Finite Group Databasebeta

← #30743151

Subgroups — #30743151

Order 17152.

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

normal subgroups only

This group has 244 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#22560280C211yesmaximal, characteristic
2#22558543C211yesmaximal, characteristic
3C67×#1464C211yesmaximal, characteristic
4#16863453—21no—
5C2×C22⋊(C67⋊C8)C411yescharacteristic
6C2×C22⋊(C67⋊C8)C411yescharacteristic
7#16863151—21no—
8#16863151—21no—
9#16863122—21no—
10#16863120C412yes—
11C2×#117×C67C2211yescharacteristic
12#432×C67—21no—
13#324×C67—21no—
14#323×C67C412yes—
15C22×C67⋊C8—21no—
16C22×C67⋊C8—21no—
17C4⋊(C67⋊C8)—41no—
18C22⋊(C67⋊C8)—22no—
19C22⋊(C67⋊C8)—41no—
20C22⋊(C67⋊C8)—41no—
21#14523954—41no—
22C23×C4×C67C2×C411yescharacteristic
23#117×C67—41no—
24#117×C67—41no—
25C2×C4⋊C4×C67D411yescharacteristic
26C2×C4⋊C4×C67Q811yescharacteristic
27C2×C22⋊C4×C67C2×C411yescharacteristic
28C2×C22⋊C4×C67C2×C411yescharacteristic
29C2×C42×C67D411yescharacteristic
30C2×C42×C67D411yescharacteristic
31C4⋊C8×C67—41no—
32C22⋊C8×C67—22no—
33#97×C67—41no—
34C2×C67⋊C8—42no—
35C2×C67⋊C8—41no—
36C2×C67⋊C8—41no—
37C2×C67⋊C8—41no—
38C2×C67⋊C8—41no—
39C2×C67⋊C8—41no—
40C2×C67⋊C8—41no—
41C24×C67C4211yescharacteristic
42C22×C4×C67—21no—
43C22×C4×C67—21no—
44C22×C4×C67—41no—
45C22×C4×C67—41no—
46C22×C4×C67C22⋊C411yescharacteristic
47C22×C4×C67C22⋊C411yescharacteristic
48C22×C4×C67C22⋊C411yescharacteristic
49C22×C4×C67C4⋊C411yescharacteristic
50C22×C4×C67C4⋊C411yescharacteristic
51C22×C4×C67C4⋊C411yescharacteristic
52C2×C8×C67—42no—
53C4⋊C4×C67—41no—
54C4⋊C4×C67—41no—
55C22⋊C4×C67—41no—
56C22⋊C4×C67—41no—
57C42×C67—21no—
58C42×C67—21no—
59C42×C67—21no—
60C42×C67—21no—
61C23×C67—21no—
62C23×C67—21no—
63C23×C67—21no—
64C23×C67—41no—
65C23×C67—41no—
66C23×C67#9511yescharacteristic
67C2×C4×C67—21no—
68C2×C4×C67—21no—
69C2×C4×C67—21no—
70C2×C4×C67—21no—
71C2×C4×C67—21no—
72C2×C4×C67—21no—
73C2×C4×C67—21no—
74C2×C4×C67—21no—
75C2×C4×C67—21no—
76C2×C4×C67—42no—
77C2×C4×C67—42no—
78C2×C4×C67—41no—
79C2×C4×C67—41no—
80C2×C4×C67—41no—
81C2×C4×C67—41no—
82C2×C4×C67#9911yescharacteristic
83C2×C4×C67#9911yescharacteristic
84C8×C67—82no—
85C67⋊C8—81no—
86C67⋊C8—81no—
87C67⋊C8—82no—
88C22×C67—21no—
89C22×C67—21no—
90C22×C67—41no—
91C22×C67—41no—
92C22×C67—41no—
93C22×C67—81no—
94C22×C67—81no—
95C22×C67#34211yescharacteristic
96C22×C67#34411yescharacteristic
97C22×C67#34411yescharacteristic
98C4×C67—21no—
99C4×C67—21no—
100C4×C67—41no—
101C4×C67—41no—
102C4×C67—41no—
103C4×C67—41no—
104C4×C67—41no—
105#7325—671nomaximal, Sylow
106C2×C67—21no—
107C2×C67—21no—
108C2×C67—81no—
109C2×C67#130611yescharacteristic
110C2×C67#139111yescharacteristic
111C2×C67#139211yescharacteristic
