Finite Group Databasebeta

← #30765252

Subgroups — #30765252

Order 17152.

446 conjugacy classes of subgroups; only the 99 normal subgroups are shown.

normal subgroups only

This group has 446 conjugacy classes of subgroups — too many to draw. Only its 99 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#22558543C212yesmaximal
2#22558157C214yesmaximal
3C67×#2296C211yesmaximal, characteristic
4#16863453—22no—
5#16863122—22no—
6#16863120C2214yes—
7C2×#117×C67C411yescharacteristic
8C2×#117×C67C2211yescharacteristic
9C22×C42×C67C411yescharacteristic
10#389×C67—22no—
11#385×C67C412yes—
12#385×C67C2212yes—
13C4×C4⋊C4×C67—22no—
14C4⋊(C67⋊C8)—42no—
15C22⋊(C67⋊C8)—24no—
16#14523954—42no—
17C23×C4×C67C2×C411yescharacteristic
18C23×C4×C67C2×C412yes—
19#117×C67—24no—
20#117×C67—42no—
21C2×C4⋊C4×C67—21no—
22C2×C4⋊C4×C67—22no—
23C2×C4⋊C4×C67—22no—
24C2×C4⋊C4×C67D412yes—
25C2×C22⋊C4×C67C2×C412yes—
26C2×C22⋊C4×C67C2×C411yescharacteristic
27C2×C22⋊C4×C67C2311yescharacteristic
28C2×C42×C67—21no—
29C2×C42×C67—21no—
30C2×C42×C67—21no—
31C2×C42×C67—42no—
32C2×C42×C67D412yes—
33#95×C67—22no—
34#95×C67—22no—
35C2×C67⋊C8—44no—
36C24×C67C22×C411yescharacteristic
37C22×C4×C67—22no—
38C22×C4×C67—22no—
39C22×C4×C67—22no—
40C22×C4×C67—22no—
41C22×C4×C67—22no—
42C22×C4×C67—44no—
43C22×C4×C67—42no—
44C22×C4×C67C22⋊C411yescharacteristic
45C22×C4×C67C22⋊C412yes—
46C22×C4×C67C22⋊C411yescharacteristic
47C22×C4×C67C2×D412yes—
48C4⋊C4×C67—24no—
49C4⋊C4×C67—42no—
50C4⋊C4×C67—44no—
51C4⋊C4×C67—44no—
52C22⋊C4×C67—22no—
53C22⋊C4×C67—41no—
54C22⋊C4×C67—41no—
55C22⋊C4×C67M1614yes—
56C42×C67—24no—
57C42×C67—22no—
58C42×C67—22no—
59C42×C67—42no—
60C42×C67—42no—
61C23×C67—21no—
62C23×C67—21no—
63C23×C67—42no—
64C23×C67C2×C22⋊C411yescharacteristic
65C23×C67C2×M1612yes—
66C2×C4×C67—28no—
67C2×C4×C67—21no—
68C2×C4×C67—21no—
69C2×C4×C67—21no—
70C2×C4×C67—21no—
71C2×C4×C67—24no—
72C2×C4×C67—24no—
73C2×C4×C67—22no—
74C2×C4×C67—22no—
75C2×C4×C67—42no—
76C2×C4×C67—42no—
77C2×C4×C67—42no—
78C2×C4×C67—44no—
79C2×C4×C67—42no—
80C2×C4×C67—42no—
81C2×C4×C67—42no—
82C2×C4×C67—44no—
83C2×C4×C67#9912yes—
84C2×C4×C67#10112yes—
85C67⋊C8—84no—
86C22×C67—22no—
87C22×C67—22no—
88C22×C67—42no—
89C22×C67—42no—
90C22×C67—42no—
91C22×C67#40711yescharacteristic
92C22×C67C2×#9911yescharacteristic
93C22×C67C2×#10111yescharacteristic
94C4×C67—22no—
95C4×C67—22no—
96C4×C67—44no—
97C4×C67—42no—
98C4×C67—42no—
99C4×C67—44no—
100#11518—671nomaximal, Sylow
101C2×C67—21no—
102C2×C67—21no—
103C2×C67—42no—
104C2×C67#146311yescharacteristic
105C2×C67#212012yes—
