411yescharacteristic
84C2×C4×C67AGL(1,ℤ/8)11yescharacteristic
85C22⋊D67—162no—
86C2×Dic67—42no—
87D268—42no—
88D268—42no—
89D268—162no—
90C4×D67—162no—
91C8×C67—21no—
92C8×C67—21no—
93C8×C67—21no—
94C8×C67—42no—
95C8×C67#10411yescharacteristic
96C8×C67#10411yescharacteristic
97C22×C67—21no—
98C22×C67—41no—
99C22×C67#41811yescharacteristic
100C2×D67—82no—
101C2×D67—82no—
102C2×D67—82no—
103C4×C67—21no—
104C4×C67—41no—
105C4×C67—41no—
106C4×C67—82no—
107C4×C67#41011yescharacteristic
108C4×C67C2×#10411yescharacteristic
109Dic67—82no—
110#11770—671nomaximal, Sylow
111C2×C67—21no—
112C2×C67—41no—
113C2×C67#150511yescharacteristic
114D67—162no—
115#3176—671no—
116#2166—671no—
117#1747D6711yescharacteristic, complemented
118#1341—672no—
119#1339—672no—
120C67#1177011yescharacteristic, complemented, Sylow
121#540—1341no—
122#499—671no—
123#493—671no—
124#490—1341no—
125#469—672no—
126#466—1341no—
127#442—672no—
128C4×D8—1341no—
129#405—21no—
130#405C2×D6711yescharacteristic
131#364—1341no—
132#349—672no—
133#347—1341no—
134#338C2×D6711yescharacteristic
135H(3,ℤ/4)—21no—
136H(3,ℤ/4)C2×D6711yescharacteristic
137C2×D8—672no—
138C2×D8—1341no—
139C2×M16—21no—
140C2×M16C22×D6711yescharacteristic
141C22×C8C22⋊D6711yescharacteristic
142#127—1341no—
143#125—1341no—
144#123—2682no—
145#121—2682no—
146#121—1342no—
147C4×D4—1342no—
148#117—21no—
149#117C22⋊D6711yescharacteristic
150M32—1342no—
151#107—1341no—
152#106—672no—
153#102—1342no—
154#102—1342no—
155#97—22no—
156#97C4×D6711yescharacteristic
157#97D26811yescharacteristic
158C4×C8—22no—
159C4×C8C4×D6711yescharacteristic
160C4×C8D26811yescharacteristic
161C2×D4—2682no—
162C2×D4—2682no—
163C2×D4—1342no—
164C22×C4—2682no—
165C22×C4C2×C22⋊D6711yescharacteristic
166D8—2682no—
167D8—2681no—
168D8—2681no—
169M16—41no—
170M16—41no—
171M16—41no—
172C2×C8—21no—
173C2×C8—21no—
174C2×C8—22no—
175C2×C8(C2×D67)⋊C411yescharacteristic
176C2×C8(C2×D67)⋊C411yescharacteristic
177C2×C8C2×C4×D6711yescharacteristic
178C2×C8C2×D26811yescharacteristic
179C4⋊C4—21no—
180C4⋊C4—41no—
181C4⋊C4—2682no—
182C4⋊C4—1342no—
183C22⋊C4—21no—
184C22⋊C4—41no—
185C22⋊C4—2682no—
186C22⋊C4—2682no—
187C42—22no—
188C42(C2×D67)⋊C411yescharacteristic
189C42(C2×D67)⋊C411yescharacteristic
190C16—2682no—
191C23—2682no—
192C23#1452406811yescharacteristic
193D4—2682no—
194D4—2682no—
195D4—10722no—
196D4—10722no—
197C2×C4—21no—
198C2×C4—21no—
199C2×C4—21no—
200C2×C4—21no—
201C2×C4—42no—
202C2×C4—2682no—
203C2×C4—10722no—
204C2×C4C2×(C2×D67)⋊C411yescharacteristic
205C2×C4#1452406811yescharacteristic
206C8—21no—
207C8—21no—
208C8—21no—
209C8—42no—
210C8#1452397111yescharacteristic
211C8#1452397111yescharacteristic
212C22—21no—
213C22—41no—
214C22—5362no—
215C22—5362no—
216C22—5362no—
217C22#1686340011yescharacteristic
218C4—21no—
219C4—41no—
220C4—41no—
221C4—82no—
222C4—5362no—
223C4#1686336911yescharacteristic
224C4C2×#1452397111yescharacteristic
225C2—21no—
226C2—41no—
227C2—10722no—
228C2#2255871611yescharacteristic
subgroupG/H
centerC2#22558716
commutatorC2×C8×C67C22×C4
FrattiniC2×M16C22×D67
FittingC67×#1747C2
socleC2×C67#1505
prime radical#22570859C2
cosocleC2×M16×C67C23
hypercenter#1747D67
hypocenterC67#11770
LCSG ▷ C2×C8×C67 ▷ C2×C4×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ C2×C4 ◁ #405 ◁ #1747

Complements

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

NH ≅ G/Ntype
2. #22562025C2semidirect
3. #22558199C2semidirect
4. #22558193C2semidirect
5. C67×#1747C2semidirect
7. #16863798C4semidirect
8. #16863773C4semidirect
12. #16863509C4semidirect
117. #1747D67semidirect
120. C67#11770semidirect

Maximal subgroups

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

subgroupindex
#225708592
#225620252
#225581992
#225581932
C67×#17472
#1177067