112#1818—671no—
113#1464—671no—
114#1464D6711yescharacteristic
115C67#732511yescharacteristic, complemented, Sylow
116C2×#117C2×D6711yescharacteristic
117#432—21no—
118#432—1341no—
119C2×C22⋊C8—671no—
120C2×C22⋊C8—671no—
121#336—1341no—
122#336—1341no—
123#324—21no—
124#324—1341no—
125#323—672no—
126#323Dic6712yes—
127C23×C4C4×D6711yescharacteristic
128C22×C8—1341no—
129C22×C8—1341no—
130#117—41no—
131#117—41no—
132C2×C4⋊C4Dic13411yescharacteristic
133C2×C4⋊C4C22⋊D6711yescharacteristic
134C2×C22⋊C4C4×D6711yescharacteristic
135C2×C22⋊C4C2×Dic6711yescharacteristic
136C2×C42D26811yescharacteristic
137C2×C42C22⋊D6711yescharacteristic
138C4⋊C8—41no—
139C4⋊C8—2681no—
140C22⋊C8—22no—
141C22⋊C8—2681no—
142C22⋊C8—2681no—
143C22⋊C8—1342no—
144#97—41no—
145#97—2681no—
146C24C4×Dic6711yescharacteristic
147C22×C4—21no—
148C22×C4—21no—
149C22×C4—41no—
150C22×C4—41no—
151C22×C4(C2×D67)⋊C411yescharacteristic
152C22×C4(C2×D67)⋊C411yescharacteristic
153C22×C4C22⋊Dic6711yescharacteristic
154C22×C4Dic67⋊C411yescharacteristic
155C22×C4Dic67⋊C411yescharacteristic
156C22×C4C4⋊Dic6711yescharacteristic
157C2×C8—42no—
158C2×C8—2682no—
159C2×C8—2681no—
160C2×C8—2681no—
161C2×C8—2681no—
162C2×C8—2681no—
163C2×C8—2681no—
164C2×C8—2681no—
165C4⋊C4—41no—
166C4⋊C4—41no—
167C22⋊C4—41no—
168C22⋊C4—41no—
169C42—21no—
170C42—21no—
171C42—21no—
172C42—21no—
173C23—21no—
174C23—21no—
175C23—21no—
176C23—41no—
177C23—41no—
178C23C67⋊#9511yescharacteristic
179C2×C4—21no—
180C2×C4—21no—
181C2×C4—21no—
182C2×C4—21no—
183C2×C4—21no—
184C2×C4—21no—
185C2×C4—21no—
186C2×C4—21no—
187C2×C4—21no—
188C2×C4—42no—
189C2×C4—42no—
190C2×C4—41no—
191C2×C4—41no—
192C2×C4—41no—
193C2×C4—41no—
194C2×C4#1452395711yescharacteristic
195C2×C4#1452395711yescharacteristic
196C8—82no—
197C8—5361no—
198C8—5361no—
199C8—5362no—
200C22—21no—
201C22—21no—
202C22—41no—
203C22—41no—
204C22—41no—
205C22—81no—
206C22—81no—
207C22#1686316211yescharacteristic
208C22#1686316711yescharacteristic
209C22#1686316811yescharacteristic
210C4—21no—
211C4—21no—
212C4—41no—
213C4—41no—
214C4—41no—
215C4—41no—
216C4—41no—
217C2—21no—
218C2—21no—
219C2—81no—
220C2#2255811811yescharacteristic
221C2#2255832911yescharacteristic
222C2#2255833111yescharacteristic
subgroupG/H
centerC22#16863162
commutator, prime radicalC24×C67C42
FrattiniC2×#117C2×D67
FittingC67×#1464C2
socleC22×C67#342
cosocleC2×#117×C67C22
hypercenter#1464D67
hypocenterC67#7325
LCSG ▷ C24×C67 ▷ C23×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #1464

Complements

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

NH ≅ G/Ntype
115. C67#7325semidirect

Maximal subgroups

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

subgroupindex
#225602802
#225585432
C67×#14642
#732567