Finite Group Databasebeta

← #30776182

Subgroups — #30776182

Order 17152.

388 conjugacy classes of subgroups; only the 79 normal subgroups are shown.

normal subgroups only

This group has 388 conjugacy classes of subgroups — too many to draw. Only its 79 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
1C2×#16863189C212yesmaximal
2#22558380C214yesmaximal, complemented
3C67×#3138C211yesmaximal, characteristic
4C2×#14523958—22no—
5#16863189—22no—
6#16863189C2214yes—
7#533×C67C411yescharacteristic, complemented
8C2×#127×C67C2211yescharacteristic
9#490×C67C412yescomplemented
10#490×C67C2212yes—
11#486×C67—22no—
12#418×C67—22no—
13C4×M16×C67C411yescharacteristic, complemented
14C2×C22⋊Dic67—42no—
15#14523958—24no—
16#14523958—42no—
17C22×D4×C67—41no—
18C22×D4×C67D412yes—
19C2×M16×C67—21no—
20C4⋊Q8×C67—21no—
21#127×C67—41no—
22#127×C67C2×C411yescharacteristic
23#127×C67D412yes—
24#127×C67C2311yescharacteristic
25#125×C67C2×C412yescomplemented
26C22⋊Q8×C67—41no—
27C4×Q8×C67—41no—
28#117×C67—21no—
29C2×C42×C67C2×C411yescharacteristic
30#102×C67—24no—
31#102×C67—24no—
32#97×C67C2×C412yescomplemented
33C4×C8×C67—21no—
34C22×Dic67—82no—
35C22⋊Dic67—44no—
36C22⋊Dic67—82no—
37C24×C67—22no—
38C2×Q8×C67—41no—
39C2×D4×C67—22no—
40C2×D4×C67—42no—
41C2×D4×C67—42no—
42C2×D4×C67—41no—
43C2×D4×C67—42no—
44C2×D4×C67—41no—
45C2×D4×C67—84no—
46C2×D4×C67C22⋊C412yes—
47C2×D4×C67C2×D412yes—
48C22×C4×C67—21no—
49C22×C4×C67C22⋊C411yescharacteristic
50M16×C67—42no—
51C2×C8×C67—22no—
52C4⋊C4×C67—22no—
53C4⋊C4×C67—42no—
54C4⋊C4×C67—41no—
55C22⋊C4×C67—41no—
56C42×C67—21no—
57C42×C67—41no—
58C42×C67C22⋊C411yescharacteristic
59C42×C67C22×C411yescharacteristic
60C23×C67—22no—
61C23×C67—22no—
62C23×C67—42no—
63C23×C67—42no—
64C23×C67—84no—
65C23×C67#9911yescharacteristic
66Q8×C67—82no—
67D4×C67—42no—
68D4×C67—42no—
69D4×C67—82no—
70D4×C67—84no—
71D4×C67—82no—
72D4×C67—82no—
73C2×C4×C67—22no—
74C2×C4×C67—21no—
75C2×C4×C67—21no—
76C2×C4×C67—42no—
77C2×C4×C67—42no—
78C2×C4×C67#9911yescharacteristic
79C2×C4×C67C2×C22⋊C411yescharacteristic
80C2×Dic67—84no—
81C2×Dic67—164no—
82C8×C67—42no—
83C22×C67—41no—
84C22×C67—42no—
85C22×C67—42no—
86C22×C67—82no—
87C22×C67—82no—
88C22×C67—84no—
89C22×C67—162no—
90C22×C67#35412yes—
91C22×C67C2×#9911yescharacteristic
92C4×C67—22no—
93C4×C67—21no—
94C4×C67—21no—
95C4×C67—41no—
96C4×C67—82no—
97Dic67—164no—
98#13595—671nomaximal, Sylow
99C2×C67—21no—
100C2×C67—22no—
101C2×C67—82no—
102C2×C67—82no—
103C2×C67C2×#35411yescharacteristic
104#3138D6711yescharacteristic
105C2×#353—672no—
106#1412—674no—
107C67#1359511yescharacteristic, complemented, Sylow
108#533Dic6711yescharacteristic, complemented