106#2296D6711yescharacteristic
107#1464—672no—
108#1326—674no—
109C67#1151811yescharacteristic, complemented, Sylow
110C2×#117Dic6711yescharacteristic
111C2×#117C2×D6711yescharacteristic
112C22×C42Dic6711yescharacteristic
113#432—1342no—
114#389—22no—
115#385Dic6712yes—
116#385C2×D6712yes—
117C4×C4⋊C4—22no—
118#324—1342no—
119#323—674no—
120C23×C4C2×Dic6711yescharacteristic
121C23×C4C2×Dic6712yes—
122#117—24no—
123#117—42no—
124C2×C4⋊C4—21no—
125C2×C4⋊C4—22no—
126C2×C4⋊C4—22no—
127C2×C4⋊C4C22⋊D6712yes—
128C2×C22⋊C4C2×Dic6712yes—
129C2×C22⋊C4C2×Dic6711yescharacteristic
130C2×C22⋊C4C22×D6711yescharacteristic
131C2×C42—21no—
132C2×C42—21no—
133C2×C42—21no—
134C2×C42—42no—
135C2×C42C22⋊D6712yes—
136C4⋊C8—2682no—
137C22⋊C8—1344no—
138#97—2682no—
139#95—22no—
140#95—22no—
141C24C22×Dic6711yescharacteristic
142C22×C4—22no—
143C22×C4—22no—
144C22×C4—22no—
145C22×C4—22no—
146C22×C4—22no—
147C22×C4—44no—
148C22×C4—42no—
149C22×C4C22⋊Dic6711yescharacteristic
150C22×C4C22⋊Dic6712yes—
151C22×C4C22⋊Dic6711yescharacteristic
152C22×C4C2×C22⋊D6712yes—
153C2×C8—2684no—
154C4⋊C4—24no—
155C4⋊C4—42no—
156C4⋊C4—44no—
157C4⋊C4—44no—
158C22⋊C4—22no—
159C22⋊C4—41no—
160C22⋊C4—41no—
161C22⋊C4#1176161214yes—
162C42—24no—
163C42—22no—
164C42—22no—
165C42—42no—
166C42—42no—
167C23—21no—
168C23—21no—
169C23—42no—
170C23C2×C22⋊Dic6711yescharacteristic
171C23C2×#1176161212yes—
172C2×C4—28no—
173C2×C4—21no—
174C2×C4—21no—
175C2×C4—21no—
176C2×C4—21no—
177C2×C4—24no—
178C2×C4—24no—
179C2×C4—22no—
180C2×C4—22no—
181C2×C4—42no—
182C2×C4—42no—
183C2×C4—42no—
184C2×C4—44no—
185C2×C4—42no—
186C2×C4—42no—
187C2×C4—42no—
188C2×C4—44no—
189C2×C4#1452395812yes—
190C2×C4#1452396212yes—
191C8—5364no—
192C22—22no—
193C22—22no—
194C22—42no—
195C22—42no—
196C22—42no—
197C22#1686336011yescharacteristic
198C22C2×#1452395811yescharacteristic
199C22C2×#1452396211yescharacteristic
200C4—22no—
201C4—22no—
202C4—44no—
203C4—42no—
204C4—44no—
205C4—42no—
206C2—21no—
207C2—21no—
208C2—42no—
209C2#2255853911yescharacteristic
210C2#2256185012yes—
subgroupG/H
centerC22C2×#14523958
commutator, prime radicalC24×C67C22×C4
FrattiniC2×C22⋊C4C22×D67
FittingC67×#2296C2
socleC22×C67C2×#99
cosocleC2×C22⋊C4×C67C23
hypercenter#2296D67
hypocenterC67#11518
LCSG ▷ C24×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C22 ◁ C22×C4 ◁ C2×#117 ◁ #2296

Complements

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

NH ≅ G/Ntype
109. C67#11518semidirect

Maximal subgroups

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

subgroupindex
#225585432
#225581572
C67×#22962
#1151867