109C2×#127C2×D6711yescharacteristic
110#490Dic6712yescomplemented
111#490C2×D6712yes—
112#486—22no—
113#418—22no—
114C2×#99—1342no—
115C4×M16Dic6711yescharacteristic, complemented
116#353—674no—
117#353—1342no—
118C22×D4—41no—
119C22×D4C22⋊D6712yes—
120C2×M16—21no—
121C4⋊Q8—21no—
122#127—41no—
123#127C2×Dic6711yescharacteristic
124#127C22⋊D6712yes—
125#127C22×D6711yescharacteristic
126#125C2×Dic6712yescomplemented
127C22⋊Q8—41no—
128C4×Q8—41no—
129#117—21no—
130C2×C22⋊C4—2682no—
131C2×C42C2×Dic6711yescharacteristic
132#102—24no—
133#102—24no—
134#99—2682no—
135#99—1344no—
136#97C2×Dic6712yescomplemented
137C4×C8—21no—
138C24—22no—
139C2×Q8—41no—
140C2×D4—22no—
141C2×D4—42no—
142C2×D4—42no—
143C2×D4—41no—
144C2×D4—42no—
145C2×D4—41no—
146C2×D4—84no—
147C2×D4C22⋊Dic6712yes—
148C2×D4C2×C22⋊D6712yes—
149C22×C4—21no—
150C22×C4—5362no—
151C22×C4C22⋊Dic6711yescharacteristic
152M16—42no—
153C2×C8—22no—
154C4⋊C4—22no—
155C4⋊C4—42no—
156C4⋊C4—41no—
157C22⋊C4—41no—
158C22⋊C4—2684no—
159C22⋊C4—5362no—
160C42—21no—
161C42—41no—
162C42C22⋊Dic6711yescharacteristic
163C42C22×Dic6711yescharacteristic
164C23—22no—
165C23—22no—
166C23—42no—
167C23—42no—
168C23—84no—
169C23#1452395811yescharacteristic
170Q8—82no—
171D4—42no—
172D4—42no—
173D4—82no—
174D4—84no—
175D4—82no—
176D4—82no—
177C2×C4—21no—
178C2×C4—22no—
179C2×C4—21no—
180C2×C4—42no—
181C2×C4—42no—
182C2×C4—5364no—
183C2×C4—10724no—
184C2×C4#1452395811yescharacteristic
185C2×C4C2×C22⋊Dic6711yescharacteristic
186C8—42no—
187C22—41no—
188C22—42no—
189C22—42no—
190C22—82no—
191C22—82no—
192C22—84no—
193C22—162no—
194C22#1686319112yes—
195C22C2×#1452395811yescharacteristic
196C4—22no—
197C4—21no—
198C4—21no—
199C4—41no—
200C4—82no—
201C4—10724no—
202C2—21no—
203C2—22no—
204C2—82no—
205C2—82no—
206C2C2×#1686319111yescharacteristic
subgroupG/H
centerC2C2×#16863191
commutatorC42×C67C22×C4
Frattini#127C22×D67
FittingC67×#3138C2
socleC2×C67C2×#354
prime radicalC2×#127×C67C22
cosocle#127×C67C23
hypercenter#3138D67
hypocenterC67#13595
LCSG ▷ C42×C67 ▷ C2×C4×C67 ▷ C22×C67 ▷ C2×C67 ▷ C67
UCSC1 ◁ C2 ◁ C23 ◁ C22×C4 ◁ C2×#127 ◁ #3138

Complements

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

NH ≅ G/Ntype
2. #22558380C2semidirect
7. #533×C67C4semidirect
9. #490×C67C4semidirect
13. C4×M16×C67C4semidirect
25. #125×C67C2×C4semidirect
32. #97×C67C2×C4semidirect
107. C67#13595semidirect
108. #533Dic67semidirect
110. #490Dic67semidirect
115. C4×M16Dic67semidirect
126. #125C2×Dic67semidirect
136. #97C2×Dic67semidirect

Maximal subgroups

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

subgroupindex
C2×#168631892
#225583802
C67×#31382
#1359567